*)\n\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Jinja/J/Progress.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"763e9530c3e0a4dd16ea32528b54e7052253e53a07ddab21979d7d53c3142a98\"\n}"}}},{"rowIdx":147,"cells":{"text":{"kind":"string","value":"(*<*)\ntheory IVSubst\n imports Syntax\nbegin\n (*>*)\n\nchapter ‹Immutable Variable Substitution›\n\ntext ‹Substitution involving immutable variables. We define a class and instances for all\nof the term forms›\n\nsection ‹Class›\n\nclass has_subst_v = fs +\n fixes subst_v :: \"'a::fs ⇒ x ⇒ v ⇒ 'a::fs\" (\"_[_::=_]⇩v\" [1000,50,50] 1000)\n assumes fresh_subst_v_if: \"y ♯ (subst_v a x v) ⟷ (atom x ♯ a ∧ y ♯ a) ∨ (y ♯ v ∧ (y ♯ a ∨ y = atom x))\" \n and forget_subst_v[simp]: \"atom x ♯ a ⟹ subst_v a x v = a\"\n and subst_v_id[simp]: \"subst_v a x (V_var x) = a\"\n and eqvt[simp,eqvt]: \"(p::perm) ∙ (subst_v a x v) = (subst_v (p ∙ a) (p ∙x) (p ∙v))\"\n and flip_subst_v[simp]: \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\"\n and subst_v_simple_commute[simp]: \"atom x ♯ c ⟹(c[z::=[x]⇧v]⇩v)[x::=b]⇩v = c[z::=b]⇩v\" \nbegin\n\nlemma subst_v_flip_eq_one:\n fixes z1::x and z2::x and x1::x and x2::x \n assumes \"[[atom z1]]lst. c1 = [[atom z2]]lst. c2\" \n and \"atom x1 ♯ (z1,z2,c1,c2)\"\n shows \"(c1[z1::=[x1]⇧v]⇩v) = (c2[z2::=[x1]⇧v]⇩v)\"\nproof (-)\n (*goal: ‹c1[z1::=[ x1 ]⇧v]⇩v = c2[z2::=[ x1 ]⇧v]⇩v›*)\n have \"(c1[z1::=[x1]⇧v]⇩v) = (x1 ↔ z1) ∙ c1\"\n using assms (*‹[[atom z1]]lst. c1 = [[atom z2]]lst. c2› ‹atom x1 ♯ (z1, z2, c1, c2)›*) flip_subst_v (*‹atom ?x ♯ ?c ⟹ (?x ↔ ?z) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) by auto\n moreover have \"(c2[z2::=[x1]⇧v]⇩v) = (x1 ↔ z2) ∙ c2\"\n using assms (*‹[[atom z1]]lst. c1 = [[atom z2]]lst. c2› ‹atom x1 ♯ (z1, z2, c1, c2)›*) flip_subst_v (*‹atom (?x::x) ♯ (?c::'a) ⟹ (?x ↔ ?z::x) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) by auto\n ultimately show \"?thesis\"\n (*goal: ‹c1[z1::=[ x1 ]⇧v]⇩v = c2[z2::=[ x1 ]⇧v]⇩v›*)\n using Abs1_eq_iff_all(3)[of z1 c1 z2 c2 z1] (*‹([[atom z1]]lst. c1 = [[atom z2]]lst. c2) = (∀c. atom c ♯ z1 ⟶ atom c ♯ (z1, z2, c1, c2) ⟶ (z1 ↔ c) ∙ c1 = (z2 ↔ c) ∙ c2)›*) assms (*‹[[atom z1]]lst. c1 = [[atom z2]]lst. c2› ‹atom x1 ♯ (z1, z2, c1, c2)›*) by (metis Abs1_eq_iff_fresh( (*‹atom ?c ♯ (?a, ?b, ?x, ?y) ⟹ ([[atom ?a]]lst. ?x = [[atom ?b]]lst. ?y) = ((?a ↔ ?c) ∙ ?x = (?b ↔ ?c) ∙ ?y)›*) 3) flip_commute (*‹(?a ↔ ?b) = (?b ↔ ?a)›*))\nqed\n\nlemma subst_v_flip_eq_two:\n fixes z1::x and z2::x and x1::x and x2::x \n assumes \"[[atom z1]]lst. c1 = [[atom z2]]lst. c2\" \n shows \"(c1[z1::=b]⇩v) = (c2[z2::=b]⇩v)\"\nproof (-)\n (*goal: ‹c1[z1::=b]⇩v = c2[z2::=b]⇩v›*)\n obtain x :: x where \"*\": \"atom x ♯ (z1,z2,c1,c2)\"\n (*goal: ‹(⋀x. atom x ♯ (z1, z2, c1, c2) ⟹ thesis) ⟹ thesis›*)\n using obtain_fresh (*‹(⋀a. atom a ♯ ?x ⟹ ?thesis) ⟹ ?thesis›*) by metis\n hence \"(c1[z1::=[x]⇧v]⇩v) = (c2[z2::=[x]⇧v]⇩v)\"\n using subst_v_flip_eq_one[OF assms, of x] (*‹atom x ♯ (z1, z2, c1, c2) ⟹ c1[z1::=[ x ]⇧v]⇩v = c2[z2::=[ x ]⇧v]⇩v›*) by metis\n hence \"(c1[z1::=[x]⇧v]⇩v)[x::=b]⇩v = (c2[z2::=[x]⇧v]⇩v)[x::=b]⇩v\"\n by auto\n thus \"?thesis\"\n (*goal: ‹c1[z1::=b]⇩v = c2[z2::=b]⇩v›*)\n using subst_v_simple_commute (*‹atom ?x ♯ ?c ⟹ ?c[?z::=[ ?x ]⇧v]⇩v[?x::=?b]⇩v = ?c[?z::=?b]⇩v›*) \"*\" (*‹atom x ♯ (z1, z2, c1, c2)›*) fresh_prod4 (*‹?x ♯ (?a, ?b, ?c, ?d) = (?x ♯ ?a ∧ ?x ♯ ?b ∧ ?x ♯ ?c ∧ ?x ♯ ?d)›*) by metis\nqed\n\nlemma subst_v_flip_eq_three:\n assumes \"[[atom z1]]lst. c1 = [[atom z1']]lst. c1'\" and \"atom x ♯ c1\" and \"atom x' ♯ (x,z1,z1', c1, c1')\"\n shows \"(x ↔ x') ∙ (c1[z1::=[x]⇧v]⇩v) = c1'[z1'::=[x']⇧v]⇩v\" \nproof (-)\n (*goal: ‹(x ↔ x') ∙ c1[z1::=[ x ]⇧v]⇩v = c1'[z1'::=[ x' ]⇧v]⇩v›*)\n have \"atom x' ♯ c1[z1::=[x]⇧v]⇩v\"\n using assms (*‹[[atom (z1::x)]]lst. (c1::'a::fs) = [[atom (z1'::x)]]lst. (c1'::'a::fs)› ‹atom x ♯ c1› ‹atom x' ♯ (x, z1, z1', c1, c1')›*) fresh_subst_v_if (*‹(?y::atom) ♯ (?a::'a)[(?x::x)::=(?v::v)]⇩v = (atom ?x ♯ ?a ∧ ?y ♯ ?a ∨ ?y ♯ ?v ∧ (?y ♯ ?a ∨ ?y = atom ?x))›*) by simp\n hence \"(x ↔ x') ∙ (c1[z1::=[x]⇧v]⇩v) = c1[z1::=[x]⇧v]⇩v[x::=[x']⇧v]⇩v\"\n using flip_subst_v[of x' \"c1[z1::=[x]⇧v]⇩v\" x] (*‹atom x' ♯ c1[z1::=[ x ]⇧v]⇩v ⟹ (x' ↔ x) ∙ c1[z1::=[ x ]⇧v]⇩v = c1[z1::=[ x ]⇧v]⇩v[x::=[ x' ]⇧v]⇩v›*) flip_commute (*‹(?a ↔ ?b) = (?b ↔ ?a)›*) by metis\n also (*calculation: ‹(x ↔ x') ∙ c1[z1::=[ x ]⇧v]⇩v = c1[z1::=[ x ]⇧v]⇩v[x::=[ x' ]⇧v]⇩v›*) have \"... = c1[z1::=[x']⇧v]⇩v\"\n using subst_v_simple_commute (*‹atom ?x ♯ ?c ⟹ ?c[?z::=[ ?x ]⇧v]⇩v[?x::=?b]⇩v = ?c[?z::=?b]⇩v›*) fresh_prod4 (*‹?x ♯ (?a, ?b, ?c, ?d) = (?x ♯ ?a ∧ ?x ♯ ?b ∧ ?x ♯ ?c ∧ ?x ♯ ?d)›*) assms (*‹[[atom (z1::x)]]lst. (c1::'a) = [[atom (z1'::x)]]lst. (c1'::'a)› ‹atom (x::x) ♯ (c1::'a)› ‹atom x' ♯ (x, z1, z1', c1, c1')›*) by auto\n also (*calculation: ‹(x ↔ x') ∙ c1[z1::=[ x ]⇧v]⇩v = c1[z1::=[ x' ]⇧v]⇩v›*) have \"... = c1'[z1'::=[x']⇧v]⇩v\"\n using subst_v_flip_eq_one[of z1 c1 z1' c1' x'] (*‹⟦[[atom z1]]lst. c1 = [[atom z1']]lst. c1'; atom x' ♯ (z1, z1', c1, c1')⟧ ⟹ c1[z1::=[ x' ]⇧v]⇩v = c1'[z1'::=[ x' ]⇧v]⇩v›*) using assms (*‹[[atom (z1::x)]]lst. (c1::'a::fs) = [[atom (z1'::x)]]lst. (c1'::'a::fs)› ‹atom x ♯ c1› ‹atom x' ♯ (x, z1, z1', c1, c1')›*) by auto\n finally (*calculation: ‹(x ↔ x') ∙ c1[z1::=[ x ]⇧v]⇩v = c1'[z1'::=[ x' ]⇧v]⇩v›*) show \"?thesis\"\n (*goal: ‹(x ↔ x') ∙ c1[z1::=[ x ]⇧v]⇩v = c1'[z1'::=[ x' ]⇧v]⇩v›*)\n by auto\nqed\n\nend\n\nsection ‹Values›\n\nnominal_function \n subst_vv :: \"v ⇒ x ⇒ v ⇒ v\" where\n \"subst_vv (V_lit l) x v = V_lit l\"\n| \"subst_vv (V_var y) x v = (if x = y then v else V_var y)\"\n| \"subst_vv (V_cons tyid c v') x v = V_cons tyid c (subst_vv v' x v)\"\n| \"subst_vv (V_consp tyid c b v') x v = V_consp tyid c b (subst_vv v' x v)\"\n| \"subst_vv (V_pair v1 v2) x v = V_pair (subst_vv v1 x v ) (subst_vv v2 x v )\"\n by(auto simp: eqvt_def subst_vv_graph_aux_def, metis v.strong_exhaust)\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_vv_abbrev :: \"v ⇒ x ⇒ v ⇒ v\" (\"_[_::=_]⇩v⇩v\" [1000,50,50] 1000)\n where \n \"v[x::=v']⇩v⇩v ≡ subst_vv v x v'\" \n\nlemma fresh_subst_vv_if [simp]:\n \"j ♯ t[i::=x]⇩v⇩v = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n using supp_l_empty (*‹supp [ ?l ]⇧v = {}›*) apply (induct t rule: v.induct (*‹⟦⋀x. ?P [ x ]⇧v; ⋀x. ?P [ x ]⇧v; ⋀x1a x2a. ⟦?P x1a; ?P x2a⟧ ⟹ ?P [ x1a , x2a ]⇧v; ⋀x1a x2a x3. ?P x3 ⟹ ?P (V_cons x1a x2a x3); ⋀x1a x2a x3 x4. ?P x4 ⟹ ?P (V_consp x1a x2a x3 x4)⟧ ⟹ ?P ?v›*), auto simp add: subst_vv.simps (*‹[ ?l ]⇧v[?x::=?v]⇩v⇩v = [ ?l ]⇧v› ‹[ ?y ]⇧v[?x::=?v]⇩v⇩v = (if ?x = ?y then ?v else [ ?y ]⇧v)› ‹(V_cons ?tyid ?c ?v')[?x::=?v]⇩v⇩v = V_cons ?tyid ?c ?v'[?x::=?v]⇩v⇩v› ‹(V_consp ?tyid ?c ?b ?v')[?x::=?v]⇩v⇩v = V_consp ?tyid ?c ?b ?v'[?x::=?v]⇩v⇩v› ‹[ ?v1.0 , ?v2.0 ]⇧v[?x::=?v]⇩v⇩v = [ ?v1.0[?x::=?v]⇩v⇩v , ?v2.0[?x::=?v]⇩v⇩v ]⇧v›*) fresh_def (*‹?a ♯ ?x ≡ ?a ∉ supp ?x›*), auto)\n (*goal: ‹j ♯ t[i::=x]⇩v⇩v = (atom i ♯ t ∧ j ♯ t ∨ j ♯ x ∧ (j ♯ t ∨ j = atom i))›*)\n by (((simp add: supp_at_base (*‹supp (?a::?'a) = {atom ?a}›*)) | (metis b.supp (*‹supp B_int = {}› ‹supp B_bool = {}› ‹supp (B_id ?list) = supp ?list› ‹supp [ ?b1.0 , ?b2.0 ]⇧b = supp ?b1.0 ∪ supp ?b2.0› ‹supp B_unit = {}› ‹supp B_bitvec = {}› ‹supp (B_var ?bv) = supp ?bv› ‹supp (B_app ?list ?b) = supp ?list ∪ supp ?b›*) supp_b_empty (*‹atom ?x ∉ supp ?b›*)))+)\n\nlemma forget_subst_vv [simp]: \"atom a ♯ tm ⟹ tm[a::=x]⇩v⇩v = tm\"\n apply (induct tm rule: v.induct (*‹⟦⋀x. ?P [ x ]⇧v; ⋀x. ?P [ x ]⇧v; ⋀x1a x2a. ⟦?P x1a; ?P x2a⟧ ⟹ ?P [ x1a , x2a ]⇧v; ⋀x1a x2a x3. ?P x3 ⟹ ?P (V_cons x1a x2a x3); ⋀x1a x2a x3 x4. ?P x4 ⟹ ?P (V_consp x1a x2a x3 x4)⟧ ⟹ ?P ?v›*))\n (*goals:\n 1. ‹⋀xa. atom a ♯ [ xa ]⇧v ⟹ [ xa ]⇧v[a::=x]⇩v⇩v = [ xa ]⇧v›\n 2. ‹⋀xa. atom a ♯ [ xa ]⇧v ⟹ [ xa ]⇧v[a::=x]⇩v⇩v = [ xa ]⇧v›\n 3. ‹⋀x1a x2a. ⟦atom a ♯ x1a ⟹ x1a[a::=x]⇩v⇩v = x1a; atom a ♯ x2a ⟹ x2a[a::=x]⇩v⇩v = x2a; atom a ♯ [ x1a , x2a ]⇧v⟧ ⟹ [ x1a , x2a ]⇧v[a::=x]⇩v⇩v = [ x1a , x2a ]⇧v›\n 4. ‹⋀x1a x2a x3. ⟦atom a ♯ x3 ⟹ x3[a::=x]⇩v⇩v = x3; atom a ♯ V_cons x1a x2a x3⟧ ⟹ (V_cons x1a x2a x3)[a::=x]⇩v⇩v = V_cons x1a x2a x3›\n 5. ‹⋀x1a x2a x3 x4. ⟦atom a ♯ x4 ⟹ x4[a::=x]⇩v⇩v = x4; atom a ♯ V_consp x1a x2a x3 x4⟧ ⟹ (V_consp x1a x2a x3 x4)[a::=x]⇩v⇩v = V_consp x1a x2a x3 x4›\n discuss goal 1*)\n apply (simp add: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))\n (*discuss goal 2*)\n apply (simp add: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))\n (*discuss goal 3*)\n apply (simp add: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))\n (*discuss goal 4*)\n apply (simp add: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))\n (*discuss goal 5*)\n apply (simp add: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))\n (*proven 5 subgoals*) .\n\nlemma subst_vv_id [simp]: \"tm[a::=V_var a]⇩v⇩v = tm\"\n apply (induct tm rule: v.induct (*‹⟦⋀x. ?P [ x ]⇧v; ⋀x. ?P [ x ]⇧v; ⋀x1a x2a. ⟦?P x1a; ?P x2a⟧ ⟹ ?P [ x1a , x2a ]⇧v; ⋀x1a x2a x3. ?P x3 ⟹ ?P (V_cons x1a x2a x3); ⋀x1a x2a x3 x4. ?P x4 ⟹ ?P (V_consp x1a x2a x3 x4)⟧ ⟹ ?P ?v›*))\n (*goals:\n 1. ‹⋀x. [ x ]⇧v[a::=[ a ]⇧v]⇩v⇩v = [ x ]⇧v›\n 2. ‹⋀x. [ x ]⇧v[a::=[ a ]⇧v]⇩v⇩v = [ x ]⇧v›\n 3. ‹⋀x1a x2a. ⟦x1a[a::=[ a ]⇧v]⇩v⇩v = x1a; x2a[a::=[ a ]⇧v]⇩v⇩v = x2a⟧ ⟹ [ x1a , x2a ]⇧v[a::=[ a ]⇧v]⇩v⇩v = [ x1a , x2a ]⇧v›\n 4. ‹⋀x1a x2a x3. x3[a::=[ a ]⇧v]⇩v⇩v = x3 ⟹ (V_cons x1a x2a x3)[a::=[ a ]⇧v]⇩v⇩v = V_cons x1a x2a x3›\n 5. ‹⋀x1a x2a x3 x4. x4[a::=[ a ]⇧v]⇩v⇩v = x4 ⟹ (V_consp x1a x2a x3 x4)[a::=[ a ]⇧v]⇩v⇩v = V_consp x1a x2a x3 x4›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*proven 5 subgoals*) .\n\nlemma subst_vv_commute [simp]:\n \"atom j ♯ tm ⟹ tm[i::=t]⇩v⇩v[j::=u]⇩v⇩v = tm[i::=t[j::=u]⇩v⇩v]⇩v⇩v \"\n apply (induct tm rule: v.induct (*‹⟦⋀x. ?P [ x ]⇧v; ⋀x. ?P [ x ]⇧v; ⋀x1a x2a. ⟦?P x1a; ?P x2a⟧ ⟹ ?P [ x1a , x2a ]⇧v; ⋀x1a x2a x3. ?P x3 ⟹ ?P (V_cons x1a x2a x3); ⋀x1a x2a x3 x4. ?P x4 ⟹ ?P (V_consp x1a x2a x3 x4)⟧ ⟹ ?P ?v›*))\n (*goals:\n 1. ‹⋀x::l. atom (j::x) ♯ [ x ]⇧v ⟹ [ x ]⇧v[(i::x)::=(t::v)]⇩v⇩v[j::=(u::v)]⇩v⇩v = [ x ]⇧v[i::=t[j::=u]⇩v⇩v]⇩v⇩v›\n 2. ‹⋀x::x. atom (j::x) ♯ [ x ]⇧v ⟹ [ x ]⇧v[(i::x)::=(t::v)]⇩v⇩v[j::=(u::v)]⇩v⇩v = [ x ]⇧v[i::=t[j::=u]⇩v⇩v]⇩v⇩v›\n 3. ‹⋀(x1a::v) x2a::v. ⟦atom (j::x) ♯ x1a ⟹ x1a[(i::x)::=(t::v)]⇩v⇩v[j::=(u::v)]⇩v⇩v = x1a[i::=t[j::=u]⇩v⇩v]⇩v⇩v; atom j ♯ x2a ⟹ x2a[i::=t]⇩v⇩v[j::=u]⇩v⇩v = x2a[i::=t[j::=u]⇩v⇩v]⇩v⇩v; atom j ♯ [ x1a , x2a ]⇧v⟧ ⟹ [ x1a , x2a ]⇧v[i::=t]⇩v⇩v[j::=u]⇩v⇩v = [ x1a , x2a ]⇧v[i::=t[j::=u]⇩v⇩v]⇩v⇩v›\n 4. ‹⋀(x1a::char list) (x2a::char list) x3::v. ⟦atom (j::x) ♯ x3 ⟹ x3[(i::x)::=(t::v)]⇩v⇩v[j::=(u::v)]⇩v⇩v = x3[i::=t[j::=u]⇩v⇩v]⇩v⇩v; atom j ♯ V_cons x1a x2a x3⟧ ⟹ (V_cons x1a x2a x3)[i::=t]⇩v⇩v[j::=u]⇩v⇩v = (V_cons x1a x2a x3)[i::=t[j::=u]⇩v⇩v]⇩v⇩v›\n 5. ‹⋀(x1a::char list) (x2a::char list) (x3::b) x4::v. ⟦atom (j::x) ♯ x4 ⟹ x4[(i::x)::=(t::v)]⇩v⇩v[j::=(u::v)]⇩v⇩v = x4[i::=t[j::=u]⇩v⇩v]⇩v⇩v; atom j ♯ V_consp x1a x2a x3 x4⟧ ⟹ (V_consp x1a x2a x3 x4)[i::=t]⇩v⇩v[j::=u]⇩v⇩v = (V_consp x1a x2a x3 x4)[i::=t[j::=u]⇩v⇩v]⇩v⇩v›\n discuss goal 1*)\n apply ((auto simp: fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*))[1])\n (*proven 5 subgoals*) .\n\nlemma subst_vv_commute_full [simp]:\n \"atom j ♯ t ⟹ atom i ♯ u ⟹ i ≠ j ⟹ tm[i::=t]⇩v⇩v[j::=u]⇩v⇩v = tm[j::=u]⇩v⇩v[i::=t]⇩v⇩v\"\n apply (induct tm rule: v.induct (*‹⟦⋀x. ?P [ x ]⇧v; ⋀x. ?P [ x ]⇧v; ⋀x1a x2a. ⟦?P x1a; ?P x2a⟧ ⟹ ?P [ x1a , x2a ]⇧v; ⋀x1a x2a x3. ?P x3 ⟹ ?P (V_cons x1a x2a x3); ⋀x1a x2a x3 x4. ?P x4 ⟹ ?P (V_consp x1a x2a x3 x4)⟧ ⟹ ?P ?v›*))\n (*goals:\n 1. ‹⋀x. ⟦atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ [ x ]⇧v[i::=t]⇩v⇩v[j::=u]⇩v⇩v = [ x ]⇧v[j::=u]⇩v⇩v[i::=t]⇩v⇩v›\n 2. ‹⋀x. ⟦atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ [ x ]⇧v[i::=t]⇩v⇩v[j::=u]⇩v⇩v = [ x ]⇧v[j::=u]⇩v⇩v[i::=t]⇩v⇩v›\n 3. ‹⋀x1a x2a. ⟦⟦atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ x1a[i::=t]⇩v⇩v[j::=u]⇩v⇩v = x1a[j::=u]⇩v⇩v[i::=t]⇩v⇩v; ⟦atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ x2a[i::=t]⇩v⇩v[j::=u]⇩v⇩v = x2a[j::=u]⇩v⇩v[i::=t]⇩v⇩v; atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ [ x1a , x2a ]⇧v[i::=t]⇩v⇩v[j::=u]⇩v⇩v = [ x1a , x2a ]⇧v[j::=u]⇩v⇩v[i::=t]⇩v⇩v›\n 4. ‹⋀x1a x2a x3. ⟦⟦atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ x3[i::=t]⇩v⇩v[j::=u]⇩v⇩v = x3[j::=u]⇩v⇩v[i::=t]⇩v⇩v; atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ (V_cons x1a x2a x3)[i::=t]⇩v⇩v[j::=u]⇩v⇩v = (V_cons x1a x2a x3)[j::=u]⇩v⇩v[i::=t]⇩v⇩v›\n 5. ‹⋀x1a x2a x3 x4. ⟦⟦atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ x4[i::=t]⇩v⇩v[j::=u]⇩v⇩v = x4[j::=u]⇩v⇩v[i::=t]⇩v⇩v; atom j ♯ t; atom i ♯ u; i ≠ j⟧ ⟹ (V_consp x1a x2a x3 x4)[i::=t]⇩v⇩v[j::=u]⇩v⇩v = (V_consp x1a x2a x3 x4)[j::=u]⇩v⇩v[i::=t]⇩v⇩v›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*proven 5 subgoals*) .\n\nlemma subst_vv_var_flip[simp]:\n fixes v::v\n assumes \"atom y ♯ v\"\n shows \"(y ↔ x) ∙ v = v [x::=V_var y]⇩v⇩v\"\n using assms (*‹atom y ♯ v›*) apply (induct v rule:v.induct (*‹⟦⋀x::l. (?P::v ⇒ bool) [ x ]⇧v; ⋀x::x. ?P [ x ]⇧v; ⋀(x1a::v) x2a::v. ⟦?P x1a; ?P x2a⟧ ⟹ ?P [ x1a , x2a ]⇧v; ⋀(x1a::char list) (x2a::char list) x3::v. ?P x3 ⟹ ?P (V_cons x1a x2a x3); ⋀(x1a::char list) (x2a::char list) (x3::b) x4::v. ?P x4 ⟹ ?P (V_consp x1a x2a x3 x4)⟧ ⟹ ?P (?v::v)›*))\n (*goal: ‹(y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v›*)\n apply auto\n (*top goal: ‹⋀xa. atom y ♯ [ xa ]⇧v ⟹ (y ↔ x) ∙ [ xa ]⇧v = [ xa ]⇧v[x::=[ y ]⇧v]⇩v⇩v› and 4 goals remain*)\n using l.fresh (*‹?a ♯ L_num ?int = ?a ♯ ?int› ‹?a ♯ L_true› ‹(?a::atom) ♯ L_false› ‹?a ♯ L_unit› ‹?a ♯ L_bitvec ?list = ?a ♯ ?list›*) l.perm_simps (*‹?p ∙ L_num ?int = L_num (?p ∙ ?int)› ‹?p ∙ L_true = L_true› ‹?p ∙ L_false = L_false› ‹?p ∙ L_unit = L_unit› ‹?p ∙ L_bitvec ?list = L_bitvec (?p ∙ ?list)›*) l.strong_exhaust (*‹⟦⋀x1. ?y = L_num x1 ⟹ ?P; ?y = L_true ⟹ ?P; ?y = L_false ⟹ ?P; ?y = L_unit ⟹ ?P; ⋀x5. ?y = L_bitvec x5 ⟹ ?P⟧ ⟹ ?P›*) supp_l_empty (*‹supp [ ?l ]⇧v = {}›*) permute_pure (*‹?p ∙ ?x = ?x›*) permute_list.simps (*‹?p ∙ [] = []› ‹?p ∙ (?x # ?xs) = ?p ∙ ?x # ?p ∙ ?xs›*) fresh_def (*‹?a ♯ ?x ≡ ?a ∉ supp ?x›*) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) apply fastforce\n (*top goal: ‹⋀xa. atom y ♯ xa ⟹ (y ↔ x) ∙ xa = xa› and 4 goals remain*)\n using permute_pure (*‹?p ∙ ?x = ?x›*) apply -\n (*goals:\n 1. ‹⋀x1a x2a x3. ⟦(y ↔ x) ∙ x3 = x3[x::=[ y ]⇧v]⇩v⇩v; atom y ♯ x1a; atom y ♯ x2a; atom y ♯ x3; ⋀p x. p ∙ x = x⟧ ⟹ (y ↔ x) ∙ x1a = x1a›\n 2. ‹⋀x1a x2a x3. ⟦(y ↔ x) ∙ x3 = x3[x::=[ y ]⇧v]⇩v⇩v; atom y ♯ x1a; atom y ♯ x2a; atom y ♯ x3; ⋀p x. p ∙ x = x⟧ ⟹ (y ↔ x) ∙ x2a = x2a›\n 3. ‹⋀x1a x2a x4. ⟦(y ↔ x) ∙ x4 = x4[x::=[ y ]⇧v]⇩v⇩v; atom y ♯ x1a; atom y ♯ x2a; atom y ♯ x4; ⋀p x. p ∙ x = x⟧ ⟹ (y ↔ x) ∙ x1a = x1a›\n 4. ‹⋀x1a x2a x4. ⟦(y ↔ x) ∙ x4 = x4[x::=[ y ]⇧v]⇩v⇩v; atom y ♯ x1a; atom y ♯ x2a; atom y ♯ x4; ⋀p x. p ∙ x = x⟧ ⟹ (y ↔ x) ∙ x2a = x2a›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*discuss goal 4*)\n apply blast\n (*proven 4 subgoals*) .\n\ninstantiation v :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_vv\"\n\ninstance proof\n fix j::atom and i::x and x::v and t::v\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n using fresh_subst_vv_if[of j t i x] subst_v_v_def by metis\n\n fix a::x and tm::v and x::v\n show \"atom a ♯ tm ⟹ subst_v tm a x = tm\"\n using forget_subst_vv subst_v_v_def by simp\n\n fix a::x and tm::v\n show \"subst_v tm a (V_var a) = tm\" using subst_vv_id subst_v_v_def by simp\n\n fix p::perm and x1::x and v::v and t1::v\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n using subst_v_v_def by simp\n\n fix x::x and c::v and z::x\n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\"\n using subst_v_v_def by simp\n\n fix x::x and c::v and z::x\n show \"atom x ♯ c ⟹ c[z::=[x]⇧v]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n using subst_v_v_def by simp\nqed\n\nend\n\nsection ‹Expressions›\n\nnominal_function subst_ev :: \"e ⇒ x ⇒ v ⇒ e\" where\n \"subst_ev ( (AE_val v') ) x v = ( (AE_val (subst_vv v' x v)) )\"\n| \"subst_ev ( (AE_app f v') ) x v = ( (AE_app f (subst_vv v' x v )) )\" \n| \"subst_ev ( (AE_appP f b v') ) x v = ( (AE_appP f b (subst_vv v' x v )) )\" \n| \"subst_ev ( (AE_op opp v1 v2) ) x v = ( (AE_op opp (subst_vv v1 x v ) (subst_vv v2 x v )) )\"\n| \"subst_ev [#1 v']⇧e x v = [#1 (subst_vv v' x v )]⇧e\"\n| \"subst_ev [#2 v']⇧e x v = [#2 (subst_vv v' x v )]⇧e\"\n| \"subst_ev ( (AE_mvar u)) x v = AE_mvar u\"\n| \"subst_ev [| v' |]⇧e x v = [| (subst_vv v' x v ) |]⇧e\"\n| \"subst_ev ( AE_concat v1 v2) x v = AE_concat (subst_vv v1 x v ) (subst_vv v2 x v )\"\n| \"subst_ev ( AE_split v1 v2) x v = AE_split (subst_vv v1 x v ) (subst_vv v2 x v )\"\n by(simp add: eqvt_def subst_ev_graph_aux_def,auto)(meson e.strong_exhaust)\n\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_ev_abbrev :: \"e ⇒ x ⇒ v ⇒ e\" (\"_[_::=_]⇩e⇩v\" [1000,50,50] 500)\n where \n \"e[x::=v']⇩e⇩v ≡ subst_ev e x v' \" \n\nlemma size_subst_ev [simp]: \"size ( subst_ev A i x) = size A\"\n apply (nominal_induct A avoiding: i x rule: e.strong_induct)\n (*goals:\n 1. ‹⋀x i xa. size ([ x ]⇧e[i::=xa]⇩e⇩v) = size [ x ]⇧e›\n 2. ‹⋀x1a x2 i x. size ([ x1a x2 ]⇧e[i::=x]⇩e⇩v) = size [ x1a x2 ]⇧e›\n 3. ‹⋀x1a x2 x3 i x. size ([x1a [ x2 ] x3 ]⇧e[i::=x]⇩e⇩v) = size [x1a [ x2 ] x3 ]⇧e›\n 4. ‹⋀x1a x2 x3 i x. size ([ x1a x2 x3 ]⇧e[i::=x]⇩e⇩v) = size [ x1a x2 x3 ]⇧e›\n 5. ‹⋀x1a x2 i x. size ([ x1a @@ x2 ]⇧e[i::=x]⇩e⇩v) = size [ x1a @@ x2 ]⇧e›\n 6. ‹⋀x i xa. size ([#1x ]⇧e[i::=xa]⇩e⇩v) = size [#1x ]⇧e›\n 7. ‹⋀x i xa. size ([#2x ]⇧e[i::=xa]⇩e⇩v) = size [#2x ]⇧e›\n 8. ‹⋀x i xa. size ([ x ]⇧e[i::=xa]⇩e⇩v) = size [ x ]⇧e›\n 9. ‹⋀x i xa. size ([| x |]⇧e[i::=xa]⇩e⇩v) = size [| x |]⇧e›\n 10. ‹⋀x1a x2 i x. size ([ x1a x2 ]⇧e[i::=x]⇩e⇩v) = size [ x1a x2 ]⇧e›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*proven 10 subgoals*) .\n\nlemma forget_subst_ev [simp]: \"atom a ♯ A ⟹ subst_ev A a x = A\"\n apply (nominal_induct A avoiding: a x rule: e.strong_induct)\n (*goals:\n 1. ‹⋀x a xa. atom a ♯ [ x ]⇧e ⟹ [ x ]⇧e[a::=xa]⇩e⇩v = [ x ]⇧e›\n 2. ‹⋀x1a x2 a x. atom a ♯ [ x1a x2 ]⇧e ⟹ [ x1a x2 ]⇧e[a::=x]⇩e⇩v = [ x1a x2 ]⇧e›\n 3. ‹⋀x1a x2 x3 a x. atom a ♯ [x1a [ x2 ] x3 ]⇧e ⟹ [x1a [ x2 ] x3 ]⇧e[a::=x]⇩e⇩v = [x1a [ x2 ] x3 ]⇧e›\n 4. ‹⋀x1a x2 x3 a x. atom a ♯ [ x1a x2 x3 ]⇧e ⟹ [ x1a x2 x3 ]⇧e[a::=x]⇩e⇩v = [ x1a x2 x3 ]⇧e›\n 5. ‹⋀x1a x2 a x. atom a ♯ [ x1a @@ x2 ]⇧e ⟹ [ x1a @@ x2 ]⇧e[a::=x]⇩e⇩v = [ x1a @@ x2 ]⇧e›\n 6. ‹⋀x a xa. atom a ♯ [#1x ]⇧e ⟹ [#1x ]⇧e[a::=xa]⇩e⇩v = [#1x ]⇧e›\n 7. ‹⋀x a xa. atom a ♯ [#2x ]⇧e ⟹ [#2x ]⇧e[a::=xa]⇩e⇩v = [#2x ]⇧e›\n 8. ‹⋀x a xa. atom a ♯ [ x ]⇧e ⟹ [ x ]⇧e[a::=xa]⇩e⇩v = [ x ]⇧e›\n 9. ‹⋀x a xa. atom a ♯ [| x |]⇧e ⟹ [| x |]⇧e[a::=xa]⇩e⇩v = [| x |]⇧e›\n 10. ‹⋀x1a x2 a x. atom a ♯ [ x1a x2 ]⇧e ⟹ [ x1a x2 ]⇧e[a::=x]⇩e⇩v = [ x1a x2 ]⇧e›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a::at_base)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a::at_base) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 7*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 8*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 9*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 10*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*proven 10 subgoals*) .\n\nlemma subst_ev_id [simp]: \"subst_ev A a (V_var a) = A\"\n apply (nominal_induct A avoiding: a rule: e.strong_induct)\n (*goals:\n 1. ‹⋀(x::v) a::x. [ x ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [ x ]⇧e›\n 2. ‹⋀(x1a::char list) (x2::v) a::x. [ x1a x2 ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [ x1a x2 ]⇧e›\n 3. ‹⋀(x1a::char list) (x2::b) (x3::v) a::x. [x1a [ x2 ] x3 ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [x1a [ x2 ] x3 ]⇧e›\n 4. ‹⋀(x1a::opp) (x2::v) (x3::v) a::x. [ x1a x2 x3 ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [ x1a x2 x3 ]⇧e›\n 5. ‹⋀(x1a::v) (x2::v) a::x. [ x1a @@ x2 ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [ x1a @@ x2 ]⇧e›\n 6. ‹⋀(x::v) a::x. [#1x ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [#1x ]⇧e›\n 7. ‹⋀(x::v) a::x. [#2x ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [#2x ]⇧e›\n 8. ‹⋀(x::u) a::x. [ x ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [ x ]⇧e›\n 9. ‹⋀(x::v) a::x. [| x |]⇧e[a::=[ a ]⇧v]⇩e⇩v = [| x |]⇧e›\n 10. ‹⋀(x1a::v) (x2::v) a::x. [ x1a x2 ]⇧e[a::=[ a ]⇧v]⇩e⇩v = [ x1a x2 ]⇧e›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a::at_base)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a::at_base) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 7*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 8*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 9*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 10*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a::at_base)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a::at_base) = (?a ≠ atom ?b)›*))[1])\n (*proven 10 subgoals*) .\n\nlemma fresh_subst_ev_if [simp]:\n \"j ♯ (subst_ev A i x ) = ((atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i)))\"\n apply (induct A rule: e.induct (*‹⟦⋀x::v. (?P::e ⇒ bool) [ x ]⇧e; ⋀(x1a::char list) x2::v. ?P [ x1a x2 ]⇧e; ⋀(x1a::char list) (x2::b) x3::v. ?P [x1a [ x2 ] x3 ]⇧e; ⋀(x1a::opp) (x2::v) x3::v. ?P [ x1a x2 x3 ]⇧e; ⋀(x1a::v) x2::v. ?P [ x1a @@ x2 ]⇧e; ⋀x::v. ?P [#1x ]⇧e; ⋀x::v. ?P [#2x ]⇧e; ⋀x::u. ?P [ x ]⇧e; ⋀x::v. ?P [| x |]⇧e; ⋀(x1a::v) x2::v. ?P [ x1a x2 ]⇧e⟧ ⟹ ?P (?e::e)›*))\n (*goal: ‹(j::atom) ♯ (A::e)[(i::x)::=(x::v)]⇩e⇩v = (atom i ♯ A ∧ j ♯ A ∨ j ♯ x ∧ (j ♯ A ∨ j = atom i))›*)\n unfolding subst_ev.simps fresh_subst_vv_if\n (*goals:\n 1. ‹⋀xa. j ♯ [ xa[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [ xa ]⇧e ∧ j ♯ [ xa ]⇧e ∨ j ♯ x ∧ (j ♯ [ xa ]⇧e ∨ j = atom i))›\n 2. ‹⋀x1a x2. j ♯ [ x1a x2[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [ x1a x2 ]⇧e ∧ j ♯ [ x1a x2 ]⇧e ∨ j ♯ x ∧ (j ♯ [ x1a x2 ]⇧e ∨ j = atom i))›\n 3. ‹⋀x1a x2 x3. j ♯ [x1a [ x2 ] x3[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [x1a [ x2 ] x3 ]⇧e ∧ j ♯ [x1a [ x2 ] x3 ]⇧e ∨ j ♯ x ∧ (j ♯ [x1a [ x2 ] x3 ]⇧e ∨ j = atom i))›\n 4. ‹⋀x1a x2 x3. j ♯ [ x1a x2[i::=x]⇩v⇩v x3[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [ x1a x2 x3 ]⇧e ∧ j ♯ [ x1a x2 x3 ]⇧e ∨ j ♯ x ∧ (j ♯ [ x1a x2 x3 ]⇧e ∨ j = atom i))›\n 5. ‹⋀x1a x2. j ♯ [ x1a[i::=x]⇩v⇩v @@ x2[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [ x1a @@ x2 ]⇧e ∧ j ♯ [ x1a @@ x2 ]⇧e ∨ j ♯ x ∧ (j ♯ [ x1a @@ x2 ]⇧e ∨ j = atom i))›\n 6. ‹⋀xa. j ♯ [#1xa[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [#1xa ]⇧e ∧ j ♯ [#1xa ]⇧e ∨ j ♯ x ∧ (j ♯ [#1xa ]⇧e ∨ j = atom i))›\n 7. ‹⋀xa. j ♯ [#2xa[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [#2xa ]⇧e ∧ j ♯ [#2xa ]⇧e ∨ j ♯ x ∧ (j ♯ [#2xa ]⇧e ∨ j = atom i))›\n 8. ‹⋀xa. j ♯ [ xa ]⇧e = (atom i ♯ [ xa ]⇧e ∧ j ♯ [ xa ]⇧e ∨ j ♯ x ∧ (j ♯ [ xa ]⇧e ∨ j = atom i))›\n 9. ‹⋀xa. j ♯ [| xa[i::=x]⇩v⇩v |]⇧e = (atom i ♯ [| xa |]⇧e ∧ j ♯ [| xa |]⇧e ∨ j ♯ x ∧ (j ♯ [| xa |]⇧e ∨ j = atom i))›\n 10. ‹⋀x1a x2. j ♯ [ x1a[i::=x]⇩v⇩v x2[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [ x1a x2 ]⇧e ∧ j ♯ [ x1a x2 ]⇧e ∨ j ♯ x ∧ (j ♯ [ x1a x2 ]⇧e ∨ j = atom i))›*)\n apply ((auto)+)\n (*top goal: ‹⋀xa. j ♯ [ xa[i::=x]⇩v⇩v ]⇧e = (atom i ♯ [ xa ]⇧e ∧ j ♯ [ xa ]⇧e ∨ j ♯ x ∧ (j ♯ [ xa ]⇧e ∨ j = atom i))› and 9 goals remain*)\n using pure_fresh (*‹?a ♯ ?x›*) fresh_opp_all (*‹(?z::atom) ♯ (?opp::opp)›*) by ((metis)+)\n\nlemma subst_ev_commute [simp]:\n \"atom j ♯ A ⟹ (A[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = A[i::=t[j::=u]⇩v⇩v]⇩e⇩v\"\n apply (nominal_induct A avoiding: i j t u rule: e.strong_induct)\n (*goals:\n 1. ‹⋀x i j t u. atom j ♯ [ x ]⇧e ⟹ ([ x ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [ x ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 2. ‹⋀x1a x2 i j t u. atom j ♯ [ x1a x2 ]⇧e ⟹ ([ x1a x2 ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [ x1a x2 ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 3. ‹⋀x1a x2 x3 i j t u. atom j ♯ [x1a [ x2 ] x3 ]⇧e ⟹ ([x1a [ x2 ] x3 ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [x1a [ x2 ] x3 ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 4. ‹⋀x1a x2 x3 i j t u. atom j ♯ [ x1a x2 x3 ]⇧e ⟹ ([ x1a x2 x3 ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [ x1a x2 x3 ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 5. ‹⋀x1a x2 i j t u. atom j ♯ [ x1a @@ x2 ]⇧e ⟹ ([ x1a @@ x2 ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [ x1a @@ x2 ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 6. ‹⋀x i j t u. atom j ♯ [#1x ]⇧e ⟹ ([#1x ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [#1x ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 7. ‹⋀x i j t u. atom j ♯ [#2x ]⇧e ⟹ ([#2x ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [#2x ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 8. ‹⋀x i j t u. atom j ♯ [ x ]⇧e ⟹ ([ x ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [ x ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 9. ‹⋀x i j t u. atom j ♯ [| x |]⇧e ⟹ ([| x |]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [| x |]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n 10. ‹⋀x1a x2 i j t u. atom j ♯ [ x1a x2 ]⇧e ⟹ ([ x1a x2 ]⇧e[i::=t]⇩e⇩v)[j::=u]⇩e⇩v = [ x1a x2 ]⇧e[i::=t[j::=u]⇩v⇩v]⇩e⇩v›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 7*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 8*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 9*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 10*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*proven 10 subgoals*) .\n\nlemma subst_ev_var_flip[simp]:\n fixes e::e and y::x and x::x\n assumes \"atom y ♯ e\"\n shows \"(y ↔ x) ∙ e = e [x::=V_var y]⇩e⇩v\"\n using assms (*‹atom (y::x) ♯ (e::e)›*) apply (nominal_induct e rule:e.strong_induct)\n (*goal: ‹(y ↔ x) ∙ e = e[x::=[ y ]⇧v]⇩e⇩v›*)\n apply (simp add: subst_v_v_def (*‹subst_v = subst_vv_abbrev›*))\n (*top goal: ‹⋀xa::v. atom (y::x) ♯ [ xa ]⇧e ⟹ (y ↔ x::x) ∙ [ xa ]⇧e = [ xa ]⇧e[x::=[ y ]⇧v]⇩e⇩v› and 9 goals remain*)\n apply (metis (mono_tags, lifting) b.eq_iff (*‹(B_int = B_int) = True› ‹(B_bool = B_bool) = True› ‹(B_id ?list = B_id ?lista) = (?list = ?lista)› ‹([ ?b1.0 , ?b2.0 ]⇧b = [ ?b1a , ?b2a ]⇧b) = (?b1.0 = ?b1a ∧ ?b2.0 = ?b2a)› ‹(B_unit = B_unit) = True› ‹(B_bitvec = B_bitvec) = True› ‹(B_var ?bv = B_var ?bva) = (?bv = ?bva)› ‹(B_app ?list ?b = B_app ?lista ?ba) = (?list = ?lista ∧ ?b = ?ba)›*) b.perm_simps (*‹?p ∙ B_int = B_int› ‹?p ∙ B_bool = B_bool› ‹?p ∙ B_id ?list = B_id (?p ∙ ?list)› ‹?p ∙ [ ?b1.0 , ?b2.0 ]⇧b = [ ?p ∙ ?b1.0 , ?p ∙ ?b2.0 ]⇧b› ‹?p ∙ B_unit = B_unit› ‹?p ∙ B_bitvec = B_bitvec› ‹?p ∙ B_var ?bv = B_var (?p ∙ ?bv)› ‹?p ∙ B_app ?list ?b = B_app (?p ∙ ?list) (?p ∙ ?b)›*) e.fresh (*‹?a ♯ [ ?v ]⇧e = ?a ♯ ?v› ‹?a ♯ [ ?list ?v ]⇧e = (?a ♯ ?list ∧ ?a ♯ ?v)› ‹?a ♯ [?list [ ?b ] ?v ]⇧e = (?a ♯ ?list ∧ ?a ♯ ?b ∧ ?a ♯ ?v)› ‹?a ♯ [ ?opp ?v1.0 ?v2.0 ]⇧e = (?a ♯ ?opp ∧ ?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)› ‹?a ♯ [ ?v1.0 @@ ?v2.0 ]⇧e = (?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)› ‹?a ♯ [#1?v ]⇧e = ?a ♯ ?v› ‹?a ♯ [#2?v ]⇧e = ?a ♯ ?v› ‹?a ♯ [ ?u ]⇧e = ?a ♯ ?u› ‹?a ♯ [| ?v |]⇧e = ?a ♯ ?v› ‹?a ♯ [ ?v1.0 ?v2.0 ]⇧e = (?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)›*) e.perm_simps (*‹?p ∙ [ ?v ]⇧e = [ ?p ∙ ?v ]⇧e› ‹?p ∙ [ ?list ?v ]⇧e = [ ?p ∙ ?list ?p ∙ ?v ]⇧e› ‹?p ∙ [?list [ ?b ] ?v ]⇧e = [?p ∙ ?list [ ?p ∙ ?b ] ?p ∙ ?v ]⇧e› ‹?p ∙ [ ?opp ?v1.0 ?v2.0 ]⇧e = [ ?p ∙ ?opp ?p ∙ ?v1.0 ?p ∙ ?v2.0 ]⇧e› ‹?p ∙ [ ?v1.0 @@ ?v2.0 ]⇧e = [ ?p ∙ ?v1.0 @@ ?p ∙ ?v2.0 ]⇧e› ‹?p ∙ [#1?v ]⇧e = [#1?p ∙ ?v ]⇧e› ‹?p ∙ [#2?v ]⇧e = [#2?p ∙ ?v ]⇧e› ‹?p ∙ [ ?u ]⇧e = [ ?p ∙ ?u ]⇧e› ‹?p ∙ [| ?v |]⇧e = [| ?p ∙ ?v |]⇧e› ‹?p ∙ [ ?v1.0 ?v2.0 ]⇧e = [ ?p ∙ ?v1.0 ?p ∙ ?v2.0 ]⇧e›*) flip_b_id (*‹(?x ↔ ?x') ∙ ?b = ?b›*) subst_ev.simps (*‹[ ?v' ]⇧e[?x::=?v]⇩e⇩v = [ ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?f ?v' ]⇧e[?x::=?v]⇩e⇩v = [ ?f ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[?f [ ?b ] ?v' ]⇧e[?x::=?v]⇩e⇩v = [?f [ ?b ] ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?opp ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?opp ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[#1?v' ]⇧e[?x::=?v]⇩e⇩v = [#1?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[#2?v' ]⇧e[?x::=?v]⇩e⇩v = [#2?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?u ]⇧e[?x::=?v]⇩e⇩v = [ ?u ]⇧e› ‹[| ?v' |]⇧e[?x::=?v]⇩e⇩v = [| ?v'[?x::=?v]⇩v⇩v |]⇧e› ‹[ ?v1.0 @@ ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v @@ ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e›*) subst_vv_var_flip (*‹atom ?y ♯ ?v ⟹ (?y ↔ ?x) ∙ ?v = ?v[?x::=[ ?y ]⇧v]⇩v⇩v›*))\n (*top goal: ‹⋀x1a x2. atom y ♯ [ x1a x2 ]⇧e ⟹ (y ↔ x) ∙ [ x1a x2 ]⇧e = [ x1a x2 ]⇧e[x::=[ y ]⇧v]⇩e⇩v› and 8 goals remain*)\n apply (metis (mono_tags, lifting) b.eq_iff (*‹(B_int = B_int) = True› ‹(B_bool = B_bool) = True› ‹(B_id ?list = B_id ?lista) = (?list = ?lista)› ‹([ ?b1.0 , ?b2.0 ]⇧b = [ ?b1a , ?b2a ]⇧b) = (?b1.0 = ?b1a ∧ ?b2.0 = ?b2a)› ‹(B_unit = B_unit) = True› ‹(B_bitvec = B_bitvec) = True› ‹(B_var ?bv = B_var ?bva) = (?bv = ?bva)› ‹(B_app ?list ?b = B_app ?lista ?ba) = (?list = ?lista ∧ ?b = ?ba)›*) b.perm_simps (*‹?p ∙ B_int = B_int› ‹?p ∙ B_bool = B_bool› ‹?p ∙ B_id ?list = B_id (?p ∙ ?list)› ‹?p ∙ [ ?b1.0 , ?b2.0 ]⇧b = [ ?p ∙ ?b1.0 , ?p ∙ ?b2.0 ]⇧b› ‹?p ∙ B_unit = B_unit› ‹?p ∙ B_bitvec = B_bitvec› ‹?p ∙ B_var ?bv = B_var (?p ∙ ?bv)› ‹?p ∙ B_app ?list ?b = B_app (?p ∙ ?list) (?p ∙ ?b)›*) e.fresh (*‹?a ♯ [ ?v ]⇧e = ?a ♯ ?v› ‹?a ♯ [ ?list ?v ]⇧e = (?a ♯ ?list ∧ ?a ♯ ?v)› ‹?a ♯ [?list [ ?b ] ?v ]⇧e = (?a ♯ ?list ∧ ?a ♯ ?b ∧ ?a ♯ ?v)› ‹?a ♯ [ ?opp ?v1.0 ?v2.0 ]⇧e = (?a ♯ ?opp ∧ ?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)› ‹?a ♯ [ ?v1.0 @@ ?v2.0 ]⇧e = (?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)› ‹?a ♯ [#1?v ]⇧e = ?a ♯ ?v› ‹?a ♯ [#2?v ]⇧e = ?a ♯ ?v› ‹?a ♯ [ ?u ]⇧e = ?a ♯ ?u› ‹?a ♯ [| ?v |]⇧e = ?a ♯ ?v› ‹?a ♯ [ ?v1.0 ?v2.0 ]⇧e = (?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)›*) e.perm_simps (*‹?p ∙ [ ?v ]⇧e = [ ?p ∙ ?v ]⇧e› ‹?p ∙ [ ?list ?v ]⇧e = [ ?p ∙ ?list ?p ∙ ?v ]⇧e› ‹?p ∙ [?list [ ?b ] ?v ]⇧e = [?p ∙ ?list [ ?p ∙ ?b ] ?p ∙ ?v ]⇧e› ‹?p ∙ [ ?opp ?v1.0 ?v2.0 ]⇧e = [ ?p ∙ ?opp ?p ∙ ?v1.0 ?p ∙ ?v2.0 ]⇧e› ‹?p ∙ [ ?v1.0 @@ ?v2.0 ]⇧e = [ ?p ∙ ?v1.0 @@ ?p ∙ ?v2.0 ]⇧e› ‹?p ∙ [#1?v ]⇧e = [#1?p ∙ ?v ]⇧e› ‹?p ∙ [#2?v ]⇧e = [#2?p ∙ ?v ]⇧e› ‹?p ∙ [ ?u ]⇧e = [ ?p ∙ ?u ]⇧e› ‹?p ∙ [| ?v |]⇧e = [| ?p ∙ ?v |]⇧e› ‹?p ∙ [ ?v1.0 ?v2.0 ]⇧e = [ ?p ∙ ?v1.0 ?p ∙ ?v2.0 ]⇧e›*) flip_b_id (*‹(?x ↔ ?x') ∙ ?b = ?b›*) subst_ev.simps (*‹[ ?v' ]⇧e[?x::=?v]⇩e⇩v = [ ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?f ?v' ]⇧e[?x::=?v]⇩e⇩v = [ ?f ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[?f [ ?b ] ?v' ]⇧e[?x::=?v]⇩e⇩v = [?f [ ?b ] ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?opp ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?opp ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[#1?v' ]⇧e[?x::=?v]⇩e⇩v = [#1?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[#2?v' ]⇧e[?x::=?v]⇩e⇩v = [#2?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?u ]⇧e[?x::=?v]⇩e⇩v = [ ?u ]⇧e› ‹[| ?v' |]⇧e[?x::=?v]⇩e⇩v = [| ?v'[?x::=?v]⇩v⇩v |]⇧e› ‹[ ?v1.0 @@ ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v @@ ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e›*) subst_vv_var_flip (*‹atom ?y ♯ ?v ⟹ (?y ↔ ?x) ∙ ?v = ?v[?x::=[ ?y ]⇧v]⇩v⇩v›*))\n (*top goal: ‹⋀x1a x2 x3. atom y ♯ [x1a [ x2 ] x3 ]⇧e ⟹ (y ↔ x) ∙ [x1a [ x2 ] x3 ]⇧e = [x1a [ x2 ] x3 ]⇧e[x::=[ y ]⇧v]⇩e⇩v› and 7 goals remain*)\n subgoal for x\n apply (rule_tac y=x in opp.strong_exhaust (*‹⟦?y = plus ⟹ ?P; ?y = leq ⟹ ?P; ?y = eq ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹atom y ♯ [ x x2_ x3_ ]⇧e ⟹ (y ↔ x) ∙ [ x x2_ x3_ ]⇧e = [ x x2_ x3_ ]⇧e[x::=[ y ]⇧v]⇩e⇩v›*)\n using subst_vv_var_flip (*‹atom (?y::x) ♯ (?v::v) ⟹ (?y ↔ ?x::x) ∙ ?v = ?v[?x::=[ ?y ]⇧v]⇩v⇩v›*) flip_def (*‹(?a::?'a ↔ ?b::?'a) = (atom ?a ⇌ atom ?b)›*) apply -\n (*goals:\n 1. ‹⟦atom y ♯ [ x x2_ x3_ ]⇧e; x = plus; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [ x x2_ x3_ ]⇧e = [ x x2_ x3_ ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n 2. ‹⟦atom y ♯ [ x x2_ x3_ ]⇧e; x = leq; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [ x x2_ x3_ ]⇧e = [ x x2_ x3_ ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n 3. ‹⟦atom y ♯ [ x x2_ x3_ ]⇧e; x = eq; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [ x x2_ x3_ ]⇧e = [ x x2_ x3_ ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n discuss goal 1*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*discuss goal 2*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*discuss goal 3*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*proven 3 subgoals*) .\n using subst_vv_var_flip (*‹atom ?y ♯ ?v ⟹ (?y ↔ ?x) ∙ ?v = ?v[?x::=[ ?y ]⇧v]⇩v⇩v›*) flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) apply -\n (*goals:\n 1. ‹⋀x1a x2. ⟦atom y ♯ [ x1a @@ x2 ]⇧e; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [ x1a @@ x2 ]⇧e = [ x1a @@ x2 ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n 2. ‹⋀xa. ⟦atom y ♯ [#1xa ]⇧e; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [#1xa ]⇧e = [#1xa ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n 3. ‹⋀xa. ⟦atom y ♯ [#2xa ]⇧e; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [#2xa ]⇧e = [#2xa ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n 4. ‹⋀xa. ⟦atom y ♯ [ xa ]⇧e; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [ xa ]⇧e = [ xa ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n 5. ‹⋀xa. ⟦atom y ♯ [| xa |]⇧e; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [| xa |]⇧e = [| xa |]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n 6. ‹⋀x1a x2. ⟦atom y ♯ [ x1a x2 ]⇧e; ⋀y v x. atom y ♯ v ⟹ (y ↔ x) ∙ v = v[x::=[ y ]⇧v]⇩v⇩v; ⋀a b. (a ↔ b) = (atom a ⇌ atom b)⟧ ⟹ (y ↔ x) ∙ [ x1a x2 ]⇧e = [ x1a x2 ]⇧e[x::=[ y ]⇧v]⇩e⇩v›\n discuss goal 1*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*discuss goal 2*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*discuss goal 3*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*discuss goal 4*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*discuss goal 5*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*discuss goal 6*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) permute_pure (*‹?p ∙ ?x = ?x›*))\n (*proven 6 subgoals*) .\n\nlemma subst_ev_flip:\n fixes e::e and ea::e and c::x\n assumes \"atom c ♯ (e, ea)\" and \"atom c ♯ (x, xa, e, ea)\" and \"(x ↔ c) ∙ e = (xa ↔ c) ∙ ea\" \n shows \"e[x::=v']⇩e⇩v = ea[xa::=v']⇩e⇩v\"\nproof (-)\n (*goal: ‹(e::e)[(x::x)::=(v'::v)]⇩e⇩v = (ea::e)[(xa::x)::=v']⇩e⇩v›*)\n have \"e[x::=v']⇩e⇩v = (e[x::=V_var c]⇩e⇩v)[c::=v']⇩e⇩v\"\n using subst_ev_commute (*‹atom ?j ♯ ?A ⟹ (?A[?i::=?t]⇩e⇩v)[?j::=?u]⇩e⇩v = ?A[?i::=?t[?j::=?u]⇩v⇩v]⇩e⇩v›*) assms (*‹atom (c::x) ♯ (e::e, ea::e)› ‹atom c ♯ (x, xa, e, ea)› ‹(x::x ↔ c::x) ∙ (e::e) = (xa::x ↔ c) ∙ (ea::e)›*) by simp\n also (*calculation: ‹e[x::=v']⇩e⇩v = (e[x::=[ c ]⇧v]⇩e⇩v)[c::=v']⇩e⇩v›*) have \"... = ((c ↔ x) ∙ e)[c::=v']⇩e⇩v\"\n using subst_ev_var_flip (*‹atom ?y ♯ ?e ⟹ (?y ↔ ?x) ∙ ?e = ?e[?x::=[ ?y ]⇧v]⇩e⇩v›*) assms (*‹atom c ♯ (e, ea)› ‹atom (c::x) ♯ (x::x, xa::x, e::e, ea::e)› ‹(x ↔ c) ∙ e = (xa ↔ c) ∙ ea›*) by simp\n also (*calculation: ‹e[x::=v']⇩e⇩v = ((c ↔ x) ∙ e)[c::=v']⇩e⇩v›*) have \"... = ((c ↔ xa) ∙ ea)[c::=v']⇩e⇩v\"\n using assms (*‹atom c ♯ (e, ea)› ‹atom c ♯ (x, xa, e, ea)› ‹(x ↔ c) ∙ e = (xa ↔ c) ∙ ea›*) flip_commute (*‹(?a ↔ ?b) = (?b ↔ ?a)›*) by metis\n also (*calculation: ‹(e::e)[(x::x)::=(v'::v)]⇩e⇩v = ((c::x ↔ xa::x) ∙ (ea::e))[c::=v']⇩e⇩v›*) have \"... = ea[xa::=v']⇩e⇩v\"\n using subst_ev_var_flip (*‹atom ?y ♯ ?e ⟹ (?y ↔ ?x) ∙ ?e = ?e[?x::=[ ?y ]⇧v]⇩e⇩v›*) assms (*‹atom c ♯ (e, ea)› ‹atom c ♯ (x, xa, e, ea)› ‹(x ↔ c) ∙ e = (xa ↔ c) ∙ ea›*) by simp\n finally (*calculation: ‹e[x::=v']⇩e⇩v = ea[xa::=v']⇩e⇩v›*) show \"?thesis\"\n (*goal: ‹e[x::=v']⇩e⇩v = ea[xa::=v']⇩e⇩v›*)\n by auto\nqed\n\nlemma subst_ev_var[simp]:\n \"(AE_val (V_var x))[x::=[z]⇧v]⇩e⇩v = AE_val (V_var z)\"\n by auto\n\ninstantiation e :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_ev\"\n\ninstance proof\n fix j::atom and i::x and x::v and t::e\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n using fresh_subst_ev_if[of j t i x] subst_v_e_def by metis\n\n fix a::x and tm::e and x::v\n show \"atom a ♯ tm ⟹ subst_v tm a x = tm\"\n using forget_subst_ev subst_v_e_def by simp\n\n fix a::x and tm::e\n show \"subst_v tm a (V_var a) = tm\" using subst_ev_id subst_v_e_def by simp\n\n fix p::perm and x1::x and v::v and t1::e\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n using subst_ev_commute subst_v_e_def by simp\n\n fix x::x and c::e and z::x\n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\"\n using subst_v_e_def by simp\n\n fix x::x and c::e and z::x\n show \"atom x ♯ c ⟹ c[z::=[x]⇧v]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n using subst_v_e_def by simp\nqed\nend\n\nlemma subst_ev_commute_full:\n fixes e::e and w::v and v::v\n assumes \"atom z ♯ v\" and \"atom x ♯ w\" and \"x ≠ z\"\n shows \"subst_ev (e[z::=w]⇩e⇩v) x v = subst_ev (e[x::=v]⇩e⇩v) z w\" \n using assms (*‹atom z ♯ v› ‹atom x ♯ w› ‹x ≠ z›*) apply (nominal_induct e rule: e.strong_induct)\n (*goals:\n 1. ‹⋀xa. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ xa ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([ xa ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 2. ‹⋀x1a x2. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ x1a x2 ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([ x1a x2 ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 3. ‹⋀x1a x2 x3. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([x1a [ x2 ] x3 ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([x1a [ x2 ] x3 ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 4. ‹⋀x1a x2 x3. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ x1a x2 x3 ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([ x1a x2 x3 ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 5. ‹⋀x1a x2. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ x1a @@ x2 ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([ x1a @@ x2 ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 6. ‹⋀xa. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([#1xa ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([#1xa ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 7. ‹⋀xa. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([#2xa ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([#2xa ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 8. ‹⋀xa. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ xa ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([ xa ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 9. ‹⋀xa. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([| xa |]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([| xa |]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n 10. ‹⋀x1a x2. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ x1a x2 ]⇧e[z::=w]⇩e⇩v)[x::=v]⇩e⇩v = ([ x1a x2 ]⇧e[x::=v]⇩e⇩v)[z::=w]⇩e⇩v›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*discuss goal 8*)\n apply simp\n (*discuss goal 9*)\n apply simp\n (*discuss goal 10*)\n apply simp\n (*proven 10 subgoals*) .\n\nlemma subst_ev_v_flip1[simp]:\n fixes e::e\n assumes \"atom z1 ♯ (z,e)\" and \"atom z1' ♯ (z,e)\"\n shows\"(z1 ↔ z1') ∙ e[z::=v]⇩e⇩v = e[z::= ((z1 ↔ z1') ∙ v)]⇩e⇩v\"\n using assms (*‹atom z1 ♯ (z, e)› ‹atom (z1'::'a) ♯ (z::x, e::e)›*) proof (nominal_induct e rule:e.strong_induct)\n (*goals:\n 1. ‹⋀x. ⟦atom z1 ♯ (z, [ x ]⇧e); atom z1' ♯ (z, [ x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x ]⇧e[z::=v]⇩e⇩v = [ x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 2. ‹⋀x1a x2. ⟦atom z1 ♯ (z, [ x1a x2 ]⇧e); atom z1' ♯ (z, [ x1a x2 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a x2 ]⇧e[z::=v]⇩e⇩v = [ x1a x2 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 3. ‹⋀x1a x2 x3. ⟦atom z1 ♯ (z, [x1a [ x2 ] x3 ]⇧e); atom z1' ♯ (z, [x1a [ x2 ] x3 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [x1a [ x2 ] x3 ]⇧e[z::=v]⇩e⇩v = [x1a [ x2 ] x3 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 4. ‹⋀x1a x2 x3. ⟦atom z1 ♯ (z, [ x1a x2 x3 ]⇧e); atom z1' ♯ (z, [ x1a x2 x3 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a x2 x3 ]⇧e[z::=v]⇩e⇩v = [ x1a x2 x3 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 5. ‹⋀x1a x2. ⟦atom z1 ♯ (z, [ x1a @@ x2 ]⇧e); atom z1' ♯ (z, [ x1a @@ x2 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a @@ x2 ]⇧e[z::=v]⇩e⇩v = [ x1a @@ x2 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 6. ‹⋀x. ⟦atom z1 ♯ (z, [#1x ]⇧e); atom z1' ♯ (z, [#1x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [#1x ]⇧e[z::=v]⇩e⇩v = [#1x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 7. ‹⋀x. ⟦atom z1 ♯ (z, [#2x ]⇧e); atom z1' ♯ (z, [#2x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [#2x ]⇧e[z::=v]⇩e⇩v = [#2x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 8. ‹⋀x. ⟦atom z1 ♯ (z, [ x ]⇧e); atom z1' ♯ (z, [ x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x ]⇧e[z::=v]⇩e⇩v = [ x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 9. ‹⋀x. ⟦atom z1 ♯ (z, [| x |]⇧e); atom z1' ♯ (z, [| x |]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [| x |]⇧e[z::=v]⇩e⇩v = [| x |]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 10. ‹⋀x1a x2. ⟦atom z1 ♯ (z, [ x1a x2 ]⇧e); atom z1' ♯ (z, [ x1a x2 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a x2 ]⇧e[z::=v]⇩e⇩v = [ x1a x2 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›*)\n qed ((simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) swap_fresh_fresh (*‹⟦?a ♯ ?x; ?b ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))+)\n (*solves the remaining goals:\n 1. ‹⋀x. ⟦atom z1 ♯ (z, [ x ]⇧e); atom z1' ♯ (z, [ x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x ]⇧e[z::=v]⇩e⇩v = [ x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 2. ‹⋀x1a x2. ⟦atom z1 ♯ (z, [ x1a x2 ]⇧e); atom z1' ♯ (z, [ x1a x2 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a x2 ]⇧e[z::=v]⇩e⇩v = [ x1a x2 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 3. ‹⋀x1a x2 x3. ⟦atom z1 ♯ (z, [x1a [ x2 ] x3 ]⇧e); atom z1' ♯ (z, [x1a [ x2 ] x3 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [x1a [ x2 ] x3 ]⇧e[z::=v]⇩e⇩v = [x1a [ x2 ] x3 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 4. ‹⋀x1a x2 x3. ⟦atom z1 ♯ (z, [ x1a x2 x3 ]⇧e); atom z1' ♯ (z, [ x1a x2 x3 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a x2 x3 ]⇧e[z::=v]⇩e⇩v = [ x1a x2 x3 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 5. ‹⋀x1a x2. ⟦atom z1 ♯ (z, [ x1a @@ x2 ]⇧e); atom z1' ♯ (z, [ x1a @@ x2 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a @@ x2 ]⇧e[z::=v]⇩e⇩v = [ x1a @@ x2 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 6. ‹⋀x. ⟦atom z1 ♯ (z, [#1x ]⇧e); atom z1' ♯ (z, [#1x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [#1x ]⇧e[z::=v]⇩e⇩v = [#1x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 7. ‹⋀x. ⟦atom z1 ♯ (z, [#2x ]⇧e); atom z1' ♯ (z, [#2x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [#2x ]⇧e[z::=v]⇩e⇩v = [#2x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 8. ‹⋀x. ⟦atom z1 ♯ (z, [ x ]⇧e); atom z1' ♯ (z, [ x ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x ]⇧e[z::=v]⇩e⇩v = [ x ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 9. ‹⋀x. ⟦atom z1 ♯ (z, [| x |]⇧e); atom z1' ♯ (z, [| x |]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [| x |]⇧e[z::=v]⇩e⇩v = [| x |]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›\n 10. ‹⋀x1a x2. ⟦atom z1 ♯ (z, [ x1a x2 ]⇧e); atom z1' ♯ (z, [ x1a x2 ]⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a x2 ]⇧e[z::=v]⇩e⇩v = [ x1a x2 ]⇧e[z::=(z1 ↔ z1') ∙ v]⇩e⇩v›*)\n\nsection ‹Expressions in Constraints›\n\nnominal_function subst_cev :: \"ce ⇒ x ⇒ v ⇒ ce\" where\n \"subst_cev ( (CE_val v') ) x v = ( (CE_val (subst_vv v' x v )) )\" \n| \"subst_cev ( (CE_op opp v1 v2) ) x v = ( (CE_op opp (subst_cev v1 x v ) (subst_cev v2 x v )) )\"\n| \"subst_cev ( (CE_fst v')) x v = CE_fst (subst_cev v' x v )\"\n| \"subst_cev ( (CE_snd v')) x v = CE_snd (subst_cev v' x v )\"\n| \"subst_cev ( (CE_len v')) x v = CE_len (subst_cev v' x v )\"\n| \"subst_cev ( CE_concat v1 v2) x v = CE_concat (subst_cev v1 x v ) (subst_cev v2 x v )\"\n apply (simp add: eqvt_def subst_cev_graph_aux_def,auto)\n by (meson ce.strong_exhaust)\n\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_cev_abbrev :: \"ce ⇒ x ⇒ v ⇒ ce\" (\"_[_::=_]⇩c⇩e⇩v\" [1000,50,50] 500)\n where \n \"e[x::=v']⇩c⇩e⇩v ≡ subst_cev e x v'\" \n\nlemma size_subst_cev [simp]: \"size ( subst_cev A i x ) = size A\"\n apply (nominal_induct A avoiding: i x rule: ce.strong_induct)\n (*goals:\n 1. ‹⋀x i xa. size ([ x ]⇧c⇧e[i::=xa]⇩c⇩e⇩v) = size [ x ]⇧c⇧e›\n 2. ‹⋀x1a x2 x3 i x. ⟦⋀b ba. size (x2[b::=ba]⇩c⇩e⇩v) = size x2; ⋀b ba. size (x3[b::=ba]⇩c⇩e⇩v) = size x3⟧ ⟹ size ([ x1a x2 x3 ]⇧c⇧e[i::=x]⇩c⇩e⇩v) = size [ x1a x2 x3 ]⇧c⇧e›\n 3. ‹⋀x1a x2 i x. ⟦⋀b ba. size (x1a[b::=ba]⇩c⇩e⇩v) = size x1a; ⋀b ba. size (x2[b::=ba]⇩c⇩e⇩v) = size x2⟧ ⟹ size ([ x1a @@ x2 ]⇧c⇧e[i::=x]⇩c⇩e⇩v) = size [ x1a @@ x2 ]⇧c⇧e›\n 4. ‹⋀x i xa. (⋀b ba. size (x[b::=ba]⇩c⇩e⇩v) = size x) ⟹ size ([#1x]⇧c⇧e[i::=xa]⇩c⇩e⇩v) = size [#1x]⇧c⇧e›\n 5. ‹⋀x i xa. (⋀b ba. size (x[b::=ba]⇩c⇩e⇩v) = size x) ⟹ size ([#2x]⇧c⇧e[i::=xa]⇩c⇩e⇩v) = size [#2x]⇧c⇧e›\n 6. ‹⋀x i xa. (⋀b ba. size (x[b::=ba]⇩c⇩e⇩v) = size x) ⟹ size ([| x |]⇧c⇧e[i::=xa]⇩c⇩e⇩v) = size [| x |]⇧c⇧e›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*proven 6 subgoals*) . \n\nlemma forget_subst_cev [simp]: \"atom a ♯ A ⟹ subst_cev A a x = A\"\n apply (nominal_induct A avoiding: a x rule: ce.strong_induct)\n (*goals:\n 1. ‹⋀x a xa. atom a ♯ [ x ]⇧c⇧e ⟹ [ x ]⇧c⇧e[a::=xa]⇩c⇩e⇩v = [ x ]⇧c⇧e›\n 2. ‹⋀x1a x2 x3 a x. ⟦⋀b ba. atom b ♯ x2 ⟹ x2[b::=ba]⇩c⇩e⇩v = x2; ⋀b ba. atom b ♯ x3 ⟹ x3[b::=ba]⇩c⇩e⇩v = x3; atom a ♯ [ x1a x2 x3 ]⇧c⇧e⟧ ⟹ [ x1a x2 x3 ]⇧c⇧e[a::=x]⇩c⇩e⇩v = [ x1a x2 x3 ]⇧c⇧e›\n 3. ‹⋀x1a x2 a x. ⟦⋀b ba. atom b ♯ x1a ⟹ x1a[b::=ba]⇩c⇩e⇩v = x1a; ⋀b ba. atom b ♯ x2 ⟹ x2[b::=ba]⇩c⇩e⇩v = x2; atom a ♯ [ x1a @@ x2 ]⇧c⇧e⟧ ⟹ [ x1a @@ x2 ]⇧c⇧e[a::=x]⇩c⇩e⇩v = [ x1a @@ x2 ]⇧c⇧e›\n 4. ‹⋀x a xa. ⟦⋀b ba. atom b ♯ x ⟹ x[b::=ba]⇩c⇩e⇩v = x; atom a ♯ [#1x]⇧c⇧e⟧ ⟹ [#1x]⇧c⇧e[a::=xa]⇩c⇩e⇩v = [#1x]⇧c⇧e›\n 5. ‹⋀x a xa. ⟦⋀b ba. atom b ♯ x ⟹ x[b::=ba]⇩c⇩e⇩v = x; atom a ♯ [#2x]⇧c⇧e⟧ ⟹ [#2x]⇧c⇧e[a::=xa]⇩c⇩e⇩v = [#2x]⇧c⇧e›\n 6. ‹⋀x a xa. ⟦⋀b ba. atom b ♯ x ⟹ x[b::=ba]⇩c⇩e⇩v = x; atom a ♯ [| x |]⇧c⇧e⟧ ⟹ [| x |]⇧c⇧e[a::=xa]⇩c⇩e⇩v = [| x |]⇧c⇧e›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a::at_base)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a::at_base) = (?a ≠ atom ?b)›*))[1])\n (*proven 6 subgoals*) .\n\nlemma subst_cev_id [simp]: \"subst_cev A a (V_var a) = A\"\n apply (nominal_induct A avoiding: a rule: ce.strong_induct)\n (*goals:\n 1. ‹⋀x a. [ x ]⇧c⇧e[a::=[ a ]⇧v]⇩c⇩e⇩v = [ x ]⇧c⇧e›\n 2. ‹⋀x1a x2 x3 a. ⟦⋀b. x2[b::=[ b ]⇧v]⇩c⇩e⇩v = x2; ⋀b. x3[b::=[ b ]⇧v]⇩c⇩e⇩v = x3⟧ ⟹ [ x1a x2 x3 ]⇧c⇧e[a::=[ a ]⇧v]⇩c⇩e⇩v = [ x1a x2 x3 ]⇧c⇧e›\n 3. ‹⋀x1a x2 a. ⟦⋀b. x1a[b::=[ b ]⇧v]⇩c⇩e⇩v = x1a; ⋀b. x2[b::=[ b ]⇧v]⇩c⇩e⇩v = x2⟧ ⟹ [ x1a @@ x2 ]⇧c⇧e[a::=[ a ]⇧v]⇩c⇩e⇩v = [ x1a @@ x2 ]⇧c⇧e›\n 4. ‹⋀x a. (⋀b. x[b::=[ b ]⇧v]⇩c⇩e⇩v = x) ⟹ [#1x]⇧c⇧e[a::=[ a ]⇧v]⇩c⇩e⇩v = [#1x]⇧c⇧e›\n 5. ‹⋀x a. (⋀b. x[b::=[ b ]⇧v]⇩c⇩e⇩v = x) ⟹ [#2x]⇧c⇧e[a::=[ a ]⇧v]⇩c⇩e⇩v = [#2x]⇧c⇧e›\n 6. ‹⋀x a. (⋀b. x[b::=[ b ]⇧v]⇩c⇩e⇩v = x) ⟹ [| x |]⇧c⇧e[a::=[ a ]⇧v]⇩c⇩e⇩v = [| x |]⇧c⇧e›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*proven 6 subgoals*) .\n\nlemma fresh_subst_cev_if [simp]:\n \"j ♯ (subst_cev A i x ) = ((atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i)))\"\nproof (nominal_induct A avoiding: i x rule: ce.strong_induct)\n (*goals:\n 1. ‹⋀(x::v) (i::x) xa::v. (j::atom) ♯ [ x ]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [ x ]⇧c⇧e ∧ j ♯ [ x ]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [ x ]⇧c⇧e ∨ j = atom i))›\n 2. ‹⋀(x1a::opp) (x2::ce) (x3::ce) (i::x) x::v. ⟦⋀(b::x) ba::v. (j::atom) ♯ x2[b::=ba]⇩c⇩e⇩v = (atom b ♯ x2 ∧ j ♯ x2 ∨ j ♯ ba ∧ (j ♯ x2 ∨ j = atom b)); ⋀(b::x) ba::v. j ♯ x3[b::=ba]⇩c⇩e⇩v = (atom b ♯ x3 ∧ j ♯ x3 ∨ j ♯ ba ∧ (j ♯ x3 ∨ j = atom b))⟧ ⟹ j ♯ [ x1a x2 x3 ]⇧c⇧e[i::=x]⇩c⇩e⇩v = (atom i ♯ [ x1a x2 x3 ]⇧c⇧e ∧ j ♯ [ x1a x2 x3 ]⇧c⇧e ∨ j ♯ x ∧ (j ♯ [ x1a x2 x3 ]⇧c⇧e ∨ j = atom i))›\n 3. ‹⋀(x1a::ce) (x2::ce) (i::x) x::v. ⟦⋀(b::x) ba::v. (j::atom) ♯ x1a[b::=ba]⇩c⇩e⇩v = (atom b ♯ x1a ∧ j ♯ x1a ∨ j ♯ ba ∧ (j ♯ x1a ∨ j = atom b)); ⋀(b::x) ba::v. j ♯ x2[b::=ba]⇩c⇩e⇩v = (atom b ♯ x2 ∧ j ♯ x2 ∨ j ♯ ba ∧ (j ♯ x2 ∨ j = atom b))⟧ ⟹ j ♯ [ x1a @@ x2 ]⇧c⇧e[i::=x]⇩c⇩e⇩v = (atom i ♯ [ x1a @@ x2 ]⇧c⇧e ∧ j ♯ [ x1a @@ x2 ]⇧c⇧e ∨ j ♯ x ∧ (j ♯ [ x1a @@ x2 ]⇧c⇧e ∨ j = atom i))›\n 4. ‹⋀(x::ce) (i::x) xa::v. (⋀(b::x) ba::v. (j::atom) ♯ x[b::=ba]⇩c⇩e⇩v = (atom b ♯ x ∧ j ♯ x ∨ j ♯ ba ∧ (j ♯ x ∨ j = atom b))) ⟹ j ♯ [#1x]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [#1x]⇧c⇧e ∧ j ♯ [#1x]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [#1x]⇧c⇧e ∨ j = atom i))›\n 5. ‹⋀(x::ce) (i::x) xa::v. (⋀(b::x) ba::v. (j::atom) ♯ x[b::=ba]⇩c⇩e⇩v = (atom b ♯ x ∧ j ♯ x ∨ j ♯ ba ∧ (j ♯ x ∨ j = atom b))) ⟹ j ♯ [#2x]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [#2x]⇧c⇧e ∧ j ♯ [#2x]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [#2x]⇧c⇧e ∨ j = atom i))›\n 6. ‹⋀(x::ce) (i::x) xa::v. (⋀(b::x) ba::v. (j::atom) ♯ x[b::=ba]⇩c⇩e⇩v = (atom b ♯ x ∧ j ♯ x ∨ j ♯ ba ∧ (j ♯ x ∨ j = atom b))) ⟹ j ♯ [| x |]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [| x |]⇧c⇧e ∧ j ♯ [| x |]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [| x |]⇧c⇧e ∨ j = atom i))›*)\n case (CE_op opp v1 v2) (*‹j ♯ v1[?b::=?ba]⇩c⇩e⇩v = (atom ?b ♯ v1 ∧ j ♯ v1 ∨ j ♯ ?ba ∧ (j ♯ v1 ∨ j = atom ?b))› ‹(j::atom) ♯ (v2::ce)[(?b::x)::=(?ba::v)]⇩c⇩e⇩v = (atom ?b ♯ v2 ∧ j ♯ v2 ∨ j ♯ ?ba ∧ (j ♯ v2 ∨ j = atom ?b))›*)\n then show \"?case\"\n (*goal: ‹j ♯ [ opp v1 v2 ]⇧c⇧e[i::=x]⇩c⇩e⇩v = (atom i ♯ [ opp v1 v2 ]⇧c⇧e ∧ j ♯ [ opp v1 v2 ]⇧c⇧e ∨ j ♯ x ∧ (j ♯ [ opp v1 v2 ]⇧c⇧e ∨ j = atom i))›*)\n using fresh_subst_vv_if (*‹(?j::atom) ♯ (?t::v)[(?i::x)::=(?x::v)]⇩v⇩v = (atom ?i ♯ ?t ∧ ?j ♯ ?t ∨ ?j ♯ ?x ∧ (?j ♯ ?t ∨ ?j = atom ?i))›*) subst_ev.simps (*‹[ ?v'::v ]⇧e[(?x::x)::=(?v::v)]⇩e⇩v = [ ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?f ?v' ]⇧e[?x::=?v]⇩e⇩v = [ ?f ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[?f [ ?b ] ?v' ]⇧e[?x::=?v]⇩e⇩v = [?f [ ?b ] ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?opp ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?opp ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[#1?v' ]⇧e[?x::=?v]⇩e⇩v = [#1?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[#2?v' ]⇧e[?x::=?v]⇩e⇩v = [#2?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?u ]⇧e[?x::=?v]⇩e⇩v = [ ?u ]⇧e› ‹[| ?v' |]⇧e[?x::=?v]⇩e⇩v = [| ?v'[?x::=?v]⇩v⇩v |]⇧e› ‹[ ?v1.0 @@ ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v @@ ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e›*) e.supp (*‹supp [ ?v ]⇧e = supp ?v› ‹supp [ ?list ?v ]⇧e = supp ?list ∪ supp ?v› ‹supp [?list [ ?b ] ?v ]⇧e = supp ?list ∪ (supp ?b ∪ supp ?v)› ‹supp [ ?opp ?v1.0 ?v2.0 ]⇧e = supp ?opp ∪ (supp ?v1.0 ∪ supp ?v2.0)› ‹supp [ ?v1.0 @@ ?v2.0 ]⇧e = supp ?v1.0 ∪ supp ?v2.0› ‹supp [#1?v ]⇧e = supp ?v› ‹supp [#2?v ]⇧e = supp ?v› ‹supp [ ?u ]⇧e = supp ?u› ‹supp [| ?v |]⇧e = supp ?v› ‹supp [ ?v1.0 ?v2.0 ]⇧e = supp ?v1.0 ∪ supp ?v2.0›*) pure_fresh (*‹?a ♯ ?x›*) opp.fresh (*‹?a ♯ plus› ‹?a ♯ leq› ‹?a ♯ eq›*) fresh_e_opp (*‹atom ?z ♯ ?v1.0 ∧ atom ?z ♯ ?v2.0 ⟹ atom ?z ♯ [ ?opp ?v1.0 ?v2.0 ]⇧e›*) using fresh_opp_all (*‹?z ♯ ?opp›*) by auto\nqed ((auto)+)\n(*solves the remaining goals:\n 1. ‹⋀(x::v) (i::x) xa::v. (j::atom) ♯ [ x ]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [ x ]⇧c⇧e ∧ j ♯ [ x ]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [ x ]⇧c⇧e ∨ j = atom i))›\n 2. ‹⋀(x1a::ce) (x2::ce) (i::x) x::v. ⟦⋀(b::x) ba::v. (j::atom) ♯ x1a[b::=ba]⇩c⇩e⇩v = (atom b ♯ x1a ∧ j ♯ x1a ∨ j ♯ ba ∧ (j ♯ x1a ∨ j = atom b)); ⋀(b::x) ba::v. j ♯ x2[b::=ba]⇩c⇩e⇩v = (atom b ♯ x2 ∧ j ♯ x2 ∨ j ♯ ba ∧ (j ♯ x2 ∨ j = atom b))⟧ ⟹ j ♯ [ x1a @@ x2 ]⇧c⇧e[i::=x]⇩c⇩e⇩v = (atom i ♯ [ x1a @@ x2 ]⇧c⇧e ∧ j ♯ [ x1a @@ x2 ]⇧c⇧e ∨ j ♯ x ∧ (j ♯ [ x1a @@ x2 ]⇧c⇧e ∨ j = atom i))›\n 3. ‹⋀(x::ce) (i::x) xa::v. (⋀(b::x) ba::v. (j::atom) ♯ x[b::=ba]⇩c⇩e⇩v = (atom b ♯ x ∧ j ♯ x ∨ j ♯ ba ∧ (j ♯ x ∨ j = atom b))) ⟹ j ♯ [#1x]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [#1x]⇧c⇧e ∧ j ♯ [#1x]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [#1x]⇧c⇧e ∨ j = atom i))›\n 4. ‹⋀(x::ce) (i::x) xa::v. (⋀(b::x) ba::v. (j::atom) ♯ x[b::=ba]⇩c⇩e⇩v = (atom b ♯ x ∧ j ♯ x ∨ j ♯ ba ∧ (j ♯ x ∨ j = atom b))) ⟹ j ♯ [#2x]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [#2x]⇧c⇧e ∧ j ♯ [#2x]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [#2x]⇧c⇧e ∨ j = atom i))›\n 5. ‹⋀(x::ce) (i::x) xa::v. (⋀(b::x) ba::v. (j::atom) ♯ x[b::=ba]⇩c⇩e⇩v = (atom b ♯ x ∧ j ♯ x ∨ j ♯ ba ∧ (j ♯ x ∨ j = atom b))) ⟹ j ♯ [| x |]⇧c⇧e[i::=xa]⇩c⇩e⇩v = (atom i ♯ [| x |]⇧c⇧e ∧ j ♯ [| x |]⇧c⇧e ∨ j ♯ xa ∧ (j ♯ [| x |]⇧c⇧e ∨ j = atom i))›*)\n\nlemma subst_cev_commute [simp]:\n \"atom j ♯ A ⟹ (subst_cev (subst_cev A i t ) j u) = subst_cev A i (subst_vv t j u )\"\n apply (nominal_induct A avoiding: i j t u rule: ce.strong_induct)\n (*goals:\n 1. ‹⋀x i j t u. atom j ♯ [ x ]⇧c⇧e ⟹ ([ x ]⇧c⇧e[i::=t]⇩c⇩e⇩v)[j::=u]⇩c⇩e⇩v = [ x ]⇧c⇧e[i::=t[j::=u]⇩v⇩v]⇩c⇩e⇩v›\n 2. ‹⋀x1a x2 x3 i j t u. ⟦⋀b ba bb bc. atom ba ♯ x2 ⟹ (x2[b::=bb]⇩c⇩e⇩v)[ba::=bc]⇩c⇩e⇩v = x2[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩e⇩v; ⋀b ba bb bc. atom ba ♯ x3 ⟹ (x3[b::=bb]⇩c⇩e⇩v)[ba::=bc]⇩c⇩e⇩v = x3[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩e⇩v; atom j ♯ [ x1a x2 x3 ]⇧c⇧e⟧ ⟹ ([ x1a x2 x3 ]⇧c⇧e[i::=t]⇩c⇩e⇩v)[j::=u]⇩c⇩e⇩v = [ x1a x2 x3 ]⇧c⇧e[i::=t[j::=u]⇩v⇩v]⇩c⇩e⇩v›\n 3. ‹⋀x1a x2 i j t u. ⟦⋀b ba bb bc. atom ba ♯ x1a ⟹ (x1a[b::=bb]⇩c⇩e⇩v)[ba::=bc]⇩c⇩e⇩v = x1a[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩e⇩v; ⋀b ba bb bc. atom ba ♯ x2 ⟹ (x2[b::=bb]⇩c⇩e⇩v)[ba::=bc]⇩c⇩e⇩v = x2[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩e⇩v; atom j ♯ [ x1a @@ x2 ]⇧c⇧e⟧ ⟹ ([ x1a @@ x2 ]⇧c⇧e[i::=t]⇩c⇩e⇩v)[j::=u]⇩c⇩e⇩v = [ x1a @@ x2 ]⇧c⇧e[i::=t[j::=u]⇩v⇩v]⇩c⇩e⇩v›\n 4. ‹⋀x i j t u. ⟦⋀b ba bb bc. atom ba ♯ x ⟹ (x[b::=bb]⇩c⇩e⇩v)[ba::=bc]⇩c⇩e⇩v = x[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩e⇩v; atom j ♯ [#1x]⇧c⇧e⟧ ⟹ ([#1x]⇧c⇧e[i::=t]⇩c⇩e⇩v)[j::=u]⇩c⇩e⇩v = [#1x]⇧c⇧e[i::=t[j::=u]⇩v⇩v]⇩c⇩e⇩v›\n 5. ‹⋀x i j t u. ⟦⋀b ba bb bc. atom ba ♯ x ⟹ (x[b::=bb]⇩c⇩e⇩v)[ba::=bc]⇩c⇩e⇩v = x[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩e⇩v; atom j ♯ [#2x]⇧c⇧e⟧ ⟹ ([#2x]⇧c⇧e[i::=t]⇩c⇩e⇩v)[j::=u]⇩c⇩e⇩v = [#2x]⇧c⇧e[i::=t[j::=u]⇩v⇩v]⇩c⇩e⇩v›\n 6. ‹⋀x i j t u. ⟦⋀b ba bb bc. atom ba ♯ x ⟹ (x[b::=bb]⇩c⇩e⇩v)[ba::=bc]⇩c⇩e⇩v = x[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩e⇩v; atom j ♯ [| x |]⇧c⇧e⟧ ⟹ ([| x |]⇧c⇧e[i::=t]⇩c⇩e⇩v)[j::=u]⇩c⇩e⇩v = [| x |]⇧c⇧e[i::=t[j::=u]⇩v⇩v]⇩c⇩e⇩v›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*proven 6 subgoals*) .\n\nlemma subst_cev_var_flip[simp]:\n fixes e::ce and y::x and x::x\n assumes \"atom y ♯ e\"\n shows \"(y ↔ x) ∙ e = e [x::=V_var y]⇩c⇩e⇩v\"\n using assms (*‹atom y ♯ e›*) proof (nominal_induct e rule:ce.strong_induct)\n (*goals:\n 1. ‹⋀xa. atom y ♯ [ xa ]⇧c⇧e ⟹ (y ↔ x) ∙ [ xa ]⇧c⇧e = [ xa ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 2. ‹⋀x1a x2 x3. ⟦atom y ♯ x2 ⟹ (y ↔ x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ x3 ⟹ (y ↔ x) ∙ x3 = x3[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [ x1a x2 x3 ]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [ x1a x2 x3 ]⇧c⇧e = [ x1a x2 x3 ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 3. ‹⋀x1a x2. ⟦atom y ♯ x1a ⟹ (y ↔ x) ∙ x1a = x1a[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ x2 ⟹ (y ↔ x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [ x1a @@ x2 ]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [ x1a @@ x2 ]⇧c⇧e = [ x1a @@ x2 ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 4. ‹⋀xa. ⟦atom y ♯ xa ⟹ (y ↔ x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [#1xa]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [#1xa]⇧c⇧e = [#1xa]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 5. ‹⋀xa. ⟦atom y ♯ xa ⟹ (y ↔ x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [#2xa]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [#2xa]⇧c⇧e = [#2xa]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 6. ‹⋀xa. ⟦atom y ♯ xa ⟹ (y ↔ x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [| xa |]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [| xa |]⇧c⇧e = [| xa |]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›*)\n case (CE_val v) (*‹atom y ♯ [ v ]⇧c⇧e›*)\n then show \"?case\"\n (*goal: ‹(y::x ↔ x::x) ∙ [ v::v ]⇧c⇧e = [ v ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›*)\n using subst_vv_var_flip (*‹atom ?y ♯ ?v ⟹ (?y ↔ ?x) ∙ ?v = ?v[?x::=[ ?y ]⇧v]⇩v⇩v›*) by auto\n next\n (*goals:\n 1. ‹⋀(x1a::opp) (x2::ce) x3::ce. ⟦atom (y::x) ♯ x2 ⟹ (y ↔ x::x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ x3 ⟹ (y ↔ x) ∙ x3 = x3[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [ x1a x2 x3 ]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [ x1a x2 x3 ]⇧c⇧e = [ x1a x2 x3 ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 2. ‹⋀(x1a::ce) x2::ce. ⟦atom (y::x) ♯ x1a ⟹ (y ↔ x::x) ∙ x1a = x1a[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ x2 ⟹ (y ↔ x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [ x1a @@ x2 ]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [ x1a @@ x2 ]⇧c⇧e = [ x1a @@ x2 ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 3. ‹⋀xa::ce. ⟦atom (y::x) ♯ xa ⟹ (y ↔ x::x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [#1xa]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [#1xa]⇧c⇧e = [#1xa]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 4. ‹⋀xa::ce. ⟦atom (y::x) ♯ xa ⟹ (y ↔ x::x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [#2xa]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [#2xa]⇧c⇧e = [#2xa]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 5. ‹⋀xa::ce. ⟦atom (y::x) ♯ xa ⟹ (y ↔ x::x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [| xa |]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [| xa |]⇧c⇧e = [| xa |]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›*)\n case (CE_op opp v1 v2) (*‹atom y ♯ v1 ⟹ (y ↔ x) ∙ v1 = v1[x::=[ y ]⇧v]⇩c⇩e⇩v› ‹atom y ♯ v2 ⟹ (y ↔ x) ∙ v2 = v2[x::=[ y ]⇧v]⇩c⇩e⇩v› ‹atom y ♯ [ opp v1 v2 ]⇧c⇧e›*)\n hence yf: \"atom y ♯ v1 ∧ atom y ♯ v2\"\n using ce.fresh (*‹(?a::atom) ♯ [ ?v::v ]⇧c⇧e = ?a ♯ ?v› ‹?a ♯ [ ?opp ?ce1.0 ?ce2.0 ]⇧c⇧e = (?a ♯ ?opp ∧ ?a ♯ ?ce1.0 ∧ ?a ♯ ?ce2.0)› ‹?a ♯ [ ?ce1.0 @@ ?ce2.0 ]⇧c⇧e = (?a ♯ ?ce1.0 ∧ ?a ♯ ?ce2.0)› ‹?a ♯ [#1?ce]⇧c⇧e = ?a ♯ ?ce› ‹?a ♯ [#2?ce]⇧c⇧e = ?a ♯ ?ce› ‹?a ♯ [| ?ce |]⇧c⇧e = ?a ♯ ?ce›*) by blast\n have \" (y ↔ x) ∙ (CE_op opp v1 v2 ) = CE_op ((y ↔ x) ∙ opp) ( (y ↔ x) ∙ v1 ) ( (y ↔ x) ∙ v2)\"\n using opp.perm_simps (*‹(?p::perm) ∙ plus = plus› ‹?p ∙ leq = leq› ‹?p ∙ eq = eq›*) ce.perm_simps (*‹?p ∙ [ ?v ]⇧c⇧e = [ ?p ∙ ?v ]⇧c⇧e› ‹?p ∙ [ ?opp ?ce1.0 ?ce2.0 ]⇧c⇧e = [ ?p ∙ ?opp ?p ∙ ?ce1.0 ?p ∙ ?ce2.0 ]⇧c⇧e› ‹?p ∙ [ ?ce1.0 @@ ?ce2.0 ]⇧c⇧e = [ ?p ∙ ?ce1.0 @@ ?p ∙ ?ce2.0 ]⇧c⇧e› ‹?p ∙ [#1?ce]⇧c⇧e = [#1?p ∙ ?ce]⇧c⇧e› ‹?p ∙ [#2?ce]⇧c⇧e = [#2?p ∙ ?ce]⇧c⇧e› ‹?p ∙ [| ?ce |]⇧c⇧e = [| ?p ∙ ?ce |]⇧c⇧e›*) permute_pure (*‹(?p::perm) ∙ (?x::?'a) = ?x›*) ce.fresh (*‹?a ♯ [ ?v ]⇧c⇧e = ?a ♯ ?v› ‹(?a::atom) ♯ [ ?opp::opp ?ce1.0::ce ?ce2.0::ce ]⇧c⇧e = (?a ♯ ?opp ∧ ?a ♯ ?ce1.0 ∧ ?a ♯ ?ce2.0)› ‹?a ♯ [ ?ce1.0 @@ ?ce2.0 ]⇧c⇧e = (?a ♯ ?ce1.0 ∧ ?a ♯ ?ce2.0)› ‹?a ♯ [#1?ce]⇧c⇧e = ?a ♯ ?ce› ‹?a ♯ [#2?ce]⇧c⇧e = ?a ♯ ?ce› ‹?a ♯ [| ?ce |]⇧c⇧e = ?a ♯ ?ce›*) opp.strong_exhaust (*‹⟦?y = plus ⟹ ?P; ?y = leq ⟹ ?P; ?y = eq ⟹ ?P⟧ ⟹ ?P›*) by presburger\n also (*calculation: ‹(y::x ↔ x::x) ∙ [ opp::opp v1::ce v2::ce ]⇧c⇧e = [ (y ↔ x) ∙ opp (y ↔ x) ∙ v1 (y ↔ x) ∙ v2 ]⇧c⇧e›*) have \"... = CE_op ((y ↔ x) ∙ opp) (v1[x::=V_var y]⇩c⇩e⇩v) (v2 [x::=V_var y]⇩c⇩e⇩v)\"\n using yf (*‹atom y ♯ v1 ∧ atom y ♯ v2›*) by (simp add: CE_op.hyps( (*‹atom (y::x) ♯ (v1::ce) ⟹ (y ↔ x::x) ∙ v1 = v1[x::=[ y ]⇧v]⇩c⇩e⇩v›*) 1) CE_op.hyps( (*‹atom (y::x) ♯ (v2::ce) ⟹ (y ↔ x::x) ∙ v2 = v2[x::=[ y ]⇧v]⇩c⇩e⇩v›*) 2))\n finally (*calculation: ‹(y ↔ x) ∙ [ opp v1 v2 ]⇧c⇧e = [ (y ↔ x) ∙ opp v1[x::=[ y ]⇧v]⇩c⇩e⇩v v2[x::=[ y ]⇧v]⇩c⇩e⇩v ]⇧c⇧e›*) show \"?case\"\n (*goal: ‹(y ↔ x) ∙ [ opp v1 v2 ]⇧c⇧e = [ opp v1 v2 ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›*)\n using subst_cev.simps (*‹[ ?v' ]⇧c⇧e[?x::=?v]⇩c⇩e⇩v = [ ?v'[?x::=?v]⇩v⇩v ]⇧c⇧e› ‹[ ?opp ?v1.0 ?v2.0 ]⇧c⇧e[?x::=?v]⇩c⇩e⇩v = [ ?opp ?v1.0[?x::=?v]⇩c⇩e⇩v ?v2.0[?x::=?v]⇩c⇩e⇩v ]⇧c⇧e› ‹[#1?v']⇧c⇧e[?x::=?v]⇩c⇩e⇩v = [#1?v'[?x::=?v]⇩c⇩e⇩v]⇧c⇧e› ‹[#2?v']⇧c⇧e[?x::=?v]⇩c⇩e⇩v = [#2?v'[?x::=?v]⇩c⇩e⇩v]⇧c⇧e› ‹[| ?v' |]⇧c⇧e[?x::=?v]⇩c⇩e⇩v = [| ?v'[?x::=?v]⇩c⇩e⇩v |]⇧c⇧e› ‹[ ?v1.0 @@ ?v2.0 ]⇧c⇧e[?x::=?v]⇩c⇩e⇩v = [ ?v1.0[?x::=?v]⇩c⇩e⇩v @@ ?v2.0[?x::=?v]⇩c⇩e⇩v ]⇧c⇧e›*) opp.perm_simps (*‹?p ∙ plus = plus› ‹?p ∙ leq = leq› ‹?p ∙ eq = eq›*) opp.strong_exhaust (*‹⟦?y = plus ⟹ ?P; ?y = leq ⟹ ?P; ?y = eq ⟹ ?P⟧ ⟹ ?P›*) by (metis (full_types))\n qed ((auto simp add: permute_pure (*‹(?p::perm) ∙ (?x::?'a) = ?x›*) subst_vv_var_flip (*‹atom (?y::x) ♯ (?v::v) ⟹ (?y ↔ ?x::x) ∙ ?v = ?v[?x::=[ ?y ]⇧v]⇩v⇩v›*))+)\n (*solves the remaining goals:\n 1. ‹⋀(x1a::ce) x2::ce. ⟦atom (y::x) ♯ x1a ⟹ (y ↔ x::x) ∙ x1a = x1a[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ x2 ⟹ (y ↔ x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [ x1a @@ x2 ]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [ x1a @@ x2 ]⇧c⇧e = [ x1a @@ x2 ]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 2. ‹⋀xa::ce. ⟦atom (y::x) ♯ xa ⟹ (y ↔ x::x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [#1xa]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [#1xa]⇧c⇧e = [#1xa]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 3. ‹⋀xa::ce. ⟦atom (y::x) ♯ xa ⟹ (y ↔ x::x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [#2xa]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [#2xa]⇧c⇧e = [#2xa]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›\n 4. ‹⋀xa::ce. ⟦atom (y::x) ♯ xa ⟹ (y ↔ x::x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩e⇩v; atom y ♯ [| xa |]⇧c⇧e⟧ ⟹ (y ↔ x) ∙ [| xa |]⇧c⇧e = [| xa |]⇧c⇧e[x::=[ y ]⇧v]⇩c⇩e⇩v›*)\n\nlemma subst_cev_flip:\n fixes e::ce and ea::ce and c::x\n assumes \"atom c ♯ (e, ea)\" and \"atom c ♯ (x, xa, e, ea)\" and \"(x ↔ c) ∙ e = (xa ↔ c) ∙ ea\" \n shows \"e[x::=v']⇩c⇩e⇩v = ea[xa::=v']⇩c⇩e⇩v\"\nproof (-)\n (*goal: ‹(e::ce)[(x::x)::=(v'::v)]⇩c⇩e⇩v = (ea::ce)[(xa::x)::=v']⇩c⇩e⇩v›*)\n have \"e[x::=v']⇩c⇩e⇩v = (e[x::=V_var c]⇩c⇩e⇩v)[c::=v']⇩c⇩e⇩v\"\n using subst_ev_commute (*‹atom ?j ♯ ?A ⟹ (?A[?i::=?t]⇩e⇩v)[?j::=?u]⇩e⇩v = ?A[?i::=?t[?j::=?u]⇩v⇩v]⇩e⇩v›*) assms (*‹atom c ♯ (e, ea)› ‹atom c ♯ (x, xa, e, ea)› ‹(x ↔ c) ∙ e = (xa ↔ c) ∙ ea›*) by simp\n also (*calculation: ‹(e::ce)[(x::x)::=(v'::v)]⇩c⇩e⇩v = (e[x::=[ c::x ]⇧v]⇩c⇩e⇩v)[c::=v']⇩c⇩e⇩v›*) have \"... = ((c ↔ x) ∙ e)[c::=v']⇩c⇩e⇩v\"\n using subst_ev_var_flip (*‹atom ?y ♯ ?e ⟹ (?y ↔ ?x) ∙ ?e = ?e[?x::=[ ?y ]⇧v]⇩e⇩v›*) assms (*‹atom c ♯ (e, ea)› ‹atom (c::x) ♯ (x::x, xa::x, e::ce, ea::ce)› ‹(x ↔ c) ∙ e = (xa ↔ c) ∙ ea›*) by simp\n also (*calculation: ‹(e::ce)[(x::x)::=(v'::v)]⇩c⇩e⇩v = ((c::x ↔ x) ∙ e)[c::=v']⇩c⇩e⇩v›*) have \"... = ((c ↔ xa) ∙ ea)[c::=v']⇩c⇩e⇩v\"\n using assms (*‹atom (c::x) ♯ (e::ce, ea::ce)› ‹atom c ♯ (x, xa, e, ea)› ‹(x ↔ c) ∙ e = (xa ↔ c) ∙ ea›*) flip_commute (*‹(?a ↔ ?b) = (?b ↔ ?a)›*) by metis\n also (*calculation: ‹(e::ce)[(x::x)::=(v'::v)]⇩c⇩e⇩v = ((c::x ↔ xa::x) ∙ (ea::ce))[c::=v']⇩c⇩e⇩v›*) have \"... = ea[xa::=v']⇩c⇩e⇩v\"\n using subst_ev_var_flip (*‹atom ?y ♯ ?e ⟹ (?y ↔ ?x) ∙ ?e = ?e[?x::=[ ?y ]⇧v]⇩e⇩v›*) assms (*‹atom c ♯ (e, ea)› ‹atom c ♯ (x, xa, e, ea)› ‹(x ↔ c) ∙ e = (xa ↔ c) ∙ ea›*) by simp\n finally (*calculation: ‹e[x::=v']⇩c⇩e⇩v = ea[xa::=v']⇩c⇩e⇩v›*) show \"?thesis\"\n (*goal: ‹(e::ce)[(x::x)::=(v'::v)]⇩c⇩e⇩v = (ea::ce)[(xa::x)::=v']⇩c⇩e⇩v›*)\n by auto\nqed\n\nlemma subst_cev_var[simp]:\n fixes z::x and x::x\n shows \"[[x]⇧v]⇧c⇧e [x::=[z]⇧v]⇩c⇩e⇩v = [[z]⇧v]⇧c⇧e\"\n by auto\n\ninstantiation ce :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_cev\"\n\ninstance proof\n fix j::atom and i::x and x::v and t::ce\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n using fresh_subst_cev_if[of j t i x] subst_v_ce_def by metis\n\n fix a::x and tm::ce and x::v\n show \"atom a ♯ tm ⟹ subst_v tm a x = tm\"\n using forget_subst_cev subst_v_ce_def by simp\n\n fix a::x and tm::ce\n show \"subst_v tm a (V_var a) = tm\" using subst_cev_id subst_v_ce_def by simp\n\n fix p::perm and x1::x and v::v and t1::ce\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n using subst_cev_commute subst_v_ce_def by simp\n\n fix x::x and c::ce and z::x \n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c [z::=V_var x]⇩v\"\n using subst_v_ce_def by simp\n\n fix x::x and c::ce and z::x\n show \"atom x ♯ c ⟹ c [z::=V_var x]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n using subst_v_ce_def by simp\nqed\n\nend\n\nlemma subst_cev_commute_full:\n fixes e::ce and w::v and v::v\n assumes \"atom z ♯ v\" and \"atom x ♯ w\" and \"x ≠ z\"\n shows \"subst_cev (e[z::=w]⇩c⇩e⇩v) x v = subst_cev (e[x::=v]⇩c⇩e⇩v) z w \" \n using assms (*‹atom (z::x) ♯ (v::v)› ‹atom x ♯ w› ‹x ≠ z›*) apply (nominal_induct e rule: ce.strong_induct)\n (*goals:\n 1. ‹⋀xa. ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ xa ]⇧c⇧e[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = ([ xa ]⇧c⇧e[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v›\n 2. ‹⋀x1a x2 x3. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x2[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = (x2[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x3[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = (x3[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ x1a x2 x3 ]⇧c⇧e[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = ([ x1a x2 x3 ]⇧c⇧e[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v›\n 3. ‹⋀x1a x2. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x1a[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = (x1a[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x2[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = (x2[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([ x1a @@ x2 ]⇧c⇧e[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = ([ x1a @@ x2 ]⇧c⇧e[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v›\n 4. ‹⋀xa. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (xa[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = (xa[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([#1xa]⇧c⇧e[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = ([#1xa]⇧c⇧e[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v›\n 5. ‹⋀xa. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (xa[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = (xa[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([#2xa]⇧c⇧e[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = ([#2xa]⇧c⇧e[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v›\n 6. ‹⋀xa. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (xa[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = (xa[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ([| xa |]⇧c⇧e[z::=w]⇩c⇩e⇩v)[x::=v]⇩c⇩e⇩v = ([| xa |]⇧c⇧e[x::=v]⇩c⇩e⇩v)[z::=w]⇩c⇩e⇩v›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*proven 6 subgoals*) .\n\n\nlemma subst_cev_v_flip1[simp]:\n fixes e::ce\n assumes \"atom z1 ♯ (z,e)\" and \"atom z1' ♯ (z,e)\"\n shows\"(z1 ↔ z1') ∙ e[z::=v]⇩c⇩e⇩v = e[z::= ((z1 ↔ z1') ∙ v)]⇩c⇩e⇩v\"\n using assms (*‹atom z1 ♯ (z, e)› ‹atom z1' ♯ (z, e)›*) apply (nominal_induct e rule:ce.strong_induct)\n (*goals:\n 1. ‹⋀x. ⟦atom z1 ♯ (z, [ x ]⇧c⇧e); atom z1' ♯ (z, [ x ]⇧c⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x ]⇧c⇧e[z::=v]⇩c⇩e⇩v = [ x ]⇧c⇧e[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v›\n 2. ‹⋀x1a x2 x3. ⟦⟦atom z1 ♯ (z, x2); atom z1' ♯ (z, x2)⟧ ⟹ (z1 ↔ z1') ∙ x2[z::=v]⇩c⇩e⇩v = x2[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v; ⟦atom z1 ♯ (z, x3); atom z1' ♯ (z, x3)⟧ ⟹ (z1 ↔ z1') ∙ x3[z::=v]⇩c⇩e⇩v = x3[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v; atom z1 ♯ (z, [ x1a x2 x3 ]⇧c⇧e); atom z1' ♯ (z, [ x1a x2 x3 ]⇧c⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a x2 x3 ]⇧c⇧e[z::=v]⇩c⇩e⇩v = [ x1a x2 x3 ]⇧c⇧e[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v›\n 3. ‹⋀x1a x2. ⟦⟦atom z1 ♯ (z, x1a); atom z1' ♯ (z, x1a)⟧ ⟹ (z1 ↔ z1') ∙ x1a[z::=v]⇩c⇩e⇩v = x1a[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v; ⟦atom z1 ♯ (z, x2); atom z1' ♯ (z, x2)⟧ ⟹ (z1 ↔ z1') ∙ x2[z::=v]⇩c⇩e⇩v = x2[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v; atom z1 ♯ (z, [ x1a @@ x2 ]⇧c⇧e); atom z1' ♯ (z, [ x1a @@ x2 ]⇧c⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [ x1a @@ x2 ]⇧c⇧e[z::=v]⇩c⇩e⇩v = [ x1a @@ x2 ]⇧c⇧e[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v›\n 4. ‹⋀x. ⟦⟦atom z1 ♯ (z, x); atom z1' ♯ (z, x)⟧ ⟹ (z1 ↔ z1') ∙ x[z::=v]⇩c⇩e⇩v = x[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v; atom z1 ♯ (z, [#1x]⇧c⇧e); atom z1' ♯ (z, [#1x]⇧c⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [#1x]⇧c⇧e[z::=v]⇩c⇩e⇩v = [#1x]⇧c⇧e[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v›\n 5. ‹⋀x. ⟦⟦atom z1 ♯ (z, x); atom z1' ♯ (z, x)⟧ ⟹ (z1 ↔ z1') ∙ x[z::=v]⇩c⇩e⇩v = x[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v; atom z1 ♯ (z, [#2x]⇧c⇧e); atom z1' ♯ (z, [#2x]⇧c⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [#2x]⇧c⇧e[z::=v]⇩c⇩e⇩v = [#2x]⇧c⇧e[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v›\n 6. ‹⋀x. ⟦⟦atom z1 ♯ (z, x); atom z1' ♯ (z, x)⟧ ⟹ (z1 ↔ z1') ∙ x[z::=v]⇩c⇩e⇩v = x[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v; atom z1 ♯ (z, [| x |]⇧c⇧e); atom z1' ♯ (z, [| x |]⇧c⇧e)⟧ ⟹ (z1 ↔ z1') ∙ [| x |]⇧c⇧e[z::=v]⇩c⇩e⇩v = [| x |]⇧c⇧e[z::=(z1 ↔ z1') ∙ v]⇩c⇩e⇩v›\n discuss goal 1*)\n apply (simp add: flip_def (*‹(?a::?'a::at_base ↔ ?b::?'a::at_base) = (atom ?a ⇌ atom ?b)›*) fresh_Pair (*‹(?a::atom) ♯ (?x::?'a::pt, ?y::?'b::pt) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) swap_fresh_fresh (*‹⟦(?a::atom) ♯ (?x::?'a::pt); (?b::atom) ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))\n (*discuss goal 2*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) swap_fresh_fresh (*‹⟦?a ♯ ?x; ?b ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))\n (*discuss goal 3*)\n apply (simp add: flip_def (*‹(?a::?'a::at_base ↔ ?b::?'a::at_base) = (atom ?a ⇌ atom ?b)›*) fresh_Pair (*‹(?a::atom) ♯ (?x::?'a::pt, ?y::?'b::pt) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) swap_fresh_fresh (*‹⟦(?a::atom) ♯ (?x::?'a::pt); (?b::atom) ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))\n (*discuss goal 4*)\n apply (simp add: flip_def (*‹(?a::?'a ↔ ?b::?'a) = (atom ?a ⇌ atom ?b)›*) fresh_Pair (*‹(?a::atom) ♯ (?x::?'a, ?y::?'b) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) swap_fresh_fresh (*‹⟦(?a::atom) ♯ (?x::?'a); (?b::atom) ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))\n (*discuss goal 5*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) swap_fresh_fresh (*‹⟦?a ♯ ?x; ?b ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))\n (*discuss goal 6*)\n apply (simp add: flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) swap_fresh_fresh (*‹⟦?a ♯ ?x; ?b ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))\n (*proven 6 subgoals*) .\n\nsection ‹Constraints›\n\nnominal_function subst_cv :: \"c ⇒ x ⇒ v ⇒ c\" where\n \"subst_cv (C_true) x v = C_true\"\n| \"subst_cv (C_false) x v = C_false\"\n| \"subst_cv (C_conj c1 c2) x v = C_conj (subst_cv c1 x v ) (subst_cv c2 x v )\"\n| \"subst_cv (C_disj c1 c2) x v = C_disj (subst_cv c1 x v ) (subst_cv c2 x v )\"\n| \"subst_cv (C_imp c1 c2) x v = C_imp (subst_cv c1 x v ) (subst_cv c2 x v )\"\n| \"subst_cv (e1 == e2) x v = ((subst_cev e1 x v ) == (subst_cev e2 x v ))\"\n| \"subst_cv (C_not c) x v = C_not (subst_cv c x v )\"\n apply (simp add: eqvt_def subst_cv_graph_aux_def,auto)\n using c.strong_exhaust by metis\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_cv_abbrev :: \"c ⇒ x ⇒ v ⇒ c\" (\"_[_::=_]⇩c⇩v\" [1000,50,50] 1000)\n where \n \"c[x::=v']⇩c⇩v ≡ subst_cv c x v'\" \n\nlemma size_subst_cv [simp]: \"size ( subst_cv A i x ) = size A\"\n apply (nominal_induct A avoiding: i x rule: c.strong_induct)\n (*goals:\n 1. ‹⋀i x. size (TRUE)[i::=x]⇩c⇩v = size (TRUE)›\n 2. ‹⋀i x. size (FALSE)[i::=x]⇩c⇩v = size (FALSE)›\n 3. ‹⋀x1 x2 i x. ⟦⋀b ba. size x1[b::=ba]⇩c⇩v = size x1; ⋀b ba. size x2[b::=ba]⇩c⇩v = size x2⟧ ⟹ size (x1 AND x2 )[i::=x]⇩c⇩v = size (x1 AND x2 )›\n 4. ‹⋀x1 x2 i x. ⟦⋀b ba. size x1[b::=ba]⇩c⇩v = size x1; ⋀b ba. size x2[b::=ba]⇩c⇩v = size x2⟧ ⟹ size (x1 OR x2 )[i::=x]⇩c⇩v = size (x1 OR x2 )›\n 5. ‹⋀x i xa. (⋀b ba. size x[b::=ba]⇩c⇩v = size x) ⟹ size (¬ x )[i::=xa]⇩c⇩v = size (¬ x )›\n 6. ‹⋀x1 x2 i x. ⟦⋀b ba. size x1[b::=ba]⇩c⇩v = size x1; ⋀b ba. size x2[b::=ba]⇩c⇩v = size x2⟧ ⟹ size (x1 IMP x2 )[i::=x]⇩c⇩v = size (x1 IMP x2 )›\n 7. ‹⋀x1 x2 i x. size (x1 == x2 )[i::=x]⇩c⇩v = size (x1 == x2 )›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*proven 7 subgoals*) . \n\nlemma forget_subst_cv [simp]: \"atom a ♯ A ⟹ subst_cv A a x = A\"\n apply (nominal_induct A avoiding: a x rule: c.strong_induct)\n (*goals:\n 1. ‹⋀a x. atom a ♯ (TRUE) ⟹ (TRUE)[a::=x]⇩c⇩v = (TRUE)›\n 2. ‹⋀a x. atom a ♯ (FALSE) ⟹ (FALSE)[a::=x]⇩c⇩v = (FALSE)›\n 3. ‹⋀x1 x2 a x. ⟦⋀b ba. atom b ♯ x1 ⟹ x1[b::=ba]⇩c⇩v = x1; ⋀b ba. atom b ♯ x2 ⟹ x2[b::=ba]⇩c⇩v = x2; atom a ♯ (x1 AND x2 )⟧ ⟹ (x1 AND x2 )[a::=x]⇩c⇩v = (x1 AND x2 )›\n 4. ‹⋀x1 x2 a x. ⟦⋀b ba. atom b ♯ x1 ⟹ x1[b::=ba]⇩c⇩v = x1; ⋀b ba. atom b ♯ x2 ⟹ x2[b::=ba]⇩c⇩v = x2; atom a ♯ (x1 OR x2 )⟧ ⟹ (x1 OR x2 )[a::=x]⇩c⇩v = (x1 OR x2 )›\n 5. ‹⋀x a xa. ⟦⋀b ba. atom b ♯ x ⟹ x[b::=ba]⇩c⇩v = x; atom a ♯ (¬ x )⟧ ⟹ (¬ x )[a::=xa]⇩c⇩v = (¬ x )›\n 6. ‹⋀x1 x2 a x. ⟦⋀b ba. atom b ♯ x1 ⟹ x1[b::=ba]⇩c⇩v = x1; ⋀b ba. atom b ♯ x2 ⟹ x2[b::=ba]⇩c⇩v = x2; atom a ♯ (x1 IMP x2 )⟧ ⟹ (x1 IMP x2 )[a::=x]⇩c⇩v = (x1 IMP x2 )›\n 7. ‹⋀x1 x2 a x. atom a ♯ (x1 == x2 ) ⟹ (x1 == x2 )[a::=x]⇩c⇩v = (x1 == x2 )›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a::at_base)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a::at_base) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 7*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*proven 7 subgoals*) .\n\nlemma subst_cv_id [simp]: \"subst_cv A a (V_var a) = A\"\n apply (nominal_induct A avoiding: a rule: c.strong_induct)\n (*goals:\n 1. ‹⋀a. (TRUE)[a::=[ a ]⇧v]⇩c⇩v = (TRUE)›\n 2. ‹⋀a. (FALSE)[a::=[ a ]⇧v]⇩c⇩v = (FALSE)›\n 3. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩c⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩c⇩v = x2⟧ ⟹ (x1 AND x2 )[a::=[ a ]⇧v]⇩c⇩v = (x1 AND x2 )›\n 4. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩c⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩c⇩v = x2⟧ ⟹ (x1 OR x2 )[a::=[ a ]⇧v]⇩c⇩v = (x1 OR x2 )›\n 5. ‹⋀x a. (⋀b. x[b::=[ b ]⇧v]⇩c⇩v = x) ⟹ (¬ x )[a::=[ a ]⇧v]⇩c⇩v = (¬ x )›\n 6. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩c⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩c⇩v = x2⟧ ⟹ (x1 IMP x2 )[a::=[ a ]⇧v]⇩c⇩v = (x1 IMP x2 )›\n 7. ‹⋀x1 x2 a. (x1 == x2 )[a::=[ a ]⇧v]⇩c⇩v = (x1 == x2 )›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a::at_base)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a::at_base) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 7*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*proven 7 subgoals*) .\n\nlemma fresh_subst_cv_if [simp]:\n \"j ♯ (subst_cv A i x ) ⟷ (atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i))\"\n apply (nominal_induct A avoiding: i x rule: c.strong_induct)\n (*goals:\n 1. ‹⋀i x. j ♯ (TRUE)[i::=x]⇩c⇩v = (atom i ♯ (TRUE) ∧ j ♯ (TRUE) ∨ j ♯ x ∧ (j ♯ (TRUE) ∨ j = atom i))›\n 2. ‹⋀i x. j ♯ (FALSE)[i::=x]⇩c⇩v = (atom i ♯ (FALSE) ∧ j ♯ (FALSE) ∨ j ♯ x ∧ (j ♯ (FALSE) ∨ j = atom i))›\n 3. ‹⋀x1 x2 i x. ⟦⋀b ba. j ♯ x1[b::=ba]⇩c⇩v = (atom b ♯ x1 ∧ j ♯ x1 ∨ j ♯ ba ∧ (j ♯ x1 ∨ j = atom b)); ⋀b ba. j ♯ x2[b::=ba]⇩c⇩v = (atom b ♯ x2 ∧ j ♯ x2 ∨ j ♯ ba ∧ (j ♯ x2 ∨ j = atom b))⟧ ⟹ j ♯ (x1 AND x2 )[i::=x]⇩c⇩v = (atom i ♯ (x1 AND x2 ) ∧ j ♯ (x1 AND x2 ) ∨ j ♯ x ∧ (j ♯ (x1 AND x2 ) ∨ j = atom i))›\n 4. ‹⋀x1 x2 i x. ⟦⋀b ba. j ♯ x1[b::=ba]⇩c⇩v = (atom b ♯ x1 ∧ j ♯ x1 ∨ j ♯ ba ∧ (j ♯ x1 ∨ j = atom b)); ⋀b ba. j ♯ x2[b::=ba]⇩c⇩v = (atom b ♯ x2 ∧ j ♯ x2 ∨ j ♯ ba ∧ (j ♯ x2 ∨ j = atom b))⟧ ⟹ j ♯ (x1 OR x2 )[i::=x]⇩c⇩v = (atom i ♯ (x1 OR x2 ) ∧ j ♯ (x1 OR x2 ) ∨ j ♯ x ∧ (j ♯ (x1 OR x2 ) ∨ j = atom i))›\n 5. ‹⋀x i xa. (⋀b ba. j ♯ x[b::=ba]⇩c⇩v = (atom b ♯ x ∧ j ♯ x ∨ j ♯ ba ∧ (j ♯ x ∨ j = atom b))) ⟹ j ♯ (¬ x )[i::=xa]⇩c⇩v = (atom i ♯ (¬ x ) ∧ j ♯ (¬ x ) ∨ j ♯ xa ∧ (j ♯ (¬ x ) ∨ j = atom i))›\n 6. ‹⋀x1 x2 i x. ⟦⋀b ba. j ♯ x1[b::=ba]⇩c⇩v = (atom b ♯ x1 ∧ j ♯ x1 ∨ j ♯ ba ∧ (j ♯ x1 ∨ j = atom b)); ⋀b ba. j ♯ x2[b::=ba]⇩c⇩v = (atom b ♯ x2 ∧ j ♯ x2 ∨ j ♯ ba ∧ (j ♯ x2 ∨ j = atom b))⟧ ⟹ j ♯ (x1 IMP x2 )[i::=x]⇩c⇩v = (atom i ♯ (x1 IMP x2 ) ∧ j ♯ (x1 IMP x2 ) ∨ j ♯ x ∧ (j ♯ (x1 IMP x2 ) ∨ j = atom i))›\n 7. ‹⋀x1 x2 i x. j ♯ (x1 == x2 )[i::=x]⇩c⇩v = (atom i ♯ (x1 == x2 ) ∧ j ♯ (x1 == x2 ) ∨ j ♯ x ∧ (j ♯ (x1 == x2 ) ∨ j = atom i))›\n discuss goal 1*)\n apply ((auto simp add: pure_fresh (*‹(?a::atom) ♯ (?x::?'a)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: pure_fresh (*‹?a ♯ ?x›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: pure_fresh (*‹?a ♯ ?x›*))[1])\n (*discuss goal 4*)\n apply ((auto simp add: pure_fresh (*‹?a ♯ ?x›*))[1])\n (*discuss goal 5*)\n apply ((auto simp add: pure_fresh (*‹?a ♯ ?x›*))[1])\n (*discuss goal 6*)\n apply ((auto simp add: pure_fresh (*‹?a ♯ ?x›*))[1])\n (*discuss goal 7*)\n apply ((auto simp add: pure_fresh (*‹?a ♯ ?x›*))[1])\n (*proven 7 subgoals*) .\n\nlemma subst_cv_commute [simp]:\n \"atom j ♯ A ⟹ (subst_cv (subst_cv A i t ) j u ) = subst_cv A i (subst_vv t j u )\"\n apply (nominal_induct A avoiding: i j t u rule: c.strong_induct)\n (*goals:\n 1. ‹⋀i j t u. atom j ♯ (TRUE) ⟹ (TRUE)[i::=t]⇩c⇩v[j::=u]⇩c⇩v = (TRUE)[i::=t[j::=u]⇩v⇩v]⇩c⇩v›\n 2. ‹⋀i j t u. atom j ♯ (FALSE) ⟹ (FALSE)[i::=t]⇩c⇩v[j::=u]⇩c⇩v = (FALSE)[i::=t[j::=u]⇩v⇩v]⇩c⇩v›\n 3. ‹⋀x1 x2 i j t u. ⟦⋀b ba bb bc. atom ba ♯ x1 ⟹ x1[b::=bb]⇩c⇩v[ba::=bc]⇩c⇩v = x1[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩v; ⋀b ba bb bc. atom ba ♯ x2 ⟹ x2[b::=bb]⇩c⇩v[ba::=bc]⇩c⇩v = x2[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩v; atom j ♯ (x1 AND x2 )⟧ ⟹ (x1 AND x2 )[i::=t]⇩c⇩v[j::=u]⇩c⇩v = (x1 AND x2 )[i::=t[j::=u]⇩v⇩v]⇩c⇩v›\n 4. ‹⋀x1 x2 i j t u. ⟦⋀b ba bb bc. atom ba ♯ x1 ⟹ x1[b::=bb]⇩c⇩v[ba::=bc]⇩c⇩v = x1[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩v; ⋀b ba bb bc. atom ba ♯ x2 ⟹ x2[b::=bb]⇩c⇩v[ba::=bc]⇩c⇩v = x2[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩v; atom j ♯ (x1 OR x2 )⟧ ⟹ (x1 OR x2 )[i::=t]⇩c⇩v[j::=u]⇩c⇩v = (x1 OR x2 )[i::=t[j::=u]⇩v⇩v]⇩c⇩v›\n 5. ‹⋀x i j t u. ⟦⋀b ba bb bc. atom ba ♯ x ⟹ x[b::=bb]⇩c⇩v[ba::=bc]⇩c⇩v = x[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩v; atom j ♯ (¬ x )⟧ ⟹ (¬ x )[i::=t]⇩c⇩v[j::=u]⇩c⇩v = (¬ x )[i::=t[j::=u]⇩v⇩v]⇩c⇩v›\n 6. ‹⋀x1 x2 i j t u. ⟦⋀b ba bb bc. atom ba ♯ x1 ⟹ x1[b::=bb]⇩c⇩v[ba::=bc]⇩c⇩v = x1[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩v; ⋀b ba bb bc. atom ba ♯ x2 ⟹ x2[b::=bb]⇩c⇩v[ba::=bc]⇩c⇩v = x2[b::=bb[ba::=bc]⇩v⇩v]⇩c⇩v; atom j ♯ (x1 IMP x2 )⟧ ⟹ (x1 IMP x2 )[i::=t]⇩c⇩v[j::=u]⇩c⇩v = (x1 IMP x2 )[i::=t[j::=u]⇩v⇩v]⇩c⇩v›\n 7. ‹⋀x1 x2 i j t u. atom j ♯ (x1 == x2 ) ⟹ (x1 == x2 )[i::=t]⇩c⇩v[j::=u]⇩c⇩v = (x1 == x2 )[i::=t[j::=u]⇩v⇩v]⇩c⇩v›\n discuss goal 1*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))[1])\n (*discuss goal 7*)\n apply ((auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a::at_base)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a::at_base) = (?a ≠ atom ?b)›*))[1])\n (*proven 7 subgoals*) .\n\nlemma let_s_size [simp]: \"size s ≤ size (AS_let x e s)\"\n apply (nominal_induct s avoiding: e x rule: s_branch_s_branch_list.strong_induct(1))\n (*goals:\n 1. ‹⋀(x::v) (e::e) xa::x. size [x]⇧s ≤ size LET xa = e IN [x]⇧s›\n 2. ‹⋀(x1::x) (x2::e) (x3::s) (e::e) x::x. ⟦atom x1 ♯ e; atom x1 ♯ x; ⋀(b::e) ba::x. size x3 ≤ size LET ba = b IN x3⟧ ⟹ size LET x1 = x2 IN x3 ≤ size LET x = e IN LET x1 = x2 IN x3›\n 3. ‹⋀(x1::x) (x2::τ) (x3::s) (x4::s) (e::e) x::x. ⟦atom x1 ♯ e; atom x1 ♯ x; ⋀(b::e) ba::x. size x3 ≤ size LET ba = b IN x3; ⋀(b::e) ba::x. size x4 ≤ size LET ba = b IN x4⟧ ⟹ size LET x1 : x2 = x3 IN x4 ≤ size LET x = e IN LET x1 : x2 = x3 IN x4›\n 4. ‹⋀(x1::v) (x2::s) (x3::s) (e::e) x::x. ⟦⋀(b::e) ba::x. size x2 ≤ size LET ba = b IN x2; ⋀(b::e) ba::x. size x3 ≤ size LET ba = b IN x3⟧ ⟹ size (IF x1 THEN x2 ELSE x3) ≤ size LET x = e IN IF x1 THEN x2 ELSE x3›\n 5. ‹⋀(x1::u) (x2::τ) (x3::v) (x4::s) (e::e) x::x. ⟦atom x1 ♯ e; atom x1 ♯ x; ⋀(b::e) ba::x. size x4 ≤ size LET ba = b IN x4⟧ ⟹ size VAR x1 : x2 = x3 IN x4 ≤ size LET x = e IN VAR x1 : x2 = x3 IN x4›\n 6. ‹⋀(x1::u) (x2::v) (e::e) x::x. size x1 ::= x2 ≤ size LET x = e IN x1 ::= x2›\n 7. ‹⋀(x1::v) (x2::branch_list) (e::e) x::x. (⋀(b::e) ba::x. (?P3.0::(unit × e) × x ⇒ branch_list ⇒ bool) (((), b), ba) x2) ⟹ size MATCH x1 WITH { x2 } ≤ size LET x = e IN MATCH x1 WITH { x2 }›\n 8. ‹⋀(x1::s) (x2::s) (e::e) x::x. ⟦⋀(b::e) ba::x. size x1 ≤ size LET ba = b IN x1; ⋀(b::e) ba::x. size x2 ≤ size LET ba = b IN x2⟧ ⟹ size (WHILE x1 DO { x2 } ) ≤ size LET x = e IN WHILE x1 DO { x2 } ›\n 9. ‹⋀(x1::s) (x2::s) (e::e) x::x. ⟦⋀(b::e) ba::x. size x1 ≤ size LET ba = b IN x1; ⋀(b::e) ba::x. size x2 ≤ size LET ba = b IN x2⟧ ⟹ size ( x1 ;; x2 ) ≤ size LET x = e IN x1 ;; x2 ›\n 10. ‹⋀(x1::c) (x2::s) (e::e) x::x. (⋀(b::e) ba::x. size x2 ≤ size LET ba = b IN x2) ⟹ size ASSERT x1 IN x2 ≤ size LET x = e IN ASSERT x1 IN x2 ›\n 11. ‹⋀(x1::char list) (x2::x) (x3::s) (e::e) x::x. ⟦atom x2 ♯ e; atom x2 ♯ x; ⋀(b::e) ba::x. size x3 ≤ size LET ba = b IN x3⟧ ⟹ (?P2.0::(unit × e) × x ⇒ branch_s ⇒ bool) (((), e), x) x1 x2 ⇒ x3 ›\n 12. ‹⋀(x::branch_s) (e::e) xa::x. (⋀(b::e) ba::x. (?P2.0::(unit × e) × x ⇒ branch_s ⇒ bool) (((), b), ba) x) ⟹ (?P3.0::(unit × e) × x ⇒ branch_list ⇒ bool) (((), e), xa) { x }›\n 13. ‹⋀(x1::branch_s) (x2::branch_list) (e::e) x::x. ⟦⋀(b::e) ba::x. (?P2.0::(unit × e) × x ⇒ branch_s ⇒ bool) (((), b), ba) x1; ⋀(b::e) ba::x. (?P3.0::(unit × e) × x ⇒ branch_list ⇒ bool) (((), b), ba) x2⟧ ⟹ ?P3.0 (((), e), x) x1 | x2 ›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*discuss goal 11*)\n apply ((auto)[1])\n (*discuss goal 12*)\n apply ((auto)[1])\n (*discuss goal 13*)\n apply ((auto)[1])\n (*proven 13 subgoals*) .\n\nlemma subst_cv_var_flip[simp]:\n fixes c::c\n assumes \"atom y ♯ c\"\n shows \"(y ↔ x) ∙ c = c[x::=V_var y]⇩c⇩v\"\n using assms (*‹atom y ♯ c›*) apply (nominal_induct c rule:c.strong_induct)\n (*goals:\n 1. ‹atom y ♯ (TRUE) ⟹ (y ↔ x) ∙ (TRUE) = (TRUE)[x::=[ y ]⇧v]⇩c⇩v›\n 2. ‹atom y ♯ (FALSE) ⟹ (y ↔ x) ∙ (FALSE) = (FALSE)[x::=[ y ]⇧v]⇩c⇩v›\n 3. ‹⋀x1 x2. ⟦atom y ♯ x1 ⟹ (y ↔ x) ∙ x1 = x1[x::=[ y ]⇧v]⇩c⇩v; atom y ♯ x2 ⟹ (y ↔ x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩v; atom y ♯ (x1 AND x2 )⟧ ⟹ (y ↔ x) ∙ (x1 AND x2 ) = (x1 AND x2 )[x::=[ y ]⇧v]⇩c⇩v›\n 4. ‹⋀x1 x2. ⟦atom y ♯ x1 ⟹ (y ↔ x) ∙ x1 = x1[x::=[ y ]⇧v]⇩c⇩v; atom y ♯ x2 ⟹ (y ↔ x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩v; atom y ♯ (x1 OR x2 )⟧ ⟹ (y ↔ x) ∙ (x1 OR x2 ) = (x1 OR x2 )[x::=[ y ]⇧v]⇩c⇩v›\n 5. ‹⋀xa. ⟦atom y ♯ xa ⟹ (y ↔ x) ∙ xa = xa[x::=[ y ]⇧v]⇩c⇩v; atom y ♯ (¬ xa )⟧ ⟹ (y ↔ x) ∙ (¬ xa ) = (¬ xa )[x::=[ y ]⇧v]⇩c⇩v›\n 6. ‹⋀x1 x2. ⟦atom y ♯ x1 ⟹ (y ↔ x) ∙ x1 = x1[x::=[ y ]⇧v]⇩c⇩v; atom y ♯ x2 ⟹ (y ↔ x) ∙ x2 = x2[x::=[ y ]⇧v]⇩c⇩v; atom y ♯ (x1 IMP x2 )⟧ ⟹ (y ↔ x) ∙ (x1 IMP x2 ) = (x1 IMP x2 )[x::=[ y ]⇧v]⇩c⇩v›\n 7. ‹⋀x1 x2. atom y ♯ (x1 == x2 ) ⟹ (y ↔ x) ∙ (x1 == x2 ) = (x1 == x2 )[x::=[ y ]⇧v]⇩c⇩v›\n discuss goal 1*)\n apply (simp add: flip_subst_v (*‹atom ?x ♯ ?c ⟹ (?x ↔ ?z) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_ce_def (*‹subst_v = subst_cev_abbrev›*))\n (*discuss goal 2*)\n apply (simp add: flip_subst_v (*‹atom (?x::x) ♯ (?c::?'a::has_subst_v) ⟹ (?x ↔ ?z::x) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_ce_def (*‹subst_v = subst_cev_abbrev›*))\n (*discuss goal 3*)\n apply (simp add: flip_subst_v (*‹atom ?x ♯ ?c ⟹ (?x ↔ ?z) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_ce_def (*‹subst_v = subst_cev_abbrev›*))\n (*discuss goal 4*)\n apply (simp add: flip_subst_v (*‹atom ?x ♯ ?c ⟹ (?x ↔ ?z) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_ce_def (*‹subst_v = subst_cev_abbrev›*))\n (*discuss goal 5*)\n apply (simp add: flip_subst_v (*‹atom ?x ♯ ?c ⟹ (?x ↔ ?z) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_ce_def (*‹subst_v = subst_cev_abbrev›*))\n (*discuss goal 6*)\n apply (simp add: flip_subst_v (*‹atom ?x ♯ ?c ⟹ (?x ↔ ?z) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_ce_def (*‹subst_v = subst_cev_abbrev›*))\n (*discuss goal 7*)\n apply (simp add: flip_subst_v (*‹atom (?x::x) ♯ (?c::?'a) ⟹ (?x ↔ ?z::x) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_ce_def (*‹subst_v = subst_cev_abbrev›*))\n (*proven 7 subgoals*) .\n\ninstantiation c :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_cv\"\n\ninstance proof\n fix j::atom and i::x and x::v and t::c\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n using fresh_subst_cv_if[of j t i x] subst_v_c_def by metis\n\n fix a::x and tm::c and x::v\n show \"atom a ♯ tm ⟹ subst_v tm a x = tm\"\n using forget_subst_cv subst_v_c_def by simp\n\n fix a::x and tm::c\n show \"subst_v tm a (V_var a) = tm\" using subst_cv_id subst_v_c_def by simp\n\n fix p::perm and x1::x and v::v and t1::c\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n using subst_cv_commute subst_v_c_def by simp\n\n fix x::x and c::c and z::x\n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\"\n using subst_cv_var_flip subst_v_c_def by simp\n\n fix x::x and c::c and z::x\n show \"atom x ♯ c ⟹ c[z::=[x]⇧v]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n using subst_cv_var_flip subst_v_c_def by simp\nqed\n\nend\n\nlemma subst_cv_var_flip1[simp]:\n fixes c::c\n assumes \"atom y ♯ c\"\n shows \"(x ↔ y) ∙ c = c[x::=V_var y]⇩c⇩v\"\n using subst_cv_var_flip (*‹atom ?y ♯ ?c ⟹ (?y ↔ ?x) ∙ ?c = ?c[?x::=[ ?y ]⇧v]⇩c⇩v›*) flip_commute (*‹(?a ↔ ?b) = (?b ↔ ?a)›*) by (metis assms (*‹atom (y::x) ♯ (c::c)›*))\n\nlemma subst_cv_v_flip3[simp]:\n fixes c::c\n assumes \"atom z1 ♯ c\" and \"atom z1' ♯ c\"\n shows\"(z1 ↔ z1') ∙ c[z::=[z1]⇧v]⇩c⇩v = c[z::=[z1']⇧v]⇩c⇩v\"\nproof (-)\n (*goal: ‹(z1::x ↔ z1'::x) ∙ (c::c)[(z::x)::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n consider \"z1' = z\" | \"z1 = z\" | \"atom z1 ♯ z ∧ atom z1' ♯ z\"\n (*goal: ‹⟦z1' = z ⟹ thesis; z1 = z ⟹ thesis; atom z1 ♯ z ∧ atom z1' ♯ z ⟹ thesis⟧ ⟹ thesis›*)\n by force\n then show \"?thesis\"\n (*goal: ‹(z1::x ↔ z1'::x) ∙ (c::c)[(z::x)::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n proof (cases)\n (*goals:\n 1. ‹z1' = z ⟹ (z1 ↔ z1') ∙ c[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›\n 2. ‹z1 = z ⟹ (z1 ↔ z1') ∙ c[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›\n 3. ‹atom z1 ♯ z ∧ atom z1' ♯ z ⟹ (z1 ↔ z1') ∙ c[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n case 1 (*‹z1' = z›*)\n then show \"?thesis\"\n (*goal: ‹(z1 ↔ z1') ∙ c[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n using \"1\" (*‹(z1'::x) = (z::x)›*) assms (*‹atom (z1::x) ♯ (c::c)› ‹atom z1' ♯ c›*) by auto\n next\n (*goals:\n 1. ‹(z1::x) = (z::x) ⟹ (z1 ↔ z1'::x) ∙ (c::c)[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›\n 2. ‹atom (z1::x) ♯ (z::x) ∧ atom (z1'::x) ♯ z ⟹ (z1 ↔ z1') ∙ (c::c)[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n case 2 (*‹z1 = z›*)\n then show \"?thesis\"\n (*goal: ‹(z1 ↔ z1') ∙ c[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n using \"2\" (*‹(z1::x) = (z::x)›*) assms (*‹atom z1 ♯ c› ‹atom z1' ♯ c›*) by auto\n next\n (*goal: ‹atom z1 ♯ z ∧ atom z1' ♯ z ⟹ (z1 ↔ z1') ∙ c[z::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n case 3 (*‹atom z1 ♯ z ∧ atom z1' ♯ z›*)\n then show \"?thesis\"\n (*goal: ‹(z1::x ↔ z1'::x) ∙ (c::c)[(z::x)::=[ z1 ]⇧v]⇩c⇩v = c[z::=[ z1' ]⇧v]⇩c⇩v›*)\n using assms (*‹atom z1 ♯ c› ‹atom z1' ♯ c›*) by auto\n qed\nqed\n\nlemma subst_cv_v_flip[simp]:\n fixes c::c\n assumes \"atom x ♯ c\"\n shows \"((x ↔ z) ∙ c)[x::=v]⇩c⇩v = c [z::=v]⇩c⇩v\"\n using assms (*‹atom x ♯ c›*) subst_v_c_def (*‹subst_v = subst_cv_abbrev›*) by auto\n\nlemma subst_cv_commute_full:\n fixes c::c\n assumes \"atom z ♯ v\" and \"atom x ♯ w\" and \"x≠z\"\n shows \"(c[z::=w]⇩c⇩v)[x::=v]⇩c⇩v = (c[x::=v]⇩c⇩v)[z::=w]⇩c⇩v\" \n using assms (*‹atom z ♯ v› ‹atom x ♯ w› ‹x ≠ z›*) proof (nominal_induct c rule: c.strong_induct)\n (*goals:\n 1. ‹⟦atom (z::x) ♯ (v::v); atom (x::x) ♯ (w::v); x ≠ z⟧ ⟹ (TRUE)[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (TRUE)[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 2. ‹⟦atom (z::x) ♯ (v::v); atom (x::x) ♯ (w::v); x ≠ z⟧ ⟹ (FALSE)[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (FALSE)[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 3. ‹⋀(x1::c) x2::c. ⟦⟦atom (z::x) ♯ (v::v); atom (x::x) ♯ (w::v); x ≠ z⟧ ⟹ x1[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x1[x::=v]⇩c⇩v[z::=w]⇩c⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x2[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x2[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x1 AND x2 )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (x1 AND x2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 4. ‹⋀(x1::c) x2::c. ⟦⟦atom (z::x) ♯ (v::v); atom (x::x) ♯ (w::v); x ≠ z⟧ ⟹ x1[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x1[x::=v]⇩c⇩v[z::=w]⇩c⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x2[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x2[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x1 OR x2 )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (x1 OR x2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 5. ‹⋀xa::c. ⟦⟦atom (z::x) ♯ (v::v); atom (x::x) ♯ (w::v); x ≠ z⟧ ⟹ xa[z::=w]⇩c⇩v[x::=v]⇩c⇩v = xa[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (¬ xa )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (¬ xa )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 6. ‹⋀(x1::c) x2::c. ⟦⟦atom (z::x) ♯ (v::v); atom (x::x) ♯ (w::v); x ≠ z⟧ ⟹ x1[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x1[x::=v]⇩c⇩v[z::=w]⇩c⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x2[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x2[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x1 IMP x2 )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (x1 IMP x2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 7. ‹⋀(x1::ce) x2::ce. ⟦atom (z::x) ♯ (v::v); atom (x::x) ♯ (w::v); x ≠ z⟧ ⟹ (x1 == x2 )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (x1 == x2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›*)\n case (C_eq e1 e2) (*‹atom z ♯ v› ‹atom x ♯ w› ‹x ≠ z›*)\n then show \"?case\"\n (*goal: ‹((e1::ce) == (e2::ce) )[(z::x)::=(w::v)]⇩c⇩v[(x::x)::=(v::v)]⇩c⇩v = (e1 == e2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›*)\n using subst_cev_commute_full (*‹⟦atom (?z::x) ♯ (?v::v); atom (?x::x) ♯ (?w::v); ?x ≠ ?z⟧ ⟹ ((?e::ce)[?z::=?w]⇩c⇩e⇩v)[?x::=?v]⇩c⇩e⇩v = (?e[?x::=?v]⇩c⇩e⇩v)[?z::=?w]⇩c⇩e⇩v›*) by simp\n qed ((force)+)\n (*solves the remaining goals:\n 1. ‹⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (TRUE)[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (TRUE)[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 2. ‹⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (FALSE)[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (FALSE)[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 3. ‹⋀x1 x2. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x1[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x1[x::=v]⇩c⇩v[z::=w]⇩c⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x2[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x2[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x1 AND x2 )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (x1 AND x2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 4. ‹⋀x1 x2. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x1[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x1[x::=v]⇩c⇩v[z::=w]⇩c⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x2[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x2[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x1 OR x2 )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (x1 OR x2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 5. ‹⋀xa. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ xa[z::=w]⇩c⇩v[x::=v]⇩c⇩v = xa[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (¬ xa )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (¬ xa )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›\n 6. ‹⋀x1 x2. ⟦⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x1[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x1[x::=v]⇩c⇩v[z::=w]⇩c⇩v; ⟦atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ x2[z::=w]⇩c⇩v[x::=v]⇩c⇩v = x2[x::=v]⇩c⇩v[z::=w]⇩c⇩v; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ (x1 IMP x2 )[z::=w]⇩c⇩v[x::=v]⇩c⇩v = (x1 IMP x2 )[x::=v]⇩c⇩v[z::=w]⇩c⇩v›*)\n\nlemma subst_cv_eq[simp]:\n assumes \"atom z1 ♯ e1\" \n shows \"(CE_val (V_var z1) == e1 )[z1::=[x]⇧v]⇩c⇩v = (CE_val (V_var x) == e1 )\" (is \"?A = ?B\")\nproof (-)\n (*goal: ‹([ [ z1::x ]⇧v ]⇧c⇧e == (e1::ce) )[z1::=[ x::x ]⇧v]⇩c⇩v = ([ [ x ]⇧v ]⇧c⇧e == e1 )›*)\n have \"?A = (((CE_val (V_var z1))[z1::=[x]⇧v]⇩c⇩e⇩v) == e1)\"\n using subst_cv.simps (*‹(TRUE)[(?x::x)::=(?v::v)]⇩c⇩v = (TRUE)› ‹(FALSE)[(?x::x)::=(?v::v)]⇩c⇩v = (FALSE)› ‹(?c1.0 AND ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v AND ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 OR ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v OR ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 IMP ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v IMP ?c2.0[?x::=?v]⇩c⇩v )› ‹((?e1.0::ce) == (?e2.0::ce) )[(?x::x)::=(?v::v)]⇩c⇩v = (?e1.0[?x::=?v]⇩c⇩e⇩v == ?e2.0[?x::=?v]⇩c⇩e⇩v )› ‹(¬ ?c )[?x::=?v]⇩c⇩v = (¬ ?c[?x::=?v]⇩c⇩v )›*) assms (*‹atom z1 ♯ e1›*) by simp\n thus \"?thesis\"\n (*goal: ‹([ [ z1 ]⇧v ]⇧c⇧e == e1 )[z1::=[ x ]⇧v]⇩c⇩v = ([ [ x ]⇧v ]⇧c⇧e == e1 )›*)\n by simp\nqed\n\nsection ‹Variable Context›\n\ntext ‹The idea of this substitution is to remove x from the context. We really want to add the condition \nthat x is fresh in v but this causes problems with proofs.›\n\nnominal_function subst_gv :: \"Γ ⇒ x ⇒ v ⇒ Γ\" where\n \"subst_gv GNil x v = GNil\"\n| \"subst_gv ((y,b,c) #⇩Γ Γ) x v = (if x = y then Γ else ((y,b,c[x::=v]⇩c⇩v)#⇩Γ (subst_gv Γ x v )))\"\nproof(goal_cases)\n case 1\n then show ?case by(simp add: eqvt_def subst_gv_graph_aux_def )\nnext\n case (3 P x)\n then show ?case by (metis neq_GNil_conv prod_cases3)\nqed(fast+)\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_gv_abbrev :: \"Γ ⇒ x ⇒ v ⇒ Γ\" (\"_[_::=_]⇩Γ⇩v\" [1000,50,50] 1000)\n where \n \"g[x::=v]⇩Γ⇩v ≡ subst_gv g x v\" \n\nlemma size_subst_gv [simp]: \"size ( subst_gv G i x ) ≤ size G\"\n apply (induct G)\n (*goals:\n 1. ‹size GNil[i::=x]⇩Γ⇩v ≤ size GNil›\n 2. ‹⋀x1 G. size G[i::=x]⇩Γ⇩v ≤ size G ⟹ size (x1 #⇩Γ G)[i::=x]⇩Γ⇩v ≤ size (x1 #⇩Γ G)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) . \n\nlemma forget_subst_gv [simp]: \"atom a ♯ G ⟹ subst_gv G a x = G\"\n apply (induct G, auto)\n (*goal: ‹atom (a::x) ♯ (G::Γ) ⟹ G[a::=(x::v)]⇩Γ⇩v = G›*)\n using fresh_GCons (*‹?a ♯ ?x #⇩Γ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) fresh_PairD(1) (*‹?a ♯ (?x, ?y) ⟹ ?a ♯ ?x›*) not_self_fresh (*‹atom ?x ♯ ?x = False›*)\n (*goals:\n 1. ‹⋀aa b G. ⟦atom a ♯ G ⟹ G[a::=x]⇩Γ⇩v = G; atom a ♯ (a, aa, b) #⇩Γ G⟧ ⟹ False›\n 2. ‹⋀aa aaa b G. ⟦atom a ♯ G ⟹ G[a::=x]⇩Γ⇩v = G; atom a ♯ (aa, aaa, b) #⇩Γ G; a ≠ aa⟧ ⟹ b[a::=x]⇩c⇩v = b›\n 3. ‹⋀aa aaa b G. ⟦atom a ♯ G ⟹ G[a::=x]⇩Γ⇩v = G; atom a ♯ (aa, aaa, b) #⇩Γ G; a ≠ aa⟧ ⟹ G[a::=x]⇩Γ⇩v = G›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply (simp add: fresh_GCons (*‹(?a::atom) ♯ (?x::x × b × c) #⇩Γ (?xs::Γ) = (?a ♯ ?x ∧ ?a ♯ ?xs)›*))\n (*discuss goal 3*)\n apply (simp add: fresh_GCons (*‹?a ♯ ?x #⇩Γ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*))\n (*proven 3 subgoals*) .\n\nlemma fresh_subst_gv: \"atom a ♯ G ⟹ atom a ♯ v ⟹ atom a ♯ subst_gv G x v\"\nproof (induct G)\n (*goals:\n 1. ‹⟦atom a ♯ GNil; atom a ♯ v⟧ ⟹ atom a ♯ GNil[x::=v]⇩Γ⇩v›\n 2. ‹⋀x1 G. ⟦⟦atom a ♯ G; atom a ♯ v⟧ ⟹ atom a ♯ G[x::=v]⇩Γ⇩v; atom a ♯ x1 #⇩Γ G; atom a ♯ v⟧ ⟹ atom a ♯ (x1 #⇩Γ G)[x::=v]⇩Γ⇩v›*)\n case GNil (*‹atom a ♯ GNil› ‹atom a ♯ v›*)\n then show \"?case\"\n (*goal: ‹atom a ♯ GNil[x::=v]⇩Γ⇩v›*)\n by auto\nnext\n (*goal: ‹⋀(x1::x × b × c) G::Γ. ⟦⟦atom (a::'a) ♯ G; atom a ♯ (v::v)⟧ ⟹ atom a ♯ G[(x::x)::=v]⇩Γ⇩v; atom a ♯ x1 #⇩Γ G; atom a ♯ v⟧ ⟹ atom a ♯ (x1 #⇩Γ G)[x::=v]⇩Γ⇩v›*)\n case (GCons xbc G) (*‹⟦atom a ♯ G; atom a ♯ v⟧ ⟹ atom a ♯ G[x::=v]⇩Γ⇩v› ‹atom a ♯ xbc #⇩Γ G› ‹atom (a::'a) ♯ (v::v)›*)\n obtain x' and b' and c' where xbc: \"xbc = (x',b',c')\"\n (*goal: ‹(⋀(x'::x) (b'::b) c'::c. (xbc::x × b × c) = (x', b', c') ⟹ thesis::bool) ⟹ thesis›*)\n using prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*) by blast\n show \"?case\"\n (*goal: ‹atom (a::'a) ♯ ((xbc::x × b × c) #⇩Γ (G::Γ))[(x::x)::=(v::v)]⇩Γ⇩v›*)\n proof (cases \"x=x'\")\n (*goals:\n 1. ‹(x::x) = (x'::x) ⟹ atom (a::'a) ♯ ((xbc::x × b × c) #⇩Γ (G::Γ))[x::=(v::v)]⇩Γ⇩v›\n 2. ‹(x::x) ≠ (x'::x) ⟹ atom (a::'a) ♯ ((xbc::x × b × c) #⇩Γ (G::Γ))[x::=(v::v)]⇩Γ⇩v›*)\n case True (*‹x = x'›*)\n have \"atom a ♯ G\"\n using GCons (*‹⟦atom a ♯ G; atom a ♯ v⟧ ⟹ atom a ♯ G[x::=v]⇩Γ⇩v› ‹atom a ♯ xbc #⇩Γ G› ‹atom a ♯ v›*) fresh_GCons (*‹?a ♯ ?x #⇩Γ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) by blast\n thus \"?thesis\"\n (*goal: ‹atom a ♯ (xbc #⇩Γ G)[x::=v]⇩Γ⇩v›*)\n using subst_gv.simps(2)[of x' b' c' G] (*‹((x', b', c') #⇩Γ G)[?x::=?v]⇩Γ⇩v = (if ?x = x' then G else (x', b', c'[?x::=?v]⇩c⇩v) #⇩Γ G[?x::=?v]⇩Γ⇩v)›*) GCons (*‹⟦atom a ♯ G; atom a ♯ v⟧ ⟹ atom a ♯ G[x::=v]⇩Γ⇩v› ‹atom a ♯ xbc #⇩Γ G› ‹atom a ♯ v›*) xbc (*‹(xbc::x × b × c) = (x'::x, b'::b, c'::c)›*) True (*‹x = x'›*) by presburger\n next\n (*goal: ‹(x::x) ≠ (x'::x) ⟹ atom (a::'a) ♯ ((xbc::x × b × c) #⇩Γ (G::Γ))[x::=(v::v)]⇩Γ⇩v›*)\n case False (*‹(x::x) ≠ (x'::x)›*)\n then show \"?thesis\"\n (*goal: ‹atom a ♯ (xbc #⇩Γ G)[x::=v]⇩Γ⇩v›*)\n using subst_gv.simps(2)[of x' b' c' G] (*‹((x', b', c') #⇩Γ G)[?x::=?v]⇩Γ⇩v = (if ?x = x' then G else (x', b', c'[?x::=?v]⇩c⇩v) #⇩Γ G[?x::=?v]⇩Γ⇩v)›*) GCons (*‹⟦atom a ♯ G; atom a ♯ v⟧ ⟹ atom a ♯ G[x::=v]⇩Γ⇩v› ‹atom a ♯ xbc #⇩Γ G› ‹atom a ♯ v›*) xbc (*‹xbc = (x', b', c')›*) False (*‹x ≠ x'›*) fresh_GCons (*‹?a ♯ ?x #⇩Γ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) by simp\n qed\nqed\n\nlemma subst_gv_flip: \n fixes x::x and xa::x and z::x and c::c and b::b and Γ::Γ\n assumes \"atom xa ♯ ((x, b, c[z::=[x]⇧v]⇩c⇩v) #⇩Γ Γ)\" and \"atom xa ♯ Γ\" and \"atom x ♯ Γ\" and \"atom x ♯ (z, c)\" and \"atom xa ♯ (z, c)\"\n shows \"(x ↔ xa) ∙ ((x, b, c[z::=[x]⇧v]⇩c⇩v) #⇩Γ Γ) = (xa, b, c[z::=V_var xa]⇩c⇩v) #⇩Γ Γ\"\nproof (-)\n (*goal: ‹(x ↔ xa) ∙ ((x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ) = (xa, b, c[z::=[ xa ]⇧v]⇩c⇩v) #⇩Γ Γ›*)\n have \"(x ↔ xa) ∙ ((x, b, c[z::=[x]⇧v]⇩c⇩v) #⇩Γ Γ) = (( (x ↔ xa) ∙ x, b, (x ↔ xa) ∙ c[z::=[x]⇧v]⇩c⇩v) #⇩Γ ((x ↔ xa) ∙ Γ))\"\n using subst (*‹⟦?s = ?t; ?P ?s⟧ ⟹ ?P ?t›*) Cons_eqvt (*‹(?p::perm) ∙ ((?x::?'a) # (?xs::?'a list)) = ?p ∙ ?x # ?p ∙ ?xs›*) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) using G_cons_flip (*‹(?x ↔ ?x') ∙ ((?x'', ?b, ?c) #⇩Γ ?Γ) = ((?x ↔ ?x') ∙ ?x'', ?b, (?x ↔ ?x') ∙ ?c) #⇩Γ (?x ↔ ?x') ∙ ?Γ›*) by simp\n also (*calculation: ‹(x ↔ xa) ∙ ((x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ) = ((x ↔ xa) ∙ x, b, (x ↔ xa) ∙ c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ (x ↔ xa) ∙ Γ›*) have \"... = ((xa, b, (x ↔ xa) ∙ c[z::=[x]⇧v]⇩c⇩v) #⇩Γ ((x ↔ xa) ∙ Γ))\"\n using assms (*‹atom xa ♯ (x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ› ‹atom xa ♯ Γ› ‹atom x ♯ Γ› ‹atom x ♯ (z, c)› ‹atom xa ♯ (z, c)›*) by fastforce\n also (*calculation: ‹(x ↔ xa) ∙ ((x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ) = (xa, b, (x ↔ xa) ∙ c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ (x ↔ xa) ∙ Γ›*) have \"... = ((xa, b, c[z::=V_var xa]⇩c⇩v) #⇩Γ ((x ↔ xa) ∙ Γ))\"\n using assms (*‹atom xa ♯ (x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ› ‹atom (xa::x) ♯ (Γ::Γ)› ‹atom x ♯ Γ› ‹atom x ♯ (z, c)› ‹atom xa ♯ (z, c)›*) subst_cv_var_flip (*‹atom ?y ♯ ?c ⟹ (?y ↔ ?x) ∙ ?c = ?c[?x::=[ ?y ]⇧v]⇩c⇩v›*) by fastforce\n also (*calculation: ‹(x ↔ xa) ∙ ((x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ) = (xa, b, c[z::=[ xa ]⇧v]⇩c⇩v) #⇩Γ (x ↔ xa) ∙ Γ›*) have \"... = ((xa, b, c[z::=V_var xa]⇩c⇩v) #⇩Γ Γ)\"\n using assms (*‹atom (xa::x) ♯ (x::x, b::b, (c::c)[(z::x)::=[ x ]⇧v]⇩c⇩v) #⇩Γ (Γ::Γ)› ‹atom xa ♯ Γ› ‹atom x ♯ Γ› ‹atom x ♯ (z, c)› ‹atom xa ♯ (z, c)›*) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) by blast\n finally (*calculation: ‹(x ↔ xa) ∙ ((x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ) = (xa, b, c[z::=[ xa ]⇧v]⇩c⇩v) #⇩Γ Γ›*) show \"?thesis\"\n (*goal: ‹(x ↔ xa) ∙ ((x, b, c[z::=[ x ]⇧v]⇩c⇩v) #⇩Γ Γ) = (xa, b, c[z::=[ xa ]⇧v]⇩c⇩v) #⇩Γ Γ›*)\n by simp\nqed\n\nsection ‹Types›\n\nnominal_function subst_tv :: \"τ ⇒ x ⇒ v ⇒ τ\" where\n \"atom z ♯ (x,v) ⟹ subst_tv ⦃ z : b | c ⦄ x v = ⦃ z : b | c[x::=v]⇩c⇩v ⦄\"\n apply (simp add: eqvt_def subst_tv_graph_aux_def )\n apply auto\n subgoal for P a aa b\n apply(rule_tac y=a and c=\"(aa,b)\" in τ.strong_exhaust)\n by (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base) \n apply (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base)\nproof -\n fix z :: x and c :: c and za :: x and xa :: x and va :: v and ca :: c and cb :: x\n assume a1: \"atom za ♯ va\" and a2: \"atom z ♯ va\" and a3: \"∀cb. atom cb ♯ c ∧ atom cb ♯ ca ⟶ cb ≠ z ∧ cb ≠ za ⟶ c[z::=V_var cb]⇩c⇩v = ca[za::=V_var cb]⇩c⇩v\"\n assume a4: \"atom cb ♯ c\" and a5: \"atom cb ♯ ca\" and a6: \"cb ≠ z\" and a7: \"cb ≠ za\" and \"atom cb ♯ va\" and a8: \"za ≠ xa\" and a9: \"z ≠ xa\"\n assume a10:\"cb ≠ xa\"\n note assms = a10 a9 a8 a7 a6 a5 a4 a3 a2 a1 \n\n have \"c[z::=V_var cb]⇩c⇩v = ca[za::=V_var cb]⇩c⇩v\" using assms by auto\n hence \"c[z::=V_var cb]⇩c⇩v[xa::=va]⇩c⇩v = ca[za::=V_var cb]⇩c⇩v[xa::=va]⇩c⇩v\" by simp\n moreover have \"c[z::=V_var cb]⇩c⇩v[xa::=va]⇩c⇩v = c[xa::=va]⇩c⇩v[z::=V_var cb]⇩c⇩v\" using subst_cv_commute_full[of z va xa \"V_var cb\" ] assms fresh_def v.supp by fastforce\n moreover have \"ca[za::=V_var cb]⇩c⇩v[xa::=va]⇩c⇩v = ca[xa::=va]⇩c⇩v[za::=V_var cb]⇩c⇩v\" \n using subst_cv_commute_full[of za va xa \"V_var cb\" ] assms fresh_def v.supp by fastforce\n\n ultimately show \"c[xa::=va]⇩c⇩v[z::=V_var cb]⇩c⇩v = ca[xa::=va]⇩c⇩v[za::=V_var cb]⇩c⇩v\" by simp\nqed\n\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_tv_abbrev :: \"τ ⇒ x ⇒ v ⇒ τ\" (\"_[_::=_]⇩τ⇩v\" [1000,50,50] 1000)\n where \n \"t[x::=v]⇩τ⇩v ≡ subst_tv t x v\" \n\nlemma size_subst_tv [simp]: \"size ( subst_tv A i x ) = size A\"\nproof (nominal_induct A avoiding: i x rule: τ.strong_induct)\n (*goal: ‹⋀x1a x2a x3a i x. ⟦atom x1a ♯ i; atom x1a ♯ x⟧ ⟹ size ⦃ x1a : x2a | x3a ⦄[i::=x]⇩τ⇩v = size ⦃ x1a : x2a | x3a ⦄›*)\n case (T_refined_type x' b' c') (*‹atom x' ♯ i› ‹atom x' ♯ x›*)\n then show \"?case\"\n (*goal: ‹size ⦃ (x'::x) : (b'::b) | c'::c ⦄[(i::x)::=(x::v)]⇩τ⇩v = size ⦃ x' : b' | c' ⦄›*)\n by auto\nqed\n\nlemma forget_subst_tv [simp]: \"atom a ♯ A ⟹ subst_tv A a x = A\"\n apply (nominal_induct A avoiding: a x rule: τ.strong_induct)\n (*goal: ‹atom a ♯ A ⟹ A[a::=x]⇩τ⇩v = A›*)\n by (auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))\n\nlemma subst_tv_id [simp]: \"subst_tv A a (V_var a) = A\"\n apply (nominal_induct A avoiding: a rule: τ.strong_induct)\n (*goal: ‹A[a::=[ a ]⇧v]⇩τ⇩v = A›*)\n by (auto simp: fresh_at_base (*‹sort_of (?a::atom) ≠ sort_of (atom (?b::?'a)) ⟹ ?a ♯ ?b› ‹(?a::atom) ♯ (?b::?'a) = (?a ≠ atom ?b)›*))\n\nlemma fresh_subst_tv_if [simp]:\n \"j ♯ (subst_tv A i x ) ⟷ (atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i))\"\n apply (nominal_induct A avoiding: i x rule: τ.strong_induct)\n (*goal: ‹j ♯ A[i::=x]⇩τ⇩v = (atom i ♯ A ∧ j ♯ A ∨ j ♯ x ∧ (j ♯ A ∨ j = atom i))›*)\n using fresh_def (*‹?a ♯ ?x ≡ ?a ∉ supp ?x›*) supp_b_empty (*‹atom ?x ∉ supp ?b›*) x_fresh_b (*‹atom (?x::x) ♯ (?b::b)›*) by auto\n\nlemma subst_tv_commute [simp]:\n \"atom y ♯ τ ⟹ (τ[x::= t]⇩τ⇩v)[y::=v]⇩τ⇩v = τ[x::= t[y::=v]⇩v⇩v]⇩τ⇩v \"\n apply (nominal_induct τ avoiding: x y t v rule: τ.strong_induct)\n (*goal: ‹atom y ♯ τ ⟹ τ[x::=t]⇩τ⇩v[y::=v]⇩τ⇩v = τ[x::=t[y::=v]⇩v⇩v]⇩τ⇩v›*)\n by (auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))\n\nlemma subst_tv_var_flip [simp]:\n fixes x::x and xa::x and τ::τ\n assumes \"atom xa ♯ τ\"\n shows \"(x ↔ xa) ∙ τ = τ[x::=V_var xa]⇩τ⇩v\"\nproof (-)\n (*goal: ‹(x ↔ xa) ∙ τ = τ[x::=[ xa ]⇧v]⇩τ⇩v›*)\n obtain z :: x and b and c where zbc: \"atom z ♯ (x,xa, V_var xa) ∧ τ = ⦃ z : b | c ⦄\"\n (*goal: ‹(⋀z b c. atom z ♯ (x, xa, [ xa ]⇧v) ∧ τ = ⦃ z : b | c ⦄ ⟹ thesis) ⟹ thesis›*)\n using obtain_fresh_z (*‹(⋀z b c. atom z ♯ ?t ∧ ?τ = ⦃ z : b | c ⦄ ⟹ ?thesis) ⟹ ?thesis›*) by (metis prod.inject (*‹((?x1.0, ?x2.0) = (?y1.0, ?y2.0)) = (?x1.0 = ?y1.0 ∧ ?x2.0 = ?y2.0)›*) subst_tv.cases (*‹(⋀z x v b c. ⟦atom z ♯ (x, v); ?x = (⦃ z : b | c ⦄, x, v)⟧ ⟹ ?P) ⟹ ?P›*))\n hence \"atom xa ∉ supp c - { atom z }\"\n using \"τ.supp\"[of z b c] (*‹supp ⦃ z : b | c ⦄ = supp c - set [atom z] ∪ supp b›*) fresh_def (*‹?a ♯ ?x ≡ ?a ∉ supp ?x›*) supp_b_empty (*‹atom ?x ∉ supp ?b›*) assms (*‹atom xa ♯ τ›*) by auto\n moreover have \"xa ≠ z\"\n using zbc (*‹atom z ♯ (x, xa, [ xa ]⇧v) ∧ τ = ⦃ z : b | c ⦄›*) fresh_prod3 (*‹?x ♯ (?a, ?b, ?c) = (?x ♯ ?a ∧ ?x ♯ ?b ∧ ?x ♯ ?c)›*) by force\n ultimately have xaf: \"atom xa ♯ c\"\n using fresh_def (*‹?a ♯ ?x ≡ ?a ∉ supp ?x›*) by auto\n have \"(x ↔ xa) ∙ τ = ⦃ z : b | (x ↔ xa) ∙ c ⦄\"\n by (metis τ.perm_simps (*‹?p ∙ ⦃ ?x : ?b | ?c ⦄ = ⦃ ?p ∙ ?x : ?p ∙ ?b | ?p ∙ ?c ⦄›*) empty_iff (*‹(?c ∈ {}) = False›*) flip_at_base_simps( (*‹⟦?a ≠ ?c; ?b ≠ ?c⟧ ⟹ (?a ↔ ?b) ∙ ?c = ?c›*) 3) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) fresh_PairD( (*‹?a ♯ (?x, ?y) ⟹ ?a ♯ ?x›*) 1) fresh_PairD( (*‹?a ♯ (?x, ?y) ⟹ ?a ♯ ?y›*) 2) fresh_def (*‹?a ♯ ?x ≡ ?a ∉ supp ?x›*) not_self_fresh (*‹atom ?x ♯ ?x = False›*) supp_b_empty (*‹atom ?x ∉ supp ?b›*) v.fresh( (*‹?a ♯ [ ?x ]⇧v = ?a ♯ ?x›*) 2) zbc (*‹atom z ♯ (x, xa, [ xa ]⇧v) ∧ τ = ⦃ z : b | c ⦄›*))\n also (*calculation: ‹(x ↔ xa) ∙ τ = ⦃ z : b | (x ↔ xa) ∙ c ⦄›*) have \"... = ⦃ z : b | c[x::=V_var xa]⇩c⇩v ⦄\"\n using subst_cv_v_flip (*‹atom (?x::x) ♯ (?c::c) ⟹ ((?x ↔ ?z::x) ∙ ?c)[?x::=(?v::v)]⇩c⇩v = ?c[?z::=?v]⇩c⇩v›*) xaf (*‹atom xa ♯ c›*) by (metis permute_flip_cancel (*‹(?a ↔ ?b) ∙ (?a ↔ ?b) ∙ ?x = ?x›*) permute_flip_cancel2 (*‹(?a ↔ ?b) ∙ (?b ↔ ?a) ∙ ?x = ?x›*) subst_cv_var_flip (*‹atom ?y ♯ ?c ⟹ (?y ↔ ?x) ∙ ?c = ?c[?x::=[ ?y ]⇧v]⇩c⇩v›*))\n finally (*calculation: ‹(x::x ↔ xa::x) ∙ (τ::τ) = ⦃ (z::x) : (b::b) | (c::c)[x::=[ xa ]⇧v]⇩c⇩v ⦄›*) show \"?thesis\"\n (*goal: ‹(x ↔ xa) ∙ τ = τ[x::=[ xa ]⇧v]⇩τ⇩v›*)\n using subst_tv.simps (*‹atom (?z::x) ♯ (?x::x, ?v::v) ⟹ ⦃ ?z : (?b::b) | ?c::c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) zbc (*‹atom (z::x) ♯ (x::x, xa::x, [ xa ]⇧v) ∧ (τ::τ) = ⦃ z : (b::b) | c::c ⦄›*) using fresh_PairD(1) (*‹?a ♯ (?x, ?y) ⟹ ?a ♯ ?x›*) not_self_fresh (*‹atom ?x ♯ ?x = False›*) by force\nqed\n\ninstantiation τ :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_tv\"\n\ninstance proof\n fix j::atom and i::x and x::v and t::τ\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n\n proof(nominal_induct t avoiding: i x rule:τ.strong_induct)\n case (T_refined_type z b c)\n hence \" j ♯ ⦃ z : b | c ⦄[i::=x]⇩v = j ♯ ⦃ z : b | c[i::=x]⇩c⇩v ⦄\" using subst_tv.simps subst_v_τ_def fresh_Pair by simp\n also have \"... = (atom i ♯ ⦃ z : b | c ⦄ ∧ j ♯ ⦃ z : b | c ⦄ ∨ j ♯ x ∧ (j ♯ ⦃ z : b | c ⦄ ∨ j = atom i))\" \n unfolding τ.fresh using subst_v_c_def fresh_subst_v_if \n using T_refined_type.hyps(1) T_refined_type.hyps(2) x_fresh_b by auto\n finally show ?case by auto\n qed\n\n fix a::x and tm::τ and x::v\n show \"atom a ♯ tm ⟹ subst_v tm a x = tm\"\n apply(nominal_induct tm avoiding: a x rule:τ.strong_induct)\n using subst_v_c_def forget_subst_v subst_tv.simps subst_v_τ_def fresh_Pair by simp\n\n fix a::x and tm::τ\n show \"subst_v tm a (V_var a) = tm\" \n apply(nominal_induct tm avoiding: a rule:τ.strong_induct)\n using subst_v_c_def forget_subst_v subst_tv.simps subst_v_τ_def fresh_Pair by simp\n\n fix p::perm and x1::x and v::v and t1::τ\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n apply(nominal_induct tm avoiding: a x rule:τ.strong_induct)\n using subst_v_c_def forget_subst_v subst_tv.simps subst_v_τ_def fresh_Pair by simp\n\n fix x::x and c::τ and z::x\n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\" \n apply(nominal_induct c avoiding: z x rule:τ.strong_induct)\n using subst_v_c_def flip_subst_v subst_tv.simps subst_v_τ_def fresh_Pair by auto\n\n fix x::x and c::τ and z::x\n show \"atom x ♯ c ⟹ c[z::=[x]⇧v]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n apply(nominal_induct c avoiding: x v z rule:τ.strong_induct)\n using subst_v_c_def subst_tv.simps subst_v_τ_def fresh_Pair \n by (metis flip_commute subst_tv_commute subst_tv_var_flip subst_v_τ_def subst_vv.simps(2))\nqed\n\nend\n\nlemma subst_tv_commute_full:\n fixes c::τ\n assumes \"atom z ♯ v\" and \"atom x ♯ w\" and \"x≠z\"\n shows \"(c[z::=w]⇩τ⇩v)[x::=v]⇩τ⇩v = (c[x::=v]⇩τ⇩v)[z::=w]⇩τ⇩v\" \n using assms (*‹atom (z::x) ♯ (v::v)› ‹atom x ♯ w› ‹x ≠ z›*) proof (nominal_induct c avoiding: x v z w rule: τ.strong_induct)\n (*goal: ‹⋀(x1a::x) (x2a::b) (x3a::c) (x::x) (v::v) (z::x) w::v. ⟦atom x1a ♯ x; atom x1a ♯ v; atom x1a ♯ z; atom x1a ♯ w; atom z ♯ v; atom x ♯ w; x ≠ z⟧ ⟹ ⦃ x1a : x2a | x3a ⦄[z::=w]⇩τ⇩v[x::=v]⇩τ⇩v = ⦃ x1a : x2a | x3a ⦄[x::=v]⇩τ⇩v[z::=w]⇩τ⇩v›*)\n case (T_refined_type x1a x2a x3a) (*‹atom x1a ♯ x› ‹atom x1a ♯ v› ‹atom (x1a::x) ♯ (z::x)› ‹atom (x1a::x) ♯ (w::v)› ‹atom (z::x) ♯ (v::v)› ‹atom x ♯ w› ‹(x::x) ≠ (z::x)›*)\n then show \"?case\"\n (*goal: ‹⦃ x1a : x2a | x3a ⦄[z::=w]⇩τ⇩v[x::=v]⇩τ⇩v = ⦃ x1a : x2a | x3a ⦄[x::=v]⇩τ⇩v[z::=w]⇩τ⇩v›*)\n using subst_cv_commute_full (*‹⟦atom ?z ♯ ?v; atom ?x ♯ ?w; ?x ≠ ?z⟧ ⟹ ?c[?z::=?w]⇩c⇩v[?x::=?v]⇩c⇩v = ?c[?x::=?v]⇩c⇩v[?z::=?w]⇩c⇩v›*) by simp\n qed\n\nlemma type_eq_subst_eq:\n fixes v::v and c1::c\n assumes \"⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄\"\n shows \"c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v\"\n using subst_v_flip_eq_two[of z1 c1 z2 c2 v] (*‹[[atom z1]]lst. c1 = [[atom z2]]lst. c2 ⟹ c1[z1::=v]⇩v = c2[z2::=v]⇩v›*) \"τ.eq_iff\" (*‹(⦃ (?x::x) : (?b::b) | ?c::c ⦄ = ⦃ (?xa::x) : (?ba::b) | ?ca::c ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) assms (*‹⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄›*) subst_v_c_def (*‹subst_v = subst_cv_abbrev›*) by simp\n\ntext ‹Extract constraint from a type. We cannot just project out the constraint as this would\nmean alpha-equivalent types give different answers ›\n\nnominal_function c_of :: \"τ ⇒ x ⇒ c\" where\n \"atom z ♯ x ⟹ c_of (T_refined_type z b c) x = c[z::=[x]⇧v]⇩c⇩v\"\nproof(goal_cases)\n case 1\n then show ?case using eqvt_def c_of_graph_aux_def by force\nnext\n case (2 x y)\n then show ?case using eqvt_def c_of_graph_aux_def by force\nnext\n case (3 P x)\n then obtain x1::τ and x2::x where *:\"x = (x1,x2)\" by force\n obtain z' and b' and c' where \"x1 = ⦃ z' : b' | c' ⦄ ∧ atom z' ♯ x2\" using obtain_fresh_z by metis\n then show ?case using 3 * by auto\nnext\n case (4 z1 x1 b1 c1 z2 x2 b2 c2)\n then show ?case using subst_v_flip_eq_two τ.eq_iff by (metis prod.inject type_eq_subst_eq)\nqed\n\nnominal_termination (eqvt) by lexicographic_order\n\nlemma c_of_eq:\n shows \"c_of ⦃ x : b | c ⦄ x = c\"\nproof (nominal_induct \"⦃ x : b | c ⦄\" avoiding: x rule: τ.strong_induct)\n (*goal: ‹⋀x1a x3a x. ⟦atom x1a ♯ x; [[atom x1a]]lst. x3a = [[atom x]]lst. c⟧ ⟹ c_of ⦃ x : b | c ⦄ x = c›*)\n case (T_refined_type x' c') (*‹atom x' ♯ x› ‹[[atom x']]lst. c' = [[atom x]]lst. c›*)\n moreover hence \"c_of ⦃ x' : b | c' ⦄ x = c'[x'::=V_var x]⇩c⇩v\"\n using c_of.simps (*‹atom (?z::x) ♯ (?x::x) ⟹ c_of ⦃ ?z : (?b::b) | ?c::c ⦄ ?x = ?c[?z::=[ ?x ]⇧v]⇩c⇩v›*) by auto\n moreover have \"⦃ x' : b | c' ⦄ = ⦃ x : b | c ⦄\"\n using T_refined_type (*‹atom x' ♯ x› ‹[[atom x']]lst. c' = [[atom x]]lst. c›*) \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) by metis\n moreover have \"c'[x'::=V_var x]⇩c⇩v = c\"\n using T_refined_type (*‹atom (x'::x) ♯ (x::x)› ‹[[atom x']]lst. c' = [[atom x]]lst. c›*) Abs1_eq_iff (*‹([{atom ?a}]set. ?x = [{atom ?b}]set. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([{atom ?a}]res. ?x = [{atom ?b}]res. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([[atom ?a]]lst. ?x = [[atom ?b]]lst. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)›*) flip_subst_v (*‹atom ?x ♯ ?c ⟹ (?x ↔ ?z) ∙ ?c = ?c[?z::=[ ?x ]⇧v]⇩v›*) subst_v_c_def (*‹subst_v = subst_cv_abbrev›*) by (metis subst_cv_id (*‹?A[?a::=[ ?a ]⇧v]⇩c⇩v = ?A›*))\n ultimately show \"?case\"\n (*goal: ‹c_of ⦃ (x::x) : (b::b) | c::c ⦄ x = c›*)\n by auto\nqed\n\nlemma obtain_fresh_z_c_of:\n fixes t::\"'b::fs\"\n obtains z where \"atom z ♯ t ∧ τ = ⦃ z : b_of τ | c_of τ z ⦄\"\nproof (-)\n (*goal: ‹(⋀z. atom z ♯ t ∧ τ = ⦃ z : b_of τ | c_of τ z ⦄ ⟹ thesis) ⟹ thesis›*)\n obtain z and c where \"atom z ♯ t ∧ τ = ⦃ z : b_of τ | c ⦄\"\n (*goal: ‹(⋀z c. atom z ♯ t ∧ τ = ⦃ z : b_of τ | c ⦄ ⟹ thesis) ⟹ thesis›*)\n using obtain_fresh_z2 (*‹(⋀z c. atom z ♯ ?t ∧ ?τ = ⦃ z : b_of ?τ | c ⦄ ⟹ ?thesis) ⟹ ?thesis›*) by metis\n moreover hence \"c = c_of τ z\"\n using c_of.simps (*‹atom ?z ♯ ?x ⟹ c_of ⦃ ?z : ?b | ?c ⦄ ?x = ?c[?z::=[ ?x ]⇧v]⇩c⇩v›*) using c_of_eq (*‹c_of ⦃ ?x : ?b | ?c ⦄ ?x = ?c›*) by metis\n ultimately show \"?thesis\"\n (*goal: ‹thesis›*)\n using that (*‹atom (?z::x) ♯ (t::'b) ∧ (τ::τ) = ⦃ ?z : b_of τ | c_of τ ?z ⦄ ⟹ thesis::bool›*) by auto\nqed\n\nlemma c_of_fresh:\n fixes x::x\n assumes \"atom x ♯ (t,z)\" \n shows \"atom x ♯ c_of t z\" \nproof (-)\n (*goal: ‹atom (x::x) ♯ c_of (t::τ) (z::x)›*)\n obtain z' and c' where z: \"t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x,z)\"\n (*goal: ‹(⋀(z'::x) c'::c. (t::τ) = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x::x, z::x) ⟹ thesis::bool) ⟹ thesis›*)\n using obtain_fresh_z_c_of (*‹(⋀z. atom z ♯ ?t ∧ ?τ = ⦃ z : b_of ?τ | c_of ?τ z ⦄ ⟹ ?thesis) ⟹ ?thesis›*) by metis\n hence \"*\": \"c_of t z = c'[z'::=V_var z]⇩c⇩v\"\n using c_of.simps (*‹atom ?z ♯ ?x ⟹ c_of ⦃ ?z : ?b | ?c ⦄ ?x = ?c[?z::=[ ?x ]⇧v]⇩c⇩v›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) by metis\n have \"(atom x ♯ c' ∨ atom x ∈ set [atom z']) ∧ atom x ♯ b_of t\"\n using \"τ.fresh\" (*‹?a ♯ ⦃ ?x : ?b | ?c ⦄ = ((?a ♯ ?c ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?b)›*) assms (*‹atom x ♯ (t, z)›*) z (*‹t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) by metis\n hence \"atom x ♯ c'\"\n using fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) z (*‹t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z)›*) fresh_at_base(2) (*‹?a ♯ ?b = (?a ≠ atom ?b)›*) by fastforce\n moreover have \"atom x ♯ V_var z\"\n using assms (*‹atom x ♯ (t, z)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) v.fresh (*‹?a ♯ [ ?l ]⇧v = ?a ♯ ?l› ‹?a ♯ [ ?x ]⇧v = ?a ♯ ?x› ‹?a ♯ [ ?v1.0 , ?v2.0 ]⇧v = (?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)› ‹?a ♯ V_cons ?list1.0 ?list2.0 ?v = (?a ♯ ?list1.0 ∧ ?a ♯ ?list2.0 ∧ ?a ♯ ?v)› ‹(?a::atom) ♯ V_consp (?list1.0::char list) (?list2.0::char list) (?b::b) (?v::v) = (?a ♯ ?list1.0 ∧ ?a ♯ ?list2.0 ∧ ?a ♯ ?b ∧ ?a ♯ ?v)›*) by metis\n ultimately show \"?thesis\"\n (*goal: ‹atom x ♯ c_of t z›*)\n using assms (*‹atom x ♯ (t, z)›*) fresh_subst_v_if[of \"atom x\" c' z' \"V_var z\"] (*‹atom x ♯ c'[z'::=[ z ]⇧v]⇩v = (atom z' ♯ c' ∧ atom x ♯ c' ∨ atom x ♯ [ z ]⇧v ∧ (atom x ♯ c' ∨ atom x = atom z'))›*) subst_v_c_def (*‹subst_v = subst_cv_abbrev›*) \"*\" (*‹c_of t z = c'[z'::=[ z ]⇧v]⇩c⇩v›*) by metis\nqed\n\nlemma c_of_switch:\n fixes z::x\n assumes \"atom z ♯ t\" \n shows \"(c_of t z)[z::=V_var x]⇩c⇩v = c_of t x\"\nproof (-)\n (*goal: ‹(c_of t z)[z::=[ x ]⇧v]⇩c⇩v = c_of t x›*)\n obtain z' and c' where z: \"t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x,z)\"\n (*goal: ‹(⋀z' c'. t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z) ⟹ thesis) ⟹ thesis›*)\n using obtain_fresh_z_c_of (*‹(⋀z::x. atom z ♯ (?t::?'b) ∧ (?τ::τ) = ⦃ z : b_of ?τ | c_of ?τ z ⦄ ⟹ ?thesis::bool) ⟹ ?thesis›*) by metis\n hence \"(atom z ♯ c' ∨ atom z ∈ set [atom z']) ∧ atom z ♯ b_of t\"\n using \"τ.fresh\"[of \"atom z\" z' \"b_of t\" c'] (*‹atom z ♯ ⦃ z' : b_of t | c' ⦄ = ((atom z ♯ c' ∨ atom z ∈ set [atom z']) ∧ atom z ♯ b_of t)›*) assms (*‹atom z ♯ t›*) by metis\n moreover have \" atom z ∉ set [atom z']\"\n using z (*‹t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) by force\n ultimately have \"**\": \"atom z ♯ c'\"\n using fresh_Pair (*‹(?a::atom) ♯ (?x::?'a, ?y::?'b) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) z (*‹t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z)›*) fresh_at_base(2) (*‹?a ♯ ?b = (?a ≠ atom ?b)›*) by metis\n have \"(c_of t z)[z::=V_var x]⇩c⇩v = c'[z'::=V_var z]⇩c⇩v[z::=V_var x]⇩c⇩v\"\n using c_of.simps (*‹atom (?z::x) ♯ (?x::x) ⟹ c_of ⦃ ?z : (?b::b) | ?c::c ⦄ ?x = ?c[?z::=[ ?x ]⇧v]⇩c⇩v›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) z (*‹t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z)›*) by metis\n also (*calculation: ‹(c_of t z)[z::=[ x ]⇧v]⇩c⇩v = c'[z'::=[ z ]⇧v]⇩c⇩v[z::=[ x ]⇧v]⇩c⇩v›*) have \"... = c'[z'::=V_var x]⇩c⇩v\"\n using subst_v_simple_commute (*‹atom (?x::x) ♯ (?c::?'a) ⟹ ?c[(?z::x)::=[ ?x ]⇧v]⇩v[?x::=(?b::v)]⇩v = ?c[?z::=?b]⇩v›*) subst_v_c_def (*‹subst_v = subst_cv_abbrev›*) assms (*‹atom z ♯ t›*) c_of.simps (*‹atom ?z ♯ ?x ⟹ c_of ⦃ ?z : ?b | ?c ⦄ ?x = ?c[?z::=[ ?x ]⇧v]⇩c⇩v›*) z (*‹t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z)›*) \"**\" (*‹atom (z::x) ♯ (c'::c)›*) by metis\n finally (*calculation: ‹(c_of t z)[z::=[ x ]⇧v]⇩c⇩v = c'[z'::=[ x ]⇧v]⇩c⇩v›*) show \"?thesis\"\n (*goal: ‹(c_of t z)[z::=[ x ]⇧v]⇩c⇩v = c_of t x›*)\n using c_of.simps[of z' x \"b_of t\" c'] (*‹atom z' ♯ x ⟹ c_of ⦃ z' : b_of t | c' ⦄ x = c'[z'::=[ x ]⇧v]⇩c⇩v›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) z (*‹t = ⦃ z' : b_of t | c' ⦄ ∧ atom z' ♯ (x, z)›*) by metis\nqed\n\nlemma type_eq_subst_eq1:\n fixes v::v and c1::c\n assumes \"⦃ z1 : b1 | c1 ⦄ = (⦃ z2 : b2 | c2 ⦄)\" and \"atom z1 ♯ c2\" \n shows \"c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v\" and \"b1=b2\" and \" c1 = (z1 ↔ z2) ∙ c2\"\nproof (-)\n (*goals:\n 1. ‹c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v›\n 2. ‹b1 = b2›\n 3. ‹c1 = (z1 ↔ z2) ∙ c2›*)\n show \"c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v\"\n using type_eq_subst_eq (*‹⦃ ?z1.0 : ?b1.0 | ?c1.0 ⦄ = ⦃ ?z2.0 : ?b2.0 | ?c2.0 ⦄ ⟹ ?c1.0[?z1.0::=?v]⇩c⇩v = ?c2.0[?z2.0::=?v]⇩c⇩v›*) assms (*‹⦃ (z1::x) : (b1::b) | c1::c ⦄ = ⦃ (z2::x) : (b2::b) | c2::c ⦄› ‹atom z1 ♯ c2›*) by blast\n show \"b1=b2\"\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) assms (*‹⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄› ‹atom z1 ♯ c2›*) by blast\n have \"z1 = z2 ∧ c1 = c2 ∨ z1 ≠ z2 ∧ c1 = (z1 ↔ z2) ∙ c2 ∧ atom z1 ♯ c2\"\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) Abs1_eq_iff[of z1 c1 z2 c2] (*‹([{atom z1}]set. c1 = [{atom z2}]set. c2) = (z1 = z2 ∧ c1 = c2 ∨ z1 ≠ z2 ∧ c1 = (z1 ↔ z2) ∙ c2 ∧ atom z1 ♯ c2)› ‹([{atom (z1::x)}]res. (c1::c) = [{atom (z2::x)}]res. (c2::c)) = (z1 = z2 ∧ c1 = c2 ∨ z1 ≠ z2 ∧ c1 = (z1 ↔ z2) ∙ c2 ∧ atom z1 ♯ c2)› ‹([[atom z1]]lst. c1 = [[atom z2]]lst. c2) = (z1 = z2 ∧ c1 = c2 ∨ z1 ≠ z2 ∧ c1 = (z1 ↔ z2) ∙ c2 ∧ atom z1 ♯ c2)›*) assms (*‹⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄› ‹atom z1 ♯ c2›*) by blast\n thus \"c1 = (z1 ↔ z2) ∙ c2\"\n by auto\nqed\n\nlemma type_eq_subst_eq2:\n fixes v::v and c1::c\n assumes \"⦃ z1 : b1 | c1 ⦄ = (⦃ z2 : b2 | c2 ⦄)\" \n shows \"c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v\" and \"b1=b2\" and \"[[atom z1]]lst. c1 = [[atom z2]]lst. c2\"\nproof (-)\n (*goals:\n 1. ‹c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v›\n 2. ‹b1 = b2›\n 3. ‹[[atom z1]]lst. c1 = [[atom z2]]lst. c2›*)\n show \"c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v\"\n using type_eq_subst_eq (*‹⦃ ?z1.0 : ?b1.0 | ?c1.0 ⦄ = ⦃ ?z2.0 : ?b2.0 | ?c2.0 ⦄ ⟹ ?c1.0[?z1.0::=?v]⇩c⇩v = ?c2.0[?z2.0::=?v]⇩c⇩v›*) assms (*‹⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄›*) by blast\n show \"b1=b2\"\n using \"τ.eq_iff\" (*‹(⦃ (?x::x) : (?b::b) | ?c::c ⦄ = ⦃ (?xa::x) : (?ba::b) | ?ca::c ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) assms (*‹⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄›*) by blast\n show \"[[atom z1]]lst. c1 = [[atom z2]]lst. c2\"\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) assms (*‹⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄›*) by auto\nqed\n\nlemma type_eq_subst_eq3:\n fixes v::v and c1::c\n assumes \"⦃ z1 : b1 | c1 ⦄ = (⦃ z2 : b2 | c2 ⦄)\" and \"atom z1 ♯ c2\" \n shows \"c1 = c2[z2::=V_var z1]⇩c⇩v\" and \"b1=b2\"\n using type_eq_subst_eq1 (*‹⟦⦃ ?z1.0 : ?b1.0 | ?c1.0 ⦄ = ⦃ ?z2.0 : ?b2.0 | ?c2.0 ⦄; atom ?z1.0 ♯ ?c2.0⟧ ⟹ ?c1.0[?z1.0::=?v]⇩c⇩v = ?c2.0[?z2.0::=?v]⇩c⇩v› ‹⟦⦃ (?z1.0::x) : (?b1.0::b) | ?c1.0::c ⦄ = ⦃ (?z2.0::x) : (?b2.0::b) | ?c2.0::c ⦄; atom ?z1.0 ♯ ?c2.0⟧ ⟹ ?b1.0 = ?b2.0› ‹⟦⦃ ?z1.0 : ?b1.0 | ?c1.0 ⦄ = ⦃ ?z2.0 : ?b2.0 | ?c2.0 ⦄; atom ?z1.0 ♯ ?c2.0⟧ ⟹ ?c1.0 = (?z1.0 ↔ ?z2.0) ∙ ?c2.0›*) assms (*‹⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄› ‹atom z1 ♯ c2›*) subst_v_c_def (*‹subst_v = subst_cv_abbrev›*) apply -\n (*goals:\n 1. ‹⟦⋀z1 b1 c1 z2 b2 c2 v. ⟦⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2⟧ ⟹ c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v; ⋀z1 b1 c1 z2 b2 c2. ⟦⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2⟧ ⟹ b1 = b2; ⋀z1 b1 c1 z2 b2 c2. ⟦⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2⟧ ⟹ c1 = (z1 ↔ z2) ∙ c2; ⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2; subst_v = subst_cv_abbrev⟧ ⟹ c1 = c2[z2::=[ z1 ]⇧v]⇩c⇩v›\n 2. ‹⟦⋀z1 b1 c1 z2 b2 c2 v. ⟦⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2⟧ ⟹ c1[z1::=v]⇩c⇩v = c2[z2::=v]⇩c⇩v; ⋀z1 b1 c1 z2 b2 c2. ⟦⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2⟧ ⟹ b1 = b2; ⋀z1 b1 c1 z2 b2 c2. ⟦⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2⟧ ⟹ c1 = (z1 ↔ z2) ∙ c2; ⦃ z1 : b1 | c1 ⦄ = ⦃ z2 : b2 | c2 ⦄; atom z1 ♯ c2; subst_v = subst_cv_abbrev⟧ ⟹ b1 = b2›\n discuss goal 1*)\n apply (metis subst_cv_var_flip (*‹atom (?y::x) ♯ (?c::c) ⟹ (?y ↔ ?x::x) ∙ ?c = ?c[?x::=[ ?y ]⇧v]⇩c⇩v›*))\n (*discuss goal 2*)\n apply (metis subst_cv_var_flip (*‹atom ?y ♯ ?c ⟹ (?y ↔ ?x) ∙ ?c = ?c[?x::=[ ?y ]⇧v]⇩c⇩v›*))\n (*proven 2 subgoals*) .\n\nlemma type_eq_flip:\n assumes \"atom x ♯ c\"\n shows \"⦃ z : b | c ⦄ = ⦃ x : b | (x ↔ z ) ∙ c ⦄\"\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) Abs1_eq_iff (*‹([{atom ?a}]set. ?x = [{atom ?b}]set. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([{atom ?a}]res. ?x = [{atom ?b}]res. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([[atom ?a]]lst. ?x = [[atom ?b]]lst. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)›*) assms (*‹atom x ♯ c›*) by (metis (no_types, lifting) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*))\n\nlemma c_of_true:\n \"c_of ⦃ z' : B_bool | TRUE ⦄ x = C_true\"\nproof (nominal_induct \"⦃ z' : B_bool | TRUE ⦄\" avoiding: x rule:τ.strong_induct)\n (*goal: ‹⋀x1a x3a x. ⟦atom x1a ♯ x; [[atom x1a]]lst. x3a = [[atom z']]lst. (TRUE)⟧ ⟹ c_of ⦃ z' : B_bool | TRUE ⦄ x = (TRUE)›*)\n case (T_refined_type x1a x3a) (*‹atom x1a ♯ x› ‹[[atom x1a]]lst. x3a = [[atom z']]lst. (TRUE)›*)\n hence \"⦃ z' : B_bool | TRUE ⦄ = ⦃ x1a : B_bool | x3a ⦄\"\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) by metis\n then show \"?case\"\n (*goal: ‹c_of ⦃ z' : B_bool | TRUE ⦄ x = (TRUE)›*)\n using subst_cv.simps (*‹(TRUE)[?x::=?v]⇩c⇩v = (TRUE)› ‹(FALSE)[?x::=?v]⇩c⇩v = (FALSE)› ‹((?c1.0::c) AND (?c2.0::c) )[(?x::x)::=(?v::v)]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v AND ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 OR ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v OR ?c2.0[?x::=?v]⇩c⇩v )› ‹((?c1.0::c) IMP (?c2.0::c) )[(?x::x)::=(?v::v)]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v IMP ?c2.0[?x::=?v]⇩c⇩v )› ‹(?e1.0 == ?e2.0 )[?x::=?v]⇩c⇩v = (?e1.0[?x::=?v]⇩c⇩e⇩v == ?e2.0[?x::=?v]⇩c⇩e⇩v )› ‹(¬ ?c )[?x::=?v]⇩c⇩v = (¬ ?c[?x::=?v]⇩c⇩v )›*) c_of.simps (*‹atom ?z ♯ ?x ⟹ c_of ⦃ ?z : ?b | ?c ⦄ ?x = ?c[?z::=[ ?x ]⇧v]⇩c⇩v›*) T_refined_type (*‹atom x1a ♯ x› ‹[[atom x1a]]lst. x3a = [[atom z']]lst. (TRUE)›*) type_eq_subst_eq3 (*‹⟦⦃ (?z1.0::x) : (?b1.0::b) | ?c1.0::c ⦄ = ⦃ (?z2.0::x) : (?b2.0::b) | ?c2.0::c ⦄; atom ?z1.0 ♯ ?c2.0⟧ ⟹ ?c1.0 = ?c2.0[?z2.0::=[ ?z1.0 ]⇧v]⇩c⇩v› ‹⟦⦃ ?z1.0 : ?b1.0 | ?c1.0 ⦄ = ⦃ ?z2.0 : ?b2.0 | ?c2.0 ⦄; atom ?z1.0 ♯ ?c2.0⟧ ⟹ ?b1.0 = ?b2.0›*) by (metis type_eq_subst_eq (*‹⦃ ?z1.0 : ?b1.0 | ?c1.0 ⦄ = ⦃ ?z2.0 : ?b2.0 | ?c2.0 ⦄ ⟹ ?c1.0[?z1.0::=?v]⇩c⇩v = ?c2.0[?z2.0::=?v]⇩c⇩v›*))\nqed\n\nlemma type_eq_subst:\n assumes \"atom x ♯ c\"\n shows \"⦃ z : b | c ⦄ = ⦃ x : b | c[z::=[x]⇧v]⇩c⇩v ⦄\"\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) Abs1_eq_iff (*‹([{atom (?a::?'b::at)}]set. (?x::?'a::fs) = [{atom (?b::?'b::at)}]set. (?y::?'a::fs)) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([{atom ?a}]res. ?x = [{atom ?b}]res. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([[atom ?a]]lst. ?x = [[atom ?b]]lst. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)›*) assms (*‹atom (x::x) ♯ (c::c)›*) using subst_cv_var_flip (*‹atom ?y ♯ ?c ⟹ (?y ↔ ?x) ∙ ?c = ?c[?x::=[ ?y ]⇧v]⇩c⇩v›*) type_eq_flip (*‹atom ?x ♯ ?c ⟹ ⦃ ?z : ?b | ?c ⦄ = ⦃ ?x : ?b | (?x ↔ ?z) ∙ ?c ⦄›*) by auto\n\nlemma type_e_subst_fresh:\n fixes x::x and z::x\n assumes \"atom z ♯ (x,v)\" and \"atom x ♯ e\" \n shows \"⦃ z : b | CE_val (V_var z) == e ⦄[x::=v]⇩τ⇩v = ⦃ z : b | CE_val (V_var z) == e ⦄\"\n using assms (*‹atom z ♯ (x, v)› ‹atom x ♯ e›*) subst_tv.simps (*‹atom (?z::x) ♯ (?x::x, ?v::v) ⟹ ⦃ ?z : (?b::b) | ?c::c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) subst_cv.simps (*‹(TRUE)[?x::=?v]⇩c⇩v = (TRUE)› ‹(FALSE)[?x::=?v]⇩c⇩v = (FALSE)› ‹(?c1.0 AND ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v AND ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 OR ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v OR ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 IMP ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v IMP ?c2.0[?x::=?v]⇩c⇩v )› ‹(?e1.0 == ?e2.0 )[?x::=?v]⇩c⇩v = (?e1.0[?x::=?v]⇩c⇩e⇩v == ?e2.0[?x::=?v]⇩c⇩e⇩v )› ‹(¬ ?c )[?x::=?v]⇩c⇩v = (¬ ?c[?x::=?v]⇩c⇩v )›*) forget_subst_cev (*‹atom ?a ♯ ?A ⟹ ?A[?a::=?x]⇩c⇩e⇩v = ?A›*) by simp\n\nlemma type_v_subst_fresh:\n fixes x::x and z::x\n assumes \"atom z ♯ (x,v)\" and \"atom x ♯ v'\" \n shows \"⦃ z : b | CE_val (V_var z) == CE_val v' ⦄[x::=v]⇩τ⇩v = ⦃ z : b | CE_val (V_var z) == CE_val v' ⦄\"\n using assms (*‹atom z ♯ (x, v)› ‹atom x ♯ v'›*) subst_tv.simps (*‹atom ?z ♯ (?x, ?v) ⟹ ⦃ ?z : ?b | ?c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) subst_cv.simps (*‹(TRUE)[?x::=?v]⇩c⇩v = (TRUE)› ‹(FALSE)[?x::=?v]⇩c⇩v = (FALSE)› ‹((?c1.0::c) AND (?c2.0::c) )[(?x::x)::=(?v::v)]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v AND ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 OR ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v OR ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 IMP ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v IMP ?c2.0[?x::=?v]⇩c⇩v )› ‹(?e1.0 == ?e2.0 )[?x::=?v]⇩c⇩v = (?e1.0[?x::=?v]⇩c⇩e⇩v == ?e2.0[?x::=?v]⇩c⇩e⇩v )› ‹(¬ ?c )[?x::=?v]⇩c⇩v = (¬ ?c[?x::=?v]⇩c⇩v )›*) by simp\n\nlemma subst_tbase_eq:\n \"b_of τ = b_of τ[x::=v]⇩τ⇩v\"\nproof (-)\n (*goal: ‹b_of τ = b_of τ[x::=v]⇩τ⇩v›*)\n obtain z and b and c where zbc: \"τ = ⦃ z:b|c⦄ ∧ atom z ♯ (x,v)\"\n (*goal: ‹(⋀z b c. τ = ⦃ z : b | c ⦄ ∧ atom z ♯ (x, v) ⟹ thesis) ⟹ thesis›*)\n using \"τ.exhaust\" (*‹(⋀x1 x2 x3. ?y = ⦃ x1 : x2 | x3 ⦄ ⟹ ?P) ⟹ ?P›*) by (metis prod.inject (*‹((?x1.0, ?x2.0) = (?y1.0, ?y2.0)) = (?x1.0 = ?y1.0 ∧ ?x2.0 = ?y2.0)›*) subst_tv.cases (*‹(⋀z x v b c. ⟦atom z ♯ (x, v); ?x = (⦃ z : b | c ⦄, x, v)⟧ ⟹ ?P) ⟹ ?P›*))\n hence \"b_of ⦃ z:b|c⦄ = b_of ⦃ z:b|c⦄[x::=v]⇩τ⇩v\"\n using subst_tv.simps (*‹atom ?z ♯ (?x, ?v) ⟹ ⦃ ?z : ?b | ?c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) by simp\n thus \"?thesis\"\n (*goal: ‹b_of τ = b_of τ[x::=v]⇩τ⇩v›*)\n using zbc (*‹τ = ⦃ z : b | c ⦄ ∧ atom z ♯ (x, v)›*) by blast\nqed\n\nlemma subst_tv_if:\n assumes \"atom z1 ♯ (x,v)\" and \"atom z' ♯ (x,v)\" \n shows \"⦃ z1 : b | CE_val (v'[x::=v]⇩v⇩v) == CE_val (V_lit l) IMP (c'[x::=v]⇩c⇩v)[z'::=[z1]⇧v]⇩c⇩v ⦄ = \n ⦃ z1 : b | CE_val v' == CE_val (V_lit l) IMP c'[z'::=[z1]⇧v]⇩c⇩v ⦄[x::=v]⇩τ⇩v\" \n using subst_cv_commute_full[of z' v x \"V_var z1\" c'] (*‹⟦atom (z'::x) ♯ (v::v); atom (x::x) ♯ [ z1::x ]⇧v; x ≠ z'⟧ ⟹ (c'::c)[z'::=[ z1 ]⇧v]⇩c⇩v[x::=v]⇩c⇩v = c'[x::=v]⇩c⇩v[z'::=[ z1 ]⇧v]⇩c⇩v›*) subst_tv.simps (*‹atom ?z ♯ (?x, ?v) ⟹ ⦃ ?z : ?b | ?c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) subst_vv.simps(1) (*‹[ ?l ]⇧v[?x::=?v]⇩v⇩v = [ ?l ]⇧v›*) subst_ev.simps (*‹[ ?v'::v ]⇧e[(?x::x)::=(?v::v)]⇩e⇩v = [ ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?f ?v' ]⇧e[?x::=?v]⇩e⇩v = [ ?f ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[?f [ ?b ] ?v' ]⇧e[?x::=?v]⇩e⇩v = [?f [ ?b ] ?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?opp ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?opp ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[#1?v' ]⇧e[?x::=?v]⇩e⇩v = [#1?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[#2?v' ]⇧e[?x::=?v]⇩e⇩v = [#2?v'[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?u ]⇧e[?x::=?v]⇩e⇩v = [ ?u ]⇧e› ‹[| ?v' |]⇧e[?x::=?v]⇩e⇩v = [| ?v'[?x::=?v]⇩v⇩v |]⇧e› ‹[ ?v1.0 @@ ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v @@ ?v2.0[?x::=?v]⇩v⇩v ]⇧e› ‹[ ?v1.0 ?v2.0 ]⇧e[?x::=?v]⇩e⇩v = [ ?v1.0[?x::=?v]⇩v⇩v ?v2.0[?x::=?v]⇩v⇩v ]⇧e›*) subst_cv.simps (*‹(TRUE)[?x::=?v]⇩c⇩v = (TRUE)› ‹(FALSE)[?x::=?v]⇩c⇩v = (FALSE)› ‹(?c1.0 AND ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v AND ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 OR ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v OR ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 IMP ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v IMP ?c2.0[?x::=?v]⇩c⇩v )› ‹(?e1.0 == ?e2.0 )[?x::=?v]⇩c⇩v = (?e1.0[?x::=?v]⇩c⇩e⇩v == ?e2.0[?x::=?v]⇩c⇩e⇩v )› ‹(¬ ?c )[?x::=?v]⇩c⇩v = (¬ ?c[?x::=?v]⇩c⇩v )›*) assms (*‹atom z1 ♯ (x, v)› ‹atom z' ♯ (x, v)›*) by simp\n\nlemma subst_tv_tid:\n assumes \"atom za ♯ (x,v)\"\n shows \"⦃ za : B_id tid | TRUE ⦄ = ⦃ za : B_id tid | TRUE ⦄[x::=v]⇩τ⇩v\"\n using assms (*‹atom za ♯ (x, v)›*) subst_tv.simps (*‹atom ?z ♯ (?x, ?v) ⟹ ⦃ ?z : ?b | ?c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) subst_cv.simps (*‹(TRUE)[(?x::x)::=(?v::v)]⇩c⇩v = (TRUE)› ‹(FALSE)[?x::=?v]⇩c⇩v = (FALSE)› ‹(?c1.0 AND ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v AND ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 OR ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v OR ?c2.0[?x::=?v]⇩c⇩v )› ‹((?c1.0::c) IMP (?c2.0::c) )[(?x::x)::=(?v::v)]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v IMP ?c2.0[?x::=?v]⇩c⇩v )› ‹(?e1.0 == ?e2.0 )[?x::=?v]⇩c⇩v = (?e1.0[?x::=?v]⇩c⇩e⇩v == ?e2.0[?x::=?v]⇩c⇩e⇩v )› ‹(¬ ?c )[?x::=?v]⇩c⇩v = (¬ ?c[?x::=?v]⇩c⇩v )›*) by presburger\n\n\nlemma b_of_subst:\n \"b_of (τ[x::=v]⇩τ⇩v) = b_of τ\"\nproof (-)\n (*goal: ‹b_of (τ::τ)[(x::x)::=(v::v)]⇩τ⇩v = b_of τ›*)\n obtain z and b and c where \"*\": \"τ = ⦃ z : b | c ⦄ ∧ atom z ♯ (x,v)\"\n (*goal: ‹(⋀z b c. τ = ⦃ z : b | c ⦄ ∧ atom z ♯ (x, v) ⟹ thesis) ⟹ thesis›*)\n using obtain_fresh_z (*‹(⋀z b c. atom z ♯ ?t ∧ ?τ = ⦃ z : b | c ⦄ ⟹ ?thesis) ⟹ ?thesis›*) by metis\n thus \"?thesis\"\n (*goal: ‹b_of τ[x::=v]⇩τ⇩v = b_of τ›*)\n using subst_tv.simps (*‹atom ?z ♯ (?x, ?v) ⟹ ⦃ ?z : ?b | ?c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) \"*\" (*‹(τ::τ) = ⦃ (z::x) : (b::b) | c::c ⦄ ∧ atom z ♯ (x::x, v::v)›*) by auto\nqed\n\nlemma subst_tv_flip:\n assumes \"τ'[x::=v]⇩τ⇩v = τ\" and \"atom x ♯ (v,τ)\" and \"atom x' ♯ (v,τ)\"\n shows \"((x' ↔ x) ∙ τ')[x'::=v]⇩τ⇩v = τ\"\nproof (-)\n (*goal: ‹((x' ↔ x) ∙ τ')[x'::=v]⇩τ⇩v = τ›*)\n have \"(x' ↔ x) ∙ v = v ∧ (x' ↔ x) ∙ τ = τ\"\n using assms (*‹τ'[x::=v]⇩τ⇩v = τ› ‹atom x ♯ (v, τ)› ‹atom x' ♯ (v, τ)›*) flip_fresh_fresh (*‹⟦atom (?a::?'b::at_base) ♯ (?x::?'a::pt); atom (?b::?'b::at_base) ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) by auto\n thus \"?thesis\"\n (*goal: ‹((x'::x ↔ x::x) ∙ (τ'::τ))[x'::=(v::v)]⇩τ⇩v = (τ::τ)›*)\n using subst_tv.eqvt[of \"(x' ↔ x)\" τ' x v] (*‹(x' ↔ x) ∙ τ'[x::=v]⇩τ⇩v = ((x' ↔ x) ∙ τ')[(x' ↔ x) ∙ x::=(x' ↔ x) ∙ v]⇩τ⇩v›*) assms (*‹τ'[x::=v]⇩τ⇩v = τ› ‹atom (x::x) ♯ (v::v, τ::τ)› ‹atom x' ♯ (v, τ)›*) by auto\nqed\n\nlemma subst_cv_true:\n \"⦃ z : B_id tid | TRUE ⦄ = ⦃ z : B_id tid | TRUE ⦄[x::=v]⇩τ⇩v\" \nproof (-)\n (*goal: ‹⦃ (z::x) : B_id (tid::char list) | TRUE ⦄ = ⦃ z : B_id tid | TRUE ⦄[(x::x)::=(v::v)]⇩τ⇩v›*)\n obtain za :: x where \"atom za ♯ (x,v)\"\n (*goal: ‹(⋀za. atom za ♯ (x, v) ⟹ thesis) ⟹ thesis›*)\n using obtain_fresh (*‹(⋀a::?'a::at_base. atom a ♯ (?x::?'b::fs) ⟹ ?thesis::bool) ⟹ ?thesis›*) by auto\n hence \"⦃ z : B_id tid | TRUE ⦄ = ⦃ za: B_id tid | TRUE ⦄\"\n using \"τ.eq_iff\" (*‹(⦃ (?x::x) : (?b::b) | ?c::c ⦄ = ⦃ (?xa::x) : (?ba::b) | ?ca::c ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) Abs1_eq_iff (*‹([{atom ?a}]set. ?x = [{atom ?b}]set. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([{atom ?a}]res. ?x = [{atom ?b}]res. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)› ‹([[atom (?a::?'b::at)]]lst. (?x::?'a::fs) = [[atom (?b::?'b::at)]]lst. (?y::?'a::fs)) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)›*) by fastforce\n moreover have \"⦃ za : B_id tid | TRUE ⦄ = ⦃ za : B_id tid | TRUE ⦄[x::=v]⇩τ⇩v\"\n using subst_cv.simps (*‹(TRUE)[?x::=?v]⇩c⇩v = (TRUE)› ‹(FALSE)[?x::=?v]⇩c⇩v = (FALSE)› ‹(?c1.0 AND ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v AND ?c2.0[?x::=?v]⇩c⇩v )› ‹(?c1.0 OR ?c2.0 )[?x::=?v]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v OR ?c2.0[?x::=?v]⇩c⇩v )› ‹((?c1.0::c) IMP (?c2.0::c) )[(?x::x)::=(?v::v)]⇩c⇩v = (?c1.0[?x::=?v]⇩c⇩v IMP ?c2.0[?x::=?v]⇩c⇩v )› ‹((?e1.0::ce) == (?e2.0::ce) )[(?x::x)::=(?v::v)]⇩c⇩v = (?e1.0[?x::=?v]⇩c⇩e⇩v == ?e2.0[?x::=?v]⇩c⇩e⇩v )› ‹(¬ ?c )[?x::=?v]⇩c⇩v = (¬ ?c[?x::=?v]⇩c⇩v )›*) subst_tv.simps (*‹atom ?z ♯ (?x, ?v) ⟹ ⦃ ?z : ?b | ?c ⦄[?x::=?v]⇩τ⇩v = ⦃ ?z : ?b | ?c[?x::=?v]⇩c⇩v ⦄›*) by (simp add: ‹atom za ♯ (x, v)›)\n ultimately show \"?thesis\"\n (*goal: ‹⦃ z : B_id tid | TRUE ⦄ = ⦃ z : B_id tid | TRUE ⦄[x::=v]⇩τ⇩v›*)\n by argo\nqed\n\nlemma t_eq_supp:\n assumes \"(⦃ z : b | c ⦄) = (⦃ z1 : b1 | c1 ⦄)\"\n shows \"supp c - { atom z } = supp c1 - { atom z1 }\"\nproof (-)\n (*goal: ‹supp c - {atom z} = supp c1 - {atom z1}›*)\n have \"supp c - { atom z } ∪ supp b = supp c1 - { atom z1 } ∪ supp b1\"\n using \"τ.supp\" (*‹supp ⦃ (?x::x) : (?b::b) | ?c::c ⦄ = supp ?c - set [atom ?x] ∪ supp ?b›*) assms (*‹⦃ z : b | c ⦄ = ⦃ z1 : b1 | c1 ⦄›*) by (metis list.set( (*‹set [] = {}›*) 1) list.simps( (*‹set (?x21.0 # ?x22.0) = insert ?x21.0 (set ?x22.0)›*) 15) sup_bot.right_neutral (*‹sup ?a bot = ?a›*) supp_b_empty (*‹atom ?x ∉ supp ?b›*))\n moreover have \"supp b = supp b1\"\n using assms (*‹⦃ z : b | c ⦄ = ⦃ z1 : b1 | c1 ⦄›*) \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) by simp\n moreover have \"atom z1 ∉ supp b1 ∧ atom z ∉ supp b\"\n using supp_b_empty (*‹atom ?x ∉ supp ?b›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹supp (c::c) - {atom (z::x)} = supp (c1::c) - {atom (z1::x)}›*)\n by (metis τ.eq_iff (*‹(⦃ (?x::x) : (?b::b) | ?c::c ⦄ = ⦃ (?xa::x) : (?ba::b) | ?ca::c ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) τ.supp (*‹supp ⦃ (?x::x) : (?b::b) | ?c::c ⦄ = supp ?c - set [atom ?x] ∪ supp ?b›*) assms (*‹⦃ (z::x) : (b::b) | c::c ⦄ = ⦃ (z1::x) : (b1::b) | c1::c ⦄›*) b.supp( (*‹supp B_int = {}›*) 1) list.set( (*‹set [] = {}›*) 1) list.set( (*‹set ((?x21.0::?'a) # (?x22.0::?'a list)) = insert ?x21.0 (set ?x22.0)›*) 2) sup_bot.right_neutral (*‹sup (?a::?'a) bot = ?a›*))\nqed\n\nlemma fresh_t_eq: \n fixes x::x\n assumes \"(⦃ z : b | c ⦄) = (⦃ zz : b | cc ⦄)\" and \"atom x ♯ c\" and \"x ≠ zz\"\n shows \"atom x ♯ cc\"\nproof (-)\n (*goal: ‹atom (x::x) ♯ (cc::c)›*)\n have \"supp c - { atom z } ∪ supp b = supp cc - { atom zz } ∪ supp b\"\n using \"τ.supp\" (*‹supp ⦃ ?x : ?b | ?c ⦄ = supp ?c - set [atom ?x] ∪ supp ?b›*) assms (*‹⦃ (z::x) : (b::b) | c::c ⦄ = ⦃ (zz::x) : b | cc::c ⦄› ‹atom x ♯ c› ‹x ≠ zz›*) by (metis list.set( (*‹set [] = {}›*) 1) list.simps( (*‹set (?x21.0 # ?x22.0) = insert ?x21.0 (set ?x22.0)›*) 15) sup_bot.right_neutral (*‹sup ?a bot = ?a›*) supp_b_empty (*‹atom ?x ∉ supp ?b›*))\n moreover have \"atom x ∉ supp c\"\n using assms (*‹⦃ (z::x) : (b::b) | c::c ⦄ = ⦃ (zz::x) : b | cc::c ⦄› ‹atom x ♯ c› ‹x ≠ zz›*) fresh_def (*‹(?a::atom) ♯ (?x::?'a::pt) ≡ ?a ∉ supp ?x›*) by blast\n ultimately have \"atom x ∉ supp cc - { atom zz } ∪ supp b\"\n by force\n hence \"atom x ∉ supp cc\"\n using assms (*‹⦃ (z::x) : (b::b) | c::c ⦄ = ⦃ (zz::x) : b | cc::c ⦄› ‹atom x ♯ c› ‹x ≠ zz›*) by simp\n thus \"?thesis\"\n (*goal: ‹atom x ♯ cc›*)\n using fresh_def (*‹(?a::atom) ♯ (?x::?'a) ≡ ?a ∉ supp ?x›*) by auto\nqed\n\nsection ‹Mutable Variable Context›\n\nnominal_function subst_dv :: \"Δ ⇒ x ⇒ v ⇒ Δ\" where\n \"subst_dv DNil x v = DNil\"\n| \"subst_dv ((u,t) #⇩Δ Δ) x v = ((u,t[x::=v]⇩τ⇩v) #⇩Δ (subst_dv Δ x v ))\"\n apply (simp add: eqvt_def subst_dv_graph_aux_def,auto )\n using delete_aux.elims by (metis Δ.exhaust surj_pair)\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_dv_abbrev :: \"Δ ⇒ x ⇒ v ⇒ Δ\" (\"_[_::=_]⇩Δ⇩v\" [1000,50,50] 1000)\n where \n \"Δ[x::=v]⇩Δ⇩v ≡ subst_dv Δ x v \" \n\nnominal_function dmap :: \"(u*τ ⇒ u*τ) ⇒ Δ ⇒ Δ\" where\n \"dmap f DNil = DNil\"\n| \"dmap f ((u,t)#⇩ΔΔ) = (f (u,t) #⇩Δ (dmap f Δ ))\"\n apply (simp add: eqvt_def dmap_graph_aux_def,auto )\n using delete_aux.elims by (metis Δ.exhaust surj_pair)\nnominal_termination (eqvt) by lexicographic_order\n\nlemma subst_dv_iff:\n \"Δ[x::=v]⇩Δ⇩v = dmap (λ(u,t). (u, t[x::=v]⇩τ⇩v)) Δ\"\n apply (induct Δ)\n (*goals:\n 1. ‹[]⇩Δ[(x::x)::=(v::v)]⇩Δ⇩v = dmap (λa::u × τ. case a of (u::u, t::τ) ⇒ (u, t[x::=v]⇩τ⇩v)) []⇩Δ›\n 2. ‹⋀(x1::u × τ) Δ::Δ. Δ[(x::x)::=(v::v)]⇩Δ⇩v = dmap (λa::u × τ. case a of (u::u, t::τ) ⇒ (u, t[x::=v]⇩τ⇩v)) Δ ⟹ (x1 #⇩Δ Δ)[x::=v]⇩Δ⇩v = dmap (λa::u × τ. case a of (u::u, t::τ) ⇒ (u, t[x::=v]⇩τ⇩v)) (x1 #⇩Δ Δ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma size_subst_dv [simp]: \"size ( subst_dv G i x) ≤ size G\"\n apply (induct G)\n (*goals:\n 1. ‹size []⇩Δ[i::=x]⇩Δ⇩v ≤ size []⇩Δ›\n 2. ‹⋀x1 G. size G[i::=x]⇩Δ⇩v ≤ size G ⟹ size (x1 #⇩Δ G)[i::=x]⇩Δ⇩v ≤ size (x1 #⇩Δ G)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) . \n\nlemma forget_subst_dv [simp]: \"atom a ♯ G ⟹ subst_dv G a x = G\"\n apply (induct G, auto)\n (*goal: ‹atom a ♯ G ⟹ G[a::=x]⇩Δ⇩v = G›*)\n using fresh_DCons (*‹?a ♯ ?x #⇩Δ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) fresh_PairD(1) (*‹?a ♯ (?x, ?y) ⟹ ?a ♯ ?x›*) not_self_fresh (*‹atom ?x ♯ ?x = False›*)\n (*goals:\n 1. ‹⋀aa b G. ⟦atom a ♯ G ⟹ G[a::=x]⇩Δ⇩v = G; atom a ♯ (aa, b) #⇩Δ G⟧ ⟹ b[a::=x]⇩τ⇩v = b›\n 2. ‹⋀aa b G. ⟦atom a ♯ G ⟹ G[a::=x]⇩Δ⇩v = G; atom a ♯ (aa, b) #⇩Δ G⟧ ⟹ G[a::=x]⇩Δ⇩v = G›\n discuss goal 1*)\n apply fastforce\n (*discuss goal 2*)\n apply (simp add: fresh_DCons (*‹?a ♯ ?x #⇩Δ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*))\n (*proven 2 subgoals*) .\n\nlemma subst_dv_member:\n assumes \"(u,τ) ∈ setD Δ\"\n shows \"(u, τ[x::=v]⇩τ⇩v) ∈ setD (Δ[x::=v]⇩Δ⇩v)\"\n using assms (*‹(u, τ) ∈ setD Δ›*) apply (induct Δ rule: Δ_induct (*‹⟦?P []⇩Δ; ⋀u t Δ'. ?P Δ' ⟹ ?P ((u, t) #⇩Δ Δ')⟧ ⟹ ?P ?Δ›*))\n (*goals:\n 1. ‹(u, τ) ∈ setD []⇩Δ ⟹ (u, τ[x::=v]⇩τ⇩v) ∈ setD []⇩Δ[x::=v]⇩Δ⇩v›\n 2. ‹⋀ua t Δ'. ⟦(u, τ) ∈ setD Δ' ⟹ (u, τ[x::=v]⇩τ⇩v) ∈ setD Δ'[x::=v]⇩Δ⇩v; (u, τ) ∈ setD ((ua, t) #⇩Δ Δ')⟧ ⟹ (u, τ[x::=v]⇩τ⇩v) ∈ setD ((ua, t) #⇩Δ Δ')[x::=v]⇩Δ⇩v›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma fresh_subst_dv:\n fixes x::x\n assumes \"atom xa ♯ Δ\" and \"atom xa ♯ v\"\n shows \"atom xa ♯Δ[x::=v]⇩Δ⇩v\" \n using assms (*‹atom xa ♯ Δ› ‹atom xa ♯ v›*) proof (induct Δ rule:Δ_induct (*‹⟦(?P::Δ ⇒ bool) []⇩Δ; ⋀(u::u) (t::τ) Δ'::Δ. ?P Δ' ⟹ ?P ((u, t) #⇩Δ Δ')⟧ ⟹ ?P (?Δ::Δ)›*))\n (*goals:\n 1. ‹⟦atom (xa::'a) ♯ []⇩Δ; atom xa ♯ (v::v)⟧ ⟹ atom xa ♯ []⇩Δ[(x::x)::=v]⇩Δ⇩v›\n 2. ‹⋀(u::u) (t::τ) Δ'::Δ. ⟦⟦atom (xa::'a) ♯ Δ'; atom xa ♯ (v::v)⟧ ⟹ atom xa ♯ Δ'[(x::x)::=v]⇩Δ⇩v; atom xa ♯ (u, t) #⇩Δ Δ'; atom xa ♯ v⟧ ⟹ atom xa ♯ ((u, t) #⇩Δ Δ')[x::=v]⇩Δ⇩v›*)\n case DNil (*‹atom xa ♯ []⇩Δ› ‹atom (xa::'a::at_base) ♯ (v::v)›*)\n then show \"?case\"\n (*goal: ‹atom xa ♯ []⇩Δ[x::=v]⇩Δ⇩v›*)\n by auto\n next\n (*goal: ‹⋀u t Δ'. ⟦⟦atom xa ♯ Δ'; atom xa ♯ v⟧ ⟹ atom xa ♯ Δ'[x::=v]⇩Δ⇩v; atom xa ♯ (u, t) #⇩Δ Δ'; atom xa ♯ v⟧ ⟹ atom xa ♯ ((u, t) #⇩Δ Δ')[x::=v]⇩Δ⇩v›*)\n case (DCons u t Δ) (*‹⟦atom xa ♯ Δ; atom xa ♯ v⟧ ⟹ atom xa ♯ Δ[x::=v]⇩Δ⇩v› ‹atom xa ♯ (u, t) #⇩Δ Δ› ‹atom xa ♯ v›*)\n then show \"?case\"\n (*goal: ‹atom xa ♯ ((u, t) #⇩Δ Δ)[x::=v]⇩Δ⇩v›*)\n using subst_dv.simps (*‹[]⇩Δ[?x::=?v]⇩Δ⇩v = []⇩Δ› ‹((?u, ?t) #⇩Δ ?Δ)[?x::=?v]⇩Δ⇩v = (?u, ?t[?x::=?v]⇩τ⇩v) #⇩Δ ?Δ[?x::=?v]⇩Δ⇩v›*) \"subst_v_τ_def\" (*‹subst_v = subst_tv_abbrev›*) fresh_DCons (*‹?a ♯ ?x #⇩Δ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) by simp\n qed\n\nlemma fresh_subst_dv_if:\n fixes j::atom and i::x and x::v and t::Δ\n assumes \"j ♯ t ∧ j ♯ x\" \n shows \"(j ♯ subst_dv t i x)\"\n using assms (*‹j ♯ t ∧ j ♯ x›*) proof (induct t rule: Δ_induct (*‹⟦?P []⇩Δ; ⋀u t Δ'. ?P Δ' ⟹ ?P ((u, t) #⇩Δ Δ')⟧ ⟹ ?P ?Δ›*))\n (*goals:\n 1. ‹(j::atom) ♯ []⇩Δ ∧ j ♯ (x::v) ⟹ j ♯ []⇩Δ[(i::x)::=x]⇩Δ⇩v›\n 2. ‹⋀(u::u) (t::τ) Δ'::Δ. ⟦(j::atom) ♯ Δ' ∧ j ♯ (x::v) ⟹ j ♯ Δ'[(i::x)::=x]⇩Δ⇩v; j ♯ (u, t) #⇩Δ Δ' ∧ j ♯ x⟧ ⟹ j ♯ ((u, t) #⇩Δ Δ')[i::=x]⇩Δ⇩v›*)\n case DNil (*‹j ♯ []⇩Δ ∧ j ♯ x›*)\n then show \"?case\"\n (*goal: ‹(j::atom) ♯ []⇩Δ[(i::x)::=(x::v)]⇩Δ⇩v›*)\n using subst_gv.simps (*‹GNil[?x::=?v]⇩Γ⇩v = GNil› ‹((?y, ?b, ?c) #⇩Γ ?Γ)[?x::=?v]⇩Γ⇩v = (if ?x = ?y then ?Γ else (?y, ?b, ?c[?x::=?v]⇩c⇩v) #⇩Γ ?Γ[?x::=?v]⇩Γ⇩v)›*) fresh_GNil (*‹?a ♯ GNil›*) by auto\n next\n (*goal: ‹⋀u t Δ'. ⟦j ♯ Δ' ∧ j ♯ x ⟹ j ♯ Δ'[i::=x]⇩Δ⇩v; j ♯ (u, t) #⇩Δ Δ' ∧ j ♯ x⟧ ⟹ j ♯ ((u, t) #⇩Δ Δ')[i::=x]⇩Δ⇩v›*)\n case (DCons u' t' D') (*‹j ♯ D' ∧ j ♯ x ⟹ j ♯ D'[i::=x]⇩Δ⇩v› ‹j ♯ (u', t') #⇩Δ D' ∧ j ♯ x›*)\n then show \"?case\"\n (*goal: ‹j ♯ ((u', t') #⇩Δ D')[i::=x]⇩Δ⇩v›*)\n unfolding subst_dv.simps\n (*goal: ‹j ♯ (u', t'[i::=x]⇩τ⇩v) #⇩Δ D'[i::=x]⇩Δ⇩v›*)\n using fresh_DCons (*‹?a ♯ ?x #⇩Δ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) fresh_subst_tv_if (*‹?j ♯ ?A[?i::=?x]⇩τ⇩v = (atom ?i ♯ ?A ∧ ?j ♯ ?A ∨ ?j ♯ ?x ∧ (?j ♯ ?A ∨ ?j = atom ?i))›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) by metis\n qed\n\nsection ‹Statements›\n\ntext ‹ Using ideas from proofs at top of AFP/Launchbury/Substitution.thy.\n Subproofs borrowed from there; hence the apply style proofs. ›\n\nnominal_function (default \"case_sum (λx. Inl undefined) (case_sum (λx. Inl undefined) (λx. Inr undefined))\")\n subst_sv :: \"s ⇒ x ⇒ v ⇒ s\"\n and subst_branchv :: \"branch_s ⇒ x ⇒ v ⇒ branch_s\" \n and subst_branchlv :: \"branch_list ⇒ x ⇒ v ⇒ branch_list\" where\n \"subst_sv ( (AS_val v') ) x v = (AS_val (subst_vv v' x v ))\"\n| \"atom y ♯ (x,v) ⟹ subst_sv (AS_let y e s) x v = (AS_let y (e[x::=v]⇩e⇩v) (subst_sv s x v ))\" \n| \"atom y ♯ (x,v) ⟹ subst_sv (AS_let2 y t s1 s2) x v = (AS_let2 y (t[x::=v]⇩τ⇩v) (subst_sv s1 x v ) (subst_sv s2 x v ))\" \n| \" subst_sv (AS_match v' cs) x v = AS_match (v'[x::=v]⇩v⇩v) (subst_branchlv cs x v )\"\n| \"subst_sv (AS_assign y v') x v = AS_assign y (subst_vv v' x v )\"\n| \"subst_sv ( (AS_if v' s1 s2) ) x v = (AS_if (subst_vv v' x v ) (subst_sv s1 x v ) (subst_sv s2 x v ) )\" \n| \"atom u ♯ (x,v) ⟹ subst_sv (AS_var u τ v' s) x v = AS_var u (subst_tv τ x v ) (subst_vv v' x v ) (subst_sv s x v ) \"\n| \"subst_sv (AS_while s1 s2) x v = AS_while (subst_sv s1 x v ) (subst_sv s2 x v )\"\n| \"subst_sv (AS_seq s1 s2) x v = AS_seq (subst_sv s1 x v ) (subst_sv s2 x v )\" \n| \"subst_sv (AS_assert c s) x v = AS_assert (subst_cv c x v) (subst_sv s x v)\"\n| \"atom x1 ♯ (x,v) ⟹ subst_branchv (AS_branch dc x1 s1 ) x v = AS_branch dc x1 (subst_sv s1 x v )\" \n\n| \"subst_branchlv (AS_final cs) x v = AS_final (subst_branchv cs x v )\"\n| \"subst_branchlv (AS_cons cs css) x v = AS_cons (subst_branchv cs x v ) (subst_branchlv css x v )\"\n apply (auto,simp add: eqvt_def subst_sv_subst_branchv_subst_branchlv_graph_aux_def )\nproof(goal_cases)\n\n have eqvt_at_proj: \"⋀ s xa va . eqvt_at subst_sv_subst_branchv_subst_branchlv_sumC (Inl (s, xa, va)) ⟹ \n eqvt_at (λa. projl (subst_sv_subst_branchv_subst_branchlv_sumC (Inl a))) (s, xa, va)\"\n apply(simp add: eqvt_at_def)\n apply(rule)\n apply(subst Projl_permute)\n apply(thin_tac _)+\n apply (simp add: subst_sv_subst_branchv_subst_branchlv_sumC_def)\n apply (simp add: THE_default_def)\n apply (case_tac \"Ex1 (subst_sv_subst_branchv_subst_branchlv_graph (Inl (s,xa,va)))\")\n apply simp\n apply(auto)[1]\n apply (erule_tac x=\"x\" in allE)\n apply simp\n apply(cases rule: subst_sv_subst_branchv_subst_branchlv_graph.cases) \n apply(assumption)\n apply(rule_tac x=\"Sum_Type.projl x\" in exI,clarify,rule the1_equality,blast,simp (no_asm) only: sum.sel)+\n apply blast +\n\n apply(simp)+ \n done\n\n {\n case (1 P x') \n then show ?case proof(cases x')\n case (Inl a) thus P \n proof(cases a)\n case (fields aa bb cc)\n thus P using Inl 1 s_branch_s_branch_list.strong_exhaust fresh_star_insert by metis\n qed\n next\n case (Inr b) thus P\n proof(cases b)\n case (Inl a) thus P proof(cases a)\n case (fields aa bb cc)\n then show ?thesis using Inr Inl 1 s_branch_s_branch_list.strong_exhaust fresh_star_insert by metis\n qed\n next\n case Inr2: (Inr b) thus P proof(cases b)\n case (fields aa bb cc)\n then show ?thesis using Inr Inr2 1 s_branch_s_branch_list.strong_exhaust fresh_star_insert by metis\n qed\n qed\n qed\n next\n case (2 y s ya xa va sa c)\n thus ?case using eqvt_triple eqvt_at_proj by blast\n next\n case (3 y s2 ya xa va s1a s2a c)\n thus ?case using eqvt_triple eqvt_at_proj by blast\n next\n case (4 u xa va s ua sa c)\n moreover have \"atom u ♯ (xa, va) ∧ atom ua ♯ (xa, va)\" \n using fresh_Pair u_fresh_xv by auto\n ultimately show ?case using eqvt_triple[of u xa va ua s sa] subst_sv_def eqvt_at_proj by metis\n next\n case (5 x1 s1 x1a xa va s1a c)\n thus ?case using eqvt_triple eqvt_at_proj by blast\n }\nqed\nnominal_termination (eqvt) by lexicographic_order\n\nabbreviation \n subst_sv_abbrev :: \"s ⇒ x ⇒ v ⇒ s\" (\"_[_::=_]⇩s⇩v\" [1000,50,50] 1000)\n where \n \"s[x::=v]⇩s⇩v ≡ subst_sv s x v\" \n\nabbreviation \n subst_branchv_abbrev :: \"branch_s ⇒ x ⇒ v ⇒ branch_s\" (\"_[_::=_]⇩s⇩v\" [1000,50,50] 1000)\n where \n \"s[x::=v]⇩s⇩v ≡ subst_branchv s x v\" \n\nlemma size_subst_sv [simp]: \"size (subst_sv A i x ) = size A\" and \"size (subst_branchv B i x ) = size B\" and \"size (subst_branchlv C i x ) = size C\"\n by (nominal_induct A and B and C avoiding: i x rule: s_branch_s_branch_list.strong_induct, auto)\n\nlemma forget_subst_sv [simp]: shows \"atom a ♯ A ⟹ subst_sv A a x = A\" and \"atom a ♯ B ⟹ subst_branchv B a x = B\" and \"atom a ♯ C ⟹ subst_branchlv C a x = C\"\n by (nominal_induct A and B and C avoiding: a x rule: s_branch_s_branch_list.strong_induct, auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))\n\nlemma subst_sv_id [simp]: \"subst_sv A a (V_var a) = A\" and \"subst_branchv B a (V_var a) = B\" and \"subst_branchlv C a (V_var a) = C\"\nproof (nominal_induct A and B and C avoiding: a rule: s_branch_s_branch_list.strong_induct)\n (*goals:\n 1. ‹⋀x a. [x]⇧s[a::=[ a ]⇧v]⇩s⇩v = [x]⇧s›\n 2. ‹⋀x1 x2 x3 a. ⟦atom x1 ♯ a; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3⟧ ⟹ LET x1 = x2 IN x3[a::=[ a ]⇧v]⇩s⇩v = LET x1 = x2 IN x3›\n 3. ‹⋀x1 x2 x3 x4 a. ⟦atom x1 ♯ a; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3; ⋀b. x4[b::=[ b ]⇧v]⇩s⇩v = x4⟧ ⟹ LET x1 : x2 = x3 IN x4[a::=[ a ]⇧v]⇩s⇩v = LET x1 : x2 = x3 IN x4›\n 4. ‹⋀x1 x2 x3 a. ⟦⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3⟧ ⟹ (IF x1 THEN x2 ELSE x3)[a::=[ a ]⇧v]⇩s⇩v = IF x1 THEN x2 ELSE x3›\n 5. ‹⋀x1 x2 x3 x4 a. ⟦atom x1 ♯ a; ⋀b. x4[b::=[ b ]⇧v]⇩s⇩v = x4⟧ ⟹ VAR x1 : x2 = x3 IN x4[a::=[ a ]⇧v]⇩s⇩v = VAR x1 : x2 = x3 IN x4›\n 6. ‹⋀x1 x2 a. x1 ::= x2[a::=[ a ]⇧v]⇩s⇩v = x1 ::= x2›\n 7. ‹⋀x1 x2 a. (⋀b. subst_branchlv x2 b [ b ]⇧v = x2) ⟹ MATCH x1 WITH { x2 }[a::=[ a ]⇧v]⇩s⇩v = MATCH x1 WITH { x2 }›\n 8. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2⟧ ⟹ (WHILE x1 DO { x2 } )[a::=[ a ]⇧v]⇩s⇩v = WHILE x1 DO { x2 } ›\n 9. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2⟧ ⟹ ( x1 ;; x2 )[a::=[ a ]⇧v]⇩s⇩v = x1 ;; x2 ›\n 10. ‹⋀x1 x2 a. (⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2) ⟹ ASSERT x1 IN x2 [a::=[ a ]⇧v]⇩s⇩v = ASSERT x1 IN x2 ›\n 11. ‹⋀x1 x2 x3 a. ⟦atom x2 ♯ a; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3⟧ ⟹ x1 x2 ⇒ x3 [a::=[ a ]⇧v]⇩s⇩v = x1 x2 ⇒ x3 ›\n 12. ‹⋀x a. (⋀b. x[b::=[ b ]⇧v]⇩s⇩v = x) ⟹ subst_branchlv { x } a [ a ]⇧v = { x }›\n 13. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. subst_branchlv x2 b [ b ]⇧v = x2⟧ ⟹ subst_branchlv x1 | x2 a [ a ]⇧v = x1 | x2 ›*)\n case (AS_let x option e s) (*‹atom x ♯ s› ‹e[?b::=[ ?b ]⇧v]⇩s⇩v = e›*)\n then show \"?case\"\n (*goal: ‹LET x = option IN e[s::=[ s ]⇧v]⇩s⇩v = LET x = option IN e›*)\n by (metis (no_types, lifting) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) not_None_eq (*‹(?x ≠ None) = (∃y. ?x = Some y)›*) subst_ev_id (*‹?A[?a::=[ ?a ]⇧v]⇩e⇩v = ?A›*) subst_sv.simps( (*‹atom ?y ♯ (?x, ?v) ⟹ LET ?y = ?e IN ?s[?x::=?v]⇩s⇩v = LET ?y = ?e[?x::=?v]⇩e⇩v IN ?s[?x::=?v]⇩s⇩v›*) 2) subst_sv.simps( (*‹atom ?y ♯ (?x, ?v) ⟹ LET ?y : ?t = ?s1.0 IN ?s2.0[?x::=?v]⇩s⇩v = LET ?y : ?t[?x::=?v]⇩τ⇩v = ?s1.0[?x::=?v]⇩s⇩v IN ?s2.0[?x::=?v]⇩s⇩v›*) 3) subst_tv_id (*‹?A[?a::=[ ?a ]⇧v]⇩τ⇩v = ?A›*) v.fresh( (*‹?a ♯ [ ?x ]⇧v = ?a ♯ ?x›*) 2))\nnext\n (*goals:\n 1. ‹⋀x a. [x]⇧s[a::=[ a ]⇧v]⇩s⇩v = [x]⇧s›\n 2. ‹⋀x1 x2 x3 x4 a. ⟦atom x1 ♯ a; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3; ⋀b. x4[b::=[ b ]⇧v]⇩s⇩v = x4⟧ ⟹ LET x1 : x2 = x3 IN x4[a::=[ a ]⇧v]⇩s⇩v = LET x1 : x2 = x3 IN x4›\n 3. ‹⋀x1 x2 x3 a. ⟦⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3⟧ ⟹ (IF x1 THEN x2 ELSE x3)[a::=[ a ]⇧v]⇩s⇩v = IF x1 THEN x2 ELSE x3›\n 4. ‹⋀x1 x2 x3 x4 a. ⟦atom x1 ♯ a; ⋀b. x4[b::=[ b ]⇧v]⇩s⇩v = x4⟧ ⟹ VAR x1 : x2 = x3 IN x4[a::=[ a ]⇧v]⇩s⇩v = VAR x1 : x2 = x3 IN x4›\n 5. ‹⋀x1 x2 a. x1 ::= x2[a::=[ a ]⇧v]⇩s⇩v = x1 ::= x2›\n 6. ‹⋀x1 x2 a. (⋀b. subst_branchlv x2 b [ b ]⇧v = x2) ⟹ MATCH x1 WITH { x2 }[a::=[ a ]⇧v]⇩s⇩v = MATCH x1 WITH { x2 }›\n 7. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2⟧ ⟹ (WHILE x1 DO { x2 } )[a::=[ a ]⇧v]⇩s⇩v = WHILE x1 DO { x2 } ›\n 8. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2⟧ ⟹ ( x1 ;; x2 )[a::=[ a ]⇧v]⇩s⇩v = x1 ;; x2 ›\n 9. ‹⋀x1 x2 a. (⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2) ⟹ ASSERT x1 IN x2 [a::=[ a ]⇧v]⇩s⇩v = ASSERT x1 IN x2 ›\n 10. ‹⋀x1 x2 x3 a. ⟦atom x2 ♯ a; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3⟧ ⟹ x1 x2 ⇒ x3 [a::=[ a ]⇧v]⇩s⇩v = x1 x2 ⇒ x3 ›\n 11. ‹⋀x a. (⋀b. x[b::=[ b ]⇧v]⇩s⇩v = x) ⟹ subst_branchlv { x } a [ a ]⇧v = { x }›\n 12. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. subst_branchlv x2 b [ b ]⇧v = x2⟧ ⟹ subst_branchlv x1 | x2 a [ a ]⇧v = x1 | x2 ›*)\n case (AS_match v branch_s) (*‹subst_branchlv branch_s ?b [ ?b ]⇧v = branch_s›*)\n then show \"?case\"\n (*goal: ‹MATCH v WITH { branch_s }[a::=[ a ]⇧v]⇩s⇩v = MATCH v WITH { branch_s }›*)\n using fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) not_None_eq (*‹(?x ≠ None) = (∃y. ?x = Some y)›*) subst_ev_id (*‹?A[?a::=[ ?a ]⇧v]⇩e⇩v = ?A›*) subst_sv.simps (*‹[?v']⇧s[?x::=?v]⇩s⇩v = [?v'[?x::=?v]⇩v⇩v]⇧s› ‹atom ?y ♯ (?x, ?v) ⟹ LET ?y = ?e IN ?s[?x::=?v]⇩s⇩v = LET ?y = ?e[?x::=?v]⇩e⇩v IN ?s[?x::=?v]⇩s⇩v› ‹atom ?y ♯ (?x, ?v) ⟹ LET ?y : ?t = ?s1.0 IN ?s2.0[?x::=?v]⇩s⇩v = LET ?y : ?t[?x::=?v]⇩τ⇩v = ?s1.0[?x::=?v]⇩s⇩v IN ?s2.0[?x::=?v]⇩s⇩v› ‹MATCH ?v'::v WITH { ?cs::branch_list }[(?x::x)::=(?v::v)]⇩s⇩v = MATCH ?v'[?x::=?v]⇩v⇩v WITH { subst_branchlv ?cs ?x ?v }› ‹?y ::= ?v'[?x::=?v]⇩s⇩v = ?y ::= ?v'[?x::=?v]⇩v⇩v› ‹(IF ?v' THEN ?s1.0 ELSE ?s2.0)[?x::=?v]⇩s⇩v = IF ?v'[?x::=?v]⇩v⇩v THEN ?s1.0[?x::=?v]⇩s⇩v ELSE ?s2.0[?x::=?v]⇩s⇩v› ‹atom ?u ♯ (?x, ?v) ⟹ VAR ?u : ?τ = ?v' IN ?s[?x::=?v]⇩s⇩v = VAR ?u : ?τ[?x::=?v]⇩τ⇩v = ?v'[?x::=?v]⇩v⇩v IN ?s[?x::=?v]⇩s⇩v› ‹(WHILE ?s1.0 DO { ?s2.0 } )[?x::=?v]⇩s⇩v = WHILE ?s1.0[?x::=?v]⇩s⇩v DO { ?s2.0[?x::=?v]⇩s⇩v } › ‹( ?s1.0 ;; ?s2.0 )[?x::=?v]⇩s⇩v = ?s1.0[?x::=?v]⇩s⇩v ;; ?s2.0[?x::=?v]⇩s⇩v › ‹ASSERT ?c IN ?s [?x::=?v]⇩s⇩v = ASSERT ?c[?x::=?v]⇩c⇩v IN ?s[?x::=?v]⇩s⇩v ›*) subst_sv.simps (*‹[?v'::v]⇧s[(?x::x)::=(?v::v)]⇩s⇩v = [?v'[?x::=?v]⇩v⇩v]⇧s› ‹atom ?y ♯ (?x, ?v) ⟹ LET ?y = ?e IN ?s[?x::=?v]⇩s⇩v = LET ?y = ?e[?x::=?v]⇩e⇩v IN ?s[?x::=?v]⇩s⇩v› ‹atom ?y ♯ (?x, ?v) ⟹ LET ?y : ?t = ?s1.0 IN ?s2.0[?x::=?v]⇩s⇩v = LET ?y : ?t[?x::=?v]⇩τ⇩v = ?s1.0[?x::=?v]⇩s⇩v IN ?s2.0[?x::=?v]⇩s⇩v› ‹MATCH ?v'::v WITH { ?cs::branch_list }[(?x::x)::=(?v::v)]⇩s⇩v = MATCH ?v'[?x::=?v]⇩v⇩v WITH { subst_branchlv ?cs ?x ?v }› ‹?y ::= ?v'[?x::=?v]⇩s⇩v = ?y ::= ?v'[?x::=?v]⇩v⇩v› ‹(IF ?v' THEN ?s1.0 ELSE ?s2.0)[?x::=?v]⇩s⇩v = IF ?v'[?x::=?v]⇩v⇩v THEN ?s1.0[?x::=?v]⇩s⇩v ELSE ?s2.0[?x::=?v]⇩s⇩v› ‹atom ?u ♯ (?x, ?v) ⟹ VAR ?u : ?τ = ?v' IN ?s[?x::=?v]⇩s⇩v = VAR ?u : ?τ[?x::=?v]⇩τ⇩v = ?v'[?x::=?v]⇩v⇩v IN ?s[?x::=?v]⇩s⇩v› ‹(WHILE ?s1.0 DO { ?s2.0 } )[?x::=?v]⇩s⇩v = WHILE ?s1.0[?x::=?v]⇩s⇩v DO { ?s2.0[?x::=?v]⇩s⇩v } › ‹( ?s1.0 ;; ?s2.0 )[?x::=?v]⇩s⇩v = ?s1.0[?x::=?v]⇩s⇩v ;; ?s2.0[?x::=?v]⇩s⇩v › ‹ASSERT ?c IN ?s [?x::=?v]⇩s⇩v = ASSERT ?c[?x::=?v]⇩c⇩v IN ?s[?x::=?v]⇩s⇩v ›*) subst_tv_id (*‹?A[?a::=[ ?a ]⇧v]⇩τ⇩v = ?A›*) v.fresh (*‹?a ♯ [ ?l ]⇧v = ?a ♯ ?l› ‹?a ♯ [ ?x ]⇧v = ?a ♯ ?x› ‹?a ♯ [ ?v1.0 , ?v2.0 ]⇧v = (?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)› ‹?a ♯ V_cons ?list1.0 ?list2.0 ?v = (?a ♯ ?list1.0 ∧ ?a ♯ ?list2.0 ∧ ?a ♯ ?v)› ‹(?a::atom) ♯ V_consp (?list1.0::char list) (?list2.0::char list) (?b::b) (?v::v) = (?a ♯ ?list1.0 ∧ ?a ♯ ?list2.0 ∧ ?a ♯ ?b ∧ ?a ♯ ?v)›*) subst_vv_id (*‹?tm[?a::=[ ?a ]⇧v]⇩v⇩v = ?tm›*) by metis\nqed ((auto)+)\n(*solves the remaining goals:\n 1. ‹⋀x a. [x]⇧s[a::=[ a ]⇧v]⇩s⇩v = [x]⇧s›\n 2. ‹⋀x1 x2 x3 x4 a. ⟦atom x1 ♯ a; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3; ⋀b. x4[b::=[ b ]⇧v]⇩s⇩v = x4⟧ ⟹ LET x1 : x2 = x3 IN x4[a::=[ a ]⇧v]⇩s⇩v = LET x1 : x2 = x3 IN x4›\n 3. ‹⋀x1 x2 x3 a. ⟦⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3⟧ ⟹ (IF x1 THEN x2 ELSE x3)[a::=[ a ]⇧v]⇩s⇩v = IF x1 THEN x2 ELSE x3›\n 4. ‹⋀x1 x2 x3 x4 a. ⟦atom x1 ♯ a; ⋀b. x4[b::=[ b ]⇧v]⇩s⇩v = x4⟧ ⟹ VAR x1 : x2 = x3 IN x4[a::=[ a ]⇧v]⇩s⇩v = VAR x1 : x2 = x3 IN x4›\n 5. ‹⋀x1 x2 a. x1 ::= x2[a::=[ a ]⇧v]⇩s⇩v = x1 ::= x2›\n 6. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2⟧ ⟹ (WHILE x1 DO { x2 } )[a::=[ a ]⇧v]⇩s⇩v = WHILE x1 DO { x2 } ›\n 7. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2⟧ ⟹ ( x1 ;; x2 )[a::=[ a ]⇧v]⇩s⇩v = x1 ;; x2 ›\n 8. ‹⋀x1 x2 a. (⋀b. x2[b::=[ b ]⇧v]⇩s⇩v = x2) ⟹ ASSERT x1 IN x2 [a::=[ a ]⇧v]⇩s⇩v = ASSERT x1 IN x2 ›\n 9. ‹⋀x1 x2 x3 a. ⟦atom x2 ♯ a; ⋀b. x3[b::=[ b ]⇧v]⇩s⇩v = x3⟧ ⟹ x1 x2 ⇒ x3 [a::=[ a ]⇧v]⇩s⇩v = x1 x2 ⇒ x3 ›\n 10. ‹⋀x a. (⋀b. x[b::=[ b ]⇧v]⇩s⇩v = x) ⟹ subst_branchlv { x } a [ a ]⇧v = { x }›\n 11. ‹⋀x1 x2 a. ⟦⋀b. x1[b::=[ b ]⇧v]⇩s⇩v = x1; ⋀b. subst_branchlv x2 b [ b ]⇧v = x2⟧ ⟹ subst_branchlv x1 | x2 a [ a ]⇧v = x1 | x2 ›*) \n\nlemma fresh_subst_sv_if_rl:\n shows \n \"(atom x ♯ s ∧ j ♯ s) ∨ (j ♯ v ∧ (j ♯ s ∨ j = atom x)) ⟹ j ♯ (subst_sv s x v )\" and\n \"(atom x ♯ cs ∧ j ♯ cs) ∨ (j ♯ v ∧ (j ♯ cs ∨ j = atom x)) ⟹ j ♯ (subst_branchv cs x v)\" and\n \"(atom x ♯ css ∧ j ♯ css) ∨ (j ♯ v ∧ (j ♯ css ∨ j = atom x)) ⟹ j ♯ (subst_branchlv css x v )\" \n apply (nominal_induct s and cs and css avoiding: v x rule: s_branch_s_branch_list.strong_induct)\n (*top goal: ‹atom (x::x) ♯ (s::s) ∧ (j::atom) ♯ s ∨ j ♯ (v::v) ∧ (j ♯ s ∨ j = atom x) ⟹ j ♯ s[x::=v]⇩s⇩v› and 2 goals remain*)\n using pure_fresh (*‹?a ♯ ?x›*) apply -\n (*goals:\n 1. ‹⋀x v xa. ⟦atom xa ♯ [x]⇧s ∧ j ♯ [x]⇧s ∨ j ♯ v ∧ (j ♯ [x]⇧s ∨ j = atom xa); ⋀a x. a ♯ x⟧ ⟹ j ♯ [x]⇧s[xa::=v]⇩s⇩v›\n 2. ‹⋀x1 x2 x3 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba) ⟹ j ♯ x3[ba::=b]⇩s⇩v; atom x ♯ LET x1 = x2 IN x3 ∧ j ♯ LET x1 = x2 IN x3 ∨ j ♯ v ∧ (j ♯ LET x1 = x2 IN x3 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ LET x1 = x2 IN x3[x::=v]⇩s⇩v›\n 3. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba) ⟹ j ♯ x3[ba::=b]⇩s⇩v; ⋀b ba. atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba) ⟹ j ♯ x4[ba::=b]⇩s⇩v; atom x ♯ LET x1 : x2 = x3 IN x4 ∧ j ♯ LET x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ LET x1 : x2 = x3 IN x4 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ LET x1 : x2 = x3 IN x4[x::=v]⇩s⇩v›\n 4. ‹⋀x1 x2 x3 v x. ⟦⋀b ba. atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba) ⟹ j ♯ x2[ba::=b]⇩s⇩v; ⋀b ba. atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba) ⟹ j ♯ x3[ba::=b]⇩s⇩v; atom x ♯ IF x1 THEN x2 ELSE x3 ∧ j ♯ IF x1 THEN x2 ELSE x3 ∨ j ♯ v ∧ (j ♯ IF x1 THEN x2 ELSE x3 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ (IF x1 THEN x2 ELSE x3)[x::=v]⇩s⇩v›\n 5. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba) ⟹ j ♯ x4[ba::=b]⇩s⇩v; atom x ♯ VAR x1 : x2 = x3 IN x4 ∧ j ♯ VAR x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ VAR x1 : x2 = x3 IN x4 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ VAR x1 : x2 = x3 IN x4[x::=v]⇩s⇩v›\n 6. ‹⋀x1 x2 v x. ⟦atom x ♯ x1 ::= x2 ∧ j ♯ x1 ::= x2 ∨ j ♯ v ∧ (j ♯ x1 ::= x2 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ x1 ::= x2[x::=v]⇩s⇩v›\n 7. ‹⋀x1 x2 v x. ⟦⋀b ba. atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba) ⟹ j ♯ subst_branchlv x2 ba b; atom x ♯ MATCH x1 WITH { x2 } ∧ j ♯ MATCH x1 WITH { x2 } ∨ j ♯ v ∧ (j ♯ MATCH x1 WITH { x2 } ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ MATCH x1 WITH { x2 }[x::=v]⇩s⇩v›\n 8. ‹⋀x1 x2 v x. ⟦⋀b ba. atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba) ⟹ j ♯ x1[ba::=b]⇩s⇩v; ⋀b ba. atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba) ⟹ j ♯ x2[ba::=b]⇩s⇩v; atom x ♯ WHILE x1 DO { x2 } ∧ j ♯ WHILE x1 DO { x2 } ∨ j ♯ v ∧ (j ♯ WHILE x1 DO { x2 } ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ (WHILE x1 DO { x2 } )[x::=v]⇩s⇩v›\n 9. ‹⋀x1 x2 v x. ⟦⋀b ba. atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba) ⟹ j ♯ x1[ba::=b]⇩s⇩v; ⋀b ba. atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba) ⟹ j ♯ x2[ba::=b]⇩s⇩v; atom x ♯ x1 ;; x2 ∧ j ♯ x1 ;; x2 ∨ j ♯ v ∧ (j ♯ x1 ;; x2 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ ( x1 ;; x2 )[x::=v]⇩s⇩v›\n 10. ‹⋀x1 x2 v x. ⟦⋀b ba. atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba) ⟹ j ♯ x2[ba::=b]⇩s⇩v; atom x ♯ ASSERT x1 IN x2 ∧ j ♯ ASSERT x1 IN x2 ∨ j ♯ v ∧ (j ♯ ASSERT x1 IN x2 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ ASSERT x1 IN x2 [x::=v]⇩s⇩v›\n 11. ‹⋀x1 x2 x3 v x. ⟦atom x2 ♯ v; atom x2 ♯ x; ⋀b ba. atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba) ⟹ j ♯ x3[ba::=b]⇩s⇩v; atom x ♯ x1 x2 ⇒ x3 ∧ j ♯ x1 x2 ⇒ x3 ∨ j ♯ v ∧ (j ♯ x1 x2 ⇒ x3 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ x1 x2 ⇒ x3 [x::=v]⇩s⇩v›\n 12. ‹⋀x v xa. ⟦⋀b ba. atom ba ♯ x ∧ j ♯ x ∨ j ♯ b ∧ (j ♯ x ∨ j = atom ba) ⟹ j ♯ x[ba::=b]⇩s⇩v; atom xa ♯ { x } ∧ j ♯ { x } ∨ j ♯ v ∧ (j ♯ { x } ∨ j = atom xa); ⋀a x. a ♯ x⟧ ⟹ j ♯ subst_branchlv { x } xa v›\n 13. ‹⋀x1 x2 v x. ⟦⋀b ba. atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba) ⟹ j ♯ x1[ba::=b]⇩s⇩v; ⋀b ba. atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba) ⟹ j ♯ subst_branchlv x2 ba b; atom x ♯ x1 | x2 ∧ j ♯ x1 | x2 ∨ j ♯ v ∧ (j ♯ x1 | x2 ∨ j = atom x); ⋀a x. a ♯ x⟧ ⟹ j ♯ subst_branchlv x1 | x2 x v›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*discuss goal 3*)\n apply force\n (*discuss goal 4*)\n apply force\n (*discuss goal 5*)\n apply force\n (*discuss goal 6*)\n apply force\n (*discuss goal 7*)\n apply force\n (*discuss goal 8*)\n apply force\n (*discuss goal 9*)\n apply force\n (*discuss goal 10*)\n apply force\n (*discuss goal 11*)\n apply force\n (*discuss goal 12*)\n apply force\n (*discuss goal 13*)\n apply force\n (*proven 13 subgoals*) .\n\nlemma fresh_subst_sv_if_lr:\n shows \"j ♯ (subst_sv s x v) ⟹ (atom x ♯ s ∧ j ♯ s) ∨ (j ♯ v ∧ (j ♯ s ∨ j = atom x))\" and\n \"j ♯ (subst_branchv cs x v) ⟹ (atom x ♯ cs ∧ j ♯ cs) ∨ (j ♯ v ∧ (j ♯ cs ∨ j = atom x))\" and \n \"j ♯ (subst_branchlv css x v ) ⟹ (atom x ♯ css ∧ j ♯ css) ∨ (j ♯ v ∧ (j ♯ css ∨ j = atom x))\"\nproof (nominal_induct s and cs and css avoiding: v x rule: s_branch_s_branch_list.strong_induct)\n (*goals:\n 1. ‹⋀x v xa. j ♯ [x]⇧s[xa::=v]⇩s⇩v ⟹ atom xa ♯ [x]⇧s ∧ j ♯ [x]⇧s ∨ j ♯ v ∧ (j ♯ [x]⇧s ∨ j = atom xa)›\n 2. ‹⋀x1 x2 x3 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); j ♯ LET x1 = x2 IN x3[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ LET x1 = x2 IN x3 ∧ j ♯ LET x1 = x2 IN x3 ∨ j ♯ v ∧ (j ♯ LET x1 = x2 IN x3 ∨ j = atom x)›\n 3. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); ⋀b ba. j ♯ x4[ba::=b]⇩s⇩v ⟹ atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba); j ♯ LET x1 : x2 = x3 IN x4[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ LET x1 : x2 = x3 IN x4 ∧ j ♯ LET x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ LET x1 : x2 = x3 IN x4 ∨ j = atom x)›\n 4. ‹⋀x1 x2 x3 v x. ⟦⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); j ♯ (IF x1 THEN x2 ELSE x3)[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ IF x1 THEN x2 ELSE x3 ∧ j ♯ IF x1 THEN x2 ELSE x3 ∨ j ♯ v ∧ (j ♯ IF x1 THEN x2 ELSE x3 ∨ j = atom x)›\n 5. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x4[ba::=b]⇩s⇩v ⟹ atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba); j ♯ VAR x1 : x2 = x3 IN x4[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ VAR x1 : x2 = x3 IN x4 ∧ j ♯ VAR x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ VAR x1 : x2 = x3 IN x4 ∨ j = atom x)›\n 6. ‹⋀x1 x2 v x. j ♯ x1 ::= x2[x::=v]⇩s⇩v ⟹ atom x ♯ x1 ::= x2 ∧ j ♯ x1 ::= x2 ∨ j ♯ v ∧ (j ♯ x1 ::= x2 ∨ j = atom x)›\n 7. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ MATCH x1 WITH { x2 }[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ MATCH x1 WITH { x2 } ∧ j ♯ MATCH x1 WITH { x2 } ∨ j ♯ v ∧ (j ♯ MATCH x1 WITH { x2 } ∨ j = atom x)›\n 8. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ (WHILE x1 DO { x2 } )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ WHILE x1 DO { x2 } ∧ j ♯ WHILE x1 DO { x2 } ∨ j ♯ v ∧ (j ♯ WHILE x1 DO { x2 } ∨ j = atom x)›\n 9. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ( x1 ;; x2 )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ x1 ;; x2 ∧ j ♯ x1 ;; x2 ∨ j ♯ v ∧ (j ♯ x1 ;; x2 ∨ j = atom x)›\n 10. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ASSERT x1 IN x2 [x::=v]⇩s⇩v⟧ ⟹ atom x ♯ ASSERT x1 IN x2 ∧ j ♯ ASSERT x1 IN x2 ∨ j ♯ v ∧ (j ♯ ASSERT x1 IN x2 ∨ j = atom x)›\n 11. ‹⋀x1 x2 x3 v x. ⟦atom x2 ♯ v; atom x2 ♯ x; ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); j ♯ x1 x2 ⇒ x3 [x::=v]⇩s⇩v⟧ ⟹ atom x ♯ x1 x2 ⇒ x3 ∧ j ♯ x1 x2 ⇒ x3 ∨ j ♯ v ∧ (j ♯ x1 x2 ⇒ x3 ∨ j = atom x)›\n 12. ‹⋀x v xa. ⟦⋀b ba. j ♯ x[ba::=b]⇩s⇩v ⟹ atom ba ♯ x ∧ j ♯ x ∨ j ♯ b ∧ (j ♯ x ∨ j = atom ba); j ♯ subst_branchlv { x } xa v⟧ ⟹ atom xa ♯ { x } ∧ j ♯ { x } ∨ j ♯ v ∧ (j ♯ { x } ∨ j = atom xa)›\n 13. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ subst_branchlv x1 | x2 x v⟧ ⟹ atom x ♯ x1 | x2 ∧ j ♯ x1 | x2 ∨ j ♯ v ∧ (j ♯ x1 | x2 ∨ j = atom x)›*)\n case (AS_branch list x s) (*‹atom xa__ ♯ v› ‹atom (xa__::x) ♯ (x::x)› ‹j ♯ s[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s ∧ j ♯ s ∨ j ♯ ?b ∧ (j ♯ s ∨ j = atom ?ba)› ‹j ♯ list xa__ ⇒ s [x::=v]⇩s⇩v›*)\n then show \"?case\"\n (*goal: ‹atom x ♯ list xa__ ⇒ s ∧ j ♯ list xa__ ⇒ s ∨ j ♯ v ∧ (j ♯ list xa__ ⇒ s ∨ j = atom x)›*)\n using s_branch_s_branch_list.fresh (*‹(?a::atom) ♯ [?v::v]⇧s = ?a ♯ ?v› ‹?a ♯ LET ?x = ?e IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?e)› ‹?a ♯ LET ?x : ?τ = ?s1.0 IN ?s2.0 = ((?a ♯ ?s2.0 ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?s1.0)› ‹?a ♯ IF ?v THEN ?s1.0 ELSE ?s2.0 = (?a ♯ ?v ∧ ?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ VAR ?u : ?τ = ?v IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?u]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?v)› ‹?a ♯ ?u ::= ?v = (?a ♯ ?u ∧ ?a ♯ ?v)› ‹?a ♯ MATCH ?v WITH { ?branch_list } = (?a ♯ ?v ∧ ?a ♯ ?branch_list)› ‹?a ♯ WHILE ?s1.0 DO { ?s2.0 } = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ ?s1.0 ;; ?s2.0 = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ ASSERT ?c IN ?s = (?a ♯ ?c ∧ ?a ♯ ?s)› ‹?a ♯ ?list ?x ⇒ ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?list)› ‹?a ♯ { ?branch_s } = ?a ♯ ?branch_s› and more 1 facts*) fresh_Pair (*‹(?a::atom) ♯ (?x::?'a, ?y::?'b) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) list.distinct(1) (*‹[] ≠ (?x21.0::?'a::type) # (?x22.0::?'a::type list)›*) list.set_cases (*‹⟦?e ∈ set ?a; ⋀z2. ?a = ?e # z2 ⟹ ?thesis; ⋀z1 z2. ⟦?a = z1 # z2; ?e ∈ set z2⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) pure_fresh (*‹?a ♯ ?x›*) set_ConsD (*‹(?y::?'a) ∈ set ((?x::?'a) # (?xs::?'a list)) ⟹ ?y = ?x ∨ ?y ∈ set ?xs›*) subst_branchv.simps (*‹atom (?x1.0::x) ♯ (?x::x, ?v::v) ⟹ ?dc::char list ?x1.0 ⇒ ?s1.0::s [?x::=?v]⇩s⇩v = ?dc ?x1.0 ⇒ ?s1.0[?x::=?v]⇩s⇩v ›*) by metis\nnext\n (*goals:\n 1. ‹⋀x v xa. j ♯ [x]⇧s[xa::=v]⇩s⇩v ⟹ atom xa ♯ [x]⇧s ∧ j ♯ [x]⇧s ∨ j ♯ v ∧ (j ♯ [x]⇧s ∨ j = atom xa)›\n 2. ‹⋀x1 x2 x3 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); j ♯ LET x1 = x2 IN x3[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ LET x1 = x2 IN x3 ∧ j ♯ LET x1 = x2 IN x3 ∨ j ♯ v ∧ (j ♯ LET x1 = x2 IN x3 ∨ j = atom x)›\n 3. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); ⋀b ba. j ♯ x4[ba::=b]⇩s⇩v ⟹ atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba); j ♯ LET x1 : x2 = x3 IN x4[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ LET x1 : x2 = x3 IN x4 ∧ j ♯ LET x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ LET x1 : x2 = x3 IN x4 ∨ j = atom x)›\n 4. ‹⋀x1 x2 x3 v x. ⟦⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); j ♯ (IF x1 THEN x2 ELSE x3)[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ IF x1 THEN x2 ELSE x3 ∧ j ♯ IF x1 THEN x2 ELSE x3 ∨ j ♯ v ∧ (j ♯ IF x1 THEN x2 ELSE x3 ∨ j = atom x)›\n 5. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x4[ba::=b]⇩s⇩v ⟹ atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba); j ♯ VAR x1 : x2 = x3 IN x4[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ VAR x1 : x2 = x3 IN x4 ∧ j ♯ VAR x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ VAR x1 : x2 = x3 IN x4 ∨ j = atom x)›\n 6. ‹⋀x1 x2 v x. j ♯ x1 ::= x2[x::=v]⇩s⇩v ⟹ atom x ♯ x1 ::= x2 ∧ j ♯ x1 ::= x2 ∨ j ♯ v ∧ (j ♯ x1 ::= x2 ∨ j = atom x)›\n 7. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ MATCH x1 WITH { x2 }[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ MATCH x1 WITH { x2 } ∧ j ♯ MATCH x1 WITH { x2 } ∨ j ♯ v ∧ (j ♯ MATCH x1 WITH { x2 } ∨ j = atom x)›\n 8. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ (WHILE x1 DO { x2 } )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ WHILE x1 DO { x2 } ∧ j ♯ WHILE x1 DO { x2 } ∨ j ♯ v ∧ (j ♯ WHILE x1 DO { x2 } ∨ j = atom x)›\n 9. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ( x1 ;; x2 )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ x1 ;; x2 ∧ j ♯ x1 ;; x2 ∨ j ♯ v ∧ (j ♯ x1 ;; x2 ∨ j = atom x)›\n 10. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ASSERT x1 IN x2 [x::=v]⇩s⇩v⟧ ⟹ atom x ♯ ASSERT x1 IN x2 ∧ j ♯ ASSERT x1 IN x2 ∨ j ♯ v ∧ (j ♯ ASSERT x1 IN x2 ∨ j = atom x)›\n 11. ‹⋀x v xa. ⟦⋀b ba. j ♯ x[ba::=b]⇩s⇩v ⟹ atom ba ♯ x ∧ j ♯ x ∨ j ♯ b ∧ (j ♯ x ∨ j = atom ba); j ♯ subst_branchlv { x } xa v⟧ ⟹ atom xa ♯ { x } ∧ j ♯ { x } ∨ j ♯ v ∧ (j ♯ { x } ∨ j = atom xa)›\n 12. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ subst_branchlv x1 | x2 x v⟧ ⟹ atom x ♯ x1 | x2 ∧ j ♯ x1 | x2 ∨ j ♯ v ∧ (j ♯ x1 | x2 ∨ j = atom x)›*)\n case (AS_let y e s') (*‹atom y ♯ v› ‹atom y ♯ x› ‹j ♯ s'[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s' ∧ j ♯ s' ∨ j ♯ ?b ∧ (j ♯ s' ∨ j = atom ?ba)› ‹j ♯ LET y = e IN s'[x::=v]⇩s⇩v›*)\n thus \"?case\"\n (*goal: ‹atom x ♯ LET y = e IN s' ∧ j ♯ LET y = e IN s' ∨ j ♯ v ∧ (j ♯ LET y = e IN s' ∨ j = atom x)›*)\n proof (cases \"atom x ♯ (AS_let y e s')\")\n (*goals:\n 1. ‹⟦atom y ♯ v; atom y ♯ x; ⋀ba b. j ♯ s'[ba::=b]⇩s⇩v ⟹ atom ba ♯ s' ∧ j ♯ s' ∨ j ♯ b ∧ (j ♯ s' ∨ j = atom ba); j ♯ LET y = e IN s'[x::=v]⇩s⇩v; atom x ♯ LET y = e IN s'⟧ ⟹ atom x ♯ LET y = e IN s' ∧ j ♯ LET y = e IN s' ∨ j ♯ v ∧ (j ♯ LET y = e IN s' ∨ j = atom x)›\n 2. ‹⟦atom y ♯ v; atom y ♯ x; ⋀ba b. j ♯ s'[ba::=b]⇩s⇩v ⟹ atom ba ♯ s' ∧ j ♯ s' ∨ j ♯ b ∧ (j ♯ s' ∨ j = atom ba); j ♯ LET y = e IN s'[x::=v]⇩s⇩v; ¬ atom x ♯ LET y = e IN s'⟧ ⟹ atom x ♯ LET y = e IN s' ∧ j ♯ LET y = e IN s' ∨ j ♯ v ∧ (j ♯ LET y = e IN s' ∨ j = atom x)›*)\n case True (*‹atom (x::x) ♯ LET y::x = e::e IN s'::s›*)\n hence \"subst_sv (AS_let y e s') x v = (AS_let y e s')\"\n using forget_subst_sv (*‹atom ?a ♯ ?A ⟹ ?A[?a::=?x]⇩s⇩v = ?A› ‹atom ?a ♯ ?B ⟹ ?B[?a::=?x]⇩s⇩v = ?B› ‹atom ?a ♯ ?C ⟹ subst_branchlv ?C ?a ?x = ?C›*) by simp\n hence \"j ♯ (AS_let y e s')\"\n using AS_let (*‹atom y ♯ v› ‹atom (y::x) ♯ (x::x)› ‹j ♯ s'[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s' ∧ j ♯ s' ∨ j ♯ ?b ∧ (j ♯ s' ∨ j = atom ?ba)› ‹j ♯ LET y = e IN s'[x::=v]⇩s⇩v›*) by argo\n then show \"?thesis\"\n (*goal: ‹atom (x::x) ♯ LET y::x = e::e IN s'::s ∧ (j::atom) ♯ LET y = e IN s' ∨ j ♯ (v::v) ∧ (j ♯ LET y = e IN s' ∨ j = atom x)›*)\n using True (*‹atom x ♯ LET y = e IN s'›*) by blast\n next\n (*goal: ‹⟦atom (y::x) ♯ (v::v); atom y ♯ (x::x); ⋀(ba::x) b::v. (j::atom) ♯ (s'::s)[ba::=b]⇩s⇩v ⟹ atom ba ♯ s' ∧ j ♯ s' ∨ j ♯ b ∧ (j ♯ s' ∨ j = atom ba); j ♯ LET y = e::e IN s'[x::=v]⇩s⇩v; ¬ atom x ♯ LET y = e IN s'⟧ ⟹ atom x ♯ LET y = e IN s' ∧ j ♯ LET y = e IN s' ∨ j ♯ v ∧ (j ♯ LET y = e IN s' ∨ j = atom x)›*)\n case False (*‹¬ atom x ♯ LET y = e IN s'›*)\n have \"subst_sv (AS_let y e s') x v = AS_let y (e[x::=v]⇩e⇩v) (s'[x::=v]⇩s⇩v)\"\n using subst_sv.simps(2) (*‹atom ?y ♯ (?x, ?v) ⟹ LET ?y = ?e IN ?s[?x::=?v]⇩s⇩v = LET ?y = ?e[?x::=?v]⇩e⇩v IN ?s[?x::=?v]⇩s⇩v›*) AS_let (*‹atom y ♯ v› ‹atom (y::x) ♯ (x::x)› ‹j ♯ s'[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s' ∧ j ♯ s' ∨ j ♯ ?b ∧ (j ♯ s' ∨ j = atom ?ba)› ‹j ♯ LET y = e IN s'[x::=v]⇩s⇩v›*) by force\n hence \"((j ♯ s'[x::=v]⇩s⇩v ∨ j ∈ set [atom y]) ∧ j ♯ None ∧ j ♯ e[x::=v]⇩e⇩v)\"\n using s_branch_s_branch_list.fresh (*‹?a ♯ [?v]⇧s = ?a ♯ ?v› ‹?a ♯ LET ?x = ?e IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?e)› ‹?a ♯ LET ?x : ?τ = ?s1.0 IN ?s2.0 = ((?a ♯ ?s2.0 ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?s1.0)› ‹?a ♯ IF ?v THEN ?s1.0 ELSE ?s2.0 = (?a ♯ ?v ∧ ?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ VAR ?u : ?τ = ?v IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?u]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?v)› ‹?a ♯ ?u ::= ?v = (?a ♯ ?u ∧ ?a ♯ ?v)› ‹(?a::atom) ♯ MATCH ?v::v WITH { ?branch_list::branch_list } = (?a ♯ ?v ∧ ?a ♯ ?branch_list)› ‹?a ♯ WHILE ?s1.0 DO { ?s2.0 } = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹(?a::atom) ♯ (?s1.0::s) ;; (?s2.0::s) = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ ASSERT ?c IN ?s = (?a ♯ ?c ∧ ?a ♯ ?s)› ‹?a ♯ ?list ?x ⇒ ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?list)› ‹?a ♯ { ?branch_s } = ?a ♯ ?branch_s› and more 1 facts*) AS_let (*‹atom y ♯ v› ‹atom y ♯ x› ‹j ♯ s'[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s' ∧ j ♯ s' ∨ j ♯ ?b ∧ (j ♯ s' ∨ j = atom ?ba)› ‹j ♯ LET y = e IN s'[x::=v]⇩s⇩v›*) by (simp add: fresh_None (*‹?a ♯ None›*))\n then show \"?thesis\"\n (*goal: ‹atom x ♯ LET y = e IN s' ∧ j ♯ LET y = e IN s' ∨ j ♯ v ∧ (j ♯ LET y = e IN s' ∨ j = atom x)›*)\n using AS_let (*‹atom (y::x) ♯ (v::v)› ‹atom y ♯ x› ‹j ♯ s'[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s' ∧ j ♯ s' ∨ j ♯ ?b ∧ (j ♯ s' ∨ j = atom ?ba)› ‹j ♯ LET y = e IN s'[x::=v]⇩s⇩v›*) fresh_None (*‹?a ♯ None›*) fresh_subst_ev_if (*‹?j ♯ ?A[?i::=?x]⇩e⇩v = (atom ?i ♯ ?A ∧ ?j ♯ ?A ∨ ?j ♯ ?x ∧ (?j ♯ ?A ∨ ?j = atom ?i))›*) list.discI (*‹?list = ?x21.0 # ?x22.0 ⟹ ?list ≠ []›*) list.set_cases (*‹⟦(?e::?'a::type) ∈ set (?a::?'a::type list); ⋀z2::?'a::type list. ?a = ?e # z2 ⟹ ?thesis::bool; ⋀(z1::?'a::type) z2::?'a::type list. ⟦?a = z1 # z2; ?e ∈ set z2⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) s_branch_s_branch_list.fresh (*‹?a ♯ [?v]⇧s = ?a ♯ ?v› ‹?a ♯ LET ?x = ?e IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?e)› ‹?a ♯ LET ?x : ?τ = ?s1.0 IN ?s2.0 = ((?a ♯ ?s2.0 ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?s1.0)› ‹?a ♯ IF ?v THEN ?s1.0 ELSE ?s2.0 = (?a ♯ ?v ∧ ?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ VAR ?u : ?τ = ?v IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?u]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?v)› ‹?a ♯ ?u ::= ?v = (?a ♯ ?u ∧ ?a ♯ ?v)› ‹(?a::atom) ♯ MATCH ?v::v WITH { ?branch_list::branch_list } = (?a ♯ ?v ∧ ?a ♯ ?branch_list)› ‹?a ♯ WHILE ?s1.0 DO { ?s2.0 } = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ ?s1.0 ;; ?s2.0 = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ ASSERT ?c IN ?s = (?a ♯ ?c ∧ ?a ♯ ?s)› ‹?a ♯ ?list ?x ⇒ ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?list)› ‹?a ♯ { ?branch_s } = ?a ♯ ?branch_s› and more 1 facts*) set_ConsD (*‹?y ∈ set (?x # ?xs) ⟹ ?y = ?x ∨ ?y ∈ set ?xs›*) by metis\n qed\nnext\n (*goals:\n 1. ‹⋀x v xa. j ♯ [x]⇧s[xa::=v]⇩s⇩v ⟹ atom xa ♯ [x]⇧s ∧ j ♯ [x]⇧s ∨ j ♯ v ∧ (j ♯ [x]⇧s ∨ j = atom xa)›\n 2. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); ⋀b ba. j ♯ x4[ba::=b]⇩s⇩v ⟹ atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba); j ♯ LET x1 : x2 = x3 IN x4[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ LET x1 : x2 = x3 IN x4 ∧ j ♯ LET x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ LET x1 : x2 = x3 IN x4 ∨ j = atom x)›\n 3. ‹⋀x1 x2 x3 v x. ⟦⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); ⋀b ba. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); j ♯ (IF x1 THEN x2 ELSE x3)[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ IF x1 THEN x2 ELSE x3 ∧ j ♯ IF x1 THEN x2 ELSE x3 ∨ j ♯ v ∧ (j ♯ IF x1 THEN x2 ELSE x3 ∨ j = atom x)›\n 4. ‹⋀x1 x2 x3 x4 v x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀b ba. j ♯ x4[ba::=b]⇩s⇩v ⟹ atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba); j ♯ VAR x1 : x2 = x3 IN x4[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ VAR x1 : x2 = x3 IN x4 ∧ j ♯ VAR x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ VAR x1 : x2 = x3 IN x4 ∨ j = atom x)›\n 5. ‹⋀x1 x2 v x. j ♯ x1 ::= x2[x::=v]⇩s⇩v ⟹ atom x ♯ x1 ::= x2 ∧ j ♯ x1 ::= x2 ∨ j ♯ v ∧ (j ♯ x1 ::= x2 ∨ j = atom x)›\n 6. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ MATCH x1 WITH { x2 }[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ MATCH x1 WITH { x2 } ∧ j ♯ MATCH x1 WITH { x2 } ∨ j ♯ v ∧ (j ♯ MATCH x1 WITH { x2 } ∨ j = atom x)›\n 7. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ (WHILE x1 DO { x2 } )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ WHILE x1 DO { x2 } ∧ j ♯ WHILE x1 DO { x2 } ∨ j ♯ v ∧ (j ♯ WHILE x1 DO { x2 } ∨ j = atom x)›\n 8. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ( x1 ;; x2 )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ x1 ;; x2 ∧ j ♯ x1 ;; x2 ∨ j ♯ v ∧ (j ♯ x1 ;; x2 ∨ j = atom x)›\n 9. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ASSERT x1 IN x2 [x::=v]⇩s⇩v⟧ ⟹ atom x ♯ ASSERT x1 IN x2 ∧ j ♯ ASSERT x1 IN x2 ∨ j ♯ v ∧ (j ♯ ASSERT x1 IN x2 ∨ j = atom x)›\n 10. ‹⋀x v xa. ⟦⋀b ba. j ♯ x[ba::=b]⇩s⇩v ⟹ atom ba ♯ x ∧ j ♯ x ∨ j ♯ b ∧ (j ♯ x ∨ j = atom ba); j ♯ subst_branchlv { x } xa v⟧ ⟹ atom xa ♯ { x } ∧ j ♯ { x } ∨ j ♯ v ∧ (j ♯ { x } ∨ j = atom xa)›\n 11. ‹⋀x1 x2 v x. ⟦⋀b ba. j ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀b ba. j ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ subst_branchlv x1 | x2 x v⟧ ⟹ atom x ♯ x1 | x2 ∧ j ♯ x1 | x2 ∨ j ♯ v ∧ (j ♯ x1 | x2 ∨ j = atom x)›*)\n case (AS_let2 y τ s1 s2) (*‹atom y ♯ v› ‹atom y ♯ x› ‹j ♯ s1[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s1 ∧ j ♯ s1 ∨ j ♯ ?b ∧ (j ♯ s1 ∨ j = atom ?ba)› ‹(j::atom) ♯ (s2::s)[(?ba::x)::=(?b::v)]⇩s⇩v ⟹ atom ?ba ♯ s2 ∧ j ♯ s2 ∨ j ♯ ?b ∧ (j ♯ s2 ∨ j = atom ?ba)› ‹j ♯ LET y : τ = s1 IN s2[x::=v]⇩s⇩v›*)\n thus \"?case\"\n (*goal: ‹atom x ♯ LET y : τ = s1 IN s2 ∧ j ♯ LET y : τ = s1 IN s2 ∨ j ♯ v ∧ (j ♯ LET y : τ = s1 IN s2 ∨ j = atom x)›*)\n proof (cases \"atom x ♯ (AS_let2 y τ s1 s2)\")\n (*goals:\n 1. ‹⟦atom y ♯ v; atom y ♯ x; ⋀ba b. j ♯ s1[ba::=b]⇩s⇩v ⟹ atom ba ♯ s1 ∧ j ♯ s1 ∨ j ♯ b ∧ (j ♯ s1 ∨ j = atom ba); ⋀ba b. j ♯ s2[ba::=b]⇩s⇩v ⟹ atom ba ♯ s2 ∧ j ♯ s2 ∨ j ♯ b ∧ (j ♯ s2 ∨ j = atom ba); j ♯ LET y : τ = s1 IN s2[x::=v]⇩s⇩v; atom x ♯ LET y : τ = s1 IN s2⟧ ⟹ atom x ♯ LET y : τ = s1 IN s2 ∧ j ♯ LET y : τ = s1 IN s2 ∨ j ♯ v ∧ (j ♯ LET y : τ = s1 IN s2 ∨ j = atom x)›\n 2. ‹⟦atom y ♯ v; atom y ♯ x; ⋀ba b. j ♯ s1[ba::=b]⇩s⇩v ⟹ atom ba ♯ s1 ∧ j ♯ s1 ∨ j ♯ b ∧ (j ♯ s1 ∨ j = atom ba); ⋀ba b. j ♯ s2[ba::=b]⇩s⇩v ⟹ atom ba ♯ s2 ∧ j ♯ s2 ∨ j ♯ b ∧ (j ♯ s2 ∨ j = atom ba); j ♯ LET y : τ = s1 IN s2[x::=v]⇩s⇩v; ¬ atom x ♯ LET y : τ = s1 IN s2⟧ ⟹ atom x ♯ LET y : τ = s1 IN s2 ∧ j ♯ LET y : τ = s1 IN s2 ∨ j ♯ v ∧ (j ♯ LET y : τ = s1 IN s2 ∨ j = atom x)›*)\n case True (*‹atom x ♯ LET y : τ = s1 IN s2›*)\n hence \"subst_sv (AS_let2 y τ s1 s2) x v = (AS_let2 y τ s1 s2)\"\n using forget_subst_sv (*‹atom (?a::x) ♯ (?A::s) ⟹ ?A[?a::=(?x::v)]⇩s⇩v = ?A› ‹atom ?a ♯ ?B ⟹ ?B[?a::=?x]⇩s⇩v = ?B› ‹atom ?a ♯ ?C ⟹ subst_branchlv ?C ?a ?x = ?C›*) by simp\n hence \"j ♯ (AS_let2 y τ s1 s2)\"\n using AS_let2 (*‹atom y ♯ v› ‹atom y ♯ x› ‹j ♯ s1[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s1 ∧ j ♯ s1 ∨ j ♯ ?b ∧ (j ♯ s1 ∨ j = atom ?ba)› ‹j ♯ s2[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s2 ∧ j ♯ s2 ∨ j ♯ ?b ∧ (j ♯ s2 ∨ j = atom ?ba)› ‹j ♯ LET y : τ = s1 IN s2[x::=v]⇩s⇩v›*) by argo\n then show \"?thesis\"\n (*goal: ‹atom x ♯ LET y : τ = s1 IN s2 ∧ j ♯ LET y : τ = s1 IN s2 ∨ j ♯ v ∧ (j ♯ LET y : τ = s1 IN s2 ∨ j = atom x)›*)\n using True (*‹atom x ♯ LET y : τ = s1 IN s2›*) by blast\n next\n (*goal: ‹⟦atom (y::x) ♯ (v::v); atom y ♯ (x::x); ⋀(ba::x) b::v. (j::atom) ♯ (s1::s)[ba::=b]⇩s⇩v ⟹ atom ba ♯ s1 ∧ j ♯ s1 ∨ j ♯ b ∧ (j ♯ s1 ∨ j = atom ba); ⋀(ba::x) b::v. j ♯ (s2::s)[ba::=b]⇩s⇩v ⟹ atom ba ♯ s2 ∧ j ♯ s2 ∨ j ♯ b ∧ (j ♯ s2 ∨ j = atom ba); j ♯ LET y : τ::τ = s1 IN s2[x::=v]⇩s⇩v; ¬ atom x ♯ LET y : τ = s1 IN s2⟧ ⟹ atom x ♯ LET y : τ = s1 IN s2 ∧ j ♯ LET y : τ = s1 IN s2 ∨ j ♯ v ∧ (j ♯ LET y : τ = s1 IN s2 ∨ j = atom x)›*)\n case False (*‹¬ atom x ♯ LET y : τ = s1 IN s2›*)\n have \"subst_sv (AS_let2 y τ s1 s2) x v = AS_let2 y (τ[x::=v]⇩τ⇩v) (s1[x::=v]⇩s⇩v) (s2[x::=v]⇩s⇩v)\"\n using subst_sv.simps (*‹[?v'::v]⇧s[(?x::x)::=(?v::v)]⇩s⇩v = [?v'[?x::=?v]⇩v⇩v]⇧s› ‹atom ?y ♯ (?x, ?v) ⟹ LET ?y = ?e IN ?s[?x::=?v]⇩s⇩v = LET ?y = ?e[?x::=?v]⇩e⇩v IN ?s[?x::=?v]⇩s⇩v› ‹atom ?y ♯ (?x, ?v) ⟹ LET ?y : ?t = ?s1.0 IN ?s2.0[?x::=?v]⇩s⇩v = LET ?y : ?t[?x::=?v]⇩τ⇩v = ?s1.0[?x::=?v]⇩s⇩v IN ?s2.0[?x::=?v]⇩s⇩v› ‹MATCH ?v'::v WITH { ?cs::branch_list }[(?x::x)::=(?v::v)]⇩s⇩v = MATCH ?v'[?x::=?v]⇩v⇩v WITH { subst_branchlv ?cs ?x ?v }› ‹?y ::= ?v'[?x::=?v]⇩s⇩v = ?y ::= ?v'[?x::=?v]⇩v⇩v› ‹(IF ?v' THEN ?s1.0 ELSE ?s2.0)[?x::=?v]⇩s⇩v = IF ?v'[?x::=?v]⇩v⇩v THEN ?s1.0[?x::=?v]⇩s⇩v ELSE ?s2.0[?x::=?v]⇩s⇩v› ‹atom ?u ♯ (?x, ?v) ⟹ VAR ?u : ?τ = ?v' IN ?s[?x::=?v]⇩s⇩v = VAR ?u : ?τ[?x::=?v]⇩τ⇩v = ?v'[?x::=?v]⇩v⇩v IN ?s[?x::=?v]⇩s⇩v› ‹(WHILE ?s1.0 DO { ?s2.0 } )[?x::=?v]⇩s⇩v = WHILE ?s1.0[?x::=?v]⇩s⇩v DO { ?s2.0[?x::=?v]⇩s⇩v } › ‹( ?s1.0 ;; ?s2.0 )[?x::=?v]⇩s⇩v = ?s1.0[?x::=?v]⇩s⇩v ;; ?s2.0[?x::=?v]⇩s⇩v › ‹ASSERT ?c IN ?s [?x::=?v]⇩s⇩v = ASSERT ?c[?x::=?v]⇩c⇩v IN ?s[?x::=?v]⇩s⇩v ›*) AS_let2 (*‹atom (y::x) ♯ (v::v)› ‹atom (y::x) ♯ (x::x)› ‹j ♯ s1[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s1 ∧ j ♯ s1 ∨ j ♯ ?b ∧ (j ♯ s1 ∨ j = atom ?ba)› ‹j ♯ s2[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s2 ∧ j ♯ s2 ∨ j ♯ ?b ∧ (j ♯ s2 ∨ j = atom ?ba)› ‹j ♯ LET y : τ = s1 IN s2[x::=v]⇩s⇩v›*) by force\n then show \"?thesis\"\n (*goal: ‹atom x ♯ LET y : τ = s1 IN s2 ∧ j ♯ LET y : τ = s1 IN s2 ∨ j ♯ v ∧ (j ♯ LET y : τ = s1 IN s2 ∨ j = atom x)›*)\n using AS_let2 (*‹atom y ♯ v› ‹atom (y::x) ♯ (x::x)› ‹j ♯ s1[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s1 ∧ j ♯ s1 ∨ j ♯ ?b ∧ (j ♯ s1 ∨ j = atom ?ba)› ‹j ♯ s2[?ba::=?b]⇩s⇩v ⟹ atom ?ba ♯ s2 ∧ j ♯ s2 ∨ j ♯ ?b ∧ (j ♯ s2 ∨ j = atom ?ba)› ‹j ♯ LET y : τ = s1 IN s2[x::=v]⇩s⇩v›*) fresh_subst_tv_if (*‹?j ♯ ?A[?i::=?x]⇩τ⇩v = (atom ?i ♯ ?A ∧ ?j ♯ ?A ∨ ?j ♯ ?x ∧ (?j ♯ ?A ∨ ?j = atom ?i))›*) list.discI (*‹?list = ?x21.0 # ?x22.0 ⟹ ?list ≠ []›*) list.set_cases (*‹⟦?e ∈ set ?a; ⋀z2. ?a = ?e # z2 ⟹ ?thesis; ⋀z1 z2. ⟦?a = z1 # z2; ?e ∈ set z2⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) s_branch_s_branch_list.fresh(4) (*‹?a ♯ IF ?v THEN ?s1.0 ELSE ?s2.0 = (?a ♯ ?v ∧ ?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)›*) set_ConsD (*‹?y ∈ set (?x # ?xs) ⟹ ?y = ?x ∨ ?y ∈ set ?xs›*) by auto\n qed\nqed ((auto)+)\n(*solves the remaining goals:\n 1. ‹⋀(x::v) (v::v) xa::x. (j::atom) ♯ [x]⇧s[xa::=v]⇩s⇩v ⟹ atom xa ♯ [x]⇧s ∧ j ♯ [x]⇧s ∨ j ♯ v ∧ (j ♯ [x]⇧s ∨ j = atom xa)›\n 2. ‹⋀(x1::v) (x2::s) (x3::s) (v::v) x::x. ⟦⋀(b::v) ba::x. (j::atom) ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); ⋀(b::v) ba::x. j ♯ x3[ba::=b]⇩s⇩v ⟹ atom ba ♯ x3 ∧ j ♯ x3 ∨ j ♯ b ∧ (j ♯ x3 ∨ j = atom ba); j ♯ (IF x1 THEN x2 ELSE x3)[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ IF x1 THEN x2 ELSE x3 ∧ j ♯ IF x1 THEN x2 ELSE x3 ∨ j ♯ v ∧ (j ♯ IF x1 THEN x2 ELSE x3 ∨ j = atom x)›\n 3. ‹⋀(x1::u) (x2::τ) (x3::v) (x4::s) (v::v) x::x. ⟦atom x1 ♯ v; atom x1 ♯ x; ⋀(b::v) ba::x. (j::atom) ♯ x4[ba::=b]⇩s⇩v ⟹ atom ba ♯ x4 ∧ j ♯ x4 ∨ j ♯ b ∧ (j ♯ x4 ∨ j = atom ba); j ♯ VAR x1 : x2 = x3 IN x4[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ VAR x1 : x2 = x3 IN x4 ∧ j ♯ VAR x1 : x2 = x3 IN x4 ∨ j ♯ v ∧ (j ♯ VAR x1 : x2 = x3 IN x4 ∨ j = atom x)›\n 4. ‹⋀(x1::u) (x2::v) (v::v) x::x. (j::atom) ♯ x1 ::= x2[x::=v]⇩s⇩v ⟹ atom x ♯ x1 ::= x2 ∧ j ♯ x1 ::= x2 ∨ j ♯ v ∧ (j ♯ x1 ::= x2 ∨ j = atom x)›\n 5. ‹⋀(x1::v) (x2::branch_list) (v::v) x::x. ⟦⋀(b::v) ba::x. (j::atom) ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ MATCH x1 WITH { x2 }[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ MATCH x1 WITH { x2 } ∧ j ♯ MATCH x1 WITH { x2 } ∨ j ♯ v ∧ (j ♯ MATCH x1 WITH { x2 } ∨ j = atom x)›\n 6. ‹⋀(x1::s) (x2::s) (v::v) x::x. ⟦⋀(b::v) ba::x. (j::atom) ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀(b::v) ba::x. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ (WHILE x1 DO { x2 } )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ WHILE x1 DO { x2 } ∧ j ♯ WHILE x1 DO { x2 } ∨ j ♯ v ∧ (j ♯ WHILE x1 DO { x2 } ∨ j = atom x)›\n 7. ‹⋀(x1::s) (x2::s) (v::v) x::x. ⟦⋀(b::v) ba::x. (j::atom) ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀(b::v) ba::x. j ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ( x1 ;; x2 )[x::=v]⇩s⇩v⟧ ⟹ atom x ♯ x1 ;; x2 ∧ j ♯ x1 ;; x2 ∨ j ♯ v ∧ (j ♯ x1 ;; x2 ∨ j = atom x)›\n 8. ‹⋀(x1::c) (x2::s) (v::v) x::x. ⟦⋀(b::v) ba::x. (j::atom) ♯ x2[ba::=b]⇩s⇩v ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ ASSERT x1 IN x2 [x::=v]⇩s⇩v⟧ ⟹ atom x ♯ ASSERT x1 IN x2 ∧ j ♯ ASSERT x1 IN x2 ∨ j ♯ v ∧ (j ♯ ASSERT x1 IN x2 ∨ j = atom x)›\n 9. ‹⋀(x::branch_s) (v::v) xa::x. ⟦⋀(b::v) ba::x. (j::atom) ♯ x[ba::=b]⇩s⇩v ⟹ atom ba ♯ x ∧ j ♯ x ∨ j ♯ b ∧ (j ♯ x ∨ j = atom ba); j ♯ subst_branchlv { x } xa v⟧ ⟹ atom xa ♯ { x } ∧ j ♯ { x } ∨ j ♯ v ∧ (j ♯ { x } ∨ j = atom xa)›\n 10. ‹⋀(x1::branch_s) (x2::branch_list) (v::v) x::x. ⟦⋀(b::v) ba::x. (j::atom) ♯ x1[ba::=b]⇩s⇩v ⟹ atom ba ♯ x1 ∧ j ♯ x1 ∨ j ♯ b ∧ (j ♯ x1 ∨ j = atom ba); ⋀(b::v) ba::x. j ♯ subst_branchlv x2 ba b ⟹ atom ba ♯ x2 ∧ j ♯ x2 ∨ j ♯ b ∧ (j ♯ x2 ∨ j = atom ba); j ♯ subst_branchlv x1 | x2 x v⟧ ⟹ atom x ♯ x1 | x2 ∧ j ♯ x1 | x2 ∨ j ♯ v ∧ (j ♯ x1 | x2 ∨ j = atom x)›*)\n\nlemma fresh_subst_sv_if[simp]:\n fixes x::x and v::v\n shows \"j ♯ (subst_sv s x v) ⟷ (atom x ♯ s ∧ j ♯ s) ∨ (j ♯ v ∧ (j ♯ s ∨ j = atom x))\" and\n \"j ♯ (subst_branchv cs x v) ⟷ (atom x ♯ cs ∧ j ♯ cs) ∨ (j ♯ v ∧ (j ♯ cs ∨ j = atom x))\"\n using fresh_subst_sv_if_lr (*‹?j ♯ ?s[?x::=?v]⇩s⇩v ⟹ atom ?x ♯ ?s ∧ ?j ♯ ?s ∨ ?j ♯ ?v ∧ (?j ♯ ?s ∨ ?j = atom ?x)› ‹?j ♯ ?cs[?x::=?v]⇩s⇩v ⟹ atom ?x ♯ ?cs ∧ ?j ♯ ?cs ∨ ?j ♯ ?v ∧ (?j ♯ ?cs ∨ ?j = atom ?x)› ‹?j ♯ subst_branchlv ?css ?x ?v ⟹ atom ?x ♯ ?css ∧ ?j ♯ ?css ∨ ?j ♯ ?v ∧ (?j ♯ ?css ∨ ?j = atom ?x)›*) fresh_subst_sv_if_rl (*‹atom ?x ♯ ?s ∧ ?j ♯ ?s ∨ ?j ♯ ?v ∧ (?j ♯ ?s ∨ ?j = atom ?x) ⟹ ?j ♯ ?s[?x::=?v]⇩s⇩v› ‹atom ?x ♯ ?cs ∧ ?j ♯ ?cs ∨ ?j ♯ ?v ∧ (?j ♯ ?cs ∨ ?j = atom ?x) ⟹ ?j ♯ ?cs[?x::=?v]⇩s⇩v› ‹atom (?x::x) ♯ (?css::branch_list) ∧ (?j::atom) ♯ ?css ∨ ?j ♯ (?v::v) ∧ (?j ♯ ?css ∨ ?j = atom ?x) ⟹ ?j ♯ subst_branchlv ?css ?x ?v›*) apply -\n (*goals:\n 1. ‹⟦⋀j s x v. j ♯ s[x::=v]⇩s⇩v ⟹ atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x); ⋀j cs x v. j ♯ cs[x::=v]⇩s⇩v ⟹ atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x); ⋀j css x v. j ♯ subst_branchlv css x v ⟹ atom x ♯ css ∧ j ♯ css ∨ j ♯ v ∧ (j ♯ css ∨ j = atom x); ⋀x s j v. atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x) ⟹ j ♯ s[x::=v]⇩s⇩v; ⋀x cs j v. atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x) ⟹ j ♯ cs[x::=v]⇩s⇩v; ⋀x css j v. atom x ♯ css ∧ j ♯ css ∨ j ♯ v ∧ (j ♯ css ∨ j = atom x) ⟹ j ♯ subst_branchlv css x v⟧ ⟹ j ♯ s[x::=v]⇩s⇩v = (atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x))›\n 2. ‹⟦⋀j s x v. j ♯ s[x::=v]⇩s⇩v ⟹ atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x); ⋀j cs x v. j ♯ cs[x::=v]⇩s⇩v ⟹ atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x); ⋀j css x v. j ♯ subst_branchlv css x v ⟹ atom x ♯ css ∧ j ♯ css ∨ j ♯ v ∧ (j ♯ css ∨ j = atom x); ⋀x s j v. atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x) ⟹ j ♯ s[x::=v]⇩s⇩v; ⋀x cs j v. atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x) ⟹ j ♯ cs[x::=v]⇩s⇩v; ⋀x css j v. atom x ♯ css ∧ j ♯ css ∨ j ♯ v ∧ (j ♯ css ∨ j = atom x) ⟹ j ♯ subst_branchlv css x v⟧ ⟹ j ♯ cs[x::=v]⇩s⇩v = (atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x))›\n discuss goal 1*)\n apply metis\n (*discuss goal 2*)\n apply metis\n (*proven 2 subgoals*) .\n\nlemma subst_sv_commute [simp]:\n fixes A::s and t::v and j::x and i::x\n shows \"atom j ♯ A ⟹ (subst_sv (subst_sv A i t) j u ) = subst_sv A i (subst_vv t j u )\" and\n \"atom j ♯ B ⟹ (subst_branchv (subst_branchv B i t ) j u ) = subst_branchv B i (subst_vv t j u )\" and\n \"atom j ♯ C ⟹ (subst_branchlv (subst_branchlv C i t) j u ) = subst_branchlv C i (subst_vv t j u ) \"\n apply (nominal_induct A and B and C avoiding: i j t u rule: s_branch_s_branch_list.strong_induct)\n (*top goal: ‹atom j ♯ A ⟹ A[i::=t]⇩s⇩v[j::=u]⇩s⇩v = A[i::=t[j::=u]⇩v⇩v]⇩s⇩v› and 2 goals remain*)\n by (auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))\n\nlemma c_eq_perm:\n assumes \"( (atom z) ⇌ (atom z') ) ∙ c = c'\" and \"atom z' ♯ c\"\n shows \"⦃ z : b | c ⦄ = ⦃ z' : b | c' ⦄\"\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) Abs1_eq_iff(3) (*‹([[atom ?a]]lst. ?x = [[atom ?b]]lst. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ ?x = (?a ↔ ?b) ∙ ?y ∧ atom ?a ♯ ?y)›*) by (metis Nominal2_Base.swap_commute (*‹(?a ⇌ ?b) = (?b ⇌ ?a)›*) assms( (*‹(atom z ⇌ atom z') ∙ c = c'›*) 1) assms( (*‹atom z' ♯ c›*) 2) flip_def (*‹(?a ↔ ?b) = (atom ?a ⇌ atom ?b)›*) swap_fresh_fresh (*‹⟦?a ♯ ?x; ?b ♯ ?x⟧ ⟹ (?a ⇌ ?b) ∙ ?x = ?x›*))\n\nlemma subst_sv_flip:\n fixes s::s and sa::s and v'::v\n assumes \"atom c ♯ (s, sa)\" and \"atom c ♯ (v',x, xa, s, sa)\" \"atom x ♯ v'\" and \"atom xa ♯ v'\" and \"(x ↔ c) ∙ s = (xa ↔ c) ∙ sa\" \n shows \"s[x::=v']⇩s⇩v = sa[xa::=v']⇩s⇩v\"\nproof (-)\n (*goal: ‹s[x::=v']⇩s⇩v = sa[xa::=v']⇩s⇩v›*)\n have \"atom x ♯ (s[x::=v']⇩s⇩v)\" and xafr: \"atom xa ♯ (sa[xa::=v']⇩s⇩v)\" and \"atom c ♯ ( s[x::=v']⇩s⇩v, sa[xa::=v']⇩s⇩v)\"\n using assms (*‹atom c ♯ (s, sa)› ‹atom c ♯ (v', x, xa, s, sa)› ‹atom x ♯ v'› ‹atom xa ♯ v'› ‹(x::x ↔ c::x) ∙ (s::s) = (xa::x ↔ c) ∙ (sa::s)›*) using fresh_subst_sv_if (*‹?j ♯ ?s[?x::=?v]⇩s⇩v = (atom ?x ♯ ?s ∧ ?j ♯ ?s ∨ ?j ♯ ?v ∧ (?j ♯ ?s ∨ ?j = atom ?x))› ‹?j ♯ ?cs[?x::=?v]⇩s⇩v = (atom ?x ♯ ?cs ∧ ?j ♯ ?cs ∨ ?j ♯ ?v ∧ (?j ♯ ?cs ∨ ?j = atom ?x))›*) assms (*‹atom c ♯ (s, sa)› ‹atom c ♯ (v', x, xa, s, sa)› ‹atom (x::x) ♯ (v'::v)› ‹atom xa ♯ v'› ‹(x::x ↔ c::x) ∙ (s::s) = (xa::x ↔ c) ∙ (sa::s)›*) apply -\n (*goals:\n 1. ‹⟦atom c ♯ (s, sa); atom c ♯ (v', x, xa, s, sa); atom x ♯ v'; atom xa ♯ v'; (x ↔ c) ∙ s = (xa ↔ c) ∙ sa; ⋀j s x v. j ♯ s[x::=v]⇩s⇩v = (atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x)); ⋀j cs x v. j ♯ cs[x::=v]⇩s⇩v = (atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x)); atom c ♯ (s, sa); atom c ♯ (v', x, xa, s, sa); atom x ♯ v'; atom xa ♯ v'; (x ↔ c) ∙ s = (xa ↔ c) ∙ sa⟧ ⟹ atom x ♯ s[x::=v']⇩s⇩v›\n 2. ‹⟦atom c ♯ (s, sa); atom c ♯ (v', x, xa, s, sa); atom x ♯ v'; atom xa ♯ v'; (x ↔ c) ∙ s = (xa ↔ c) ∙ sa; ⋀j s x v. j ♯ s[x::=v]⇩s⇩v = (atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x)); ⋀j cs x v. j ♯ cs[x::=v]⇩s⇩v = (atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x)); atom c ♯ (s, sa); atom c ♯ (v', x, xa, s, sa); atom x ♯ v'; atom xa ♯ v'; (x ↔ c) ∙ s = (xa ↔ c) ∙ sa⟧ ⟹ atom xa ♯ sa[xa::=v']⇩s⇩v›\n 3. ‹⟦atom c ♯ (s, sa); atom c ♯ (v', x, xa, s, sa); atom x ♯ v'; atom xa ♯ v'; (x ↔ c) ∙ s = (xa ↔ c) ∙ sa; ⋀j s x v. j ♯ s[x::=v]⇩s⇩v = (atom x ♯ s ∧ j ♯ s ∨ j ♯ v ∧ (j ♯ s ∨ j = atom x)); ⋀j cs x v. j ♯ cs[x::=v]⇩s⇩v = (atom x ♯ cs ∧ j ♯ cs ∨ j ♯ v ∧ (j ♯ cs ∨ j = atom x)); atom c ♯ (s, sa); atom c ♯ (v', x, xa, s, sa); atom x ♯ v'; atom xa ♯ v'; (x ↔ c) ∙ s = (xa ↔ c) ∙ sa⟧ ⟹ atom c ♯ (s[x::=v']⇩s⇩v, sa[xa::=v']⇩s⇩v)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply force\n (*proven 3 subgoals*) .\n hence \"s[x::=v']⇩s⇩v = (x ↔ c) ∙ (s[x::=v']⇩s⇩v)\"\n by (simp add: flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*))\n also (*calculation: ‹s[x::=v']⇩s⇩v = (x ↔ c) ∙ s[x::=v']⇩s⇩v›*) have \" ... = ((x ↔ c) ∙ s)[ ((x ↔ c) ∙ x) ::= ((x ↔ c) ∙ v') ]⇩s⇩v\"\n using subst_sv_subst_branchv_subst_branchlv.eqvt (*‹?p ∙ ?a[?aa::=?ab]⇩s⇩v = (?p ∙ ?a)[?p ∙ ?aa::=?p ∙ ?ab]⇩s⇩v› ‹(?p::perm) ∙ (?b::branch_s)[(?ba::x)::=(?bb::v)]⇩s⇩v = (?p ∙ ?b)[?p ∙ ?ba::=?p ∙ ?bb]⇩s⇩v› ‹?p ∙ subst_branchlv ?c ?ca ?cb = subst_branchlv (?p ∙ ?c) (?p ∙ ?ca) (?p ∙ ?cb)›*) by blast\n also (*calculation: ‹s[x::=v']⇩s⇩v = ((x ↔ c) ∙ s)[(x ↔ c) ∙ x::=(x ↔ c) ∙ v']⇩s⇩v›*) have \"... = ((xa ↔ c) ∙ sa)[ ((x ↔ c) ∙ x) ::= ((x ↔ c) ∙ v') ]⇩s⇩v\"\n using assms (*‹atom c ♯ (s, sa)› ‹atom c ♯ (v', x, xa, s, sa)› ‹atom x ♯ v'› ‹atom xa ♯ v'› ‹(x ↔ c) ∙ s = (xa ↔ c) ∙ sa›*) by presburger\n also (*calculation: ‹s[x::=v']⇩s⇩v = ((xa ↔ c) ∙ sa)[(x ↔ c) ∙ x::=(x ↔ c) ∙ v']⇩s⇩v›*) have \"... = ((xa ↔ c) ∙ sa)[ ((xa ↔ c) ∙ xa) ::= ((xa ↔ c) ∙ v') ]⇩s⇩v\"\n using assms (*‹atom (c::x) ♯ (s::s, sa::s)› ‹atom c ♯ (v', x, xa, s, sa)› ‹atom x ♯ v'› ‹atom xa ♯ v'› ‹(x ↔ c) ∙ s = (xa ↔ c) ∙ sa›*) by (metis flip_at_simps( (*‹(?a ↔ ?b) ∙ ?a = ?b›*) 1) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) fresh_PairD( (*‹?a ♯ (?x, ?y) ⟹ ?a ♯ ?x›*) 1))\n also (*calculation: ‹s[x::=v']⇩s⇩v = ((xa ↔ c) ∙ sa)[(xa ↔ c) ∙ xa::=(xa ↔ c) ∙ v']⇩s⇩v›*) have \"... = (xa ↔ c) ∙ (sa[xa::=v']⇩s⇩v)\"\n using subst_sv_subst_branchv_subst_branchlv.eqvt (*‹?p ∙ ?a[?aa::=?ab]⇩s⇩v = (?p ∙ ?a)[?p ∙ ?aa::=?p ∙ ?ab]⇩s⇩v› ‹?p ∙ ?b[?ba::=?bb]⇩s⇩v = (?p ∙ ?b)[?p ∙ ?ba::=?p ∙ ?bb]⇩s⇩v› ‹?p ∙ subst_branchlv ?c ?ca ?cb = subst_branchlv (?p ∙ ?c) (?p ∙ ?ca) (?p ∙ ?cb)›*) by presburger\n also (*calculation: ‹s[x::=v']⇩s⇩v = (xa ↔ c) ∙ sa[xa::=v']⇩s⇩v›*) have \"... = sa[xa::=v']⇩s⇩v\"\n using xafr (*‹atom xa ♯ sa[xa::=v']⇩s⇩v›*) assms (*‹atom c ♯ (s, sa)› ‹atom c ♯ (v', x, xa, s, sa)› ‹atom x ♯ v'› ‹atom xa ♯ v'› ‹(x ↔ c) ∙ s = (xa ↔ c) ∙ sa›*) by (simp add: flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*))\n finally (*calculation: ‹s[x::=v']⇩s⇩v = sa[xa::=v']⇩s⇩v›*) show \"?thesis\"\n (*goal: ‹s[x::=v']⇩s⇩v = sa[xa::=v']⇩s⇩v›*)\n by simp\nqed\n\nlemma if_type_eq:\n fixes Γ::Γ and v::v and z1::x\n assumes \"atom z1' ♯ (v, ca, (x, b, c) #⇩Γ Γ, (CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1]⇧v]⇩c⇩v ))\" and \"atom z1 ♯ v\" \n and \"atom z1 ♯ (za,ca)\" and \"atom z1' ♯ (za,ca)\"\n shows \"(⦃ z1' : ba | CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1']⇧v]⇩c⇩v ⦄) = ⦃ z1 : ba | CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1]⇧v]⇩c⇩v ⦄\"\nproof (-)\n (*goal: ‹⦃ z1' : ba | [ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v ⦄ = ⦃ z1 : ba | [ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ⦄›*)\n have \"atom z1' ♯ (CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1]⇧v]⇩c⇩v )\"\n using assms (*‹atom z1' ♯ (v, ca, (x, b, c) #⇩Γ Γ, [ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v )› ‹atom z1 ♯ v› ‹atom z1 ♯ (za, ca)› ‹atom z1' ♯ (za, ca)›*) fresh_prod4 (*‹?x ♯ (?a, ?b, ?c, ?d) = (?x ♯ ?a ∧ ?x ♯ ?b ∧ ?x ♯ ?c ∧ ?x ♯ ?d)›*) by blast\n moreover hence \"(CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1']⇧v]⇩c⇩v) = (z1' ↔ z1) ∙ (CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1]⇧v]⇩c⇩v )\"\n proof (-)\n (*goal: ‹atom z1' ♯ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ) ⟹ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v ) = (z1' ↔ z1) ∙ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v )›*)\n have \"(z1' ↔ z1) ∙ (CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1]⇧v]⇩c⇩v ) = ( (z1' ↔ z1) ∙ (CE_val v == CE_val (V_lit ll)) IMP ((z1' ↔ z1) ∙ ca[za::=[z1]⇧v]⇩c⇩v ))\"\n by auto\n also (*calculation: ‹(z1' ↔ z1) ∙ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ) = ((z1' ↔ z1) ∙ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e ) IMP (z1' ↔ z1) ∙ ca[za::=[ z1 ]⇧v]⇩c⇩v )›*) have \"... = ((CE_val v == CE_val (V_lit ll)) IMP ((z1' ↔ z1) ∙ ca[za::=[z1]⇧v]⇩c⇩v ))\"\n using ‹atom z1 ♯ v› (*‹atom z1 ♯ v›*) assms (*‹atom z1' ♯ (v, ca, (x, b, c) #⇩Γ Γ, [ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v )› ‹atom z1 ♯ v› ‹atom z1 ♯ (za, ca)› ‹atom z1' ♯ (za, ca)›*) by (metis (mono_tags) ‹atom z1' ♯ (CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1]⇧v]⇩c⇩v )› c.fresh( (*‹(?a::atom) ♯ ((?c1.0::c) IMP (?c2.0::c) ) = (?a ♯ ?c1.0 ∧ ?a ♯ ?c2.0)›*) 6) c.fresh( (*‹(?a::atom) ♯ ((?ce1.0::ce) == (?ce2.0::ce) ) = (?a ♯ ?ce1.0 ∧ ?a ♯ ?ce2.0)›*) 7) ce.fresh( (*‹(?a::atom) ♯ [ ?v::v ]⇧c⇧e = ?a ♯ ?v›*) 1) flip_at_simps( (*‹(?a::?'a ↔ ?b::?'a) ∙ ?b = ?a›*) 2) flip_fresh_fresh (*‹⟦atom (?a::?'b) ♯ (?x::?'a); atom (?b::?'b) ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) fresh_at_base_permute_iff (*‹atom ((?p::perm) ∙ (?a::?'a)) ♯ ?p ∙ (?x::?'b) = atom ?a ♯ ?x›*) fresh_def (*‹(?a::atom) ♯ (?x::?'a) ≡ ?a ∉ supp ?x›*) supp_l_empty (*‹supp [ ?l::l ]⇧v = {}›*) v.fresh( (*‹(?a::atom) ♯ [ ?l::l ]⇧v = ?a ♯ ?l›*) 1))\n also (*calculation: ‹(z1'::x ↔ z1::x) ∙ ([ v::v ]⇧c⇧e == [ [ ll::l ]⇧v ]⇧c⇧e IMP (ca::c)[(za::x)::=[ z1 ]⇧v]⇩c⇩v ) = ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP (z1' ↔ z1) ∙ ca[za::=[ z1 ]⇧v]⇩c⇩v )›*) have \"... = ((CE_val v == CE_val (V_lit ll)) IMP (ca[za::=[z1']⇧v]⇩c⇩v ))\"\n using assms (*‹atom z1' ♯ (v, ca, (x, b, c) #⇩Γ Γ, [ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v )› ‹atom z1 ♯ v› ‹atom (z1::x) ♯ (za::x, ca::c)› ‹atom z1' ♯ (za, ca)›*) by fastforce\n finally (*calculation: ‹(z1' ↔ z1) ∙ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ) = ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v )›*) show \"?thesis\"\n (*goal: ‹([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v ) = (z1' ↔ z1) ∙ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v )›*)\n by auto\n qed\n ultimately show \"?thesis\"\n (*goal: ‹⦃ z1' : ba | [ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v ⦄ = ⦃ z1 : ba | [ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ⦄›*)\n using \"τ.eq_iff\" (*‹(⦃ ?x : ?b | ?c ⦄ = ⦃ ?xa : ?ba | ?ca ⦄) = ([[atom ?x]]lst. ?c = [[atom ?xa]]lst. ?ca ∧ ?b = ?ba)›*) Abs1_eq_iff(3)[of z1' \"CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1']⇧v]⇩c⇩v\" z1 \"CE_val v == CE_val (V_lit ll) IMP ca[za::=[z1]⇧v]⇩c⇩v\"] (*‹([[atom z1']]lst. ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v ) = [[atom z1]]lst. ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v )) = (z1' = z1 ∧ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v ) = ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ) ∨ z1' ≠ z1 ∧ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1' ]⇧v]⇩c⇩v ) = (z1' ↔ z1) ∙ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ) ∧ atom z1' ♯ ([ v ]⇧c⇧e == [ [ ll ]⇧v ]⇧c⇧e IMP ca[za::=[ z1 ]⇧v]⇩c⇩v ))›*) by blast\nqed \n\nlemma subst_sv_var_flip:\n fixes x::x and s::s and z::x\n shows \"atom x ♯ s ⟹ ((x ↔ z) ∙ s) = s[z::=[x]⇧v]⇩s⇩v\" and \n \"atom x ♯ cs ⟹ ((x ↔ z) ∙ cs) = subst_branchv cs z [x]⇧v\" and\n \"atom x ♯ css ⟹ ((x ↔ z) ∙ css) = subst_branchlv css z [x]⇧v\"\n apply (nominal_induct s and cs and css avoiding: z x rule: s_branch_s_branch_list.strong_induct)\n (*top goal: ‹atom x ♯ s ⟹ (x ↔ z) ∙ s = s[z::=[ x ]⇧v]⇩s⇩v› and 2 goals remain*)\n using [simproc del : alpha_lst] (*‹TERM _›*) apply auto\n (*top goal: ‹⋀(x::v) (z::x) xa::x. atom xa ♯ [x]⇧s ⟹ (xa ↔ z) ∙ [x]⇧s = [x]⇧s[z::=[ xa ]⇧v]⇩s⇩v› and 12 goals remain*)\n using subst_tv_var_flip (*‹atom ?xa ♯ ?τ ⟹ (?x ↔ ?xa) ∙ ?τ = ?τ[?x::=[ ?xa ]⇧v]⇩τ⇩v›*) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) v.fresh (*‹(?a::atom) ♯ [ ?l::l ]⇧v = ?a ♯ ?l› ‹?a ♯ [ ?x ]⇧v = ?a ♯ ?x› ‹?a ♯ [ ?v1.0 , ?v2.0 ]⇧v = (?a ♯ ?v1.0 ∧ ?a ♯ ?v2.0)› ‹?a ♯ V_cons ?list1.0 ?list2.0 ?v = (?a ♯ ?list1.0 ∧ ?a ♯ ?list2.0 ∧ ?a ♯ ?v)› ‹?a ♯ V_consp ?list1.0 ?list2.0 ?b ?v = (?a ♯ ?list1.0 ∧ ?a ♯ ?list2.0 ∧ ?a ♯ ?b ∧ ?a ♯ ?v)›*) s_branch_s_branch_list.fresh (*‹?a ♯ [?v]⇧s = ?a ♯ ?v› ‹?a ♯ LET ?x = ?e IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?e)› ‹?a ♯ LET ?x : ?τ = ?s1.0 IN ?s2.0 = ((?a ♯ ?s2.0 ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?s1.0)› ‹?a ♯ IF ?v THEN ?s1.0 ELSE ?s2.0 = (?a ♯ ?v ∧ ?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ VAR ?u : ?τ = ?v IN ?s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?u]) ∧ ?a ♯ ?τ ∧ ?a ♯ ?v)› ‹(?a::atom) ♯ ?u::u ::= ?v::v = (?a ♯ ?u ∧ ?a ♯ ?v)› ‹?a ♯ MATCH ?v WITH { ?branch_list } = (?a ♯ ?v ∧ ?a ♯ ?branch_list)› ‹?a ♯ WHILE ?s1.0 DO { ?s2.0 } = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ ?s1.0 ;; ?s2.0 = (?a ♯ ?s1.0 ∧ ?a ♯ ?s2.0)› ‹?a ♯ ASSERT ?c IN ?s = (?a ♯ ?c ∧ ?a ♯ ?s)› ‹(?a::atom) ♯ ?list::char list ?x::x ⇒ ?s::s = ((?a ♯ ?s ∨ ?a ∈ set [atom ?x]) ∧ ?a ♯ ?list)› ‹?a ♯ { ?branch_s } = ?a ♯ ?branch_s› and more 1 facts*) \"subst_v_τ_def\" (*‹subst_v = subst_tv_abbrev›*) subst_v_v_def (*‹subst_v = subst_vv_abbrev›*) subst_vv_var_flip (*‹atom ?y ♯ ?v ⟹ (?y ↔ ?x) ∙ ?v = ?v[?x::=[ ?y ]⇧v]⇩v⇩v›*) subst_v_e_def (*‹subst_v = subst_ev_abbrev›*) subst_ev_var_flip (*‹atom ?y ♯ ?e ⟹ (?y ↔ ?x) ∙ ?e = ?e[?x::=[ ?y ]⇧v]⇩e⇩v›*) pure_fresh (*‹?a ♯ ?x›*) apply auto\n (*top goal: ‹⋀x z xa. atom xa ♯ x ⟹ x[z::=[ xa ]⇧v]⇩v = x[z::=[ xa ]⇧v]⇩v⇩v› and 11 goals remain*)\n defer 1\n (*top goal: ‹⋀x1 x2 z x. ⟦atom x ♯ x2; ⋀a x b. ⟦atom a ♯ x; atom b ♯ x⟧ ⟹ (a ↔ b) ∙ x = x; subst_v = subst_tv_abbrev; subst_v = subst_vv_abbrev; subst_v = subst_ev_abbrev; ⋀a x. a ♯ x⟧ ⟹ (x ↔ z) ∙ x1 = x1› and 3 goals remain*)\n using x_fresh_u (*‹atom ?x ♯ ?u›*) apply blast\n (*top goal: ‹⋀x1 x2 z x. ⟦atom x ♯ x2; ⋀a x b. ⟦atom a ♯ x; atom b ♯ x⟧ ⟹ (a ↔ b) ∙ x = x; subst_v = subst_tv_abbrev; subst_v = subst_vv_abbrev; subst_v = subst_ev_abbrev; ⋀a x. a ♯ x⟧ ⟹ (x ↔ z) ∙ x1 = x1› and 3 goals remain*)\n defer 1\n (*top goal: ‹⋀x1 x2 x3 z x. ⟦atom x2 ♯ z; atom x2 ♯ x; ⋀b ba. atom ba ♯ x3 ⟹ (ba ↔ b) ∙ x3 = x3[b::=[ ba ]⇧v]⇩s⇩v; atom x ♯ x1; atom x ♯ x3; ⋀a x b. ⟦atom a ♯ x; atom b ♯ x⟧ ⟹ (a ↔ b) ∙ x = x; subst_v = subst_tv_abbrev; subst_v = subst_vv_abbrev; subst_v = subst_ev_abbrev; ⋀a x. a ♯ x⟧ ⟹ (x ↔ z) ∙ x1 = x1› and 2 goals remain*)\n using x_fresh_u (*‹atom ?x ♯ ?u›*) apply blast\n (*top goal: ‹⋀x1 x2 x3 z x. ⟦atom x2 ♯ z; atom x2 ♯ x; ⋀b ba. atom ba ♯ x3 ⟹ (ba ↔ b) ∙ x3 = x3[b::=[ ba ]⇧v]⇩s⇩v; atom x ♯ x1; atom x ♯ x3; ⋀a x b. ⟦atom a ♯ x; atom b ♯ x⟧ ⟹ (a ↔ b) ∙ x = x; subst_v = subst_tv_abbrev; subst_v = subst_vv_abbrev; subst_v = subst_ev_abbrev; ⋀a x. a ♯ x⟧ ⟹ (x ↔ z) ∙ x1 = x1› and 2 goals remain*)\n defer 1\n (*top goal: ‹⋀x1 x2 z x. ⟦⋀b ba. atom ba ♯ x2 ⟹ (ba ↔ b) ∙ x2 = x2[b::=[ ba ]⇧v]⇩s⇩v; atom x ♯ x1; atom x ♯ x2; ⋀a x b. ⟦atom a ♯ x; atom b ♯ x⟧ ⟹ (a ↔ b) ∙ x = x; subst_v = subst_tv_abbrev; subst_v = subst_vv_abbrev; subst_v = subst_ev_abbrev; ⋀a x. a ♯ x⟧ ⟹ x1[z::=[ x ]⇧v]⇩v = x1[z::=[ x ]⇧v]⇩c⇩v› and 1 goal remains*)\n using x_fresh_u (*‹atom ?x ♯ ?u›*) Abs1_eq_iff'(3) (*‹([[atom (?a::?'b)]]lst. (?x::?'a) = [[atom (?b::?'b)]]lst. (?y::?'a)) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ (?b ↔ ?a) ∙ ?x = ?y ∧ atom ?b ♯ ?x)›*) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) apply (simp add: subst_v_c_def (*‹subst_v = subst_cv_abbrev›*))\n (*top goal: ‹⋀x1 x2 z x. ⟦⋀b ba. atom ba ♯ x2 ⟹ (ba ↔ b) ∙ x2 = x2[b::=[ ba ]⇧v]⇩s⇩v; atom x ♯ x1; atom x ♯ x2; ⋀a x b. ⟦atom a ♯ x; atom b ♯ x⟧ ⟹ (a ↔ b) ∙ x = x; subst_v = subst_tv_abbrev; subst_v = subst_vv_abbrev; subst_v = subst_ev_abbrev; ⋀a x. a ♯ x⟧ ⟹ x1[z::=[ x ]⇧v]⇩v = x1[z::=[ x ]⇧v]⇩c⇩v› and 1 goal remains*)\n using x_fresh_u (*‹atom ?x ♯ ?u›*) Abs1_eq_iff'(3) (*‹([[atom ?a]]lst. ?x = [[atom ?b]]lst. ?y) = (?a = ?b ∧ ?x = ?y ∨ ?a ≠ ?b ∧ (?b ↔ ?a) ∙ ?x = ?y ∧ atom ?b ♯ ?x)›*) flip_fresh_fresh (*‹⟦atom ?a ♯ ?x; atom ?b ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*) by (simp add: flip_fresh_fresh (*‹⟦atom (?a::?'b) ♯ (?x::?'a); atom (?b::?'b) ♯ ?x⟧ ⟹ (?a ↔ ?b) ∙ ?x = ?x›*))\n\ninstantiation s :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_sv\"\n\ninstance proof\n fix j::atom and i::x and x::v and t::s\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n using fresh_subst_sv_if subst_v_s_def by auto\n\n fix a::x and tm::s and x::v\n show \"atom a ♯ tm ⟹ subst_v tm a x = tm\"\n using forget_subst_sv subst_v_s_def by simp\n\n fix a::x and tm::s\n show \"subst_v tm a (V_var a) = tm\" using subst_sv_id subst_v_s_def by simp\n\n fix p::perm and x1::x and v::v and t1::s\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n using subst_sv_commute subst_v_s_def by simp\n\n fix x::x and c::s and z::x\n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\"\n using subst_sv_var_flip subst_v_s_def by simp\n\n fix x::x and c::s and z::x\n show \"atom x ♯ c ⟹ c[z::=[x]⇧v]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n using subst_sv_var_flip subst_v_s_def by simp\nqed\nend\n\nsection ‹Type Definition›\n\nnominal_function subst_ft_v :: \"fun_typ ⇒ x ⇒ v ⇒ fun_typ\" where\n \"atom z ♯ (x,v) ⟹ subst_ft_v ( AF_fun_typ z b c t (s::s)) x v = AF_fun_typ z b c[x::=v]⇩c⇩v t[x::=v]⇩τ⇩v s[x::=v]⇩s⇩v\"\n apply(simp add: eqvt_def subst_ft_v_graph_aux_def )\n apply(simp add:fun_typ.strong_exhaust )\n apply(auto) \n apply(rule_tac y=a and c=\"(aa,b)\" in fun_typ.strong_exhaust)\n apply (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base)\n\nproof(goal_cases)\n case (1 z xa va c t s za ca ta sa cb)\n hence \"c[z::=[ cb ]⇧v]⇩c⇩v = ca[za::=[ cb ]⇧v]⇩c⇩v\" \n by (metis flip_commute subst_cv_var_flip)\n hence \"c[z::=[ cb ]⇧v]⇩c⇩v[xa::=va]⇩c⇩v = ca[za::=[ cb ]⇧v]⇩c⇩v[xa::=va]⇩c⇩v\" by auto\n then show ?case using subst_cv_commute atom_eq_iff fresh_atom fresh_atom_at_base subst_cv_commute_full v.fresh \n using 1 subst_cv_var_flip flip_commute by metis\nnext\n case (2 z xa va c t s za ca ta sa cb)\n hence \"t[z::=[ cb ]⇧v]⇩τ⇩v = ta[za::=[ cb ]⇧v]⇩τ⇩v\" by metis\n hence \"t[z::=[ cb ]⇧v]⇩τ⇩v[xa::=va]⇩τ⇩v = ta[za::=[ cb ]⇧v]⇩τ⇩v[xa::=va]⇩τ⇩v\" by auto\n then show ?case using subst_tv_commute_full 2 \n by (metis atom_eq_iff fresh_atom fresh_atom_at_base v.fresh(2))\nqed\n\nnominal_termination (eqvt) by lexicographic_order\n\nnominal_function subst_ftq_v :: \"fun_typ_q ⇒ x ⇒ v ⇒ fun_typ_q\" where\n \"atom bv ♯ (x,v) ⟹ subst_ftq_v (AF_fun_typ_some bv ft) x v = (AF_fun_typ_some bv (subst_ft_v ft x v))\" \n| \"subst_ftq_v (AF_fun_typ_none ft) x v = (AF_fun_typ_none (subst_ft_v ft x v))\" \n apply(simp add: eqvt_def subst_ftq_v_graph_aux_def )\n apply(simp add:fun_typ_q.strong_exhaust )\n apply(auto) \n apply(rule_tac y=a and c=\"(aa,b)\" in fun_typ_q.strong_exhaust)\n apply (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base)\nproof(goal_cases)\n case (1 bv ft bva fta xa va c)\n then show ?case using subst_ft_v.simps by (simp add: flip_fresh_fresh)\nqed\nnominal_termination (eqvt) by lexicographic_order\n\nlemma size_subst_ft[simp]: \"size (subst_ft_v A x v) = size A\" \n apply (nominal_induct A avoiding: x v rule: fun_typ.strong_induct)\n (*goal: ‹size (subst_ft_v A x v) = size A›*)\n by auto\n\nlemma forget_subst_ft [simp]: shows \"atom x ♯ A ⟹ subst_ft_v A x a = A\" \n apply (nominal_induct A avoiding: a x rule: fun_typ.strong_induct)\n (*goal: ‹atom (x::x) ♯ (A::fun_typ) ⟹ subst_ft_v A x (a::v) = A›*)\n by (auto simp: fresh_at_base (*‹sort_of ?a ≠ sort_of (atom ?b) ⟹ ?a ♯ ?b› ‹?a ♯ ?b = (?a ≠ atom ?b)›*))\n\nlemma subst_ft_id [simp]: \"subst_ft_v A a (V_var a) = A\"\n apply (nominal_induct A avoiding: a rule: fun_typ.strong_induct)\n (*goal: ‹subst_ft_v (A::fun_typ) (a::x) [ a ]⇧v = A›*)\n by auto \n\ninstantiation fun_typ :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_ft_v\"\n\ninstance proof\n\n fix j::atom and i::x and x::v and t::fun_typ\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n apply(nominal_induct t avoiding: i x rule:fun_typ.strong_induct)\n apply(simp only: subst_v_fun_typ_def subst_ft_v.simps )\n using fun_typ.fresh fresh_subst_v_if apply simp\n by auto\n\n fix a::x and tm::fun_typ and x::v\n show \"atom a ♯ tm ⟹ subst_v tm a x = tm\"\n proof(nominal_induct tm avoiding: a x rule:fun_typ.strong_induct)\n case (AF_fun_typ x1a x2a x3a x4a x5a)\n then show ?case unfolding subst_ft_v.simps subst_v_fun_typ_def fun_typ.fresh using forget_subst_v subst_ft_v.simps subst_v_c_def forget_subst_sv subst_v_τ_def by fastforce\n qed\n\n fix a::x and tm::fun_typ\n show \"subst_v tm a (V_var a) = tm\" \n proof(nominal_induct tm avoiding: a x rule:fun_typ.strong_induct)\n case (AF_fun_typ x1a x2a x3a x4a x5a)\n then show ?case unfolding subst_ft_v.simps subst_v_fun_typ_def fun_typ.fresh using forget_subst_v subst_ft_v.simps subst_v_c_def forget_subst_sv subst_v_τ_def by fastforce\n qed\n\n fix p::perm and x1::x and v::v and t1::fun_typ\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n proof(nominal_induct t1 avoiding: x1 v rule:fun_typ.strong_induct)\n case (AF_fun_typ x1a x2a x3a x4a x5a)\n then show ?case unfolding subst_ft_v.simps subst_v_fun_typ_def fun_typ.fresh using forget_subst_v subst_ft_v.simps subst_v_c_def forget_subst_sv subst_v_τ_def by fastforce\n qed\n\n fix x::x and c::fun_typ and z::x\n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\"\n apply(nominal_induct c avoiding: x z rule:fun_typ.strong_induct)\n by (auto simp add: subst_v_c_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_def)\n\n fix x::x and c::fun_typ and z::x\n show \"atom x ♯ c ⟹ c[z::=[x]⇧v]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n apply(nominal_induct c avoiding: z x v rule:fun_typ.strong_induct)\n apply auto\n by (auto simp add: subst_v_c_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_def )\nqed\nend\n\ninstantiation fun_typ_q :: has_subst_v\nbegin\n\ndefinition \n \"subst_v = subst_ftq_v\"\n\ninstance proof\n fix j::atom and i::x and x::v and t::fun_typ_q\n show \"(j ♯ subst_v t i x) = ((atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i)))\"\n apply(nominal_induct t avoiding: i x rule:fun_typ_q.strong_induct,auto)\n apply(auto simp add: subst_v_fun_typ_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_q_def fresh_subst_v_if ) \n by (metis (no_types) fresh_subst_v_if subst_v_fun_typ_def)+\n\n fix i::x and t::fun_typ_q and x::v\n show \"atom i ♯ t ⟹ subst_v t i x = t\"\n apply(nominal_induct t avoiding: i x rule:fun_typ_q.strong_induct,auto)\n by(auto simp add: subst_v_fun_typ_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_q_def fresh_subst_v_if ) \n\n fix i::x and t::fun_typ_q\n show \"subst_v t i (V_var i) = t\" using subst_cv_id subst_v_fun_typ_def \n apply(nominal_induct t avoiding: i x rule:fun_typ_q.strong_induct,auto)\n by(auto simp add: subst_v_fun_typ_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_q_def fresh_subst_v_if ) \n\n fix p::perm and x1::x and v::v and t1::fun_typ_q\n show \"p ∙ subst_v t1 x1 v = subst_v (p ∙ t1) (p ∙ x1) (p ∙ v)\" \n apply(nominal_induct t1 avoiding: v x1 rule:fun_typ_q.strong_induct,auto)\n by(auto simp add: subst_v_fun_typ_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_q_def fresh_subst_v_if ) \n\n fix x::x and c::fun_typ_q and z::x\n show \"atom x ♯ c ⟹ ((x ↔ z) ∙ c) = c[z::=[x]⇧v]⇩v\"\n apply(nominal_induct c avoiding: x z rule:fun_typ_q.strong_induct,auto)\n by(auto simp add: subst_v_fun_typ_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_q_def fresh_subst_v_if ) \n\n fix x::x and c::fun_typ_q and z::x\n show \"atom x ♯ c ⟹ c[z::=[x]⇧v]⇩v[x::=v]⇩v = c[z::=v]⇩v\"\n apply(nominal_induct c avoiding: z x v rule:fun_typ_q.strong_induct,auto)\n apply(auto simp add: subst_v_fun_typ_def subst_v_s_def subst_v_τ_def subst_v_fun_typ_q_def fresh_subst_v_if ) \n by (metis subst_v_fun_typ_def flip_bv_x_cancel subst_ft_v.eqvt subst_v_simple_commute v.perm_simps )+\nqed\n\nend\n\nsection ‹Variable Context›\n\nlemma subst_dv_fst_eq:\n \"fst ` setD (Δ[x::=v]⇩Δ⇩v) = fst ` setD Δ\" \n apply (induct Δ rule: Δ_induct (*‹⟦(?P::Δ ⇒ bool) []⇩Δ; ⋀(u::u) (t::τ) Δ'::Δ. ?P Δ' ⟹ ?P ((u, t) #⇩Δ Δ')⟧ ⟹ ?P (?Δ::Δ)›*))\n (*goals:\n 1. ‹fst ` setD []⇩Δ[x::=v]⇩Δ⇩v = fst ` setD []⇩Δ›\n 2. ‹⋀u t Δ'. fst ` setD Δ'[x::=v]⇩Δ⇩v = fst ` setD Δ' ⟹ fst ` setD ((u, t) #⇩Δ Δ')[x::=v]⇩Δ⇩v = fst ` setD ((u, t) #⇩Δ Δ')›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*) .\n\nlemma subst_gv_member_iff:\n fixes x'::x and x::x and v::v and c'::c\n assumes \"(x',b',c') ∈ toSet Γ\" and \"atom x ∉ atom_dom Γ\" \n shows \"(x',b',c'[x::=v]⇩c⇩v) ∈ toSet Γ[x::=v]⇩Γ⇩v\"\nproof (-)\n (*goal: ‹(x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ[x::=v]⇩Γ⇩v›*)\n have \"x' ≠ x\"\n using assms (*‹(x'::x, b'::b, c'::c) ∈ toSet (Γ::Γ)› ‹atom x ∉ atom_dom Γ›*) fresh_dom_free2 (*‹atom (?x::x) ∉ atom_dom (?Γ::Γ) ⟹ (?x, ?b::b, ?c::c) ∉ toSet ?Γ›*) by metis\n then show \"?thesis\"\n (*goal: ‹(x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ[x::=v]⇩Γ⇩v›*)\n using assms (*‹(x', b', c') ∈ toSet Γ› ‹atom x ∉ atom_dom Γ›*) proof (induct Γ rule: Γ_induct (*‹⟦?P GNil; ⋀x b c Γ'. ?P Γ' ⟹ ?P ((x, b, c) #⇩Γ Γ')⟧ ⟹ ?P ?Γ›*))\n (*goals:\n 1. ‹⟦x' ≠ x; (x', b', c') ∈ toSet GNil; atom x ∉ atom_dom GNil⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet GNil[x::=v]⇩Γ⇩v›\n 2. ‹⋀xa b c Γ'. ⟦⟦x' ≠ x; (x', b', c') ∈ toSet Γ'; atom x ∉ atom_dom Γ'⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v; x' ≠ x; (x', b', c') ∈ toSet ((xa, b, c) #⇩Γ Γ'); atom x ∉ atom_dom ((xa, b, c) #⇩Γ Γ')⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet ((xa, b, c) #⇩Γ Γ')[x::=v]⇩Γ⇩v›*)\n case GNil (*‹x' ≠ x› ‹(x', b', c') ∈ toSet GNil› ‹atom x ∉ atom_dom GNil›*)\n then show \"?case\"\n (*goal: ‹(x'::x, b'::b, (c'::c)[(x::x)::=(v::v)]⇩c⇩v) ∈ toSet GNil[x::=v]⇩Γ⇩v›*)\n by auto\n next\n (*goal: ‹⋀xa b c Γ'. ⟦⟦x' ≠ x; (x', b', c') ∈ toSet Γ'; atom x ∉ atom_dom Γ'⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v; x' ≠ x; (x', b', c') ∈ toSet ((xa, b, c) #⇩Γ Γ'); atom x ∉ atom_dom ((xa, b, c) #⇩Γ Γ')⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet ((xa, b, c) #⇩Γ Γ')[x::=v]⇩Γ⇩v›*)\n case (GCons x1 b1 c1 Γ') (*‹⟦x' ≠ x; (x', b', c') ∈ toSet Γ'; atom x ∉ atom_dom Γ'⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v› ‹x' ≠ x› ‹(x', b', c') ∈ toSet ((x1, b1, c1) #⇩Γ Γ')› ‹atom x ∉ atom_dom ((x1, b1, c1) #⇩Γ Γ')›*)\n show \"?case\"\n (*goal: ‹(x', b', c'[x::=v]⇩c⇩v) ∈ toSet ((x1, b1, c1) #⇩Γ Γ')[x::=v]⇩Γ⇩v›*)\n proof (cases \"(x',b',c') = (x1,b1,c1)\")\n (*goals:\n 1. ‹(x'::x, b'::b, c'::c) = (x1::x, b1::b, c1::c) ⟹ (x', b', c'[(x::x)::=(v::v)]⇩c⇩v) ∈ toSet ((x1, b1, c1) #⇩Γ (Γ'::Γ))[x::=v]⇩Γ⇩v›\n 2. ‹(x'::x, b'::b, c'::c) ≠ (x1::x, b1::b, c1::c) ⟹ (x', b', c'[(x::x)::=(v::v)]⇩c⇩v) ∈ toSet ((x1, b1, c1) #⇩Γ (Γ'::Γ))[x::=v]⇩Γ⇩v›*)\n case True (*‹(x'::x, b'::b, c'::c) = (x1::x, b1::b, c1::c)›*)\n hence \"((x1, b1, c1) #⇩Γ Γ')[x::=v]⇩Γ⇩v = ((x1, b1, c1[x::=v]⇩c⇩v) #⇩Γ (Γ'[x::=v]⇩Γ⇩v))\"\n using subst_gv.simps (*‹GNil[?x::=?v]⇩Γ⇩v = GNil› ‹((?y, ?b, ?c) #⇩Γ ?Γ)[?x::=?v]⇩Γ⇩v = (if ?x = ?y then ?Γ else (?y, ?b, ?c[?x::=?v]⇩c⇩v) #⇩Γ ?Γ[?x::=?v]⇩Γ⇩v)›*) ‹x'≠x› (*‹(x'::x) ≠ (x::x)›*) by auto\n then show \"?thesis\"\n (*goal: ‹(x', b', c'[x::=v]⇩c⇩v) ∈ toSet ((x1, b1, c1) #⇩Γ Γ')[x::=v]⇩Γ⇩v›*)\n using True (*‹(x', b', c') = (x1, b1, c1)›*) by auto\n next\n (*goal: ‹(x', b', c') ≠ (x1, b1, c1) ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet ((x1, b1, c1) #⇩Γ Γ')[x::=v]⇩Γ⇩v›*)\n case False (*‹(x', b', c') ≠ (x1, b1, c1)›*)\n have \"x1≠x\"\n using fresh_def (*‹?a ♯ ?x ≡ ?a ∉ supp ?x›*) fresh_GCons (*‹?a ♯ ?x #⇩Γ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) fresh_Pair (*‹?a ♯ (?x, ?y) = (?a ♯ ?x ∧ ?a ♯ ?y)›*) supp_at_base (*‹supp ?a = {atom ?a}›*) GCons (*‹⟦x' ≠ x; (x', b', c') ∈ toSet Γ'; atom x ∉ atom_dom Γ'⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v› ‹x' ≠ x› ‹(x'::x, b'::b, c'::c) ∈ toSet ((x1::x, b1::b, c1::c) #⇩Γ (Γ'::Γ))› ‹atom x ∉ atom_dom ((x1, b1, c1) #⇩Γ Γ')›*) fresh_dom_free2 (*‹atom ?x ∉ atom_dom ?Γ ⟹ (?x, ?b, ?c) ∉ toSet ?Γ›*) by auto\n hence \"(x', b', c') ∈ toSet Γ'\"\n using GCons (*‹⟦x' ≠ x; (x', b', c') ∈ toSet Γ'; atom x ∉ atom_dom Γ'⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v› ‹x' ≠ x› ‹(x'::x, b'::b, c'::c) ∈ toSet ((x1::x, b1::b, c1::c) #⇩Γ (Γ'::Γ))› ‹atom x ∉ atom_dom ((x1, b1, c1) #⇩Γ Γ')›*) False (*‹(x', b', c') ≠ (x1, b1, c1)›*) toSet.simps (*‹toSet GNil = {}› ‹toSet (?xbc #⇩Γ ?G) = {?xbc} ∪ toSet ?G›*) by auto\n moreover have \"atom x ∉ atom_dom Γ'\"\n using fresh_GCons (*‹?a ♯ ?x #⇩Γ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) GCons (*‹⟦x' ≠ x; (x', b', c') ∈ toSet Γ'; atom x ∉ atom_dom Γ'⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v› ‹x' ≠ x› ‹(x', b', c') ∈ toSet ((x1, b1, c1) #⇩Γ Γ')› ‹atom x ∉ atom_dom ((x1, b1, c1) #⇩Γ Γ')›*) dom.simps (*‹Syntax.dom ?Γ = fst ` toSet ?Γ›*) toSet.simps (*‹toSet GNil = {}› ‹toSet (?xbc #⇩Γ ?G) = {?xbc} ∪ toSet ?G›*) by simp\n ultimately have \"(x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v\"\n using GCons (*‹⟦x' ≠ x; (x', b', c') ∈ toSet Γ'; atom x ∉ atom_dom Γ'⟧ ⟹ (x', b', c'[x::=v]⇩c⇩v) ∈ toSet Γ'[x::=v]⇩Γ⇩v› ‹(x'::x) ≠ (x::x)› ‹(x'::x, b'::b, c'::c) ∈ toSet ((x1::x, b1::b, c1::c) #⇩Γ (Γ'::Γ))› ‹atom (x::x) ∉ atom_dom ((x1::x, b1::b, c1::c) #⇩Γ (Γ'::Γ))›*) by auto\n hence \"(x', b', c'[x::=v]⇩c⇩v) ∈ toSet ((x1, b1, c1[x::=v]⇩c⇩v) #⇩Γ (Γ'[x::=v]⇩Γ⇩v))\"\n by auto\n then show \"?thesis\"\n (*goal: ‹(x', b', c'[x::=v]⇩c⇩v) ∈ toSet ((x1, b1, c1) #⇩Γ Γ')[x::=v]⇩Γ⇩v›*)\n using subst_gv.simps (*‹GNil[?x::=?v]⇩Γ⇩v = GNil› ‹((?y, ?b, ?c) #⇩Γ ?Γ)[?x::=?v]⇩Γ⇩v = (if ?x = ?y then ?Γ else (?y, ?b, ?c[?x::=?v]⇩c⇩v) #⇩Γ ?Γ[?x::=?v]⇩Γ⇩v)›*) ‹x1≠x› (*‹x1 ≠ x›*) by auto\n qed\n qed\nqed\n\nlemma fresh_subst_gv_if:\n fixes j::atom and i::x and x::v and t::Γ\n assumes \"j ♯ t ∧ j ♯ x\" \n shows \"(j ♯ subst_gv t i x)\"\n using assms (*‹j ♯ t ∧ j ♯ x›*) proof (induct t rule: Γ_induct (*‹⟦(?P::Γ ⇒ bool) GNil; ⋀(x::x) (b::b) (c::c) Γ'::Γ. ?P Γ' ⟹ ?P ((x, b, c) #⇩Γ Γ')⟧ ⟹ ?P (?Γ::Γ)›*))\n (*goals:\n 1. ‹j ♯ GNil ∧ j ♯ x ⟹ j ♯ GNil[i::=x]⇩Γ⇩v›\n 2. ‹⋀xa b c Γ'. ⟦j ♯ Γ' ∧ j ♯ x ⟹ j ♯ Γ'[i::=x]⇩Γ⇩v; j ♯ (xa, b, c) #⇩Γ Γ' ∧ j ♯ x⟧ ⟹ j ♯ ((xa, b, c) #⇩Γ Γ')[i::=x]⇩Γ⇩v›*)\n case GNil (*‹j ♯ GNil ∧ j ♯ x›*)\n then show \"?case\"\n (*goal: ‹(j::atom) ♯ GNil[(i::x)::=(x::v)]⇩Γ⇩v›*)\n using subst_gv.simps (*‹GNil[?x::=?v]⇩Γ⇩v = GNil› ‹((?y, ?b, ?c) #⇩Γ ?Γ)[?x::=?v]⇩Γ⇩v = (if ?x = ?y then ?Γ else (?y, ?b, ?c[?x::=?v]⇩c⇩v) #⇩Γ ?Γ[?x::=?v]⇩Γ⇩v)›*) fresh_GNil (*‹(?a::atom) ♯ GNil›*) by auto\n next\n (*goal: ‹⋀xa b c Γ'. ⟦j ♯ Γ' ∧ j ♯ x ⟹ j ♯ Γ'[i::=x]⇩Γ⇩v; j ♯ (xa, b, c) #⇩Γ Γ' ∧ j ♯ x⟧ ⟹ j ♯ ((xa, b, c) #⇩Γ Γ')[i::=x]⇩Γ⇩v›*)\n case (GCons x' b' c' Γ') (*‹j ♯ Γ' ∧ j ♯ x ⟹ j ♯ Γ'[i::=x]⇩Γ⇩v› ‹j ♯ (x', b', c') #⇩Γ Γ' ∧ j ♯ x›*)\n then show \"?case\"\n (*goal: ‹j ♯ ((x', b', c') #⇩Γ Γ')[i::=x]⇩Γ⇩v›*)\n unfolding subst_gv.simps\n (*goal: ‹j ♯ (if i = x' then Γ' else (x', b', c'[i::=x]⇩c⇩v) #⇩Γ Γ'[i::=x]⇩Γ⇩v)›*)\n using fresh_GCons (*‹?a ♯ ?x #⇩Γ ?xs = (?a ♯ ?x ∧ ?a ♯ ?xs)›*) fresh_subst_cv_if (*‹?j ♯ ?A[?i::=?x]⇩c⇩v = (atom ?i ♯ ?A ∧ ?j ♯ ?A ∨ ?j ♯ ?x ∧ (?j ♯ ?A ∨ ?j = atom ?i))›*) by auto\n qed\n\nsection ‹Lookup›\n\nlemma set_GConsD: \"y ∈ toSet (x #⇩Γ xs) ⟹ y=x ∨ y ∈ toSet xs\"\n by auto\n\nlemma subst_g_assoc_cons:\n assumes \"x ≠ x'\"\n shows \"(((x', b', c') #⇩Γ Γ')[x::=v]⇩Γ⇩v @ G) = ((x', b', c'[x::=v]⇩c⇩v) #⇩Γ ((Γ'[x::=v]⇩Γ⇩v) @ G))\"\n using subst_gv.simps (*‹GNil[(?x::x)::=(?v::v)]⇩Γ⇩v = GNil› ‹((?y, ?b, ?c) #⇩Γ ?Γ)[?x::=?v]⇩Γ⇩v = (if ?x = ?y then ?Γ else (?y, ?b, ?c[?x::=?v]⇩c⇩v) #⇩Γ ?Γ[?x::=?v]⇩Γ⇩v)›*) append_g.simps (*‹GNil @ ?g = ?g› ‹(?xbc #⇩Γ ?g1.0) @ ?g2.0 = ?xbc #⇩Γ ?g1.0 @ ?g2.0›*) assms (*‹(x::x) ≠ (x'::x)›*) by auto\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/MiniSail/IVSubst.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"d8c0cb0bd2b647f2575563f314e8ee4041234a9f8b8135e0366eabd1d4a10cb2\"\n}"}}},{"rowIdx":148,"cells":{"text":{"kind":"string","value":"(* Title: variants/d_fwdrreqs/Aodv_Data.thy\n License: BSD 2-Clause. See LICENSE.\n Author: Timothy Bourke, Inria\n*)\n\nsection \"Predicates and functions used in the AODV model\"\n\ntheory D_Aodv_Data\nimports D_Fwdrreqs\nbegin\n\nsubsection \"Sequence Numbers\"\n\ntext ‹Sequence numbers approximate the relative freshness of routing information.›\n\ndefinition inc :: \"sqn ⇒ sqn\"\n where \"inc sn ≡ if sn = 0 then sn else sn + 1\"\n\nlemma less_than_inc [simp]: \"x ≤ inc x\"\n unfolding inc_def\n (*goal: ‹x ≤ (if x = 0 then x else x + 1)›*)\n by simp\n\nlemma inc_minus_suc_0 [simp]:\n \"inc x - Suc 0 = x\"\n unfolding inc_def\n (*goal: ‹(if x = 0 then x else x + 1) - Suc 0 = x›*)\n by simp\n\nlemma inc_never_one' [simp, intro]: \"inc x ≠ Suc 0\"\n unfolding inc_def\n (*goal: ‹(if x = 0 then x else x + 1) ≠ Suc 0›*)\n by simp\n\nlemma inc_never_one [simp, intro]: \"inc x ≠ 1\"\n by simp\n\nsubsection \"Modelling Routes\"\n\ntext ‹\n A route is a 6-tuple, @{term \"(dsn, dsk, flag, hops, nhip, pre)\"} where\n @{term dsn} is the `destination sequence number', @{term dsk} is the\n `destination-sequence-number status', @{term flag} is the route status,\n @{term hops} is the number of hops to the destination, @{term nhip} is the\n next hop toward the destination, and @{term pre} is the set of `precursor nodes'--those\n interested in hearing about changes to the route.\n›\n\ntype_synonym r = \"sqn × k × f × nat × ip × ip set\"\n\ndefinition proj2 :: \"r ⇒ sqn\" (\"π₂\")\n where \"π₂ ≡ λ(dsn, _, _, _, _, _). dsn\"\n\ndefinition proj3 :: \"r ⇒ k\" (\"π₃\")\n where \"π₃ ≡ λ(_, dsk, _, _, _, _). dsk\"\n\ndefinition proj4 :: \"r ⇒ f\" (\"π₄\")\n where \"π₄ ≡ λ(_, _, flag, _, _, _). flag\"\n\ndefinition proj5 :: \"r ⇒ nat\" (\"π₅\")\n where \"π₅ ≡ λ(_, _, _, hops, _, _). hops\"\n\ndefinition proj6 :: \"r ⇒ ip\" (\"π₆\")\n where \"π₆ ≡ λ(_, _, _, _, nhip, _). nhip\"\n\ndefinition proj7 :: \"r ⇒ ip set\" (\"π₇\")\n where \"π₇ ≡ λ(_, _, _, _, _, pre). pre\"\n\nlemma projs [simp]:\n \"π₂(dsn, dsk, flag, hops, nhip, pre) = dsn\"\n \"π₃(dsn, dsk, flag, hops, nhip, pre) = dsk\"\n \"π₄(dsn, dsk, flag, hops, nhip, pre) = flag\"\n \"π₅(dsn, dsk, flag, hops, nhip, pre) = hops\"\n \"π₆(dsn, dsk, flag, hops, nhip, pre) = nhip\"\n \"π₇(dsn, dsk, flag, hops, nhip, pre) = pre\"\n \n (*goals:\n 1. ‹π₂ (dsn, dsk, flag, hops, nhip, pre) = dsn›\n 2. ‹π₃ (dsn, dsk, flag, hops, nhip, pre) = dsk›\n 3. ‹π₄ (dsn, dsk, flag, hops, nhip, pre) = flag›\n 4. ‹π₅ (dsn, dsk, flag, hops, nhip, pre) = hops›\n 5. ‹π₆ (dsn, dsk, flag, hops, nhip, pre) = nhip›\n 6. ‹π₇ (dsn, dsk, flag, hops, nhip, pre) = pre›\n discuss goal 1*)\n apply (clarsimp simp: proj2_def (*‹π₂ ≡ λ(dsn, uu_, uu_, uu_, uu_, uu_). dsn›*) proj3_def (*‹π₃ ≡ λ(uu_, dsk, uu_, uu_, uu_, uu_). dsk›*) proj4_def (*‹π₄ ≡ λ(uu_, uu_, flag, uu_, uu_, uu_). flag›*) proj5_def (*‹π₅ ≡ λ(uu_, uu_, uu_, hops, uu_, uu_). hops›*) proj6_def (*‹π₆ ≡ λ(uu_, uu_, uu_, uu_, nhip, uu_). nhip›*) proj7_def (*‹π₇ ≡ λ(uu_, uu_, uu_, uu_, uu_, pre). pre›*))\n (*discuss goal 2*)\n apply (clarsimp simp: proj2_def (*‹π₂ ≡ λ(dsn, uu_, uu_, uu_, uu_, uu_). dsn›*) proj3_def (*‹π₃ ≡ λ(uu_, dsk, uu_, uu_, uu_, uu_). dsk›*) proj4_def (*‹π₄ ≡ λ(uu_, uu_, flag, uu_, uu_, uu_). flag›*) proj5_def (*‹π₅ ≡ λ(uu_, uu_, uu_, hops, uu_, uu_). hops›*) proj6_def (*‹π₆ ≡ λ(uu_, uu_, uu_, uu_, nhip, uu_). nhip›*) proj7_def (*‹π₇ ≡ λ(uu_, uu_, uu_, uu_, uu_, pre). pre›*))\n (*discuss goal 3*)\n apply (clarsimp simp: proj2_def (*‹π₂ ≡ λ(dsn, uu_, uu_, uu_, uu_, uu_). dsn›*) proj3_def (*‹π₃ ≡ λ(uu_, dsk, uu_, uu_, uu_, uu_). dsk›*) proj4_def (*‹π₄ ≡ λ(uu_, uu_, flag, uu_, uu_, uu_). flag›*) proj5_def (*‹π₅ ≡ λ(uu_, uu_, uu_, hops, uu_, uu_). hops›*) proj6_def (*‹π₆ ≡ λ(uu_, uu_, uu_, uu_, nhip, uu_). nhip›*) proj7_def (*‹π₇ ≡ λ(uu_, uu_, uu_, uu_, uu_, pre). pre›*))\n (*discuss goal 4*)\n apply (clarsimp simp: proj2_def (*‹π₂ ≡ λ(dsn, uu_, uu_, uu_, uu_, uu_). dsn›*) proj3_def (*‹π₃ ≡ λ(uu_, dsk, uu_, uu_, uu_, uu_). dsk›*) proj4_def (*‹π₄ ≡ λ(uu_, uu_, flag, uu_, uu_, uu_). flag›*) proj5_def (*‹π₅ ≡ λ(uu_, uu_, uu_, hops, uu_, uu_). hops›*) proj6_def (*‹π₆ ≡ λ(uu_, uu_, uu_, uu_, nhip, uu_). nhip›*) proj7_def (*‹π₇ ≡ λ(uu_, uu_, uu_, uu_, uu_, pre). pre›*))\n (*discuss goal 5*)\n apply (clarsimp simp: proj2_def (*‹π₂ ≡ λ(dsn, uu_, uu_, uu_, uu_, uu_). dsn›*) proj3_def (*‹π₃ ≡ λ(uu_, dsk, uu_, uu_, uu_, uu_). dsk›*) proj4_def (*‹π₄ ≡ λ(uu_, uu_, flag, uu_, uu_, uu_). flag›*) proj5_def (*‹π₅ ≡ λ(uu_, uu_, uu_, hops, uu_, uu_). hops›*) proj6_def (*‹π₆ ≡ λ(uu_, uu_, uu_, uu_, nhip, uu_). nhip›*) proj7_def (*‹π₇ ≡ λ(uu_, uu_, uu_, uu_, uu_, pre). pre›*))\n (*discuss goal 6*)\n apply (clarsimp simp: proj2_def (*‹π₂ ≡ λ(dsn, uu_, uu_, uu_, uu_, uu_). dsn›*) proj3_def (*‹π₃ ≡ λ(uu_, dsk, uu_, uu_, uu_, uu_). dsk›*) proj4_def (*‹π₄ ≡ λ(uu_, uu_, flag, uu_, uu_, uu_). flag›*) proj5_def (*‹π₅ ≡ λ(uu_, uu_, uu_, hops, uu_, uu_). hops›*) proj6_def (*‹π₆ ≡ λ(uu_, uu_, uu_, uu_, nhip, uu_). nhip›*) proj7_def (*‹π₇ ≡ λ(uu_, uu_, uu_, uu_, uu_, pre). pre›*))\n (*proven 6 subgoals*) .\n\nlemma proj3_pred [intro]: \"⟦ P kno; P unk ⟧ ⟹ P (π₃ x)\"\n apply (rule k.induct (*‹⟦?P kno; ?P unk⟧ ⟹ ?P ?k›*))\n (*goals:\n 1. ‹⟦(P::k ⇒ bool) kno; P unk⟧ ⟹ P kno›\n 2. ‹⟦(P::k ⇒ bool) kno; P unk⟧ ⟹ P unk›\n discuss goal 1*)\n apply ((assumption)[1])\n (*discuss goal 2*) .\n (*proven 2 subgoals*)\n\nlemma proj4_pred [intro]: \"⟦ P val; P inv ⟧ ⟹ P (π₄ x)\"\n apply (rule f.induct (*‹⟦(?P::f ⇒ bool) val; ?P Aodv_Basic.inv⟧ ⟹ ?P (?f::f)›*))\n (*goals:\n 1. ‹⟦P val; P Aodv_Basic.inv⟧ ⟹ P val›\n 2. ‹⟦P val; P Aodv_Basic.inv⟧ ⟹ P Aodv_Basic.inv›\n discuss goal 1*)\n apply ((assumption)[1])\n (*discuss goal 2*) .\n (*proven 2 subgoals*)\n\nlemma proj6_pair_snd [simp]:\n fixes dsn' r\n shows \"π₆ (dsn', snd (r)) = π₆(r)\"\n apply (cases r)\n (*goal: ‹π₆ (dsn'::nat, snd (r::nat × k × f × nat × nat × nat set)) = π₆ r›*)\n by simp\n\nsubsection \"Routing Tables\"\n\ntext ‹Routing tables map ip addresses to route entries.›\n\ntype_synonym rt = \"ip ⇀ r\"\n\nsyntax\n \"_Sigma_route\" :: \"rt ⇒ ip ⇀ r\" (\"σ⇘route⇙'(_, _')\")\n\ntranslations\n \"σ⇘route⇙(rt, dip)\" => \"rt dip\" \n\ndefinition sqn :: \"rt ⇒ ip ⇒ sqn\"\n where \"sqn rt dip ≡ case σ⇘route⇙(rt, dip) of Some r ⇒ π₂(r) | None ⇒ 0\"\n\ndefinition sqnf :: \"rt ⇒ ip ⇒ k\"\n where \"sqnf rt dip ≡ case σ⇘route⇙(rt, dip) of Some r ⇒ π₃(r) | None ⇒ unk\"\n\nabbreviation flag :: \"rt ⇒ ip ⇀ f\"\n where \"flag rt dip ≡ map_option π₄ (σ⇘route⇙(rt, dip))\"\n\nabbreviation dhops :: \"rt ⇒ ip ⇀ nat\"\n where \"dhops rt dip ≡ map_option π₅ (σ⇘route⇙(rt, dip))\"\n\nabbreviation nhop :: \"rt ⇒ ip ⇀ ip\"\n where \"nhop rt dip ≡ map_option π₆ (σ⇘route⇙(rt, dip))\"\n\nabbreviation precs :: \"rt ⇒ ip ⇀ ip set\"\n where \"precs rt dip ≡ map_option π₇ (σ⇘route⇙(rt, dip))\"\n\ndefinition vD :: \"rt ⇒ ip set\"\n where \"vD rt ≡ {dip. flag rt dip = Some val}\"\n\ndefinition iD :: \"rt ⇒ ip set\"\n where \"iD rt ≡ {dip. flag rt dip = Some inv}\"\n\ndefinition kD :: \"rt ⇒ ip set\"\n where \"kD rt ≡ {dip. rt dip ≠ None}\"\n\nlemma kD_is_vD_and_iD: \"kD rt = vD rt ∪ iD rt\"\n unfolding kD_def vD_def iD_def\n (*goal: ‹{dip. rt dip ≠ None} = {dip. flag rt dip = Some val} ∪ {dip. flag rt dip = Some Aodv_Basic.inv}›*)\n by auto\n\nlemma vD_iD_gives_kD [simp]:\n \"⋀ip rt. ip ∈ vD rt ⟹ ip ∈ kD rt\"\n \"⋀ip rt. ip ∈ iD rt ⟹ ip ∈ kD rt\"\n unfolding kD_is_vD_and_iD\n (*goals:\n 1. ‹⋀ip rt. ip ∈ vD rt ⟹ ip ∈ vD rt ∪ iD rt›\n 2. ‹⋀ip rt. ip ∈ iD rt ⟹ ip ∈ vD rt ∪ iD rt›*)\n (*goals:\n 1. ‹⋀ip rt. ip ∈ vD rt ⟹ ip ∈ vD rt ∪ iD rt›\n 2. ‹⋀ip rt. ip ∈ iD rt ⟹ ip ∈ vD rt ∪ iD rt›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma kD_Some [dest]:\n fixes dip rt\n assumes \"dip ∈ kD rt\"\n shows \"∃dsn dsk flag hops nhip pre.\n σ⇘route⇙(rt, dip) = Some (dsn, dsk, flag, hops, nhip, pre)\"\n using assms (*‹dip ∈ kD rt›*) unfolding kD_def\n (*goal: ‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*)\n by simp\n\nlemma kD_None [dest]:\n fixes dip rt\n assumes \"dip ∉ kD rt\"\n shows \"σ⇘route⇙(rt, dip) = None\"\n using assms (*‹dip ∉ kD rt›*) unfolding kD_def\n (*goal: ‹rt dip = None›*)\n by (metis (mono_tags) mem_Collect_eq (*‹((?a::?'a) ∈ Collect (?P::?'a ⇒ bool)) = ?P ?a›*))\n\nlemma vD_Some [dest]:\n fixes dip rt\n assumes \"dip ∈ vD rt\"\n shows \"∃dsn dsk hops nhip pre.\n σ⇘route⇙(rt, dip) = Some (dsn, dsk, val, hops, nhip, pre)\"\n using assms (*‹dip ∈ vD rt›*) unfolding vD_def\n (*goal: ‹∃(dsn::nat) (dsk::k) (hops::nat) (nhip::nat) pre::nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) = Some (dsn, dsk, val, hops, nhip, pre)›*)\n by simp\n\nlemma vD_empty [simp]: \"vD Map.empty = {}\"\n unfolding vD_def\n (*goal: ‹{dip::nat. map_option π₄ None = Some val} = {}›*)\n by simp\n\nlemma iD_Some [dest]:\n fixes dip rt\n assumes \"dip ∈ iD rt\"\n shows \"∃dsn dsk hops nhip pre.\n σ⇘route⇙(rt, dip) = Some (dsn, dsk, inv, hops, nhip, pre)\"\n using assms (*‹dip ∈ iD rt›*) unfolding iD_def\n (*goal: ‹∃dsn dsk hops nhip pre. rt dip = Some (dsn, dsk, Aodv_Basic.inv, hops, nhip, pre)›*)\n by simp\n\nlemma val_is_vD [elim]:\n fixes ip rt\n assumes \"ip∈kD(rt)\"\n and \"the (flag rt ip) = val\"\n shows \"ip∈vD(rt)\"\n using assms (*‹ip ∈ kD rt› ‹the (flag rt ip) = val›*) unfolding vD_def\n (*goal: ‹ip ∈ {dip. flag rt dip = Some val}›*)\n by auto\n\nlemma inv_is_iD [elim]:\n fixes ip rt\n assumes \"ip∈kD(rt)\"\n and \"the (flag rt ip) = inv\"\n shows \"ip∈iD(rt)\"\n using assms (*‹ip ∈ kD rt› ‹the (flag rt ip) = Aodv_Basic.inv›*) unfolding iD_def\n (*goal: ‹ip ∈ {dip. flag rt dip = Some Aodv_Basic.inv}›*)\n by auto\n\nlemma iD_flag_is_inv [elim, simp]:\n fixes ip rt\n assumes \"ip∈iD(rt)\"\n shows \"the (flag rt ip) = inv\"\n proof (-)\n (*goal: ‹the (flag rt ip) = Aodv_Basic.inv›*)\n from ‹ip∈iD(rt)› (*‹ip ∈ iD rt›*) have \"ip∈kD(rt)\"\n by auto\n with assms (*‹ip ∈ iD rt›*) show \"?thesis\"\n (*goal: ‹the (flag rt ip) = Aodv_Basic.inv›*)\n unfolding iD_def\n (*goal: ‹the (flag rt ip) = Aodv_Basic.inv›*)\n by auto\n qed\n\nlemma kD_but_not_vD_is_iD [elim]:\n fixes ip rt\n assumes \"ip∈kD(rt)\"\n and \"ip∉vD(rt)\"\n shows \"ip∈iD(rt)\"\n proof (-)\n (*goal: ‹ip ∈ iD rt›*)\n from ‹ip∈kD(rt)› (*‹ip ∈ kD rt›*) obtain dsn and dsk and f and hops and nhop and pre where rtip: \"rt ip = Some (dsn, dsk, f, hops, nhop, pre)\"\n (*goal: ‹(⋀dsn dsk f hops nhop pre. rt ip = Some (dsn, dsk, f, hops, nhop, pre) ⟹ thesis) ⟹ thesis›*)\n by (metis kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n from ‹ip∉vD(rt)› (*‹ip ∉ vD rt›*) have \"f ≠ val\"\n proof (rule contrapos_nn (*‹⟦¬ ?Q; ?P ⟹ ?Q⟧ ⟹ ¬ ?P›*))\n (*goal: ‹f = val ⟹ ip ∈ vD rt›*)\n assume \"f = val\" (*‹(f::f) = val›*)\n with rtip (*‹rt ip = Some (dsn, dsk, f, hops, nhop, pre)›*) have \"the (flag rt ip) = val\"\n by simp\n with ‹ip∈kD(rt)› (*‹ip ∈ kD rt›*) show \"ip∈vD(rt)\"\n by standard\n qed\n with rtip (*‹rt ip = Some (dsn, dsk, f, hops, nhop, pre)›*) have \"the (flag rt ip)= inv\"\n by simp\n with ‹ip∈kD(rt)› (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)›*) show \"ip∈iD(rt)\"\n by standard\n qed\n\nlemma vD_or_iD [elim]:\n fixes ip rt\n assumes \"ip∈kD(rt)\"\n and \"ip∈vD(rt) ⟹ P rt ip\"\n and \"ip∈iD(rt) ⟹ P rt ip\"\n shows \"P rt ip\"\n proof (-)\n (*goal: ‹(P::(nat ⇒ (nat × k × f × nat × nat × nat set) option) ⇒ nat ⇒ bool) (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat)›*)\n from ‹ip∈kD(rt)› (*‹ip ∈ kD rt›*) have \"ip∈vD(rt) ∪ iD(rt)\"\n by (simp add: kD_is_vD_and_iD (*‹kD (?rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) = vD ?rt ∪ iD ?rt›*))\n thus \"?thesis\"\n (*goal: ‹P rt ip›*)\n by (auto elim: assms( (*‹ip ∈ vD rt ⟹ P rt ip› ‹ip ∈ iD rt ⟹ P rt ip›*) 2-3))\n qed\n\nlemma proj5_eq_dhops: \"⋀dip rt. dip∈kD(rt) ⟹ π₅(the (rt dip)) = the (dhops rt dip)\"\n unfolding sqn_def\n (*goal: ‹⋀(dip::nat) rt::nat ⇒ (nat × k × f × nat × nat × nat set) option. dip ∈ kD rt ⟹ π₅ (the (rt dip)) = the (dhops rt dip)›*)\n apply (drule kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n (*goal: ‹⋀dip rt. dip ∈ kD rt ⟹ π₅ (the (rt dip)) = the (dhops rt dip)›*)\n by clarsimp\n\nlemma proj4_eq_flag: \"⋀dip rt. dip∈kD(rt) ⟹ π₄(the (rt dip)) = the (flag rt dip)\"\n unfolding sqn_def\n (*goal: ‹⋀(dip::nat) rt::nat ⇒ (nat × k × f × nat × nat × nat set) option. dip ∈ kD rt ⟹ π₄ (the (rt dip)) = the (flag rt dip)›*)\n apply (drule kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n (*goal: ‹⋀dip rt. dip ∈ kD rt ⟹ π₄ (the (rt dip)) = the (flag rt dip)›*)\n by clarsimp\n\nlemma proj2_eq_sqn: \"⋀dip rt. dip∈kD(rt) ⟹ π₂(the (rt dip)) = sqn rt dip\"\n unfolding sqn_def\n (*goal: ‹⋀dip rt. dip ∈ kD rt ⟹ π₂ (the (rt dip)) = (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r)›*)\n apply (drule kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n (*goal: ‹⋀(dip::nat) rt::nat ⇒ (nat × k × f × nat × nat × nat set) option. dip ∈ kD rt ⟹ π₂ (the (rt dip)) = (case rt dip of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₂ r)›*)\n by clarsimp\n\nlemma kD_sqnf_is_proj3 [simp]:\n \"⋀ip rt. ip∈kD(rt) ⟹ sqnf rt ip = π₃(the (rt ip))\"\n unfolding sqnf_def\n (*goal: ‹⋀(ip::nat) rt::nat ⇒ (nat × k × f × nat × nat × nat set) option. ip ∈ kD rt ⟹ (case rt ip of None ⇒ unk | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₃ r) = π₃ (the (rt ip))›*)\n by auto\n\nlemma vD_flag_val [simp]:\n \"⋀dip rt. dip ∈ vD (rt) ⟹ the (flag rt dip) = val\"\n unfolding vD_def\n (*goal: ‹⋀dip rt. dip ∈ {dip. flag rt dip = Some val} ⟹ the (flag rt dip) = val›*)\n by clarsimp\n\nlemma kD_update [simp]:\n \"⋀rt nip v. kD (rt(nip ↦ v)) = insert nip (kD rt)\"\n unfolding kD_def\n (*goal: ‹⋀(rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (nip::nat) v::nat × k × f × nat × nat × nat set. {dip::nat. (rt(nip ↦ v)) dip ≠ None} = insert nip {dip::nat. rt dip ≠ None}›*)\n by auto\n\nlemma kD_empty [simp]: \"kD Map.empty = {}\"\n unfolding kD_def\n (*goal: ‹{dip::nat. None ≠ None} = {}›*)\n by simp\n\nlemma ip_equal_or_known [elim]:\n fixes rt ip ip'\n assumes \"ip = ip' ∨ ip∈kD(rt)\"\n and \"ip = ip' ⟹ P rt ip ip'\"\n and \"⟦ ip ≠ ip'; ip∈kD(rt)⟧ ⟹ P rt ip ip'\"\n shows \"P rt ip ip'\"\n using assms (*‹ip = ip' ∨ ip ∈ kD rt› ‹ip = ip' ⟹ P rt ip ip'› ‹⟦ip ≠ ip'; ip ∈ kD rt⟧ ⟹ P rt ip ip'›*) by auto\n\nsubsection \"Updating Routing Tables\"\n\ntext ‹Routing table entries are modified through explicit functions.\n The properties of these functions are important in invariant proofs.›\n\nsubsubsection \"Updating Precursor Lists\"\n\ndefinition addpre :: \"r ⇒ ip set ⇒ r\"\n where \"addpre r npre ≡ let (dsn, dsk, flag, hops, nhip, pre) = r in\n (dsn, dsk, flag, hops, nhip, pre ∪ npre)\"\n\nlemma proj2_addpre:\n fixes v pre\n shows \"π₂(addpre v pre) = π₂(v)\"\n unfolding addpre_def\n (*goal: ‹π₂ (let (dsn, dsk, flag, hops, nhip, prea) = v in (dsn, dsk, flag, hops, nhip, prea ∪ pre)) = π₂ v›*)\n apply (cases v)\n (*goal: ‹π₂ (let (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, prea::nat set) = v::nat × k × f × nat × nat × nat set in (dsn, dsk, flag, hops, nhip, prea ∪ (pre::nat set))) = π₂ v›*)\n by simp\n\nlemma proj3_addpre:\n fixes v pre\n shows \"π₃(addpre v pre) = π₃(v)\"\n unfolding addpre_def\n (*goal: ‹π₃ (let (dsn, dsk, flag, hops, nhip, prea) = v in (dsn, dsk, flag, hops, nhip, prea ∪ pre)) = π₃ v›*)\n apply (cases v)\n (*goal: ‹π₃ (let (dsn, dsk, flag, hops, nhip, prea) = v in (dsn, dsk, flag, hops, nhip, prea ∪ pre)) = π₃ v›*)\n by simp\n\nlemma proj4_addpre:\n fixes v pre\n shows \"π₄(addpre v pre) = π₄(v)\"\n unfolding addpre_def\n (*goal: ‹π₄ (let (dsn, dsk, flag, hops, nhip, prea) = v in (dsn, dsk, flag, hops, nhip, prea ∪ pre)) = π₄ v›*)\n apply (cases v)\n (*goal: ‹π₄ (let (dsn, dsk, flag, hops, nhip, prea) = v in (dsn, dsk, flag, hops, nhip, prea ∪ pre)) = π₄ v›*)\n by simp\n\nlemma proj5_addpre:\n fixes v pre\n shows \"π₅(addpre v pre) = π₅(v)\"\n unfolding addpre_def\n (*goal: ‹π₅ (let (dsn, dsk, flag, hops, nhip, prea) = v in (dsn, dsk, flag, hops, nhip, prea ∪ pre)) = π₅ v›*)\n apply (cases v)\n (*goal: ‹π₅ (let (dsn, dsk, flag, hops, nhip, prea) = v in (dsn, dsk, flag, hops, nhip, prea ∪ pre)) = π₅ v›*)\n by simp\n\nlemma proj6_addpre:\n fixes dsn dsk flag hops nhip pre npre\n shows \"π₆(addpre v npre) = π₆(v)\"\n unfolding addpre_def\n (*goal: ‹π₆ (let (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set) = v::nat × k × f × nat × nat × nat set in (dsn, dsk, flag, hops, nhip, pre ∪ (npre::nat set))) = π₆ v›*)\n apply (cases v)\n (*goal: ‹π₆ (let (dsn, dsk, flag, hops, nhip, pre) = v in (dsn, dsk, flag, hops, nhip, pre ∪ npre)) = π₆ v›*)\n by simp\n\nlemma proj7_addpre:\n fixes dsn dsk flag hops nhip pre npre\n shows \"π₇(addpre v npre) = π₇(v) ∪ npre\"\n unfolding addpre_def\n (*goal: ‹π₇ (let (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set) = v::nat × k × f × nat × nat × nat set in (dsn, dsk, flag, hops, nhip, pre ∪ (npre::nat set))) = π₇ v ∪ npre›*)\n apply (cases v)\n (*goal: ‹π₇ (let (dsn, dsk, flag, hops, nhip, pre) = v in (dsn, dsk, flag, hops, nhip, pre ∪ npre)) = π₇ v ∪ npre›*)\n by simp\n\nlemma addpre_empty: \"addpre r {} = r\"\n unfolding addpre_def\n (*goal: ‹(let (dsn, dsk, flag, hops, nhip, pre) = r in (dsn, dsk, flag, hops, nhip, pre ∪ {})) = r›*)\n by simp\n\nlemma addpre_r:\n \"addpre (dsn, dsk, fl, hops, nhip, pre) npre = (dsn, dsk, fl, hops, nhip, pre ∪ npre)\"\n unfolding addpre_def\n (*goal: ‹(let (dsn, dsk, flag, hops, nhip, pre) = (dsn, dsk, fl, hops, nhip, pre) in (dsn, dsk, flag, hops, nhip, pre ∪ npre)) = (dsn, dsk, fl, hops, nhip, pre ∪ npre)›*)\n by simp\n\nlemmas addpre_simps [simp] = proj2_addpre proj3_addpre proj4_addpre proj5_addpre\n proj6_addpre proj7_addpre addpre_empty addpre_r\n\ndefinition addpreRT :: \"rt ⇒ ip ⇒ ip set ⇀ rt\"\n where \"addpreRT rt dip npre ≡\n map_option (λs. rt (dip ↦ addpre s npre)) (σ⇘route⇙(rt, dip))\"\n\nlemma snd_addpre [simp]:\n \"⋀dsn dsn' v pre. (dsn, snd(addpre (dsn', v) pre)) = addpre (dsn, v) pre\"\n unfolding addpre_def\n (*goal: ‹⋀dsn dsn' v pre. (dsn, snd (let (dsn, dsk, flag, hops, nhip, prea) = (dsn', v) in (dsn, dsk, flag, hops, nhip, prea ∪ pre))) = (let (dsn, dsk, flag, hops, nhip, prea) = (dsn, v) in (dsn, dsk, flag, hops, nhip, prea ∪ pre))›*)\n by clarsimp\n\nlemma proj2_addpreRT [simp]:\n fixes ip rt ip' npre\n assumes \"ip∈kD rt\"\n and \"ip'∈kD rt\"\n shows \"π₂(the (the (addpreRT rt ip' npre) ip)) = π₂(the (rt ip))\"\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt ip = Some (dsn, dsk, flag, hops, nhip, pre)› ‹∃(dsn::nat) (dsk::k) (flag::f) (hops::nat) (nhip::nat) pre::nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip'::nat) = Some (dsn, dsk, flag, hops, nhip, pre)›*) unfolding addpreRT_def\n (*goal: ‹π₂ (the (the (map_option (λs::nat × k × f × nat × nat × nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)(ip'::nat ↦ addpre s (npre::nat set))) (rt ip')) (ip::nat))) = π₂ (the (rt ip))›*)\n by clarsimp\n\nlemma proj3_addpreRT [simp]:\n fixes ip rt ip' npre\n assumes \"ip∈kD rt\"\n and \"ip'∈kD rt\"\n shows \"π₃(the (the (addpreRT rt ip' npre) ip)) = π₃(the (rt ip))\"\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt ip = Some (dsn, dsk, flag, hops, nhip, pre)› ‹∃dsn dsk flag hops nhip pre. rt ip' = Some (dsn, dsk, flag, hops, nhip, pre)›*) unfolding addpreRT_def\n (*goal: ‹π₃ (the (the (map_option (λs::nat × k × f × nat × nat × nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)(ip'::nat ↦ addpre s (npre::nat set))) (rt ip')) (ip::nat))) = π₃ (the (rt ip))›*)\n by clarsimp\n\nlemma proj5_addpreRT [simp]:\n \"⋀rt dip ip npre. dip∈kD(rt) ⟹ π₅(the (the (addpreRT rt dip npre) ip)) = π₅(the (rt ip))\"\n unfolding addpreRT_def\n (*goal: ‹⋀rt dip ip npre. dip ∈ kD rt ⟹ π₅ (the (the (map_option (λs. rt(dip ↦ addpre s npre)) (rt dip)) ip)) = π₅ (the (rt ip))›*)\n by auto\n\nlemma flag_addpreRT [simp]:\n fixes rt pre ip dip\n assumes \"dip ∈ kD rt\"\n shows \"flag (the (addpreRT rt dip pre)) ip = flag rt ip\"\n unfolding addpreRT_def\n (*goal: ‹flag (the (map_option (λs::nat × k × f × nat × nat × nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)(dip::nat ↦ addpre s (pre::nat set))) (rt dip))) (ip::nat) = flag rt ip›*)\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*) by clarsimp\n\n lemma kD_addpreRT [simp]:\n fixes rt dip npre\n assumes \"dip ∈ kD rt\"\n shows \"kD (the (addpreRT rt dip npre)) = kD rt\"\n unfolding kD_def addpreRT_def\n (*goal: ‹{dipa. the (map_option (λs. rt(dip ↦ addpre s npre)) (rt dip)) dipa ≠ None} = {dip. rt dip ≠ None}›*)\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*) apply clarsimp\n (*goal: ‹{dipa. the (map_option (λs. rt(dip ↦ addpre s npre)) (rt dip)) dipa ≠ None} = {dip. rt dip ≠ None}›*)\n by blast\n\nlemma vD_addpreRT [simp]:\n fixes rt dip npre\n assumes \"dip ∈ kD rt\"\n shows \"vD (the (addpreRT rt dip npre)) = vD rt\"\n unfolding vD_def addpreRT_def\n (*goal: ‹{dipa::nat. flag (the (map_option (λs::nat × k × f × nat × nat × nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)(dip::nat ↦ addpre s (npre::nat set))) (rt dip))) dipa = Some val} = {dip::nat. flag rt dip = Some val}›*)\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*) apply clarsimp\n (*goal: ‹{dipa. flag (the (map_option (λs. rt(dip ↦ addpre s npre)) (rt dip))) dipa = Some val} = {dip. flag rt dip = Some val}›*)\n by auto\n\nlemma iD_addpreRT [simp]:\n fixes rt dip npre\n assumes \"dip ∈ kD rt\"\n shows \"iD (the (addpreRT rt dip npre)) = iD rt\"\n unfolding iD_def addpreRT_def\n (*goal: ‹{dipa. flag (the (map_option (λs. rt(dip ↦ addpre s npre)) (rt dip))) dipa = Some Aodv_Basic.inv} = {dip. flag rt dip = Some Aodv_Basic.inv}›*)\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*) apply clarsimp\n (*goal: ‹{dipa. flag (the (map_option (λs. rt(dip ↦ addpre s npre)) (rt dip))) dipa = Some Aodv_Basic.inv} = {dip. flag rt dip = Some Aodv_Basic.inv}›*)\n by auto\n\nlemma nhop_addpreRT [simp]:\n fixes rt pre ip dip\n assumes \"dip ∈ kD rt\"\n shows \"nhop (the (addpreRT rt dip pre)) ip = nhop rt ip\"\n unfolding sqn_def addpreRT_def\n (*goal: ‹nhop (the (map_option (λs::nat × k × f × nat × nat × nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)(dip::nat ↦ addpre s (pre::nat set))) (rt dip))) (ip::nat) = nhop rt ip›*)\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*) by clarsimp\n\nlemma sqn_addpreRT [simp]:\n fixes rt pre ip dip\n assumes \"dip ∈ kD rt\"\n shows \"sqn (the (addpreRT rt dip pre)) ip = sqn rt ip\"\n unfolding sqn_def addpreRT_def\n (*goal: ‹(case the (map_option (λs. rt(dip ↦ addpre s pre)) (rt dip)) ip of None ⇒ 0 | Some r ⇒ π₂ r) = (case rt ip of None ⇒ 0 | Some r ⇒ π₂ r)›*)\n using assms[THEN kD_Some] (*‹∃(dsn::nat) (dsk::k) (flag::f) (hops::nat) (nhip::nat) pre::nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) = Some (dsn, dsk, flag, hops, nhip, pre)›*) by clarsimp\n\nlemma dhops_addpreRT [simp]:\n fixes rt pre ip dip\n assumes \"dip ∈ kD rt\"\n shows \"dhops (the (addpreRT rt dip pre)) ip = dhops rt ip\"\n unfolding addpreRT_def\n (*goal: ‹dhops (the (map_option (λs. rt(dip ↦ addpre s pre)) (rt dip))) ip = dhops rt ip›*)\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*) by clarsimp\n\nlemma sqnf_addpreRT [simp]:\n \"⋀ip dip. ip∈kD(rt ξ) ⟹ sqnf (the (addpreRT (rt ξ) ip npre)) dip = sqnf (rt ξ) dip\"\n unfolding sqnf_def addpreRT_def\n (*goal: ‹⋀ip dip. ip ∈ kD (rt ξ) ⟹ (case the (map_option (λs. (rt ξ)(ip ↦ addpre s npre)) (rt ξ ip)) dip of None ⇒ unk | Some r ⇒ π₃ r) = (case rt ξ dip of None ⇒ unk | Some r ⇒ π₃ r)›*)\n by auto\n\nsubsubsection \"Updating route entries\"\n\nlemma in_kD_case [simp]:\n fixes dip rt\n assumes \"dip ∈ kD(rt)\"\n shows \"(case rt dip of None ⇒ en | Some r ⇒ es r) = es (the (rt dip))\"\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre)›*) by auto\n\nlemma not_in_kD_case [simp]:\n fixes dip rt\n assumes \"dip ∉ kD(rt)\"\n shows \"(case rt dip of None ⇒ en | Some r ⇒ es r) = en\"\n using assms[THEN kD_None] (*‹rt dip = None›*) by auto\n\nlemma rt_Some_sqn [dest]:\n fixes rt and ip dsn dsk flag hops nhip pre\n assumes \"rt ip = Some (dsn, dsk, flag, hops, nhip, pre)\"\n shows \"sqn rt ip = dsn\"\n unfolding sqn_def\n (*goal: ‹(case rt ip of None ⇒ 0 | Some r ⇒ π₂ r) = dsn›*)\n using assms (*‹rt ip = Some (dsn, dsk, flag, hops, nhip, pre)›*) by simp\n\nlemma not_kD_sqn [simp]:\n fixes dip rt\n assumes \"dip ∉ kD(rt)\"\n shows \"sqn rt dip = 0\"\n using assms (*‹dip ∉ kD rt›*) unfolding sqn_def\n (*goal: ‹(case rt dip of None ⇒ 0 | Some r ⇒ π₂ r) = 0›*)\n by simp\n\ndefinition update_arg_wf :: \"r ⇒ bool\"\nwhere \"update_arg_wf r ≡ π₄(r) = val ∧\n (π₂(r) = 0) = (π₃(r) = unk) ∧\n (π₃(r) = unk ⟶ π₅(r) = 1)\"\n\nlemma update_arg_wf_gives_cases:\n \"⋀r. update_arg_wf r ⟹ (π₂(r) = 0) = (π₃(r) = unk)\"\n unfolding update_arg_wf_def\n (*goal: ‹⋀r. π₄ r = val ∧ (π₂ r = 0) = (π₃ r = unk) ∧ (π₃ r = unk ⟶ π₅ r = 1) ⟹ (π₂ r = 0) = (π₃ r = unk)›*)\n by simp\n\nlemma update_arg_wf_tuples [simp]:\n \"⋀nhip pre. update_arg_wf (0, unk, val, Suc 0, nhip, pre)\"\n \"⋀n hops nhip pre. update_arg_wf (Suc n, kno, val, hops, nhip, pre)\"\n unfolding update_arg_wf_def\n (*goals:\n 1. ‹⋀(nhip::nat) pre::nat set. π₄ (0::nat, unk, val, Suc (0::nat), nhip, pre) = val ∧ (π₂ (0::nat, unk, val, Suc (0::nat), nhip, pre) = (0::nat)) = (π₃ (0::nat, unk, val, Suc (0::nat), nhip, pre) = unk) ∧ (π₃ (0::nat, unk, val, Suc (0::nat), nhip, pre) = unk ⟶ π₅ (0::nat, unk, val, Suc (0::nat), nhip, pre) = (1::nat))›\n 2. ‹⋀(n::nat) (hops::nat) (nhip::nat) pre::nat set. π₄ (Suc n, kno, val, hops, nhip, pre) = val ∧ (π₂ (Suc n, kno, val, hops, nhip, pre) = (0::nat)) = (π₃ (Suc n, kno, val, hops, nhip, pre) = unk) ∧ (π₃ (Suc n, kno, val, hops, nhip, pre) = unk ⟶ π₅ (Suc n, kno, val, hops, nhip, pre) = (1::nat))›*)\n (*goals:\n 1. ‹⋀nhip pre. π₄ (0, unk, val, Suc 0, nhip, pre) = val ∧ (π₂ (0, unk, val, Suc 0, nhip, pre) = 0) = (π₃ (0, unk, val, Suc 0, nhip, pre) = unk) ∧ (π₃ (0, unk, val, Suc 0, nhip, pre) = unk ⟶ π₅ (0, unk, val, Suc 0, nhip, pre) = 1)›\n 2. ‹⋀n hops nhip pre. π₄ (Suc n, kno, val, hops, nhip, pre) = val ∧ (π₂ (Suc n, kno, val, hops, nhip, pre) = 0) = (π₃ (Suc n, kno, val, hops, nhip, pre) = unk) ∧ (π₃ (Suc n, kno, val, hops, nhip, pre) = unk ⟶ π₅ (Suc n, kno, val, hops, nhip, pre) = 1)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma update_arg_wf_tuples' [elim]:\n \"⋀n hops nhip pre. Suc 0 ≤ n ⟹ update_arg_wf (n, kno, val, hops, nhip, pre)\"\n unfolding update_arg_wf_def\n (*goal: ‹⋀(n::nat) (hops::nat) (nhip::nat) pre::nat set. Suc (0::nat) ≤ n ⟹ π₄ (n, kno, val, hops, nhip, pre) = val ∧ (π₂ (n, kno, val, hops, nhip, pre) = (0::nat)) = (π₃ (n, kno, val, hops, nhip, pre) = unk) ∧ (π₃ (n, kno, val, hops, nhip, pre) = unk ⟶ π₅ (n, kno, val, hops, nhip, pre) = (1::nat))›*)\n by auto\n\nlemma wf_r_cases [intro]:\n fixes P r\n assumes \"update_arg_wf r\"\n and c1: \"⋀nhip pre. P (0, unk, val, Suc 0, nhip, pre)\"\n and c2: \"⋀dsn hops nhip pre. dsn > 0 ⟹ P (dsn, kno, val, hops, nhip, pre)\"\n shows \"P r\"\n proof (-)\n (*goal: ‹P r›*)\n obtain dsn and dsk and flag and hops and nhip and pre where \"*\": \"r = (dsn, dsk, flag, hops, nhip, pre)\"\n (*goal: ‹(⋀dsn dsk flag hops nhip pre. r = (dsn, dsk, flag, hops, nhip, pre) ⟹ thesis) ⟹ thesis›*)\n by (cases r)\n with ‹update_arg_wf r› (*‹update_arg_wf (r::nat × k × f × nat × nat × nat set)›*) have wf1: \"flag = val\" and wf2: \"(dsn = 0) = (dsk = unk)\" and wf3: \"dsk = unk ⟶ (hops = 1)\"\n unfolding update_arg_wf_def\n (*goals:\n 1. ‹flag = val›\n 2. ‹(dsn = 0) = (dsk = unk)›\n 3. ‹dsk = unk ⟶ hops = 1›*)\n apply -\n (*goals:\n 1. ‹⟦π₄ r = val ∧ (π₂ r = 0) = (π₃ r = unk) ∧ (π₃ r = unk ⟶ π₅ r = 1); r = (dsn, dsk, flag, hops, nhip, pre)⟧ ⟹ flag = val›\n 2. ‹⟦π₄ r = val ∧ (π₂ r = 0) = (π₃ r = unk) ∧ (π₃ r = unk ⟶ π₅ r = 1); r = (dsn, dsk, flag, hops, nhip, pre)⟧ ⟹ (dsn = 0) = (dsk = unk)›\n 3. ‹⟦π₄ r = val ∧ (π₂ r = 0) = (π₃ r = unk) ∧ (π₃ r = unk ⟶ π₅ r = 1); r = (dsn, dsk, flag, hops, nhip, pre)⟧ ⟹ dsk = unk ⟶ hops = 1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n have \"P (dsn, dsk, flag, hops, nhip, pre)\"\n proof (cases dsk)\n (*goals:\n 1. ‹(dsk::k) = kno ⟹ (P::nat × k × f × nat × nat × nat set ⇒ bool) (dsn::nat, dsk, flag::f, hops::nat, nhip::nat, pre::nat set)›\n 2. ‹(dsk::k) = unk ⟹ (P::nat × k × f × nat × nat × nat set ⇒ bool) (dsn::nat, dsk, flag::f, hops::nat, nhip::nat, pre::nat set)›*)\n assume \"dsk = unk\" (*‹(dsk::k) = unk›*)\n moreover with wf2 (*‹(dsn = 0) = (dsk = unk)›*) wf3 (*‹dsk = unk ⟶ hops = 1›*) have \"dsn = 0\" and \"hops = Suc 0\"\n apply -\n (*goals:\n 1. ‹⟦((dsn::nat) = (0::nat)) = ((dsk::k) = unk); dsk = unk ⟶ (hops::nat) = (1::nat); dsk = unk⟧ ⟹ dsn = (0::nat)›\n 2. ‹⟦((dsn::nat) = (0::nat)) = ((dsk::k) = unk); dsk = unk ⟶ (hops::nat) = (1::nat); dsk = unk⟧ ⟹ hops = Suc (0::nat)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n ultimately show \"?thesis\"\n (*goal: ‹P (dsn, dsk, flag, hops, nhip, pre)›*)\n using ‹flag = val› (*‹flag = val›*) apply simp\n (*goal: ‹P (dsn, dsk, flag, hops, nhip, pre)›*)\n by (rule c1 (*‹P (0, unk, val, Suc 0, ?nhip, ?pre)›*))\n next\n (*goal: ‹dsk = kno ⟹ P (dsn, dsk, flag, hops, nhip, pre)›*)\n assume \"dsk = kno\" (*‹(dsk::k) = kno›*)\n moreover with wf2 (*‹(dsn = 0) = (dsk = unk)›*) have \"dsn > 0\"\n by simp\n ultimately show \"?thesis\"\n (*goal: ‹P (dsn, dsk, flag, hops, nhip, pre)›*)\n using ‹flag = val› (*‹flag = val›*) apply simp\n (*goal: ‹P (dsn, dsk, flag, hops, nhip, pre)›*)\n by (rule c2 (*‹0 < ?dsn ⟹ P (?dsn, kno, val, ?hops, ?nhip, ?pre)›*))\n qed\n with \"*\" (*‹r = (dsn, dsk, flag, hops, nhip, pre)›*) show \"P r\"\n by simp\n qed\n\ndefinition update :: \"rt ⇒ ip ⇒ r ⇒ rt\"\n where\n \"update rt ip r ≡\n case σ⇘route⇙(rt, ip) of\n None ⇒ rt (ip ↦ r)\n | Some s ⇒\n if π₂(s) < π₂(r) then rt (ip ↦ addpre r (π₇(s)))\n else if π₂(s) = π₂(r) ∧ (π₅(s) > π₅(r) ∨ π₄(s) = inv)\n then rt (ip ↦ addpre r (π₇(s)))\n else if π₃(r) = unk\n then rt (ip ↦ (π₂(s), snd (addpre r (π₇(s)))))\n else rt (ip ↦ addpre s (π₇(r)))\"\n\nlemma update_simps [simp]:\n fixes r s nrt nr nr' ns rt ip\n defines \"s ≡ the σ⇘route⇙(rt, ip)\"\n and \"nr ≡ addpre r (π₇(s))\"\n and \"nr' ≡ (π₂(s), π₃(nr), π₄(nr), π₅(nr), π₆(nr), π₇(nr))\"\n and \"ns ≡ addpre s (π₇(r))\"\n shows\n \"⟦ip ∉ kD(rt)⟧ ⟹ update rt ip r = rt (ip ↦ r)\"\n \"⟦ip ∈ kD(rt); sqn rt ip < π₂(r)⟧ ⟹ update rt ip r = rt (ip ↦ nr)\"\n \"⟦ip ∈ kD(rt); sqn rt ip = π₂(r);\n the (dhops rt ip) > π₅(r)⟧ ⟹ update rt ip r = rt (ip ↦ nr)\"\n \"⟦ip ∈ kD(rt); sqn rt ip = π₂(r);\n flag rt ip = Some inv⟧ ⟹ update rt ip r = rt (ip ↦ nr)\"\n \"⟦ip ∈ kD(rt); π₃(r) = unk; (π₂(r) = 0) = (π₃(r) = unk)⟧ ⟹ update rt ip r = rt (ip ↦ nr')\"\n \"⟦ip ∈ kD(rt); sqn rt ip ≥ π₂(r); π₃(r) = kno;\n sqn rt ip = π₂(r) ⟹ the (dhops rt ip) ≤ π₅(r) ∧ the (flag rt ip) = val ⟧\n ⟹ update rt ip r = rt (ip ↦ ns)\"\n proof (-)\n (*goals:\n 1. ‹ip ∉ kD rt ⟹ update rt ip r = rt(ip ↦ r)›\n 2. ‹⟦ip ∈ kD rt; sqn rt ip < π₂ r⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 3. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; π₅ r < the (dhops rt ip)⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 4. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; flag rt ip = Some Aodv_Basic.inv⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 5. ‹⟦ip ∈ kD rt; π₃ r = unk; (π₂ r = 0) = (π₃ r = unk)⟧ ⟹ update rt ip r = rt(ip ↦ nr')›\n 6. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ update rt ip r = rt(ip ↦ ns)›*)\n assume \"ip∉kD(rt)\" (*‹(ip::nat) ∉ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)›*)\n hence \"σ⇘route⇙(rt, ip) = None\"\n apply -\n (*goal: ‹rt ip = None›*)\n apply standard\n (*goals:\n 1. ‹ip ∉ kD rt ⟹ update_arg_wf ?r1›\n 2. ‹⋀nhip pre. ip ∉ kD rt ⟹ rt ip = None›\n 3. ‹⋀dsn hops nhip pre. ⟦ip ∉ kD rt; 0 < dsn⟧ ⟹ rt ip = None›\n discuss goal 1*)\n apply ((msorry)[1])\n (*discuss goal 2*)\n apply ((msorry)[1])\n (*discuss goal 3*)\n apply ((msorry)[1])\n (*proven 3 subgoals*) .\n thus \"update rt ip r = rt (ip ↦ r)\"\n unfolding update_def\n (*goal: ‹(case (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) of None ⇒ rt(ip ↦ r::nat × k × f × nat × nat × nat set) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r))) = rt(ip ↦ r)›*)\n by simp\n next\n (*goals:\n 1. ‹⟦ip ∈ kD rt; sqn rt ip < π₂ r⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 2. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; π₅ r < the (dhops rt ip)⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 3. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; flag rt ip = Some Aodv_Basic.inv⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 4. ‹⟦ip ∈ kD rt; π₃ r = unk; (π₂ r = 0) = (π₃ r = unk)⟧ ⟹ update rt ip r = rt(ip ↦ nr')›\n 5. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ update rt ip r = rt(ip ↦ ns)›*)\n assume \"ip ∈ kD(rt)\" and \"sqn rt ip < π₂(r)\" (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)› ‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) < π₂ (r::nat × k × f × nat × nat × nat set)›*)\n from this(1) (*‹ip ∈ kD rt›*) obtain dsn and dsk and fl and hops and nhip and pre where \"rt ip = Some (dsn, dsk, fl, hops, nhip, pre)\"\n (*goal: ‹(⋀dsn dsk fl hops nhip pre. rt ip = Some (dsn, dsk, fl, hops, nhip, pre) ⟹ thesis) ⟹ thesis›*)\n by (metis kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n with ‹sqn rt ip < π₂(r)› (*‹sqn rt ip < π₂ r›*) show \"update rt ip r = rt (ip ↦ nr)\"\n unfolding update_def nr_def s_def\n (*goal: ‹(case (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) of None ⇒ rt(ip ↦ r::nat × k × f × nat × nat × nat set) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r))) = rt(ip ↦ addpre r (π₇ (the (rt ip))))›*)\n by auto\n next\n (*goals:\n 1. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; π₅ r < the (dhops rt ip)⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 2. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; flag rt ip = Some Aodv_Basic.inv⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 3. ‹⟦ip ∈ kD rt; π₃ r = unk; (π₂ r = 0) = (π₃ r = unk)⟧ ⟹ update rt ip r = rt(ip ↦ nr')›\n 4. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ update rt ip r = rt(ip ↦ ns)›*)\n assume \"ip ∈ kD(rt)\" and \"sqn rt ip = π₂(r)\" and \"the (dhops rt ip) > π₅(r)\" (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)› ‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = π₂ (r::nat × k × f × nat × nat × nat set)› ‹π₅ (r::nat × k × f × nat × nat × nat set) < the (dhops (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat))›*)\n from this(1) (*‹ip ∈ kD rt›*) obtain dsn and dsk and fl and hops and nhip and pre where \"rt ip = Some (dsn, dsk, fl, hops, nhip, pre)\"\n (*goal: ‹(⋀(dsn::nat) (dsk::k) (fl::f) (hops::nat) (nhip::nat) pre::nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = Some (dsn, dsk, fl, hops, nhip, pre) ⟹ thesis::bool) ⟹ thesis›*)\n by (metis kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n with ‹sqn rt ip = π₂(r)› (*‹sqn rt ip = π₂ r›*) ‹the (dhops rt ip) > π₅(r)› (*‹π₅ r < the (dhops rt ip)›*) show \"update rt ip r = rt (ip ↦ nr)\"\n unfolding update_def nr_def s_def\n (*goal: ‹(case rt ip of None ⇒ rt(ip ↦ r) | Some s ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r))) = rt(ip ↦ addpre r (π₇ (the (rt ip))))›*)\n by auto\n next\n (*goals:\n 1. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; flag rt ip = Some Aodv_Basic.inv⟧ ⟹ update rt ip r = rt(ip ↦ nr)›\n 2. ‹⟦ip ∈ kD rt; π₃ r = unk; (π₂ r = 0) = (π₃ r = unk)⟧ ⟹ update rt ip r = rt(ip ↦ nr')›\n 3. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ update rt ip r = rt(ip ↦ ns)›*)\n assume \"ip ∈ kD(rt)\" and \"sqn rt ip = π₂(r)\" and \"flag rt ip = Some inv\" (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)› ‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = π₂ (r::nat × k × f × nat × nat × nat set)› ‹flag (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = Some Aodv_Basic.inv›*)\n from this(1) (*‹ip ∈ kD rt›*) obtain dsn and dsk and fl and hops and nhip and pre where \"rt ip = Some (dsn, dsk, fl, hops, nhip, pre)\"\n (*goal: ‹(⋀dsn dsk fl hops nhip pre. rt ip = Some (dsn, dsk, fl, hops, nhip, pre) ⟹ thesis) ⟹ thesis›*)\n by (metis kD_Some (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) ⟹ ∃(dsn::nat) (dsk::k) (flag::f) (hops::nat) (nhip::nat) pre::nat set. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n with ‹sqn rt ip = π₂(r)› (*‹sqn rt ip = π₂ r›*) ‹flag rt ip = Some inv› (*‹flag rt ip = Some Aodv_Basic.inv›*) show \"update rt ip r = rt (ip ↦ nr)\"\n unfolding update_def nr_def s_def\n (*goal: ‹(case rt ip of None ⇒ rt(ip ↦ r) | Some s ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r))) = rt(ip ↦ addpre r (π₇ (the (rt ip))))›*)\n by auto\n next\n (*goals:\n 1. ‹⟦ip ∈ kD rt; π₃ r = unk; (π₂ r = 0) = (π₃ r = unk)⟧ ⟹ update rt ip r = rt(ip ↦ nr')›\n 2. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ update rt ip r = rt(ip ↦ ns)›*)\n assume \"ip ∈ kD(rt)\" and \"π₃(r) = unk\" and \"(π₂(r) = 0) = (π₃(r) = unk)\" (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)› ‹π₃ (r::nat × k × f × nat × nat × nat set) = unk› ‹(π₂ (r::nat × k × f × nat × nat × nat set) = (0::nat)) = (π₃ r = unk)›*)\n from this(1) (*‹ip ∈ kD rt›*) obtain dsn and dsk and fl and hops and nhip and pre where \"rt ip = Some (dsn, dsk, fl, hops, nhip, pre)\"\n (*goal: ‹(⋀dsn dsk fl hops nhip pre. rt ip = Some (dsn, dsk, fl, hops, nhip, pre) ⟹ thesis) ⟹ thesis›*)\n by (metis kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n with ‹(π₂(r) = 0) = (π₃(r) = unk)› (*‹(π₂ r = 0) = (π₃ r = unk)›*) ‹π₃(r) = unk› (*‹π₃ r = unk›*) show \"update rt ip r = rt (ip ↦ nr')\"\n unfolding update_def nr'_def nr_def s_def\n (*goal: ‹(case rt ip of None ⇒ rt(ip ↦ r) | Some s ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r))) = rt(ip ↦ (π₂ (the (rt ip)), π₃ (addpre r (π₇ (the (rt ip)))), π₄ (addpre r (π₇ (the (rt ip)))), π₅ (addpre r (π₇ (the (rt ip)))), π₆ (addpre r (π₇ (the (rt ip)))), π₇ (addpre r (π₇ (the (rt ip))))))›*)\n apply (cases r)\n (*goal: ‹(case rt ip of None ⇒ rt(ip ↦ r) | Some s ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r))) = rt(ip ↦ (π₂ (the (rt ip)), π₃ (addpre r (π₇ (the (rt ip)))), π₄ (addpre r (π₇ (the (rt ip)))), π₅ (addpre r (π₇ (the (rt ip)))), π₆ (addpre r (π₇ (the (rt ip)))), π₇ (addpre r (π₇ (the (rt ip))))))›*)\n by simp\n next\n (*goal: ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ update rt ip r = rt(ip ↦ ns)›*)\n assume \"ip ∈ kD(rt)\" and otherassms: \"sqn rt ip ≥ π₂(r)\" \"π₃(r) = kno\" \"sqn rt ip = π₂(r) ⟹ the (dhops rt ip) ≤ π₅(r) ∧ the (flag rt ip) = val\" (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)› ‹π₂ (r::nat × k × f × nat × nat × nat set) ≤ sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat)› ‹π₃ (r::nat × k × f × nat × nat × nat set) = kno› ‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = π₂ (r::nat × k × f × nat × nat × nat set) ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val›*)\n from this(1) (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)›*) obtain dsn and dsk and fl and hops and nhip and pre where \"rt ip = Some (dsn, dsk, fl, hops, nhip, pre)\"\n (*goal: ‹(⋀dsn dsk fl hops nhip pre. rt ip = Some (dsn, dsk, fl, hops, nhip, pre) ⟹ thesis) ⟹ thesis›*)\n by (metis kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n with otherassms (*‹π₂ r ≤ sqn rt ip› ‹π₃ r = kno› ‹sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val›*) show \"update rt ip r = rt (ip ↦ ns)\"\n unfolding update_def ns_def s_def\n (*goal: ‹(case rt ip of None ⇒ rt(ip ↦ r) | Some s ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r))) = rt(ip ↦ addpre (the (rt ip)) (π₇ r))›*)\n by auto\n qed\n\nlemma update_cases [elim]:\n assumes \"(π₂(r) = 0) = (π₃(r) = unk)\"\n and c1: \"⟦ip ∉ kD(rt)⟧ ⟹ P (rt (ip ↦ r))\"\n\n and c2: \"⟦ip ∈ kD(rt); sqn rt ip < π₂(r)⟧\n ⟹ P (rt (ip ↦ addpre r (π₇(the σ⇘route⇙(rt, ip)))))\"\n and c3: \"⟦ip ∈ kD(rt); sqn rt ip = π₂(r); the (dhops rt ip) > π₅(r)⟧\n ⟹ P (rt (ip ↦ addpre r (π₇(the σ⇘route⇙(rt, ip)))))\"\n and c4: \"⟦ip ∈ kD(rt); sqn rt ip = π₂(r); the (flag rt ip) = inv⟧\n ⟹ P (rt (ip ↦ addpre r (π₇(the σ⇘route⇙(rt, ip)))))\"\n and c5: \"⟦ip ∈ kD(rt); π₃(r) = unk⟧\n ⟹ P (rt (ip ↦ (π₂(the σ⇘route⇙(rt, ip)), π₃(r),\n π₄(r), π₅(r), π₆(r), π₇(addpre r (π₇(the σ⇘route⇙(rt, ip)))))))\"\n and c6: \"⟦ip ∈ kD(rt); sqn rt ip ≥ π₂(r); π₃(r) = kno;\n sqn rt ip = π₂(r) ⟹ the (dhops rt ip) ≤ π₅(r) ∧ the (flag rt ip) = val⟧\n ⟹ P (rt (ip ↦ addpre (the σ⇘route⇙(rt, ip)) (π₇(r))))\"\n shows \"(P (update rt ip r))\"\n proof (cases \"ip ∈ kD(rt)\")\n (*goals:\n 1. ‹ip ∈ kD rt ⟹ P (update rt ip r)›\n 2. ‹ip ∉ kD rt ⟹ P (update rt ip r)›*)\n assume \"ip ∉ kD(rt)\" (*‹(ip::nat) ∉ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)›*)\n with c1 (*‹ip ∉ kD rt ⟹ P (rt(ip ↦ r))›*) show \"?thesis\"\n (*goal: ‹P (update rt ip r)›*)\n by simp\n next\n (*goal: ‹ip ∈ kD rt ⟹ P (update rt ip r)›*)\n assume \"ip ∈ kD(rt)\" (*‹(ip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)›*)\n moreover then obtain dsn and dsk and fl and hops and nhip and pre where rteq: \"rt ip = Some (dsn, dsk, fl, hops, nhip, pre)\"\n (*goal: ‹(⋀(dsn::nat) (dsk::k) (fl::f) (hops::nat) (nhip::nat) pre::nat set. (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = Some (dsn, dsk, fl, hops, nhip, pre) ⟹ thesis::bool) ⟹ thesis›*)\n by (metis kD_Some (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) ⟹ ∃(dsn::nat) (dsk::k) (flag::f) (hops::nat) (nhip::nat) pre::nat set. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n moreover obtain dsn' and dsk' and fl' and hops' and nhip' and pre' where req: \"r = (dsn', dsk', fl', hops', nhip', pre')\"\n (*goal: ‹(⋀(dsn'::nat) (dsk'::k) (fl'::f) (hops'::nat) (nhip'::nat) pre'::nat set. (r::nat × k × f × nat × nat × nat set) = (dsn', dsk', fl', hops', nhip', pre') ⟹ thesis::bool) ⟹ thesis›*)\n apply (cases r)\n (*goal: ‹(⋀dsn' dsk' fl' hops' nhip' pre'. r = (dsn', dsk', fl', hops', nhip', pre') ⟹ thesis) ⟹ thesis›*)\n by metis\n ultimately show \"?thesis\"\n (*goal: ‹P (update rt ip r)›*)\n using ‹(π₂(r) = 0) = (π₃(r) = unk)› (*‹(π₂ r = 0) = (π₃ r = unk)›*) c2[OF ‹ip∈kD(rt)›] (*‹sqn rt ip < π₂ r ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›*) c3[OF ‹ip∈kD(rt)›] (*‹⟦sqn rt ip = π₂ r; π₅ r < the (dhops rt ip)⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›*) c4[OF ‹ip∈kD(rt)›] (*‹⟦sqn rt ip = π₂ r; the (flag rt ip) = Aodv_Basic.inv⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›*) c5[OF ‹ip∈kD(rt)›] (*‹π₃ r = unk ⟹ P (rt(ip ↦ (π₂ (the (rt ip)), π₃ r, π₄ r, π₅ r, π₆ r, π₇ (addpre r (π₇ (the (rt ip)))))))›*) c6[OF ‹ip∈kD(rt)›] (*‹⟦π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ P (rt(ip ↦ addpre (the (rt ip)) (π₇ r)))›*) unfolding update_def sqn_def\n (*goal: ‹P (case rt ip of None ⇒ rt(ip ↦ r) | Some s ⇒ if π₂ s < π₂ r then rt(ip ↦ addpre r (π₇ s)) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre r (π₇ s)) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd (addpre r (π₇ s)))) else rt(ip ↦ addpre s (π₇ r)))›*)\n by auto\n qed\n\nlemma update_cases_kD:\n assumes \"(π₂(r) = 0) = (π₃(r) = unk)\"\n and \"ip ∈ kD(rt)\"\n and c2: \"sqn rt ip < π₂(r) ⟹ P (rt (ip ↦ addpre r (π₇(the σ⇘route⇙(rt, ip)))))\"\n and c3: \"⟦sqn rt ip = π₂(r); the (dhops rt ip) > π₅(r)⟧\n ⟹ P (rt (ip ↦ addpre r (π₇(the σ⇘route⇙(rt, ip)))))\"\n and c4: \"⟦sqn rt ip = π₂(r); the (flag rt ip) = inv⟧\n ⟹ P (rt (ip ↦ addpre r (π₇(the σ⇘route⇙(rt, ip)))))\"\n and c5: \"π₃(r) = unk ⟹ P (rt (ip ↦ (π₂(the σ⇘route⇙(rt, ip)), π₃(r),\n π₄(r), π₅(r), π₆(r),\n π₇(addpre r (π₇(the σ⇘route⇙(rt, ip)))))))\"\n and c6: \"⟦sqn rt ip ≥ π₂(r); π₃(r) = kno;\n sqn rt ip = π₂(r) ⟹ the (dhops rt ip) ≤ π₅(r) ∧ the (flag rt ip) = val⟧\n ⟹ P (rt (ip ↦ addpre (the σ⇘route⇙(rt, ip)) (π₇(r))))\"\n shows \"(P (update rt ip r))\"\n using assms(1) (*‹(π₂ r = 0) = (π₃ r = unk)›*) proof (rule update_cases (*‹⟦(π₂ ?r = 0) = (π₃ ?r = unk); ?ip ∉ kD ?rt ⟹ ?P (?rt(?ip ↦ ?r)); ⟦?ip ∈ kD ?rt; sqn ?rt ?ip < π₂ ?r⟧ ⟹ ?P (?rt(?ip ↦ addpre ?r (π₇ (the (?rt ?ip))))); ⟦?ip ∈ kD ?rt; sqn ?rt ?ip = π₂ ?r; π₅ ?r < the (dhops ?rt ?ip)⟧ ⟹ ?P (?rt(?ip ↦ addpre ?r (π₇ (the (?rt ?ip))))); ⟦?ip ∈ kD ?rt; sqn ?rt ?ip = π₂ ?r; the (flag ?rt ?ip) = Aodv_Basic.inv⟧ ⟹ ?P (?rt(?ip ↦ addpre ?r (π₇ (the (?rt ?ip))))); ⟦?ip ∈ kD ?rt; π₃ ?r = unk⟧ ⟹ ?P (?rt(?ip ↦ (π₂ (the (?rt ?ip)), π₃ ?r, π₄ ?r, π₅ ?r, π₆ ?r, π₇ (addpre ?r (π₇ (the (?rt ?ip))))))); ⟦?ip ∈ kD ?rt; π₂ ?r ≤ sqn ?rt ?ip; π₃ ?r = kno; sqn ?rt ?ip = π₂ ?r ⟹ the (dhops ?rt ?ip) ≤ π₅ ?r ∧ the (flag ?rt ?ip) = val⟧ ⟹ ?P (?rt(?ip ↦ addpre (the (?rt ?ip)) (π₇ ?r)))⟧ ⟹ ?P (update ?rt ?ip ?r)›*))\n (*goals:\n 1. ‹ip ∉ kD rt ⟹ P (rt(ip ↦ r))›\n 2. ‹⟦ip ∈ kD rt; sqn rt ip < π₂ r⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›\n 3. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; π₅ r < the (dhops rt ip)⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›\n 4. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; the (flag rt ip) = Aodv_Basic.inv⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›\n 5. ‹⟦ip ∈ kD rt; π₃ r = unk⟧ ⟹ P (rt(ip ↦ (π₂ (the (rt ip)), π₃ r, π₄ r, π₅ r, π₆ r, π₇ (addpre r (π₇ (the (rt ip)))))))›\n 6. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ P (rt(ip ↦ addpre (the (rt ip)) (π₇ r)))›*)\n assume \"sqn rt ip < π₂(r)\" (*‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) < π₂ (r::nat × k × f × nat × nat × nat set)›*)\n thus \"P (rt(ip ↦ addpre r (π₇(the (rt ip)))))\"\n by (rule c2 (*‹sqn rt ip < π₂ r ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›*))\n next\n (*goals:\n 1. ‹ip ∉ kD rt ⟹ P (rt(ip ↦ r))›\n 2. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; π₅ r < the (dhops rt ip)⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›\n 3. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; the (flag rt ip) = Aodv_Basic.inv⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›\n 4. ‹⟦ip ∈ kD rt; π₃ r = unk⟧ ⟹ P (rt(ip ↦ (π₂ (the (rt ip)), π₃ r, π₄ r, π₅ r, π₆ r, π₇ (addpre r (π₇ (the (rt ip)))))))›\n 5. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ P (rt(ip ↦ addpre (the (rt ip)) (π₇ r)))›*)\n assume \"sqn rt ip = π₂(r)\" and \"the (dhops rt ip) > π₅(r)\" (*‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = π₂ (r::nat × k × f × nat × nat × nat set)› ‹π₅ (r::nat × k × f × nat × nat × nat set) < the (dhops (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat))›*)\n thus \"P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))\"\n by (rule c3 (*‹⟦sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = π₂ (r::nat × k × f × nat × nat × nat set); π₅ r < the (dhops rt ip)⟧ ⟹ (P::(nat ⇒ (nat × k × f × nat × nat × nat set) option) ⇒ bool) (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›*))\n next\n (*goals:\n 1. ‹ip ∉ kD rt ⟹ P (rt(ip ↦ r))›\n 2. ‹⟦ip ∈ kD rt; sqn rt ip = π₂ r; the (flag rt ip) = Aodv_Basic.inv⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›\n 3. ‹⟦ip ∈ kD rt; π₃ r = unk⟧ ⟹ P (rt(ip ↦ (π₂ (the (rt ip)), π₃ r, π₄ r, π₅ r, π₆ r, π₇ (addpre r (π₇ (the (rt ip)))))))›\n 4. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ P (rt(ip ↦ addpre (the (rt ip)) (π₇ r)))›*)\n assume \"sqn rt ip = π₂(r)\" and \"the (flag rt ip) = inv\" (*‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = π₂ (r::nat × k × f × nat × nat × nat set)› ‹the (flag (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat)) = Aodv_Basic.inv›*)\n thus \"P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))\"\n by (rule c4 (*‹⟦sqn rt ip = π₂ r; the (flag rt ip) = Aodv_Basic.inv⟧ ⟹ P (rt(ip ↦ addpre r (π₇ (the (rt ip)))))›*))\n next\n (*goals:\n 1. ‹ip ∉ kD rt ⟹ P (rt(ip ↦ r))›\n 2. ‹⟦ip ∈ kD rt; π₃ r = unk⟧ ⟹ P (rt(ip ↦ (π₂ (the (rt ip)), π₃ r, π₄ r, π₅ r, π₆ r, π₇ (addpre r (π₇ (the (rt ip)))))))›\n 3. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ P (rt(ip ↦ addpre (the (rt ip)) (π₇ r)))›*)\n assume \"π₃(r) = unk\" (*‹π₃ (r::nat × k × f × nat × nat × nat set) = unk›*)\n thus \"P (rt (ip ↦ (π₂(the σ⇘route⇙(rt, ip)), π₃(r), π₄(r), π₅(r), π₆(r),\n π₇(addpre r (π₇(the (rt ip)))))))\"\n by (rule c5 (*‹π₃ (r::nat × k × f × nat × nat × nat set) = unk ⟹ (P::(nat ⇒ (nat × k × f × nat × nat × nat set) option) ⇒ bool) ((rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)(ip::nat ↦ (π₂ (the (rt ip)), π₃ r, π₄ r, π₅ r, π₆ r, π₇ (addpre r (π₇ (the (rt ip)))))))›*))\n next\n (*goals:\n 1. ‹ip ∉ kD rt ⟹ P (rt(ip ↦ r))›\n 2. ‹⟦ip ∈ kD rt; π₂ r ≤ sqn rt ip; π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ P (rt(ip ↦ addpre (the (rt ip)) (π₇ r)))›*)\n assume \"sqn rt ip ≥ π₂(r)\" and \"π₃(r) = kno\" and \"sqn rt ip = π₂(r) ⟹ the (dhops rt ip) ≤ π₅(r) ∧ the (flag rt ip) = val\" (*‹π₂ (r::nat × k × f × nat × nat × nat set) ≤ sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat)› ‹π₃ (r::nat × k × f × nat × nat × nat set) = kno› ‹sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) = π₂ (r::nat × k × f × nat × nat × nat set) ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val›*)\n thus \"P (rt (ip ↦ addpre (the (rt ip)) (π₇(r))))\"\n by (rule c6 (*‹⟦π₂ (r::nat × k × f × nat × nat × nat set) ≤ sqn (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat); π₃ r = kno; sqn rt ip = π₂ r ⟹ the (dhops rt ip) ≤ π₅ r ∧ the (flag rt ip) = val⟧ ⟹ (P::(nat ⇒ (nat × k × f × nat × nat × nat set) option) ⇒ bool) (rt(ip ↦ addpre (the (rt ip)) (π₇ r)))›*))\n qed (simp add: ‹ip ∈ kD(rt)›)\n (*solved the remaining goal: ‹ip ∉ kD rt ⟹ P (rt(ip ↦ r))›*)\n\nlemma in_kD_after_update [simp]:\n fixes rt nip dsn dsk flag hops nhip pre\n shows \"kD (update rt nip (dsn, dsk, flag, hops, nhip, pre)) = insert nip (kD rt)\"\n unfolding update_def\n (*goal: ‹kD (case rt nip of None ⇒ rt(nip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(nip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(nip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(nip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(nip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) = insert nip (kD rt)›*)\n apply (cases \"rt nip\")\n (*goals:\n 1. ‹rt nip = None ⟹ kD (case rt nip of None ⇒ rt(nip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(nip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(nip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(nip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(nip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) = insert nip (kD rt)›\n 2. ‹⋀a. rt nip = Some a ⟹ kD (case rt nip of None ⇒ rt(nip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(nip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(nip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(nip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(nip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) = insert nip (kD rt)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma nhop_of_update [simp]:\n fixes rt dip dsn dsk flag hops nhip\n assumes \"rt ≠ update rt dip (dsn, dsk, flag, hops, nhip, {})\"\n shows \"the (nhop (update rt dip (dsn, dsk, flag, hops, nhip, {})) dip) = nhip\"\n proof (-)\n (*goal: ‹the (nhop (update rt dip (dsn, dsk, flag, hops, nhip, {})) dip) = nhip›*)\n from assms (*‹rt ≠ update rt dip (dsn, dsk, flag, hops, nhip, {})›*) have update_neq: \"⋀v. rt dip = Some v ⟹\n update rt dip (dsn, dsk, flag, hops, nhip, {})\n ≠ rt(dip ↦ addpre (the (rt dip)) (π₇ (dsn, dsk, flag, hops, nhip, {})))\"\n by auto\n show \"?thesis\"\n (*goal: ‹the (nhop (update rt dip (dsn, dsk, flag, hops, nhip, {})) dip) = nhip›*)\n proof (cases \"rt dip = None\")\n (*goals:\n 1. ‹rt dip = None ⟹ the (nhop (update rt dip (dsn, dsk, flag, hops, nhip, {})) dip) = nhip›\n 2. ‹rt dip ≠ None ⟹ the (nhop (update rt dip (dsn, dsk, flag, hops, nhip, {})) dip) = nhip›*)\n assume \"rt dip = None\" (*‹(rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) = None›*)\n thus \"?thesis\"\n (*goal: ‹the (nhop (update rt dip (dsn, dsk, flag, hops, nhip, {})) dip) = nhip›*)\n unfolding update_def\n (*goal: ‹the (nhop (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flag, hops, nhip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, {}) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, {}) ∧ (π₅ (dsn, dsk, flag, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, {}) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, {})))) dip) = nhip›*)\n by clarsimp\n next\n (*goal: ‹(rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) ≠ None ⟹ the (nhop (update rt dip (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, {})) dip) = nhip›*)\n assume \"rt dip ≠ None\" (*‹(rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) ≠ None›*)\n then obtain v where \"rt dip = Some v\"\n (*goal: ‹(⋀v. rt dip = Some v ⟹ thesis) ⟹ thesis›*)\n by (metis not_None_eq (*‹(?x ≠ None) = (∃y. ?x = Some y)›*))\n with update_neq[OF this] (*‹update (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, {}) ≠ rt(dip ↦ addpre (the (rt dip)) (π₇ (dsn, dsk, flag, hops, nhip, {})))›*) show \"?thesis\"\n (*goal: ‹the (nhop (update (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, {})) dip) = nhip›*)\n unfolding update_def\n (*goal: ‹the (nhop (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flag, hops, nhip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, {}) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, {}) ∧ (π₅ (dsn, dsk, flag, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, {}) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, {})))) dip) = nhip›*)\n by auto\n qed\n qed\n\nlemma sqn_if_updated:\n fixes rip v rt ip\n shows \"sqn (λx. if x = rip then Some v else rt x) ip\n = (if ip = rip then π₂(v) else sqn rt ip)\"\n unfolding sqn_def\n (*goal: ‹(case if (ip::nat) = (rip::nat) then Some (v::nat × k × f × nat × nat × nat set) else (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) ip of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₂ r) = (if ip = rip then π₂ v else case rt ip of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₂ r)›*)\n by simp\n\nlemma update_sqn [simp]:\n fixes rt dip rip dsn dsk hops nhip pre\n assumes \"(dsn = 0) = (dsk = unk)\"\n shows \"sqn rt dip ≤ sqn (update rt rip (dsn, dsk, val, hops, nhip, pre)) dip\"\n proof (rule update_cases (*‹⟦(π₂ ?r = 0) = (π₃ ?r = unk); ?ip ∉ kD ?rt ⟹ ?P (?rt(?ip ↦ ?r)); ⟦?ip ∈ kD ?rt; sqn ?rt ?ip < π₂ ?r⟧ ⟹ ?P (?rt(?ip ↦ addpre ?r (π₇ (the (?rt ?ip))))); ⟦?ip ∈ kD ?rt; sqn ?rt ?ip = π₂ ?r; π₅ ?r < the (dhops ?rt ?ip)⟧ ⟹ ?P (?rt(?ip ↦ addpre ?r (π₇ (the (?rt ?ip))))); ⟦?ip ∈ kD ?rt; sqn ?rt ?ip = π₂ ?r; the (flag ?rt ?ip) = Aodv_Basic.inv⟧ ⟹ ?P (?rt(?ip ↦ addpre ?r (π₇ (the (?rt ?ip))))); ⟦?ip ∈ kD ?rt; π₃ ?r = unk⟧ ⟹ ?P (?rt(?ip ↦ (π₂ (the (?rt ?ip)), π₃ ?r, π₄ ?r, π₅ ?r, π₆ ?r, π₇ (addpre ?r (π₇ (the (?rt ?ip))))))); ⟦?ip ∈ kD ?rt; π₂ ?r ≤ sqn ?rt ?ip; π₃ ?r = kno; sqn ?rt ?ip = π₂ ?r ⟹ the (dhops ?rt ?ip) ≤ π₅ ?r ∧ the (flag ?rt ?ip) = val⟧ ⟹ ?P (?rt(?ip ↦ addpre (the (?rt ?ip)) (π₇ ?r)))⟧ ⟹ ?P (update ?rt ?ip ?r)›*))\n (*goals:\n 1. ‹(π₂ (dsn, dsk, val, hops, nhip, pre) = 0) = (π₃ (dsn, dsk, val, hops, nhip, pre) = unk)›\n 2. ‹rip ∉ kD rt ⟹ sqn rt dip ≤ sqn (rt(rip ↦ (dsn, dsk, val, hops, nhip, pre))) dip›\n 3. ‹⟦rip ∈ kD rt; sqn rt rip < π₂ (dsn, dsk, val, hops, nhip, pre)⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))) dip›\n 4. ‹⟦rip ∈ kD rt; sqn rt rip = π₂ (dsn, dsk, val, hops, nhip, pre); π₅ (dsn, dsk, val, hops, nhip, pre) < the (dhops rt rip)⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))) dip›\n 5. ‹⟦rip ∈ kD rt; sqn rt rip = π₂ (dsn, dsk, val, hops, nhip, pre); the (flag rt rip) = Aodv_Basic.inv⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))) dip›\n 6. ‹⟦rip ∈ kD rt; π₃ (dsn, dsk, val, hops, nhip, pre) = unk⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ (π₂ (the (rt rip)), π₃ (dsn, dsk, val, hops, nhip, pre), π₄ (dsn, dsk, val, hops, nhip, pre), π₅ (dsn, dsk, val, hops, nhip, pre), π₆ (dsn, dsk, val, hops, nhip, pre), π₇ (addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))))) dip›\n 7. ‹⟦rip ∈ kD rt; π₂ (dsn, dsk, val, hops, nhip, pre) ≤ sqn rt rip; π₃ (dsn, dsk, val, hops, nhip, pre) = kno; sqn rt rip = π₂ (dsn, dsk, val, hops, nhip, pre) ⟹ the (dhops rt rip) ≤ π₅ (dsn, dsk, val, hops, nhip, pre) ∧ the (flag rt rip) = val⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (the (rt rip)) (π₇ (dsn, dsk, val, hops, nhip, pre)))) dip›*)\n show \"(π₂ (dsn, dsk, val, hops, nhip, pre) = 0) = (π₃ (dsn, dsk, val, hops, nhip, pre) = unk)\"\n apply simp\n (*goal: ‹(π₂ (dsn, dsk, val, hops, nhip, pre) = 0) = (π₃ (dsn, dsk, val, hops, nhip, pre) = unk)›*)\n by (rule assms (*‹(dsn = 0) = (dsk = unk)›*))\n qed ((clarsimp simp: sqn_if_updated (*‹sqn (λx. if x = ?rip then Some ?v else ?rt x) ?ip = (if ?ip = ?rip then π₂ ?v else sqn ?rt ?ip)›*) sqn_def (*‹sqn ?rt ?dip ≡ case ?rt ?dip of None ⇒ 0 | Some r ⇒ π₂ r›*))+)\n (*solves the remaining goals:\n 1. ‹rip ∉ kD rt ⟹ sqn rt dip ≤ sqn (rt(rip ↦ (dsn, dsk, val, hops, nhip, pre))) dip›\n 2. ‹⟦rip ∈ kD rt; sqn rt rip < π₂ (dsn, dsk, val, hops, nhip, pre)⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))) dip›\n 3. ‹⟦rip ∈ kD rt; sqn rt rip = π₂ (dsn, dsk, val, hops, nhip, pre); π₅ (dsn, dsk, val, hops, nhip, pre) < the (dhops rt rip)⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))) dip›\n 4. ‹⟦rip ∈ kD rt; sqn rt rip = π₂ (dsn, dsk, val, hops, nhip, pre); the (flag rt rip) = Aodv_Basic.inv⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))) dip›\n 5. ‹⟦rip ∈ kD rt; π₃ (dsn, dsk, val, hops, nhip, pre) = unk⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ (π₂ (the (rt rip)), π₃ (dsn, dsk, val, hops, nhip, pre), π₄ (dsn, dsk, val, hops, nhip, pre), π₅ (dsn, dsk, val, hops, nhip, pre), π₆ (dsn, dsk, val, hops, nhip, pre), π₇ (addpre (dsn, dsk, val, hops, nhip, pre) (π₇ (the (rt rip))))))) dip›\n 6. ‹⟦rip ∈ kD rt; π₂ (dsn, dsk, val, hops, nhip, pre) ≤ sqn rt rip; π₃ (dsn, dsk, val, hops, nhip, pre) = kno; sqn rt rip = π₂ (dsn, dsk, val, hops, nhip, pre) ⟹ the (dhops rt rip) ≤ π₅ (dsn, dsk, val, hops, nhip, pre) ∧ the (flag rt rip) = val⟧ ⟹ sqn rt dip ≤ sqn (rt(rip ↦ addpre (the (rt rip)) (π₇ (dsn, dsk, val, hops, nhip, pre)))) dip›*)\n\nlemma sqn_update_bigger [simp]:\n fixes rt ip ip' dsn dsk flag hops nhip pre\n assumes \"1 ≤ hops\"\n shows \"sqn rt ip ≤ sqn (update rt ip' (dsn, dsk, flag, hops, nhip, pre)) ip\"\n using assms (*‹1 ≤ hops›*) unfolding update_def sqn_def\n (*goal: ‹(case rt ip of None ⇒ 0 | Some r ⇒ π₂ r) ≤ (case (case rt ip' of None ⇒ rt(ip' ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(ip' ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip' ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(ip' ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(ip' ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip of None ⇒ 0 | Some r ⇒ π₂ r)›*)\n apply (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n (*goal: ‹(case (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ip::nat) of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₂ r) ≤ (case (case rt (ip'::nat) of None ⇒ rt(ip' ↦ (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set)) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(ip' ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip' ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(ip' ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(ip' ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₂ r)›*)\n by auto\n\nlemma dhops_update [intro]:\n fixes rt dsn dsk flag hops ip rip nhip pre\n assumes ex: \"∀ip∈kD rt. the (dhops rt ip) ≥ 1\"\n and ip: \"(ip = rip ∧ Suc 0 ≤ hops) ∨ (ip ≠ rip ∧ ip∈kD rt)\"\n shows \"Suc 0 ≤ the (dhops (update rt rip (dsn, dsk, flag, hops, nhip, pre)) ip)\"\n using ip (*‹ip = rip ∧ Suc 0 ≤ hops ∨ ip ≠ rip ∧ ip ∈ kD rt›*) proof (standard)\n (*goals:\n 1. ‹(ip::nat) = (rip::nat) ∧ Suc (0::nat) ≤ (hops::nat) ⟹ Suc (0::nat) ≤ the (dhops (update (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) rip (dsn::nat, dsk::k, flag::f, hops, nhip::nat, pre::nat set)) ip)›\n 2. ‹(ip::nat) ≠ (rip::nat) ∧ ip ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) ⟹ Suc (0::nat) ≤ the (dhops (update rt rip (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set)) ip)›*)\n assume \"ip = rip ∧ Suc 0 ≤ hops\" (*‹(ip::nat) = (rip::nat) ∧ Suc (0::nat) ≤ (hops::nat)›*)\n thus \"?thesis\"\n (*goal: ‹Suc 0 ≤ the (dhops (update rt rip (dsn, dsk, flag, hops, nhip, pre)) ip)›*)\n unfolding update_def\n (*goal: ‹Suc (0::nat) ≤ the (dhops (case (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (rip::nat) of None ⇒ rt(rip ↦ (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set)) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(rip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(rip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) (ip::nat))›*)\n using ex (*‹∀ip::nat∈kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option). (1::nat) ≤ the (dhops rt ip)›*) apply (cases \"rip ∈ kD rt\")\n (*goals:\n 1. ‹⟦ip = rip ∧ Suc 0 ≤ hops; ∀ip∈kD rt. 1 ≤ the (dhops rt ip); rip ∈ kD rt⟧ ⟹ Suc 0 ≤ the (dhops (case rt rip of None ⇒ rt(rip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(rip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(rip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip)›\n 2. ‹⟦ip = rip ∧ Suc 0 ≤ hops; ∀ip∈kD rt. 1 ≤ the (dhops rt ip); rip ∉ kD rt⟧ ⟹ Suc 0 ≤ the (dhops (case rt rip of None ⇒ rt(rip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(rip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(rip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip)›\n discuss goal 1*)\n apply (drule(1) bspec (*‹⟦∀x∈?A. ?P x; ?x ∈ ?A⟧ ⟹ ?P ?x›*))\n (*top goal: ‹⟦ip = rip ∧ Suc 0 ≤ hops; ∀ip∈kD rt. 1 ≤ the (dhops rt ip); rip ∈ kD rt⟧ ⟹ Suc 0 ≤ the (dhops (case rt rip of None ⇒ rt(rip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(rip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(rip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n next\n (*goal: ‹ip ≠ rip ∧ ip ∈ kD rt ⟹ Suc 0 ≤ the (dhops (update rt rip (dsn, dsk, flag, hops, nhip, pre)) ip)›*)\n assume \"ip ≠ rip ∧ ip∈kD rt\" (*‹(ip::nat) ≠ (rip::nat) ∧ ip ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)›*)\n thus \"?thesis\"\n (*goal: ‹Suc 0 ≤ the (dhops (update rt rip (dsn, dsk, flag, hops, nhip, pre)) ip)›*)\n using ex (*‹∀ip∈kD rt. 1 ≤ the (dhops rt ip)›*) unfolding update_def\n (*goal: ‹Suc 0 ≤ the (dhops (case rt rip of None ⇒ rt(rip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(rip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(rip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip)›*)\n apply (cases \"rip∈kD rt\")\n (*goals:\n 1. ‹⟦ip ≠ rip ∧ ip ∈ kD rt; ∀ip∈kD rt. 1 ≤ the (dhops rt ip); rip ∈ kD rt⟧ ⟹ Suc 0 ≤ the (dhops (case rt rip of None ⇒ rt(rip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(rip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(rip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip)›\n 2. ‹⟦ip ≠ rip ∧ ip ∈ kD rt; ∀ip∈kD rt. 1 ≤ the (dhops rt ip); rip ∉ kD rt⟧ ⟹ Suc 0 ≤ the (dhops (case rt rip of None ⇒ rt(rip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(rip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(rip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(rip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n qed\n\nlemma update_another [simp]:\n fixes dip ip rt dsn dsk flag hops nhip pre\n assumes \"ip ≠ dip\"\n shows \"(update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = rt ip\"\n using assms (*‹ip ≠ dip›*) unfolding update_def\n (*goal: ‹(case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip = rt ip›*)\n by (clarsimp split: option.split (*‹(?P::?'b::type ⇒ bool) (case ?option::?'a::type option of None ⇒ ?f1.0::?'b::type | Some (x::?'a::type) ⇒ (?f2.0::?'a::type ⇒ ?'b::type) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a::type. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma nhop_update_another [simp]:\n fixes dip ip rt dsn dsk flag hops nhip pre\n assumes \"ip ≠ dip\"\n shows \"nhop (update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = nhop rt ip\"\n using assms (*‹ip ≠ dip›*) unfolding update_def\n (*goal: ‹nhop (case (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) of None ⇒ rt(dip ↦ (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set)) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) (ip::nat) = nhop rt ip›*)\n by (clarsimp split: option.split (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma dhops_update_another [simp]:\n fixes dip ip rt dsn dsk flag hops nhip pre\n assumes \"ip ≠ dip\"\n shows \"dhops (update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = dhops rt ip\"\n using assms (*‹(ip::nat) ≠ (dip::nat)›*) unfolding update_def\n (*goal: ‹dhops (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip = dhops rt ip›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma sqn_update_same [simp]:\n \"⋀rt ip dsn dsk flag hops nhip pre. sqn (rt(ip ↦ v)) ip = π₂(v)\"\n unfolding sqn_def\n (*goal: ‹⋀rt ip dsn dsk flag hops nhip pre. (case (rt(ip ↦ v)) ip of None ⇒ 0 | Some r ⇒ π₂ r) = π₂ v›*)\n by simp\n\nlemma dhops_update_changed [simp]:\n fixes rt dip osn hops nhip\n assumes \"rt ≠ update rt dip (osn, kno, val, hops, nhip, {})\"\n shows \"the (dhops (update rt dip (osn, kno, val, hops, nhip, {})) dip) = hops\"\n using assms (*‹rt ≠ update rt dip (osn, kno, val, hops, nhip, {})›*) unfolding update_def\n (*goal: ‹the (dhops (case rt dip of None ⇒ rt(dip ↦ (osn, kno, val, hops, nhip, {})) | Some s ⇒ if π₂ s < π₂ (osn, kno, val, hops, nhip, {}) then rt(dip ↦ addpre (osn, kno, val, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (osn, kno, val, hops, nhip, {}) ∧ (π₅ (osn, kno, val, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (osn, kno, val, hops, nhip, {}) (π₇ s)) else if π₃ (osn, kno, val, hops, nhip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (osn, kno, val, hops, nhip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (osn, kno, val, hops, nhip, {})))) dip) = hops›*)\n apply (clarsimp split: option.split_asm (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*) if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹the (dhops (case (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) of None ⇒ rt(dip ↦ (osn::nat, kno, val, hops::nat, nhip::nat, {})) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ (osn, kno, val, hops, nhip, {}) then rt(dip ↦ addpre (osn, kno, val, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (osn, kno, val, hops, nhip, {}) ∧ (π₅ (osn, kno, val, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (osn, kno, val, hops, nhip, {}) (π₇ s)) else if π₃ (osn, kno, val, hops, nhip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (osn, kno, val, hops, nhip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (osn, kno, val, hops, nhip, {})))) dip) = hops›*)\n by auto\n\nlemma nhop_update_unk_val [simp]:\n \"⋀rt dip ip dsn hops npre.\n the (nhop (update rt dip (dsn, unk, val, hops, ip, npre)) dip) = ip\"\n unfolding update_def\n (*goal: ‹⋀rt dip ip dsn hops npre. the (nhop (case rt dip of None ⇒ rt(dip ↦ (dsn, unk, val, hops, ip, npre)) | Some s ⇒ if π₂ s < π₂ (dsn, unk, val, hops, ip, npre) then rt(dip ↦ addpre (dsn, unk, val, hops, ip, npre) (π₇ s)) else if π₂ s = π₂ (dsn, unk, val, hops, ip, npre) ∧ (π₅ (dsn, unk, val, hops, ip, npre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, unk, val, hops, ip, npre) (π₇ s)) else if π₃ (dsn, unk, val, hops, ip, npre) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, unk, val, hops, ip, npre) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, unk, val, hops, ip, npre)))) dip) = ip›*)\n by (clarsimp split: option.split (*‹(?P::?'b::type ⇒ bool) (case ?option::?'a::type option of None ⇒ ?f1.0::?'b::type | Some (x::?'a::type) ⇒ (?f2.0::?'a::type ⇒ ?'b::type) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a::type. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma nhop_update_changed [simp]:\n fixes rt dip dsn dsk flg hops sip\n assumes \"update rt dip (dsn, dsk, flg, hops, sip, {}) ≠ rt\"\n shows \"the (nhop (update rt dip (dsn, dsk, flg, hops, sip, {})) dip) = sip\"\n using assms (*‹update rt dip (dsn, dsk, flg, hops, sip, {}) ≠ rt›*) unfolding update_def\n (*goal: ‹the (nhop (case (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) of None ⇒ rt(dip ↦ (dsn::nat, dsk::k, flg::f, hops::nat, sip::nat, {})) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip) = sip›*)\n apply (clarsimp split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹the (nhop (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip) = sip›*)\n by auto\n\nlemma update_rt_split_asm:\n \"⋀rt ip dsn dsk flag hops sip.\n P (update rt ip (dsn, dsk, flag, hops, sip, {}))\n =\n (¬(rt = update rt ip (dsn, dsk, flag, hops, sip, {}) ∧ ¬P rt\n ∨ rt ≠ update rt ip (dsn, dsk, flag, hops, sip, {})\n ∧ ¬P (update rt ip (dsn, dsk, flag, hops, sip, {}))))\"\n by auto\n\nlemma sqn_update [simp]: \"⋀rt dip dsn flg hops sip.\n rt ≠ update rt dip (dsn, kno, flg, hops, sip, {})\n ⟹ sqn (update rt dip (dsn, kno, flg, hops, sip, {})) dip = dsn\"\n unfolding update_def\n (*goal: ‹⋀rt dip dsn flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) ⟹ sqn (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) dip = dsn›*)\n apply (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*) if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀(rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat) (dsn::nat) (flg::f) (hops::nat) sip::nat. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) ⟹ sqn (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some (s::nat × k × f × nat × nat × nat set) ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) dip = dsn›*)\n by auto\n\nlemma sqnf_update [simp]: \"⋀rt dip dsn dsk flg hops sip.\n rt ≠ update rt dip (dsn, dsk, flg, hops, sip, {})\n ⟹ sqnf (update rt dip (dsn, dsk, flg, hops, sip, {})) dip = dsk\"\n unfolding update_def sqnf_def\n (*goal: ‹⋀rt dip dsn dsk flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) ⟹ (case (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip of None ⇒ unk | Some r ⇒ π₃ r) = dsk›*)\n apply (clarsimp split: option.splits (*‹(?P::?'b::type ⇒ bool) (case ?option::?'a::type option of None ⇒ ?f1.0::?'b::type | Some (x::?'a::type) ⇒ (?f2.0::?'a::type ⇒ ?'b::type) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a::type. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b::type ⇒ bool) (case ?option::?'a::type option of None ⇒ ?f1.0::?'b::type | Some (x::?'a::type) ⇒ (?f2.0::?'a::type ⇒ ?'b::type) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a::type. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) if_split_asm (*‹(?P::?'a::type ⇒ bool) (if ?Q::bool then ?x::?'a::type else (?y::?'a::type)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀rt dip dsn dsk flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) ⟹ (case (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip of None ⇒ unk | Some r ⇒ π₃ r) = dsk›*)\n by auto\n\nlemma update_kno_dsn_greater_zero:\n \"⋀rt dip ip dsn hops npre. 1 ≤ dsn ⟹ 1 ≤ (sqn (update rt dip (dsn, kno, val, hops, ip, npre)) dip)\"\n unfolding update_def\n (*goal: ‹⋀rt dip ip dsn hops npre. 1 ≤ dsn ⟹ 1 ≤ sqn (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, val, hops, ip, npre)) | Some s ⇒ if π₂ s < π₂ (dsn, kno, val, hops, ip, npre) then rt(dip ↦ addpre (dsn, kno, val, hops, ip, npre) (π₇ s)) else if π₂ s = π₂ (dsn, kno, val, hops, ip, npre) ∧ (π₅ (dsn, kno, val, hops, ip, npre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, val, hops, ip, npre) (π₇ s)) else if π₃ (dsn, kno, val, hops, ip, npre) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, val, hops, ip, npre) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, val, hops, ip, npre)))) dip›*)\n by (clarsimp split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n\nlemma proj3_update [simp]: \"⋀rt dip dsn dsk flg hops sip.\n rt ≠ update rt dip (dsn, dsk, flg, hops, sip, {})\n ⟹ π₃(the (update rt dip (dsn, dsk, flg, hops, sip, {}) dip)) = dsk\"\n unfolding update_def sqnf_def\n (*goal: ‹⋀rt dip dsn dsk flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) ⟹ π₃ (the ((case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip)) = dsk›*)\n apply (clarsimp split: option.splits (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) if_split_asm (*‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀rt dip dsn dsk flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) ⟹ π₃ (the ((case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip)) = dsk›*)\n by auto\n\nlemma nhop_update_changed_kno_val [simp]: \"⋀rt ip dsn dsk hops nhip.\n rt ≠ update rt ip (dsn, kno, val, hops, nhip, {})\n ⟹ the (nhop (update rt ip (dsn, kno, val, hops, nhip, {})) ip) = nhip\"\n unfolding update_def\n (*goal: ‹⋀rt ip dsn dsk hops nhip. rt ≠ (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip, {}) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip, {}) ∧ (π₅ (dsn, kno, val, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₃ (dsn, kno, val, hops, nhip, {}) = unk then rt(ip ↦ (π₂ s, snd (addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)))) else rt(ip ↦ addpre s (π₇ (dsn, kno, val, hops, nhip, {})))) ⟹ the (nhop (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip, {}) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip, {}) ∧ (π₅ (dsn, kno, val, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₃ (dsn, kno, val, hops, nhip, {}) = unk then rt(ip ↦ (π₂ s, snd (addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)))) else rt(ip ↦ addpre s (π₇ (dsn, kno, val, hops, nhip, {})))) ip) = nhip›*)\n apply (clarsimp split: option.split_asm (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*) if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀rt ip dsn dsk hops nhip. rt ≠ (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip, {}) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip, {}) ∧ (π₅ (dsn, kno, val, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₃ (dsn, kno, val, hops, nhip, {}) = unk then rt(ip ↦ (π₂ s, snd (addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)))) else rt(ip ↦ addpre s (π₇ (dsn, kno, val, hops, nhip, {})))) ⟹ the (nhop (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip, {}) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip, {}) ∧ (π₅ (dsn, kno, val, hops, nhip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)) else if π₃ (dsn, kno, val, hops, nhip, {}) = unk then rt(ip ↦ (π₂ s, snd (addpre (dsn, kno, val, hops, nhip, {}) (π₇ s)))) else rt(ip ↦ addpre s (π₇ (dsn, kno, val, hops, nhip, {})))) ip) = nhip›*)\n by auto\n\nlemma flag_update [simp]: \"⋀rt dip dsn flg hops sip.\n rt ≠ update rt dip (dsn, kno, flg, hops, sip, {})\n ⟹ the (flag (update rt dip (dsn, kno, flg, hops, sip, {})) dip) = flg\"\n unfolding update_def\n (*goal: ‹⋀rt dip dsn flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) ⟹ the (flag (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) dip) = flg›*)\n apply (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*) if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀rt dip dsn flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) ⟹ the (flag (case rt dip of None ⇒ rt(dip ↦ (dsn, kno, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, kno, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, kno, flg, hops, sip, {}) ∧ (π₅ (dsn, kno, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, kno, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, kno, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, kno, flg, hops, sip, {})))) dip) = flg›*)\n by auto\n\nlemma the_flag_Some [dest!]:\n fixes ip rt\n assumes \"the (flag rt ip) = x\"\n and \"ip ∈ kD rt\"\n shows \"flag rt ip = Some x\"\n using assms (*‹the (flag rt ip) = x› ‹ip ∈ kD rt›*) by auto\n\nlemma kD_update_unchanged [dest]:\n fixes rt dip dsn dsk flag hops nhip pre\n assumes \"rt = update rt dip (dsn, dsk, flag, hops, nhip, pre)\"\n shows \"dip∈kD(rt)\"\n proof (-)\n (*goal: ‹dip ∈ kD rt›*)\n have \"dip∈kD(update rt dip (dsn, dsk, flag, hops, nhip, pre))\"\n by simp\n with assms (*‹rt = update rt dip (dsn, dsk, flag, hops, nhip, pre)›*) show \"?thesis\"\n (*goal: ‹dip ∈ kD rt›*)\n by simp\n qed\n\nlemma nhop_update [simp]: \"⋀rt dip dsn dsk flg hops sip.\n rt ≠ update rt dip (dsn, dsk, flg, hops, sip, {})\n ⟹ the (nhop (update rt dip (dsn, dsk, flg, hops, sip, {})) dip) = sip\"\n unfolding update_def sqnf_def\n (*goal: ‹⋀rt dip dsn dsk flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) ⟹ the (nhop (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip) = sip›*)\n apply (clarsimp split: option.splits (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) if_split_asm (*‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀rt dip dsn dsk flg hops sip. rt ≠ (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) ⟹ the (nhop (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flg, hops, sip, {})) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flg, hops, sip, {}) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flg, hops, sip, {}) ∧ (π₅ (dsn, dsk, flg, hops, sip, {}) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)) else if π₃ (dsn, dsk, flg, hops, sip, {}) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flg, hops, sip, {}) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flg, hops, sip, {})))) dip) = sip›*)\n by auto\n\nlemma sqn_update_another [simp]:\n fixes dip ip rt dsn dsk flag hops nhip pre\n assumes \"ip ≠ dip\"\n shows \"sqn (update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = sqn rt ip\"\n using assms (*‹ip ≠ dip›*) unfolding update_def sqn_def\n (*goal: ‹(case (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip of None ⇒ 0 | Some r ⇒ π₂ r) = (case rt ip of None ⇒ 0 | Some r ⇒ π₂ r)›*)\n by (msorry)\n\nlemma sqnf_update_another [simp]:\n fixes dip ip rt dsn dsk flag hops nhip pre\n assumes \"ip ≠ dip\"\n shows \"sqnf (update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = sqnf rt ip\"\n using assms (*‹(ip::nat) ≠ (dip::nat)›*) unfolding update_def sqnf_def\n (*goal: ‹(case (case rt dip of None ⇒ rt(dip ↦ (dsn, dsk, flag, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, flag, hops, nhip, pre) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, flag, hops, nhip, pre) ∧ (π₅ (dsn, dsk, flag, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip ↦ addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, flag, hops, nhip, pre) = unk then rt(dip ↦ (π₂ s, snd (addpre (dsn, dsk, flag, hops, nhip, pre) (π₇ s)))) else rt(dip ↦ addpre s (π₇ (dsn, dsk, flag, hops, nhip, pre)))) ip of None ⇒ unk | Some r ⇒ π₃ r) = (case rt ip of None ⇒ unk | Some r ⇒ π₃ r)›*)\n by (msorry)\n\nlemma vD_update_val [dest]:\n \"⋀dip rt dip' dsn dsk hops nhip pre.\n dip ∈ vD(update rt dip' (dsn, dsk, val, hops, nhip, pre)) ⟹ (dip∈vD(rt) ∨ dip=dip')\"\n unfolding update_def vD_def\n (*goal: ‹⋀dip rt dip' dsn dsk hops nhip pre. dip ∈ {dip. flag (case rt dip' of None ⇒ rt(dip' ↦ (dsn, dsk, val, hops, nhip, pre)) | Some s ⇒ if π₂ s < π₂ (dsn, dsk, val, hops, nhip, pre) then rt(dip' ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ s)) else if π₂ s = π₂ (dsn, dsk, val, hops, nhip, pre) ∧ (π₅ (dsn, dsk, val, hops, nhip, pre) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(dip' ↦ addpre (dsn, dsk, val, hops, nhip, pre) (π₇ s)) else if π₃ (dsn, dsk, val, hops, nhip, pre) = unk then rt(dip' ↦ (π₂ s, snd (addpre (dsn, dsk, val, hops, nhip, pre) (π₇ s)))) else rt(dip' ↦ addpre s (π₇ (dsn, dsk, val, hops, nhip, pre)))) dip = Some val} ⟹ dip ∈ {dip. flag rt dip = Some val} ∨ dip = dip'›*)\n by (clarsimp split: option.split_asm (*‹(?P::?'b::type ⇒ bool) (case ?option::?'a::type option of None ⇒ ?f1.0::?'b::type | Some (x::?'a::type) ⇒ (?f2.0::?'a::type ⇒ ?'b::type) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a::type. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) if_split_asm (*‹(?P::?'a::type ⇒ bool) (if ?Q::bool then ?x::?'a::type else (?y::?'a::type)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n\nsubsubsection \"Invalidating route entries\"\n\ndefinition invalidate :: \"rt ⇒ (ip ⇀ sqn) ⇒ rt\"\nwhere \"invalidate rt dests ≡\n λip. case (rt ip, dests ip) of\n (None, _) ⇒ None\n | (Some s, None) ⇒ Some s\n | (Some (_, dsk, _, hops, nhip, pre), Some rsn) ⇒\n Some (rsn, dsk, inv, hops, nhip, pre)\"\n\nlemma proj3_invalidate [simp]:\n \"⋀dip. π₃(the ((invalidate rt dests) dip)) = π₃(the (rt dip))\"\n unfolding invalidate_def\n (*goal: ‹⋀dip. π₃ (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba))) = π₃ (the (rt dip))›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma proj5_invalidate [simp]:\n \"⋀dip. π₅(the ((invalidate rt dests) dip)) = π₅(the (rt dip))\"\n unfolding invalidate_def\n (*goal: ‹⋀dip. π₅ (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba))) = π₅ (the (rt dip))›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma proj6_invalidate [simp]:\n \"⋀dip. π₆(the ((invalidate rt dests) dip)) = π₆(the (rt dip))\"\n unfolding invalidate_def\n (*goal: ‹⋀dip::nat. π₆ (the (case ((rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) dip, (dests::nat ⇒ nat option) dip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba))) = π₆ (the (rt dip))›*)\n by (clarsimp split: option.split (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma proj7_invalidate [simp]:\n \"⋀dip. π₇(the ((invalidate rt dests) dip)) = π₇(the (rt dip))\"\n unfolding invalidate_def\n (*goal: ‹⋀dip::nat. π₇ (the (case ((rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) dip, (dests::nat ⇒ nat option) dip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba))) = π₇ (the (rt dip))›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma invalidate_kD_inv [simp]:\n \"⋀rt dests. kD (invalidate rt dests) = kD rt\"\n unfolding invalidate_def kD_def\n (*goal: ‹⋀(rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) dests::nat ⇒ nat option. {dip::nat. (case (rt dip, dests dip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) ≠ None} = {dip::nat. rt dip ≠ None}›*)\n by (simp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma invalidate_sqn:\n fixes rt dip dests\n assumes \"∀rsn. dests dip = Some rsn ⟶ sqn rt dip ≤ rsn\"\n shows \"sqn rt dip ≤ sqn (invalidate rt dests) dip\"\n proof (cases \"dip ∉ kD(rt)\")\n (*goals:\n 1. ‹(dip::nat) ∉ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) ⟹ sqn rt dip ≤ sqn (invalidate rt (dests::nat ⇒ nat option)) dip›\n 2. ‹¬ (dip::nat) ∉ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) ⟹ sqn rt dip ≤ sqn (invalidate rt (dests::nat ⇒ nat option)) dip›*)\n assume \"¬ dip ∉ kD(rt)\" (*‹¬ (dip::nat) ∉ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option)›*)\n hence \"dip∈kD(rt)\"\n by simp\n then obtain dsn and dsk and flag and hops and nhip and pre where \"rt dip = Some (dsn, dsk, flag, hops, nhip, pre)\"\n (*goal: ‹(⋀dsn dsk flag hops nhip pre. rt dip = Some (dsn, dsk, flag, hops, nhip, pre) ⟹ thesis) ⟹ thesis›*)\n by (metis kD_Some (*‹?dip ∈ kD ?rt ⟹ ∃dsn dsk flag hops nhip pre. ?rt ?dip = Some (dsn, dsk, flag, hops, nhip, pre)›*))\n with assms (*‹∀rsn. dests dip = Some rsn ⟶ sqn rt dip ≤ rsn›*) show \"sqn rt dip ≤ sqn (invalidate rt dests) dip\"\n apply (cases \"dests dip\")\n (*goals:\n 1. ‹⟦∀rsn. dests dip = Some rsn ⟶ sqn rt dip ≤ rsn; rt dip = Some (dsn, dsk, flag, hops, nhip, pre); dests dip = None⟧ ⟹ sqn rt dip ≤ sqn (invalidate rt dests) dip›\n 2. ‹⋀a. ⟦∀rsn. dests dip = Some rsn ⟶ sqn rt dip ≤ rsn; rt dip = Some (dsn, dsk, flag, hops, nhip, pre); dests dip = Some a⟧ ⟹ sqn rt dip ≤ sqn (invalidate rt dests) dip›\n discuss goal 1*)\n apply ((auto simp add: invalidate_def (*‹invalidate ?rt ?dests ≡ λip. case (?rt ip, ?dests ip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)›*) sqn_def (*‹sqn ?rt ?dip ≡ case ?rt ?dip of None ⇒ 0 | Some r ⇒ π₂ r›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: invalidate_def (*‹invalidate (?rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (?dests::nat ⇒ nat option) ≡ λip::nat. case (?rt ip, ?dests ip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)›*) sqn_def (*‹sqn (?rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (?dip::nat) ≡ case ?rt ?dip of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₂ r›*))[1])\n (*proven 2 subgoals*) .\n qed (simp)\n (*solved the remaining goal: ‹(dip::nat) ∉ kD (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) ⟹ sqn rt dip ≤ sqn (invalidate rt (dests::nat ⇒ nat option)) dip›*)\n\nlemma sqn_invalidate_in_dests [simp]:\n fixes dests ipa rsn rt\n assumes \"dests ipa = Some rsn\"\n and \"ipa∈kD(rt)\"\n shows \"sqn (invalidate rt dests) ipa = rsn\"\n unfolding invalidate_def sqn_def\n (*goal: ‹(case case ((rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (ipa::nat), (dests::nat ⇒ nat option) ipa) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba) of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat × nat set) ⇒ π₂ r) = (rsn::nat)›*)\n using assms(1) (*‹dests ipa = Some rsn›*) assms(2)[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt ipa = Some (dsn, dsk, flag, hops, nhip, pre)›*) by clarsimp\n\nlemma dhops_invalidate [simp]:\n \"⋀dip. the (dhops (invalidate rt dests) dip) = the (dhops rt dip)\"\n unfolding invalidate_def\n (*goal: ‹⋀dip. the (map_option π₅ (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba))) = the (dhops rt dip)›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma sqnf_invalidate [simp]:\n \"⋀dip. sqnf (invalidate (rt ξ) (dests ξ)) dip = sqnf (rt ξ) dip\"\n unfolding sqnf_def invalidate_def\n (*goal: ‹⋀dip. (case case (rt ξ dip, dests ξ dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba) of None ⇒ unk | Some r ⇒ π₃ r) = (case rt ξ dip of None ⇒ unk | Some r ⇒ π₃ r)›*)\n by (clarsimp split: option.split (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma nhop_invalidate [simp]:\n \"⋀dip. the (nhop (invalidate (rt ξ) (dests ξ)) dip) = the (nhop (rt ξ) dip)\"\n unfolding invalidate_def\n (*goal: ‹⋀dip. the (map_option π₆ (case (rt ξ dip, dests ξ dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba))) = the (nhop (rt ξ) dip)›*)\n by (clarsimp split: option.split (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma invalidate_other [simp]:\n fixes rt dests dip\n assumes \"dip∉dom(dests)\"\n shows \"invalidate rt dests dip = rt dip\"\n using assms (*‹dip ∉ dom dests›*) unfolding invalidate_def\n (*goal: ‹(case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = rt dip›*)\n by (clarsimp split: option.split_asm (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n\nlemma invalidate_none [simp]:\n fixes rt dests dip\n assumes \"dip∉kD(rt)\"\n shows \"invalidate rt dests dip = None\"\n using assms (*‹dip ∉ kD rt›*) unfolding invalidate_def\n (*goal: ‹(case ((rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat), (dests::nat ⇒ nat option) dip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = None›*)\n by clarsimp\n\nlemma vD_invalidate_vD_not_dests:\n \"⋀dip rt dests. dip∈vD(invalidate rt dests) ⟹ dip∈vD(rt) ∧ dests dip = None\"\n unfolding invalidate_def vD_def\n (*goal: ‹⋀dip rt dests. dip ∈ {dip. map_option π₄ (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = Some val} ⟹ dip ∈ {dip. flag rt dip = Some val} ∧ dests dip = None›*)\n by (clarsimp split: option.split_asm (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n\nlemma sqn_invalidate_not_in_dests [simp]:\n fixes dests dip rt\n assumes \"dip∉dom(dests)\"\n shows \"sqn (invalidate rt dests) dip = sqn rt dip\"\n using assms (*‹(dip::nat) ∉ dom (dests::nat ⇒ nat option)›*) unfolding sqn_def\n (*goal: ‹(case invalidate rt dests dip of None ⇒ 0 | Some r ⇒ π₂ r) = (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r)›*)\n by simp\n\nlemma invalidate_changes:\n fixes rt dests dip dsn dsk flag hops nhip pre\n assumes \"invalidate rt dests dip = Some (dsn, dsk, flag, hops, nhip, pre)\"\n shows \" dsn = (case dests dip of None ⇒ π₂(the (rt dip)) | Some rsn ⇒ rsn)\n ∧ dsk = π₃(the (rt dip))\n ∧ flag = (if dests dip = None then π₄(the (rt dip)) else inv)\n ∧ hops = π₅(the (rt dip))\n ∧ nhip = π₆(the (rt dip))\n ∧ pre = π₇(the (rt dip))\"\n using assms (*‹invalidate (rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dests::nat ⇒ nat option) (dip::nat) = Some (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set)›*) unfolding invalidate_def\n (*goal: ‹dsn = (case dests dip of None ⇒ π₂ (the (rt dip)) | Some rsn ⇒ rsn) ∧ dsk = π₃ (the (rt dip)) ∧ flag = (if dests dip = None then π₄ (the (rt dip)) else Aodv_Basic.inv) ∧ hops = π₅ (the (rt dip)) ∧ nhip = π₆ (the (rt dip)) ∧ pre = π₇ (the (rt dip))›*)\n apply (cases \"rt dip\")\n (*goals:\n 1. ‹⟦(case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = Some (dsn, dsk, flag, hops, nhip, pre); rt dip = None⟧ ⟹ dsn = (case dests dip of None ⇒ π₂ (the (rt dip)) | Some rsn ⇒ rsn) ∧ dsk = π₃ (the (rt dip)) ∧ flag = (if dests dip = None then π₄ (the (rt dip)) else Aodv_Basic.inv) ∧ hops = π₅ (the (rt dip)) ∧ nhip = π₆ (the (rt dip)) ∧ pre = π₇ (the (rt dip))›\n 2. ‹⋀a. ⟦(case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = Some (dsn, dsk, flag, hops, nhip, pre); rt dip = Some a⟧ ⟹ dsn = (case dests dip of None ⇒ π₂ (the (rt dip)) | Some rsn ⇒ rsn) ∧ dsk = π₃ (the (rt dip)) ∧ flag = (if dests dip = None then π₄ (the (rt dip)) else Aodv_Basic.inv) ∧ hops = π₅ (the (rt dip)) ∧ nhip = π₆ (the (rt dip)) ∧ pre = π₇ (the (rt dip))›\n discuss goal 1*)\n apply clarsimp\n (*discuss goal 2*)\n apply (cases \"dests dip\")\n (*goals:\n 1. ‹⋀a::nat × k × f × nat × nat × nat set. ⟦(case ((rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat), (dests::nat ⇒ nat option) dip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = Some (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set); rt dip = Some a; dests dip = None⟧ ⟹ dsn = (case dests dip of None ⇒ π₂ (the (rt dip)) | Some (rsn::nat) ⇒ rsn) ∧ dsk = π₃ (the (rt dip)) ∧ flag = (if dests dip = None then π₄ (the (rt dip)) else Aodv_Basic.inv) ∧ hops = π₅ (the (rt dip)) ∧ nhip = π₆ (the (rt dip)) ∧ pre = π₇ (the (rt dip))›\n 2. ‹⋀(a::nat × k × f × nat × nat × nat set) aa::nat. ⟦(case ((rt::nat ⇒ (nat × k × f × nat × nat × nat set) option) (dip::nat), (dests::nat ⇒ nat option) dip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a::nat, ab::k, ad::f, af::nat, ah::nat, ba::nat set), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = Some (dsn::nat, dsk::k, flag::f, hops::nat, nhip::nat, pre::nat set); rt dip = Some a; dests dip = Some aa⟧ ⟹ dsn = (case dests dip of None ⇒ π₂ (the (rt dip)) | Some (rsn::nat) ⇒ rsn) ∧ dsk = π₃ (the (rt dip)) ∧ flag = (if dests dip = None then π₄ (the (rt dip)) else Aodv_Basic.inv) ∧ hops = π₅ (the (rt dip)) ∧ nhip = π₆ (the (rt dip)) ∧ pre = π₇ (the (rt dip))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n\nlemma proj3_inv: \"⋀dip rt dests. dip∈kD (rt)\n ⟹ π₃(the (invalidate rt dests dip)) = π₃(the (rt dip))\"\n by (clarsimp simp: invalidate_def (*‹invalidate ?rt ?dests ≡ λip. case (?rt ip, ?dests ip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)›*) kD_def (*‹kD ?rt ≡ {dip. ?rt dip ≠ None}›*) split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma dests_iD_invalidate [simp]:\n assumes \"dests ip = Some rsn\"\n and \"ip∈kD(rt)\"\n shows \"ip∈iD(invalidate rt dests)\"\n using assms(1) (*‹dests ip = Some rsn›*) assms(2)[THEN kD_Some] (*‹∃dsn dsk flag hops nhip pre. rt ip = Some (dsn, dsk, flag, hops, nhip, pre)›*) unfolding invalidate_def iD_def\n (*goal: ‹ip ∈ {dip. map_option π₄ (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, ah, ba), None) ⇒ Some (a, ab, ad, af, ah, ba) | (Some (a, ab, ad, af, ah, ba), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, ah, ba)) = Some Aodv_Basic.inv}›*)\n by (clarsimp split: option.split (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nsubsection \"Route Requests\"\n\ntext ‹Generate a fresh route request identifier.›\n\ndefinition nrreqid :: \"(ip × rreqid) set ⇒ ip ⇒ rreqid\"\n where \"nrreqid rreqs ip ≡ Max ({n. (ip, n) ∈ rreqs} ∪ {0}) + 1\"\n\nsubsection \"Queued Packets\"\n\ntext ‹Functions for sending data packets.›\n\ntype_synonym store = \"ip ⇀ (p × data list)\"\n\ndefinition sigma_queue :: \"store ⇒ ip ⇒ data list\" (\"σ⇘queue⇙'(_, _')\")\n where \"σ⇘queue⇙(store, dip) ≡ case store dip of None ⇒ [] | Some (p, q) ⇒ q\"\n\ndefinition qD :: \"store ⇒ ip set\"\n where \"qD ≡ dom\"\n\ndefinition add :: \"data ⇒ ip ⇒ store ⇒ store\"\n where \"add d dip store ≡ case store dip of\n None ⇒ store (dip ↦ (req, [d]))\n | Some (p, q) ⇒ store (dip ↦ (p, q @ [d]))\"\n\nlemma qD_add [simp]:\n fixes d dip store\n shows \"qD(add d dip store) = insert dip (qD store)\"\n unfolding add_def Let_def qD_def\n (*goal: ‹dom (case store dip of None ⇒ store(dip ↦ (req, [d])) | Some (p, q) ⇒ store(dip ↦ (p, q @ [d]))) = insert dip (dom store)›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\ndefinition drop :: \"ip ⇒ store ⇀ store\"\n where \"drop dip store ≡\n map_option (λ(p, q). if tl q = [] then store (dip := None)\n else store (dip ↦ (p, tl q))) (store dip)\"\n\ndefinition sigma_p_flag :: \"store ⇒ ip ⇀ p\" (\"σ⇘p-flag⇙'(_, _')\")\n where \"σ⇘p-flag⇙(store, dip) ≡ map_option fst (store dip)\"\n\ndefinition unsetRRF :: \"store ⇒ ip ⇒ store\"\n where \"unsetRRF store dip ≡ case store dip of\n None ⇒ store\n | Some (p, q) ⇒ store (dip ↦ (noreq, q))\"\n\ndefinition setRRF :: \"store ⇒ (ip ⇀ sqn) ⇒ store\"\n where \"setRRF store dests ≡ λdip. if dests dip = None then store dip\n else map_option (λ(_, q). (req, q)) (store dip)\"\n\nsubsection \"Comparison with the original technical report\"\n\ntext ‹\n The major differences with the AODV technical report of Fehnker et al are:\n \\begin{enumerate}\n \\item @{term nhop} is partial, thus a `@{term the}' is needed, similarly for @{term dhops}\n and @{term addpreRT}.\n \\item @{term precs} is partial.\n \\item @{term \"σ⇘p-flag⇙(store, dip)\"} is partial.\n \\item The routing table (@{typ rt}) is modelled as a map (@{typ \"ip ⇒ r option\"})\n rather than a set of 7-tuples, likewise, the @{typ r} is a 6-tuple rather than\n a 7-tuple, i.e., the destination ip-address (@{term \"dip\"}) is taken from the\n argument to the function, rather than a part of the result. Well-definedness then\n follows from the structure of the type and more related facts are available\n automatically, rather than having to be acquired through tedious proofs.\n \\item Similar remarks hold for the dests mapping passed to @{term \"invalidate\"},\n and @{term \"store\"}.\n \\end{enumerate}\n›\n\nend\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/AODV/variants/d_fwdrreqs/D_Aodv_Data.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"4cd0bd4f1775252ecd1644968440bc76d20520ec3c3ab74a42f99dd7a7eca54d\"\n}"}}},{"rowIdx":149,"cells":{"text":{"kind":"string","value":"(* Copyright 2021 (C) Mihails Milehins *)\n\nsection‹Initial and terminal objects as limits and colimits›\ntheory CZH_UCAT_Limit_IT\n imports \n CZH_UCAT_Limit\n CZH_Elementary_Categories.CZH_ECAT_Comma\nbegin\n\n\n\nsubsection‹Initial and terminal objects as limits/colimits of an empty diagram›\n\n\nsubsubsection‹Definition and elementary properties›\n\ntext‹\nSee \n\\<^cite>‹\"noauthor_nlab_nodate\"›\\footnote{\n\\url{https://ncatlab.org/nlab/show/initial+object}\n}, \\<^cite>‹\"noauthor_nlab_nodate\"›\\footnote{\n\\url{https://ncatlab.org/nlab/show/terminal+object}\n} and Chapter X-1 in \\<^cite>‹\"mac_lane_categories_2010\"›.\n›\n\nlocale is_cat_obj_empty_terminal = is_cat_limit α cat_0 ℭ ‹cf_0 ℭ› z ℨ \n for α ℭ z ℨ\n\nsyntax \"_is_cat_obj_empty_terminal\" :: \"V ⇒ V ⇒ V ⇒ V ⇒ bool\"\n (‹(_ :/ _ <⇩C⇩F⇩.₁ 0⇩C⇩F :/ 0⇩C ↦↦⇩Cı _)› [51, 51] 51)\ntranslations \"ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ\" ⇌ \n \"CONST is_cat_obj_empty_terminal α ℭ z ℨ\"\n\nlocale is_cat_obj_empty_initial = is_cat_colimit α cat_0 ℭ ‹cf_0 ℭ› z ℨ \n for α ℭ z ℨ\n\nsyntax \"_is_cat_obj_empty_initial\" :: \"V ⇒ V ⇒ V ⇒ V ⇒ bool\"\n (‹(_ :/ 0⇩C⇩F >⇩C⇩F⇩.₀ _ :/ 0⇩C ↦↦⇩Cı _)› [51, 51] 51)\ntranslations \"ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ ℭ\" ⇌ \n \"CONST is_cat_obj_empty_initial α ℭ z ℨ\"\n\n\ntext‹Rules.›\n\nlemma (in is_cat_obj_empty_terminal) \n is_cat_obj_empty_terminal_axioms'[cat_lim_cs_intros]:\n assumes \"α' = α\" and \"z' = z\" and \"ℭ' = ℭ\" \n shows \"ℨ : z' <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α'⇙ ℭ'\"\n unfolding assms\n (*goal: ‹ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*)\n by (rule is_cat_obj_empty_terminal_axioms (*‹ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*))\n\nmk_ide rf is_cat_obj_empty_terminal_def\n |intro is_cat_obj_empty_terminalI|\n |dest is_cat_obj_empty_terminalD[dest]|\n |elim is_cat_obj_empty_terminalE[elim]|\n\nlemmas [cat_lim_cs_intros] = is_cat_obj_empty_terminalD\n\nlemma (in is_cat_obj_empty_initial) \n is_cat_obj_empty_initial_axioms'[cat_lim_cs_intros]:\n assumes \"α' = α\" and \"z' = z\" and \"ℭ' = ℭ\" \n shows \"ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ z' : 0⇩C ↦↦⇩C⇘α'⇙ ℭ'\"\n unfolding assms\n (*goal: ‹ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*)\n by (rule is_cat_obj_empty_initial_axioms (*‹(ℨ::V) : 0⇩C⇩F >⇩C⇩F⇩.₀ (z::V) : 0⇩C ↦↦⇩C⇘α::V⇙ ℭ::V›*))\n\nmk_ide rf is_cat_obj_empty_initial_def\n |intro is_cat_obj_empty_initialI|\n |dest is_cat_obj_empty_initialD[dest]|\n |elim is_cat_obj_empty_initialE[elim]|\n\nlemmas [cat_lim_cs_intros] = is_cat_obj_empty_initialD\n\n\ntext‹Duality.›\n\nlemma (in is_cat_obj_empty_terminal) is_cat_obj_empty_initial_op:\n \"op_ntcf ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ op_cat ℭ\"\n apply (intro is_cat_obj_empty_initialI (*‹?ℨ : cf_0 ?ℭ >⇩C⇩F⇩.⇩c⇩o⇩l⇩i⇩m ?z : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ ?z : 0⇩C ↦↦⇩C⇘?α⇙ ?ℭ›*))\n (*goal: ‹op_ntcf ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ op_cat ℭ›*)\n by (cs_concl cs_shallow cs_simp: cat_op_simps op_cf_cf_0 cs_intro: cat_cs_intros cat_op_intros)\n\nlemma (in is_cat_obj_empty_terminal) is_cat_obj_empty_initial_op'[cat_op_intros]:\n assumes \"ℭ' = op_cat ℭ\"\n shows \"op_ntcf ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ ℭ'\"\n unfolding assms\n (*goal: ‹op_ntcf (ℨ::V) : 0⇩C⇩F >⇩C⇩F⇩.₀ (z::V) : 0⇩C ↦↦⇩C⇘α::V⇙ op_cat (ℭ::V)›*)\n by (rule is_cat_obj_empty_initial_op (*‹op_ntcf (ℨ::V) : 0⇩C⇩F >⇩C⇩F⇩.₀ (z::V) : 0⇩C ↦↦⇩C⇘α::V⇙ op_cat (ℭ::V)›*))\n\nlemmas [cat_op_intros] = is_cat_obj_empty_terminal.is_cat_obj_empty_initial_op'\n\nlemma (in is_cat_obj_empty_initial) is_cat_obj_empty_terminal_op:\n \"op_ntcf ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ op_cat ℭ\"\n apply (intro is_cat_obj_empty_terminalI (*‹?ℨ : ?z <⇩C⇩F⇩.⇩l⇩i⇩m cf_0 ?ℭ : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?ℨ : ?z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘?α⇙ ?ℭ›*))\n (*goal: ‹op_ntcf ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ op_cat ℭ›*)\n by (cs_concl cs_shallow cs_simp: cat_op_simps op_cf_cf_0 cs_intro: cat_cs_intros cat_op_intros)\n\nlemma (in is_cat_obj_empty_initial) is_cat_obj_empty_terminal_op'[cat_op_intros]:\n assumes \"ℭ' = op_cat ℭ\"\n shows \"op_ntcf ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ'\"\n unfolding assms\n (*goal: ‹op_ntcf ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ op_cat ℭ›*)\n by (rule is_cat_obj_empty_terminal_op (*‹op_ntcf ℨ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ op_cat ℭ›*))\n\nlemmas [cat_op_intros] = is_cat_obj_empty_initial.is_cat_obj_empty_terminal_op'\n\n\ntext‹Elementary properties.›\n\nlemma (in is_cat_obj_empty_terminal) cat_oet_ntcf_0: \"ℨ = ntcf_0 ℭ\"\n apply (rule is_ntcf_is_ntcf_0_if_cat_0 (*‹?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔑 = ntcf_0 ?ℭ› ‹?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔉 = cf_0 ?ℭ› ‹?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔊 = cf_0 ?ℭ›*))\n (*goal: ‹(ℨ::V) = ntcf_0 (ℭ::V)›*)\n by (cs_concl cs_shallow cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n\nlemma (in is_cat_obj_empty_initial) cat_oei_ntcf_0: \"ℨ = ntcf_0 ℭ\"\n apply (rule is_ntcf_is_ntcf_0_if_cat_0 (*‹?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔑 = ntcf_0 ?ℭ› ‹?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔉 = cf_0 ?ℭ› ‹?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔊 = cf_0 ?ℭ›*))\n (*goal: ‹ℨ = ntcf_0 ℭ›*)\n by (cs_concl cs_shallow cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n\n\nsubsubsection‹\nInitial and terminal objects as limits/colimits of an empty diagram \nare initial and terminal objects\n›\n\nlemma (in category) cat_obj_terminal_is_cat_obj_empty_terminal:\n assumes \"obj_terminal ℭ z\"\n shows \"ntcf_0 ℭ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ\"\nproof (-)\n (*goal: ‹ntcf_0 ℭ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*)\n from assms (*‹obj_terminal (ℭ::V) (z::V)›*) have z: \"z ∈⇩∘ ℭ⦇Obj⦈\"\n by auto\n from z (*‹z ∈⇩∘ ℭ⦇Obj⦈›*) have [cat_cs_simps]: \"cf_const cat_0 ℭ z = cf_0 ℭ\"\n apply (intro is_functor_is_cf_0_if_cat_0 (*‹(?𝔉::V) : cat_0 ↦↦⇩C⇘?α::V⇙ (?ℭ::V) ⟹ ?𝔉 = cf_0 ?ℭ›*))\n (*goal: ‹dghm_const cat_0 ℭ z (ℭ⦇CId⦈⦇z⦈) = cf_0 ℭ›*)\n by (cs_concl cs_intro: cat_cs_intros)\n note obj_terminalD = obj_terminalD[OF assms] (*‹z ∈⇩∘ ℭ⦇Obj⦈› ‹(?a::V) ∈⇩∘ (ℭ::V)⦇Obj⦈ ⟹ Ex1 (is_arr ℭ ?a (z::V))›*)\n show \"?thesis\"\n (*goal: ‹ntcf_0 ℭ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*)\n proof (intro is_cat_obj_empty_terminalI (*‹?ℨ : ?z <⇩C⇩F⇩.⇩l⇩i⇩m cf_0 ?ℭ : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?ℨ : ?z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘?α⇙ ?ℭ›*) is_cat_limitI (*‹⟦?u : ?r <⇩C⇩F⇩.⇩c⇩o⇩n⇩e ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ; ⋀u' r'. u' : r' <⇩C⇩F⇩.⇩c⇩o⇩n⇩e ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ∃!f'. f' : r' ↦⇘?ℭ⇙ ?r ∧ u' = ?u ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ?𝔍 ?ℭ f'⟧ ⟹ ?u : ?r <⇩C⇩F⇩.⇩l⇩i⇩m ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ›*) is_cat_coneI (*‹⟦?𝔑 : dghm_const ?𝔍 ?ℭ ?c (?ℭ⦇CId⦈⦇?c⦈) ↦⇩C⇩F ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ; ?c ∈⇩∘ ?ℭ⦇Obj⦈⟧ ⟹ ?𝔑 : ?c <⇩C⇩F⇩.⇩c⇩o⇩n⇩e ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ›*), unfold cat_cs_simps (*‹category ?α ?ℭ ⟹ vcard ?ℭ = 6⇩ℕ› ‹category ?α ?ℭ ⟹ 𝒟⇩∘ (?ℭ⦇CId⦈) = ?ℭ⦇Obj⦈› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇘?ℭ⇙ ?b⟧ ⟹ ?ℭ⦇CId⦈⦇?b⦈ ∘⇩A⇘?ℭ⇙ ?f = ?f› ‹⟦category ?α ?ℭ; ?f : ?b ↦⇘?ℭ⇙ ?c⟧ ⟹ ?f ∘⇩A⇘?ℭ⇙ ?ℭ⦇CId⦈⦇?b⦈ = ?f› ‹?f : ↦⇩i⇩d⇩e⇘?ℭ⇙ ?b ⟹ ?f ∘⇩A⇘?ℭ⇙ ?f = ?f› ‹⟦category ?α ?ℭ; ?h : ?c ↦⇘?ℭ⇙ ?d; ?g : ?b ↦⇘?ℭ⇙ ?c; ?f : ?a ↦⇘?ℭ⇙ ?b⟧ ⟹ ?h ∘⇩A⇘?ℭ⇙ ?g ∘⇩A⇘?ℭ⇙ ?f = ?h ∘⇩A⇘?ℭ⇙ (?g ∘⇩A⇘?ℭ⇙ ?f)› ‹⟦category ?α ?ℭ; ?g : ?b ↦⇩i⇩s⇩o⇘?ℭ⇙ ?c; ?f : ?a ↦⇩i⇩s⇩o⇘?ℭ⇙ ?b⟧ ⟹ (?g ∘⇩A⇘?ℭ⇙ ?f)¯⇩C⇘?ℭ⇙ = ?f¯⇩C⇘?ℭ⇙ ∘⇩A⇘?ℭ⇙ ?g¯⇩C⇘?ℭ⇙› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇩i⇩s⇩o⇘?ℭ⇙ ?b⟧ ⟹ ?f¯⇩C⇘?ℭ⇙ ∘⇩A⇘?ℭ⇙ ?f = ?ℭ⦇CId⦈⦇?a⦈› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇩i⇩s⇩o⇘?ℭ⇙ ?b⟧ ⟹ ?f ∘⇩A⇘?ℭ⇙ ?f¯⇩C⇘?ℭ⇙ = ?ℭ⦇CId⦈⦇?b⦈› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇩i⇩s⇩o⇘?ℭ⇙ ?b⟧ ⟹ (?f¯⇩C⇘?ℭ⇙)¯⇩C⇘?ℭ⇙ = ?f› ‹?f : ?a ↦⇘?ℭ⇙ ?b ⟹ ?ℭ⦇Dom⦈⦇?f⦈ = ?a› ‹?f : ?a ↦⇘?ℭ⇙ ?b ⟹ ?ℭ⦇Cod⦈⦇?f⦈ = ?b› and more 355 facts*))\n (*goals:\n 1. ‹ntcf_0 ℭ : cf_0 ℭ ↦⇩C⇩F cf_0 ℭ : cat_0 ↦↦⇩C⇘α⇙ ℭ›\n 2. ‹z ∈⇩∘ ℭ⦇Obj⦈›\n 3. ‹⋀u' r'. u' : r' <⇩C⇩F⇩.⇩c⇩o⇩n⇩e cf_0 ℭ : cat_0 ↦↦⇩C⇘α⇙ ℭ ⟹ ∃!f'. f' : r' ↦⇘ℭ⇙ z ∧ u' = ntcf_0 ℭ ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ f'›*)\n show \"∃!f'. f' : r' ↦⇘ℭ⇙ z ∧ u' = ntcf_0 ℭ ∙⇩N⇩T⇩C⇩F ntcf_const cat_0 ℭ f'\" if \"u' : r' <⇩C⇩F⇩.⇩c⇩o⇩n⇩e cf_0 ℭ : cat_0 ↦↦⇩C⇘α⇙ ℭ\" for u' and r'\n proof (-)\n (*goal: ‹∃!f'. f' : r' ↦⇘ℭ⇙ z ∧ u' = ntcf_0 ℭ ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ f'›*)\n interpret u': is_cat_cone \"α\" r' cat_0 \"ℭ\" \"cf_0 ℭ\" u'\n by (rule that (*‹u' : r' <⇩C⇩F⇩.⇩c⇩o⇩n⇩e cf_0 ℭ : cat_0 ↦↦⇩C⇘α⇙ ℭ›*))\n from z (*‹z ∈⇩∘ ℭ⦇Obj⦈›*) have [cat_cs_simps]: \"cf_const cat_0 ℭ r' = cf_0 ℭ\"\n apply (intro is_functor_is_cf_0_if_cat_0 (*‹?𝔉 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔉 = cf_0 ?ℭ›*))\n (*goal: ‹dghm_const cat_0 (ℭ::V) (r'::V) (ℭ⦇CId⦈⦇r'⦈) = cf_0 ℭ›*)\n by (cs_concl cs_shallow cs_intro: cat_cs_intros)\n have u'_def: \"u' = ntcf_0 ℭ\"\n by (rule is_ntcf_is_ntcf_0_if_cat_0[ OF u'.is_ntcf_axioms, unfolded cat_cs_simps (*‹u' = ntcf_0 ℭ› ‹cf_0 ℭ = cf_0 ℭ› ‹cf_0 ℭ = cf_0 ℭ›*) ])\n from obj_terminalD(2)[OF u'.cat_cone_obj] (*‹Ex1 (is_arr (ℭ::V) (r'::V) (z::V))›*) obtain f' where f': \"f' : r' ↦⇘ℭ⇙ z\" and f'_unique: \"f'' : r' ↦⇘ℭ⇙ z ⟹ f'' = f'\" for f''\n (*goal: ‹(⋀f'. ⟦f' : r' ↦⇘ℭ⇙ z; ⋀f''. f'' : r' ↦⇘ℭ⇙ z ⟹ f'' = f'⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n from f' (*‹f' : r' ↦⇘ℭ⇙ z›*) have [cat_cs_simps]: \"ntcf_const cat_0 ℭ f' = ntcf_0 ℭ\"\n apply (intro is_ntcf_is_ntcf_0_if_cat_0( (*‹?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔑 = ntcf_0 ?ℭ›*) 1))\n (*goal: ‹ntcf_const cat_0 ℭ f' = ntcf_0 ℭ›*)\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n show \"?thesis\"\n (*goal: ‹∃!f'. f' : r' ↦⇘ℭ⇙ z ∧ u' = ntcf_0 ℭ ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ f'›*)\n proof (intro ex1I (*‹⟦(?P::?'a ⇒ bool) (?a::?'a); ⋀x::?'a. ?P x ⟹ x = ?a⟧ ⟹ ∃!x::?'a. ?P x›*) conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*); (elim conjE (*‹⟦?P ∧ ?Q; ⟦?P; ?Q⟧ ⟹ ?R⟧ ⟹ ?R›*))?)\n (*goals:\n 1. ‹?a1 : r' ↦⇘ℭ⇙ z›\n 2. ‹u' = ntcf_0 ℭ ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ ?a1›\n 3. ‹⋀f'. ⟦f' : r' ↦⇘ℭ⇙ z; u' = ntcf_0 ℭ ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ f'⟧ ⟹ f' = ?a1›*)\n show \"u' = ntcf_0 ℭ ∙⇩N⇩T⇩C⇩F ntcf_const cat_0 ℭ f'\"\n by (cs_concl cs_shallow cs_simp: u'_def cat_cs_simps cs_intro: cat_cs_intros)\n fix f''\n assume prems: \"f'' : r' ↦⇘ℭ⇙ z\" \"u' = ntcf_0 ℭ ∙⇩N⇩T⇩C⇩F ntcf_const cat_0 ℭ f''\" (*‹(f''::V) : (r'::V) ↦⇘ℭ::V⇙ (z::V)› ‹(u'::V) = ntcf_0 (ℭ::V) ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ (f''::V)›*)\n show \"f'' = f'\"\n by (rule f'_unique[OF prems(1)] (*‹f'' = f'›*))\n qed (rule f' (*‹(f'::V) : (r'::V) ↦⇘ℭ::V⇙ (z::V)›*))\n (*solved the remaining goal: ‹f' : r' ↦⇘ℭ⇙ z›*)\n qed\n qed (cs_concl cs_simp: cat_cs_simps cs_intro: z cat_cs_intros)\n (*solves the remaining goals:\n 1. ‹ntcf_0 ℭ : cf_0 ℭ ↦⇩C⇩F cf_0 ℭ : cat_0 ↦↦⇩C⇘α⇙ ℭ›\n 2. ‹z ∈⇩∘ ℭ⦇Obj⦈›*)\nqed\n\nlemma (in category) cat_obj_initial_is_cat_obj_empty_initial:\n assumes \"obj_initial ℭ z\"\n shows \"ntcf_0 ℭ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ ℭ\"\nproof (-)\n (*goal: ‹ntcf_0 ℭ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*)\n have z: \"obj_terminal (op_cat ℭ) z\"\n unfolding cat_op_simps\n (*goal: ‹obj_initial ℭ z›*)\n by (rule assms (*‹obj_initial ℭ z›*))\n show \"?thesis\"\n (*goal: ‹ntcf_0 ℭ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*)\n by (rule is_cat_obj_empty_terminal.is_cat_obj_empty_initial_op [ OF category.cat_obj_terminal_is_cat_obj_empty_terminal[ OF category_op z, folded op_ntcf_ntcf_0 ], unfolded cat_op_simps op_ntcf_ntcf_0 (*‹ntcf_0 ℭ : 0⇩C⇩F >⇩C⇩F⇩.₀ (z::V) : 0⇩C ↦↦⇩C⇘α::V⇙ ℭ::V›*) ])\nqed\n\nlemma (in is_cat_obj_empty_terminal) cat_oet_obj_terminal: \"obj_terminal ℭ z\"\nproof (-)\n (*goal: ‹obj_terminal ℭ z›*)\n show \"obj_terminal ℭ z\"\n proof (rule obj_terminalI (*‹⟦?t ∈⇩∘ ?ℭ⦇Obj⦈; ⋀a. a ∈⇩∘ ?ℭ⦇Obj⦈ ⟹ Ex1 (is_arr ?ℭ a ?t)⟧ ⟹ obj_terminal ?ℭ ?t›*))\n (*goals:\n 1. ‹z ∈⇩∘ ℭ⦇Obj⦈›\n 2. ‹⋀a. a ∈⇩∘ ℭ⦇Obj⦈ ⟹ Ex1 (is_arr ℭ a z)›*)\n fix a\n assume prems: \"a ∈⇩∘ ℭ⦇Obj⦈\" (*‹(a::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*)\n have [cat_cs_simps]: \"cf_const cat_0 ℭ a = cf_0 ℭ\"\n apply (rule is_functor_is_cf_0_if_cat_0 (*‹?𝔉 : cat_0 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?𝔉 = cf_0 ?ℭ›*))\n (*goal: ‹dghm_const cat_0 ℭ a (ℭ⦇CId⦈⦇a⦈) = cf_0 ℭ›*)\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros prems)\n from prems (*‹a ∈⇩∘ ℭ⦇Obj⦈›*) have \"ntcf_0 ℭ : a <⇩C⇩F⇩.⇩c⇩o⇩n⇩e cf_0 ℭ : cat_0 ↦↦⇩C⇘α⇙ ℭ\"\n apply (intro is_cat_coneI (*‹⟦?𝔑 : dghm_const ?𝔍 ?ℭ ?c (?ℭ⦇CId⦈⦇?c⦈) ↦⇩C⇩F ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ; ?c ∈⇩∘ ?ℭ⦇Obj⦈⟧ ⟹ ?𝔑 : ?c <⇩C⇩F⇩.⇩c⇩o⇩n⇩e ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ›*))\n (*goals:\n 1. ‹a ∈⇩∘ ℭ⦇Obj⦈ ⟹ ntcf_0 ℭ : dghm_const cat_0 ℭ a (ℭ⦇CId⦈⦇a⦈) ↦⇩C⇩F cf_0 ℭ : cat_0 ↦↦⇩C⇘α⇙ ℭ›\n 2. ‹a ∈⇩∘ ℭ⦇Obj⦈ ⟹ a ∈⇩∘ ℭ⦇Obj⦈›\n discuss goal 1*)\n apply (cs_concl cs_shallow cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n (*discuss goal 2*) .\n (*proven 2 subgoals*)\n from cat_lim_ua_fo[OF this] (*‹∃!f'::V. f' : (a::V) ↦⇘ℭ::V⇙ (z::V) ∧ ntcf_0 ℭ = (ℨ::V) ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ f'›*) obtain f' where f': \"f' : a ↦⇘ℭ⇙ z\" and \"ntcf_0 ℭ = ℨ ∙⇩N⇩T⇩C⇩F ntcf_const cat_0 ℭ f'\" and f'_unique: \"⟦ f'' : a ↦⇘ℭ⇙ z; ntcf_0 ℭ = ℨ ∙⇩N⇩T⇩C⇩F ntcf_const cat_0 ℭ f'' ⟧ ⟹\n f'' = f'\" for f''\n (*goal: ‹(⋀f'. ⟦f' : a ↦⇘ℭ⇙ z; ntcf_0 ℭ = ℨ ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ f'; ⋀f''. ⟦f'' : a ↦⇘ℭ⇙ z; ntcf_0 ℭ = ℨ ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const cat_0 ℭ f''⟧ ⟹ f'' = f'⟧ ⟹ thesis) ⟹ thesis›*)\n by metis\n show \"∃!f'. f' : a ↦⇘ℭ⇙ z\"\n proof (intro ex1I (*‹⟦(?P::?'a ⇒ bool) (?a::?'a); ⋀x::?'a. ?P x ⟹ x = ?a⟧ ⟹ ∃!x::?'a. ?P x›*))\n (*goals:\n 1. ‹(?a::V) : (a::V) ↦⇘ℭ::V⇙ (z::V)›\n 2. ‹⋀f'::V. f' : (a::V) ↦⇘ℭ::V⇙ (z::V) ⟹ f' = (?a::V)›*)\n fix f''\n assume prems': \"f'' : a ↦⇘ℭ⇙ z\" (*‹(f''::V) : (a::V) ↦⇘ℭ::V⇙ (z::V)›*)\n from prems' (*‹f'' : a ↦⇘ℭ⇙ z›*) have \"ntcf_0 ℭ = ntcf_0 ℭ ∙⇩N⇩T⇩C⇩F ntcf_const cat_0 ℭ f''\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n from f'_unique[OF prems', unfolded cat_oet_ntcf_0, OF this] (*‹f'' = f'›*) show \"f'' = f'\" .\n qed (rule f' (*‹f' : a ↦⇘ℭ⇙ z›*))\n (*solved the remaining goal: ‹(f'::V) : (a::V) ↦⇘ℭ::V⇙ (z::V)›*)\n qed (rule cat_cone_obj (*‹(z::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*))\n (*solved the remaining goal: ‹z ∈⇩∘ ℭ⦇Obj⦈›*)\nqed\n\nlemma (in is_cat_obj_empty_initial) cat_oei_obj_initial: \"obj_initial ℭ z\"\n by (rule is_cat_obj_empty_terminal.cat_oet_obj_terminal[ OF is_cat_obj_empty_initial.is_cat_obj_empty_terminal_op[ OF is_cat_obj_empty_initial_axioms ], unfolded cat_op_simps (*‹obj_initial ℭ z›*) ])\n\nlemma (in category) cat_is_cat_obj_empty_terminal_obj_terminal_iff:\n \"(ntcf_0 ℭ : z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ) ⟷ obj_terminal ℭ z\"\n using cat_obj_terminal_is_cat_obj_empty_terminal (*‹obj_terminal ℭ ?z ⟹ ntcf_0 ℭ : ?z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*) is_cat_obj_empty_terminal.cat_oet_obj_terminal (*‹?ℨ : ?z <⇩C⇩F⇩.₁ 0⇩C⇩F : 0⇩C ↦↦⇩C⇘?α⇙ ?ℭ ⟹ obj_terminal ?ℭ ?z›*) by auto\n\nlemma (in category) cat_is_cat_obj_empty_initial_obj_initial_iff:\n \"(ntcf_0 ℭ : 0⇩C⇩F >⇩C⇩F⇩.₀ z : 0⇩C ↦↦⇩C⇘α⇙ ℭ) ⟷ obj_initial ℭ z\"\n using cat_obj_initial_is_cat_obj_empty_initial (*‹obj_initial ℭ ?z ⟹ ntcf_0 ℭ : 0⇩C⇩F >⇩C⇩F⇩.₀ ?z : 0⇩C ↦↦⇩C⇘α⇙ ℭ›*) is_cat_obj_empty_initial.cat_oei_obj_initial (*‹?ℨ : 0⇩C⇩F >⇩C⇩F⇩.₀ ?z : 0⇩C ↦↦⇩C⇘?α⇙ ?ℭ ⟹ obj_initial ?ℭ ?z›*) by auto\n\n\n\nsubsection‹Initial cone and terminal cocone›\n\n\nsubsubsection‹Definitions and elementary properties›\n\ndefinition ntcf_initial :: \"V ⇒ V ⇒ V\"\n where \"ntcf_initial ℭ z =\n [\n (λb∈⇩∘ℭ⦇Obj⦈. THE f. f : z ↦⇘ℭ⇙ b),\n cf_const ℭ ℭ z,\n cf_id ℭ,\n ℭ,\n ℭ\n ]⇩∘\"\n\ndefinition ntcf_terminal :: \"V ⇒ V ⇒ V\"\n where \"ntcf_terminal ℭ z =\n [\n (λb∈⇩∘ℭ⦇Obj⦈. THE f. f : b ↦⇘ℭ⇙ z),\n cf_id ℭ,\n cf_const ℭ ℭ z,\n ℭ,\n ℭ\n ]⇩∘\"\n\n\ntext‹Components.›\n\nlemma ntcf_initial_components:\n shows \"ntcf_initial ℭ z⦇NTMap⦈ = (λc∈⇩∘ℭ⦇Obj⦈. THE f. f : z ↦⇘ℭ⇙ c)\"\n and \"ntcf_initial ℭ z⦇NTDom⦈ = cf_const ℭ ℭ z\"\n and \"ntcf_initial ℭ z⦇NTCod⦈ = cf_id ℭ\"\n and \"ntcf_initial ℭ z⦇NTDGDom⦈ = ℭ\"\n and \"ntcf_initial ℭ z⦇NTDGCod⦈ = ℭ\"\n unfolding ntcf_initial_def nt_field_simps\n (*goals:\n 1. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇[]⇩∘⦈ = (λc∈⇩∘ℭ⦇Obj⦈. THE f. f : z ↦⇘ℭ⇙ c)›\n 2. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇1⇩ℕ⦈ = dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)›\n 3. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇2⇩ℕ⦈ = dghm_id ℭ›\n 4. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇3⇩ℕ⦈ = ℭ›\n 5. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇4⇩ℕ⦈ = ℭ›*)\n (*goals:\n 1. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇[]⇩∘⦈ = (λc∈⇩∘ℭ⦇Obj⦈. THE f. f : z ↦⇘ℭ⇙ c)›\n 2. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇1⇩ℕ⦈ = dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)›\n 3. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇2⇩ℕ⦈ = dghm_id ℭ›\n 4. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇3⇩ℕ⦈ = ℭ›\n 5. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇4⇩ℕ⦈ = ℭ›\n discuss goal 1*)\n apply (simp add: nat_omega_simps (*‹(?a::nat)⇩ℕ + (?b::nat)⇩ℕ = (?a + ?b)⇩ℕ› ‹(?a::nat)⇩ℕ * (?b::nat)⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ ((?a::nat)⇩ℕ) = Suc ?a⇩ℕ› ‹(?m::nat)⇩ℕ ⊕ (?n::nat)⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc ((?a::nat)⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = (1::nat)⇩ℕ› ‹(1::V) = (1::nat)⇩ℕ›*))\n (*discuss goal 2*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*discuss goal 3*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*discuss goal 4*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*discuss goal 5*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*proven 5 subgoals*) .\n\nlemmas [cat_lim_cs_simps] = ntcf_initial_components(2-5)\n \nlemma ntcf_terminal_components:\n shows \"ntcf_terminal ℭ z⦇NTMap⦈ = (λc∈⇩∘ℭ⦇Obj⦈. THE f. f : c ↦⇘ℭ⇙ z)\"\n and \"ntcf_terminal ℭ z⦇NTDom⦈ = cf_id ℭ\"\n and \"ntcf_terminal ℭ z⦇NTCod⦈ = cf_const ℭ ℭ z\"\n and \"ntcf_terminal ℭ z⦇NTDGDom⦈ = ℭ\"\n and \"ntcf_terminal ℭ z⦇NTDGCod⦈ = ℭ\"\n unfolding ntcf_terminal_def nt_field_simps\n (*goals:\n 1. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇[]⇩∘⦈ = (λc∈⇩∘ℭ⦇Obj⦈. THE f. f : c ↦⇘ℭ⇙ z)›\n 2. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇1⇩ℕ⦈ = dghm_id ℭ›\n 3. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇2⇩ℕ⦈ = dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)›\n 4. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇3⇩ℕ⦈ = ℭ›\n 5. ‹[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇4⇩ℕ⦈ = ℭ›*)\n (*goals:\n 1. ‹[λb::V∈⇩∘(ℭ::V)⦇Obj⦈. The (is_arr ℭ b (z::V)), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇[]⇩∘⦈ = (λc::V∈⇩∘ℭ⦇Obj⦈. THE f::V. f : c ↦⇘ℭ⇙ z)›\n 2. ‹[λb::V∈⇩∘(ℭ::V)⦇Obj⦈. The (is_arr ℭ b (z::V)), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇(1::nat)⇩ℕ⦈ = dghm_id ℭ›\n 3. ‹[λb::V∈⇩∘(ℭ::V)⦇Obj⦈. The (is_arr ℭ b (z::V)), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇(2::nat)⇩ℕ⦈ = dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)›\n 4. ‹[λb::V∈⇩∘(ℭ::V)⦇Obj⦈. The (is_arr ℭ b (z::V)), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇(3::nat)⇩ℕ⦈ = ℭ›\n 5. ‹[λb::V∈⇩∘(ℭ::V)⦇Obj⦈. The (is_arr ℭ b (z::V)), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇(4::nat)⇩ℕ⦈ = ℭ›\n discuss goal 1*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*discuss goal 2*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*discuss goal 3*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*discuss goal 4*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*discuss goal 5*)\n apply (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n (*proven 5 subgoals*) .\n\nlemmas [cat_lim_cs_simps] = ntcf_terminal_components(2-5)\n\n\ntext‹Duality.›\n\nlemma ntcf_initial_op[cat_op_simps]:\n \"op_ntcf (ntcf_initial ℭ z) = ntcf_terminal (op_cat ℭ) z\"\n unfolding ntcf_initial_def ntcf_terminal_def op_ntcf_def nt_field_simps cat_op_simps\n (*goal: ‹[[λb::V∈⇩∘(ℭ::V)⦇Obj⦈. The (is_arr ℭ (z::V) b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇[]⇩∘⦈, op_cf ([λb::V∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇(2::nat)⇩ℕ⦈), op_cf ([λb::V∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇(1::nat)⇩ℕ⦈), op_cat ([λb::V∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇(3::nat)⇩ℕ⦈), op_cat ([λb::V∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘⦇(4::nat)⇩ℕ⦈)]⇩∘ = [λb::V∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_id (op_cat ℭ), dghm_const (op_cat ℭ) (op_cat ℭ) z (ℭ⦇CId⦈⦇z⦈), op_cat ℭ, op_cat ℭ]⇩∘›*)\n by (auto simp: nat_omega_simps (*‹(?a::nat)⇩ℕ + (?b::nat)⇩ℕ = (?a + ?b)⇩ℕ› ‹(?a::nat)⇩ℕ * (?b::nat)⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ ((?a::nat)⇩ℕ) = Suc ?a⇩ℕ› ‹(?m::nat)⇩ℕ ⊕ (?n::nat)⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc ((?a::nat)⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = (1::nat)⇩ℕ› ‹(1::V) = (1::nat)⇩ℕ›*) cat_op_simps (*‹op_cat (?ℭ::V)⦇Obj⦈ = ?ℭ⦇Obj⦈› ‹op_cat (?ℭ::V)⦇Arr⦈ = ?ℭ⦇Arr⦈› ‹op_cat (?ℭ::V)⦇Dom⦈ = ?ℭ⦇Cod⦈› ‹op_cat (?ℭ::V)⦇Cod⦈ = ?ℭ⦇Dom⦈› ‹op_cat (?ℭ::V)⦇CId⦈ = ?ℭ⦇CId⦈› ‹(?f::V) : (?b::V) ↦⇘op_cat (?ℭ::V)⇙ (?a::V) = ?f : ?a ↦⇘?ℭ⇙ ?b› ‹Hom (op_cat (?ℭ::V)) (?a::V) (?b::V) = Hom ?ℭ ?b ?a› ‹obj_initial (op_cat (?ℭ::V)) (?a::V) = obj_terminal ?ℭ ?a› ‹obj_terminal (op_cat (?ℭ::V)) (?a::V) = obj_initial ?ℭ ?a› ‹obj_null (op_cat (?ℭ::V)) (?a::V) = obj_null ?ℭ ?a› ‹category (?α::V) (?ℭ::V) ⟹ ℛ⇩∘ (op_cat ?ℭ⦇Comp⦈) = ℛ⇩∘ (?ℭ⦇Comp⦈)› ‹⟦category (?α::V) (?ℭ::V); (?f::V) : (?b::V) ↦⇘?ℭ⇙ (?c::V); (?g::V) : (?a::V) ↦⇘?ℭ⇙ ?b⟧ ⟹ ?g ∘⇩A⇘op_cat ?ℭ⇙ ?f = ?f ∘⇩A⇘?ℭ⇙ ?g› and more 48 facts*)) \n\nlemma ntcf_cone_terminal_op[cat_op_simps]:\n \"op_ntcf (ntcf_terminal ℭ z) = ntcf_initial (op_cat ℭ) z\"\n unfolding ntcf_initial_def ntcf_terminal_def op_ntcf_def nt_field_simps cat_op_simps\n (*goal: ‹[[λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇[]⇩∘⦈, op_cf ([λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇2⇩ℕ⦈), op_cf ([λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇1⇩ℕ⦈), op_cat ([λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇3⇩ℕ⦈), op_cat ([λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_id ℭ, dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), ℭ, ℭ]⇩∘⦇4⇩ℕ⦈)]⇩∘ = [λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ b z), dghm_const (op_cat ℭ) (op_cat ℭ) z (ℭ⦇CId⦈⦇z⦈), dghm_id (op_cat ℭ), op_cat ℭ, op_cat ℭ]⇩∘›*)\n by (auto simp: nat_omega_simps (*‹(?a::nat)⇩ℕ + (?b::nat)⇩ℕ = (?a + ?b)⇩ℕ› ‹(?a::nat)⇩ℕ * (?b::nat)⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ ((?a::nat)⇩ℕ) = Suc ?a⇩ℕ› ‹(?m::nat)⇩ℕ ⊕ (?n::nat)⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc ((?a::nat)⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = (1::nat)⇩ℕ› ‹(1::V) = (1::nat)⇩ℕ›*) cat_op_simps (*‹op_cat (?ℭ::V)⦇Obj⦈ = ?ℭ⦇Obj⦈› ‹op_cat (?ℭ::V)⦇Arr⦈ = ?ℭ⦇Arr⦈› ‹op_cat (?ℭ::V)⦇Dom⦈ = ?ℭ⦇Cod⦈› ‹op_cat (?ℭ::V)⦇Cod⦈ = ?ℭ⦇Dom⦈› ‹op_cat (?ℭ::V)⦇CId⦈ = ?ℭ⦇CId⦈› ‹(?f::V) : (?b::V) ↦⇘op_cat (?ℭ::V)⇙ (?a::V) = ?f : ?a ↦⇘?ℭ⇙ ?b› ‹Hom (op_cat (?ℭ::V)) (?a::V) (?b::V) = Hom ?ℭ ?b ?a› ‹obj_initial (op_cat (?ℭ::V)) (?a::V) = obj_terminal ?ℭ ?a› ‹obj_terminal (op_cat (?ℭ::V)) (?a::V) = obj_initial ?ℭ ?a› ‹obj_null (op_cat (?ℭ::V)) (?a::V) = obj_null ?ℭ ?a› ‹category (?α::V) (?ℭ::V) ⟹ ℛ⇩∘ (op_cat ?ℭ⦇Comp⦈) = ℛ⇩∘ (?ℭ⦇Comp⦈)› ‹⟦category (?α::V) (?ℭ::V); (?f::V) : (?b::V) ↦⇘?ℭ⇙ (?c::V); (?g::V) : (?a::V) ↦⇘?ℭ⇙ ?b⟧ ⟹ ?g ∘⇩A⇘op_cat ?ℭ⇙ ?f = ?f ∘⇩A⇘?ℭ⇙ ?g› and more 49 facts*))\n\n\nsubsubsection‹Natural transformation map›\n\nmk_VLambda ntcf_initial_components(1)\n |vsv ntcf_initial_vsv[cat_lim_cs_intros]|\n |vdomain ntcf_initial_vdomain[cat_lim_cs_simps]|\n |app ntcf_initial_app|\n\nmk_VLambda ntcf_terminal_components(1)\n |vsv ntcf_terminal_vsv[cat_lim_cs_intros]|\n |vdomain ntcf_terminal_vdomain[cat_lim_cs_simps]|\n |app ntcf_terminal_app|\n\nlemma (in category)\n assumes \"obj_initial ℭ z\" and \"c ∈⇩∘ ℭ⦇Obj⦈\"\n shows ntcf_initial_NTMap_app_is_arr: \n \"ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c\"\n and ntcf_initial_NTMap_app_unique: \n \"⋀f'. f' : z ↦⇘ℭ⇙ c ⟹ f' = ntcf_initial ℭ z⦇NTMap⦈⦇c⦈\"\nproof (-)\n (*goals:\n 1. ‹ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c›\n 2. ‹⋀f'. f' : z ↦⇘ℭ⇙ c ⟹ f' = ntcf_initial ℭ z⦇NTMap⦈⦇c⦈›*)\n from obj_initialD(2)[OF assms ( 1 , 2 )] (*‹∃!f. f : z ↦⇘ℭ⇙ c›*) obtain f where f: \"f : z ↦⇘ℭ⇙ c\" and f_unique: \"f' : z ↦⇘ℭ⇙ c ⟹ f' = f\" for f'\n (*goal: ‹(⋀f. ⟦f : z ↦⇘ℭ⇙ c; ⋀f'. f' : z ↦⇘ℭ⇙ c ⟹ f' = f⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n show is_arr: \"ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c\"\n proof (cs_concl_step ntcf_initial_app, rule assms( (*‹(c::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*) 2), rule theI (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ?P (THE x. ?P x)›*))\n (*goals:\n 1. ‹?a1 : z ↦⇘ℭ⇙ c›\n 2. ‹⋀x. x : z ↦⇘ℭ⇙ c ⟹ x = ?a1›*)\n fix f'\n assume \"f' : z ↦⇘ℭ⇙ c\" (*‹(f'::V) : (z::V) ↦⇘ℭ::V⇙ (c::V)›*)\n from f_unique[OF this] (*‹(f'::V) = (f::V)›*) show \"f' = f\" .\n qed (rule f (*‹f : z ↦⇘ℭ⇙ c›*))\n (*solved the remaining goal: ‹f : z ↦⇘ℭ⇙ c›*)\n fix f'\n assume \"f' : z ↦⇘ℭ⇙ c\" (*‹(f'::V) : (z::V) ↦⇘ℭ::V⇙ (c::V)›*)\n from f_unique[OF this, folded f_unique [ OF is_arr ]] (*‹f' = ntcf_initial ℭ z⦇NTMap⦈⦇c⦈›*) show \"f' = ntcf_initial ℭ z⦇NTMap⦈⦇c⦈\" .\nqed\n\nlemma (in category) ntcf_initial_NTMap_app_is_arr'[cat_lim_cs_intros]:\n assumes \"obj_initial ℭ z\" \n and \"c ∈⇩∘ ℭ⦇Obj⦈\" \n and \"ℭ' = ℭ\" \n and \"z' = z\"\n and \"c' = c\"\n shows \"ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ : z' ↦⇘ℭ'⇙ c'\"\n using assms(1,2) (*‹obj_initial ℭ z› ‹c ∈⇩∘ ℭ⦇Obj⦈›*) unfolding assms(3-5)\n (*goal: ‹ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c›*)\n by (rule ntcf_initial_NTMap_app_is_arr (*‹⟦obj_initial ℭ ?z; ?c ∈⇩∘ ℭ⦇Obj⦈⟧ ⟹ ntcf_initial ℭ ?z⦇NTMap⦈⦇?c⦈ : ?z ↦⇘ℭ⇙ ?c›*))\n\nlemma (in category)\n assumes \"obj_terminal ℭ z\" and \"c ∈⇩∘ ℭ⦇Obj⦈\"\n shows ntcf_terminal_NTMap_app_is_arr: \n \"ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈ : c ↦⇘ℭ⇙ z\"\n and ntcf_terminal_NTMap_app_unique: \n \"⋀f'. f' : c ↦⇘ℭ⇙ z ⟹ f' = ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈\"\nproof (-)\n (*goals:\n 1. ‹ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈ : (c::V) ↦⇘ℭ::V⇙ (z::V)›\n 2. ‹⋀f'::V. f' : (c::V) ↦⇘ℭ::V⇙ (z::V) ⟹ f' = ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈›*)\n from obj_terminalD(2)[OF assms ( 1 , 2 )] (*‹Ex1 (is_arr ℭ c z)›*) obtain f where f: \"f : c ↦⇘ℭ⇙ z\" and f_unique: \"f' : c ↦⇘ℭ⇙ z ⟹ f' = f\" for f'\n (*goal: ‹(⋀f::V. ⟦f : (c::V) ↦⇘ℭ::V⇙ (z::V); ⋀f'::V. f' : c ↦⇘ℭ⇙ z ⟹ f' = f⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n show is_arr: \"ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈ : c ↦⇘ℭ⇙ z\"\n proof (cs_concl_step ntcf_terminal_app, rule assms( (*‹(c::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*) 2), rule theI (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ?P (THE x. ?P x)›*))\n (*goals:\n 1. ‹?a1 : c ↦⇘ℭ⇙ z›\n 2. ‹⋀x. x : c ↦⇘ℭ⇙ z ⟹ x = ?a1›*)\n fix f'\n assume \"f' : c ↦⇘ℭ⇙ z\" (*‹(f'::V) : (c::V) ↦⇘ℭ::V⇙ (z::V)›*)\n from f_unique[OF this] (*‹f' = f›*) show \"f' = f\" .\n qed (rule f (*‹f : c ↦⇘ℭ⇙ z›*))\n (*solved the remaining goal: ‹(f::V) : (c::V) ↦⇘ℭ::V⇙ (z::V)›*)\n fix f'\n assume \"f' : c ↦⇘ℭ⇙ z\" (*‹(f'::V) : (c::V) ↦⇘ℭ::V⇙ (z::V)›*)\n from f_unique[OF this, folded f_unique [ OF is_arr ]] (*‹f' = ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈›*) show \"f' = ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈\" .\nqed\n\nlemma (in category) ntcf_terminal_NTMap_app_is_arr'[cat_lim_cs_intros]:\n assumes \"obj_terminal ℭ z\" \n and \"c ∈⇩∘ ℭ⦇Obj⦈\" \n and \"ℭ' = ℭ\" \n and \"z' = z\"\n and \"c' = c\"\n shows \"ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈ : c' ↦⇘ℭ'⇙ z'\"\n using assms(1,2) (*‹obj_terminal ℭ z› ‹c ∈⇩∘ ℭ⦇Obj⦈›*) unfolding assms(3-5)\n (*goal: ‹ntcf_terminal ℭ z⦇NTMap⦈⦇c⦈ : (c::V) ↦⇘ℭ::V⇙ (z::V)›*)\n by (rule ntcf_terminal_NTMap_app_is_arr (*‹⟦obj_terminal ℭ ?z; ?c ∈⇩∘ ℭ⦇Obj⦈⟧ ⟹ ntcf_terminal ℭ ?z⦇NTMap⦈⦇?c⦈ : ?c ↦⇘ℭ⇙ ?z›*))\n\n\n\nsubsection‹\nInitial and terminal objects as limits/colimits of the identity functor\n›\n\n\nsubsubsection‹Definition and elementary properties›\n\ntext‹\nSee \n\\<^cite>‹\"noauthor_nlab_nodate\"›\\footnote{\n\\url{https://ncatlab.org/nlab/show/initial+object}\n}, \\<^cite>‹\"noauthor_nlab_nodate\"›\\footnote{\n\\url{https://ncatlab.org/nlab/show/terminal+object}\n} and Chapter X-1 in \\<^cite>‹\"mac_lane_categories_2010\"›.\n›\n\nlocale is_cat_obj_id_initial = is_cat_limit α ℭ ℭ ‹cf_id ℭ› z ℨ for α ℭ z ℨ\n\nsyntax \"_is_cat_obj_id_initial\" :: \"V ⇒ V ⇒ V ⇒ V ⇒ bool\"\n (‹(_ :/ _ <⇩C⇩F⇩.₀ id⇩C :/ ↦↦⇩Cı _)› [51, 51, 51] 51)\ntranslations \"ℨ : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ ℭ\" ⇌ \n \"CONST is_cat_obj_id_initial α ℭ z ℨ\"\n\nlocale is_cat_obj_id_terminal = is_cat_colimit α ℭ ℭ ‹cf_id ℭ› z ℨ for α ℭ z ℨ\n\nsyntax \"_is_cat_obj_id_terminal\" :: \"V ⇒ V ⇒ V ⇒ V ⇒ bool\"\n (‹(_ :/ id⇩C >⇩C⇩F⇩.₁ _ :/ ↦↦⇩Cı _)› [51, 51, 51] 51)\ntranslations \"ℨ : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ ℭ\" ⇌ \n \"CONST is_cat_obj_id_terminal α ℭ z ℨ\"\n\n\ntext‹Rules.›\n\nlemma (in is_cat_obj_id_initial)\n is_cat_obj_id_initial_axioms'[cat_lim_cs_intros]:\n assumes \"α' = α\" and \"z' = z\" and \"ℭ' = ℭ\" \n shows \"ℨ : z' <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ ℭ'\"\n unfolding assms\n (*goal: ‹ℨ : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ ℭ›*)\n by (rule is_cat_obj_id_initial_axioms (*‹ℨ : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ ℭ›*))\n\nmk_ide rf is_cat_obj_id_initial_def\n |intro is_cat_obj_id_initialI|\n |dest is_cat_obj_id_initialD[dest]|\n |elim is_cat_obj_id_initialE[elim]|\n\nlemmas [cat_lim_cs_intros] = is_cat_obj_id_initialD\n\nlemma (in is_cat_obj_id_terminal) \n is_cat_obj_id_terminal_axioms'[cat_lim_cs_intros]:\n assumes \"α' = α\" and \"z' = z\" and \"ℭ' = ℭ\" \n shows \"ℨ : id⇩C >⇩C⇩F⇩.₁ z' : ↦↦⇩C⇘α'⇙ ℭ'\"\n unfolding assms\n (*goal: ‹(ℨ::V) : id⇩C >⇩C⇩F⇩.₁ (z::V) : ↦↦⇩C⇘α::V⇙ (ℭ::V)›*)\n by (rule is_cat_obj_id_terminal_axioms (*‹(ℨ::V) : id⇩C >⇩C⇩F⇩.₁ (z::V) : ↦↦⇩C⇘α::V⇙ (ℭ::V)›*))\n\nmk_ide rf is_cat_obj_id_terminal_def\n |intro is_cat_obj_id_terminalI|\n |dest is_cat_obj_id_terminalD[dest]|\n |elim is_cat_obj_id_terminalE[elim]|\n\nlemmas [cat_lim_cs_intros] = is_cat_obj_id_terminalD\n\n\ntext‹Duality.›\n\nlemma (in is_cat_obj_id_initial) is_cat_obj_id_terminal_op:\n \"op_ntcf ℨ : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ op_cat ℭ\"\n apply (intro is_cat_obj_id_terminalI (*‹?ℨ : dghm_id ?ℭ >⇩C⇩F⇩.⇩c⇩o⇩l⇩i⇩m ?z : ?ℭ ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?ℨ : id⇩C >⇩C⇩F⇩.₁ ?z : ↦↦⇩C⇘?α⇙ ?ℭ›*))\n (*goal: ‹op_ntcf ℨ : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ op_cat ℭ›*)\n by (cs_concl cs_shallow cs_simp: cat_op_simps cs_intro: cat_op_intros)\n\nlemma (in is_cat_obj_id_initial) is_cat_obj_id_terminal_op'[cat_op_intros]:\n assumes \"ℭ' = op_cat ℭ\"\n shows \"op_ntcf ℨ : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ ℭ'\"\n unfolding assms\n (*goal: ‹op_ntcf ℨ : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ op_cat ℭ›*)\n by (rule is_cat_obj_id_terminal_op (*‹op_ntcf ℨ : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ op_cat ℭ›*))\n\nlemmas [cat_op_intros] = is_cat_obj_id_initial.is_cat_obj_id_terminal_op'\n\nlemma (in is_cat_obj_id_terminal) is_cat_obj_id_initial_op:\n \"op_ntcf ℨ : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ op_cat ℭ\"\n apply (intro is_cat_obj_id_initialI (*‹?ℨ : ?z <⇩C⇩F⇩.⇩l⇩i⇩m dghm_id ?ℭ : ?ℭ ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?ℨ : ?z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘?α⇙ ?ℭ›*))\n (*goal: ‹op_ntcf (ℨ::V) : (z::V) <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α::V⇙ op_cat (ℭ::V)›*)\n by (cs_concl cs_shallow cs_simp: cat_op_simps cs_intro: cat_op_intros)\n\nlemma (in is_cat_obj_id_terminal) is_cat_obj_id_initial_op'[cat_op_intros]:\n assumes \"ℭ' = op_cat ℭ\"\n shows \"op_ntcf ℨ : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ ℭ'\"\n unfolding assms\n (*goal: ‹op_ntcf ℨ : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ op_cat ℭ›*)\n by (rule is_cat_obj_id_initial_op (*‹op_ntcf ℨ : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ op_cat ℭ›*))\n\nlemmas [cat_op_intros] = is_cat_obj_id_terminal.is_cat_obj_id_initial_op'\n\n\nsubsubsection‹\nInitial and terminal objects as limits/colimits are initial and terminal objects\n›\n\nlemma (in category) cat_obj_initial_is_cat_obj_id_initial:\n assumes \"obj_initial ℭ z\"\n shows \"ntcf_initial ℭ z : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ ℭ\"\nproof (intro is_cat_obj_id_initialI (*‹?ℨ : ?z <⇩C⇩F⇩.⇩l⇩i⇩m dghm_id ?ℭ : ?ℭ ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ?ℨ : ?z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘?α⇙ ?ℭ›*) is_cat_limitI (*‹⟦?u : ?r <⇩C⇩F⇩.⇩c⇩o⇩n⇩e ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ; ⋀u' r'. u' : r' <⇩C⇩F⇩.⇩c⇩o⇩n⇩e ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ ⟹ ∃!f'. f' : r' ↦⇘?ℭ⇙ ?r ∧ u' = ?u ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ?𝔍 ?ℭ f'⟧ ⟹ ?u : ?r <⇩C⇩F⇩.⇩l⇩i⇩m ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ›*))\n (*goals:\n 1. ‹ntcf_initial ℭ z : z <⇩C⇩F⇩.⇩c⇩o⇩n⇩e dghm_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ›\n 2. ‹⋀u' r'. u' : r' <⇩C⇩F⇩.⇩c⇩o⇩n⇩e dghm_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ ⟹ ∃!f'. f' : r' ↦⇘ℭ⇙ z ∧ u' = ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ f'›*)\n from assms (*‹obj_initial ℭ z›*) have z: \"z ∈⇩∘ ℭ⦇Obj⦈\"\n by auto\n note obj_initialD = obj_initialD[OF assms] (*‹z ∈⇩∘ ℭ⦇Obj⦈› ‹?b ∈⇩∘ ℭ⦇Obj⦈ ⟹ ∃!f. f : z ↦⇘ℭ⇙ ?b›*)\n show \"ntcf_initial ℭ z : z <⇩C⇩F⇩.⇩c⇩o⇩n⇩e cf_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ\"\n proof (intro is_cat_coneI (*‹⟦?𝔑 : dghm_const ?𝔍 ?ℭ ?c (?ℭ⦇CId⦈⦇?c⦈) ↦⇩C⇩F ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ; ?c ∈⇩∘ ?ℭ⦇Obj⦈⟧ ⟹ ?𝔑 : ?c <⇩C⇩F⇩.⇩c⇩o⇩n⇩e ?𝔉 : ?𝔍 ↦↦⇩C⇘?α⇙ ?ℭ›*) is_ntcfI' (*‹⟦𝒵 ?α; vfsequence ?𝔑; vcard ?𝔑 = 5⇩ℕ; ?𝔉 : ?𝔄 ↦↦⇩C⇘?α⇙ ?𝔅; ?𝔊 : ?𝔄 ↦↦⇩C⇘?α⇙ ?𝔅; ?𝔑⦇NTDom⦈ = ?𝔉; ?𝔑⦇NTCod⦈ = ?𝔊; ?𝔑⦇NTDGDom⦈ = ?𝔄; ?𝔑⦇NTDGCod⦈ = ?𝔅; vsv (?𝔑⦇NTMap⦈); 𝒟⇩∘ (?𝔑⦇NTMap⦈) = ?𝔄⦇Obj⦈; ⋀a. a ∈⇩∘ ?𝔄⦇Obj⦈ ⟹ ?𝔑⦇NTMap⦈⦇a⦈ : ?𝔉⦇ObjMap⦈⦇a⦈ ↦⇘?𝔅⇙ ?𝔊⦇ObjMap⦈⦇a⦈; ⋀a b f. f : a ↦⇘?𝔄⇙ b ⟹ ?𝔑⦇NTMap⦈⦇b⦈ ∘⇩A⇘?𝔅⇙ ?𝔉⦇ArrMap⦈⦇f⦈ = ?𝔊⦇ArrMap⦈⦇f⦈ ∘⇩A⇘?𝔅⇙ ?𝔑⦇NTMap⦈⦇a⦈⟧ ⟹ ?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : ?𝔄 ↦↦⇩C⇘?α⇙ ?𝔅›*), unfold cat_lim_cs_simps (*‹ntcf_initial ?ℭ ?z⦇NTDom⦈ = dghm_const ?ℭ ?ℭ ?z (?ℭ⦇CId⦈⦇?z⦈)› ‹ntcf_initial ?ℭ ?z⦇NTCod⦈ = dghm_id ?ℭ› ‹ntcf_initial ?ℭ ?z⦇NTDGDom⦈ = ?ℭ› ‹ntcf_initial ?ℭ ?z⦇NTDGCod⦈ = ?ℭ› ‹ntcf_terminal ?ℭ ?z⦇NTDom⦈ = dghm_id ?ℭ› ‹ntcf_terminal ?ℭ ?z⦇NTCod⦈ = dghm_const ?ℭ ?ℭ ?z (?ℭ⦇CId⦈⦇?z⦈)› ‹ntcf_terminal ?ℭ ?z⦇NTDGDom⦈ = ?ℭ› ‹ntcf_terminal ?ℭ ?z⦇NTDGCod⦈ = ?ℭ› ‹𝒟⇩∘ (ntcf_initial ?ℭ ?z⦇NTMap⦈) = ?ℭ⦇Obj⦈› ‹𝒟⇩∘ (ntcf_terminal ?ℭ ?z⦇NTMap⦈) = ?ℭ⦇Obj⦈›*))\n (*goals:\n 1. ‹𝒵 α›\n 2. ‹vfsequence (ntcf_initial ℭ z)›\n 3. ‹vcard (ntcf_initial ℭ z) = 5⇩ℕ›\n 4. ‹dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈) : ℭ ↦↦⇩C⇘α⇙ ℭ›\n 5. ‹dghm_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ›\n 6. ‹dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈) = dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)›\n 7. ‹dghm_id ℭ = dghm_id ℭ›\n 8. ‹ℭ = ℭ›\n 9. ‹ℭ = ℭ›\n 10. ‹vsv (ntcf_initial ℭ z⦇NTMap⦈)›\n 11. ‹ℭ⦇Obj⦈ = ℭ⦇Obj⦈›\n 12. ‹⋀a. a ∈⇩∘ ℭ⦇Obj⦈ ⟹ ntcf_initial ℭ z⦇NTMap⦈⦇a⦈ : dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)⦇ObjMap⦈⦇a⦈ ↦⇘ℭ⇙ dghm_id ℭ⦇ObjMap⦈⦇a⦈›\n 13. ‹⋀a b f. f : a ↦⇘ℭ⇙ b ⟹ ntcf_initial ℭ z⦇NTMap⦈⦇b⦈ ∘⇩A⇘ℭ⇙ dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)⦇ArrMap⦈⦇f⦈ = dghm_id ℭ⦇ArrMap⦈⦇f⦈ ∘⇩A⇘ℭ⇙ ntcf_initial ℭ z⦇NTMap⦈⦇a⦈›\n 14. ‹z ∈⇩∘ ℭ⦇Obj⦈›*)\n show \"vfsequence (ntcf_initial ℭ z)\"\n unfolding ntcf_initial_def\n (*goal: ‹vfsequence [λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘›*)\n by auto\n show \"vcard (ntcf_initial ℭ z) = 5⇩ℕ\"\n unfolding ntcf_initial_def\n (*goal: ‹vcard [λb∈⇩∘ℭ⦇Obj⦈. The (is_arr ℭ z b), dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈), dghm_id ℭ, ℭ, ℭ]⇩∘ = 5⇩ℕ›*)\n by (simp add: nat_omega_simps (*‹?a⇩ℕ + ?b⇩ℕ = (?a + ?b)⇩ℕ› ‹?a⇩ℕ * ?b⇩ℕ = (?a * ?b)⇩ℕ› ‹ZFC_in_HOL.succ (?a⇩ℕ) = Suc ?a⇩ℕ› ‹?m⇩ℕ ⊕ ?n⇩ℕ = (?m + ?n)⇩ℕ› ‹csucc (?a⇩ℕ) = ZFC_in_HOL.succ (?a⇩ℕ)› ‹ZFC_in_HOL.succ []⇩∘ = 1⇩ℕ› ‹1 = 1⇩ℕ›*))\n show \"ntcf_initial ℭ z⦇NTMap⦈⦇a⦈ :\n cf_const ℭ ℭ z⦇ObjMap⦈⦇a⦈ ↦⇘ℭ⇙ cf_id ℭ⦇ObjMap⦈⦇a⦈\" if \"a ∈⇩∘ ℭ⦇Obj⦈\" for a\n using that (*‹a ∈⇩∘ ℭ⦇Obj⦈›*) assms(1) (*‹obj_initial ℭ z›*) by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros cat_lim_cs_intros)\n show \"ntcf_initial ℭ z⦇NTMap⦈⦇b⦈ ∘⇩A⇘ℭ⇙ cf_const ℭ ℭ z⦇ArrMap⦈⦇f⦈ =\n cf_id ℭ⦇ArrMap⦈⦇f⦈ ∘⇩A⇘ℭ⇙ ntcf_initial ℭ z⦇NTMap⦈⦇a⦈\" if \"f : a ↦⇘ℭ⇙ b\" for a and b and f\n proof (-)\n (*goal: ‹ntcf_initial ℭ z⦇NTMap⦈⦇b⦈ ∘⇩A⇘ℭ⇙ dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)⦇ArrMap⦈⦇f⦈ = dghm_id ℭ⦇ArrMap⦈⦇f⦈ ∘⇩A⇘ℭ⇙ ntcf_initial ℭ z⦇NTMap⦈⦇a⦈›*)\n from that (*‹(f::V) : (a::V) ↦⇘ℭ::V⇙ (b::V)›*) assms(1) (*‹obj_initial ℭ z›*) have \"f ∘⇩A⇘ℭ⇙ ntcf_initial ℭ z⦇NTMap⦈⦇a⦈ : z ↦⇘ℭ⇙ b\"\n by (cs_concl cs_intro: cat_cs_intros cat_lim_cs_intros)\n note [cat_cs_simps] = ntcf_initial_NTMap_app_unique[OF assms ( 1 ) cat_is_arrD ( 3 ) [ OF that ] this] (*‹f ∘⇩A⇘ℭ⇙ ntcf_initial ℭ z⦇NTMap⦈⦇a⦈ = ntcf_initial ℭ z⦇NTMap⦈⦇b⦈›*)\n from that (*‹(f::V) : (a::V) ↦⇘ℭ::V⇙ (b::V)›*) assms(1) (*‹obj_initial (ℭ::V) (z::V)›*) show \"?thesis\"\n (*goal: ‹ntcf_initial ℭ z⦇NTMap⦈⦇b⦈ ∘⇩A⇘ℭ⇙ dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)⦇ArrMap⦈⦇f⦈ = dghm_id ℭ⦇ArrMap⦈⦇f⦈ ∘⇩A⇘ℭ⇙ ntcf_initial ℭ z⦇NTMap⦈⦇a⦈›*)\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros cat_lim_cs_intros)\n qed\n qed ((use z in ‹cs_concl cs_intro: cat_cs_intros cat_lim_cs_intros›)+)\n (*solves the remaining goals:\n 1. ‹𝒵 α›\n 2. ‹dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈) : ℭ ↦↦⇩C⇘α⇙ ℭ›\n 3. ‹dghm_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ›\n 4. ‹dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈) = dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)›\n 5. ‹dghm_id ℭ = dghm_id ℭ›\n 6. ‹ℭ = ℭ›\n 7. ‹ℭ = ℭ›\n 8. ‹vsv (ntcf_initial ℭ z⦇NTMap⦈)›\n 9. ‹ℭ⦇Obj⦈ = ℭ⦇Obj⦈›\n 10. ‹z ∈⇩∘ ℭ⦇Obj⦈›*)\n then interpret i: is_cat_cone \"α\" z \"ℭ\" \"ℭ\" \"cf_id ℭ\" \"ntcf_initial ℭ z\" .\n fix u and r\n assume \"u : r <⇩C⇩F⇩.⇩c⇩o⇩n⇩e cf_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ\" (*‹(u::V) : (r::V) <⇩C⇩F⇩.⇩c⇩o⇩n⇩e dghm_id ℭ : (ℭ::V) ↦↦⇩C⇘α::V⇙ ℭ›*)\n then interpret u: is_cat_cone \"α\" r \"ℭ\" \"ℭ\" \"cf_id ℭ\" u .\n from obj_initialD(2)[OF u.cat_cone_obj] (*‹∃!f. f : z ↦⇘ℭ⇙ r›*) obtain f where f: \"f : z ↦⇘ℭ⇙ r\" and f_unique: \"f' : z ↦⇘ℭ⇙ r ⟹ f' = f\" for f'\n (*goal: ‹(⋀f. ⟦f : z ↦⇘ℭ⇙ r; ⋀f'. f' : z ↦⇘ℭ⇙ r ⟹ f' = f⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n note u.cat_cone_Comp_commute[cat_cs_simps del] (*‹?f : ?a ↦⇘ℭ⇙ ?b ⟹ dghm_id ℭ⦇ArrMap⦈⦇?f⦈ ∘⇩A⇘ℭ⇙ u⦇NTMap⦈⦇?a⦈ = u⦇NTMap⦈⦇?b⦈›*)\n from u.ntcf_Comp_commute[OF f] (*‹u⦇NTMap⦈⦇r⦈ ∘⇩A⇘ℭ⇙ dghm_const ℭ ℭ r (ℭ⦇CId⦈⦇r⦈)⦇ArrMap⦈⦇f⦈ = dghm_id ℭ⦇ArrMap⦈⦇f⦈ ∘⇩A⇘ℭ⇙ u⦇NTMap⦈⦇z⦈›*) f (*‹f : z ↦⇘ℭ⇙ r›*) have \"u⦇NTMap⦈⦇r⦈ = f ∘⇩A⇘ℭ⇙ u⦇NTMap⦈⦇z⦈\"\n by (cs_prems cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n show \"∃!f'.\n f' : r ↦⇘ℭ⇙ z ∧\n u = ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ f'\"\n proof (intro ex1I (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ∃!x. ?P x›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*); (elim conjE (*‹⟦?P ∧ ?Q; ⟦?P; ?Q⟧ ⟹ ?R⟧ ⟹ ?R›*))?)\n (*goals:\n 1. ‹?a1 : r ↦⇘ℭ⇙ z›\n 2. ‹u = ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ ?a1›\n 3. ‹⋀f'. ⟦f' : r ↦⇘ℭ⇙ z; u = ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ f'⟧ ⟹ f' = ?a1›*)\n from f (*‹f : z ↦⇘ℭ⇙ r›*) show \"u⦇NTMap⦈⦇z⦈ : r ↦⇘ℭ⇙ z\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n show \"u = ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈)\"\n proof (rule ntcf_eqI (*‹⟦?𝔑 : ?𝔉 ↦⇩C⇩F ?𝔊 : ?𝔄 ↦↦⇩C⇘?α⇙ ?𝔅; ?𝔑' : ?𝔉' ↦⇩C⇩F ?𝔊' : ?𝔄' ↦↦⇩C⇘?α⇙ ?𝔅'; ?𝔑⦇NTMap⦈ = ?𝔑'⦇NTMap⦈; ?𝔉 = ?𝔉'; ?𝔊 = ?𝔊'; ?𝔄 = ?𝔄'; ?𝔅 = ?𝔅'⟧ ⟹ ?𝔑 = ?𝔑'›*), rule u.is_ntcf_axioms (*‹u : dghm_const ℭ ℭ r (ℭ⦇CId⦈⦇r⦈) ↦⇩C⇩F dghm_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ›*))\n (*goals:\n 1. ‹ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈) : ?𝔉' ↦⇩C⇩F ?𝔊' : ?𝔄' ↦↦⇩C⇘α⇙ ?𝔅'›\n 2. ‹u⦇NTMap⦈ = (ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈›\n 3. ‹dghm_const ℭ ℭ r (ℭ⦇CId⦈⦇r⦈) = ?𝔉'›\n 4. ‹dghm_id ℭ = ?𝔊'›\n 5. ‹ℭ = ?𝔄'›\n 6. ‹ℭ = ?𝔅'›*)\n show \"ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈) :\n cf_const ℭ ℭ r ↦⇩C⇩F cf_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n from z (*‹(z::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*) have dom_rhs: \"𝒟⇩∘ ((ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈) =\n ℭ⦇Obj⦈\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n show \"u⦇NTMap⦈ =\n (ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈\"\n proof (rule vsv_eqI (*‹⟦vsv (?r::V); vsv (?s::V); 𝒟⇩∘ ?r = 𝒟⇩∘ ?s; ⋀a::V. a ∈⇩∘ 𝒟⇩∘ ?r ⟹ ?r⦇a⦈ = ?s⦇a⦈⟧ ⟹ ?r = ?s›*), unfold dom_rhs (*‹𝒟⇩∘ ((ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈) = ℭ⦇Obj⦈›*) u.ntcf_NTMap_vdomain (*‹𝒟⇩∘ (u⦇NTMap⦈) = ℭ⦇Obj⦈›*))\n (*goals:\n 1. ‹vsv (u⦇NTMap⦈)›\n 2. ‹vsv ((ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈)›\n 3. ‹ℭ⦇Obj⦈ = ℭ⦇Obj⦈›\n 4. ‹⋀a. a ∈⇩∘ ℭ⦇Obj⦈ ⟹ u⦇NTMap⦈⦇a⦈ = (ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈⦇a⦈›*)\n fix c\n assume prems: \"c ∈⇩∘ ℭ⦇Obj⦈\" (*‹(c::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*)\n then have ic: \"ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n from u.ntcf_Comp_commute[OF ic] (*‹u⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ dghm_const ℭ ℭ r (ℭ⦇CId⦈⦇r⦈)⦇ArrMap⦈⦇ntcf_initial ℭ z⦇NTMap⦈⦇c⦈⦈ = dghm_id ℭ⦇ArrMap⦈⦇ntcf_initial ℭ z⦇NTMap⦈⦇c⦈⦈ ∘⇩A⇘ℭ⇙ u⦇NTMap⦈⦇z⦈›*) ic (*‹ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c›*) have [cat_cs_simps]: \"ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ u⦇NTMap⦈⦇z⦈ = u⦇NTMap⦈⦇c⦈\"\n apply (cs_prems cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n (*goal: ‹ntcf_initial ℭ z⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ u⦇NTMap⦈⦇z⦈ = u⦇NTMap⦈⦇c⦈›*)\n by simp\n from prems (*‹c ∈⇩∘ ℭ⦇Obj⦈›*) z (*‹z ∈⇩∘ ℭ⦇Obj⦈›*) show \"u⦇NTMap⦈⦇c⦈ =\n (ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈⦇c⦈\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n qed (auto intro: cat_cs_intros (*‹⟦category ?α ?ℭ; ?a ∈⇩∘ ?ℭ⦇Obj⦈; ?b = ?a; ?c = ?a; ?ℭ' = ?ℭ⟧ ⟹ ?ℭ⦇CId⦈⦇?a⦈ : ?b ↦⇘?ℭ'⇙ ?c› ‹⟦category ?α ?ℭ; ?a ∈⇩∘ ?ℭ⦇Obj⦈; ?f = ?ℭ⦇CId⦈⦇?a⦈⟧ ⟹ ?f : ?a ↦⇘?ℭ⇙ ?a› ‹⟦category ?α ?ℭ; ?g : ?b ↦⇘?ℭ⇙ ?c; ?f : ?a ↦⇘?ℭ⇙ ?b; ?gf = [?g, ?f]⇩∘⟧ ⟹ ?gf ∈⇩∘ 𝒟⇩∘ (?ℭ⦇Comp⦈)› ‹⟦category ?α ?ℭ; ?a ∈⇩∘ ?ℭ⦇Obj⦈; ?b ∈⇩∘ ?ℭ⦇Obj⦈⟧ ⟹ Hom ?ℭ ?a ?b ∈⇩∘ Vset ?α› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇘?ℭ⇙ ?b⟧ ⟹ ?f ∈⇩∘ ?ℭ⦇Arr⦈› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇘?ℭ⇙ ?b⟧ ⟹ ?a ∈⇩∘ ?ℭ⦇Obj⦈› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇘?ℭ⇙ ?b⟧ ⟹ ?b ∈⇩∘ ?ℭ⦇Obj⦈› ‹⟦category ?α ?ℭ; ?g : ?b ↦⇘?ℭ⇙ ?c; ?f : ?a ↦⇘?ℭ⇙ ?b; ?ℭ' = ?ℭ⟧ ⟹ ?g ∘⇩A⇘?ℭ⇙ ?f : ?a ↦⇘?ℭ'⇙ ?c› ‹⟦category ?α ?ℭ; ?g : ?b ↦⇘?ℭ⇙ ?c; ?f : ?a ↦⇘?ℭ⇙ ?b⟧ ⟹ ?g ∘⇩A⇘?ℭ⇙ ?f : ?a ↦⇘?ℭ⇙ ?c› ‹⟦category ?α ?ℭ; ?g : ?b ↦⇘?ℭ⇙ ?c; ?f : ?a ↦⇘?ℭ⇙ ?b; is_inverse ?ℭ ?g' ?g; is_inverse ?ℭ ?f' ?f; ?f'g' = ?f' ∘⇩A⇘?ℭ⇙ ?g'; ?gf = ?g ∘⇩A⇘?ℭ⇙ ?f⟧ ⟹ is_inverse ?ℭ ?f'g' ?gf› ‹⟦category ?α ?ℭ; ?f : ?a ↦⇩i⇩s⇩o⇘?ℭ⇙ ?b; ?ℭ' = ?ℭ⟧ ⟹ ?f¯⇩C⇘?ℭ⇙ : ?b ↦⇩i⇩s⇩o⇘?ℭ'⇙ ?a› ‹?f : ?a ↦⇘?ℭ⇙ ?b ⟹ ?f ∈⇩∘ ?ℭ⦇Arr⦈› and more 230 facts*))\n (*solves the remaining goals:\n 1. ‹vsv (u⦇NTMap⦈)›\n 2. ‹vsv ((ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ (u⦇NTMap⦈⦇z⦈))⦇NTMap⦈)›\n 3. ‹ℭ⦇Obj⦈ = ℭ⦇Obj⦈›*)\n qed (simp_all)\n (*solves the remaining goals:\n 1. ‹dghm_const ℭ ℭ r (ℭ⦇CId⦈⦇r⦈) = dghm_const ℭ ℭ r (ℭ⦇CId⦈⦇r⦈)›\n 2. ‹dghm_id ℭ = dghm_id ℭ›\n 3. ‹ℭ = ℭ›\n 4. ‹ℭ = ℭ›*)\n fix f'\n assume prems: \"f' : r ↦⇘ℭ⇙ z\" \"u = ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ f'\" (*‹(f'::V) : (r::V) ↦⇘ℭ::V⇙ (z::V)› ‹(u::V) = ntcf_initial (ℭ::V) (z::V) ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ (f'::V)›*)\n from z (*‹z ∈⇩∘ ℭ⦇Obj⦈›*) have \"ntcf_initial ℭ z⦇NTMap⦈⦇z⦈ : z ↦⇘ℭ⇙ z\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n note [cat_cs_simps] = cat_obj_initial_CId[OF assms this, symmetric] (*‹ntcf_initial ℭ z⦇NTMap⦈⦇z⦈ = ℭ⦇CId⦈⦇z⦈›*)\n from prems(2) (*‹u = ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ f'›*) have \"u⦇NTMap⦈⦇z⦈ = (ntcf_initial ℭ z ∙⇩N⇩T⇩C⇩F ntcf_const ℭ ℭ f')⦇NTMap⦈⦇z⦈\"\n by simp\n from this (*‹u⦇NTMap⦈⦇z⦈ = (ntcf_initial ℭ z ∙⇩N⇩T⇩S⇩M⇩C⇩F ntcf_const ℭ ℭ f')⦇NTMap⦈⦇z⦈›*) prems(1) (*‹f' : r ↦⇘ℭ⇙ z›*) show \"f' = u⦇NTMap⦈⦇z⦈\"\n apply (cs_prems cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n (*goal: ‹f' = u⦇NTMap⦈⦇z⦈›*)\n by simp\n qed\nqed\n\nlemma (in category) cat_obj_terminal_is_cat_obj_id_terminal:\n assumes \"obj_terminal ℭ z\"\n shows \"ntcf_terminal ℭ z : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ ℭ\"\n by (rule is_cat_obj_id_initial.is_cat_obj_id_terminal_op [ OF category.cat_obj_initial_is_cat_obj_id_initial[ OF category_op op_cat_obj_initial[THEN iffD2, OF assms(1)] ], unfolded cat_op_simps (*‹ntcf_terminal ℭ z : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ ℭ›*) ])\n\nlemma cat_cone_CId_obj_initial:\n assumes \"ℨ : z <⇩C⇩F⇩.⇩c⇩o⇩n⇩e cf_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ\" and \"ℨ⦇NTMap⦈⦇z⦈ = ℭ⦇CId⦈⦇z⦈\"\n shows \"obj_initial ℭ z\"\nproof (intro obj_initialI (*‹⟦?a ∈⇩∘ ?ℭ⦇Obj⦈; ⋀b. b ∈⇩∘ ?ℭ⦇Obj⦈ ⟹ ∃!f. f : ?a ↦⇘?ℭ⇙ b⟧ ⟹ obj_initial ?ℭ ?a›*))\n (*goals:\n 1. ‹(z::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›\n 2. ‹⋀b::V. b ∈⇩∘ (ℭ::V)⦇Obj⦈ ⟹ ∃!f::V. f : (z::V) ↦⇘ℭ⇙ b›*)\n interpret ℨ: is_cat_cone \"α\" z \"ℭ\" \"ℭ\" \"cf_id ℭ\" \"ℨ\"\n by (rule assms( (*‹ℨ : z <⇩C⇩F⇩.⇩c⇩o⇩n⇩e dghm_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ›*) 1))\n show \"z ∈⇩∘ ℭ⦇Obj⦈\"\n by (cs_concl cs_intro: cat_cs_intros)\n fix c\n assume prems: \"c ∈⇩∘ ℭ⦇Obj⦈\" (*‹(c::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*)\n show \"∃!f. f : z ↦⇘ℭ⇙ c\"\n proof (intro ex1I (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ∃!x. ?P x›*))\n (*goals:\n 1. ‹(?a::V) : (z::V) ↦⇘ℭ::V⇙ (c::V)›\n 2. ‹⋀f::V. f : (z::V) ↦⇘ℭ::V⇙ (c::V) ⟹ f = (?a::V)›*)\n from prems (*‹(c::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*) show \"ℨc\": \"ℨ⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n fix f\n assume prems': \"f : z ↦⇘ℭ⇙ c\" (*‹(f::V) : (z::V) ↦⇘ℭ::V⇙ (c::V)›*)\n from \"ℨ.ntcf_Comp_commute\"[OF prems'] (*‹ℨ⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)⦇ArrMap⦈⦇f⦈ = dghm_id ℭ⦇ArrMap⦈⦇f⦈ ∘⇩A⇘ℭ⇙ ℨ⦇NTMap⦈⦇z⦈›*) prems' (*‹f : z ↦⇘ℭ⇙ c›*) \"ℨc\" (*‹ℨ⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c›*) show \"f = ℨ⦇NTMap⦈⦇c⦈\"\n apply (cs_prems cs_simp: cat_cs_simps assms(2) cs_intro: cat_cs_intros)\n (*goal: ‹f = ℨ⦇NTMap⦈⦇c⦈›*)\n by simp\n qed\nqed\n\nlemma cat_cocone_CId_obj_terminal:\n assumes \"ℨ : cf_id ℭ >⇩C⇩F⇩.⇩c⇩o⇩c⇩o⇩n⇩e z : ℭ ↦↦⇩C⇘α⇙ ℭ\" and \"ℨ⦇NTMap⦈⦇z⦈ = ℭ⦇CId⦈⦇z⦈\"\n shows \"obj_terminal ℭ z\"\nproof (-)\n (*goal: ‹obj_terminal (ℭ::V) (z::V)›*)\n interpret ℨ: is_cat_cocone \"α\" z \"ℭ\" \"ℭ\" \"cf_id ℭ\" \"ℨ\"\n by (rule assms( (*‹ℨ : dghm_id ℭ >⇩C⇩F⇩.⇩c⇩o⇩c⇩o⇩n⇩e z : ℭ ↦↦⇩C⇘α⇙ ℭ›*) 1))\n show \"?thesis\"\n (*goal: ‹obj_terminal (ℭ::V) (z::V)›*)\n by (rule cat_cone_CId_obj_initial [ OF ℨ.is_cat_cone_op[unfolded cat_op_simps], unfolded cat_op_simps, OF assms(2) (*‹obj_terminal ℭ z›*) ])\nqed\n \nlemma (in is_cat_obj_id_initial) cat_oii_obj_initial: \"obj_initial ℭ z\"\nproof (rule cat_cone_CId_obj_initial (*‹⟦?ℨ : ?z <⇩C⇩F⇩.⇩c⇩o⇩n⇩e dghm_id ?ℭ : ?ℭ ↦↦⇩C⇘?α⇙ ?ℭ; ?ℨ⦇NTMap⦈⦇?z⦈ = ?ℭ⦇CId⦈⦇?z⦈⟧ ⟹ obj_initial ?ℭ ?z›*), rule is_cat_cone_axioms (*‹ℨ : z <⇩C⇩F⇩.⇩c⇩o⇩n⇩e dghm_id ℭ : ℭ ↦↦⇩C⇘α⇙ ℭ›*))\n (*goal: ‹ℨ⦇NTMap⦈⦇z⦈ = ℭ⦇CId⦈⦇z⦈›*)\n from cat_lim_unique_cone'[OF is_cat_cone_axioms] (*‹∃!f'. f' : z ↦⇘ℭ⇙ z ∧ (∀j∈elts (ℭ⦇Obj⦈). ℨ⦇NTMap⦈⦇j⦈ = ℨ⦇NTMap⦈⦇j⦈ ∘⇩A⇘ℭ⇙ f')›*) obtain f where f: \"f : z ↦⇘ℭ⇙ z\" and \"ℨ'j\": \"⋀j. j ∈⇩∘ ℭ⦇Obj⦈ ⟹ ℨ⦇NTMap⦈⦇j⦈ = ℨ⦇NTMap⦈⦇j⦈ ∘⇩A⇘ℭ⇙ f\" and f_unique: \"⟦\n f' : z ↦⇘ℭ⇙ z;\n ⋀j. j ∈⇩∘ ℭ⦇Obj⦈ ⟹ ℨ⦇NTMap⦈⦇j⦈ = ℨ⦇NTMap⦈⦇j⦈ ∘⇩A⇘ℭ⇙ f'\n ⟧ ⟹ f' = f\" for f'\n (*goal: ‹(⋀f. ⟦f : z ↦⇘ℭ⇙ z; ⋀j. j ∈⇩∘ ℭ⦇Obj⦈ ⟹ ℨ⦇NTMap⦈⦇j⦈ = ℨ⦇NTMap⦈⦇j⦈ ∘⇩A⇘ℭ⇙ f; ⋀f'. ⟦f' : z ↦⇘ℭ⇙ z; ⋀j. j ∈⇩∘ ℭ⦇Obj⦈ ⟹ ℨ⦇NTMap⦈⦇j⦈ = ℨ⦇NTMap⦈⦇j⦈ ∘⇩A⇘ℭ⇙ f'⟧ ⟹ f' = f⟧ ⟹ thesis) ⟹ thesis›*)\n by metis\n have CId_z: \"ℭ⦇CId⦈⦇z⦈ : z ↦⇘ℭ⇙ z\"\n by (cs_concl cs_intro: cat_cs_intros)\n have \"ℨ⦇NTMap⦈⦇j⦈ = ℨ⦇NTMap⦈⦇j⦈ ∘⇩A⇘ℭ⇙ ℭ⦇CId⦈⦇z⦈\" if \"j ∈⇩∘ ℭ⦇Obj⦈\" for j\n using that (*‹(j::V) ∈⇩∘ (ℭ::V)⦇Obj⦈›*) by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n from f_unique[OF CId_z this] (*‹(⋀j. j ∈⇩∘ ℭ⦇Obj⦈ ⟹ j ∈⇩∘ ℭ⦇Obj⦈) ⟹ ℭ⦇CId⦈⦇z⦈ = f›*) have CId_f: \"ℭ⦇CId⦈⦇z⦈ = f\" .\n have \"ℨz\": \"ℨ⦇NTMap⦈⦇z⦈ : z ↦⇘ℭ⇙ z\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n have \"ℨ⦇NTMap⦈⦇c⦈ = ℨ⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ ℨ⦇NTMap⦈⦇z⦈\" if \"c ∈⇩∘ ℭ⦇Obj⦈\" for c\n proof (-)\n (*goal: ‹ℨ⦇NTMap⦈⦇c⦈ = ℨ⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ ℨ⦇NTMap⦈⦇z⦈›*)\n from that (*‹c ∈⇩∘ ℭ⦇Obj⦈›*) have \"ℨc\": \"ℨ⦇NTMap⦈⦇c⦈ : z ↦⇘ℭ⇙ c\"\n by (cs_concl cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n note cat_cone_Comp_commute[cat_cs_simps del] (*‹?f : ?a ↦⇘ℭ⇙ ?b ⟹ dghm_id ℭ⦇ArrMap⦈⦇?f⦈ ∘⇩A⇘ℭ⇙ ℨ⦇NTMap⦈⦇?a⦈ = ℨ⦇NTMap⦈⦇?b⦈›*)\n from ntcf_Comp_commute[OF ℨc] (*‹ℨ⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ dghm_const ℭ ℭ z (ℭ⦇CId⦈⦇z⦈)⦇ArrMap⦈⦇ℨ⦇NTMap⦈⦇c⦈⦈ = dghm_id ℭ⦇ArrMap⦈⦇ℨ⦇NTMap⦈⦇c⦈⦈ ∘⇩A⇘ℭ⇙ ℨ⦇NTMap⦈⦇z⦈›*) \"ℨc\" (*‹(ℨ::V)⦇NTMap⦈⦇c⦈ : (z::V) ↦⇘ℭ::V⇙ (c::V)›*) show \"ℨ⦇NTMap⦈⦇c⦈ = ℨ⦇NTMap⦈⦇c⦈ ∘⇩A⇘ℭ⇙ ℨ⦇NTMap⦈⦇z⦈\"\n by (cs_prems cs_simp: cat_cs_simps cs_intro: cat_cs_intros)\n qed\n from f_unique[OF ℨz this] (*‹(⋀j::V. j ∈⇩∘ (ℭ::V)⦇Obj⦈ ⟹ j ∈⇩∘ ℭ⦇Obj⦈) ⟹ (ℨ::V)⦇NTMap⦈⦇(z::V)⦈ = (f::V)›*) have \"ℨ⦇NTMap⦈⦇z⦈ = f\" .\n with CId_f (*‹ℭ⦇CId⦈⦇z⦈ = f›*) show \"ℨ⦇NTMap⦈⦇z⦈ = ℭ⦇CId⦈⦇z⦈\"\n by simp\nqed\n\nlemma (in is_cat_obj_id_terminal) cat_oit_obj_terminal: \"obj_terminal ℭ z\"\n by (rule is_cat_obj_id_initial.cat_oii_obj_initial[ OF is_cat_obj_id_initial_op, unfolded cat_op_simps (*‹obj_terminal (ℭ::V) (z::V)›*) ])\n\nlemma (in category) cat_is_cat_obj_id_initial_obj_initial_iff:\n \"(ntcf_initial ℭ z : z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α⇙ ℭ) ⟷ obj_initial ℭ z\"\n using cat_obj_initial_is_cat_obj_id_initial (*‹obj_initial (ℭ::V) (?z::V) ⟹ ntcf_initial ℭ ?z : ?z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘α::V⇙ ℭ›*) is_cat_obj_id_initial.cat_oii_obj_initial (*‹?ℨ : ?z <⇩C⇩F⇩.₀ id⇩C : ↦↦⇩C⇘?α⇙ ?ℭ ⟹ obj_initial ?ℭ ?z›*) by auto\n\nlemma (in category) cat_is_cat_obj_id_terminal_obj_terminal_iff:\n \"(ntcf_terminal ℭ z : id⇩C >⇩C⇩F⇩.₁ z : ↦↦⇩C⇘α⇙ ℭ) ⟷ obj_terminal ℭ z\"\n using cat_obj_terminal_is_cat_obj_id_terminal (*‹obj_terminal ℭ ?z ⟹ ntcf_terminal ℭ ?z : id⇩C >⇩C⇩F⇩.₁ ?z : ↦↦⇩C⇘α⇙ ℭ›*) is_cat_obj_id_terminal.cat_oit_obj_terminal (*‹?ℨ : id⇩C >⇩C⇩F⇩.₁ ?z : ↦↦⇩C⇘?α⇙ ?ℭ ⟹ obj_terminal ?ℭ ?z›*) by auto\n\ntext‹\\newpage›\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/CZH_Universal_Constructions/czh_ucategories/CZH_UCAT_Limit_IT.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"3ab73f87b0360396be50d8630c3bb55314962b907de42343f80485fa249f1745\"\n}"}}},{"rowIdx":150,"cells":{"text":{"kind":"string","value":"(*\n Author: Salomon Sickert\n Author: Benedikt Seidl\n Author: Alexander Schimpf (original entry: CAVA/LTL.thy)\n Author: Stephan Merz (original entry: Stuttering_Equivalence/PLTL.thy)\n License: BSD\n*)\n\nsection ‹Linear Temporal Logic›\n\ntheory LTL\nimports\n Main \"HOL-Library.Omega_Words_Fun\"\nbegin\n\ntext ‹This theory provides a formalisation of linear temporal logic. It provides three variants:\n \\begin{enumerate}\n \\item LTL with syntactic sugar. This variant is the semantic reference and the included parser\n generates ASTs of this datatype.\n \\item LTL in negation normal form without syntactic sugar. This variant is used by the included\n rewriting engine and is used for the translation to automata (implemented in other entries).\n \\item LTL in restricted negation normal form without the rather uncommon operators ``weak until''\n and ``strong release''. It is used by the formalization of Gerth's algorithm.\n \\item PLTL. A variant with a reduced set of operators.\n \\end{enumerate}\n This theory subsumes (and partly reuses) the existing formalisation found in LTL\\_to\\_GBA and\n Stuttering\\_Equivalence and unifies them.›\n\nsubsection ‹LTL with Syntactic Sugar›\n\ntext ‹In this section, we provide a formulation of LTL with explicit syntactic sugar deeply embedded.\n This formalization serves as a reference semantics.›\n\nsubsubsection ‹Syntax›\n\ndatatype (atoms_ltlc: 'a) ltlc =\n True_ltlc (\"true⇩c\")\n | False_ltlc (\"false⇩c\")\n | Prop_ltlc 'a (\"prop⇩c'(_')\")\n | Not_ltlc \"'a ltlc\" (\"not⇩c _\" [85] 85)\n | And_ltlc \"'a ltlc\" \"'a ltlc\" (\"_ and⇩c _\" [82,82] 81)\n | Or_ltlc \"'a ltlc\" \"'a ltlc\" (\"_ or⇩c _\" [81,81] 80)\n | Implies_ltlc \"'a ltlc\" \"'a ltlc\" (\"_ implies⇩c _\" [81,81] 80)\n | Next_ltlc \"'a ltlc\" (\"X⇩c _\" [88] 87)\n | Final_ltlc \"'a ltlc\" (\"F⇩c _\" [88] 87)\n | Global_ltlc \"'a ltlc\" (\"G⇩c _\" [88] 87)\n | Until_ltlc \"'a ltlc\" \"'a ltlc\" (\"_ U⇩c _\" [84,84] 83)\n | Release_ltlc \"'a ltlc\" \"'a ltlc\" (\"_ R⇩c _\" [84,84] 83)\n | WeakUntil_ltlc \"'a ltlc\" \"'a ltlc\" (\"_ W⇩c _\" [84,84] 83)\n | StrongRelease_ltlc \"'a ltlc\" \"'a ltlc\" (\"_ M⇩c _\" [84,84] 83)\n\ndefinition Iff_ltlc (\"_ iff⇩c _\" [81,81] 80)\nwhere\n \"φ iff⇩c ψ ≡ (φ implies⇩c ψ) and⇩c (ψ implies⇩c φ)\"\n\nsubsubsection ‹Semantics›\n\nprimrec semantics_ltlc :: \"['a set word, 'a ltlc] ⇒ bool\" (\"_ ⊨⇩c _\" [80,80] 80)\nwhere\n \"ξ ⊨⇩c true⇩c = True\"\n| \"ξ ⊨⇩c false⇩c = False\"\n| \"ξ ⊨⇩c prop⇩c(q) = (q∈ξ 0)\"\n| \"ξ ⊨⇩c not⇩c φ = (¬ ξ ⊨⇩c φ)\"\n| \"ξ ⊨⇩c φ and⇩c ψ = (ξ ⊨⇩c φ ∧ ξ ⊨⇩c ψ)\"\n| \"ξ ⊨⇩c φ or⇩c ψ = (ξ ⊨⇩c φ ∨ ξ ⊨⇩c ψ)\"\n| \"ξ ⊨⇩c φ implies⇩c ψ = (ξ ⊨⇩c φ ⟶ ξ ⊨⇩c ψ)\"\n| \"ξ ⊨⇩c X⇩c φ = (suffix 1 ξ ⊨⇩c φ)\"\n| \"ξ ⊨⇩c F⇩c φ = (∃i. suffix i ξ ⊨⇩c φ)\"\n| \"ξ ⊨⇩c G⇩c φ = (∀i. suffix i ξ ⊨⇩c φ)\"\n| \"ξ ⊨⇩c φ U⇩c ψ = (∃i. suffix i ξ ⊨⇩c ψ ∧ (∀j<i. suffix j ξ ⊨⇩c φ))\"\n| \"ξ ⊨⇩c φ R⇩c ψ = (∀i. suffix i ξ ⊨⇩c ψ ∨ (∃j<i. suffix j ξ ⊨⇩c φ))\"\n| \"ξ ⊨⇩c φ W⇩c ψ = (∀i. suffix i ξ ⊨⇩c φ ∨ (∃j≤i. suffix j ξ ⊨⇩c ψ))\"\n| \"ξ ⊨⇩c φ M⇩c ψ = (∃i. suffix i ξ ⊨⇩c φ ∧ (∀j≤i. suffix j ξ ⊨⇩c ψ))\"\n\nlemma semantics_ltlc_sugar [simp]:\n \"ξ ⊨⇩c φ iff⇩c ψ = (ξ ⊨⇩c φ ⟷ ξ ⊨⇩c ψ)\"\n \"ξ ⊨⇩c F⇩c φ = ξ ⊨⇩c (true⇩c U⇩c φ)\"\n \"ξ ⊨⇩c G⇩c φ = ξ ⊨⇩c (false⇩c R⇩c φ)\"\n \n (*goals:\n 1. ‹ξ ⊨⇩c φ iff⇩c ψ = (ξ ⊨⇩c φ = ξ ⊨⇩c ψ)›\n 2. ‹ξ ⊨⇩c F⇩c φ = ξ ⊨⇩c true⇩c U⇩c φ›\n 3. ‹ξ ⊨⇩c G⇩c φ = ξ ⊨⇩c false⇩c R⇩c φ›\n discuss goal 1*)\n apply ((auto simp add: Iff_ltlc_def (*‹?φ iff⇩c ?ψ ≡ (?φ implies⇩c ?ψ) and⇩c (?ψ implies⇩c ?φ)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: Iff_ltlc_def (*‹?φ iff⇩c ?ψ ≡ (?φ implies⇩c ?ψ) and⇩c (?ψ implies⇩c ?φ)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: Iff_ltlc_def (*‹?φ iff⇩c ?ψ ≡ (?φ implies⇩c ?ψ) and⇩c (?ψ implies⇩c ?φ)›*))[1])\n (*proven 3 subgoals*) .\n\ndefinition \"language_ltlc φ ≡ {ξ. ξ ⊨⇩c φ}\"\n\nlemma language_ltlc_negate[simp]:\n \"language_ltlc (not⇩c φ) = - language_ltlc φ\"\n unfolding language_ltlc_def\n (*goal: ‹{ξ. ξ ⊨⇩c not⇩c φ} = - {ξ. ξ ⊨⇩c φ}›*)\n by auto\n\nlemma ltl_true_or_con[simp]:\n \"ξ ⊨⇩c prop⇩c(p) or⇩c (not⇩c prop⇩c(p))\"\n by auto\n\nlemma ltl_false_true_con[simp]:\n \"ξ ⊨⇩c not⇩c true⇩c ⟷ ξ ⊨⇩c false⇩c\"\n by auto\n\nlemma ltl_Next_Neg_con[simp]:\n \"ξ ⊨⇩c X⇩c (not⇩c φ) ⟷ ξ ⊨⇩c not⇩c X⇩c φ\"\n by auto<close>\n\nlemma ltl_Until_Release_con:\n \"ξ ⊨⇩c φ R⇩c ψ ⟷ (¬ ξ ⊨⇩c (not⇩c φ) U⇩c (not⇩c ψ))\"\n \"ξ ⊨⇩c φ U⇩c ψ ⟷ (¬ ξ ⊨⇩c (not⇩c φ) R⇩c (not⇩c ψ))\"\n \n (*goals:\n 1. ‹ξ ⊨⇩c φ R⇩c ψ = (¬ ξ ⊨⇩c not⇩c φ U⇩c not⇩c ψ)›\n 2. ‹ξ ⊨⇩c φ U⇩c ψ = (¬ ξ ⊨⇩c not⇩c φ R⇩c not⇩c ψ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma ltl_WeakUntil_StrongRelease_con:\n \"ξ ⊨⇩c φ W⇩c ψ ⟷ (¬ ξ ⊨⇩c (not⇩c φ) M⇩c (not⇩c ψ))\"\n \"ξ ⊨⇩c φ M⇩c ψ ⟷ (¬ ξ ⊨⇩c (not⇩c φ) W⇩c (not⇩c ψ))\"\n \n (*goals:\n 1. ‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) W⇩c (ψ::'a ltlc) = (¬ ξ ⊨⇩c not⇩c φ M⇩c not⇩c ψ)›\n 2. ‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) M⇩c (ψ::'a ltlc) = (¬ ξ ⊨⇩c not⇩c φ W⇩c not⇩c ψ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .<close>\n\nlemma ltl_Release_StrongRelease_con:\n \"ξ ⊨⇩c φ R⇩c ψ ⟷ ξ ⊨⇩c (φ M⇩c ψ) or⇩c (G⇩c ψ)\"\n \"ξ ⊨⇩c φ M⇩c ψ ⟷ ξ ⊨⇩c (φ R⇩c ψ) and⇩c (F⇩c φ)\"\nproof (safe)\n (*goals:\n 1. ‹ξ ⊨⇩c φ R⇩c ψ ⟹ ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ›\n 2. ‹ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ ⟹ ξ ⊨⇩c φ R⇩c ψ›\n 3. ‹ξ ⊨⇩c φ M⇩c ψ ⟹ ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ›\n 4. ‹ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ ⟹ ξ ⊨⇩c φ M⇩c ψ›*)\n assume asm: \"ξ ⊨⇩c φ R⇩c ψ\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) R⇩c (ψ::'a ltlc)›*)\n show \"ξ ⊨⇩c (φ M⇩c ψ) or⇩c (G⇩c ψ)\"\n proof (cases \"ξ ⊨⇩c G⇩c ψ\")\n (*goals:\n 1. ‹ξ ⊨⇩c G⇩c ψ ⟹ ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ›\n 2. ‹¬ ξ ⊨⇩c G⇩c ψ ⟹ ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ›*)\n case False (*‹¬ ξ ⊨⇩c G⇩c ψ›*)\n then obtain i where \"¬ suffix i ξ ⊨⇩c ψ\" and \"∀j<i. suffix j ξ ⊨⇩c ψ\"\n (*goal: ‹(⋀i. ⟦¬ suffix i ξ ⊨⇩c ψ; ∀j<i. suffix j ξ ⊨⇩c ψ⟧ ⟹ thesis) ⟹ thesis›*)\n using exists_least_iff[of \"λi. ¬ suffix i ξ ⊨⇩c ψ\"] (*‹(∃n. ¬ suffix n ξ ⊨⇩c ψ) = (∃n. ¬ suffix n ξ ⊨⇩c ψ ∧ (∀m<n. ¬ ¬ suffix m ξ ⊨⇩c ψ))›*) by force\n then show \"?thesis\"\n (*goal: ‹ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ›*)\n using asm (*‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) R⇩c (ψ::'a ltlc)›*) by force\n qed (simp)\n (*solved the remaining goal: ‹ξ ⊨⇩c G⇩c ψ ⟹ ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ›*)\nnext\n (*goals:\n 1. ‹ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ ⟹ ξ ⊨⇩c φ R⇩c ψ›\n 2. ‹ξ ⊨⇩c φ M⇩c ψ ⟹ ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ›\n 3. ‹ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ ⟹ ξ ⊨⇩c φ M⇩c ψ›*)\n assume asm: \"ξ ⊨⇩c (φ R⇩c ψ) and⇩c (F⇩c φ)\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) R⇩c (ψ::'a ltlc) and⇩c F⇩c φ›*)\n then show \"ξ ⊨⇩c φ M⇩c ψ\"\n proof (cases \"ξ ⊨⇩c F⇩c φ\")\n (*goals:\n 1. ‹⟦ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ; ξ ⊨⇩c F⇩c φ⟧ ⟹ ξ ⊨⇩c φ M⇩c ψ›\n 2. ‹⟦ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ; ¬ ξ ⊨⇩c F⇩c φ⟧ ⟹ ξ ⊨⇩c φ M⇩c ψ›*)\n case True (*‹ξ ⊨⇩c F⇩c φ›*)\n then obtain i where \"suffix i ξ ⊨⇩c φ\" and \"∀j<i. ¬ suffix j ξ ⊨⇩c φ\"\n (*goal: ‹(⋀i::nat. ⟦suffix i (ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc); ∀j<i. ¬ suffix j ξ ⊨⇩c φ⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using exists_least_iff[of \"λi. suffix i ξ ⊨⇩c φ\"] (*‹(∃n::nat. suffix n (ξ::nat ⇒ 'a::type set) ⊨⇩c (φ::'a::type ltlc)) = (∃n::nat. suffix n ξ ⊨⇩c φ ∧ (∀m<n. ¬ suffix m ξ ⊨⇩c φ))›*) by force\n then show \"?thesis\"\n (*goal: ‹ξ ⊨⇩c φ M⇩c ψ›*)\n using asm (*‹(ξ::nat ⇒ 'a::type set) ⊨⇩c (φ::'a::type ltlc) R⇩c (ψ::'a::type ltlc) and⇩c F⇩c φ›*) by force\n qed (simp)\n (*solved the remaining goal: ‹⟦ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ; ¬ ξ ⊨⇩c F⇩c φ⟧ ⟹ ξ ⊨⇩c φ M⇩c ψ›*)\nqed ((unfold semantics_ltlc.simps (*‹?ξ ⊨⇩c true⇩c = True› ‹?ξ ⊨⇩c false⇩c = False› ‹?ξ ⊨⇩c prop⇩c(?q) = (?q ∈ ?ξ 0)› ‹?ξ ⊨⇩c not⇩c ?φ = (¬ ?ξ ⊨⇩c ?φ)› ‹?ξ ⊨⇩c ?φ and⇩c ?ψ = (?ξ ⊨⇩c ?φ ∧ ?ξ ⊨⇩c ?ψ)› ‹?ξ ⊨⇩c ?φ or⇩c ?ψ = (?ξ ⊨⇩c ?φ ∨ ?ξ ⊨⇩c ?ψ)› ‹?ξ ⊨⇩c ?φ implies⇩c ?ψ = (?ξ ⊨⇩c ?φ ⟶ ?ξ ⊨⇩c ?ψ)› ‹?ξ ⊨⇩c X⇩c ?φ = suffix 1 ?ξ ⊨⇩c ?φ› ‹?ξ ⊨⇩c F⇩c ?φ = (∃i. suffix i ?ξ ⊨⇩c ?φ)› ‹?ξ ⊨⇩c G⇩c ?φ = (∀i. suffix i ?ξ ⊨⇩c ?φ)› ‹?ξ ⊨⇩c ?φ U⇩c ?ψ = (∃i. suffix i ?ξ ⊨⇩c ?ψ ∧ (∀j<i. suffix j ?ξ ⊨⇩c ?φ))› ‹?ξ ⊨⇩c ?φ R⇩c ?ψ = (∀i. suffix i ?ξ ⊨⇩c ?ψ ∨ (∃j<i. suffix j ?ξ ⊨⇩c ?φ))› and more 2 facts*); insert not_less (*‹(¬ ?x < ?y) = (?y ≤ ?x)›*), blast)+)\n(*solves the remaining goals:\n 1. ‹ξ ⊨⇩c φ M⇩c ψ or⇩c G⇩c ψ ⟹ ξ ⊨⇩c φ R⇩c ψ›\n 2. ‹ξ ⊨⇩c φ M⇩c ψ ⟹ ξ ⊨⇩c φ R⇩c ψ and⇩c F⇩c φ›*)\n\nlemma ltl_Until_WeakUntil_con:\n \"ξ ⊨⇩c φ U⇩c ψ ⟷ ξ ⊨⇩c (φ W⇩c ψ) and⇩c (F⇩c ψ)\"\n \"ξ ⊨⇩c φ W⇩c ψ ⟷ ξ ⊨⇩c (φ U⇩c ψ) or⇩c (G⇩c φ)\"\nproof (safe)\n (*goals:\n 1. ‹ξ ⊨⇩c φ U⇩c ψ ⟹ ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ›\n 2. ‹ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ ⟹ ξ ⊨⇩c φ U⇩c ψ›\n 3. ‹ξ ⊨⇩c φ W⇩c ψ ⟹ ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ›\n 4. ‹ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ ⟹ ξ ⊨⇩c φ W⇩c ψ›*)\n assume asm: \"ξ ⊨⇩c (φ W⇩c ψ) and⇩c (F⇩c ψ)\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) W⇩c (ψ::'a ltlc) and⇩c F⇩c ψ›*)\n then show \"ξ ⊨⇩c φ U⇩c ψ\"\n proof (cases \"ξ ⊨⇩c F⇩c ψ\")\n (*goals:\n 1. ‹⟦ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ; ξ ⊨⇩c F⇩c ψ⟧ ⟹ ξ ⊨⇩c φ U⇩c ψ›\n 2. ‹⟦ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ; ¬ ξ ⊨⇩c F⇩c ψ⟧ ⟹ ξ ⊨⇩c φ U⇩c ψ›*)\n case True (*‹ξ ⊨⇩c F⇩c ψ›*)\n then obtain i where \"suffix i ξ ⊨⇩c ψ\" and \"∀j<i. ¬ suffix j ξ ⊨⇩c ψ\"\n (*goal: ‹(⋀i. ⟦suffix i ξ ⊨⇩c ψ; ∀j<i. ¬ suffix j ξ ⊨⇩c ψ⟧ ⟹ thesis) ⟹ thesis›*)\n using exists_least_iff[of \"λi. suffix i ξ ⊨⇩c ψ\"] (*‹(∃n. suffix n ξ ⊨⇩c ψ) = (∃n. suffix n ξ ⊨⇩c ψ ∧ (∀m<n. ¬ suffix m ξ ⊨⇩c ψ))›*) by force\n then show \"?thesis\"\n (*goal: ‹ξ ⊨⇩c φ U⇩c ψ›*)\n using asm (*‹ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ›*) by force\n qed (simp)\n (*solved the remaining goal: ‹⟦ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ; ¬ ξ ⊨⇩c F⇩c ψ⟧ ⟹ ξ ⊨⇩c φ U⇩c ψ›*)\nnext\n (*goals:\n 1. ‹ξ ⊨⇩c φ U⇩c ψ ⟹ ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ›\n 2. ‹ξ ⊨⇩c φ W⇩c ψ ⟹ ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ›\n 3. ‹ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ ⟹ ξ ⊨⇩c φ W⇩c ψ›*)\n assume asm: \"ξ ⊨⇩c φ W⇩c ψ\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) W⇩c (ψ::'a ltlc)›*)\n then show \"ξ ⊨⇩c (φ U⇩c ψ) or⇩c (G⇩c φ)\"\n proof (cases \"ξ ⊨⇩c G⇩c φ\")\n (*goals:\n 1. ‹⟦ξ ⊨⇩c φ W⇩c ψ; ξ ⊨⇩c G⇩c φ⟧ ⟹ ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ›\n 2. ‹⟦ξ ⊨⇩c φ W⇩c ψ; ¬ ξ ⊨⇩c G⇩c φ⟧ ⟹ ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ›*)\n case False (*‹¬ ξ ⊨⇩c G⇩c φ›*)\n then obtain i where \"¬ suffix i ξ ⊨⇩c φ\" and \"∀j<i. suffix j ξ ⊨⇩c φ\"\n (*goal: ‹(⋀i. ⟦¬ suffix i ξ ⊨⇩c φ; ∀j<i. suffix j ξ ⊨⇩c φ⟧ ⟹ thesis) ⟹ thesis›*)\n using exists_least_iff[of \"λi. ¬ suffix i ξ ⊨⇩c φ\"] (*‹(∃n. ¬ suffix n ξ ⊨⇩c φ) = (∃n. ¬ suffix n ξ ⊨⇩c φ ∧ (∀m<n. ¬ ¬ suffix m ξ ⊨⇩c φ))›*) by force\n then show \"?thesis\"\n (*goal: ‹ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ›*)\n using asm (*‹ξ ⊨⇩c φ W⇩c ψ›*) by force\n qed (simp)\n (*solved the remaining goal: ‹⟦ξ ⊨⇩c φ W⇩c ψ; ξ ⊨⇩c G⇩c φ⟧ ⟹ ξ ⊨⇩c φ U⇩c ψ or⇩c G⇩c φ›*)\nqed ((unfold semantics_ltlc.simps (*‹?ξ ⊨⇩c true⇩c = True› ‹?ξ ⊨⇩c false⇩c = False› ‹?ξ ⊨⇩c prop⇩c(?q) = (?q ∈ ?ξ 0)› ‹?ξ ⊨⇩c not⇩c ?φ = (¬ ?ξ ⊨⇩c ?φ)› ‹?ξ ⊨⇩c ?φ and⇩c ?ψ = (?ξ ⊨⇩c ?φ ∧ ?ξ ⊨⇩c ?ψ)› ‹?ξ ⊨⇩c ?φ or⇩c ?ψ = (?ξ ⊨⇩c ?φ ∨ ?ξ ⊨⇩c ?ψ)› ‹?ξ ⊨⇩c ?φ implies⇩c ?ψ = (?ξ ⊨⇩c ?φ ⟶ ?ξ ⊨⇩c ?ψ)› ‹?ξ ⊨⇩c X⇩c ?φ = suffix 1 ?ξ ⊨⇩c ?φ› ‹?ξ ⊨⇩c F⇩c ?φ = (∃i. suffix i ?ξ ⊨⇩c ?φ)› ‹?ξ ⊨⇩c G⇩c ?φ = (∀i. suffix i ?ξ ⊨⇩c ?φ)› ‹?ξ ⊨⇩c ?φ U⇩c ?ψ = (∃i. suffix i ?ξ ⊨⇩c ?ψ ∧ (∀j<i. suffix j ?ξ ⊨⇩c ?φ))› ‹?ξ ⊨⇩c ?φ R⇩c ?ψ = (∀i. suffix i ?ξ ⊨⇩c ?ψ ∨ (∃j<i. suffix j ?ξ ⊨⇩c ?φ))› and more 2 facts*); insert not_less (*‹(¬ (?x::?'a) < (?y::?'a)) = (?y ≤ ?x)›*), blast)+)\n(*solves the remaining goals:\n 1. ‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) U⇩c (ψ::'a ltlc) ⟹ ξ ⊨⇩c φ W⇩c ψ and⇩c F⇩c ψ›\n 2. ‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) U⇩c (ψ::'a ltlc) or⇩c G⇩c φ ⟹ ξ ⊨⇩c φ W⇩c ψ›*)\n\nlemma ltl_StrongRelease_Until_con:\n \"ξ ⊨⇩c φ M⇩c ψ ⟷ ξ ⊨⇩c ψ U⇩c (φ and⇩c ψ)\"\n using order.order_iff_strict (*‹(?a ≤ ?b) = (?a < ?b ∨ ?a = ?b)›*) by auto\n\nlemma ltl_WeakUntil_Release_con:\n \"ξ ⊨⇩c φ R⇩c ψ ⟷ ξ ⊨⇩c ψ W⇩c (φ and⇩c ψ)\"\n by (meson ltl_Release_StrongRelease_con( (*‹?ξ ⊨⇩c ?φ R⇩c ?ψ = ?ξ ⊨⇩c ?φ M⇩c ?ψ or⇩c G⇩c ?ψ›*) 1) ltl_StrongRelease_Until_con (*‹?ξ ⊨⇩c ?φ M⇩c ?ψ = ?ξ ⊨⇩c ?ψ U⇩c (?φ and⇩c ?ψ)›*) ltl_Until_WeakUntil_con( (*‹?ξ ⊨⇩c ?φ W⇩c ?ψ = ?ξ ⊨⇩c ?φ U⇩c ?ψ or⇩c G⇩c ?φ›*) 2) semantics_ltlc.simps( (*‹?ξ ⊨⇩c ?φ or⇩c ?ψ = (?ξ ⊨⇩c ?φ ∨ ?ξ ⊨⇩c ?ψ)›*) 6))\n\n\ndefinition \"pw_eq_on S w w' ≡ ∀i. w i ∩ S = w' i ∩ S\"\n\nlemma pw_eq_on_refl[simp]: \"pw_eq_on S w w\"\n and pw_eq_on_sym: \"pw_eq_on S w w' ⟹ pw_eq_on S w' w\"\n and pw_eq_on_trans[trans]: \"⟦pw_eq_on S w w'; pw_eq_on S w' w''⟧ ⟹ pw_eq_on S w w''\"\n unfolding pw_eq_on_def\n (*goals:\n 1. ‹∀i. w i ∩ S = w i ∩ S›\n 2. ‹∀i. w i ∩ S = w' i ∩ S ⟹ ∀i. w' i ∩ S = w i ∩ S›\n 3. ‹⟦∀i. w i ∩ S = w' i ∩ S; ∀i. w' i ∩ S = w'' i ∩ S⟧ ⟹ ∀i. w i ∩ S = w'' i ∩ S›*)\n (*goals:\n 1. ‹∀i. w i ∩ S = w i ∩ S›\n 2. ‹∀i. w i ∩ S = w' i ∩ S ⟹ ∀i. w' i ∩ S = w i ∩ S›\n 3. ‹⟦∀i. w i ∩ S = w' i ∩ S; ∀i. w' i ∩ S = w'' i ∩ S⟧ ⟹ ∀i. w i ∩ S = w'' i ∩ S›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n\nlemma pw_eq_on_suffix:\n \"pw_eq_on S w w' ⟹ pw_eq_on S (suffix k w) (suffix k w')\"\n by (simp add: pw_eq_on_def (*‹pw_eq_on ?S ?w ?w' ≡ ∀i. ?w i ∩ ?S = ?w' i ∩ ?S›*))\n\nlemma pw_eq_on_subset:\n \"S ⊆ S' ⟹ pw_eq_on S' w w' ⟹ pw_eq_on S w w'\"\n by (auto simp add: pw_eq_on_def (*‹pw_eq_on (?S::?'a set) (?w::?'b ⇒ ?'a set) (?w'::?'b ⇒ ?'a set) ≡ ∀i::?'b. ?w i ∩ ?S = ?w' i ∩ ?S›*))\n\nlemma ltlc_eq_on_aux:\n \"pw_eq_on (atoms_ltlc φ) w w' ⟹ w ⊨⇩c φ ⟹ w' ⊨⇩c φ\"\nproof (induction φ arbitrary: w w')\n (*goals:\n 1. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀x w w'. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 and⇩c φ2)) w w'; w ⊨⇩c φ1 and⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 and⇩c φ2›\n 6. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 or⇩c φ2)) w w'; w ⊨⇩c φ1 or⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 or⇩c φ2›\n 7. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 implies⇩c φ2)) w w'; w ⊨⇩c φ1 implies⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 implies⇩c φ2›\n 8. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 9. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 10. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›\n 11. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 U⇩c φ2)) w w'; w ⊨⇩c φ1 U⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 U⇩c φ2›\n 12. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 R⇩c φ2)) w w'; w ⊨⇩c φ1 R⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 R⇩c φ2›\n 13. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 W⇩c φ2)) w w'; w ⊨⇩c φ1 W⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 W⇩c φ2›\n 14. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 M⇩c φ2)) w w'; w ⊨⇩c φ1 M⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 M⇩c φ2›*)\n case Until_ltlc (*‹⟦pw_eq_on (atoms_ltlc φ1_) ?w ?w'; ?w ⊨⇩c φ1_⟧ ⟹ ?w' ⊨⇩c φ1_› ‹⟦pw_eq_on (atoms_ltlc φ2_) ?w ?w'; ?w ⊨⇩c φ2_⟧ ⟹ ?w' ⊨⇩c φ2_› ‹pw_eq_on (atoms_ltlc (φ1_ U⇩c φ2_)) w w'› ‹w ⊨⇩c φ1_ U⇩c φ2_›*)\n thus \"?case\"\n (*goal: ‹w' ⊨⇩c φ1_ U⇩c φ2_›*)\n apply simp\n (*goal: ‹w' ⊨⇩c φ1_ U⇩c φ2_›*)\n by (meson Un_upper1 (*‹?A ⊆ ?A ∪ ?B›*) Un_upper2 (*‹?B ⊆ ?A ∪ ?B›*) pw_eq_on_subset (*‹⟦?S ⊆ ?S'; pw_eq_on ?S' ?w ?w'⟧ ⟹ pw_eq_on ?S ?w ?w'›*) pw_eq_on_suffix (*‹pw_eq_on ?S ?w ?w' ⟹ pw_eq_on ?S (suffix ?k ?w) (suffix ?k ?w')›*))\nnext\n (*goals:\n 1. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀x w w'. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 and⇩c φ2)) w w'; w ⊨⇩c φ1 and⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 and⇩c φ2›\n 6. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 or⇩c φ2)) w w'; w ⊨⇩c φ1 or⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 or⇩c φ2›\n 7. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 implies⇩c φ2)) w w'; w ⊨⇩c φ1 implies⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 implies⇩c φ2›\n 8. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 9. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 10. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›\n 11. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 R⇩c φ2)) w w'; w ⊨⇩c φ1 R⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 R⇩c φ2›\n 12. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 W⇩c φ2)) w w'; w ⊨⇩c φ1 W⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 W⇩c φ2›\n 13. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 M⇩c φ2)) w w'; w ⊨⇩c φ1 M⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 M⇩c φ2›*)\n case Release_ltlc (*‹⟦pw_eq_on (atoms_ltlc φ1_) ?w ?w'; ?w ⊨⇩c φ1_⟧ ⟹ ?w' ⊨⇩c φ1_› ‹⟦pw_eq_on (atoms_ltlc (φ2_::'a ltlc)) (?w::nat ⇒ 'a set) (?w'::nat ⇒ 'a set); ?w ⊨⇩c φ2_⟧ ⟹ ?w' ⊨⇩c φ2_› ‹pw_eq_on (atoms_ltlc (φ1_ R⇩c φ2_)) w w'› ‹w ⊨⇩c φ1_ R⇩c φ2_›*)\n thus \"?case\"\n (*goal: ‹w' ⊨⇩c φ1_ R⇩c φ2_›*)\n apply simp\n (*goal: ‹w' ⊨⇩c φ1_ R⇩c φ2_›*)\n by (metis Un_upper1 (*‹?A ⊆ ?A ∪ ?B›*) pw_eq_on_subset (*‹⟦?S ⊆ ?S'; pw_eq_on ?S' ?w ?w'⟧ ⟹ pw_eq_on ?S ?w ?w'›*) pw_eq_on_suffix (*‹pw_eq_on ?S ?w ?w' ⟹ pw_eq_on ?S (suffix ?k ?w) (suffix ?k ?w')›*) sup_commute (*‹sup ?x ?y = sup ?y ?x›*))\nnext\n (*goals:\n 1. ‹⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀(x::'a) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 and⇩c φ2)) w w'; w ⊨⇩c φ1 and⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 and⇩c φ2›\n 6. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 or⇩c φ2)) w w'; w ⊨⇩c φ1 or⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 or⇩c φ2›\n 7. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 implies⇩c φ2)) w w'; w ⊨⇩c φ1 implies⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 implies⇩c φ2›\n 8. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 9. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 10. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›\n 11. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 W⇩c φ2)) w w'; w ⊨⇩c φ1 W⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 W⇩c φ2›\n 12. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 M⇩c φ2)) w w'; w ⊨⇩c φ1 M⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 M⇩c φ2›*)\n case WeakUntil_ltlc (*‹⟦pw_eq_on (atoms_ltlc φ1_) ?w ?w'; ?w ⊨⇩c φ1_⟧ ⟹ ?w' ⊨⇩c φ1_› ‹⟦pw_eq_on (atoms_ltlc φ2_) ?w ?w'; ?w ⊨⇩c φ2_⟧ ⟹ ?w' ⊨⇩c φ2_› ‹pw_eq_on (atoms_ltlc (φ1_ W⇩c φ2_)) w w'› ‹w ⊨⇩c φ1_ W⇩c φ2_›*)\n thus \"?case\"\n (*goal: ‹(w'::nat ⇒ 'a set) ⊨⇩c (φ1_::'a ltlc) W⇩c (φ2_::'a ltlc)›*)\n apply simp\n (*goal: ‹w' ⊨⇩c φ1_ W⇩c φ2_›*)\n by (meson pw_eq_on_subset (*‹⟦?S ⊆ ?S'; pw_eq_on ?S' ?w ?w'⟧ ⟹ pw_eq_on ?S ?w ?w'›*) pw_eq_on_suffix (*‹pw_eq_on ?S ?w ?w' ⟹ pw_eq_on ?S (suffix ?k ?w) (suffix ?k ?w')›*) sup.cobounded1 (*‹?a ≤ sup ?a ?b›*) sup_ge2 (*‹?y ≤ sup ?x ?y›*))\nnext\n (*goals:\n 1. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀x w w'. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 and⇩c φ2)) w w'; w ⊨⇩c φ1 and⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 and⇩c φ2›\n 6. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 or⇩c φ2)) w w'; w ⊨⇩c φ1 or⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 or⇩c φ2›\n 7. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 implies⇩c φ2)) w w'; w ⊨⇩c φ1 implies⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 implies⇩c φ2›\n 8. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 9. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 10. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›\n 11. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 M⇩c φ2)) w w'; w ⊨⇩c φ1 M⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 M⇩c φ2›*)\n case StrongRelease_ltlc (*‹⟦pw_eq_on (atoms_ltlc φ1_) ?w ?w'; ?w ⊨⇩c φ1_⟧ ⟹ ?w' ⊨⇩c φ1_› ‹⟦pw_eq_on (atoms_ltlc (φ2_::'a ltlc)) (?w::nat ⇒ 'a set) (?w'::nat ⇒ 'a set); ?w ⊨⇩c φ2_⟧ ⟹ ?w' ⊨⇩c φ2_› ‹pw_eq_on (atoms_ltlc ((φ1_::'a ltlc) M⇩c (φ2_::'a ltlc))) (w::nat ⇒ 'a set) (w'::nat ⇒ 'a set)› ‹w ⊨⇩c φ1_ M⇩c φ2_›*)\n thus \"?case\"\n (*goal: ‹w' ⊨⇩c φ1_ M⇩c φ2_›*)\n apply simp\n (*goal: ‹w' ⊨⇩c φ1_ M⇩c φ2_›*)\n by (metis Un_upper1 (*‹(?A::?'a set) ⊆ ?A ∪ (?B::?'a set)›*) pw_eq_on_subset (*‹⟦(?S::?'a set) ⊆ (?S'::?'a set); pw_eq_on ?S' (?w::?'b ⇒ ?'a set) (?w'::?'b ⇒ ?'a set)⟧ ⟹ pw_eq_on ?S ?w ?w'›*) pw_eq_on_suffix (*‹pw_eq_on (?S::?'a set) (?w::nat ⇒ ?'a set) (?w'::nat ⇒ ?'a set) ⟹ pw_eq_on ?S (suffix (?k::nat) ?w) (suffix ?k ?w')›*) pw_eq_on_sym (*‹pw_eq_on (?S::?'a set) (?w::?'b ⇒ ?'a set) (?w'::?'b ⇒ ?'a set) ⟹ pw_eq_on ?S ?w' ?w›*) sup_ge2 (*‹(?y::?'a) ≤ sup (?x::?'a) ?y›*))\nnext\n (*goals:\n 1. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀x w w'. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 and⇩c φ2)) w w'; w ⊨⇩c φ1 and⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 and⇩c φ2›\n 6. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 or⇩c φ2)) w w'; w ⊨⇩c φ1 or⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 or⇩c φ2›\n 7. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 implies⇩c φ2)) w w'; w ⊨⇩c φ1 implies⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 implies⇩c φ2›\n 8. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 9. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 10. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›*)\n case (And_ltlc φ ψ) (*‹⟦pw_eq_on (atoms_ltlc φ) ?w ?w'; ?w ⊨⇩c φ⟧ ⟹ ?w' ⊨⇩c φ› ‹⟦pw_eq_on (atoms_ltlc (ψ::'a ltlc)) (?w::nat ⇒ 'a set) (?w'::nat ⇒ 'a set); ?w ⊨⇩c ψ⟧ ⟹ ?w' ⊨⇩c ψ› ‹pw_eq_on (atoms_ltlc (φ and⇩c ψ)) w w'› ‹(w::nat ⇒ 'a::type set) ⊨⇩c (φ::'a::type ltlc) and⇩c (ψ::'a::type ltlc)›*)\n thus \"?case\"\n (*goal: ‹w' ⊨⇩c φ and⇩c ψ›*)\n apply simp\n (*goal: ‹w' ⊨⇩c φ and⇩c ψ›*)\n by (meson Un_upper1 (*‹?A ⊆ ?A ∪ ?B›*) inf_sup_ord( (*‹?y ≤ sup ?x ?y›*) 4) pw_eq_on_subset (*‹⟦?S ⊆ ?S'; pw_eq_on ?S' ?w ?w'⟧ ⟹ pw_eq_on ?S ?w ?w'›*))\nnext\n (*goals:\n 1. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀x w w'. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 or⇩c φ2)) w w'; w ⊨⇩c φ1 or⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 or⇩c φ2›\n 6. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 implies⇩c φ2)) w w'; w ⊨⇩c φ1 implies⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 implies⇩c φ2›\n 7. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 8. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 9. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›*)\n case (Or_ltlc φ ψ) (*‹⟦pw_eq_on (atoms_ltlc φ) ?w ?w'; ?w ⊨⇩c φ⟧ ⟹ ?w' ⊨⇩c φ› ‹⟦pw_eq_on (atoms_ltlc (ψ::'a ltlc)) (?w::nat ⇒ 'a set) (?w'::nat ⇒ 'a set); ?w ⊨⇩c ψ⟧ ⟹ ?w' ⊨⇩c ψ› ‹pw_eq_on (atoms_ltlc ((φ::'a ltlc) or⇩c (ψ::'a ltlc))) (w::nat ⇒ 'a set) (w'::nat ⇒ 'a set)› ‹w ⊨⇩c φ or⇩c ψ›*)\n thus \"?case\"\n (*goal: ‹w' ⊨⇩c φ or⇩c ψ›*)\n apply simp\n (*goal: ‹w' ⊨⇩c φ or⇩c ψ›*)\n by (meson Un_upper2 (*‹?B ⊆ ?A ∪ ?B›*) pw_eq_on_subset (*‹⟦?S ⊆ ?S'; pw_eq_on ?S' ?w ?w'⟧ ⟹ pw_eq_on ?S ?w ?w'›*) sup_ge1 (*‹?x ≤ sup ?x ?y›*))\nnext\n (*goals:\n 1. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀w w'. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀x w w'. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀φ1 φ2 w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ1) w w'; w ⊨⇩c φ1⟧ ⟹ w' ⊨⇩c φ1; ⋀w w'. ⟦pw_eq_on (atoms_ltlc φ2) w w'; w ⊨⇩c φ2⟧ ⟹ w' ⊨⇩c φ2; pw_eq_on (atoms_ltlc (φ1 implies⇩c φ2)) w w'; w ⊨⇩c φ1 implies⇩c φ2⟧ ⟹ w' ⊨⇩c φ1 implies⇩c φ2›\n 6. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 7. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 8. ‹⋀φ w w'. ⟦⋀w w'. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›*)\n case (Implies_ltlc φ ψ) (*‹⟦pw_eq_on (atoms_ltlc φ) ?w ?w'; ?w ⊨⇩c φ⟧ ⟹ ?w' ⊨⇩c φ› ‹⟦pw_eq_on (atoms_ltlc ψ) ?w ?w'; ?w ⊨⇩c ψ⟧ ⟹ ?w' ⊨⇩c ψ› ‹pw_eq_on (atoms_ltlc (φ implies⇩c ψ)) w w'› ‹w ⊨⇩c φ implies⇩c ψ›*)\n thus \"?case\"\n (*goal: ‹(w'::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) implies⇩c (ψ::'a ltlc)›*)\n apply simp\n (*goal: ‹(w'::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) implies⇩c (ψ::'a ltlc)›*)\n by (meson Un_upper1 (*‹?A ⊆ ?A ∪ ?B›*) Un_upper2 (*‹?B ⊆ ?A ∪ ?B›*) pw_eq_on_subset[of \"atoms_ltlc _\" \"atoms_ltlc φ ∪ atoms_ltlc ψ\"] (*‹⟦atoms_ltlc ?uu4 ⊆ atoms_ltlc φ ∪ atoms_ltlc ψ; pw_eq_on (atoms_ltlc φ ∪ atoms_ltlc ψ) ?w ?w'⟧ ⟹ pw_eq_on (atoms_ltlc ?uu4) ?w ?w'›*) pw_eq_on_sym (*‹pw_eq_on ?S ?w ?w' ⟹ pw_eq_on ?S ?w' ?w›*))\nqed ((auto simp add: pw_eq_on_def (*‹pw_eq_on ?S ?w ?w' ≡ ∀i. ?w i ∩ ?S = ?w' i ∩ ?S›*); metis suffix_nth (*‹suffix ?k ?x ?n = ?x (?k + ?n)›*))+)\n(*solves the remaining goals:\n 1. ‹⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc true⇩c) w w'; w ⊨⇩c true⇩c⟧ ⟹ w' ⊨⇩c true⇩c›\n 2. ‹⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc false⇩c) w w'; w ⊨⇩c false⇩c⟧ ⟹ w' ⊨⇩c false⇩c›\n 3. ‹⋀(x::'a) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc prop⇩c(x)) w w'; w ⊨⇩c prop⇩c(x)⟧ ⟹ w' ⊨⇩c prop⇩c(x)›\n 4. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (not⇩c φ)) w w'; w ⊨⇩c not⇩c φ⟧ ⟹ w' ⊨⇩c not⇩c φ›\n 5. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (X⇩c φ)) w w'; w ⊨⇩c X⇩c φ⟧ ⟹ w' ⊨⇩c X⇩c φ›\n 6. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (F⇩c φ)) w w'; w ⊨⇩c F⇩c φ⟧ ⟹ w' ⊨⇩c F⇩c φ›\n 7. ‹⋀(φ::'a ltlc) (w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦⋀(w::nat ⇒ 'a set) w'::nat ⇒ 'a set. ⟦pw_eq_on (atoms_ltlc φ) w w'; w ⊨⇩c φ⟧ ⟹ w' ⊨⇩c φ; pw_eq_on (atoms_ltlc (G⇩c φ)) w w'; w ⊨⇩c G⇩c φ⟧ ⟹ w' ⊨⇩c G⇩c φ›*)\n\nlemma ltlc_eq_on:\n \"pw_eq_on (atoms_ltlc φ) w w' ⟹ w ⊨⇩c φ ⟷ w' ⊨⇩c φ\"\n using ltlc_eq_on_aux (*‹⟦pw_eq_on (atoms_ltlc ?φ) ?w ?w'; ?w ⊨⇩c ?φ⟧ ⟹ ?w' ⊨⇩c ?φ›*) pw_eq_on_sym (*‹pw_eq_on ?S ?w ?w' ⟹ pw_eq_on ?S ?w' ?w›*) by blast\n\nlemma suffix_comp: \"(λi. f (suffix k w i)) = suffix k (f o w)\"\n by auto\n\nlemma suffix_range: \"⋃(range ξ) ⊆ APs ⟹ ⋃(range (suffix k ξ)) ⊆ APs\"\n by auto\n\nlemma map_semantics_ltlc_aux:\n assumes \"inj_on f APs\"\n assumes \"⋃(range w) ⊆ APs\"\n assumes \"atoms_ltlc φ ⊆ APs\"\n shows \"w ⊨⇩c φ ⟷ (λi. f ` w i) ⊨⇩c map_ltlc f φ\"\n using assms(2,3) (*‹⋃ (range w) ⊆ APs› ‹atoms_ltlc φ ⊆ APs›*) proof (induction φ arbitrary: w)\n (*goals:\n 1. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀x w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc prop⇩c(x) ⊆ APs⟧ ⟹ w ⊨⇩c prop⇩c(x) = (λi. f ` w i) ⊨⇩c map_ltlc f prop⇩c(x)›\n 4. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 7. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 8. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (X⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c X⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (X⇩c φ)›\n 9. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (F⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c F⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (F⇩c φ)›\n 10. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (G⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c G⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (G⇩c φ)›\n 11. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 U⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 U⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 U⇩c φ2)›\n 12. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 R⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 R⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 R⇩c φ2)›\n 13. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 W⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 W⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 W⇩c φ2)›\n 14. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case (Prop_ltlc x) (*‹⋃ (range (w::nat ⇒ 'a set)) ⊆ (APs::'a set)› ‹atoms_ltlc prop⇩c(x::'a) ⊆ (APs::'a set)›*)\n thus \"?case\"\n (*goal: ‹w ⊨⇩c prop⇩c(x) = (λi. f ` w i) ⊨⇩c map_ltlc f prop⇩c(x)›*)\n using assms(1) (*‹inj_on f APs›*) by (simp add: SUP_le_iff (*‹(Sup (?f ` ?A) ≤ ?u) = (∀i∈?A. ?f i ≤ ?u)›*) inj_on_image_mem_iff (*‹⟦inj_on ?f ?B; ?a ∈ ?B; ?A ⊆ ?B⟧ ⟹ (?f ?a ∈ ?f ` ?A) = (?a ∈ ?A)›*))\n next\n (*goals:\n 1. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 7. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (X⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c X⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (X⇩c φ)›\n 8. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (F⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c F⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (F⇩c φ)›\n 9. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (G⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c G⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (G⇩c φ)›\n 10. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 U⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 U⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 U⇩c φ2)›\n 11. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 R⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 R⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 R⇩c φ2)›\n 12. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 W⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 W⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 W⇩c φ2)›\n 13. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case (Next_ltlc φ) (*‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ› ‹⋃ (range w) ⊆ APs› ‹atoms_ltlc (X⇩c φ) ⊆ APs›*)\n show \"?case\"\n (*goal: ‹w ⊨⇩c X⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (X⇩c φ)›*)\n using Next_ltlc(1)[of \"suffix 1 w\", unfolded suffix_comp comp_def] (*‹⟦⋃ (range (suffix 1 w)) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ suffix 1 w ⊨⇩c φ = suffix 1 (λx. f ` w x) ⊨⇩c map_ltlc f φ›*) Next_ltlc(2,3) (*‹⋃ (range (w::nat ⇒ 'a set)) ⊆ (APs::'a set)› ‹atoms_ltlc (X⇩c φ) ⊆ APs›*) apply simp\n (*goal: ‹w ⊨⇩c X⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (X⇩c φ)›*)\n by (metis Next_ltlc.prems( (*‹⋃ (range w) ⊆ APs›*) 1) One_nat_def (*‹1 = Suc 0›*) ‹⟦⋃(range (suffix 1 w)) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ suffix 1 w ⊨⇩c φ = suffix 1 (λx. f ` w x) ⊨⇩c map_ltlc f φ› suffix_range (*‹⋃ (range ?ξ) ⊆ ?APs ⟹ ⋃ (range (suffix ?k ?ξ)) ⊆ ?APs›*))\n next\n (*goals:\n 1. ‹⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀(φ::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 7. ‹⋀(φ::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (F⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c F⇩c φ = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (F⇩c φ)›\n 8. ‹⋀(φ::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (G⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c G⇩c φ = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (G⇩c φ)›\n 9. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 U⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 U⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 U⇩c φ2)›\n 10. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 R⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 R⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 R⇩c φ2)›\n 11. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 W⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 W⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 W⇩c φ2)›\n 12. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case (Final_ltlc φ) (*‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ› ‹⋃ (range w) ⊆ APs› ‹atoms_ltlc (F⇩c (φ::'a ltlc)) ⊆ (APs::'a set)›*)\n thus \"?case\"\n (*goal: ‹w ⊨⇩c F⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (F⇩c φ)›*)\n using Final_ltlc(1)[of \"suffix _ _\", unfolded suffix_comp comp_def, OF suffix_range] (*‹⟦⋃ (range ?uua3) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ suffix ?uu3 ?uua3 ⊨⇩c φ = suffix ?uu3 (λx. f ` ?uua3 x) ⊨⇩c map_ltlc f φ›*) by fastforce\n next\n (*goals:\n 1. ‹⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀(φ::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 7. ‹⋀(φ::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (G⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c G⇩c φ = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (G⇩c φ)›\n 8. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 U⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 U⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 U⇩c φ2)›\n 9. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 R⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 R⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 R⇩c φ2)›\n 10. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 W⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 W⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 W⇩c φ2)›\n 11. ‹⋀(φ1::'a ltlc) (φ2::'a ltlc) w::nat ⇒ 'a set. ⟦⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ (APs::'a set); atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f φ1; ⋀w::nat ⇒ 'a set. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi::nat. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case Global_ltlc (*‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ_› ‹⋃ (range w) ⊆ APs› ‹atoms_ltlc (G⇩c φ_) ⊆ APs›*)\n thus \"?case\"\n (*goal: ‹(w::nat ⇒ 'a set) ⊨⇩c G⇩c (φ_::'a ltlc) = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f (G⇩c φ_)›*)\n using Global_ltlc(1)[of \"suffix _ w\", unfolded suffix_comp comp_def, OF suffix_range] (*‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ_›*) by fastforce\n next\n (*goals:\n 1. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 7. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 U⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 U⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 U⇩c φ2)›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 R⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 R⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 R⇩c φ2)›\n 9. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 W⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 W⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 W⇩c φ2)›\n 10. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case Until_ltlc (*‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ1_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range (?w::nat ⇒ 'a set)) ⊆ (APs::'a set); atoms_ltlc (φ2_::'a ltlc) ⊆ APs⟧ ⟹ ?w ⊨⇩c φ2_ = (λi::nat. (f::'a ⇒ 'b) ` ?w i) ⊨⇩c map_ltlc f φ2_› ‹⋃ (range (w::nat ⇒ 'a::type set)) ⊆ (APs::'a::type set)› ‹atoms_ltlc ((φ1_::'a ltlc) U⇩c (φ2_::'a ltlc)) ⊆ (APs::'a set)›*)\n thus \"?case\"\n (*goal: ‹(w::nat ⇒ 'a set) ⊨⇩c (φ1_::'a ltlc) U⇩c (φ2_::'a ltlc) = (λi::nat. (f::'a ⇒ 'b) ` w i) ⊨⇩c map_ltlc f (φ1_ U⇩c φ2_)›*)\n using Until_ltlc(1,2)[of \"suffix _ w\", unfolded suffix_comp comp_def, OF suffix_range] (*‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ1_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ2_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ2_›*) by fastforce\n next\n (*goals:\n 1. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 7. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 R⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 R⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 R⇩c φ2)›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 W⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 W⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 W⇩c φ2)›\n 9. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case Release_ltlc (*‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ1_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ2_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ2_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ2_› ‹⋃ (range w) ⊆ APs› ‹atoms_ltlc (φ1_ R⇩c φ2_) ⊆ APs›*)\n thus \"?case\"\n (*goal: ‹w ⊨⇩c φ1_ R⇩c φ2_ = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1_ R⇩c φ2_)›*)\n using Release_ltlc(1,2)[of \"suffix _ w\", unfolded suffix_comp comp_def, OF suffix_range] (*‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ1_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range (w::nat ⇒ 'a set)) ⊆ (APs::'a set); atoms_ltlc (φ2_::'a ltlc) ⊆ APs⟧ ⟹ suffix (?uu3::nat) w ⊨⇩c φ2_ = suffix ?uu3 (λx::nat. (f::'a ⇒ 'b) ` w x) ⊨⇩c map_ltlc f φ2_›*) by fastforce\n next\n (*goals:\n 1. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 7. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 W⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 W⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 W⇩c φ2)›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case WeakUntil_ltlc (*‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ1_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ2_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ2_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ2_› ‹⋃ (range (w::nat ⇒ 'a::type set)) ⊆ (APs::'a::type set)› ‹atoms_ltlc (φ1_ W⇩c φ2_) ⊆ APs›*)\n thus \"?case\"\n (*goal: ‹(w::nat ⇒ 'a::type set) ⊨⇩c (φ1_::'a::type ltlc) W⇩c (φ2_::'a::type ltlc) = (λi::nat. (f::'a::type ⇒ 'b::type) ` w i) ⊨⇩c map_ltlc f (φ1_ W⇩c φ2_)›*)\n using WeakUntil_ltlc(1,2)[of \"suffix _ w\", unfolded suffix_comp comp_def, OF suffix_range] (*‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ1_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ2_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ2_›*) by fastforce\n next\n (*goals:\n 1. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›\n 7. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 M⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 M⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 M⇩c φ2)›*)\n case StrongRelease_ltlc (*‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ1_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range ?w) ⊆ APs; atoms_ltlc φ2_ ⊆ APs⟧ ⟹ ?w ⊨⇩c φ2_ = (λi. f ` ?w i) ⊨⇩c map_ltlc f φ2_› ‹⋃ (range w) ⊆ APs› ‹atoms_ltlc (φ1_ M⇩c φ2_) ⊆ APs›*)\n thus \"?case\"\n (*goal: ‹w ⊨⇩c φ1_ M⇩c φ2_ = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1_ M⇩c φ2_)›*)\n using StrongRelease_ltlc(1,2)[of \"suffix _ w\", unfolded suffix_comp comp_def, OF suffix_range] (*‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ1_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ1_› ‹⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2_ ⊆ APs⟧ ⟹ suffix ?uu3 w ⊨⇩c φ2_ = suffix ?uu3 (λx. f ` w x) ⊨⇩c map_ltlc f φ2_›*) by fastforce\n qed ((simp)+)\n (*solves the remaining goals:\n 1. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc true⇩c ⊆ APs⟧ ⟹ w ⊨⇩c true⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f true⇩c›\n 2. ‹⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc false⇩c ⊆ APs⟧ ⟹ w ⊨⇩c false⇩c = (λi. f ` w i) ⊨⇩c map_ltlc f false⇩c›\n 3. ‹⋀φ w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ ⊆ APs⟧ ⟹ w ⊨⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f φ; ⋃ (range w) ⊆ APs; atoms_ltlc (not⇩c φ) ⊆ APs⟧ ⟹ w ⊨⇩c not⇩c φ = (λi. f ` w i) ⊨⇩c map_ltlc f (not⇩c φ)›\n 4. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 and⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 and⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 and⇩c φ2)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 or⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 or⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 or⇩c φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ1 ⊆ APs⟧ ⟹ w ⊨⇩c φ1 = (λi. f ` w i) ⊨⇩c map_ltlc f φ1; ⋀w. ⟦⋃ (range w) ⊆ APs; atoms_ltlc φ2 ⊆ APs⟧ ⟹ w ⊨⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f φ2; ⋃ (range w) ⊆ APs; atoms_ltlc (φ1 implies⇩c φ2) ⊆ APs⟧ ⟹ w ⊨⇩c φ1 implies⇩c φ2 = (λi. f ` w i) ⊨⇩c map_ltlc f (φ1 implies⇩c φ2)›*)\n\ndefinition \"map_props f APs ≡ {i. ∃p∈APs. f p = Some i}\"\n\nlemma map_semantics_ltlc:\n assumes INJ: \"inj_on f (dom f)\" and DOM: \"atoms_ltlc φ ⊆ dom f\"\n shows \"ξ ⊨⇩c φ ⟷ (map_props f o ξ) ⊨⇩c map_ltlc (the o f) φ\"\nproof (-)\n (*goal: ‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) = (map_props (f::'a ⇒ 'b option) ∘ ξ) ⊨⇩c map_ltlc (the ∘ f) φ›*)\n let ?ξr = \"λi. ξ i ∩ atoms_ltlc φ\"\n let ?ξr' = \"λi. ξ i ∩ dom f\"\n have 1: \"⋃(range ?ξr) ⊆ atoms_ltlc φ\"\n by auto\n have INJ_the_dom: \"inj_on (the o f) (dom f)\"\n using assms (*‹inj_on (f::'a ⇒ 'b option) (dom f)› ‹atoms_ltlc (φ::'a ltlc) ⊆ dom (f::'a ⇒ 'b option)›*) by (auto simp: inj_on_def (*‹inj_on (?f::?'a ⇒ ?'b) (?A::?'a set) = (∀x::?'a∈?A. ∀y::?'a∈?A. ?f x = ?f y ⟶ x = y)›*) domIff (*‹((?a::?'a) ∈ dom (?m::?'a ⇒ ?'b option)) = (?m ?a ≠ None)›*))\n note 2 = subset_inj_on[OF this DOM] (*‹inj_on (the ∘ f) (atoms_ltlc φ)›*)\n have 3: \"(λi. (the o f) ` ?ξr' i) = map_props f o ξ\"\n using DOM (*‹atoms_ltlc φ ⊆ dom f›*) INJ (*‹inj_on (f::'a ⇒ 'b option) (dom f)›*) apply (auto intro!: ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*) simp: map_props_def (*‹map_props ?f ?APs ≡ {i. ∃p∈?APs. ?f p = Some i}›*) domIff (*‹(?a ∈ dom ?m) = (?m ?a ≠ None)›*) image_iff (*‹(?z ∈ ?f ` ?A) = (∃x∈?A. ?z = ?f x)›*))\n (*goal: ‹(λi. (the ∘ f) ` (ξ i ∩ dom f)) = map_props f ∘ ξ›*)\n by (metis Int_iff (*‹(?c ∈ ?A ∩ ?B) = (?c ∈ ?A ∧ ?c ∈ ?B)›*) domI (*‹?m ?a = Some ?b ⟹ ?a ∈ dom ?m›*) option.sel (*‹the (Some ?x2.0) = ?x2.0›*))\n have \"ξ ⊨⇩c φ ⟷ ?ξr ⊨⇩c φ\"\n apply (rule ltlc_eq_on (*‹pw_eq_on (atoms_ltlc ?φ) ?w ?w' ⟹ ?w ⊨⇩c ?φ = ?w' ⊨⇩c ?φ›*))\n (*goal: ‹ξ ⊨⇩c φ = (λi. ξ i ∩ atoms_ltlc φ) ⊨⇩c φ›*)\n by (auto simp: pw_eq_on_def (*‹pw_eq_on ?S ?w ?w' ≡ ∀i. ?w i ∩ ?S = ?w' i ∩ ?S›*))\n also (*calculation: ‹ξ ⊨⇩c φ = (λi. ξ i ∩ atoms_ltlc φ) ⊨⇩c φ›*) from map_semantics_ltlc_aux[OF 2 1 subset_refl] (*‹(λi. ξ i ∩ atoms_ltlc φ) ⊨⇩c φ = (λi. (the ∘ f) ` (ξ i ∩ atoms_ltlc φ)) ⊨⇩c map_ltlc (the ∘ f) φ›*) have \"… ⟷ (λi. (the o f) ` ?ξr i) ⊨⇩c map_ltlc (the o f) φ\" .\n also (*calculation: ‹ξ ⊨⇩c φ = (λi. (the ∘ f) ` (ξ i ∩ atoms_ltlc φ)) ⊨⇩c map_ltlc (the ∘ f) φ›*) have \"… ⟷ (λi. (the o f) ` ?ξr' i) ⊨⇩c map_ltlc (the o f) φ\"\n apply (rule ltlc_eq_on (*‹pw_eq_on (atoms_ltlc ?φ) ?w ?w' ⟹ ?w ⊨⇩c ?φ = ?w' ⊨⇩c ?φ›*))\n (*goal: ‹(λi. (the ∘ f) ` (ξ i ∩ atoms_ltlc φ)) ⊨⇩c map_ltlc (the ∘ f) φ = (λi. (the ∘ f) ` (ξ i ∩ dom f)) ⊨⇩c map_ltlc (the ∘ f) φ›*)\n using DOM (*‹atoms_ltlc φ ⊆ dom f›*) INJ (*‹inj_on (f::'a ⇒ 'b option) (dom f)›*) apply (auto simp: pw_eq_on_def (*‹pw_eq_on ?S ?w ?w' ≡ ∀i. ?w i ∩ ?S = ?w' i ∩ ?S›*) ltlc.set_map (*‹atoms_ltlc (map_ltlc ?f ?v) = ?f ` atoms_ltlc ?v›*) domIff (*‹(?a ∈ dom ?m) = (?m ?a ≠ None)›*) image_iff (*‹(?z ∈ ?f ` ?A) = (∃x∈?A. ?z = ?f x)›*))\n (*goal: ‹pw_eq_on (atoms_ltlc (map_ltlc (the ∘ (f::'a ⇒ 'b option)) (φ::'a ltlc))) (λi::nat. (the ∘ f) ` ((ξ::nat ⇒ 'a set) i ∩ atoms_ltlc φ)) (λi::nat. (the ∘ f) ` (ξ i ∩ dom f))›*)\n by (metis Int_iff (*‹(?c ∈ ?A ∩ ?B) = (?c ∈ ?A ∧ ?c ∈ ?B)›*) contra_subsetD (*‹⟦?A ⊆ ?B; ?c ∉ ?B⟧ ⟹ ?c ∉ ?A›*) domD (*‹?a ∈ dom ?m ⟹ ∃b. ?m ?a = Some b›*) domI (*‹?m ?a = Some ?b ⟹ ?a ∈ dom ?m›*) inj_on_eq_iff (*‹⟦inj_on ?f ?A; ?x ∈ ?A; ?y ∈ ?A⟧ ⟹ (?f ?x = ?f ?y) = (?x = ?y)›*) option.sel (*‹the (Some ?x2.0) = ?x2.0›*))\n also (*calculation: ‹(ξ::nat ⇒ 'a set) ⊨⇩c (φ::'a ltlc) = (λi::nat. (the ∘ (f::'a ⇒ 'b option)) ` (ξ i ∩ dom f)) ⊨⇩c map_ltlc (the ∘ f) φ›*) note \"3\" (*‹(λi. (the ∘ f) ` (ξ i ∩ dom f)) = map_props f ∘ ξ›*)\n finally (*calculation: ‹ξ ⊨⇩c φ = (map_props f ∘ ξ) ⊨⇩c map_ltlc (the ∘ f) φ›*) show \"?thesis\"\n (*goal: ‹ξ ⊨⇩c φ = (map_props f ∘ ξ) ⊨⇩c map_ltlc (the ∘ f) φ›*) .\nqed\n\nlemma map_semantics_ltlc_inv:\n assumes INJ: \"inj_on f (dom f)\" and DOM: \"atoms_ltlc φ ⊆ dom f\"\n shows \"ξ ⊨⇩c map_ltlc (the o f) φ ⟷ (λi. (the o f) -` ξ i) ⊨⇩c φ\"\n using map_semantics_ltlc[OF assms] (*‹?ξ ⊨⇩c φ = (map_props f ∘ ?ξ) ⊨⇩c map_ltlc (the ∘ f) φ›*) apply simp\n (*goal: ‹ξ ⊨⇩c map_ltlc (the ∘ f) φ = (λi. (the ∘ f) -` ξ i) ⊨⇩c φ›*)\n apply (intro ltlc_eq_on (*‹pw_eq_on (atoms_ltlc ?φ) ?w ?w' ⟹ ?w ⊨⇩c ?φ = ?w' ⊨⇩c ?φ›*))\n (*goal: ‹(⋀ξ. ξ ⊨⇩c φ = (map_props f ∘ ξ) ⊨⇩c map_ltlc (the ∘ f) φ) ⟹ ξ ⊨⇩c map_ltlc (the ∘ f) φ = (map_props f ∘ (λi. (the ∘ f) -` ξ i)) ⊨⇩c map_ltlc (the ∘ f) φ›*)\n apply (auto simp add: pw_eq_on_def (*‹pw_eq_on ?S ?w ?w' ≡ ∀i. ?w i ∩ ?S = ?w' i ∩ ?S›*) ltlc.set_map (*‹atoms_ltlc (map_ltlc ?f ?v) = ?f ` atoms_ltlc ?v›*) map_props_def (*‹map_props ?f ?APs ≡ {i. ∃p∈?APs. ?f p = Some i}›*))\n (*goal: ‹(⋀ξ. ξ ⊨⇩c φ = (map_props f ∘ ξ) ⊨⇩c map_ltlc (the ∘ f) φ) ⟹ pw_eq_on (atoms_ltlc (map_ltlc (the ∘ f) φ)) ξ (map_props f ∘ (λi. (the ∘ f) -` ξ i))›*)\n by (metis DOM (*‹atoms_ltlc (φ::'a ltlc) ⊆ dom (f::'a ⇒ 'b option)›*) comp_apply (*‹((?f::?'b ⇒ ?'a) ∘ (?g::?'c ⇒ ?'b)) (?x::?'c) = ?f (?g ?x)›*) contra_subsetD (*‹⟦(?A::?'a set) ⊆ (?B::?'a set); (?c::?'a) ∉ ?B⟧ ⟹ ?c ∉ ?A›*) domD (*‹(?a::?'a) ∈ dom (?m::?'a ⇒ ?'b option) ⟹ ∃b::?'b. ?m ?a = Some b›*) option.sel (*‹the (Some (?x2.0::?'a)) = ?x2.0›*) vimage_eq (*‹((?a::?'a) ∈ (?f::?'a ⇒ ?'b) -` (?B::?'b set)) = (?f ?a ∈ ?B)›*))\n\n\n\n\nsubsection ‹LTL in Negation Normal Form›\n\ntext ‹We define a type of LTL formula in negation normal form (NNF).›\n\nsubsubsection ‹Syntax›\n\ndatatype (atoms_ltln: 'a) ltln =\n True_ltln (\"true⇩n\")\n | False_ltln (\"false⇩n\")\n | Prop_ltln 'a (\"prop⇩n'(_')\")\n | Nprop_ltln 'a (\"nprop⇩n'(_')\")\n | And_ltln \"'a ltln\" \"'a ltln\" (\"_ and⇩n _\" [82,82] 81)\n | Or_ltln \"'a ltln\" \"'a ltln\" (\"_ or⇩n _\" [84,84] 83)\n | Next_ltln \"'a ltln\" (\"X⇩n _\" [88] 87)\n | Until_ltln \"'a ltln\" \"'a ltln\" (\"_ U⇩n _\" [84,84] 83)\n | Release_ltln \"'a ltln\" \"'a ltln\" (\"_ R⇩n _\" [84,84] 83)\n | WeakUntil_ltln \"'a ltln\" \"'a ltln\" (\"_ W⇩n _\" [84,84] 83)\n | StrongRelease_ltln \"'a ltln\" \"'a ltln\" (\"_ M⇩n _\" [84,84] 83)\n\nabbreviation finally⇩n :: \"'a ltln ⇒ 'a ltln\" (\"F⇩n _\" [88] 87)\nwhere\n \"F⇩n φ ≡ true⇩n U⇩n φ\"\n\nnotation (input) finally⇩n (\"♢⇩n _\" [88] 87)\n\nabbreviation globally⇩n :: \"'a ltln ⇒ 'a ltln\" (\"G⇩n _\" [88] 87)\nwhere\n \"G⇩n φ ≡ false⇩n R⇩n φ\"\n\nnotation (input) globally⇩n (\"□⇩n _\" [88] 87)\n\nsubsubsection ‹Semantics›\n\nprimrec semantics_ltln :: \"['a set word, 'a ltln] ⇒ bool\" (\"_ ⊨⇩n _\" [80,80] 80)\nwhere\n \"ξ ⊨⇩n true⇩n = True\"\n| \"ξ ⊨⇩n false⇩n = False\"\n| \"ξ ⊨⇩n prop⇩n(q) = (q ∈ ξ 0)\"\n| \"ξ ⊨⇩n nprop⇩n(q) = (q ∉ ξ 0)\"\n| \"ξ ⊨⇩n φ and⇩n ψ = (ξ ⊨⇩n φ ∧ ξ ⊨⇩n ψ)\"\n| \"ξ ⊨⇩n φ or⇩n ψ = (ξ ⊨⇩n φ ∨ ξ ⊨⇩n ψ)\"\n| \"ξ ⊨⇩n X⇩n φ = (suffix 1 ξ ⊨⇩n φ)\"\n| \"ξ ⊨⇩n φ U⇩n ψ = (∃i. suffix i ξ ⊨⇩n ψ ∧ (∀j<i. suffix j ξ ⊨⇩n φ))\"\n| \"ξ ⊨⇩n φ R⇩n ψ = (∀i. suffix i ξ ⊨⇩n ψ ∨ (∃j<i. suffix j ξ ⊨⇩n φ))\"\n| \"ξ ⊨⇩n φ W⇩n ψ = (∀i. suffix i ξ ⊨⇩n φ ∨ (∃j≤i. suffix j ξ ⊨⇩n ψ))\"\n| \"ξ ⊨⇩n φ M⇩n ψ = (∃i. suffix i ξ ⊨⇩n φ ∧ (∀j≤i. suffix j ξ ⊨⇩n ψ))\"\n\ndefinition \"language_ltln φ ≡ {ξ. ξ ⊨⇩n φ}\"\n\nlemma semantics_ltln_ite_simps[simp]:\n \"w ⊨⇩n (if P then true⇩n else false⇩n) = P\"\n \"w ⊨⇩n (if P then false⇩n else true⇩n) = (¬P)\"\n \n (*goals:\n 1. ‹w ⊨⇩n (if P then true⇩n else false⇩n) = P›\n 2. ‹w ⊨⇩n (if P then false⇩n else true⇩n) = (¬ P)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nsubsubsection ‹Conversion›\n\nfun ltlc_to_ltln' :: \"bool ⇒ 'a ltlc ⇒ 'a ltln\"\nwhere\n \"ltlc_to_ltln' False true⇩c = true⇩n\"\n| \"ltlc_to_ltln' False false⇩c = false⇩n\"\n| \"ltlc_to_ltln' False prop⇩c(q) = prop⇩n(q)\"\n| \"ltlc_to_ltln' False (φ and⇩c ψ) = (ltlc_to_ltln' False φ) and⇩n (ltlc_to_ltln' False ψ)\"\n| \"ltlc_to_ltln' False (φ or⇩c ψ) = (ltlc_to_ltln' False φ) or⇩n (ltlc_to_ltln' False ψ)\"\n| \"ltlc_to_ltln' False (φ implies⇩c ψ) = (ltlc_to_ltln' True φ) or⇩n (ltlc_to_ltln' False ψ)\"\n| \"ltlc_to_ltln' False (F⇩c φ) = true⇩n U⇩n (ltlc_to_ltln' False φ)\"\n| \"ltlc_to_ltln' False (G⇩c φ) = false⇩n R⇩n (ltlc_to_ltln' False φ)\"\n| \"ltlc_to_ltln' False (φ U⇩c ψ) = (ltlc_to_ltln' False φ) U⇩n (ltlc_to_ltln' False ψ)\"\n| \"ltlc_to_ltln' False (φ R⇩c ψ) = (ltlc_to_ltln' False φ) R⇩n (ltlc_to_ltln' False ψ)\"\n| \"ltlc_to_ltln' False (φ W⇩c ψ) = (ltlc_to_ltln' False φ) W⇩n (ltlc_to_ltln' False ψ)\"\n| \"ltlc_to_ltln' False (φ M⇩c ψ) = (ltlc_to_ltln' False φ) M⇩n (ltlc_to_ltln' False ψ)\"\n| \"ltlc_to_ltln' True true⇩c = false⇩n\"\n| \"ltlc_to_ltln' True false⇩c = true⇩n\"\n| \"ltlc_to_ltln' True prop⇩c(q) = nprop⇩n(q)\"\n| \"ltlc_to_ltln' True (φ and⇩c ψ) = (ltlc_to_ltln' True φ) or⇩n (ltlc_to_ltln' True ψ)\"\n| \"ltlc_to_ltln' True (φ or⇩c ψ) = (ltlc_to_ltln' True φ) and⇩n (ltlc_to_ltln' True ψ)\"\n| \"ltlc_to_ltln' True (φ implies⇩c ψ) = (ltlc_to_ltln' False φ) and⇩n (ltlc_to_ltln' True ψ)\"\n| \"ltlc_to_ltln' True (F⇩c φ) = false⇩n R⇩n (ltlc_to_ltln' True φ)\"\n| \"ltlc_to_ltln' True (G⇩c φ) = true⇩n U⇩n (ltlc_to_ltln' True φ)\"\n| \"ltlc_to_ltln' True (φ U⇩c ψ) = (ltlc_to_ltln' True φ) R⇩n (ltlc_to_ltln' True ψ)\"\n| \"ltlc_to_ltln' True (φ R⇩c ψ) = (ltlc_to_ltln' True φ) U⇩n (ltlc_to_ltln' True ψ)\"\n| \"ltlc_to_ltln' True (φ W⇩c ψ) = (ltlc_to_ltln' True φ) M⇩n (ltlc_to_ltln' True ψ)\"\n| \"ltlc_to_ltln' True (φ M⇩c ψ) = (ltlc_to_ltln' True φ) W⇩n (ltlc_to_ltln' True ψ)\"\n| \"ltlc_to_ltln' b (not⇩c φ) = ltlc_to_ltln' (¬ b) φ\"\n| \"ltlc_to_ltln' b (X⇩c φ) = X⇩n (ltlc_to_ltln' b φ)\"\n\nfun ltlc_to_ltln :: \"'a ltlc ⇒ 'a ltln\"\nwhere\n \"ltlc_to_ltln φ = ltlc_to_ltln' False φ\"\n\nfun ltln_to_ltlc :: \"'a ltln ⇒ 'a ltlc\"\nwhere\n \"ltln_to_ltlc true⇩n = true⇩c\"\n| \"ltln_to_ltlc false⇩n = false⇩c\"\n| \"ltln_to_ltlc prop⇩n(q) = prop⇩c(q)\"\n| \"ltln_to_ltlc nprop⇩n(q) = not⇩c (prop⇩c(q))\"\n| \"ltln_to_ltlc (φ and⇩n ψ) = (ltln_to_ltlc φ and⇩c ltln_to_ltlc ψ)\"\n| \"ltln_to_ltlc (φ or⇩n ψ) = (ltln_to_ltlc φ or⇩c ltln_to_ltlc ψ)\"\n| \"ltln_to_ltlc (X⇩n φ) = (X⇩c ltln_to_ltlc φ)\"\n| \"ltln_to_ltlc (φ U⇩n ψ) = (ltln_to_ltlc φ U⇩c ltln_to_ltlc ψ)\"\n| \"ltln_to_ltlc (φ R⇩n ψ) = (ltln_to_ltlc φ R⇩c ltln_to_ltlc ψ)\"\n| \"ltln_to_ltlc (φ W⇩n ψ) = (ltln_to_ltlc φ W⇩c ltln_to_ltlc ψ)\"\n| \"ltln_to_ltlc (φ M⇩n ψ) = (ltln_to_ltlc φ M⇩c ltln_to_ltlc ψ)\"\n\nlemma ltlc_to_ltln'_correct:\n \"w ⊨⇩n (ltlc_to_ltln' True φ) ⟷ ¬ w ⊨⇩c φ\"\n \"w ⊨⇩n (ltlc_to_ltln' False φ) ⟷ w ⊨⇩c φ\"\n \"size (ltlc_to_ltln' True φ) ≤ 2 * size φ\"\n \"size (ltlc_to_ltln' False φ) ≤ 2 * size φ\"\n apply (induction φ arbitrary: w)\n (*top goal: ‹(w::nat ⇒ 'a set) ⊨⇩n ltlc_to_ltln' True (φ::'a ltlc) = (¬ w ⊨⇩c φ)› and 3 goals remain*)\n by ((simp)+)\n\nlemma ltlc_to_ltln_semantics [simp]:\n \"w ⊨⇩n ltlc_to_ltln φ ⟷ w ⊨⇩c φ\"\n using ltlc_to_ltln'_correct (*‹(?w::nat ⇒ ?'a set) ⊨⇩n ltlc_to_ltln' True (?φ::?'a ltlc) = (¬ ?w ⊨⇩c ?φ)› ‹?w ⊨⇩n ltlc_to_ltln' False ?φ = ?w ⊨⇩c ?φ› ‹size (ltlc_to_ltln' True (?φ::?'a ltlc)) ≤ (2::nat) * size ?φ› ‹size (ltlc_to_ltln' False ?φ) ≤ 2 * size ?φ›*) by auto\n\nlemma ltlc_to_ltln_size:\n \"size (ltlc_to_ltln φ) ≤ 2 * size φ\"\n using ltlc_to_ltln'_correct (*‹?w ⊨⇩n ltlc_to_ltln' True ?φ = (¬ ?w ⊨⇩c ?φ)› ‹(?w::nat ⇒ ?'a set) ⊨⇩n ltlc_to_ltln' False (?φ::?'a ltlc) = ?w ⊨⇩c ?φ› ‹size (ltlc_to_ltln' True ?φ) ≤ 2 * size ?φ› ‹size (ltlc_to_ltln' False ?φ) ≤ 2 * size ?φ›*) by simp\n\nlemma ltln_to_ltlc_semantics [simp]:\n \"w ⊨⇩c ltln_to_ltlc φ ⟷ w ⊨⇩n φ\"\n apply (induction φ arbitrary: w)\n (*goals:\n 1. ‹⋀w. w ⊨⇩c ltln_to_ltlc true⇩n = w ⊨⇩n true⇩n›\n 2. ‹⋀w. w ⊨⇩c ltln_to_ltlc false⇩n = w ⊨⇩n false⇩n›\n 3. ‹⋀x w. w ⊨⇩c ltln_to_ltlc prop⇩n(x) = w ⊨⇩n prop⇩n(x)›\n 4. ‹⋀x w. w ⊨⇩c ltln_to_ltlc nprop⇩n(x) = w ⊨⇩n nprop⇩n(x)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩c ltln_to_ltlc φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩c ltln_to_ltlc φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩c ltln_to_ltlc (φ1 and⇩n φ2) = w ⊨⇩n φ1 and⇩n φ2›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩c ltln_to_ltlc φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩c ltln_to_ltlc φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩c ltln_to_ltlc (φ1 or⇩n φ2) = w ⊨⇩n φ1 or⇩n φ2›\n 7. ‹⋀φ w. (⋀w. w ⊨⇩c ltln_to_ltlc φ = w ⊨⇩n φ) ⟹ w ⊨⇩c ltln_to_ltlc (X⇩n φ) = w ⊨⇩n X⇩n φ›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩c ltln_to_ltlc φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩c ltln_to_ltlc φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩c ltln_to_ltlc (φ1 U⇩n φ2) = w ⊨⇩n φ1 U⇩n φ2›\n 9. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩c ltln_to_ltlc φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩c ltln_to_ltlc φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩c ltln_to_ltlc (φ1 R⇩n φ2) = w ⊨⇩n φ1 R⇩n φ2›\n 10. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩c ltln_to_ltlc φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩c ltln_to_ltlc φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩c ltln_to_ltlc (φ1 W⇩n φ2) = w ⊨⇩n φ1 W⇩n φ2›\n 11. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩c ltln_to_ltlc φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩c ltln_to_ltlc φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩c ltln_to_ltlc (φ1 M⇩n φ2) = w ⊨⇩n φ1 M⇩n φ2›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*discuss goal 8*)\n apply simp\n (*discuss goal 9*)\n apply simp\n (*discuss goal 10*)\n apply simp\n (*discuss goal 11*)\n apply simp\n (*proven 11 subgoals*) .\n\nlemma ltlc_to_ltln_atoms:\n \"atoms_ltln (ltlc_to_ltln φ) = atoms_ltlc φ\"\nproof (-)\n (*goal: ‹atoms_ltln (ltlc_to_ltln φ) = atoms_ltlc φ›*)\n have \"atoms_ltln (ltlc_to_ltln' True φ) = atoms_ltlc φ\" \"atoms_ltln (ltlc_to_ltln' False φ) = atoms_ltlc φ\"\n apply (induction φ)\n (*top goal: ‹atoms_ltln (ltlc_to_ltln' True φ) = atoms_ltlc φ› and 1 goal remains*)\n by ((simp)+)\n thus \"?thesis\"\n (*goal: ‹atoms_ltln (ltlc_to_ltln φ) = atoms_ltlc φ›*)\n by simp\nqed\n\nsubsubsection ‹Negation›\n\nfun not⇩n\nwhere\n \"not⇩n true⇩n = false⇩n\"\n| \"not⇩n false⇩n = true⇩n\"\n| \"not⇩n prop⇩n(a) = nprop⇩n(a)\"\n| \"not⇩n nprop⇩n(a) = prop⇩n(a)\"\n| \"not⇩n (φ and⇩n ψ) = (not⇩n φ) or⇩n (not⇩n ψ)\"\n| \"not⇩n (φ or⇩n ψ) = (not⇩n φ) and⇩n (not⇩n ψ)\"\n| \"not⇩n (X⇩n φ) = X⇩n (not⇩n φ)\"\n| \"not⇩n (φ U⇩n ψ) = (not⇩n φ) R⇩n (not⇩n ψ)\"\n| \"not⇩n (φ R⇩n ψ) = (not⇩n φ) U⇩n (not⇩n ψ)\"\n| \"not⇩n (φ W⇩n ψ) = (not⇩n φ) M⇩n (not⇩n ψ)\"\n| \"not⇩n (φ M⇩n ψ) = (not⇩n φ) W⇩n (not⇩n ψ)\"\n\nlemma not⇩n_semantics[simp]:\n \"w ⊨⇩n not⇩n φ ⟷ ¬ w ⊨⇩n φ\"\n apply (induction φ arbitrary: w)\n (*goals:\n 1. ‹⋀w. w ⊨⇩n not⇩n true⇩n = (¬ w ⊨⇩n true⇩n)›\n 2. ‹⋀w. w ⊨⇩n not⇩n false⇩n = (¬ w ⊨⇩n false⇩n)›\n 3. ‹⋀x w. w ⊨⇩n not⇩n prop⇩n(x) = (¬ w ⊨⇩n prop⇩n(x))›\n 4. ‹⋀x w. w ⊨⇩n not⇩n nprop⇩n(x) = (¬ w ⊨⇩n nprop⇩n(x))›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n not⇩n φ1 = (¬ w ⊨⇩n φ1); ⋀w. w ⊨⇩n not⇩n φ2 = (¬ w ⊨⇩n φ2)⟧ ⟹ w ⊨⇩n not⇩n (φ1 and⇩n φ2) = (¬ w ⊨⇩n φ1 and⇩n φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n not⇩n φ1 = (¬ w ⊨⇩n φ1); ⋀w. w ⊨⇩n not⇩n φ2 = (¬ w ⊨⇩n φ2)⟧ ⟹ w ⊨⇩n not⇩n (φ1 or⇩n φ2) = (¬ w ⊨⇩n φ1 or⇩n φ2)›\n 7. ‹⋀φ w. (⋀w. w ⊨⇩n not⇩n φ = (¬ w ⊨⇩n φ)) ⟹ w ⊨⇩n not⇩n (X⇩n φ) = (¬ w ⊨⇩n X⇩n φ)›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n not⇩n φ1 = (¬ w ⊨⇩n φ1); ⋀w. w ⊨⇩n not⇩n φ2 = (¬ w ⊨⇩n φ2)⟧ ⟹ w ⊨⇩n not⇩n (φ1 U⇩n φ2) = (¬ w ⊨⇩n φ1 U⇩n φ2)›\n 9. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n not⇩n φ1 = (¬ w ⊨⇩n φ1); ⋀w. w ⊨⇩n not⇩n φ2 = (¬ w ⊨⇩n φ2)⟧ ⟹ w ⊨⇩n not⇩n (φ1 R⇩n φ2) = (¬ w ⊨⇩n φ1 R⇩n φ2)›\n 10. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n not⇩n φ1 = (¬ w ⊨⇩n φ1); ⋀w. w ⊨⇩n not⇩n φ2 = (¬ w ⊨⇩n φ2)⟧ ⟹ w ⊨⇩n not⇩n (φ1 W⇩n φ2) = (¬ w ⊨⇩n φ1 W⇩n φ2)›\n 11. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n not⇩n φ1 = (¬ w ⊨⇩n φ1); ⋀w. w ⊨⇩n not⇩n φ2 = (¬ w ⊨⇩n φ2)⟧ ⟹ w ⊨⇩n not⇩n (φ1 M⇩n φ2) = (¬ w ⊨⇩n φ1 M⇩n φ2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*discuss goal 11*)\n apply ((auto)[1])\n (*proven 11 subgoals*) .\n\nlemma not⇩n_size:\n \"size (not⇩n φ) = size φ\"\n apply (induction φ)\n (*goals:\n 1. ‹size (not⇩n true⇩n) = size true⇩n›\n 2. ‹size (not⇩n false⇩n) = size false⇩n›\n 3. ‹⋀x. size (not⇩n prop⇩n(x)) = size prop⇩n(x)›\n 4. ‹⋀x. size (not⇩n nprop⇩n(x)) = size nprop⇩n(x)›\n 5. ‹⋀φ1 φ2. ⟦size (not⇩n φ1) = size φ1; size (not⇩n φ2) = size φ2⟧ ⟹ size (not⇩n (φ1 and⇩n φ2)) = size (φ1 and⇩n φ2)›\n 6. ‹⋀φ1 φ2. ⟦size (not⇩n φ1) = size φ1; size (not⇩n φ2) = size φ2⟧ ⟹ size (not⇩n (φ1 or⇩n φ2)) = size (φ1 or⇩n φ2)›\n 7. ‹⋀φ. size (not⇩n φ) = size φ ⟹ size (not⇩n (X⇩n φ)) = size (X⇩n φ)›\n 8. ‹⋀φ1 φ2. ⟦size (not⇩n φ1) = size φ1; size (not⇩n φ2) = size φ2⟧ ⟹ size (not⇩n (φ1 U⇩n φ2)) = size (φ1 U⇩n φ2)›\n 9. ‹⋀φ1 φ2. ⟦size (not⇩n φ1) = size φ1; size (not⇩n φ2) = size φ2⟧ ⟹ size (not⇩n (φ1 R⇩n φ2)) = size (φ1 R⇩n φ2)›\n 10. ‹⋀φ1 φ2. ⟦size (not⇩n φ1) = size φ1; size (not⇩n φ2) = size φ2⟧ ⟹ size (not⇩n (φ1 W⇩n φ2)) = size (φ1 W⇩n φ2)›\n 11. ‹⋀φ1 φ2. ⟦size (not⇩n φ1) = size φ1; size (not⇩n φ2) = size φ2⟧ ⟹ size (not⇩n (φ1 M⇩n φ2)) = size (φ1 M⇩n φ2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*discuss goal 11*)\n apply ((auto)[1])\n (*proven 11 subgoals*) .\n\n\nsubsubsection ‹Subformulas›\n\nfun subfrmlsn :: \"'a ltln ⇒ 'a ltln set\"\nwhere\n \"subfrmlsn (φ and⇩n ψ) = {φ and⇩n ψ} ∪ subfrmlsn φ ∪ subfrmlsn ψ\"\n| \"subfrmlsn (φ or⇩n ψ) = {φ or⇩n ψ} ∪ subfrmlsn φ ∪ subfrmlsn ψ\"\n| \"subfrmlsn (X⇩n φ) = {X⇩n φ} ∪ subfrmlsn φ\"\n| \"subfrmlsn (φ U⇩n ψ) = {φ U⇩n ψ} ∪ subfrmlsn φ ∪ subfrmlsn ψ\"\n| \"subfrmlsn (φ R⇩n ψ) = {φ R⇩n ψ} ∪ subfrmlsn φ ∪ subfrmlsn ψ\"\n| \"subfrmlsn (φ W⇩n ψ) = {φ W⇩n ψ} ∪ subfrmlsn φ ∪ subfrmlsn ψ\"\n| \"subfrmlsn (φ M⇩n ψ) = {φ M⇩n ψ} ∪ subfrmlsn φ ∪ subfrmlsn ψ\"\n| \"subfrmlsn φ = {φ}\"\n\nlemma subfrmlsn_id[simp]:\n \"φ ∈ subfrmlsn φ\"\n apply (induction φ)\n (*goals:\n 1. ‹true⇩n ∈ subfrmlsn true⇩n›\n 2. ‹false⇩n ∈ subfrmlsn false⇩n›\n 3. ‹⋀x. prop⇩n(x) ∈ subfrmlsn prop⇩n(x)›\n 4. ‹⋀x. nprop⇩n(x) ∈ subfrmlsn nprop⇩n(x)›\n 5. ‹⋀φ1 φ2. ⟦φ1 ∈ subfrmlsn φ1; φ2 ∈ subfrmlsn φ2⟧ ⟹ φ1 and⇩n φ2 ∈ subfrmlsn (φ1 and⇩n φ2)›\n 6. ‹⋀φ1 φ2. ⟦φ1 ∈ subfrmlsn φ1; φ2 ∈ subfrmlsn φ2⟧ ⟹ φ1 or⇩n φ2 ∈ subfrmlsn (φ1 or⇩n φ2)›\n 7. ‹⋀φ. φ ∈ subfrmlsn φ ⟹ X⇩n φ ∈ subfrmlsn (X⇩n φ)›\n 8. ‹⋀φ1 φ2. ⟦φ1 ∈ subfrmlsn φ1; φ2 ∈ subfrmlsn φ2⟧ ⟹ φ1 U⇩n φ2 ∈ subfrmlsn (φ1 U⇩n φ2)›\n 9. ‹⋀φ1 φ2. ⟦φ1 ∈ subfrmlsn φ1; φ2 ∈ subfrmlsn φ2⟧ ⟹ φ1 R⇩n φ2 ∈ subfrmlsn (φ1 R⇩n φ2)›\n 10. ‹⋀φ1 φ2. ⟦φ1 ∈ subfrmlsn φ1; φ2 ∈ subfrmlsn φ2⟧ ⟹ φ1 W⇩n φ2 ∈ subfrmlsn (φ1 W⇩n φ2)›\n 11. ‹⋀φ1 φ2. ⟦φ1 ∈ subfrmlsn φ1; φ2 ∈ subfrmlsn φ2⟧ ⟹ φ1 M⇩n φ2 ∈ subfrmlsn (φ1 M⇩n φ2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*discuss goal 11*)\n apply ((auto)[1])\n (*proven 11 subgoals*) .\n\nlemma subfrmlsn_finite:\n \"finite (subfrmlsn φ)\"\n apply (induction φ)\n (*goals:\n 1. ‹finite (subfrmlsn true⇩n)›\n 2. ‹finite (subfrmlsn false⇩n)›\n 3. ‹⋀x. finite (subfrmlsn prop⇩n(x))›\n 4. ‹⋀x. finite (subfrmlsn nprop⇩n(x))›\n 5. ‹⋀φ1 φ2. ⟦finite (subfrmlsn φ1); finite (subfrmlsn φ2)⟧ ⟹ finite (subfrmlsn (φ1 and⇩n φ2))›\n 6. ‹⋀φ1 φ2. ⟦finite (subfrmlsn φ1); finite (subfrmlsn φ2)⟧ ⟹ finite (subfrmlsn (φ1 or⇩n φ2))›\n 7. ‹⋀φ. finite (subfrmlsn φ) ⟹ finite (subfrmlsn (X⇩n φ))›\n 8. ‹⋀φ1 φ2. ⟦finite (subfrmlsn φ1); finite (subfrmlsn φ2)⟧ ⟹ finite (subfrmlsn (φ1 U⇩n φ2))›\n 9. ‹⋀φ1 φ2. ⟦finite (subfrmlsn φ1); finite (subfrmlsn φ2)⟧ ⟹ finite (subfrmlsn (φ1 R⇩n φ2))›\n 10. ‹⋀φ1 φ2. ⟦finite (subfrmlsn φ1); finite (subfrmlsn φ2)⟧ ⟹ finite (subfrmlsn (φ1 W⇩n φ2))›\n 11. ‹⋀φ1 φ2. ⟦finite (subfrmlsn φ1); finite (subfrmlsn φ2)⟧ ⟹ finite (subfrmlsn (φ1 M⇩n φ2))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*discuss goal 11*)\n apply ((auto)[1])\n (*proven 11 subgoals*) .\n\nlemma subfrmlsn_card:\n \"card (subfrmlsn φ) ≤ size φ\"\n apply (induction φ)\n (*goals:\n 1. ‹card (subfrmlsn true⇩n) ≤ size true⇩n›\n 2. ‹card (subfrmlsn false⇩n) ≤ size false⇩n›\n 3. ‹⋀x. card (subfrmlsn prop⇩n(x)) ≤ size prop⇩n(x)›\n 4. ‹⋀x. card (subfrmlsn nprop⇩n(x)) ≤ size nprop⇩n(x)›\n 5. ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 and⇩n φ2)) ≤ size (φ1 and⇩n φ2)›\n 6. ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 or⇩n φ2)) ≤ size (φ1 or⇩n φ2)›\n 7. ‹⋀φ. card (subfrmlsn φ) ≤ size φ ⟹ card (subfrmlsn (X⇩n φ)) ≤ size (X⇩n φ)›\n 8. ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 U⇩n φ2)) ≤ size (φ1 U⇩n φ2)›\n 9. ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 R⇩n φ2)) ≤ size (φ1 R⇩n φ2)›\n 10. ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 W⇩n φ2)) ≤ size (φ1 W⇩n φ2)›\n 11. ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 M⇩n φ2)) ≤ size (φ1 M⇩n φ2)›\n discuss goal 1*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*discuss goal 2*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*discuss goal 3*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*discuss goal 4*)\n apply (simp add: card_insert_if (*‹finite (?A::?'a set) ⟹ card (insert (?x::?'a) ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn (?φ::?'a ltln))›*))\n (*discuss goal 5*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*top goal: ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 and⇩n φ2)) ≤ size (φ1 and⇩n φ2)› and 6 goals remain*)\n apply (meson add_le_mono (*‹⟦?i ≤ ?j; ?k ≤ ?l⟧ ⟹ ?i + ?k ≤ ?j + ?l›*) card_Un_le (*‹card (?A ∪ ?B) ≤ card ?A + card ?B›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) le_SucI (*‹?m ≤ ?n ⟹ ?m ≤ Suc ?n›*))\n (*discuss goal 6*)\n apply (simp add: card_insert_if (*‹finite (?A::?'a set) ⟹ card (insert (?x::?'a) ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn (?φ::?'a ltln))›*))\n (*top goal: ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 or⇩n φ2)) ≤ size (φ1 or⇩n φ2)› and 5 goals remain*)\n apply (meson add_le_mono (*‹⟦?i ≤ ?j; ?k ≤ ?l⟧ ⟹ ?i + ?k ≤ ?j + ?l›*) card_Un_le (*‹card (?A ∪ ?B) ≤ card ?A + card ?B›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) le_SucI (*‹?m ≤ ?n ⟹ ?m ≤ Suc ?n›*))\n (*discuss goal 7*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*discuss goal 8*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*top goal: ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 U⇩n φ2)) ≤ size (φ1 U⇩n φ2)› and 3 goals remain*)\n apply (meson add_le_mono (*‹⟦?i ≤ ?j; ?k ≤ ?l⟧ ⟹ ?i + ?k ≤ ?j + ?l›*) card_Un_le (*‹card (?A ∪ ?B) ≤ card ?A + card ?B›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) le_SucI (*‹?m ≤ ?n ⟹ ?m ≤ Suc ?n›*))\n (*discuss goal 9*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*top goal: ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 R⇩n φ2)) ≤ size (φ1 R⇩n φ2)› and 2 goals remain*)\n apply (meson add_le_mono (*‹⟦?i ≤ ?j; ?k ≤ ?l⟧ ⟹ ?i + ?k ≤ ?j + ?l›*) card_Un_le (*‹card (?A ∪ ?B) ≤ card ?A + card ?B›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) le_SucI (*‹?m ≤ ?n ⟹ ?m ≤ Suc ?n›*))\n (*discuss goal 10*)\n apply (simp add: card_insert_if (*‹finite ?A ⟹ card (insert ?x ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn ?φ)›*))\n (*top goal: ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 W⇩n φ2)) ≤ size (φ1 W⇩n φ2)› and 1 goal remains*)\n apply (meson add_le_mono (*‹⟦?i ≤ ?j; ?k ≤ ?l⟧ ⟹ ?i + ?k ≤ ?j + ?l›*) card_Un_le (*‹card (?A ∪ ?B) ≤ card ?A + card ?B›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) le_SucI (*‹?m ≤ ?n ⟹ ?m ≤ Suc ?n›*))\n (*discuss goal 11*)\n apply (simp add: card_insert_if (*‹finite (?A::?'a set) ⟹ card (insert (?x::?'a) ?A) = (if ?x ∈ ?A then card ?A else Suc (card ?A))›*) subfrmlsn_finite (*‹finite (subfrmlsn (?φ::?'a ltln))›*))\n (*goal: ‹⋀φ1 φ2. ⟦card (subfrmlsn φ1) ≤ size φ1; card (subfrmlsn φ2) ≤ size φ2⟧ ⟹ card (subfrmlsn (φ1 M⇩n φ2)) ≤ size (φ1 M⇩n φ2)›*)\n apply (meson add_le_mono (*‹⟦?i ≤ ?j; ?k ≤ ?l⟧ ⟹ ?i + ?k ≤ ?j + ?l›*) card_Un_le (*‹card (?A ∪ ?B) ≤ card ?A + card ?B›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) le_SucI (*‹?m ≤ ?n ⟹ ?m ≤ Suc ?n›*))\n (*proven 11 subgoals*) .\n\nlemma subfrmlsn_subset:\n \"ψ ∈ subfrmlsn φ ⟹ subfrmlsn ψ ⊆ subfrmlsn φ\"\n apply (induction φ)\n (*goals:\n 1. ‹ψ ∈ subfrmlsn true⇩n ⟹ subfrmlsn ψ ⊆ subfrmlsn true⇩n›\n 2. ‹ψ ∈ subfrmlsn false⇩n ⟹ subfrmlsn ψ ⊆ subfrmlsn false⇩n›\n 3. ‹⋀x. ψ ∈ subfrmlsn prop⇩n(x) ⟹ subfrmlsn ψ ⊆ subfrmlsn prop⇩n(x)›\n 4. ‹⋀x. ψ ∈ subfrmlsn nprop⇩n(x) ⟹ subfrmlsn ψ ⊆ subfrmlsn nprop⇩n(x)›\n 5. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ1; ψ ∈ subfrmlsn φ2 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ2; ψ ∈ subfrmlsn (φ1 and⇩n φ2)⟧ ⟹ subfrmlsn ψ ⊆ subfrmlsn (φ1 and⇩n φ2)›\n 6. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ1; ψ ∈ subfrmlsn φ2 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ2; ψ ∈ subfrmlsn (φ1 or⇩n φ2)⟧ ⟹ subfrmlsn ψ ⊆ subfrmlsn (φ1 or⇩n φ2)›\n 7. ‹⋀φ. ⟦ψ ∈ subfrmlsn φ ⟹ subfrmlsn ψ ⊆ subfrmlsn φ; ψ ∈ subfrmlsn (X⇩n φ)⟧ ⟹ subfrmlsn ψ ⊆ subfrmlsn (X⇩n φ)›\n 8. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ1; ψ ∈ subfrmlsn φ2 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ2; ψ ∈ subfrmlsn (φ1 U⇩n φ2)⟧ ⟹ subfrmlsn ψ ⊆ subfrmlsn (φ1 U⇩n φ2)›\n 9. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ1; ψ ∈ subfrmlsn φ2 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ2; ψ ∈ subfrmlsn (φ1 R⇩n φ2)⟧ ⟹ subfrmlsn ψ ⊆ subfrmlsn (φ1 R⇩n φ2)›\n 10. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ1; ψ ∈ subfrmlsn φ2 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ2; ψ ∈ subfrmlsn (φ1 W⇩n φ2)⟧ ⟹ subfrmlsn ψ ⊆ subfrmlsn (φ1 W⇩n φ2)›\n 11. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ1; ψ ∈ subfrmlsn φ2 ⟹ subfrmlsn ψ ⊆ subfrmlsn φ2; ψ ∈ subfrmlsn (φ1 M⇩n φ2)⟧ ⟹ subfrmlsn ψ ⊆ subfrmlsn (φ1 M⇩n φ2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*discuss goal 11*)\n apply ((auto)[1])\n (*proven 11 subgoals*) .\n\nlemma subfrmlsn_size:\n \"ψ ∈ subfrmlsn φ ⟹ size ψ < size φ ∨ ψ = φ\"\n apply (induction φ)\n (*goals:\n 1. ‹ψ ∈ subfrmlsn true⇩n ⟹ size ψ < size true⇩n ∨ ψ = true⇩n›\n 2. ‹ψ ∈ subfrmlsn false⇩n ⟹ size ψ < size false⇩n ∨ ψ = false⇩n›\n 3. ‹⋀x. ψ ∈ subfrmlsn prop⇩n(x) ⟹ size ψ < size prop⇩n(x) ∨ ψ = prop⇩n(x)›\n 4. ‹⋀x. ψ ∈ subfrmlsn nprop⇩n(x) ⟹ size ψ < size nprop⇩n(x) ∨ ψ = nprop⇩n(x)›\n 5. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsn φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsn (φ1 and⇩n φ2)⟧ ⟹ size ψ < size (φ1 and⇩n φ2) ∨ ψ = φ1 and⇩n φ2›\n 6. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsn φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsn (φ1 or⇩n φ2)⟧ ⟹ size ψ < size (φ1 or⇩n φ2) ∨ ψ = φ1 or⇩n φ2›\n 7. ‹⋀φ. ⟦ψ ∈ subfrmlsn φ ⟹ size ψ < size φ ∨ ψ = φ; ψ ∈ subfrmlsn (X⇩n φ)⟧ ⟹ size ψ < size (X⇩n φ) ∨ ψ = X⇩n φ›\n 8. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsn φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsn (φ1 U⇩n φ2)⟧ ⟹ size ψ < size (φ1 U⇩n φ2) ∨ ψ = φ1 U⇩n φ2›\n 9. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsn φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsn (φ1 R⇩n φ2)⟧ ⟹ size ψ < size (φ1 R⇩n φ2) ∨ ψ = φ1 R⇩n φ2›\n 10. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsn φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsn (φ1 W⇩n φ2)⟧ ⟹ size ψ < size (φ1 W⇩n φ2) ∨ ψ = φ1 W⇩n φ2›\n 11. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsn φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsn φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsn (φ1 M⇩n φ2)⟧ ⟹ size ψ < size (φ1 M⇩n φ2) ∨ ψ = φ1 M⇩n φ2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*discuss goal 11*)\n apply ((auto)[1])\n (*proven 11 subgoals*) .\n\nabbreviation\n \"size_set S ≡ sum (λx. 2 * size x + 1) S\"\n\nlemma size_set_diff:\n \"finite S ⟹ S' ⊆ S ⟹ size_set (S - S') = size_set S - size_set S'\"\n using sum_diff_nat (*‹⟦finite ?B; ?B ⊆ ?A⟧ ⟹ sum ?f (?A - ?B) = sum ?f ?A - sum ?f ?B›*) finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) by metis\n\n\nsubsubsection ‹Constant Folding›\n\nlemma U_consts[intro, simp]:\n \"w ⊨⇩n φ U⇩n true⇩n\"\n \"¬ (w ⊨⇩n φ U⇩n false⇩n)\"\n \"(w ⊨⇩n false⇩n U⇩n φ) = (w ⊨⇩n φ)\"\n \n (*goals:\n 1. ‹(w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) U⇩n true⇩n›\n 2. ‹¬ (w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) U⇩n false⇩n›\n 3. ‹(w::nat ⇒ 'a set) ⊨⇩n false⇩n U⇩n (φ::'a ltln) = w ⊨⇩n φ›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*discuss goal 3*)\n apply force\n (*proven 3 subgoals*) .\n\nlemma R_consts[intro, simp]:\n \"w ⊨⇩n φ R⇩n true⇩n\"\n \"¬ (w ⊨⇩n φ R⇩n false⇩n)\"\n \"(w ⊨⇩n true⇩n R⇩n φ) = (w ⊨⇩n φ)\"\n \n (*goals:\n 1. ‹w ⊨⇩n φ R⇩n true⇩n›\n 2. ‹¬ w ⊨⇩n φ R⇩n false⇩n›\n 3. ‹w ⊨⇩n true⇩n R⇩n φ = w ⊨⇩n φ›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*discuss goal 3*)\n apply force\n (*proven 3 subgoals*) .\n\nlemma W_consts[intro, simp]:\n \"w ⊨⇩n true⇩n W⇩n φ\"\n \"w ⊨⇩n φ W⇩n true⇩n\"\n \"(w ⊨⇩n false⇩n W⇩n φ) = (w ⊨⇩n φ)\"\n \"(w ⊨⇩n φ W⇩n false⇩n) = (w ⊨⇩n G⇩n φ)\"\n \n (*goals:\n 1. ‹w ⊨⇩n true⇩n W⇩n φ›\n 2. ‹w ⊨⇩n φ W⇩n true⇩n›\n 3. ‹w ⊨⇩n false⇩n W⇩n φ = w ⊨⇩n φ›\n 4. ‹w ⊨⇩n φ W⇩n false⇩n = w ⊨⇩n G⇩n φ›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*discuss goal 3*)\n apply force\n (*discuss goal 4*)\n apply force\n (*proven 4 subgoals*) .\n\nlemma M_consts[intro, simp]:\n \"¬ (w ⊨⇩n false⇩n M⇩n φ)\"\n \"¬ (w ⊨⇩n φ M⇩n false⇩n)\"\n \"(w ⊨⇩n true⇩n M⇩n φ) = (w ⊨⇩n φ)\"\n \"(w ⊨⇩n φ M⇩n true⇩n) = (w ⊨⇩n F⇩n φ)\"\n \n (*goals:\n 1. ‹¬ (w::nat ⇒ 'a::type set) ⊨⇩n false⇩n M⇩n (φ::'a::type ltln)›\n 2. ‹¬ (w::nat ⇒ 'a::type set) ⊨⇩n (φ::'a::type ltln) M⇩n false⇩n›\n 3. ‹(w::nat ⇒ 'a::type set) ⊨⇩n true⇩n M⇩n (φ::'a::type ltln) = w ⊨⇩n φ›\n 4. ‹(w::nat ⇒ 'a::type set) ⊨⇩n (φ::'a::type ltln) M⇩n true⇩n = w ⊨⇩n F⇩n φ›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*discuss goal 3*)\n apply force\n (*discuss goal 4*)\n apply force\n (*proven 4 subgoals*) .\n\n\nsubsubsection ‹Distributivity›\n\nlemma until_and_left_distrib:\n \"w ⊨⇩n (φ₁ and⇩n φ₂) U⇩n ψ ⟷ w ⊨⇩n (φ₁ U⇩n ψ) and⇩n (φ₂ U⇩n ψ)\"\nproof (standard)\n (*goals:\n 1. ‹w ⊨⇩n (φ₁ and⇩n φ₂) U⇩n ψ ⟹ w ⊨⇩n φ₁ U⇩n ψ and⇩n φ₂ U⇩n ψ›\n 2. ‹w ⊨⇩n φ₁ U⇩n ψ and⇩n φ₂ U⇩n ψ ⟹ w ⊨⇩n (φ₁ and⇩n φ₂) U⇩n ψ›*)\n assume \"w ⊨⇩n φ₁ U⇩n ψ and⇩n φ₂ U⇩n ψ\" (*‹(w::nat ⇒ 'a set) ⊨⇩n (φ₁::'a ltln) U⇩n (ψ::'a ltln) and⇩n (φ₂::'a ltln) U⇩n ψ›*)\n then obtain i1 and i2 where \"suffix i1 w ⊨⇩n ψ ∧ (∀j<i1. suffix j w ⊨⇩n φ₁)\" and \"suffix i2 w ⊨⇩n ψ ∧ (∀j<i2. suffix j w ⊨⇩n φ₂)\"\n (*goal: ‹(⋀i1 i2. ⟦suffix i1 w ⊨⇩n ψ ∧ (∀j<i1. suffix j w ⊨⇩n φ₁); suffix i2 w ⊨⇩n ψ ∧ (∀j<i2. suffix j w ⊨⇩n φ₂)⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n then have \"suffix (min i1 i2) w ⊨⇩n ψ ∧ (∀j<min i1 i2. suffix j w ⊨⇩n φ₁ and⇩n φ₂)\"\n by (simp add: min_def (*‹min ?a ?b = (if ?a ≤ ?b then ?a else ?b)›*))\n then show \"w ⊨⇩n (φ₁ and⇩n φ₂) U⇩n ψ\"\n by force\nqed (auto)\n(*solved the remaining goal: ‹w ⊨⇩n (φ₁ and⇩n φ₂) U⇩n ψ ⟹ w ⊨⇩n φ₁ U⇩n ψ and⇩n φ₂ U⇩n ψ›*)\n\nlemma until_or_right_distrib:\n \"w ⊨⇩n φ U⇩n (ψ₁ or⇩n ψ₂) ⟷ w ⊨⇩n (φ U⇩n ψ₁) or⇩n (φ U⇩n ψ₂)\"\n by auto\n\nlemma release_and_right_distrib:\n \"w ⊨⇩n φ R⇩n (ψ₁ and⇩n ψ₂) ⟷ w ⊨⇩n (φ R⇩n ψ₁) and⇩n (φ R⇩n ψ₂)\"\n by auto\n\nlemma release_or_left_distrib:\n \"w ⊨⇩n (φ₁ or⇩n φ₂) R⇩n ψ ⟷ w ⊨⇩n (φ₁ R⇩n ψ) or⇩n (φ₂ R⇩n ψ)\"\n by (metis not⇩n.simps( (*‹not⇩n (?φ or⇩n ?ψ) = not⇩n ?φ and⇩n not⇩n ?ψ›*) 6) not⇩n.simps( (*‹not⇩n (?φ R⇩n ?ψ) = not⇩n ?φ U⇩n not⇩n ?ψ›*) 9) not⇩n_semantics (*‹?w ⊨⇩n not⇩n ?φ = (¬ ?w ⊨⇩n ?φ)›*) until_and_left_distrib (*‹?w ⊨⇩n (?φ₁ and⇩n ?φ₂) U⇩n ?ψ = ?w ⊨⇩n ?φ₁ U⇩n ?ψ and⇩n ?φ₂ U⇩n ?ψ›*))\n\nlemma strong_release_and_right_distrib:\n \"w ⊨⇩n φ M⇩n (ψ₁ and⇩n ψ₂) ⟷ w ⊨⇩n (φ M⇩n ψ₁) and⇩n (φ M⇩n ψ₂)\"\nproof (standard)\n (*goals:\n 1. ‹w ⊨⇩n φ M⇩n (ψ₁ and⇩n ψ₂) ⟹ w ⊨⇩n φ M⇩n ψ₁ and⇩n φ M⇩n ψ₂›\n 2. ‹w ⊨⇩n φ M⇩n ψ₁ and⇩n φ M⇩n ψ₂ ⟹ w ⊨⇩n φ M⇩n (ψ₁ and⇩n ψ₂)›*)\n assume \"w ⊨⇩n (φ M⇩n ψ₁) and⇩n (φ M⇩n ψ₂)\" (*‹(w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) M⇩n (ψ₁::'a ltln) and⇩n φ M⇩n (ψ₂::'a ltln)›*)\n then obtain i1 and i2 where \"suffix i1 w ⊨⇩n φ ∧ (∀j≤i1. suffix j w ⊨⇩n ψ₁)\" and \"suffix i2 w ⊨⇩n φ ∧ (∀j≤i2. suffix j w ⊨⇩n ψ₂)\"\n (*goal: ‹(⋀i1 i2. ⟦suffix i1 w ⊨⇩n φ ∧ (∀j≤i1. suffix j w ⊨⇩n ψ₁); suffix i2 w ⊨⇩n φ ∧ (∀j≤i2. suffix j w ⊨⇩n ψ₂)⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n then have \"suffix (min i1 i2) w ⊨⇩n φ ∧ (∀j≤min i1 i2. suffix j w ⊨⇩n ψ₁ and⇩n ψ₂)\"\n by (simp add: min_def (*‹min ?a ?b = (if ?a ≤ ?b then ?a else ?b)›*))\n then show \"w ⊨⇩n φ M⇩n (ψ₁ and⇩n ψ₂)\"\n by force\nqed (auto)\n(*solved the remaining goal: ‹w ⊨⇩n φ M⇩n (ψ₁ and⇩n ψ₂) ⟹ w ⊨⇩n φ M⇩n ψ₁ and⇩n φ M⇩n ψ₂›*)\n\nlemma strong_release_or_left_distrib:\n \"w ⊨⇩n (φ₁ or⇩n φ₂) M⇩n ψ ⟷ w ⊨⇩n (φ₁ M⇩n ψ) or⇩n (φ₂ M⇩n ψ)\"\n by auto\n\nlemma weak_until_and_left_distrib:\n \"w ⊨⇩n (φ₁ and⇩n φ₂) W⇩n ψ ⟷ w ⊨⇩n (φ₁ W⇩n ψ) and⇩n (φ₂ W⇩n ψ)\"\n by auto\n\nlemma weak_until_or_right_distrib:\n \"w ⊨⇩n φ W⇩n (ψ₁ or⇩n ψ₂) ⟷ w ⊨⇩n (φ W⇩n ψ₁) or⇩n (φ W⇩n ψ₂)\"\n by (metis not⇩n.simps( (*‹not⇩n (?φ W⇩n ?ψ) = not⇩n ?φ M⇩n not⇩n ?ψ›*) 10) not⇩n.simps( (*‹not⇩n (?φ or⇩n ?ψ) = not⇩n ?φ and⇩n not⇩n ?ψ›*) 6) not⇩n_semantics (*‹?w ⊨⇩n not⇩n ?φ = (¬ ?w ⊨⇩n ?φ)›*) strong_release_and_right_distrib (*‹?w ⊨⇩n ?φ M⇩n (?ψ₁ and⇩n ?ψ₂) = ?w ⊨⇩n ?φ M⇩n ?ψ₁ and⇩n ?φ M⇩n ?ψ₂›*))\n\n\nlemma next_until_distrib:\n \"w ⊨⇩n X⇩n (φ U⇩n ψ) ⟷ w ⊨⇩n (X⇩n φ) U⇩n (X⇩n ψ)\"\n by auto\n\nlemma next_release_distrib:\n \"w ⊨⇩n X⇩n (φ R⇩n ψ) ⟷ w ⊨⇩n (X⇩n φ) R⇩n (X⇩n ψ)\"\n by auto\n\nlemma next_weak_until_distrib:\n \"w ⊨⇩n X⇩n (φ W⇩n ψ) ⟷ w ⊨⇩n (X⇩n φ) W⇩n (X⇩n ψ)\"\n by auto\n\nlemma next_strong_release_distrib:\n \"w ⊨⇩n X⇩n (φ M⇩n ψ) ⟷ w ⊨⇩n (X⇩n φ) M⇩n (X⇩n ψ)\"\n by auto\n\n\nsubsubsection ‹Nested operators›\n\nlemma finally_until[simp]:\n \"w ⊨⇩n F⇩n (φ U⇩n ψ) ⟷ w ⊨⇩n F⇩n ψ\"\n apply auto\n (*goal: ‹w ⊨⇩n F⇩n (φ U⇩n ψ) = w ⊨⇩n F⇩n ψ›*)\n by force\n\nlemma globally_release[simp]:\n \"w ⊨⇩n G⇩n (φ R⇩n ψ) ⟷ w ⊨⇩n G⇩n ψ\"\n apply auto\n (*goal: ‹w ⊨⇩n G⇩n (φ R⇩n ψ) = w ⊨⇩n G⇩n ψ›*)\n by force\n\nlemma globally_weak_until[simp]:\n \"w ⊨⇩n G⇩n (φ W⇩n ψ) ⟷ w ⊨⇩n G⇩n (φ or⇩n ψ)\"\n apply auto\n (*goal: ‹w ⊨⇩n G⇩n (φ W⇩n ψ) = w ⊨⇩n G⇩n (φ or⇩n ψ)›*)\n by force\n\nlemma finally_strong_release[simp]:\n \"w ⊨⇩n F⇩n (φ M⇩n ψ) ⟷ w ⊨⇩n F⇩n (φ and⇩n ψ)\"\n apply auto\n (*goal: ‹(w::nat ⇒ 'a set) ⊨⇩n F⇩n ((φ::'a ltln) M⇩n (ψ::'a ltln)) = w ⊨⇩n F⇩n (φ and⇩n ψ)›*)\n by force\n\n\nsubsubsection ‹Weak and strong operators›\n\nlemma ltln_weak_strong:\n \"w ⊨⇩n φ W⇩n ψ ⟷ w ⊨⇩n (G⇩n φ) or⇩n (φ U⇩n ψ)\"\n \"w ⊨⇩n φ R⇩n ψ ⟷ w ⊨⇩n (G⇩n ψ) or⇩n (φ M⇩n ψ)\"\nproof (auto)\n (*goals:\n 1. ‹⋀i. ⟦∀i. suffix i w ⊨⇩n φ ∨ (∃j≤i. suffix j w ⊨⇩n ψ); ∀i. suffix i w ⊨⇩n ψ ⟶ (∃j<i. ¬ suffix j w ⊨⇩n φ)⟧ ⟹ suffix i w ⊨⇩n φ›\n 2. ‹⋀i ia. ⟦∀j≤i. ¬ suffix j w ⊨⇩n ψ; suffix ia w ⊨⇩n ψ; ∀j<ia. suffix j w ⊨⇩n φ⟧ ⟹ suffix i w ⊨⇩n φ›\n 3. ‹⋀i. ⟦∀i. suffix i w ⊨⇩n ψ ∨ (∃j<i. suffix j w ⊨⇩n φ); ∀i. suffix i w ⊨⇩n φ ⟶ (∃j≤i. ¬ suffix j w ⊨⇩n ψ)⟧ ⟹ suffix i w ⊨⇩n ψ›\n 4. ‹⋀i ia. ⟦∀j<i. ¬ suffix j w ⊨⇩n φ; suffix ia w ⊨⇩n φ; ∀j≤ia. suffix j w ⊨⇩n ψ⟧ ⟹ suffix i w ⊨⇩n ψ›*)\n fix i\n assume \"∀i. suffix i w ⊨⇩n φ ∨ (∃j≤i. suffix j w ⊨⇩n ψ)\" and \"∀i. suffix i w ⊨⇩n ψ ⟶ (∃j<i. ¬ suffix j w ⊨⇩n φ)\" (*‹∀i::nat. suffix i (w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) ∨ (∃j≤i. suffix j w ⊨⇩n (ψ::'a ltln))› ‹∀i::nat. suffix i (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln) ⟶ (∃j<i. ¬ suffix j w ⊨⇩n (φ::'a ltln))›*)\n then show \"suffix i w ⊨⇩n φ\"\n apply (induction i rule: less_induct (*‹(⋀x. (⋀y. y < x ⟹ ?P y) ⟹ ?P x) ⟹ ?P ?a›*))\n (*goal: ‹suffix (i::nat) (w::nat ⇒ 'a::type set) ⊨⇩n (φ::'a::type ltln)›*)\n by force\nnext\n (*goals:\n 1. ‹⋀i ia. ⟦∀j≤i. ¬ suffix j w ⊨⇩n ψ; suffix ia w ⊨⇩n ψ; ∀j<ia. suffix j w ⊨⇩n φ⟧ ⟹ suffix i w ⊨⇩n φ›\n 2. ‹⋀i. ⟦∀i. suffix i w ⊨⇩n ψ ∨ (∃j<i. suffix j w ⊨⇩n φ); ∀i. suffix i w ⊨⇩n φ ⟶ (∃j≤i. ¬ suffix j w ⊨⇩n ψ)⟧ ⟹ suffix i w ⊨⇩n ψ›\n 3. ‹⋀i ia. ⟦∀j<i. ¬ suffix j w ⊨⇩n φ; suffix ia w ⊨⇩n φ; ∀j≤ia. suffix j w ⊨⇩n ψ⟧ ⟹ suffix i w ⊨⇩n ψ›*)\n fix i and k\n assume \"∀j≤i. ¬ suffix j w ⊨⇩n ψ\" and \"suffix k w ⊨⇩n ψ\" and \"∀j<k. suffix j w ⊨⇩n φ\" (*‹∀j≤i::nat. ¬ suffix j (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln)› ‹suffix (k::nat) (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln)› ‹∀j<k::nat. suffix j (w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln)›*)\n then show \"suffix i w ⊨⇩n φ\"\n apply (cases \"i < k\")\n (*goals:\n 1. ‹⟦∀j≤i::nat. ¬ suffix j (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); suffix (k::nat) w ⊨⇩n ψ; ∀j<k. suffix j w ⊨⇩n (φ::'a ltln); i < k⟧ ⟹ suffix i w ⊨⇩n φ›\n 2. ‹⟦∀j≤i::nat. ¬ suffix j (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); suffix (k::nat) w ⊨⇩n ψ; ∀j<k. suffix j w ⊨⇩n (φ::'a ltln); ¬ i < k⟧ ⟹ suffix i w ⊨⇩n φ›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\nnext\n (*goals:\n 1. ‹⋀i. ⟦∀i. suffix i w ⊨⇩n ψ ∨ (∃j<i. suffix j w ⊨⇩n φ); ∀i. suffix i w ⊨⇩n φ ⟶ (∃j≤i. ¬ suffix j w ⊨⇩n ψ)⟧ ⟹ suffix i w ⊨⇩n ψ›\n 2. ‹⋀i ia. ⟦∀j<i. ¬ suffix j w ⊨⇩n φ; suffix ia w ⊨⇩n φ; ∀j≤ia. suffix j w ⊨⇩n ψ⟧ ⟹ suffix i w ⊨⇩n ψ›*)\n fix i\n assume \"∀i. suffix i w ⊨⇩n ψ ∨ (∃j<i. suffix j w ⊨⇩n φ)\" and \"∀i. suffix i w ⊨⇩n φ ⟶ (∃j≤i. ¬ suffix j w ⊨⇩n ψ)\" (*‹∀i::nat. suffix i (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln) ∨ (∃j<i. suffix j w ⊨⇩n (φ::'a ltln))› ‹∀i::nat. suffix i (w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) ⟶ (∃j≤i. ¬ suffix j w ⊨⇩n (ψ::'a ltln))›*)\n then show \"suffix i w ⊨⇩n ψ\"\n apply (induction i rule: less_induct (*‹(⋀x::?'a::wellorder. (⋀y::?'a::wellorder. y < x ⟹ (?P::?'a::wellorder ⇒ bool) y) ⟹ ?P x) ⟹ ?P (?a::?'a::wellorder)›*))\n (*goal: ‹suffix i w ⊨⇩n ψ›*)\n by force\nnext\n (*goal: ‹⋀i ia. ⟦∀j<i. ¬ suffix j w ⊨⇩n φ; suffix ia w ⊨⇩n φ; ∀j≤ia. suffix j w ⊨⇩n ψ⟧ ⟹ suffix i w ⊨⇩n ψ›*)\n fix i and k\n assume \"∀j<i. ¬ suffix j w ⊨⇩n φ\" and \"suffix k w ⊨⇩n φ\" and \"∀j≤k. suffix j w ⊨⇩n ψ\" (*‹∀j<i::nat. ¬ suffix j (w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln)› ‹suffix (k::nat) (w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln)› ‹∀j≤k::nat. suffix j (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln)›*)\n then show \"suffix i w ⊨⇩n ψ\"\n apply (cases \"i ≤ k\")\n (*goals:\n 1. ‹⟦∀j<i. ¬ suffix j w ⊨⇩n φ; suffix k w ⊨⇩n φ; ∀j≤k. suffix j w ⊨⇩n ψ; i ≤ k⟧ ⟹ suffix i w ⊨⇩n ψ›\n 2. ‹⟦∀j<i. ¬ suffix j w ⊨⇩n φ; suffix k w ⊨⇩n φ; ∀j≤k. suffix j w ⊨⇩n ψ; ¬ i ≤ k⟧ ⟹ suffix i w ⊨⇩n ψ›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\nqed\n\nlemma ltln_strong_weak:\n \"w ⊨⇩n φ U⇩n ψ ⟷ w ⊨⇩n (F⇩n ψ) and⇩n (φ W⇩n ψ)\"\n \"w ⊨⇩n φ M⇩n ψ ⟷ w ⊨⇩n (F⇩n φ) and⇩n (φ R⇩n ψ)\"\n \n (*goals:\n 1. ‹w ⊨⇩n φ U⇩n ψ = w ⊨⇩n F⇩n ψ and⇩n φ W⇩n ψ›\n 2. ‹w ⊨⇩n φ M⇩n ψ = w ⊨⇩n F⇩n φ and⇩n φ R⇩n ψ›\n discuss goal 1*)\n apply (metis ltln_weak_strong (*‹?w ⊨⇩n ?φ W⇩n ?ψ = ?w ⊨⇩n G⇩n ?φ or⇩n (?φ U⇩n ?ψ)› ‹?w ⊨⇩n ?φ R⇩n ?ψ = ?w ⊨⇩n G⇩n ?ψ or⇩n (?φ M⇩n ?ψ)›*) not⇩n.simps( (*‹not⇩n true⇩n = false⇩n› ‹not⇩n (?φ and⇩n ?ψ) = not⇩n ?φ or⇩n not⇩n ?ψ› ‹not⇩n (?φ U⇩n ?ψ) = not⇩n ?φ R⇩n not⇩n ?ψ› ‹not⇩n (?φ R⇩n ?ψ) = not⇩n ?φ U⇩n not⇩n ?ψ› ‹not⇩n (?φ W⇩n ?ψ) = not⇩n ?φ M⇩n not⇩n ?ψ› ‹not⇩n (?φ M⇩n ?ψ) = not⇩n ?φ W⇩n not⇩n ?ψ›*) 1,5,8-11) not⇩n_semantics (*‹?w ⊨⇩n not⇩n ?φ = (¬ ?w ⊨⇩n ?φ)›*))\n (*discuss goal 2*)\n apply (metis ltln_weak_strong (*‹?w ⊨⇩n ?φ W⇩n ?ψ = ?w ⊨⇩n G⇩n ?φ or⇩n (?φ U⇩n ?ψ)› ‹?w ⊨⇩n ?φ R⇩n ?ψ = ?w ⊨⇩n G⇩n ?ψ or⇩n (?φ M⇩n ?ψ)›*) not⇩n.simps( (*‹not⇩n true⇩n = false⇩n› ‹not⇩n (?φ and⇩n ?ψ) = not⇩n ?φ or⇩n not⇩n ?ψ› ‹not⇩n (?φ U⇩n ?ψ) = not⇩n ?φ R⇩n not⇩n ?ψ› ‹not⇩n (?φ R⇩n ?ψ) = not⇩n ?φ U⇩n not⇩n ?ψ› ‹not⇩n (?φ W⇩n ?ψ) = not⇩n ?φ M⇩n not⇩n ?ψ› ‹not⇩n (?φ M⇩n ?ψ) = not⇩n ?φ W⇩n not⇩n ?ψ›*) 1,5,8-11) not⇩n_semantics (*‹?w ⊨⇩n not⇩n ?φ = (¬ ?w ⊨⇩n ?φ)›*))\n (*proven 2 subgoals*) .\n\nlemma ltln_strong_to_weak:\n \"w ⊨⇩n φ U⇩n ψ ⟹ w ⊨⇩n φ W⇩n ψ\"\n \"w ⊨⇩n φ M⇩n ψ ⟹ w ⊨⇩n φ R⇩n ψ\"\n using ltln_weak_strong (*‹?w ⊨⇩n ?φ W⇩n ?ψ = ?w ⊨⇩n G⇩n ?φ or⇩n (?φ U⇩n ?ψ)› ‹(?w::nat ⇒ ?'a set) ⊨⇩n (?φ::?'a ltln) R⇩n (?ψ::?'a ltln) = ?w ⊨⇩n G⇩n ?ψ or⇩n (?φ M⇩n ?ψ)›*) apply -\n (*goals:\n 1. ‹⟦(w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) U⇩n (ψ::'a ltln); ⋀(w::nat ⇒ ?'a4 set) (φ::?'a4 ltln) ψ::?'a4 ltln. w ⊨⇩n φ W⇩n ψ = w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); ⋀(w::nat ⇒ ?'a6 set) (φ::?'a6 ltln) ψ::?'a6 ltln. w ⊨⇩n φ R⇩n ψ = w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ)⟧ ⟹ w ⊨⇩n φ W⇩n ψ›\n 2. ‹⟦(w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) M⇩n (ψ::'a ltln); ⋀(w::nat ⇒ ?'a set) (φ::?'a ltln) ψ::?'a ltln. w ⊨⇩n φ W⇩n ψ = w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); ⋀(w::nat ⇒ ?'a2 set) (φ::?'a2 ltln) ψ::?'a2 ltln. w ⊨⇩n φ R⇩n ψ = w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ)⟧ ⟹ w ⊨⇩n φ R⇩n ψ›\n discuss goal 1*)\n apply simp\n (*top goal: ‹⟦(w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) U⇩n (ψ::'a ltln); ⋀(w::nat ⇒ ?'a4 set) (φ::?'a4 ltln) ψ::?'a4 ltln. w ⊨⇩n φ W⇩n ψ = w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); ⋀(w::nat ⇒ ?'a6 set) (φ::?'a6 ltln) ψ::?'a6 ltln. w ⊨⇩n φ R⇩n ψ = w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ)⟧ ⟹ w ⊨⇩n φ W⇩n ψ› and 1 goal remains*)\n apply blast\n (*discuss goal 2*)\n apply simp\n (*goal: ‹⟦w ⊨⇩n φ M⇩n ψ; ⋀w φ ψ. w ⊨⇩n φ W⇩n ψ = w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); ⋀w φ ψ. w ⊨⇩n φ R⇩n ψ = w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ)⟧ ⟹ w ⊨⇩n φ R⇩n ψ›*)\n apply blast\n (*proven 2 subgoals*) .\n\nlemma ltln_weak_to_strong:\n \"⟦w ⊨⇩n φ W⇩n ψ; ¬ w ⊨⇩n G⇩n φ⟧ ⟹ w ⊨⇩n φ U⇩n ψ\"\n \"⟦w ⊨⇩n φ W⇩n ψ; w ⊨⇩n F⇩n ψ⟧ ⟹ w ⊨⇩n φ U⇩n ψ\"\n \"⟦w ⊨⇩n φ R⇩n ψ; ¬ w ⊨⇩n G⇩n ψ⟧ ⟹ w ⊨⇩n φ M⇩n ψ\"\n \"⟦w ⊨⇩n φ R⇩n ψ; w ⊨⇩n F⇩n φ⟧ ⟹ w ⊨⇩n φ M⇩n ψ\"\n unfolding ltln_weak_strong[of w φ ψ]\n (*goals:\n 1. ‹⟦w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); ¬ w ⊨⇩n G⇩n φ⟧ ⟹ w ⊨⇩n φ U⇩n ψ›\n 2. ‹⟦w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); w ⊨⇩n F⇩n ψ⟧ ⟹ w ⊨⇩n φ U⇩n ψ›\n 3. ‹⟦w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ); ¬ w ⊨⇩n G⇩n ψ⟧ ⟹ w ⊨⇩n φ M⇩n ψ›\n 4. ‹⟦w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ); w ⊨⇩n F⇩n φ⟧ ⟹ w ⊨⇩n φ M⇩n ψ›*)\n (*goals:\n 1. ‹⟦w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); ¬ w ⊨⇩n G⇩n φ⟧ ⟹ w ⊨⇩n φ U⇩n ψ›\n 2. ‹⟦w ⊨⇩n G⇩n φ or⇩n (φ U⇩n ψ); w ⊨⇩n F⇩n ψ⟧ ⟹ w ⊨⇩n φ U⇩n ψ›\n 3. ‹⟦w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ); ¬ w ⊨⇩n G⇩n ψ⟧ ⟹ w ⊨⇩n φ M⇩n ψ›\n 4. ‹⟦w ⊨⇩n G⇩n ψ or⇩n (φ M⇩n ψ); w ⊨⇩n F⇩n φ⟧ ⟹ w ⊨⇩n φ M⇩n ψ›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n\n\nlemma ltln_StrongRelease_to_Until:\n \"w ⊨⇩n φ M⇩n ψ ⟷ w ⊨⇩n ψ U⇩n (φ and⇩n ψ)\"\n using order.order_iff_strict (*‹(?a ≤ ?b) = (?a < ?b ∨ ?a = ?b)›*) by auto\n\nlemma ltln_Release_to_WeakUntil:\n \"w ⊨⇩n φ R⇩n ψ ⟷ w ⊨⇩n ψ W⇩n (φ and⇩n ψ)\"\n by (meson ltln_StrongRelease_to_Until (*‹(?w::nat ⇒ ?'a::type set) ⊨⇩n (?φ::?'a::type ltln) M⇩n (?ψ::?'a::type ltln) = ?w ⊨⇩n ?ψ U⇩n (?φ and⇩n ?ψ)›*) ltln_weak_strong (*‹(?w::nat ⇒ ?'a::type set) ⊨⇩n (?φ::?'a::type ltln) W⇩n (?ψ::?'a::type ltln) = ?w ⊨⇩n G⇩n ?φ or⇩n (?φ U⇩n ?ψ)› ‹(?w::nat ⇒ ?'a::type set) ⊨⇩n (?φ::?'a::type ltln) R⇩n (?ψ::?'a::type ltln) = ?w ⊨⇩n G⇩n ?ψ or⇩n (?φ M⇩n ?ψ)›*) semantics_ltln.simps( (*‹(?ξ::nat ⇒ ?'a::type set) ⊨⇩n (?φ::?'a::type ltln) or⇩n (?ψ::?'a::type ltln) = (?ξ ⊨⇩n ?φ ∨ ?ξ ⊨⇩n ?ψ)›*) 6))\n\nlemma ltln_WeakUntil_to_Release:\n \"w ⊨⇩n φ W⇩n ψ ⟷ w ⊨⇩n ψ R⇩n (φ or⇩n ψ)\"\n by (metis ltln_StrongRelease_to_Until (*‹?w ⊨⇩n ?φ M⇩n ?ψ = ?w ⊨⇩n ?ψ U⇩n (?φ and⇩n ?ψ)›*) not⇩n.simps( (*‹not⇩n (?φ or⇩n ?ψ) = not⇩n ?φ and⇩n not⇩n ?ψ› ‹not⇩n (?φ R⇩n ?ψ) = not⇩n ?φ U⇩n not⇩n ?ψ› ‹not⇩n (?φ W⇩n ?ψ) = not⇩n ?φ M⇩n not⇩n ?ψ›*) 6,9,10) not⇩n_semantics (*‹?w ⊨⇩n not⇩n ?φ = (¬ ?w ⊨⇩n ?φ)›*))\n\nlemma ltln_Until_to_StrongRelease:\n \"w ⊨⇩n φ U⇩n ψ ⟷ w ⊨⇩n ψ M⇩n (φ or⇩n ψ)\"\n by (metis ltln_Release_to_WeakUntil (*‹?w ⊨⇩n ?φ R⇩n ?ψ = ?w ⊨⇩n ?ψ W⇩n (?φ and⇩n ?ψ)›*) not⇩n.simps( (*‹not⇩n (?φ or⇩n ?ψ) = not⇩n ?φ and⇩n not⇩n ?ψ› ‹not⇩n (?φ U⇩n ?ψ) = not⇩n ?φ R⇩n not⇩n ?ψ› ‹not⇩n (?φ M⇩n ?ψ) = not⇩n ?φ W⇩n not⇩n ?ψ›*) 6,8,11) not⇩n_semantics (*‹?w ⊨⇩n not⇩n ?φ = (¬ ?w ⊨⇩n ?φ)›*))\n\n\nsubsubsection ‹GF and FG semantics›\n\nlemma GF_suffix:\n \"suffix i w ⊨⇩n G⇩n (F⇩n ψ) ⟷ w ⊨⇩n G⇩n (F⇩n ψ)\"\n apply auto\n (*goal: ‹suffix i w ⊨⇩n G⇩n (F⇩n ψ) = w ⊨⇩n G⇩n (F⇩n ψ)›*)\n by (metis ab_semigroup_add_class.add_ac( (*‹?a + ?b + ?c = ?a + (?b + ?c)›*) 1) add.left_commute (*‹?b + (?a + ?c) = ?a + (?b + ?c)›*))\n\nlemma FG_suffix:\n \"suffix i w ⊨⇩n F⇩n (G⇩n ψ) ⟷ w ⊨⇩n F⇩n (G⇩n ψ)\"\n apply (auto simp: algebra_simps (*‹?a + ?b + ?c = ?a + (?b + ?c)› ‹?a + ?b = ?b + ?a› ‹?b + (?a + ?c) = ?a + (?b + ?c)› ‹?a * ?b * ?c = ?a * (?b * ?c)› ‹?a * ?b = ?b * ?a› ‹?b * (?a * ?c) = ?a * (?b * ?c)› ‹?a - ?b - ?c = ?a - (?b + ?c)› ‹?a + (?b - ?c) = ?a + ?b - ?c› ‹(?a - ?b = ?c) = (?a = ?c + ?b)› ‹(?a = ?c - ?b) = (?a + ?b = ?c)› ‹?a - (?b - ?c) = ?a + ?c - ?b› ‹?a - ?b + ?c = ?a + ?c - ?b› and more 11 facts*))\n (*goal: ‹suffix i w ⊨⇩n F⇩n (G⇩n ψ) = w ⊨⇩n F⇩n (G⇩n ψ)›*)\n by (metis add.commute (*‹?a + ?b = ?b + ?a›*) add.left_commute (*‹?b + (?a + ?c) = ?a + (?b + ?c)›*))\n\nlemma GF_Inf_many:\n \"w ⊨⇩n G⇩n (F⇩n φ) ⟷ (∃⇩∞ i. suffix i w ⊨⇩n φ)\"\n unfolding INFM_nat_le\n (*goal: ‹w ⊨⇩n G⇩n (F⇩n φ) = (∀m. ∃n≥m. suffix n w ⊨⇩n φ)›*)\n apply simp\n (*goal: ‹(w::nat ⇒ 'a set) ⊨⇩n G⇩n (F⇩n (φ::'a ltln)) = (∀m::nat. ∃n≥m. suffix n w ⊨⇩n φ)›*)\n by (blast dest: le_Suc_ex (*‹?k ≤ ?l ⟹ ∃n. ?l = ?k + n›*) intro: le_add1 (*‹?n ≤ ?n + ?m›*))\n\nlemma FG_Alm_all:\n \"w ⊨⇩n F⇩n (G⇩n φ) ⟷ (∀⇩∞ i. suffix i w ⊨⇩n φ)\"\n unfolding MOST_nat_le\n (*goal: ‹w ⊨⇩n F⇩n (G⇩n φ) = (∃m. ∀n≥m. suffix n w ⊨⇩n φ)›*)\n apply simp\n (*goal: ‹w ⊨⇩n F⇩n (G⇩n φ) = (∃m. ∀n≥m. suffix n w ⊨⇩n φ)›*)\n by (blast dest: le_Suc_ex (*‹?k ≤ ?l ⟹ ∃n. ?l = ?k + n›*) intro: le_add1 (*‹?n ≤ ?n + ?m›*))\n\n\n(* TODO: move to Infinite_Set.thy ?? *)\nlemma MOST_nat_add:\n \"(∀⇩∞i::nat. P i) ⟷ (∀⇩∞i. P (i + j))\"\n by (simp add: cofinite_eq_sequentially (*‹cofinite = sequentially›*))\n\nlemma INFM_nat_add:\n \"(∃⇩∞i::nat. P i) ⟷ (∃⇩∞i. P (i + j))\"\n using MOST_nat_add (*‹(∀⇩∞i. ?P i) = (∀⇩∞i. ?P (i + ?j))›*) not_MOST (*‹(¬ (∀⇩∞x. ?P x)) = (∃⇩∞x. ¬ ?P x)›*) not_INFM (*‹(¬ (∃⇩∞x. ?P x)) = (∀⇩∞x. ¬ ?P x)›*) by blast\n\n\nlemma FG_suffix_G:\n \"w ⊨⇩n F⇩n (G⇩n φ) ⟹ ∀⇩∞i. suffix i w ⊨⇩n G⇩n φ\"\nproof (-)\n (*goal: ‹(w::nat ⇒ 'a set) ⊨⇩n F⇩n (G⇩n (φ::'a ltln)) ⟹ ∀⇩∞i::nat. suffix i w ⊨⇩n G⇩n φ›*)\n assume \"w ⊨⇩n F⇩n (G⇩n φ)\" (*‹(w::nat ⇒ 'a set) ⊨⇩n F⇩n (G⇩n (φ::'a ltln))›*)\n then have \"w ⊨⇩n F⇩n (G⇩n (G⇩n φ))\"\n by (meson globally_release (*‹(?w::nat ⇒ ?'a::type set) ⊨⇩n G⇩n ((?φ::?'a::type ltln) R⇩n (?ψ::?'a::type ltln)) = ?w ⊨⇩n G⇩n ?ψ›*) semantics_ltln.simps( (*‹(?ξ::nat ⇒ ?'a::type set) ⊨⇩n (?φ::?'a::type ltln) U⇩n (?ψ::?'a::type ltln) = (∃i::nat. suffix i ?ξ ⊨⇩n ?ψ ∧ (∀j<i. suffix j ?ξ ⊨⇩n ?φ))›*) 8))\n then show \"∀⇩∞i. suffix i w ⊨⇩n G⇩n φ\"\n unfolding FG_Alm_all\n (*goal: ‹∀⇩∞i. suffix i w ⊨⇩n G⇩n φ›*) .\nqed\n\nlemma Alm_all_GF_F:\n \"∀⇩∞i. suffix i w ⊨⇩n G⇩n (F⇩n ψ) ⟷ suffix i w ⊨⇩n F⇩n ψ\"\n unfolding MOST_nat\n (*goal: ‹∃m::nat. ∀n>m. suffix n (w::nat ⇒ 'a set) ⊨⇩n G⇩n (F⇩n (ψ::'a ltln)) = suffix n w ⊨⇩n F⇩n ψ›*)\n proof ((standard)+)\n (*goals:\n 1. ‹⋀n. ⟦?m < n; suffix n w ⊨⇩n G⇩n (F⇩n ψ)⟧ ⟹ suffix n w ⊨⇩n F⇩n ψ›\n 2. ‹⋀n. ⟦?m < n; suffix n w ⊨⇩n F⇩n ψ⟧ ⟹ suffix n w ⊨⇩n G⇩n (F⇩n ψ)›*)\n fix i :: nat\n assume \"suffix i w ⊨⇩n G⇩n (F⇩n ψ)\" (*‹suffix (i::nat) (w::nat ⇒ 'a set) ⊨⇩n G⇩n (F⇩n (ψ::'a ltln))›*)\n then show \"suffix i w ⊨⇩n F⇩n ψ\"\n unfolding GF_Inf_many INFM_nat\n (*goal: ‹suffix i w ⊨⇩n F⇩n ψ›*)\n by fastforce\n next\n (*goal: ‹⋀n. ⟦?m < n; suffix n w ⊨⇩n F⇩n ψ⟧ ⟹ suffix n w ⊨⇩n G⇩n (F⇩n ψ)›*)\n fix i :: nat\n assume suffix: \"suffix i w ⊨⇩n F⇩n ψ\" (*‹suffix (i::nat) (w::nat ⇒ 'a set) ⊨⇩n F⇩n (ψ::'a ltln)›*)\n assume max: \"i > Max {i. suffix i w ⊨⇩n ψ}\" (*‹Max {i::nat. suffix i (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln)} < (i::nat)›*)\n with suffix (*‹suffix i w ⊨⇩n F⇩n ψ›*) obtain j where \"j ≥ i\" and j_suffix: \"suffix j w ⊨⇩n ψ\"\n (*goal: ‹(⋀j. ⟦i ≤ j; suffix j w ⊨⇩n ψ⟧ ⟹ thesis) ⟹ thesis›*)\n apply simp\n (*goal: ‹(⋀j. ⟦i ≤ j; suffix j w ⊨⇩n ψ⟧ ⟹ thesis) ⟹ thesis›*)\n by (blast intro: le_add1 (*‹(?n::nat) ≤ ?n + (?m::nat)›*))\n with max (*‹Max {i. suffix i w ⊨⇩n ψ} < i›*) have j_max: \"j > Max {i. suffix i w ⊨⇩n ψ}\"\n by fastforce\n show \"suffix i w ⊨⇩n G⇩n (F⇩n ψ)\"\n proof (cases \"w ⊨⇩n G⇩n (F⇩n ψ)\")\n (*goals:\n 1. ‹(w::nat ⇒ 'a set) ⊨⇩n G⇩n (F⇩n (ψ::'a ltln)) ⟹ suffix (i::nat) w ⊨⇩n G⇩n (F⇩n ψ)›\n 2. ‹¬ (w::nat ⇒ 'a set) ⊨⇩n G⇩n (F⇩n (ψ::'a ltln)) ⟹ suffix (i::nat) w ⊨⇩n G⇩n (F⇩n ψ)›*)\n case False (*‹¬ w ⊨⇩n G⇩n (F⇩n ψ)›*)\n then have \"¬ (∃⇩∞i. suffix i w ⊨⇩n ψ)\"\n unfolding GF_Inf_many\n (*goal: ‹¬ (∃⇩∞i. suffix i w ⊨⇩n ψ)›*)\n by simp\n then have \"finite {i. suffix i w ⊨⇩n ψ}\"\n by (simp add: INFM_iff_infinite (*‹(∃⇩∞x. ?P x) = infinite {x. ?P x}›*))\n then have \"∀i>Max {i. suffix i w ⊨⇩n ψ}. ¬ suffix i w ⊨⇩n ψ\"\n using Max_ge (*‹⟦finite ?A; ?x ∈ ?A⟧ ⟹ ?x ≤ Max ?A›*) not_le (*‹(¬ ?x ≤ ?y) = (?y < ?x)›*) by auto\n then show \"?thesis\"\n (*goal: ‹suffix (i::nat) (w::nat ⇒ 'a set) ⊨⇩n G⇩n (F⇩n (ψ::'a ltln))›*)\n using j_suffix (*‹suffix j w ⊨⇩n ψ›*) j_max (*‹Max {i. suffix i w ⊨⇩n ψ} < j›*) by blast\n qed (force)\n (*solved the remaining goal: ‹w ⊨⇩n G⇩n (F⇩n ψ) ⟹ suffix i w ⊨⇩n G⇩n (F⇩n ψ)›*)\n qed\n\nlemma Alm_all_FG_G:\n \"∀⇩∞i. suffix i w ⊨⇩n F⇩n (G⇩n ψ) ⟷ suffix i w ⊨⇩n G⇩n ψ\"\n unfolding MOST_nat\n (*goal: ‹∃m::nat. ∀n>m. suffix n (w::nat ⇒ 'a::type set) ⊨⇩n F⇩n (G⇩n (ψ::'a::type ltln)) = suffix n w ⊨⇩n G⇩n ψ›*)\n proof ((standard)+)\n (*goals:\n 1. ‹⋀n. ⟦?m < n; suffix n w ⊨⇩n F⇩n (G⇩n ψ)⟧ ⟹ suffix n w ⊨⇩n G⇩n ψ›\n 2. ‹⋀n. ⟦?m < n; suffix n w ⊨⇩n G⇩n ψ⟧ ⟹ suffix n w ⊨⇩n F⇩n (G⇩n ψ)›*)\n fix i :: nat\n assume \"suffix i w ⊨⇩n G⇩n ψ\" (*‹suffix (i::nat) (w::nat ⇒ 'a set) ⊨⇩n G⇩n (ψ::'a ltln)›*)\n then show \"suffix i w ⊨⇩n F⇩n (G⇩n ψ)\"\n unfolding FG_Alm_all INFM_nat\n (*goal: ‹∀⇩∞ia::nat. suffix ia (suffix (i::nat) (w::nat ⇒ 'a set)) ⊨⇩n (ψ::'a ltln)›*)\n by fastforce\n next\n (*goal: ‹⋀n. ⟦?m < n; suffix n w ⊨⇩n F⇩n (G⇩n ψ)⟧ ⟹ suffix n w ⊨⇩n G⇩n ψ›*)\n fix i :: nat\n assume suffix: \"suffix i w ⊨⇩n F⇩n (G⇩n ψ)\" (*‹suffix (i::nat) (w::nat ⇒ 'a set) ⊨⇩n F⇩n (G⇩n (ψ::'a ltln))›*)\n assume max: \"i > Max {i. ¬ suffix i w ⊨⇩n G⇩n ψ}\" (*‹Max {i::nat. ¬ suffix i (w::nat ⇒ 'a set) ⊨⇩n G⇩n (ψ::'a ltln)} < (i::nat)›*)\n with suffix (*‹suffix i w ⊨⇩n F⇩n (G⇩n ψ)›*) have \"∀⇩∞j. suffix (i + j) w ⊨⇩n G⇩n ψ\"\n using FG_suffix_G[of \"suffix i w\"] (*‹suffix i w ⊨⇩n F⇩n (G⇩n ?φ) ⟹ ∀⇩∞ia. suffix ia (suffix i w) ⊨⇩n G⇩n ?φ›*) suffix_suffix (*‹suffix ?m (suffix ?k ?x) = suffix (?k + ?m) ?x›*) by fastforce\n then have \"¬ (∃⇩∞j. ¬ suffix j w ⊨⇩n G⇩n ψ)\"\n using MOST_nat_add[of \"λi. suffix i w ⊨⇩n G⇩n ψ\" i] (*‹(∀⇩∞i. suffix i w ⊨⇩n G⇩n ψ) = (∀⇩∞ia. suffix (ia + i) w ⊨⇩n G⇩n ψ)›*) by (simp add: algebra_simps (*‹?a + ?b + ?c = ?a + (?b + ?c)› ‹?a + ?b = ?b + ?a› ‹?b + (?a + ?c) = ?a + (?b + ?c)› ‹?a * ?b * ?c = ?a * (?b * ?c)› ‹?a * ?b = ?b * ?a› ‹?b * (?a * ?c) = ?a * (?b * ?c)› ‹?a - ?b - ?c = ?a - (?b + ?c)› ‹?a + (?b - ?c) = ?a + ?b - ?c› ‹(?a - ?b = ?c) = (?a = ?c + ?b)› ‹(?a = ?c - ?b) = (?a + ?b = ?c)› ‹?a - (?b - ?c) = ?a + ?c - ?b› ‹?a - ?b + ?c = ?a + ?c - ?b› and more 11 facts*))\n then have \"finite {i. ¬ suffix i w ⊨⇩n G⇩n ψ}\"\n by (simp add: INFM_iff_infinite (*‹(∃⇩∞x::?'a. (?P::?'a ⇒ bool) x) = infinite {x::?'a. ?P x}›*))\n with max (*‹Max {i. ¬ suffix i w ⊨⇩n G⇩n ψ} < i›*) show \"suffix i w ⊨⇩n G⇩n ψ\"\n using Max_ge (*‹⟦finite ?A; ?x ∈ ?A⟧ ⟹ ?x ≤ Max ?A›*) leD (*‹?y ≤ ?x ⟹ ¬ ?x < ?y›*) by blast\n qed\n\n\nsubsubsection ‹Expansion›\n\nlemma ltln_expand_Until:\n \"ξ ⊨⇩n φ U⇩n ψ ⟷ (ξ ⊨⇩n ψ or⇩n (φ and⇩n (X⇩n (φ U⇩n ψ))))\"\n (is \"?lhs = ?rhs\")\nproof (standard)\n (*goals:\n 1. ‹ξ ⊨⇩n φ U⇩n ψ ⟹ ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ U⇩n ψ))›\n 2. ‹ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ U⇩n ψ)) ⟹ ξ ⊨⇩n φ U⇩n ψ›*)\n assume \"?lhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) U⇩n (ψ::'a ltln)›*)\n then obtain i where \"suffix i ξ ⊨⇩n ψ\" and \"∀j<i. suffix j ξ ⊨⇩n φ\"\n (*goal: ‹(⋀i. ⟦suffix i ξ ⊨⇩n ψ; ∀j<i. suffix j ξ ⊨⇩n φ⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n thus \"?rhs\"\n apply (cases i)\n (*goals:\n 1. ‹⟦suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); ∀j<i. suffix j ξ ⊨⇩n (φ::'a ltln); i = (0::nat)⟧ ⟹ ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ U⇩n ψ))›\n 2. ‹⋀nat::nat. ⟦suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); ∀j<i. suffix j ξ ⊨⇩n (φ::'a ltln); i = Suc nat⟧ ⟹ ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ U⇩n ψ))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nnext\n (*goal: ‹ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ U⇩n ψ)) ⟹ ξ ⊨⇩n φ U⇩n ψ›*)\n assume \"?rhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln) or⇩n ((φ::'a ltln) and⇩n X⇩n (φ U⇩n ψ))›*)\n show \"?lhs\"\n proof (cases \"ξ ⊨⇩n ψ\")\n (*goals:\n 1. ‹ξ ⊨⇩n ψ ⟹ ξ ⊨⇩n φ U⇩n ψ›\n 2. ‹¬ ξ ⊨⇩n ψ ⟹ ξ ⊨⇩n φ U⇩n ψ›*)\n case False (*‹¬ ξ ⊨⇩n ψ›*)\n then have \"ξ ⊨⇩n φ\" and \"ξ ⊨⇩n X⇩n (φ U⇩n ψ)\"\n using ‹?rhs› (*‹ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ U⇩n ψ))›*) apply -\n (*goals:\n 1. ‹⟦¬ (ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); ξ ⊨⇩n ψ or⇩n ((φ::'a ltln) and⇩n X⇩n (φ U⇩n ψ))⟧ ⟹ ξ ⊨⇩n φ›\n 2. ‹⟦¬ (ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); ξ ⊨⇩n ψ or⇩n ((φ::'a ltln) and⇩n X⇩n (φ U⇩n ψ))⟧ ⟹ ξ ⊨⇩n X⇩n (φ U⇩n ψ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thus \"?lhs\"\n using less_Suc_eq_0_disj (*‹(?m < Suc ?n) = (?m = 0 ∨ (∃j. ?m = Suc j ∧ j < ?n))›*) suffix_singleton_suffix (*‹?w ?i ## suffix (Suc ?i) ?w = suffix ?i ?w›*) by force\n qed (force)\n (*solved the remaining goal: ‹ξ ⊨⇩n ψ ⟹ ξ ⊨⇩n φ U⇩n ψ›*)\nqed\n\nlemma ltln_expand_Release:\n \"ξ ⊨⇩n φ R⇩n ψ ⟷ (ξ ⊨⇩n ψ and⇩n (φ or⇩n (X⇩n (φ R⇩n ψ))))\"\n (is \"?lhs = ?rhs\")\nproof (standard)\n (*goals:\n 1. ‹ξ ⊨⇩n φ R⇩n ψ ⟹ ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ R⇩n ψ)›\n 2. ‹ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ R⇩n ψ) ⟹ ξ ⊨⇩n φ R⇩n ψ›*)\n assume \"?lhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) R⇩n (ψ::'a ltln)›*)\n thus \"?rhs\"\n using less_Suc_eq_0_disj (*‹(?m < Suc ?n) = (?m = 0 ∨ (∃j. ?m = Suc j ∧ j < ?n))›*) by force\nnext\n (*goal: ‹ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ R⇩n ψ) ⟹ ξ ⊨⇩n φ R⇩n ψ›*)\n assume \"?rhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln) and⇩n (φ::'a ltln) or⇩n X⇩n (φ R⇩n ψ)›*)\n {\n fix i\n assume \"¬ suffix i ξ ⊨⇩n ψ\" (*‹¬ suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln)›*)\n then have \"∃j<i. suffix j ξ ⊨⇩n φ\"\n using ‹?rhs› (*‹ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ R⇩n ψ)›*) apply (cases i)\n (*goals:\n 1. ‹⟦¬ suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); ξ ⊨⇩n ψ and⇩n (φ::'a ltln) or⇩n X⇩n (φ R⇩n ψ); i = (0::nat)⟧ ⟹ ∃j<i. suffix j ξ ⊨⇩n φ›\n 2. ‹⋀nat::nat. ⟦¬ suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln); ξ ⊨⇩n ψ and⇩n (φ::'a ltln) or⇩n X⇩n (φ R⇩n ψ); i = Suc nat⟧ ⟹ ∃j<i. suffix j ξ ⊨⇩n φ›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*) .\n }\n thus \"?lhs\"\n by auto\nqed\n\nlemma ltln_expand_WeakUntil:\n \"ξ ⊨⇩n φ W⇩n ψ ⟷ (ξ ⊨⇩n ψ or⇩n (φ and⇩n (X⇩n (φ W⇩n ψ))))\"\n (is \"?lhs = ?rhs\")\nproof (standard)\n (*goals:\n 1. ‹(ξ::nat ⇒ 'a::type set) ⊨⇩n (φ::'a::type ltln) W⇩n (ψ::'a::type ltln) ⟹ ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ W⇩n ψ))›\n 2. ‹(ξ::nat ⇒ 'a::type set) ⊨⇩n (ψ::'a::type ltln) or⇩n ((φ::'a::type ltln) and⇩n X⇩n (φ W⇩n ψ)) ⟹ ξ ⊨⇩n φ W⇩n ψ›*)\n assume \"?lhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) W⇩n (ψ::'a ltln)›*)\n thus \"?rhs\"\n by (metis ltln_expand_Release (*‹(?ξ::nat ⇒ ?'a set) ⊨⇩n (?φ::?'a ltln) R⇩n (?ψ::?'a ltln) = ?ξ ⊨⇩n ?ψ and⇩n ?φ or⇩n X⇩n (?φ R⇩n ?ψ)›*) ltln_expand_Until (*‹(?ξ::nat ⇒ ?'a set) ⊨⇩n (?φ::?'a ltln) U⇩n (?ψ::?'a ltln) = ?ξ ⊨⇩n ?ψ or⇩n (?φ and⇩n X⇩n (?φ U⇩n ?ψ))›*) ltln_weak_strong( (*‹(?w::nat ⇒ ?'a set) ⊨⇩n (?φ::?'a ltln) W⇩n (?ψ::?'a ltln) = ?w ⊨⇩n G⇩n ?φ or⇩n (?φ U⇩n ?ψ)›*) 1) semantics_ltln.simps( (*‹(?ξ::nat ⇒ ?'a set) ⊨⇩n false⇩n = False› ‹(?ξ::nat ⇒ ?'a set) ⊨⇩n (?φ::?'a ltln) and⇩n (?ψ::?'a ltln) = (?ξ ⊨⇩n ?φ ∧ ?ξ ⊨⇩n ?ψ)› ‹(?ξ::nat ⇒ ?'a set) ⊨⇩n (?φ::?'a ltln) or⇩n (?ψ::?'a ltln) = (?ξ ⊨⇩n ?φ ∨ ?ξ ⊨⇩n ?ψ)› ‹(?ξ::nat ⇒ ?'a set) ⊨⇩n X⇩n (?φ::?'a ltln) = suffix (1::nat) ?ξ ⊨⇩n ?φ›*) 2,5,6,7))\nnext\n (*goal: ‹ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ W⇩n ψ)) ⟹ ξ ⊨⇩n φ W⇩n ψ›*)\n assume \"?rhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln) or⇩n ((φ::'a ltln) and⇩n X⇩n (φ W⇩n ψ))›*)\n {\n fix i\n assume \"¬ suffix i ξ ⊨⇩n φ\" (*‹¬ suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln)›*)\n then have \"∃j≤i. suffix j ξ ⊨⇩n ψ\"\n using ‹?rhs› (*‹ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ W⇩n ψ))›*) apply (cases i)\n (*goals:\n 1. ‹⟦¬ suffix i ξ ⊨⇩n φ; ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ W⇩n ψ)); i = 0⟧ ⟹ ∃j≤i. suffix j ξ ⊨⇩n ψ›\n 2. ‹⋀nat. ⟦¬ suffix i ξ ⊨⇩n φ; ξ ⊨⇩n ψ or⇩n (φ and⇩n X⇩n (φ W⇩n ψ)); i = Suc nat⟧ ⟹ ∃j≤i. suffix j ξ ⊨⇩n ψ›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*) .\n }\n thus \"?lhs\"\n by auto\nqed\n\nlemma ltln_expand_StrongRelease:\n \"ξ ⊨⇩n φ M⇩n ψ ⟷ (ξ ⊨⇩n ψ and⇩n (φ or⇩n (X⇩n (φ M⇩n ψ))))\"\n (is \"?lhs = ?rhs\")\nproof (standard)\n (*goals:\n 1. ‹ξ ⊨⇩n φ M⇩n ψ ⟹ ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ M⇩n ψ)›\n 2. ‹ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ M⇩n ψ) ⟹ ξ ⊨⇩n φ M⇩n ψ›*)\n assume \"?lhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) M⇩n (ψ::'a ltln)›*)\n then obtain i where \"suffix i ξ ⊨⇩n φ\" and \"∀j≤i. suffix j ξ ⊨⇩n ψ\"\n (*goal: ‹(⋀i. ⟦suffix i ξ ⊨⇩n φ; ∀j≤i. suffix j ξ ⊨⇩n ψ⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n thus \"?rhs\"\n apply (cases i)\n (*goals:\n 1. ‹⟦suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln); ∀j≤i. suffix j ξ ⊨⇩n (ψ::'a ltln); i = (0::nat)⟧ ⟹ ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ M⇩n ψ)›\n 2. ‹⋀nat::nat. ⟦suffix (i::nat) (ξ::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln); ∀j≤i. suffix j ξ ⊨⇩n (ψ::'a ltln); i = Suc nat⟧ ⟹ ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ M⇩n ψ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nnext\n (*goal: ‹ξ ⊨⇩n ψ and⇩n φ or⇩n X⇩n (φ M⇩n ψ) ⟹ ξ ⊨⇩n φ M⇩n ψ›*)\n assume \"?rhs\" (*‹(ξ::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln) and⇩n (φ::'a ltln) or⇩n X⇩n (φ M⇩n ψ)›*)\n show \"?lhs\"\n proof (cases \"ξ ⊨⇩n φ\")\n (*goals:\n 1. ‹ξ ⊨⇩n φ ⟹ ξ ⊨⇩n φ M⇩n ψ›\n 2. ‹¬ ξ ⊨⇩n φ ⟹ ξ ⊨⇩n φ M⇩n ψ›*)\n case True (*‹ξ ⊨⇩n φ›*)\n thus \"?lhs\"\n using ‹?rhs› (*‹(ξ::nat ⇒ 'a::type set) ⊨⇩n (ψ::'a::type ltln) and⇩n (φ::'a::type ltln) or⇩n X⇩n (φ M⇩n ψ)›*) ltln_expand_WeakUntil (*‹?ξ ⊨⇩n ?φ W⇩n ?ψ = ?ξ ⊨⇩n ?ψ or⇩n (?φ and⇩n X⇩n (?φ W⇩n ?ψ))›*) by fastforce\n next\n (*goal: ‹¬ ξ ⊨⇩n φ ⟹ ξ ⊨⇩n φ M⇩n ψ›*)\n case False (*‹¬ ξ ⊨⇩n φ›*)\n thus \"?lhs\"\n by (metis ‹?rhs› ltln_expand_WeakUntil (*‹(?ξ::nat ⇒ ?'a set) ⊨⇩n (?φ::?'a ltln) W⇩n (?ψ::?'a ltln) = ?ξ ⊨⇩n ?ψ or⇩n (?φ and⇩n X⇩n (?φ W⇩n ?ψ))›*) not⇩n.simps( (*‹not⇩n ((?φ::?'a ltln) and⇩n (?ψ::?'a ltln)) = not⇩n ?φ or⇩n not⇩n ?ψ› ‹not⇩n ((?φ::?'a ltln) or⇩n (?ψ::?'a ltln)) = not⇩n ?φ and⇩n not⇩n ?ψ› ‹not⇩n (X⇩n (?φ::?'a ltln)) = X⇩n not⇩n ?φ› ‹not⇩n ((?φ::?'a ltln) M⇩n (?ψ::?'a ltln)) = not⇩n ?φ W⇩n not⇩n ?ψ›*) 5,6,7,11) not⇩n_semantics (*‹(?w::nat ⇒ ?'a set) ⊨⇩n not⇩n (?φ::?'a ltln) = (¬ ?w ⊨⇩n ?φ)›*))\n qed\nqed\n\nlemma ltln_Release_alterdef:\n \"w ⊨⇩n φ R⇩n ψ ⟷ w ⊨⇩n (G⇩n ψ) or⇩n (ψ U⇩n (φ and⇩n ψ))\"\nproof (cases \"∃i. ¬suffix i w ⊨⇩n ψ\")\n (*goals:\n 1. ‹∃i. ¬ suffix i w ⊨⇩n ψ ⟹ w ⊨⇩n φ R⇩n ψ = w ⊨⇩n G⇩n ψ or⇩n (ψ U⇩n (φ and⇩n ψ))›\n 2. ‹∄i. ¬ suffix i w ⊨⇩n ψ ⟹ w ⊨⇩n φ R⇩n ψ = w ⊨⇩n G⇩n ψ or⇩n (ψ U⇩n (φ and⇩n ψ))›*)\n case True (*‹∃i. ¬ suffix i w ⊨⇩n ψ›*)\n define i where \"i ≡ Least (λi. ¬suffix i w ⊨⇩n ψ)\"\n have \"⋀j. j < i ⟹ suffix j w ⊨⇩n ψ\" and \"¬ suffix i w ⊨⇩n ψ\"\n using True (*‹∃i::nat. ¬ suffix i (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln)›*) LeastI (*‹?P ?k ⟹ ?P (LEAST x. ?P x)›*) not_less_Least (*‹?k < (LEAST x. ?P x) ⟹ ¬ ?P ?k›*) unfolding i_def\n (*goals:\n 1. ‹⋀j. j < (LEAST i. ¬ suffix i w ⊨⇩n ψ) ⟹ suffix j w ⊨⇩n ψ›\n 2. ‹¬ suffix (LEAST i. ¬ suffix i w ⊨⇩n ψ) w ⊨⇩n ψ›*)\n apply -\n (*goals:\n 1. ‹⋀j. ⟦j < (LEAST i. ¬ suffix i w ⊨⇩n ψ); ∃i. ¬ suffix i w ⊨⇩n ψ; ⋀P k. P k ⟹ P (LEAST x. P x); ⋀k P. k < (LEAST x. P x) ⟹ ¬ P k⟧ ⟹ suffix j w ⊨⇩n ψ›\n 2. ‹⟦∃i. ¬ suffix i w ⊨⇩n ψ; ⋀P k. P k ⟹ P (LEAST x. P x); ⋀k P. k < (LEAST x. P x) ⟹ ¬ P k⟧ ⟹ ¬ suffix (LEAST i. ¬ suffix i w ⊨⇩n ψ) w ⊨⇩n ψ›\n discuss goal 1*)\n apply fast\n (*discuss goal 2*)\n apply fast\n (*proven 2 subgoals*) .\n hence \"*\": \"∀i. suffix i w ⊨⇩n ψ ∨ (∃j<i. suffix j w ⊨⇩n φ) ⟹ (∃i. (suffix i w ⊨⇩n ψ ∧ suffix i w ⊨⇩n φ) ∧ (∀j<i. suffix j w ⊨⇩n ψ))\"\n by fastforce\n hence \"∃i. (suffix i w ⊨⇩n ψ ∧ suffix i w ⊨⇩n φ) ∧ (∀j<i. suffix j w ⊨⇩n ψ) ⟹ (∀i. suffix i w ⊨⇩n ψ ∨ (∃j<i. suffix j w ⊨⇩n φ))\"\n using linorder_cases (*‹⟦?x < ?y ⟹ ?P; ?x = ?y ⟹ ?P; ?y < ?x ⟹ ?P⟧ ⟹ ?P›*) by blast\n thus \"?thesis\"\n (*goal: ‹(w::nat ⇒ 'a set) ⊨⇩n (φ::'a ltln) R⇩n (ψ::'a ltln) = w ⊨⇩n G⇩n ψ or⇩n (ψ U⇩n (φ and⇩n ψ))›*)\n using True (*‹∃i. ¬ suffix i w ⊨⇩n ψ›*) \"*\" (*‹∀i. suffix i w ⊨⇩n ψ ∨ (∃j<i. suffix j w ⊨⇩n φ) ⟹ ∃i. (suffix i w ⊨⇩n ψ ∧ suffix i w ⊨⇩n φ) ∧ (∀j<i. suffix j w ⊨⇩n ψ)›*) by auto\nqed (auto)\n(*solved the remaining goal: ‹∄i::nat. ¬ suffix i (w::nat ⇒ 'a set) ⊨⇩n (ψ::'a ltln) ⟹ w ⊨⇩n (φ::'a ltln) R⇩n ψ = w ⊨⇩n G⇩n ψ or⇩n (ψ U⇩n (φ and⇩n ψ))›*)\n\n\n\n\nsubsection ‹LTL in restricted Negation Normal Form›\n\ntext ‹Some algorithms do not handle the operators W and M,\n hence we also provide a datatype without these two operators.›\n\nsubsubsection ‹Syntax›\n\ndatatype (atoms_ltlr: 'a) ltlr =\n True_ltlr (\"true⇩r\")\n | False_ltlr (\"false⇩r\")\n | Prop_ltlr 'a (\"prop⇩r'(_')\")\n | Nprop_ltlr 'a (\"nprop⇩r'(_')\")\n | And_ltlr \"'a ltlr\" \"'a ltlr\" (\"_ and⇩r _\" [82,82] 81)\n | Or_ltlr \"'a ltlr\" \"'a ltlr\" (\"_ or⇩r _\" [84,84] 83)\n | Next_ltlr \"'a ltlr\" (\"X⇩r _\" [88] 87)\n | Until_ltlr \"'a ltlr\" \"'a ltlr\" (\"_ U⇩r _\" [84,84] 83)\n | Release_ltlr \"'a ltlr\" \"'a ltlr\" (\"_ R⇩r _\" [84,84] 83)\n\n\nsubsubsection ‹Semantics›\n\nprimrec semantics_ltlr :: \"['a set word, 'a ltlr] ⇒ bool\" (\"_ ⊨⇩r _\" [80,80] 80)\nwhere\n \"ξ ⊨⇩r true⇩r = True\"\n| \"ξ ⊨⇩r false⇩r = False\"\n| \"ξ ⊨⇩r prop⇩r(q) = (q ∈ ξ 0)\"\n| \"ξ ⊨⇩r nprop⇩r(q) = (q ∉ ξ 0)\"\n| \"ξ ⊨⇩r φ and⇩r ψ = (ξ ⊨⇩r φ ∧ ξ ⊨⇩r ψ)\"\n| \"ξ ⊨⇩r φ or⇩r ψ = (ξ ⊨⇩r φ ∨ ξ ⊨⇩r ψ)\"\n| \"ξ ⊨⇩r X⇩r φ = (suffix 1 ξ ⊨⇩r φ)\"\n| \"ξ ⊨⇩r φ U⇩r ψ = (∃i. suffix i ξ ⊨⇩r ψ ∧ (∀j<i. suffix j ξ ⊨⇩r φ))\"\n| \"ξ ⊨⇩r φ R⇩r ψ = (∀i. suffix i ξ ⊨⇩r ψ ∨ (∃j<i. suffix j ξ ⊨⇩r φ))\"\n\n\nsubsubsection ‹Conversion›\n\nfun ltln_to_ltlr :: \"'a ltln ⇒ 'a ltlr\"\nwhere\n \"ltln_to_ltlr true⇩n = true⇩r\"\n| \"ltln_to_ltlr false⇩n = false⇩r\"\n| \"ltln_to_ltlr prop⇩n(a) = prop⇩r(a)\"\n| \"ltln_to_ltlr nprop⇩n(a) = nprop⇩r(a)\"\n| \"ltln_to_ltlr (φ and⇩n ψ) = (ltln_to_ltlr φ) and⇩r (ltln_to_ltlr ψ)\"\n| \"ltln_to_ltlr (φ or⇩n ψ) = (ltln_to_ltlr φ) or⇩r (ltln_to_ltlr ψ)\"\n| \"ltln_to_ltlr (X⇩n φ) = X⇩r (ltln_to_ltlr φ)\"\n| \"ltln_to_ltlr (φ U⇩n ψ) = (ltln_to_ltlr φ) U⇩r (ltln_to_ltlr ψ)\"\n| \"ltln_to_ltlr (φ R⇩n ψ) = (ltln_to_ltlr φ) R⇩r (ltln_to_ltlr ψ)\"\n| \"ltln_to_ltlr (φ W⇩n ψ) = (ltln_to_ltlr ψ) R⇩r ((ltln_to_ltlr φ) or⇩r (ltln_to_ltlr ψ))\"\n| \"ltln_to_ltlr (φ M⇩n ψ) = (ltln_to_ltlr ψ) U⇩r ((ltln_to_ltlr φ) and⇩r (ltln_to_ltlr ψ))\"\n\nfun ltlr_to_ltln :: \"'a ltlr ⇒ 'a ltln\"\nwhere\n \"ltlr_to_ltln true⇩r = true⇩n\"\n| \"ltlr_to_ltln false⇩r = false⇩n\"\n| \"ltlr_to_ltln prop⇩r(a) = prop⇩n(a)\"\n| \"ltlr_to_ltln nprop⇩r(a) = nprop⇩n(a)\"\n| \"ltlr_to_ltln (φ and⇩r ψ) = (ltlr_to_ltln φ) and⇩n (ltlr_to_ltln ψ)\"\n| \"ltlr_to_ltln (φ or⇩r ψ) = (ltlr_to_ltln φ) or⇩n (ltlr_to_ltln ψ)\"\n| \"ltlr_to_ltln (X⇩r φ) = X⇩n (ltlr_to_ltln φ)\"\n| \"ltlr_to_ltln (φ U⇩r ψ) = (ltlr_to_ltln φ) U⇩n (ltlr_to_ltln ψ)\"\n| \"ltlr_to_ltln (φ R⇩r ψ) = (ltlr_to_ltln φ) R⇩n (ltlr_to_ltln ψ)\"\n\nlemma ltln_to_ltlr_semantics:\n \"w ⊨⇩r ltln_to_ltlr φ ⟷ w ⊨⇩n φ\"\n apply (induction φ arbitrary: w)\n (*goals:\n 1. ‹⋀w. w ⊨⇩r ltln_to_ltlr true⇩n = w ⊨⇩n true⇩n›\n 2. ‹⋀w. w ⊨⇩r ltln_to_ltlr false⇩n = w ⊨⇩n false⇩n›\n 3. ‹⋀x w. w ⊨⇩r ltln_to_ltlr prop⇩n(x) = w ⊨⇩n prop⇩n(x)›\n 4. ‹⋀x w. w ⊨⇩r ltln_to_ltlr nprop⇩n(x) = w ⊨⇩n nprop⇩n(x)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 and⇩n φ2) = w ⊨⇩n φ1 and⇩n φ2›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 or⇩n φ2) = w ⊨⇩n φ1 or⇩n φ2›\n 7. ‹⋀φ w. (⋀w. w ⊨⇩r ltln_to_ltlr φ = w ⊨⇩n φ) ⟹ w ⊨⇩r ltln_to_ltlr (X⇩n φ) = w ⊨⇩n X⇩n φ›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 U⇩n φ2) = w ⊨⇩n φ1 U⇩n φ2›\n 9. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 R⇩n φ2) = w ⊨⇩n φ1 R⇩n φ2›\n 10. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 W⇩n φ2) = w ⊨⇩n φ1 W⇩n φ2›\n 11. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 M⇩n φ2) = w ⊨⇩n φ1 M⇩n φ2›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*discuss goal 8*)\n apply simp\n (*discuss goal 9*)\n apply simp\n (*discuss goal 10*)\n apply ((unfold ltln_WeakUntil_to_Release (*‹?w ⊨⇩n ?φ W⇩n ?ψ = ?w ⊨⇩n ?ψ R⇩n (?φ or⇩n ?ψ)›*) ltln_StrongRelease_to_Until (*‹?w ⊨⇩n ?φ M⇩n ?ψ = ?w ⊨⇩n ?ψ U⇩n (?φ and⇩n ?ψ)›*))[1])\n (*top goal: ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 W⇩n φ2) = w ⊨⇩n φ1 W⇩n φ2› and 1 goal remains*)\n apply simp\n (*discuss goal 11*)\n apply ((unfold ltln_WeakUntil_to_Release (*‹?w ⊨⇩n ?φ W⇩n ?ψ = ?w ⊨⇩n ?ψ R⇩n (?φ or⇩n ?ψ)›*) ltln_StrongRelease_to_Until (*‹?w ⊨⇩n ?φ M⇩n ?ψ = ?w ⊨⇩n ?ψ U⇩n (?φ and⇩n ?ψ)›*))[1])\n (*goal: ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r ltln_to_ltlr φ1 = w ⊨⇩n φ1; ⋀w. w ⊨⇩r ltln_to_ltlr φ2 = w ⊨⇩n φ2⟧ ⟹ w ⊨⇩r ltln_to_ltlr (φ1 M⇩n φ2) = w ⊨⇩n φ1 M⇩n φ2›*)\n apply simp\n (*proven 11 subgoals*) .\n\nlemma ltlr_to_ltln_semantics:\n \"w ⊨⇩n ltlr_to_ltln φ ⟷ w ⊨⇩r φ\"\n apply (induction φ arbitrary: w)\n (*goals:\n 1. ‹⋀w. w ⊨⇩n ltlr_to_ltln true⇩r = w ⊨⇩r true⇩r›\n 2. ‹⋀w. w ⊨⇩n ltlr_to_ltln false⇩r = w ⊨⇩r false⇩r›\n 3. ‹⋀x w. w ⊨⇩n ltlr_to_ltln prop⇩r(x) = w ⊨⇩r prop⇩r(x)›\n 4. ‹⋀x w. w ⊨⇩n ltlr_to_ltln nprop⇩r(x) = w ⊨⇩r nprop⇩r(x)›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n ltlr_to_ltln φ1 = w ⊨⇩r φ1; ⋀w. w ⊨⇩n ltlr_to_ltln φ2 = w ⊨⇩r φ2⟧ ⟹ w ⊨⇩n ltlr_to_ltln (φ1 and⇩r φ2) = w ⊨⇩r φ1 and⇩r φ2›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n ltlr_to_ltln φ1 = w ⊨⇩r φ1; ⋀w. w ⊨⇩n ltlr_to_ltln φ2 = w ⊨⇩r φ2⟧ ⟹ w ⊨⇩n ltlr_to_ltln (φ1 or⇩r φ2) = w ⊨⇩r φ1 or⇩r φ2›\n 7. ‹⋀φ w. (⋀w. w ⊨⇩n ltlr_to_ltln φ = w ⊨⇩r φ) ⟹ w ⊨⇩n ltlr_to_ltln (X⇩r φ) = w ⊨⇩r X⇩r φ›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n ltlr_to_ltln φ1 = w ⊨⇩r φ1; ⋀w. w ⊨⇩n ltlr_to_ltln φ2 = w ⊨⇩r φ2⟧ ⟹ w ⊨⇩n ltlr_to_ltln (φ1 U⇩r φ2) = w ⊨⇩r φ1 U⇩r φ2›\n 9. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩n ltlr_to_ltln φ1 = w ⊨⇩r φ1; ⋀w. w ⊨⇩n ltlr_to_ltln φ2 = w ⊨⇩r φ2⟧ ⟹ w ⊨⇩n ltlr_to_ltln (φ1 R⇩r φ2) = w ⊨⇩r φ1 R⇩r φ2›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*discuss goal 8*)\n apply simp\n (*discuss goal 9*)\n apply simp\n (*proven 9 subgoals*) .\n\n\nsubsubsection ‹Negation›\n\nfun not⇩r\nwhere\n \"not⇩r true⇩r = false⇩r\"\n| \"not⇩r false⇩r = true⇩r\"\n| \"not⇩r prop⇩r(a) = nprop⇩r(a)\"\n| \"not⇩r nprop⇩r(a) = prop⇩r(a)\"\n| \"not⇩r (φ and⇩r ψ) = (not⇩r φ) or⇩r (not⇩r ψ)\"\n| \"not⇩r (φ or⇩r ψ) = (not⇩r φ) and⇩r (not⇩r ψ)\"\n| \"not⇩r (X⇩r φ) = X⇩r (not⇩r φ)\"\n| \"not⇩r (φ U⇩r ψ) = (not⇩r φ) R⇩r (not⇩r ψ)\"\n| \"not⇩r (φ R⇩r ψ) = (not⇩r φ) U⇩r (not⇩r ψ)\"\n\nlemma not⇩r_semantics [simp]:\n \"w ⊨⇩r not⇩r φ ⟷ ¬ w ⊨⇩r φ\"\n apply (induction φ arbitrary: w)\n (*goals:\n 1. ‹⋀w. w ⊨⇩r not⇩r true⇩r = (¬ w ⊨⇩r true⇩r)›\n 2. ‹⋀w. w ⊨⇩r not⇩r false⇩r = (¬ w ⊨⇩r false⇩r)›\n 3. ‹⋀x w. w ⊨⇩r not⇩r prop⇩r(x) = (¬ w ⊨⇩r prop⇩r(x))›\n 4. ‹⋀x w. w ⊨⇩r not⇩r nprop⇩r(x) = (¬ w ⊨⇩r nprop⇩r(x))›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r not⇩r φ1 = (¬ w ⊨⇩r φ1); ⋀w. w ⊨⇩r not⇩r φ2 = (¬ w ⊨⇩r φ2)⟧ ⟹ w ⊨⇩r not⇩r (φ1 and⇩r φ2) = (¬ w ⊨⇩r φ1 and⇩r φ2)›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r not⇩r φ1 = (¬ w ⊨⇩r φ1); ⋀w. w ⊨⇩r not⇩r φ2 = (¬ w ⊨⇩r φ2)⟧ ⟹ w ⊨⇩r not⇩r (φ1 or⇩r φ2) = (¬ w ⊨⇩r φ1 or⇩r φ2)›\n 7. ‹⋀φ w. (⋀w. w ⊨⇩r not⇩r φ = (¬ w ⊨⇩r φ)) ⟹ w ⊨⇩r not⇩r (X⇩r φ) = (¬ w ⊨⇩r X⇩r φ)›\n 8. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r not⇩r φ1 = (¬ w ⊨⇩r φ1); ⋀w. w ⊨⇩r not⇩r φ2 = (¬ w ⊨⇩r φ2)⟧ ⟹ w ⊨⇩r not⇩r (φ1 U⇩r φ2) = (¬ w ⊨⇩r φ1 U⇩r φ2)›\n 9. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩r not⇩r φ1 = (¬ w ⊨⇩r φ1); ⋀w. w ⊨⇩r not⇩r φ2 = (¬ w ⊨⇩r φ2)⟧ ⟹ w ⊨⇩r not⇩r (φ1 R⇩r φ2) = (¬ w ⊨⇩r φ1 R⇩r φ2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*proven 9 subgoals*) .\n\n\nsubsubsection ‹Subformulas›\n\nfun subfrmlsr :: \"'a ltlr ⇒ 'a ltlr set\"\nwhere\n \"subfrmlsr (φ and⇩r ψ) = {φ and⇩r ψ} ∪ subfrmlsr φ ∪ subfrmlsr ψ\"\n| \"subfrmlsr (φ or⇩r ψ) = {φ or⇩r ψ} ∪ subfrmlsr φ ∪ subfrmlsr ψ\"\n| \"subfrmlsr (φ U⇩r ψ) = {φ U⇩r ψ} ∪ subfrmlsr φ ∪ subfrmlsr ψ\"\n| \"subfrmlsr (φ R⇩r ψ) = {φ R⇩r ψ} ∪ subfrmlsr φ ∪ subfrmlsr ψ\"\n| \"subfrmlsr (X⇩r φ) = {X⇩r φ} ∪ subfrmlsr φ\"\n| \"subfrmlsr x = {x}\"\n\nlemma subfrmlsr_id[simp]:\n \"φ ∈ subfrmlsr φ\"\n apply (induction φ)\n (*goals:\n 1. ‹true⇩r ∈ subfrmlsr true⇩r›\n 2. ‹false⇩r ∈ subfrmlsr false⇩r›\n 3. ‹⋀x::'a. prop⇩r(x) ∈ subfrmlsr prop⇩r(x)›\n 4. ‹⋀x::'a. nprop⇩r(x) ∈ subfrmlsr nprop⇩r(x)›\n 5. ‹⋀(φ1::'a ltlr) φ2::'a ltlr. ⟦φ1 ∈ subfrmlsr φ1; φ2 ∈ subfrmlsr φ2⟧ ⟹ φ1 and⇩r φ2 ∈ subfrmlsr (φ1 and⇩r φ2)›\n 6. ‹⋀(φ1::'a ltlr) φ2::'a ltlr. ⟦φ1 ∈ subfrmlsr φ1; φ2 ∈ subfrmlsr φ2⟧ ⟹ φ1 or⇩r φ2 ∈ subfrmlsr (φ1 or⇩r φ2)›\n 7. ‹⋀φ::'a ltlr. φ ∈ subfrmlsr φ ⟹ X⇩r φ ∈ subfrmlsr (X⇩r φ)›\n 8. ‹⋀(φ1::'a ltlr) φ2::'a ltlr. ⟦φ1 ∈ subfrmlsr φ1; φ2 ∈ subfrmlsr φ2⟧ ⟹ φ1 U⇩r φ2 ∈ subfrmlsr (φ1 U⇩r φ2)›\n 9. ‹⋀(φ1::'a ltlr) φ2::'a ltlr. ⟦φ1 ∈ subfrmlsr φ1; φ2 ∈ subfrmlsr φ2⟧ ⟹ φ1 R⇩r φ2 ∈ subfrmlsr (φ1 R⇩r φ2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*proven 9 subgoals*) .\n\nlemma subfrmlsr_finite:\n \"finite (subfrmlsr φ)\"\n apply (induction φ)\n (*goals:\n 1. ‹finite (subfrmlsr true⇩r)›\n 2. ‹finite (subfrmlsr false⇩r)›\n 3. ‹⋀x. finite (subfrmlsr prop⇩r(x))›\n 4. ‹⋀x. finite (subfrmlsr nprop⇩r(x))›\n 5. ‹⋀φ1 φ2. ⟦finite (subfrmlsr φ1); finite (subfrmlsr φ2)⟧ ⟹ finite (subfrmlsr (φ1 and⇩r φ2))›\n 6. ‹⋀φ1 φ2. ⟦finite (subfrmlsr φ1); finite (subfrmlsr φ2)⟧ ⟹ finite (subfrmlsr (φ1 or⇩r φ2))›\n 7. ‹⋀φ. finite (subfrmlsr φ) ⟹ finite (subfrmlsr (X⇩r φ))›\n 8. ‹⋀φ1 φ2. ⟦finite (subfrmlsr φ1); finite (subfrmlsr φ2)⟧ ⟹ finite (subfrmlsr (φ1 U⇩r φ2))›\n 9. ‹⋀φ1 φ2. ⟦finite (subfrmlsr φ1); finite (subfrmlsr φ2)⟧ ⟹ finite (subfrmlsr (φ1 R⇩r φ2))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*proven 9 subgoals*) .\n\nlemma subfrmlsr_subset:\n \"ψ ∈ subfrmlsr φ ⟹ subfrmlsr ψ ⊆ subfrmlsr φ\"\n apply (induction φ)\n (*goals:\n 1. ‹(ψ::'a::type ltlr) ∈ subfrmlsr true⇩r ⟹ subfrmlsr ψ ⊆ subfrmlsr true⇩r›\n 2. ‹(ψ::'a::type ltlr) ∈ subfrmlsr false⇩r ⟹ subfrmlsr ψ ⊆ subfrmlsr false⇩r›\n 3. ‹⋀x::'a::type. (ψ::'a::type ltlr) ∈ subfrmlsr prop⇩r(x) ⟹ subfrmlsr ψ ⊆ subfrmlsr prop⇩r(x)›\n 4. ‹⋀x::'a::type. (ψ::'a::type ltlr) ∈ subfrmlsr nprop⇩r(x) ⟹ subfrmlsr ψ ⊆ subfrmlsr nprop⇩r(x)›\n 5. ‹⋀(φ1::'a::type ltlr) φ2::'a::type ltlr. ⟦(ψ::'a::type ltlr) ∈ subfrmlsr φ1 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ1; ψ ∈ subfrmlsr φ2 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ2; ψ ∈ subfrmlsr (φ1 and⇩r φ2)⟧ ⟹ subfrmlsr ψ ⊆ subfrmlsr (φ1 and⇩r φ2)›\n 6. ‹⋀(φ1::'a::type ltlr) φ2::'a::type ltlr. ⟦(ψ::'a::type ltlr) ∈ subfrmlsr φ1 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ1; ψ ∈ subfrmlsr φ2 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ2; ψ ∈ subfrmlsr (φ1 or⇩r φ2)⟧ ⟹ subfrmlsr ψ ⊆ subfrmlsr (φ1 or⇩r φ2)›\n 7. ‹⋀φ::'a::type ltlr. ⟦(ψ::'a::type ltlr) ∈ subfrmlsr φ ⟹ subfrmlsr ψ ⊆ subfrmlsr φ; ψ ∈ subfrmlsr (X⇩r φ)⟧ ⟹ subfrmlsr ψ ⊆ subfrmlsr (X⇩r φ)›\n 8. ‹⋀(φ1::'a::type ltlr) φ2::'a::type ltlr. ⟦(ψ::'a::type ltlr) ∈ subfrmlsr φ1 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ1; ψ ∈ subfrmlsr φ2 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ2; ψ ∈ subfrmlsr (φ1 U⇩r φ2)⟧ ⟹ subfrmlsr ψ ⊆ subfrmlsr (φ1 U⇩r φ2)›\n 9. ‹⋀(φ1::'a::type ltlr) φ2::'a::type ltlr. ⟦(ψ::'a::type ltlr) ∈ subfrmlsr φ1 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ1; ψ ∈ subfrmlsr φ2 ⟹ subfrmlsr ψ ⊆ subfrmlsr φ2; ψ ∈ subfrmlsr (φ1 R⇩r φ2)⟧ ⟹ subfrmlsr ψ ⊆ subfrmlsr (φ1 R⇩r φ2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*proven 9 subgoals*) .\n\nlemma subfrmlsr_size:\n \"ψ ∈ subfrmlsr φ ⟹ size ψ < size φ ∨ ψ = φ\"\n apply (induction φ)\n (*goals:\n 1. ‹ψ ∈ subfrmlsr true⇩r ⟹ size ψ < size true⇩r ∨ ψ = true⇩r›\n 2. ‹ψ ∈ subfrmlsr false⇩r ⟹ size ψ < size false⇩r ∨ ψ = false⇩r›\n 3. ‹⋀x. ψ ∈ subfrmlsr prop⇩r(x) ⟹ size ψ < size prop⇩r(x) ∨ ψ = prop⇩r(x)›\n 4. ‹⋀x. ψ ∈ subfrmlsr nprop⇩r(x) ⟹ size ψ < size nprop⇩r(x) ∨ ψ = nprop⇩r(x)›\n 5. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsr φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsr φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsr (φ1 and⇩r φ2)⟧ ⟹ size ψ < size (φ1 and⇩r φ2) ∨ ψ = φ1 and⇩r φ2›\n 6. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsr φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsr φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsr (φ1 or⇩r φ2)⟧ ⟹ size ψ < size (φ1 or⇩r φ2) ∨ ψ = φ1 or⇩r φ2›\n 7. ‹⋀φ. ⟦ψ ∈ subfrmlsr φ ⟹ size ψ < size φ ∨ ψ = φ; ψ ∈ subfrmlsr (X⇩r φ)⟧ ⟹ size ψ < size (X⇩r φ) ∨ ψ = X⇩r φ›\n 8. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsr φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsr φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsr (φ1 U⇩r φ2)⟧ ⟹ size ψ < size (φ1 U⇩r φ2) ∨ ψ = φ1 U⇩r φ2›\n 9. ‹⋀φ1 φ2. ⟦ψ ∈ subfrmlsr φ1 ⟹ size ψ < size φ1 ∨ ψ = φ1; ψ ∈ subfrmlsr φ2 ⟹ size ψ < size φ2 ∨ ψ = φ2; ψ ∈ subfrmlsr (φ1 R⇩r φ2)⟧ ⟹ size ψ < size (φ1 R⇩r φ2) ∨ ψ = φ1 R⇩r φ2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*proven 9 subgoals*) .\n\n\nsubsubsection ‹Expansion lemmas›\n\nlemma ltlr_expand_Until:\n \"ξ ⊨⇩r φ U⇩r ψ ⟷ (ξ ⊨⇩r ψ or⇩r (φ and⇩r (X⇩r (φ U⇩r ψ))))\"\n by (metis ltln_expand_Until (*‹?ξ ⊨⇩n ?φ U⇩n ?ψ = ?ξ ⊨⇩n ?ψ or⇩n (?φ and⇩n X⇩n (?φ U⇩n ?ψ))›*) ltlr_to_ltln.simps( (*‹ltlr_to_ltln (?φ and⇩r ?ψ) = ltlr_to_ltln ?φ and⇩n ltlr_to_ltln ?ψ› ‹ltlr_to_ltln (?φ or⇩r ?ψ) = ltlr_to_ltln ?φ or⇩n ltlr_to_ltln ?ψ› ‹ltlr_to_ltln (X⇩r ?φ) = X⇩n ltlr_to_ltln ?φ› ‹ltlr_to_ltln (?φ U⇩r ?ψ) = ltlr_to_ltln ?φ U⇩n ltlr_to_ltln ?ψ›*) 5-8) ltlr_to_ltln_semantics (*‹?w ⊨⇩n ltlr_to_ltln ?φ = ?w ⊨⇩r ?φ›*))\n\nlemma ltlr_expand_Release:\n \"ξ ⊨⇩r φ R⇩r ψ ⟷ (ξ ⊨⇩r ψ and⇩r (φ or⇩r (X⇩r (φ R⇩r ψ))))\"\n sorry\n\n\n\n\nsubsection ‹Propositional LTL›\n\ntext ‹We define the syntax and semantics of propositional linear-time\n temporal logic PLTL.\n PLTL formulas are built from atomic formulas, propositional connectives,\n and the temporal operators ``next'' and ``until''. The following data\n type definition is parameterized by the type of states over which\n formulas are evaluated.›\n\nsubsubsection ‹Syntax›\n\ndatatype 'a pltl =\n False_ltlp (\"false⇩p\")\n | Atom_ltlp \"'a ⇒ bool\" (\"atom⇩p'(_')\")\n | Implies_ltlp \"'a pltl\" \"'a pltl\" (\"_ implies⇩p _\" [81,81] 80)\n | Next_ltlp \"'a pltl\" (\"X⇩p _\" [88] 87)\n | Until_ltlp \"'a pltl\" \"'a pltl\" (\"_ U⇩p _\" [84,84] 83)\n\n― ‹Further connectives of PLTL can be defined in terms of the existing syntax.›\n\ndefinition Not_ltlp (\"not⇩p _\" [85] 85)\nwhere\n \"not⇩p φ ≡ φ implies⇩p false⇩p\"\n\ndefinition True_ltlp (\"true⇩p\")\nwhere\n \"true⇩p ≡ not⇩p false⇩p\"\n\ndefinition Or_ltlp (\"_ or⇩p _\" [81,81] 80)\nwhere\n \"φ or⇩p ψ ≡ (not⇩p φ) implies⇩p ψ\"\n\ndefinition And_ltlp (\"_ and⇩p _\" [82,82] 81)\nwhere\n \"φ and⇩p ψ ≡ not⇩p ((not⇩p φ) or⇩p (not⇩p ψ))\"\n\ndefinition Eventually_ltlp (\"F⇩p _\" [88] 87)\nwhere\n \"F⇩p φ ≡ true⇩p U⇩p φ\"\n\ndefinition Always_ltlp (\"G⇩p _\" [88] 87)\nwhere\n \"G⇩p φ ≡ not⇩p (F⇩p (not⇩p φ))\"\n\ndefinition Release_ltlp (\"_ R⇩p _\" [84,84] 83)\nwhere\n \"φ R⇩p ψ ≡ not⇩p ((not⇩p φ) U⇩p (not⇩p ψ))\"\n\ndefinition WeakUntil_ltlp (\"_ W⇩p _\" [84,84] 83)\nwhere\n \"φ W⇩p ψ ≡ ψ R⇩p (φ or⇩p ψ)\"\n\ndefinition StrongRelease_ltlp (\"_ M⇩p _\" [84,84] 83)\nwhere\n \"φ M⇩p ψ ≡ ψ U⇩p (φ and⇩p ψ)\"\n\n\nsubsubsection ‹Semantics›\n\nfun semantics_pltl :: \"['a word, 'a pltl] ⇒ bool\" (\"_ ⊨⇩p _\" [80,80] 80)\nwhere\n \"w ⊨⇩p false⇩p = False\"\n| \"w ⊨⇩p atom⇩p(p) = (p (w 0))\"\n| \"w ⊨⇩p φ implies⇩p ψ = (w ⊨⇩p φ ⟶ w ⊨⇩p ψ)\"\n| \"w ⊨⇩p X⇩p φ = (suffix 1 w ⊨⇩p φ)\"\n| \"w ⊨⇩p φ U⇩p ψ = (∃i. suffix i w ⊨⇩p ψ ∧ (∀j<i. suffix j w ⊨⇩p φ))\"\n\nlemma semantics_pltl_sugar [simp]:\n \"w ⊨⇩p not⇩p φ = (¬w ⊨⇩p φ)\"\n \"w ⊨⇩p true⇩p = True\"\n \"w ⊨⇩p φ or⇩p ψ = (w ⊨⇩p φ ∨ w ⊨⇩p ψ)\"\n \"w ⊨⇩p φ and⇩p ψ = (w ⊨⇩p φ ∧ w ⊨⇩p ψ)\"\n \"w ⊨⇩p F⇩p φ = (∃i. suffix i w ⊨⇩p φ)\"\n \"w ⊨⇩p G⇩p φ = (∀i. suffix i w ⊨⇩p φ)\"\n \"w ⊨⇩p φ R⇩p ψ = (∀i. suffix i w ⊨⇩p ψ ∨ (∃j<i. suffix j w ⊨⇩p φ))\"\n \"w ⊨⇩p φ W⇩p ψ = (∀i. suffix i w ⊨⇩p φ ∨ (∃j≤i. suffix j w ⊨⇩p ψ))\"\n \"w ⊨⇩p φ M⇩p ψ = (∃i. suffix i w ⊨⇩p φ ∧ (∀j≤i. suffix j w ⊨⇩p ψ))\"\n \n (*goals:\n 1. ‹w ⊨⇩p not⇩p φ = (¬ w ⊨⇩p φ)›\n 2. ‹w ⊨⇩p true⇩p = True›\n 3. ‹w ⊨⇩p φ or⇩p ψ = (w ⊨⇩p φ ∨ w ⊨⇩p ψ)›\n 4. ‹w ⊨⇩p φ and⇩p ψ = (w ⊨⇩p φ ∧ w ⊨⇩p ψ)›\n 5. ‹w ⊨⇩p F⇩p φ = (∃i. suffix i w ⊨⇩p φ)›\n 6. ‹w ⊨⇩p G⇩p φ = (∀i. suffix i w ⊨⇩p φ)›\n 7. ‹w ⊨⇩p φ R⇩p ψ = (∀i. suffix i w ⊨⇩p ψ ∨ (∃j<i. suffix j w ⊨⇩p φ))›\n 8. ‹w ⊨⇩p φ W⇩p ψ = (∀i. suffix i w ⊨⇩p φ ∨ (∃j≤i. suffix j w ⊨⇩p ψ))›\n 9. ‹w ⊨⇩p φ M⇩p ψ = (∃i. suffix i w ⊨⇩p φ ∧ (∀j≤i. suffix j w ⊨⇩p ψ))›\n discuss goal 1*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p (?φ::?'a pltl) ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹(?φ::?'a pltl) or⇩p (?ψ::?'a pltl) ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹(?φ::?'a pltl) and⇩p (?ψ::?'a pltl) ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p (?φ::?'a pltl) ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p (?φ::?'a pltl) ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹(?φ::?'a pltl) R⇩p (?ψ::?'a pltl) ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹(?φ::?'a pltl) W⇩p (?ψ::?'a pltl) ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹(?φ::?'a pltl) M⇩p (?ψ::?'a pltl) ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*discuss goal 7*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*discuss goal 8*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*top goal: ‹w ⊨⇩p φ W⇩p ψ = (∀i. suffix i w ⊨⇩p φ ∨ (∃j≤i. suffix j w ⊨⇩p ψ))› and 1 goal remains*)\n apply ((insert le_neq_implies_less (*‹⟦?m ≤ ?n; ?m ≠ ?n⟧ ⟹ ?m < ?n›*))[1])\n (*top goal: ‹⋀i. ⟦∀i. suffix i w ⊨⇩p φ ∨ (∃j≤i. suffix j w ⊨⇩p ψ); ¬ suffix i w ⊨⇩p φ; ∀j<i. ¬ suffix j w ⊨⇩p ψ⟧ ⟹ suffix i w ⊨⇩p ψ› and 1 goal remains*)\n apply blast\n (*discuss goal 9*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*) Release_ltlp_def (*‹?φ R⇩p ?ψ ≡ not⇩p (not⇩p ?φ U⇩p not⇩p ?ψ)›*) WeakUntil_ltlp_def (*‹?φ W⇩p ?ψ ≡ ?ψ R⇩p (?φ or⇩p ?ψ)›*) StrongRelease_ltlp_def (*‹?φ M⇩p ?ψ ≡ ?ψ U⇩p (?φ and⇩p ?ψ)›*))[1])\n (*goal: ‹w ⊨⇩p φ M⇩p ψ = (∃i. suffix i w ⊨⇩p φ ∧ (∀j≤i. suffix j w ⊨⇩p ψ))›*)\n apply ((insert le_neq_implies_less (*‹⟦?m ≤ ?n; ?m ≠ ?n⟧ ⟹ ?m < ?n›*))[1])\n (*goal: ‹⋀i. ⟦suffix i w ⊨⇩p φ; suffix i w ⊨⇩p ψ; ∀j<i. suffix j w ⊨⇩p ψ⟧ ⟹ ∃i. suffix i w ⊨⇩p φ ∧ (∀j≤i. suffix j w ⊨⇩p ψ)›*)\n apply (insert le_neq_implies_less (*‹⟦?m ≤ ?n; ?m ≠ ?n⟧ ⟹ ?m < ?n›*))\n (*goal: ‹⋀i. ⟦suffix i w ⊨⇩p φ; suffix i w ⊨⇩p ψ; ∀j<i. suffix j w ⊨⇩p ψ; ⋀m n. ⟦m ≤ n; m ≠ n⟧ ⟹ m < n⟧ ⟹ ∃i. suffix i w ⊨⇩p φ ∧ (∀j≤i. suffix j w ⊨⇩p ψ)›*)\n apply blast\n (*proven 9 subgoals*) .\n\ndefinition \"language_ltlp φ ≡ {ξ. ξ ⊨⇩p φ}\"\n\n\nsubsubsection ‹Conversion›\n\nfun ltlc_to_pltl :: \"'a ltlc ⇒ 'a set pltl\"\nwhere\n \"ltlc_to_pltl true⇩c = true⇩p\"\n| \"ltlc_to_pltl false⇩c = false⇩p\"\n| \"ltlc_to_pltl (prop⇩c(q)) = atom⇩p((∈) q)\"\n| \"ltlc_to_pltl (not⇩c φ) = not⇩p (ltlc_to_pltl φ)\"\n| \"ltlc_to_pltl (φ and⇩c ψ) = (ltlc_to_pltl φ) and⇩p (ltlc_to_pltl ψ)\"\n| \"ltlc_to_pltl (φ or⇩c ψ) = (ltlc_to_pltl φ) or⇩p (ltlc_to_pltl ψ)\"\n| \"ltlc_to_pltl (φ implies⇩c ψ) = (ltlc_to_pltl φ) implies⇩p (ltlc_to_pltl ψ)\"\n| \"ltlc_to_pltl (X⇩c φ) = X⇩p (ltlc_to_pltl φ)\"\n| \"ltlc_to_pltl (F⇩c φ) = F⇩p (ltlc_to_pltl φ)\"\n| \"ltlc_to_pltl (G⇩c φ) = G⇩p (ltlc_to_pltl φ)\"\n| \"ltlc_to_pltl (φ U⇩c ψ) = (ltlc_to_pltl φ) U⇩p (ltlc_to_pltl ψ)\"\n| \"ltlc_to_pltl (φ R⇩c ψ) = (ltlc_to_pltl φ) R⇩p (ltlc_to_pltl ψ)\"\n| \"ltlc_to_pltl (φ W⇩c ψ) = (ltlc_to_pltl φ) W⇩p (ltlc_to_pltl ψ)\"\n| \"ltlc_to_pltl (φ M⇩c ψ) = (ltlc_to_pltl φ) M⇩p (ltlc_to_pltl ψ)\"\n\nlemma ltlc_to_pltl_semantics [simp]:\n \"w ⊨⇩p (ltlc_to_pltl φ) ⟷ w ⊨⇩c φ\"\n apply (induction φ arbitrary: w)\n (*goals:\n 1. ‹⋀w. w ⊨⇩p ltlc_to_pltl true⇩c = w ⊨⇩c true⇩c›\n 2. ‹⋀w. w ⊨⇩p ltlc_to_pltl false⇩c = w ⊨⇩c false⇩c›\n 3. ‹⋀x w. w ⊨⇩p ltlc_to_pltl prop⇩c(x) = w ⊨⇩c prop⇩c(x)›\n 4. ‹⋀φ w. (⋀w. w ⊨⇩p ltlc_to_pltl φ = w ⊨⇩c φ) ⟹ w ⊨⇩p ltlc_to_pltl (not⇩c φ) = w ⊨⇩c not⇩c φ›\n 5. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩p ltlc_to_pltl φ1 = w ⊨⇩c φ1; ⋀w. w ⊨⇩p ltlc_to_pltl φ2 = w ⊨⇩c φ2⟧ ⟹ w ⊨⇩p ltlc_to_pltl (φ1 and⇩c φ2) = w ⊨⇩c φ1 and⇩c φ2›\n 6. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩p ltlc_to_pltl φ1 = w ⊨⇩c φ1; ⋀w. w ⊨⇩p ltlc_to_pltl φ2 = w ⊨⇩c φ2⟧ ⟹ w ⊨⇩p ltlc_to_pltl (φ1 or⇩c φ2) = w ⊨⇩c φ1 or⇩c φ2›\n 7. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩p ltlc_to_pltl φ1 = w ⊨⇩c φ1; ⋀w. w ⊨⇩p ltlc_to_pltl φ2 = w ⊨⇩c φ2⟧ ⟹ w ⊨⇩p ltlc_to_pltl (φ1 implies⇩c φ2) = w ⊨⇩c φ1 implies⇩c φ2›\n 8. ‹⋀φ w. (⋀w. w ⊨⇩p ltlc_to_pltl φ = w ⊨⇩c φ) ⟹ w ⊨⇩p ltlc_to_pltl (X⇩c φ) = w ⊨⇩c X⇩c φ›\n 9. ‹⋀φ w. (⋀w. w ⊨⇩p ltlc_to_pltl φ = w ⊨⇩c φ) ⟹ w ⊨⇩p ltlc_to_pltl (F⇩c φ) = w ⊨⇩c F⇩c φ›\n 10. ‹⋀φ w. (⋀w. w ⊨⇩p ltlc_to_pltl φ = w ⊨⇩c φ) ⟹ w ⊨⇩p ltlc_to_pltl (G⇩c φ) = w ⊨⇩c G⇩c φ›\n 11. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩p ltlc_to_pltl φ1 = w ⊨⇩c φ1; ⋀w. w ⊨⇩p ltlc_to_pltl φ2 = w ⊨⇩c φ2⟧ ⟹ w ⊨⇩p ltlc_to_pltl (φ1 U⇩c φ2) = w ⊨⇩c φ1 U⇩c φ2›\n 12. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩p ltlc_to_pltl φ1 = w ⊨⇩c φ1; ⋀w. w ⊨⇩p ltlc_to_pltl φ2 = w ⊨⇩c φ2⟧ ⟹ w ⊨⇩p ltlc_to_pltl (φ1 R⇩c φ2) = w ⊨⇩c φ1 R⇩c φ2›\n 13. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩p ltlc_to_pltl φ1 = w ⊨⇩c φ1; ⋀w. w ⊨⇩p ltlc_to_pltl φ2 = w ⊨⇩c φ2⟧ ⟹ w ⊨⇩p ltlc_to_pltl (φ1 W⇩c φ2) = w ⊨⇩c φ1 W⇩c φ2›\n 14. ‹⋀φ1 φ2 w. ⟦⋀w. w ⊨⇩p ltlc_to_pltl φ1 = w ⊨⇩c φ1; ⋀w. w ⊨⇩p ltlc_to_pltl φ2 = w ⊨⇩c φ2⟧ ⟹ w ⊨⇩p ltlc_to_pltl (φ1 M⇩c φ2) = w ⊨⇩c φ1 M⇩c φ2›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*discuss goal 8*)\n apply simp\n (*discuss goal 9*)\n apply simp\n (*discuss goal 10*)\n apply simp\n (*discuss goal 11*)\n apply simp\n (*discuss goal 12*)\n apply simp\n (*discuss goal 13*)\n apply simp\n (*discuss goal 14*)\n apply simp\n (*proven 14 subgoals*) .\n\n\nsubsubsection ‹Atoms›\n\nfun atoms_pltl :: \"'a pltl ⇒ ('a ⇒ bool) set\"\nwhere\n \"atoms_pltl false⇩p = {}\"\n| \"atoms_pltl atom⇩p(p) = {p}\"\n| \"atoms_pltl (φ implies⇩p ψ) = atoms_pltl φ ∪ atoms_pltl ψ\"\n| \"atoms_pltl (X⇩p φ) = atoms_pltl φ\"\n| \"atoms_pltl (φ U⇩p ψ) = atoms_pltl φ ∪ atoms_pltl ψ\"\n\nlemma atoms_finite [iff]:\n \"finite (atoms_pltl φ)\"\n apply (induct φ)\n (*goals:\n 1. ‹finite (atoms_pltl false⇩p)›\n 2. ‹⋀x. finite (atoms_pltl atom⇩p(x))›\n 3. ‹⋀φ1 φ2. ⟦finite (atoms_pltl φ1); finite (atoms_pltl φ2)⟧ ⟹ finite (atoms_pltl (φ1 implies⇩p φ2))›\n 4. ‹⋀φ. finite (atoms_pltl φ) ⟹ finite (atoms_pltl (X⇩p φ))›\n 5. ‹⋀φ1 φ2. ⟦finite (atoms_pltl φ1); finite (atoms_pltl φ2)⟧ ⟹ finite (atoms_pltl (φ1 U⇩p φ2))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*proven 5 subgoals*) .\n\nlemma atoms_pltl_sugar [simp]:\n \"atoms_pltl (not⇩p φ) = atoms_pltl φ\"\n \"atoms_pltl true⇩p = {}\"\n \"atoms_pltl (φ or⇩p ψ) = atoms_pltl φ ∪ atoms_pltl ψ\"\n \"atoms_pltl (φ and⇩p ψ) = atoms_pltl φ ∪ atoms_pltl ψ\"\n \"atoms_pltl (F⇩p φ) = atoms_pltl φ\"\n \"atoms_pltl (G⇩p φ) = atoms_pltl φ\"\n \n (*goals:\n 1. ‹atoms_pltl (not⇩p φ) = atoms_pltl φ›\n 2. ‹atoms_pltl true⇩p = {}›\n 3. ‹atoms_pltl (φ or⇩p ψ) = atoms_pltl φ ∪ atoms_pltl ψ›\n 4. ‹atoms_pltl (φ and⇩p ψ) = atoms_pltl φ ∪ atoms_pltl ψ›\n 5. ‹atoms_pltl (F⇩p φ) = atoms_pltl φ›\n 6. ‹atoms_pltl (G⇩p φ) = atoms_pltl φ›\n discuss goal 1*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p (?φ::?'a pltl) ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹(?φ::?'a pltl) or⇩p (?ψ::?'a pltl) ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹(?φ::?'a pltl) and⇩p (?ψ::?'a pltl) ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p (?φ::?'a pltl) ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p (?φ::?'a pltl) ≡ not⇩p F⇩p (not⇩p ?φ)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*))[1])\n (*discuss goal 5*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p ?φ ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹?φ or⇩p ?ψ ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹?φ and⇩p ?ψ ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p ?φ ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p ?φ ≡ not⇩p F⇩p (not⇩p ?φ)›*))[1])\n (*discuss goal 6*)\n apply ((auto simp: Not_ltlp_def (*‹not⇩p (?φ::?'a pltl) ≡ ?φ implies⇩p false⇩p›*) True_ltlp_def (*‹true⇩p ≡ not⇩p false⇩p›*) Or_ltlp_def (*‹(?φ::?'a pltl) or⇩p (?ψ::?'a pltl) ≡ not⇩p ?φ implies⇩p ?ψ›*) And_ltlp_def (*‹(?φ::?'a pltl) and⇩p (?ψ::?'a pltl) ≡ not⇩p (not⇩p ?φ or⇩p not⇩p ?ψ)›*) Eventually_ltlp_def (*‹F⇩p (?φ::?'a pltl) ≡ true⇩p U⇩p ?φ›*) Always_ltlp_def (*‹G⇩p (?φ::?'a pltl) ≡ not⇩p F⇩p (not⇩p ?φ)›*))[1])\n (*proven 6 subgoals*) .\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/LTL/LTL.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"43f7e16beae168f43354e662bf56c26ca00814faeb6d778ac08ee11042c4f72f\"\n}"}}},{"rowIdx":151,"cells":{"text":{"kind":"string","value":"theory Post_COMPOSE2\nimports\n \"Post_ISSUER\"\n \"Post_RECEIVER\"\n \"BD_Security_Compositional.Composing_Security\"\nbegin\n\nsubsection ‹Confidentiality for the (binary) issuer-receiver composition›\n\ntype_synonym ttrans = \"(state, act, out) trans\"\ntype_synonym value1 = Post_ISSUER.value type_synonym value2 = Post_RECEIVER.value\ntype_synonym obs1 = Post_Observation_Setup_ISSUER.obs\ntype_synonym obs2 = Post_Observation_Setup_RECEIVER.obs\n\n(* irrelevant for the security conditions: *)\ndatatype cval = PValC post\ntype_synonym cobs = \"obs1 × obs2\"\n\nlocale Post_COMPOSE2 =\n Iss: Post_ISSUER UIDs PID +\n Rcv: Post_RECEIVER UIDs2 PID AID1\nfor UIDs :: \"userID set\" and UIDs2 :: \"userID set\" and\n AID1 :: \"apiID\" and PID :: \"postID\"\n+ fixes AID2 :: \"apiID\"\nbegin\n\nabbreviation \"φ1 ≡ Iss.φ\" abbreviation \"f1 ≡ Iss.f\" abbreviation \"γ1 ≡ Iss.γ\" abbreviation \"g1 ≡ Iss.g\"\n abbreviation \"T1 ≡ Iss.T\" abbreviation \"B1 ≡ Iss.B\"\nabbreviation \"φ2 ≡ Rcv.φ\" abbreviation \"f2 ≡ Rcv.f\" abbreviation \"γ2 ≡ Rcv.γ\" abbreviation \"g2 ≡ Rcv.g\"\n abbreviation \"T2 ≡ Rcv.T\" abbreviation \"B2 ≡ Rcv.B\"\n\n(* Recall that we assume that the system prevents communication if error occurs: *)\nfun isCom1 :: \"ttrans ⇒ bool\" where\n \"isCom1 (Trans s (COMact ca1) ou1 s') = (ou1 ≠ outErr)\"\n|\"isCom1 _ = False\"\n\nfun isCom2 :: \"ttrans ⇒ bool\" where\n \"isCom2 (Trans s (COMact ca2) ou2 s') = (ou2 ≠ outErr)\"\n|\"isCom2 _ = False\"\n\nfun isComV1 :: \"value1 ⇒ bool\" where\n \"isComV1 (Iss.PValS aid1 txt1) = True\"\n|\"isComV1 _ = False\"\n\nfun isComV2 :: \"value2 ⇒ bool\" where\n \"isComV2 (Rcv.PValR txt2) = True\"\n(* |\"isComV2 _ = False\" *)\n\nfun syncV :: \"value1 ⇒ value2 ⇒ bool\" where\n \"syncV (Iss.PValS aid1 txt1) (Rcv.PValR txt2) = (txt1 = txt2)\"\n|\"syncV _ _ = False\"\n\n(* irrelevant for the security conditions: *)\nfun cmpV :: \"value1 ⇒ value2 ⇒ cval\" where\n \"cmpV (Iss.PValS aid1 txt1) (Rcv.PValR txt2) = PValC txt1\"\n|\"cmpV _ _ = undefined\"\n\nfun isComO1 :: \"obs1 ⇒ bool\" where\n \"isComO1 (COMact ca1, ou1) = (ou1 ≠ outErr)\"\n|\"isComO1 _ = False\"\n\nfun isComO2 :: \"obs2 ⇒ bool\" where\n \"isComO2 (COMact ca2, ou2) = (ou2 ≠ outErr)\"\n|\"isComO2 _ = False\"\n\nfun comSyncOA :: \"out ⇒ comActt ⇒ bool\" where\n \"comSyncOA (O_sendServerReq (aid2, reqInfo1)) (comReceiveClientReq aid1 reqInfo2) =\n (aid1 = AID1 ∧ aid2 = AID2 ∧ reqInfo1 = reqInfo2)\"\n|\"comSyncOA (O_connectClient (aid2, sp1)) (comConnectServer aid1 sp2) =\n (aid1 = AID1 ∧ aid2 = AID2 ∧ sp1 = sp2)\"\n|\"comSyncOA (O_sendPost (aid2, sp1, pid1, pst1, uid1, vis1)) (comReceivePost aid1 sp2 pid2 pst2 uid2 vis2) =\n (aid1 = AID1 ∧ aid2 = AID2 ∧ (pid1, pst1, uid1, vis1) = (pid2, pst2, uid2, vis2))\"\n|\"comSyncOA (O_sendCreateOFriend (aid2, sp1, uid1, uid1')) (comReceiveCreateOFriend aid1 sp2 uid2 uid2') =\n (aid1 = AID1 ∧ aid2 = AID2 ∧ (uid1, uid1') = (uid2, uid2'))\"\n|\"comSyncOA (O_sendDeleteOFriend (aid2, sp1, uid1, uid1')) (comReceiveDeleteOFriend aid1 sp2 uid2 uid2') =\n (aid1 = AID1 ∧ aid2 = AID2 ∧ (uid1, uid1') = (uid2, uid2'))\"\n|\"comSyncOA _ _ = False\"\n\nfun syncO :: \"obs1 ⇒ obs2 ⇒ bool\" where\n \"syncO (COMact ca1, ou1) (COMact ca2, ou2) =\n (ou1 ≠ outErr ∧ ou2 ≠ outErr ∧\n (comSyncOA ou1 ca2 ∨ comSyncOA ou2 ca1)\n )\"\n|\"syncO _ _ = False\"\n\nfun sync :: \"ttrans ⇒ ttrans ⇒ bool\" where\n\"sync (Trans s1 a1 ou1 s1') (Trans s2 a2 ou2 s2') = syncO (a1, ou1) (a2, ou2)\"\n\n(* irrelevant for the security conditions: *)\ndefinition cmpO :: \"obs1 ⇒ obs2 ⇒ cobs\" where\n\"cmpO o1 o2 ≡ (o1,o2)\"\n\n\n(* *)\n\nlemma isCom1_isComV1:\nassumes \"validTrans trn1\" and \"reach (srcOf trn1)\" and \"φ1 trn1\"\nshows \"isCom1 trn1 ⟷ isComV1 (f1 trn1)\"\nusing assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹φ1 trn1›*) apply (cases trn1)\n(*goal: ‹isCom1 trn1 = isComV1 (f1 trn1)›*)\nby (auto simp: Iss.φ_def2 (*‹φ1 (Trans ?s ?a ?ou ?s') = ((∃uid p pst. ?a = Uact (uPost uid p PID pst) ∧ ?ou = outOK) ∨ (∃uid p aid. ?a = COMact (comSendPost uid p aid PID) ∧ ?ou ≠ outErr))›*) split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))\n\nlemma isCom1_isComO1:\nassumes \"validTrans trn1\" and \"reach (srcOf trn1)\" and \"γ1 trn1\"\nshows \"isCom1 trn1 ⟷ isComO1 (g1 trn1)\"\nusing assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹γ1 trn1›*) apply (cases trn1 rule: isCom1.cases (*‹⟦⋀s ca1 ou1 s'. ?x = Trans s (COMact ca1) ou1 s' ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Sact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Cact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Dact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Uact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Ract vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Lact vd) vb vc ⟹ ?P⟧ ⟹ ?P›*))\n(*goals:\n 1. ‹⋀s ca1 ou1 s'. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1; trn1 = Trans s (COMact ca1) ou1 s'⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1)›\n 2. ‹⋀v vd vb vc. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1; trn1 = Trans v (Sact vd) vb vc⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1)›\n 3. ‹⋀v vd vb vc. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1; trn1 = Trans v (Cact vd) vb vc⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1)›\n 4. ‹⋀v vd vb vc. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1; trn1 = Trans v (Dact vd) vb vc⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1)›\n 5. ‹⋀v vd vb vc. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1; trn1 = Trans v (Uact vd) vb vc⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1)›\n 6. ‹⋀v vd vb vc. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1; trn1 = Trans v (Ract vd) vb vc⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1)›\n 7. ‹⋀v vd vb vc. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1; trn1 = Trans v (Lact vd) vb vc⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1)›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*discuss goal 4*)\n apply ((auto)[1])\n(*discuss goal 5*)\n apply ((auto)[1])\n(*discuss goal 6*)\n apply ((auto)[1])\n(*discuss goal 7*)\n apply ((auto)[1])\n(*proven 7 subgoals*) .\n\nlemma isCom2_isComV2:\nassumes \"validTrans trn2\" and \"reach (srcOf trn2)\" and \"φ2 trn2\"\nshows \"isCom2 trn2 ⟷ isComV2 (f2 trn2)\"\nusing assms (*‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹φ2 trn2›*) apply (cases trn2)\n(*goal: ‹isCom2 trn2 = isComV2 (f2 trn2)›*)\nby (auto simp: Rcv.φ_def2 (*‹φ2 (Trans ?s ?a ?ou ?s') = (∃uid p pst vs. ?a = COMact (comReceivePost AID1 p PID pst uid vs) ∧ ?ou = outOK)›*) split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))\n\nlemma isCom2_isComO2:\nassumes \"validTrans trn2\" and \"reach (srcOf trn2)\" and \"γ2 trn2\"\nshows \"isCom2 trn2 ⟷ isComO2 (g2 trn2)\"\nusing assms (*‹Iss.validTrans trn2› ‹Iss.reach (srcOf (trn2::(state, act, out) trans))› ‹γ2 trn2›*) apply (cases trn2 rule: isCom2.cases (*‹⟦⋀s ca2 ou2 s'. ?x = Trans s (COMact ca2) ou2 s' ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Sact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Cact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Dact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Uact vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Ract vd) vb vc ⟹ ?P; ⋀v vd vb vc. ?x = Trans v (Lact vd) vb vc ⟹ ?P⟧ ⟹ ?P›*))\n(*goals:\n 1. ‹⋀s ca2 ou2 s'. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2; trn2 = Trans s (COMact ca2) ou2 s'⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2)›\n 2. ‹⋀v vd vb vc. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2; trn2 = Trans v (Sact vd) vb vc⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2)›\n 3. ‹⋀v vd vb vc. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2; trn2 = Trans v (Cact vd) vb vc⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2)›\n 4. ‹⋀v vd vb vc. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2; trn2 = Trans v (Dact vd) vb vc⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2)›\n 5. ‹⋀v vd vb vc. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2; trn2 = Trans v (Uact vd) vb vc⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2)›\n 6. ‹⋀v vd vb vc. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2; trn2 = Trans v (Ract vd) vb vc⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2)›\n 7. ‹⋀v vd vb vc. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2; trn2 = Trans v (Lact vd) vb vc⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2)›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*discuss goal 4*)\n apply ((auto)[1])\n(*discuss goal 5*)\n apply ((auto)[1])\n(*discuss goal 6*)\n apply ((auto)[1])\n(*discuss goal 7*)\n apply ((auto)[1])\n(*proven 7 subgoals*) .\n\nlemma sync_syncV:\nassumes \"validTrans trn1\" and \"reach (srcOf trn1)\"\nand \"validTrans trn2\" and \"reach (srcOf trn2)\"\nand \"isCom1 trn1\" and \"isCom2 trn2\" and \"φ1 trn1\" and \"φ2 trn2\"\nand \"sync trn1 trn2\"\nshows \"syncV (f1 trn1) (f2 trn2)\"\nusing assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom1 (trn1::(state, act, out) trans)› ‹isCom2 trn2› ‹φ1 trn1› ‹φ2 (trn2::(state, act, out) trans)› ‹sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›*) apply (cases trn1)\n(*goal: ‹syncV (f1 (trn1::(state, act, out) trans)) (f2 (trn2::(state, act, out) trans))›*)\napply (cases trn2)\n(*goal: ‹⋀(x1::state) (x2::act) (x3::out) x4::state. ⟦Iss.validTrans (trn1::(state, act, out) trans); Iss.reach (srcOf trn1); Iss.validTrans (trn2::(state, act, out) trans); Iss.reach (srcOf trn2); isCom1 trn1; isCom2 trn2; φ1 trn1; φ2 trn2; sync trn1 trn2; trn1 = Trans x1 x2 x3 x4⟧ ⟹ syncV (f1 trn1) (f2 trn2)›*)\nby (auto simp: Iss.φ_def2 (*‹φ1 (Trans ?s ?a ?ou ?s') = ((∃uid p pst. ?a = Uact (uPost uid p PID pst) ∧ ?ou = outOK) ∨ (∃uid p aid. ?a = COMact (comSendPost uid p aid PID) ∧ ?ou ≠ outErr))›*) Rcv.φ_def2 (*‹φ2 (Trans ?s ?a ?ou ?s') = (∃uid p pst vs. ?a = COMact (comReceivePost AID1 p PID pst uid vs) ∧ ?ou = outOK)›*) split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))\n\nlemma sync_syncO:\nassumes \"validTrans trn1\" and \"reach (srcOf trn1)\"\nand \"validTrans trn2\" and \"reach (srcOf trn2)\"\nand \"isCom1 trn1\" and \"isCom2 trn2\" and \"γ1 trn1\" and \"γ2 trn2\"\nand \"sync trn1 trn2\"\nshows \"syncO (g1 trn1) (g2 trn2)\"\nproof (cases trn1)\n (*goal: ‹⋀x1 x2 x3 x4. trn1 = Trans x1 x2 x3 x4 ⟹ syncO (g1 trn1) (g2 trn2)›*)\n case (Trans s1 a1 ou1 s1') (*‹trn1 = Trans s1 a1 ou1 s1'›*)\n note trn1 = Trans (*‹trn1 = Trans s1 a1 ou1 s1'›*)\n show \"?thesis\"\n (*goal: ‹syncO (g1 trn1) (g2 trn2)›*)\n proof (cases trn2)\n (*goal: ‹⋀x1 x2 x3 x4. trn2 = Trans x1 x2 x3 x4 ⟹ syncO (g1 trn1) (g2 trn2)›*)\n case (Trans s2 a2 ou2 s2') (*‹trn2 = Trans s2 a2 ou2 s2'›*)\n note trn2 = Trans (*‹trn2 = Trans s2 a2 ou2 s2'›*)\n show \"?thesis\"\n (*goal: ‹syncO (g1 trn1) (g2 trn2)›*)\n proof (cases a1)\n (*goals:\n 1. ‹⋀x1. a1 = Sact x1 ⟹ syncO (g1 trn1) (g2 trn2)›\n 2. ‹⋀x2. a1 = Cact x2 ⟹ syncO (g1 trn1) (g2 trn2)›\n 3. ‹⋀x3. a1 = Dact x3 ⟹ syncO (g1 trn1) (g2 trn2)›\n 4. ‹⋀x4. a1 = Uact x4 ⟹ syncO (g1 trn1) (g2 trn2)›\n 5. ‹⋀x5. a1 = Ract x5 ⟹ syncO (g1 trn1) (g2 trn2)›\n 6. ‹⋀x6. a1 = Lact x6 ⟹ syncO (g1 trn1) (g2 trn2)›\n 7. ‹⋀x7. a1 = COMact x7 ⟹ syncO (g1 trn1) (g2 trn2)›*)\n case (COMact ca1) (*‹a1 = COMact ca1›*)\n note a1 = COMact (*‹a1 = COMact ca1›*)\n show \"?thesis\"\n (*goal: ‹syncO (g1 trn1) (g2 trn2)›*)\n proof (cases a2)\n (*goals:\n 1. ‹⋀x1::sActt. (a2::act) = Sact x1 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 2. ‹⋀x2::cActt. (a2::act) = Cact x2 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 3. ‹⋀x3::dActt. (a2::act) = Dact x3 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 4. ‹⋀x4::uActt. (a2::act) = Uact x4 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 5. ‹⋀x5::rActt. (a2::act) = Ract x5 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 6. ‹⋀x6::lActt. (a2::act) = Lact x6 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 7. ‹⋀x7::comActt. (a2::act) = COMact x7 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›*)\n case (COMact ca2) (*‹a2 = COMact ca2›*)\n note a2 = COMact (*‹(a2::act) = COMact (ca2::comActt)›*)\n show \"?thesis\"\n (*goal: ‹syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›*)\n using assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf (trn2::(state, act, out) trans))› ‹isCom1 trn1› ‹isCom2 trn2› ‹γ1 trn1› ‹γ2 trn2› ‹sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›*) unfolding trn1 trn2 a1 a2\n (*goal: ‹syncO (g1 (Trans (s1::state) (COMact (ca1::comActt)) (ou1::out) (s1'::state))) (g2 (Trans (s2::state) (COMact (ca2::comActt)) (ou2::out) (s2'::state)))›*)\n apply (cases ca1)\n (*goals:\n 1. ‹⋀x11 x12 x13 x14. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 2. ‹⋀x21 x22. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comReceiveClientReq x21 x22⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 3. ‹⋀x31 x32 x33 x34. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comConnectClient x31 x32 x33 x34⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 4. ‹⋀x41 x42. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comConnectServer x41 x42⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 5. ‹⋀x51 x52 x53 x54 x55 x56. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 6. ‹⋀x61 x62 x63 x64. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendPost x61 x62 x63 x64⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 7. ‹⋀x71 x72 x73 x74. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 8. ‹⋀x81 x82 x83 x84. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 9. ‹⋀x91 x92 x93 x94. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 10. ‹⋀x101 x102 x103 x104. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n discuss goal 1*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x11 x12 x13 x14 x11a x12a x13a x14a. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendServerReq x11a x12a x13a x14a⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 2. ‹⋀x11 x12 x13 x14 x21 x22. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceiveClientReq x21 x22⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 3. ‹⋀x11 x12 x13 x14 x31 x32 x33 x34. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 4. ‹⋀x11 x12 x13 x14 x41 x42. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comConnectServer x41 x42⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 5. ‹⋀x11 x12 x13 x14 x51 x52 x53 x54 x55 x56. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 6. ‹⋀x11 x12 x13 x14 x61 x62 x63 x64. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendPost x61 x62 x63 x64⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 7. ‹⋀x11 x12 x13 x14 x71 x72 x73 x74. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 8. ‹⋀x11 x12 x13 x14 x81 x82 x83 x84. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 9. ‹⋀x11 x12 x13 x14 x91 x92 x93 x94. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n 10. ‹⋀x11 x12 x13 x14 x101 x102 x103 x104. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›\n discuss goal 1*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 3*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 4*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 5*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 6*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 7*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 8*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 9*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 10*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*proven 10 subgoals*)\n (*discuss goal 2*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 3*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*top goal: ‹⋀x31 x32 x33 x34. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comConnectClient x31 x32 x33 x34⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))› and 7 goals remain*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x31 x32 x33 x34 a b x11 x12 x13 x14. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 2. ‹⋀x31 x32 x33 x34 a b x21 x22. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comReceiveClientReq x21 x22⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 3. ‹⋀x31 x32 x33 x34 a b x31a x32a x33a x34a. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comConnectClient x31a x32a x33a x34a⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 4. ‹⋀x31 x32 x33 x34 a b x41 x42. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comConnectServer x41 x42⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 5. ‹⋀x31 x32 x33 x34 a b x51 x52 x53 x54 x55 x56. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 6. ‹⋀x31 x32 x33 x34 a b x61 x62 x63 x64. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comSendPost x61 x62 x63 x64⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 7. ‹⋀x31 x32 x33 x34 a b x71 x72 x73 x74. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 8. ‹⋀x31 x32 x33 x34 a b x81 x82 x83 x84. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 9. ‹⋀x31 x32 x33 x34 a b x91 x92 x93 x94. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n 10. ‹⋀x31 x32 x33 x34 a b x101 x102 x103 x104. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_connectClient (a, b)) ca2; ca1 = comConnectClient x31 x32 x33 x34; e_connectClient s1 x31 x32 x33 x34; connectClient s1 x31 x32 x33 x34 = ((a, b), s1'); ou1 = O_connectClient (a, b); ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ comSyncOA (O_connectClient (a, b)) (Rcv.comPurge ca2)›\n discuss goal 1*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 3*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 4*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 5*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 6*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 7*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 8*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 9*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 10*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*proven 10 subgoals*)\n (*discuss goal 4*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 5*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 6*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*goals:\n 1. ‹⋀x61 x62 x63 x64 a aa ac ad b. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b)⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 2. ‹⋀x61 x62 x63 x64 a aa ab ac ad b. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ PID⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n discuss goal 1*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x61 x62 x63 x64 a aa ac ad b x11 x12 x13 x14. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 2. ‹⋀x61 x62 x63 x64 a aa ac ad b x21 x22. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comReceiveClientReq x21 x22⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 3. ‹⋀x61 x62 x63 x64 a aa ac ad b x31 x32 x33 x34. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 4. ‹⋀x61 x62 x63 x64 a aa ac ad b x41 x42. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comConnectServer x41 x42⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 5. ‹⋀x61 x62 x63 x64 a aa ac ad b x51 x52 x53 x54 x55 x56. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 6. ‹⋀x61 x62 x63 x64 a aa ac ad b x61a x62a x63a x64a. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comSendPost x61a x62a x63a x64a⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 7. ‹⋀x61 x62 x63 x64 a aa ac ad b x71 x72 x73 x74. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 8. ‹⋀x61 x62 x63 x64 a aa ac ad b x81 x82 x83 x84. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 9. ‹⋀x61 x62 x63 x64 a aa ac ad b x91 x92 x93 x94. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n 10. ‹⋀x61 x62 x63 x64 a aa ac ad b x101 x102 x103 x104. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, PID, ac, ad, b)) ca2; ca1 = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, PID, ac, ad, b), s1'); ou1 = O_sendPost (a, aa, PID, ac, ad, b); ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ comSyncOA (O_sendPost (a, aa, PID, emptyPost, ad, b)) (Rcv.comPurge ca2)›\n discuss goal 1*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 3*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 4*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 5*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 6*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 7*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 8*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 9*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 10*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*proven 10 subgoals*)\n (*discuss goal 2*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x11::userID) (x12::password) (x13::apiID) x14::requestInfo. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 2. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x21::apiID) x22::requestInfo. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comReceiveClientReq x21 x22⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 3. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x31::userID) (x32::password) (x33::apiID) x34::password. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 4. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x41::apiID) x42::password. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comConnectServer x41 x42⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 5. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x51::apiID) (x52::password) (x53::postID) (x54::post) (x55::userID) x56::vis. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 6. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x61a::userID) (x62a::password) (x63a::apiID) x64a::postID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comSendPost x61a x62a x63a x64a⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 7. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x71::apiID) (x72::password) (x73::userID) x74::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 8. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x81::userID) (x82::password) (x83::apiID) x84::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 9. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x91::apiID) (x92::password) (x93::userID) x94::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n 10. ‹⋀(x61::userID) (x62::password) (x63::apiID) (x64::postID) (a::apiID) (aa::password) (ab::postID) (ac::post) (ad::userID) (b::vis) (x101::userID) (x102::password) (x103::apiID) x104::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) ca2; (ca1::comActt) = comSendPost x61 x62 x63 x64; e_sendPost s1 x61 x62 x63 x64; sendPost s1 x61 x62 x63 x64 = ((a, aa, ab, ac, ad, b), s1'::state); (ou1::out) = O_sendPost (a, aa, ab, ac, ad, b); ab ≠ (PID::postID); ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ comSyncOA (O_sendPost (a, aa, ab, ac, ad, b)) (Rcv.comPurge ca2)›\n discuss goal 1*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 3*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 4*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 5*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 6*)\n apply ((auto split: prod.splits (*‹(?P::?'c::type ⇒ bool) (case ?prod::?'a::type × ?'b::type of (x::?'a::type, xa::?'b::type) ⇒ (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) x xa) = (∀(x1::?'a::type) x2::?'b::type. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c::type ⇒ bool) (case ?prod::?'a::type × ?'b::type of (x::?'a::type, xa::?'b::type) ⇒ (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) x xa) = (∄(x1::?'a::type) x2::?'b::type. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 7*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 8*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 9*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 10*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*proven 10 subgoals*)\n (*proven 2 subgoals*)\n (*discuss goal 7*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 8*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*top goal: ‹⋀x81 x82 x83 x84. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))› and 2 goals remain*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x81 x82 x83 x84 a aa ab b x11 x12 x13 x14. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 2. ‹⋀x81 x82 x83 x84 a aa ab b x21 x22. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comReceiveClientReq x21 x22⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 3. ‹⋀x81 x82 x83 x84 a aa ab b x31 x32 x33 x34. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 4. ‹⋀x81 x82 x83 x84 a aa ab b x41 x42. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comConnectServer x41 x42⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 5. ‹⋀x81 x82 x83 x84 a aa ab b x51 x52 x53 x54 x55 x56. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 6. ‹⋀x81 x82 x83 x84 a aa ab b x61 x62 x63 x64. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comSendPost x61 x62 x63 x64⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 7. ‹⋀x81 x82 x83 x84 a aa ab b x71 x72 x73 x74. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 8. ‹⋀x81 x82 x83 x84 a aa ab b x81a x82a x83a x84a. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comSendCreateOFriend x81a x82a x83a x84a⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 9. ‹⋀x81 x82 x83 x84 a aa ab b x91 x92 x93 x94. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 10. ‹⋀x81 x82 x83 x84 a aa ab b x101 x102 x103 x104. ⟦Iss.reach s1; comStep s2 ca2 = (ou2, s2'); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) ca2; ca1 = comSendCreateOFriend x81 x82 x83 x84; e_sendCreateOFriend s1 x81 x82 x83 x84; sendCreateOFriend s1 x81 x82 x83 x84 = ((a, aa, ab, b), s1'); ou1 = O_sendCreateOFriend (a, aa, ab, b); ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ comSyncOA (O_sendCreateOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n discuss goal 1*)\n apply ((auto split: prod.splits (*‹(?P::?'c::type ⇒ bool) (case ?prod::?'a::type × ?'b::type of (x::?'a::type, xa::?'b::type) ⇒ (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) x xa) = (∀(x1::?'a::type) x2::?'b::type. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c::type ⇒ bool) (case ?prod::?'a::type × ?'b::type of (x::?'a::type, xa::?'b::type) ⇒ (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) x xa) = (∄(x1::?'a::type) x2::?'b::type. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 3*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 4*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 5*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 6*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 7*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 8*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 9*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 10*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*proven 10 subgoals*)\n (*discuss goal 9*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 10*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*goal: ‹⋀x101 x102 x103 x104. ⟦Iss.validTrans (Trans s1 (COMact ca1) ou1 s1'); Iss.reach (srcOf (Trans s1 (COMact ca1) ou1 s1')); Iss.validTrans (Trans s2 (COMact ca2) ou2 s2'); Iss.reach (srcOf (Trans s2 (COMact ca2) ou2 s2')); isCom1 (Trans s1 (COMact ca1) ou1 s1'); isCom2 (Trans s2 (COMact ca2) ou2 s2'); γ1 (Trans s1 (COMact ca1) ou1 s1'); γ2 (Trans s2 (COMact ca2) ou2 s2'); sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); ca1 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ syncO (g1 (Trans s1 (COMact ca1) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2'))›*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x11::userID) (x12::password) (x13::apiID) x14::requestInfo. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 2. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x21::apiID) x22::requestInfo. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comReceiveClientReq x21 x22⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 3. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x31::userID) (x32::password) (x33::apiID) x34::password. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 4. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x41::apiID) x42::password. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comConnectServer x41 x42⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 5. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x51::apiID) (x52::password) (x53::postID) (x54::post) (x55::userID) x56::vis. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 6. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x61::userID) (x62::password) (x63::apiID) x64::postID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comSendPost x61 x62 x63 x64⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 7. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x71::apiID) (x72::password) (x73::userID) x74::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 8. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x81::userID) (x82::password) (x83::apiID) x84::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 9. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x91::apiID) (x92::password) (x93::userID) x94::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n 10. ‹⋀(x101::userID) (x102::password) (x103::apiID) (x104::userID) (a::apiID) (aa::password) (ab::userID) (b::userID) (x101a::userID) (x102a::password) (x103a::apiID) x104a::userID. ⟦Iss.reach (s1::state); comStep (s2::state) (ca2::comActt) = (ou2::out, s2'::state); Iss.reach s2; ou2 ≠ outErr; comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) ca2; (ca1::comActt) = comSendDeleteOFriend x101 x102 x103 x104; e_sendDeleteOFriend s1 x101 x102 x103 x104; sendDeleteOFriend s1 x101 x102 x103 x104 = ((a, aa, ab, b), s1'::state); (ou1::out) = O_sendDeleteOFriend (a, aa, ab, b); ca2 = comSendDeleteOFriend x101a x102a x103a x104a⟧ ⟹ comSyncOA (O_sendDeleteOFriend (a, aa, ab, b)) (Rcv.comPurge ca2)›\n discuss goal 1*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 3*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 4*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 5*)\n apply ((auto split: prod.splits (*‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∀(x1::?'a) x2::?'b. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c ⇒ bool) (case ?prod::?'a × ?'b of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = (∄(x1::?'a) x2::?'b. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 6*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 7*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 8*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 9*)\n apply ((auto split: prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*discuss goal 10*)\n apply ((auto split: prod.splits (*‹(?P::?'c::type ⇒ bool) (case ?prod::?'a::type × ?'b::type of (x::?'a::type, xa::?'b::type) ⇒ (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) x xa) = (∀(x1::?'a::type) x2::?'b::type. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹(?P::?'c::type ⇒ bool) (case ?prod::?'a::type × ?'b::type of (x::?'a::type, xa::?'b::type) ⇒ (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) x xa) = (∄(x1::?'a::type) x2::?'b::type. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))[1])\n (*proven 10 subgoals*)\n (*proven 10 subgoals*) .\n qed (insert assms (*‹Iss.validTrans (trn1::(state, act, out) trans)› ‹Iss.reach (srcOf (trn1::(state, act, out) trans))› ‹Iss.validTrans (trn2::(state, act, out) trans)› ‹Iss.reach (srcOf (trn2::(state, act, out) trans))› ‹isCom1 (trn1::(state, act, out) trans)› ‹isCom2 (trn2::(state, act, out) trans)› ‹γ1 (trn1::(state, act, out) trans)› ‹γ2 (trn2::(state, act, out) trans)› ‹sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›*), unfold trn1 (*‹trn1 = Trans s1 a1 ou1 s1'›*) trn2 (*‹trn2 = Trans s2 a2 ou2 s2'›*), auto)\n (*solves the remaining goals:\n 1. ‹⋀x1. a2 = Sact x1 ⟹ syncO (g1 trn1) (g2 trn2)›\n 2. ‹⋀x2. a2 = Cact x2 ⟹ syncO (g1 trn1) (g2 trn2)›\n 3. ‹⋀x3. a2 = Dact x3 ⟹ syncO (g1 trn1) (g2 trn2)›\n 4. ‹⋀x4. a2 = Uact x4 ⟹ syncO (g1 trn1) (g2 trn2)›\n 5. ‹⋀x5. a2 = Ract x5 ⟹ syncO (g1 trn1) (g2 trn2)›\n 6. ‹⋀x6. a2 = Lact x6 ⟹ syncO (g1 trn1) (g2 trn2)›*)\n qed (insert assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom1 trn1› ‹isCom2 trn2› ‹γ1 trn1› ‹γ2 trn2› ‹sync trn1 trn2›*), unfold trn1 (*‹trn1 = Trans s1 a1 ou1 s1'›*) trn2 (*‹trn2 = Trans s2 a2 ou2 s2'›*), auto)\n (*solves the remaining goals:\n 1. ‹⋀x1::sActt. (a1::act) = Sact x1 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 2. ‹⋀x2::cActt. (a1::act) = Cact x2 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 3. ‹⋀x3::dActt. (a1::act) = Dact x3 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 4. ‹⋀x4::uActt. (a1::act) = Uact x4 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 5. ‹⋀x5::rActt. (a1::act) = Ract x5 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›\n 6. ‹⋀x6::lActt. (a1::act) = Lact x6 ⟹ syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))›*)\n qed\nqed\n\nlemma sync_φ1_φ2:\nassumes v1: \"validTrans trn1\" and r1: \"reach (srcOf trn1)\"\nand v2: \"validTrans trn2\" and s2: \"reach (srcOf trn2)\"\nand c1: \"isCom1 trn1\" and c2: \"isCom2 trn2\"\nand sn: \"sync trn1 trn2\"\nshows \"φ1 trn1 ⟷ φ2 trn2\" (is \"?A ⟷ ?B\")\nproof (cases trn1)\n (*goal: ‹⋀x1 x2 x3 x4. trn1 = Trans x1 x2 x3 x4 ⟹ φ1 trn1 = φ2 trn2›*)\n case (Trans s1 a1 ou1 s1') (*‹trn1 = Trans s1 a1 ou1 s1'›*)\n note trn1 = Trans (*‹trn1 = Trans s1 a1 ou1 s1'›*)\n hence step1: \"step s1 a1 = (ou1,s1')\"\n using v1 (*‹Iss.validTrans (trn1::(state, act, out) trans)›*) by auto\n show \"?thesis\"\n (*goal: ‹φ1 trn1 = φ2 trn2›*)\n proof (cases trn2)\n (*goal: ‹⋀x1 x2 x3 x4. trn2 = Trans x1 x2 x3 x4 ⟹ φ1 trn1 = φ2 trn2›*)\n case (Trans s2 a2 ou2 s2') (*‹trn2 = Trans s2 a2 ou2 s2'›*)\n note trn2 = Trans (*‹(trn2::(state, act, out) trans) = Trans (s2::state) (a2::act) (ou2::out) (s2'::state)›*)\n hence step2: \"step s2 a2 = (ou2,s2')\"\n using v2 (*‹Iss.validTrans (trn2::(state, act, out) trans)›*) by auto\n show \"?thesis\"\n (*goal: ‹φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›*)\n proof (cases a1)\n (*goals:\n 1. ‹⋀x1. a1 = Sact x1 ⟹ φ1 trn1 = φ2 trn2›\n 2. ‹⋀x2. a1 = Cact x2 ⟹ φ1 trn1 = φ2 trn2›\n 3. ‹⋀x3. a1 = Dact x3 ⟹ φ1 trn1 = φ2 trn2›\n 4. ‹⋀x4. a1 = Uact x4 ⟹ φ1 trn1 = φ2 trn2›\n 5. ‹⋀x5. a1 = Ract x5 ⟹ φ1 trn1 = φ2 trn2›\n 6. ‹⋀x6. a1 = Lact x6 ⟹ φ1 trn1 = φ2 trn2›\n 7. ‹⋀x7. a1 = COMact x7 ⟹ φ1 trn1 = φ2 trn2›*)\n case (COMact ca1) (*‹a1 = COMact ca1›*)\n note a1 = COMact (*‹a1 = COMact ca1›*)\n show \"?thesis\"\n (*goal: ‹φ1 trn1 = φ2 trn2›*)\n proof (cases a2)\n (*goals:\n 1. ‹⋀x1. a2 = Sact x1 ⟹ φ1 trn1 = φ2 trn2›\n 2. ‹⋀x2. a2 = Cact x2 ⟹ φ1 trn1 = φ2 trn2›\n 3. ‹⋀x3. a2 = Dact x3 ⟹ φ1 trn1 = φ2 trn2›\n 4. ‹⋀x4. a2 = Uact x4 ⟹ φ1 trn1 = φ2 trn2›\n 5. ‹⋀x5. a2 = Ract x5 ⟹ φ1 trn1 = φ2 trn2›\n 6. ‹⋀x6. a2 = Lact x6 ⟹ φ1 trn1 = φ2 trn2›\n 7. ‹⋀x7. a2 = COMact x7 ⟹ φ1 trn1 = φ2 trn2›*)\n case (COMact ca2) (*‹a2 = COMact ca2›*)\n note a2 = COMact (*‹a2 = COMact ca2›*)\n have \"?A ⟷ (∃aid1. ca1 =\n (comSendPost (admin s1) (pass s1 (admin s1)) aid1\n PID) ∧\n ou1 =\n O_sendPost\n (aid1, clientPass s1 aid1, PID, post s1 PID,\n owner s1 PID, vis s1 PID))\"\n using c1 (*‹isCom1 trn1›*) unfolding trn1 \"Iss.φ_def3\"[OF step1]\n (*goal: ‹((∃pst. a1 = Uact (uPost (owner s1 PID) (pass s1 (owner s1 PID)) PID pst) ∧ ou1 = outOK) ∨ (∃aid. a1 = COMact (comSendPost (admin s1) (pass s1 (admin s1)) aid PID) ∧ ou1 = O_sendPost (aid, clientPass s1 aid, PID, post s1 PID, owner s1 PID, vis s1 PID))) = (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID))›*)\n unfolding a1\n (*goal: ‹((∃pst. COMact ca1 = Uact (uPost (owner s1 PID) (pass s1 (owner s1 PID)) PID pst) ∧ ou1 = outOK) ∨ (∃aid. COMact ca1 = COMact (comSendPost (admin s1) (pass s1 (admin s1)) aid PID) ∧ ou1 = O_sendPost (aid, clientPass s1 aid, PID, post s1 PID, owner s1 PID, vis s1 PID))) = (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID))›*)\n by auto\n also (*calculation: ‹φ1 trn1 = (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID))›*) have \"… ⟷ (∃uid2 pst2 vs2.\n ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)\"\n using sn (*‹sync trn1 trn2›*) step1 (*‹step s1 a1 = (ou1, s1')›*) step2 (*‹step s2 a2 = (ou2, s2')›*) unfolding trn1 trn2 a1 a2\n (*goal: ‹(∃aid1::apiID. (ca1::comActt) = comSendPost (admin (s1::state)) (pass s1 (admin s1)) aid1 (PID::postID) ∧ (ou1::out) = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃(uid2::userID) (pst2::post) vs2::vis. (ca2::comActt) = comReceivePost (AID1::apiID) (serverPass (s2::state) AID1) PID pst2 uid2 vs2 ∧ (ou2::out) = outOK)›*)\n apply (cases ca1)\n (*goals:\n 1. ‹⋀x11 x12 x13 x14. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 2. ‹⋀x21 x22. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comReceiveClientReq x21 x22⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 3. ‹⋀x31 x32 x33 x34. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comConnectClient x31 x32 x33 x34⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 4. ‹⋀x41 x42. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comConnectServer x41 x42⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 5. ‹⋀x51 x52 x53 x54 x55 x56. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 6. ‹⋀x61 x62 x63 x64. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendPost x61 x62 x63 x64⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 7. ‹⋀x71 x72 x73 x74. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 8. ‹⋀x81 x82 x83 x84. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 9. ‹⋀x91 x92 x93 x94. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 10. ‹⋀x101 x102 x103 x104. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n discuss goal 1*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x11 x12 x13 x14 x11a x12a x13a x14a. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendServerReq x11a x12a x13a x14a⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 2. ‹⋀x11 x12 x13 x14 x21 x22. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceiveClientReq x21 x22⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 3. ‹⋀x11 x12 x13 x14 x31 x32 x33 x34. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 4. ‹⋀x11 x12 x13 x14 x41 x42. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comConnectServer x41 x42⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 5. ‹⋀x11 x12 x13 x14 x51 x52 x53 x54 x55 x56. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 6. ‹⋀x11 x12 x13 x14 x61 x62 x63 x64. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendPost x61 x62 x63 x64⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 7. ‹⋀x11 x12 x13 x14 x71 x72 x73 x74. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 8. ‹⋀x11 x12 x13 x14 x81 x82 x83 x84. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 9. ‹⋀x11 x12 x13 x14 x91 x92 x93 x94. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n 10. ‹⋀x11 x12 x13 x14 x101 x102 x103 x104. ⟦sync (Trans s1 (COMact ca1) ou1 s1') (Trans s2 (COMact ca2) ou2 s2'); step s1 (COMact ca1) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca1 = comSendServerReq x11 x12 x13 x14; ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ (∃aid1. ca1 = comSendPost (admin s1) (pass s1 (admin s1)) aid1 PID ∧ ou1 = O_sendPost (aid1, clientPass s1 aid1, PID, post s1 PID, owner s1 PID, vis s1 PID)) = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›\n discuss goal 1*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 2*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 3*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 4*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 5*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 6*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 7*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 8*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 9*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 10*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*proven 10 subgoals*)\n (*discuss goal 2*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 3*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 4*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 5*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 6*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*goals:\n 1. ‹⋀x63. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 2. ‹⋀x63. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈⟧ ⟹ ou2 = outOK›\n discuss goal 1*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x63 x11 x12 x13 x14. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 2. ‹⋀x63 x21 x22. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceiveClientReq x21 x22⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 3. ‹⋀x63 x31 x32 x33 x34. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 4. ‹⋀x63 x41 x42. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comConnectServer x41 x42⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 5. ‹⋀x63 x51 x52 x53 x54 x55 x56. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 6. ‹⋀x63 x61 x62 x63a x64. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendPost x61 x62 x63a x64⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 7. ‹⋀x63 x71 x72 x73 x74. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 8. ‹⋀x63 x81 x82 x83 x84. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 9. ‹⋀x63 x91 x92 x93 x94. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n 10. ‹⋀x63 x101 x102 x103 x104. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ ∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2›\n discuss goal 1*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 2*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 3*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 4*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 5*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 6*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 7*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 8*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 9*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 10*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*proven 10 subgoals*)\n (*discuss goal 2*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x63 x11 x12 x13 x14. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ ou2 = outOK›\n 2. ‹⋀x63 x21 x22. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceiveClientReq x21 x22⟧ ⟹ ou2 = outOK›\n 3. ‹⋀x63 x31 x32 x33 x34. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ ou2 = outOK›\n 4. ‹⋀x63 x41 x42. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comConnectServer x41 x42⟧ ⟹ ou2 = outOK›\n 5. ‹⋀x63 x51 x52 x53 x54 x55 x56. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ ou2 = outOK›\n 6. ‹⋀x63 x61 x62 x63a x64. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendPost x61 x62 x63a x64⟧ ⟹ ou2 = outOK›\n 7. ‹⋀x63 x71 x72 x73 x74. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ ou2 = outOK›\n 8. ‹⋀x63 x81 x82 x83 x84. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ ou2 = outOK›\n 9. ‹⋀x63 x91 x92 x93 x94. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ ou2 = outOK›\n 10. ‹⋀x63 x101 x102 x103 x104. ⟦comStep s2 ca2 = (ou2, s2'); ca1 = comSendPost (admin s1) (pass s1 (admin s1)) x63 PID; admin s1 ∈∈ userIDs s1; ou2 ≠ outErr; comSyncOA (O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID)) ca2; PID ∈∈ postIDs s1; x63 ∈∈ clientApiIDs s1; ou1 = O_sendPost (x63, clientPass s1 x63, PID, post s1 PID, owner s1 PID, vis s1 PID); s1' = s1⦇sharedWith := (sharedWith s1)(PID := insert2 x63 True (sharedWith s1 PID))⦈; ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ ou2 = outOK›\n discuss goal 1*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 2*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 3*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 4*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 5*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 6*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 7*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 8*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 9*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 10*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*proven 10 subgoals*)\n (*proven 2 subgoals*)\n (*discuss goal 7*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 8*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 9*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 10*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*proven 10 subgoals*) .\n also (*calculation: ‹φ1 trn1 = (∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK)›*) have \"… ⟷ ?B\"\n using c2 (*‹isCom2 trn2›*) unfolding trn2 \"Rcv.φ_def3\"[OF step2]\n (*goal: ‹(∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK) = (∃uid pst vs. a2 = COMact (comReceivePost AID1 (serverPass s2 AID1) PID pst uid vs) ∧ ou2 = outOK)›*)\n unfolding a2\n (*goal: ‹(∃uid2 pst2 vs2. ca2 = comReceivePost AID1 (serverPass s2 AID1) PID pst2 uid2 vs2 ∧ ou2 = outOK) = (∃uid pst vs. COMact ca2 = COMact (comReceivePost AID1 (serverPass s2 AID1) PID pst uid vs) ∧ ou2 = outOK)›*)\n by auto\n finally (*calculation: ‹φ1 trn1 = φ2 trn2›*) show \"?thesis\"\n (*goal: ‹φ1 trn1 = φ2 trn2›*) .\n qed (insert assms (*‹Iss.validTrans (trn1::(state, act, out) trans)› ‹Iss.reach (srcOf (trn1::(state, act, out) trans))› ‹Iss.validTrans (trn2::(state, act, out) trans)› ‹Iss.reach (srcOf (trn2::(state, act, out) trans))› ‹isCom1 (trn1::(state, act, out) trans)› ‹isCom2 (trn2::(state, act, out) trans)› ‹sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›*), unfold trn1 (*‹(trn1::(state, act, out) trans) = Trans (s1::state) (a1::act) (ou1::out) (s1'::state)›*) trn2 (*‹(trn2::(state, act, out) trans) = Trans (s2::state) (a2::act) (ou2::out) (s2'::state)›*), auto)\n (*solves the remaining goals:\n 1. ‹⋀x1::sActt. (a2::act) = Sact x1 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 2. ‹⋀x2::cActt. (a2::act) = Cact x2 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 3. ‹⋀x3::dActt. (a2::act) = Dact x3 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 4. ‹⋀x4::uActt. (a2::act) = Uact x4 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 5. ‹⋀x5::rActt. (a2::act) = Ract x5 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 6. ‹⋀x6::lActt. (a2::act) = Lact x6 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›*)\n qed (insert assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom1 trn1› ‹isCom2 trn2› ‹sync trn1 trn2›*), unfold trn1 (*‹trn1 = Trans s1 a1 ou1 s1'›*) trn2 (*‹trn2 = Trans s2 a2 ou2 s2'›*), auto)\n (*solves the remaining goals:\n 1. ‹⋀x1::sActt. (a1::act) = Sact x1 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 2. ‹⋀x2::cActt. (a1::act) = Cact x2 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 3. ‹⋀x3::dActt. (a1::act) = Dact x3 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 4. ‹⋀x4::uActt. (a1::act) = Uact x4 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 5. ‹⋀x5::rActt. (a1::act) = Ract x5 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›\n 6. ‹⋀x6::lActt. (a1::act) = Lact x6 ⟹ φ1 (trn1::(state, act, out) trans) = φ2 (trn2::(state, act, out) trans)›*)\n qed\nqed\n\nlemma textPost_textPost_cong[intro]:\nassumes \"textPost pst1 = textPost pst2\"\nand \"setTextPost pst1 emptyText = setTextPost pst2 emptyText\"\nshows \"pst1 = pst2\"\nusing assms (*‹textPost (pst1::post) = textPost (pst2::post)› ‹setTextPost pst1 emptyText = setTextPost pst2 emptyText›*) apply (cases pst1)\n(*goal: ‹pst1 = pst2›*)\napply (cases pst2)\n(*goal: ‹⋀x1 x2 x3. ⟦textPost pst1 = textPost pst2; setTextPost pst1 emptyText = setTextPost pst2 emptyText; pst1 = Pst x1 x2 x3⟧ ⟹ pst1 = pst2›*)\nby auto\n\nlemma sync_φ_γ:\nassumes \"validTrans trn1\" and \"reach (srcOf trn1)\"\nand \"validTrans trn2\" and \"reach (srcOf trn2)\"\nand \"isCom1 trn1\" and \"isCom2 trn2\"\nand \"γ1 trn1\" and \"γ2 trn2\"\nand so: \"syncO (g1 trn1) (g2 trn2)\"\nand \"φ1 trn1 ⟹ φ2 trn2 ⟹ syncV (f1 trn1) (f2 trn2)\"\nshows \"sync trn1 trn2\"\napply (cases trn1)\n(*goal: ‹sync trn1 trn2›*)\nproof (cases trn2)\n (*goal: ‹⋀x1 x2 x3 x4 x1a x2a x3a x4a. ⟦trn1 = Trans x1 x2 x3 x4; trn2 = Trans x1a x2a x3a x4a⟧ ⟹ sync trn1 trn2›*)\n fix s1 and a1 and ou1 and s1' and s2 and a2 and ou2 and s2'\n assume trn1: \"trn1 = Trans s1 a1 ou1 s1'\" and trn2: \"trn2 = Trans s2 a2 ou2 s2'\" (*‹(trn1::(state, act, out) trans) = Trans (s1::state) (a1::act) (ou1::out) (s1'::state)› ‹(trn2::(state, act, out) trans) = Trans (s2::state) (a2::act) (ou2::out) (s2'::state)›*)\n hence step1: \"step s1 a1 = (ou1,s1')\" and step2: \"step s2 a2 = (ou2,s2')\"\n using assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom1 (trn1::(state, act, out) trans)› ‹isCom2 trn2› ‹γ1 (trn1::(state, act, out) trans)› ‹γ2 trn2› ‹syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))› ‹⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)›*) apply -\n (*goals:\n 1. ‹⟦trn1 = Trans s1 a1 ou1 s1'; trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom1 trn1; isCom2 trn2; γ1 trn1; γ2 trn2; syncO (g1 trn1) (g2 trn2); ⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)⟧ ⟹ step s1 a1 = (ou1, s1')›\n 2. ‹⟦trn1 = Trans s1 a1 ou1 s1'; trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom1 trn1; isCom2 trn2; γ1 trn1; γ2 trn2; syncO (g1 trn1) (g2 trn2); ⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)⟧ ⟹ step s2 a2 = (ou2, s2')›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"?thesis\"\n (*goal: ‹sync trn1 trn2›*)\n proof (cases a1)\n (*goals:\n 1. ‹⋀x1. a1 = Sact x1 ⟹ sync trn1 trn2›\n 2. ‹⋀x2. a1 = Cact x2 ⟹ sync trn1 trn2›\n 3. ‹⋀x3. a1 = Dact x3 ⟹ sync trn1 trn2›\n 4. ‹⋀x4. a1 = Uact x4 ⟹ sync trn1 trn2›\n 5. ‹⋀x5. a1 = Ract x5 ⟹ sync trn1 trn2›\n 6. ‹⋀x6. a1 = Lact x6 ⟹ sync trn1 trn2›\n 7. ‹⋀x7. a1 = COMact x7 ⟹ sync trn1 trn2›*)\n case (COMact ca1) (*‹(a1::act) = COMact (ca1::comActt)›*)\n note a1 = COMact (*‹(a1::act) = COMact (ca1::comActt)›*)\n show \"?thesis\"\n (*goal: ‹sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›*)\n proof (cases a2)\n (*goals:\n 1. ‹⋀x1::sActt. (a2::act) = Sact x1 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 2. ‹⋀x2::cActt. (a2::act) = Cact x2 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 3. ‹⋀x3::dActt. (a2::act) = Dact x3 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 4. ‹⋀x4::uActt. (a2::act) = Uact x4 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 5. ‹⋀x5::rActt. (a2::act) = Ract x5 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 6. ‹⋀x6::lActt. (a2::act) = Lact x6 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 7. ‹⋀x7::comActt. (a2::act) = COMact x7 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›*)\n case (COMact ca2) (*‹a2 = COMact ca2›*)\n note a2 = COMact (*‹a2 = COMact ca2›*)\n show \"?thesis\"\n (*goal: ‹sync trn1 trn2›*)\n proof (cases ca1)\n (*goals:\n 1. ‹⋀x11 x12 x13 x14. ca1 = comSendServerReq x11 x12 x13 x14 ⟹ sync trn1 trn2›\n 2. ‹⋀x21 x22. ca1 = comReceiveClientReq x21 x22 ⟹ sync trn1 trn2›\n 3. ‹⋀x31 x32 x33 x34. ca1 = comConnectClient x31 x32 x33 x34 ⟹ sync trn1 trn2›\n 4. ‹⋀x41 x42. ca1 = comConnectServer x41 x42 ⟹ sync trn1 trn2›\n 5. ‹⋀x51 x52 x53 x54 x55 x56. ca1 = comReceivePost x51 x52 x53 x54 x55 x56 ⟹ sync trn1 trn2›\n 6. ‹⋀x61 x62 x63 x64. ca1 = comSendPost x61 x62 x63 x64 ⟹ sync trn1 trn2›\n 7. ‹⋀x71 x72 x73 x74. ca1 = comReceiveCreateOFriend x71 x72 x73 x74 ⟹ sync trn1 trn2›\n 8. ‹⋀x81 x82 x83 x84. ca1 = comSendCreateOFriend x81 x82 x83 x84 ⟹ sync trn1 trn2›\n 9. ‹⋀x91 x92 x93 x94. ca1 = comReceiveDeleteOFriend x91 x92 x93 x94 ⟹ sync trn1 trn2›\n 10. ‹⋀x101 x102 x103 x104. ca1 = comSendDeleteOFriend x101 x102 x103 x104 ⟹ sync trn1 trn2›*)\n term comReceivePost\n case (comSendPost uid1 p1 aid1 pid) (*‹ca1 = comSendPost uid1 p1 aid1 pid›*)\n note ca1 = comSendPost (*‹ca1 = comSendPost uid1 p1 aid1 pid›*)\n then obtain pst where p1: \"p1 = pass s1 (admin s1)\" and aid1: \"aid1 = AID2\" and ou2: \"ou2 = outOK\" and ou1: \"ou1 ≠ outErr\" and ca2: \"ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)\"\n (*goal: ‹(⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis) ⟹ thesis›*)\n using so (*‹syncO (g1 trn1) (g2 trn2)›*) step1 (*‹step s1 a1 = (ou1, s1')›*) step2 (*‹step s2 a2 = (ou2, s2')›*) unfolding trn1 trn2 a1 a2 ca1\n (*goal: ‹(⋀pst::post. ⟦(p1::password) = pass (s1::state) (admin s1); (aid1::apiID) = (AID2::apiID); (ou2::out) = outOK; (ou1::out) ≠ outErr; (ca2::comActt) = comReceivePost (AID1::apiID) (serverPass (s2::state) AID1) (pid::postID) pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n apply (cases ca2)\n (*goals:\n 1. ‹⋀x11 x12 x13 x14. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ thesis›\n 2. ‹⋀x21 x22. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comReceiveClientReq x21 x22⟧ ⟹ thesis›\n 3. ‹⋀x31 x32 x33 x34. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ thesis›\n 4. ‹⋀x41 x42. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comConnectServer x41 x42⟧ ⟹ thesis›\n 5. ‹⋀x51 x52 x53 x54 x55 x56. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ thesis›\n 6. ‹⋀x61 x62 x63 x64. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comSendPost x61 x62 x63 x64⟧ ⟹ thesis›\n 7. ‹⋀x71 x72 x73 x74. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ thesis›\n 8. ‹⋀x81 x82 x83 x84. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ thesis›\n 9. ‹⋀x91 x92 x93 x94. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ thesis›\n 10. ‹⋀x101 x102 x103 x104. ⟦⋀pst. ⟦p1 = pass s1 (admin s1); aid1 = AID2; ou2 = outOK; ou1 ≠ outErr; ca2 = comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid)⟧ ⟹ thesis; comSendPost uid1 p1 aid1 pid = comSendPost uid1 p1 aid1 pid; syncO (g1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1')) (g2 (Trans s2 (COMact ca2) ou2 s2')); step s1 (COMact (comSendPost uid1 p1 aid1 pid)) = (ou1, s1'); step s2 (COMact ca2) = (ou2, s2'); ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ thesis›\n discuss goal 1*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 2*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 3*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 4*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 5*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 6*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 7*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 8*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 9*)\n apply ((auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*discuss goal 10*)\n apply ((auto simp: all_defs (*‹e_startSys (?s::state) (?uID::userID) (?p::password) ≡ userIDs ?s = []› ‹startSys (?s::state) (?uID::userID) (?p::password) ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq (?s::state) (?uID::userID) (?requestInfo::requestInfo) ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq (?s::state) (?uID::userID) (?reqInfo::requestInfo) ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?p'::password) ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost (?s::state) (?uID::userID) (?p::password) (?pID::postID) ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq (?s::state) (?uID::userID) (?p::password) (?uID'::userID) (?req::requestInfo) ≡ let pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend (?s::state) (?uID::userID) (?p::password) (?uID'::userID) ≡ let fr::userID ⇒ userID list = friendIDs ?s; pfr::userID ⇒ userID list = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))[1])\n (*proven 10 subgoals*) .\n have ou1: \"ou1 = O_sendPost (AID2,clientPass s1 AID2,pid, post s1 pid, owner s1 pid, vis s1 pid)\"\n using step1 (*‹step s1 a1 = (ou1, s1')›*) ou1 (*‹ou1 ≠ outErr›*) unfolding a1 ca1 aid1\n (*goal: ‹ou1 = O_sendPost (AID2, clientPass s1 AID2, pid, post s1 pid, owner s1 pid, vis s1 pid)›*)\n by (auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))\n show \"?thesis\"\n (*goal: ‹sync trn1 trn2›*)\n proof (cases \"pid = PID\")\n (*goals:\n 1. ‹pid = PID ⟹ sync trn1 trn2›\n 2. ‹pid ≠ PID ⟹ sync trn1 trn2›*)\n case False (*‹pid ≠ PID›*)\n thus \"?thesis\"\n (*goal: ‹sync trn1 trn2›*)\n using so (*‹syncO (g1 trn1) (g2 trn2)›*) step1 (*‹step s1 a1 = (ou1, s1')›*) step2 (*‹step s2 a2 = (ou2, s2')›*) unfolding trn1 trn2 a1 a2 ca1 ca2\n (*goal: ‹sync (Trans (s1::state) (COMact (comSendPost (uid1::userID) (p1::password) (aid1::apiID) (pid::postID))) (ou1::out) (s1'::state)) (Trans (s2::state) (COMact (comReceivePost (AID1::apiID) (serverPass s2 AID1) pid (pst::post) (owner s1 pid) (vis s1 pid))) (ou2::out) (s2'::state))›*)\n by (auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))\n next\n (*goal: ‹pid = PID ⟹ sync trn1 trn2›*)\n case True (*‹pid = PID›*)\n note pid = True (*‹pid = PID›*)\n hence \"φ1 trn1 ∧ φ2 trn2\"\n using ou1 (*‹(ou1::out) = O_sendPost (AID2::apiID, clientPass (s1::state) AID2, pid::postID, post s1 pid, owner s1 pid, vis s1 pid)›*) ou2 (*‹ou2 = outOK›*) unfolding trn1 trn2 a1 a2 ca1 ca2\n (*goal: ‹φ1 (Trans s1 (COMact (comSendPost uid1 p1 aid1 pid)) ou1 s1') ∧ φ2 (Trans s2 (COMact (comReceivePost AID1 (serverPass s2 AID1) pid pst (owner s1 pid) (vis s1 pid))) ou2 s2')›*)\n by auto\n hence \"syncV (f1 trn1) (f2 trn2)\"\n using assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom1 trn1› ‹isCom2 trn2› ‹γ1 trn1› ‹γ2 trn2› ‹syncO (g1 (trn1::(state, act, out) trans)) (g2 (trn2::(state, act, out) trans))› ‹⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)›*) by simp\n hence pst: \"pst = post s1 PID\"\n using pid (*‹(pid::postID) = (PID::postID)›*) unfolding trn1 trn2 a1 a2 ca1 ca2 aid1 ou1\n (*goal: ‹(pst::post) = post (s1::state) (PID::postID)›*)\n by auto\n show \"?thesis\"\n (*goal: ‹sync trn1 trn2›*)\n unfolding trn1 trn2 a1 a2 ca1 ca2 ou1 ou2 pst pid\n (*goal: ‹sync (Trans s1 (COMact (comSendPost uid1 p1 aid1 PID)) (O_sendPost (AID2, clientPass s1 AID2, PID, post s1 PID, owner s1 PID, vis s1 PID)) s1') (Trans s2 (COMact (comReceivePost AID1 (serverPass s2 AID1) PID (post s1 PID) (owner s1 PID) (vis s1 PID))) outOK s2')›*)\n by auto\n qed\n qed (insert so (*‹syncO (g1 trn1) (g2 trn2)›*) step1 (*‹step s1 a1 = (ou1, s1')›*) step2 (*‹step s2 a2 = (ou2, s2')›*), unfold trn1 (*‹trn1 = Trans s1 a1 ou1 s1'›*) trn2 (*‹trn2 = Trans s2 a2 ou2 s2'›*) a1 (*‹a1 = COMact ca1›*) a2 (*‹a2 = COMact ca2›*), (cases ca2, auto simp: all_defs (*‹e_startSys ?s ?uID ?p ≡ userIDs ?s = []› ‹startSys ?s ?uID ?p ≡ ?s⦇admin := ?uID, userIDs := [?uID], user := (user ?s)(?uID := emptyUser), pass := (pass ?s)(?uID := ?p)⦈› ‹e_createNUReq ?s ?uID ?requestInfo ≡ admin ?s ∈∈ userIDs ?s ∧ ?uID ∉ set (userIDs ?s) ∧ ?uID ∉ set (pendingUReqs ?s)› ‹createNUReq ?s ?uID ?reqInfo ≡ ?s⦇pendingUReqs := pendingUReqs ?s ## ?uID, userReq := (userReq ?s)(?uID := ?reqInfo)⦈› ‹e_createUser ?s ?uID ?p ?uID' ?p' ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID = admin ?s ∧ ?uID' ∈∈ pendingUReqs ?s› ‹createUser ?s ?uID ?p ?uID' ?p' ≡ ?s⦇userIDs := ?uID' # userIDs ?s, user := (user ?s)(?uID' := emptyUser), pass := (pass ?s)(?uID' := ?p'), pendingUReqs := remove1 ?uID' (pendingUReqs ?s), userReq := (userReq ?s)(?uID := emptyRequestInfo)⦈› ‹e_createPost ?s ?uID ?p ?pID ≡ IDsOK ?s [?uID] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?pID ∉ set (postIDs ?s)› ‹createPost ?s ?uID ?p ?pID ≡ ?s⦇postIDs := ?pID # postIDs ?s, post := (post ?s)(?pID := emptyPost), owner := (owner ?s)(?pID := ?uID)⦈› ‹e_createFriendReq ?s ?uID ?p ?uID' ?req ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID ∉ set (pendingFReqs ?s ?uID') ∧ ?uID ∉ set (friendIDs ?s ?uID')› ‹createFriendReq ?s ?uID ?p ?uID' ?req ≡ let pfr = pendingFReqs ?s in ?s⦇pendingFReqs := pfr(?uID' := pfr ?uID' ## ?uID), friendReq := fun_upd2 (friendReq ?s) ?uID ?uID' ?req⦈› ‹e_createFriend ?s ?uID ?p ?uID' ≡ IDsOK ?s [?uID, ?uID'] [] [] [] ∧ pass ?s ?uID = ?p ∧ ?uID' ∈∈ pendingFReqs ?s ?uID› ‹createFriend ?s ?uID ?p ?uID' ≡ let fr = friendIDs ?s; pfr = pendingFReqs ?s in ?s⦇friendIDs := fr(?uID := fr ?uID ## ?uID', ?uID' := fr ?uID' ## ?uID), pendingFReqs := pfr(?uID := remove1 ?uID' (pfr ?uID), ?uID' := remove1 ?uID (pfr ?uID')), friendReq := fun_upd2 (fun_upd2 (friendReq ?s) ?uID' ?uID emptyRequestInfo) ?uID ?uID' emptyRequestInfo⦈› and more 78 facts*))+)\n (*solves the remaining goals:\n 1. ‹⋀x11 x12 x13 x14. ca1 = comSendServerReq x11 x12 x13 x14 ⟹ sync trn1 trn2›\n 2. ‹⋀x21 x22. ca1 = comReceiveClientReq x21 x22 ⟹ sync trn1 trn2›\n 3. ‹⋀x31 x32 x33 x34. ca1 = comConnectClient x31 x32 x33 x34 ⟹ sync trn1 trn2›\n 4. ‹⋀x41 x42. ca1 = comConnectServer x41 x42 ⟹ sync trn1 trn2›\n 5. ‹⋀x51 x52 x53 x54 x55 x56. ca1 = comReceivePost x51 x52 x53 x54 x55 x56 ⟹ sync trn1 trn2›\n 6. ‹⋀x71 x72 x73 x74. ca1 = comReceiveCreateOFriend x71 x72 x73 x74 ⟹ sync trn1 trn2›\n 7. ‹⋀x81 x82 x83 x84. ca1 = comSendCreateOFriend x81 x82 x83 x84 ⟹ sync trn1 trn2›\n 8. ‹⋀x91 x92 x93 x94. ca1 = comReceiveDeleteOFriend x91 x92 x93 x94 ⟹ sync trn1 trn2›\n 9. ‹⋀x101 x102 x103 x104. ca1 = comSendDeleteOFriend x101 x102 x103 x104 ⟹ sync trn1 trn2›*)\n qed (insert assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom1 trn1› ‹isCom2 trn2› ‹γ1 trn1› ‹γ2 trn2› ‹syncO (g1 trn1) (g2 trn2)› ‹⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)›*), unfold trn1 (*‹(trn1::(state, act, out) trans) = Trans (s1::state) (a1::act) (ou1::out) (s1'::state)›*) trn2 (*‹(trn2::(state, act, out) trans) = Trans (s2::state) (a2::act) (ou2::out) (s2'::state)›*), auto)\n (*solves the remaining goals:\n 1. ‹⋀x1::sActt. (a2::act) = Sact x1 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 2. ‹⋀x2::cActt. (a2::act) = Cact x2 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 3. ‹⋀x3::dActt. (a2::act) = Dact x3 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 4. ‹⋀x4::uActt. (a2::act) = Uact x4 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 5. ‹⋀x5::rActt. (a2::act) = Ract x5 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›\n 6. ‹⋀x6::lActt. (a2::act) = Lact x6 ⟹ sync (trn1::(state, act, out) trans) (trn2::(state, act, out) trans)›*)\n qed (insert assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom1 trn1› ‹isCom2 trn2› ‹γ1 trn1› ‹γ2 trn2› ‹syncO (g1 trn1) (g2 trn2)› ‹⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)›*), unfold trn1 (*‹(trn1::(state, act, out) trans) = Trans (s1::state) (a1::act) (ou1::out) (s1'::state)›*) trn2 (*‹(trn2::(state, act, out) trans) = Trans (s2::state) (a2::act) (ou2::out) (s2'::state)›*), auto)\n (*solves the remaining goals:\n 1. ‹⋀x1. a1 = Sact x1 ⟹ sync trn1 trn2›\n 2. ‹⋀x2. a1 = Cact x2 ⟹ sync trn1 trn2›\n 3. ‹⋀x3. a1 = Dact x3 ⟹ sync trn1 trn2›\n 4. ‹⋀x4. a1 = Uact x4 ⟹ sync trn1 trn2›\n 5. ‹⋀x5. a1 = Ract x5 ⟹ sync trn1 trn2›\n 6. ‹⋀x6. a1 = Lact x6 ⟹ sync trn1 trn2›*)\nqed\n\nlemma isCom1_γ1:\nassumes \"validTrans trn1\" and \"reach (srcOf trn1)\" and \"isCom1 trn1\"\nshows \"γ1 trn1\"\nproof (cases trn1)\n (*goal: ‹⋀x1 x2 x3 x4. trn1 = Trans x1 x2 x3 x4 ⟹ γ1 trn1›*)\n case (Trans s1 a1 ou1 s1') (*‹trn1 = Trans s1 a1 ou1 s1'›*)\n thus \"?thesis\"\n (*goal: ‹γ1 (trn1::(state, act, out) trans)›*)\n using assms (*‹Iss.validTrans trn1› ‹Iss.reach (srcOf trn1)› ‹isCom1 (trn1::(state, act, out) trans)›*) apply (cases a1)\n (*goals:\n 1. ‹⋀x1. ⟦trn1 = Trans s1 a1 ou1 s1'; Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; a1 = Sact x1⟧ ⟹ γ1 trn1›\n 2. ‹⋀x2. ⟦trn1 = Trans s1 a1 ou1 s1'; Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; a1 = Cact x2⟧ ⟹ γ1 trn1›\n 3. ‹⋀x3. ⟦trn1 = Trans s1 a1 ou1 s1'; Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; a1 = Dact x3⟧ ⟹ γ1 trn1›\n 4. ‹⋀x4. ⟦trn1 = Trans s1 a1 ou1 s1'; Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; a1 = Uact x4⟧ ⟹ γ1 trn1›\n 5. ‹⋀x5. ⟦trn1 = Trans s1 a1 ou1 s1'; Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; a1 = Ract x5⟧ ⟹ γ1 trn1›\n 6. ‹⋀x6. ⟦trn1 = Trans s1 a1 ou1 s1'; Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; a1 = Lact x6⟧ ⟹ γ1 trn1›\n 7. ‹⋀x7. ⟦trn1 = Trans s1 a1 ou1 s1'; Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; a1 = COMact x7⟧ ⟹ γ1 trn1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*proven 7 subgoals*) .\nqed\n\nlemma isCom2_γ2:\nassumes \"validTrans trn2\" and \"reach (srcOf trn2)\" and \"isCom2 trn2\"\nshows \"γ2 trn2\"\nproof (cases trn2)\n (*goal: ‹⋀x1 x2 x3 x4. trn2 = Trans x1 x2 x3 x4 ⟹ γ2 trn2›*)\n case (Trans s2 a2 ou2 s2') (*‹trn2 = Trans s2 a2 ou2 s2'›*)\n thus \"?thesis\"\n (*goal: ‹γ2 trn2›*)\n using assms (*‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹isCom2 trn2›*) apply (cases a2)\n (*goals:\n 1. ‹⋀x1. ⟦trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; a2 = Sact x1⟧ ⟹ γ2 trn2›\n 2. ‹⋀x2. ⟦trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; a2 = Cact x2⟧ ⟹ γ2 trn2›\n 3. ‹⋀x3. ⟦trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; a2 = Dact x3⟧ ⟹ γ2 trn2›\n 4. ‹⋀x4. ⟦trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; a2 = Uact x4⟧ ⟹ γ2 trn2›\n 5. ‹⋀x5. ⟦trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; a2 = Ract x5⟧ ⟹ γ2 trn2›\n 6. ‹⋀x6. ⟦trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; a2 = Lact x6⟧ ⟹ γ2 trn2›\n 7. ‹⋀x7. ⟦trn2 = Trans s2 a2 ou2 s2'; Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; a2 = COMact x7⟧ ⟹ γ2 trn2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*proven 7 subgoals*) .\nqed\n\nlemma isCom2_V2:\nassumes \"validTrans trn2\" and \"reach (srcOf trn2)\" and \"φ2 trn2\"\nshows \"isCom2 trn2\"\nproof (cases trn2)\n (*goal: ‹⋀x1 x2 x3 x4. trn2 = Trans x1 x2 x3 x4 ⟹ isCom2 trn2›*)\n case (Trans s2 a2 ou2 s2') (*‹trn2 = Trans s2 a2 ou2 s2'›*)\n note trn2 = Trans (*‹trn2 = Trans s2 a2 ou2 s2'›*)\n show \"?thesis\"\n (*goal: ‹isCom2 trn2›*)\n proof (cases a2)\n (*goals:\n 1. ‹⋀x1. a2 = Sact x1 ⟹ isCom2 trn2›\n 2. ‹⋀x2. a2 = Cact x2 ⟹ isCom2 trn2›\n 3. ‹⋀x3. a2 = Dact x3 ⟹ isCom2 trn2›\n 4. ‹⋀x4. a2 = Uact x4 ⟹ isCom2 trn2›\n 5. ‹⋀x5. a2 = Ract x5 ⟹ isCom2 trn2›\n 6. ‹⋀x6. a2 = Lact x6 ⟹ isCom2 trn2›\n 7. ‹⋀x7. a2 = COMact x7 ⟹ isCom2 trn2›*)\n case (COMact ca2) (*‹a2 = COMact ca2›*)\n thus \"?thesis\"\n (*goal: ‹isCom2 trn2›*)\n using assms (*‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹φ2 (trn2::(state, act, out) trans)›*) trn2 (*‹trn2 = Trans s2 a2 ou2 s2'›*) apply (cases ca2)\n (*goals:\n 1. ‹⋀x11 x12 x13 x14. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comSendServerReq x11 x12 x13 x14⟧ ⟹ isCom2 trn2›\n 2. ‹⋀x21 x22. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comReceiveClientReq x21 x22⟧ ⟹ isCom2 trn2›\n 3. ‹⋀x31 x32 x33 x34. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comConnectClient x31 x32 x33 x34⟧ ⟹ isCom2 trn2›\n 4. ‹⋀x41 x42. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comConnectServer x41 x42⟧ ⟹ isCom2 trn2›\n 5. ‹⋀x51 x52 x53 x54 x55 x56. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comReceivePost x51 x52 x53 x54 x55 x56⟧ ⟹ isCom2 trn2›\n 6. ‹⋀x61 x62 x63 x64. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comSendPost x61 x62 x63 x64⟧ ⟹ isCom2 trn2›\n 7. ‹⋀x71 x72 x73 x74. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comReceiveCreateOFriend x71 x72 x73 x74⟧ ⟹ isCom2 trn2›\n 8. ‹⋀x81 x82 x83 x84. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comSendCreateOFriend x81 x82 x83 x84⟧ ⟹ isCom2 trn2›\n 9. ‹⋀x91 x92 x93 x94. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comReceiveDeleteOFriend x91 x92 x93 x94⟧ ⟹ isCom2 trn2›\n 10. ‹⋀x101 x102 x103 x104. ⟦a2 = COMact ca2; Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; trn2 = Trans s2 a2 ou2 s2'; ca2 = comSendDeleteOFriend x101 x102 x103 x104⟧ ⟹ isCom2 trn2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*discuss goal 9*)\n apply ((auto)[1])\n (*discuss goal 10*)\n apply ((auto)[1])\n (*proven 10 subgoals*) .\n qed (insert assms (*‹Iss.validTrans trn2› ‹Iss.reach (srcOf trn2)› ‹φ2 trn2›*) trn2 (*‹trn2 = Trans s2 a2 ou2 s2'›*), auto)\n (*solves the remaining goals:\n 1. ‹⋀x1. a2 = Sact x1 ⟹ isCom2 trn2›\n 2. ‹⋀x2. a2 = Cact x2 ⟹ isCom2 trn2›\n 3. ‹⋀x3. a2 = Dact x3 ⟹ isCom2 trn2›\n 4. ‹⋀x4. a2 = Uact x4 ⟹ isCom2 trn2›\n 5. ‹⋀x5. a2 = Ract x5 ⟹ isCom2 trn2›\n 6. ‹⋀x6. a2 = Lact x6 ⟹ isCom2 trn2›*)\nqed\n\nend (* context Post_COMPOSE2 *)\n\n\nsublocale Post_COMPOSE2 < BD_Security_TS_Comp where\n istate1 = istate and validTrans1 = validTrans and srcOf1 = srcOf and tgtOf1 = tgtOf\n and φ1 = φ1 and f1 = f1 and γ1 = γ1 and g1 = g1 and T1 = T1 and B1 = B1\n and\n istate2 = istate and validTrans2 = validTrans and srcOf2 = srcOf and tgtOf2 = tgtOf\n and φ2 = φ2 and f2 = f2 and γ2 = γ2 and g2 = g2 and T2 = T2 and B2 = B2\n and isCom1 = isCom1 and isCom2 = isCom2 and sync = sync\n and isComV1 = isComV1 and isComV2 = isComV2 and syncV = syncV\n and isComO1 = isComO1 and isComO2 = isComO2 and syncO = syncO\n (*and cmpV = cmpV and cmpO = cmpO *)\napply standard\n(*goal: ‹BD_Security_TS_Comp istate Iss.validTrans srcOf tgtOf φ1 f1 γ1 g1 T1 Iss.B istate Iss.validTrans srcOf tgtOf φ2 f2 γ2 g2 T2 Rcv.B isCom1 isCom2 sync isComV1 isComV2 syncV isComO1 isComO2 syncO›*)\nusing isCom1_isComV1 (*‹⟦Iss.validTrans ?trn1.0; Iss.reach (srcOf ?trn1.0); φ1 ?trn1.0⟧ ⟹ isCom1 ?trn1.0 = isComV1 (f1 ?trn1.0)›*) isCom1_isComO1 (*‹⟦Iss.validTrans ?trn1.0; Iss.reach (srcOf ?trn1.0); γ1 ?trn1.0⟧ ⟹ isCom1 ?trn1.0 = isComO1 (g1 ?trn1.0)›*) isCom2_isComV2 (*‹⟦Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); φ2 ?trn2.0⟧ ⟹ isCom2 ?trn2.0 = isComV2 (f2 ?trn2.0)›*) isCom2_isComO2 (*‹⟦Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); γ2 ?trn2.0⟧ ⟹ isCom2 ?trn2.0 = isComO2 (g2 ?trn2.0)›*) sync_syncV (*‹⟦Iss.validTrans ?trn1.0; Iss.reach (srcOf ?trn1.0); Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); isCom1 ?trn1.0; isCom2 ?trn2.0; φ1 ?trn1.0; φ2 ?trn2.0; sync ?trn1.0 ?trn2.0⟧ ⟹ syncV (f1 ?trn1.0) (f2 ?trn2.0)›*) sync_syncO (*‹⟦Iss.validTrans ?trn1.0; Iss.reach (srcOf ?trn1.0); Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); isCom1 ?trn1.0; isCom2 ?trn2.0; γ1 ?trn1.0; γ2 ?trn2.0; sync ?trn1.0 ?trn2.0⟧ ⟹ syncO (g1 ?trn1.0) (g2 ?trn2.0)›*) apply auto\n(*top goal: ‹⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); φ1 trn1⟧ ⟹ isCom1 trn1 = isComV1 (f1 trn1)› and 11 goals remain*)\napply (meson sync_φ1_φ2 (*‹⟦Iss.validTrans ?trn1.0; Iss.reach (srcOf ?trn1.0); Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); isCom1 ?trn1.0; isCom2 ?trn2.0; sync ?trn1.0 ?trn2.0⟧ ⟹ φ1 ?trn1.0 = φ2 ?trn2.0›*), meson sync_φ1_φ2 (*‹⟦Iss.validTrans (?trn1.0::(state, act, out) trans); Iss.reach (srcOf ?trn1.0); Iss.validTrans (?trn2.0::(state, act, out) trans); Iss.reach (srcOf ?trn2.0); isCom1 ?trn1.0; isCom2 ?trn2.0; sync ?trn1.0 ?trn2.0⟧ ⟹ φ1 ?trn1.0 = φ2 ?trn2.0›*))\n(*top goal: ‹⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComV1 (f1 trn1); isCom2 trn2; sync trn1 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); φ1 trn1⟧ ⟹ isCom1 trn1 = isComV1 (f1 trn1); ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2 = isComV2 (f2 trn2); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComV1 (f1 trn1); isComV2 (f2 trn2); φ1 trn1; φ2 trn2; sync trn1 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; sync trn1 trn2⟧ ⟹ syncO (g1 trn1) (g2 trn2); φ1 trn1⟧ ⟹ φ2 trn2› and 5 goals remain*)\nusing \"sync_φ_γ\" (*‹⟦Iss.validTrans ?trn1.0; Iss.reach (srcOf ?trn1.0); Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); isCom1 ?trn1.0; isCom2 ?trn2.0; γ1 ?trn1.0; γ2 ?trn2.0; syncO (g1 ?trn1.0) (g2 ?trn2.0); ⟦φ1 ?trn1.0; φ2 ?trn2.0⟧ ⟹ syncV (f1 ?trn1.0) (f2 ?trn2.0)⟧ ⟹ sync ?trn1.0 ?trn2.0›*) apply auto\n(*top goal: ‹⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; syncO (g1 trn1) (g2 trn2); ⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2); ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); φ1 trn1⟧ ⟹ isCom1 trn1 = isComV1 (f1 trn1); ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2 = isComV2 (f2 trn2); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComV1 (f1 trn1); isComV2 (f2 trn2); φ1 trn1; φ2 trn2; sync trn1 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; sync trn1 trn2⟧ ⟹ syncO (g1 trn1) (g2 trn2)⟧ ⟹ sync trn1 trn2› and 3 goals remain*)\nusing \"isCom1_γ1\" (*‹⟦Iss.validTrans ?trn1.0; Iss.reach (srcOf ?trn1.0); isCom1 ?trn1.0⟧ ⟹ γ1 ?trn1.0›*) \"isCom2_γ2\" (*‹⟦Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); isCom2 ?trn2.0⟧ ⟹ γ2 ?trn2.0›*) isCom2_V2 (*‹⟦Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); φ2 ?trn2.0⟧ ⟹ isCom2 ?trn2.0›*) apply -\n(*goals:\n 1. ‹⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); φ1 trn1⟧ ⟹ isCom1 trn1 = isComV1 (f1 trn1); ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2 = isComV2 (f2 trn2); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComV1 (f1 trn1); isComV2 (f2 trn2); φ1 trn1; φ2 trn2; sync trn1 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; sync trn1 trn2⟧ ⟹ syncO (g1 trn1) (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; syncO (g1 trn1) (g2 trn2); ⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)⟧ ⟹ sync trn1 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1⟧ ⟹ γ1 trn1; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2⟧ ⟹ γ2 trn2; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2⟧ ⟹ γ1 trn1›\n 2. ‹⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); φ1 trn1⟧ ⟹ isCom1 trn1 = isComV1 (f1 trn1); ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2 = isComV2 (f2 trn2); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComV1 (f1 trn1); isComV2 (f2 trn2); φ1 trn1; φ2 trn2; sync trn1 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; sync trn1 trn2⟧ ⟹ syncO (g1 trn1) (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; syncO (g1 trn1) (g2 trn2); ⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)⟧ ⟹ sync trn1 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1⟧ ⟹ γ1 trn1; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2⟧ ⟹ γ2 trn2; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2⟧ ⟹ γ2 trn2›\n 3. ‹⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); φ1 trn1⟧ ⟹ isCom1 trn1 = isComV1 (f1 trn1); ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2 = isComV2 (f2 trn2); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComV1 (f1 trn1); isComV2 (f2 trn2); φ1 trn1; φ2 trn2; sync trn1 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; sync trn1 trn2⟧ ⟹ syncO (g1 trn1) (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; syncO (g1 trn1) (g2 trn2); ⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)⟧ ⟹ sync trn1 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1⟧ ⟹ γ1 trn1; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2⟧ ⟹ γ2 trn2; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2⟧ ⟹ isComV2 (f2 trn2)›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n (*goal: ‹⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); φ1 trn1⟧ ⟹ isCom1 trn1 = isComV1 (f1 trn1); ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); γ1 trn1⟧ ⟹ isCom1 trn1 = isComO1 (g1 trn1); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2 = isComV2 (f2 trn2); ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); γ2 trn2⟧ ⟹ isCom2 trn2 = isComO2 (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComV1 (f1 trn1); isComV2 (f2 trn2); φ1 trn1; φ2 trn2; sync trn1 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; sync trn1 trn2⟧ ⟹ syncO (g1 trn1) (g2 trn2); ⋀trn1 trn2. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); Iss.validTrans trn2; Iss.reach (srcOf trn2); isComO1 (g1 trn1); isComO2 (g2 trn2); γ1 trn1; γ2 trn2; syncO (g1 trn1) (g2 trn2); ⟦φ1 trn1; φ2 trn2⟧ ⟹ syncV (f1 trn1) (f2 trn2)⟧ ⟹ sync trn1 trn2; ⋀trn1. ⟦Iss.validTrans trn1; Iss.reach (srcOf trn1); isCom1 trn1⟧ ⟹ γ1 trn1; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); isCom2 trn2⟧ ⟹ γ2 trn2; ⋀trn2. ⟦Iss.validTrans trn2; Iss.reach (srcOf trn2); φ2 trn2⟧ ⟹ isCom2 trn2⟧ ⟹ isComV2 (f2 trn2)›*)\n apply (meson isCom2_V2 (*‹⟦Iss.validTrans ?trn2.0; Iss.reach (srcOf ?trn2.0); φ2 ?trn2.0⟧ ⟹ isCom2 ?trn2.0›*))\n(*proven 3 subgoals*) .\n\n\ncontext Post_COMPOSE2\nbegin\n\ntheorem secure: \"secure\"\n using secure1_secure2_secure[OF Iss.Post_secure Rcv.Post_secure] (*‹K.secure›*) .\n\nend (* context Post_COMPOSE2 *)\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/CoSMeDis/Post_Confidentiality/Post_COMPOSE2.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"e840644ea5ca13293720ba7c4bd5189672ea53eb4432e676df56aabdde72c1f6\"\n}"}}},{"rowIdx":152,"cells":{"text":{"kind":"string","value":"(* Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com> *)\n\nsection ‹Injections from A to B up to a Permutation of A›\n\ntheory Twelvefold_Way_Entry5\nimports\n Equiv_Relations_on_Functions\nbegin\n\nsubsection ‹Definition of Bijections›\n\ndefinition subset_of :: \"'a set ⇒ ('a ⇒ 'b) set ⇒ 'b set\"\nwhere\n \"subset_of A F = univ (λf. f ` A) F\"\n\ndefinition functions_of :: \"'a set ⇒ 'b set ⇒ ('a ⇒ 'b) set\"\nwhere\n \"functions_of A B = {f ∈ A →⇩E B. f ` A = B}\"\n\nsubsection ‹Properties for Bijections›\n\nlemma functions_of_eq:\n assumes \"finite A\"\n assumes \"f ∈ {f ∈ A →⇩E B. inj_on f A}\"\n shows \"functions_of A (f ` A) = domain_permutation A B `` {f}\"\nproof (standard)\n (*goals:\n 1. ‹functions_of (A::'a set) ((f::'a ⇒ 'b) ` A) ⊆ domain_permutation A (B::'b set) `` {f}›\n 2. ‹domain_permutation (A::'a set) (B::'b set) `` {f::'a ⇒ 'b} ⊆ functions_of A (f ` A)›*)\n have bij: \"bij_betw f A (f ` A)\"\n using assms (*‹finite A› ‹f ∈ {f ∈ A →⇩E B. inj_on f A}›*) by (simp add: bij_betw_imageI (*‹⟦inj_on ?f ?A; ?f ` ?A = ?B⟧ ⟹ bij_betw ?f ?A ?B›*))\n show \"functions_of A (f ` A) ⊆ domain_permutation A B `` {f}\"\n proof (standard)\n (*goal: ‹⋀x. x ∈ functions_of A (f ` A) ⟹ x ∈ domain_permutation A B `` {f}›*)\n fix f'\n assume \"f' ∈ functions_of A (f ` A)\" (*‹(f'::'a ⇒ 'b) ∈ functions_of (A::'a set) ((f::'a ⇒ 'b) ` A)›*)\n from this (*‹f' ∈ functions_of A (f ` A)›*) have \"f' ∈ A →⇩E f ` A\" and \"f' ` A = f ` A\"\n unfolding functions_of_def\n (*goals:\n 1. ‹f' ∈ A →⇩E f ` A›\n 2. ‹f' ` A = f ` A›*)\n apply -\n (*goals:\n 1. ‹f' ∈ {fa ∈ A →⇩E f ` A. fa ` A = f ` A} ⟹ f' ∈ A →⇩E f ` A›\n 2. ‹f' ∈ {fa ∈ A →⇩E f ` A. fa ` A = f ` A} ⟹ f' ` A = f ` A›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from this (*‹f' ∈ A →⇩E f ` A› ‹f' ` A = f ` A›*) assms (*‹finite (A::'a::type set)› ‹f ∈ {f ∈ A →⇩E B. inj_on f A}›*) have \"f' ∈ A →⇩E B\" and \"inj_on f A\"\n using PiE_mem (*‹⟦?f ∈ Pi⇩E ?S ?T; ?x ∈ ?S⟧ ⟹ ?f ?x ∈ ?T ?x›*) apply -\n (*goals:\n 1. ‹⟦f' ∈ A →⇩E f ` A; f' ` A = f ` A; finite A; f ∈ {f ∈ A →⇩E B. inj_on f A}; ⋀f S T x. ⟦f ∈ Pi⇩E S T; x ∈ S⟧ ⟹ f x ∈ T x⟧ ⟹ f' ∈ A →⇩E B›\n 2. ‹⟦f' ∈ A →⇩E f ` A; f' ` A = f ` A; finite A; f ∈ {f ∈ A →⇩E B. inj_on f A}; ⋀f S T x. ⟦f ∈ Pi⇩E S T; x ∈ S⟧ ⟹ f x ∈ T x⟧ ⟹ inj_on f A›\n discuss goal 1*)\n apply fastforce\n (*discuss goal 2*)\n apply fastforce\n (*proven 2 subgoals*) .\n moreover have \"∃p. p permutes A ∧ (∀x∈A. f x = f' (p x))\"\n proof (standard)\n (*goal: ‹?p permutes A ∧ (∀x∈A. f x = f' (?p x))›*)\n let ?p = \"λx. if x ∈ A then inv_into A f' (f x) else x\"\n show \"?p permutes A ∧ (∀x∈A. f x = f' (?p x))\"\n proof (standard)\n (*goals:\n 1. ‹(λx. if x ∈ A then inv_into A f' (f x) else x) permutes A›\n 2. ‹∀x∈A. f x = f' (if x ∈ A then inv_into A f' (f x) else x)›*)\n show \"?p permutes A\"\n proof (rule bij_imp_permutes (*‹⟦bij_betw ?p ?S ?S; ⋀x. x ∉ ?S ⟹ ?p x = x⟧ ⟹ ?p permutes ?S›*))\n (*goals:\n 1. ‹bij_betw (λx. if x ∈ A then inv_into A f' (f x) else x) A A›\n 2. ‹⋀x. x ∉ A ⟹ (if x ∈ A then inv_into A f' (f x) else x) = x›*)\n show \"bij_betw ?p A A\"\n proof (rule bij_betw_imageI (*‹⟦inj_on ?f ?A; ?f ` ?A = ?B⟧ ⟹ bij_betw ?f ?A ?B›*))\n (*goals:\n 1. ‹inj_on (λx. if x ∈ A then inv_into A f' (f x) else x) A›\n 2. ‹(λx. if x ∈ A then inv_into A f' (f x) else x) ` A = A›*)\n show \"inj_on ?p A\"\n proof (rule inj_onI (*‹(⋀x y. ⟦x ∈ ?A; y ∈ ?A; ?f x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*))\n (*goal: ‹⋀x y. ⟦x ∈ A; y ∈ A; (if x ∈ A then inv_into A f' (f x) else x) = (if y ∈ A then inv_into A f' (f y) else y)⟧ ⟹ x = y›*)\n fix a and a'\n assume \"a ∈ A\" \"a' ∈ A\" \"?p a = ?p a'\" (*‹(a::'a) ∈ (A::'a set)› ‹(a'::'a) ∈ (A::'a set)› ‹(if (a::'a) ∈ (A::'a set) then inv_into A (f'::'a ⇒ 'b) ((f::'a ⇒ 'b) a) else a) = (if (a'::'a) ∈ A then inv_into A f' (f a') else a')›*)\n from this (*‹a ∈ A› ‹a' ∈ A› ‹(if a ∈ A then inv_into A f' (f a) else a) = (if a' ∈ A then inv_into A f' (f a') else a')›*) have \"inv_into A f' (f a) = inv_into A f' (f a')\"\n by auto\n from this (*‹inv_into A f' (f a) = inv_into A f' (f a')›*) ‹a ∈ A› (*‹a ∈ A›*) ‹a' ∈ A› (*‹a' ∈ A›*) ‹f' ` A = f ` A› (*‹f' ` A = f ` A›*) have \"f a = f a'\"\n using inv_into_injective (*‹⟦inv_into ?A ?f ?x = inv_into ?A ?f ?y; ?x ∈ ?f ` ?A; ?y ∈ ?f ` ?A⟧ ⟹ ?x = ?y›*) by fastforce\n from this (*‹f a = f a'›*) ‹a ∈ A› (*‹a ∈ A›*) ‹a' ∈ A› (*‹a' ∈ A›*) show \"a = a'\"\n by (metis bij (*‹bij_betw f A (f ` A)›*) bij_betw_inv_into_left (*‹⟦bij_betw ?f ?A ?A'; ?a ∈ ?A⟧ ⟹ inv_into ?A ?f (?f ?a) = ?a›*))\n qed\n next\n (*goal: ‹(λx. if x ∈ A then inv_into A f' (f x) else x) ` A = A›*)\n show \"?p ` A = A\"\n proof (standard)\n (*goals:\n 1. ‹(λx. if x ∈ A then inv_into A f' (f x) else x) ` A ⊆ A›\n 2. ‹A ⊆ (λx. if x ∈ A then inv_into A f' (f x) else x) ` A›*)\n show \"?p ` A ⊆ A\"\n using ‹f' ` A = f ` A› (*‹(f'::'a::type ⇒ 'b::type) ` (A::'a::type set) = (f::'a::type ⇒ 'b::type) ` A›*) by (simp add: image_subsetI (*‹(⋀x. x ∈ ?A ⟹ ?f x ∈ ?B) ⟹ ?f ` ?A ⊆ ?B›*) inv_into_into (*‹?x ∈ ?f ` ?A ⟹ inv_into ?A ?f ?x ∈ ?A›*))\n next\n (*goal: ‹(A::'a set) ⊆ (λx::'a. if x ∈ A then inv_into A (f'::'a ⇒ 'b) ((f::'a ⇒ 'b) x) else x) ` A›*)\n show \"A ⊆ ?p ` A\"\n proof (standard)\n (*goal: ‹⋀x::'a. x ∈ (A::'a set) ⟹ x ∈ (λx::'a. if x ∈ A then inv_into A (f'::'a ⇒ 'b) ((f::'a ⇒ 'b) x) else x) ` A›*)\n fix a\n assume \"a ∈ A\" (*‹(a::'a) ∈ (A::'a set)›*)\n have \"inj_on f' A\"\n using ‹finite A› (*‹finite A›*) ‹f' ` A = f ` A› (*‹f' ` A = f ` A›*) ‹inj_on f A› (*‹inj_on f A›*) by (simp add: card_image (*‹inj_on (?f::?'a ⇒ ?'b) (?A::?'a set) ⟹ card (?f ` ?A) = card ?A›*) eq_card_imp_inj_on (*‹⟦finite (?A::?'a set); card ((?f::?'a ⇒ ?'b) ` ?A) = card ?A⟧ ⟹ inj_on ?f ?A›*))\n from ‹a ∈ A› (*‹a ∈ A›*) ‹f' ` A = f ` A› (*‹f' ` A = f ` A›*) have \"inv_into A f (f' a) ∈ A\"\n by (metis image_eqI (*‹⟦?b = ?f ?x; ?x ∈ ?A⟧ ⟹ ?b ∈ ?f ` ?A›*) inv_into_into (*‹?x ∈ ?f ` ?A ⟹ inv_into ?A ?f ?x ∈ ?A›*))\n moreover have \"a = inv_into A f' (f (inv_into A f (f' a)))\"\n using ‹a ∈ A› (*‹a ∈ A›*) ‹f' ` A = f ` A› (*‹f' ` A = f ` A›*) ‹inj_on f' A› (*‹inj_on f' A›*) by (metis f_inv_into_f (*‹?y ∈ ?f ` ?A ⟹ ?f (inv_into ?A ?f ?y) = ?y›*) image_eqI (*‹⟦?b = ?f ?x; ?x ∈ ?A⟧ ⟹ ?b ∈ ?f ` ?A›*) inv_into_f_f (*‹⟦inj_on ?f ?A; ?x ∈ ?A⟧ ⟹ inv_into ?A ?f (?f ?x) = ?x›*))\n ultimately show \"a ∈ ?p ` A\"\n by auto\n qed\n qed\n qed\n next\n (*goal: ‹⋀x::'a::type. x ∉ (A::'a::type set) ⟹ (if x ∈ A then inv_into A (f'::'a::type ⇒ 'b::type) ((f::'a::type ⇒ 'b::type) x) else x) = x›*)\n fix x\n assume \"x ∉ A\" (*‹(x::'a) ∉ (A::'a set)›*)\n from this (*‹(x::'a::type) ∉ (A::'a::type set)›*) show \"?p x = x\"\n by simp\n qed\n next\n (*goal: ‹∀x∈A. f x = f' (if x ∈ A then inv_into A f' (f x) else x)›*)\n from ‹f' ` A = f ` A› (*‹(f'::'a::type ⇒ 'b::type) ` (A::'a::type set) = (f::'a::type ⇒ 'b::type) ` A›*) show \"∀x∈A. f x = f' (?p x)\"\n by (simp add: f_inv_into_f (*‹?y ∈ ?f ` ?A ⟹ ?f (inv_into ?A ?f ?y) = ?y›*))\n qed\n qed\n moreover have \"f ∈ A →⇩E B\"\n using assms (*‹finite A› ‹f ∈ {f ∈ A →⇩E B. inj_on f A}›*) by auto\n ultimately show \"f' ∈ domain_permutation A B `` {f}\"\n unfolding domain_permutation_def\n (*goal: ‹f' ∈ {(f, f'). (f, f') ∈ (A →⇩E B) × (A →⇩E B) ∧ (∃p. p permutes A ∧ (∀x∈A. f x = f' (p x)))} `` {f}›*)\n by auto\n qed\nnext\n (*goal: ‹domain_permutation A B `` {f} ⊆ functions_of A (f ` A)›*)\n show \"domain_permutation A B `` {f} ⊆ functions_of A (f ` A)\"\n proof (standard)\n (*goal: ‹⋀x::'a ⇒ 'b. x ∈ domain_permutation (A::'a set) (B::'b set) `` {f::'a ⇒ 'b} ⟹ x ∈ functions_of A (f ` A)›*)\n fix f'\n assume \"f' ∈ domain_permutation A B `` {f}\" (*‹(f'::'a ⇒ 'b) ∈ domain_permutation (A::'a set) (B::'b set) `` {f::'a ⇒ 'b}›*)\n from this (*‹(f'::'a::type ⇒ 'b::type) ∈ domain_permutation (A::'a::type set) (B::'b::type set) `` {f::'a::type ⇒ 'b::type}›*) obtain p where p: \"p permutes A\" \"∀x∈A. f x = f' (p x)\" and \"f ∈ A →⇩E B\" \"f' ∈ A →⇩E B\"\n (*goal: ‹(⋀p. ⟦p permutes A; ∀x∈A. f x = f' (p x); f ∈ A →⇩E B; f' ∈ A →⇩E B⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding domain_permutation_def\n (*goal: ‹(⋀p::'a::type ⇒ 'a::type. ⟦p permutes (A::'a::type set); ∀x::'a::type∈A. (f::'a::type ⇒ 'b::type) x = (f'::'a::type ⇒ 'b::type) (p x); f ∈ A →⇩E (B::'b::type set); f' ∈ A →⇩E B⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n have \"f' ` A = f ` A\"\n proof (standard)\n (*goals:\n 1. ‹f' ` A ⊆ f ` A›\n 2. ‹f ` A ⊆ f' ` A›*)\n show \"f' ` A ⊆ f ` A\"\n proof (standard)\n (*goal: ‹⋀x. x ∈ f' ` A ⟹ x ∈ f ` A›*)\n fix x\n assume \"x ∈ f' ` A\" (*‹(x::'b) ∈ (f'::'a ⇒ 'b) ` (A::'a set)›*)\n from this (*‹x ∈ f' ` A›*) obtain x' where \"x = f' x'\" and \"x' ∈ A\"\n (*goal: ‹(⋀x'. ⟦x = f' x'; x' ∈ A⟧ ⟹ thesis) ⟹ thesis›*)\n by standard\n from this (*‹x = f' x'› ‹x' ∈ A›*) have \"x = f (inv p x')\"\n using p (*‹p permutes A› ‹∀x∈A. f x = f' (p x)›*) by (metis (mono_tags, lifting) permutes_in_image (*‹?p permutes ?S ⟹ (?p ?x ∈ ?S) = (?x ∈ ?S)›*) permutes_inverses( (*‹?p permutes ?S ⟹ ?p (inv ?p ?x) = ?x›*) 1))\n moreover have \"inv p x' ∈ A\"\n using p (*‹p permutes A› ‹∀x∈A. f x = f' (p x)›*) ‹x' ∈ A› (*‹x' ∈ A›*) by (simp add: permutes_in_image (*‹?p permutes ?S ⟹ (?p ?x ∈ ?S) = (?x ∈ ?S)›*) permutes_inv (*‹?p permutes ?S ⟹ inv ?p permutes ?S›*))\n ultimately show \"x ∈ f ` A\"\n by standard\n qed\n next\n (*goal: ‹f ` A ⊆ f' ` A›*)\n show \"f ` A ⊆ f' ` A\"\n using p (*‹(p::'a ⇒ 'a) permutes (A::'a set)› ‹∀x::'a::type∈A::'a::type set. (f::'a::type ⇒ 'b::type) x = (f'::'a::type ⇒ 'b::type) ((p::'a::type ⇒ 'a::type) x)›*) permutes_in_image (*‹(?p::?'a ⇒ ?'a) permutes (?S::?'a set) ⟹ (?p (?x::?'a) ∈ ?S) = (?x ∈ ?S)›*) by fastforce\n qed\n moreover from this (*‹f' ` A = f ` A›*) ‹f' ∈ A →⇩E B› (*‹f' ∈ A →⇩E B›*) have \"f' ∈ A →⇩E f ` A\"\n by auto\n ultimately show \"f' ∈ functions_of A (f ` A)\"\n unfolding functions_of_def\n (*goal: ‹f' ∈ {fa ∈ A →⇩E f ` A. fa ` A = f ` A}›*)\n by auto\n qed\nqed\n\nlemma subset_of:\n assumes \"F ∈ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B\"\n shows \"subset_of A F ⊆ B\" and \"card (subset_of A F) = card A\"\nproof (-)\n (*goals:\n 1. ‹subset_of (A::'a set) (F::('a ⇒ 'b) set) ⊆ (B::'b set)›\n 2. ‹card (subset_of (A::'a set) (F::('a ⇒ 'b) set)) = card A›*)\n from assms (*‹F ∈ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B›*) obtain f where F_eq: \"F = (domain_permutation A B) `` {f}\" and f: \"f ∈ A →⇩E B\" \"inj_on f A\"\n (*goal: ‹(⋀f. ⟦F = domain_permutation A B `` {f}; f ∈ A →⇩E B; inj_on f A⟧ ⟹ thesis) ⟹ thesis›*)\n using mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) quotientE (*‹⟦?X ∈ ?A // ?r; ⋀x. ⟦?X = ?r `` {x}; x ∈ ?A⟧ ⟹ ?P⟧ ⟹ ?P›*) by force\n from this (*‹F = domain_permutation A B `` {f}› ‹f ∈ A →⇩E B› ‹inj_on f A›*) have \"subset_of A (domain_permutation A B `` {f}) = f ` A\"\n using equiv_domain_permutation (*‹equiv (?A →⇩E ?B) (domain_permutation ?A ?B)›*) image_respects_domain_permutation (*‹(λf. f ` ?A) respects domain_permutation ?A ?B›*) unfolding subset_of_def\n (*goal: ‹univ (λf. f ` A) (domain_permutation A B `` {f}) = f ` A›*)\n apply (intro univ_commute' (*‹⟦equiv ?A ?r; ?f respects ?r; ?x ∈ ?A⟧ ⟹ univ ?f (?r `` {?x}) = ?f ?x›*))\n (*goal: ‹univ (λf::'a ⇒ 'b. f ` (A::'a set)) (domain_permutation A (B::'b set) `` {f::'a ⇒ 'b}) = f ` A›*)\n by auto\n from this (*‹subset_of (A::'a set) (domain_permutation A (B::'b set) `` {f::'a ⇒ 'b}) = f ` A›*) f (*‹f ∈ A →⇩E B› ‹inj_on (f::'a ⇒ 'b) (A::'a set)›*) F_eq (*‹F = domain_permutation A B `` {f}›*) show \"subset_of A F ⊆ B\" and \"card (subset_of A F) = card A\"\n apply -\n (*goals:\n 1. ‹⟦subset_of A (domain_permutation A B `` {f}) = f ` A; f ∈ A →⇩E B; inj_on f A; F = domain_permutation A B `` {f}⟧ ⟹ subset_of A F ⊆ B›\n 2. ‹⟦subset_of A (domain_permutation A B `` {f}) = f ` A; f ∈ A →⇩E B; inj_on f A; F = domain_permutation A B `` {f}⟧ ⟹ card (subset_of A F) = card A›\n discuss goal 1*)\n apply ((auto simp add: card_image (*‹inj_on ?f ?A ⟹ card (?f ` ?A) = card ?A›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: card_image (*‹inj_on ?f ?A ⟹ card (?f ` ?A) = card ?A›*))[1])\n (*proven 2 subgoals*) .\nqed\n\nlemma functions_of:\n assumes \"finite A\" \"finite B\" \"X ⊆ B\" \"card X = card A\"\n shows \"functions_of A X ∈ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B\"\nproof (-)\n (*goal: ‹functions_of A X ∈ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B›*)\n from assms (*‹finite A› ‹finite B› ‹X ⊆ B› ‹card X = card A›*) obtain f where f: \"f ∈ A →⇩E X ∧ bij_betw f A X\"\n (*goal: ‹(⋀f. f ∈ A →⇩E X ∧ bij_betw f A X ⟹ thesis) ⟹ thesis›*)\n using ‹finite A› (*‹finite A›*) ‹finite B› (*‹finite B›*) by (metis finite_same_card_bij_on_ext_funcset (*‹⟦finite ?A; finite ?B; card ?A = card ?B⟧ ⟹ ∃f. f ∈ ?A →⇩E ?B ∧ bij_betw f ?A ?B›*) finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*))\n from this (*‹(f::'a ⇒ 'b) ∈ (A::'a set) →⇩E (X::'b set) ∧ bij_betw f A X›*) have \"X = f ` A\"\n by (simp add: bij_betw_def (*‹bij_betw ?f ?A ?B = (inj_on ?f ?A ∧ ?f ` ?A = ?B)›*))\n from f (*‹(f::'a ⇒ 'b) ∈ (A::'a set) →⇩E (X::'b set) ∧ bij_betw f A X›*) ‹X ⊆ B› (*‹X ⊆ B›*) have \"f ∈ {f ∈ A →⇩E B. inj_on f A}\"\n by (auto simp add: bij_betw_imp_inj_on (*‹bij_betw ?f ?A ?B ⟹ inj_on ?f ?A›*))\n have \"functions_of A X = domain_permutation A B `` {f}\"\n using ‹finite A› (*‹finite A›*) ‹X = f ` A› (*‹(X::'b::type set) = (f::'a::type ⇒ 'b::type) ` (A::'a::type set)›*) ‹f ∈ {f ∈ A →⇩E B. inj_on f A}› (*‹f ∈ {f ∈ A →⇩E B. inj_on f A}›*) by (simp add: functions_of_eq (*‹⟦finite ?A; ?f ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A}⟧ ⟹ functions_of ?A (?f ` ?A) = domain_permutation ?A ?B `` {?f}›*))\n from this (*‹functions_of (A::'a set) (X::'b set) = domain_permutation A (B::'b set) `` {f::'a ⇒ 'b}›*) show \"functions_of A X ∈ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B\"\n using ‹f ∈ {f ∈ A →⇩E B. inj_on f A}› (*‹f ∈ {f ∈ A →⇩E B. inj_on f A}›*) by (auto intro: quotientI (*‹(?x::?'a) ∈ (?A::?'a set) ⟹ (?r::(?'a × ?'a) set) `` {?x} ∈ ?A // ?r›*))\nqed\n\nlemma subset_of_functions_of:\n assumes \"finite A\" \"finite X\" \"card A = card X\"\n shows \"subset_of A (functions_of A X) = X\"\nproof (-)\n (*goal: ‹subset_of A (functions_of A X) = X›*)\n from assms (*‹finite A› ‹finite X› ‹card (A::'a set) = card (X::'b set)›*) obtain f where \"f ∈ A →⇩E X\" and \"bij_betw f A X\"\n (*goal: ‹(⋀f. ⟦f ∈ A →⇩E X; bij_betw f A X⟧ ⟹ thesis) ⟹ thesis›*)\n using finite_same_card_bij_on_ext_funcset (*‹⟦finite ?A; finite ?B; card ?A = card ?B⟧ ⟹ ∃f. f ∈ ?A →⇩E ?B ∧ bij_betw f ?A ?B›*) by blast\n from this (*‹f ∈ A →⇩E X› ‹bij_betw (f::'a::type ⇒ 'b::type) (A::'a::type set) (X::'b::type set)›*) have subset_of: \"subset_of A (domain_permutation A X `` {f}) = f ` A\"\n using equiv_domain_permutation (*‹equiv (?A →⇩E ?B) (domain_permutation ?A ?B)›*) image_respects_domain_permutation (*‹(λf. f ` ?A) respects domain_permutation ?A ?B›*) unfolding subset_of_def\n (*goal: ‹univ (λf::'a ⇒ 'b. f ` (A::'a set)) (domain_permutation A (X::'b set) `` {f::'a ⇒ 'b}) = f ` A›*)\n apply (intro univ_commute' (*‹⟦equiv (?A::?'a set) (?r::(?'a × ?'a) set); (?f::?'a ⇒ ?'b) respects ?r; (?x::?'a) ∈ ?A⟧ ⟹ univ ?f (?r `` {?x}) = ?f ?x›*))\n (*goal: ‹univ (λf. f ` A) (domain_permutation A X `` {f}) = f ` A›*)\n by auto\n from ‹bij_betw f A X› (*‹bij_betw f A X›*) have \"inj_on f A\" and \"f ` A = X\"\n apply -\n (*goals:\n 1. ‹bij_betw f A X ⟹ inj_on f A›\n 2. ‹bij_betw f A X ⟹ f ` A = X›\n discuss goal 1*)\n apply ((auto simp add: bij_betw_def (*‹bij_betw ?f ?A ?B = (inj_on ?f ?A ∧ ?f ` ?A = ?B)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: bij_betw_def (*‹bij_betw ?f ?A ?B = (inj_on ?f ?A ∧ ?f ` ?A = ?B)›*))[1])\n (*proven 2 subgoals*) .\n have \"subset_of A (functions_of A X) = subset_of A (functions_of A (f ` A))\"\n using ‹f ` A = X› (*‹f ` A = X›*) by simp\n also (*calculation: ‹subset_of A (functions_of A X) = subset_of A (functions_of A (f ` A))›*) have \"… = subset_of A (domain_permutation A X `` {f})\"\n using ‹finite A› (*‹finite (A::'a set)›*) ‹inj_on f A› (*‹inj_on f A›*) ‹f ∈ A →⇩E X› (*‹f ∈ A →⇩E X›*) by (auto simp add: functions_of_eq (*‹⟦finite ?A; ?f ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A}⟧ ⟹ functions_of ?A (?f ` ?A) = domain_permutation ?A ?B `` {?f}›*))\n also (*calculation: ‹subset_of A (functions_of A X) = subset_of A (domain_permutation A X `` {f})›*) have \"… = f ` A\"\n using ‹inj_on f A› (*‹inj_on f A›*) ‹f ∈ A →⇩E X› (*‹f ∈ A →⇩E X›*) by (simp add: subset_of (*‹subset_of A (domain_permutation A X `` {f}) = f ` A›*))\n also (*calculation: ‹subset_of A (functions_of A X) = f ` A›*) have \"… = X\"\n using ‹f ` A = X› (*‹f ` A = X›*) by simp\n finally (*calculation: ‹subset_of (A::'a set) (functions_of A (X::'b set)) = X›*) show \"?thesis\"\n (*goal: ‹subset_of (A::'a set) (functions_of A (X::'b set)) = X›*) .\nqed\n\nlemma functions_of_subset_of:\n assumes \"finite A\"\n assumes \"F ∈ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B\"\n shows \"functions_of A (subset_of A F) = F\"\nsorry\n\nsubsection ‹Bijections›\n\nlemma bij_betw_subset_of:\n assumes \"finite A\" \"finite B\"\n shows \"bij_betw (subset_of A) ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) {X. X ⊆ B ∧ card X = card A}\"\nproof (rule bij_betw_byWitness[where f'=\"functions_of A\"] (*‹⟦∀a∈?A. functions_of A (?f a) = a; ∀a'∈?A'. ?f (functions_of A a') = a'; ?f ` ?A ⊆ ?A'; functions_of A ` ?A' ⊆ ?A⟧ ⟹ bij_betw ?f ?A ?A'›*))\n (*goals:\n 1. ‹∀a∈{f ∈ A →⇩E B. inj_on f A} // domain_permutation A B. functions_of A (subset_of A a) = a›\n 2. ‹∀a'∈{X. X ⊆ B ∧ card X = card A}. subset_of A (functions_of A a') = a'›\n 3. ‹subset_of A ` {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B ⊆ {X. X ⊆ B ∧ card X = card A}›\n 4. ‹functions_of A ` {X. X ⊆ B ∧ card X = card A} ⊆ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B›*)\n show \"∀F∈{f ∈ A →⇩E B. inj_on f A} // domain_permutation A B. functions_of A (subset_of A F) = F\"\n using ‹finite A› (*‹finite A›*) functions_of_subset_of (*‹⟦finite ?A; ?F ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B⟧ ⟹ functions_of ?A (subset_of ?A ?F) = ?F›*) by auto\n show \"∀X∈{X. X ⊆ B ∧ card X = card A}. subset_of A (functions_of A X) = X\"\n using subset_of_functions_of (*‹⟦finite (?A::?'a::type set); finite (?X::?'b::type set); card ?A = card ?X⟧ ⟹ subset_of ?A (functions_of ?A ?X) = ?X›*) ‹finite A› (*‹finite A›*) ‹finite B› (*‹finite (B::'b set)›*) by (metis (mono_tags) finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*))\n show \"subset_of A ` ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) ⊆ {X. X ⊆ B ∧ card X = card A}\"\n using subset_of (*‹?F ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B ⟹ subset_of ?A ?F ⊆ ?B› ‹?F ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B ⟹ card (subset_of ?A ?F) = card ?A›*) by fastforce\n show \"functions_of A ` {X. X ⊆ B ∧ card X = card A} ⊆ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B\"\n using ‹finite A› (*‹finite A›*) ‹finite B› (*‹finite B›*) functions_of (*‹⟦finite ?A; finite ?B; ?X ⊆ ?B; card ?X = card ?A⟧ ⟹ functions_of ?A ?X ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B›*) by auto\nqed\n\nlemma bij_betw_functions_of:\n assumes \"finite A\" \"finite B\"\n shows \"bij_betw (functions_of A) {X. X ⊆ B ∧ card X = card A} ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B)\"\nproof (rule bij_betw_byWitness[where f'=\"subset_of A\"] (*‹⟦∀a∈?A. subset_of A (?f a) = a; ∀a'∈?A'. ?f (subset_of A a') = a'; ?f ` ?A ⊆ ?A'; subset_of A ` ?A' ⊆ ?A⟧ ⟹ bij_betw ?f ?A ?A'›*))\n (*goals:\n 1. ‹∀a∈{X. X ⊆ B ∧ card X = card A}. subset_of A (functions_of A a) = a›\n 2. ‹∀a'∈{f ∈ A →⇩E B. inj_on f A} // domain_permutation A B. functions_of A (subset_of A a') = a'›\n 3. ‹functions_of A ` {X. X ⊆ B ∧ card X = card A} ⊆ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B›\n 4. ‹subset_of A ` {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B ⊆ {X. X ⊆ B ∧ card X = card A}›*)\n show \"∀F∈{f ∈ A →⇩E B. inj_on f A} // domain_permutation A B. functions_of A (subset_of A F) = F\"\n using ‹finite A› (*‹finite A›*) functions_of_subset_of (*‹⟦finite (?A::?'a set); (?F::(?'a ⇒ ?'b) set) ∈ {f::?'a ⇒ ?'b ∈ ?A →⇩E (?B::?'b set). inj_on f ?A} // domain_permutation ?A ?B⟧ ⟹ functions_of ?A (subset_of ?A ?F) = ?F›*) by auto\n show \"∀X∈{X. X ⊆ B ∧ card X = card A}. subset_of A (functions_of A X) = X\"\n using subset_of_functions_of (*‹⟦finite ?A; finite ?X; card ?A = card ?X⟧ ⟹ subset_of ?A (functions_of ?A ?X) = ?X›*) ‹finite A› (*‹finite (A::'a set)›*) ‹finite B› (*‹finite B›*) by (metis (mono_tags) finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*))\n show \"subset_of A ` ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) ⊆ {X. X ⊆ B ∧ card X = card A}\"\n using subset_of (*‹?F ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B ⟹ subset_of ?A ?F ⊆ ?B› ‹?F ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B ⟹ card (subset_of ?A ?F) = card ?A›*) by fastforce\n show \"functions_of A ` {X. X ⊆ B ∧ card X = card A} ⊆ {f ∈ A →⇩E B. inj_on f A} // domain_permutation A B\"\n using ‹finite A› (*‹finite A›*) ‹finite B› (*‹finite B›*) functions_of (*‹⟦finite ?A; finite ?B; ?X ⊆ ?B; card ?X = card ?A⟧ ⟹ functions_of ?A ?X ∈ {f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B›*) by auto\nqed\n\nlemma bij_betw_mset_set:\n shows \"bij_betw mset_set {A. finite A} {M. ∀x. count M x ≤ 1}\"\nproof (rule bij_betw_byWitness[where f'=\"set_mset\"] (*‹⟦∀a∈?A. set_mset (?f a) = a; ∀a'∈?A'. ?f (set_mset a') = a'; ?f ` ?A ⊆ ?A'; set_mset ` ?A' ⊆ ?A⟧ ⟹ bij_betw ?f ?A ?A'›*))\n (*goals:\n 1. ‹∀a∈{A. finite A}. set_mset (mset_set a) = a›\n 2. ‹∀a'∈{M. ∀x. count M x ≤ 1}. mset_set (set_mset a') = a'›\n 3. ‹mset_set ` {A. finite A} ⊆ {M. ∀x. count M x ≤ 1}›\n 4. ‹set_mset ` {M. ∀x. count M x ≤ 1} ⊆ {A. finite A}›*)\n show \"∀A∈{A. finite A}. set_mset (mset_set A) = A\"\n by auto\n show \"∀M∈{M. ∀x. count M x ≤ 1}. mset_set (set_mset M) = M\"\n by (auto simp add: mset_set_set_mset' (*‹∀x. count ?M x ≤ 1 ⟹ mset_set (set_mset ?M) = ?M›*))\n show \"mset_set ` {A. finite A} ⊆ {M. ∀x. count M x ≤ 1}\"\n using nat_le_linear (*‹(?m::nat) ≤ (?n::nat) ∨ ?n ≤ ?m›*) by fastforce\n show \"set_mset ` {M. ∀x. count M x ≤ 1} ⊆ {A. finite A}\"\n by auto\nqed\n\nlemma bij_betw_mset_set_card:\n assumes \"finite A\"\n shows \"bij_betw mset_set {X. X ⊆ A ∧ card X = k} {M. M ⊆# mset_set A ∧ size M = k}\"\nproof (rule bij_betw_byWitness[where f'=\"set_mset\"] (*‹⟦∀a∈?A. set_mset (?f a) = a; ∀a'∈?A'. ?f (set_mset a') = a'; ?f ` ?A ⊆ ?A'; set_mset ` ?A' ⊆ ?A⟧ ⟹ bij_betw ?f ?A ?A'›*))\n (*goals:\n 1. ‹∀a∈{X. X ⊆ A ∧ card X = k}. set_mset (mset_set a) = a›\n 2. ‹∀a'∈{M. M ⊆# mset_set A ∧ size M = k}. mset_set (set_mset a') = a'›\n 3. ‹mset_set ` {X. X ⊆ A ∧ card X = k} ⊆ {M. M ⊆# mset_set A ∧ size M = k}›\n 4. ‹set_mset ` {M. M ⊆# mset_set A ∧ size M = k} ⊆ {X. X ⊆ A ∧ card X = k}›*)\n show \"∀X∈{X. X ⊆ A ∧ card X = k}. set_mset (mset_set X) = X\"\n using ‹finite A› (*‹finite A›*) rev_finite_subset[of A] (*‹⟦finite A; ?A ⊆ A⟧ ⟹ finite ?A›*) by auto\n show \"∀M∈{M. M ⊆# mset_set A ∧ size M = k}. mset_set (set_mset M) = M\"\n by (auto simp add: mset_set_set_mset (*‹?M ⊆# mset_set ?A ⟹ mset_set (set_mset ?M) = ?M›*))\n show \"mset_set ` {X. X ⊆ A ∧ card X = k} ⊆ {M. M ⊆# mset_set A ∧ size M = k}\"\n using ‹finite A› (*‹finite A›*) rev_finite_subset[of A] (*‹⟦finite A; ?A ⊆ A⟧ ⟹ finite ?A›*) by (auto simp add: mset_set_subseteq_mset_set (*‹⟦finite ?B; ?A ⊆ ?B⟧ ⟹ mset_set ?A ⊆# mset_set ?B›*))\n show \"set_mset ` {M. M ⊆# mset_set A ∧ size M = k} ⊆ {X. X ⊆ A ∧ card X = k}\"\n using assms (*‹finite A›*) mset_subset_eqD (*‹⟦?A ⊆# ?B; ?x ∈# ?A⟧ ⟹ ?x ∈# ?B›*) card_set_mset (*‹?M ⊆# mset_set ?A ⟹ card (set_mset ?M) = size ?M›*) by fastforce\nqed\n\nlemma bij_betw_mset_set_card':\n assumes \"finite A\"\n shows \"bij_betw mset_set {X. X ⊆ A ∧ card X = k} {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)}\"\nproof (rule bij_betw_byWitness[where f'=\"set_mset\"] (*‹⟦∀a∈?A. set_mset (?f a) = a; ∀a'∈?A'. ?f (set_mset a') = a'; ?f ` ?A ⊆ ?A'; set_mset ` ?A' ⊆ ?A⟧ ⟹ bij_betw ?f ?A ?A'›*))\n (*goals:\n 1. ‹∀a∈{X. X ⊆ A ∧ card X = k}. set_mset (mset_set a) = a›\n 2. ‹∀a'∈{M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)}. mset_set (set_mset a') = a'›\n 3. ‹mset_set ` {X. X ⊆ A ∧ card X = k} ⊆ {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)}›\n 4. ‹set_mset ` {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)} ⊆ {X. X ⊆ A ∧ card X = k}›*)\n show \"∀X∈{X. X ⊆ A ∧ card X = k}. set_mset (mset_set X) = X\"\n using ‹finite A› (*‹finite A›*) rev_finite_subset[of A] (*‹⟦finite A; ?A ⊆ A⟧ ⟹ finite ?A›*) by auto\n show \"∀M∈{M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)}. mset_set (set_mset M) = M\"\n by (auto simp add: mset_set_set_mset' (*‹∀x. count ?M x ≤ 1 ⟹ mset_set (set_mset ?M) = ?M›*))\n show \"mset_set ` {X. X ⊆ A ∧ card X = k} ⊆ {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)}\"\n using ‹finite A› (*‹finite A›*) rev_finite_subset[of A] (*‹⟦finite A; ?A ⊆ A⟧ ⟹ finite ?A›*) by (auto simp add: count_mset_set_leq' (*‹finite ?A ⟹ count (mset_set ?A) ?x ≤ Suc 0›*))\n show \"set_mset ` {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)} ⊆ {X. X ⊆ A ∧ card X = k}\"\n by (auto simp add: card_set_mset' (*‹∀x. count ?M x ≤ 1 ⟹ card (set_mset ?M) = size ?M›*))\nqed\n\nsubsection ‹Cardinality›\n\nlemma card_injective_functions_domain_permutation:\n assumes \"finite A\" \"finite B\"\n shows \"card ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) = card B choose card A\"\nproof (-)\n (*goal: ‹card ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) = card B choose card A›*)\n have \"bij_betw (subset_of A) ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) {X. X ⊆ B ∧ card X = card A}\"\n using ‹finite A› (*‹finite A›*) ‹finite B› (*‹finite B›*) by (rule bij_betw_subset_of (*‹⟦finite ?A; finite ?B⟧ ⟹ bij_betw (subset_of ?A) ({f ∈ ?A →⇩E ?B. inj_on f ?A} // domain_permutation ?A ?B) {X. X ⊆ ?B ∧ card X = card ?A}›*))\n from this (*‹bij_betw (subset_of (A::'a set)) ({f::'a ⇒ 'b ∈ A →⇩E (B::'b set). inj_on f A} // domain_permutation A B) {X::'b set. X ⊆ B ∧ card X = card A}›*) have \"card ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) = card {X. X ⊆ B ∧ card X = card A}\"\n by (rule bij_betw_same_card (*‹bij_betw ?f ?A ?B ⟹ card ?A = card ?B›*))\n also (*calculation: ‹card ({f::'a::type ⇒ 'b::type ∈ (A::'a::type set) →⇩E (B::'b::type set). inj_on f A} // domain_permutation A B) = card {X::'b::type set. X ⊆ B ∧ card X = card A}›*) have \"card {X. X ⊆ B ∧ card X = card A} = card B choose card A\"\n using ‹finite B› (*‹finite B›*) by (rule n_subsets (*‹finite ?A ⟹ card {B. B ⊆ ?A ∧ card B = ?k} = card ?A choose ?k›*))\n finally (*calculation: ‹card ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) = card B choose card A›*) show \"?thesis\"\n (*goal: ‹card ({f ∈ A →⇩E B. inj_on f A} // domain_permutation A B) = card B choose card A›*) .\nqed\n\nlemma card_multiset_only_sets:\n assumes \"finite A\"\n shows \"card {M. M ⊆# mset_set A ∧ size M = k} = card A choose k\"\nproof (-)\n (*goal: ‹card {M. M ⊆# mset_set A ∧ size M = k} = card A choose k›*)\n have \"bij_betw mset_set {X. X ⊆ A ∧ card X = k} {M. M ⊆# mset_set A ∧ size M = k}\"\n using ‹finite A› (*‹finite A›*) by (rule bij_betw_mset_set_card (*‹finite ?A ⟹ bij_betw mset_set {X. X ⊆ ?A ∧ card X = ?k} {M. M ⊆# mset_set ?A ∧ size M = ?k}›*))\n from this (*‹bij_betw mset_set {X. X ⊆ A ∧ card X = k} {M. M ⊆# mset_set A ∧ size M = k}›*) have \"card {M. M ⊆# mset_set A ∧ size M = k} = card {X. X ⊆ A ∧ card X = k}\"\n by (simp add: bij_betw_same_card (*‹bij_betw ?f ?A ?B ⟹ card ?A = card ?B›*))\n also (*calculation: ‹card {M. M ⊆# mset_set A ∧ size M = k} = card {X. X ⊆ A ∧ card X = k}›*) have \" card {X. X ⊆ A ∧ card X = k} = card A choose k\"\n using ‹finite A› (*‹finite (A::'a set)›*) by (rule n_subsets (*‹finite ?A ⟹ card {B. B ⊆ ?A ∧ card B = ?k} = card ?A choose ?k›*))\n finally (*calculation: ‹card {M. M ⊆# mset_set A ∧ size M = k} = card A choose k›*) show \"?thesis\"\n (*goal: ‹card {M::'a multiset. M ⊆# mset_set (A::'a set) ∧ size M = (k::nat)} = card A choose k›*) .\nqed\n\nlemma card_multiset_only_sets':\n assumes \"finite A\"\n shows \"card {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)} = card A choose k\"\nproof (-)\n (*goal: ‹card {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)} = card A choose k›*)\n from ‹finite A› (*‹finite A›*) have \"{M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)} =\n {M. M ⊆# mset_set A ∧ size M = k}\"\n using msubset_mset_set_iff (*‹finite ?A ⟹ (set_mset ?M ⊆ ?A ∧ (∀x. count ?M x ≤ 1)) = (?M ⊆# mset_set ?A)›*) by auto\n from this (*‹{M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)} = {M. M ⊆# mset_set A ∧ size M = k}›*) ‹finite A› (*‹finite (A::'a set)›*) card_multiset_only_sets (*‹finite ?A ⟹ card {M. M ⊆# mset_set ?A ∧ size M = ?k} = card ?A choose ?k›*) show \"?thesis\"\n (*goal: ‹card {M. set_mset M ⊆ A ∧ size M = k ∧ (∀x. count M x ≤ 1)} = card A choose k›*)\n by simp\nqed\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Twelvefold_Way/Twelvefold_Way_Entry5.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"cbba1cf7ac9948639e040a1ae68c90e71291076d49f6f712c7b390204ccb4be8\"\n}"}}},{"rowIdx":153,"cells":{"text":{"kind":"string","value":"theory Key\n imports\n \"../Fused_Resource\"\nbegin\n\n\nsection ‹Key specification›\n\nlocale ideal_key =\n fixes valid_keys :: \"'key set\"\nbegin\n\nsubsection ‹Data-types for Parties, State, Events, Input, and Output›\n\ndatatype party = Alice | Bob\n\ntype_synonym s_shell = \"party set\"\ndatatype 'key' s_kernel = PState_Store | State_Store 'key'\ntype_synonym 'key' state = \"'key' s_kernel × s_shell\"\n\ndatatype event = Event_Shell party | Event_Kernel\n\ndatatype iadv = Inp_Adversary\n\ndatatype iusr_alice = Inp_Alice\ndatatype iusr_bob = Inp_Bob\ntype_synonym iusr = \"iusr_alice + iusr_bob\"\n\ndatatype oadv = Out_Adversary\n\ndatatype 'key' ousr_alice = Out_Alice 'key'\ndatatype 'key' ousr_bob = Out_Bob 'key'\ntype_synonym 'key' ousr = \"'key' ousr_alice + 'key' ousr_bob\"\n\n\nsubsubsection ‹Basic lemmas for automated handling of party sets (i.e. @{text s_shell})›\n\nlemma Alice_neq_iff [simp]: \"Alice ≠ x ⟷ x = Bob\"\n apply (cases x)\n (*goals:\n 1. ‹x = Alice ⟹ (Alice ≠ x) = (x = Bob)›\n 2. ‹x = Bob ⟹ (Alice ≠ x) = (x = Bob)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma neq_Alice_iff [simp]: \"x ≠ Alice ⟷ x = Bob\"\n apply (cases x)\n (*goals:\n 1. ‹x = Alice ⟹ (x ≠ Alice) = (x = Bob)›\n 2. ‹x = Bob ⟹ (x ≠ Alice) = (x = Bob)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma Bob_neq_iff [simp]: \"Bob ≠ x ⟷ x = Alice\"\n apply (cases x)\n (*goals:\n 1. ‹x = Alice ⟹ (Bob ≠ x) = (x = Alice)›\n 2. ‹x = Bob ⟹ (Bob ≠ x) = (x = Alice)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma neq_Bob_iff [simp]: \"x ≠ Bob ⟷ x = Alice\"\n apply (cases x)\n (*goals:\n 1. ‹x = Alice ⟹ (x ≠ Bob) = (x = Alice)›\n 2. ‹x = Bob ⟹ (x ≠ Bob) = (x = Alice)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma Alice_in_iff_nonempty: \"Alice ∈ A ⟷ A ≠ {}\" if \"Bob ∉ A\"\n using that (*‹Bob ∉ A›*) apply auto\n (*goal: ‹(Alice ∈ (A::party set)) = (A ≠ {})›*)\n by (metis (full_types) party.exhaust (*‹⟦?y = Alice ⟹ ?P; ?y = Bob ⟹ ?P⟧ ⟹ ?P›*))\n\nlemma Bob_in_iff_nonempty: \"Bob ∈ A ⟷ A ≠ {}\" if \"Alice ∉ A\"\n using that (*‹Alice ∉ (A::party set)›*) apply auto\n (*goal: ‹(Bob ∈ A) = (A ≠ {})›*)\n by (metis (full_types) party.exhaust (*‹⟦?y = Alice ⟹ ?P; ?y = Bob ⟹ ?P⟧ ⟹ ?P›*))\n\n\nsubsection ‹Defining the event handler›\n\nfun poke :: \"('key state, event) handler\"\n where\n \"poke (s_kernel, parties) (Event_Shell party) = \n (if party ∈ parties then \n return_pmf None\n else \n return_spmf (s_kernel, insert party parties))\"\n | \"poke (PState_Store, s_shell) (Event_Kernel) = do {\n key ← spmf_of_set valid_keys;\n return_spmf (State_Store key, s_shell) } \"\n | \"poke _ _ = return_pmf None\"\n\nlemma in_set_spmf_poke:\n \"s' ∈ set_spmf (poke s x) ⟷\n (∃s_kernel parties party. s = (s_kernel, parties) ∧ x = Event_Shell party ∧ party ∉ parties ∧ s' = (s_kernel, insert party parties)) ∨\n (∃s_shell key. s = (PState_Store, s_shell) ∧ x = Event_Kernel ∧ key ∈ valid_keys ∧ finite valid_keys ∧ s' = (State_Store key, s_shell))\"\n apply (cases \"(s, x)\" rule: poke.cases (*‹⟦⋀s_kernel parties party. ?x = ((s_kernel, parties), Event_Shell party) ⟹ ?P; ⋀s_shell. ?x = ((PState_Store, s_shell), Event_Kernel) ⟹ ?P; ⋀vb va. ?x = ((State_Store vb, va), Event_Kernel) ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀(s_kernel::'key s_kernel) (parties::party set) party::party. (s::'key s_kernel × party set, x::event) = ((s_kernel, parties), Event_Shell party) ⟹ ((s'::'key s_kernel × party set) ∈ set_spmf (poke s x)) = ((∃(s_kernel::'key s_kernel) (parties::party set) party::party. s = (s_kernel, parties) ∧ x = Event_Shell party ∧ party ∉ parties ∧ s' = (s_kernel, insert party parties)) ∨ (∃(s_shell::party set) key::'key. s = (PState_Store, s_shell) ∧ x = Event_Kernel ∧ key ∈ (valid_keys::'key set) ∧ finite valid_keys ∧ s' = (State_Store key, s_shell)))›\n 2. ‹⋀s_shell::party set. (s::'key s_kernel × party set, x::event) = ((PState_Store, s_shell), Event_Kernel) ⟹ ((s'::'key s_kernel × party set) ∈ set_spmf (poke s x)) = ((∃(s_kernel::'key s_kernel) (parties::party set) party::party. s = (s_kernel, parties) ∧ x = Event_Shell party ∧ party ∉ parties ∧ s' = (s_kernel, insert party parties)) ∨ (∃(s_shell::party set) key::'key. s = (PState_Store, s_shell) ∧ x = Event_Kernel ∧ key ∈ (valid_keys::'key set) ∧ finite valid_keys ∧ s' = (State_Store key, s_shell)))›\n 3. ‹⋀(vb::'key) va::party set. (s::'key s_kernel × party set, x::event) = ((State_Store vb, va), Event_Kernel) ⟹ ((s'::'key s_kernel × party set) ∈ set_spmf (poke s x)) = ((∃(s_kernel::'key s_kernel) (parties::party set) party::party. s = (s_kernel, parties) ∧ x = Event_Shell party ∧ party ∉ parties ∧ s' = (s_kernel, insert party parties)) ∨ (∃(s_shell::party set) key::'key. s = (PState_Store, s_shell) ∧ x = Event_Kernel ∧ key ∈ (valid_keys::'key set) ∧ finite valid_keys ∧ s' = (State_Store key, s_shell)))›\n discuss goal 1*)\n apply ((auto simp add: set_spmf_of_set (*‹set_spmf (spmf_of_set ?A) = (if finite ?A then ?A else {})›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: set_spmf_of_set (*‹set_spmf (spmf_of_set ?A) = (if finite ?A then ?A else {})›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: set_spmf_of_set (*‹set_spmf (spmf_of_set ?A) = (if finite ?A then ?A else {})›*))[1])\n (*proven 3 subgoals*) .\n\nlemma foldl_poke_invar:\n \"⟦ (s_kernel', parties') ∈ set_spmf (foldl_spmf poke p events); ∀(s_kernel, parties)∈set_spmf p. set_s_kernel s_kernel ⊆ valid_keys ⟧\n ⟹ set_s_kernel s_kernel' ⊆ valid_keys\"\n apply (induction events arbitrary: parties' rule: rev_induct)\n (*goals:\n 1. ‹⋀parties'. ⟦(s_kernel', parties') ∈ set_spmf (foldl_spmf poke p []); ∀(s_kernel, parties)∈set_spmf p. set_s_kernel s_kernel ⊆ valid_keys⟧ ⟹ set_s_kernel s_kernel' ⊆ valid_keys›\n 2. ‹⋀x xs parties'. ⟦⋀parties'. ⟦(s_kernel', parties') ∈ set_spmf (foldl_spmf poke p xs); ∀(s_kernel, parties)∈set_spmf p. set_s_kernel s_kernel ⊆ valid_keys⟧ ⟹ set_s_kernel s_kernel' ⊆ valid_keys; (s_kernel', parties') ∈ set_spmf (foldl_spmf poke p (xs @ [x])); ∀(s_kernel, parties)∈set_spmf p. set_s_kernel s_kernel ⊆ valid_keys⟧ ⟹ set_s_kernel s_kernel' ⊆ valid_keys›\n discuss goal 1*)\n apply ((auto 4 3 simp add: split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) foldl_spmf_append (*‹foldl_spmf ?f ?p (?xs @ ?ys) = foldl_spmf ?f (foldl_spmf ?f ?p ?xs) ?ys›*) in_set_spmf_poke (*‹(?s' ∈ set_spmf (poke ?s ?x)) = ((∃s_kernel parties party. ?s = (s_kernel, parties) ∧ ?x = Event_Shell party ∧ party ∉ parties ∧ ?s' = (s_kernel, insert party parties)) ∨ (∃s_shell key. ?s = (PState_Store, s_shell) ∧ ?x = Event_Kernel ∧ key ∈ valid_keys ∧ finite valid_keys ∧ ?s' = (State_Store key, s_shell)))›*) dest: bspec (*‹⟦∀x∈?A. ?P x; ?x ∈ ?A⟧ ⟹ ?P ?x›*))[1])\n (*discuss goal 2*)\n apply ((auto 4 3 simp add: split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) foldl_spmf_append (*‹foldl_spmf ?f ?p (?xs @ ?ys) = foldl_spmf ?f (foldl_spmf ?f ?p ?xs) ?ys›*) in_set_spmf_poke (*‹(?s' ∈ set_spmf (poke ?s ?x)) = ((∃s_kernel parties party. ?s = (s_kernel, parties) ∧ ?x = Event_Shell party ∧ party ∉ parties ∧ ?s' = (s_kernel, insert party parties)) ∨ (∃s_shell key. ?s = (PState_Store, s_shell) ∧ ?x = Event_Kernel ∧ key ∈ valid_keys ∧ finite valid_keys ∧ ?s' = (State_Store key, s_shell)))›*) dest: bspec (*‹⟦∀x∈?A. ?P x; ?x ∈ ?A⟧ ⟹ ?P ?x›*))[1])\n (*proven 2 subgoals*) .\n\nsubsection ‹Defining the adversary interface›\n\nfun iface_adv :: \"('key state, iadv, oadv) oracle'\"\n where\n \"iface_adv state _ = return_spmf (Out_Adversary, state)\"\n\n\nsubsection ‹Defining the user interfaces›\n\ncontext\nbegin\n\nprivate fun iface_usr_func :: \"party ⇒ _ ⇒ _ ⇒ 'inp ⇒ ('wrap_key × 'key state) spmf\"\n where\n \"iface_usr_func party wrap (State_Store key, parties) inp = \n (if party ∈ parties then\n return_spmf (wrap key, State_Store key, parties)\n else\n return_pmf None)\"\n | \"iface_usr_func _ _ _ _ = return_pmf None\"\n\nabbreviation iface_alice :: \"('key state, iusr_alice, 'key ousr_alice) oracle'\"\n where\n \"iface_alice ≡ iface_usr_func Alice Out_Alice\"\n\nabbreviation iface_bob :: \"('key state, iusr_bob, 'key ousr_bob) oracle'\"\n where\n \"iface_bob ≡ iface_usr_func Bob Out_Bob\"\n\nabbreviation iface_usr :: \"('key state, iusr, 'key ousr) oracle'\"\n where\n \"iface_usr ≡ plus_oracle iface_alice iface_bob\"\n\nlemma in_set_iface_usr_func [simp]:\n \"x ∈ set_spmf (iface_usr_func party wrap state inp) ⟷\n (∃key parties. state = (State_Store key, parties) ∧ party ∈ parties ∧ x = (wrap key, State_Store key, parties))\"\n apply (cases \"(party, wrap, state, inp)\" rule: iface_usr_func.cases (*‹⟦⋀party wrap key parties inp. ?x = (party, wrap, (State_Store key, parties), inp) ⟹ ?P; ⋀uu_ uv_ va ux_. ?x = (uu_, uv_, (PState_Store, va), ux_) ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀partya wrapa key parties inpa. (party, wrap, state, inp) = (partya, wrapa, (State_Store key, parties), inpa) ⟹ (x ∈ set_spmf (iface_usr_func party wrap state inp)) = (∃key parties. state = (State_Store key, parties) ∧ party ∈ parties ∧ x = (wrap key, State_Store key, parties))›\n 2. ‹⋀uu_ uv_ va ux_. (party, wrap, state, inp) = (uu_, uv_, (PState_Store, va), ux_) ⟹ (x ∈ set_spmf (iface_usr_func party wrap state inp)) = (∃key parties. state = (State_Store key, parties) ∧ party ∈ parties ∧ x = (wrap key, State_Store key, parties))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nend\n\n\nsubsection ‹Defining the Fuse Resource›\n\nprimcorec core :: \"('key state, event, iadv, iusr, oadv,'key ousr) core\" \n where\n \"cpoke core = poke\"\n | \"cfunc_adv core = iface_adv\"\n | \"cfunc_usr core = iface_usr\"\n\nsublocale fused_resource core \"(PState_Store, {})\" .\n\n\nsubsubsection ‹Lemma showing that the resulting resource is well-typed›\n\nlemma WT_core [WT_intro]: \n \"WT_core ℐ_full (ℐ_uniform UNIV (Out_Alice ` valid_keys) ⊕⇩ℐ ℐ_uniform UNIV (Out_Bob ` valid_keys))\n (pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True)) core\"\n apply (rule WT_core.intros (*‹⟦⋀s e s'. ⟦s' ∈ set_spmf (cpoke ?core s e); ?I s⟧ ⟹ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_adv ?core s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_usr ?core s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'⟧ ⟹ WT_core ?ℐ_adv ?ℐ_usr ?I ?core›*))\n (*goal: ‹WT_core ℐ_full (ℐ_uniform UNIV (Out_Alice ` valid_keys) ⊕⇩ℐ ℐ_uniform UNIV (Out_Bob ` valid_keys)) (pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True)) core›*)\n subgoal for s and e and s'\n apply (cases \"(s, e)\" rule: poke.cases (*‹⟦⋀s_kernel parties party. ?x = ((s_kernel, parties), Event_Shell party) ⟹ ?P; ⋀s_shell. ?x = ((PState_Store, s_shell), Event_Kernel) ⟹ ?P; ⋀vb va. ?x = ((State_Store vb, va), Event_Kernel) ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀s_kernel parties party. ⟦s' ∈ set_spmf (cpoke core s e); pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s; (s, e) = ((s_kernel, parties), Event_Shell party)⟧ ⟹ pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s'›\n 2. ‹⋀s_shell. ⟦s' ∈ set_spmf (cpoke core s e); pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s; (s, e) = ((PState_Store, s_shell), Event_Kernel)⟧ ⟹ pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s'›\n 3. ‹⋀vb va. ⟦s' ∈ set_spmf (cpoke core s e); pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s; (s, e) = ((State_Store vb, va), Event_Kernel)⟧ ⟹ pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s'›\n discuss goal 1*)\n apply ((auto split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*) simp add: set_spmf_of_set (*‹set_spmf (spmf_of_set ?A) = (if finite ?A then ?A else {})›*))[1])\n (*discuss goal 2*)\n apply ((auto split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*) simp add: set_spmf_of_set (*‹set_spmf (spmf_of_set ?A) = (if finite ?A then ?A else {})›*))[1])\n (*discuss goal 3*)\n apply ((auto split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*) simp add: set_spmf_of_set (*‹set_spmf (spmf_of_set ?A) = (if finite ?A then ?A else {})›*))[1])\n (*proven 3 subgoals*) .\n (*goals:\n 1. ‹⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_adv core s x); x ∈ outs_ℐ ℐ_full; pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s⟧ ⟹ y ∈ responses_ℐ ℐ_full x ∧ pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s'›\n 2. ‹⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_usr core s x); x ∈ outs_ℐ (ℐ_uniform UNIV (Out_Alice ` valid_keys) ⊕⇩ℐ ℐ_uniform UNIV (Out_Bob ` valid_keys)); pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s⟧ ⟹ y ∈ responses_ℐ (ℐ_uniform UNIV (Out_Alice ` valid_keys) ⊕⇩ℐ ℐ_uniform UNIV (Out_Bob ` valid_keys)) x ∧ pred_prod (pred_s_kernel (λkey. key ∈ valid_keys)) (λ_. True) s'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma WT_fuse [WT_intro]: \n assumes [WT_intro]: \"WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest\"\n shows \"(ℐ_full ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ ((ℐ_uniform UNIV (Out_Alice ` valid_keys) ⊕⇩ℐ ℐ_uniform UNIV (Out_Bob ` valid_keys)) ⊕⇩ℐ ℐ_usr_rest) ⊢res resource rest √\"\n apply (rule WT_intro (*‹map_ℐ ?g ?h ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv' ?f ?g ?h ?gpv √› ‹map_ℐ ?g id ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv ?f ?g ?gpv √› ‹⟦?ℐ1.0 ⊢res ?res1.0 √; ?ℐ2.0 ⊢res ?res2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0 ⊢res ?res1.0 ∥ ?res2.0 √› ‹⟦map_ℐ (inv ?f) (inv ?g) ?ℐ, map_ℐ ?f' ?g' ?ℐ' ⊢⇩C ?conv √; inj ?f; surj ?g⟧ ⟹ ?ℐ, ?ℐ' ⊢⇩C map_converter ?f ?g ?f' ?g' ?conv √› ‹?ℐ, ?ℐ ⊢⇩C 1⇩C √› ‹?ℐ, ?ℐ' ⊢⇩C ⊥⇩C √› ‹⟦?ℐ1.0, ?ℐ2.0 ⊢⇩C ?conv1.0 √; ?ℐ1', ?ℐ2' ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ1', ?ℐ2.0 ⊕⇩ℐ ?ℐ2' ⊢⇩C ?conv1.0 |₌ ?conv2.0 √› ‹⟦?ℐ1.0, ?ℐ ⊢⇩C ?conv1.0 √; ?ℐ2.0, ?ℐ ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0, ?ℐ ⊢⇩C ?conv1.0 |⇩∝ ?conv2.0 √› ‹?ℐ ⊢res ?res √ ⟹ ?ℐ, ?ℐ' ⊢⇩C converter_of_resource ?res √› ‹?ℐ, ?ℐ' ⊢⇩C ?cnv √ ⟹ ?ℐ, ?ℐ' ⊢⇩C restrict_converter ?A ?ℐ' ?cnv √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ' ⊢res ?res √⟧ ⟹ ?ℐ ⊢res ?conv ⊳ ?res √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ', ?ℐ'' ⊢⇩C ?conv' √⟧ ⟹ ?ℐ, ?ℐ'' ⊢⇩C ?conv ⊙ ?conv' √› and more 22 facts*))\n (*goals:\n 1. ‹WT_core ℐ_full (ℐ_uniform UNIV (Out_Alice ` valid_keys) ⊕⇩ℐ ℐ_uniform UNIV (Out_Bob ` valid_keys)) ?I_core core›\n 2. ‹WT_rest ℐ_adv_rest ℐ_usr_rest ?I_rest rest›\n 3. ‹?I_core (PState_Store, {})›\n discuss goal 1*)\n apply (rule WT_intro (*‹map_ℐ ?g ?h ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv' ?f ?g ?h ?gpv √› ‹map_ℐ ?g id ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv ?f ?g ?gpv √› ‹⟦?ℐ1.0 ⊢res ?res1.0 √; ?ℐ2.0 ⊢res ?res2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0 ⊢res ?res1.0 ∥ ?res2.0 √› ‹⟦map_ℐ (inv ?f) (inv ?g) ?ℐ, map_ℐ ?f' ?g' ?ℐ' ⊢⇩C ?conv √; inj ?f; surj ?g⟧ ⟹ ?ℐ, ?ℐ' ⊢⇩C map_converter ?f ?g ?f' ?g' ?conv √› ‹?ℐ, ?ℐ ⊢⇩C 1⇩C √› ‹?ℐ, ?ℐ' ⊢⇩C ⊥⇩C √› ‹⟦?ℐ1.0, ?ℐ2.0 ⊢⇩C ?conv1.0 √; ?ℐ1', ?ℐ2' ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ1', ?ℐ2.0 ⊕⇩ℐ ?ℐ2' ⊢⇩C ?conv1.0 |₌ ?conv2.0 √› ‹⟦?ℐ1.0, ?ℐ ⊢⇩C ?conv1.0 √; ?ℐ2.0, ?ℐ ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0, ?ℐ ⊢⇩C ?conv1.0 |⇩∝ ?conv2.0 √› ‹?ℐ ⊢res ?res √ ⟹ ?ℐ, ?ℐ' ⊢⇩C converter_of_resource ?res √› ‹?ℐ, ?ℐ' ⊢⇩C ?cnv √ ⟹ ?ℐ, ?ℐ' ⊢⇩C restrict_converter ?A ?ℐ' ?cnv √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ' ⊢res ?res √⟧ ⟹ ?ℐ ⊢res ?conv ⊳ ?res √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ', ?ℐ'' ⊢⇩C ?conv' √⟧ ⟹ ?ℐ, ?ℐ'' ⊢⇩C ?conv ⊙ ?conv' √› and more 22 facts*))\n (*discuss goal 2*)\n apply (rule WT_intro (*‹map_ℐ ?g ?h ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv' ?f ?g ?h ?gpv √› ‹map_ℐ ?g id ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv ?f ?g ?gpv √› ‹⟦?ℐ1.0 ⊢res ?res1.0 √; ?ℐ2.0 ⊢res ?res2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0 ⊢res ?res1.0 ∥ ?res2.0 √› ‹⟦map_ℐ (inv ?f) (inv ?g) ?ℐ, map_ℐ ?f' ?g' ?ℐ' ⊢⇩C ?conv √; inj ?f; surj ?g⟧ ⟹ ?ℐ, ?ℐ' ⊢⇩C map_converter ?f ?g ?f' ?g' ?conv √› ‹?ℐ, ?ℐ ⊢⇩C 1⇩C √› ‹?ℐ, ?ℐ' ⊢⇩C ⊥⇩C √› ‹⟦?ℐ1.0, ?ℐ2.0 ⊢⇩C ?conv1.0 √; ?ℐ1', ?ℐ2' ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ1', ?ℐ2.0 ⊕⇩ℐ ?ℐ2' ⊢⇩C ?conv1.0 |₌ ?conv2.0 √› ‹⟦?ℐ1.0, ?ℐ ⊢⇩C ?conv1.0 √; ?ℐ2.0, ?ℐ ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0, ?ℐ ⊢⇩C ?conv1.0 |⇩∝ ?conv2.0 √› ‹?ℐ ⊢res ?res √ ⟹ ?ℐ, ?ℐ' ⊢⇩C converter_of_resource ?res √› ‹?ℐ, ?ℐ' ⊢⇩C ?cnv √ ⟹ ?ℐ, ?ℐ' ⊢⇩C restrict_converter ?A ?ℐ' ?cnv √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ' ⊢res ?res √⟧ ⟹ ?ℐ ⊢res ?conv ⊳ ?res √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ', ?ℐ'' ⊢⇩C ?conv' √⟧ ⟹ ?ℐ, ?ℐ'' ⊢⇩C ?conv ⊙ ?conv' √› and more 22 facts*))\n (*discuss goal 3*)\n apply simp\n (*proven 3 subgoals*) .\n \nend\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Constructive_Cryptography_CM/Specifications/Key.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"49321e2c7b0c8ec56b1cc761d02802d392a4bd71af60041136f6baab5a0dfba0\"\n}"}}},{"rowIdx":154,"cells":{"text":{"kind":"string","value":"(*\n * Copyright 2016, Data61, CSIRO\n *\n * This software may be distributed and modified according to the terms of\n * the BSD 2-Clause license. Note that NO WARRANTY is provided.\n * See \"LICENSE_BSD2.txt\" for details.\n *\n * @TAG(DATA61_BSD)\n *)\n\nsection ‹The COMPLX Syntax›\n\ntheory Language\nimports Main\nbegin\n\nsubsection ‹The Core Language›\n\ntext ‹We use a shallow embedding of boolean expressions as well as assertions\nas sets of states. \n›\n\ntype_synonym 's bexp = \"'s set\"\ntype_synonym 's assn = \"'s set\"\n\ndatatype ('s, 'p, 'f) com =\n Skip\n | Basic \"'s ⇒ 's\"\n | Spec \"('s × 's) set\"\n | Seq \"('s ,'p, 'f) com\" \"('s,'p, 'f) com\" \n | Cond \"'s bexp\" \"('s,'p,'f) com\" \"('s,'p,'f) com\"\n | While \"'s bexp\" \"('s,'p,'f) com\"\n | Call \"'p\"\n | DynCom \"'s ⇒ ('s,'p,'f) com\" \n | Guard \"'f\" \"'s bexp\" \"('s,'p,'f) com\" \n | Throw\n | Catch \"('s,'p,'f) com\" \"('s,'p,'f) com\"\n | Parallel \"('s, 'p, 'f) com list\"\n | Await \"'s bexp\" \"('s,'p,'f) com\"\n\n\nsubsection ‹Derived Language Constructs›\n\ntext ‹We inherit the remainder of derived language constructs from SIMPL›\n\ndefinition\n raise:: \"('s ⇒ 's) ⇒ ('s,'p,'f) com\" where\n \"raise f = Seq (Basic f) Throw\"\n\ndefinition\n condCatch:: \"('s,'p,'f) com ⇒ 's bexp ⇒ ('s,'p,'f) com ⇒ ('s,'p,'f) com\" where\n \"condCatch c₁ b c₂ = Catch c₁ (Cond b c₂ Throw)\"\n\ndefinition\n bind:: \"('s ⇒ 'v) ⇒ ('v ⇒ ('s,'p,'f) com) ⇒ ('s,'p,'f) com\" where\n \"bind e c = DynCom (λs. c (e s))\"\n\ndefinition\n bseq:: \"('s,'p,'f) com ⇒ ('s,'p,'f) com ⇒ ('s,'p,'f) com\" where\n \"bseq = Seq\"\n \ndefinition\n block:: \"['s⇒'s ,('s,'p,'f) com,'s⇒'s⇒'s,'s⇒'s⇒('s,'p,'f) com]⇒('s,'p,'f) com\"\nwhere\n \"block init bdy restore return =\n DynCom (λs. (Seq (Catch (Seq (Basic init) bdy) (Seq (Basic (restore s)) Throw)) \n (DynCom (λt. Seq (Basic (restore s)) (return s t))))\n )\" \n\ndefinition\n call:: \"('s⇒'s) ⇒ 'p ⇒ ('s ⇒ 's ⇒ 's)⇒('s⇒'s⇒('s,'p,'f) com)⇒('s,'p,'f)com\" where\n \"call init p restore return = block init (Call p) restore return\"\n\ndefinition\n dynCall:: \"('s⇒'s) ⇒ ('s ⇒ 'p) ⇒ ('s ⇒ 's ⇒ 's) ⇒\n ('s ⇒ 's ⇒ ('s,'p,'f) com) ⇒ ('s,'p,'f) com\" where\n \"dynCall init p restore return = DynCom (λs. call init (p s) restore return)\"\n\ndefinition\n fcall:: \"('s⇒'s) ⇒ 'p ⇒ ('s ⇒ 's ⇒ 's)⇒('s ⇒ 'v) ⇒ ('v⇒('s,'p,'f) com)\n ⇒('s,'p,'f) com\" where\n \"fcall init p restore result return = call init p restore (λs t. return (result t))\"\n\ndefinition\n lem:: \"'x ⇒ ('s,'p,'f)com ⇒('s,'p,'f)com\" where\n \"lem x c = c\"\n\nprimrec switch:: \"('s ⇒ 'v) ⇒ ('v set × ('s,'p,'f) com) list ⇒ ('s,'p,'f) com\" \nwhere\n\"switch v [] = Skip\" |\n\"switch v (Vc#vs) = Cond {s. v s ∈ fst Vc} (snd Vc) (switch v vs)\"\n\ndefinition guaranteeStrip:: \"'f ⇒ 's set ⇒ ('s,'p,'f) com ⇒ ('s,'p,'f) com\"\n where \"guaranteeStrip f g c = Guard f g c\"\n\ndefinition guaranteeStripPair:: \"'f ⇒ 's set ⇒ ('f × 's set)\"\n where \"guaranteeStripPair f g = (f,g)\"\n\nprimrec guards:: \"('f × 's set ) list ⇒ ('s,'p,'f) com ⇒ ('s,'p,'f) com\"\nwhere\n\"guards [] c = c\" |\n\"guards (g#gs) c = Guard (fst g) (snd g) (guards gs c)\"\n\ndefinition\n while:: \"('f × 's set) list ⇒ 's bexp ⇒ ('s,'p,'f) com ⇒ ('s, 'p, 'f) com\"\nwhere\n \"while gs b c = guards gs (While b (Seq c (guards gs Skip)))\"\n\ndefinition\n whileAnno:: \n \"'s bexp ⇒ 's assn ⇒ ('s × 's) assn ⇒ ('s,'p,'f) com ⇒ ('s,'p,'f) com\" where\n \"whileAnno b I V c = While b c\"\n\ndefinition\n whileAnnoG:: \n \"('f × 's set) list ⇒ 's bexp ⇒ 's assn ⇒ ('s × 's) assn ⇒ \n ('s,'p,'f) com ⇒ ('s,'p,'f) com\" where\n \"whileAnnoG gs b I V c = while gs b c\"\n \ndefinition\n specAnno:: \"('a ⇒ 's assn) ⇒ ('a ⇒ ('s,'p,'f) com) ⇒ \n ('a ⇒ 's assn) ⇒ ('a ⇒ 's assn) ⇒ ('s,'p,'f) com\"\n where \"specAnno P c Q A = (c undefined)\"\n\ndefinition\n whileAnnoFix:: \n \"'s bexp ⇒ ('a ⇒ 's assn) ⇒ ('a ⇒ ('s × 's) assn) ⇒ ('a ⇒ ('s,'p,'f) com) ⇒ \n ('s,'p,'f) com\" where\n \"whileAnnoFix b I V c = While b (c undefined)\"\n\ndefinition\n whileAnnoGFix:: \n \"('f × 's set) list ⇒ 's bexp ⇒ ('a ⇒ 's assn) ⇒ ('a ⇒ ('s × 's) assn) ⇒ \n ('a ⇒ ('s,'p,'f) com) ⇒ ('s,'p,'f) com\" where\n \"whileAnnoGFix gs b I V c = while gs b (c undefined)\"\n\ndefinition if_rel::\"('s ⇒ bool) ⇒ ('s ⇒ 's) ⇒ ('s ⇒ 's) ⇒ ('s ⇒ 's) ⇒ ('s × 's) set\" \n where \"if_rel b f g h = {(s,t). if b s then t = f s else t = g s ∨ t = h s}\"\n\nlemma fst_guaranteeStripPair: \"fst (guaranteeStripPair f g) = f\"\n by (simp add: guaranteeStripPair_def (*‹guaranteeStripPair ?f ?g = (?f, ?g)›*))\n\nlemma snd_guaranteeStripPair: \"snd (guaranteeStripPair f g) = g\"\n by (simp add: guaranteeStripPair_def (*‹guaranteeStripPair ?f ?g = (?f, ?g)›*))\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Complx/Language.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"a19b24d158cca83e921a98f40c557d8c2d70dd500e803e59151f18981b46a915\"\n}"}}},{"rowIdx":155,"cells":{"text":{"kind":"string","value":"(* The definitions for a challenge suggested by Adam Chlipala *)\n\ntheory Compile\nimports \"HOL-Nominal.Nominal\"\nbegin\n\natom_decl name \n\nnominal_datatype data = \n DNat\n | DProd \"data\" \"data\"\n | DSum \"data\" \"data\"\n\nnominal_datatype ty = \n Data \"data\"\n | Arrow \"ty\" \"ty\" (\"_→_\" [100,100] 100)\n\nnominal_datatype trm = \n Var \"name\"\n | Lam \"«name»trm\" (\"Lam [_]._\" [100,100] 100)\n | App \"trm\" \"trm\"\n | Const \"nat\"\n | Pr \"trm\" \"trm\"\n | Fst \"trm\"\n | Snd \"trm\"\n | InL \"trm\"\n | InR \"trm\"\n | Case \"trm\" \"«name»trm\" \"«name»trm\" \n (\"Case _ of inl _ → _ | inr _ → _\" [100,100,100,100,100] 100)\n\nnominal_datatype dataI = OneI | NatI\n\nnominal_datatype tyI = \n DataI \"dataI\"\n | ArrowI \"tyI\" \"tyI\" (\"_→_\" [100,100] 100)\n\nnominal_datatype trmI = \n IVar \"name\"\n | ILam \"«name»trmI\" (\"ILam [_]._\" [100,100] 100)\n | IApp \"trmI\" \"trmI\"\n | IUnit\n | INat \"nat\"\n | ISucc \"trmI\"\n | IAss \"trmI\" \"trmI\" (\"_↦_\" [100,100] 100)\n | IRef \"trmI\" \n | ISeq \"trmI\" \"trmI\" (\"_;;_\" [100,100] 100)\n | Iif \"trmI\" \"trmI\" \"trmI\"\n\ntext ‹valid contexts›\n\ninductive \n valid :: \"(name×'a::pt_name) list ⇒ bool\"\nwhere\n v1[intro]: \"valid []\"\n| v2[intro]: \"⟦valid Γ;a♯Γ⟧⟹ valid ((a,σ)#Γ)\" (* maybe dom of Γ *)\n\ntext ‹typing judgements for trms›\n\ninductive \n typing :: \"(name×ty) list⇒trm⇒ty⇒bool\" (\" _ ⊢ _ : _ \" [80,80,80] 80)\nwhere\n t0[intro]: \"⟦valid Γ; (x,τ)∈set Γ⟧⟹ Γ ⊢ Var x : τ\"\n| t1[intro]: \"⟦Γ ⊢ e1 : τ1→τ2; Γ ⊢ e2 : τ1⟧⟹ Γ ⊢ App e1 e2 : τ2\"\n| t2[intro]: \"⟦x♯Γ;((x,τ1)#Γ) ⊢ t : τ2⟧ ⟹ Γ ⊢ Lam [x].t : τ1→τ2\"\n| t3[intro]: \"valid Γ ⟹ Γ ⊢ Const n : Data(DNat)\"\n| t4[intro]: \"⟦Γ ⊢ e1 : Data(σ1); Γ ⊢ e2 : Data(σ2)⟧ ⟹ Γ ⊢ Pr e1 e2 : Data (DProd σ1 σ2)\"\n| t5[intro]: \"⟦Γ ⊢ e : Data(DProd σ1 σ2)⟧ ⟹ Γ ⊢ Fst e : Data(σ1)\"\n| t6[intro]: \"⟦Γ ⊢ e : Data(DProd σ1 σ2)⟧ ⟹ Γ ⊢ Snd e : Data(σ2)\"\n| t7[intro]: \"⟦Γ ⊢ e : Data(σ1)⟧ ⟹ Γ ⊢ InL e : Data(DSum σ1 σ2)\"\n| t8[intro]: \"⟦Γ ⊢ e : Data(σ2)⟧ ⟹ Γ ⊢ InR e : Data(DSum σ1 σ2)\"\n| t9[intro]: \"⟦x1♯Γ; x2♯Γ; Γ ⊢ e: Data(DSum σ1 σ2); \n ((x1,Data(σ1))#Γ) ⊢ e1 : τ; ((x2,Data(σ2))#Γ) ⊢ e2 : τ⟧ \n ⟹ Γ ⊢ (Case e of inl x1 → e1 | inr x2 → e2) : τ\"\n\ntext ‹typing judgements for Itrms›\n\ninductive \n Ityping :: \"(name×tyI) list⇒trmI⇒tyI⇒bool\" (\" _ I⊢ _ : _ \" [80,80,80] 80)\nwhere\n t0[intro]: \"⟦valid Γ; (x,τ)∈set Γ⟧⟹ Γ I⊢ IVar x : τ\"\n| t1[intro]: \"⟦Γ I⊢ e1 : τ1→τ2; Γ I⊢ e2 : τ1⟧⟹ Γ I⊢ IApp e1 e2 : τ2\"\n| t2[intro]: \"⟦x♯Γ;((x,τ1)#Γ) I⊢ t : τ2⟧ ⟹ Γ I⊢ ILam [x].t : τ1→τ2\"\n| t3[intro]: \"valid Γ ⟹ Γ I⊢ IUnit : DataI(OneI)\"\n| t4[intro]: \"valid Γ ⟹ Γ I⊢ INat(n) : DataI(NatI)\"\n| t5[intro]: \"Γ I⊢ e : DataI(NatI) ⟹ Γ I⊢ ISucc(e) : DataI(NatI)\"\n| t6[intro]: \"⟦Γ I⊢ e : DataI(NatI)⟧ ⟹ Γ I⊢ IRef e : DataI (NatI)\"\n| t7[intro]: \"⟦Γ I⊢ e1 : DataI(NatI); Γ I⊢ e2 : DataI(NatI)⟧ ⟹ Γ I⊢ e1↦e2 : DataI(OneI)\"\n| t8[intro]: \"⟦Γ I⊢ e1 : DataI(NatI); Γ I⊢ e2 : τ⟧ ⟹ Γ I⊢ e1;;e2 : τ\"\n| t9[intro]: \"⟦Γ I⊢ e: DataI(NatI); Γ I⊢ e1 : τ; Γ I⊢ e2 : τ⟧ ⟹ Γ I⊢ Iif e e1 e2 : τ\"\n\ntext ‹capture-avoiding substitution›\n\nclass subst =\n fixes subst :: \"'a ⇒ name ⇒ 'a ⇒ 'a\" (\"_[_::=_]\" [100,100,100] 100)\n\ninstantiation trm :: subst\nbegin\n\nnominal_primrec subst_trm\nwhere\n \"(Var x)[y::=t'] = (if x=y then t' else (Var x))\"\n| \"(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])\"\n| \"⟦x♯y; x♯t'⟧ ⟹ (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])\"\n| \"(Const n)[y::=t'] = Const n\"\n| \"(Pr e1 e2)[y::=t'] = Pr (e1[y::=t']) (e2[y::=t'])\"\n| \"(Fst e)[y::=t'] = Fst (e[y::=t'])\"\n| \"(Snd e)[y::=t'] = Snd (e[y::=t'])\"\n| \"(InL e)[y::=t'] = InL (e[y::=t'])\"\n| \"(InR e)[y::=t'] = InR (e[y::=t'])\"\n| \"⟦z≠x; x♯y; x♯e; x♯e2; z♯y; z♯e; z♯e1; x♯t'; z♯t'⟧ ⟹\n (Case e of inl x → e1 | inr z → e2)[y::=t'] =\n (Case (e[y::=t']) of inl x → (e1[y::=t']) | inr z → (e2[y::=t']))\"\n apply(finite_guess)+\n apply(rule TrueI)+\n apply(simp add: abs_fresh)+\n apply(fresh_guess)+\n done\n\ninstance ..\n\nend\n\ninstantiation trmI :: subst\nbegin\n\nnominal_primrec subst_trmI\nwhere\n \"(IVar x)[y::=t'] = (if x=y then t' else (IVar x))\"\n| \"(IApp t1 t2)[y::=t'] = IApp (t1[y::=t']) (t2[y::=t'])\"\n| \"⟦x♯y; x♯t'⟧ ⟹ (ILam [x].t)[y::=t'] = ILam [x].(t[y::=t'])\"\n| \"(INat n)[y::=t'] = INat n\"\n| \"(IUnit)[y::=t'] = IUnit\"\n| \"(ISucc e)[y::=t'] = ISucc (e[y::=t'])\"\n| \"(IAss e1 e2)[y::=t'] = IAss (e1[y::=t']) (e2[y::=t'])\"\n| \"(IRef e)[y::=t'] = IRef (e[y::=t'])\"\n| \"(ISeq e1 e2)[y::=t'] = ISeq (e1[y::=t']) (e2[y::=t'])\"\n| \"(Iif e e1 e2)[y::=t'] = Iif (e[y::=t']) (e1[y::=t']) (e2[y::=t'])\"\n apply(finite_guess)+\n apply(rule TrueI)+\n apply(simp add: abs_fresh)+\n apply(fresh_guess)+\n done\n\ninstance ..\n\nend\n\nlemma Isubst_eqvt[eqvt]:\n fixes pi::\"name prm\"\n and t1::\"trmI\"\n and t2::\"trmI\"\n and x::\"name\"\n shows \"pi∙(t1[x::=t2]) = ((pi∙t1)[(pi∙x)::=(pi∙t2)])\"\n apply (nominal_induct t1 avoiding: x t2 rule: trmI.strong_induct)\n (*goals:\n 1. ‹⋀name x t2. pi ∙ IVar name[x::=t2] = (pi ∙ IVar name)[(pi ∙ x)::=(pi ∙ t2)]›\n 2. ‹⋀name trmI x t2. ⟦name ♯ x; name ♯ t2; ⋀b ba. pi ∙ trmI[b::=ba] = (pi ∙ trmI)[(pi ∙ b)::=(pi ∙ ba)]⟧ ⟹ pi ∙ ILam [name].trmI[x::=t2] = (pi ∙ ILam [name].trmI)[(pi ∙ x)::=(pi ∙ t2)]›\n 3. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. pi ∙ trmI1[b::=ba] = (pi ∙ trmI1)[(pi ∙ b)::=(pi ∙ ba)]; ⋀b ba. pi ∙ trmI2[b::=ba] = (pi ∙ trmI2)[(pi ∙ b)::=(pi ∙ ba)]⟧ ⟹ pi ∙ IApp trmI1 trmI2[x::=t2] = (pi ∙ IApp trmI1 trmI2)[(pi ∙ x)::=(pi ∙ t2)]›\n 4. ‹⋀x t2. pi ∙ IUnit[x::=t2] = (pi ∙ IUnit)[(pi ∙ x)::=(pi ∙ t2)]›\n 5. ‹⋀nat x t2. pi ∙ INat nat[x::=t2] = (pi ∙ INat nat)[(pi ∙ x)::=(pi ∙ t2)]›\n 6. ‹⋀trmI x t2. (⋀b ba. pi ∙ trmI[b::=ba] = (pi ∙ trmI)[(pi ∙ b)::=(pi ∙ ba)]) ⟹ pi ∙ ISucc trmI[x::=t2] = (pi ∙ ISucc trmI)[(pi ∙ x)::=(pi ∙ t2)]›\n 7. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. pi ∙ trmI1[b::=ba] = (pi ∙ trmI1)[(pi ∙ b)::=(pi ∙ ba)]; ⋀b ba. pi ∙ trmI2[b::=ba] = (pi ∙ trmI2)[(pi ∙ b)::=(pi ∙ ba)]⟧ ⟹ pi ∙ trmI1↦trmI2[x::=t2] = (pi ∙ trmI1↦trmI2)[(pi ∙ x)::=(pi ∙ t2)]›\n 8. ‹⋀trmI x t2. (⋀b ba. pi ∙ trmI[b::=ba] = (pi ∙ trmI)[(pi ∙ b)::=(pi ∙ ba)]) ⟹ pi ∙ IRef trmI[x::=t2] = (pi ∙ IRef trmI)[(pi ∙ x)::=(pi ∙ t2)]›\n 9. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. pi ∙ trmI1[b::=ba] = (pi ∙ trmI1)[(pi ∙ b)::=(pi ∙ ba)]; ⋀b ba. pi ∙ trmI2[b::=ba] = (pi ∙ trmI2)[(pi ∙ b)::=(pi ∙ ba)]⟧ ⟹ pi ∙ trmI1;;trmI2[x::=t2] = (pi ∙ trmI1;;trmI2)[(pi ∙ x)::=(pi ∙ t2)]›\n 10. ‹⋀trmI1 trmI2 trmI3 x t2. ⟦⋀b ba. pi ∙ trmI1[b::=ba] = (pi ∙ trmI1)[(pi ∙ b)::=(pi ∙ ba)]; ⋀b ba. pi ∙ trmI2[b::=ba] = (pi ∙ trmI2)[(pi ∙ b)::=(pi ∙ ba)]; ⋀b ba. pi ∙ trmI3[b::=ba] = (pi ∙ trmI3)[(pi ∙ b)::=(pi ∙ ba)]⟧ ⟹ pi ∙ Iif trmI1 trmI2 trmI3[x::=t2] = (pi ∙ Iif trmI1 trmI2 trmI3)[(pi ∙ x)::=(pi ∙ t2)]›\n discuss goal 1*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 2*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 3*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 4*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 5*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 6*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 7*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 8*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 9*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*discuss goal 10*)\n apply (simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) eqvts (*‹?pi ∙ Iif ?x3.0 ?x2.0 ?x1.0 = Iif (?pi ∙ ?x3.0) (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0;;?x1.0 = (?pi ∙ ?x2.0);;(?pi ∙ ?x1.0)› ‹?pi ∙ IRef ?x1.0 = IRef (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0↦?x1.0 = (?pi ∙ ?x2.0)↦(?pi ∙ ?x1.0)› ‹?pi ∙ ISucc ?x1.0 = ISucc (?pi ∙ ?x1.0)› ‹?pi ∙ INat ?x1.0 = INat (?pi ∙ ?x1.0)› ‹?pi ∙ IUnit = IUnit› ‹?pi ∙ IApp ?x2.0 ?x1.0 = IApp (?pi ∙ ?x2.0) (?pi ∙ ?x1.0)› ‹?pi ∙ ILam [?x1.0].?x2.0 = ILam [(?pi ∙ ?x1.0)].(?pi ∙ ?x2.0)› ‹?pi ∙ IVar ?x1.0 = IVar (?pi ∙ ?x1.0)› ‹?pi ∙ ?x2.0→?x1.0 = (?pi ∙ ?x2.0)→(?pi ∙ ?x1.0)› ‹?pi ∙ DataI ?x1.0 = DataI (?pi ∙ ?x1.0)› and more 72 facts*) fresh_bij (*‹?pi ∙ ?a ♯ ?pi ∙ ?x = ?a ♯ ?x›*))\n (*proven 10 subgoals*) .\n\nlemma Isubst_supp:\n fixes t1::\"trmI\"\n and t2::\"trmI\"\n and x::\"name\"\n shows \"((supp (t1[x::=t2]))::name set) ⊆ (supp t2)∪((supp t1)-{x})\"\n apply (nominal_induct t1 avoiding: x t2 rule: trmI.strong_induct)\n (*goals:\n 1. ‹⋀name x t2. supp (IVar name[x::=t2]) ⊆ supp t2 ∪ (supp (IVar name) - {x})›\n 2. ‹⋀name trmI x t2. ⟦name ♯ x; name ♯ t2; ⋀b ba. supp (trmI[b::=ba]) ⊆ supp ba ∪ (supp trmI - {b})⟧ ⟹ supp (ILam [name].trmI[x::=t2]) ⊆ supp t2 ∪ (supp (ILam [name].trmI) - {x})›\n 3. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b})⟧ ⟹ supp (IApp trmI1 trmI2[x::=t2]) ⊆ supp t2 ∪ (supp (IApp trmI1 trmI2) - {x})›\n 4. ‹⋀x t2. supp (IUnit[x::=t2]) ⊆ supp t2 ∪ (supp IUnit - {x})›\n 5. ‹⋀nat x t2. supp (INat nat[x::=t2]) ⊆ supp t2 ∪ (supp (INat nat) - {x})›\n 6. ‹⋀trmI x t2. (⋀b ba. supp (trmI[b::=ba]) ⊆ supp ba ∪ (supp trmI - {b})) ⟹ supp (ISucc trmI[x::=t2]) ⊆ supp t2 ∪ (supp (ISucc trmI) - {x})›\n 7. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b})⟧ ⟹ supp (trmI1↦trmI2[x::=t2]) ⊆ supp t2 ∪ (supp (trmI1↦trmI2) - {x})›\n 8. ‹⋀trmI x t2. (⋀b ba. supp (trmI[b::=ba]) ⊆ supp ba ∪ (supp trmI - {b})) ⟹ supp (IRef trmI[x::=t2]) ⊆ supp t2 ∪ (supp (IRef trmI) - {x})›\n 9. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b})⟧ ⟹ supp (trmI1;;trmI2[x::=t2]) ⊆ supp t2 ∪ (supp (trmI1;;trmI2) - {x})›\n 10. ‹⋀trmI1 trmI2 trmI3 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); ⋀b ba. supp (trmI3[b::=ba]) ⊆ supp ba ∪ (supp trmI3 - {b})⟧ ⟹ supp (Iif trmI1 trmI2 trmI3[x::=t2]) ⊆ supp t2 ∪ (supp (Iif trmI1 trmI2 trmI3) - {x})›\n discuss goal 1*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*goals:\n 1. ‹⋀trmI1 trmI2 x t2 xa. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); xa ∈ supp (trmI2[x::=t2]); xa ∉ supp t2; xa ∉ supp trmI2⟧ ⟹ xa ∈ supp trmI1›\n 2. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); x ∈ supp (trmI2[x::=t2]); x ∉ supp t2⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*)\n (*discuss goal 4*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*discuss goal 5*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar (?x::name)[(?y::name)::=(?t'::trmI)] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp (?t1.0::trmI) (?t2.0::trmI)[(?y::name)::=(?t'::trmI)] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦(?x::name) ♯ (?y::name); ?x ♯ (?t'::trmI)⟧ ⟹ ILam [?x].(?t::trmI)[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat (?n::nat)[(?y::name)::=(?t'::trmI)] = INat ?n› ‹IUnit[(?y::name)::=(?t'::trmI)] = IUnit› ‹ISucc (?e::trmI)[(?y::name)::=(?t'::trmI)] = ISucc (?e[?y::=?t'])› ‹(?e1.0::trmI)↦(?e2.0::trmI)[(?y::name)::=(?t'::trmI)] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef (?e::trmI)[(?y::name)::=(?t'::trmI)] = IRef (?e[?y::=?t'])› ‹(?e1.0::trmI);;(?e2.0::trmI)[(?y::name)::=(?t'::trmI)] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif (?e::trmI) (?e1.0::trmI) (?e2.0::trmI)[(?y::name)::=(?t'::trmI)] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar (?x1.0::name)) = supp ?x1.0› ‹supp (ILam [(?x1.0::name)].(?x2.0::trmI)) = supp ([?x1.0].?x2.0)› ‹supp (IApp (?x2.0::trmI) (?x1.0::trmI)) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat (?x1.0::nat)) = supp ?x1.0› ‹supp (ISucc (?x1.0::trmI)) = supp ?x1.0› ‹supp ((?x2.0::trmI)↦(?x1.0::trmI)) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef (?x1.0::trmI)) = supp ?x1.0› ‹supp ((?x2.0::trmI);;(?x1.0::trmI)) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif (?x3.0::trmI) (?x2.0::trmI) (?x1.0::trmI)) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp (?x::name) = {?x}›*) abs_supp (*‹finite (supp (?x::?'x2)) ⟹ supp ([(?a::name)].?x) = supp ?x - {?a}› ‹supp ([(?a::name)].(?x::?'x4)) = supp ?x - {?a}›*) supp_nat (*‹supp (?n::nat) = {}›*))[1])\n (*discuss goal 6*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*discuss goal 7*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*goals:\n 1. ‹⋀trmI1 trmI2 x t2 xa. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); xa ∈ supp (trmI2[x::=t2]); xa ∉ supp t2; xa ∉ supp trmI2⟧ ⟹ xa ∈ supp trmI1›\n 2. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); x ∈ supp (trmI2[x::=t2]); x ∉ supp t2⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*)\n (*discuss goal 8*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*discuss goal 9*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*goals:\n 1. ‹⋀trmI1 trmI2 x t2 xa. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); xa ∈ supp (trmI2[x::=t2]); xa ∉ supp t2; xa ∉ supp trmI2⟧ ⟹ xa ∈ supp trmI1›\n 2. ‹⋀trmI1 trmI2 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); x ∈ supp (trmI2[x::=t2]); x ∉ supp t2⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*)\n (*discuss goal 10*)\n apply ((auto simp add: subst_trmI.simps (*‹IVar ?x[?y::=?t'] = (if ?x = ?y then ?t' else IVar ?x)› ‹IApp ?t1.0 ?t2.0[?y::=?t'] = IApp (?t1.0[?y::=?t']) (?t2.0[?y::=?t'])› ‹⟦?x ♯ ?y; ?x ♯ ?t'⟧ ⟹ ILam [?x].?t[?y::=?t'] = ILam [?x].?t[?y::=?t']› ‹INat ?n[?y::=?t'] = INat ?n› ‹IUnit[?y::=?t'] = IUnit› ‹ISucc ?e[?y::=?t'] = ISucc (?e[?y::=?t'])› ‹?e1.0↦?e2.0[?y::=?t'] = ?e1.0[?y::=?t']↦?e2.0[?y::=?t']› ‹IRef ?e[?y::=?t'] = IRef (?e[?y::=?t'])› ‹?e1.0;;?e2.0[?y::=?t'] = ?e1.0[?y::=?t'];;?e2.0[?y::=?t']› ‹Iif ?e ?e1.0 ?e2.0[?y::=?t'] = Iif (?e[?y::=?t']) (?e1.0[?y::=?t']) (?e2.0[?y::=?t'])›*) trmI.supp (*‹supp (IVar ?x1.0) = supp ?x1.0› ‹supp (ILam [?x1.0].?x2.0) = supp ([?x1.0].?x2.0)› ‹supp (IApp ?x2.0 ?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp IUnit = {}› ‹supp (INat ?x1.0) = supp ?x1.0› ‹supp (ISucc ?x1.0) = supp ?x1.0› ‹supp (?x2.0↦?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (IRef ?x1.0) = supp ?x1.0› ‹supp (?x2.0;;?x1.0) = supp ?x2.0 ∪ supp ?x1.0› ‹supp (Iif ?x3.0 ?x2.0 ?x1.0) = supp ?x3.0 ∪ (supp ?x2.0 ∪ supp ?x1.0)›*) supp_atm (*‹supp ?x = {?x}›*) abs_supp (*‹finite (supp ?x) ⟹ supp ([?a].?x) = supp ?x - {?a}› ‹supp ([?a].?x) = supp ?x - {?a}›*) supp_nat (*‹supp ?n = {}›*))[1])\n (*goals:\n 1. ‹⋀trmI1 trmI2 trmI3 x t2 xa. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); ⋀b ba. supp (trmI3[b::=ba]) ⊆ supp ba ∪ (supp trmI3 - {b}); xa ∈ supp (trmI2[x::=t2]); xa ∉ supp t2; xa ∉ supp trmI1; xa ∉ supp trmI3⟧ ⟹ xa ∈ supp trmI2›\n 2. ‹⋀trmI1 trmI2 trmI3 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); ⋀b ba. supp (trmI3[b::=ba]) ⊆ supp ba ∪ (supp trmI3 - {b}); x ∈ supp (trmI2[x::=t2]); x ∉ supp t2⟧ ⟹ False›\n 3. ‹⋀trmI1 trmI2 trmI3 x t2 xa. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); ⋀b ba. supp (trmI3[b::=ba]) ⊆ supp ba ∪ (supp trmI3 - {b}); xa ∈ supp (trmI3[x::=t2]); xa ∉ supp t2; xa ∉ supp trmI1; xa ∉ supp trmI3⟧ ⟹ xa ∈ supp trmI2›\n 4. ‹⋀trmI1 trmI2 trmI3 x t2. ⟦⋀b ba. supp (trmI1[b::=ba]) ⊆ supp ba ∪ (supp trmI1 - {b}); ⋀b ba. supp (trmI2[b::=ba]) ⊆ supp ba ∪ (supp trmI2 - {b}); ⋀b ba. supp (trmI3[b::=ba]) ⊆ supp ba ∪ (supp trmI3 - {b}); x ∈ supp (trmI3[x::=t2]); x ∉ supp t2⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*discuss goal 4*)\n apply blast\n (*proven 4 subgoals*)\n (*proven 10 subgoals*) .\n\nlemma Isubst_fresh:\n fixes x::\"name\"\n and y::\"name\"\n and t1::\"trmI\"\n and t2::\"trmI\"\n assumes a: \"x♯[y].t1\" \"x♯t2\"\n shows \"x♯(t1[y::=t2])\"\nusing a (*‹x ♯ [y].t1› ‹x ♯ t2›*) apply (auto simp add: fresh_def (*‹(?a::?'x) ♯ (?x::?'a) = (?a ∉ supp ?x)›*) Isubst_supp (*‹supp ((?t1.0::trmI)[(?x::name)::=(?t2.0::trmI)]) ⊆ supp ?t2.0 ∪ (supp ?t1.0 - {?x})›*))\n(*goal: ‹x ♯ t1[y::=t2]›*)\napply (drule rev_subsetD (*‹⟦?c ∈ ?A; ?A ⊆ ?B⟧ ⟹ ?c ∈ ?B›*))\n(*goals:\n 1. ‹⟦x ∉ supp ([y].t1); x ∉ supp t2⟧ ⟹ supp (t1[y::=t2]) ⊆ ?B3›\n 2. ‹⟦x ∉ supp ([y].t1); x ∉ supp t2; x ∈ ?B3⟧ ⟹ False›\ndiscuss goal 1*)\n apply (rule Isubst_supp (*‹supp (?t1.0[?x::=?t2.0]) ⊆ supp ?t2.0 ∪ (supp ?t1.0 - {?x})›*))\n(*discuss goal 2*)\n apply (simp add: abs_supp (*‹finite (supp (?x::?'x2::pt_name)) ⟹ supp ([(?a::name)].?x) = supp ?x - {?a}› ‹supp ([(?a::name)].(?x::?'x4::fs_name)) = supp ?x - {?a}›*))\n(*proven 2 subgoals*) .\n\ntext ‹big-step evaluation for trms›\n\ninductive \n big :: \"trm⇒trm⇒bool\" (\"_ ⇓ _\" [80,80] 80)\nwhere\n b0[intro]: \"Lam [x].e ⇓ Lam [x].e\"\n| b1[intro]: \"⟦e1⇓Lam [x].e; e2⇓e2'; e[x::=e2']⇓e'⟧ ⟹ App e1 e2 ⇓ e'\"\n| b2[intro]: \"Const n ⇓ Const n\"\n| b3[intro]: \"⟦e1⇓e1'; e2⇓e2'⟧ ⟹ Pr e1 e2 ⇓ Pr e1' e2'\"\n| b4[intro]: \"e⇓Pr e1 e2 ⟹ Fst e⇓e1\"\n| b5[intro]: \"e⇓Pr e1 e2 ⟹ Snd e⇓e2\"\n| b6[intro]: \"e⇓e' ⟹ InL e ⇓ InL e'\"\n| b7[intro]: \"e⇓e' ⟹ InR e ⇓ InR e'\"\n| b8[intro]: \"⟦e⇓InL e'; e1[x::=e']⇓e''⟧ ⟹ Case e of inl x1 → e1 | inr x2 → e2 ⇓ e''\"\n| b9[intro]: \"⟦e⇓InR e'; e2[x::=e']⇓e''⟧ ⟹ Case e of inl x1 → e1 | inr x2 → e2 ⇓ e''\"\n\ninductive \n Ibig :: \"((nat⇒nat)×trmI)⇒((nat⇒nat)×trmI)⇒bool\" (\"_ I⇓ _\" [80,80] 80)\nwhere\n m0[intro]: \"(m,ILam [x].e) I⇓ (m,ILam [x].e)\"\n| m1[intro]: \"⟦(m,e1)I⇓(m',ILam [x].e); (m',e2)I⇓(m'',e3); (m'',e[x::=e3])I⇓(m''',e4)⟧ \n ⟹ (m,IApp e1 e2) I⇓ (m''',e4)\"\n| m2[intro]: \"(m,IUnit) I⇓ (m,IUnit)\"\n| m3[intro]: \"(m,INat(n))I⇓(m,INat(n))\"\n| m4[intro]: \"(m,e)I⇓(m',INat(n)) ⟹ (m,ISucc(e))I⇓(m',INat(n+1))\"\n| m5[intro]: \"(m,e)I⇓(m',INat(n)) ⟹ (m,IRef(e))I⇓(m',INat(m' n))\"\n| m6[intro]: \"⟦(m,e1)I⇓(m',INat(n1)); (m',e2)I⇓(m'',INat(n2))⟧ ⟹ (m,e1↦e2)I⇓(m''(n1:=n2),IUnit)\"\n| m7[intro]: \"⟦(m,e1)I⇓(m',IUnit); (m',e2)I⇓(m'',e)⟧ ⟹ (m,e1;;e2)I⇓(m'',e)\"\n| m8[intro]: \"⟦(m,e)I⇓(m',INat(n)); n≠0; (m',e1)I⇓(m'',e)⟧ ⟹ (m,Iif e e1 e2)I⇓(m'',e)\"\n| m9[intro]: \"⟦(m,e)I⇓(m',INat(0)); (m',e2)I⇓(m'',e)⟧ ⟹ (m,Iif e e1 e2)I⇓(m'',e)\"\n\ntext ‹Translation functions›\n\nnominal_primrec\n trans :: \"trm ⇒ trmI\"\nwhere\n \"trans (Var x) = (IVar x)\"\n| \"trans (App e1 e2) = IApp (trans e1) (trans e2)\"\n| \"trans (Lam [x].e) = ILam [x].(trans e)\"\n| \"trans (Const n) = INat n\"\n| \"trans (Pr e1 e2) = \n (let limit = IRef(INat 0) in \n let v1 = (trans e1) in \n let v2 = (trans e2) in \n (((ISucc limit)↦v1);;(ISucc(ISucc limit)↦v2));;(INat 0 ↦ ISucc(ISucc(limit))))\"\n| \"trans (Fst e) = IRef (ISucc (trans e))\"\n| \"trans (Snd e) = IRef (ISucc (ISucc (trans e)))\"\n| \"trans (InL e) = \n (let limit = IRef(INat 0) in \n let v = (trans e) in \n (((ISucc limit)↦INat(0));;(ISucc(ISucc limit)↦v));;(INat 0 ↦ ISucc(ISucc(limit))))\"\n| \"trans (InR e) = \n (let limit = IRef(INat 0) in \n let v = (trans e) in \n (((ISucc limit)↦INat(1));;(ISucc(ISucc limit)↦v));;(INat 0 ↦ ISucc(ISucc(limit))))\"\n| \"⟦x2≠x1; x1♯e; x1♯e2; x2♯e; x2♯e1⟧ ⟹ \n trans (Case e of inl x1 → e1 | inr x2 → e2) =\n (let v = (trans e) in\n let v1 = (trans e1) in\n let v2 = (trans e2) in \n Iif (IRef (ISucc v)) (v2[x2::=IRef (ISucc (ISucc v))]) (v1[x1::=IRef (ISucc (ISucc v))]))\"\n apply(finite_guess add: Let_def)+\n apply(rule TrueI)+\n apply(simp add: abs_fresh Isubst_fresh)+\n apply(fresh_guess add: Let_def)+\n done\n\nnominal_primrec\n trans_type :: \"ty ⇒ tyI\"\nwhere\n \"trans_type (Data σ) = DataI(NatI)\"\n| \"trans_type (τ1→τ2) = (trans_type τ1)→(trans_type τ2)\"\n by (rule TrueI)+\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/Nominal/Examples/Compile.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"c215a05ae71813549b4b3df1d8d5db9a9145e24414f6fb221208852fb7ae9cc2\"\n}"}}},{"rowIdx":156,"cells":{"text":{"kind":"string","value":"section ‹Characteristic of Rings\\label{sec:ring_char}›\n\ntheory Ring_Characteristic\n imports\n \"Finite_Fields_Factorization_Ext\"\n \"HOL-Algebra.IntRing\"\n \"HOL-Algebra.Embedded_Algebras\"\nbegin\n\nlocale finite_field = field +\n assumes finite_carrier: \"finite (carrier R)\"\nbegin\n\nlemma finite_field_min_order:\n \"order R > 1\"\nproof (rule ccontr (*‹(¬ (?P::bool) ⟹ False) ⟹ ?P›*))\n (*goal: ‹¬ 1 < order R ⟹ False›*)\n assume a: \"¬(1 < order R)\" (*‹¬ (1::nat) < order R›*)\n have \"{𝟬⇘R⇙,𝟭⇘R⇙} ⊆ carrier R\"\n by auto\n hence \"card {𝟬⇘R⇙,𝟭⇘R⇙} ≤ card (carrier R)\"\n using card_mono (*‹⟦finite ?B; ?A ⊆ ?B⟧ ⟹ card ?A ≤ card ?B›*) finite_carrier (*‹finite (carrier R)›*) by blast\n also (*calculation: ‹card {𝟬, 𝟭} ≤ card (carrier R)›*) have \"... ≤ 1\"\n using a (*‹¬ 1 < order R›*) by (simp add:order_def (*‹order ?S = card (carrier ?S)›*))\n finally (*calculation: ‹card {𝟬, 𝟭} ≤ 1›*) have \"card {𝟬⇘R⇙,𝟭⇘R⇙} ≤ 1\"\n by blast\n thus False\n by simp\nqed\n\nlemma (in finite_field) order_pow_eq_self:\n assumes \"x ∈ carrier R\"\n shows \"x [^] (order R) = x\"\nproof (cases \"x = 𝟬\")\n (*goals:\n 1. ‹x = 𝟬 ⟹ x [^] order R = x›\n 2. ‹x ≠ 𝟬 ⟹ x [^] order R = x›*)\n case True (*‹(x::'a) = 𝟬›*)\n have \"order R > 0\"\n using assms(1) (*‹x ∈ carrier R›*) order_gt_0_iff_finite (*‹(0 < order R) = finite (carrier R)›*) finite_carrier (*‹finite (carrier R)›*) by simp\n then obtain n where n_def: \"order R = Suc n\"\n (*goal: ‹(⋀n. order R = Suc n ⟹ thesis) ⟹ thesis›*)\n using lessE (*‹⟦?i < ?k; ?k = Suc ?i ⟹ ?P; ⋀j. ⟦?i < j; ?k = Suc j⟧ ⟹ ?P⟧ ⟹ ?P›*) by blast\n have \"x [^] (order R) = 𝟬\"\n unfolding n_def\n (*goal: ‹x [^] Suc n = 𝟬›*)\n using True (*‹x = 𝟬›*) apply (subst nat_pow_Suc (*‹?x [^] Suc ?n = ?x [^] ?n ⊗ ?x›*))\n (*goal: ‹(x::'a) [^] Suc (n::nat) = 𝟬›*)\n by simp\n thus \"?thesis\"\n (*goal: ‹x [^] order R = x›*)\n using True (*‹x = 𝟬›*) by simp\nnext\n (*goal: ‹(x::'a) ≠ 𝟬 ⟹ x [^] order R = x›*)\n case False (*‹x ≠ 𝟬›*)\n have x_carr: \"x ∈ carrier (mult_of R)\"\n using False (*‹x ≠ 𝟬›*) assms (*‹x ∈ carrier R›*) by simp\n have carr_non_empty: \"card (carrier R) > 0\"\n using order_gt_0_iff_finite (*‹(0 < order R) = finite (carrier R)›*) finite_carrier (*‹finite (carrier R)›*) unfolding order_def\n (*goal: ‹0 < card (carrier R)›*)\n by simp\n have \"x [^] (order R) = x [^]⇘mult_of R⇙ (order R)\"\n by (simp add:nat_pow_mult_of (*‹([^]⇘Ring_Divisibility.mult_of ?R⇙) = ([^]⇘?R⇙)›*))\n also (*calculation: ‹(x::'a) [^] order R = x [^]⇘Ring_Divisibility.mult_of R⇙ order R›*) have \"... = x [^]⇘mult_of R⇙ (order (mult_of R)+1)\"\n using carr_non_empty (*‹(0::nat) < card (carrier R)›*) unfolding order_def\n (*goal: ‹(x::'a) [^]⇘Ring_Divisibility.mult_of R⇙ card (carrier R) = x [^]⇘Ring_Divisibility.mult_of R⇙ (card (carrier (Ring_Divisibility.mult_of R)) + (1::nat))›*)\n apply (intro arg_cong[where f=\"λt. x [^]⇘mult_of R⇙ t\"] (*‹?x = ?y ⟹ x [^]⇘Ring_Divisibility.mult_of R⇙ ?x = x [^]⇘Ring_Divisibility.mult_of R⇙ ?y›*))\n (*goal: ‹x [^]⇘Ring_Divisibility.mult_of R⇙ card (carrier R) = x [^]⇘Ring_Divisibility.mult_of R⇙ (card (carrier (Ring_Divisibility.mult_of R)) + 1)›*)\n by simp\n also (*calculation: ‹x [^] order R = x [^]⇘Ring_Divisibility.mult_of R⇙ (order (Ring_Divisibility.mult_of R) + 1)›*) have \"... = x\"\n using x_carr (*‹x ∈ carrier (Ring_Divisibility.mult_of R)›*) by (simp add:mult_of.pow_order_eq_1 (*‹(?a::'a) ∈ carrier (Ring_Divisibility.mult_of R) ⟹ ?a [^]⇘Ring_Divisibility.mult_of R⇙ order (Ring_Divisibility.mult_of R) = 𝟭›*))\n finally (*calculation: ‹x [^] order R = x›*) show \"x [^] (order R) = x\"\n by simp\nqed\n\nlemma (in finite_field) order_pow_eq_self':\n assumes \"x ∈ carrier R\"\n shows \"x [^] (order R ^ d) = x\"\nproof (induction d)\n (*goals:\n 1. ‹(x::'a) [^] order R ^ (0::nat) = x›\n 2. ‹⋀d::nat. (x::'a) [^] order R ^ d = x ⟹ x [^] order R ^ Suc d = x›*)\n case 0 (*no hyothesis introduced yet*)\n then show \"?case\"\n (*goal: ‹x [^] order R ^ 0 = x›*)\n using assms (*‹(x::'a) ∈ carrier R›*) by simp\nnext\n (*goal: ‹⋀d. x [^] order R ^ d = x ⟹ x [^] order R ^ Suc d = x›*)\n case (Suc d) (*‹x [^] order R ^ d = x›*)\n have \"x [^] order R ^ (Suc d) = x [^] (order R ^ d * order R)\"\n by (simp add:mult.commute (*‹(?a::?'a) * (?b::?'a) = ?b * ?a›*))\n also (*calculation: ‹x [^] order R ^ Suc d = x [^] (order R ^ d * order R)›*) have \"... = (x [^] (order R ^ d)) [^] order R\"\n using assms (*‹(x::'a) ∈ carrier R›*) by (simp add: nat_pow_pow (*‹?x ∈ carrier R ⟹ (?x [^] ?n) [^] ?m = ?x [^] (?n * ?m)›*))\n also (*calculation: ‹x [^] order R ^ Suc d = (x [^] order R ^ d) [^] order R›*) have \"... = (x [^] (order R ^ d))\"\n using order_pow_eq_self (*‹?x ∈ carrier R ⟹ ?x [^] order R = ?x›*) assms (*‹x ∈ carrier R›*) by simp\n also (*calculation: ‹(x::'a) [^] order R ^ Suc (d::nat) = x [^] order R ^ d›*) have \"... = x\"\n using Suc (*‹x [^] order R ^ d = x›*) by simp\n finally (*calculation: ‹x [^] order R ^ Suc d = x›*) show \"?case\"\n (*goal: ‹x [^] order R ^ Suc d = x›*)\n by simp\nqed\n\nend\n\nlemma finite_fieldI:\n assumes \"field R\"\n assumes \"finite (carrier R)\"\n shows \"finite_field R\"\n using assms (*‹field R› ‹finite (carrier (R::('a, 'b) ring_scheme))›*) unfolding finite_field_def finite_field_axioms_def\n (*goal: ‹field R ∧ finite (carrier R)›*)\n by auto\n\nlemma (in domain) finite_domain_units:\n assumes \"finite (carrier R)\"\n shows \"Units R = carrier R - {𝟬}\" (is \"?lhs = ?rhs\")\nproof (standard)\n (*goals:\n 1. ‹Units R ⊆ carrier R - {𝟬}›\n 2. ‹carrier R - {𝟬} ⊆ Units R›*)\n have \"Units R ⊆ carrier R\"\n by (simp add:Units_def (*‹Units ?G = {y ∈ carrier ?G. ∃x∈carrier ?G. x ⊗⇘?G⇙ y = 𝟭⇘?G⇙ ∧ y ⊗⇘?G⇙ x = 𝟭⇘?G⇙}›*))\n moreover have \"𝟬 ∉ Units R\"\n by (meson zero_is_prime( (*‹Divisibility.prime R 𝟬›*) 1) primeE (*‹⟦Divisibility.prime (?G::(?'a, ?'b) monoid_scheme) (?p::?'a); ⟦?p ∉ Units ?G; ∀a::?'a∈carrier ?G. ∀b::?'a∈carrier ?G. ?p divides⇘?G⇙ a ⊗⇘?G⇙ b ⟶ ?p divides⇘?G⇙ a ∨ ?p divides⇘?G⇙ b⟧ ⟹ ?P::bool⟧ ⟹ ?P›*))\n ultimately show \"Units R ⊆ carrier R - {𝟬}\"\n by blast\nnext\n (*goal: ‹carrier R - {𝟬} ⊆ Units R›*)\n have \"x ∈ Units R\" if a: \"x ∈ carrier R - {𝟬}\" for x\n proof (-)\n (*goal: ‹x ∈ Units R›*)\n have x_carr: \"x ∈ carrier R\"\n using a (*‹x ∈ carrier R - {𝟬}›*) by blast\n define f where \"f = (λy. y ⊗⇘R⇙ x)\"\n have \"inj_on f (carrier R)\"\n unfolding f_def\n (*goal: ‹inj_on (λy. y ⊗ x) (carrier R)›*)\n apply (rule inj_onI (*‹(⋀(x::?'a) y::?'a. ⟦x ∈ (?A::?'a set); y ∈ ?A; (?f::?'a ⇒ ?'b) x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*))\n (*goal: ‹inj_on (λy. y ⊗ x) (carrier R)›*)\n by (metis DiffD1 (*‹(?c::?'a) ∈ (?A::?'a set) - (?B::?'a set) ⟹ ?c ∈ ?A›*) DiffD2 (*‹⟦(?c::?'a) ∈ (?A::?'a set) - (?B::?'a set); ?c ∈ ?B⟧ ⟹ ?P::bool›*) a (*‹(x::'a) ∈ carrier R - {𝟬}›*) m_rcancel (*‹⟦(?a::'a) ≠ 𝟬; ?a ∈ carrier R; (?b::'a) ∈ carrier R; (?c::'a) ∈ carrier R⟧ ⟹ (?b ⊗ ?a = ?c ⊗ ?a) = (?b = ?c)›*) insertI1 (*‹(?a::?'a) ∈ insert ?a (?B::?'a set)›*))\n hence \"card (carrier R) = card (f ` carrier R)\"\n by (metis card_image (*‹inj_on ?f ?A ⟹ card (?f ` ?A) = card ?A›*))\n moreover have \"f ` carrier R ⊆ carrier R\"\n unfolding f_def\n (*goal: ‹(λy. y ⊗ x) ` carrier R ⊆ carrier R›*)\n apply (rule image_subsetI (*‹(⋀x::?'a. x ∈ (?A::?'a set) ⟹ (?f::?'a ⇒ ?'b) x ∈ (?B::?'b set)) ⟹ ?f ` ?A ⊆ ?B›*))\n (*goal: ‹(λy. y ⊗ x) ` carrier R ⊆ carrier R›*)\n by (simp add: ring.ring_simprules (*‹⟦ring ?R; ?x ∈ carrier ?R; ?y ∈ carrier ?R⟧ ⟹ ?x ⊕⇘?R⇙ ?y ∈ carrier ?R› ‹ring ?R ⟹ 𝟬⇘?R⇙ ∈ carrier ?R› ‹⟦ring ?R; ?x ∈ carrier ?R⟧ ⟹ ⊖⇘?R⇙ ?x ∈ carrier ?R› ‹⟦ring ?R; ?x ∈ carrier ?R; ?y ∈ carrier ?R⟧ ⟹ ?x ⊖⇘?R⇙ ?y ∈ carrier ?R› ‹⟦ring ?R; ?x ∈ carrier ?R; ?y ∈ carrier ?R⟧ ⟹ ?x ⊗⇘?R⇙ ?y ∈ carrier ?R› ‹ring ?R ⟹ 𝟭⇘?R⇙ ∈ carrier ?R› ‹⟦ring ?R; ?x ∈ carrier ?R; ?y ∈ carrier ?R; ?z ∈ carrier ?R⟧ ⟹ ?x ⊕⇘?R⇙ ?y ⊕⇘?R⇙ ?z = ?x ⊕⇘?R⇙ (?y ⊕⇘?R⇙ ?z)› ‹⟦ring ?R; ?x ∈ carrier ?R⟧ ⟹ 𝟬⇘?R⇙ ⊕⇘?R⇙ ?x = ?x› ‹⟦ring ?R; ?x ∈ carrier ?R⟧ ⟹ ⊖⇘?R⇙ ?x ⊕⇘?R⇙ ?x = 𝟬⇘?R⇙› ‹⟦ring ?R; ?x ∈ carrier ?R; ?y ∈ carrier ?R⟧ ⟹ ?x ⊕⇘?R⇙ ?y = ?y ⊕⇘?R⇙ ?x› ‹⟦ring ?R; ?x ∈ carrier ?R; ?y ∈ carrier ?R; ?z ∈ carrier ?R⟧ ⟹ ?x ⊗⇘?R⇙ ?y ⊗⇘?R⇙ ?z = ?x ⊗⇘?R⇙ (?y ⊗⇘?R⇙ ?z)› ‹⟦ring ?R; ?x ∈ carrier ?R⟧ ⟹ 𝟭⇘?R⇙ ⊗⇘?R⇙ ?x = ?x› and more 15 facts*) x_carr (*‹x ∈ carrier R›*))\n ultimately have \"f ` carrier R = carrier R\"\n using card_subset_eq (*‹⟦finite ?B; ?A ⊆ ?B; card ?A = card ?B⟧ ⟹ ?A = ?B›*) assms (*‹finite (carrier R)›*) by metis\n moreover have \"𝟭⇘R⇙ ∈ carrier R\"\n by simp\n ultimately have \"∃y ∈ carrier R. f y = 𝟭⇘R⇙\"\n by (metis image_iff (*‹(?z ∈ ?f ` ?A) = (∃x∈?A. ?z = ?f x)›*))\n then obtain y where y_carrier: \"y ∈ carrier R\" and y_left_inv: \"y ⊗⇘R⇙ x = 𝟭⇘R⇙\"\n (*goal: ‹(⋀y. ⟦y ∈ carrier R; y ⊗ x = 𝟭⟧ ⟹ thesis) ⟹ thesis›*)\n using f_def (*‹f = (λy. y ⊗ x)›*) by blast\n hence y_right_inv: \"x ⊗⇘R⇙ y = 𝟭⇘R⇙\"\n by (metis DiffD1 (*‹?c ∈ ?A - ?B ⟹ ?c ∈ ?A›*) a (*‹x ∈ carrier R - {𝟬}›*) cring_simprules( (*‹⟦?x ∈ carrier R; ?y ∈ carrier R⟧ ⟹ ?x ⊗ ?y = ?y ⊗ ?x›*) 14))\n show \"x ∈ Units R\"\n using y_carrier (*‹y ∈ carrier R›*) y_left_inv (*‹(y::'a) ⊗ (x::'a) = 𝟭›*) y_right_inv (*‹(x::'a) ⊗ (y::'a) = 𝟭›*) by (metis DiffD1 (*‹(?c::?'a) ∈ (?A::?'a set) - (?B::?'a set) ⟹ ?c ∈ ?A›*) a (*‹(x::'a) ∈ carrier R - {𝟬}›*) divides_one (*‹(?a::'a) ∈ carrier R ⟹ ?a divides 𝟭 = (?a ∈ Units R)›*) factor_def (*‹(?a::?'a) divides⇘?G::(?'a, ?'b) monoid_scheme⇙ (?b::?'a) = (∃c::?'a∈carrier ?G. ?b = ?a ⊗⇘?G⇙ c)›*))\n qed\n thus \"?rhs ⊆ ?lhs\"\n by auto\nqed\n\ntext ‹The following theorem can be found in Lidl and Niederreiter~\\<^cite>‹‹Theorem 1.31› in \"lidl1986\"›.›\n\ntheorem finite_domains_are_fields:\n assumes \"domain R\"\n assumes \"finite (carrier R)\"\n shows \"finite_field R\"\nproof (-)\n (*goal: ‹finite_field R›*)\n interpret domain R\n using assms (*‹domain R› ‹finite (carrier R)›*) by auto\n have \"Units R = carrier R - {𝟬⇘R⇙}\"\n using finite_domain_units[OF assms ( 2 )] (*‹Units R = carrier R - {𝟬⇘R⇙}›*) by simp\n then have \"field R\"\n by (simp add: assms( (*‹domain R›*) 1) field.intro (*‹⟦domain ?R; field_axioms ?R⟧ ⟹ field ?R›*) field_axioms.intro (*‹Units ?R = carrier ?R - {𝟬⇘?R⇙} ⟹ field_axioms ?R›*))\n thus \"?thesis\"\n (*goal: ‹finite_field R›*)\n using assms(2) (*‹finite (carrier (R::('a, 'b) ring_scheme))›*) finite_fieldI (*‹⟦field ?R; finite (carrier ?R)⟧ ⟹ finite_field ?R›*) by auto\nqed\n\ndefinition zfact_iso :: \"nat ⇒ nat ⇒ int set\" where\n \"zfact_iso p k = Idl⇘𝒵⇙ {int p} +>⇘𝒵⇙ (int k)\"\n\ncontext\n fixes n :: nat\n assumes n_gt_0: \"n > 0\"\nbegin\n\nprivate abbreviation I where \"I ≡ Idl⇘𝒵⇙ {int n}\"\n\nprivate lemma ideal_I: \"ideal I 𝒵\"\n by (simp add: int.genideal_ideal (*‹?S ⊆ UNIV ⟹ ideal (Idl⇘𝒵⇙ ?S) 𝒵›*))\n\nlemma int_cosetI:\n assumes \"u mod (int n) = v mod (int n)\"\n shows \"Idl⇘𝒵⇙ {int n} +>⇘𝒵⇙ u = Idl⇘𝒵⇙ {int n} +>⇘𝒵⇙ v\"\nproof (-)\n (*goal: ‹I +>⇘𝒵⇙ u = I +>⇘𝒵⇙ v›*)\n have \"u - v ∈ I\"\n by (metis Idl_subset_eq_dvd (*‹(Idl⇘𝒵⇙ {?k::int} ⊆ Idl⇘𝒵⇙ {?l::int}) = (?l dvd ?k)›*) assms (*‹(u::int) mod int (n::nat) = (v::int) mod int n›*) int_Idl_subset_ideal (*‹(Idl⇘𝒵⇙ {?k::int} ⊆ Idl⇘𝒵⇙ {?l::int}) = (?k ∈ Idl⇘𝒵⇙ {?l})›*) mod_eq_dvd_iff (*‹((?a::?'a::euclidean_ring_cancel) mod (?c::?'a::euclidean_ring_cancel) = (?b::?'a::euclidean_ring_cancel) mod ?c) = (?c dvd ?a - ?b)›*))\n thus \"?thesis\"\n (*goal: ‹I +>⇘𝒵⇙ u = I +>⇘𝒵⇙ v›*)\n using ideal_I (*‹ideal I 𝒵›*) int.quotient_eq_iff_same_a_r_cos (*‹⟦ideal (?I::int set) 𝒵; (?a::int) ∈ UNIV; (?b::int) ∈ UNIV⟧ ⟹ (?a - ?b ∈ ?I) = (?I +>⇘𝒵⇙ ?a = ?I +>⇘𝒵⇙ ?b)›*) by simp\nqed\n\nlemma zfact_iso_inj:\n \"inj_on (zfact_iso n) {..<n}\"\nproof (rule inj_onI (*‹(⋀x y. ⟦x ∈ ?A; y ∈ ?A; ?f x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*))\n (*goal: ‹⋀(x::nat) y::nat. ⟦x ∈ {..<n::nat}; y ∈ {..<n}; zfact_iso n x = zfact_iso n y⟧ ⟹ x = y›*)\n fix x and y\n assume a: \"x ∈ {..<n}\" \"y ∈ {..<n}\" (*‹(x::nat) ∈ {..<n::nat}› ‹(y::nat) ∈ {..<n::nat}›*)\n assume \"zfact_iso n x = zfact_iso n y\" (*‹zfact_iso (n::nat) (x::nat) = zfact_iso n (y::nat)›*)\n hence \"I +>⇘𝒵⇙ (int x) = I +>⇘𝒵⇙ (int y)\"\n by (simp add:zfact_iso_def (*‹zfact_iso ?p ?k = ??.Ring_Characteristic.I ?p +>⇘𝒵⇙ int ?k›*))\n hence \"int x - int y ∈ I\"\n apply (subst int.quotient_eq_iff_same_a_r_cos[OF ideal_I] (*‹⟦?a ∈ UNIV; ?b ∈ UNIV⟧ ⟹ (?a - ?b ∈ I) = (I +>⇘𝒵⇙ ?a = I +>⇘𝒵⇙ ?b)›*))\n (*goals:\n 1. ‹I +>⇘𝒵⇙ int x = I +>⇘𝒵⇙ int y ⟹ int x ∈ UNIV›\n 2. ‹I +>⇘𝒵⇙ int x = I +>⇘𝒵⇙ int y ⟹ int y ∈ UNIV›\n 3. ‹I +>⇘𝒵⇙ int x = I +>⇘𝒵⇙ int y ⟹ I +>⇘𝒵⇙ int x = I +>⇘𝒵⇙ int y›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n hence \"int x mod int n = int y mod int n\"\n by (meson Idl_subset_eq_dvd (*‹(Idl⇘𝒵⇙ {?k} ⊆ Idl⇘𝒵⇙ {?l}) = (?l dvd ?k)›*) int_Idl_subset_ideal (*‹(Idl⇘𝒵⇙ {?k} ⊆ Idl⇘𝒵⇙ {?l}) = (?k ∈ Idl⇘𝒵⇙ {?l})›*) mod_eq_dvd_iff (*‹(?a mod ?c = ?b mod ?c) = (?c dvd ?a - ?b)›*))\n thus \"x = y\"\n using a (*‹x ∈ {..<n}› ‹(y::nat) ∈ {..<n::nat}›*) by simp\nqed\n\nlemma zfact_iso_ran:\n \"zfact_iso n ` {..<n} = carrier (ZFact (int n))\"\nproof (-)\n (*goal: ‹zfact_iso n ` {..<n} = carrier (ZFact (int n))›*)\n have \"zfact_iso n ` {..<n} ⊆ carrier (ZFact (int n))\"\n unfolding zfact_iso_def ZFact_def FactRing_simps\n (*goal: ‹(λk. I +>⇘𝒵⇙ int k) ` {..<n} ⊆ carrier ⦇carrier = ⋃a∈carrier (add_monoid 𝒵). {I +>⇘𝒵⇙ a}, monoid.mult = rcoset_mult 𝒵 I, one = I +>⇘𝒵⇙ 𝟭⇘𝒵⇙, zero = I, add = (<+>⇘𝒵⇙)⦈›*)\n using int.a_rcosetsI (*‹⟦?H ⊆ UNIV; ?x ∈ UNIV⟧ ⟹ ?H +>⇘𝒵⇙ ?x ∈ a_rcosets⇘𝒵⇙ ?H›*) by auto\n moreover have \"x ∈ zfact_iso n ` {..<n}\" if a: \"x ∈ carrier (ZFact (int n))\" for x\n proof (-)\n (*goal: ‹x ∈ zfact_iso n ` {..<n}›*)\n obtain y where y_def: \"x = I +>⇘𝒵⇙ y\"\n (*goal: ‹(⋀y. x = I +>⇘𝒵⇙ y ⟹ thesis) ⟹ thesis›*)\n using a (*‹x ∈ carrier (ZFact (int n))›*) unfolding ZFact_def FactRing_simps\n (*goal: ‹(⋀y. x = I +>⇘𝒵⇙ y ⟹ thesis) ⟹ thesis›*)\n by auto\n define z where \"z = nat (y mod int n)\"\n with n_gt_0 (*‹0 < n›*) have z_def: \"int z mod int n = y mod int n\" \"z < n\"\n apply -\n (*goals:\n 1. ‹⟦0 < n; z = nat (y mod int n)⟧ ⟹ int z mod int n = y mod int n›\n 2. ‹⟦0 < n; z = nat (y mod int n)⟧ ⟹ z < n›\n discuss goal 1*)\n apply (simp add: z_def (*‹z = nat (y mod int n)›*) nat_less_iff (*‹0 ≤ ?w ⟹ (nat ?w < ?m) = (?w < int ?m)›*))\n (*discuss goal 2*)\n apply (simp add: z_def (*‹z = nat (y mod int n)›*) nat_less_iff (*‹0 ≤ ?w ⟹ (nat ?w < ?m) = (?w < int ?m)›*))\n (*proven 2 subgoals*) .\n have \"x = I +>⇘𝒵⇙ y\"\n by (simp add:y_def (*‹(x::int set) = I +>⇘𝒵⇙ (y::int)›*))\n also (*calculation: ‹x = I +>⇘𝒵⇙ y›*) have \"... = I +>⇘𝒵⇙ (int z)\"\n apply (intro int_cosetI (*‹?u mod int n = ?v mod int n ⟹ I +>⇘𝒵⇙ ?u = I +>⇘𝒵⇙ ?v›*))\n (*goal: ‹I +>⇘𝒵⇙ y = I +>⇘𝒵⇙ int z›*)\n by (simp add:z_def (*‹int z mod int n = y mod int n› ‹z < n›*))\n also (*calculation: ‹x = I +>⇘𝒵⇙ int z›*) have \"... = zfact_iso n z\"\n by (simp add:zfact_iso_def (*‹zfact_iso ?p ?k = ??.Ring_Characteristic.I ?p +>⇘𝒵⇙ int ?k›*))\n finally (*calculation: ‹(x::int set) = zfact_iso (n::nat) (z::nat)›*) have \"x = zfact_iso n z\"\n by simp\n thus \"x ∈ zfact_iso n ` {..<n}\"\n using z_def(2) (*‹z < n›*) by blast\n qed\n ultimately show \"?thesis\"\n (*goal: ‹zfact_iso n ` {..<n} = carrier (ZFact (int n))›*)\n by auto\nqed\n\nlemma zfact_iso_bij:\n \"bij_betw (zfact_iso n) {..<n} (carrier (ZFact (int n)))\"\n using bij_betw_def (*‹bij_betw (?f::?'a ⇒ ?'b) (?A::?'a set) (?B::?'b set) = (inj_on ?f ?A ∧ ?f ` ?A = ?B)›*) zfact_iso_inj (*‹inj_on (zfact_iso n) {..<n}›*) zfact_iso_ran (*‹zfact_iso (n::nat) ` {..<n} = carrier (ZFact (int n))›*) by blast\n\nlemma card_zfact_carr: \"card (carrier (ZFact (int n))) = n\"\n using bij_betw_same_card[OF zfact_iso_bij] (*‹card {..<n::nat} = card (carrier (ZFact (int n)))›*) by simp\n\nlemma fin_zfact: \"finite (carrier (ZFact (int n)))\"\n using card_zfact_carr (*‹card (carrier (ZFact (int n))) = n›*) n_gt_0 (*‹0 < n›*) card_ge_0_finite (*‹0 < card ?A ⟹ finite ?A›*) by force\n\nend\n\nlemma zfact_prime_is_finite_field:\n assumes \"Factorial_Ring.prime p\"\n shows \"finite_field (ZFact (int p))\"\nproof (-)\n (*goal: ‹finite_field (ZFact (int p))›*)\n have p_gt_0: \"p > 0\"\n using assms(1) (*‹normalization_semidom_class.prime p›*) prime_gt_0_nat (*‹normalization_semidom_class.prime ?p ⟹ 0 < ?p›*) by simp\n have \"Factorial_Ring.prime (int p)\"\n using assms (*‹normalization_semidom_class.prime p›*) by simp\n moreover have \"finite (carrier (ZFact (int p)))\"\n using fin_zfact[OF p_gt_0] (*‹finite (carrier (ZFact (int p)))›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹finite_field (ZFact (int p))›*)\n apply (intro finite_domains_are_fields (*‹⟦domain ?R; finite (carrier ?R)⟧ ⟹ finite_field ?R›*) ZFact_prime_is_domain (*‹normalization_semidom_class.prime ?p ⟹ domain (ZFact ?p)›*))\n (*goals:\n 1. ‹⟦normalization_semidom_class.prime (int (p::nat)); finite (carrier (ZFact (int p)))⟧ ⟹ normalization_semidom_class.prime (int p)›\n 2. ‹⟦normalization_semidom_class.prime (int (p::nat)); finite (carrier (ZFact (int p)))⟧ ⟹ finite (carrier (ZFact (int p)))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nqed\n\ndefinition int_embed :: \"_ ⇒ int ⇒ _\" where\n \"int_embed R k = add_pow R k 𝟭⇘R⇙\"\n\nlemma (in ring) add_pow_consistent:\n fixes i :: \"int\"\n assumes \"subring K R\"\n assumes \"k ∈ K\"\n shows \"add_pow R i k = add_pow (R ⦇ carrier := K ⦈) i k\"\n (is \"?lhs = ?rhs\")\nproof (-)\n (*goal: ‹[i] ⋅ k = [i] ⋅⇘R⦇carrier := K⦈⇙ k›*)\n have a: \"subgroup K (add_monoid R)\"\n using assms(1) (*‹subring K R›*) subring.axioms (*‹subring (?H::?'a set) (?R::(?'a, ?'b) ring_scheme) ⟹ subgroup ?H (add_monoid ?R)› ‹subring ?H ?R ⟹ submonoid ?H ?R›*) by auto\n have \"add_pow R i k = k [^]⇘add_monoid R⦇carrier := K⦈⇙ i\"\n using add.int_pow_consistent[OF a assms ( 2 )] (*‹[?n] ⋅ k = k [^]⇘add_monoid R⦇carrier := K⦈⇙ ?n›*) by simp\n also (*calculation: ‹[i] ⋅ k = k [^]⇘add_monoid R⦇carrier := K⦈⇙ i›*) have \"... = ?rhs\"\n unfolding add_pow_def\n (*goal: ‹(k::'a) [^]⇘add_monoid R⦇carrier := K::'a set⦈⇙ (i::int) = k [^]⇘add_monoid (R⦇carrier := K⦈)⇙ i›*)\n by simp\n finally (*calculation: ‹[(i::int)] ⋅ (k::'a::type) = [i] ⋅⇘R⦇carrier := K::'a::type set⦈⇙ k›*) show \"?thesis\"\n (*goal: ‹[i] ⋅ k = [i] ⋅⇘R⦇carrier := K⦈⇙ k›*)\n by simp\nqed\n\nlemma (in ring) int_embed_consistent:\n assumes \"subring K R\"\n shows \"int_embed R i = int_embed (R ⦇ carrier := K ⦈) i\"\nproof (-)\n (*goal: ‹int_embed R i = int_embed (R⦇carrier := K⦈) i›*)\n have a: \"𝟭 = 𝟭⇘R ⦇ carrier := K ⦈⇙\"\n by simp\n have b: \"𝟭⇘R⦇carrier := K⦈⇙ ∈ K\"\n using assms (*‹subring (K::'a::type set) R›*) subringE(3) (*‹subring ?H ?R ⟹ 𝟭⇘?R⇙ ∈ ?H›*) by auto\n show \"?thesis\"\n (*goal: ‹int_embed R i = int_embed (R⦇carrier := K⦈) i›*)\n unfolding int_embed_def a\n (*goal: ‹[i] ⋅ 𝟭⇘R⦇carrier := K⦈⇙ = [i] ⋅⇘R⦇carrier := K⦈⇙ 𝟭⇘R⦇carrier := K⦈⇙›*)\n using b (*‹𝟭⇘R⦇carrier := K⦈⇙ ∈ K›*) add_pow_consistent[OF assms ( 1 )] (*‹?k ∈ K ⟹ [?i] ⋅ ?k = [?i] ⋅⇘R⦇carrier := K⦈⇙ ?k›*) by simp\nqed\n\nlemma (in ring) int_embed_closed:\n \"int_embed R k ∈ carrier R\"\n unfolding int_embed_def\n (*goal: ‹[k] ⋅ 𝟭 ∈ carrier R›*)\n using add.int_pow_closed (*‹?x ∈ carrier R ⟹ [?i] ⋅ ?x ∈ carrier R›*) by simp\n\nlemma (in ring) int_embed_range:\n assumes \"subring K R\"\n shows \"int_embed R k ∈ K\"\nproof (-)\n (*goal: ‹int_embed R k ∈ K›*)\n let ?R' = \"R ⦇ carrier := K ⦈\"\n interpret x: ring \"?R'\"\n using subring_is_ring[OF assms] (*‹ring (R⦇carrier := K::'a::type set⦈)›*) by simp\n have \"int_embed R k = int_embed ?R' k\"\n using int_embed_consistent[OF assms] (*‹int_embed R ?i = int_embed (R⦇carrier := K⦈) ?i›*) by simp\n also (*calculation: ‹int_embed R k = int_embed (R⦇carrier := K⦈) k›*) have \"... ∈ K\"\n using x.int_embed_closed (*‹int_embed (R⦇carrier := K⦈) ?k ∈ carrier (R⦇carrier := K⦈)›*) by simp\n finally (*calculation: ‹int_embed R (k::int) ∈ (K::'a set)›*) show \"?thesis\"\n (*goal: ‹int_embed R k ∈ K›*)\n by simp\nqed\n\nlemma (in ring) int_embed_zero:\n \"int_embed R 0 = 𝟬⇘R⇙\"\n by (simp add:int_embed_def (*‹int_embed ?R ?k = [?k] ⋅⇘?R⇙ 𝟭⇘?R⇙›*) add_pow_def (*‹[?k] ⋅⇘?R⇙ ?a = ?a [^]⇘add_monoid ?R⇙ ?k›*))\n\nlemma (in ring) int_embed_one:\n \"int_embed R 1 = 𝟭⇘R⇙\"\n by (simp add:int_embed_def (*‹int_embed (?R::(?'a, ?'b) ring_scheme) (?k::int) = [?k] ⋅⇘?R⇙ 𝟭⇘?R⇙›*))\n\nlemma (in ring) int_embed_add:\n \"int_embed R (x+y) = int_embed R x ⊕⇘R⇙ int_embed R y\"\n by (simp add:int_embed_def (*‹int_embed ?R ?k = [?k] ⋅⇘?R⇙ 𝟭⇘?R⇙›*) add.int_pow_mult (*‹?x ∈ carrier R ⟹ [(?i + ?j)] ⋅ ?x = [?i] ⋅ ?x ⊕ [?j] ⋅ ?x›*))\n\nlemma (in ring) int_embed_inv:\n \"int_embed R (-x) = ⊖⇘R⇙ int_embed R x\" (is \"?lhs = ?rhs\")\nproof (-)\n (*goal: ‹int_embed R (- x) = ⊖ int_embed R x›*)\n have \"?lhs = int_embed R (-x) ⊕ (int_embed R x ⊖ int_embed R x)\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) by simp\n also (*calculation: ‹int_embed R (- (x::int)) = int_embed R (- x) ⊕ (int_embed R x ⊖ int_embed R x)›*) have \"... = int_embed R (-x) ⊕ int_embed R x ⊕ (⊖ int_embed R x)\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) apply (subst a_minus_def (*‹?x ⊖⇘?R⇙ ?y = ?x ⊕⇘?R⇙ ⊖⇘?R⇙ ?y›*))\n (*goal: ‹int_embed R (- (x::int)) ⊕ (int_embed R x ⊖ int_embed R x) = int_embed R (- x) ⊕ int_embed R x ⊕ ⊖ int_embed R x›*)\n apply (subst a_assoc (*‹⟦?x ∈ carrier R; ?y ∈ carrier R; ?z ∈ carrier R⟧ ⟹ ?x ⊕ ?y ⊕ ?z = ?x ⊕ (?y ⊕ ?z)›*))\n (*goals:\n 1. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ int_embed R (- x) ∈ carrier R›\n 2. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ int_embed R x ∈ carrier R›\n 3. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ ⊖ int_embed R x ∈ carrier R›\n 4. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ int_embed R (- x) ⊕ (int_embed R x ⊕ ⊖ int_embed R x) = int_embed R (- x) ⊕ (int_embed R x ⊕ ⊖ int_embed R x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n also (*calculation: ‹int_embed R (- (x::int)) = int_embed R (- x) ⊕ int_embed R x ⊕ ⊖ int_embed R x›*) have \"... = int_embed R (-x +x) ⊕ (⊖ int_embed R x)\"\n apply (subst int_embed_add (*‹int_embed R (?x + ?y) = int_embed R ?x ⊕ int_embed R ?y›*))\n (*goal: ‹int_embed R (- x) ⊕ int_embed R x ⊕ ⊖ int_embed R x = int_embed R (- x + x) ⊕ ⊖ int_embed R x›*)\n by simp\n also (*calculation: ‹int_embed R (- (x::int)) = int_embed R (- x + x) ⊕ ⊖ int_embed R x›*) have \"... = ?rhs\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) by (simp add:int_embed_zero (*‹int_embed R 0 = 𝟬›*))\n finally (*calculation: ‹int_embed R (- (x::int)) = ⊖ int_embed R x›*) show \"?thesis\"\n (*goal: ‹int_embed R (- x) = ⊖ int_embed R x›*)\n by simp\nqed\n\nlemma (in ring) int_embed_diff:\n \"int_embed R (x-y) = int_embed R x ⊖⇘R⇙ int_embed R y\"\n (is \"?lhs = ?rhs\")\nproof (-)\n (*goal: ‹int_embed R (x - y) = int_embed R x ⊖ int_embed R y›*)\n have \"?lhs = int_embed R (x + (-y))\"\n by simp\n also (*calculation: ‹int_embed R (x - y) = int_embed R (x + - y)›*) have \"... = ?rhs\"\n apply (subst int_embed_add (*‹int_embed R ((?x::int) + (?y::int)) = int_embed R ?x ⊕ int_embed R ?y›*))\n (*goal: ‹int_embed R (x + - y) = int_embed R x ⊖ int_embed R y›*)\n by (simp add:a_minus_def (*‹?x ⊖⇘?R⇙ ?y = ?x ⊕⇘?R⇙ ⊖⇘?R⇙ ?y›*) int_embed_inv (*‹int_embed R (- ?x) = ⊖ int_embed R ?x›*))\n finally (*calculation: ‹int_embed R (x - y) = int_embed R x ⊖ int_embed R y›*) show \"?thesis\"\n (*goal: ‹int_embed R ((x::int) - (y::int)) = int_embed R x ⊖ int_embed R y›*)\n by simp\nqed\n\nlemma (in ring) int_embed_mult_aux:\n \"int_embed R (x*int y) = int_embed R x ⊗ int_embed R y\"\nproof (induction y)\n (*goals:\n 1. ‹int_embed R (x * int 0) = int_embed R x ⊗ int_embed R (int 0)›\n 2. ‹⋀y. int_embed R (x * int y) = int_embed R x ⊗ int_embed R (int y) ⟹ int_embed R (x * int (Suc y)) = int_embed R x ⊗ int_embed R (int (Suc y))›*)\n case 0 (*no hyothesis introduced yet*)\n then show \"?case\"\n (*goal: ‹int_embed R (x * int 0) = int_embed R x ⊗ int_embed R (int 0)›*)\n by (simp add:int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) int_embed_zero (*‹int_embed R 0 = 𝟬›*))\nnext\n (*goal: ‹⋀y. int_embed R (x * int y) = int_embed R x ⊗ int_embed R (int y) ⟹ int_embed R (x * int (Suc y)) = int_embed R x ⊗ int_embed R (int (Suc y))›*)\n case (Suc y) (*‹int_embed R (x * int y) = int_embed R x ⊗ int_embed R (int y)›*)\n have \"int_embed R (x * int (Suc y)) = int_embed R (x + x * int y)\"\n by (simp add:algebra_simps (*‹(?a::?'a::semigroup_add) + (?b::?'a::semigroup_add) + (?c::?'a::semigroup_add) = ?a + (?b + ?c)› ‹(?a::?'a::ab_semigroup_add) + (?b::?'a::ab_semigroup_add) = ?b + ?a› ‹(?b::?'a::ab_semigroup_add) + ((?a::?'a::ab_semigroup_add) + (?c::?'a::ab_semigroup_add)) = ?a + (?b + ?c)› ‹(?a::?'a::semigroup_mult) * (?b::?'a::semigroup_mult) * (?c::?'a::semigroup_mult) = ?a * (?b * ?c)› ‹(?a::?'a::ab_semigroup_mult) * (?b::?'a::ab_semigroup_mult) = ?b * ?a› ‹(?b::?'a::ab_semigroup_mult) * ((?a::?'a::ab_semigroup_mult) * (?c::?'a::ab_semigroup_mult)) = ?a * (?b * ?c)› ‹(?a::?'a::cancel_ab_semigroup_add) - (?b::?'a::cancel_ab_semigroup_add) - (?c::?'a::cancel_ab_semigroup_add) = ?a - (?b + ?c)› ‹(?a::?'a::group_add) + ((?b::?'a::group_add) - (?c::?'a::group_add)) = ?a + ?b - ?c› ‹((?a::?'a::group_add) - (?b::?'a::group_add) = (?c::?'a::group_add)) = (?a = ?c + ?b)› ‹((?a::?'a::group_add) = (?c::?'a::group_add) - (?b::?'a::group_add)) = (?a + ?b = ?c)› ‹(?a::?'a::group_add) - ((?b::?'a::group_add) - (?c::?'a::group_add)) = ?a + ?c - ?b› ‹(?a::?'a::ab_group_add) - (?b::?'a::ab_group_add) + (?c::?'a::ab_group_add) = ?a + ?c - ?b› and more 15 facts*))\n also (*calculation: ‹int_embed R (x * int (Suc y)) = int_embed R (x + x * int y)›*) have \"... = int_embed R x ⊕ int_embed R (x * int y)\"\n apply (subst int_embed_add (*‹int_embed R (?x + ?y) = int_embed R ?x ⊕ int_embed R ?y›*))\n (*goal: ‹int_embed R ((x::int) + x * int (y::nat)) = int_embed R x ⊕ int_embed R (x * int y)›*)\n by simp\n also (*calculation: ‹int_embed R (x * int (Suc y)) = int_embed R x ⊕ int_embed R (x * int y)›*) have \"... = int_embed R x ⊗ 𝟭 ⊕ int_embed R x ⊗ int_embed R y\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) apply (subst Suc (*‹int_embed R (x * int y) = int_embed R x ⊗ int_embed R (int y)›*))\n (*goal: ‹int_embed R x ⊕ int_embed R (x * int y) = int_embed R x ⊗ 𝟭 ⊕ int_embed R x ⊗ int_embed R (int y)›*)\n by simp\n also (*calculation: ‹int_embed R (x * int (Suc y)) = int_embed R x ⊗ 𝟭 ⊕ int_embed R x ⊗ int_embed R (int y)›*) have \"... = int_embed R x ⊗ (int_embed R 1 ⊕ int_embed R y)\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) apply (subst r_distr (*‹⟦?x ∈ carrier R; ?y ∈ carrier R; ?z ∈ carrier R⟧ ⟹ ?z ⊗ (?x ⊕ ?y) = ?z ⊗ ?x ⊕ ?z ⊗ ?y›*))\n (*goals:\n 1. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ int_embed R 1 ∈ carrier R›\n 2. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ int_embed R (int y) ∈ carrier R›\n 3. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ int_embed R x ∈ carrier R›\n 4. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ int_embed R x ⊗ 𝟭 ⊕ int_embed R x ⊗ int_embed R (int y) = int_embed R x ⊗ int_embed R 1 ⊕ int_embed R x ⊗ int_embed R (int y)›\n discuss goal 1*)\n apply (simp add:int_embed_one (*‹int_embed R 1 = 𝟭›*))\n (*discuss goal 2*)\n apply (simp add:int_embed_one (*‹int_embed R 1 = 𝟭›*))\n (*discuss goal 3*)\n apply (simp add:int_embed_one (*‹int_embed R 1 = 𝟭›*))\n (*discuss goal 4*)\n apply (simp add:int_embed_one (*‹int_embed R 1 = 𝟭›*))\n (*proven 4 subgoals*) .\n also (*calculation: ‹int_embed R (x * int (Suc y)) = int_embed R x ⊗ (int_embed R 1 ⊕ int_embed R (int y))›*) have \"... = int_embed R x ⊗ int_embed R (1+int y)\"\n apply (subst int_embed_add (*‹int_embed R (?x + ?y) = int_embed R ?x ⊕ int_embed R ?y›*))\n (*goal: ‹int_embed R x ⊗ (int_embed R 1 ⊕ int_embed R (int y)) = int_embed R x ⊗ int_embed R (1 + int y)›*)\n by simp\n also (*calculation: ‹int_embed R ((x::int) * int (Suc (y::nat))) = int_embed R x ⊗ int_embed R ((1::int) + int y)›*) have \"... = int_embed R x ⊗ int_embed R (Suc y)\"\n by simp\n finally (*calculation: ‹int_embed R (x * int (Suc y)) = int_embed R x ⊗ int_embed R (int (Suc y))›*) show \"?case\"\n (*goal: ‹int_embed R (x * int (Suc y)) = int_embed R x ⊗ int_embed R (int (Suc y))›*)\n by simp\nqed\n\nlemma (in ring) int_embed_mult:\n \"int_embed R (x*y) = int_embed R x ⊗⇘R⇙ int_embed R y\"\nproof (cases \"y ≥ 0\")\n (*goals:\n 1. ‹(0::int) ≤ (y::int) ⟹ int_embed R ((x::int) * y) = int_embed R x ⊗ int_embed R y›\n 2. ‹¬ (0::int) ≤ (y::int) ⟹ int_embed R ((x::int) * y) = int_embed R x ⊗ int_embed R y›*)\n case True (*‹(0::int) ≤ (y::int)›*)\n then obtain y' where y_def: \"y = int y'\"\n (*goal: ‹(⋀y'. y = int y' ⟹ thesis) ⟹ thesis›*)\n using nonneg_int_cases (*‹⟦(0::int) ≤ (?k::int); ⋀n::nat. ?k = int n ⟹ ?thesis::bool⟧ ⟹ ?thesis›*) by auto\n have \"int_embed R (x * y) = int_embed R (x * int y')\"\n unfolding y_def\n (*goal: ‹int_embed R ((x::int) * int (y'::nat)) = int_embed R (x * int y')›*)\n by simp\n also (*calculation: ‹int_embed R ((x::int) * (y::int)) = int_embed R (x * int (y'::nat))›*) have \"... = int_embed R x ⊗ int_embed R y'\"\n apply (subst int_embed_mult_aux (*‹int_embed R ((?x::int) * int (?y::nat)) = int_embed R ?x ⊗ int_embed R (int ?y)›*))\n (*goal: ‹int_embed R ((x::int) * int (y'::nat)) = int_embed R x ⊗ int_embed R (int y')›*)\n by simp\n also (*calculation: ‹int_embed R (x * y) = int_embed R x ⊗ int_embed R (int y')›*) have \"... = int_embed R x ⊗ int_embed R y\"\n unfolding y_def\n (*goal: ‹int_embed R x ⊗ int_embed R (int y') = int_embed R x ⊗ int_embed R (int y')›*)\n by simp\n finally (*calculation: ‹int_embed R (x * y) = int_embed R x ⊗ int_embed R y›*) show \"?thesis\"\n (*goal: ‹int_embed R (x * y) = int_embed R x ⊗ int_embed R y›*)\n by simp\nnext\n (*goal: ‹¬ (0::int) ≤ (y::int) ⟹ int_embed R ((x::int) * y) = int_embed R x ⊗ int_embed R y›*)\n case False (*‹¬ 0 ≤ y›*)\n then obtain y' where y_def: \"y = - int y'\"\n (*goal: ‹(⋀y'. y = - int y' ⟹ thesis) ⟹ thesis›*)\n by (meson nle_le (*‹(¬ ?a ≤ ?b) = (?b ≤ ?a ∧ ?b ≠ ?a)›*) nonpos_int_cases (*‹⟦?k ≤ 0; ⋀n. ?k = - int n ⟹ ?thesis⟧ ⟹ ?thesis›*))\n have \"int_embed R (x * y) = int_embed R (-(x * int y'))\"\n unfolding y_def\n (*goal: ‹int_embed R ((x::int) * - int (y'::nat)) = int_embed R (- (x * int y'))›*)\n by simp\n also (*calculation: ‹int_embed R ((x::int) * (y::int)) = int_embed R (- (x * int (y'::nat)))›*) have \"... = ⊖ (int_embed R (x * int y'))\"\n apply (subst int_embed_inv (*‹int_embed R (- ?x) = ⊖ int_embed R ?x›*))\n (*goal: ‹int_embed R (- (x * int y')) = ⊖ int_embed R (x * int y')›*)\n by simp\n also (*calculation: ‹int_embed R (x * y) = ⊖ int_embed R (x * int y')›*) have \"... = ⊖ (int_embed R x ⊗ int_embed R y')\"\n apply (subst int_embed_mult_aux (*‹int_embed R (?x * int ?y) = int_embed R ?x ⊗ int_embed R (int ?y)›*))\n (*goal: ‹⊖ int_embed R ((x::int) * int (y'::nat)) = ⊖ (int_embed R x ⊗ int_embed R (int y'))›*)\n by simp\n also (*calculation: ‹int_embed R (x * y) = ⊖ (int_embed R x ⊗ int_embed R (int y'))›*) have \"... = int_embed R x ⊗ ⊖ int_embed R y'\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) by algebra\n also (*calculation: ‹int_embed R (x * y) = int_embed R x ⊗ ⊖ int_embed R (int y')›*) have \"... = int_embed R x ⊗ int_embed R (-y')\"\n apply (subst int_embed_inv (*‹int_embed R (- ?x) = ⊖ int_embed R ?x›*))\n (*goal: ‹int_embed R x ⊗ ⊖ int_embed R (int y') = int_embed R x ⊗ int_embed R (- int y')›*)\n by simp\n also (*calculation: ‹int_embed R (x * y) = int_embed R x ⊗ int_embed R (- int y')›*) have \"... = int_embed R x ⊗ int_embed R y\"\n unfolding y_def\n (*goal: ‹int_embed R x ⊗ int_embed R (- int y') = int_embed R x ⊗ int_embed R (- int y')›*)\n by simp\n finally (*calculation: ‹int_embed R (x * y) = int_embed R x ⊗ int_embed R y›*) show \"?thesis\"\n (*goal: ‹int_embed R (x * y) = int_embed R x ⊗ int_embed R y›*)\n by simp\nqed\n\nlemma (in ring) int_embed_ring_hom:\n \"ring_hom_ring int_ring R (int_embed R)\"\nproof (rule ring_hom_ringI (*‹⟦ring ?R; ring ?S; ⋀x. x ∈ carrier ?R ⟹ ?h x ∈ carrier ?S; ⋀x y. ⟦x ∈ carrier ?R; y ∈ carrier ?R⟧ ⟹ ?h (x ⊗⇘?R⇙ y) = ?h x ⊗⇘?S⇙ ?h y; ⋀x y. ⟦x ∈ carrier ?R; y ∈ carrier ?R⟧ ⟹ ?h (x ⊕⇘?R⇙ y) = ?h x ⊕⇘?S⇙ ?h y; ?h 𝟭⇘?R⇙ = 𝟭⇘?S⇙⟧ ⟹ ring_hom_ring ?R ?S ?h›*))\n (*goals:\n 1. ‹ring 𝒵›\n 2. ‹ring R›\n 3. ‹⋀x. x ∈ carrier 𝒵 ⟹ int_embed R x ∈ carrier R›\n 4. ‹⋀x y. ⟦x ∈ carrier 𝒵; y ∈ carrier 𝒵⟧ ⟹ int_embed R (x ⊗⇘𝒵⇙ y) = int_embed R x ⊗ int_embed R y›\n 5. ‹⋀x y. ⟦x ∈ carrier 𝒵; y ∈ carrier 𝒵⟧ ⟹ int_embed R (x ⊕⇘𝒵⇙ y) = int_embed R x ⊕ int_embed R y›\n 6. ‹int_embed R 𝟭⇘𝒵⇙ = 𝟭›*)\n show \"ring int_ring\"\n using int.ring_axioms (*‹ring 𝒵›*) by simp\n show \"ring R\"\n using ring_axioms (*‹ring R›*) by simp\n show \"int_embed R x ∈ carrier R\" if \"x ∈ carrier 𝒵\" for x\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) by simp\n show \"int_embed R (x⊗⇘𝒵⇙y) = int_embed R x ⊗ int_embed R y\" if \"x ∈ carrier 𝒵\" \"y ∈ carrier 𝒵\" for x and y\n using int_embed_mult (*‹int_embed R (?x * ?y) = int_embed R ?x ⊗ int_embed R ?y›*) by simp\n show \"int_embed R (x⊕⇘𝒵⇙y) = int_embed R x ⊕ int_embed R y\" if \"x ∈ carrier 𝒵\" \"y ∈ carrier 𝒵\" for x and y\n using int_embed_add (*‹int_embed R (?x + ?y) = int_embed R ?x ⊕ int_embed R ?y›*) by simp\n show \"int_embed R 𝟭⇘𝒵⇙ = 𝟭\"\n by (simp add:int_embed_one (*‹int_embed R (1::int) = 𝟭›*))\nqed\n\nabbreviation char_subring where\n \"char_subring R ≡ int_embed R ` UNIV\"\n\ndefinition char where\n \"char R = card (char_subring R)\"\n\ntext ‹This is a non-standard definition for the characteristic of a ring.\n\nCommonly~\\<^cite>‹‹Definition 1.43› in \"lidl1986\"› it is defined to be the smallest natural number $n$ such\nthat n-times repeated addition of any number is zero. If no such number exists then it is defined\nto be $0$. In the case of rings with unit elements --- not that the locale @{locale \"ring\"} requires\nunit elements --- the above definition can be simplified to the number of times the unit elements\nneeds to be repeatedly added to reach $0$.\n\nThe following three lemmas imply that the definition of the characteristic here coincides with the\nlatter definition.›\n\nlemma (in ring) char_bound:\n assumes \"x > 0\"\n assumes \"int_embed R (int x) = 𝟬\"\n shows \"char R ≤ x\" \"char R > 0\"\nproof (-)\n (*goals:\n 1. ‹char R ≤ x›\n 2. ‹0 < char R›*)\n have \"char_subring R ⊆ int_embed R ` ({0..<int x})\"\n proof (rule image_subsetI (*‹(⋀x. x ∈ ?A ⟹ ?f x ∈ ?B) ⟹ ?f ` ?A ⊆ ?B›*))\n (*goal: ‹⋀xa::int. xa ∈ UNIV ⟹ int_embed R xa ∈ int_embed R ` {0::int..<int (x::nat)}›*)\n fix y :: int\n assume \"y ∈ UNIV\" (*‹(y::int) ∈ UNIV›*)\n define u where \"u = y div (int x)\"\n define v where \"v = y mod (int x)\"\n have \"int x > 0\"\n using assms (*‹0 < x› ‹int_embed R (int x) = 𝟬›*) by simp\n hence y_exp: \"y = u * int x + v\" \"v ≥ 0\" \"v < int x\"\n unfolding u_def v_def\n (*goals:\n 1. ‹y = y div int x * int x + y mod int x›\n 2. ‹0 ≤ y mod int x›\n 3. ‹y mod int x < int x›*)\n apply -\n (*goals:\n 1. ‹0 < int x ⟹ y = y div int x * int x + y mod int x›\n 2. ‹0 < int x ⟹ 0 ≤ y mod int x›\n 3. ‹0 < int x ⟹ y mod int x < int x›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*proven 3 subgoals*) .\n have \"int_embed R y = int_embed R v\"\n using int_embed_closed (*‹int_embed R (?k::int) ∈ carrier R›*) unfolding y_exp\n (*goal: ‹int_embed R (u * int x + v) = int_embed R v›*)\n by (simp add:int_embed_mult (*‹int_embed R ((?x::int) * (?y::int)) = int_embed R ?x ⊗ int_embed R ?y›*) int_embed_add (*‹int_embed R ((?x::int) + (?y::int)) = int_embed R ?x ⊕ int_embed R ?y›*) assms( (*‹int_embed R (int (x::nat)) = 𝟬›*) 2))\n also (*calculation: ‹int_embed R y = int_embed R v›*) have \"... ∈ int_embed R ` ({0..<int x})\"\n using y_exp(2,3) (*‹0 ≤ v› ‹v < int x›*) by simp\n finally (*calculation: ‹int_embed R y ∈ int_embed R ` {0..<int x}›*) show \"int_embed R y ∈ int_embed R ` {0..<int x}\"\n by simp\n qed\n hence a: \"char_subring R = int_embed R ` {0..<int x}\"\n by auto\n hence \"char R = card (int_embed R ` ({0..<int x}))\"\n unfolding char_def a\n (*goal: ‹card (int_embed R ` {0..<int x}) = card (int_embed R ` {0..<int x})›*)\n by simp\n also (*calculation: ‹char R = card (int_embed R ` {0..<int x})›*) have \"... ≤ card {0..<int x}\"\n apply (intro card_image_le (*‹finite ?A ⟹ card (?f ` ?A) ≤ card ?A›*))\n (*goal: ‹card (int_embed R ` {0..<int x}) ≤ card {0..<int x}›*)\n by simp\n also (*calculation: ‹char R ≤ card {0..<int x}›*) have \"... = x\"\n by simp\n finally (*calculation: ‹char R ≤ (x::nat)›*) show \"char R ≤ x\"\n by simp\n have \"1 = card {int_embed R 0}\"\n by simp\n also (*calculation: ‹1 = card {int_embed R 0}›*) have \"... ≤ card (int_embed R ` {0..<int x})\"\n using assms(1) (*‹0 < x›*) apply (intro card_mono (*‹⟦finite (?B::?'a::type set); (?A::?'a::type set) ⊆ ?B⟧ ⟹ card ?A ≤ card ?B›*) finite_imageI (*‹finite (?F::?'a::type set) ⟹ finite ((?h::?'a::type ⇒ ?'b::type) ` ?F)›*))\n (*goals:\n 1. ‹0 < x ⟹ finite {0..<int x}›\n 2. ‹0 < x ⟹ {int_embed R 0} ⊆ int_embed R ` {0..<int x}›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n also (*calculation: ‹1 ≤ card (int_embed R ` {0..<int x})›*) have \"... = char R\"\n unfolding char_def a\n (*goal: ‹card (int_embed R ` {0..<int x}) = card (int_embed R ` {0..<int x})›*)\n by simp\n finally (*calculation: ‹1 ≤ char R›*) show \"char R > 0\"\n by simp\nqed\n\nlemma (in ring) embed_char_eq_0:\n \"int_embed R (int (char R)) = 𝟬\"\nproof (cases \"finite (char_subring R)\")\n (*goals:\n 1. ‹finite (char_subring R) ⟹ int_embed R (int (char R)) = 𝟬›\n 2. ‹infinite (char_subring R) ⟹ int_embed R (int (char R)) = 𝟬›*)\n case True (*‹finite (char_subring R)›*)\n interpret h: ring_hom_ring int_ring R \"(int_embed R)\"\n using int_embed_ring_hom (*‹ring_hom_ring 𝒵 R (int_embed R)›*) by simp\n define A where \"A = {0..int (char R)}\"\n have \"card (int_embed R ` A) ≤ card (char_subring R)\"\n apply (intro card_mono[OF True] (*‹?A ⊆ char_subring R ⟹ card ?A ≤ card (char_subring R)›*) image_subsetI (*‹(⋀x. x ∈ ?A ⟹ ?f x ∈ ?B) ⟹ ?f ` ?A ⊆ ?B›*))\n (*goal: ‹card (int_embed R ` A) ≤ card (char_subring R)›*)\n by simp\n also (*calculation: ‹card (int_embed R ` A) ≤ card (char_subring R)›*) have \"... = char R\"\n unfolding char_def\n (*goal: ‹card (char_subring R) = card (char_subring R)›*)\n by simp\n also (*calculation: ‹card (int_embed R ` A) ≤ char R›*) have \"... < card A\"\n unfolding A_def\n (*goal: ‹char R < card {0..int (char R)}›*)\n by simp\n finally (*calculation: ‹card (int_embed R ` A) < card A›*) have \"card (int_embed R ` A) < card A\"\n by simp\n hence \"¬inj_on (int_embed R) A\"\n using pigeonhole (*‹card (?f ` ?A) < card ?A ⟹ ¬ inj_on ?f ?A›*) by simp\n then obtain x and y where xy: \"x ∈ A\" \"y ∈ A\" \"x ≠ y\" \"int_embed R x = int_embed R y\"\n (*goal: ‹(⋀x y. ⟦x ∈ A; y ∈ A; x ≠ y; int_embed R x = int_embed R y⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding inj_on_def\n (*goal: ‹(⋀x y. ⟦x ∈ A; y ∈ A; x ≠ y; int_embed R x = int_embed R y⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n define v where \"v = nat (max x y - min x y)\"\n have a: \"int_embed R v = 𝟬\"\n using xy (*‹(x::int) ∈ (A::int set)› ‹y ∈ A› ‹x ≠ y› ‹int_embed R x = int_embed R y›*) int_embed_closed (*‹int_embed R (?k::int) ∈ carrier R›*) apply (cases \"x < y\")\n (*goals:\n 1. ‹⟦x ∈ A; y ∈ A; x ≠ y; int_embed R x = int_embed R y; ⋀k. int_embed R k ∈ carrier R; x < y⟧ ⟹ int_embed R (int v) = 𝟬›\n 2. ‹⟦x ∈ A; y ∈ A; x ≠ y; int_embed R x = int_embed R y; ⋀k. int_embed R k ∈ carrier R; ¬ x < y⟧ ⟹ int_embed R (int v) = 𝟬›\n discuss goal 1*)\n apply (simp add:int_embed_diff (*‹int_embed R (?x - ?y) = int_embed R ?x ⊖ int_embed R ?y›*) v_def (*‹v = nat (max x y - min x y)›*))\n (*discuss goal 2*)\n apply (simp add:int_embed_diff (*‹int_embed R (?x - ?y) = int_embed R ?x ⊖ int_embed R ?y›*) v_def (*‹v = nat (max x y - min x y)›*))\n (*proven 2 subgoals*) .\n moreover have \"v > 0\"\n using xy (*‹x ∈ A› ‹y ∈ A› ‹x ≠ y› ‹int_embed R (x::int) = int_embed R (y::int)›*) apply (cases \"x < y\")\n (*goals:\n 1. ‹⟦(x::int) ∈ (A::int set); (y::int) ∈ A; x ≠ y; int_embed R x = int_embed R y; x < y⟧ ⟹ (0::nat) < (v::nat)›\n 2. ‹⟦(x::int) ∈ (A::int set); (y::int) ∈ A; x ≠ y; int_embed R x = int_embed R y; ¬ x < y⟧ ⟹ (0::nat) < (v::nat)›\n discuss goal 1*)\n apply (simp add:v_def (*‹v = nat (max x y - min x y)›*))\n (*discuss goal 2*)\n apply (simp add:v_def (*‹(v::nat) = nat (max (x::int) (y::int) - min x y)›*))\n (*proven 2 subgoals*) .\n ultimately have \"char R ≤ v\"\n using char_bound (*‹⟦0 < ?x; int_embed R (int ?x) = 𝟬⟧ ⟹ char R ≤ ?x› ‹⟦0 < ?x; int_embed R (int ?x) = 𝟬⟧ ⟹ 0 < char R›*) by simp\n moreover have \"v ≤ char R\"\n using xy (*‹x ∈ A› ‹(y::int) ∈ (A::int set)› ‹(x::int) ≠ (y::int)› ‹int_embed R (x::int) = int_embed R (y::int)›*) v_def (*‹(v::nat) = nat (max (x::int) (y::int) - min x y)›*) A_def (*‹(A::int set) = {0::int..int (char R)}›*) apply (cases \"x < y\")\n (*goals:\n 1. ‹⟦x ∈ A; y ∈ A; x ≠ y; int_embed R x = int_embed R y; v = nat (max x y - min x y); A = {0..int (char R)}; x < y⟧ ⟹ v ≤ char R›\n 2. ‹⟦x ∈ A; y ∈ A; x ≠ y; int_embed R x = int_embed R y; v = nat (max x y - min x y); A = {0..int (char R)}; ¬ x < y⟧ ⟹ v ≤ char R›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n ultimately have \"char R = v\"\n by simp\n then show \"?thesis\"\n (*goal: ‹int_embed R (int (char R)) = 𝟬›*)\n using a (*‹int_embed R (int v) = 𝟬›*) by simp\nnext\n (*goal: ‹infinite (char_subring R) ⟹ int_embed R (int (char R)) = 𝟬›*)\n case False (*‹infinite (char_subring R)›*)\n hence \"char R = 0\"\n unfolding char_def\n (*goal: ‹card (char_subring R) = (0::nat)›*)\n by simp\n then show \"?thesis\"\n (*goal: ‹int_embed R (int (char R)) = 𝟬›*)\n by (simp add:int_embed_zero (*‹int_embed R (0::int) = 𝟬›*))\nqed\n\nlemma (in ring) embed_char_eq_0_iff:\n fixes n :: int\n shows \"int_embed R n = 𝟬 ⟷ char R dvd n\"\nproof (cases \"char R > 0\")\n (*goals:\n 1. ‹0 < char R ⟹ (int_embed R n = 𝟬) = (int (char R) dvd n)›\n 2. ‹¬ 0 < char R ⟹ (int_embed R n = 𝟬) = (int (char R) dvd n)›*)\n case True (*‹0 < char R›*)\n define r where \"r = n mod char R\"\n define s where \"s = n div char R\"\n have rs: \"r < char R\" \"r ≥ 0\" \"n = r + s * char R\"\n using True (*‹0 < char R›*) apply -\n (*goals:\n 1. ‹0 < char R ⟹ r < int (char R)›\n 2. ‹0 < char R ⟹ 0 ≤ r›\n 3. ‹0 < char R ⟹ n = r + s * int (char R)›\n discuss goal 1*)\n apply (simp add:r_def (*‹r = n mod int (char R)›*) s_def (*‹s = n div int (char R)›*))\n (*discuss goal 2*)\n apply (simp add:r_def (*‹r = n mod int (char R)›*) s_def (*‹s = n div int (char R)›*))\n (*discuss goal 3*)\n apply (simp add:r_def (*‹(r::int) = (n::int) mod int (char R)›*) s_def (*‹(s::int) = (n::int) div int (char R)›*))\n (*proven 3 subgoals*) .\n have \"int_embed R n = int_embed R r\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) unfolding rs(3)\n (*goal: ‹int_embed R (r + s * int (char R)) = int_embed R r›*)\n by (simp add: int_embed_add (*‹int_embed R (?x + ?y) = int_embed R ?x ⊕ int_embed R ?y›*) int_embed_mult (*‹int_embed R (?x * ?y) = int_embed R ?x ⊗ int_embed R ?y›*) embed_char_eq_0 (*‹int_embed R (int (char R)) = 𝟬›*))\n moreover have \"nat r < char R\"\n using rs (*‹r < int (char R)› ‹0 ≤ r› ‹n = r + s * int (char R)›*) by simp\n hence \"int_embed R (nat r) ≠ 𝟬 ∨ nat r = 0\"\n using True (*‹(0::nat) < char R›*) char_bound (*‹⟦0 < ?x; int_embed R (int ?x) = 𝟬⟧ ⟹ char R ≤ ?x› ‹⟦0 < ?x; int_embed R (int ?x) = 𝟬⟧ ⟹ 0 < char R›*) not_less (*‹(¬ (?x::?'a) < (?y::?'a)) = (?y ≤ ?x)›*) by blast\n hence \"int_embed R r ≠ 𝟬 ∨ r = 0\"\n using rs (*‹r < int (char R)› ‹0 ≤ r› ‹(n::int) = (r::int) + (s::int) * int (char R)›*) by simp\n ultimately have \"int_embed R n = 𝟬 ⟷ r = 0\"\n using int_embed_zero (*‹int_embed R 0 = 𝟬›*) by auto\n also (*calculation: ‹(int_embed R n = 𝟬) = (r = 0)›*) have \"r = 0 ⟷ char R dvd n\"\n using r_def (*‹r = n mod int (char R)›*) by auto\n finally (*calculation: ‹(int_embed R n = 𝟬) = (int (char R) dvd n)›*) show \"?thesis\"\n (*goal: ‹(int_embed R n = 𝟬) = (int (char R) dvd n)›*)\n by simp\nnext\n (*goal: ‹¬ 0 < char R ⟹ (int_embed R n = 𝟬) = (int (char R) dvd n)›*)\n case False (*‹¬ (0::nat) < char R›*)\n hence \"char R = 0\"\n by simp\n hence a: \"x > 0 ⟹ int_embed R (int x) ≠ 𝟬\" for x\n using char_bound (*‹⟦0 < ?x; int_embed R (int ?x) = 𝟬⟧ ⟹ char R ≤ ?x› ‹⟦(0::nat) < (?x::nat); int_embed R (int ?x) = 𝟬⟧ ⟹ (0::nat) < char R›*) by auto\n have c: \"int_embed R (abs x) ≠ 𝟬 ⟷ int_embed R x ≠ 𝟬\" for x\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) apply (cases \"x > 0\")\n (*goals:\n 1. ‹⟦⋀k. int_embed R k ∈ carrier R; 0 < x⟧ ⟹ (int_embed R ¦x¦ ≠ 𝟬) = (int_embed R x ≠ 𝟬)›\n 2. ‹⟦⋀k. int_embed R k ∈ carrier R; ¬ 0 < x⟧ ⟹ (int_embed R ¦x¦ ≠ 𝟬) = (int_embed R x ≠ 𝟬)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply (simp add:int_embed_inv (*‹int_embed R (- ?x) = ⊖ int_embed R ?x›*))\n (*proven 2 subgoals*) .\n have \"int_embed R x ≠ 𝟬\" if b: \"x ≠ 0\" for x\n proof (-)\n (*goal: ‹int_embed R x ≠ 𝟬›*)\n have \"nat (abs x) > 0\"\n using b (*‹x ≠ 0›*) by simp\n hence \"int_embed R (nat (abs x)) ≠ 𝟬\"\n using a (*‹0 < ?x ⟹ int_embed R (int ?x) ≠ 𝟬›*) by blast\n hence \"int_embed R (abs x) ≠ 𝟬\"\n by simp\n thus \"?thesis\"\n (*goal: ‹int_embed R x ≠ 𝟬›*)\n using c (*‹(int_embed R ¦?x¦ ≠ 𝟬) = (int_embed R ?x ≠ 𝟬)›*) by simp\n qed\n hence \"int_embed R n = 𝟬 ⟷ n = 0\"\n using int_embed_zero (*‹int_embed R 0 = 𝟬›*) by auto\n also (*calculation: ‹(int_embed R n = 𝟬) = (n = 0)›*) have \"n = 0 ⟷ char R dvd n\"\n using False (*‹¬ (0::nat) < char R›*) by simp\n finally (*calculation: ‹(int_embed R n = 𝟬) = (int (char R) dvd n)›*) show \"?thesis\"\n (*goal: ‹(int_embed R n = 𝟬) = (int (char R) dvd n)›*)\n by simp\nqed\n\ntext ‹This result can be found in \\<^cite>‹‹Theorem 1.44› in \"lidl1986\"›.›\n\nlemma (in domain) characteristic_is_prime:\n assumes \"char R > 0\"\n shows \"prime (char R)\"\nproof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹¬ normalization_semidom_class.prime (char R) ⟹ False›*)\n have \"¬(char R = 1)\"\n using embed_char_eq_0 (*‹int_embed R (int (char R)) = 𝟬›*) int_embed_one (*‹int_embed R 1 = 𝟭›*) by auto\n hence \"¬(char R dvd 1)\"\n using assms(1) (*‹0 < char R›*) by simp\n moreover assume \"¬(prime (char R))\" (*‹¬ normalization_semidom_class.prime (char R)›*)\n hence \"¬(irreducible (char R))\"\n using irreducible_imp_prime_elem_gcd (*‹comm_semiring_1_class.irreducible ?x ⟹ prime_elem ?x›*) prime_elem_nat_iff (*‹prime_elem (?n::nat) = normalization_semidom_class.prime ?n›*) by blast\n ultimately obtain p and q where pq_def: \"p * q = char R\" \"p > 1\" \"q > 1\"\n (*goal: ‹(⋀p q. ⟦p * q = char R; 1 < p; 1 < q⟧ ⟹ thesis) ⟹ thesis›*)\n using assms (*‹(0::nat) < char R›*) unfolding Factorial_Ring.irreducible_def\n (*goal: ‹(⋀p q. ⟦p * q = char R; 1 < p; 1 < q⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n have \"int_embed R p ⊗ int_embed R q = 𝟬\"\n using embed_char_eq_0 (*‹int_embed R (int (char R)) = 𝟬›*) pq_def (*‹p * q = char R› ‹1 < p› ‹1 < q›*) apply (subst int_embed_mult[symmetric] (*‹int_embed R (?x::int) ⊗ int_embed R (?y::int) = int_embed R (?x * ?y)›*))\n (*goal: ‹int_embed R (int p) ⊗ int_embed R (int q) = 𝟬›*)\n by (metis of_nat_mult (*‹of_nat (?m * ?n) = of_nat ?m * of_nat ?n›*))\n hence \"int_embed R p = 𝟬 ∨ int_embed R q = 𝟬\"\n using integral (*‹⟦(?a::'a::type) ⊗ (?b::'a::type) = 𝟬; ?a ∈ carrier R; ?b ∈ carrier R⟧ ⟹ ?a = 𝟬 ∨ ?b = 𝟬›*) int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) by simp\n hence \"p*q ≤ p ∨ p*q ≤ q\"\n using char_bound (*‹⟦0 < ?x; int_embed R (int ?x) = 𝟬⟧ ⟹ char R ≤ ?x› ‹⟦0 < ?x; int_embed R (int ?x) = 𝟬⟧ ⟹ 0 < char R›*) pq_def (*‹p * q = char R› ‹1 < p› ‹1 < q›*) by auto\n thus False\n using pq_def(2,3) (*‹1 < p› ‹1 < q›*) by simp\nqed\n\nlemma (in ring) char_ring_is_subring:\n \"subring (char_subring R) R\"\nproof (-)\n (*goal: ‹subring (char_subring R) R›*)\n have \"subring (int_embed R ` carrier int_ring) R\"\n by (intro ring.carrier_is_subring (*‹ring ?R ⟹ subring (carrier ?R) ?R›*) int.ring_axioms (*‹ring 𝒵›*) ring_hom_ring.img_is_subring[OF int_embed_ring_hom] (*‹subring ?K 𝒵 ⟹ subring (int_embed R ` ?K) R›*))\n thus \"?thesis\"\n (*goal: ‹subring (char_subring R) R›*)\n by simp\nqed\n\nlemma (in cring) char_ring_is_subcring:\n \"subcring (char_subring R) R\"\n using subcringI'[OF char_ring_is_subring] (*‹subcring (char_subring R) R›*) by auto\n\nlemma (in domain) char_ring_is_subdomain:\n \"subdomain (char_subring R) R\"\n using subdomainI'[OF char_ring_is_subring] (*‹subdomain (char_subring R) R›*) by auto\n\nlemma image_set_eqI:\n assumes \"⋀x. x ∈ A ⟹ f x ∈ B\"\n assumes \"⋀x. x ∈ B ⟹ g x ∈ A ∧ f (g x) = x\"\n shows \"f ` A = B\"\n using assms (*‹?x ∈ A ⟹ f ?x ∈ B› ‹?x ∈ B ⟹ g ?x ∈ A ∧ f (g ?x) = ?x›*) by force\n\ntext ‹This is the binomial expansion theorem for commutative rings.›\n\nlemma (in cring) binomial_expansion:\n fixes n :: nat\n assumes [simp]: \"x ∈ carrier R\" \"y ∈ carrier R\"\n shows \"(x ⊕ y) [^] n =\n (⨁k ∈ {..n}. int_embed R (n choose k) ⊗ x [^] k ⊗ y [^] (n-k))\"\nproof (-)\n (*goal: ‹(x ⊕ y) [^] n = (⨁k∈{..n}. int_embed R (int (n choose k)) ⊗ x [^] k ⊗ y [^] (n - k))›*)\n define A where \"A = (λk. {A. A ⊆ {..<n} ∧ card A = k})\"\n have fin_A: \"finite (A i)\" for i\n unfolding A_def\n (*goal: ‹finite {A. A ⊆ {..<n} ∧ card A = i}›*)\n by simp\n have disj_A: \"pairwise (λi j. disjnt (A i) (A j)) {..n}\"\n unfolding pairwise_def disjnt_def A_def\n (*goal: ‹∀x∈{..n}. ∀y∈{..n}. x ≠ y ⟶ {A. A ⊆ {..<n} ∧ card A = x} ∩ {A. A ⊆ {..<n} ∧ card A = y} = {}›*)\n by auto\n have card_A: \"B ∈ A i ⟹ card B = i\" if \" i ∈ {..n}\" for i and B\n unfolding A_def\n (*goal: ‹B ∈ {A. A ⊆ {..<n} ∧ card A = i} ⟹ card B = i›*)\n by simp\n have card_A2: \"card (A i) = (n choose i)\" if \"i ∈ {..n}\" for i\n unfolding A_def\n (*goal: ‹card {A. A ⊆ {..<n} ∧ card A = i} = n choose i›*)\n using n_subsets[where A = \"{..<n}\"] (*‹finite {..<n} ⟹ card {B. B ⊆ {..<n} ∧ card B = ?k} = card {..<n} choose ?k›*) by simp\n have card_bound: \"card A ≤ n\" if \"A ⊆ {..<n}\" for n and A\n by (metis card_lessThan (*‹card {..<?u} = ?u›*) finite_lessThan (*‹finite {..<?k}›*) card_mono (*‹⟦finite ?B; ?A ⊆ ?B⟧ ⟹ card ?A ≤ card ?B›*) that (*‹A ⊆ {..<n}›*))\n have card_insert: \"card (insert n A) = card A + 1\" if \"A ⊆ {..<(n::nat)}\" for n and A\n using finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) that (*‹A ⊆ {..<n}›*) apply (subst card_insert_disjoint (*‹⟦finite ?A; ?x ∉ ?A⟧ ⟹ card (insert ?x ?A) = Suc (card ?A)›*))\n (*goals:\n 1. ‹⟦⋀A B. ⟦A ⊆ B; finite B⟧ ⟹ finite A; A ⊆ {..<n}⟧ ⟹ finite A›\n 2. ‹⟦⋀A B. ⟦A ⊆ B; finite B⟧ ⟹ finite A; A ⊆ {..<n}⟧ ⟹ n ∉ A›\n 3. ‹⟦⋀A B. ⟦A ⊆ B; finite B⟧ ⟹ finite A; A ⊆ {..<n}⟧ ⟹ Suc (card A) = card A + 1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n have embed_distr: \"[m] ⋅ y = int_embed R (int m) ⊗ y\" if \"y ∈ carrier R\" for m and y\n unfolding int_embed_def add_pow_def\n (*goal: ‹y [^]⇘add_monoid R⇙ m = 𝟭 [^]⇘add_monoid R⇙ int m ⊗ y›*)\n using that (*‹y ∈ carrier R›*) by (simp add:add_pow_def[symmetric] (*‹?a [^]⇘add_monoid ?R⇙ ?k = [?k] ⋅⇘?R⇙ ?a›*) int_pow_int (*‹?x [^]⇘?G⇙ int ?n = ?x [^]⇘?G⇙ ?n›*) add_pow_ldistr (*‹⟦?a ∈ carrier R; ?b ∈ carrier R⟧ ⟹ [?k] ⋅ ?a ⊗ ?b = [?k] ⋅ (?a ⊗ ?b)›*))\n have \"(x ⊕ y) [^] n =\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n-card A))\"\n proof (induction n)\n (*goals:\n 1. ‹(x ⊕ y) [^] 0 = (⨁A∈Pow {..<0}. x [^] card A ⊗ y [^] (0 - card A))›\n 2. ‹⋀n. (x ⊕ y) [^] n = (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⟹ (x ⊕ y) [^] Suc n = (⨁A∈Pow {..<Suc n}. x [^] card A ⊗ y [^] (Suc n - card A))›*)\n case 0 (*no hyothesis introduced yet*)\n then show \"?case\"\n (*goal: ‹(x ⊕ y) [^] 0 = (⨁A∈Pow {..<0}. x [^] card A ⊗ y [^] (0 - card A))›*)\n by simp\n next\n (*goal: ‹⋀n::nat. ((x::'a::type) ⊕ (y::'a::type)) [^] n = (⨁A::nat set∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⟹ (x ⊕ y) [^] Suc n = (⨁A::nat set∈Pow {..<Suc n}. x [^] card A ⊗ y [^] (Suc n - card A))›*)\n case (Suc n) (*‹((x::'a::type) ⊕ (y::'a::type)) [^] (n::nat) = (⨁A::nat set∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A))›*)\n have s1: \"insert n ` Pow {..<n} = {A. A ⊆ {..<n+1} ∧ n ∈ A}\"\n apply (intro image_set_eqI[where g=\"λx. x ∩ {..<n}\"] (*‹⟦⋀x. x ∈ ?A ⟹ ?f x ∈ ?B; ⋀x. x ∈ ?B ⟹ x ∩ {..<n} ∈ ?A ∧ ?f (x ∩ {..<n}) = x⟧ ⟹ ?f ` ?A = ?B›*))\n (*goals:\n 1. ‹⋀x::nat set. x ∈ Pow {..<n::nat} ⟹ insert n x ∈ {A::nat set. A ⊆ {..<n + (1::nat)} ∧ n ∈ A}›\n 2. ‹⋀x::nat set. x ∈ {A::nat set. A ⊆ {..<(n::nat) + (1::nat)} ∧ n ∈ A} ⟹ x ∩ {..<n} ∈ Pow {..<n} ∧ insert n (x ∩ {..<n}) = x›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have s2: \"Pow {..<n} = {A. A ⊆ {..<n+1} ∧ n ∉ A}\"\n using lessThan_Suc (*‹{..<Suc ?k} = insert ?k {..<?k}›*) by auto\n have \"(x ⊕ y) [^] Suc n = (x ⊕ y) [^] n ⊗ (x ⊕ y)\"\n by simp\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (x ⊕ y) [^] n ⊗ (x ⊕ y)›*) have \"... =\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n-card A)) ⊗\n (x ⊕ y)\"\n apply (subst Suc (*‹(x ⊕ y) [^] n = (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A))›*))\n (*goal: ‹(x ⊕ y) [^] n ⊗ (x ⊕ y) = (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ (x ⊕ y)›*)\n by simp\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ (x ⊕ y)›*) have \"... =\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n-card A)) ⊗ x ⊕\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n-card A)) ⊗ y\"\n apply (subst r_distr (*‹⟦?x ∈ carrier R; ?y ∈ carrier R; ?z ∈ carrier R⟧ ⟹ ?z ⊗ (?x ⊕ ?y) = ?z ⊗ ?x ⊕ ?z ⊗ ?y›*))\n (*goals:\n 1. ‹x ∈ carrier R›\n 2. ‹y ∈ carrier R›\n 3. ‹(⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ∈ carrier R›\n 4. ‹(⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ x ⊕ (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ y = (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ x ⊕ (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ y›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ x ⊕ (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A)) ⊗ y›*) have \"... =\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n-card A) ⊗ x) ⊕\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n-card A) ⊗ y)\"\n by (simp add:finsum_ldistr (*‹⟦finite ?A; ?a ∈ carrier R; ?f ∈ ?A → carrier R⟧ ⟹ finsum R ?f ?A ⊗ ?a = (⨁i∈?A. ?f i ⊗ ?a)›*))\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A) ⊗ x) ⊕ (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A) ⊗ y)›*) have \"... =\n (⨁A ∈ Pow {..<n}. x [^] (card A+1) ⊗ y [^] (n-card A)) ⊕\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n-card A+1))\"\n using m_assoc (*‹⟦?x ∈ carrier R; ?y ∈ carrier R; ?z ∈ carrier R⟧ ⟹ ?x ⊗ ?y ⊗ ?z = ?x ⊗ (?y ⊗ ?z)›*) m_comm (*‹⟦?x ∈ carrier R; ?y ∈ carrier R⟧ ⟹ ?x ⊗ ?y = ?y ⊗ ?x›*) apply (intro arg_cong2[where f=\"(⊕)\"] (*‹⟦?a = ?b; ?c = ?d⟧ ⟹ ?a ⊕ ?c = ?b ⊕ ?d›*) finsum_cong' (*‹⟦?A = ?B; ?g ∈ ?B → carrier R; ⋀i. i ∈ ?B ⟹ ?f i = ?g i⟧ ⟹ finsum R ?f ?A = finsum R ?g ?B›*))\n (*goals:\n 1. ‹⟦⋀(x::'a) (y::'a) z::'a. ⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z); ⋀(x::'a) y::'a. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ⊗ y = y ⊗ x⟧ ⟹ Pow {..<n::nat} = Pow {..<n}›\n 2. ‹⟦⋀(x::'a) (y::'a) z::'a. ⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z); ⋀(x::'a) y::'a. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ⊗ y = y ⊗ x⟧ ⟹ (λA::nat set. (x::'a) [^] (card A + (1::nat)) ⊗ (y::'a) [^] ((n::nat) - card A)) ∈ Pow {..<n} → carrier R›\n 3. ‹⋀i::nat set. ⟦⋀(x::'a) (y::'a) z::'a. ⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z); ⋀(x::'a) y::'a. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ⊗ y = y ⊗ x; i ∈ Pow {..<n::nat}⟧ ⟹ (x::'a) [^] card i ⊗ (y::'a) [^] (n - card i) ⊗ x = x [^] (card i + (1::nat)) ⊗ y [^] (n - card i)›\n 4. ‹⟦⋀(x::'a) (y::'a) z::'a. ⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z); ⋀(x::'a) y::'a. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ⊗ y = y ⊗ x⟧ ⟹ Pow {..<n::nat} = Pow {..<n}›\n 5. ‹⟦⋀(x::'a) (y::'a) z::'a. ⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z); ⋀(x::'a) y::'a. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ⊗ y = y ⊗ x⟧ ⟹ (λA::nat set. (x::'a) [^] card A ⊗ (y::'a) [^] ((n::nat) - card A + (1::nat))) ∈ Pow {..<n} → carrier R›\n 6. ‹⋀i::nat set. ⟦⋀(x::'a) (y::'a) z::'a. ⟦x ∈ carrier R; y ∈ carrier R; z ∈ carrier R⟧ ⟹ x ⊗ y ⊗ z = x ⊗ (y ⊗ z); ⋀(x::'a) y::'a. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ⊗ y = y ⊗ x; i ∈ Pow {..<n::nat}⟧ ⟹ (x::'a) [^] card i ⊗ (y::'a) [^] (n - card i) ⊗ y = x [^] card i ⊗ y [^] (n - card i + (1::nat))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*proven 6 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (⨁A∈Pow {..<n}. x [^] (card A + 1) ⊗ y [^] (n - card A)) ⊕ (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A + 1))›*) have \"... =\n (⨁A ∈ Pow {..<n}. x [^] (card (insert n A))\n ⊗ y [^] (n+1-card (insert n A))) ⊕\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n+1-card A))\"\n using finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) card_bound (*‹(?A::nat set) ⊆ {..<?n::nat} ⟹ card ?A ≤ ?n›*) card_insert (*‹?A ⊆ {..<?n} ⟹ card (insert ?n ?A) = card ?A + 1›*) Suc_diff_le (*‹?n ≤ ?m ⟹ Suc ?m - ?n = Suc (?m - ?n)›*) apply (intro arg_cong2[where f=\"(⊕)\"] (*‹⟦?a = ?b; ?c = ?d⟧ ⟹ ?a ⊕ ?c = ?b ⊕ ?d›*) finsum_cong' (*‹⟦?A = ?B; ?g ∈ ?B → carrier R; ⋀i. i ∈ ?B ⟹ ?f i = ?g i⟧ ⟹ finsum R ?f ?A = finsum R ?g ?B›*))\n (*goals:\n 1. ‹⟦⋀(A::?'a set) B::?'a set. ⟦A ⊆ B; finite B⟧ ⟹ finite A; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card A ≤ n; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card (insert n A) = card A + (1::nat); ⋀(n::nat) m::nat. n ≤ m ⟹ Suc m - n = Suc (m - n)⟧ ⟹ Pow {..<n::nat} = Pow {..<n}›\n 2. ‹⟦⋀(A::?'a set) B::?'a set. ⟦A ⊆ B; finite B⟧ ⟹ finite A; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card A ≤ n; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card (insert n A) = card A + (1::nat); ⋀(n::nat) m::nat. n ≤ m ⟹ Suc m - n = Suc (m - n)⟧ ⟹ (λA::nat set. (x::'a) [^] card (insert (n::nat) A) ⊗ (y::'a) [^] (n + (1::nat) - card (insert n A))) ∈ Pow {..<n} → carrier R›\n 3. ‹⋀i::nat set. ⟦⋀(A::?'a set) B::?'a set. ⟦A ⊆ B; finite B⟧ ⟹ finite A; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card A ≤ n; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card (insert n A) = card A + (1::nat); ⋀(n::nat) m::nat. n ≤ m ⟹ Suc m - n = Suc (m - n); i ∈ Pow {..<n::nat}⟧ ⟹ (x::'a) [^] (card i + (1::nat)) ⊗ (y::'a) [^] (n - card i) = x [^] card (insert n i) ⊗ y [^] (n + (1::nat) - card (insert n i))›\n 4. ‹⟦⋀(A::?'a set) B::?'a set. ⟦A ⊆ B; finite B⟧ ⟹ finite A; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card A ≤ n; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card (insert n A) = card A + (1::nat); ⋀(n::nat) m::nat. n ≤ m ⟹ Suc m - n = Suc (m - n)⟧ ⟹ Pow {..<n::nat} = Pow {..<n}›\n 5. ‹⟦⋀(A::?'a set) B::?'a set. ⟦A ⊆ B; finite B⟧ ⟹ finite A; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card A ≤ n; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card (insert n A) = card A + (1::nat); ⋀(n::nat) m::nat. n ≤ m ⟹ Suc m - n = Suc (m - n)⟧ ⟹ (λA::nat set. (x::'a) [^] card A ⊗ (y::'a) [^] ((n::nat) + (1::nat) - card A)) ∈ Pow {..<n} → carrier R›\n 6. ‹⋀i::nat set. ⟦⋀(A::?'a set) B::?'a set. ⟦A ⊆ B; finite B⟧ ⟹ finite A; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card A ≤ n; ⋀(A::nat set) n::nat. A ⊆ {..<n} ⟹ card (insert n A) = card A + (1::nat); ⋀(n::nat) m::nat. n ≤ m ⟹ Suc m - n = Suc (m - n); i ∈ Pow {..<n::nat}⟧ ⟹ (x::'a) [^] card i ⊗ (y::'a) [^] (n - card i + (1::nat)) = x [^] card i ⊗ y [^] (n + (1::nat) - card i)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*proven 6 subgoals*) .\n also (*calculation: ‹((x::'a::type) ⊕ (y::'a::type)) [^] Suc (n::nat) = (⨁A::nat set∈Pow {..<n}. x [^] card (insert n A) ⊗ y [^] (n + (1::nat) - card (insert n A))) ⊕ (⨁A::nat set∈Pow {..<n}. x [^] card A ⊗ y [^] (n + (1::nat) - card A))›*) have \"... =\n (⨁A ∈ insert n ` Pow {..<n}. x [^] (card A)\n ⊗ y [^] (n+1-card A)) ⊕\n (⨁A ∈ Pow {..<n}. x [^] (card A) ⊗ y [^] (n+1-card A))\"\n apply (subst finsum_reindex (*‹⟦?f ∈ ?h ` ?A → carrier R; inj_on ?h ?A⟧ ⟹ finsum R ?f (?h ` ?A) = (⨁x∈?A. ?f (?h x))›*))\n (*goals:\n 1. ‹(λA::nat set. (x::'a) [^] card A ⊗ (y::'a) [^] ((n::nat) + (1::nat) - card A)) ∈ insert n ` Pow {..<n} → carrier R›\n 2. ‹inj_on (insert (n::nat)) (Pow {..<n})›\n 3. ‹(⨁A::nat set∈Pow {..<n}. (x::'a) [^] card (insert (n::nat) A) ⊗ (y::'a) [^] (n + (1::nat) - card (insert n A))) ⊕ (⨁A::nat set∈Pow {..<n}. x [^] card A ⊗ y [^] (n + (1::nat) - card A)) = (⨁xa::nat set∈Pow {..<n}. x [^] card (insert n xa) ⊗ y [^] (n + (1::nat) - card (insert n xa))) ⊕ (⨁A::nat set∈Pow {..<n}. x [^] card A ⊗ y [^] (n + (1::nat) - card A))›\n discuss goal 1*)\n apply ((auto simp add:inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add:inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add:inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*))[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (⨁A∈insert n ` Pow {..<n}. x [^] card A ⊗ y [^] (n + 1 - card A)) ⊕ (⨁A∈Pow {..<n}. x [^] card A ⊗ y [^] (n + 1 - card A))›*) have \"... =\n (⨁A ∈ {A. A ⊆ {..<n+1} ∧ n ∈ A}.\n x [^] (card A) ⊗ y [^] (n+1-card A)) ⊕\n (⨁A ∈ {A. A ⊆ {..<n+1} ∧ n ∉ A}.\n x [^] (card A) ⊗ y [^] (n+1-card A))\"\n apply (intro arg_cong2[where f=\"(⊕)\"] (*‹⟦?a = ?b; ?c = ?d⟧ ⟹ ?a ⊕ ?c = ?b ⊕ ?d›*) finsum_cong' (*‹⟦?A = ?B; ?g ∈ ?B → carrier R; ⋀i. i ∈ ?B ⟹ ?f i = ?g i⟧ ⟹ finsum R ?f ?A = finsum R ?g ?B›*) s1 (*‹insert n ` Pow {..<n} = {A. A ⊆ {..<n + 1} ∧ n ∈ A}›*) s2 (*‹Pow {..<n} = {A. A ⊆ {..<n + 1} ∧ n ∉ A}›*))\n (*goals:\n 1. ‹(λA. x [^] card A ⊗ y [^] (n + 1 - card A)) ∈ {A. A ⊆ {..<n + 1} ∧ n ∈ A} → carrier R›\n 2. ‹⋀i. i ∈ {A. A ⊆ {..<n + 1} ∧ n ∈ A} ⟹ x [^] card i ⊗ y [^] (n + 1 - card i) = x [^] card i ⊗ y [^] (n + 1 - card i)›\n 3. ‹(λA. x [^] card A ⊗ y [^] (n + 1 - card A)) ∈ {A. A ⊆ {..<n + 1} ∧ n ∉ A} → carrier R›\n 4. ‹⋀i. i ∈ {A. A ⊆ {..<n + 1} ∧ n ∉ A} ⟹ x [^] card i ⊗ y [^] (n + 1 - card i) = x [^] card i ⊗ y [^] (n + 1 - card i)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*proven 4 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (⨁A∈{A. A ⊆ {..<n + 1} ∧ n ∈ A}. x [^] card A ⊗ y [^] (n + 1 - card A)) ⊕ (⨁A∈{A. A ⊆ {..<n + 1} ∧ n ∉ A}. x [^] card A ⊗ y [^] (n + 1 - card A))›*) have \"... = (⨁A ∈\n {A. A ⊆ {..<n+1} ∧ n ∈ A} ∪ {A. A ⊆ {..<n+1} ∧ n ∉ A}.\n x [^] (card A) ⊗ y [^] (n+1-card A))\"\n apply (subst finsum_Un_disjoint (*‹⟦finite ?A; finite ?B; ?A ∩ ?B = {}; ?g ∈ ?A → carrier R; ?g ∈ ?B → carrier R⟧ ⟹ finsum R ?g (?A ∪ ?B) = finsum R ?g ?A ⊕ finsum R ?g ?B›*))\n (*goals:\n 1. ‹finite {A. A ⊆ {..<n + 1} ∧ n ∈ A}›\n 2. ‹finite {A. A ⊆ {..<n + 1} ∧ n ∉ A}›\n 3. ‹{A. A ⊆ {..<n + 1} ∧ n ∈ A} ∩ {A. A ⊆ {..<n + 1} ∧ n ∉ A} = {}›\n 4. ‹(λA. x [^] card A ⊗ y [^] (n + 1 - card A)) ∈ {A. A ⊆ {..<n + 1} ∧ n ∈ A} → carrier R›\n 5. ‹(λA. x [^] card A ⊗ y [^] (n + 1 - card A)) ∈ {A. A ⊆ {..<n + 1} ∧ n ∉ A} → carrier R›\n 6. ‹(⨁A∈{A. A ⊆ {..<n + 1} ∧ n ∈ A}. x [^] card A ⊗ y [^] (n + 1 - card A)) ⊕ (⨁A∈{A. A ⊆ {..<n + 1} ∧ n ∉ A}. x [^] card A ⊗ y [^] (n + 1 - card A)) = (⨁A∈{A. A ⊆ {..<n + 1} ∧ n ∈ A}. x [^] card A ⊗ y [^] (n + 1 - card A)) ⊕ (⨁A∈{A. A ⊆ {..<n + 1} ∧ n ∉ A}. x [^] card A ⊗ y [^] (n + 1 - card A))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*proven 6 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] Suc n = (⨁A∈{A. A ⊆ {..<n + 1} ∧ n ∈ A} ∪ {A. A ⊆ {..<n + 1} ∧ n ∉ A}. x [^] card A ⊗ y [^] (n + 1 - card A))›*) have \"... =\n (⨁A ∈ Pow {..<n+1}. x [^] (card A) ⊗ y [^] (n+1-card A))\"\n apply (intro finsum_cong' (*‹⟦?A = ?B; ?g ∈ ?B → carrier R; ⋀i. i ∈ ?B ⟹ ?f i = ?g i⟧ ⟹ finsum R ?f ?A = finsum R ?g ?B›*))\n (*goals:\n 1. ‹{A. A ⊆ {..<n + 1} ∧ n ∈ A} ∪ {A. A ⊆ {..<n + 1} ∧ n ∉ A} = Pow {..<n + 1}›\n 2. ‹(λA. x [^] card A ⊗ y [^] (n + 1 - card A)) ∈ Pow {..<n + 1} → carrier R›\n 3. ‹⋀i. i ∈ Pow {..<n + 1} ⟹ x [^] card i ⊗ y [^] (n + 1 - card i) = x [^] card i ⊗ y [^] (n + 1 - card i)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n finally (*calculation: ‹((x::'a) ⊕ (y::'a)) [^] Suc (n::nat) = (⨁A::nat set∈Pow {..<n + (1::nat)}. x [^] card A ⊗ y [^] (n + (1::nat) - card A))›*) show \"?case\"\n (*goal: ‹(x ⊕ y) [^] Suc n = (⨁A∈Pow {..<Suc n}. x [^] card A ⊗ y [^] (Suc n - card A))›*)\n by simp\n qed\n also (*calculation: ‹((x::'a) ⊕ (y::'a)) [^] (n::nat) = (⨁A::nat set∈Pow {..<n}. x [^] card A ⊗ y [^] (n - card A))›*) have \"... =\n (⨁A ∈ (⋃ (A ` {..n})). x [^] (card A) ⊗ y [^] (n-card A))\"\n using card_bound (*‹?A ⊆ {..<?n} ⟹ card ?A ≤ ?n›*) apply (intro finsum_cong' (*‹⟦?A = ?B; ?g ∈ ?B → carrier R; ⋀i. i ∈ ?B ⟹ ?f i = ?g i⟧ ⟹ finsum R ?f ?A = finsum R ?g ?B›*))\n (*goals:\n 1. ‹(⋀A n. A ⊆ {..<n} ⟹ card A ≤ n) ⟹ Pow {..<n} = ⋃ (A ` {..n})›\n 2. ‹(⋀A n. A ⊆ {..<n} ⟹ card A ≤ n) ⟹ (λA. x [^] card A ⊗ y [^] (n - card A)) ∈ ⋃ (A ` {..n}) → carrier R›\n 3. ‹⋀i. ⟦⋀A n. A ⊆ {..<n} ⟹ card A ≤ n; i ∈ ⋃ (A ` {..n})⟧ ⟹ x [^] card i ⊗ y [^] (n - card i) = x [^] card i ⊗ y [^] (n - card i)›\n discuss goal 1*)\n apply ((auto simp add:A_def (*‹A = (λk. {A. A ⊆ {..<n} ∧ card A = k})›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add:A_def (*‹A = (λk. {A. A ⊆ {..<n} ∧ card A = k})›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add:A_def (*‹A = (λk. {A. A ⊆ {..<n} ∧ card A = k})›*))[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] n = (⨁A∈⋃ (A ` {..n}). x [^] card A ⊗ y [^] (n - card A))›*) have \"... =\n (⨁ k ∈ {..n}. (⨁ A ∈ A k. x [^] (card A) ⊗ y [^] (n-card A)))\"\n using fin_A (*‹finite (A ?i)›*) disj_A (*‹pairwise (λi j. disjnt (A i) (A j)) {..n}›*) apply (subst add.finprod_UN_disjoint (*‹⟦finite ?I; ⋀i. i ∈ ?I ⟹ finite (?A i); pairwise (λi j. disjnt (?A i) (?A j)) ?I; ⋀i x. ⟦i ∈ ?I; x ∈ ?A i⟧ ⟹ ?g x ∈ carrier R⟧ ⟹ finsum R ?g (⋃ (?A ` ?I)) = (⨁i∈?I. finsum R ?g (?A i))›*))\n (*goals:\n 1. ‹⟦⋀i::nat. finite ((A::nat ⇒ nat set set) i); pairwise (λ(i::nat) j::nat. disjnt (A i) (A j)) {..n::nat}⟧ ⟹ finite {..n}›\n 2. ‹⋀i::nat. ⟦⋀i::nat. finite ((A::nat ⇒ nat set set) i); pairwise (λ(i::nat) j::nat. disjnt (A i) (A j)) {..n::nat}; i ∈ {..n}⟧ ⟹ finite (A i)›\n 3. ‹⟦⋀i::nat. finite ((A::nat ⇒ nat set set) i); pairwise (λ(i::nat) j::nat. disjnt (A i) (A j)) {..n::nat}⟧ ⟹ pairwise (λ(i::nat) j::nat. disjnt (A i) (A j)) {..n}›\n 4. ‹⋀(i::nat) xa::nat set. ⟦⋀i::nat. finite ((A::nat ⇒ nat set set) i); pairwise (λ(i::nat) j::nat. disjnt (A i) (A j)) {..n::nat}; i ∈ {..n}; xa ∈ A i⟧ ⟹ (x::'a) [^] card xa ⊗ (y::'a) [^] (n - card xa) ∈ carrier R›\n 5. ‹⟦⋀i::nat. finite ((A::nat ⇒ nat set set) i); pairwise (λ(i::nat) j::nat. disjnt (A i) (A j)) {..n::nat}⟧ ⟹ (⨁i::nat∈{..n}. ⨁A::nat set∈A i. (x::'a) [^] card A ⊗ (y::'a) [^] (n - card A)) = (⨁k::nat∈{..n}. ⨁A::nat set∈A k. x [^] card A ⊗ y [^] (n - card A))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*proven 5 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] n = (⨁k∈{..n}. ⨁A∈A k. x [^] card A ⊗ y [^] (n - card A))›*) have \"... = (⨁ k ∈ {..n}. (⨁ A ∈ A k. x [^] k ⊗ y [^] (n-k)))\"\n using card_A (*‹⟦?i ∈ {..n}; ?B ∈ A ?i⟧ ⟹ card ?B = ?i›*) apply (intro finsum_cong' (*‹⟦?A = ?B; ?g ∈ ?B → carrier R; ⋀i. i ∈ ?B ⟹ ?f i = ?g i⟧ ⟹ finsum R ?f ?A = finsum R ?g ?B›*))\n (*goals:\n 1. ‹(⋀(i::nat) B::nat set. ⟦i ∈ {..n::nat}; B ∈ (A::nat ⇒ nat set set) i⟧ ⟹ card B = i) ⟹ {..n} = {..n}›\n 2. ‹(⋀(i::nat) B::nat set. ⟦i ∈ {..n::nat}; B ∈ (A::nat ⇒ nat set set) i⟧ ⟹ card B = i) ⟹ (λk::nat. ⨁A::nat set∈A k. (x::'a::type) [^] k ⊗ (y::'a::type) [^] (n - k)) ∈ {..n} → carrier R›\n 3. ‹⋀i::nat. ⟦⋀(i::nat) B::nat set. ⟦i ∈ {..n::nat}; B ∈ (A::nat ⇒ nat set set) i⟧ ⟹ card B = i; i ∈ {..n}⟧ ⟹ A i = A i›\n 4. ‹⋀i::nat. ⟦⋀(i::nat) B::nat set. ⟦i ∈ {..n::nat}; B ∈ (A::nat ⇒ nat set set) i⟧ ⟹ card B = i; i ∈ {..n}⟧ ⟹ (λA::nat set. (x::'a::type) [^] i ⊗ (y::'a::type) [^] (n - i)) ∈ A i → carrier R›\n 5. ‹⋀(i::nat) ia::nat set. ⟦⋀(i::nat) B::nat set. ⟦i ∈ {..n::nat}; B ∈ (A::nat ⇒ nat set set) i⟧ ⟹ card B = i; i ∈ {..n}; ia ∈ A i⟧ ⟹ (x::'a::type) [^] card ia ⊗ (y::'a::type) [^] (n - card ia) = x [^] i ⊗ y [^] (n - i)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*proven 5 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] n = (⨁k∈{..n}. ⨁A∈A k. x [^] k ⊗ y [^] (n - k))›*) have \"... =\n (⨁ k ∈ {..n}. int_embed R (card (A k)) ⊗ x [^] k ⊗ y [^] (n-k))\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) apply (subst add.finprod_const (*‹?a ∈ carrier R ⟹ (⨁x∈?A. ?a) = [card ?A] ⋅ ?a›*))\n (*goals:\n 1. ‹⋀k. (⋀k. int_embed R k ∈ carrier R) ⟹ x [^] k ⊗ y [^] (n - k) ∈ carrier R›\n 2. ‹(⋀k. int_embed R k ∈ carrier R) ⟹ (⨁k∈{..n}. [card (A k)] ⋅ (x [^] k ⊗ y [^] (n - k))) = (⨁k∈{..n}. int_embed R (int (card (A k))) ⊗ x [^] k ⊗ y [^] (n - k))›\n discuss goal 1*)\n apply (simp add:embed_distr (*‹?y ∈ carrier R ⟹ [?m] ⋅ ?y = int_embed R (int ?m) ⊗ ?y›*) m_assoc (*‹⟦?x ∈ carrier R; ?y ∈ carrier R; ?z ∈ carrier R⟧ ⟹ ?x ⊗ ?y ⊗ ?z = ?x ⊗ (?y ⊗ ?z)›*))\n (*discuss goal 2*)\n apply (simp add:embed_distr (*‹?y ∈ carrier R ⟹ [?m] ⋅ ?y = int_embed R (int ?m) ⊗ ?y›*) m_assoc (*‹⟦?x ∈ carrier R; ?y ∈ carrier R; ?z ∈ carrier R⟧ ⟹ ?x ⊗ ?y ⊗ ?z = ?x ⊗ (?y ⊗ ?z)›*))\n (*proven 2 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] n = (⨁k∈{..n}. int_embed R (int (card (A k))) ⊗ x [^] k ⊗ y [^] (n - k))›*) have \"... =\n (⨁ k ∈ {..n}. int_embed R (n choose k) ⊗ x [^] k ⊗ y [^] (n-k))\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) card_A2 (*‹(?i::nat) ∈ {..n::nat} ⟹ card ((A::nat ⇒ nat set set) ?i) = n choose ?i›*) apply (intro finsum_cong' (*‹⟦?A = ?B; ?g ∈ ?B → carrier R; ⋀i. i ∈ ?B ⟹ ?f i = ?g i⟧ ⟹ finsum R ?f ?A = finsum R ?g ?B›*))\n (*goals:\n 1. ‹⟦⋀k. int_embed R k ∈ carrier R; ⋀i. i ∈ {..n} ⟹ card (A i) = n choose i⟧ ⟹ {..n} = {..n}›\n 2. ‹⟦⋀k. int_embed R k ∈ carrier R; ⋀i. i ∈ {..n} ⟹ card (A i) = n choose i⟧ ⟹ (λk. int_embed R (int (n choose k)) ⊗ x [^] k ⊗ y [^] (n - k)) ∈ {..n} → carrier R›\n 3. ‹⋀i. ⟦⋀k. int_embed R k ∈ carrier R; ⋀i. i ∈ {..n} ⟹ card (A i) = n choose i; i ∈ {..n}⟧ ⟹ int_embed R (int (card (A i))) ⊗ x [^] i ⊗ y [^] (n - i) = int_embed R (int (n choose i)) ⊗ x [^] i ⊗ y [^] (n - i)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*proven 3 subgoals*) .\n finally (*calculation: ‹(x ⊕ y) [^] n = (⨁k∈{..n}. int_embed R (int (n choose k)) ⊗ x [^] k ⊗ y [^] (n - k))›*) show \"?thesis\"\n (*goal: ‹(x ⊕ y) [^] n = (⨁k∈{..n}. int_embed R (int (n choose k)) ⊗ x [^] k ⊗ y [^] (n - k))›*)\n by simp\nqed\n\nlemma bin_prime_factor:\n assumes \"prime p\"\n assumes \"k > 0\" \"k < p\"\n shows \"p dvd (p choose k)\"\nproof (-)\n (*goal: ‹(p::nat) dvd p choose (k::nat)›*)\n have \"p dvd fact p\"\n using assms(1) (*‹normalization_semidom_class.prime p›*) prime_dvd_fact_iff (*‹normalization_semidom_class.prime ?p ⟹ (?p dvd fact ?n) = (?p ≤ ?n)›*) by auto\n hence \"p dvd fact k * fact (p - k) * (p choose k)\"\n using binomial_fact_lemma (*‹(?k::nat) ≤ (?n::nat) ⟹ fact ?k * fact (?n - ?k) * (?n choose ?k) = fact ?n›*) assms (*‹normalization_semidom_class.prime p› ‹0 < k› ‹k < p›*) by simp\n hence \"p dvd fact k ∨ p dvd fact (p-k) ∨ p dvd (p choose k)\"\n by (simp add: assms( (*‹normalization_semidom_class.prime p›*) 1) prime_dvd_mult_eq_nat (*‹normalization_semidom_class.prime ?p ⟹ (?p dvd ?a * ?b) = (?p dvd ?a ∨ ?p dvd ?b)›*))\n thus \"p dvd (p choose k)\"\n using assms(1,2,3) (*‹normalization_semidom_class.prime p› ‹0 < k› ‹k < p›*) prime_dvd_fact_iff (*‹normalization_semidom_class.prime ?p ⟹ (?p dvd fact ?n) = (?p ≤ ?n)›*) by auto\nqed\n\ntheorem (in domain) freshmans_dream:\n assumes \"char R > 0\"\n assumes [simp]: \"x ∈ carrier R\" \"y ∈ carrier R\"\n shows \"(x ⊕ y) [^] (char R) = x [^] char R ⊕ y [^] char R\"\n (is \"?lhs = ?rhs\")\nproof (-)\n (*goal: ‹((x::'a) ⊕ (y::'a)) [^] char R = x [^] char R ⊕ y [^] char R›*)\n have c: \"prime (char R)\"\n using assms(1) (*‹0 < char R›*) characteristic_is_prime (*‹0 < char R ⟹ normalization_semidom_class.prime (char R)›*) by auto\n have a: \"int_embed R (char R choose i) = 𝟬\" if \"i ∈ {..char R} - {0, char R}\" for i\n proof (-)\n (*goal: ‹int_embed R (int (char R choose i)) = 𝟬›*)\n have \"i > 0\" \"i < char R\"\n using that (*‹i ∈ {..char R} - {0, char R}›*) apply -\n (*goals:\n 1. ‹(i::nat) ∈ {..char R} - {0::nat, char R} ⟹ (0::nat) < i›\n 2. ‹(i::nat) ∈ {..char R} - {0::nat, char R} ⟹ i < char R›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence \"char R dvd char R choose i\"\n using c (*‹normalization_semidom_class.prime (char R)›*) bin_prime_factor (*‹⟦normalization_semidom_class.prime ?p; 0 < ?k; ?k < ?p⟧ ⟹ ?p dvd ?p choose ?k›*) by simp\n thus \"?thesis\"\n (*goal: ‹int_embed R (int (char R choose (i::nat))) = 𝟬›*)\n using embed_char_eq_0_iff (*‹(int_embed R ?n = 𝟬) = (int (char R) dvd ?n)›*) by simp\n qed\n have \"?lhs = (⨁k ∈ {..char R}. int_embed R (char R choose k)\n ⊗ x [^] k ⊗ y [^] (char R-k))\"\n using binomial_expansion[OF assms ( 2 , 3 )] (*‹(x ⊕ y) [^] ?n = (⨁k∈{..?n}. int_embed R (int (?n choose k)) ⊗ x [^] k ⊗ y [^] (?n - k))›*) by simp\n also (*calculation: ‹(x ⊕ y) [^] char R = (⨁k∈{..char R}. int_embed R (int (char R choose k)) ⊗ x [^] k ⊗ y [^] (char R - k))›*) have \"... = (⨁k ∈ {0,char R}.int_embed R (char R choose k)\n ⊗ x [^] k ⊗ y [^] (char R-k))\"\n using a (*‹(?i::nat) ∈ {..char R} - {0::nat, char R} ⟹ int_embed R (int (char R choose ?i)) = 𝟬›*) int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) apply (intro add.finprod_mono_neutral_cong_right (*‹⟦finite ?B; ?A ⊆ ?B; ⋀i. i ∈ ?B - ?A ⟹ ?g i = 𝟬; ⋀x. x ∈ ?A ⟹ ?g x = ?h x; ?g ∈ ?B → carrier R⟧ ⟹ finsum R ?g ?B = finsum R ?h ?A›*))\n (*goals:\n 1. ‹⟦⋀i. i ∈ {..char R} - {0, char R} ⟹ int_embed R (int (char R choose i)) = 𝟬; ⋀k. int_embed R k ∈ carrier R⟧ ⟹ finite {..char R}›\n 2. ‹⟦⋀i. i ∈ {..char R} - {0, char R} ⟹ int_embed R (int (char R choose i)) = 𝟬; ⋀k. int_embed R k ∈ carrier R⟧ ⟹ {0, char R} ⊆ {..char R}›\n 3. ‹⋀i. ⟦⋀i. i ∈ {..char R} - {0, char R} ⟹ int_embed R (int (char R choose i)) = 𝟬; ⋀k. int_embed R k ∈ carrier R; i ∈ {..char R} - {0, char R}⟧ ⟹ int_embed R (int (char R choose i)) ⊗ x [^] i ⊗ y [^] (char R - i) = 𝟬›\n 4. ‹⋀xa. ⟦⋀i. i ∈ {..char R} - {0, char R} ⟹ int_embed R (int (char R choose i)) = 𝟬; ⋀k. int_embed R k ∈ carrier R; xa ∈ {0, char R}⟧ ⟹ int_embed R (int (char R choose xa)) ⊗ x [^] xa ⊗ y [^] (char R - xa) = int_embed R (int (char R choose xa)) ⊗ x [^] xa ⊗ y [^] (char R - xa)›\n 5. ‹⟦⋀i. i ∈ {..char R} - {0, char R} ⟹ int_embed R (int (char R choose i)) = 𝟬; ⋀k. int_embed R k ∈ carrier R⟧ ⟹ (λk. int_embed R (int (char R choose k)) ⊗ x [^] k ⊗ y [^] (char R - k)) ∈ {..char R} → carrier R›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*proven 5 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] char R = (⨁k∈{0, char R}. int_embed R (int (char R choose k)) ⊗ x [^] k ⊗ y [^] (char R - k))›*) have \"... = ?rhs\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) assms(1) (*‹0 < char R›*) by (simp add:int_embed_one (*‹int_embed R 1 = 𝟭›*) a_comm (*‹⟦?x ∈ carrier R; ?y ∈ carrier R⟧ ⟹ ?x ⊕ ?y = ?y ⊕ ?x›*))\n finally (*calculation: ‹(x ⊕ y) [^] char R = x [^] char R ⊕ y [^] char R›*) show \"?thesis\"\n (*goal: ‹(x ⊕ y) [^] char R = x [^] char R ⊕ y [^] char R›*)\n by simp\nqed\n\ntext ‹The following theorem is somtimes called Freshman's dream for obvious reasons,\nit can be found in Lidl and Niederreiter~\\<^cite>‹‹Theorem 1.46› in \"lidl1986\"›.›\n\nlemma (in domain) freshmans_dream_ext:\n fixes m\n assumes \"char R > 0\"\n assumes [simp]: \"x ∈ carrier R\" \"y ∈ carrier R\"\n defines \"n ≡ char R^m\"\n shows \"(x ⊕ y) [^] n = x [^] n ⊕ y [^] n\"\n (is \"?lhs = ?rhs\")\n unfolding n_def\n (*goal: ‹(x ⊕ y) [^] char R ^ m = x [^] char R ^ m ⊕ y [^] char R ^ m›*)\n proof (induction m)\n (*goals:\n 1. ‹(x ⊕ y) [^] char R ^ 0 = x [^] char R ^ 0 ⊕ y [^] char R ^ 0›\n 2. ‹⋀m. (x ⊕ y) [^] char R ^ m = x [^] char R ^ m ⊕ y [^] char R ^ m ⟹ (x ⊕ y) [^] char R ^ Suc m = x [^] char R ^ Suc m ⊕ y [^] char R ^ Suc m›*)\n case 0 (*no hyothesis introduced yet*)\n then show \"?case\"\n (*goal: ‹(x ⊕ y) [^] char R ^ 0 = x [^] char R ^ 0 ⊕ y [^] char R ^ 0›*)\n by simp\n next\n (*goal: ‹⋀m::nat. ((x::'a::type) ⊕ (y::'a::type)) [^] char R ^ m = x [^] char R ^ m ⊕ y [^] char R ^ m ⟹ (x ⊕ y) [^] char R ^ Suc m = x [^] char R ^ Suc m ⊕ y [^] char R ^ Suc m›*)\n case (Suc m) (*‹(x ⊕ y) [^] char R ^ m = x [^] char R ^ m ⊕ y [^] char R ^ m›*)\n have \"(x ⊕ y) [^] (char R^(m+1)) =\n (x ⊕ y) [^] (char R^m * char R)\"\n by (simp add:mult.commute (*‹?a * ?b = ?b * ?a›*))\n also (*calculation: ‹((x::'a::type) ⊕ (y::'a::type)) [^] char R ^ ((m::nat) + (1::nat)) = (x ⊕ y) [^] (char R ^ m * char R)›*) have \"... = ((x ⊕ y) [^] (char R^m)) [^] char R\"\n using nat_pow_pow (*‹?x ∈ carrier R ⟹ (?x [^] ?n) [^] ?m = ?x [^] (?n * ?m)›*) by simp\n also (*calculation: ‹(x ⊕ y) [^] char R ^ (m + 1) = ((x ⊕ y) [^] char R ^ m) [^] char R›*) have \"... = (x [^] (char R^m) ⊕ y [^] (char R^m)) [^] char R\"\n apply (subst Suc (*‹(x ⊕ y) [^] char R ^ m = x [^] char R ^ m ⊕ y [^] char R ^ m›*))\n (*goal: ‹(((x::'a::type) ⊕ (y::'a::type)) [^] char R ^ (m::nat)) [^] char R = (x [^] char R ^ m ⊕ y [^] char R ^ m) [^] char R›*)\n by simp\n also (*calculation: ‹(x ⊕ y) [^] char R ^ (m + 1) = (x [^] char R ^ m ⊕ y [^] char R ^ m) [^] char R›*) have \"... =\n (x [^] (char R^m)) [^] char R ⊕ (y [^] (char R^m)) [^] char R\"\n apply (subst freshmans_dream[OF assms(1), symmetric] (*‹⟦?x ∈ carrier R; ?y ∈ carrier R⟧ ⟹ ?x [^] char R ⊕ ?y [^] char R = (?x ⊕ ?y) [^] char R›*))\n (*goals:\n 1. ‹(x::'a::type) [^] char R ^ (m::nat) ∈ carrier R›\n 2. ‹(y::'a::type) [^] char R ^ (m::nat) ∈ carrier R›\n 3. ‹((x::'a::type) [^] char R ^ (m::nat) ⊕ (y::'a::type) [^] char R ^ m) [^] char R = (x [^] char R ^ m ⊕ y [^] char R ^ m) [^] char R›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*proven 3 subgoals*) .\n also (*calculation: ‹(x ⊕ y) [^] char R ^ (m + 1) = (x [^] char R ^ m) [^] char R ⊕ (y [^] char R ^ m) [^] char R›*) have \"... =\n x [^] (char R^m * char R) ⊕ y [^] (char R^m * char R)\"\n by (simp add:nat_pow_pow (*‹?x ∈ carrier R ⟹ (?x [^] ?n) [^] ?m = ?x [^] (?n * ?m)›*))\n also (*calculation: ‹(x ⊕ y) [^] char R ^ (m + 1) = x [^] (char R ^ m * char R) ⊕ y [^] (char R ^ m * char R)›*) have \"... = x [^] (char R^Suc m) ⊕ y [^] (char R^Suc m)\"\n by (simp add:mult.commute (*‹?a * ?b = ?b * ?a›*))\n finally (*calculation: ‹((x::'a) ⊕ (y::'a)) [^] char R ^ ((m::nat) + (1::nat)) = x [^] char R ^ Suc m ⊕ y [^] char R ^ Suc m›*) show \"?case\"\n (*goal: ‹(x ⊕ y) [^] char R ^ Suc m = x [^] char R ^ Suc m ⊕ y [^] char R ^ Suc m›*)\n by simp\n qed\n\ntext ‹The following is a generalized version of the Frobenius homomorphism. The classic version\nof the theorem is the case where @{term \"(k::nat) = 1\"}.›\n\ntheorem (in domain) frobenius_hom:\n assumes \"char R > 0\"\n assumes \"m = char R ^ k\"\n shows \"ring_hom_cring R R (λx. x [^] m)\"\nproof (-)\n (*goal: ‹ring_hom_cring R R (λx::'a. x [^] (m::nat))›*)\n have a: \"(x ⊗ y) [^] m = x [^] m ⊗ y [^] m\" if b: \"x ∈ carrier R\" \"y ∈ carrier R\" for x and y\n using b (*‹(x::'a) ∈ carrier R› ‹y ∈ carrier R›*) nat_pow_distrib (*‹⟦(?x::'a) ∈ carrier R; (?y::'a) ∈ carrier R⟧ ⟹ (?x ⊗ ?y) [^] (?n::nat) = ?x [^] ?n ⊗ ?y [^] ?n›*) by simp\n have b: \"(x ⊕ y) [^] m = x [^] m ⊕ y [^] m\" if b: \"x ∈ carrier R\" \"y ∈ carrier R\" for x and y\n unfolding assms(2) freshmans_dream_ext[OF assms ( 1 ) b]\n (*goal: ‹x [^] char R ^ k ⊕ y [^] char R ^ k = x [^] char R ^ k ⊕ y [^] char R ^ k›*)\n by simp\n have \"ring_hom_ring R R (λx. x [^] m)\"\n apply (intro ring_hom_ringI (*‹⟦ring ?R; ring ?S; ⋀x. x ∈ carrier ?R ⟹ ?h x ∈ carrier ?S; ⋀x y. ⟦x ∈ carrier ?R; y ∈ carrier ?R⟧ ⟹ ?h (x ⊗⇘?R⇙ y) = ?h x ⊗⇘?S⇙ ?h y; ⋀x y. ⟦x ∈ carrier ?R; y ∈ carrier ?R⟧ ⟹ ?h (x ⊕⇘?R⇙ y) = ?h x ⊕⇘?S⇙ ?h y; ?h 𝟭⇘?R⇙ = 𝟭⇘?S⇙⟧ ⟹ ring_hom_ring ?R ?S ?h›*) a (*‹⟦?x ∈ carrier R; ?y ∈ carrier R⟧ ⟹ (?x ⊗ ?y) [^] m = ?x [^] m ⊗ ?y [^] m›*) b (*‹⟦?x ∈ carrier R; ?y ∈ carrier R⟧ ⟹ (?x ⊕ ?y) [^] m = ?x [^] m ⊕ ?y [^] m›*) ring_axioms (*‹ring R›*))\n (*goals:\n 1. ‹⋀x. x ∈ carrier R ⟹ x [^] m ∈ carrier R›\n 2. ‹⋀x y. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ∈ carrier R›\n 3. ‹⋀x y. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ y ∈ carrier R›\n 4. ‹⋀x y. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ x ∈ carrier R›\n 5. ‹⋀x y. ⟦x ∈ carrier R; y ∈ carrier R⟧ ⟹ y ∈ carrier R›\n 6. ‹𝟭 [^] m = 𝟭›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*proven 6 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹ring_hom_cring R R (λx. x [^] m)›*)\n using RingHom.ring_hom_cringI (*‹⟦ring_hom_ring ?R ?S ?h; cring ?R; cring ?S⟧ ⟹ ring_hom_cring ?R ?S ?h›*) is_cring (*‹cring R›*) by blast\nqed\n\nlemma (in domain) char_ring_is_subfield:\n assumes \"char R > 0\"\n shows \"subfield (char_subring R) R\"\nproof (-)\n (*goal: ‹subfield (char_subring R) R›*)\n interpret d: domain \"R ⦇ carrier := char_subring R ⦈\"\n using char_ring_is_subdomain (*‹subdomain (char_subring R) R›*) subdomain_is_domain (*‹subdomain ?H R ⟹ domain (R⦇carrier := ?H⦈)›*) by simp\n have \"finite (char_subring R)\"\n using char_def (*‹char ?R = card (char_subring ?R)›*) assms (*‹(0::nat) < char R›*) by (metis card_ge_0_finite (*‹0 < card ?A ⟹ finite ?A›*))\n hence \"Units (R ⦇ carrier := char_subring R ⦈)\n = char_subring R - {𝟬}\"\n using d.finite_domain_units (*‹finite (carrier (R⦇carrier := char_subring R⦈)) ⟹ Units (R⦇carrier := char_subring R⦈) = carrier (R⦇carrier := char_subring R⦈) - {𝟬⇘R⦇carrier := char_subring R⦈⇙}›*) by simp\n thus \"?thesis\"\n (*goal: ‹subfield (char_subring R) R›*)\n using subfieldI[OF char_ring_is_subcring] (*‹Units (R⦇carrier := char_subring R⦈) = char_subring R - {𝟬} ⟹ subfield (char_subring R) R›*) by simp\nqed\n\nlemma card_lists_length_eq':\n fixes A :: \"'a set\"\n shows \"card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n\"\nproof (cases \"finite A\")\n (*goals:\n 1. ‹finite (A::'a set) ⟹ card {xs::'a list. set xs ⊆ A ∧ length xs = (n::nat)} = card A ^ n›\n 2. ‹infinite (A::'a set) ⟹ card {xs::'a list. set xs ⊆ A ∧ length xs = (n::nat)} = card A ^ n›*)\n case True (*‹finite A›*)\n then show \"?thesis\"\n (*goal: ‹card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›*)\n using card_lists_length_eq (*‹finite ?A ⟹ card {xs. set xs ⊆ ?A ∧ length xs = ?n} = card ?A ^ ?n›*) by auto\nnext\n (*goal: ‹infinite A ⟹ card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›*)\n case False (*‹infinite A›*)\n hence inf_A: \"infinite A\"\n by simp\n show \"?thesis\"\n (*goal: ‹card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›*)\n proof (cases \"n = 0\")\n (*goals:\n 1. ‹n = 0 ⟹ card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›\n 2. ‹n ≠ 0 ⟹ card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›*)\n case True (*‹n = 0›*)\n hence \"card {xs. set xs ⊆ A ∧ length xs = n} = card {([] :: 'a list)}\"\n apply (intro arg_cong[where f=\"card\"] (*‹?x = ?y ⟹ card ?x = card ?y›*))\n (*goal: ‹card {xs. set xs ⊆ A ∧ length xs = n} = card {[]}›*)\n by (auto simp add:set_eq_iff (*‹((?A::?'a set) = (?B::?'a set)) = (∀x::?'a. (x ∈ ?A) = (x ∈ ?B))›*))\n also (*calculation: ‹card {xs::'a list. set xs ⊆ (A::'a set) ∧ length xs = (n::nat)} = card {[]}›*) have \"... = 1\"\n by simp\n also (*calculation: ‹card {xs. set xs ⊆ A ∧ length xs = n} = 1›*) have \"... = card A^n\"\n using True (*‹n = 0›*) inf_A (*‹infinite A›*) by simp\n finally (*calculation: ‹card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›*) show \"?thesis\"\n (*goal: ‹card {xs::'a list. set xs ⊆ (A::'a set) ∧ length xs = (n::nat)} = card A ^ n›*)\n by simp\n next\n (*goal: ‹n ≠ 0 ⟹ card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›*)\n case False (*‹n ≠ 0›*)\n hence \"inj (replicate n)\"\n by (meson inj_onI (*‹(⋀x y. ⟦x ∈ ?A; y ∈ ?A; ?f x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*) replicate_eq_replicate (*‹(replicate ?m ?x = replicate ?n ?y) = (?m = ?n ∧ (?m ≠ 0 ⟶ ?x = ?y))›*))\n hence \"inj_on (replicate n) A\"\n using inj_on_subset (*‹⟦inj_on ?f ?A; ?B ⊆ ?A⟧ ⟹ inj_on ?f ?B›*) by (metis subset_UNIV (*‹?A ⊆ UNIV›*))\n hence \"infinite (replicate n ` A)\"\n using inf_A (*‹infinite (A::'a set)›*) finite_image_iff (*‹inj_on ?f ?A ⟹ finite (?f ` ?A) = finite ?A›*) by auto\n moreover have \"replicate n ` A ⊆ {xs. set xs ⊆ A ∧ length xs = n}\"\n apply (intro image_subsetI (*‹(⋀x. x ∈ ?A ⟹ ?f x ∈ ?B) ⟹ ?f ` ?A ⊆ ?B›*))\n (*goal: ‹replicate (n::nat) ` (A::'a::type set) ⊆ {xs::'a::type list. set xs ⊆ A ∧ length xs = n}›*)\n by auto\n ultimately have \"infinite {xs. set xs ⊆ A ∧ length xs = n}\"\n using infinite_super (*‹⟦?S ⊆ ?T; infinite ?S⟧ ⟹ infinite ?T›*) by auto\n hence \"card {xs. set xs ⊆ A ∧ length xs = n} = 0\"\n by simp\n then show \"?thesis\"\n (*goal: ‹card {xs. set xs ⊆ A ∧ length xs = n} = card A ^ n›*)\n using inf_A (*‹infinite A›*) False (*‹n ≠ 0›*) by simp\n qed\nqed\n\nlemma (in ring) card_span:\n assumes \"subfield K R\"\n assumes \"independent K w\"\n assumes \"set w ⊆ carrier R\"\n shows \"card (Span K w) = card K^(length w)\"\nproof (-)\n (*goal: ‹card (Span K w) = card K ^ length w›*)\n define A where \"A = {x. set x ⊆ K ∧ length x = length w}\"\n define f where \"f = (λx. combine x w)\"\n have \"x ∈ f ` A\" if a: \"x ∈ Span K w\" for x\n proof (-)\n (*goal: ‹x ∈ f ` A›*)\n obtain y where \"y ∈ A\" \"x = f y\"\n (*goal: ‹(⋀y. ⟦y ∈ A; x = f y⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding A_def f_def\n (*goal: ‹(⋀y::'a list. ⟦y ∈ {x::'a list. set x ⊆ (K::'a set) ∧ length x = length (w::'a list)}; (x::'a) = combine y w⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using unique_decomposition[OF assms ( 1 , 2 ) a] (*‹∃!Ks. set Ks ⊆ K ∧ length Ks = length w ∧ x = combine Ks w›*) by auto\n thus \"?thesis\"\n (*goal: ‹(x::'a) ∈ (f::'a list ⇒ 'a) ` (A::'a list set)›*)\n by simp\n qed\n moreover have \"f x ∈ Span K w\" if a: \"x ∈ A\" for x\n using Span_eq_combine_set[OF assms ( 1 , 3 )] (*‹Span K w = {combine Ks w |Ks. set Ks ⊆ K}›*) a (*‹x ∈ A›*) unfolding A_def f_def\n (*goal: ‹combine x w ∈ Span K w›*)\n by auto\n ultimately have b: \"Span K w = f ` A\"\n by auto\n have False if a: \"x ∈ A\" \"y ∈ A\" \"f x = f y\" \"x ≠ y\" for x and y\n proof (-)\n (*goal: ‹False›*)\n have \"f x ∈ Span K w\"\n using b (*‹Span K w = f ` A›*) a (*‹x ∈ A› ‹y ∈ A› ‹f x = f y› ‹x ≠ y›*) by simp\n thus False\n using a (*‹x ∈ A› ‹y ∈ A› ‹f x = f y› ‹x ≠ y›*) unique_decomposition[OF assms ( 1 , 2 )] (*‹?a ∈ Span K w ⟹ ∃!Ks. set Ks ⊆ K ∧ length Ks = length w ∧ ?a = combine Ks w›*) unfolding f_def A_def\n (*goal: ‹False›*)\n by blast\n qed\n hence f_inj: \"inj_on f A\"\n unfolding inj_on_def\n (*goal: ‹∀x∈A. ∀y∈A. f x = f y ⟶ x = y›*)\n by auto\n have \"card (Span K w) = card (f ` A)\"\n using b (*‹Span (K::'a set) (w::'a list) = (f::'a list ⇒ 'a) ` (A::'a list set)›*) by simp\n also (*calculation: ‹card (Span K w) = card (f ` A)›*) have \"... = card A\"\n by (intro card_image (*‹inj_on ?f ?A ⟹ card (?f ` ?A) = card ?A›*) f_inj (*‹inj_on f A›*))\n also (*calculation: ‹card (Span K w) = card A›*) have \"... = card K^length w\"\n unfolding A_def\n (*goal: ‹card {x::'a list. set x ⊆ (K::'a set) ∧ length x = length (w::'a list)} = card K ^ length w›*)\n by (intro card_lists_length_eq' (*‹card {xs. set xs ⊆ ?A ∧ length xs = ?n} = card ?A ^ ?n›*))\n finally (*calculation: ‹card (Span K w) = card K ^ length w›*) show \"?thesis\"\n (*goal: ‹card (Span K w) = card K ^ length w›*)\n by simp\nqed\n\nlemma (in ring) finite_carr_imp_char_ge_0:\n assumes \"finite (carrier R)\"\n shows \"char R > 0\"\nproof (-)\n (*goal: ‹0 < char R›*)\n have \"char_subring R ⊆ carrier R\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) by auto\n hence \"finite (char_subring R)\"\n using finite_subset (*‹⟦(?A::?'a::type set) ⊆ (?B::?'a::type set); finite ?B⟧ ⟹ finite ?A›*) assms (*‹finite (carrier R)›*) by auto\n hence \"card (char_subring R) > 0\"\n using card_range_greater_zero (*‹finite (range ?f) ⟹ 0 < card (range ?f)›*) by simp\n thus \"char R > 0\"\n unfolding char_def\n (*goal: ‹0 < card (char_subring R)›*)\n by simp\nqed\n\nlemma (in ring) char_consistent:\n assumes \"subring H R\"\n shows \"char (R ⦇ carrier := H ⦈) = char R\"\nproof (-)\n (*goal: ‹char (R⦇carrier := H⦈) = char R›*)\n show \"?thesis\"\n (*goal: ‹char (R⦇carrier := H⦈) = char R›*)\n using int_embed_consistent[OF assms ( 1 )] (*‹int_embed R ?i = int_embed (R⦇carrier := H⦈) ?i›*) unfolding char_def\n (*goal: ‹card (char_subring (R⦇carrier := H::'a set⦈)) = card (char_subring R)›*)\n by simp\nqed\n\nlemma (in ring_hom_ring) char_consistent:\n assumes \"inj_on h (carrier R)\"\n shows \"char R = char S\"\nproof (-)\n (*goal: ‹char R = char S›*)\n have a: \"h (int_embed R (int n)) = int_embed S (int n)\" for n\n using R.int_embed_range[OF R.carrier_is_subring] (*‹int_embed R ?k ∈ carrier R›*) using R.int_embed_range[OF R.carrier_is_subring] (*‹int_embed R ?k ∈ carrier R›*) using S.int_embed_one (*‹int_embed S 1 = 𝟭⇘S⇙›*) R.int_embed_one (*‹int_embed R 1 = 𝟭›*) using S.int_embed_zero (*‹int_embed S 0 = 𝟬⇘S⇙›*) R.int_embed_zero (*‹int_embed R 0 = 𝟬›*) using S.int_embed_add (*‹int_embed S (?x + ?y) = int_embed S ?x ⊕⇘S⇙ int_embed S ?y›*) R.int_embed_add (*‹int_embed R (?x + ?y) = int_embed R ?x ⊕ int_embed R ?y›*) apply (induction n)\n (*goals:\n 1. ‹⟦⋀k. int_embed R k ∈ carrier R; ⋀k. int_embed R k ∈ carrier R; int_embed S 1 = 𝟭⇘S⇙; int_embed R 1 = 𝟭; int_embed S 0 = 𝟬⇘S⇙; int_embed R 0 = 𝟬; ⋀x y. int_embed S (x + y) = int_embed S x ⊕⇘S⇙ int_embed S y; ⋀x y. int_embed R (x + y) = int_embed R x ⊕ int_embed R y⟧ ⟹ h (int_embed R (int 0)) = int_embed S (int 0)›\n 2. ‹⋀n. ⟦⟦⋀k. int_embed R k ∈ carrier R; ⋀k. int_embed R k ∈ carrier R; int_embed S 1 = 𝟭⇘S⇙; int_embed R 1 = 𝟭; int_embed S 0 = 𝟬⇘S⇙; int_embed R 0 = 𝟬; ⋀x y. int_embed S (x + y) = int_embed S x ⊕⇘S⇙ int_embed S y; ⋀x y. int_embed R (x + y) = int_embed R x ⊕ int_embed R y⟧ ⟹ h (int_embed R (int n)) = int_embed S (int n); ⋀k. int_embed R k ∈ carrier R; ⋀k. int_embed R k ∈ carrier R; int_embed S 1 = 𝟭⇘S⇙; int_embed R 1 = 𝟭; int_embed S 0 = 𝟬⇘S⇙; int_embed R 0 = 𝟬; ⋀x y. int_embed S (x + y) = int_embed S x ⊕⇘S⇙ int_embed S y; ⋀x y. int_embed R (x + y) = int_embed R x ⊕ int_embed R y⟧ ⟹ h (int_embed R (int (Suc n))) = int_embed S (int (Suc n))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n have b: \"h (int_embed R (-(int n))) = int_embed S (-(int n))\" for n\n using R.int_embed_range[OF R.carrier_is_subring] (*‹int_embed R ?k ∈ carrier R›*) using S.int_embed_range[OF S.carrier_is_subring] (*‹int_embed (S::('c, 'd) ring_scheme) (?k::int) ∈ carrier S›*) a (*‹h (int_embed R (int ?n)) = int_embed S (int ?n)›*) by (simp add:R.int_embed_inv (*‹int_embed R (- (?x::int)) = ⊖ int_embed R ?x›*) S.int_embed_inv (*‹int_embed (S::('c, 'd) ring_scheme) (- (?x::int)) = ⊖⇘S⇙ int_embed S ?x›*))\n have c: \"h (int_embed R n) = int_embed S n\" for n\n proof (cases \"n ≥ 0\")\n (*goals:\n 1. ‹(0::int) ≤ (n::int) ⟹ (h::'a ⇒ 'c) (int_embed R n) = int_embed (S::('c, 'd) ring_scheme) n›\n 2. ‹¬ (0::int) ≤ (n::int) ⟹ (h::'a ⇒ 'c) (int_embed R n) = int_embed (S::('c, 'd) ring_scheme) n›*)\n case True (*‹0 ≤ n›*)\n then obtain m where \"n = int m\"\n (*goal: ‹(⋀m. n = int m ⟹ thesis) ⟹ thesis›*)\n using nonneg_int_cases (*‹⟦0 ≤ ?k; ⋀n. ?k = int n ⟹ ?thesis⟧ ⟹ ?thesis›*) by auto\n then show \"?thesis\"\n (*goal: ‹h (int_embed R n) = int_embed S n›*)\n by (simp add:a (*‹h (int_embed R (int ?n)) = int_embed S (int ?n)›*))\n next\n (*goal: ‹¬ 0 ≤ n ⟹ h (int_embed R n) = int_embed S n›*)\n case False (*‹¬ 0 ≤ n›*)\n hence \"n ≤ 0\"\n by simp\n then obtain m where \"n = -int m\"\n (*goal: ‹(⋀m. n = - int m ⟹ thesis) ⟹ thesis›*)\n using nonpos_int_cases (*‹⟦?k ≤ 0; ⋀n. ?k = - int n ⟹ ?thesis⟧ ⟹ ?thesis›*) by auto\n then show \"?thesis\"\n (*goal: ‹h (int_embed R n) = int_embed S n›*)\n by (simp add:b (*‹(h::'a::type ⇒ 'c::type) (int_embed R (- int (?n::nat))) = int_embed (S::('c, 'd) ring_scheme) (- int ?n)›*))\n qed\n have \"char S = card (h ` char_subring R)\"\n unfolding char_def image_image c\n (*goal: ‹card (char_subring S) = card (char_subring S)›*)\n by simp\n also (*calculation: ‹char (S::('c, 'd) ring_scheme) = card ((h::'a::type ⇒ 'c::type) ` char_subring R)›*) have \"... = card (char_subring R)\"\n using R.int_embed_range[OF R.carrier_is_subring] (*‹int_embed R (?k::int) ∈ carrier R›*) apply (intro card_image (*‹inj_on ?f ?A ⟹ card (?f ` ?A) = card ?A›*) inj_on_subset[OF assms(1)] (*‹?B ⊆ carrier R ⟹ inj_on h ?B›*))\n (*goal: ‹card (h ` char_subring R) = card (char_subring R)›*)\n by auto\n also (*calculation: ‹char S = card (char_subring R)›*) have \"... = char R\"\n unfolding char_def\n (*goal: ‹card (char_subring R) = card (char_subring R)›*)\n by simp\n finally (*calculation: ‹char S = char R›*) show \"?thesis\"\n (*goal: ‹char R = char S›*)\n by simp\nqed\n\ndefinition char_iso :: \"_ ⇒ int set ⇒ 'a\"\n where \"char_iso R x = the_elem (int_embed R ` x)\"\n\ntext ‹The function @{term \"char_iso R\"} denotes the isomorphism between @{term \"ZFact (char R)\"} and\nthe characteristic subring.›\n\nlemma (in ring) char_iso: \"char_iso R ∈\n ring_iso (ZFact (char R)) (R⦇carrier := char_subring R⦈)\"\nproof (-)\n (*goal: ‹char_iso R ∈ ring_iso (ZFact (int (char R))) (R⦇carrier := char_subring R⦈)›*)\n interpret h: ring_hom_ring int_ring R \"int_embed R\"\n using int_embed_ring_hom (*‹ring_hom_ring 𝒵 R (int_embed R)›*) by simp\n have \"a_kernel 𝒵 R (int_embed R) = {x. int_embed R x = 𝟬}\"\n unfolding a_kernel_def kernel_def\n (*goal: ‹{x::int ∈ carrier (add_monoid 𝒵). int_embed R x = 𝟭⇘add_monoid R⇙} = {x::int. int_embed R x = 𝟬}›*)\n by simp\n also (*calculation: ‹a_kernel 𝒵 R (int_embed R) = {x. int_embed R x = 𝟬}›*) have \"... = {x. char R dvd x}\"\n using embed_char_eq_0_iff (*‹(int_embed R ?n = 𝟬) = (int (char R) dvd ?n)›*) by simp\n also (*calculation: ‹a_kernel 𝒵 R (int_embed R) = {x. int (char R) dvd x}›*) have \"... = PIdl⇘𝒵⇙ (int (char R))\"\n unfolding cgenideal_def\n (*goal: ‹{x. int (char R) dvd x} = {x ⊗⇘𝒵⇙ int (char R) |x. x ∈ carrier 𝒵}›*)\n by auto\n also (*calculation: ‹a_kernel 𝒵 R (int_embed R) = PIdl⇘𝒵⇙ int (char R)›*) have \"... = Idl⇘𝒵⇙ {int (char R)}\"\n using int.cgenideal_eq_genideal (*‹?i ∈ UNIV ⟹ PIdl⇘𝒵⇙ ?i = Idl⇘𝒵⇙ {?i}›*) by simp\n finally (*calculation: ‹a_kernel 𝒵 R (int_embed R) = ??.Ring_Characteristic.I (char R)›*) have a: \"a_kernel 𝒵 R (int_embed R) = Idl⇘𝒵⇙ {int (char R)}\"\n by simp\n show \"?thesis\"\n (*goal: ‹char_iso R ∈ ring_iso (ZFact (int (char R))) (R⦇carrier := char_subring R⦈)›*)\n unfolding char_iso_def ZFact_def a[symmetric]\n (*goal: ‹(λx::int set. the_elem (int_embed R ` x)) ∈ ring_iso (𝒵 Quot a_kernel 𝒵 R (int_embed R)) (R⦇carrier := char_subring R⦈)›*)\n by (intro h.FactRing_iso_set_aux (*‹(λX::int set. the_elem (int_embed R ` X)) ∈ ring_iso (𝒵 Quot a_kernel 𝒵 R (int_embed R)) (R⦇carrier := char_subring R⦈)›*))\nqed\n\ntext ‹The size of a finite field must be a prime power.\nThis can be found in Ireland and Rosen~\\<^cite>‹‹Proposition 7.1.3› in \"ireland1982\"›.›\n\ntheorem (in finite_field) finite_field_order:\n \"∃n. order R = char R ^ n ∧ n > 0\"\nproof (-)\n (*goal: ‹∃n. order R = char R ^ n ∧ 0 < n›*)\n have a: \"char R > 0\"\n using finite_carr_imp_char_ge_0[OF finite_carrier] (*‹0 < char R›*) by simp\n let ?CR = \"char_subring R\"\n obtain v where v_def: \"set v = carrier R\"\n (*goal: ‹(⋀v. set v = carrier R ⟹ thesis) ⟹ thesis›*)\n using finite_carrier (*‹finite (carrier R)›*) finite_list (*‹finite ?A ⟹ ∃xs. set xs = ?A›*) by auto\n hence b: \"set v ⊆ carrier R\"\n by auto\n have \"carrier R = set v\"\n using v_def (*‹set v = carrier R›*) by simp\n also (*calculation: ‹carrier R = set (v::'a list)›*) have \"... ⊆ Span ?CR v\"\n using Span_base_incl[OF char_ring_is_subfield [ OF a ] b] (*‹set v ⊆ Span (char_subring R) v›*) by simp\n finally (*calculation: ‹carrier R ⊆ Span (char_subring R) (v::'a list)›*) have \"carrier R ⊆ Span ?CR v\"\n by simp\n moreover have \"Span ?CR v ⊆ carrier R\"\n using int_embed_closed (*‹int_embed R ?k ∈ carrier R›*) v_def (*‹set (v::'a list) = carrier R›*) apply (intro Span_in_carrier (*‹⟦?K ⊆ carrier R; set ?Us ⊆ carrier R⟧ ⟹ Span ?K ?Us ⊆ carrier R›*))\n (*goals:\n 1. ‹⟦⋀k. int_embed R k ∈ carrier R; set v = carrier R⟧ ⟹ char_subring R ⊆ carrier R›\n 2. ‹⟦⋀k. int_embed R k ∈ carrier R; set v = carrier R⟧ ⟹ set v ⊆ carrier R›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n ultimately have Span_v: \"Span ?CR v = carrier R\"\n by simp\n obtain w where w_def: \"set w ⊆ carrier R\" \"independent ?CR w\" \"Span ?CR v = Span ?CR w\"\n (*goal: ‹(⋀w. ⟦set w ⊆ carrier R; local.independent (char_subring R) w; Span (char_subring R) v = Span (char_subring R) w⟧ ⟹ thesis) ⟹ thesis›*)\n using b (*‹set v ⊆ carrier R›*) filter_base[OF char_ring_is_subfield [ OF a ]] (*‹⟦set ?Us ⊆ carrier R; ⋀Vs. ⟦set Vs ⊆ carrier R; local.independent (char_subring R) Vs; Span (char_subring R) Vs = Span (char_subring R) ?Us⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) by metis\n have Span_w: \"Span ?CR w = carrier R\"\n using w_def(3) (*‹Span (char_subring R) v = Span (char_subring R) w›*) Span_v (*‹Span (char_subring R) v = carrier R›*) by simp\n hence \"order R = card (Span ?CR w)\"\n by (simp add:order_def (*‹order ?S = card (carrier ?S)›*))\n also (*calculation: ‹order R = card (Span (char_subring R) (w::'a list))›*) have \"... = card ?CR^length w\"\n by (intro card_span (*‹⟦subfield ?K R; local.independent ?K ?w; set ?w ⊆ carrier R⟧ ⟹ card (Span ?K ?w) = card ?K ^ length ?w›*) char_ring_is_subfield[OF a] (*‹subfield (char_subring R) R›*) w_def( (*‹set w ⊆ carrier R› ‹local.independent (char_subring R) w›*) 1,2))\n finally (*calculation: ‹order R = card (char_subring R) ^ length w›*) have c: \"order R = char R^(length w)\"\n by (simp add:char_def (*‹char ?R = card (char_subring ?R)›*))\n have \"length w > 0\"\n using finite_field_min_order (*‹1 < order R›*) c (*‹order R = char R ^ length w›*) by auto\n thus \"?thesis\"\n (*goal: ‹∃n. order R = char R ^ n ∧ 0 < n›*)\n using c (*‹order R = char R ^ length w›*) by auto\nqed\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Finite_Fields/Ring_Characteristic.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"20f12a2b661260ec420b2d7d092093d37eddeedaba1f25ac16bbd87e777bc09a\"\n}"}}},{"rowIdx":157,"cells":{"text":{"kind":"string","value":"section ‹Zeckendorf's Theorem›\n\ntheory Zeckendorf\n\nimports \n Main\n \"HOL-Number_Theory.Number_Theory\"\n\nbegin\n\nsubsection ‹Definitions›\ntext ‹\n Formulate auxiliary definitions. An increasing sequence is a predicate of a function $f$\n together with a set $I$. $f$ is an increasing sequence on $I$, if $f(x)+1 < f(x+1)$ \n for all $x \\in I$. This definition is used to ensure that the Fibonacci numbers in the sum are\n non-consecutive.\n›\ndefinition is_fib :: \"nat ⇒ bool\" where\n \"is_fib n = (∃ i. n = fib i)\"\n\ndefinition le_fib_idx_set :: \"nat ⇒ nat set\" where\n \"le_fib_idx_set n = {i .fib i < n}\"\n\ndefinition inc_seq_on :: \"(nat ⇒ nat) ⇒ nat set ⇒ bool\" where\n \"inc_seq_on f I = (∀ n ∈ I. f(Suc n) > Suc(f n))\"\n\ndefinition fib_idx_set :: \"nat ⇒ nat set\" where\n \"fib_idx_set n = {i. fib i = n}\"\n\nsubsection ‹Auxiliary Lemmas›\n\nlemma fib_values[simp]:\n \"fib 3 = 2\"\n \"fib 4 = 3\"\n \"fib 5 = 5\"\n \"fib 6 = 8\"\n \n (*goals:\n 1. ‹fib 3 = 2›\n 2. ‹fib 4 = 3›\n 3. ‹fib 5 = 5›\n 4. ‹fib 6 = 8›\n discuss goal 1*)\n apply ((auto simp: numeral_Bit0 (*‹numeral (num.Bit0 ?n) = numeral ?n + numeral ?n›*) numeral_eq_Suc (*‹numeral ?k = Suc (pred_numeral ?k)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: numeral_Bit0 (*‹numeral (num.Bit0 ?n) = numeral ?n + numeral ?n›*) numeral_eq_Suc (*‹numeral ?k = Suc (pred_numeral ?k)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: numeral_Bit0 (*‹numeral (num.Bit0 ?n) = numeral ?n + numeral ?n›*) numeral_eq_Suc (*‹numeral ?k = Suc (pred_numeral ?k)›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: numeral_Bit0 (*‹numeral (num.Bit0 ?n) = numeral ?n + numeral ?n›*) numeral_eq_Suc (*‹numeral ?k = Suc (pred_numeral ?k)›*))[1])\n (*proven 4 subgoals*) .\n\nlemma fib_strict_mono: \"i ≥ 2 ⟹ fib i < fib (Suc i)\"\n using fib_mono (*‹(?m::nat) ≤ (?n::nat) ⟹ fib ?m ≤ fib ?n›*) apply (induct i)\n (*goals:\n 1. ‹⟦(2::nat) ≤ (0::nat); ⋀(m::nat) n::nat. m ≤ n ⟹ fib m ≤ fib n⟧ ⟹ fib (0::nat) < fib (Suc (0::nat))›\n 2. ‹⋀i::nat. ⟦⟦(2::nat) ≤ i; ⋀(m::nat) n::nat. m ≤ n ⟹ fib m ≤ fib n⟧ ⟹ fib i < fib (Suc i); (2::nat) ≤ Suc i; ⋀(m::nat) n::nat. m ≤ n ⟹ fib m ≤ fib n⟧ ⟹ fib (Suc i) < fib (Suc (Suc i))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply fastforce\n (*proven 2 subgoals*) .\n\nlemma smaller_index_implies_fib_le: \"i < j ⟹ fib(Suc i) ≤ fib j\"\n using fib_mono (*‹(?m::nat) ≤ (?n::nat) ⟹ fib ?m ≤ fib ?n›*) apply (induct j)\n (*goals:\n 1. ‹⟦i < 0; ⋀m n. m ≤ n ⟹ fib m ≤ fib n⟧ ⟹ fib (Suc i) ≤ fib 0›\n 2. ‹⋀j. ⟦⟦i < j; ⋀m n. m ≤ n ⟹ fib m ≤ fib n⟧ ⟹ fib (Suc i) ≤ fib j; i < Suc j; ⋀m n. m ≤ n ⟹ fib m ≤ fib n⟧ ⟹ fib (Suc i) ≤ fib (Suc j)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma fib_index_strict_mono : \"i ≥ 2 ⟹ j > i ⟹ fib j > fib i\"\n apply (induct i)\n (*goals:\n 1. ‹⟦2 ≤ 0; 0 < j⟧ ⟹ fib 0 < fib j›\n 2. ‹⋀i. ⟦⟦2 ≤ i; i < j⟧ ⟹ fib i < fib j; 2 ≤ Suc i; Suc i < j⟧ ⟹ fib (Suc i) < fib j›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply (metis Suc_leI (*‹?m < ?n ⟹ Suc ?m ≤ ?n›*) fib_mono (*‹?m ≤ ?n ⟹ fib ?m ≤ fib ?n›*) fib_strict_mono (*‹2 ≤ ?i ⟹ fib ?i < fib (Suc ?i)›*) nle_le (*‹(¬ ?a ≤ ?b) = (?b ≤ ?a ∧ ?b ≠ ?a)›*) nless_le (*‹(¬ ?a < ?b) = (¬ ?a ≤ ?b ∨ ?a = ?b)›*))\n (*proven 2 subgoals*) .\n\nlemma fib_implies_is_fib: \"fib i = n ⟹ is_fib n\"\n using is_fib_def (*‹is_fib ?n = (∃i. ?n = fib i)›*) by auto\n\nlemma zero_fib_unique_idx: \"n = fib i ⟹ n = fib 0 ⟹ i = 0\"\n using fib_neq_0_nat (*‹0 < ?n ⟹ 0 < fib ?n›*) fib_idx_set_def (*‹fib_idx_set (?n::nat) = {i::nat. fib i = ?n}›*) by fastforce\n\nlemma zero_fib_equiv: \"fib i = 0 ⟷ i = 0\"\n using zero_fib_unique_idx (*‹⟦?n = fib ?i; ?n = fib 0⟧ ⟹ ?i = 0›*) by auto\n\nlemma one_fib_idxs: \"fib i = Suc 0 ⟹ i = Suc 0 ∨ i = Suc(Suc 0)\"\n using Fib.fib0 (*‹fib 0 = 0›*) One_nat_def (*‹1 = Suc 0›*) Suc_1 (*‹Suc (1::nat) = (2::nat)›*) eq_imp_le (*‹?m = ?n ⟹ ?m ≤ ?n›*) fib_2 (*‹fib 2 = 1›*) fib_index_strict_mono (*‹⟦2 ≤ ?i; ?i < ?j⟧ ⟹ fib ?i < fib ?j›*) less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*) nat_neq_iff (*‹(?m ≠ ?n) = (?m < ?n ∨ ?n < ?m)›*) by metis\n\nlemma ge_two_eq_fib_implies_eq_idx: \"n ≥ 2 ⟹ n = fib i ⟹ n = fib j ⟹ i = j\"\n using fib_index_strict_mono (*‹⟦(2::nat) ≤ (?i::nat); ?i < (?j::nat)⟧ ⟹ fib ?i < fib ?j›*) fib_mono (*‹(?m::nat) ≤ (?n::nat) ⟹ fib ?m ≤ fib ?n›*) Suc_1 (*‹Suc 1 = 2›*) fib_2 (*‹fib 2 = 1›*) nle_le (*‹(¬ ?a ≤ ?b) = (?b ≤ ?a ∧ ?b ≠ ?a)›*) nless_le (*‹(¬ ?a < ?b) = (¬ ?a ≤ ?b ∨ ?a = ?b)›*) not_less_eq (*‹(¬ ?m < ?n) = (?n < Suc ?m)›*) by metis\n\nlemma ge_two_fib_unique_idx: \"fib i ≥ 2 ⟹ fib i = fib j ⟹ i = j\"\n using ge_two_eq_fib_implies_eq_idx (*‹⟦2 ≤ ?n; ?n = fib ?i; ?n = fib ?j⟧ ⟹ ?i = ?j›*) by auto\n\nlemma no_fib_lower_bound: \"¬ is_fib n ⟹ n ≥ 4\"\nproof (rule ccontr (*‹(¬ (?P::bool) ⟹ False) ⟹ ?P›*))\n (*goal: ‹⟦¬ is_fib n; ¬ 4 ≤ n⟧ ⟹ False›*)\n assume \"¬ is_fib n\" \"¬ 4 ≤ n\" (*‹¬ is_fib (n::nat)› ‹¬ (4::nat) ≤ (n::nat)›*)\n hence \"n ∈ {0,1,2,3}\"\n by auto\n have \"is_fib 0\" \"is_fib 1\" \"is_fib 2\" \"is_fib 3\"\n using fib0 (*‹fib 0 = 0›*) fib1 (*‹fib (Suc (0::nat)) = (1::nat)›*) fib_values (*‹fib 3 = 2› ‹fib (4::nat) = (3::nat)› ‹fib (5::nat) = (5::nat)› ‹fib 6 = 8›*) fib_implies_is_fib (*‹fib ?i = ?n ⟹ is_fib ?n›*) apply -\n (*goals:\n 1. ‹⟦fib 0 = 0; fib (Suc 0) = 1; fib 3 = 2; fib 4 = 3; fib 5 = 5; fib 6 = 8; ⋀i n. fib i = n ⟹ is_fib n⟧ ⟹ is_fib 0›\n 2. ‹⟦fib 0 = 0; fib (Suc 0) = 1; fib 3 = 2; fib 4 = 3; fib 5 = 5; fib 6 = 8; ⋀i n. fib i = n ⟹ is_fib n⟧ ⟹ is_fib 1›\n 3. ‹⟦fib 0 = 0; fib (Suc 0) = 1; fib 3 = 2; fib 4 = 3; fib 5 = 5; fib 6 = 8; ⋀i n. fib i = n ⟹ is_fib n⟧ ⟹ is_fib 2›\n 4. ‹⟦fib 0 = 0; fib (Suc 0) = 1; fib 3 = 2; fib 4 = 3; fib 5 = 5; fib 6 = 8; ⋀i n. fib i = n ⟹ is_fib n⟧ ⟹ is_fib 3›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*discuss goal 4*)\n apply blast\n (*proven 4 subgoals*) .\n then show False\n using ‹¬ is_fib n› (*‹¬ is_fib (n::nat)›*) ‹n ∈ {0,1,2,3}› (*‹n ∈ {0, 1, 2, 3}›*) by blast\nqed\n\nlemma pos_fib_has_idx_ge_two: \"n > 0 ⟹ is_fib n ⟹ (∃ i. i ≥ 2 ∧ fib i = n)\"\n unfolding is_fib_def\n (*goal: ‹⟦0 < n; ∃i. n = fib i⟧ ⟹ ∃i≥2. fib i = n›*)\n by (metis One_nat_def (*‹1 = Suc 0›*) fib_2 (*‹fib 2 = 1›*) fib_mono (*‹?m ≤ ?n ⟹ fib ?m ≤ fib ?n›*) less_eq_Suc_le (*‹(?n < ?m) = (Suc ?n ≤ ?m)›*) nle_le (*‹(¬ ?a ≤ ?b) = (?b ≤ ?a ∧ ?b ≠ ?a)›*))\n\nlemma finite_fib0_idx: \"finite({i. fib i = 0})\"\n using zero_fib_unique_idx (*‹⟦?n = fib ?i; ?n = fib 0⟧ ⟹ ?i = 0›*) finite_nat_set_iff_bounded (*‹finite ?N = (∃m. ∀n∈?N. n < m)›*) by auto\n\nlemma finite_fib1_idx: \"finite({i. fib i = 1})\"\n using one_fib_idxs (*‹fib ?i = Suc 0 ⟹ ?i = Suc 0 ∨ ?i = Suc (Suc 0)›*) finite_nat_set_iff_bounded (*‹finite ?N = (∃m. ∀n∈?N. n < m)›*) by auto\n\nlemma finite_fib_ge_two_idx: \"n ≥ 2 ⟹ finite({i. fib i = n})\"\n using ge_two_fib_unique_idx (*‹⟦2 ≤ fib ?i; fib ?i = fib ?j⟧ ⟹ ?i = ?j›*) finite_nat_set_iff_bounded (*‹finite ?N = (∃m. ∀n∈?N. n < m)›*) by auto\n\nlemma finite_fib_index: \"finite({i. fib i = n})\"\n using finite_fib0_idx (*‹finite {i::nat. fib i = (0::nat)}›*) finite_fib1_idx (*‹finite {i. fib i = 1}›*) finite_fib_ge_two_idx (*‹2 ≤ ?n ⟹ finite {i. fib i = ?n}›*) apply (rule nat_induct2 (*‹⟦?P 0; ?P 1; ⋀n. ?P n ⟹ ?P (n + 2)⟧ ⟹ ?P ?n›*))\n (*goal: ‹finite {i. fib i = n}›*)\n by auto\n\nlemma no_fib_implies_zero_in_le_idx_set: \"¬ is_fib n ⟹ 0 ∈ {i. fib i < n}\"\n using no_fib_lower_bound (*‹¬ is_fib ?n ⟹ 4 ≤ ?n›*) by fastforce\n\nlemma no_fib_implies_le_fib_idx_set: \"¬ is_fib n ⟹ {i. fib i < n} ≠ {}\"\n using no_fib_implies_zero_in_le_idx_set (*‹¬ is_fib ?n ⟹ 0 ∈ {i. fib i < ?n}›*) by blast\n\nlemma finite_smaller_fibs: \"finite({i. fib i < n})\"\nproof (induct n)\n (*goals:\n 1. ‹finite {i::nat. fib i < (0::nat)}›\n 2. ‹⋀n::nat. finite {i::nat. fib i < n} ⟹ finite {i::nat. fib i < Suc n}›*)\n case (Suc n) (*‹finite {i. fib i < n}›*)\n moreover have \"{i. fib i < Suc n} = {i. fib i < n} ∪ {i. fib i = n}\"\n by auto\n moreover have \"finite({i. fib i = n})\"\n using finite_fib_index (*‹finite {i. fib i = ?n}›*) by auto\n ultimately show \"?case\"\n (*goal: ‹finite {i. fib i < Suc n}›*)\n by auto\nqed (simp)\n(*solved the remaining goal: ‹finite {i. fib i < 0}›*)\n\nlemma nat_ge_2_fib_idx_bound: \"2 ≤ n ⟹ fib i ≤ n ⟹ n < fib (Suc i) ⟹ 2 ≤ i\"\n by (metis One_nat_def (*‹1 = Suc 0›*) fib_1 (*‹fib 1 = 1›*) fib_2 (*‹fib 2 = 1›*) le_Suc_eq (*‹(?m ≤ Suc ?n) = (?m ≤ ?n ∨ ?m = Suc ?n)›*) less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*) linorder_not_le (*‹(¬ ?x ≤ ?y) = (?y < ?x)›*) not_less_eq (*‹(¬ ?m < ?n) = (?n < Suc ?m)›*))\n\nlemma inc_seq_on_aux: \"inc_seq_on c {0..k - 1} ⟹ n - fib i < fib (i-1) ⟹ fib (c k) < fib i ⟹ \n (n - fib i) = (∑ i=0..k. fib (c i)) ⟹ Suc (c k) < i\"\n by (metis fib_mono (*‹?m ≤ ?n ⟹ fib ?m ≤ fib ?n›*) bot_nat_0.extremum (*‹0 ≤ ?a›*) diff_Suc_1 (*‹Suc ?n - 1 = ?n›*) leD (*‹?y ≤ ?x ⟹ ¬ ?x < ?y›*) le_SucE (*‹⟦?m ≤ Suc ?n; ?m ≤ ?n ⟹ ?R; ?m = Suc ?n ⟹ ?R⟧ ⟹ ?R›*) linorder_le_less_linear (*‹?x ≤ ?y ∨ ?y < ?x›*) not_add_less1 (*‹¬ ?i + ?j < ?i›*) sum.last_plus (*‹?m ≤ ?n ⟹ sum ?g {?m..?n} = ?g ?n + sum ?g {?m..<?n}›*))\n\nlemma inc_seq_zero_at_start: \"inc_seq_on c {0..k-1} ⟹ c k = 0 ⟹ k = 0\"\n unfolding inc_seq_on_def\n (*goal: ‹⟦∀n∈{0..k - 1}. Suc (c n) < c (Suc n); c k = 0⟧ ⟹ k = 0›*)\n by (metis One_nat_def (*‹1 = Suc 0›*) Suc_pred (*‹0 < ?n ⟹ Suc (?n - Suc 0) = ?n›*) atLeast0AtMost (*‹{0..?n} = {..?n}›*) atMost_iff (*‹(?i ∈ {..?k}) = (?i ≤ ?k)›*) less_nat_zero_code (*‹(?n < 0) = False›*) not_gr_zero (*‹(¬ 0 < ?n) = (?n = 0)›*) order.refl (*‹?a ≤ ?a›*))\n\nlemma fib_sum_zero_equiv: \"(∑ i=n..m::nat . fib (c i)) = 0 ⟷ (∀ i∈{n..m}. c i = 0)\"\n using finite_atLeastAtMost (*‹finite {?l::nat..?u::nat}›*) sum_eq_0_iff (*‹finite ?F ⟹ (sum ?f ?F = 0) = (∀a∈?F. ?f a = 0)›*) zero_fib_equiv (*‹(fib ?i = 0) = (?i = 0)›*) by auto\n\nlemma fib_idx_ge_two_fib_sum_not_zero: \"n ≤ m ⟹ ∀i∈{n..m::nat}. c i ≥ 2 ⟹ ¬ (∑ i=n..m. fib (c i)) = 0\"\n apply (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹⟦n ≤ m; ∀i∈{n..m}. 2 ≤ c i⟧ ⟹ (∑i = n..m. fib (c i)) ≠ 0›*)\n by (simp add: fib_sum_zero_equiv (*‹((∑i = ?n..?m. fib (?c i)) = 0) = (∀i∈{?n..?m}. ?c i = 0)›*))\n\nlemma one_unique_fib_sum: \"inc_seq_on c {0..k-1} ⟹ ∀i∈{0..k}. c i ≥ 2 ⟹ (∑ i=0..k. fib (c i)) = 1 ⟷ k = 0 ∧ c 0 = 2\"\nproof (standard)\n (*goals:\n 1. ‹⟦inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; (∑i::nat = 0::nat..k. fib (c i)) = (1::nat)⟧ ⟹ k = (0::nat) ∧ c (0::nat) = (2::nat)›\n 2. ‹⟦inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; k = (0::nat) ∧ c (0::nat) = (2::nat)⟧ ⟹ (∑i::nat = 0::nat..k. fib (c i)) = (1::nat)›*)\n assume assms: \"(∑i = 0..k. fib (c i)) = 1\" \"inc_seq_on c {0..k-1}\" \"∀i∈{0..k}. c i ≥ 2\" (*‹(∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i)) = (1::nat)› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i›*)\n hence \"fib (c 0) + (∑i = 1..k. fib (c i)) = 1\"\n by (simp add: sum.atLeast_Suc_atMost (*‹?m ≤ ?n ⟹ sum ?g {?m..?n} = ?g ?m + sum ?g {Suc ?m..?n}›*))\n moreover have \"fib (c 0) ≥ 1\"\n using assms (*‹(∑i = 0..k. fib (c i)) = 1› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*) fib_neq_0_nat[of \"c 0\"] (*‹0 < c 0 ⟹ 0 < fib (c 0)›*) by force\n ultimately show \"k = 0 ∧ c 0 = 2\"\n using fib_idx_ge_two_fib_sum_not_zero[of 1 k c] (*‹⟦1 ≤ k; ∀i∈{1..k}. 2 ≤ c i⟧ ⟹ (∑i = 1..k. fib (c i)) ≠ 0›*) assms (*‹(∑i = 0..k. fib (c i)) = 1› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i∈{0..k}. 2 ≤ c i›*) add_is_1 (*‹(?m + ?n = Suc 0) = (?m = Suc 0 ∧ ?n = 0 ∨ ?m = 0 ∧ ?n = Suc 0)›*) one_fib_idxs (*‹fib ?i = Suc 0 ⟹ ?i = Suc 0 ∨ ?i = Suc (Suc 0)›*) apply (cases \"k=0\")\n (*goals:\n 1. ‹⟦fib ((c::nat ⇒ nat) (0::nat)) + (∑i::nat = 1::nat..k::nat. fib (c i)) = (1::nat); (1::nat) ≤ fib (c (0::nat)); ⟦(1::nat) ≤ k; ∀i::nat∈{1::nat..k}. (2::nat) ≤ c i⟧ ⟹ (∑i::nat = 1::nat..k. fib (c i)) ≠ (0::nat); (∑i::nat = 0::nat..k. fib (c i)) = (1::nat); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; ⋀(m::nat) n::nat. (m + n = Suc (0::nat)) = (m = Suc (0::nat) ∧ n = (0::nat) ∨ m = (0::nat) ∧ n = Suc (0::nat)); ⋀i::nat. fib i = Suc (0::nat) ⟹ i = Suc (0::nat) ∨ i = Suc (Suc (0::nat)); k = (0::nat)⟧ ⟹ k = (0::nat) ∧ c (0::nat) = (2::nat)›\n 2. ‹⟦fib ((c::nat ⇒ nat) (0::nat)) + (∑i::nat = 1::nat..k::nat. fib (c i)) = (1::nat); (1::nat) ≤ fib (c (0::nat)); ⟦(1::nat) ≤ k; ∀i::nat∈{1::nat..k}. (2::nat) ≤ c i⟧ ⟹ (∑i::nat = 1::nat..k. fib (c i)) ≠ (0::nat); (∑i::nat = 0::nat..k. fib (c i)) = (1::nat); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; ⋀(m::nat) n::nat. (m + n = Suc (0::nat)) = (m = Suc (0::nat) ∧ n = (0::nat) ∨ m = (0::nat) ∧ n = Suc (0::nat)); ⋀i::nat. fib i = Suc (0::nat) ⟹ i = Suc (0::nat) ∨ i = Suc (Suc (0::nat)); k ≠ (0::nat)⟧ ⟹ k = (0::nat) ∧ c (0::nat) = (2::nat)›\n discuss goal 1*)\n apply fastforce\n (*discuss goal 2*)\n apply auto\n (*proven 2 subgoals*) .\nqed (simp)\n(*solved the remaining goal: ‹⟦inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; k = 0 ∧ c 0 = 2⟧ ⟹ (∑i = 0..k. fib (c i)) = 1›*)\n\nlemma no_fib_betw_fibs: \n assumes \"¬ is_fib n\"\n shows \"∃ i. fib i < n ∧ n < fib (Suc i)\"\nproof (-)\n (*goal: ‹∃i::nat. fib i < (n::nat) ∧ n < fib (Suc i)›*)\n have finite_le_fib: \"finite {i. fib i < n}\"\n using finite_smaller_fibs (*‹finite {i. fib i < ?n}›*) by auto\n obtain i where max_def: \"i = Max {i. fib i < n}\"\n (*goal: ‹(⋀i::nat. i = Max {i::nat. fib i < (n::nat)} ⟹ thesis::bool) ⟹ thesis›*)\n by blast\n show \"∃ i. fib i < n ∧ n < fib (Suc i)\"\n proof (intro exI (*‹?P ?x ⟹ ∃x. ?P x›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹fib ?i < n›\n 2. ‹n < fib (Suc ?i)›*)\n have \"(Suc i) ∉ {i. fib i < n}\"\n using max_def (*‹i = Max {i. fib i < n}›*) Max_ge (*‹⟦finite ?A; ?x ∈ ?A⟧ ⟹ ?x ≤ Max ?A›*) Suc_n_not_le_n (*‹¬ Suc ?n ≤ ?n›*) finite_le_fib (*‹finite {i. fib i < n}›*) by blast\n thus \"fib (Suc i) > n\"\n using ‹¬ is_fib n› (*‹¬ is_fib (n::nat)›*) fib_implies_is_fib (*‹fib ?i = ?n ⟹ is_fib ?n›*) linorder_less_linear (*‹?x < ?y ∨ ?x = ?y ∨ ?y < ?x›*) by blast\n qed (insert max_def (*‹i = Max {i. fib i < n}›*) Max_in (*‹⟦finite ?A; ?A ≠ {}⟧ ⟹ Max ?A ∈ ?A›*) ‹¬ is_fib n› finite_le_fib (*‹finite {i. fib i < n}›*) no_fib_implies_le_fib_idx_set (*‹¬ is_fib ?n ⟹ {i. fib i < ?n} ≠ {}›*), auto)\n (*solved the remaining goal: ‹fib i < n›*)\nqed\n\nlemma betw_fibs: \n shows \"∃ i. fib i ≤ n ∧ fib(Suc i) > n\" \nproof (cases \"is_fib n\")\n (*goals:\n 1. ‹is_fib n ⟹ ∃i. fib i ≤ n ∧ n < fib (Suc i)›\n 2. ‹¬ is_fib n ⟹ ∃i. fib i ≤ n ∧ n < fib (Suc i)›*)\n case True (*‹is_fib (n::nat)›*)\n then obtain i where a: \"n = fib i\"\n (*goal: ‹(⋀i::nat. (n::nat) = fib i ⟹ thesis::bool) ⟹ thesis›*)\n unfolding is_fib_def\n (*goal: ‹(⋀i. n = fib i ⟹ thesis) ⟹ thesis›*)\n by auto\n then show \"?thesis\"\n (*goal: ‹∃i::nat. fib i ≤ (n::nat) ∧ n < fib (Suc i)›*)\n by (metis fib1 (*‹fib (Suc 0) = 1›*) Suc_le_eq (*‹(Suc ?m ≤ ?n) = (?m < ?n)›*) fib_2 (*‹fib 2 = 1›*) fib_mono (*‹?m ≤ ?n ⟹ fib ?m ≤ fib ?n›*) fib_strict_mono (*‹2 ≤ ?i ⟹ fib ?i < fib (Suc ?i)›*) le0 (*‹0 ≤ ?n›*) le_eq_less_or_eq (*‹(?m ≤ ?n) = (?m < ?n ∨ ?m = ?n)›*) not_less_eq_eq (*‹(¬ ?m ≤ ?n) = (Suc ?n ≤ ?m)›*))\nqed (insert no_fib_betw_fibs (*‹¬ is_fib ?n ⟹ ∃i. fib i < ?n ∧ ?n < fib (Suc i)›*), force)\n(*solved the remaining goal: ‹¬ is_fib n ⟹ ∃i. fib i ≤ n ∧ n < fib (Suc i)›*)\n\ntext ‹\n Proof that the sum of non-consecutive Fibonacci numbers with largest member $F_i$ is strictly\n less then $F_{i+1}$. This lemma is used for the uniqueness proof.\n›\nlemma fib_sum_upper_bound:\n assumes \"inc_seq_on c {0..k-1}\" \"∀i∈{0..k}. c i ≥ 2\"\n shows \"(∑ i=0..k. fib (c i)) < fib (Suc (c k))\"\napply (insert assms (*‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*))\n(*goal: ‹(∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\nproof (induct \"c k\" arbitrary: k rule: nat_less_induct)\n (*goal: ‹⋀k. ⟦∀m<c k. ∀x. m = c x ⟶ inc_seq_on c {0..x - 1} ⟶ (∀i∈{0..x}. 2 ≤ c i) ⟶ (∑i = 0..x. fib (c i)) < fib (Suc (c x)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i⟧ ⟹ (∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\n case 1 (*‹∀m<c k. ∀x. m = c x ⟶ inc_seq_on c {0..x - 1} ⟶ (∀i∈{0..x}. 2 ≤ c i) ⟶ (∑i = 0..x. fib (c i)) < fib (Suc (c x))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*)\n then show \"?case\"\n (*goal: ‹(∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\n proof (cases \"c k\")\n (*goals:\n 1. ‹⟦∀m<(c::nat ⇒ nat) (k::nat). ∀x::nat. m = c x ⟶ inc_seq_on c {0::nat..x - (1::nat)} ⟶ (∀i::nat∈{0::nat..x}. (2::nat) ≤ c i) ⟶ (∑i::nat = 0::nat..x. fib (c i)) < fib (Suc (c x)); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; c k = (0::nat)⟧ ⟹ (∑i::nat = 0::nat..k. fib (c i)) < fib (Suc (c k))›\n 2. ‹⋀nat::nat. ⟦∀m<(c::nat ⇒ nat) (k::nat). ∀x::nat. m = c x ⟶ inc_seq_on c {0::nat..x - (1::nat)} ⟶ (∀i::nat∈{0::nat..x}. (2::nat) ≤ c i) ⟶ (∑i::nat = 0::nat..x. fib (c i)) < fib (Suc (c x)); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; c k = Suc nat⟧ ⟹ (∑i::nat = 0::nat..k. fib (c i)) < fib (Suc (c k))›*)\n case (Suc_) (*‹c k = Suc nat_›*)\n show \"?thesis\"\n (*goal: ‹(∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\n proof (cases k)\n (*goals:\n 1. ‹k = 0 ⟹ (∑i = 0..k. fib (c i)) < fib (Suc (c k))›\n 2. ‹⋀nat. k = Suc nat ⟹ (∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\n case k_Suc: (Suc_) (*‹k = Suc nat_›*)\n hence ck_bounds: \"c(k-1) + 1 < c k\" \"c(k-1) < c k\"\n using \"1\"(2) (*‹inc_seq_on c {0..k - 1}›*) unfolding inc_seq_on_def\n (*goals:\n 1. ‹c (k - 1) + 1 < c k›\n 2. ‹c (k - 1) < c k›*)\n apply -\n (*goals:\n 1. ‹⟦k = Suc nat_; ∀n∈{0..k - 1}. Suc (c n) < c (Suc n)⟧ ⟹ c (k - 1) + 1 < c k›\n 2. ‹⟦k = Suc nat_; ∀n∈{0..k - 1}. Suc (c n) < c (Suc n)⟧ ⟹ c (k - 1) < c k›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*) .\n moreover have \"(∑i = 0..k. fib (c i)) = fib(c k) + (∑i = 0..k-1. fib (c i))\"\n using k_Suc (*‹k = Suc nat_›*) by simp\n moreover have \"(∑i = 0..(k-1). fib (c i)) < fib (Suc (c (k-1)))\"\n using ck_bounds(2) (*‹c (k - 1) < c k›*) \"1\" (*‹∀m<c k. ∀x. m = c x ⟶ inc_seq_on c {0..x - 1} ⟶ (∀i∈{0..x}. 2 ≤ c i) ⟶ (∑i = 0..x. fib (c i)) < fib (Suc (c x))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*) unfolding inc_seq_on_def\n (*goal: ‹(∑i = 0..k - 1. fib (c i)) < fib (Suc (c (k - 1)))›*)\n by auto\n ultimately show \"?thesis\"\n (*goal: ‹(∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\n using Suc (*‹c k = Suc nat___›*) smaller_index_implies_fib_le (*‹(?i::nat) < (?j::nat) ⟹ fib (Suc ?i) ≤ fib ?j›*) by fastforce\n qed (simp add: fib_index_strict_mono (*‹⟦2 ≤ ?i; ?i < ?j⟧ ⟹ fib ?i < fib ?j›*) assms( (*‹∀i∈{0..k}. 2 ≤ c i›*) 2))\n (*solved the remaining goal: ‹k = 0 ⟹ (∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\n qed (insert inc_seq_zero_at_start[OF 1(2)] (*‹c k = 0 ⟹ k = 0›*), auto)\n (*solved the remaining goal: ‹⟦∀m<c k. ∀x. m = c x ⟶ inc_seq_on c {0..x - 1} ⟶ (∀i∈{0..x}. 2 ≤ c i) ⟶ (∑i = 0..x. fib (c i)) < fib (Suc (c x)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; c k = 0⟧ ⟹ (∑i = 0..k. fib (c i)) < fib (Suc (c k))›*)\nqed\n\nlemma last_fib_sum_index_constraint:\n assumes \"n ≥ 2\" \"n = (∑ i=0..k. fib (c i))\" \"inc_seq_on c {0..k-1}\" \n assumes \"∀i∈{0..k}. c i ≥ 2\" \"fib i ≤ n\" \"fib(Suc i) > n\"\n shows \"c k = i\"\nproof (-)\n (*goal: ‹c k = i›*)\n have \"2 ≤ i\"\n using assms(1,5,6) (*‹2 ≤ n› ‹fib i ≤ n› ‹n < fib (Suc i)›*) nat_ge_2_fib_idx_bound (*‹⟦2 ≤ ?n; fib ?i ≤ ?n; ?n < fib (Suc ?i)⟧ ⟹ 2 ≤ ?i›*) by simp\n have \"c k > i ⟶ False\"\n using smaller_index_implies_fib_le (*‹?i < ?j ⟹ fib (Suc ?i) ≤ fib ?j›*) assms (*‹2 ≤ n› ‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i∈{0..k}. 2 ≤ c i› ‹fib i ≤ n› ‹n < fib (Suc i)›*) by (metis bot_nat_0.extremum (*‹0 ≤ ?a›*) leD (*‹?y ≤ ?x ⟹ ¬ ?x < ?y›*) sum.last_plus (*‹?m ≤ ?n ⟹ sum ?g {?m..?n} = ?g ?n + sum ?g {?m..<?n}›*) trans_le_add1 (*‹?i ≤ ?j ⟹ ?i ≤ ?j + ?m›*))\n moreover have \"c k < i ⟶ False\"\n proof (standard)\n (*goal: ‹c k < i ⟹ False›*)\n assume \"c k < i\" (*‹(c::nat ⇒ nat) (k::nat) < (i::nat)›*)\n have seq: \"inc_seq_on c {0..k - 1 - 1}\" \"∀i∈{0..k-1}. 2 ≤ c i\"\n using assms (*‹2 ≤ n› ‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i∈{0..k}. 2 ≤ c i› ‹fib i ≤ n› ‹n < fib (Suc i)›*) unfolding inc_seq_on_def\n (*goals:\n 1. ‹∀n∈{0..k - 1 - 1}. Suc (c n) < c (Suc n)›\n 2. ‹∀i∈{0..k - 1}. 2 ≤ c i›*)\n apply -\n (*goals:\n 1. ‹⟦2 ≤ n; n = (∑i = 0..k. fib (c i)); ∀n∈{0..k - 1}. Suc (c n) < c (Suc n); ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i)⟧ ⟹ ∀n∈{0..k - 1 - 1}. Suc (c n) < c (Suc n)›\n 2. ‹⟦2 ≤ n; n = (∑i = 0..k. fib (c i)); ∀n∈{0..k - 1}. Suc (c n) < c (Suc n); ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i)⟧ ⟹ ∀i∈{0..k - 1}. 2 ≤ c i›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n have \"k > 0\"\n apply (rule ccontr (*‹(¬ (?P::bool) ⟹ False) ⟹ ?P›*))\n (*goal: ‹0 < k›*)\n apply (insert ‹c k < i› assms (*‹2 ≤ n› ‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i› ‹fib i ≤ n› ‹n < fib (Suc i)›*) fib_index_strict_mono (*‹⟦2 ≤ ?i; ?i < ?j⟧ ⟹ fib ?i < fib ?j›*) leD (*‹?y ≤ ?x ⟹ ¬ ?x < ?y›*))\n (*goal: ‹¬ 0 < k ⟹ False›*)\n by auto\n hence \"c (k-1) + 1 < c k\" \"c (k-1) + 3 ≤ i\"\n using ‹c k < i› (*‹c k < i›*) assms (*‹2 ≤ n› ‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i› ‹fib i ≤ n› ‹n < fib (Suc i)›*) unfolding inc_seq_on_def\n (*goals:\n 1. ‹c (k - 1) + 1 < c k›\n 2. ‹c (k - 1) + 3 ≤ i›*)\n apply -\n (*goals:\n 1. ‹⟦(0::nat) < (k::nat); (c::nat ⇒ nat) k < (i::nat); (2::nat) ≤ (n::nat); n = (∑i::nat = 0::nat..k. fib (c i)); ∀n::nat∈{0::nat..k - (1::nat)}. Suc (c n) < c (Suc n); ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; fib i ≤ n; n < fib (Suc i)⟧ ⟹ c (k - (1::nat)) + (1::nat) < c k›\n 2. ‹⟦(0::nat) < (k::nat); (c::nat ⇒ nat) k < (i::nat); (2::nat) ≤ (n::nat); n = (∑i::nat = 0::nat..k. fib (c i)); ∀n::nat∈{0::nat..k - (1::nat)}. Suc (c n) < c (Suc n); ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; fib i ≤ n; n < fib (Suc i)⟧ ⟹ c (k - (1::nat)) + (3::nat) ≤ i›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*) .\n have \"(∑i = 0..k-1. fib (c i)) + fib (c k) = (∑i = 0..k. fib (c i))\"\n using sum.atLeast0_atMost_Suc (*‹sum ?g {0..Suc ?n} = sum ?g {0..?n} + ?g (Suc ?n)›*) Suc_pred'[OF ‹k > 0›] (*‹k = Suc (k - 1)›*) by metis\n moreover have \"fib (Suc (c (k-1))) ≤ fib (i-2)\"\n using ‹c k < i› (*‹c k < i›*) ‹c (k-1) + 1 < c k› (*‹c (k - 1) + 1 < c k›*) by (simp add: fib_mono (*‹?m ≤ ?n ⟹ fib ?m ≤ fib ?n›*))\n moreover have \"fib (c k) ≤ fib (i-1)\"\n using ‹c k < i› (*‹c k < i›*) fib_mono (*‹(?m::nat) ≤ (?n::nat) ⟹ fib ?m ≤ fib ?n›*) by fastforce\n ultimately have \"(∑i = 0..k. fib (c i)) < fib (i-1) + fib (i-2)\"\n using assms (*‹2 ≤ n› ‹(n::nat) = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i› ‹fib i ≤ n› ‹n < fib (Suc i)›*) ‹c k < i› (*‹c k < i›*) ‹k > 0› (*‹0 < k›*) fib_sum_upper_bound[OF seq ( 1 ) seq ( 2 )] (*‹(∑i = 0..k - 1. fib (c i)) < fib (Suc (c (k - 1)))›*) by simp\n hence \"(∑i = 0..k. fib (c i)) < fib i\"\n using fib.simps(3)[of \"i-2\"] (*‹fib (Suc (Suc ((i::nat) - (2::nat)))) = fib (Suc (i - (2::nat))) + fib (i - (2::nat))›*) assms(4) (*‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i›*) ‹c k < i› (*‹c k < i›*) by (metis add_2_eq_Suc (*‹(2::nat) + (?n::nat) = Suc (Suc ?n)›*) diff_Suc_1 (*‹Suc (?n::nat) - (1::nat) = ?n›*) ‹2 ≤ i› le_add_diff_inverse (*‹(?b::?'a::linordered_semidom) ≤ (?a::?'a::linordered_semidom) ⟹ ?b + (?a - ?b) = ?a›*))\n then show False\n using assms (*‹2 ≤ n› ‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i› ‹fib i ≤ n› ‹n < fib (Suc i)›*) by simp\n qed\n ultimately show \"?thesis\"\n (*goal: ‹c k = i›*)\n by simp\nqed\n\nsubsection ‹Theorem›\ntext ‹\n Now, both parts of Zeckendorf's Theorem can be proven. Firstly, the existence of an increasing \n sequence for a positive integer $N$ such that the corresponding Fibonacci numbers sum up to $N$ \n is proven. Then, the uniqueness of such an increasing sequence is proven.\n›\nlemma fib_implies_zeckendorf:\n assumes \"is_fib n\" \"n > 0\"\n shows \"∃ c k. n = (∑ i=0..k. fib(c i)) ∧ inc_seq_on c {0..k-1} ∧ (∀ i∈{0..k}. c i ≥ 2)\" \nproof (-)\n (*goal: ‹∃c k. n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›*)\n from assms (*‹is_fib n› ‹0 < n›*) obtain i where i_def: \"fib i = n\" \"i ≥ 2\"\n (*goal: ‹(⋀i. ⟦fib i = n; 2 ≤ i⟧ ⟹ thesis) ⟹ thesis›*)\n using pos_fib_has_idx_ge_two (*‹⟦0 < ?n; is_fib ?n⟧ ⟹ ∃i≥2. fib i = ?n›*) by auto\n define c where c_def: \"(c :: nat ⇒ nat) = (λ n::nat. if n = 0 then i else i + 3)\"\n from i_def (*‹fib i = n› ‹2 ≤ i›*) have \"n = (∑i = 0..0. fib (c i))\"\n by (simp add: c_def (*‹(c::nat ⇒ nat) = (λn::nat. if n = (0::nat) then i::nat else i + (3::nat))›*))\n moreover have \"inc_seq_on c {0..0}\"\n by (simp add: c_def (*‹c = (λn. if n = 0 then i else i + 3)›*) inc_seq_on_def (*‹inc_seq_on ?f ?I = (∀n∈?I. Suc (?f n) < ?f (Suc n))›*))\n ultimately show \"∃ c k. n = (∑ i=0..k. fib(c i)) ∧ inc_seq_on c {0..k-1} ∧ (∀ i∈{0..k}. c i ≥ 2)\"\n using i_def (*‹fib i = n› ‹2 ≤ i›*) c_def (*‹(c::nat ⇒ nat) = (λn::nat. if n = (0::nat) then i::nat else i + (3::nat))›*) by fastforce\nqed\n\ntheorem zeckendorf_existence:\n assumes \"n > 0\"\n shows \"∃ c k. n = (∑ i=0..k. fib (c i)) ∧ inc_seq_on c {0..k-1} ∧ (∀i∈{0..k}. c i ≥ 2)\" \n using assms (*‹0 < n›*) proof (induct n rule: nat_less_induct (*‹(⋀n. ∀m<n. ?P m ⟹ ?P n) ⟹ ?P ?n›*))\n (*goal: ‹⋀n. ⟦∀m<n. 0 < m ⟶ (∃c k. m = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)); 0 < n⟧ ⟹ ∃c k. n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›*)\n case (1 n) (*‹∀m<n. 0 < m ⟶ (∃c k. m = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i))› ‹0 < n›*)\n then show \"?case\"\n (*goal: ‹∃c k. n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›*)\n proof (cases \"is_fib n\")\n (*goals:\n 1. ‹⟦∀m<n. 0 < m ⟶ (∃c k. m = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)); 0 < n; is_fib n⟧ ⟹ ∃c k. n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›\n 2. ‹⟦∀m<n. 0 < m ⟶ (∃c k. m = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)); 0 < n; ¬ is_fib n⟧ ⟹ ∃c k. n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›*)\n case False (*‹¬ is_fib n›*)\n obtain i where bounds: \"fib i < n\" \"n < fib (Suc i)\" \"i > 0\"\n (*goal: ‹(⋀i. ⟦fib i < n; n < fib (Suc i); 0 < i⟧ ⟹ thesis) ⟹ thesis›*)\n using no_fib_betw_fibs (*‹¬ is_fib ?n ⟹ ∃i. fib i < ?n ∧ ?n < fib (Suc i)›*) \"1\"(2) (*‹0 < n›*) False (*‹¬ is_fib n›*) by force\n then obtain c and k where seq: \"(n - fib i) = (∑ i=0..k. fib (c i))\" \"inc_seq_on c {0..k-1}\" \"∀ i∈{0..k}. c i ≥ 2\"\n (*goal: ‹(⋀c k. ⟦n - fib i = (∑i = 0..k. fib (c i)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i⟧ ⟹ thesis) ⟹ thesis›*)\n using \"1\" (*‹∀m<n::nat. (0::nat) < m ⟶ (∃(c::nat ⇒ nat) k::nat. m = (∑i::nat = 0::nat..k. fib (c i)) ∧ inc_seq_on c {0::nat..k - (1::nat)} ∧ (∀i::nat∈{0::nat..k}. (2::nat) ≤ c i))› ‹0 < n›*) fib_neq_0_nat (*‹(0::nat) < (?n::nat) ⟹ (0::nat) < fib ?n›*) zero_less_diff (*‹(0 < ?n - ?m) = (?m < ?n)›*) diff_less (*‹⟦0 < ?n; 0 < ?m⟧ ⟹ ?m - ?n < ?m›*) by metis\n let ?c' = \"(λ n. if n = k+1 then i else c n)\"\n have diff_le_fib: \"n - fib i < fib(i-1)\"\n using bounds (*‹fib (i::nat) < (n::nat)› ‹(n::nat) < fib (Suc (i::nat))› ‹0 < i›*) fib2 (*‹fib (Suc (Suc ?n)) = fib (Suc ?n) + fib ?n›*) not0_implies_Suc[of i] (*‹i ≠ 0 ⟹ ∃m. i = Suc m›*) by auto\n hence ck_lt_fib: \"fib (c k) < fib i\"\n using fib_Suc_mono[of \"i-1\"] (*‹fib (i - 1) ≤ fib (Suc (i - 1))›*) bounds (*‹fib (i::nat) < (n::nat)› ‹n < fib (Suc i)› ‹(0::nat) < (i::nat)›*) by (simp add: sum.last_plus (*‹?m ≤ ?n ⟹ sum ?g {?m..?n} = ?g ?n + sum ?g {?m..<?n}›*) seq (*‹n - fib i = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*))\n have \"inc_seq_on ?c' {0..k}\"\n using inc_seq_on_aux[OF seq ( 2 ) diff_le_fib ck_lt_fib seq ( 1 )] (*‹Suc (c k) < i›*) One_nat_def (*‹1 = Suc 0›*) inc_seq_on_def (*‹inc_seq_on (?f::nat ⇒ nat) (?I::nat set) = (∀n::nat∈?I. Suc (?f n) < ?f (Suc n))›*) leI (*‹¬ ?x < ?y ⟹ ?y ≤ ?x›*) seq (*‹n - fib i = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*) by force\n moreover have \"n = (∑ i=0..k+1. fib (?c' i))\"\n using bounds (*‹fib i < n› ‹n < fib (Suc i)› ‹0 < i›*) seq (*‹n - fib i = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i›*) by simp\n moreover have \"∀ i ∈ {0..k+1}. ?c' i ≥ 2\"\n using seq (*‹(n::nat) - fib (i::nat) = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*) bounds (*‹fib i < n› ‹n < fib (Suc i)› ‹0 < i›*) fib2 (*‹fib (Suc (Suc ?n)) = fib (Suc ?n) + fib ?n›*) not0_implies_Suc[of i] (*‹i ≠ 0 ⟹ ∃m. i = Suc m›*) atLeastAtMost_iff (*‹(?i ∈ {?l..?u}) = (?l ≤ ?i ∧ ?i ≤ ?u)›*) diff_is_0_eq' (*‹?m ≤ ?n ⟹ ?m - ?n = 0›*) less_nat_zero_code (*‹(?n < 0) = False›*) not_less_eq_eq (*‹(¬ ?m ≤ ?n) = (Suc ?n ≤ ?m)›*) by fastforce\n ultimately show \"?thesis\"\n (*goal: ‹∃c k. n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›*)\n by fastforce\n qed (insert fib_implies_zeckendorf (*‹⟦is_fib ?n; 0 < ?n⟧ ⟹ ∃c k. ?n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›*), auto)\n (*solved the remaining goal: ‹⟦∀m<n. 0 < m ⟶ (∃c k. m = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)); 0 < n; is_fib n⟧ ⟹ ∃c k. n = (∑i = 0..k. fib (c i)) ∧ inc_seq_on c {0..k - 1} ∧ (∀i∈{0..k}. 2 ≤ c i)›*)\n qed\n\nlemma fib_unique_fib_sum:\n fixes k :: nat\n assumes \"n ≥ 2\" \"inc_seq_on c {0..k-1}\" \"∀i∈{0..k}. c i ≥ 2\" \n assumes \"n = fib i\"\n shows \"n = (∑i=0..k. fib (c i)) ⟷ k = 0 ∧ c 0 = i\"\nproof (standard)\n (*goals:\n 1. ‹n = (∑i = 0..k. fib (c i)) ⟹ k = 0 ∧ c 0 = i›\n 2. ‹k = 0 ∧ c 0 = i ⟹ n = (∑i = 0..k. fib (c i))›*)\n assume ass: \"n = (∑i = 0..k. fib (c i))\" (*‹(n::nat) = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i))›*)\n obtain j where bounds: \"fib j ≤ n\" \"fib(Suc j) > n\" \"j ≥ 2\"\n (*goal: ‹(⋀j. ⟦fib j ≤ n; n < fib (Suc j); 2 ≤ j⟧ ⟹ thesis) ⟹ thesis›*)\n using betw_fibs (*‹∃i. fib i ≤ ?n ∧ ?n < fib (Suc i)›*) assms (*‹2 ≤ n› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i› ‹n = fib i›*) nat_ge_2_fib_idx_bound (*‹⟦2 ≤ ?n; fib ?i ≤ ?n; ?n < fib (Suc ?i)⟧ ⟹ 2 ≤ ?i›*) by blast\n have idx_eq: \"c k = j\"\n using last_fib_sum_index_constraint (*‹⟦2 ≤ ?n; ?n = (∑i = 0..?k. fib (?c i)); inc_seq_on ?c {0..?k - 1}; ∀i∈{0..?k}. 2 ≤ ?c i; fib ?i ≤ ?n; ?n < fib (Suc ?i)⟧ ⟹ ?c ?k = ?i›*) assms(1-3) (*‹2 ≤ n› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*) ass (*‹n = (∑i = 0..k. fib (c i))›*) bounds (*‹fib j ≤ n› ‹n < fib (Suc j)› ‹2 ≤ j›*) by simp\n have \"i = j\"\n using bounds (*‹fib j ≤ n› ‹(n::nat) < fib (Suc (j::nat))› ‹2 ≤ j›*) ass (*‹n = (∑i = 0..k. fib (c i))›*) assms (*‹2 ≤ n› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i∈{0..k}. 2 ≤ c i› ‹n = fib i›*) by (metis Suc_leI (*‹?m < ?n ⟹ Suc ?m ≤ ?n›*) fib_mono (*‹?m ≤ ?n ⟹ fib ?m ≤ fib ?n›*) ge_two_fib_unique_idx (*‹⟦2 ≤ fib ?i; fib ?i = fib ?j⟧ ⟹ ?i = ?j›*) le_neq_implies_less (*‹⟦?m ≤ ?n; ?m ≠ ?n⟧ ⟹ ?m < ?n›*) linorder_not_le (*‹(¬ ?x ≤ ?y) = (?y < ?x)›*))\n have \"k > 0 ⟶ fib i = fib i + (∑i = 0..k-1. fib (c i))\"\n using ass (*‹n = (∑i = 0..k. fib (c i))›*) assms (*‹2 ≤ n› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i∈{0..k}. 2 ≤ c i› ‹n = fib i›*) by (metis idx_eq (*‹c k = j›*) One_nat_def (*‹1 = Suc 0›*) Suc_pred (*‹0 < ?n ⟹ Suc (?n - Suc 0) = ?n›*) ‹i = j› add.commute (*‹?a + ?b = ?b + ?a›*) sum.atLeast0_atMost_Suc (*‹sum ?g {0..Suc ?n} = sum ?g {0..?n} + ?g (Suc ?n)›*))\n hence \"k > 0 ⟶ False\"\n using fib_idx_ge_two_fib_sum_not_zero[of 0 \"k-1\" c] (*‹⟦0 ≤ k - 1; ∀i∈{0..k - 1}. 2 ≤ c i⟧ ⟹ (∑i = 0..k - 1. fib (c i)) ≠ 0›*) assms (*‹(2::nat) ≤ (n::nat)› ‹inc_seq_on c {0..k - 1}› ‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i› ‹(n::nat) = fib (i::nat)›*) by auto\n then show \"k = 0 ∧ c 0 = i\"\n using ‹i = j› (*‹i = j›*) idx_eq (*‹c k = j›*) by simp\nqed (auto simp: assms (*‹(2::nat) ≤ (n::nat)› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i› ‹(n::nat) = fib (i::nat)›*))\n(*solved the remaining goal: ‹k = 0 ∧ c 0 = i ⟹ n = (∑i = 0..k. fib (c i))›*)\n\ntheorem zeckendorf_unique:\n assumes \"n > 0\"\n assumes \"n = (∑ i=0..k. fib (c i))\" \"inc_seq_on c {0..k-1}\" \"∀i∈{0..k}. c i ≥ 2\" \n assumes \"n = (∑ i=0..k'. fib (c' i))\" \"inc_seq_on c' {0..k'-1}\" \"∀i∈{0..k'}. c' i ≥ 2\"\n shows \"k = k' ∧ (∀ i ∈ {0..k}. c i = c' i)\"\n using assms (*‹0 < n› ‹(n::nat) = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i))› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i∈{0..k}. 2 ≤ c i› ‹n = (∑i = 0..k'. fib (c' i))› ‹inc_seq_on c' {0..k' - 1}› ‹∀i∈{0..k'}. 2 ≤ c' i›*) proof (induct n arbitrary: k k' rule: nat_less_induct)\n (*goal: ‹⋀(n::nat) (k::nat) k'::nat. ⟦∀m<n. (0::nat) < m ⟶ (∀x::nat. m = (∑i::nat = 0::nat..x. fib ((c::nat ⇒ nat) i)) ⟶ inc_seq_on c {0::nat..x - (1::nat)} ⟶ (∀i::nat∈{0::nat..x}. (2::nat) ≤ c i) ⟶ (∀xa::nat. m = (∑i::nat = 0::nat..xa. fib ((c'::nat ⇒ nat) i)) ⟶ inc_seq_on c' {0::nat..xa - (1::nat)} ⟶ (∀i::nat∈{0::nat..xa}. (2::nat) ≤ c' i) ⟶ x = xa ∧ (∀i::nat∈{0::nat..x}. c i = c' i))); (0::nat) < n; n = (∑i::nat = 0::nat..k. fib (c i)); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = (∑i::nat = 0::nat..k'. fib (c' i)); inc_seq_on c' {0::nat..k' - (1::nat)}; ∀i::nat∈{0::nat..k'}. (2::nat) ≤ c' i⟧ ⟹ k = k' ∧ (∀i::nat∈{0::nat..k}. c i = c' i)›*)\n case IH: (1 n) (*‹∀m<n::nat. (0::nat) < m ⟶ (∀x::nat. m = (∑i::nat = 0::nat..x. fib ((c::nat ⇒ nat) i)) ⟶ inc_seq_on c {0::nat..x - (1::nat)} ⟶ (∀i::nat∈{0::nat..x}. (2::nat) ≤ c i) ⟶ (∀xa::nat. m = (∑i::nat = 0::nat..xa. fib ((c'::nat ⇒ nat) i)) ⟶ inc_seq_on c' {0::nat..xa - (1::nat)} ⟶ (∀i::nat∈{0::nat..xa}. (2::nat) ≤ c' i) ⟶ x = xa ∧ (∀i::nat∈{0::nat..x}. c i = c' i)))› ‹0 < n› ‹(n::nat) = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i))› ‹inc_seq_on c {0..k - 1}› ‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i› ‹n = (∑i = 0..k'. fib (c' i))› ‹inc_seq_on c' {0..k' - 1}› ‹∀i∈{0..k'}. 2 ≤ c' i›*)\n consider \"n = 0\" | \"n = 1\" | \"n ≥ 2\"\n (*goal: ‹⟦(n::nat) = (0::nat) ⟹ thesis::bool; n = (1::nat) ⟹ thesis; (2::nat) ≤ n ⟹ thesis⟧ ⟹ thesis›*)\n by linarith\n then show \"?case\"\n (*goal: ‹k = k' ∧ (∀i∈{0..k}. c i = c' i)›*)\n proof (cases)\n (*goals:\n 1. ‹n = 0 ⟹ k = k' ∧ (∀i∈{0..k}. c i = c' i)›\n 2. ‹n = 1 ⟹ k = k' ∧ (∀i∈{0..k}. c i = c' i)›\n 3. ‹2 ≤ n ⟹ k = k' ∧ (∀i∈{0..k}. c i = c' i)›*)\n case 3 (*‹2 ≤ n›*)\n obtain i where bounds: \"fib i ≤ n\" \"fib(Suc i) > n\" \"2 ≤ i\"\n (*goal: ‹(⋀i. ⟦fib i ≤ n; n < fib (Suc i); 2 ≤ i⟧ ⟹ thesis) ⟹ thesis›*)\n using betw_fibs (*‹∃i. fib i ≤ ?n ∧ ?n < fib (Suc i)›*) nat_ge_2_fib_idx_bound (*‹⟦2 ≤ ?n; fib ?i ≤ ?n; ?n < fib (Suc ?i)⟧ ⟹ 2 ≤ ?i›*) \"3\" (*‹2 ≤ n›*) by blast\n have last_idx_eq: \"c' k' = i\" \"c k = i\" \"c' k' = c k\"\n using last_fib_sum_index_constraint[OF 3] (*‹⟦n = (∑i = 0..?k. fib (?c i)); inc_seq_on ?c {0..?k - 1}; ∀i∈{0..?k}. 2 ≤ ?c i; fib ?i ≤ n; n < fib (Suc ?i)⟧ ⟹ ?c ?k = ?i›*) IH(6-8) (*‹n = (∑i = 0..k'. fib (c' i))› ‹inc_seq_on (c'::nat ⇒ nat) {0::nat..(k'::nat) - (1::nat)}› ‹∀i∈{0..k'}. 2 ≤ c' i›*) IH(3-5) (*‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*) bounds (*‹fib (i::nat) ≤ (n::nat)› ‹n < fib (Suc i)› ‹2 ≤ i›*) apply -\n (*goals:\n 1. ‹⟦⋀c k i. ⟦n = (∑i = 0..k. fib (c i)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i)⟧ ⟹ c k = i; n = (∑i = 0..k'. fib (c' i)); inc_seq_on c' {0..k' - 1}; ∀i∈{0..k'}. 2 ≤ c' i; n = (∑i = 0..k. fib (c i)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i); 2 ≤ i⟧ ⟹ c' k' = i›\n 2. ‹⟦⋀c k i. ⟦n = (∑i = 0..k. fib (c i)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i)⟧ ⟹ c k = i; n = (∑i = 0..k'. fib (c' i)); inc_seq_on c' {0..k' - 1}; ∀i∈{0..k'}. 2 ≤ c' i; n = (∑i = 0..k. fib (c i)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i); 2 ≤ i⟧ ⟹ c k = i›\n 3. ‹⟦⋀c k i. ⟦n = (∑i = 0..k. fib (c i)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i)⟧ ⟹ c k = i; n = (∑i = 0..k'. fib (c' i)); inc_seq_on c' {0..k' - 1}; ∀i∈{0..k'}. 2 ≤ c' i; n = (∑i = 0..k. fib (c i)); inc_seq_on c {0..k - 1}; ∀i∈{0..k}. 2 ≤ c i; fib i ≤ n; n < fib (Suc i); 2 ≤ i⟧ ⟹ c' k' = c k›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*) .\n then show \"?thesis\"\n (*goal: ‹k = k' ∧ (∀i∈{0..k}. c i = c' i)›*)\n proof (cases \"is_fib n\")\n (*goals:\n 1. ‹⟦c' k' = i; c k = i; c' k' = c k; is_fib n⟧ ⟹ k = k' ∧ (∀i∈{0..k}. c i = c' i)›\n 2. ‹⟦c' k' = i; c k = i; c' k' = c k; ¬ is_fib n⟧ ⟹ k = k' ∧ (∀i∈{0..k}. c i = c' i)›*)\n case True (*‹is_fib (n::nat)›*)\n hence \"fib i = n\"\n unfolding is_fib_def\n (*goal: ‹fib i = n›*)\n using bounds (*‹fib i ≤ n› ‹n < fib (Suc i)› ‹(2::nat) ≤ (i::nat)›*) IH(2-8) (*‹(0::nat) < (n::nat)› ‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i› ‹n = (∑i = 0..k'. fib (c' i))› ‹inc_seq_on (c'::nat ⇒ nat) {0::nat..(k'::nat) - (1::nat)}› ‹∀i::nat∈{0::nat..k'::nat}. (2::nat) ≤ (c'::nat ⇒ nat) i›*) fib_mono (*‹?m ≤ ?n ⟹ fib ?m ≤ fib ?n›*) leD (*‹?y ≤ ?x ⟹ ¬ ?x < ?y›*) nle_le (*‹(¬ ?a ≤ ?b) = (?b ≤ ?a ∧ ?b ≠ ?a)›*) not_less_eq_eq (*‹(¬ ?m ≤ ?n) = (Suc ?n ≤ ?m)›*) by metis\n hence \"k = 0\" \"c 0 = i\" \"k' = 0\" \"c' 0 = i\"\n using fib_unique_fib_sum (*‹⟦(2::nat) ≤ (?n::nat); inc_seq_on (?c::nat ⇒ nat) {0::nat..(?k::nat) - (1::nat)}; ∀i::nat∈{0::nat..?k}. (2::nat) ≤ ?c i; ?n = fib (?i::nat)⟧ ⟹ (?n = (∑i::nat = 0::nat..?k. fib (?c i))) = (?k = (0::nat) ∧ ?c (0::nat) = ?i)›*) \"3\" (*‹2 ≤ n›*) IH(3-8) (*‹n = (∑i = 0..k. fib (c i))› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i› ‹n = (∑i = 0..k'. fib (c' i))› ‹inc_seq_on c' {0..k' - 1}› ‹∀i::nat∈{0::nat..k'::nat}. (2::nat) ≤ (c'::nat ⇒ nat) i›*) apply -\n (*goals:\n 1. ‹⟦fib (i::nat) = (n::nat); ⋀(n::nat) (c::nat ⇒ nat) (k::nat) i::nat. ⟦(2::nat) ≤ n; inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = fib i⟧ ⟹ (n = (∑i::nat = 0::nat..k. fib (c i))) = (k = (0::nat) ∧ c (0::nat) = i); (2::nat) ≤ n; n = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i)); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = (∑i::nat = 0::nat..k'::nat. fib ((c'::nat ⇒ nat) i)); inc_seq_on c' {0::nat..k' - (1::nat)}; ∀i::nat∈{0::nat..k'}. (2::nat) ≤ c' i⟧ ⟹ k = (0::nat)›\n 2. ‹⟦fib (i::nat) = (n::nat); ⋀(n::nat) (c::nat ⇒ nat) (k::nat) i::nat. ⟦(2::nat) ≤ n; inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = fib i⟧ ⟹ (n = (∑i::nat = 0::nat..k. fib (c i))) = (k = (0::nat) ∧ c (0::nat) = i); (2::nat) ≤ n; n = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i)); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = (∑i::nat = 0::nat..k'::nat. fib ((c'::nat ⇒ nat) i)); inc_seq_on c' {0::nat..k' - (1::nat)}; ∀i::nat∈{0::nat..k'}. (2::nat) ≤ c' i⟧ ⟹ c (0::nat) = i›\n 3. ‹⟦fib (i::nat) = (n::nat); ⋀(n::nat) (c::nat ⇒ nat) (k::nat) i::nat. ⟦(2::nat) ≤ n; inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = fib i⟧ ⟹ (n = (∑i::nat = 0::nat..k. fib (c i))) = (k = (0::nat) ∧ c (0::nat) = i); (2::nat) ≤ n; n = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i)); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = (∑i::nat = 0::nat..k'::nat. fib ((c'::nat ⇒ nat) i)); inc_seq_on c' {0::nat..k' - (1::nat)}; ∀i::nat∈{0::nat..k'}. (2::nat) ≤ c' i⟧ ⟹ k' = (0::nat)›\n 4. ‹⟦fib (i::nat) = (n::nat); ⋀(n::nat) (c::nat ⇒ nat) (k::nat) i::nat. ⟦(2::nat) ≤ n; inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = fib i⟧ ⟹ (n = (∑i::nat = 0::nat..k. fib (c i))) = (k = (0::nat) ∧ c (0::nat) = i); (2::nat) ≤ n; n = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i)); inc_seq_on c {0::nat..k - (1::nat)}; ∀i::nat∈{0::nat..k}. (2::nat) ≤ c i; n = (∑i::nat = 0::nat..k'::nat. fib ((c'::nat ⇒ nat) i)); inc_seq_on c' {0::nat..k' - (1::nat)}; ∀i::nat∈{0::nat..k'}. (2::nat) ≤ c' i⟧ ⟹ c' (0::nat) = i›\n discuss goal 1*)\n apply metis\n (*discuss goal 2*)\n apply metis\n (*discuss goal 3*)\n apply metis\n (*discuss goal 4*)\n apply metis\n (*proven 4 subgoals*) .\n then show \"?thesis\"\n (*goal: ‹k = k' ∧ (∀i∈{0..k}. c i = c' i)›*)\n by simp\n next\n (*goal: ‹⟦(c'::nat ⇒ nat) (k'::nat) = (i::nat); (c::nat ⇒ nat) (k::nat) = i; c' k' = c k; ¬ is_fib (n::nat)⟧ ⟹ k = k' ∧ (∀i::nat∈{0::nat..k}. c i = c' i)›*)\n case False (*‹¬ is_fib n›*)\n have \"k > 0\"\n using IH(3) (*‹n = (∑i = 0..k. fib (c i))›*) False (*‹¬ is_fib n›*) unfolding is_fib_def\n (*goal: ‹0 < k›*)\n by fastforce\n have \"k' > 0\"\n using IH(6) (*‹n = (∑i = 0..k'. fib (c' i))›*) False (*‹¬ is_fib n›*) unfolding is_fib_def\n (*goal: ‹(0::nat) < (k'::nat)›*)\n by fastforce\n have \"0 < n - fib (c k)\"\n using False (*‹¬ is_fib n›*) bounds (*‹fib i ≤ n› ‹(n::nat) < fib (Suc (i::nat))› ‹2 ≤ i›*) last_idx_eq(2) (*‹c k = i›*) unfolding is_fib_def\n (*goal: ‹0 < n - fib (c k)›*)\n by fastforce\n moreover have \"n - fib (c k) < n\"\n using bounds (*‹fib i ≤ n› ‹n < fib (Suc i)› ‹(2::nat) ≤ (i::nat)›*) last_idx_eq (*‹c' k' = i› ‹c k = i› ‹c' k' = c k›*) by (simp add: dual_order.strict_trans1 (*‹⟦?b ≤ ?a; ?c < ?b⟧ ⟹ ?c < ?a›*) fib_neq_0_nat (*‹0 < ?n ⟹ 0 < fib ?n›*))\n moreover have \"n - fib (c k) = (∑i = 0..k-1. fib (c i))\"\n using sum.atLeast0_atMost_Suc[of \"λ i. fib (c i)\" \"k-1\"] (*‹(∑i = 0..Suc (k - 1). fib (c i)) = (∑i = 0..k - 1. fib (c i)) + fib (c (Suc (k - 1)))›*) Suc_diff_1 (*‹0 < ?n ⟹ Suc (?n - 1) = ?n›*) ‹k > 0› (*‹0 < k›*) IH(3) (*‹n = (∑i = 0..k. fib (c i))›*) by simp\n moreover have \"n - fib (c' k' ) = (∑i = 0..k'-1. fib (c' i))\"\n using sum.atLeast0_atMost_Suc[of \"λ i. fib (c' i)\" \"k'-1\"] (*‹(∑i = 0..Suc (k' - 1). fib (c' i)) = (∑i = 0..k' - 1. fib (c' i)) + fib (c' (Suc (k' - 1)))›*) Suc_diff_1 (*‹0 < ?n ⟹ Suc (?n - 1) = ?n›*) ‹k' > 0› (*‹(0::nat) < (k'::nat)›*) IH(6) (*‹(n::nat) = (∑i::nat = 0::nat..k'::nat. fib ((c'::nat ⇒ nat) i))›*) by simp\n moreover have \"inc_seq_on c {0..k-1 - 1}\" \"∀i∈{0..k-1}. 2 ≤ c i\"\n using IH(4,5) (*‹inc_seq_on c {0..k - 1}› ‹∀i∈{0..k}. 2 ≤ c i›*) unfolding inc_seq_on_def\n (*goals:\n 1. ‹∀n∈{0..k - 1 - 1}. Suc (c n) < c (Suc n)›\n 2. ‹∀i∈{0..k - 1}. 2 ≤ c i›*)\n apply -\n (*goals:\n 1. ‹⟦∀n∈{0..k - 1}. Suc (c n) < c (Suc n); ∀i∈{0..k}. 2 ≤ c i⟧ ⟹ ∀n∈{0..k - 1 - 1}. Suc (c n) < c (Suc n)›\n 2. ‹⟦∀n∈{0..k - 1}. Suc (c n) < c (Suc n); ∀i∈{0..k}. 2 ≤ c i⟧ ⟹ ∀i∈{0..k - 1}. 2 ≤ c i›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n moreover have \"inc_seq_on c' {0..k'-1 - 1}\" \"∀i∈{0..k'-1}. 2 ≤ c' i\"\n using IH(7,8) (*‹inc_seq_on c' {0..k' - 1}› ‹∀i∈{0..k'}. 2 ≤ c' i›*) unfolding inc_seq_on_def\n (*goals:\n 1. ‹∀n::nat∈{0::nat..(k'::nat) - (1::nat) - (1::nat)}. Suc ((c'::nat ⇒ nat) n) < c' (Suc n)›\n 2. ‹∀i::nat∈{0::nat..(k'::nat) - (1::nat)}. (2::nat) ≤ (c'::nat ⇒ nat) i›*)\n apply -\n (*goals:\n 1. ‹⟦∀n∈{0..k' - 1}. Suc (c' n) < c' (Suc n); ∀i∈{0..k'}. 2 ≤ c' i⟧ ⟹ ∀n∈{0..k' - 1 - 1}. Suc (c' n) < c' (Suc n)›\n 2. ‹⟦∀n∈{0..k' - 1}. Suc (c' n) < c' (Suc n); ∀i∈{0..k'}. 2 ≤ c' i⟧ ⟹ ∀i∈{0..k' - 1}. 2 ≤ c' i›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n ultimately have \"k-1 = k'-1 ∧ (∀i∈{0..k-1}. c i = c' i)\"\n using IH(1) (*‹∀m<n. 0 < m ⟶ (∀x. m = (∑i = 0..x. fib (c i)) ⟶ inc_seq_on c {0..x - 1} ⟶ (∀i∈{0..x}. 2 ≤ c i) ⟶ (∀xa. m = (∑i = 0..xa. fib (c' i)) ⟶ inc_seq_on c' {0..xa - 1} ⟶ (∀i∈{0..xa}. 2 ≤ c' i) ⟶ x = xa ∧ (∀i∈{0..x}. c i = c' i)))›*) unfolding last_idx_eq\n (*goal: ‹(k::nat) - (1::nat) = (k'::nat) - (1::nat) ∧ (∀i::nat∈{0::nat..k - (1::nat)}. (c::nat ⇒ nat) i = (c'::nat ⇒ nat) i)›*)\n by blast\n then show \"?thesis\"\n (*goal: ‹k = k' ∧ (∀i∈{0..k}. c i = c' i)›*)\n using IH(1) (*‹∀m<n. 0 < m ⟶ (∀x. m = (∑i = 0..x. fib (c i)) ⟶ inc_seq_on c {0..x - 1} ⟶ (∀i∈{0..x}. 2 ≤ c i) ⟶ (∀xa. m = (∑i = 0..xa. fib (c' i)) ⟶ inc_seq_on c' {0..xa - 1} ⟶ (∀i∈{0..xa}. 2 ≤ c' i) ⟶ x = xa ∧ (∀i∈{0..x}. c i = c' i)))›*) last_idx_eq (*‹(c'::nat ⇒ nat) (k'::nat) = (i::nat)› ‹c k = i› ‹(c'::nat ⇒ nat) (k'::nat) = (c::nat ⇒ nat) (k::nat)›*) by (metis One_nat_def (*‹(1::nat) = Suc (0::nat)›*) Suc_pred (*‹(0::nat) < (?n::nat) ⟹ Suc (?n - Suc (0::nat)) = ?n›*) ‹0 < k'› ‹0 < k› atLeastAtMost_iff (*‹((?i::?'a) ∈ {?l::?'a..?u::?'a}) = (?l ≤ ?i ∧ ?i ≤ ?u)›*) le_Suc_eq (*‹((?m::nat) ≤ Suc (?n::nat)) = (?m ≤ ?n ∨ ?m = Suc ?n)›*))\n qed\n qed (insert IH (*‹∀m<n::nat. (0::nat) < m ⟶ (∀x::nat. m = (∑i::nat = 0::nat..x. fib ((c::nat ⇒ nat) i)) ⟶ inc_seq_on c {0::nat..x - (1::nat)} ⟶ (∀i::nat∈{0::nat..x}. (2::nat) ≤ c i) ⟶ (∀xa::nat. m = (∑i::nat = 0::nat..xa. fib ((c'::nat ⇒ nat) i)) ⟶ inc_seq_on c' {0::nat..xa - (1::nat)} ⟶ (∀i::nat∈{0::nat..xa}. (2::nat) ≤ c' i) ⟶ x = xa ∧ (∀i::nat∈{0::nat..x}. c i = c' i)))› ‹(0::nat) < (n::nat)› ‹(n::nat) = (∑i::nat = 0::nat..k::nat. fib ((c::nat ⇒ nat) i))› ‹inc_seq_on (c::nat ⇒ nat) {0::nat..(k::nat) - (1::nat)}› ‹∀i::nat∈{0::nat..k::nat}. (2::nat) ≤ (c::nat ⇒ nat) i› ‹(n::nat) = (∑i::nat = 0::nat..k'::nat. fib ((c'::nat ⇒ nat) i))› ‹inc_seq_on (c'::nat ⇒ nat) {0::nat..(k'::nat) - (1::nat)}› ‹∀i::nat∈{0::nat..k'::nat}. (2::nat) ≤ (c'::nat ⇒ nat) i›*) one_unique_fib_sum (*‹⟦inc_seq_on (?c::nat ⇒ nat) {0::nat..(?k::nat) - (1::nat)}; ∀i::nat∈{0::nat..?k}. (2::nat) ≤ ?c i⟧ ⟹ ((∑i::nat = 0::nat..?k. fib (?c i)) = (1::nat)) = (?k = (0::nat) ∧ ?c (0::nat) = (2::nat))›*), auto)\n (*solves the remaining goals:\n 1. ‹n = 0 ⟹ k = k' ∧ (∀i∈{0..k}. c i = c' i)›\n 2. ‹n = 1 ⟹ k = k' ∧ (∀i∈{0..k}. c i = c' i)›*)\n qed\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Zeckendorf/Zeckendorf.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"2eb81b20926ee2f654efb524698512c9a896d70fa7f4c965eb04e60d0087ee61\"\n}"}}},{"rowIdx":158,"cells":{"text":{"kind":"string","value":"✐‹creator \"Kevin Kappelmann\"›\nsection ‹Transport For Functions›\nsubsection ‹Basic Setup›\ntheory Transport_Functions_Base\n imports\n Monotone_Function_Relator\n Transport_Base\nbegin\n\nparagraph ‹Summary›\ntext ‹Basic setup for closure proofs. We introduce locales for the syntax,\nthe dependent relator, the non-dependent relator, the monotone dependent relator,\nand the monotone non-dependent relator.›\n\n\ndefinition \"flip2 f x1 x2 x3 x4 ≡ f x2 x1 x4 x3\"\n\nlemma flip2_eq: \"flip2 f x1 x2 x3 x4 = f x2 x1 x4 x3\"\n unfolding flip2_def\n (*goal: ‹(f::'b ⇒ 'c ⇒ 'd ⇒ 'e ⇒ 'a) (x2::'b) (x1::'c) (x4::'d) (x3::'e) = f x2 x1 x4 x3›*)\n by simp\n\nlemma flip2_eq_rel_inv [simp]: \"flip2 R x y = (R y x)¯\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹flip2 (R::'c ⇒ 'd ⇒ 'b ⇒ 'a ⇒ bool) (x::'d) (y::'c) = (R y x)¯›*)\n by (simp only: flip2_eq (*‹flip2 ?f ?x1.0 ?x2.0 ?x3.0 ?x4.0 = ?f ?x2.0 ?x1.0 ?x4.0 ?x3.0›*) rel_inv_iff_rel (*‹?R¯ ?x ?y = ?R ?y ?x›*))\n\nlemma flip2_flip2_eq_self [simp]: \"flip2 (flip2 f) = f\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹flip2 (flip2 (f::'a::type ⇒ 'b::type ⇒ 'c::type ⇒ 'd::type ⇒ 'e::type)) = f›*)\n by (simp add: flip2_eq (*‹flip2 ?f ?x1.0 ?x2.0 ?x3.0 ?x4.0 = ?f ?x2.0 ?x1.0 ?x4.0 ?x3.0›*))\n\nlemma flip2_eq_flip2_iff_eq [iff]: \"flip2 f = flip2 g ⟷ f = g\"\n unfolding flip2_def\n (*goal: ‹((λx1 x2 x3 x4. f x2 x1 x4 x3) = (λx1 x2 x3 x4. g x2 x1 x4 x3)) = (f = g)›*)\n apply (intro iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*) ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goals:\n 1. ‹⋀x xa xb xc. (λx1 x2 x3 x4. f x2 x1 x4 x3) = (λx1 x2 x3 x4. g x2 x1 x4 x3) ⟹ f x xa xb xc = g x xa xb xc›\n 2. ‹⋀x1 x2 x3 x4. f = g ⟹ f x2 x1 x4 x3 = g x2 x1 x4 x3›\n discuss goal 1*)\n apply ((auto dest: fun_cong (*‹?f = ?g ⟹ ?f ?x = ?g ?x›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: fun_cong (*‹(?f::?'a::type ⇒ ?'b::type) = (?g::?'a::type ⇒ ?'b::type) ⟹ ?f (?x::?'a::type) = ?g ?x›*))[1])\n (*proven 2 subgoals*) .\n\n\nparagraph ‹Dependent Function Relator›\n\nlocale transport_Dep_Fun_Rel_syntax =\n t1 : transport L1 R1 l1 r1 +\n dfro1 : hom_Dep_Fun_Rel_orders L1 L2 +\n dfro2 : hom_Dep_Fun_Rel_orders R1 R2\n for L1 :: \"'a1 ⇒ 'a1 ⇒ bool\"\n and R1 :: \"'a2 ⇒ 'a2 ⇒ bool\"\n and l1 :: \"'a1 ⇒ 'a2\"\n and r1 :: \"'a2 ⇒ 'a1\"\n and L2 :: \"'a1 ⇒ 'a1 ⇒ 'b1 ⇒ 'b1 ⇒ bool\"\n and R2 :: \"'a2 ⇒ 'a2 ⇒ 'b2 ⇒ 'b2 ⇒ bool\"\n and l2 :: \"'a2 ⇒ 'a1 ⇒ 'b1 ⇒ 'b2\"\n and r2 :: \"'a1 ⇒ 'a2 ⇒ 'b2 ⇒ 'b1\"\nbegin\n\nnotation L1 (infix \"≤⇘L1⇙\" 50)\nnotation R1 (infix \"≤⇘R1⇙\" 50)\n\nnotation t1.ge_left (infix \"≥⇘L1⇙\" 50)\nnotation t1.ge_right (infix \"≥⇘R1⇙\" 50)\n\nnotation t1.left_Galois (infix \"⇘L1⇙⪅\" 50)\nnotation t1.ge_Galois_left (infix \"⪆⇘L1⇙\" 50)\nnotation t1.right_Galois (infix \"⇘R1⇙⪅\" 50)\nnotation t1.ge_Galois_right (infix \"⪆⇘R1⇙\" 50)\nnotation t1.right_ge_Galois (infix \"⇘R1⇙⪆\" 50)\nnotation t1.Galois_right (infix \"⪅⇘R1⇙\" 50)\nnotation t1.left_ge_Galois (infix \"⇘L1⇙⪆\" 50)\nnotation t1.Galois_left (infix \"⪅⇘L1⇙\" 50)\n\nnotation t1.unit (\"η₁\")\nnotation t1.counit (\"ε₁\")\n\nnotation L2 (\"(≤⇘L2 (_) (_)⇙)\" 50)\nnotation R2 (\"(≤⇘R2 (_) (_)⇙)\" 50)\n\nnotation dfro1.right_infix (\"(_) ≤⇘L2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\nnotation dfro2.right_infix (\"(_) ≤⇘R2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\n\nnotation dfro1.o.ge_right (\"(≥⇘L2 (_) (_)⇙)\" 50)\nnotation dfro2.o.ge_right (\"(≥⇘R2 (_) (_)⇙)\" 50)\n\nnotation dfro1.ge_right_infix (\"(_) ≥⇘L2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\nnotation dfro2.ge_right_infix (\"(_) ≥⇘R2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\n\nnotation l2 (\"l2⇘(_) (_)⇙\")\nnotation r2 (\"r2⇘(_) (_)⇙\")\n\nsublocale t2 : transport \"(≤⇘L2 x (r1 x')⇙)\" \"(≤⇘R2 (l1 x) x'⇙)\" \"l2⇘x' x⇙\" \"r2⇘x x'⇙\" for x x' .\n\nnotation t2.left_Galois (\"(⇘L2 (_) (_)⇙⪅)\" 50)\nnotation t2.right_Galois (\"(⇘R2 (_) (_)⇙⪅)\" 50)\n\nabbreviation \"left2_Galois_infix y x x' y' ≡ (⇘L2 x x'⇙⪅) y y'\"\nnotation left2_Galois_infix (\"(_) ⇘L2 (_) (_)⇙⪅ (_)\" [51,51,51,51] 50)\nabbreviation \"right2_Galois_infix y' x x' y ≡ (⇘R2 x x'⇙⪅) y' y\"\nnotation right2_Galois_infix (\"(_) ⇘R2 (_) (_)⇙⪅ (_)\" [51,51,51,51] 50)\n\nnotation t2.ge_Galois_left (\"(⪆⇘L2 (_) (_)⇙)\" 50)\nnotation t2.ge_Galois_right (\"(⪆⇘R2 (_) (_)⇙)\" 50)\n\nabbreviation (input) \"ge_Galois_left_left2_infix y' x x' y ≡ (⪆⇘L2 x x'⇙) y' y\"\nnotation ge_Galois_left_left2_infix (\"(_) ⪆⇘L2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\nabbreviation (input) \"ge_Galois_left_right2_infix y x x' y' ≡ (⪆⇘R2 x x'⇙) y y'\"\nnotation ge_Galois_left_right2_infix (\"(_) ⪆⇘R2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\n\nnotation t2.right_ge_Galois (\"(⇘R2 (_) (_)⇙⪆)\" 50)\nnotation t2.left_ge_Galois (\"(⇘L2 (_) (_)⇙⪆)\" 50)\n\nabbreviation \"left2_ge_Galois_left_infix y x x' y' ≡ (⇘L2 x x'⇙⪆) y y'\"\nnotation left2_ge_Galois_left_infix (\"(_) ⇘L2 (_) (_)⇙⪆ (_)\" [51,51,51,51] 50)\nabbreviation \"right2_ge_Galois_left_infix y' x x' y ≡ (⇘R2 x x'⇙⪆) y' y\"\nnotation right2_ge_Galois_left_infix (\"(_) ⇘R2 (_) (_)⇙⪆ (_)\" [51,51,51,51] 50)\n\nnotation t2.Galois_right (\"(⪅⇘R2 (_) (_)⇙)\" 50)\nnotation t2.Galois_left (\"(⪅⇘L2 (_) (_)⇙)\" 50)\n\nabbreviation (input) \"Galois_left2_infix y' x x' y ≡ (⪅⇘L2 x x'⇙) y' y\"\nnotation Galois_left2_infix (\"(_) ⪅⇘L2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\nabbreviation (input) \"Galois_right2_infix y x x' y' ≡ (⪅⇘R2 x x'⇙) y y'\"\nnotation Galois_right2_infix (\"(_) ⪅⇘R2 (_) (_)⇙ (_)\" [51,51,51,51] 50)\n\nabbreviation \"t2_unit x x' ≡ t2.unit x' x\"\nnotation t2_unit (\"η⇘2 (_) (_)⇙\")\nabbreviation \"t2_counit x x' ≡ t2.counit x' x\"\nnotation t2_counit (\"ε⇘2 (_) (_)⇙\")\n\nend\n\nlocale transport_Dep_Fun_Rel =\n transport_Dep_Fun_Rel_syntax L1 R1 l1 r1 L2 R2 l2 r2\n for L1 :: \"'a1 ⇒ 'a1 ⇒ bool\"\n and R1 :: \"'a2 ⇒ 'a2 ⇒ bool\"\n and l1 :: \"'a1 ⇒ 'a2\"\n and r1 :: \"'a2 ⇒ 'a1\"\n and L2 :: \"'a1 ⇒ 'a1 ⇒ 'b1 ⇒ 'b1 ⇒ bool\"\n and R2 :: \"'a2 ⇒ 'a2 ⇒ 'b2 ⇒ 'b2 ⇒ bool\"\n and l2 :: \"'a2 ⇒ 'a1 ⇒ 'b1 ⇒ 'b2\"\n and r2 :: \"'a1 ⇒ 'a2 ⇒ 'b2 ⇒ 'b1\"\nbegin\n\ndefinition \"L ≡ (x1 x2 ∷ (≤⇘L1⇙)) ⇛ (≤⇘L2 x1 x2⇙)\"\n\nlemma left_rel_eq_Dep_Fun_Rel: \"L = ((x1 x2 ∷ (≤⇘L1⇙)) ⇛ (≤⇘L2 x1 x2⇙))\"\n unfolding L_def\n (*goal: ‹Dep_Fun_Rel (≤⇘L1⇙) L2 = Dep_Fun_Rel (≤⇘L1⇙) L2›*)\n by standard\n\ndefinition \"l ≡ ((x' : r1) ↝ l2 x')\"\n\nlemma left_eq_dep_fun_map: \"l = ((x' : r1) ↝ l2 x')\"\n unfolding l_def\n (*goal: ‹dep_fun_map r1 l2 = dep_fun_map r1 l2›*)\n by standard\n\nlemma left_eq [simp]: \"l f x' = l2⇘x' (r1 x')⇙ (f (r1 x'))\"\n unfolding left_eq_dep_fun_map\n (*goal: ‹dep_fun_map r1 l2 f x' = l2⇘x' r1 x'⇙ (f (r1 x'))›*)\n by simp\n\ncontext\nbegin\n\ninterpretation flip : transport_Dep_Fun_Rel R1 L1 r1 l1 R2 L2 r2 l2 .\n\nabbreviation \"R ≡ flip.L\"\nabbreviation \"r ≡ flip.l\"\n\nlemma right_rel_eq_Dep_Fun_Rel: \"R = ((x1' x2' ∷ (≤⇘R1⇙)) ⇛ (≤⇘R2 x1' x2'⇙))\"\n unfolding flip.L_def\n (*goal: ‹Dep_Fun_Rel (≤⇘R1⇙) R2 = Dep_Fun_Rel (≤⇘R1⇙) R2›*)\n by standard\n\nlemma right_eq_dep_fun_map: \"r = ((x : l1) ↝ r2 x)\"\n unfolding flip.l_def\n (*goal: ‹dep_fun_map l1 r2 = dep_fun_map l1 r2›*)\n by standard\n\nend\n\nlemma right_eq [simp]: \"r g x = r2⇘x (l1 x)⇙ (g (l1 x))\"\n unfolding right_eq_dep_fun_map\n (*goal: ‹dep_fun_map (l1::'a1::type ⇒ 'a2::type) r2 (g::'a2::type ⇒ 'b2::type) (x::'a1::type) = r2⇘x l1 x⇙ (g (l1 x))›*)\n by simp\n\nlemmas transport_defs = left_rel_eq_Dep_Fun_Rel left_eq_dep_fun_map\n right_rel_eq_Dep_Fun_Rel right_eq_dep_fun_map\n\nsublocale transport L R l r .\n\n(*FIXME: somehow the notation for the fixed parameters L and R, defined in\nOrder_Functions_Base.thy, is lost. We hence re-declare it here.*)\nnotation L (infix \"≤⇘L⇙\" 50)\nnotation R (infix \"≤⇘R⇙\" 50)\n\nlemma left_relI [intro]:\n assumes \"⋀x1 x2. x1 ≤⇘L1⇙ x2 ⟹ f x1 ≤⇘L2 x1 x2⇙ f' x2\"\n shows \"f ≤⇘L⇙ f'\"\n unfolding left_rel_eq_Dep_Fun_Rel\n (*goal: ‹Dep_Fun_Rel (≤⇘L1⇙) L2 f f'›*)\n using assms (*‹?x1.0 ≤⇘L1⇙ ?x2.0 ⟹ f ?x1.0 ≤⇘L2 ?x1.0 ?x2.0⇙ f' ?x2.0›*) by blast\n\nlemma left_relE [elim]:\n assumes \"f ≤⇘L⇙ f'\"\n and \"x1 ≤⇘L1⇙ x2\"\n obtains \"f x1 ≤⇘L2 x1 x2⇙ f' x2\"\n using assms (*‹f ≤⇘L⇙ f'› ‹x1 ≤⇘L1⇙ x2›*) unfolding left_rel_eq_Dep_Fun_Rel\n (*goal: ‹(f x1 ≤⇘L2 x1 x2⇙ f' x2 ⟹ thesis) ⟹ thesis›*)\n by blast\n\ninterpretation flip_inv :\n transport_Dep_Fun_Rel \"(≥⇘R1⇙)\" \"(≥⇘L1⇙)\" r1 l1 \"flip2 R2\" \"flip2 L2\" r2 l2 .\n\nlemma flip_inv_right_eq_ge_left: \"flip_inv.R = (≥⇘L⇙)\"\n unfolding left_rel_eq_Dep_Fun_Rel flip_inv.right_rel_eq_Dep_Fun_Rel\n (*goal: ‹Dep_Fun_Rel (≤⇘L1⇙)¯ (flip2 L2) = (Dep_Fun_Rel (≤⇘L1⇙) L2)¯›*)\n by (simp only: rel_inv_Dep_Fun_Rel_rel_eq (*‹(Dep_Fun_Rel ?R ?S)¯ = ((y x ∷ ?R¯) ⇛ (?S x y)¯)›*) flip2_eq_rel_inv[symmetric, of \"L2\"] (*‹(≤⇘L2 ?y ?x⇙)¯ = flip2 L2 ?x ?y›*))\n\ninterpretation flip : transport_Dep_Fun_Rel R1 L1 r1 l1 R2 L2 r2 l2 .\n\nlemma flip_inv_left_eq_ge_right: \"flip_inv.L ≡ (≥⇘R⇙)\"\n unfolding flip.flip_inv_right_eq_ge_left\n (*goal: ‹flip.L¯ ≡ flip.L¯›*) .\n\n\nsubparagraph ‹Useful Rewritings for Dependent Relation›\n\nlemma left_rel2_unit_eqs_left_rel2I:\n assumes \"⋀x1 x2. x1 ≤⇘L1⇙ x2 ⟹ (≤⇘L2 x2 x2⇙) ≤ (≤⇘L2 x1 x2⇙)\"\n and \"⋀x. x ≤⇘L1⇙ x ⟹ (≤⇘L2 (η₁ x) x⇙) ≤ (≤⇘L2 x x⇙)\"\n and \"⋀x1 x2. x1 ≤⇘L1⇙ x2 ⟹ (≤⇘L2 x1 x1⇙) ≤ (≤⇘L2 x1 x2⇙)\"\n and \"⋀x. x ≤⇘L1⇙ x ⟹ (≤⇘L2 x (η₁ x)⇙) ≤ (≤⇘L2 x x⇙)\"\n and \"x ≤⇘L1⇙ x\"\n and \"x ≡⇘L1⇙ η₁ x\"\n shows \"(≤⇘L2 (η₁ x) x⇙) = (≤⇘L2 x x⇙)\"\n and \"(≤⇘L2 x (η₁ x)⇙) = (≤⇘L2 x x⇙)\"\n using assms (*‹?x1.0 ≤⇘L1⇙ ?x2.0 ⟹ (≤⇘L2 ?x2.0 ?x2.0⇙) ≤ (≤⇘L2 ?x1.0 ?x2.0⇙)› ‹(?x::'a1) ≤⇘L1⇙ ?x ⟹ (≤⇘L2 η₁ ?x ?x⇙) ≤ (≤⇘L2 ?x ?x⇙)› ‹?x1.0 ≤⇘L1⇙ ?x2.0 ⟹ (≤⇘L2 ?x1.0 ?x1.0⇙) ≤ (≤⇘L2 ?x1.0 ?x2.0⇙)› ‹?x ≤⇘L1⇙ ?x ⟹ (≤⇘L2 ?x η₁ ?x⇙) ≤ (≤⇘L2 ?x ?x⇙)› ‹x ≤⇘L1⇙ x› ‹x ≡⇘(≤⇘L1⇙)⇙ η₁ x›*) apply -\n (*goals:\n 1. ‹⟦⋀(x1::'a1) x2::'a1. x1 ≤⇘L1⇙ x2 ⟹ (≤⇘L2 x2 x2⇙) ≤ (≤⇘L2 x1 x2⇙); ⋀x::'a1. x ≤⇘L1⇙ x ⟹ (≤⇘L2 η₁ x x⇙) ≤ (≤⇘L2 x x⇙); ⋀(x1::'a1) x2::'a1. x1 ≤⇘L1⇙ x2 ⟹ (≤⇘L2 x1 x1⇙) ≤ (≤⇘L2 x1 x2⇙); ⋀x::'a1. x ≤⇘L1⇙ x ⟹ (≤⇘L2 x η₁ x⇙) ≤ (≤⇘L2 x x⇙); (x::'a1) ≤⇘L1⇙ x; x ≡⇘(≤⇘L1⇙)⇙ η₁ x⟧ ⟹ (≤⇘L2 η₁ x x⇙) = (≤⇘L2 x x⇙)›\n 2. ‹⟦⋀(x1::'a1) x2::'a1. x1 ≤⇘L1⇙ x2 ⟹ (≤⇘L2 x2 x2⇙) ≤ (≤⇘L2 x1 x2⇙); ⋀x::'a1. x ≤⇘L1⇙ x ⟹ (≤⇘L2 η₁ x x⇙) ≤ (≤⇘L2 x x⇙); ⋀(x1::'a1) x2::'a1. x1 ≤⇘L1⇙ x2 ⟹ (≤⇘L2 x1 x1⇙) ≤ (≤⇘L2 x1 x2⇙); ⋀x::'a1. x ≤⇘L1⇙ x ⟹ (≤⇘L2 x η₁ x⇙) ≤ (≤⇘L2 x x⇙); (x::'a1) ≤⇘L1⇙ x; x ≡⇘(≤⇘L1⇙)⇙ η₁ x⟧ ⟹ (≤⇘L2 x η₁ x⇙) = (≤⇘L2 x x⇙)›\n discuss goal 1*)\n apply ((auto intro!: antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: antisym (*‹⟦(?a::?'a) ≤ (?b::?'a); ?b ≤ ?a⟧ ⟹ ?a = ?b›*))[1])\n (*proven 2 subgoals*) .\n\nlemma left2_eq_if_bi_related_if_monoI:\n assumes mono_L2: \"((x1 x2 ∷ (≥⇘L1⇙)) ⇛⇩m (x3 x4 ∷ (≤⇘L1⇙) | x1 ≤⇘L1⇙ x3) ⇛ (≤)) L2\"\n and \"x1 ≤⇘L1⇙ x2\"\n and \"x1 ≡⇘L1⇙ x3\"\n and \"x2 ≡⇘L1⇙ x4\"\n and trans_L1: \"transitive (≤⇘L1⇙)\"\n shows \"(≤⇘L2 x1 x2⇙) = (≤⇘L2 x3 x4⇙)\"\nproof (intro antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹(≤⇘L2 x1::'a1 x2::'a1⇙) ≤ (≤⇘L2 x3::'a1 x4::'a1⇙)›\n 2. ‹(≤⇘L2 x3::'a1 x4::'a1⇙) ≤ (≤⇘L2 x1::'a1 x2::'a1⇙)›*)\n from ‹x1 ≡⇘L1⇙ x3› (*‹x1 ≡⇘(≤⇘L1⇙)⇙ x3›*) ‹x2 ≡⇘L1⇙ x4› (*‹x2 ≡⇘(≤⇘L1⇙)⇙ x4›*) have \"x3 ≤⇘L1⇙ x1\" \"x2 ≤⇘L1⇙ x4\"\n apply -\n (*goals:\n 1. ‹⟦x1 ≡⇘(≤⇘L1⇙)⇙ x3; x2 ≡⇘(≤⇘L1⇙)⇙ x4⟧ ⟹ x3 ≤⇘L1⇙ x1›\n 2. ‹⟦x1 ≡⇘(≤⇘L1⇙)⇙ x3; x2 ≡⇘(≤⇘L1⇙)⇙ x4⟧ ⟹ x2 ≤⇘L1⇙ x4›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n with ‹x1 ≤⇘L1⇙ x2› (*‹x1 ≤⇘L1⇙ x2›*) mono_L2 (*‹((x1 x2 ∷ (≤⇘L1⇙)¯) ⇛⇩m (x3 x4 ∷ (≤⇘L1⇙)) ⇛ x1 ≤⇘L1⇙ x3 ⟶ (≤)) L2›*) show \"(≤⇘L2 x1 x2⇙) ≤ (≤⇘L2 x3 x4⇙)\"\n by blast\n from ‹x1 ≡⇘L1⇙ x3› (*‹(x1::'a1) ≡⇘(≤⇘L1⇙)⇙ (x3::'a1)›*) ‹x2 ≡⇘L1⇙ x4› (*‹x2 ≡⇘(≤⇘L1⇙)⇙ x4›*) have \"x1 ≤⇘L1⇙ x3\" \"x4 ≤⇘L1⇙ x2\"\n apply -\n (*goals:\n 1. ‹⟦x1 ≡⇘(≤⇘L1⇙)⇙ x3; x2 ≡⇘(≤⇘L1⇙)⇙ x4⟧ ⟹ x1 ≤⇘L1⇙ x3›\n 2. ‹⟦x1 ≡⇘(≤⇘L1⇙)⇙ x3; x2 ≡⇘(≤⇘L1⇙)⇙ x4⟧ ⟹ x4 ≤⇘L1⇙ x2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n moreover from ‹x3 ≤⇘L1⇙ x1› (*‹(x3::'a1) ≤⇘L1⇙ (x1::'a1)›*) ‹x1 ≤⇘L1⇙ x2› (*‹x1 ≤⇘L1⇙ x2›*) ‹x2 ≤⇘L1⇙ x4› (*‹x2 ≤⇘L1⇙ x4›*) have \"x3 ≤⇘L1⇙ x4\"\n using trans_L1 (*‹transitive (≤⇘L1⇙)›*) by blast\n ultimately show \"(≤⇘L2 x3 x4⇙) ≤ (≤⇘L2 x1 x2⇙)\"\n using mono_L2 (*‹((x1 x2 ∷ (≤⇘L1⇙)¯) ⇛⇩m (x3 x4 ∷ (≤⇘L1⇙)) ⇛ x1 ≤⇘L1⇙ x3 ⟶ (≤)) L2›*) by blast\nqed\n\nend\n\nparagraph ‹Function Relator›\n\nlocale transport_Fun_Rel_syntax =\n tdfrs : transport_Dep_Fun_Rel_syntax L1 R1 l1 r1 \"λ_ _. L2\" \"λ_ _. R2\"\n \"λ_ _. l2\" \"λ_ _. r2\"\n for L1 :: \"'a1 ⇒ 'a1 ⇒ bool\"\n and R1 :: \"'a2 ⇒ 'a2 ⇒ bool\"\n and l1 :: \"'a1 ⇒ 'a2\"\n and r1 :: \"'a2 ⇒ 'a1\"\n and L2 :: \"'b1 ⇒ 'b1 ⇒ bool\"\n and R2 :: \"'b2 ⇒ 'b2 ⇒ bool\"\n and l2 :: \"'b1 ⇒ 'b2\"\n and r2 :: \"'b2 ⇒ 'b1\"\nbegin\n\nnotation L1 (infix \"≤⇘L1⇙\" 50)\nnotation R1 (infix \"≤⇘R1⇙\" 50)\n\nnotation tdfrs.t1.ge_left (infix \"≥⇘L1⇙\" 50)\nnotation tdfrs.t1.ge_right (infix \"≥⇘R1⇙\" 50)\n\nnotation tdfrs.t1.left_Galois (infix \"⇘L1⇙⪅\" 50)\nnotation tdfrs.t1.ge_Galois_left (infix \"⪆⇘L1⇙\" 50)\nnotation tdfrs.t1.right_Galois (infix \"⇘R1⇙⪅\" 50)\nnotation tdfrs.t1.ge_Galois_right (infix \"⪆⇘R1⇙\" 50)\nnotation tdfrs.t1.right_ge_Galois (infix \"⇘R1⇙⪆\" 50)\nnotation tdfrs.t1.Galois_right (infix \"⪅⇘R1⇙\" 50)\nnotation tdfrs.t1.left_ge_Galois (infix \"⇘L1⇙⪆\" 50)\nnotation tdfrs.t1.Galois_left (infix \"⪅⇘L1⇙\" 50)\n\nnotation tdfrs.t1.unit (\"η₁\")\nnotation tdfrs.t1.counit (\"ε₁\")\n\nnotation L2 (infix \"≤⇘L2⇙\" 50)\nnotation R2 (infix \"≤⇘R2⇙\" 50)\n\nnotation tdfrs.t2.ge_left (infix \"≥⇘L2⇙\" 50)\nnotation tdfrs.t2.ge_right (infix \"≥⇘R2⇙\" 50)\n\nnotation tdfrs.t2.left_Galois (infix \"⇘L2⇙⪅\" 50)\nnotation tdfrs.t2.ge_Galois_left (infix \"⪆⇘L2⇙\" 50)\nnotation tdfrs.t2.right_Galois (infix \"⇘R2⇙⪅\" 50)\nnotation tdfrs.t2.ge_Galois_right (infix \"⪆⇘R2⇙\" 50)\nnotation tdfrs.t2.right_ge_Galois (infix \"⇘R2⇙⪆\" 50)\nnotation tdfrs.t2.Galois_right (infix \"⪅⇘R2⇙\" 50)\nnotation tdfrs.t2.left_ge_Galois (infix \"⇘L2⇙⪆\" 50)\nnotation tdfrs.t2.Galois_left (infix \"⪅⇘L2⇙\" 50)\n\nnotation tdfrs.t2.unit (\"η₂\")\nnotation tdfrs.t2.counit (\"ε₂\")\n\nend\n\nlocale transport_Fun_Rel =\n transport_Fun_Rel_syntax L1 R1 l1 r1 L2 R2 l2 r2 +\n tdfr : transport_Dep_Fun_Rel L1 R1 l1 r1 \"λ_ _. L2\" \"λ_ _. R2\"\n \"λ_ _. l2\" \"λ_ _. r2\"\n for L1 :: \"'a1 ⇒ 'a1 ⇒ bool\"\n and R1 :: \"'a2 ⇒ 'a2 ⇒ bool\"\n and l1 :: \"'a1 ⇒ 'a2\"\n and r1 :: \"'a2 ⇒ 'a1\"\n and L2 :: \"'b1 ⇒ 'b1 ⇒ bool\"\n and R2 :: \"'b2 ⇒ 'b2 ⇒ bool\"\n and l2 :: \"'b1 ⇒ 'b2\"\n and r2 :: \"'b2 ⇒ 'b1\"\nbegin\n\n(*FIXME: we have to repeat the Galois syntax here since tdfr already contains\na Galois instance, blocking a galois sublocale interpretation here*)\nnotation tdfr.L (\"L\")\nnotation tdfr.R (\"R\")\n\nabbreviation \"l ≡ tdfr.l\"\nabbreviation \"r ≡ tdfr.r\"\n\nnotation tdfr.L (infix \"≤⇘L⇙\" 50)\nnotation tdfr.R (infix \"≤⇘R⇙\" 50)\n\nnotation tdfr.ge_left (infix \"≥⇘L⇙\" 50)\nnotation tdfr.ge_right (infix \"≥⇘R⇙\" 50)\n\nnotation tdfr.left_Galois (infix \"⇘L⇙⪅\" 50)\nnotation tdfr.ge_Galois_left (infix \"⪆⇘L⇙\" 50)\nnotation tdfr.right_Galois (infix \"⇘R⇙⪅\" 50)\nnotation tdfr.ge_Galois_right (infix \"⪆⇘R⇙\" 50)\nnotation tdfr.right_ge_Galois (infix \"⇘R⇙⪆\" 50)\nnotation tdfr.Galois_right (infix \"⪅⇘R⇙\" 50)\nnotation tdfr.left_ge_Galois (infix \"⇘L⇙⪆\" 50)\nnotation tdfr.Galois_left (infix \"⪅⇘L⇙\" 50)\n\nnotation tdfr.unit (\"η\")\nnotation tdfr.counit (\"ε\")\n\nlemma left_rel_eq_Fun_Rel: \"(≤⇘L⇙) = ((≤⇘L1⇙) ⇛ (≤⇘L2⇙))\"\n by (urule tdfr.left_rel_eq_Dep_Fun_Rel)\n\nlemma left_eq_fun_map: \"l = (r1 ↝ l2)\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹l = (r1 ↝ l2)›*)\n by simp\n\ninterpretation flip : transport_Fun_Rel R1 L1 r1 l1 R2 L2 r2 l2 .\n\nlemma right_rel_eq_Fun_Rel: \"(≤⇘R⇙) = ((≤⇘R1⇙) ⇛ (≤⇘R2⇙))\"\n unfolding flip.left_rel_eq_Fun_Rel\n (*goal: ‹((≤⇘R1⇙) ⇛ (≤⇘R2⇙)) = ((≤⇘R1⇙) ⇛ (≤⇘R2⇙))›*)\n by standard\n\nlemma right_eq_fun_map: \"r = (l1 ↝ r2)\"\n unfolding flip.left_eq_fun_map\n (*goal: ‹(l1 ↝ r2) = (l1 ↝ r2)›*)\n by standard\n\nlemmas transport_defs = left_rel_eq_Fun_Rel right_rel_eq_Fun_Rel\n left_eq_fun_map right_eq_fun_map\n\nend\n\n\nparagraph ‹Monotone Dependent Function Relator›\n\nlocale transport_Mono_Dep_Fun_Rel =\n transport_Dep_Fun_Rel_syntax L1 R1 l1 r1 L2 R2 l2 r2\n + tdfr : transport_Dep_Fun_Rel L1 R1 l1 r1 L2 R2 l2 r2\n for L1 :: \"'a1 ⇒ 'a1 ⇒ bool\"\n and R1 :: \"'a2 ⇒ 'a2 ⇒ bool\"\n and l1 :: \"'a1 ⇒ 'a2\"\n and r1 :: \"'a2 ⇒ 'a1\"\n and L2 :: \"'a1 ⇒ 'a1 ⇒ 'b1 ⇒ 'b1 ⇒ bool\"\n and R2 :: \"'a2 ⇒ 'a2 ⇒ 'b2 ⇒ 'b2 ⇒ bool\"\n and l2 :: \"'a2 ⇒ 'a1 ⇒ 'b1 ⇒ 'b2\"\n and r2 :: \"'a1 ⇒ 'a2 ⇒ 'b2 ⇒ 'b1\"\nbegin\n\ndefinition \"L ≡ tdfr.L⇧⊕\"\n\nlemma left_rel_eq_tdfr_left_Refl_Rel: \"L = tdfr.L⇧⊕\"\n unfolding L_def\n (*goal: ‹tdfr.L⇧⊕ = tdfr.L⇧⊕›*)\n by standard\n\nlemma left_rel_eq_Mono_Dep_Fun_Rel: \"L = ((x1 x2 ∷ (≤⇘L1⇙)) ⇛⊕ (≤⇘L2 x1 x2⇙))\"\n unfolding left_rel_eq_tdfr_left_Refl_Rel tdfr.left_rel_eq_Dep_Fun_Rel\n (*goal: ‹(Dep_Fun_Rel (≤⇘L1⇙) L2)⇧⊕ = (Dep_Fun_Rel (≤⇘L1⇙) L2)⇧⊕›*)\n by simp\n\nlemma left_rel_eq_tdfr_left_rel_if_reflexive_on:\n assumes \"reflexive_on (in_field tdfr.L) tdfr.L\"\n shows \"L = tdfr.L\"\n unfolding left_rel_eq_tdfr_left_Refl_Rel\n (*goal: ‹tdfr.L⇧⊕ = tdfr.L›*)\n using assms (*‹reflexive_on (in_field tdfr.L) tdfr.L›*) by (rule Refl_Rel_eq_self_if_reflexive_on (*‹reflexive_on (in_field ?R) ?R ⟹ ?R⇧⊕ = ?R›*))\n\nabbreviation \"l ≡ tdfr.l\"\n\nlemma left_eq_tdfr_left: \"l = tdfr.l\" by standard\n\ninterpretation flip : transport_Mono_Dep_Fun_Rel R1 L1 r1 l1 R2 L2 r2 l2 .\n\nabbreviation \"R ≡ flip.L\"\n\nlemma right_rel_eq_tdfr_right_Refl_Rel: \"R = tdfr.R⇧⊕\"\n unfolding flip.left_rel_eq_tdfr_left_Refl_Rel\n (*goal: ‹flip.tdfr.L⇧⊕ = flip.tdfr.L⇧⊕›*)\n by standard\n\nlemma right_rel_eq_Mono_Dep_Fun_Rel: \"R = ((y1 y2 ∷ (≤⇘R1⇙)) ⇛⊕ (≤⇘R2 y1 y2⇙))\"\n unfolding flip.left_rel_eq_Mono_Dep_Fun_Rel\n (*goal: ‹(Dep_Fun_Rel (≤⇘R1⇙) R2)⇧⊕ = (Dep_Fun_Rel (≤⇘R1⇙) R2)⇧⊕›*)\n by standard\n\nlemma right_rel_eq_tdfr_right_rel_if_reflexive_on:\n assumes \"reflexive_on (in_field tdfr.R) tdfr.R\"\n shows \"R = tdfr.R\"\n using assms (*‹reflexive_on (in_field flip.tdfr.L) flip.tdfr.L›*) by (rule flip.left_rel_eq_tdfr_left_rel_if_reflexive_on (*‹reflexive_on (in_field flip.tdfr.L) flip.tdfr.L ⟹ R = flip.tdfr.L›*))\n\nabbreviation \"r ≡ tdfr.r\"\n\nlemma right_eq_tdfr_right: \"r = tdfr.r\" by standard\n\nlemmas transport_defs = left_rel_eq_tdfr_left_Refl_Rel\n right_rel_eq_tdfr_right_Refl_Rel\n\nsublocale transport L R l r .\n\n(*FIXME: somehow the notation for the fixed parameters L and R, defined in\nOrder_Functions_Base.thy, is lost. We hence re-declare it here.*)\nnotation L (infix \"≤⇘L⇙\" 50)\nnotation R (infix \"≤⇘R⇙\" 50)\n\nend\n\n\nparagraph ‹Monotone Function Relator›\n\nlocale transport_Mono_Fun_Rel =\n transport_Fun_Rel_syntax L1 R1 l1 r1 L2 R2 l2 r2 +\n tfr : transport_Fun_Rel L1 R1 l1 r1 L2 R2 l2 r2 +\n tpdfr : transport_Mono_Dep_Fun_Rel L1 R1 l1 r1 \"λ_ _. L2\" \"λ_ _. R2\"\n \"λ_ _. l2\" \"λ_ _. r2\"\n for L1 :: \"'a1 ⇒ 'a1 ⇒ bool\"\n and R1 :: \"'a2 ⇒ 'a2 ⇒ bool\"\n and l1 :: \"'a1 ⇒ 'a2\"\n and r1 :: \"'a2 ⇒ 'a1\"\n and L2 :: \"'b1 ⇒ 'b1 ⇒ bool\"\n and R2 :: \"'b2 ⇒ 'b2 ⇒ bool\"\n and l2 :: \"'b1 ⇒ 'b2\"\n and r2 :: \"'b2 ⇒ 'b1\"\nbegin\n\n(*FIXME: we have to repeat the Galois syntax here since tdfr already contains\na Galois instance, blocking a galois sublocale interpretation here*)\nnotation tpdfr.L (\"L\")\nnotation tpdfr.R (\"R\")\n\nabbreviation \"l ≡ tpdfr.l\"\nabbreviation \"r ≡ tpdfr.r\"\n\nnotation tpdfr.L (infix \"≤⇘L⇙\" 50)\nnotation tpdfr.R (infix \"≤⇘R⇙\" 50)\n\nnotation tpdfr.ge_left (infix \"≥⇘L⇙\" 50)\nnotation tpdfr.ge_right (infix \"≥⇘R⇙\" 50)\n\nnotation tpdfr.left_Galois (infix \"⇘L⇙⪅\" 50)\nnotation tpdfr.ge_Galois_left (infix \"⪆⇘L⇙\" 50)\nnotation tpdfr.right_Galois (infix \"⇘R⇙⪅\" 50)\nnotation tpdfr.ge_Galois_right (infix \"⪆⇘R⇙\" 50)\nnotation tpdfr.right_ge_Galois (infix \"⇘R⇙⪆\" 50)\nnotation tpdfr.Galois_right (infix \"⪅⇘R⇙\" 50)\nnotation tpdfr.left_ge_Galois (infix \"⇘L⇙⪆\" 50)\nnotation tpdfr.Galois_left (infix \"⪅⇘L⇙\" 50)\n\nnotation tpdfr.unit (\"η\")\nnotation tpdfr.counit (\"ε\")\n\nlemma left_rel_eq_Mono_Fun_Rel: \"(≤⇘L⇙) = ((≤⇘L1⇙) ⇛⊕ (≤⇘L2⇙))\"\n unfolding tpdfr.left_rel_eq_Mono_Dep_Fun_Rel\n (*goal: ‹((x y ∷ (≤⇘L1⇙)) ⇛ (≤⇘L2⇙))⇧⊕ = ((x y ∷ (≤⇘L1⇙)) ⇛ (≤⇘L2⇙))⇧⊕›*)\n by simp\n\nlemma left_eq_fun_map: \"l = (r1 ↝ l2)\"\n unfolding tfr.left_eq_fun_map\n (*goal: ‹((r1::'a2 ⇒ 'a1) ↝ (l2::'b1 ⇒ 'b2)) = (r1 ↝ l2)›*)\n by standard\n\ninterpretation flip : transport_Mono_Fun_Rel R1 L1 r1 l1 R2 L2 r2 l2 .\n\nlemma right_rel_eq_Mono_Fun_Rel: \"(≤⇘R⇙) = ((≤⇘R1⇙) ⇛⊕ (≤⇘R2⇙))\"\n unfolding flip.left_rel_eq_Mono_Fun_Rel\n (*goal: ‹((x y ∷ (≤⇘R1⇙)) ⇛ (≤⇘R2⇙))⇧⊕ = ((x y ∷ (≤⇘R1⇙)) ⇛ (≤⇘R2⇙))⇧⊕›*)\n by standard\n\nlemma right_eq_fun_map: \"r = (l1 ↝ r2)\"\n unfolding flip.left_eq_fun_map\n (*goal: ‹(l1 ↝ r2) = (l1 ↝ r2)›*)\n by standard\n\nlemmas transport_defs = tpdfr.transport_defs\n\nend\n\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Transport/Transport/Functions/Transport_Functions_Base.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"7f5fd31aadb52ff63e82d905d5ae0497436cf086782514d6fb9e71de9c42ed9d\"\n}"}}},{"rowIdx":159,"cells":{"text":{"kind":"string","value":"(* Title: statecharts/DataSpace/Update.thy\n\n Author: Steffen Helke, Software Engineering Group\n Copyright 2010 Technische Universitaet Berlin\n*)\n\nsection ‹Update-Functions on Data Spaces›\ntheory Update\nimports Data\nbegin\n\nsubsection ‹Total update-functions›\n\ndefinition\n Update :: \"(('d data) => ('d data)) => bool\" where \n \"Update U = (∀ d. Data.DataSpace d = DataSpace (U d))\"\n\nlemma Update_EmptySet: \n \"(% d. d) ∈ { L | L. Update L}\" \napply (unfold Update_def (*‹Update ?U = (∀d. Data.DataSpace d = Data.DataSpace (?U d))›*))\n(*goal: ‹(λd. d) ∈ {L |L. Update L}›*)\nby auto\n\ndefinition\n \"update = { L | (L::(('d data) => ('d data))). Update L}\"\n\ntypedef 'd update = \"update :: ('d data => 'd data) set\"\n unfolding update_def\n apply (rule exI)\n apply (rule Update_EmptySet)\n done\n \ndefinition\n UpdateApply :: \"['d update, 'd data] => 'd data\" (\"(_ !!!/ _)\" [10,11]10) where\n \"UpdateApply U D == Rep_update U D\" \n\ndefinition\n DefaultUpdate :: \"('d update)\" where\n \"DefaultUpdate == Abs_update (λ D. D)\"\n\nsubsubsection ‹Basic lemmas›\n\nlemma Update_select:\n \"Update (Rep_update U)\"\napply (cut_tac x=U in Rep_update (*‹Rep_update ?x ∈ update›*))\n(*goal: ‹Update (Rep_update (U::'a update))›*)\napply (unfold update_def (*‹update = {L |L. Update L}›*))\n(*goal: ‹Rep_update U ∈ update ⟹ Update (Rep_update U)›*)\nby auto\n\nlemma DataSpace_DataSpace_Update [simp]:\n \"Data.DataSpace (Rep_update U DP) = Data.DataSpace DP\"\napply (cut_tac U=U in Update_select (*‹Update (Rep_update ?U)›*))\n(*goal: ‹Data.DataSpace (Rep_update U DP) = Data.DataSpace DP›*)\napply (unfold Update_def (*‹Update ?U = (∀d. Data.DataSpace d = Data.DataSpace (?U d))›*))\n(*goal: ‹Update (Rep_update (U::'a update)) ⟹ Data.DataSpace (Rep_update U (DP::'a data)) = Data.DataSpace DP›*)\nby auto\n\nsubsubsection ‹‹DefaultUpdate››\n\nlemma Update_DefaultUpdate [simp]:\n \"Update (λ D. D)\"\napply (unfold Update_def (*‹Update ?U = (∀d. Data.DataSpace d = Data.DataSpace (?U d))›*))\n(*goal: ‹Update (λD. D)›*)\nby auto\n\nlemma update_DefaultUpdate [simp]:\n \"(λ D. D) ∈ update\"\napply (unfold update_def (*‹update = {L |L. Update L}›*))\n(*goal: ‹(λD. D) ∈ update›*)\nby auto\n\nlemma DataSpace_UpdateApply [simp]:\n \"Data.DataSpace (U !!! D) = Data.DataSpace D\"\napply (unfold UpdateApply_def (*‹?U !!! ?D ≡ Rep_update ?U ?D›*))\n(*goal: ‹Data.DataSpace (U !!! D) = Data.DataSpace D›*)\nby auto\n\nsubsection ‹Partial update-functions›\n\ndefinition\n PUpdate :: \"(('d data) => ('d pdata)) => bool\" where\n \"PUpdate U = (∀ d. Data.DataSpace d = PDataSpace (U d))\"\n\nlemma PUpdate_EmptySet:\n \"(% d. Data2PData d) ∈ { L | L. PUpdate L}\" \napply (unfold PUpdate_def (*‹PUpdate (?U::?'d data ⇒ ?'d pdata) = (∀d::?'d data. Data.DataSpace d = PDataSpace (?U d))›*))\n(*goal: ‹Data2PData ∈ {L |L::'a data ⇒ 'a pdata. PUpdate L}›*)\nby auto\n\ndefinition \"pupdate = { L | (L::(('d data) => ('d pdata))). PUpdate L}\"\n\ntypedef 'd pupdate = \"pupdate :: ('d data => 'd pdata) set\" \n unfolding pupdate_def\n apply (rule exI)\n apply (rule PUpdate_EmptySet)\n done\n \ndefinition\n PUpdateApply :: \"['d pupdate, 'd data] => 'd pdata\" (\"(_ !!/ _)\" [10,11]10) where\n \"PUpdateApply U D = Rep_pupdate U D\" \n\ndefinition\n DefaultPUpdate :: \"('d pupdate)\" where\n \"DefaultPUpdate = Abs_pupdate (λ D. DefaultPData (Data.DataSpace D))\"\n \nsubsubsection ‹Basic lemmas›\n\nlemma PUpdate_select:\n \"PUpdate (Rep_pupdate U)\"\napply (cut_tac x=U in Rep_pupdate (*‹Rep_pupdate ?x ∈ pupdate›*))\n(*goal: ‹PUpdate (Rep_pupdate (U::'a pupdate))›*)\napply (unfold pupdate_def (*‹pupdate = {L |L. PUpdate L}›*))\n(*goal: ‹Rep_pupdate U ∈ pupdate ⟹ PUpdate (Rep_pupdate U)›*)\nby auto\n\nlemma DataSpace_PDataSpace_PUpdate [simp]:\n \"PDataSpace (Rep_pupdate U DP) = Data.DataSpace DP\"\napply (cut_tac U=U in PUpdate_select (*‹PUpdate (Rep_pupdate (?U::?'a::type pupdate))›*))\n(*goal: ‹PDataSpace (Rep_pupdate (U::'a pupdate) (DP::'a data)) = Data.DataSpace DP›*)\napply (unfold PUpdate_def (*‹PUpdate (?U::?'d data ⇒ ?'d pdata) = (∀d::?'d data. Data.DataSpace d = PDataSpace (?U d))›*))\n(*goal: ‹PUpdate (Rep_pupdate U) ⟹ PDataSpace (Rep_pupdate U DP) = Data.DataSpace DP›*)\nby auto\n\nsubsubsection ‹‹Data2PData››\n\nlemma PUpdate_Data2PData [simp]: \n \"PUpdate Data2PData\"\napply (unfold PUpdate_def (*‹PUpdate (?U::?'d data ⇒ ?'d pdata) = (∀d::?'d data. Data.DataSpace d = PDataSpace (?U d))›*))\n(*goal: ‹PUpdate Data2PData›*)\nby auto\n\nlemma pupdate_Data2PData [simp]:\n \"Data2PData ∈ pupdate\"\napply (unfold pupdate_def (*‹pupdate = {L |L. PUpdate L}›*))\n(*goal: ‹Data2PData ∈ pupdate›*)\nby auto\n\nsubsubsection ‹‹PUpdate››\n\nlemma PUpdate_DefaultPUpdate [simp]:\n \"PUpdate (λ D. DefaultPData (Data.DataSpace D))\"\napply (unfold PUpdate_def (*‹PUpdate (?U::?'d data ⇒ ?'d pdata) = (∀d::?'d data. Data.DataSpace d = PDataSpace (?U d))›*))\n(*goal: ‹PUpdate (λD. DefaultPData (Data.DataSpace D))›*)\nby auto\n\nlemma pupdate_DefaultPUpdate [simp]:\n \"(λ D. DefaultPData (Data.DataSpace D)) ∈ pupdate\"\napply (unfold pupdate_def (*‹pupdate = {L |L. PUpdate L}›*))\n(*goal: ‹(λD. DefaultPData (Data.DataSpace D)) ∈ pupdate›*)\nby auto\n\nlemma DefaultPUpdate_None [simp]:\n \"(DefaultPUpdate !! D) = DefaultPData (DataSpace D)\"\napply (unfold DefaultPUpdate_def (*‹DefaultPUpdate = Abs_pupdate (λD. DefaultPData (Data.DataSpace D))›*) PUpdateApply_def (*‹(?U !! ?D) = Rep_pupdate ?U ?D›*))\n(*goal: ‹(DefaultPUpdate !! D) = DefaultPData (Data.DataSpace D)›*)\napply (subst Abs_pupdate_inverse (*‹?y ∈ pupdate ⟹ Rep_pupdate (Abs_pupdate ?y) = ?y›*))\n(*goals:\n 1. ‹(λD. DefaultPData (Data.DataSpace D)) ∈ pupdate›\n 2. ‹DefaultPData (Data.DataSpace D) = DefaultPData (Data.DataSpace D)›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\nsubsubsection ‹‹SequentialRacing››\n\ndefinition\n UpdateOverride :: \"['d pupdate, 'd update] => \n 'd update\" (\"(_ [U+]/ _)\" [10,11]10) where\n \"UpdateOverride U P = Abs_update (λ DA . (U !! DA) [D+] (P !!! DA))\"\n \n\n(* -------------------------------------------------------------- *)\n(* We use our own FoldSet operator simular to the definition *)\n(* of Isabelle 2002. Note, it is different to the definition *)\n(* in Isabelle 2009. Basically we express \"f (g x)\" by \"h x\" *)\n(* -------------------------------------------------------------- *)\n\ninductive\n FoldSet :: \"('b => 'a => 'a) => 'a => 'b set => 'a => bool\"\n for h :: \"'b => 'a => 'a\"\n and z :: 'a\nwhere\n emptyI [intro]: \"FoldSet h z {} z\"\n| insertI [intro]:\n \"⟦ x ∉ A; FoldSet h z A y ⟧\n ⟹ FoldSet h z (insert x A) (h x y)\"\n\ndefinition\nSequentialRacing :: \"('d pupdate set) => ('d update set)\" where\n \"SequentialRacing U = \n {u. FoldSet UpdateOverride DefaultUpdate U u}\"\n\nlemma FoldSet_imp_finite:\n \"FoldSet h z A x ⟹ finite A\"\napply (induct set: FoldSet)\n(*goals:\n 1. ‹finite {}›\n 2. ‹⋀x A y. ⟦x ∉ A; FoldSet h z A y; finite A⟧ ⟹ finite (insert x A)›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\nlemma finite_imp_FoldSet:\n \"finite A ⟹ ∃ x. FoldSet h z A x\"\napply (induct set: finite)\n(*goals:\n 1. ‹∃a. FoldSet h z {} a›\n 2. ‹⋀x F. ⟦finite F; x ∉ F; ∃a. FoldSet h z F a⟧ ⟹ ∃a. FoldSet h z (insert x F) a›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\nlemma finite_SequentialRacing:\n \"finite US ⟹ (SOME u. u ∈ SequentialRacing US) ∈ SequentialRacing US\"\napply (unfold SequentialRacing_def (*‹SequentialRacing ?U = {u. FoldSet UpdateOverride DefaultUpdate ?U u}›*))\n(*goal: ‹finite US ⟹ (SOME u. u ∈ SequentialRacing US) ∈ SequentialRacing US›*)\napply auto\n(*goal: ‹finite US ⟹ (SOME u. u ∈ Collect (FoldSet UpdateOverride DefaultUpdate US)) ∈ Collect (FoldSet UpdateOverride DefaultUpdate US)›*)\napply (drule_tac h=UpdateOverride and z=DefaultUpdate in finite_imp_FoldSet (*‹finite (?A::?'a set) ⟹ ∃x::?'b. FoldSet (?h::?'a ⇒ ?'b ⇒ ?'b) (?z::?'b) ?A x›*))\n(*goal: ‹finite US ⟹ FoldSet UpdateOverride DefaultUpdate US (Eps (FoldSet UpdateOverride DefaultUpdate US))›*)\napply auto\n(*goal: ‹∃x. FoldSet UpdateOverride DefaultUpdate US x ⟹ FoldSet UpdateOverride DefaultUpdate US (Eps (FoldSet UpdateOverride DefaultUpdate US))›*)\napply (rule someI (*‹?P ?x ⟹ ?P (Eps ?P)›*))\n(*goal: ‹⋀x. FoldSet UpdateOverride DefaultUpdate US x ⟹ FoldSet UpdateOverride DefaultUpdate US (Eps (FoldSet UpdateOverride DefaultUpdate US))›*)\nby auto\n\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Statecharts/Update.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"f71a6947463fd50f6b244c9ab3b616b62c5bd991ef2618ad402421a9cf145b39\"\n}"}}},{"rowIdx":160,"cells":{"text":{"kind":"string","value":"section ‹Storjohann's basis reduction algorithm (concrete implementation)›\n\ntext ‹We refine the abstract algorithm into a more efficient executable one.›\n\ntheory Storjohann_Impl\n imports \n Storjohann\nbegin\n\nsubsection ‹Implementation›\n\ntext ‹We basically store four components:\n ▪ The $f$-basis (as list, all values taken modulo $p$)\n ▪ The $d\\mu$-matrix (as nested arrays, all values taken modulo $d_id_{i+1}p$)\n ▪ The $d$-values (as array)\n ▪ The modulo-values $d_id_{i+1}p$ (as array)\n›\n\ntype_synonym state_impl = \"int vec list × int iarray iarray × int iarray × int iarray\" \n\nfun di_of :: \"state_impl ⇒ int iarray\" where\n \"di_of (mfsi, dmui, di, mods) = di\" \n\ncontext LLL\nbegin\n\nfun state_impl_inv :: \"_ ⇒ _ ⇒ _ ⇒ state_impl ⇒ bool\" where \n \"state_impl_inv p mfs dmu (mfsi, dmui, di, mods) = (mfsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m)\n ∧ dmui = IArray.of_fun (λ i. IArray.of_fun (λ j. dmu $$ (i,j)) i) m\n ∧ mods = IArray.of_fun (λ j. p * di !! j * di !! (Suc j)) (m - 1))\" \n\ndefinition state_iso_inv :: \"(int × int) iarray ⇒ int iarray ⇒ bool\" where\n \"state_iso_inv prods di = (prods = IArray.of_fun \n (λ i. (di !! (i+1) * di !! (i+1), di !! (i+2) * di !! i)) (m - 1))\" \n\ndefinition perform_add_row :: \"int ⇒ state_impl ⇒ nat ⇒ nat ⇒ int ⇒ int iarray ⇒ int ⇒ int ⇒ state_impl\" where\n \"perform_add_row p state i j c rowi muij dij1 = (let\n (mfsi, dmui, di, mods) = state;\n fsj = mfsi ! j;\n rowj = dmui !! j\n in \n (case split_at i mfsi of (start, fsi # end) ⇒ start @ vec n (λ k. (fsi $ k - c * fsj $ k) symmod p) # end,\n IArray.of_fun (λ ii. if i = ii then \n IArray.of_fun (λ jj. if jj < j then \n (rowi !! jj - c * rowj !! jj) symmod (mods !! jj)\n else if jj = j then muij - c * dij1 \n else rowi !! jj) i\n else dmui !! ii) m, \n di, mods))\" \n\ndefinition LLL_add_row :: \"int ⇒ state_impl ⇒ nat ⇒ nat ⇒ state_impl\" where\n \"LLL_add_row p state i j = (let\n (_, dmui, di, _) = state;\n rowi = dmui !! i;\n dij1 = di !! (Suc j);\n muij = rowi !! j;\n c = round_num_denom muij dij1\n in if c = 0 then state\n else perform_add_row p state i j c rowi muij dij1)\" \n\n\ndefinition LLL_swap_row :: \"int ⇒ state_impl ⇒ nat ⇒ state_impl\" where\n \"LLL_swap_row p state k = (case state of (mfsi, dmui, di, mods) ⇒ let \n k1 = k - 1;\n kS1 = Suc k;\n muk = dmui !! k; \n muk1 = dmui !! k1;\n mukk1 = muk !! k1;\n dk1 = di !! k1;\n dkS1 = di !! kS1;\n dk = di !! k;\n dk' = (dkS1 * dk1 + mukk1 * mukk1) div dk;\n mod1 = p * dk1 * dk';\n modk = p * dk' * dkS1\n in \n (case split_at k1 mfsi\n of (start, fsk1 # fsk # end) ⇒ start @ fsk # fsk1 # end,\n IArray.of_fun (λ i. \n if i < k1 then dmui !! i\n else if i > k then \n let row_i = dmui !! i; muik = row_i !! k; muik1 = row_i !! k1 in IArray.of_fun \n (λ j. if j = k1 then ((mukk1 * muik1 + muik * dk1) div dk) symmod mod1 \n else if j = k then ((dkS1 * muik1 - mukk1 * muik) div dk) symmod modk \n else row_i !! j) i\n else if i = k then IArray.of_fun (λ j. if j = k1 then mukk1 symmod mod1 else muk1 !! j) i\n else IArray.of_fun ((!!) muk) i \n ) m, \n IArray.of_fun (λ i. if i = k then dk' else di !! i) (Suc m), \n IArray.of_fun (λ j. if j = k1 then mod1 else if j = k then modk else mods !! j) (m - 1)))\"\n\ndefinition perform_swap_add where \"perform_swap_add p state k k1 c row_k mukk1 dk = \n(let (fs, dmu, dd, mods) = state; \n row_k1 = dmu !! k1; \n kS1 = Suc k;\n mukk1' = mukk1 - c * dk;\n dk1 = dd !! k1; \n dkS1 = dd !! kS1; \n dk' = (dkS1 * dk1 + mukk1' * mukk1') div dk; \n mod1 = p * dk1 * dk'; \n modk = p * dk' * dkS1\n in \n (case split_at k1 fs of (start, fsk1 # fsk # end) ⇒ \n start @ vec n (λk. (fsk $ k - c * fsk1 $ k) symmod p) # fsk1 # end,\n IArray.of_fun\n (λi. if i < k1\n then dmu !! i\n else if k < i\n then let row_i = dmu !! i;\n muik1 = row_i !! k1;\n muik = row_i !! k\n in IArray.of_fun\n (λj. if j = k1 then (mukk1' * muik1 + muik * dk1) div dk symmod mod1\n else if j = k then (dkS1 * muik1 - mukk1' * muik) div dk symmod modk \n else row_i !! j)\n i\n else if i = k then IArray.of_fun (λj. if j = k1 then mukk1' symmod mod1 else row_k1 !! j) k \n else IArray.of_fun (λj. (row_k !! j - c * row_k1 !! j) symmod mods !! j) i)\n m,\n IArray.of_fun (λi. if i = k then dk' else dd !! i) (Suc m), \n IArray.of_fun (λj. if j = k1 then mod1 else if j = k then modk else mods !! j) (m - 1)))\" \n\n\ndefinition LLL_swap_add where\n \"LLL_swap_add p state i = (let\n i1 = i - 1;\n (_, dmui, di, _) = state;\n rowi = dmui !! i;\n dii = di !! i;\n muij = rowi !! i1;\n c = round_num_denom muij dii\n in if c = 0 then LLL_swap_row p state i\n else perform_swap_add p state i i1 c rowi muij dii)\"\n\ndefinition LLL_max_gso_norm_di :: \"bool ⇒ int iarray ⇒ rat × nat\" where\n \"LLL_max_gso_norm_di first di = \n (if first then (of_int (di !! 1), 0) \n else case max_list_rats_with_index (map (λ i. (di !! (Suc i), di !! i, i)) [0 ..< m ])\n of (num, denom, i) ⇒ (of_int num / of_int denom, i))\" \n\ndefinition LLL_max_gso_quot:: \"(int * int) iarray ⇒ (int * int * nat)\" where\n \"LLL_max_gso_quot di_prods = max_list_rats_with_index \n (map (λi. case di_prods !! i of (l,r) ⇒ (l, r, Suc i)) [0..<(m-1)])\"\n\n\ndefinition LLL_max_gso_norm :: \"bool ⇒ state_impl ⇒ rat × nat\" where\n \"LLL_max_gso_norm first state = (case state of (_, _, di, mods) ⇒ LLL_max_gso_norm_di first di)\" \n\ndefinition perform_adjust_mod :: \"int ⇒ state_impl ⇒ state_impl\" where\n \"perform_adjust_mod p state = (case state of (mfsi, dmui, di, _) ⇒ \n let mfsi' = map (map_vec (λx. x symmod p)) mfsi;\n mods = IArray.of_fun (λ j. p * di !! j * di !! (Suc j)) (m - 1);\n dmui' = IArray.of_fun (λ i. let row = dmui !! i in IArray.of_fun (λ j. row !! j symmod (mods !! j)) i) m\n in \n ((mfsi', dmui', di, mods)))\" \n\ndefinition mod_of_gso_norm :: \"bool ⇒ rat ⇒ int\" where\n \"mod_of_gso_norm first mn = log_base ^ (log_ceiling log_base (max 2 (\n root_rat_ceiling 2 (mn * (rat_of_nat (if first then 4 else m + 3))) + 1)))\"\n\ndefinition LLL_adjust_mod :: \"int ⇒ bool ⇒ state_impl ⇒ int × state_impl × nat\" where\n \"LLL_adjust_mod p first state = ( \n let (b', g_idx) = LLL_max_gso_norm first state;\n p' = mod_of_gso_norm first b'\n in if p' < p then (p', perform_adjust_mod p' state, g_idx)\n else (p, state, g_idx) \n )\" \n\ndefinition LLL_adjust_swap_add where\n \"LLL_adjust_swap_add p first state g_idx i = (\n let state1 = LLL_swap_add p state i\n in if i - 1 = g_idx then\n LLL_adjust_mod p first state1 else (p, state1, g_idx))\" \n\n\ndefinition LLL_step :: \"int ⇒ bool ⇒ state_impl ⇒ nat ⇒ nat ⇒ int ⇒ (int × state_impl × nat) × nat × int\" where\n \"LLL_step p first state g_idx i j = (if i = 0 then ((p, state, g_idx), Suc i, j)\n else let \n i1 = i - 1; \n iS = Suc i;\n (_, _, di, _) = state;\n (num, denom) = quotient_of α;\n d_i = di !! i;\n d_i1 = di !! i1;\n d_Si = di !! iS\n in if d_i * d_i * denom ≤ num * d_i1 * d_Si then\n ((p, state, g_idx), iS, j) \n else (LLL_adjust_swap_add p first state g_idx i, i1, j + 1))\" \n\npartial_function (tailrec) LLL_main :: \"int ⇒ bool ⇒ state_impl ⇒ nat ⇒ nat ⇒ int ⇒ int × state_impl\"\n where\n \"LLL_main p first state g_idx i (j :: int) = (\n (if i < m \n then case LLL_step p first state g_idx i j of\n ((p', state', g_idx'), i', j') ⇒ \n LLL_main p' first state' g_idx' i' j'\n else\n (p, state)))\"\n\npartial_function (tailrec) LLL_iso_main_inner where\n \"LLL_iso_main_inner p first state di_prods g_idx (j :: int) = (\n case state of (_, _, di, _) ⇒\n (\n (let (max_gso_num, max_gso_denum, indx) = LLL_max_gso_quot di_prods;\n (num, denum) = quotient_of α in\n (if max_gso_num * denum > num * max_gso_denum then\n case LLL_adjust_swap_add p first state g_idx indx of \n (p', state', g_idx') ⇒ case state' of (_, _, di', _) ⇒ \n let di_prods' = IArray.of_fun (λ i. case di_prods !! i of lr ⇒ \n if i > indx ∨ i + 2 < indx then lr\n else case lr of (l,r) \n ⇒ if i + 1 = indx then let d_idx = di' !! indx in (d_idx * d_idx, r) else (l, di' !! (i + 2) * di' !! i)) (m - 1)\n in LLL_iso_main_inner p' first state' di_prods' g_idx' (j + 1)\n else\n (p, state)))))\"\n\ndefinition LLL_iso_main where\n \"LLL_iso_main p first state g_idx j = (if m > 1 then \n case state of (_, _, di, _) ⇒\n let di_prods = IArray.of_fun (λ i. (di !! (i+1) * di !! (i+1), di !! (i+2) * di !! i)) (m - 1)\n in LLL_iso_main_inner p first state di_prods g_idx j else (p,state))\" \n\n\ndefinition LLL_initial :: \"bool ⇒ int × state_impl × nat\" where\n \"LLL_initial first = (let init = dμ_impl fs_init;\n di = IArray.of_fun (λ i. if i = 0 then 1 else let i1 = i - 1 in init !! i1 !! i1) (Suc m);\n (b,g_idx) = LLL_max_gso_norm_di first di;\n p = mod_of_gso_norm first b;\n mods = IArray.of_fun (λ j. p * di !! j * di !! (Suc j)) (m - 1);\n dmui = IArray.of_fun (λ i. let row = init !! i in IArray.of_fun (λ j. row !! j symmod (mods !! j)) i) m\n in (p, (compute_initial_mfs p, dmui, di, mods), g_idx))\" \n\nfun LLL_add_rows_loop where\n \"LLL_add_rows_loop p state i 0 = state\"\n| \"LLL_add_rows_loop p state i (Suc j) = (\n let state' = LLL_add_row p state i j\n in LLL_add_rows_loop p state' i j)\" \n\nprimrec LLL_add_rows_outer_loop where\n \"LLL_add_rows_outer_loop p state 0 = state\" |\n \"LLL_add_rows_outer_loop p state (Suc i) = \n (let state' = LLL_add_rows_outer_loop p state i in\n LLL_add_rows_loop p state' (Suc i) (Suc i))\"\n\ndefinition \n \"LLL_reduce_basis = (if m = 0 then [] else\n let first = False;\n (p0, state0, g_idx0) = LLL_initial first;\n (p, state) = LLL_main p0 first state0 g_idx0 0 0;\n (mfs,_,_,_) = LLL_add_rows_outer_loop p state (m - 1)\n in mfs)\"\n\ndefinition \n \"LLL_reduce_basis_iso = (if m = 0 then [] else\n let first = False;\n (p0, state0, g_idx0) = LLL_initial first;\n (p, state) = LLL_iso_main p0 first state0 g_idx0 0;\n (mfs,_,_,_) = LLL_add_rows_outer_loop p state (m - 1)\n in mfs)\"\n\ndefinition \n \"LLL_short_vector = (\n let first = True;\n (p0, state0, g_idx0) = LLL_initial first;\n (p, (mfs,_,_,_)) = LLL_main p0 first state0 g_idx0 0 0\n in hd mfs)\"\n\ndefinition \n \"LLL_short_vector_iso = (\n let first = True;\n (p0, state0, g_idx0) = LLL_initial first;\n (p, (mfs,_,_,_)) = LLL_iso_main p0 first state0 g_idx0 0 \n in hd mfs)\"\n\nend\n\ndeclare LLL.LLL_short_vector_def[code]\ndeclare LLL.LLL_short_vector_iso_def[code]\ndeclare LLL.LLL_reduce_basis_def[code]\ndeclare LLL.LLL_reduce_basis_iso_def[code]\ndeclare LLL.LLL_iso_main_def[code]\ndeclare LLL.LLL_iso_main_inner.simps[code]\ndeclare LLL.LLL_add_rows_outer_loop.simps[code]\ndeclare LLL.LLL_add_rows_loop.simps[code]\ndeclare LLL.LLL_initial_def[code]\ndeclare LLL.LLL_main.simps[code]\ndeclare LLL.LLL_adjust_mod_def[code]\ndeclare LLL.LLL_max_gso_norm_def[code]\ndeclare LLL.perform_adjust_mod_def[code]\ndeclare LLL.LLL_max_gso_norm_di_def[code]\ndeclare LLL.LLL_max_gso_quot_def[code]\ndeclare LLL.LLL_step_def[code]\ndeclare LLL.LLL_add_row_def[code]\ndeclare LLL.perform_add_row_def[code]\ndeclare LLL.LLL_swap_row_def[code]\ndeclare LLL.LLL_swap_add_def[code]\ndeclare LLL.LLL_adjust_swap_add_def[code]\ndeclare LLL.perform_swap_add_def[code]\ndeclare LLL.mod_of_gso_norm_def[code]\ndeclare LLL.compute_initial_mfs_def[code]\ndeclare LLL.log_base_def[code]\n\n\nsubsection ‹Towards soundness proof of implementation›\n\ncontext LLL\nbegin\nlemma perform_swap_add: assumes k: \"k ≠ 0\" \"k < m\" and fs: \"length fs = m\"\n shows \"LLL_swap_row p (perform_add_row p (fs, dmu, di, mods) k (k - 1) c (dmu !! k) (dmu !! k !! (k - 1)) (di !! k)) k\n = perform_swap_add p (fs, dmu, di, mods) k (k - 1) c (dmu !! k) (dmu !! k !! (k - 1)) (di !! k)\"\nproof (-)\n (*goal: ‹LLL_swap_row p (perform_add_row p (fs, dmu, di, mods) k (k - 1) c (dmu !! k) (dmu !! k !! (k - 1)) (di !! k)) k = perform_swap_add p (fs, dmu, di, mods) k (k - 1) c (dmu !! k) (dmu !! k !! (k - 1)) (di !! k)›*)\n from k[folded fs] (*‹k ≠ 0› ‹k < length fs›*) have drop: \"drop k fs = fs ! k # drop (Suc k) fs\"\n by (simp add: Cons_nth_drop_Suc (*‹?i < length ?xs ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*))\n obtain v where v: \"vec n (λka. (fs ! k $ ka - c * fs ! (k - 1) $ ka) symmod p) = v\"\n (*goal: ‹(⋀v. vec n (λka. (fs ! k $ ka - c * fs ! (k - 1) $ ka) symmod p) = v ⟹ thesis) ⟹ thesis›*)\n by auto\n from k[folded fs] (*‹k ≠ 0› ‹k < length fs›*) have drop1: \"drop (k - 1) (take k fs @ v # drop (Suc k) fs) = fs ! (k - 1) # v # drop (Suc k) fs\"\n apply (simp add: Cons_nth_drop_Suc (*‹(?i::nat) < length (?xs::?'a list) ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*))\n (*goal: ‹drop ((k::nat) - (1::nat)) (take k (fs::int vec list) @ (v::int vec) # drop (Suc k) fs) = fs ! (k - (1::nat)) # v # drop (Suc k) fs›*)\n by (smt Cons_nth_drop_Suc (*‹(?i::nat) < length (?xs::?'a::type list) ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*) Suc_diff_Suc (*‹(?n::nat) < (?m::nat) ⟹ Suc (?m - Suc ?n) = ?m - ?n›*) Suc_less_eq (*‹(Suc (?m::nat) < Suc (?n::nat)) = (?m < ?n)›*) Suc_pred (*‹(0::nat) < (?n::nat) ⟹ Suc (?n - Suc (0::nat)) = ?n›*) diff_Suc_less (*‹(0::nat) < (?n::nat) ⟹ ?n - Suc (?i::nat) < ?n›*) diff_self_eq_0 (*‹(?m::nat) - ?m = (0::nat)›*) drop_take (*‹drop (?n::nat) (take (?m::nat) (?xs::?'a::type list)) = take (?m - ?n) (drop ?n ?xs)›*) less_SucI (*‹(?m::nat) < (?n::nat) ⟹ ?m < Suc ?n›*) take_Suc_Cons (*‹take (Suc (?n::nat)) ((?x::?'a::type) # (?xs::?'a::type list)) = ?x # take ?n ?xs›*) take_eq_Nil (*‹(take (?n::nat) (?xs::?'a::type list) = []) = (?n = (0::nat) ∨ ?xs = [])›*))\n from k[folded fs] (*‹k ≠ 0› ‹k < length fs›*) have drop2: \"drop (k - 1) fs = fs ! (k - 1) # fs ! k # drop (Suc k) fs\"\n by (metis Cons_nth_drop_Suc (*‹?i < length ?xs ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*) One_nat_def (*‹1 = Suc 0›*) Suc_less_eq (*‹(Suc ?m < Suc ?n) = (?m < ?n)›*) Suc_pred (*‹0 < ?n ⟹ Suc (?n - Suc 0) = ?n›*) less_SucI (*‹?m < ?n ⟹ ?m < Suc ?n›*) neq0_conv (*‹(?n ≠ 0) = (0 < ?n)›*))\n have take: \"take (k - 1) (take k fs @ xs) = take (k - 1) fs\" for xs\n using k[folded fs] (*‹k ≠ 0› ‹k < length fs›*) by auto\n obtain rowk where rowk: \"IArray.of_fun\n (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj\n else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k = rowk\"\n (*goal: ‹(⋀rowk. IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k = rowk ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain mukk1' where mukk1': \"(di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k = mukk1'\"\n (*goal: ‹(⋀mukk1'. (di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k = mukk1' ⟹ thesis) ⟹ thesis›*)\n by auto\n have kk1: \"k - 1 < k\"\n using k (*‹(k::nat) ≠ (0::nat)› ‹k < m›*) by auto\n have mukk1'': \"(di !! Suc k * di !! (k - 1) +\n (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div\n di !! k = mukk1'\"\n unfolding mukk1'[symmetric] rowk[symmetric] IArray.of_fun_nth[OF kk1]\n (*goal: ‹(di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k = (di !! Suc k * di !! (k - 1) + (if k - 1 < k - 1 then (dmu !! k !! (k - 1) - c * dmu !! (k - 1) !! (k - 1)) symmod mods !! (k - 1) else if k - 1 = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! (k - 1)) * (if k - 1 < k - 1 then (dmu !! k !! (k - 1) - c * dmu !! (k - 1) !! (k - 1)) symmod mods !! (k - 1) else if k - 1 = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! (k - 1))) div di !! k›*)\n by auto\n have id: \"(k = k) = True\"\n by simp\n have rowk1: \"dmu !! k !! (k - 1) - c * di !! k = rowk !! (k - 1)\"\n unfolding rowk[symmetric] IArray.of_fun_nth[OF kk1]\n (*goal: ‹dmu !! k !! (k - 1) - c * di !! k = (if k - 1 < k - 1 then (dmu !! k !! (k - 1) - c * dmu !! (k - 1) !! (k - 1)) symmod mods !! (k - 1) else if k - 1 = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! (k - 1))›*)\n by simp\n show \"?thesis\"\n (*goal: ‹LLL_swap_row p (perform_add_row p (fs, dmu, di, mods) k (k - 1) c (dmu !! k) (dmu !! k !! (k - 1)) (di !! k)) k = perform_swap_add p (fs, dmu, di, mods) k (k - 1) c (dmu !! k) (dmu !! k !! (k - 1)) (di !! k)›*)\n unfolding perform_swap_add_def split perform_add_row_def Let_def split LLL_swap_row_def split_at_def\n (*goal: ‹(case drop (k - 1) (case drop k fs of fsi # end ⇒ take k fs @ vec n (λka. (fsi $ ka - c * fs ! (k - 1) $ ka) symmod p) # end) of fsk1 # fsk # end ⇒ take (k - 1) (case drop k fs of fsi # end ⇒ take k fs @ vec n (λka. (fsi $ ka - c * fs ! (k - 1) $ ka) symmod p) # end) @ fsk # fsk1 # end, IArray.of_fun (λi. if i < k - 1 then IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! i !! (k - 1) + IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1)) div di !! k)) else if j = k then (di !! Suc k * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! i !! (k - 1) - IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1)) div di !! k) * di !! Suc k) else IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1)) div di !! k)) else IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! (k - 1) !! j) i else IArray.of_fun ((!!) (IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k)) i) m, IArray.of_fun (λi. if i = k then (di !! Suc k * di !! (k - 1) + IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1)) div di !! k else di !! i) (Suc m), IArray.of_fun (λj. if j = k - 1 then p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1)) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - 1) + IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1) * IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmu !! k !! jj - c * dmu !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! jj) k else dmu !! ii) m !! k !! (k - 1)) div di !! k) * di !! Suc k else mods !! j) (m - 1)) = (case drop (k - 1) fs of fsk1 # fsk # end ⇒ take (k - 1) fs @ vec n (λk. (fsk $ k - c * fsk1 $ k) symmod p) # fsk1 # end, IArray.of_fun (λi. if i < k - 1 then dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k) * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else dmu !! (k - 1) !! j) k else IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i) m, IArray.of_fun (λi. if i = k then (di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k else di !! i) (Suc m), IArray.of_fun (λj. if j = k - 1 then p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k) * di !! Suc k else mods !! j) (m - 1))›*)\n unfolding drop list.simps v drop1 take prod.inject drop2 rowk IArray.of_fun_nth[OF ‹k < m›] id if_True\n (*goal: ‹take ((k::nat) - (1::nat)) (fs::int vec list) @ (v::int vec) # fs ! (k - (1::nat)) # drop (Suc k) fs = take (k - (1::nat)) fs @ v # fs ! (k - (1::nat)) # drop (Suc k) fs ∧ IArray.of_fun (λi::nat. if i < k - (1::nat) then IArray.of_fun (λii::nat. if k = ii then rowk::int iarray else (dmu::int iarray iarray) !! ii) (m::nat) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then (rowk !! (k - (1::nat)) * IArray.of_fun (λii::nat. if k = ii then rowk else dmu !! ii) m !! i !! (k - (1::nat)) + IArray.of_fun (λii::nat. if k = ii then rowk else dmu !! ii) m !! i !! k * (di::int iarray) !! (k - (1::nat))) div di !! k symmod ((p::int) * di !! (k - (1::nat)) * ((di !! Suc k * di !! (k - (1::nat)) + rowk !! (k - (1::nat)) * rowk !! (k - (1::nat))) div di !! k)) else if j = k then (di !! Suc k * IArray.of_fun (λii::nat. if k = ii then rowk else dmu !! ii) m !! i !! (k - (1::nat)) - rowk !! (k - (1::nat)) * IArray.of_fun (λii::nat. if k = ii then rowk else dmu !! ii) m !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - (1::nat)) + rowk !! (k - (1::nat)) * rowk !! (k - (1::nat))) div di !! k) * di !! Suc k) else IArray.of_fun (λii::nat. if k = ii then rowk else dmu !! ii) m !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then rowk !! (k - (1::nat)) symmod (p * di !! (k - (1::nat)) * ((di !! Suc k * di !! (k - (1::nat)) + rowk !! (k - (1::nat)) * rowk !! (k - (1::nat))) div di !! k)) else IArray.of_fun (λii::nat. if k = ii then rowk else dmu !! ii) m !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) rowk) i) m = IArray.of_fun (λi::nat. if i < k - (1::nat) then dmu !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu !! k !! (k - (1::nat)) - (c::int) * di !! k) * dmu !! i !! (k - (1::nat)) + dmu !! i !! k * di !! (k - (1::nat))) div di !! k symmod (p * di !! (k - (1::nat)) * ((di !! Suc k * di !! (k - (1::nat)) + (dmu !! k !! (k - (1::nat)) - c * di !! k) * (dmu !! k !! (k - (1::nat)) - c * di !! k)) div di !! k)) else if j = k then (di !! Suc k * dmu !! i !! (k - (1::nat)) - (dmu !! k !! (k - (1::nat)) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - (1::nat)) + (dmu !! k !! (k - (1::nat)) - c * di !! k) * (dmu !! k !! (k - (1::nat)) - c * di !! k)) div di !! k) * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then (dmu !! k !! (k - (1::nat)) - c * di !! k) symmod (p * di !! (k - (1::nat)) * ((di !! Suc k * di !! (k - (1::nat)) + (dmu !! k !! (k - (1::nat)) - c * di !! k) * (dmu !! k !! (k - (1::nat)) - c * di !! k)) div di !! k)) else dmu !! (k - (1::nat)) !! j) k else IArray.of_fun (λj::nat. (dmu !! k !! j - c * dmu !! (k - (1::nat)) !! j) symmod (mods::int iarray) !! j) i) m ∧ IArray.of_fun (λi::nat. if i = k then (di !! Suc k * di !! (k - (1::nat)) + rowk !! (k - (1::nat)) * rowk !! (k - (1::nat))) div di !! k else di !! i) (Suc m) = IArray.of_fun (λi::nat. if i = k then (di !! Suc k * di !! (k - (1::nat)) + (dmu !! k !! (k - (1::nat)) - c * di !! k) * (dmu !! k !! (k - (1::nat)) - c * di !! k)) div di !! k else di !! i) (Suc m) ∧ IArray.of_fun (λj::nat. if j = k - (1::nat) then p * di !! (k - (1::nat)) * ((di !! Suc k * di !! (k - (1::nat)) + rowk !! (k - (1::nat)) * rowk !! (k - (1::nat))) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - (1::nat)) + rowk !! (k - (1::nat)) * rowk !! (k - (1::nat))) div di !! k) * di !! Suc k else mods !! j) (m - (1::nat)) = IArray.of_fun (λj::nat. if j = k - (1::nat) then p * di !! (k - (1::nat)) * ((di !! Suc k * di !! (k - (1::nat)) + (dmu !! k !! (k - (1::nat)) - c * di !! k) * (dmu !! k !! (k - (1::nat)) - c * di !! k)) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - (1::nat)) + (dmu !! k !! (k - (1::nat)) - c * di !! k) * (dmu !! k !! (k - (1::nat)) - c * di !! k)) div di !! k) * di !! Suc k else mods !! j) (m - (1::nat))›*)\n unfolding rowk1\n (*goal: ‹take (k - 1) fs @ v # fs ! (k - 1) # drop (Suc k) fs = take (k - 1) fs @ v # fs ! (k - 1) # drop (Suc k) fs ∧ IArray.of_fun (λi. if i < k - 1 then IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (rowk !! (k - 1) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) + IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else if j = k then (di !! Suc k * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) - rowk !! (k - 1) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then rowk !! (k - 1) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i else IArray.of_fun ((!!) rowk) i) m = IArray.of_fun (λi. if i < k - 1 then dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (rowk !! (k - 1) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - rowk !! (k - 1) * dmu !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then rowk !! (k - 1) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else dmu !! (k - 1) !! j) k else IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i) m ∧ IArray.of_fun (λi. if i = k then (di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k else di !! i) (Suc m) = IArray.of_fun (λi. if i = k then (di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k else di !! i) (Suc m) ∧ IArray.of_fun (λj. if j = k - 1 then p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k else mods !! j) (m - 1) = IArray.of_fun (λj. if j = k - 1 then p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k else mods !! j) (m - 1)›*)\n apply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) refl (*‹?t = ?t›*) iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*))\n (*goal: ‹take (k - 1) fs @ v # fs ! (k - 1) # drop (Suc k) fs = take (k - 1) fs @ v # fs ! (k - 1) # drop (Suc k) fs ∧ IArray.of_fun (λi. if i < k - 1 then IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (rowk !! (k - 1) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) + IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else if j = k then (di !! Suc k * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) - rowk !! (k - 1) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then rowk !! (k - 1) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i else IArray.of_fun ((!!) rowk) i) m = IArray.of_fun (λi. if i < k - 1 then dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (rowk !! (k - 1) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - rowk !! (k - 1) * dmu !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then rowk !! (k - 1) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else dmu !! (k - 1) !! j) k else IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i) m ∧ IArray.of_fun (λi. if i = k then (di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k else di !! i) (Suc m) = IArray.of_fun (λi. if i = k then (di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k else di !! i) (Suc m) ∧ IArray.of_fun (λj. if j = k - 1 then p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k else mods !! j) (m - 1) = IArray.of_fun (λj. if j = k - 1 then p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) else if j = k then p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k else mods !! j) (m - 1)›*)\n apply (unfold rowk1[symmetric] (*‹rowk !! (k - 1) = dmu !! k !! (k - 1) - c * di !! k›*))\n (*goal: ‹⋀i. i < m ⟹ (if i < k - 1 then IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (rowk !! (k - 1) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) + IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else if j = k then (di !! Suc k * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) - rowk !! (k - 1) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then rowk !! (k - 1) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i else IArray.of_fun ((!!) rowk) i) = (if i < k - 1 then dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (rowk !! (k - 1) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - rowk !! (k - 1) * dmu !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k) * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then rowk !! (k - 1) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + rowk !! (k - 1) * rowk !! (k - 1)) div di !! k)) else dmu !! (k - 1) !! j) k else IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i)›*)\n proof (goal_cases)\n (*goal: ‹⋀i. i < m ⟹ (if i < k - 1 then IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) + IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else if j = k then (di !! Suc k * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k) * di !! Suc k) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i else IArray.of_fun ((!!) rowk) i) = (if i < k - 1 then dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k) * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else dmu !! (k - 1) !! j) k else IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i)›*)\n case i: (1 i) (*‹i < m›*)\n show \"?case\"\n (*goal: ‹(if i < k - 1 then IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) + IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else if j = k then (di !! Suc k * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k) * di !! Suc k) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i else IArray.of_fun ((!!) rowk) i) = (if i < k - 1 then dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k) * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * ((di !! Suc k * di !! (k - 1) + (dmu !! k !! (k - 1) - c * di !! k) * (dmu !! k !! (k - 1) - c * di !! k)) div di !! k)) else dmu !! (k - 1) !! j) k else IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i)›*)\n unfolding IArray.of_fun_nth[OF i] IArray.of_fun_nth[OF ‹k < m›] id if_True mukk1' mukk1'' rowk1[symmetric]\n (*goal: ‹(if i < k - 1 then if k = i then rowk else dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! (k - 1) + (if k = i then rowk else dmu !! i) !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * (if k = i then rowk else dmu !! i) !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else (if k = i then rowk else dmu !! i) !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i else IArray.of_fun ((!!) rowk) i) = (if i < k - 1 then dmu !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else dmu !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else dmu !! (k - 1) !! j) k else IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i)›*)\n proof (intro if_cong[OF refl] (*‹⟦?b ⟹ ?x = ?u; ¬ ?b ⟹ ?y = ?v⟧ ⟹ (if ?b then ?x else ?y) = (if ?b then ?u else ?v)›*), force, goal_cases)\n (*goals:\n 1. ‹⟦¬ i < k - 1; k < i⟧ ⟹ IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! (k - 1) + (if k = i then rowk else dmu !! i) !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * (if k = i then rowk else dmu !! i) !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else (if k = i then rowk else dmu !! i) !! j) i = IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else dmu !! i !! j) i›\n 2. ‹⟦¬ i < k - 1; ¬ k < i; i = k⟧ ⟹ IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i = IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else dmu !! (k - 1) !! j) k›\n 3. ‹⟦¬ i < k - 1; ¬ k < i; i ≠ k⟧ ⟹ IArray.of_fun ((!!) rowk) i = IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i›*)\n case 3 (*‹¬ (i::nat) < (k::nat) - (1::nat)› ‹¬ k < i› ‹(i::nat) ≠ (k::nat)›*)\n hence i: \"i = k - 1\"\n by auto\n show \"?case\"\n (*goal: ‹IArray.of_fun ((!!) rowk) i = IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) i›*)\n unfolding i\n (*goal: ‹IArray.of_fun ((!!) (rowk::int iarray)) ((k::nat) - (1::nat)) = IArray.of_fun (λj::nat. ((dmu::int iarray iarray) !! k !! j - (c::int) * dmu !! (k - (1::nat)) !! j) symmod (mods::int iarray) !! j) (k - (1::nat))›*)\n apply (intro iarray_cong[OF refl] (*‹(⋀i. i < ?m ⟹ ?f i = ?g i) ⟹ IArray.of_fun ?f ?m = IArray.of_fun ?g ?m›*))\n (*goal: ‹IArray.of_fun ((!!) rowk) (k - 1) = IArray.of_fun (λj. (dmu !! k !! j - c * dmu !! (k - 1) !! j) symmod mods !! j) (k - 1)›*)\n apply (unfold rowk[symmetric] (*‹(rowk::int iarray) = IArray.of_fun (λjj::nat. if jj < (k::nat) - (1::nat) then ((dmu::int iarray iarray) !! k !! jj - (c::int) * dmu !! (k - (1::nat)) !! jj) symmod (mods::int iarray) !! jj else if jj = k - (1::nat) then dmu !! k !! (k - (1::nat)) - c * (di::int iarray) !! k else dmu !! k !! jj) k›*))\n (*goal: ‹⋀i. i < k - 1 ⟹ rowk !! i = (dmu !! k !! i - c * dmu !! (k - 1) !! i) symmod mods !! i›*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⋀i. i < k - 1 ⟹ i < k›\n 2. ‹⋀i. i < k - 1 ⟹ (if i < k - 1 then (dmu !! k !! i - c * dmu !! (k - 1) !! i) symmod mods !! i else if i = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! i) = (dmu !! k !! i - c * dmu !! (k - 1) !! i) symmod mods !! i›\n discuss goal 1*)\n apply ((insert k (*‹k ≠ 0› ‹k < m›*))[1])\n (*top goal: ‹⋀i::nat. i < (k::nat) - (1::nat) ⟹ i < k› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert k (*‹k ≠ 0› ‹k < m›*))[1])\n (*goal: ‹⋀i. i < k - 1 ⟹ (if i < k - 1 then (dmu !! k !! i - c * dmu !! (k - 1) !! i) symmod mods !! i else if i = k - 1 then dmu !! k !! (k - 1) - c * di !! k else dmu !! k !! i) = (dmu !! k !! i - c * dmu !! (k - 1) !! i) symmod mods !! i›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n next\n (*goals:\n 1. ‹⟦¬ i < k - 1; k < i⟧ ⟹ IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! (k - 1) + (if k = i then rowk else dmu !! i) !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * (if k = i then rowk else dmu !! i) !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else (if k = i then rowk else dmu !! i) !! j) i = IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else dmu !! i !! j) i›\n 2. ‹⟦¬ i < k - 1; ¬ k < i; i = k⟧ ⟹ IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i = IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else dmu !! (k - 1) !! j) k›*)\n case ki: 1 (*‹¬ (i::nat) < (k::nat) - (1::nat)› ‹k < i›*)\n hence id: \"(k = i) = False\"\n by auto\n show \"?case\"\n (*goal: ‹IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! (k - 1) + (if k = i then rowk else dmu !! i) !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * (if k = i then rowk else dmu !! i) !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * (if k = i then rowk else dmu !! i) !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else (if k = i then rowk else dmu !! i) !! j) i = IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else dmu !! i !! j) i›*)\n unfolding id if_False rowk\n (*goal: ‹IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else dmu !! i !! j) i = IArray.of_fun (λj. if j = k - 1 then ((dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! (k - 1) + dmu !! i !! k * di !! (k - 1)) div di !! k symmod (p * di !! (k - 1) * mukk1') else if j = k then (di !! Suc k * dmu !! i !! (k - 1) - (dmu !! k !! (k - 1) - c * di !! k) * dmu !! i !! k) div di !! k symmod (p * mukk1' * di !! Suc k) else dmu !! i !! j) i›*)\n by (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) if_cong (*‹⟦?b = ?c; ?c ⟹ ?x = ?u; ¬ ?c ⟹ ?y = ?v⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*) refl (*‹?t = ?t›*))\n next\n (*goal: ‹⟦¬ i < k - 1; ¬ k < i; i = k⟧ ⟹ IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i = IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else dmu !! (k - 1) !! j) k›*)\n case 2 (*‹¬ i < k - 1› ‹¬ k < i› ‹(i::nat) = (k::nat)›*)\n show \"?case\"\n (*goal: ‹IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else IArray.of_fun (λii. if k = ii then rowk else dmu !! ii) m !! (k - 1) !! j) i = IArray.of_fun (λj. if j = k - 1 then (dmu !! k !! (k - 1) - c * di !! k) symmod (p * di !! (k - 1) * mukk1') else dmu !! (k - 1) !! j) k›*)\n unfolding \"2\"\n (*goal: ‹IArray.of_fun (λj::nat. if j = (k::nat) - (1::nat) then ((dmu::int iarray iarray) !! k !! (k - (1::nat)) - (c::int) * (di::int iarray) !! k) symmod ((p::int) * di !! (k - (1::nat)) * (mukk1'::int)) else IArray.of_fun (λii::nat. if k = ii then rowk::int iarray else dmu !! ii) (m::nat) !! (k - (1::nat)) !! j) k = IArray.of_fun (λj::nat. if j = k - (1::nat) then (dmu !! k !! (k - (1::nat)) - c * di !! k) symmod (p * di !! (k - (1::nat)) * mukk1') else dmu !! (k - (1::nat)) !! j) k›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) if_cong (*‹⟦?b = ?c; ?c ⟹ ?x = ?u; ¬ ?c ⟹ ?y = ?v⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λj::nat. if j = (k::nat) - (1::nat) then ((dmu::int iarray iarray) !! k !! (k - (1::nat)) - (c::int) * (di::int iarray) !! k) symmod ((p::int) * di !! (k - (1::nat)) * (mukk1'::int)) else IArray.of_fun (λii::nat. if k = ii then rowk::int iarray else dmu !! ii) (m::nat) !! (k - (1::nat)) !! j) k = IArray.of_fun (λj::nat. if j = k - (1::nat) then (dmu !! k !! (k - (1::nat)) - c * di !! k) symmod (p * di !! (k - (1::nat)) * mukk1') else dmu !! (k - (1::nat)) !! j) k›*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⋀i. ⟦i < k; i ≠ k - 1⟧ ⟹ k - 1 < m›\n 2. ‹⋀i. ⟦i < k; i ≠ k - 1⟧ ⟹ (if k = k - 1 then rowk else dmu !! (k - 1)) !! i = dmu !! (k - 1) !! i›\n discuss goal 1*)\n apply ((insert k (*‹(k::nat) ≠ (0::nat)› ‹(k::nat) < (m::nat)›*))[1])\n (*top goal: ‹⋀i. ⟦i < k; i ≠ k - 1⟧ ⟹ k - 1 < m› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert k (*‹(k::nat) ≠ (0::nat)› ‹(k::nat) < (m::nat)›*))[1])\n (*goal: ‹⋀i. ⟦i < k; i ≠ k - 1⟧ ⟹ (if k = k - 1 then rowk else dmu !! (k - 1)) !! i = dmu !! (k - 1) !! i›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n qed\n qed\nqed\n \n\nlemma LLL_swap_add_eq: assumes i: \"i ≠ 0\" \"i < m\" and fs: \"length fs = m\" \n shows \"LLL_swap_add p (fs,dmu,di,mods) i = (LLL_swap_row p (LLL_add_row p (fs,dmu,di,mods) i (i - 1)) i)\" \nproof (-)\n (*goal: ‹LLL_swap_add p (fs, dmu, di, mods) i = LLL_swap_row p (LLL_add_row p (fs, dmu, di, mods) i (i - 1)) i›*)\n define c where \"c = round_num_denom (dmu !! i !! (i - 1)) (di !! i)\"\n from i (*‹i ≠ 0› ‹i < m›*) have si1: \"Suc (i - 1) = i\"\n by auto\n note res1 = LLL_swap_add_def[of p \"(fs,dmu,di,mods)\" i, unfolded split Let_def c_def [ symmetric ]] (*‹LLL_swap_add p (fs, dmu, di, mods) i = (if c = 0 then LLL_swap_row p (fs, dmu, di, mods) i else perform_swap_add p (fs, dmu, di, mods) i (i - 1) c (dmu !! i) (dmu !! i !! (i - 1)) (di !! i))›*)\n show \"?thesis\"\n (*goal: ‹LLL_swap_add p (fs, dmu, di, mods) i = LLL_swap_row p (LLL_add_row p (fs, dmu, di, mods) i (i - 1)) i›*)\n proof (cases \"c = 0\")\n (*goals:\n 1. ‹(c::int) = (0::int) ⟹ LLL_swap_add (p::int) (fs::int vec list, dmu::int iarray iarray, di::int iarray, mods::int iarray) (i::nat) = LLL_swap_row p (LLL_add_row p (fs, dmu, di, mods) i (i - (1::nat))) i›\n 2. ‹(c::int) ≠ (0::int) ⟹ LLL_swap_add (p::int) (fs::int vec list, dmu::int iarray iarray, di::int iarray, mods::int iarray) (i::nat) = LLL_swap_row p (LLL_add_row p (fs, dmu, di, mods) i (i - (1::nat))) i›*)\n case True (*‹(c::int) = (0::int)›*)\n thus \"?thesis\"\n (*goal: ‹LLL_swap_add p (fs, dmu, di, mods) i = LLL_swap_row p (LLL_add_row p (fs, dmu, di, mods) i (i - 1)) i›*)\n using i (*‹i ≠ 0› ‹i < m›*) unfolding res1 LLL_add_row_def split id c_def Let_def\n (*goal: ‹(if round_num_denom (dmu !! i !! (i - 1)) (di !! i) = 0 then LLL_swap_row p (fs, dmu, di, mods) i else perform_swap_add p (fs, dmu, di, mods) i (i - 1) (round_num_denom (dmu !! i !! (i - 1)) (di !! i)) (dmu !! i) (dmu !! i !! (i - 1)) (di !! i)) = LLL_swap_row p (if round_num_denom (dmu !! i !! (i - 1)) (di !! Suc (i - 1)) = 0 then (fs, dmu, di, mods) else perform_add_row p (fs, dmu, di, mods) i (i - 1) (round_num_denom (dmu !! i !! (i - 1)) (di !! Suc (i - 1))) (dmu !! i) (dmu !! i !! (i - 1)) (di !! Suc (i - 1))) i›*)\n by auto\n next\n (*goal: ‹c ≠ 0 ⟹ LLL_swap_add p (fs, dmu, di, mods) i = LLL_swap_row p (LLL_add_row p (fs, dmu, di, mods) i (i - 1)) i›*)\n case False (*‹c ≠ 0›*)\n hence c: \"(c = 0) = False\"\n by simp\n have add: \"LLL_add_row p (fs, dmu, di, mods) i (i - 1) = \n perform_add_row p (fs, dmu, di, mods) i (i - 1) c (dmu !! i) (dmu !! i !! (i - 1)) (di !! i)\"\n unfolding LLL_add_row_def Let_def split si1 c_def[symmetric] c\n (*goal: ‹(if False then (fs, dmu, di, mods) else perform_add_row p (fs, dmu, di, mods) i (i - 1) c (dmu !! i) (dmu !! i !! (i - 1)) (di !! i)) = perform_add_row p (fs, dmu, di, mods) i (i - 1) c (dmu !! i) (dmu !! i !! (i - 1)) (di !! i)›*)\n by auto\n show \"?thesis\"\n (*goal: ‹LLL_swap_add p (fs, dmu, di, mods) i = LLL_swap_row p (LLL_add_row p (fs, dmu, di, mods) i (i - 1)) i›*)\n unfolding res1 c if_False add\n (*goal: ‹perform_swap_add (p::int) (fs::int vec list, dmu::int iarray iarray, di::int iarray, mods::int iarray) (i::nat) (i - (1::nat)) (c::int) (dmu !! i) (dmu !! i !! (i - (1::nat))) (di !! i) = LLL_swap_row p (perform_add_row p (fs, dmu, di, mods) i (i - (1::nat)) c (dmu !! i) (dmu !! i !! (i - (1::nat))) (di !! i)) i›*)\n apply (subst perform_swap_add[OF assms] (*‹LLL_swap_row ?p (perform_add_row ?p (fs, ?dmu, ?di, ?mods) i (i - 1) ?c (?dmu !! i) (?dmu !! i !! (i - 1)) (?di !! i)) i = perform_swap_add ?p (fs, ?dmu, ?di, ?mods) i (i - 1) ?c (?dmu !! i) (?dmu !! i !! (i - 1)) (?di !! i)›*))\n (*goal: ‹perform_swap_add p (fs, dmu, di, mods) i (i - 1) c (dmu !! i) (dmu !! i !! (i - 1)) (di !! i) = LLL_swap_row p (perform_add_row p (fs, dmu, di, mods) i (i - 1) c (dmu !! i) (dmu !! i !! (i - 1)) (di !! i)) i›*)\n by simp\n qed\nqed\nend\n\n\ncontext LLL_with_assms\nbegin\n\nlemma LLL_mod_inv_to_weak: \"LLL_invariant_mod fs mfs dmu p first b i ⟹ LLL_invariant_mod_weak fs mfs dmu p first b\" \n unfolding LLL_invariant_mod_def LLL_invariant_mod_weak_def\n (*goal: ‹length (fs::int vec list) = (m::nat) ∧ length (mfs::int vec list) = m ∧ (i::nat) ≤ m ∧ lattice_of fs = L ∧ gs.lin_indpt_list (map of_int_hom.vec_hom fs) ∧ gram_schmidt_fs.weakly_reduced (n::nat) (map of_int_hom.vec_hom fs) (α::rat) i ∧ map (map_vec (λx::int. x symmod (p::int))) fs = mfs ∧ (∀i'<m. ∀j'<i'. ¦dμ fs i' j'¦ < p * d fs j' * d fs (Suc j')) ∧ (∀i'<m. ∀j'<m. dμ fs i' j' = (dmu::int mat) $$ (i', j')) ∧ (1::int) < p ∧ g_bnd_mode (first::bool) (b::rat) fs ∧ mod_invariant b p first ⟹ length fs = m ∧ length mfs = m ∧ lattice_of fs = L ∧ gs.lin_indpt_list (map of_int_hom.vec_hom fs) ∧ map (map_vec (λx::int. x symmod p)) fs = mfs ∧ (∀i'<m. ∀j'<i'. ¦dμ fs i' j'¦ < p * d fs j' * d fs (Suc j')) ∧ (∀i'<m. ∀j'<m. dμ fs i' j' = dmu $$ (i', j')) ∧ (1::int) < p ∧ g_bnd_mode first b fs ∧ mod_invariant b p first›*)\n by auto\n\ndeclare IArray.of_fun_def[simp del]\n\nlemma LLL_swap_row: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod_weak fs mfs dmu p first b\"\n and res: \"basis_reduction_mod_swap p mfs dmu k = (mfs', dmu')\" \n and res': \"LLL_swap_row p state k = state'\" \n and k: \"k < m\" \"k ≠ 0\" \nshows \"state_impl_inv p mfs' dmu' state'\" \nproof (-)\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n note inv = LLL_invD_modw[OF Linv] (*‹length (mfs::int vec list) = (m::nat)› ‹length fs = m› ‹lattice_of (fs::int vec list) = L› ‹gs.lin_indpt_list (map of_int_hom.vec_hom (fs::int vec list))› ‹map (map_vec (λx. x symmod p)) fs = mfs› ‹∀i'<m::nat. ∀j'<i'. ¦dμ (fs::int vec list) i' j'¦ < (p::int) * d fs j' * d fs (Suc j')› ‹∀i'<m. ∀j'<m. dμ fs i' j' = dmu $$ (i', j')› ‹?i < m ⟹ fs ! ?i ∈ carrier_vec n› ‹set fs ⊆ carrier_vec n› ‹?i < m ⟹ gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) ?i ∈ Rn› ‹?i < m ⟹ mfs ! ?i ∈ carrier_vec n› ‹set mfs ⊆ carrier_vec n› and more 3 facts*)\n obtain fsi and dmui and di and mods where state: \"state = (fsi, dmui, di, mods)\"\n (*goal: ‹(⋀fsi dmui di mods. state = (fsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n apply (cases state)\n (*goal: ‹(⋀fsi dmui di mods. state = (fsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain fsi' and dmui' and di' and mods' where state': \"state' = (fsi', dmui', di', mods')\"\n (*goal: ‹(⋀fsi' dmui' di' mods'. state' = (fsi', dmui', di', mods') ⟹ thesis) ⟹ thesis›*)\n apply (cases state')\n (*goal: ‹(⋀fsi' dmui' di' mods'. state' = (fsi', dmui', di', mods') ⟹ thesis) ⟹ thesis›*)\n by auto\n from impl[unfolded state, simplified] (*‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0)›*) have id: \"fsi = mfs\" \"di = IArray.of_fun (d_of dmu) (Suc m)\" \"dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m\" \"mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)\"\n apply -\n (*goals:\n 1. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ fsi = mfs›\n 2. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ di = IArray.of_fun (d_of dmu) (Suc m)›\n 3. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m›\n 4. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n have kk1: \"dmui !! k !! (k - 1) = dmu $$ (k, k - 1)\"\n using k (*‹k < m› ‹k ≠ 0›*) unfolding id IArray.of_fun_nth[OF k ( 1 )]\n (*goal: ‹IArray.of_fun (λj. dmu $$ (k, j)) k !! (k - 1) = dmu $$ (k, k - 1)›*)\n apply (subst IArray.of_fun_nth (*‹(?i::nat) < (?n::nat) ⟹ IArray.of_fun (?f::nat ⇒ ?'a::type) ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦k < m; k ≠ 0⟧ ⟹ k - 1 < k›\n 2. ‹⟦k < m; k ≠ 0⟧ ⟹ dmu $$ (k, k - 1) = dmu $$ (k, k - 1)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have di: \"i ≤ m ⟹ di !! i = d_of dmu i\" for i\n unfolding id\n (*goal: ‹i ≤ m ⟹ IArray.of_fun (d_of dmu) (Suc m) !! i = d_of dmu i›*)\n apply (subst IArray.of_fun_nth (*‹(?i::nat) < (?n::nat) ⟹ IArray.of_fun (?f::nat ⇒ ?'a) ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹i ≤ m ⟹ i < Suc m›\n 2. ‹i ≤ m ⟹ d_of dmu i = d_of dmu i›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have dS1: \"di !! Suc k = d_of dmu (Suc k)\"\n using di (*‹(?i2::nat) ≤ (m::nat) ⟹ (di::int iarray) !! ?i2 = d_of (dmu::int mat) ?i2›*) k (*‹k < m› ‹k ≠ 0›*) by auto\n have d1: \"di !! (k - 1) = d_of dmu (k - 1)\"\n using di (*‹?i2 ≤ m ⟹ di !! ?i2 = d_of dmu ?i2›*) k (*‹k < m› ‹k ≠ 0›*) by auto\n have dk: \"di !! k = d_of dmu k\"\n using di (*‹?i2 ≤ m ⟹ di !! ?i2 = d_of dmu ?i2›*) k (*‹k < m› ‹k ≠ 0›*) by auto\n define dk' where \"dk' = (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k\"\n define mod1 where \"mod1 = p * d_of dmu (k - 1) * dk'\"\n define modk where \"modk = p * dk' * d_of dmu (Suc k)\"\n define dmu'' where \"dmu'' = (mat m m\n (λ(i, j).\n if j < i\n then if i = k - 1 then dmu $$ (k, j)\n else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j)\n else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k\n else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j)\n else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i)\n else dmu $$ (i, j)))\"\n have drop: \"drop (k - 1) fsi = mfs ! (k - 1) # mfs ! k # drop (Suc k) mfs\"\n unfolding id\n (*goal: ‹drop (k - 1) mfs = mfs ! (k - 1) # mfs ! k # drop (Suc k) mfs›*)\n using ‹length mfs = m› (*‹length mfs = m›*) k (*‹k < m› ‹k ≠ 0›*) by (metis Cons_nth_drop_Suc (*‹?i < length ?xs ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*) One_nat_def (*‹1 = Suc 0›*) Suc_less_eq (*‹(Suc ?m < Suc ?n) = (?m < ?n)›*) Suc_pred (*‹0 < ?n ⟹ Suc (?n - Suc 0) = ?n›*) less_SucI (*‹?m < ?n ⟹ ?m < Suc ?n›*) linorder_neqE_nat (*‹⟦?x ≠ ?y; ?x < ?y ⟹ ?R; ?y < ?x ⟹ ?R⟧ ⟹ ?R›*) not_less0 (*‹¬ ?n < 0›*))\n have dk': \"dk' = d_of dmu'' k\"\n unfolding dk'_def d_of_def dmu''_def\n (*goal: ‹((if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) * (if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1)) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) = (if k = 0 then 1 else mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then ((if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * (if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1))) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else dmu $$ (i, j) else if i = j then if i = k - 1 then ((if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) * (if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1)) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else if Suc i = 0 then 1 else dmu $$ (Suc i - 1, Suc i - 1) else dmu $$ (i, j)) $$ (k - 1, k - 1))›*)\n using k (*‹(k::nat) < (m::nat)› ‹k ≠ 0›*) by auto\n have mod1: \"mod1 = p * d_of dmu'' (k - 1) * d_of dmu'' k\"\n unfolding mod1_def dk'\n (*goal: ‹p * d_of dmu (k - 1) * d_of dmu'' k = p * d_of dmu'' (k - 1) * d_of dmu'' k›*)\n using k (*‹k < m› ‹k ≠ 0›*) by (auto simp: dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*) d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))\n have modk: \"modk = p * d_of dmu'' k * d_of dmu'' (Suc k)\"\n unfolding modk_def dk'\n (*goal: ‹p * d_of dmu'' k * d_of dmu (Suc k) = p * d_of dmu'' k * d_of dmu'' (Suc k)›*)\n using k (*‹k < m› ‹k ≠ 0›*) by (auto simp: dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*) d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))\n note res = res[unfolded basis_reduction_mod_swap_def, folded dmu''_def, symmetric] (*‹(mfs', dmu') = (mfs[k := mfs ! (k - 1), k - 1 := mfs ! k], basis_reduction_mod_swap_dmu_mod p dmu'' k)›*)\n note res' = res'[unfolded state state' split_at_def drop list.simps split LLL_swap_row_def Let_def kk1 dS1 d1 dk, folded dk'_def mod1_def modk_def, symmetric] (*‹(fsi'::int vec list, dmui'::int iarray iarray, di'::int iarray, mods'::int iarray) = (take ((k::nat) - (1::nat)) (fsi::int vec list) @ (mfs::int vec list) ! k # mfs ! (k - (1::nat)) # drop (Suc k) mfs, IArray.of_fun (λi::nat. if i < k - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) (m::nat), IArray.of_fun (λi::nat. if i = k then dk'::int else (di::int iarray) !! i) (Suc m), IArray.of_fun (λj::nat. if j = k - (1::nat) then mod1 else if j = k then modk else (mods::int iarray) !! j) (m - (1::nat)))›*)\n from res' (*‹(fsi', dmui', di', mods') = (take (k - 1) fsi @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs, IArray.of_fun (λi. if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) m, IArray.of_fun (λi. if i = k then dk' else di !! i) (Suc m), IArray.of_fun (λj. if j = k - 1 then mod1 else if j = k then modk else mods !! j) (m - 1))›*) have fsi': \"fsi' = take (k - 1) mfs @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs\"\n unfolding id\n (*goal: ‹(fsi'::int vec list) = take ((k::nat) - (1::nat)) (mfs::int vec list) @ mfs ! k # mfs ! (k - (1::nat)) # drop (Suc k) mfs›*)\n by simp\n from res' (*‹(fsi', dmui', di', mods') = (take (k - 1) fsi @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs, IArray.of_fun (λi. if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) m, IArray.of_fun (λi. if i = k then dk' else di !! i) (Suc m), IArray.of_fun (λj. if j = k - 1 then mod1 else if j = k then modk else mods !! j) (m - 1))›*) have di': \"di' = IArray.of_fun (λii. if ii = k then dk' else di !! ii) (Suc m)\"\n by simp\n from res' (*‹(fsi'::int vec list, dmui'::int iarray iarray, di'::int iarray, mods'::int iarray) = (take ((k::nat) - (1::nat)) (fsi::int vec list) @ (mfs::int vec list) ! k # mfs ! (k - (1::nat)) # drop (Suc k) mfs, IArray.of_fun (λi::nat. if i < k - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) (m::nat), IArray.of_fun (λi::nat. if i = k then dk'::int else (di::int iarray) !! i) (Suc m), IArray.of_fun (λj::nat. if j = k - (1::nat) then mod1 else if j = k then modk else (mods::int iarray) !! j) (m - (1::nat)))›*) have dmui': \"dmui' = IArray.of_fun\n (λi. if i < k - 1 then dmui !! i\n else if k < i then IArray.of_fun\n (λj. if j = k - 1\n then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) \n div d_of dmu k symmod mod1\n else if j = k\n then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) \n div d_of dmu k symmod modk \n else dmui !! i !! j)\n i\n else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 \n else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i)\n m\"\n by auto\n from res' (*‹(fsi'::int vec list, dmui'::int iarray iarray, di'::int iarray, mods'::int iarray) = (take ((k::nat) - (1::nat)) (fsi::int vec list) @ (mfs::int vec list) ! k # mfs ! (k - (1::nat)) # drop (Suc k) mfs, IArray.of_fun (λi::nat. if i < k - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) (m::nat), IArray.of_fun (λi::nat. if i = k then dk'::int else (di::int iarray) !! i) (Suc m), IArray.of_fun (λj::nat. if j = k - (1::nat) then mod1 else if j = k then modk else (mods::int iarray) !! j) (m - (1::nat)))›*) have mods': \"mods' = IArray.of_fun (λjj. if jj = k - 1 then mod1 else if jj = k then modk else mods !! jj) (m - 1)\"\n by auto\n from res (*‹(mfs', dmu') = (mfs[k := mfs ! (k - 1), k - 1 := mfs ! k], basis_reduction_mod_swap_dmu_mod p dmu'' k)›*) have dmu': \"dmu' = basis_reduction_mod_swap_dmu_mod p dmu'' k\"\n by auto\n show \"?thesis\"\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n unfolding state' state_impl_inv.simps\n (*goal: ‹fsi' = mfs' ∧ di' = IArray.of_fun (d_of dmu') (Suc m) ∧ dmui' = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m ∧ mods' = IArray.of_fun (λj. p * di' !! j * di' !! Suc j) (m - 1)›*)\n proof (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹fsi' = mfs'›\n 2. ‹di' = IArray.of_fun (d_of dmu') (Suc m)›\n 3. ‹dmui' = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m›\n 4. ‹mods' = IArray.of_fun (λj. p * di' !! j * di' !! Suc j) (m - 1)›*)\n from res (*‹(mfs'::int vec list, dmu'::int mat) = ((mfs::int vec list)[k::nat := mfs ! (k - (1::nat)), k - (1::nat) := mfs ! k], basis_reduction_mod_swap_dmu_mod (p::int) (dmu''::int mat) k)›*) have mfs': \"mfs' = mfs[k := mfs ! (k - 1), k - 1 := mfs ! k]\"\n by simp\n show \"fsi' = mfs'\"\n unfolding fsi' mfs'\n (*goal: ‹take (k - 1) mfs @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs = mfs[k := mfs ! (k - 1), k - 1 := mfs ! k]›*)\n using ‹length mfs = m› (*‹length mfs = m›*) k (*‹(k::nat) < (m::nat)› ‹k ≠ 0›*) proof (intro nth_equalityI (*‹⟦length ?xs = length ?ys; ⋀i. i < length ?xs ⟹ ?xs ! i = ?ys ! i⟧ ⟹ ?xs = ?ys›*), force, goal_cases)\n (*goal: ‹⋀i. ⟦length mfs = m; k < m; k ≠ 0; i < length (take (k - 1) mfs @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs)⟧ ⟹ (take (k - 1) mfs @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs) ! i = mfs[k := mfs ! (k - 1), k - 1 := mfs ! k] ! i›*)\n case (1 j) (*‹length mfs = m› ‹k < m› ‹k ≠ 0› ‹(j::nat) < length (take ((k::nat) - (1::nat)) (mfs::int vec list) @ mfs ! k # mfs ! (k - (1::nat)) # drop (Suc k) mfs)›*)\n have choice: \"j = k - 1 ∨ j = k ∨ j < k - 1 ∨ j > k\"\n by linarith\n have \"min (length mfs) (k - 1) = k - 1\"\n using \"1\" (*‹length (mfs::int vec list) = (m::nat)› ‹k < m› ‹(k::nat) ≠ (0::nat)› ‹j < length (take (k - 1) mfs @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs)›*) by auto\n with \"1\" (*‹length (mfs::int vec list) = (m::nat)› ‹k < m› ‹(k::nat) ≠ (0::nat)› ‹j < length (take (k - 1) mfs @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs)›*) choice (*‹(j::nat) = (k::nat) - (1::nat) ∨ j = k ∨ j < k - (1::nat) ∨ k < j›*) show \"?case\"\n (*goal: ‹(take (k - 1) mfs @ mfs ! k # mfs ! (k - 1) # drop (Suc k) mfs) ! j = mfs[k := mfs ! (k - 1), k - 1 := mfs ! k] ! j›*)\n by (auto simp: nth_append (*‹(?xs @ ?ys) ! ?n = (if ?n < length ?xs then ?xs ! ?n else ?ys ! (?n - length ?xs))›*))\n qed\n show \"di' = IArray.of_fun (d_of dmu') (Suc m)\"\n unfolding di'\n (*goal: ‹IArray.of_fun (λii. if ii = k then dk' else di !! ii) (Suc m) = IArray.of_fun (d_of dmu') (Suc m)›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λii. if ii = k then dk' else di !! ii) (Suc m) = IArray.of_fun (d_of dmu') (Suc m)›*)\n proof (goal_cases)\n (*goal: ‹⋀i::nat. i < Suc (m::nat) ⟹ (if i = (k::nat) then dk'::int else (di::int iarray) !! i) = d_of (dmu'::int mat) i›*)\n case i: (1 i) (*‹(i::nat) < Suc (m::nat)›*)\n hence \"d_of dmu' i = d_of dmu'' i\"\n unfolding dmu' basis_reduction_mod_swap_dmu_mod_def d_of_def\n (*goal: ‹(if i = 0 then 1 else mat m m (λ(i, j). if j < i ∧ (j = k ∨ j = k - 1) then dmu'' $$ (i, j) symmod (p * (if j = 0 then 1 else dmu'' $$ (j - 1, j - 1)) * (if Suc j = 0 then 1 else dmu'' $$ (Suc j - 1, Suc j - 1))) else dmu'' $$ (i, j)) $$ (i - 1, i - 1)) = (if i = 0 then 1 else dmu'' $$ (i - 1, i - 1))›*)\n apply (intro if_cong (*‹⟦?b = ?c; ?c ⟹ ?x = ?u; ¬ ?c ⟹ ?y = ?v⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*))\n (*goals:\n 1. ‹(i::nat) < Suc (m::nat) ⟹ (i = (0::nat)) = (i = (0::nat))›\n 2. ‹⟦(i::nat) < Suc (m::nat); i = (0::nat)⟧ ⟹ (1::int) = (1::int)›\n 3. ‹⟦(i::nat) < Suc (m::nat); i ≠ (0::nat)⟧ ⟹ mat m m (λ(i::nat, j::nat). if j < i ∧ (j = (k::nat) ∨ j = k - (1::nat)) then (dmu''::int mat) $$ (i, j) symmod ((p::int) * (if j = (0::nat) then 1::int else dmu'' $$ (j - (1::nat), j - (1::nat))) * (if Suc j = (0::nat) then 1::int else dmu'' $$ (Suc j - (1::nat), Suc j - (1::nat)))) else dmu'' $$ (i, j)) $$ (i - (1::nat), i - (1::nat)) = dmu'' $$ (i - (1::nat), i - (1::nat))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹d_of dmu' i = d_of dmu'' i›*) have \"… = ((if i = k then dk' else di !! i))\"\n proof (cases \"i = k\")\n (*goals:\n 1. ‹(i::nat) = (k::nat) ⟹ d_of (dmu''::int mat) i = (if i = k then dk'::int else (di::int iarray) !! i)›\n 2. ‹(i::nat) ≠ (k::nat) ⟹ d_of (dmu''::int mat) i = (if i = k then dk'::int else (di::int iarray) !! i)›*)\n case False (*‹i ≠ k›*)\n hence \"d_of dmu'' i = d_of dmu i\"\n unfolding dmu''_def d_of_def\n (*goal: ‹(if (i::nat) = (0::nat) then 1::int else mat (m::nat) m (λ(i::nat, j::nat). if j < i then if i = (k::nat) - (1::nat) then (dmu::int mat) $$ (k, j) else if i = k ∧ j ≠ k - (1::nat) then dmu $$ (k - (1::nat), j) else if k < i ∧ j = k then ((if Suc k = (0::nat) then 1::int else dmu $$ (Suc k - (1::nat), Suc k - (1::nat))) * dmu $$ (i, k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmu $$ (i, j)) div (if k = (0::nat) then 1::int else dmu $$ (k - (1::nat), k - (1::nat))) else if k < i ∧ j = k - (1::nat) then (dmu $$ (k, k - (1::nat)) * dmu $$ (i, j) + dmu $$ (i, k) * (if k - (1::nat) = (0::nat) then 1::int else dmu $$ (k - (1::nat) - (1::nat), k - (1::nat) - (1::nat)))) div (if k = (0::nat) then 1::int else dmu $$ (k - (1::nat), k - (1::nat))) else dmu $$ (i, j) else if i = j then if i = k - (1::nat) then ((if Suc k = (0::nat) then 1::int else dmu $$ (Suc k - (1::nat), Suc k - (1::nat))) * (if k - (1::nat) = (0::nat) then 1::int else dmu $$ (k - (1::nat) - (1::nat), k - (1::nat) - (1::nat))) + dmu $$ (k, k - (1::nat)) * dmu $$ (k, k - (1::nat))) div (if k = (0::nat) then 1::int else dmu $$ (k - (1::nat), k - (1::nat))) else if Suc i = (0::nat) then 1::int else dmu $$ (Suc i - (1::nat), Suc i - (1::nat)) else dmu $$ (i, j)) $$ (i - (1::nat), i - (1::nat))) = (if i = (0::nat) then 1::int else dmu $$ (i - (1::nat), i - (1::nat)))›*)\n using i (*‹i < Suc m›*) k (*‹k < m› ‹k ≠ 0›*) apply (intro if_cong (*‹⟦?b = ?c; ?c ⟹ ?x = ?u; ¬ ?c ⟹ ?y = ?v⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*) refl (*‹?t = ?t›*))\n (*goal: ‹(if (i::nat) = (0::nat) then 1::int else mat (m::nat) m (λ(i::nat, j::nat). if j < i then if i = (k::nat) - (1::nat) then (dmu::int mat) $$ (k, j) else if i = k ∧ j ≠ k - (1::nat) then dmu $$ (k - (1::nat), j) else if k < i ∧ j = k then ((if Suc k = (0::nat) then 1::int else dmu $$ (Suc k - (1::nat), Suc k - (1::nat))) * dmu $$ (i, k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmu $$ (i, j)) div (if k = (0::nat) then 1::int else dmu $$ (k - (1::nat), k - (1::nat))) else if k < i ∧ j = k - (1::nat) then (dmu $$ (k, k - (1::nat)) * dmu $$ (i, j) + dmu $$ (i, k) * (if k - (1::nat) = (0::nat) then 1::int else dmu $$ (k - (1::nat) - (1::nat), k - (1::nat) - (1::nat)))) div (if k = (0::nat) then 1::int else dmu $$ (k - (1::nat), k - (1::nat))) else dmu $$ (i, j) else if i = j then if i = k - (1::nat) then ((if Suc k = (0::nat) then 1::int else dmu $$ (Suc k - (1::nat), Suc k - (1::nat))) * (if k - (1::nat) = (0::nat) then 1::int else dmu $$ (k - (1::nat) - (1::nat), k - (1::nat) - (1::nat))) + dmu $$ (k, k - (1::nat)) * dmu $$ (k, k - (1::nat))) div (if k = (0::nat) then 1::int else dmu $$ (k - (1::nat), k - (1::nat))) else if Suc i = (0::nat) then 1::int else dmu $$ (Suc i - (1::nat), Suc i - (1::nat)) else dmu $$ (i, j)) $$ (i - (1::nat), i - (1::nat))) = (if i = (0::nat) then 1::int else dmu $$ (i - (1::nat), i - (1::nat)))›*)\n by auto\n thus \"?thesis\"\n (*goal: ‹d_of dmu'' i = (if i = k then dk' else di !! i)›*)\n using False (*‹i ≠ k›*) i (*‹i < Suc m›*) k (*‹k < m› ‹k ≠ 0›*) unfolding id\n (*goal: ‹d_of dmu'' i = (if i = k then dk' else IArray.of_fun (d_of dmu) (Suc m) !! i)›*)\n by (metis iarray_of_fun_sub (*‹(?i::nat) < (?n::nat) ⟹ IArray.of_fun (?f::nat ⇒ ?'a) ?n !! ?i = ?f ?i›*))\n next\n (*goal: ‹i = k ⟹ d_of dmu'' i = (if i = k then dk' else di !! i)›*)\n case True (*‹i = k›*)\n thus \"?thesis\"\n (*goal: ‹d_of dmu'' i = (if i = k then dk' else di !! i)›*)\n using dk' (*‹(dk'::int) = d_of (dmu''::int mat) (k::nat)›*) by auto\n qed\n finally (*calculation: ‹d_of dmu' i = (if i = k then dk' else di !! i)›*) show \"?case\"\n (*goal: ‹(if i = k then dk' else di !! i) = d_of dmu' i›*)\n by simp\n qed\n have dkS1: \"d_of dmu (Suc k) = d_of dmu'' (Suc k)\"\n unfolding dmu''_def d_of_def\n (*goal: ‹(if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) = (if Suc k = 0 then 1 else mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then ((if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * (if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1))) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else dmu $$ (i, j) else if i = j then if i = k - 1 then ((if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) * (if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1)) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else if Suc i = 0 then 1 else dmu $$ (Suc i - 1, Suc i - 1) else dmu $$ (i, j)) $$ (Suc k - 1, Suc k - 1))›*)\n using k (*‹k < m› ‹k ≠ 0›*) by auto\n have dk1: \"d_of dmu (k - 1) = d_of dmu'' (k - 1)\"\n unfolding dmu''_def d_of_def\n (*goal: ‹(if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1)) = (if k - 1 = 0 then 1 else mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then ((if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * (if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1))) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else dmu $$ (i, j) else if i = j then if i = k - 1 then ((if Suc k = 0 then 1 else dmu $$ (Suc k - 1, Suc k - 1)) * (if k - 1 = 0 then 1 else dmu $$ (k - 1 - 1, k - 1 - 1)) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div (if k = 0 then 1 else dmu $$ (k - 1, k - 1)) else if Suc i = 0 then 1 else dmu $$ (Suc i - 1, Suc i - 1) else dmu $$ (i, j)) $$ (k - 1 - 1, k - 1 - 1))›*)\n using k (*‹k < m› ‹k ≠ 0›*) by auto\n show \"dmui' = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m\"\n unfolding dmui'\n (*goal: ‹IArray.of_fun (λi. if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) m = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λi. if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) m = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m›*)\n proof (goal_cases)\n (*goal: ‹⋀i::nat. i < (m::nat) ⟹ (if i < (k::nat) - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i›*)\n case i: (1 i) (*‹(i::nat) < (m::nat)›*)\n consider (1)\"i < k - 1\" | (2)\"i = k - 1\" | (3)\"i = k\" | (4)\"i > k\"\n (*goal: ‹⟦i < k - 1 ⟹ thesis; i = k - 1 ⟹ thesis; i = k ⟹ thesis; k < i ⟹ thesis⟧ ⟹ thesis›*)\n by linarith\n thus \"?case\"\n (*goal: ‹(if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n proof (cases)\n (*goals:\n 1. ‹i < k - 1 ⟹ (if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›\n 2. ‹i = k - 1 ⟹ (if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›\n 3. ‹i = k ⟹ (if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›\n 4. ‹k < i ⟹ (if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n case 1 (*‹i < k - 1›*)\n hence \"*\": \"(i < k - 1) = True\"\n by simp\n show \"?thesis\"\n (*goal: ‹(if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n unfolding \"*\" if_True id IArray.of_fun_nth[OF i]\n (*goal: ‹IArray.of_fun (λj. dmu $$ (i, j)) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n using i (*‹i < m›*) k (*‹k < m› ‹k ≠ 0›*) \"1\" (*‹i < k - 1›*) apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λj. dmu $$ (i, j)) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n apply (auto simp: dmu' (*‹(dmu'::int mat) = basis_reduction_mod_swap_dmu_mod (p::int) (dmu''::int mat) (k::nat)›*) basis_reduction_mod_swap_dmu_mod_def (*‹basis_reduction_mod_swap_dmu_mod (?p::int) (?dmu::int mat) (?k::nat) = mat (m::nat) m (λ(i::nat, j::nat). if j < i ∧ (j = ?k ∨ j = ?k - (1::nat)) then ?dmu $$ (i, j) symmod (?p * d_of ?dmu j * d_of ?dmu (Suc j)) else ?dmu $$ (i, j))›*))\n (*goal: ‹⋀ia::nat. ⟦(i::nat) < (m::nat); (k::nat) < m; k ≠ (0::nat); i < k - (1::nat); ia < i⟧ ⟹ (dmu::int mat) $$ (i, ia) = (dmu'::int mat) $$ (i, ia)›*)\n by (auto simp: dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))\n next\n (*goals:\n 1. ‹(i::nat) = (k::nat) - (1::nat) ⟹ (if i < k - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i›\n 2. ‹(i::nat) = (k::nat) ⟹ (if i < k - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i›\n 3. ‹(k::nat) < (i::nat) ⟹ (if i < k - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i›*)\n case 2 (*‹i = k - 1›*)\n hence \"*\": \"(i < k - 1) = False\" \"(k < i) = False\" \"(i = k) = False\"\n using k (*‹k < m› ‹k ≠ 0›*) apply -\n (*goals:\n 1. ‹⟦i = k - 1; k < m; k ≠ 0⟧ ⟹ (i < k - 1) = False›\n 2. ‹⟦i = k - 1; k < m; k ≠ 0⟧ ⟹ (k < i) = False›\n 3. ‹⟦i = k - 1; k < m; k ≠ 0⟧ ⟹ (i = k) = False›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n show \"?thesis\"\n (*goal: ‹(if (i::nat) < (k::nat) - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i›*)\n unfolding \"*\" if_False id\n (*goal: ‹IArray.of_fun ((!!) (IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. (dmu::int mat) $$ (i, j)) i) (m::nat) !! (k::nat))) (i::nat) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i›*)\n using i (*‹i < m›*) k (*‹(k::nat) < (m::nat)› ‹k ≠ 0›*) \"2\" (*‹i = k - 1›*) unfolding IArray.of_fun_nth[OF k ( 1 )]\n (*goal: ‹IArray.of_fun ((!!) (IArray.of_fun (λj. dmu $$ (k, j)) k)) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun ((!!) (IArray.of_fun (λj. dmu $$ (k, j)) k)) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n apply (subst IArray.of_fun_nth (*‹(?i::nat) < (?n::nat) ⟹ IArray.of_fun (?f::nat ⇒ ?'a) ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⋀ia. ⟦i < m; k < m; k ≠ 0; i = k - 1; ia < i⟧ ⟹ ia < k›\n 2. ‹⋀ia. ⟦i < m; k < m; k ≠ 0; i = k - 1; ia < i⟧ ⟹ dmu $$ (k, ia) = dmu' $$ (i, ia)›\n discuss goal 1*)\n apply ((auto simp: dmu' (*‹dmu' = basis_reduction_mod_swap_dmu_mod p dmu'' k›*) basis_reduction_mod_swap_dmu_mod_def (*‹basis_reduction_mod_swap_dmu_mod ?p ?dmu ?k = mat m m (λ(i, j). if j < i ∧ (j = ?k ∨ j = ?k - 1) then ?dmu $$ (i, j) symmod (?p * d_of ?dmu j * d_of ?dmu (Suc j)) else ?dmu $$ (i, j))›*) dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: dmu' (*‹dmu' = basis_reduction_mod_swap_dmu_mod p dmu'' k›*) basis_reduction_mod_swap_dmu_mod_def (*‹basis_reduction_mod_swap_dmu_mod ?p ?dmu ?k = mat m m (λ(i, j). if j < i ∧ (j = ?k ∨ j = ?k - 1) then ?dmu $$ (i, j) symmod (?p * d_of ?dmu j * d_of ?dmu (Suc j)) else ?dmu $$ (i, j))›*) dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))[1])\n (*proven 2 subgoals*) .\n next\n (*goals:\n 1. ‹i = k ⟹ (if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›\n 2. ‹k < i ⟹ (if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n case 3 (*‹i = k›*)\n hence \"*\": \"(i < k - 1) = False\" \"(k < i) = False\" \"(i = k) = True\"\n using k (*‹k < m› ‹k ≠ 0›*) apply -\n (*goals:\n 1. ‹⟦i = k; k < m; k ≠ 0⟧ ⟹ (i < k - 1) = False›\n 2. ‹⟦i = k; k < m; k ≠ 0⟧ ⟹ (k < i) = False›\n 3. ‹⟦i = k; k < m; k ≠ 0⟧ ⟹ (i = k) = True›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n show \"?thesis\"\n (*goal: ‹(if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n unfolding \"*\" if_False if_True id IArray.of_fun_nth[OF k ( 1 )]\n (*goal: ‹IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n proof (goal_cases)\n (*goal: ‹⋀ia::nat. ia < (i::nat) ⟹ (if ia = (k::nat) - (1::nat) then (dmu::int mat) $$ (k, k - (1::nat)) symmod (mod1::int) else IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. dmu $$ (i, j)) i) (m::nat) !! (k - (1::nat)) !! ia) = (dmu'::int mat) $$ (i, ia)›*)\n case j: (1 j) (*‹j < i›*)\n show \"?case\"\n (*goal: ‹(if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j) = dmu' $$ (i, j)›*)\n proof (cases \"j = k - 1\")\n (*goals:\n 1. ‹j = k - 1 ⟹ (if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j) = dmu' $$ (i, j)›\n 2. ‹j ≠ k - 1 ⟹ (if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j) = dmu' $$ (i, j)›*)\n case False (*‹(j::nat) ≠ (k::nat) - (1::nat)›*)\n hence \"*\": \"(j = k - 1) = False\"\n by auto\n show \"?thesis\"\n (*goal: ‹(if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j) = dmu' $$ (i, j)›*)\n unfolding \"*\" if_False\n (*goal: ‹IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j = dmu' $$ (i, j)›*)\n using False (*‹j ≠ k - 1›*) j (*‹j < i›*) k (*‹k < m› ‹(k::nat) ≠ (0::nat)›*) i (*‹i < m›*) \"3\" (*‹i = k›*) apply (subst IArray.of_fun_nth (*‹(?i::nat) < (?n::nat) ⟹ IArray.of_fun (?f::nat ⇒ ?'a) ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦j ≠ k - 1; j < i; k < m; k ≠ 0; i < m; i = k⟧ ⟹ k - 1 < m›\n 2. ‹⟦j ≠ k - 1; j < i; k < m; k ≠ 0; i < m; i = k⟧ ⟹ IArray.of_fun (λj. dmu $$ (k - 1, j)) (k - 1) !! j = dmu' $$ (i, j)›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦j ≠ k - 1; j < i; k < m; k ≠ 0; i < m; i = k⟧ ⟹ j < k - 1›\n 2. ‹⟦j ≠ k - 1; j < i; k < m; k ≠ 0; i < m; i = k⟧ ⟹ dmu $$ (k - 1, j) = dmu' $$ (i, j)›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply (auto simp: dmu' (*‹dmu' = basis_reduction_mod_swap_dmu_mod p dmu'' k›*) basis_reduction_mod_swap_dmu_mod_def (*‹basis_reduction_mod_swap_dmu_mod ?p ?dmu ?k = mat m m (λ(i, j). if j < i ∧ (j = ?k ∨ j = ?k - 1) then ?dmu $$ (i, j) symmod (?p * d_of ?dmu j * d_of ?dmu (Suc j)) else ?dmu $$ (i, j))›*) dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n next\n (*goal: ‹j = k - 1 ⟹ (if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! (k - 1) !! j) = dmu' $$ (i, j)›*)\n case True (*‹(j::nat) = (k::nat) - (1::nat)›*)\n hence \"*\": \"(j = k - 1) = True\"\n by auto\n show \"?thesis\"\n (*goal: ‹(if (j::nat) = (k::nat) - (1::nat) then (dmu::int mat) $$ (k, k - (1::nat)) symmod (mod1::int) else IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. dmu $$ (i, j)) i) (m::nat) !! (k - (1::nat)) !! j) = (dmu'::int mat) $$ (i::nat, j)›*)\n unfolding \"*\" if_True\n (*goal: ‹dmu $$ (k, k - 1) symmod mod1 = dmu' $$ (i, j)›*)\n unfolding True \"3\"\n (*goal: ‹dmu $$ (k, k - 1) symmod mod1 = dmu' $$ (k, k - 1)›*)\n using k (*‹k < m› ‹k ≠ 0›*) by (auto simp: basis_reduction_mod_swap_dmu_mod_def (*‹basis_reduction_mod_swap_dmu_mod ?p ?dmu ?k = mat m m (λ(i, j). if j < i ∧ (j = ?k ∨ j = ?k - 1) then ?dmu $$ (i, j) symmod (?p * d_of ?dmu j * d_of ?dmu (Suc j)) else ?dmu $$ (i, j))›*) dmu' (*‹dmu' = basis_reduction_mod_swap_dmu_mod p dmu'' k›*) dk' (*‹dk' = d_of dmu'' k›*) mod1 (*‹mod1 = p * d_of dmu'' (k - 1) * d_of dmu'' k›*) dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))\n qed\n qed\n next\n (*goal: ‹k < i ⟹ (if i < k - 1 then dmui !! i else if k < i then IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * dmui !! i !! (k - 1) + dmui !! i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - 1) - dmu $$ (k, k - 1) * dmui !! i !! k) div d_of dmu k symmod modk else dmui !! i !! j) i else if i = k then IArray.of_fun (λj. if j = k - 1 then dmu $$ (k, k - 1) symmod mod1 else dmui !! (k - 1) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n case 4 (*‹k < i›*)\n hence \"*\": \"(i < k - 1) = False\" \"(k < i) = True\"\n using k (*‹k < m› ‹k ≠ 0›*) apply -\n (*goals:\n 1. ‹⟦k < i; k < m; k ≠ 0⟧ ⟹ (i < k - 1) = False›\n 2. ‹⟦k < i; k < m; k ≠ 0⟧ ⟹ (k < i) = True›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"?thesis\"\n (*goal: ‹(if (i::nat) < (k::nat) - (1::nat) then (dmui::int iarray iarray) !! i else if k < i then IArray.of_fun (λj::nat. if j = k - (1::nat) then ((dmu::int mat) $$ (k, k - (1::nat)) * dmui !! i !! (k - (1::nat)) + dmui !! i !! k * d_of dmu (k - (1::nat))) div d_of dmu k symmod (mod1::int) else if j = k then (d_of dmu (Suc k) * dmui !! i !! (k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmui !! i !! k) div d_of dmu k symmod (modk::int) else dmui !! i !! j) i else if i = k then IArray.of_fun (λj::nat. if j = k - (1::nat) then dmu $$ (k, k - (1::nat)) symmod mod1 else dmui !! (k - (1::nat)) !! j) i else IArray.of_fun ((!!) (dmui !! k)) i) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i›*)\n unfolding \"*\" if_False if_True id IArray.of_fun_nth[OF k ( 1 )] IArray.of_fun_nth[OF ‹i < m›]\n (*goal: ‹IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) + IArray.of_fun (λj. dmu $$ (i, j)) i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) - dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! k) div d_of dmu k symmod modk else IArray.of_fun (λj. dmu $$ (i, j)) i !! j) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λj. if j = k - 1 then (dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) + IArray.of_fun (λj. dmu $$ (i, j)) i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) - dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! k) div d_of dmu k symmod modk else IArray.of_fun (λj. dmu $$ (i, j)) i !! j) i = IArray.of_fun (λj. dmu' $$ (i, j)) i›*)\n proof (goal_cases)\n (*goal: ‹⋀ia. ia < i ⟹ (if ia = k - 1 then (dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) + IArray.of_fun (λj. dmu $$ (i, j)) i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if ia = k then (d_of dmu (Suc k) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) - dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! k) div d_of dmu k symmod modk else IArray.of_fun (λj. dmu $$ (i, j)) i !! ia) = dmu' $$ (i, ia)›*)\n case j: (1 j) (*‹j < i›*)\n from \"4\" (*‹k < i›*) have k1: \"k - 1 < i\"\n by auto\n show \"?case\"\n (*goal: ‹(if j = k - 1 then (dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) + IArray.of_fun (λj. dmu $$ (i, j)) i !! k * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * IArray.of_fun (λj. dmu $$ (i, j)) i !! (k - 1) - dmu $$ (k, k - 1) * IArray.of_fun (λj. dmu $$ (i, j)) i !! k) div d_of dmu k symmod modk else IArray.of_fun (λj. dmu $$ (i, j)) i !! j) = dmu' $$ (i, j)›*)\n unfolding IArray.of_fun_nth[OF j] IArray.of_fun_nth[OF 4] IArray.of_fun_nth[OF k1]\n (*goal: ‹(if j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, k - 1) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k symmod mod1 else if j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, k)) div d_of dmu k symmod modk else dmu $$ (i, j)) = dmu' $$ (i, j)›*)\n unfolding mod1 modk dmu' basis_reduction_mod_swap_dmu_mod_def\n (*goal: ‹(if j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, k - 1) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k symmod (p * d_of dmu'' (k - 1) * d_of dmu'' k) else if j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, k)) div d_of dmu k symmod (p * d_of dmu'' k * d_of dmu'' (Suc k)) else dmu $$ (i, j)) = mat m m (λ(i, j). if j < i ∧ (j = k ∨ j = k - 1) then dmu'' $$ (i, j) symmod (p * d_of dmu'' j * d_of dmu'' (Suc j)) else dmu'' $$ (i, j)) $$ (i, j)›*)\n using i (*‹(i::nat) < (m::nat)›*) j (*‹j < i›*) \"4\" (*‹k < i›*) k (*‹k < m› ‹k ≠ 0›*) apply (auto intro!: arg_cong[of _ _ \"λ x. x symmod _\"] (*‹?x = ?y ⟹ ?x symmod ?uu5 = ?y symmod ?uu5›*))\n (*goals:\n 1. ‹⟦i < m; k < i; 0 < k; k ≠ k - Suc 0; j = k⟧ ⟹ (d_of dmu (Suc k) * dmu $$ (i, k - Suc 0) - dmu $$ (k, k - Suc 0) * dmu $$ (i, k)) div d_of dmu k = dmu'' $$ (i, k)›\n 2. ‹⟦i < m; k - Suc 0 < i; k < i; 0 < k; k ≠ k - Suc 0; j = k - Suc 0⟧ ⟹ (dmu $$ (k, k - Suc 0) * dmu $$ (i, k - Suc 0) + dmu $$ (i, k) * d_of dmu (k - Suc 0)) div d_of dmu k = dmu'' $$ (i, k - Suc 0)›\n 3. ‹⟦i < m; j < i; k < i; 0 < k; k ≠ k - Suc 0; j ≠ k; j ≠ k - Suc 0⟧ ⟹ dmu $$ (i, j) = dmu'' $$ (i, j)›\n discuss goal 1*)\n apply ((auto simp: dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: dmu''_def (*‹(dmu''::int mat) = mat (m::nat) m (λ(i::nat, j::nat). if j < i then if i = (k::nat) - (1::nat) then (dmu::int mat) $$ (k, j) else if i = k ∧ j ≠ k - (1::nat) then dmu $$ (k - (1::nat), j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - (1::nat)) - dmu $$ (k, k - (1::nat)) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - (1::nat) then (dmu $$ (k, k - (1::nat)) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - (1::nat))) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - (1::nat) then (d_of dmu (Suc k) * d_of dmu (k - (1::nat)) + dmu $$ (k, k - (1::nat)) * dmu $$ (k, k - (1::nat))) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: dmu''_def (*‹dmu'' = mat m m (λ(i, j). if j < i then if i = k - 1 then dmu $$ (k, j) else if i = k ∧ j ≠ k - 1 then dmu $$ (k - 1, j) else if k < i ∧ j = k then (d_of dmu (Suc k) * dmu $$ (i, k - 1) - dmu $$ (k, k - 1) * dmu $$ (i, j)) div d_of dmu k else if k < i ∧ j = k - 1 then (dmu $$ (k, k - 1) * dmu $$ (i, j) + dmu $$ (i, k) * d_of dmu (k - 1)) div d_of dmu k else dmu $$ (i, j) else if i = j then if i = k - 1 then (d_of dmu (Suc k) * d_of dmu (k - 1) + dmu $$ (k, k - 1) * dmu $$ (k, k - 1)) div d_of dmu k else d_of dmu (Suc i) else dmu $$ (i, j))›*))[1])\n (*proven 3 subgoals*) .\n qed\n qed\n qed\n show \"mods' = IArray.of_fun (λj. p * di' !! j * di' !! Suc j) (m - 1)\"\n unfolding mods' di' dk' mod1 modk\n (*goal: ‹IArray.of_fun (λjj. if jj = k - 1 then p * d_of dmu'' (k - 1) * d_of dmu'' k else if jj = k then p * d_of dmu'' k * d_of dmu'' (Suc k) else mods !! jj) (m - 1) = IArray.of_fun (λj. p * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! j * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! Suc j) (m - 1)›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λjj. if jj = k - 1 then p * d_of dmu'' (k - 1) * d_of dmu'' k else if jj = k then p * d_of dmu'' k * d_of dmu'' (Suc k) else mods !! jj) (m - 1) = IArray.of_fun (λj. p * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! j * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! Suc j) (m - 1)›*)\n proof (goal_cases)\n (*goal: ‹⋀i. i < m - 1 ⟹ (if i = k - 1 then p * d_of dmu'' (k - 1) * d_of dmu'' k else if i = k then p * d_of dmu'' k * d_of dmu'' (Suc k) else mods !! i) = p * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! i * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! Suc i›*)\n case (1 j) (*‹j < m - 1›*)\n hence j: \"j < Suc m\" \"Suc j < Suc m\"\n apply -\n (*goals:\n 1. ‹j < m - 1 ⟹ j < Suc m›\n 2. ‹j < m - 1 ⟹ Suc j < Suc m›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"?case\"\n (*goal: ‹(if j = k - 1 then p * d_of dmu'' (k - 1) * d_of dmu'' k else if j = k then p * d_of dmu'' k * d_of dmu'' (Suc k) else mods !! j) = p * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! j * IArray.of_fun (λii. if ii = k then d_of dmu'' k else di !! ii) (Suc m) !! Suc j›*)\n unfolding IArray.of_fun_nth[OF 1] IArray.of_fun_nth[OF j ( 1 )] IArray.of_fun_nth[OF j ( 2 )] id(4)\n (*goal: ‹(if j = k - 1 then p * d_of dmu'' (k - 1) * d_of dmu'' k else if j = k then p * d_of dmu'' k * d_of dmu'' (Suc k) else p * di !! j * di !! Suc j) = p * (if j = k then d_of dmu'' k else di !! j) * (if Suc j = k then d_of dmu'' k else di !! Suc j)›*)\n using k (*‹k < m› ‹k ≠ 0›*) di (*‹?i2 ≤ m ⟹ di !! ?i2 = d_of dmu ?i2›*) dk1 (*‹d_of dmu (k - 1) = d_of dmu'' (k - 1)›*) dkS1 (*‹d_of dmu (Suc k) = d_of dmu'' (Suc k)›*) by auto\n qed\n qed\nqed\n\n\nlemma LLL_add_row: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod_weak fs mfs dmu p first b\" \n and res: \"basis_reduction_mod_add_row p mfs dmu i j = (mfs', dmu')\" \n and res': \"LLL_add_row p state i j = state'\" \n and i: \"i < m\"\n and j: \"j < i\"\nshows \"state_impl_inv p mfs' dmu' state'\"\nproof (-)\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n note inv = LLL_invD_modw[OF Linv] (*‹length (mfs::int vec list) = (m::nat)› ‹length fs = m› ‹lattice_of fs = L› ‹gs.lin_indpt_list (map of_int_hom.vec_hom fs)› ‹map (map_vec (λx::int. x symmod (p::int))) (fs::int vec list) = (mfs::int vec list)› ‹∀i'<m. ∀j'<i'. ¦dμ fs i' j'¦ < p * d fs j' * d fs (Suc j')› ‹∀i'<m. ∀j'<m. dμ fs i' j' = dmu $$ (i', j')› ‹?i < m ⟹ fs ! ?i ∈ carrier_vec n› ‹set fs ⊆ carrier_vec n› ‹?i < m ⟹ gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) ?i ∈ Rn› ‹(?i::nat) < (m::nat) ⟹ (mfs::int vec list) ! ?i ∈ carrier_vec (n::nat)› ‹set (mfs::int vec list) ⊆ carrier_vec (n::nat)› and more 3 facts*)\n obtain fsi and dmui and di and mods where state: \"state = (fsi, dmui, di, mods)\"\n (*goal: ‹(⋀fsi dmui di mods. state = (fsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n apply (cases state)\n (*goal: ‹(⋀fsi dmui di mods. state = (fsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain fsi' and dmui' and di' and mods' where state': \"state' = (fsi', dmui', di', mods')\"\n (*goal: ‹(⋀fsi' dmui' di' mods'. state' = (fsi', dmui', di', mods') ⟹ thesis) ⟹ thesis›*)\n apply (cases state')\n (*goal: ‹(⋀fsi' dmui' di' mods'. state' = (fsi', dmui', di', mods') ⟹ thesis) ⟹ thesis›*)\n by auto\n from impl[unfolded state, simplified] (*‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0)›*) have id: \"fsi = mfs\" \"di = IArray.of_fun (d_of dmu) (Suc m)\" \"dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m\" \"mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)\"\n apply -\n (*goals:\n 1. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ fsi = mfs›\n 2. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ di = IArray.of_fun (d_of dmu) (Suc m)›\n 3. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m›\n 4. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * IArray.list_of di ! j * IArray.list_of di ! Suc j) (m - Suc 0) ⟹ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n let ?c = \"round_num_denom (dmu $$ (i, j)) (d_of dmu (Suc j))\"\n let ?c' = \"round_num_denom (dmui !! i !! j) (di !! Suc j)\"\n obtain c where c: \"?c = c\"\n (*goal: ‹(⋀c. round_num_denom (dmu $$ (i, j)) (d_of dmu (Suc j)) = c ⟹ thesis) ⟹ thesis›*)\n by auto\n have c': \"?c' = c\"\n unfolding id c[symmetric]\n (*goal: ‹round_num_denom (IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! i !! j) (IArray.of_fun (d_of dmu) (Suc m) !! Suc j) = round_num_denom (dmu $$ (i, j)) (d_of dmu (Suc j))›*)\n using i (*‹(i::nat) < (m::nat)›*) j (*‹j < i›*) apply (subst (1 2) IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦i < m; j < i⟧ ⟹ Suc j < Suc m›\n 2. ‹⟦i < m; j < i⟧ ⟹ i < m›\n 3. ‹⟦i < m; j < i⟧ ⟹ round_num_denom (IArray.of_fun (λj. dmu $$ (i, j)) i !! j) (d_of dmu (Suc j)) = round_num_denom (dmu $$ (i, j)) (d_of dmu (Suc j))›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*discuss goal 3*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦i < m; j < i⟧ ⟹ j < i›\n 2. ‹⟦i < m; j < i⟧ ⟹ round_num_denom (dmu $$ (i, j)) (d_of dmu (Suc j)) = round_num_denom (dmu $$ (i, j)) (d_of dmu (Suc j))›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*)\n (*proven 3 subgoals*) .\n have drop: \"drop i fsi = mfs ! i # drop (Suc i) mfs\"\n unfolding id\n (*goal: ‹drop i mfs = mfs ! i # drop (Suc i) mfs›*)\n using ‹length mfs = m› (*‹length mfs = m›*) i (*‹i < m›*) by (metis Cons_nth_drop_Suc (*‹?i < length ?xs ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*))\n note res = res[unfolded basis_reduction_mod_add_row_def Let_def c, symmetric] (*‹(mfs', dmu') = (if c = 0 then (mfs, dmu) else (mfs[i := map_vec (λx. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j)], mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j'))))›*)\n note res' = res'[unfolded state state' split LLL_add_row_def Let_def c', symmetric] (*‹(fsi', dmui', di', mods') = (if c = 0 then (fsi, dmui, di, mods) else perform_add_row p (fsi, dmui, di, mods) i j c (dmui !! i) (dmui !! i !! j) (di !! Suc j))›*)\n show \"?thesis\"\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n proof (cases \"c = 0\")\n (*goals:\n 1. ‹c = 0 ⟹ state_impl_inv p mfs' dmu' state'›\n 2. ‹c ≠ 0 ⟹ state_impl_inv p mfs' dmu' state'›*)\n case True (*‹c = 0›*)\n from res[unfolded True] (*‹(mfs', dmu') = (if 0 = 0 then (mfs, dmu) else (mfs[i := map_vec (λx. x symmod p) (mfs ! i - 0 ⋅⇩v mfs ! j)], mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - 0 * dmu $$ (j, j') else (dmu $$ (i, j') - 0 * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j'))))›*) res'[unfolded True] (*‹(fsi', dmui', di', mods') = (if 0 = 0 then (fsi, dmui, di, mods) else perform_add_row p (fsi, dmui, di, mods) i j 0 (dmui !! i) (dmui !! i !! j) (di !! Suc j))›*) show \"?thesis\"\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n unfolding state'\n (*goal: ‹state_impl_inv p mfs' dmu' (fsi', dmui', di', mods')›*)\n using id (*‹fsi = mfs› ‹di = IArray.of_fun (d_of dmu) (Suc m)› ‹(dmui::int iarray iarray) = IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. (dmu::int mat) $$ (i, j)) i) (m::nat)› ‹mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*) by auto\n next\n (*goal: ‹c ≠ 0 ⟹ state_impl_inv p mfs' dmu' state'›*)\n case False (*‹c ≠ 0›*)\n hence False: \"(c = 0) = False\"\n by simp\n note res = res[unfolded Let_def False if_False] (*‹(mfs', dmu') = (mfs[i := map_vec (λx. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j)], mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')))›*)\n from res (*‹(mfs', dmu') = (mfs[i := map_vec (λx. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j)], mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')))›*) have mfs': \"mfs' = mfs[i := map_vec (λx. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j)]\"\n by auto\n from res (*‹(mfs', dmu') = (mfs[i := map_vec (λx. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j)], mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')))›*) have dmu': \"dmu' = mat m m (λ(i', j').\n if i' = i ∧ j' ≤ j\n then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j')\n else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j'))\n else dmu $$ (i', j'))\"\n by auto\n note res' = res'[unfolded Let_def False if_False perform_add_row_def drop list.simps split_at_def split] (*‹(fsi', dmui', di', mods') = (take i fsi @ vec n (λk. (mfs ! i $ k - c * fsi ! j $ k) symmod p) # drop (Suc i) mfs, IArray.of_fun (λii. if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) m, di, mods)›*)\n from res' (*‹(fsi', dmui', di', mods') = (take i fsi @ vec n (λk. (mfs ! i $ k - c * fsi ! j $ k) symmod p) # drop (Suc i) mfs, IArray.of_fun (λii. if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) m, di, mods)›*) have fsi': \"fsi' = take i fsi @ vec n (λk. (mfs ! i $ k - c * mfs ! j $ k) symmod p) # drop (Suc i) mfs\"\n by (auto simp: id (*‹(fsi::int vec list) = (mfs::int vec list)› ‹(di::int iarray) = IArray.of_fun (d_of (dmu::int mat)) (Suc (m::nat))› ‹(dmui::int iarray iarray) = IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. (dmu::int mat) $$ (i, j)) i) (m::nat)› ‹(mods::int iarray) = IArray.of_fun (λj::nat. (p::int) * (di::int iarray) !! j * di !! Suc j) ((m::nat) - (1::nat))›*))\n from res' (*‹(fsi', dmui', di', mods') = (take i fsi @ vec n (λk. (mfs ! i $ k - c * fsi ! j $ k) symmod p) # drop (Suc i) mfs, IArray.of_fun (λii. if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) m, di, mods)›*) have di': \"di' = di\" and mods': \"mods' = mods\"\n apply -\n (*goals:\n 1. ‹(fsi', dmui', di', mods') = (take i fsi @ vec n (λk. (mfs ! i $ k - c * fsi ! j $ k) symmod p) # drop (Suc i) mfs, IArray.of_fun (λii. if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) m, di, mods) ⟹ di' = di›\n 2. ‹(fsi', dmui', di', mods') = (take i fsi @ vec n (λk. (mfs ! i $ k - c * fsi ! j $ k) symmod p) # drop (Suc i) mfs, IArray.of_fun (λii. if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) m, di, mods) ⟹ mods' = mods›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from res' (*‹(fsi', dmui', di', mods') = (take i fsi @ vec n (λk. (mfs ! i $ k - c * fsi ! j $ k) symmod p) # drop (Suc i) mfs, IArray.of_fun (λii. if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) m, di, mods)›*) have dmui': \"dmui' = IArray.of_fun (λii. if i = ii\n then IArray.of_fun\n (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod (mods !! jj)\n else if jj = j then dmui !! i !! j - c * di !! (Suc j) else dmui !! i !! jj)\n i\n else dmui !! ii) m\"\n by auto\n show \"?thesis\"\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n unfolding state' state_impl_inv.simps\n (*goal: ‹fsi' = mfs' ∧ di' = IArray.of_fun (d_of dmu') (Suc m) ∧ dmui' = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m ∧ mods' = IArray.of_fun (λj. p * di' !! j * di' !! Suc j) (m - 1)›*)\n proof (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹fsi' = mfs'›\n 2. ‹di' = IArray.of_fun (d_of dmu') (Suc m)›\n 3. ‹dmui' = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m›\n 4. ‹mods' = IArray.of_fun (λj. p * di' !! j * di' !! Suc j) (m - 1)›*)\n from inv(11) (*‹?i < m ⟹ mfs ! ?i ∈ carrier_vec n›*) i (*‹i < m›*) j (*‹j < i›*) have vec: \"mfs ! i ∈ carrier_vec n\" \"mfs ! j ∈ carrier_vec n\"\n apply -\n (*goals:\n 1. ‹⟦⋀i. i < m ⟹ mfs ! i ∈ carrier_vec n; i < m; j < i⟧ ⟹ mfs ! i ∈ carrier_vec n›\n 2. ‹⟦⋀i. i < m ⟹ mfs ! i ∈ carrier_vec n; i < m; j < i⟧ ⟹ mfs ! j ∈ carrier_vec n›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence id': \"map_vec (λx. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j) = vec n (λk. (mfs ! i $ k - c * mfs ! j $ k) symmod p)\"\n apply (intro eq_vecI (*‹⟦⋀i. i < dim_vec ?w ⟹ ?v $ i = ?w $ i; dim_vec ?v = dim_vec ?w⟧ ⟹ ?v = ?w›*))\n (*goals:\n 1. ‹⋀ia::nat. ⟦(mfs::int vec list) ! (i::nat) ∈ carrier_vec (n::nat); mfs ! (j::nat) ∈ carrier_vec n; ia < dim_vec (vec n (λk::nat. (mfs ! i $ k - (c::int) * mfs ! j $ k) symmod (p::int)))⟧ ⟹ map_vec (λx::int. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j) $ ia = vec n (λk::nat. (mfs ! i $ k - c * mfs ! j $ k) symmod p) $ ia›\n 2. ‹⟦(mfs::int vec list) ! (i::nat) ∈ carrier_vec (n::nat); mfs ! (j::nat) ∈ carrier_vec n⟧ ⟹ dim_vec (map_vec (λx::int. x symmod (p::int)) (mfs ! i - (c::int) ⋅⇩v mfs ! j)) = dim_vec (vec n (λk::nat. (mfs ! i $ k - c * mfs ! j $ k) symmod p))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"mods' = IArray.of_fun (λj. p * di' !! j * di' !! Suc j) (m - 1)\"\n using id (*‹fsi = mfs› ‹(di::int iarray) = IArray.of_fun (d_of (dmu::int mat)) (Suc (m::nat))› ‹dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m› ‹mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*) unfolding mods' di'\n (*goal: ‹mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*)\n by auto\n show \"fsi' = mfs'\"\n unfolding fsi' mfs' id\n (*goal: ‹take i mfs @ vec n (λk. (mfs ! i $ k - c * mfs ! j $ k) symmod p) # drop (Suc i) mfs = mfs[i := map_vec (λx. x symmod p) (mfs ! i - c ⋅⇩v mfs ! j)]›*)\n unfolding id'\n (*goal: ‹take i mfs @ vec n (λk. (mfs ! i $ k - c * mfs ! j $ k) symmod p) # drop (Suc i) mfs = mfs[i := vec n (λk. (mfs ! i $ k - c * mfs ! j $ k) symmod p)]›*)\n using ‹length mfs = m› (*‹length mfs = m›*) i (*‹(i::nat) < (m::nat)›*) by (simp add: upd_conv_take_nth_drop (*‹(?i::nat) < length (?xs::?'a::type list) ⟹ ?xs[?i := ?a::?'a::type] = take ?i ?xs @ ?a # drop (Suc ?i) ?xs›*))\n show \"di' = IArray.of_fun (d_of dmu') (Suc m)\"\n unfolding dmu' di' id d_of_def\n (*goal: ‹IArray.of_fun (λi. if i = 0 then 1 else dmu $$ (i - 1, i - 1)) (Suc m) = IArray.of_fun (λia. if ia = 0 then 1 else mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * (if j' = 0 then 1 else dmu $$ (j' - 1, j' - 1)) * (if Suc j' = 0 then 1 else dmu $$ (Suc j' - 1, Suc j' - 1))) else dmu $$ (i', j')) $$ (ia - 1, ia - 1)) (Suc m)›*)\n apply (intro iarray_cong (*‹⟦(?n::nat) = (?m::nat); ⋀i::nat. i < ?m ⟹ (?f::nat ⇒ ?'a::type) i = (?g::nat ⇒ ?'a::type) i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) if_cong (*‹⟦(?b::bool) = (?c::bool); ?c ⟹ (?x::?'a::type) = (?u::?'a::type); ¬ ?c ⟹ (?y::?'a::type) = (?v::?'a::type)⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*) refl (*‹(?t::?'a::type) = ?t›*))\n (*goal: ‹IArray.of_fun (λi. if i = 0 then 1 else dmu $$ (i - 1, i - 1)) (Suc m) = IArray.of_fun (λia. if ia = 0 then 1 else mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * (if j' = 0 then 1 else dmu $$ (j' - 1, j' - 1)) * (if Suc j' = 0 then 1 else dmu $$ (Suc j' - 1, Suc j' - 1))) else dmu $$ (i', j')) $$ (ia - 1, ia - 1)) (Suc m)›*)\n apply (insert i (*‹i < m›*) j (*‹j < i›*))\n (*goal: ‹⋀ia::nat. ⟦ia < Suc (m::nat); ia ≠ (0::nat)⟧ ⟹ (dmu::int mat) $$ (ia - (1::nat), ia - (1::nat)) = mat m m (λ(i'::nat, j'::nat). if i' = (i::nat) ∧ j' ≤ (j::nat) then if j' = j then dmu $$ (i, j') - (c::int) * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod ((p::int) * (if j' = (0::nat) then 1::int else dmu $$ (j' - (1::nat), j' - (1::nat))) * (if Suc j' = (0::nat) then 1::int else dmu $$ (Suc j' - (1::nat), Suc j' - (1::nat)))) else dmu $$ (i', j')) $$ (ia - (1::nat), ia - (1::nat))›*)\n by auto\n show \"dmui' = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m\"\n unfolding dmui'\n (*goal: ‹IArray.of_fun (λii::nat. if (i::nat) = ii then IArray.of_fun (λjj::nat. if jj < (j::nat) then ((dmui::int iarray iarray) !! i !! jj - (c::int) * dmui !! j !! jj) symmod (mods::int iarray) !! jj else if jj = j then dmui !! i !! j - c * (di::int iarray) !! Suc j else dmui !! i !! jj) i else dmui !! ii) (m::nat) = IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. (dmu'::int mat) $$ (i, j)) i) m›*)\n proof (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹⋀ia. ia < m ⟹ (if i = ia then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ia) = IArray.of_fun (λj. dmu' $$ (ia, j)) ia›*)\n fix ii\n assume ii: \"ii < m\" (*‹(ii::nat) < (m::nat)›*)\n show \"(if i = ii\n then IArray.of_fun\n (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod (mods !! jj)\n else if jj = j then dmui !! i !! j - c * di !! (Suc j) else dmui !! i !! jj)\n i\n else dmui !! ii) =\n IArray.of_fun (λj. dmu' $$ (ii, j)) ii\"\n proof (cases \"i = ii\")\n (*goals:\n 1. ‹i = ii ⟹ (if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) = IArray.of_fun (λj. dmu' $$ (ii, j)) ii›\n 2. ‹i ≠ ii ⟹ (if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) = IArray.of_fun (λj. dmu' $$ (ii, j)) ii›*)\n case False (*‹i ≠ ii›*)\n hence \"*\": \"(i = ii) = False\"\n by auto\n show \"?thesis\"\n (*goal: ‹(if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) = IArray.of_fun (λj. dmu' $$ (ii, j)) ii›*)\n unfolding \"*\" if_False id dmu'\n (*goal: ‹IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m !! ii = IArray.of_fun (λja. mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (ii, ja)) ii›*)\n using False (*‹(i::nat) ≠ (ii::nat)›*) i (*‹i < m›*) j (*‹j < i›*) ii (*‹(ii::nat) < (m::nat)›*) unfolding IArray.of_fun_nth[OF ii]\n (*goal: ‹IArray.of_fun (λj. dmu $$ (ii, j)) ii = IArray.of_fun (λja. mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (ii, ja)) ii›*)\n apply (intro iarray_cong (*‹⟦(?n::nat) = (?m::nat); ⋀i::nat. i < ?m ⟹ (?f::nat ⇒ ?'a) i = (?g::nat ⇒ ?'a) i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹(?t::?'a) = ?t›*))\n (*goal: ‹IArray.of_fun (λj. dmu $$ (ii, j)) ii = IArray.of_fun (λja. mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (ii, ja)) ii›*)\n by auto\n next\n (*goal: ‹i = ii ⟹ (if i = ii then IArray.of_fun (λjj. if jj < j then (dmui !! i !! jj - c * dmui !! j !! jj) symmod mods !! jj else if jj = j then dmui !! i !! j - c * di !! Suc j else dmui !! i !! jj) i else dmui !! ii) = IArray.of_fun (λj. dmu' $$ (ii, j)) ii›*)\n case True (*‹i = ii›*)\n hence \"*\": \"(i = ii) = True\"\n by auto\n from i (*‹i < m›*) j (*‹(j::nat) < (i::nat)›*) have \"j < m\"\n by simp\n show \"?thesis\"\n (*goal: ‹(if (i::nat) = (ii::nat) then IArray.of_fun (λjj::nat. if jj < (j::nat) then ((dmui::int iarray iarray) !! i !! jj - (c::int) * dmui !! j !! jj) symmod (mods::int iarray) !! jj else if jj = j then dmui !! i !! j - c * (di::int iarray) !! Suc j else dmui !! i !! jj) i else dmui !! ii) = IArray.of_fun (λj::nat. (dmu'::int mat) $$ (ii, j)) ii›*)\n unfolding \"*\" if_True dmu' id IArray.of_fun_nth[OF i] IArray.of_fun_nth[OF ‹j < m›]\n (*goal: ‹IArray.of_fun (λjj::nat. if jj < (j::nat) then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i::nat, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc (m::nat)) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) i = IArray.of_fun (λja::nat. mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (ii::nat, ja)) ii›*)\n unfolding True[symmetric]\n (*goal: ‹IArray.of_fun (λjj. if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) i = IArray.of_fun (λja. mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, ja)) i›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λjj. if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) i = IArray.of_fun (λja. mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, ja)) i›*)\n proof (goal_cases)\n (*goal: ‹⋀ia. ia < i ⟹ (if ia < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! ia - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! ia) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! ia else if ia = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! ia) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, ia)›*)\n case jj: (1 jj) (*‹(jj::nat) < (i::nat)›*)\n consider (1)\"jj < j\" | (2)\"jj = j\" | (3)\"jj > j\"\n (*goal: ‹⟦jj < j ⟹ thesis; jj = j ⟹ thesis; j < jj ⟹ thesis⟧ ⟹ thesis›*)\n by linarith\n thus \"?case\"\n (*goal: ‹(if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n proof (cases)\n (*goals:\n 1. ‹(jj::nat) < (j::nat) ⟹ (if jj < j then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i::nat, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc (m::nat)) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›\n 2. ‹(jj::nat) = (j::nat) ⟹ (if jj < j then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i::nat, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc (m::nat)) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›\n 3. ‹(j::nat) < (jj::nat) ⟹ (if jj < j then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i::nat, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc (m::nat)) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n case 1 (*‹jj < j›*)\n thus \"?thesis\"\n (*goal: ‹(if (jj::nat) < (j::nat) then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i::nat, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc (m::nat)) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n using jj (*‹(jj::nat) < (i::nat)›*) i (*‹i < m›*) j (*‹j < i›*) unfolding id(4)\n (*goal: ‹(if (jj::nat) < (j::nat) then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i::nat, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc (m::nat)) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n apply (subst (1 2 3 4 5 6) IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦(jj::nat) < (j::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ j < i›\n 2. ‹⟦(jj::nat) < (j::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ jj < m - (1::nat)›\n 3. ‹⟦(jj::nat) < (j::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ Suc jj < Suc m›\n 4. ‹⟦(jj::nat) < (j::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ jj < Suc m›\n 5. ‹⟦(jj::nat) < (j::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ jj < j›\n 6. ‹⟦(jj::nat) < (j::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ jj < i›\n 7. ‹⟦(jj::nat) < (j::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ (if jj < j then ((dmu::int mat) $$ (i, jj) - (c::int) * dmu $$ (j, jj)) symmod ((p::int) * d_of dmu jj * d_of dmu (Suc jj)) else if jj = j then dmu $$ (i, j) - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*proven 7 subgoals*) .\n next\n (*goals:\n 1. ‹jj = j ⟹ (if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›\n 2. ‹j < jj ⟹ (if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n case 2 (*‹jj = j›*)\n thus \"?thesis\"\n (*goal: ‹(if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n using jj (*‹(jj::nat) < (i::nat)›*) i (*‹i < m›*) j (*‹j < i›*) apply -\n (*goal: ‹(if (jj::nat) < (j::nat) then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i::nat, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc (m::nat)) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n apply (subst (5 6) IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦jj = j; jj < i; i < m; j < i⟧ ⟹ j < i›\n 2. ‹⟦jj = j; jj < i; i < m; j < i⟧ ⟹ jj < m - 1›\n 3. ‹⟦jj = j; jj < i; i < m; j < i⟧ ⟹ (if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod (p * IArray.of_fun (d_of dmu) (Suc m) !! jj * IArray.of_fun (d_of dmu) (Suc m) !! Suc jj) else if jj = j then dmu $$ (i, j) - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›\n discuss goal 1*)\n apply ((auto simp: d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))[1])\n (*goal: ‹⟦jj = j; jj < i; i < m; j < i⟧ ⟹ (if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod (p * IArray.of_fun (d_of dmu) (Suc m) !! jj * IArray.of_fun (d_of dmu) (Suc m) !! Suc jj) else if jj = j then dmu $$ (i, j) - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n apply (msorry)\n (*proven 3 subgoals*) .\n next\n (*goal: ‹j < jj ⟹ (if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n case 3 (*‹(j::nat) < (jj::nat)›*)\n thus \"?thesis\"\n (*goal: ‹(if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n using jj (*‹jj < i›*) i (*‹i < m›*) j (*‹(j::nat) < (i::nat)›*) apply -\n (*goal: ‹(if jj < j then (IArray.of_fun (λj. dmu $$ (i, j)) i !! jj - c * IArray.of_fun (λja. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj. p * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - 1) !! jj else if jj = j then IArray.of_fun (λj. dmu $$ (i, j)) i !! j - c * IArray.of_fun (d_of dmu) (Suc m) !! Suc j else IArray.of_fun (λj. dmu $$ (i, j)) i !! jj) = mat m m (λ(i', j'). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n apply (subst (7) IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦(j::nat) < (jj::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ Suc j < Suc m›\n 2. ‹⟦(j::nat) < (jj::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ (if jj < j then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * d_of dmu (Suc j) else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›\n discuss goal 1*)\n apply ((auto simp: d_of_def (*‹d_of (?dmu::int mat) (?i::nat) = (if ?i = (0::nat) then 1::int else ?dmu $$ (?i - (1::nat), ?i - (1::nat)))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))[1])\n (*goal: ‹⟦(j::nat) < (jj::nat); jj < (i::nat); i < (m::nat); j < i⟧ ⟹ (if jj < j then (IArray.of_fun (λj::nat. (dmu::int mat) $$ (i, j)) i !! jj - (c::int) * IArray.of_fun (λja::nat. dmu $$ (j, ja)) j !! jj) symmod IArray.of_fun (λj::nat. (p::int) * IArray.of_fun (d_of dmu) (Suc m) !! j * IArray.of_fun (d_of dmu) (Suc m) !! Suc j) (m - (1::nat)) !! jj else if jj = j then IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! j - c * d_of dmu (Suc j) else IArray.of_fun (λj::nat. dmu $$ (i, j)) i !! jj) = mat m m (λ(i'::nat, j'::nat). if i' = i ∧ j' ≤ j then if j' = j then dmu $$ (i, j') - c * dmu $$ (j, j') else (dmu $$ (i, j') - c * dmu $$ (j, j')) symmod (p * d_of dmu j' * d_of dmu (Suc j')) else dmu $$ (i', j')) $$ (i, jj)›*)\n apply (msorry)\n (*proven 2 subgoals*) .\n qed\n qed\n qed\n qed\n qed\n qed\nqed\n\n\nlemma LLL_max_gso_norm_di: assumes di: \"di = IArray.of_fun (d_of dmu) (Suc m)\"\n and m: \"m ≠ 0\" \nshows \"LLL_max_gso_norm_di first di = compute_max_gso_norm first dmu\"\nproof (-)\n (*goal: ‹LLL_max_gso_norm_di first di = compute_max_gso_norm first dmu›*)\n have di: \"j ≤ m ⟹ di !! j = d_of dmu j\" for j\n unfolding di\n (*goal: ‹j ≤ m ⟹ IArray.of_fun (d_of dmu) (Suc m) !! j = d_of dmu j›*)\n apply (subst IArray.of_fun_nth (*‹(?i::nat) < (?n::nat) ⟹ IArray.of_fun (?f::nat ⇒ ?'a) ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹j ≤ m ⟹ j < Suc m›\n 2. ‹j ≤ m ⟹ d_of dmu j = d_of dmu j›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have id: \"(m = 0) = False\"\n using m (*‹m ≠ 0›*) by auto\n show \"?thesis\"\n (*goal: ‹LLL_max_gso_norm_di first di = compute_max_gso_norm first dmu›*)\n proof (cases first)\n (*goals:\n 1. ‹first ⟹ LLL_max_gso_norm_di first di = compute_max_gso_norm first dmu›\n 2. ‹¬ first ⟹ LLL_max_gso_norm_di first di = compute_max_gso_norm first dmu›*)\n case False (*‹¬ first›*)\n hence id': \"first = False\"\n by auto\n show \"?thesis\"\n (*goal: ‹LLL_max_gso_norm_di first di = compute_max_gso_norm first dmu›*)\n unfolding LLL_max_gso_norm_di_def compute_max_gso_norm_def id id' if_False\n (*goal: ‹(case max_list_rats_with_index (map (λi. (di !! Suc i, di !! i, i)) [0..<m]) of (num, denom, xa) ⇒ (rat_of_int num / rat_of_int denom, xa)) = (case max_list_rats_with_index (map (λi. (d_of dmu (Suc i), d_of dmu i, i)) [0..<m]) of (num, denom, xa) ⇒ (rat_of_int num / rat_of_int denom, xa))›*)\n apply (intro if_cong (*‹⟦(?b::bool) = (?c::bool); ?c ⟹ (?x::?'a::type) = (?u::?'a::type); ¬ ?c ⟹ (?y::?'a::type) = (?v::?'a::type)⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*) refl (*‹(?t::?'a::type) = ?t›*) arg_cong[of _ _ \"λ xs. case max_list_rats_with_index xs of (num, denom, i) ⇒ (rat_of_int num / rat_of_int denom, i)\"] (*‹(?x::(int × int × nat) list) = (?y::(int × int × nat) list) ⟹ (case max_list_rats_with_index ?x of (num::int, denom::int, i::nat) ⇒ (rat_of_int num / rat_of_int denom, i)) = (case max_list_rats_with_index ?y of (num::int, denom::int, i::nat) ⇒ (rat_of_int num / rat_of_int denom, i))›*))\n (*goal: ‹(case max_list_rats_with_index (map (λi. (di !! Suc i, di !! i, i)) [0..<m]) of (num, denom, xa) ⇒ (rat_of_int num / rat_of_int denom, xa)) = (case max_list_rats_with_index (map (λi. (d_of dmu (Suc i), d_of dmu i, i)) [0..<m]) of (num, denom, xa) ⇒ (rat_of_int num / rat_of_int denom, xa))›*)\n apply (unfold map_eq_conv (*‹(map ?f ?xs = map ?g ?xs) = (∀x∈set ?xs. ?f x = ?g x)›*))\n (*goal: ‹map (λi. (di !! Suc i, di !! i, i)) [0..<m] = map (λi. (d_of dmu (Suc i), d_of dmu i, i)) [0..<m]›*)\n apply (intro ballI (*‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*))\n (*goal: ‹∀x∈set [0..<m]. (di !! Suc x, di !! x, x) = (d_of dmu (Suc x), d_of dmu x, x)›*)\n apply (subst (1 2) di (*‹?j2 ≤ m ⟹ di !! ?j2 = d_of dmu ?j2›*))\n (*goals:\n 1. ‹⋀x::nat. x ∈ set [0::nat..<m::nat] ⟹ x ≤ m›\n 2. ‹⋀x::nat. x ∈ set [0::nat..<m::nat] ⟹ Suc x ≤ m›\n 3. ‹⋀x::nat. x ∈ set [0::nat..<m::nat] ⟹ (d_of (dmu::int mat) (Suc x), d_of dmu x, x) = (d_of dmu (Suc x), d_of dmu x, x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n next\n (*goal: ‹first::bool ⟹ LLL_max_gso_norm_di first (di::int iarray) = compute_max_gso_norm first (dmu::int mat)›*)\n case True (*‹first›*)\n hence id': \"first = True\"\n by auto\n show \"?thesis\"\n (*goal: ‹LLL_max_gso_norm_di first di = compute_max_gso_norm first dmu›*)\n unfolding LLL_max_gso_norm_di_def compute_max_gso_norm_def id id' if_False if_True\n (*goal: ‹(rat_of_int (di !! 1), 0) = (case max_list_rats_with_index (map (λi. (d_of dmu (Suc i), d_of dmu i, i)) [0..<1]) of (num, denom, xa) ⇒ (rat_of_int num / rat_of_int denom, xa))›*)\n using m (*‹m ≠ 0›*) di[of 1] (*‹(1::nat) ≤ (m::nat) ⟹ (di::int iarray) !! (1::nat) = d_of (dmu::int mat) (1::nat)›*) by (simp add: d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))\n qed\nqed\n\nlemma LLL_max_gso_quot: assumes di: \"di = IArray.of_fun (d_of dmu) (Suc m)\"\n and prods: \"state_iso_inv di_prods di\" \nshows \"LLL_max_gso_quot di_prods = compute_max_gso_quot dmu\"\nproof (-)\n (*goal: ‹LLL_max_gso_quot di_prods = compute_max_gso_quot dmu›*)\n have di: \"j ≤ m ⟹ di !! j = d_of dmu j\" for j\n unfolding di\n (*goal: ‹j ≤ m ⟹ IArray.of_fun (d_of dmu) (Suc m) !! j = d_of dmu j›*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹j ≤ m ⟹ j < Suc m›\n 2. ‹j ≤ m ⟹ d_of dmu j = d_of dmu j›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"?thesis\"\n (*goal: ‹LLL_max_gso_quot (di_prods::(int × int) iarray) = compute_max_gso_quot (dmu::int mat)›*)\n unfolding LLL_max_gso_quot_def compute_max_gso_quot_def prods[unfolded state_iso_inv_def]\n (*goal: ‹max_list_rats_with_index (map (λi. case IArray.of_fun (λi. (di !! (i + 1) * di !! (i + 1), di !! (i + 2) * di !! i)) (m - 1) !! i of (l, r) ⇒ (l, r, Suc i)) [0..<m - 1]) = max_list_rats_with_index (map (λi. (d_of dmu (i + 1) * d_of dmu (i + 1), d_of dmu (i + 2) * d_of dmu i, Suc i)) [0..<m - 1])›*)\n apply (intro if_cong (*‹⟦?b = ?c; ?c ⟹ ?x = ?u; ¬ ?c ⟹ ?y = ?v⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*) refl (*‹?t = ?t›*) arg_cong[of _ _ max_list_rats_with_index] (*‹?x = ?y ⟹ max_list_rats_with_index ?x = max_list_rats_with_index ?y›*))\n (*goal: ‹max_list_rats_with_index (map (λi. case IArray.of_fun (λi. (di !! (i + 1) * di !! (i + 1), di !! (i + 2) * di !! i)) (m - 1) !! i of (l, r) ⇒ (l, r, Suc i)) [0..<m - 1]) = max_list_rats_with_index (map (λi. (d_of dmu (i + 1) * d_of dmu (i + 1), d_of dmu (i + 2) * d_of dmu i, Suc i)) [0..<m - 1])›*)\n apply (unfold map_eq_conv (*‹(map ?f ?xs = map ?g ?xs) = (∀x∈set ?xs. ?f x = ?g x)›*) Let_def (*‹Let ?s ?f ≡ ?f ?s›*))\n (*goal: ‹map (λi. case IArray.of_fun (λi. (di !! (i + 1) * di !! (i + 1), di !! (i + 2) * di !! i)) (m - 1) !! i of (l, r) ⇒ (l, r, Suc i)) [0..<m - 1] = map (λi. (d_of dmu (i + 1) * d_of dmu (i + 1), d_of dmu (i + 2) * d_of dmu i, Suc i)) [0..<m - 1]›*)\n apply (intro ballI (*‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*))\n (*goal: ‹∀x∈set [0..<m - 1]. (case IArray.of_fun (λi. (di !! (i + 1) * di !! (i + 1), di !! (i + 2) * di !! i)) (m - 1) !! x of (l, r) ⇒ (l, r, Suc x)) = (d_of dmu (x + 1) * d_of dmu (x + 1), d_of dmu (x + 2) * d_of dmu x, Suc x)›*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⋀x. x ∈ set [0..<m - 1] ⟹ x < m - 1›\n 2. ‹⋀x. x ∈ set [0..<m - 1] ⟹ (case (di !! (x + 1) * di !! (x + 1), di !! (x + 2) * di !! x) of (l, r) ⇒ (l, r, Suc x)) = (d_of dmu (x + 1) * d_of dmu (x + 1), d_of dmu (x + 2) * d_of dmu x, Suc x)›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply (unfold split (*‹(case (?a::?'b, ?b::?'c) of (c::?'b, d::?'c) ⇒ (?f::?'b ⇒ ?'c ⇒ ?'a) c d) = ?f ?a ?b›*))\n (*goal: ‹⋀x. x ∈ set [0..<m - 1] ⟹ (case (di !! (x + 1) * di !! (x + 1), di !! (x + 2) * di !! x) of (l, r) ⇒ (l, r, Suc x)) = (d_of dmu (x + 1) * d_of dmu (x + 1), d_of dmu (x + 2) * d_of dmu x, Suc x)›*)\n apply (subst (1 2 3 4) di (*‹(?j2::nat) ≤ (m::nat) ⟹ (di::int iarray) !! ?j2 = d_of (dmu::int mat) ?j2›*))\n (*goals:\n 1. ‹⋀x. x ∈ set [0..<m - 1] ⟹ x ≤ m›\n 2. ‹⋀x. x ∈ set [0..<m - 1] ⟹ x + 2 ≤ m›\n 3. ‹⋀x. x ∈ set [0..<m - 1] ⟹ x + 1 ≤ m›\n 4. ‹⋀x. x ∈ set [0..<m - 1] ⟹ (d_of dmu (x + 1) * d_of dmu (x + 1), d_of dmu (x + 2) * d_of dmu x, Suc x) = (d_of dmu (x + 1) * d_of dmu (x + 1), d_of dmu (x + 2) * d_of dmu x, Suc x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*)\n (*proven 2 subgoals*) .\nqed\n\nlemma LLL_max_gso_norm: assumes impl: \"state_impl_inv p mfs dmu state\" \n and m: \"m ≠ 0\" \nshows \"LLL_max_gso_norm first state = compute_max_gso_norm first dmu\"\nproof (-)\n (*goal: ‹LLL_max_gso_norm first state = compute_max_gso_norm first dmu›*)\n obtain mfsi and dmui and di and mods where state: \"state = (mfsi, dmui, di,mods)\"\n (*goal: ‹(⋀mfsi dmui di mods. state = (mfsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n from impl[unfolded state state_impl_inv.simps] (*‹mfsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*) have di: \"di = IArray.of_fun (d_of dmu) (Suc m)\"\n by auto\n show \"?thesis\"\n (*goal: ‹LLL_max_gso_norm first state = compute_max_gso_norm first dmu›*)\n using LLL_max_gso_norm_di[OF di m] (*‹LLL_max_gso_norm_di ?first di = compute_max_gso_norm ?first dmu›*) unfolding LLL_max_gso_norm_def state split\n (*goal: ‹LLL_max_gso_norm_di (first::bool) (di::int iarray) = compute_max_gso_norm first (dmu::int mat)›*) .\nqed\n\nlemma mod_of_gso_norm: \"m ≠ 0 ⟹ mod_of_gso_norm first mn =\n compute_mod_of_max_gso_norm first mn\" \n unfolding mod_of_gso_norm_def compute_mod_of_max_gso_norm_def bound_number_def\n (*goal: ‹(m::nat) ≠ (0::nat) ⟹ log_base ^ log_ceiling log_base (max (2::int) (root_rat_ceiling (2::nat) ((mn::rat) * rat_of_nat (if first::bool then 4::nat else m + (3::nat))) + (1::int))) = log_base ^ log_ceiling log_base (max (2::int) (root_rat_ceiling (2::nat) (mn * (rat_of_nat (if first ∧ m ≠ (0::nat) then 1::nat else m) + (3::rat))) + (1::int)))›*)\n by auto\n\nlemma LLL_adjust_mod: assumes impl: \"state_impl_inv p mfs dmu state\" \n and res: \"basis_reduction_adjust_mod p first mfs dmu = (p', mfs', dmu', g_idx)\" \n and res': \"LLL_adjust_mod p first state = (p'', state', g_idx')\" \n and m: \"m ≠ 0\" \nshows \"state_impl_inv p' mfs' dmu' state' ∧ p'' = p' ∧ g_idx' = g_idx\"\nproof (-)\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ p'' = p' ∧ g_idx' = g_idx›*)\n from LLL_max_gso_norm[OF impl m] (*‹LLL_max_gso_norm ?first state = compute_max_gso_norm ?first dmu›*) have id: \"LLL_max_gso_norm first state = compute_max_gso_norm first dmu\"\n by auto\n obtain b and gi where norm: \"compute_max_gso_norm first dmu = (b, gi)\"\n (*goal: ‹(⋀b gi. compute_max_gso_norm first dmu = (b, gi) ⟹ thesis) ⟹ thesis›*)\n by force\n obtain P where P: \"compute_mod_of_max_gso_norm first b = P\"\n (*goal: ‹(⋀P. compute_mod_of_max_gso_norm first b = P ⟹ thesis) ⟹ thesis›*)\n by auto\n note res = res[unfolded basis_reduction_adjust_mod.simps Let_def P norm split] (*‹(if P < p then (P, map (map_vec (λx. x symmod P)) mfs, mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi) else (p, mfs, dmu, gi)) = (p', mfs', dmu', g_idx)›*)\n note res' = res'[unfolded LLL_adjust_mod_def id Let_def P norm split mod_of_gso_norm [ OF m ]] (*‹(if P < p then (P, perform_adjust_mod P state, gi) else (p, state, gi)) = (p'', state', g_idx')›*)\n show \"?thesis\"\n (*goal: ‹state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (p''::int) = p' ∧ (g_idx'::nat) = (g_idx::nat)›*)\n proof (cases \"P < p\")\n (*goals:\n 1. ‹P < p ⟹ state_impl_inv p' mfs' dmu' state' ∧ p'' = p' ∧ g_idx' = g_idx›\n 2. ‹¬ P < p ⟹ state_impl_inv p' mfs' dmu' state' ∧ p'' = p' ∧ g_idx' = g_idx›*)\n case False (*‹¬ P < p›*)\n thus \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ p'' = p' ∧ g_idx' = g_idx›*)\n using res (*‹(if P < p then (P, map (map_vec (λx. x symmod P)) mfs, mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi) else (p, mfs, dmu, gi)) = (p', mfs', dmu', g_idx)›*) res' (*‹(if P < p then (P, perform_adjust_mod P state, gi) else (p, state, gi)) = (p'', state', g_idx')›*) impl (*‹state_impl_inv p mfs dmu state›*) by (auto split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n next\n (*goal: ‹(P::int) < (p::int) ⟹ state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (p''::int) = p' ∧ (g_idx'::nat) = (g_idx::nat)›*)\n case True (*‹P < p›*)\n hence id: \"(P < p) = True\"\n by auto\n obtain fsi and dmui and di and mods where state: \"state = (fsi, dmui, di, mods)\"\n (*goal: ‹(⋀fsi dmui di mods. state = (fsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n from impl[unfolded state state_impl_inv.simps] (*‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*) have impl: \"fsi = mfs\" \"di = IArray.of_fun (d_of dmu) (Suc m)\" \"dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m\"\n apply -\n (*goals:\n 1. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1) ⟹ fsi = mfs›\n 2. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1) ⟹ di = IArray.of_fun (d_of dmu) (Suc m)›\n 3. ‹fsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1) ⟹ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n note res = res[unfolded id if_True] (*‹(P, map (map_vec (λx. x symmod P)) mfs, mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi) = (p', mfs', dmu', g_idx)›*)\n from res (*‹(P::int, map (map_vec (λx::int. x symmod P)) (mfs::int vec list), mat (m::nat) m (λ(i::nat, j::nat). if j < i then (dmu::int mat) $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi::nat) = (p'::int, mfs'::int vec list, dmu'::int mat, g_idx::nat)›*) have mfs': \"mfs' = map (map_vec (λx. x symmod P)) mfs\" and p': \"p' = P\" and dmu': \"dmu' = mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j))\" and gidx: \"g_idx = gi\"\n apply -\n (*goals:\n 1. ‹(P, map (map_vec (λx. x symmod P)) mfs, mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi) = (p', mfs', dmu', g_idx) ⟹ mfs' = map (map_vec (λx. x symmod P)) mfs›\n 2. ‹(P, map (map_vec (λx. x symmod P)) mfs, mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi) = (p', mfs', dmu', g_idx) ⟹ p' = P›\n 3. ‹(P, map (map_vec (λx. x symmod P)) mfs, mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi) = (p', mfs', dmu', g_idx) ⟹ dmu' = mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j))›\n 4. ‹(P, map (map_vec (λx. x symmod P)) mfs, mat m m (λ(i, j). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)), gi) = (p', mfs', dmu', g_idx) ⟹ g_idx = gi›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n let ?mods = \"IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)\"\n let ?dmu = \"IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod ?mods !! j) i) m\"\n note res' = res'[unfolded id if_True state split impl ( 1 ) perform_adjust_mod_def Let_def] (*‹(P::int, (map (map_vec (λx::int. x symmod P)) (mfs::int vec list), IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. (dmui::int iarray iarray) !! i !! j symmod IArray.of_fun (λj::nat. P * (di::int iarray) !! j * di !! Suc j) ((m::nat) - (1::nat)) !! j) i) m, di, IArray.of_fun (λj::nat. P * di !! j * di !! Suc j) (m - (1::nat))), gi::nat) = (p''::int, state'::int vec list × int iarray iarray × int iarray × int iarray, g_idx'::nat)›*)\n from res' (*‹(P, (map (map_vec (λx. x symmod P)) mfs, IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p'', state', g_idx')›*) have p'': \"p'' = P\" and state': \"state' = (map (map_vec (λx. x symmod P)) mfs, ?dmu, di, ?mods)\" and gidx': \"g_idx' = gi\"\n apply -\n (*goals:\n 1. ‹(P, (map (map_vec (λx. x symmod P)) mfs, IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p'', state', g_idx') ⟹ p'' = P›\n 2. ‹(P, (map (map_vec (λx. x symmod P)) mfs, IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p'', state', g_idx') ⟹ state' = (map (map_vec (λx. x symmod P)) mfs, IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1))›\n 3. ‹(P, (map (map_vec (λx. x symmod P)) mfs, IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p'', state', g_idx') ⟹ g_idx' = gi›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n show \"?thesis\"\n (*goal: ‹state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (p''::int) = p' ∧ (g_idx'::nat) = (g_idx::nat)›*)\n unfolding state' state_impl_inv.simps mfs' p'' p' gidx gidx'\n (*goal: ‹(map (map_vec (λx::int. x symmod (P::int))) (mfs::int vec list) = map (map_vec (λx::int. x symmod P)) mfs ∧ (di::int iarray) = IArray.of_fun (d_of (dmu'::int mat)) (Suc (m::nat)) ∧ IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. (dmui::int iarray iarray) !! i !! j symmod IArray.of_fun (λj::nat. P * di !! j * di !! Suc j) (m - (1::nat)) !! j) i) m = IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. dmu' $$ (i, j)) i) m ∧ IArray.of_fun (λj::nat. P * di !! j * di !! Suc j) (m - (1::nat)) = IArray.of_fun (λj::nat. P * di !! j * di !! Suc j) (m - (1::nat))) ∧ P = P ∧ (gi::nat) = gi›*)\n proof (intro conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*) refl (*‹(?t::?'a) = ?t›*))\n (*goals:\n 1. ‹di = IArray.of_fun (d_of dmu') (Suc m)›\n 2. ‹IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m›*)\n show \"di = IArray.of_fun (d_of dmu') (Suc m)\"\n unfolding impl\n (*goal: ‹IArray.of_fun (d_of dmu) (Suc m) = IArray.of_fun (d_of dmu') (Suc m)›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (d_of dmu) (Suc m) = IArray.of_fun (d_of dmu') (Suc m)›*)\n by (auto simp: dmu' (*‹(dmu'::int mat) = mat (m::nat) m (λ(i::nat, j::nat). if j < i then (dmu::int mat) $$ (i, j) symmod ((P::int) * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j))›*) d_of_def (*‹d_of (?dmu::int mat) (?i::nat) = (if ?i = (0::nat) then 1::int else ?dmu $$ (?i - (1::nat), ?i - (1::nat)))›*))\n show \"?dmu = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m\"\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m = IArray.of_fun (λi. IArray.of_fun (λj. dmu' $$ (i, j)) i) m›*)\n proof (goal_cases)\n (*goal: ‹⋀i ia. ⟦i < m; ia < i⟧ ⟹ dmui !! i !! ia symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! ia = dmu' $$ (i, ia)›*)\n case (1 i j) (*‹i < m› ‹j < i›*)\n hence \"j < m\" \"Suc j < Suc m\" \"j < Suc m\" \"j < m - 1\"\n apply -\n (*goals:\n 1. ‹⟦i < m; j < i⟧ ⟹ j < m›\n 2. ‹⟦i < m; j < i⟧ ⟹ Suc j < Suc m›\n 3. ‹⟦i < m; j < i⟧ ⟹ j < Suc m›\n 4. ‹⟦i < m; j < i⟧ ⟹ j < m - 1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n show \"?case\"\n (*goal: ‹dmui !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j = dmu' $$ (i, j)›*)\n unfolding dmu' impl IArray.of_fun_nth[OF ‹i < m›] IArray.of_fun_nth[OF ‹j < i›] IArray.of_fun_nth[OF ‹j < m›] IArray.of_fun_nth[OF ‹Suc j < Suc m›] IArray.of_fun_nth[OF ‹j < Suc m›] IArray.of_fun_nth[OF ‹j < m - 1›]\n (*goal: ‹(dmu::int mat) $$ (i::nat, j::nat) symmod ((P::int) * d_of dmu j * d_of dmu (Suc j)) = mat (m::nat) m (λ(i::nat, j::nat). if j < i then dmu $$ (i, j) symmod (P * vec (Suc m) (d_of dmu) $ j * vec (Suc m) (d_of dmu) $ Suc j) else dmu $$ (i, j)) $$ (i, j)›*)\n using \"1\" (*‹i < m› ‹j < i›*) by auto\n qed\n qed\n qed\nqed\n\nlemma LLL_adjust_swap_add: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod_weak fs mfs dmu p first b\"\n and res: \"basis_reduction_adjust_swap_add_step p first mfs dmu g_idx k = (p', mfs', dmu', g_idx')\" \n and res': \"LLL_adjust_swap_add p first state g_idx k = (p'',state', G_idx')\" \n and k: \"k < m\" and k0: \"k ≠ 0\" \nshows \"state_impl_inv p' mfs' dmu' state'\" \"p'' = p'\" \"G_idx' = g_idx'\" \n \"i ≤ m ⟹ i ≠ k ⟹ di_of state' !! i = di_of state !! i\" \nproof (atomize(full), goal_cases)\n (*goal: ‹(state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (p''::int) = p') ∧ (G_idx'::nat) = (g_idx'::nat) ∧ ((i::nat) ≤ (m::nat) ⟶ i ≠ (k::nat) ⟶ di_of state' !! i = di_of (state::int vec list × int iarray iarray × int iarray × int iarray) !! i)›*)\n case 1 (*no hyothesis introduced yet*)\n from k (*‹k < m›*) have m: \"m ≠ 0\"\n by auto\n obtain mfsi and dmui and di and mods where state: \"state = (mfsi, dmui, di, mods)\"\n (*goal: ‹(⋀mfsi dmui di mods. state = (mfsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀(a::?'a) (b::?'b) c::?'c. (?y::?'a × ?'b × ?'c) = (a, b, c) ⟹ ?thesis::bool) ⟹ ?thesis›*))\n obtain state'' where add': \"LLL_add_row p state k (k - 1) = state''\"\n (*goal: ‹(⋀state''. LLL_add_row p state k (k - 1) = state'' ⟹ thesis) ⟹ thesis›*)\n by blast\n obtain mfs'' and dmu'' where add: \"basis_reduction_mod_add_row p mfs dmu k (k - 1) = (mfs'', dmu'')\"\n (*goal: ‹(⋀mfs'' dmu''. basis_reduction_mod_add_row p mfs dmu k (k - 1) = (mfs'', dmu'') ⟹ thesis) ⟹ thesis›*)\n by force\n obtain mfs3 and dmu3 where swap: \"basis_reduction_mod_swap p mfs'' dmu'' k = (mfs3, dmu3)\"\n (*goal: ‹(⋀mfs3 dmu3. basis_reduction_mod_swap p mfs'' dmu'' k = (mfs3, dmu3) ⟹ thesis) ⟹ thesis›*)\n by force\n obtain state3 where swap': \"LLL_swap_row p state'' k = state3\"\n (*goal: ‹(⋀state3. LLL_swap_row p state'' k = state3 ⟹ thesis) ⟹ thesis›*)\n by blast\n obtain mfsi2 and dmui2 and di2 and mods2 where state2: \"state'' = (mfsi2, dmui2, di2, mods2)\"\n (*goal: ‹(⋀mfsi2 dmui2 di2 mods2. state'' = (mfsi2, dmui2, di2, mods2) ⟹ thesis) ⟹ thesis›*)\n apply (cases state'')\n (*goal: ‹(⋀mfsi2 dmui2 di2 mods2. state'' = (mfsi2, dmui2, di2, mods2) ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain mfsi3 and dmui3 and di3 and mods3 where state3: \"state3 = (mfsi3, dmui3, di3, mods3)\"\n (*goal: ‹(⋀mfsi3 dmui3 di3 mods3. state3 = (mfsi3, dmui3, di3, mods3) ⟹ thesis) ⟹ thesis›*)\n apply (cases state3)\n (*goal: ‹(⋀(mfsi3::int vec list) (dmui3::int iarray iarray) (di3::int iarray) mods3::int iarray. (state3::int vec list × int iarray iarray × int iarray × int iarray) = (mfsi3, dmui3, di3, mods3) ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n have \"length mfsi = m\"\n using impl[unfolded state state_impl_inv.simps] (*‹mfsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*) LLL_invD_modw[OF Linv] (*‹length mfs = m› ‹length fs = m› ‹lattice_of (fs::int vec list) = L› ‹gs.lin_indpt_list (map of_int_hom.vec_hom fs)› ‹map (map_vec (λx. x symmod p)) fs = mfs› ‹∀i'<m. ∀j'<i'. ¦dμ fs i' j'¦ < p * d fs j' * d fs (Suc j')› ‹∀i'<m. ∀j'<m. dμ fs i' j' = dmu $$ (i', j')› ‹?i < m ⟹ fs ! ?i ∈ carrier_vec n› ‹set fs ⊆ carrier_vec n› ‹?i < m ⟹ gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) ?i ∈ Rn› ‹?i < m ⟹ mfs ! ?i ∈ carrier_vec n› ‹set mfs ⊆ carrier_vec n› and more 3 facts*) by auto\n note res' = res'[unfolded state LLL_adjust_swap_add_def LLL_swap_add_eq [ OF k0 k this ], folded state, unfolded add' swap' Let_def] (*‹(if (k::nat) - (1::nat) = (g_idx::nat) then LLL_adjust_mod (p::int) (first::bool) (state3::int vec list × int iarray iarray × int iarray × int iarray) else (p, state3, g_idx)) = (p''::int, state'::int vec list × int iarray iarray × int iarray × int iarray, G_idx'::nat)›*)\n note res = res[unfolded basis_reduction_adjust_swap_add_step_def Let_def add split swap] (*‹(if (k::nat) - (1::nat) = (g_idx::nat) then basis_reduction_adjust_mod (p::int) (first::bool) (mfs3::int vec list) (dmu3::int mat) else (p, mfs3, dmu3, g_idx)) = (p'::int, mfs'::int vec list, dmu'::int mat, g_idx'::nat)›*)\n from LLL_add_row[OF impl Linv add add' k] (*‹k - 1 < k ⟹ state_impl_inv p mfs'' dmu'' state''›*) k0 (*‹k ≠ 0›*) have impl': \"state_impl_inv p mfs'' dmu'' state''\"\n by auto\n from basis_reduction_mod_add_row[OF Linv add k _ k0] (*‹k - 1 < k ⟹ ∃fs'. LLL_invariant_mod_weak fs' mfs'' dmu'' p first b ∧ LLL_measure k fs' = LLL_measure k fs ∧ (μ_small_row k fs (Suc (k - 1)) ⟶ μ_small_row k fs' (k - 1)) ∧ ¦gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs') k (k - 1)¦ ≤ 1 / 2 ∧ (∀i' j'. i' < k ⟶ j' ≤ i' ⟶ gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs') i' j' = gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs) i' j') ∧ (LLL_invariant_mod fs mfs dmu p first b k ⟶ LLL_invariant_mod fs' mfs'' dmu'' p first b k) ∧ (∀ii≤m. d fs' ii = d fs ii)›*) k0 (*‹k ≠ 0›*) obtain fs'' where Linv': \"LLL_invariant_mod_weak fs'' mfs'' dmu'' p first b\"\n (*goal: ‹(⋀fs''. LLL_invariant_mod_weak fs'' mfs'' dmu'' p first b ⟹ thesis) ⟹ thesis›*)\n by auto\n from LLL_swap_row[OF impl' Linv' swap swap' k k0] (*‹state_impl_inv p mfs3 dmu3 state3›*) have impl3: \"state_impl_inv p mfs3 dmu3 state3\" .\n have di2: \"di2 = di\"\n using add'[unfolded state LLL_add_row_def Let_def split perform_add_row_def state2] (*‹(if round_num_denom (dmui !! k !! (k - 1)) (di !! Suc (k - 1)) = 0 then (mfsi, dmui, di, mods) else (case split_at k mfsi of (start, fsi # end) ⇒ start @ vec n (λka. (fsi $ ka - round_num_denom (dmui !! k !! (k - 1)) (di !! Suc (k - 1)) * mfsi ! (k - 1) $ ka) symmod p) # end, IArray.of_fun (λii. if k = ii then IArray.of_fun (λjj. if jj < k - 1 then (dmui !! k !! jj - round_num_denom (dmui !! k !! (k - 1)) (di !! Suc (k - 1)) * dmui !! (k - 1) !! jj) symmod mods !! jj else if jj = k - 1 then dmui !! k !! (k - 1) - round_num_denom (dmui !! k !! (k - 1)) (di !! Suc (k - 1)) * di !! Suc (k - 1) else dmui !! k !! jj) k else dmui !! ii) m, di, mods)) = (mfsi2, dmui2, di2, mods2)›*) by (auto split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n have di3: \"di3 = IArray.of_fun (λi. if i = k then (di2 !! Suc k * di2 !! (k - 1) + dmui2 !! k !! (k - 1) * dmui2 !! k !! (k - 1)) div di2 !! k else di2 !! i) (Suc m)\"\n using swap'[unfolded state2 state3] (*‹LLL_swap_row p (mfsi2, dmui2, di2, mods2) k = (mfsi3, dmui3, di3, mods3)›*) unfolding LLL_swap_row_def Let_def\n (*goal: ‹di3 = IArray.of_fun (λi. if i = k then (di2 !! Suc k * di2 !! (k - 1) + dmui2 !! k !! (k - 1) * dmui2 !! k !! (k - 1)) div di2 !! k else di2 !! i) (Suc m)›*)\n by simp\n have di3: \"i ≤ m ⟹ i ≠ k ⟹ di3 !! i = di !! i\"\n unfolding di2[symmetric] di3\n (*goal: ‹⟦i ≤ m; i ≠ k⟧ ⟹ IArray.of_fun (λi. if i = k then (di2 !! Suc k * di2 !! (k - 1) + dmui2 !! k !! (k - 1) * dmui2 !! k !! (k - 1)) div di2 !! k else di2 !! i) (Suc m) !! i = di2 !! i›*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⟦i ≤ m; i ≠ k⟧ ⟹ i < Suc m›\n 2. ‹⟦i ≤ m; i ≠ k⟧ ⟹ (if i = k then (di2 !! Suc k * di2 !! (k - 1) + dmui2 !! k !! (k - 1) * dmui2 !! k !! (k - 1)) div di2 !! k else di2 !! i) = di2 !! i›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"?case\"\n (*goal: ‹(state_impl_inv p' mfs' dmu' state' ∧ p'' = p') ∧ G_idx' = g_idx' ∧ (i ≤ m ⟶ i ≠ k ⟶ di_of state' !! i = di_of state !! i)›*)\n proof (cases \"k - 1 = g_idx\")\n (*goals:\n 1. ‹k - 1 = g_idx ⟹ (state_impl_inv p' mfs' dmu' state' ∧ p'' = p') ∧ G_idx' = g_idx' ∧ (i ≤ m ⟶ i ≠ k ⟶ di_of state' !! i = di_of state !! i)›\n 2. ‹k - 1 ≠ g_idx ⟹ (state_impl_inv p' mfs' dmu' state' ∧ p'' = p') ∧ G_idx' = g_idx' ∧ (i ≤ m ⟶ i ≠ k ⟶ di_of state' !! i = di_of state !! i)›*)\n case True (*‹k - 1 = g_idx›*)\n hence id: \"(k - 1 = g_idx) = True\"\n by simp\n note res = res[unfolded id if_True] (*‹basis_reduction_adjust_mod p first mfs3 dmu3 = (p', mfs', dmu', g_idx')›*)\n note res' = res'[unfolded id if_True] (*‹LLL_adjust_mod (p::int) (first::bool) (state3::int vec list × int iarray iarray × int iarray × int iarray) = (p''::int, state'::int vec list × int iarray iarray × int iarray × int iarray, G_idx'::nat)›*)\n obtain mfsi4 and dmui4 and di4 and mods4 where state': \"state' = (mfsi4, dmui4, di4, mods4)\"\n (*goal: ‹(⋀mfsi4 dmui4 di4 mods4. state' = (mfsi4, dmui4, di4, mods4) ⟹ thesis) ⟹ thesis›*)\n apply (cases state')\n (*goal: ‹(⋀mfsi4 dmui4 di4 mods4. state' = (mfsi4, dmui4, di4, mods4) ⟹ thesis) ⟹ thesis›*)\n by auto\n from res'[unfolded state3 state' LLL_adjust_mod_def Let_def perform_adjust_mod_def] (*‹(case LLL_max_gso_norm first (mfsi3, dmui3, di3, mods3) of (b', g_idx) ⇒ if mod_of_gso_norm first b' < p then (mod_of_gso_norm first b', case (mfsi3, dmui3, di3, mods3) of (mfsi, dmui, di, x) ⇒ (map (map_vec (λx. x symmod mod_of_gso_norm first b')) mfsi, IArray.of_fun (λi. IArray.of_fun (λj. dmui !! i !! j symmod IArray.of_fun (λj. mod_of_gso_norm first b' * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. mod_of_gso_norm first b' * di !! j * di !! Suc j) (m - 1)), g_idx) else (p, (mfsi3, dmui3, di3, mods3), g_idx)) = (p'', (mfsi4, dmui4, di4, mods4), G_idx')›*) have di4: \"di4 = di3\"\n by (auto split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*) prod.splits (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))› ‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∄x1 x2. ?prod = (x1, x2) ∧ ¬ ?P (?f x1 x2))›*))\n from LLL_adjust_mod[OF impl3 res res' m] (*‹state_impl_inv p' mfs' dmu' state' ∧ p'' = p' ∧ G_idx' = g_idx'›*) di3 (*‹⟦i ≤ m; i ≠ k⟧ ⟹ di3 !! i = di !! i›*) state (*‹state = (mfsi, dmui, di, mods)›*) state' (*‹(state'::int vec list × int iarray iarray × int iarray × int iarray) = (mfsi4::int vec list, dmui4::int iarray iarray, di4::int iarray, mods4::int iarray)›*) di4 (*‹(di4::int iarray) = (di3::int iarray)›*) res' (*‹LLL_adjust_mod (p::int) (first::bool) (state3::int vec list × int iarray iarray × int iarray × int iarray) = (p''::int, state'::int vec list × int iarray iarray × int iarray × int iarray, G_idx'::nat)›*) show \"?thesis\"\n (*goal: ‹(state_impl_inv p' mfs' dmu' state' ∧ p'' = p') ∧ G_idx' = g_idx' ∧ (i ≤ m ⟶ i ≠ k ⟶ di_of state' !! i = di_of state !! i)›*)\n by auto\n next\n (*goal: ‹k - 1 ≠ g_idx ⟹ (state_impl_inv p' mfs' dmu' state' ∧ p'' = p') ∧ G_idx' = g_idx' ∧ (i ≤ m ⟶ i ≠ k ⟶ di_of state' !! i = di_of state !! i)›*)\n case False (*‹k - 1 ≠ g_idx›*)\n hence id: \"(k - 1 = g_idx) = False\"\n by simp\n note res = res[unfolded id if_False] (*‹(p, mfs3, dmu3, g_idx) = (p', mfs', dmu', g_idx')›*)\n note res' = res'[unfolded id if_False] (*‹(p::int, state3::int vec list × int iarray iarray × int iarray × int iarray, g_idx::nat) = (p''::int, state'::int vec list × int iarray iarray × int iarray × int iarray, G_idx'::nat)›*)\n from impl3 (*‹state_impl_inv p mfs3 dmu3 state3›*) res (*‹(p, mfs3, dmu3, g_idx) = (p', mfs', dmu', g_idx')›*) res' (*‹(p, state3, g_idx) = (p'', state', G_idx')›*) di3 (*‹⟦i ≤ m; i ≠ k⟧ ⟹ di3 !! i = di !! i›*) state (*‹state = (mfsi, dmui, di, mods)›*) state3 (*‹state3 = (mfsi3, dmui3, di3, mods3)›*) show \"?thesis\"\n (*goal: ‹(state_impl_inv p' mfs' dmu' state' ∧ p'' = p') ∧ G_idx' = g_idx' ∧ (i ≤ m ⟶ i ≠ k ⟶ di_of state' !! i = di_of state !! i)›*)\n by auto\n qed\nqed\n \n\n\nlemma LLL_step: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod_weak fs mfs dmu p first b\"\n and res: \"basis_reduction_mod_step p first mfs dmu g_idx k j = (p', mfs', dmu', g_idx', k', j')\" \n and res': \"LLL_step p first state g_idx k j = ((p'',state', g_idx''), k'', j'')\" \n and k: \"k < m\" \nshows \"state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'\"\nproof (cases \"k = 0\")\n (*goals:\n 1. ‹k = 0 ⟹ state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›\n 2. ‹k ≠ 0 ⟹ state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n case True (*‹k = 0›*)\n thus \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n using res (*‹basis_reduction_mod_step p first mfs dmu g_idx k j = (p', mfs', dmu', g_idx', k', j')›*) res' (*‹LLL_step p first state g_idx k j = ((p'', state', g_idx''), k'', j'')›*) impl (*‹state_impl_inv p mfs dmu state›*) unfolding LLL_step_def basis_reduction_mod_step_def\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n by auto\nnext\n (*goal: ‹k ≠ 0 ⟹ state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n case k0: False (*‹(k::nat) ≠ (0::nat)›*)\n hence id: \"(k = 0) = False\"\n by simp\n note res = res[unfolded basis_reduction_mod_step_def id if_False] (*‹(let di = d_of dmu k; (num, denom) = quotient_of α in if di * di * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k) then (p, mfs, dmu, g_idx, Suc k, j) else let (p', mfs', dmu', g_idx') = basis_reduction_adjust_swap_add_step p first mfs dmu g_idx k in (p', mfs', dmu', g_idx', k - 1, j + 1)) = (p', mfs', dmu', g_idx', k', j')›*)\n obtain num and denom where alph: \"quotient_of α = (num,denom)\"\n (*goal: ‹(⋀num denom. quotient_of α = (num, denom) ⟹ thesis) ⟹ thesis›*)\n by force\n obtain mfsi and dmui and di and mods where state: \"state = (mfsi, dmui, di, mods)\"\n (*goal: ‹(⋀mfsi dmui di mods. state = (mfsi, dmui, di, mods) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀(a::?'a) (b::?'b) c::?'c. (?y::?'a × ?'b × ?'c) = (a, b, c) ⟹ ?thesis::bool) ⟹ ?thesis›*))\n note res' = res'[unfolded LLL_step_def id if_False Let_def state split alph, folded state] (*‹(if di !! k * di !! k * denom ≤ num * di !! (k - 1) * di !! Suc k then ((p, state, g_idx), Suc k, j) else (LLL_adjust_swap_add p first state g_idx k, k - 1, j + 1)) = ((p'', state', g_idx''), k'', j'')›*)\n from k0 (*‹k ≠ 0›*) have kk1: \"k - 1 < k\"\n by auto\n note res = res[unfolded Let_def alph split] (*‹(if d_of (dmu::int mat) (k::nat) * d_of dmu k * (denom::int) ≤ (num::int) * d_of dmu (k - (1::nat)) * d_of dmu (Suc k) then (p::int, mfs::int vec list, dmu, g_idx::nat, Suc k, j::int) else case basis_reduction_adjust_swap_add_step p (first::bool) mfs dmu g_idx k of (p'::int, mfs'::int vec list, dmu'::int mat, g_idx'::nat) ⇒ (p', mfs', dmu', g_idx', k - (1::nat), j + (1::int))) = (p'::int, mfs'::int vec list, dmu'::int mat, g_idx'::nat, k'::nat, j'::int)›*)\n obtain state'' where addi: \"LLL_swap_add p state k = state''\"\n (*goal: ‹(⋀state''. LLL_swap_add p state k = state'' ⟹ thesis) ⟹ thesis›*)\n by auto\n from impl[unfolded state state_impl_inv.simps] (*‹mfsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*) have di: \"di = IArray.of_fun (d_of dmu) (Suc m)\"\n by auto\n have id: \"di !! k = d_of dmu k\" \"di !! (Suc k) = d_of dmu (Suc k)\" \"di !! (k - 1) = d_of dmu (k - 1)\"\n unfolding di\n (*goals:\n 1. ‹IArray.of_fun (d_of (dmu::int mat)) (Suc (m::nat)) !! (k::nat) = d_of dmu k›\n 2. ‹IArray.of_fun (d_of (dmu::int mat)) (Suc (m::nat)) !! Suc (k::nat) = d_of dmu (Suc k)›\n 3. ‹IArray.of_fun (d_of (dmu::int mat)) (Suc (m::nat)) !! ((k::nat) - (1::nat)) = d_of dmu (k - (1::nat))›*)\n using k (*‹k < m›*) apply -\n (*goals:\n 1. ‹k < m ⟹ IArray.of_fun (d_of dmu) (Suc m) !! k = d_of dmu k›\n 2. ‹k < m ⟹ IArray.of_fun (d_of dmu) (Suc m) !! Suc k = d_of dmu (Suc k)›\n 3. ‹k < m ⟹ IArray.of_fun (d_of dmu) (Suc m) !! (k - 1) = d_of dmu (k - 1)›\n discuss goal 1*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹k < m ⟹ k < Suc m›\n 2. ‹k < m ⟹ d_of dmu k = d_of dmu k›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹k < m ⟹ Suc k < Suc m›\n 2. ‹k < m ⟹ d_of dmu (Suc k) = d_of dmu (Suc k)›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*)\n (*discuss goal 3*)\n apply (subst IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹k < m ⟹ k - 1 < Suc m›\n 2. ‹k < m ⟹ d_of dmu (k - 1) = d_of dmu (k - 1)›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*)\n (*proven 3 subgoals*) .\n have \"length mfsi = m\"\n using impl[unfolded state state_impl_inv.simps] (*‹mfsi = mfs ∧ di = IArray.of_fun (d_of dmu) (Suc m) ∧ dmui = IArray.of_fun (λi. IArray.of_fun (λj. dmu $$ (i, j)) i) m ∧ mods = IArray.of_fun (λj. p * di !! j * di !! Suc j) (m - 1)›*) LLL_invD_modw[OF Linv] (*‹length mfs = m› ‹length (fs::int vec list) = (m::nat)› ‹lattice_of fs = L› ‹gs.lin_indpt_list (map of_int_hom.vec_hom fs)› ‹map (map_vec (λx. x symmod p)) fs = mfs› ‹∀i'<m. ∀j'<i'. ¦dμ fs i' j'¦ < p * d fs j' * d fs (Suc j')› ‹∀i'<m::nat. ∀j'<m. dμ (fs::int vec list) i' j' = (dmu::int mat) $$ (i', j')› ‹?i < m ⟹ fs ! ?i ∈ carrier_vec n› ‹set fs ⊆ carrier_vec n› ‹(?i::nat) < (m::nat) ⟹ gram_schmidt_fs.gso (n::nat) (map of_int_hom.vec_hom (fs::int vec list)) ?i ∈ Rn› ‹?i < m ⟹ mfs ! ?i ∈ carrier_vec n› ‹set mfs ⊆ carrier_vec n› and more 3 facts*) by auto\n note res' = res'[unfolded id] (*‹(if d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k) then ((p, state, g_idx), Suc k, j) else (LLL_adjust_swap_add p first state g_idx k, k - 1, j + 1)) = ((p'', state', g_idx''), k'', j'')›*)\n let ?cond = \"d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k)\"\n show \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n proof (cases ?cond)\n (*goals:\n 1. ‹d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k) ⟹ state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›\n 2. ‹¬ d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k) ⟹ state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n case True (*‹d_of (dmu::int mat) (k::nat) * d_of dmu k * (denom::int) ≤ (num::int) * d_of dmu (k - (1::nat)) * d_of dmu (Suc k)›*)\n from True (*‹d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k)›*) res (*‹(if d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k) then (p, mfs, dmu, g_idx, Suc k, j) else case basis_reduction_adjust_swap_add_step p first mfs dmu g_idx k of (p', mfs', dmu', g_idx') ⇒ (p', mfs', dmu', g_idx', k - 1, j + 1)) = (p', mfs', dmu', g_idx', k', j')›*) res' (*‹(if d_of (dmu::int mat) (k::nat) * d_of dmu k * (denom::int) ≤ (num::int) * d_of dmu (k - (1::nat)) * d_of dmu (Suc k) then ((p::int, state::int vec list × int iarray iarray × int iarray × int iarray, g_idx::nat), Suc k, j::int) else (LLL_adjust_swap_add p (first::bool) state g_idx k, k - (1::nat), j + (1::int))) = ((p''::int, state'::int vec list × int iarray iarray × int iarray × int iarray, g_idx''::nat), k''::nat, j''::int)›*) state (*‹state = (mfsi, dmui, di, mods)›*) show \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n using impl (*‹state_impl_inv p mfs dmu state›*) by auto\n next\n (*goal: ‹¬ d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k) ⟹ state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n case False (*‹¬ d_of dmu k * d_of dmu k * denom ≤ num * d_of dmu (k - 1) * d_of dmu (Suc k)›*)\n hence cond: \"?cond = False\"\n by simp\n note res = res[unfolded cond if_False] (*‹(case basis_reduction_adjust_swap_add_step (p::int) (first::bool) (mfs::int vec list) (dmu::int mat) (g_idx::nat) (k::nat) of (p'::int, mfs'::int vec list, dmu'::int mat, g_idx'::nat) ⇒ (p', mfs', dmu', g_idx', k - (1::nat), (j::int) + (1::int))) = (p'::int, mfs'::int vec list, dmu'::int mat, g_idx'::nat, k'::nat, j'::int)›*)\n note res' = res'[unfolded cond if_False] (*‹(LLL_adjust_swap_add p first state g_idx k, k - 1, j + 1) = ((p'', state', g_idx''), k'', j'')›*)\n let ?step = \"basis_reduction_adjust_swap_add_step p first mfs dmu g_idx k\"\n let ?step' = \"LLL_adjust_swap_add p first state g_idx k\"\n from res (*‹(case basis_reduction_adjust_swap_add_step p first mfs dmu g_idx k of (p', mfs', dmu', g_idx') ⇒ (p', mfs', dmu', g_idx', k - 1, j + 1)) = (p', mfs', dmu', g_idx', k', j')›*) have step: \"?step = (p', mfs', dmu', g_idx')\"\n apply (cases ?step)\n (*goal: ‹basis_reduction_adjust_swap_add_step p first mfs dmu g_idx k = (p', mfs', dmu', g_idx')›*)\n by auto\n note res = res[unfolded step split] (*‹(p', mfs', dmu', g_idx', k - 1, j + 1) = (p', mfs', dmu', g_idx', k', j')›*)\n from res' (*‹(LLL_adjust_swap_add p first state g_idx k, k - 1, j + 1) = ((p'', state', g_idx''), k'', j'')›*) have step': \"?step' = (p'',state', g_idx'')\"\n by auto\n note res' = res'[unfolded step'] (*‹((p''::int, state'::int vec list × int iarray iarray × int iarray × int iarray, g_idx''::nat), (k::nat) - (1::nat), (j::int) + (1::int)) = ((p'', state', g_idx''), k''::nat, j''::int)›*)\n from LLL_adjust_swap_add[OF impl Linv step step' k k0] (*‹state_impl_inv p' mfs' dmu' state'› ‹p'' = p'› ‹g_idx'' = g_idx'› ‹⟦?i ≤ m; ?i ≠ k⟧ ⟹ di_of state' !! ?i = di_of state !! ?i›*) show \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ k'' = k' ∧ p'' = p' ∧ j'' = j' ∧ g_idx'' = g_idx'›*)\n using res (*‹(p', mfs', dmu', g_idx', k - 1, j + 1) = (p', mfs', dmu', g_idx', k', j')›*) res' (*‹((p'', state', g_idx''), k - 1, j + 1) = ((p'', state', g_idx''), k'', j'')›*) by auto\n qed\nqed\n\n\nlemma LLL_main: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod fs mfs dmu p first b i\"\n and res: \"basis_reduction_mod_main p first mfs dmu g_idx i k = (p', mfs', dmu')\" \n and res': \"LLL_main p first state g_idx i k = (pi', state')\" \nshows \"state_impl_inv p' mfs' dmu' state' ∧ pi' = p'\"\n using assms (*‹state_impl_inv p mfs dmu state› ‹LLL_invariant_mod fs mfs dmu p first b i› ‹basis_reduction_mod_main p first mfs dmu g_idx i k = (p', mfs', dmu')› ‹LLL_main p first state g_idx i k = (pi', state')›*) proof (induct \"LLL_measure i fs\" arbitrary: mfs dmu state fs p b k i g_idx rule: less_induct)\n (*goal: ‹⋀(fs::int vec list) (i::nat) (mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray) (p::int) (b::rat) (k::int) g_idx::nat. ⟦⋀(fsa::int vec list) (ia::nat) (mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray) (p::int) (b::rat) (k::int) g_idx::nat. ⟦LLL_measure ia fsa < LLL_measure i fs; state_impl_inv p mfs dmu state; LLL_invariant_mod fsa mfs dmu p (first::bool) b ia; basis_reduction_mod_main p first mfs dmu g_idx ia k = (p'::int, mfs'::int vec list, dmu'::int mat); LLL_main p first state g_idx ia k = (pi'::int, state'::int vec list × int iarray iarray × int iarray × int iarray)⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'; state_impl_inv p mfs dmu state; LLL_invariant_mod fs mfs dmu p first b i; basis_reduction_mod_main p first mfs dmu g_idx i k = (p', mfs', dmu'); LLL_main p first state g_idx i k = (pi', state')⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n case (less fs i mfs dmu state p b k g_idx) (*‹⟦LLL_measure (?i2::nat) (?fs2::int vec list) < LLL_measure (i::nat) (fs::int vec list); state_impl_inv (?p2::int) (?mfs2::int vec list) (?dmu2::int mat) (?state2::int vec list × int iarray iarray × int iarray × int iarray); LLL_invariant_mod ?fs2 ?mfs2 ?dmu2 ?p2 (first::bool) (?b2::rat) ?i2; basis_reduction_mod_main ?p2 first ?mfs2 ?dmu2 (?g_idx2::nat) ?i2 (?k2::int) = (p'::int, mfs'::int vec list, dmu'::int mat); LLL_main ?p2 first ?state2 ?g_idx2 ?i2 ?k2 = (pi'::int, state'::int vec list × int iarray iarray × int iarray × int iarray)⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'› ‹state_impl_inv p mfs dmu state› ‹LLL_invariant_mod fs mfs dmu p first b i› ‹basis_reduction_mod_main p first mfs dmu g_idx i k = (p', mfs', dmu')› ‹LLL_main p first state g_idx i k = (pi', state')›*)\n note impl = less(2) (*‹state_impl_inv p mfs dmu state›*)\n note Linv = less(3) (*‹LLL_invariant_mod fs mfs dmu p first b i›*)\n note res = less(4) (*‹basis_reduction_mod_main p first mfs dmu g_idx i k = (p', mfs', dmu')›*)\n note res' = less(5) (*‹LLL_main p first state g_idx i k = (pi', state')›*)\n note IH = less(1) (*‹⟦LLL_measure ?i2 ?fs2 < LLL_measure i fs; state_impl_inv ?p2 ?mfs2 ?dmu2 ?state2; LLL_invariant_mod ?fs2 ?mfs2 ?dmu2 ?p2 first ?b2 ?i2; basis_reduction_mod_main ?p2 first ?mfs2 ?dmu2 ?g_idx2 ?i2 ?k2 = (p', mfs', dmu'); LLL_main ?p2 first ?state2 ?g_idx2 ?i2 ?k2 = (pi', state')⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n note res = res[unfolded basis_reduction_mod_main.simps [ of _ _ _ _ _ _ k ]] (*‹(if i < m then case basis_reduction_mod_step p first mfs dmu g_idx i k of (p', mfs', dmu', g_idx', x, xb) ⇒ basis_reduction_mod_main p' first mfs' dmu' g_idx' x xb else (p, mfs, dmu)) = (p', mfs', dmu')›*)\n note res' = res'[unfolded LLL_main.simps [ of _ _ _ _ _ k ]] (*‹(if i < m then case LLL_step p first state g_idx i k of ((p', state', g_idx'), x, xa) ⇒ LLL_main p' first state' g_idx' x xa else (p, state)) = (pi', state')›*)\n note Linvw = LLL_mod_inv_to_weak[OF Linv] (*‹LLL_invariant_mod_weak fs mfs dmu p first b›*)\n show \"?case\"\n (*goal: ‹state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (pi'::int) = p'›*)\n proof (cases \"i < m\")\n (*goals:\n 1. ‹i < m ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›\n 2. ‹¬ i < m ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n case False (*‹¬ i < m›*)\n thus \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n using res (*‹(if i < m then case basis_reduction_mod_step p first mfs dmu g_idx i k of (p', mfs', dmu', g_idx', x, xb) ⇒ basis_reduction_mod_main p' first mfs' dmu' g_idx' x xb else (p, mfs, dmu)) = (p', mfs', dmu')›*) res' (*‹(if i < m then case LLL_step p first state g_idx i k of ((p', state', g_idx'), x, xa) ⇒ LLL_main p' first state' g_idx' x xa else (p, state)) = (pi', state')›*) impl (*‹state_impl_inv p mfs dmu state›*) by auto\n next\n (*goal: ‹i < m ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n case i: True (*‹i < m›*)\n hence id: \"(i < m) = True\"\n by simp\n obtain P'' and state'' and I'' and K'' and G_idx'' where step': \"LLL_step p first state g_idx i k = ((P'', state'', G_idx''), I'', K'')\"\n (*goal: ‹(⋀P'' state'' G_idx'' I'' K''. LLL_step p first state g_idx i k = ((P'', state'', G_idx''), I'', K'') ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀(a::?'a::type) (b::?'b::type) c::?'c::type. (?y::?'a::type × ?'b::type × ?'c::type) = (a, b, c) ⟹ ?thesis::bool) ⟹ ?thesis›*))\n obtain p'' and mfs'' and dmu'' and i'' and k'' and g_idx'' where step: \"basis_reduction_mod_step p first mfs dmu g_idx i k = (p'', mfs'', dmu'', g_idx'', i'', k'')\"\n (*goal: ‹(⋀p'' mfs'' dmu'' g_idx'' i'' k''. basis_reduction_mod_step p first mfs dmu g_idx i k = (p'', mfs'', dmu'', g_idx'', i'', k'') ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀(a::?'a::type) (b::?'b::type) c::?'c::type. (?y::?'a::type × ?'b::type × ?'c::type) = (a, b, c) ⟹ ?thesis::bool) ⟹ ?thesis›*))\n from LLL_step[OF impl Linvw step step' i] (*‹state_impl_inv p'' mfs'' dmu'' state'' ∧ I'' = i'' ∧ P'' = p'' ∧ K'' = k'' ∧ G_idx'' = g_idx''›*) have impl'': \"state_impl_inv p'' mfs'' dmu'' state''\" and ID: \"I'' = i''\" \"K'' = k''\" \"P'' = p''\" \"G_idx'' = g_idx''\"\n apply -\n (*goals:\n 1. ‹state_impl_inv (p''::int) (mfs''::int vec list) (dmu''::int mat) (state''::int vec list × int iarray iarray × int iarray × int iarray) ∧ (I''::nat) = (i''::nat) ∧ (P''::int) = p'' ∧ (K''::int) = (k''::int) ∧ (G_idx''::nat) = (g_idx''::nat) ⟹ state_impl_inv p'' mfs'' dmu'' state''›\n 2. ‹state_impl_inv (p''::int) (mfs''::int vec list) (dmu''::int mat) (state''::int vec list × int iarray iarray × int iarray × int iarray) ∧ (I''::nat) = (i''::nat) ∧ (P''::int) = p'' ∧ (K''::int) = (k''::int) ∧ (G_idx''::nat) = (g_idx''::nat) ⟹ I'' = i''›\n 3. ‹state_impl_inv (p''::int) (mfs''::int vec list) (dmu''::int mat) (state''::int vec list × int iarray iarray × int iarray × int iarray) ∧ (I''::nat) = (i''::nat) ∧ (P''::int) = p'' ∧ (K''::int) = (k''::int) ∧ (G_idx''::nat) = (g_idx''::nat) ⟹ K'' = k''›\n 4. ‹state_impl_inv (p''::int) (mfs''::int vec list) (dmu''::int mat) (state''::int vec list × int iarray iarray × int iarray × int iarray) ∧ (I''::nat) = (i''::nat) ∧ (P''::int) = p'' ∧ (K''::int) = (k''::int) ∧ (G_idx''::nat) = (g_idx''::nat) ⟹ P'' = p''›\n 5. ‹state_impl_inv (p''::int) (mfs''::int vec list) (dmu''::int mat) (state''::int vec list × int iarray iarray × int iarray × int iarray) ∧ (I''::nat) = (i''::nat) ∧ (P''::int) = p'' ∧ (K''::int) = (k''::int) ∧ (G_idx''::nat) = (g_idx''::nat) ⟹ G_idx'' = g_idx''›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*proven 5 subgoals*) .\n from basis_reduction_mod_step[OF Linv step i] (*‹∃(fs'::int vec list) b'::rat. LLL_measure (i''::nat) fs' < LLL_measure (i::nat) (fs::int vec list) ∧ LLL_invariant_mod fs' (mfs''::int vec list) (dmu''::int mat) (p''::int) (first::bool) b' i''›*) obtain fs'' and b'' where Linv'': \"LLL_invariant_mod fs'' mfs'' dmu'' p'' first b'' i''\" and decr: \"LLL_measure i'' fs'' < LLL_measure i fs\"\n (*goal: ‹(⋀fs'' b''. ⟦LLL_invariant_mod fs'' mfs'' dmu'' p'' first b'' i''; LLL_measure i'' fs'' < LLL_measure i fs⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n note res = res[unfolded id if_True step split] (*‹basis_reduction_mod_main p'' first mfs'' dmu'' g_idx'' i'' k'' = (p', mfs', dmu')›*)\n note res' = res'[unfolded id if_True step' split ID] (*‹LLL_main p'' first state'' g_idx'' i'' k'' = (pi', state')›*)\n show \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n by (rule IH[OF decr impl'' Linv'' res res'] (*‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*))\n qed\n qed\n\nlemma LLL_iso_main_inner: assumes impl: \"state_impl_inv p mfs dmu state\" \n and di_prods: \"state_iso_inv di_prods (di_of state)\" \n and Linv: \"LLL_invariant_mod_weak fs mfs dmu p first b\"\n and res: \"basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu')\" \n and res': \"LLL_iso_main_inner p first state di_prods g_idx k = (pi', state')\" \n and m: \"m > 1\" \nshows \"state_impl_inv p' mfs' dmu' state' ∧ pi' = p'\"\n using assms(1-5) (*‹state_impl_inv p mfs dmu state› ‹state_iso_inv di_prods (di_of state)› ‹LLL_invariant_mod_weak (fs::int vec list) (mfs::int vec list) (dmu::int mat) (p::int) (first::bool) (b::rat)› ‹basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu')› ‹LLL_iso_main_inner p first state di_prods g_idx k = (pi', state')›*) proof (induct \"LLL_measure (m - 1) fs\" arbitrary: mfs dmu state fs p b k di_prods g_idx rule: less_induct)\n (*goal: ‹⋀fs mfs dmu state p b k di_prods g_idx. ⟦⋀fsa mfs dmu state p b k di_prods g_idx. ⟦LLL_measure (m - 1) fsa < LLL_measure (m - 1) fs; state_impl_inv p mfs dmu state; state_iso_inv di_prods (di_of state); LLL_invariant_mod_weak fsa mfs dmu p first b; basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu'); LLL_iso_main_inner p first state di_prods g_idx k = (pi', state')⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'; state_impl_inv p mfs dmu state; state_iso_inv di_prods (di_of state); LLL_invariant_mod_weak fs mfs dmu p first b; basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu'); LLL_iso_main_inner p first state di_prods g_idx k = (pi', state')⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n case (less fs mfs dmu state p b k di_prods g_idx) (*‹⟦LLL_measure (m - 1) ?fs2 < LLL_measure (m - 1) fs; state_impl_inv ?p2 ?mfs2 ?dmu2 ?state2; state_iso_inv ?di_prods2 (di_of ?state2); LLL_invariant_mod_weak ?fs2 ?mfs2 ?dmu2 ?p2 first ?b2; basis_reduction_iso_main ?p2 first ?mfs2 ?dmu2 ?g_idx2 ?k2 = (p', mfs', dmu'); LLL_iso_main_inner ?p2 first ?state2 ?di_prods2 ?g_idx2 ?k2 = (pi', state')⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'› ‹state_impl_inv p mfs dmu state› ‹state_iso_inv di_prods (di_of state)› ‹LLL_invariant_mod_weak fs mfs dmu p first b› ‹basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu')› ‹LLL_iso_main_inner (p::int) (first::bool) (state::int vec list × int iarray iarray × int iarray × int iarray) (di_prods::(int × int) iarray) (g_idx::nat) (k::int) = (pi'::int, state'::int vec list × int iarray iarray × int iarray × int iarray)›*)\n note impl = less(2) (*‹state_impl_inv (p::int) (mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray)›*)\n note di_prods = less(3) (*‹state_iso_inv di_prods (di_of state)›*)\n note Linv = less(4) (*‹LLL_invariant_mod_weak fs mfs dmu p first b›*)\n note res = less(5) (*‹basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu')›*)\n note res' = less(6) (*‹LLL_iso_main_inner p first state di_prods g_idx k = (pi', state')›*)\n note IH = less(1) (*‹⟦LLL_measure (m - 1) ?fs2 < LLL_measure (m - 1) fs; state_impl_inv ?p2 ?mfs2 ?dmu2 ?state2; state_iso_inv ?di_prods2 (di_of ?state2); LLL_invariant_mod_weak ?fs2 ?mfs2 ?dmu2 ?p2 first ?b2; basis_reduction_iso_main ?p2 first ?mfs2 ?dmu2 ?g_idx2 ?k2 = (p', mfs', dmu'); LLL_iso_main_inner ?p2 first ?state2 ?di_prods2 ?g_idx2 ?k2 = (pi', state')⟧ ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n obtain mfsi and dmui and di and mods where state: \"state = (mfsi, dmui, di, mods)\"\n (*goal: ‹(⋀(mfsi::int vec list) (dmui::int iarray iarray) (di::int iarray) mods::int iarray. (state::int vec list × int iarray iarray × int iarray × int iarray) = (mfsi, dmui, di, mods) ⟹ thesis::bool) ⟹ thesis›*)\n by (metis prod_cases4 (*‹(⋀(a::?'a) (b::?'b) (c::?'c) d::?'d. (?y::?'a × ?'b × ?'c × ?'d) = (a, b, c, d) ⟹ ?thesis::bool) ⟹ ?thesis›*))\n from di_prods (*‹state_iso_inv di_prods (di_of state)›*) state (*‹state = (mfsi, dmui, di, mods)›*) have di_prods: \"state_iso_inv di_prods di\"\n by auto\n obtain num and denom and idx where quot': \"LLL_max_gso_quot di_prods = (num, denom, idx)\"\n (*goal: ‹(⋀num denom idx. LLL_max_gso_quot di_prods = (num, denom, idx) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n note inv = LLL_invD_modw[OF Linv] (*‹length mfs = m› ‹length fs = m› ‹lattice_of fs = L› ‹gs.lin_indpt_list (map of_int_hom.vec_hom fs)› ‹map (map_vec (λx. x symmod p)) fs = mfs› ‹∀i'<m. ∀j'<i'. ¦dμ fs i' j'¦ < p * d fs j' * d fs (Suc j')› ‹∀i'<m. ∀j'<m. dμ fs i' j' = dmu $$ (i', j')› ‹?i < m ⟹ fs ! ?i ∈ carrier_vec n› ‹set fs ⊆ carrier_vec n› ‹?i < m ⟹ gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) ?i ∈ Rn› ‹?i < m ⟹ mfs ! ?i ∈ carrier_vec n› ‹set mfs ⊆ carrier_vec n› and more 3 facts*)\n obtain na and da where alph: \"quotient_of α = (na,da)\"\n (*goal: ‹(⋀(na::int) da::int. quotient_of (α::rat) = (na, da) ⟹ thesis::bool) ⟹ thesis›*)\n by force\n from impl[unfolded state] (*‹state_impl_inv p mfs dmu (mfsi, dmui, di, mods)›*) have di: \"di = IArray.of_fun (d_of dmu) (Suc m)\"\n by auto\n from LLL_max_gso_quot[OF di di_prods] (*‹LLL_max_gso_quot di_prods = compute_max_gso_quot dmu›*) have quot: \"compute_max_gso_quot dmu = LLL_max_gso_quot di_prods\"\n by standard\n obtain cmp where cmp: \"(na * denom < num * da) = cmp\"\n (*goal: ‹(⋀cmp. (na * denom < num * da) = cmp ⟹ thesis) ⟹ thesis›*)\n by force\n have \"(m > 1) = True\"\n using m (*‹(1::nat) < (m::nat)›*) by auto\n note res = res[unfolded basis_reduction_iso_main.simps [ of _ _ _ _ _ k ] this if_True Let_def quot quot' split alph cmp] (*‹(if cmp then case basis_reduction_adjust_swap_add_step p first mfs dmu g_idx idx of (p', mfs', dmu', g_idx') ⇒ basis_reduction_iso_main p' first mfs' dmu' g_idx' (k + 1) else (p, mfs, dmu)) = (p', mfs', dmu')›*)\n note res' = res'[unfolded LLL_iso_main_inner.simps [ of _ _ _ _ _ k ] state split Let_def quot' alph cmp, folded state] (*‹(if cmp then case LLL_adjust_swap_add p first state g_idx idx of (p', state', g_idx') ⇒ case state' of (x, xaa, di', xba) ⇒ LLL_iso_main_inner p' first state' (IArray.of_fun (λi. if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di' !! idx * di' !! idx, r) else (l, di' !! (i + 2) * di' !! i)) (m - 1)) g_idx' (k + 1) else (p, state)) = (pi', state')›*)\n note cmp = compute_max_gso_quot_alpha[OF Linv quot [ unfolded quot' ] alph cmp m] (*‹cmp::bool ⟹ (idx::nat) ≠ (0::nat) ∧ idx < (m::nat) ∧ ¬ d_of (dmu::int mat) idx * d_of dmu idx * (da::int) ≤ (na::int) * d_of dmu (idx - (1::nat)) * d_of dmu (Suc idx)› ‹¬ cmp ⟹ LLL_invariant_mod fs mfs dmu p first b m›*)\n show \"?case\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n proof (cases cmp)\n (*goals:\n 1. ‹cmp::bool ⟹ state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (pi'::int) = p'›\n 2. ‹¬ (cmp::bool) ⟹ state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (pi'::int) = p'›*)\n case False (*‹¬ cmp›*)\n thus \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n using res (*‹(if cmp then case basis_reduction_adjust_swap_add_step p first mfs dmu g_idx idx of (p', mfs', dmu', g_idx') ⇒ basis_reduction_iso_main p' first mfs' dmu' g_idx' (k + 1) else (p, mfs, dmu)) = (p', mfs', dmu')›*) res' (*‹(if cmp then case LLL_adjust_swap_add p first state g_idx idx of (p', state', g_idx') ⇒ case state' of (x, xaa, di', xba) ⇒ LLL_iso_main_inner p' first state' (IArray.of_fun (λi. if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di' !! idx * di' !! idx, r) else (l, di' !! (i + 2) * di' !! i)) (m - 1)) g_idx' (k + 1) else (p, state)) = (pi', state')›*) impl (*‹state_impl_inv p mfs dmu state›*) by auto\n next\n (*goal: ‹cmp::bool ⟹ state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (pi'::int) = p'›*)\n case True (*‹cmp›*)\n hence id: \"cmp = True\"\n by simp\n note cmp = cmp(1)[OF True] (*‹(idx::nat) ≠ (0::nat) ∧ idx < (m::nat) ∧ ¬ d_of (dmu::int mat) idx * d_of dmu idx * (da::int) ≤ (na::int) * d_of dmu (idx - (1::nat)) * d_of dmu (Suc idx)›*)\n obtain state'' and P'' and G_idx'' where step': \"LLL_adjust_swap_add p first state g_idx idx = (P'',state'', G_idx'')\"\n (*goal: ‹(⋀P'' state'' G_idx''. LLL_adjust_swap_add p first state g_idx idx = (P'', state'', G_idx'') ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n obtain mfs'' and dmu'' and p'' and g_idx'' where step: \"basis_reduction_adjust_swap_add_step p first mfs dmu g_idx idx = (p'', mfs'', dmu'', g_idx'')\"\n (*goal: ‹(⋀p'' mfs'' dmu'' g_idx''. basis_reduction_adjust_swap_add_step p first mfs dmu g_idx idx = (p'', mfs'', dmu'', g_idx'') ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n obtain mfsi2 and dmui2 and di2 and mods2 where state2: \"state'' = (mfsi2, dmui2, di2, mods2)\"\n (*goal: ‹(⋀mfsi2 dmui2 di2 mods2. state'' = (mfsi2, dmui2, di2, mods2) ⟹ thesis) ⟹ thesis›*)\n apply (cases state'')\n (*goal: ‹(⋀mfsi2 dmui2 di2 mods2. state'' = (mfsi2, dmui2, di2, mods2) ⟹ thesis) ⟹ thesis›*)\n by auto\n note res = res[unfolded id if_True step split] (*‹basis_reduction_iso_main p'' first mfs'' dmu'' g_idx'' (k + 1) = (p', mfs', dmu')›*)\n note res' = res'[unfolded id if_True step' state2 split, folded state2] (*‹LLL_iso_main_inner P'' first state'' (IArray.of_fun (λi. if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i)) (m - 1)) G_idx'' (k + 1) = (pi', state')›*)\n from cmp (*‹idx ≠ 0 ∧ idx < m ∧ ¬ d_of dmu idx * d_of dmu idx * da ≤ na * d_of dmu (idx - 1) * d_of dmu (Suc idx)›*) have idx0: \"idx ≠ 0\" and idx: \"idx < m\" and ineq: \"¬ d_of dmu idx * d_of dmu idx * da ≤ na * d_of dmu (idx - 1) * d_of dmu (Suc idx)\"\n apply -\n (*goals:\n 1. ‹idx ≠ 0 ∧ idx < m ∧ ¬ d_of dmu idx * d_of dmu idx * da ≤ na * d_of dmu (idx - 1) * d_of dmu (Suc idx) ⟹ idx ≠ 0›\n 2. ‹idx ≠ 0 ∧ idx < m ∧ ¬ d_of dmu idx * d_of dmu idx * da ≤ na * d_of dmu (idx - 1) * d_of dmu (Suc idx) ⟹ idx < m›\n 3. ‹idx ≠ 0 ∧ idx < m ∧ ¬ d_of dmu idx * d_of dmu idx * da ≤ na * d_of dmu (idx - 1) * d_of dmu (Suc idx) ⟹ ¬ d_of dmu idx * d_of dmu idx * da ≤ na * d_of dmu (idx - 1) * d_of dmu (Suc idx)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n from basis_reduction_adjust_swap_add_step[OF Linv step alph ineq idx idx0] (*‹∃(fs'::int vec list) b'::rat. LLL_invariant_mod_weak fs' (mfs''::int vec list) (dmu''::int mat) (p''::int) (first::bool) b' ∧ LLL_measure ((idx::nat) - (1::nat)) fs' < LLL_measure idx (fs::int vec list) ∧ LLL_measure ((m::nat) - (1::nat)) fs' < LLL_measure (m - (1::nat)) fs ∧ (LLL_invariant_mod fs (mfs::int vec list) (dmu::int mat) (p::int) first (b::rat) idx ⟶ LLL_invariant_mod fs' mfs'' dmu'' p'' first b' (idx - (1::nat)))›*) obtain fs'' and b'' where Linv'': \"LLL_invariant_mod_weak fs'' mfs'' dmu'' p'' first b''\" and meas: \"LLL_measure (m - 1) fs'' < LLL_measure (m - 1) fs\"\n (*goal: ‹(⋀fs'' b''. ⟦LLL_invariant_mod_weak fs'' mfs'' dmu'' p'' first b''; LLL_measure (m - 1) fs'' < LLL_measure (m - 1) fs⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n from LLL_adjust_swap_add[OF impl Linv step step' idx idx0] (*‹state_impl_inv p'' mfs'' dmu'' state''› ‹P'' = p''› ‹G_idx'' = g_idx''› ‹⟦?i ≤ m; ?i ≠ idx⟧ ⟹ di_of state'' !! ?i = di_of state !! ?i›*) have impl'': \"state_impl_inv p'' mfs'' dmu'' state''\" and P'': \"P'' = p''\" \"G_idx'' = g_idx''\" and di_prod_upd: \"⋀ i. i ≤ m ⟹ i ≠ idx ⟹ di2 !! i = di !! i\"\n using state (*‹(state::int vec list × int iarray iarray × int iarray × int iarray) = (mfsi::int vec list, dmui::int iarray iarray, di::int iarray, mods::int iarray)›*) state2 (*‹(state''::int vec list × int iarray iarray × int iarray × int iarray) = (mfsi2::int vec list, dmui2::int iarray iarray, di2::int iarray, mods2::int iarray)›*) apply -\n (*goals:\n 1. ‹⟦state_impl_inv p'' mfs'' dmu'' state''; P'' = p''; G_idx'' = g_idx''; ⋀i. ⟦i ≤ m; i ≠ idx⟧ ⟹ di_of state'' !! i = di_of state !! i; state = (mfsi, dmui, di, mods); state'' = (mfsi2, dmui2, di2, mods2)⟧ ⟹ state_impl_inv p'' mfs'' dmu'' state''›\n 2. ‹⟦state_impl_inv p'' mfs'' dmu'' state''; P'' = p''; G_idx'' = g_idx''; ⋀i. ⟦i ≤ m; i ≠ idx⟧ ⟹ di_of state'' !! i = di_of state !! i; state = (mfsi, dmui, di, mods); state'' = (mfsi2, dmui2, di2, mods2)⟧ ⟹ P'' = p''›\n 3. ‹⟦state_impl_inv p'' mfs'' dmu'' state''; P'' = p''; G_idx'' = g_idx''; ⋀i. ⟦i ≤ m; i ≠ idx⟧ ⟹ di_of state'' !! i = di_of state !! i; state = (mfsi, dmui, di, mods); state'' = (mfsi2, dmui2, di2, mods2)⟧ ⟹ G_idx'' = g_idx''›\n 4. ‹⋀i. ⟦i ≤ m; i ≠ idx; state_impl_inv p'' mfs'' dmu'' state''; P'' = p''; G_idx'' = g_idx''; ⋀i. ⟦i ≤ m; i ≠ idx⟧ ⟹ di_of state'' !! i = di_of state !! i; state = (mfsi, dmui, di, mods); state'' = (mfsi2, dmui2, di2, mods2)⟧ ⟹ di2 !! i = di !! i›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n have di_prods: \"state_iso_inv (IArray.of_fun\n (λi. if idx < i ∨ i + 2 < idx then di_prods !! i\n else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i))\n (m - 1)) di2\"\n unfolding state_iso_inv_def\n (*goal: ‹IArray.of_fun (λi. if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i)) (m - 1) = IArray.of_fun (λi. (di2 !! (i + 1) * di2 !! (i + 1), di2 !! (i + 2) * di2 !! i)) (m - 1)›*)\n apply (intro iarray_cong' (*‹(⋀i. i < ?n ⟹ ?f i = ?g i) ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?n›*))\n (*goal: ‹IArray.of_fun (λi. if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i)) (m - 1) = IArray.of_fun (λi. (di2 !! (i + 1) * di2 !! (i + 1), di2 !! (i + 2) * di2 !! i)) (m - 1)›*)\n apply (insert di_prod_upd (*‹⟦?i2 ≤ m; ?i2 ≠ idx⟧ ⟹ di2 !! ?i2 = di !! ?i2›*))\n (*goal: ‹⋀i. i < m - 1 ⟹ (if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i)) = (di2 !! (i + 1) * di2 !! (i + 1), di2 !! (i + 2) * di2 !! i)›*)\n apply (unfold di_prods[unfolded state_iso_inv_def] (*‹di_prods = IArray.of_fun (λi. (di !! (i + 1) * di !! (i + 1), di !! (i + 2) * di !! i)) (m - 1)›*))\n (*goal: ‹⋀i. ⟦i < m - 1; ⋀i. ⟦i ≤ m; i ≠ idx⟧ ⟹ di2 !! i = di !! i⟧ ⟹ (if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i)) = (di2 !! (i + 1) * di2 !! (i + 1), di2 !! (i + 2) * di2 !! i)›*)\n apply (subst (1 2) IArray.of_fun_nth (*‹?i < ?n ⟹ IArray.of_fun ?f ?n !! ?i = ?f ?i›*))\n (*goals:\n 1. ‹⋀i. ⟦i < m - 1; ⋀i. ⟦i ≤ m; i ≠ idx⟧ ⟹ di2 !! i = di !! i⟧ ⟹ i < m - 1›\n 2. ‹⋀i. ⟦i < m - 1; ⋀i. ⟦i ≤ m; i ≠ idx⟧ ⟹ di2 !! i = di !! i⟧ ⟹ (if idx < i ∨ i + 2 < idx then (di !! (i + 1) * di !! (i + 1), di !! (i + 2) * di !! i) else case (di !! (i + 1) * di !! (i + 1), di !! (i + 2) * di !! i) of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i)) = (di2 !! (i + 1) * di2 !! (i + 1), di2 !! (i + 2) * di2 !! i)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"?thesis\"\n (*goal: ‹state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (pi'::int) = p'›*)\n apply (rule IH[OF meas impl'' _ Linv'' res res'[unfolded step' P'']] (*‹state_iso_inv (IArray.of_fun (λi::nat. if (idx::nat) < i ∨ i + (2::nat) < idx then (di_prods::(int × int) iarray) !! i else case di_prods !! i of (l::int, r::int) ⇒ if i + (1::nat) = idx then ((di2::int iarray) !! idx * di2 !! idx, r) else (l, di2 !! (i + (2::nat)) * di2 !! i)) ((m::nat) - (1::nat))) (di_of (state''::int vec list × int iarray iarray × int iarray × int iarray)) ⟹ state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) (state'::int vec list × int iarray iarray × int iarray × int iarray) ∧ (pi'::int) = p'›*))\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n apply (insert di_prods (*‹state_iso_inv (IArray.of_fun (λi::nat. if (idx::nat) < i ∨ i + (2::nat) < idx then (di_prods::(int × int) iarray) !! i else case di_prods !! i of (l::int, r::int) ⇒ if i + (1::nat) = idx then ((di2::int iarray) !! idx * di2 !! idx, r) else (l, di2 !! (i + (2::nat)) * di2 !! i)) ((m::nat) - (1::nat))) di2›*) state2 (*‹(state''::int vec list × int iarray iarray × int iarray × int iarray) = (mfsi2::int vec list, dmui2::int iarray iarray, di2::int iarray, mods2::int iarray)›*))\n (*goal: ‹state_iso_inv (IArray.of_fun (λi. if idx < i ∨ i + 2 < idx then di_prods !! i else case di_prods !! i of (l, r) ⇒ if i + 1 = idx then (di2 !! idx * di2 !! idx, r) else (l, di2 !! (i + 2) * di2 !! i)) (m - 1)) (di_of state'')›*)\n by auto\n qed\n qed\n\nlemma LLL_iso_main: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod_weak fs mfs dmu p first b\"\n and res: \"basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu')\" \n and res': \"LLL_iso_main p first state g_idx k = (pi', state')\" \nshows \"state_impl_inv p' mfs' dmu' state' ∧ pi' = p'\"\nproof (cases \"m > 1\")\n (*goals:\n 1. ‹1 < m ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›\n 2. ‹¬ 1 < m ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n case True (*‹1 < m›*)\n from LLL_iso_main_inner[OF impl _ Linv res _ True, unfolded state_iso_inv_def, OF refl, of pi' state'] (*‹LLL_iso_main_inner (p::int) (first::bool) (state::int vec list × int iarray iarray × int iarray × int iarray) (IArray.of_fun (λi::nat. (di_of state !! (i + (1::nat)) * di_of state !! (i + (1::nat)), di_of state !! (i + (2::nat)) * di_of state !! i)) ((m::nat) - (1::nat))) (g_idx::nat) (k::int) = (pi'::int, state'::int vec list × int iarray iarray × int iarray × int iarray) ⟹ state_impl_inv (p'::int) (mfs'::int vec list) (dmu'::int mat) state' ∧ pi' = p'›*) res' (*‹LLL_iso_main (p::int) (first::bool) (state::int vec list × int iarray iarray × int iarray × int iarray) (g_idx::nat) (k::int) = (pi'::int, state'::int vec list × int iarray iarray × int iarray × int iarray)›*) True (*‹1 < m›*) show \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n unfolding LLL_iso_main_def\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n apply (cases state)\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n by auto\nnext\n (*goal: ‹¬ 1 < m ⟹ state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n case False (*‹¬ 1 < m›*)\n thus \"?thesis\"\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n using res (*‹basis_reduction_iso_main p first mfs dmu g_idx k = (p', mfs', dmu')›*) res' (*‹LLL_iso_main p first state g_idx k = (pi', state')›*) impl (*‹state_impl_inv p mfs dmu state›*) unfolding LLL_iso_main_def basis_reduction_iso_main.simps[of _ _ _ _ _ k]\n (*goal: ‹state_impl_inv p' mfs' dmu' state' ∧ pi' = p'›*)\n by auto\nqed\n\nlemma LLL_initial: assumes res: \"compute_initial_state first = (p, mfs, dmu, g_idx)\" \n and res': \"LLL_initial first = (p', state, g_idx')\" \n and m: \"m ≠ 0\" \nshows \"state_impl_inv p mfs dmu state ∧ p' = p ∧ g_idx' = g_idx\"\nproof (-)\n (*goal: ‹state_impl_inv p mfs dmu state ∧ p' = p ∧ g_idx' = g_idx›*)\n obtain b and gi where norm: \"compute_max_gso_norm first dmu_initial = (b,gi)\"\n (*goal: ‹(⋀b gi. compute_max_gso_norm first dmu_initial = (b, gi) ⟹ thesis) ⟹ thesis›*)\n by force\n obtain P where P: \"compute_mod_of_max_gso_norm first b = P\"\n (*goal: ‹(⋀P::int. compute_mod_of_max_gso_norm (first::bool) (b::rat) = P ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n define di where \"di = IArray.of_fun (λi. if i = 0 then 1 else dμ_impl fs_init !! (i - 1) !! (i - 1)) (Suc m)\"\n note res = res[unfolded compute_initial_state_def Let_def P norm split] (*‹(P::int, compute_initial_mfs P, compute_initial_dmu P dmu_initial, gi::nat) = (p::int, mfs::int vec list, dmu::int mat, g_idx::nat)›*)\n have di: \"di = IArray.of_fun (d_of dmu_initial) (Suc m)\"\n unfolding di_def dmu_initial_def Let_def d_of_def\n (*goal: ‹IArray.of_fun (λi. if i = 0 then 1 else dμ_impl fs_init !! (i - 1) !! (i - 1)) (Suc m) = IArray.of_fun (λi. if i = 0 then 1 else mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0) $$ (i - 1, i - 1)) (Suc m)›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*) if_cong (*‹⟦?b = ?c; ?c ⟹ ?x = ?u; ¬ ?c ⟹ ?y = ?v⟧ ⟹ (if ?b then ?x else ?y) = (if ?c then ?u else ?v)›*))\n (*goal: ‹IArray.of_fun (λi. if i = 0 then 1 else dμ_impl fs_init !! (i - 1) !! (i - 1)) (Suc m) = IArray.of_fun (λi. if i = 0 then 1 else mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0) $$ (i - 1, i - 1)) (Suc m)›*)\n by auto\n note norm' = LLL_max_gso_norm_di[OF di m, of first, unfolded norm] (*‹LLL_max_gso_norm_di first di = (b, gi)›*)\n note res' = res'[unfolded LLL_initial_def Let_def, folded di_def, unfolded norm' P split mod_of_gso_norm [ OF m ]] (*‹(P, (compute_initial_mfs P, IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p', state, g_idx')›*)\n from res (*‹(P, compute_initial_mfs P, compute_initial_dmu P dmu_initial, gi) = (p, mfs, dmu, g_idx)›*) have p: \"p = P\" and mfs: \"mfs = compute_initial_mfs p\" and dmu: \"dmu = compute_initial_dmu P dmu_initial\" and g_idx: \"g_idx = gi\"\n apply -\n (*goals:\n 1. ‹(P, compute_initial_mfs P, compute_initial_dmu P dmu_initial, gi) = (p, mfs, dmu, g_idx) ⟹ p = P›\n 2. ‹(P, compute_initial_mfs P, compute_initial_dmu P dmu_initial, gi) = (p, mfs, dmu, g_idx) ⟹ mfs = compute_initial_mfs p›\n 3. ‹(P, compute_initial_mfs P, compute_initial_dmu P dmu_initial, gi) = (p, mfs, dmu, g_idx) ⟹ dmu = compute_initial_dmu P dmu_initial›\n 4. ‹(P, compute_initial_mfs P, compute_initial_dmu P dmu_initial, gi) = (p, mfs, dmu, g_idx) ⟹ g_idx = gi›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n let ?mods = \"IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)\"\n have di': \"di = IArray.of_fun (d_of (compute_initial_dmu P dmu_initial)) (Suc m)\"\n unfolding di\n (*goal: ‹IArray.of_fun (d_of dmu_initial) (Suc (m::nat)) = IArray.of_fun (d_of (compute_initial_dmu (P::int) dmu_initial)) (Suc m)›*)\n apply (intro iarray_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*) refl (*‹?t = ?t›*))\n (*goal: ‹IArray.of_fun (d_of dmu_initial) (Suc m) = IArray.of_fun (d_of (compute_initial_dmu P dmu_initial)) (Suc m)›*)\n by (auto simp: compute_initial_dmu_def (*‹compute_initial_dmu ?p ?dmu = mat m m (λ(i', j'). if j' < i' then ?dmu $$ (i', j') symmod (?p * d_of ?dmu j' * d_of ?dmu (Suc j')) else ?dmu $$ (i', j'))›*) d_of_def (*‹d_of ?dmu ?i = (if ?i = 0 then 1 else ?dmu $$ (?i - 1, ?i - 1))›*))\n from res' (*‹(P, (compute_initial_mfs P, IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p', state, g_idx')›*) have p': \"p' = P\" and g_idx': \"g_idx' = gi\" and state: \"state = (compute_initial_mfs P, IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod ?mods !! j) i) m, di, ?mods)\"\n apply -\n (*goals:\n 1. ‹(P, (compute_initial_mfs P, IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p', state, g_idx') ⟹ p' = P›\n 2. ‹(P, (compute_initial_mfs P, IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p', state, g_idx') ⟹ g_idx' = gi›\n 3. ‹(P, (compute_initial_mfs P, IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)), gi) = (p', state, g_idx') ⟹ state = (compute_initial_mfs P, IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m, di, IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n show \"?thesis\"\n (*goal: ‹state_impl_inv (p::int) (mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray) ∧ (p'::int) = p ∧ (g_idx'::nat) = (g_idx::nat)›*)\n unfolding mfs p state p' dmu state_impl_inv.simps g_idx' g_idx\n (*goal: ‹(compute_initial_mfs P = compute_initial_mfs P ∧ di = IArray.of_fun (d_of (compute_initial_dmu P dmu_initial)) (Suc m) ∧ IArray.of_fun (λi. IArray.of_fun (λj. dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j) i) m = IArray.of_fun (λi. IArray.of_fun (λj. compute_initial_dmu P dmu_initial $$ (i, j)) i) m ∧ IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) = IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1)) ∧ P = P ∧ gi = gi›*)\n apply (intro conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*) refl (*‹(?t::?'a) = ?t›*) di' (*‹(di::int iarray) = IArray.of_fun (d_of (compute_initial_dmu (P::int) dmu_initial)) (Suc (m::nat))›*) iarray_cong (*‹⟦(?n::nat) = (?m::nat); ⋀i::nat. i < ?m ⟹ (?f::nat ⇒ ?'a) i = (?g::nat ⇒ ?'a) i⟧ ⟹ IArray.of_fun ?f ?n = IArray.of_fun ?g ?m›*))\n (*goal: ‹(compute_initial_mfs (P::int) = compute_initial_mfs P ∧ (di::int iarray) = IArray.of_fun (d_of (compute_initial_dmu P dmu_initial)) (Suc (m::nat)) ∧ IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. dμ_impl (fs_init::int vec list) !! i !! j symmod IArray.of_fun (λj::nat. P * di !! j * di !! Suc j) (m - (1::nat)) !! j) i) m = IArray.of_fun (λi::nat. IArray.of_fun (λj::nat. compute_initial_dmu P dmu_initial $$ (i, j)) i) m ∧ IArray.of_fun (λj::nat. P * di !! j * di !! Suc j) (m - (1::nat)) = IArray.of_fun (λj::nat. P * di !! j * di !! Suc j) (m - (1::nat))) ∧ P = P ∧ (gi::nat) = gi›*)\n proof (goal_cases)\n (*goal: ‹⋀i ia. ⟦i < m; ia < i⟧ ⟹ dμ_impl fs_init !! i !! ia symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! ia = compute_initial_dmu P dmu_initial $$ (i, ia)›*)\n case (1 i j) (*‹i < m› ‹j < i›*)\n hence \"j < m\" \"Suc j < Suc m\" \"j < Suc m\" \"j < m - 1\"\n apply -\n (*goals:\n 1. ‹⟦i < m; j < i⟧ ⟹ j < m›\n 2. ‹⟦i < m; j < i⟧ ⟹ Suc j < Suc m›\n 3. ‹⟦i < m; j < i⟧ ⟹ j < Suc m›\n 4. ‹⟦i < m; j < i⟧ ⟹ j < m - 1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n thus \"?case\"\n (*goal: ‹dμ_impl fs_init !! i !! j symmod IArray.of_fun (λj. P * di !! j * di !! Suc j) (m - 1) !! j = compute_initial_dmu P dmu_initial $$ (i, j)›*)\n unfolding compute_initial_dmu_def di IArray.of_fun_nth[OF ‹j < m›] IArray.of_fun_nth[OF ‹Suc j < Suc m›] IArray.of_fun_nth[OF ‹j < Suc m›] IArray.of_fun_nth[OF ‹j < m - 1›]\n (*goal: ‹dμ_impl fs_init !! i !! j symmod (P * d_of dmu_initial j * d_of dmu_initial (Suc j)) = mat m m (λ(i', j'). if j' < i' then dmu_initial $$ (i', j') symmod (P * d_of dmu_initial j' * d_of dmu_initial (Suc j')) else dmu_initial $$ (i', j')) $$ (i, j)›*)\n unfolding dmu_initial_def Let_def\n (*goal: ‹dμ_impl fs_init !! i !! j symmod (P * d_of (mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0)) j * d_of (mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0)) (Suc j)) = mat m m (λ(i', j'). if j' < i' then mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0) $$ (i', j') symmod (P * d_of (mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0)) j' * d_of (mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0)) (Suc j')) else mat m m (λ(i, j). if j ≤ i then dμ_impl fs_init !! i !! j else 0) $$ (i', j')) $$ (i, j)›*)\n using \"1\" (*‹i < m› ‹j < i›*) by auto\n qed\nqed \n\nlemma LLL_add_rows_loop: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod fs mfs dmu p b first i\"\n and res: \"basis_reduction_mod_add_rows_loop p mfs dmu i j = (mfs', dmu')\" \n and res': \"LLL_add_rows_loop p state i j = state'\" \n and j: \"j ≤ i\" \n and i: \"i < m\" \nshows \"state_impl_inv p mfs' dmu' state'\"\n using assms(1-5) (*‹state_impl_inv p mfs dmu state› ‹LLL_invariant_mod fs mfs dmu p b first i› ‹basis_reduction_mod_add_rows_loop (p::int) (mfs::int vec list) (dmu::int mat) (i::nat) (j::nat) = (mfs'::int vec list, dmu'::int mat)› ‹LLL_add_rows_loop p state i j = state'› ‹j ≤ i›*) proof (induct j arbitrary: fs mfs dmu state)\n (*goals:\n 1. ‹⋀(fs::int vec list) (mfs::int vec list) (dmu::int mat) state::int vec list × int iarray iarray × int iarray × int iarray. ⟦state_impl_inv (p::int) mfs dmu state; LLL_invariant_mod fs mfs dmu p (b::bool) (first::rat) (i::nat); basis_reduction_mod_add_rows_loop p mfs dmu i (0::nat) = (mfs'::int vec list, dmu'::int mat); LLL_add_rows_loop p state i (0::nat) = (state'::int vec list × int iarray iarray × int iarray × int iarray); (0::nat) ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'›\n 2. ‹⋀(j::nat) (fs::int vec list) (mfs::int vec list) (dmu::int mat) state::int vec list × int iarray iarray × int iarray × int iarray. ⟦⋀(fs::int vec list) (mfs::int vec list) (dmu::int mat) state::int vec list × int iarray iarray × int iarray × int iarray. ⟦state_impl_inv (p::int) mfs dmu state; LLL_invariant_mod fs mfs dmu p (b::bool) (first::rat) (i::nat); basis_reduction_mod_add_rows_loop p mfs dmu i j = (mfs'::int vec list, dmu'::int mat); LLL_add_rows_loop p state i j = (state'::int vec list × int iarray iarray × int iarray × int iarray); j ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'; state_impl_inv p mfs dmu state; LLL_invariant_mod fs mfs dmu p b first i; basis_reduction_mod_add_rows_loop p mfs dmu i (Suc j) = (mfs', dmu'); LLL_add_rows_loop p state i (Suc j) = state'; Suc j ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'›*)\n case (Suc j) (*‹⟦state_impl_inv p ?mfs2 ?dmu2 ?state2; LLL_invariant_mod ?fs2 ?mfs2 ?dmu2 p b first i; basis_reduction_mod_add_rows_loop p ?mfs2 ?dmu2 i j = (mfs', dmu'); LLL_add_rows_loop p ?state2 i j = state'; j ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'› ‹state_impl_inv (p::int) (mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray)› ‹LLL_invariant_mod fs mfs dmu p b first i› ‹basis_reduction_mod_add_rows_loop (p::int) (mfs::int vec list) (dmu::int mat) (i::nat) (Suc (j::nat)) = (mfs'::int vec list, dmu'::int mat)› ‹LLL_add_rows_loop (p::int) (state::int vec list × int iarray iarray × int iarray × int iarray) (i::nat) (Suc (j::nat)) = (state'::int vec list × int iarray iarray × int iarray × int iarray)› ‹Suc j ≤ i›*)\n note impl = Suc(2) (*‹state_impl_inv p mfs dmu state›*)\n note Linv = Suc(3) (*‹LLL_invariant_mod fs mfs dmu p b first i›*)\n note res = Suc(4) (*‹basis_reduction_mod_add_rows_loop (p::int) (mfs::int vec list) (dmu::int mat) (i::nat) (Suc (j::nat)) = (mfs'::int vec list, dmu'::int mat)›*)\n note res' = Suc(5) (*‹LLL_add_rows_loop p state i (Suc j) = state'›*)\n note IH = Suc(1) (*‹⟦state_impl_inv p ?mfs2 ?dmu2 ?state2; LLL_invariant_mod ?fs2 ?mfs2 ?dmu2 p b first i; basis_reduction_mod_add_rows_loop p ?mfs2 ?dmu2 i j = (mfs', dmu'); LLL_add_rows_loop p ?state2 i j = state'; j ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'›*)\n from Suc (*‹⟦state_impl_inv p ?mfs2 ?dmu2 ?state2; LLL_invariant_mod ?fs2 ?mfs2 ?dmu2 p b first i; basis_reduction_mod_add_rows_loop p ?mfs2 ?dmu2 i j = (mfs', dmu'); LLL_add_rows_loop p ?state2 i j = state'; j ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'› ‹state_impl_inv p mfs dmu state› ‹LLL_invariant_mod fs mfs dmu p b first i› ‹basis_reduction_mod_add_rows_loop p mfs dmu i (Suc j) = (mfs', dmu')› ‹LLL_add_rows_loop (p::int) (state::int vec list × int iarray iarray × int iarray × int iarray) (i::nat) (Suc (j::nat)) = (state'::int vec list × int iarray iarray × int iarray × int iarray)› ‹Suc j ≤ i›*) have j: \"j < i\" and ji: \"j ≤ i\"\n apply -\n (*goals:\n 1. ‹⟦⋀(mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray) fs::int vec list. ⟦state_impl_inv (p::int) mfs dmu state; LLL_invariant_mod fs mfs dmu p (b::bool) (first::rat) (i::nat); basis_reduction_mod_add_rows_loop p mfs dmu i (j::nat) = (mfs'::int vec list, dmu'::int mat); LLL_add_rows_loop p state i j = (state'::int vec list × int iarray iarray × int iarray × int iarray); j ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'; state_impl_inv p (mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray); LLL_invariant_mod (fs::int vec list) mfs dmu p b first i; basis_reduction_mod_add_rows_loop p mfs dmu i (Suc j) = (mfs', dmu'); LLL_add_rows_loop p state i (Suc j) = state'; Suc j ≤ i⟧ ⟹ j < i›\n 2. ‹⟦⋀(mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray) fs::int vec list. ⟦state_impl_inv (p::int) mfs dmu state; LLL_invariant_mod fs mfs dmu p (b::bool) (first::rat) (i::nat); basis_reduction_mod_add_rows_loop p mfs dmu i (j::nat) = (mfs'::int vec list, dmu'::int mat); LLL_add_rows_loop p state i j = (state'::int vec list × int iarray iarray × int iarray × int iarray); j ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'; state_impl_inv p (mfs::int vec list) (dmu::int mat) (state::int vec list × int iarray iarray × int iarray × int iarray); LLL_invariant_mod (fs::int vec list) mfs dmu p b first i; basis_reduction_mod_add_rows_loop p mfs dmu i (Suc j) = (mfs', dmu'); LLL_add_rows_loop p state i (Suc j) = state'; Suc j ≤ i⟧ ⟹ j ≤ i›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n obtain mfs1 and dmu1 where add: \"basis_reduction_mod_add_row p mfs dmu i j = (mfs1, dmu1)\"\n (*goal: ‹(⋀mfs1 dmu1. basis_reduction_mod_add_row p mfs dmu i j = (mfs1, dmu1) ⟹ thesis) ⟹ thesis›*)\n by force\n note res = res[unfolded basis_reduction_mod_add_rows_loop.simps Let_def add split] (*‹basis_reduction_mod_add_rows_loop p mfs1 dmu1 i j = (mfs', dmu')›*)\n obtain state1 where add': \"LLL_add_row p state i j = state1\"\n (*goal: ‹(⋀state1. LLL_add_row p state i j = state1 ⟹ thesis) ⟹ thesis›*)\n by auto\n note res' = res'[unfolded LLL_add_rows_loop.simps Let_def add'] (*‹LLL_add_rows_loop p state1 i j = state'›*)\n note Linvw = LLL_mod_inv_to_weak[OF Linv] (*‹LLL_invariant_mod_weak fs mfs dmu p b first›*)\n from LLL_add_row[OF impl Linvw add add' i j] (*‹state_impl_inv p mfs1 dmu1 state1›*) have impl1: \"state_impl_inv p mfs1 dmu1 state1\" .\n from basis_reduction_mod_add_row[OF Linvw add i j] (*‹i ≠ 0 ⟹ ∃fs'. LLL_invariant_mod_weak fs' mfs1 dmu1 p b first ∧ LLL_measure i fs' = LLL_measure i fs ∧ (μ_small_row i fs (Suc j) ⟶ μ_small_row i fs' j) ∧ ¦gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs') i j¦ ≤ 1 / 2 ∧ (∀i' j'. i' < i ⟶ j' ≤ i' ⟶ gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs') i' j' = gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs) i' j') ∧ (LLL_invariant_mod fs mfs dmu p b first i ⟶ LLL_invariant_mod fs' mfs1 dmu1 p b first i) ∧ (∀ii≤m. d fs' ii = d fs ii)›*) Linv (*‹LLL_invariant_mod fs mfs dmu p b first i›*) j (*‹j < i›*) obtain fs1 where Linv1: \"LLL_invariant_mod fs1 mfs1 dmu1 p b first i\"\n (*goal: ‹(⋀fs1. LLL_invariant_mod fs1 mfs1 dmu1 p b first i ⟹ thesis) ⟹ thesis›*)\n by auto\n show \"?case\"\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n using IH[OF impl1 Linv1 res res' ji] (*‹state_impl_inv p mfs' dmu' state'›*) .\n qed (auto)\n (*solved the remaining goal: ‹⋀fs mfs dmu state. ⟦state_impl_inv p mfs dmu state; LLL_invariant_mod fs mfs dmu p b first i; basis_reduction_mod_add_rows_loop p mfs dmu i 0 = (mfs', dmu'); LLL_add_rows_loop p state i 0 = state'; 0 ≤ i⟧ ⟹ state_impl_inv p mfs' dmu' state'›*)\n\nlemma LLL_add_rows_outer_loop: assumes impl: \"state_impl_inv p mfs dmu state\" \n and Linv: \"LLL_invariant_mod fs mfs dmu p first b m\"\n and res: \"basis_reduction_mod_add_rows_outer_loop p mfs dmu i = (mfs', dmu')\" \n and res': \"LLL_add_rows_outer_loop p state i = state'\" \n and i: \"i ≤ m - 1\" \nshows \"state_impl_inv p mfs' dmu' state'\"\n using assms (*‹state_impl_inv p mfs dmu state› ‹LLL_invariant_mod fs mfs dmu p first b m› ‹basis_reduction_mod_add_rows_outer_loop p mfs dmu i = (mfs', dmu')› ‹LLL_add_rows_outer_loop p state i = state'› ‹i ≤ m - 1›*) proof (induct i arbitrary: fs mfs dmu state mfs' dmu' state')\n (*goals:\n 1. ‹⋀fs mfs dmu state mfs' dmu' state'. ⟦state_impl_inv p mfs dmu state; LLL_invariant_mod fs mfs dmu p first b m; basis_reduction_mod_add_rows_outer_loop p mfs dmu 0 = (mfs', dmu'); LLL_add_rows_outer_loop p state 0 = state'; 0 ≤ m - 1⟧ ⟹ state_impl_inv p mfs' dmu' state'›\n 2. ‹⋀i fs mfs dmu state mfs' dmu' state'. ⟦⋀fs mfs dmu state mfs' dmu' state'. ⟦state_impl_inv p mfs dmu state; LLL_invariant_mod fs mfs dmu p first b m; basis_reduction_mod_add_rows_outer_loop p mfs dmu i = (mfs', dmu'); LLL_add_rows_outer_loop p state i = state'; i ≤ m - 1⟧ ⟹ state_impl_inv p mfs' dmu' state'; state_impl_inv p mfs dmu state; LLL_invariant_mod fs mfs dmu p first b m; basis_reduction_mod_add_rows_outer_loop p mfs dmu (Suc i) = (mfs', dmu'); LLL_add_rows_outer_loop p state (Suc i) = state'; Suc i ≤ m - 1⟧ ⟹ state_impl_inv p mfs' dmu' state'›*)\n case (Suc i) (*‹⟦state_impl_inv p ?mfs2 ?dmu2 ?state2; LLL_invariant_mod ?fs2 ?mfs2 ?dmu2 p first b m; basis_reduction_mod_add_rows_outer_loop p ?mfs2 ?dmu2 i = (?mfs'2, ?dmu'2); LLL_add_rows_outer_loop p ?state2 i = ?state'2; i ≤ m - 1⟧ ⟹ state_impl_inv p ?mfs'2 ?dmu'2 ?state'2› ‹state_impl_inv p mfs dmu state› ‹LLL_invariant_mod fs mfs dmu p first b m› ‹basis_reduction_mod_add_rows_outer_loop p mfs dmu (Suc i) = (mfs', dmu')› ‹LLL_add_rows_outer_loop p state (Suc i) = state'› ‹Suc i ≤ m - 1›*)\n note impl = Suc(2) (*‹state_impl_inv p mfs dmu state›*)\n note Linv = Suc(3) (*‹LLL_invariant_mod fs mfs dmu p first b m›*)\n note res = Suc(4) (*‹basis_reduction_mod_add_rows_outer_loop p mfs dmu (Suc i) = (mfs', dmu')›*)\n note res' = Suc(5) (*‹LLL_add_rows_outer_loop p state (Suc i) = state'›*)\n note i = Suc(6) (*‹Suc i ≤ m - 1›*)\n note IH = Suc(1) (*‹⟦state_impl_inv p ?mfs2 ?dmu2 ?state2; LLL_invariant_mod ?fs2 ?mfs2 ?dmu2 p first b m; basis_reduction_mod_add_rows_outer_loop p ?mfs2 ?dmu2 i = (?mfs'2, ?dmu'2); LLL_add_rows_outer_loop p ?state2 i = ?state'2; i ≤ m - 1⟧ ⟹ state_impl_inv p ?mfs'2 ?dmu'2 ?state'2›*)\n from i (*‹Suc i ≤ m - 1›*) have im: \"i < m\" \"i ≤ m - 1\" \"Suc i < m\"\n apply -\n (*goals:\n 1. ‹Suc i ≤ m - 1 ⟹ i < m›\n 2. ‹Suc i ≤ m - 1 ⟹ i ≤ m - 1›\n 3. ‹Suc i ≤ m - 1 ⟹ Suc i < m›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n obtain mfs1 and dmu1 where add: \"basis_reduction_mod_add_rows_outer_loop p mfs dmu i = (mfs1, dmu1)\"\n (*goal: ‹(⋀mfs1 dmu1. basis_reduction_mod_add_rows_outer_loop p mfs dmu i = (mfs1, dmu1) ⟹ thesis) ⟹ thesis›*)\n by force\n note res = res[unfolded basis_reduction_mod_add_rows_outer_loop.simps Let_def add split] (*‹basis_reduction_mod_add_rows_loop p mfs1 dmu1 (Suc i) (Suc i) = (mfs', dmu')›*)\n obtain state1 where add': \"LLL_add_rows_outer_loop p state i = state1\"\n (*goal: ‹(⋀state1. LLL_add_rows_outer_loop p state i = state1 ⟹ thesis) ⟹ thesis›*)\n by auto\n note res' = res'[unfolded LLL_add_rows_outer_loop.simps Let_def add'] (*‹LLL_add_rows_loop p state1 (Suc i) (Suc i) = state'›*)\n from IH[OF impl Linv add add' im ( 2 )] (*‹state_impl_inv p mfs1 dmu1 state1›*) have impl1: \"state_impl_inv p mfs1 dmu1 state1\" .\n from basis_reduction_mod_add_rows_outer_loop_inv[OF Linv add [ symmetric ] im ( 1 )] (*‹∃fs'. LLL_invariant_mod fs' mfs1 dmu1 p first b m ∧ (∀j≤i. μ_small fs' j)›*) obtain fs1 where Linv1: \"LLL_invariant_mod fs1 mfs1 dmu1 p first b m\"\n (*goal: ‹(⋀fs1. LLL_invariant_mod fs1 mfs1 dmu1 p first b m ⟹ thesis) ⟹ thesis›*)\n by auto\n from basis_reduction_mod_add_rows_loop_inv'[OF Linv1 res im ( 3 )] (*‹∃fs'. LLL_invariant_mod fs' mfs' dmu' p first b m ∧ (∀i' j'. i' < Suc i ⟶ j' ≤ i' ⟶ gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs1) i' j' = gram_schmidt_fs.μ n (map of_int_hom.vec_hom fs') i' j') ∧ μ_small fs' (Suc i)›*) obtain fs' where Linv': \"LLL_invariant_mod fs' mfs' dmu' p first b m\"\n (*goal: ‹(⋀fs'::int vec list. LLL_invariant_mod fs' (mfs'::int vec list) (dmu'::int mat) (p::int) (first::bool) (b::rat) (m::nat) ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n from LLL_add_rows_loop[OF impl1 LLL_invariant_mod_to_weak_m_to_i ( 1 ) [ OF Linv1 ] res res' le_refl im ( 3 )] (*‹Suc i ≤ m ⟹ state_impl_inv p mfs' dmu' state'›*) i (*‹Suc i ≤ m - 1›*) show \"?case\"\n (*goal: ‹state_impl_inv p mfs' dmu' state'›*)\n by auto\n qed (auto)\n (*solved the remaining goal: ‹⋀fs mfs dmu state mfs' dmu' state'. ⟦state_impl_inv p mfs dmu state; LLL_invariant_mod fs mfs dmu p first b m; basis_reduction_mod_add_rows_outer_loop p mfs dmu 0 = (mfs', dmu'); LLL_add_rows_outer_loop p state 0 = state'; 0 ≤ m - 1⟧ ⟹ state_impl_inv p mfs' dmu' state'›*)\n\nsubsection ‹Soundness of implementation›\n\ntext ‹We just prove that the concrete implementations have the same input-output-behaviour as\n the abstract versions of Storjohann's algorithms.›\n\nlemma LLL_reduce_basis: \"LLL_reduce_basis = reduce_basis_mod\" \nproof (cases \"m = 0\")\n (*goals:\n 1. ‹m = 0 ⟹ LLL_reduce_basis = reduce_basis_mod›\n 2. ‹m ≠ 0 ⟹ LLL_reduce_basis = reduce_basis_mod›*)\n case True (*‹m = 0›*)\n from LLL_invD[OF reduce_basis_mod_inv [ OF refl ]] (*‹gs.lin_indpt_list (map of_int_hom.vec_hom reduce_basis_mod)› ‹length (map of_int_hom.vec_hom reduce_basis_mod) = m› ‹set reduce_basis_mod ⊆ carrier_vec n› ‹?i < m ⟹ reduce_basis_mod ! ?i ∈ carrier_vec n› ‹?i < m ⟹ gram_schmidt_fs.gso n (map of_int_hom.vec_hom reduce_basis_mod) ?i ∈ Rn› ‹length reduce_basis_mod = m› ‹lattice_of reduce_basis_mod = L› ‹gram_schmidt_fs.weakly_reduced (n::nat) (map of_int_hom.vec_hom reduce_basis_mod) (α::rat) (m::nat)› ‹m ≤ m› ‹gram_schmidt_fs.reduced n (map of_int_hom.vec_hom reduce_basis_mod) α m› ‹True ∨ μ_small reduce_basis_mod m›*) True (*‹m = 0›*) have \"reduce_basis_mod = []\"\n by auto\n thus \"?thesis\"\n (*goal: ‹LLL_reduce_basis = reduce_basis_mod›*)\n using True (*‹m = 0›*) unfolding LLL_reduce_basis_def\n (*goal: ‹(if m = 0 then [] else let first = False; (p0, state0, g_idx0) = LLL_initial first; (p, state) = LLL_main p0 first state0 g_idx0 0 0; (mfs, uu_, uu_, uu_) = LLL_add_rows_outer_loop p state (m - 1) in mfs) = reduce_basis_mod›*)\n by auto\nnext\n (*goal: ‹(m::nat) ≠ (0::nat) ⟹ LLL_reduce_basis = reduce_basis_mod›*)\n case False (*‹m ≠ 0›*)\n hence idm: \"(m = 0) = False\"\n by auto\n let ?first = False\n obtain p1 and mfs1 and dmu1 and g_idx1 where init: \"compute_initial_state ?first = (p1, mfs1, dmu1,g_idx1)\"\n (*goal: ‹(⋀p1 mfs1 dmu1 g_idx1. compute_initial_state False = (p1, mfs1, dmu1, g_idx1) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n obtain p1' and state1 and g_idx1' where init': \"LLL_initial ?first = (p1', state1, g_idx1')\"\n (*goal: ‹(⋀p1' state1 g_idx1'. LLL_initial False = (p1', state1, g_idx1') ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n from LLL_initial[OF init init' False] (*‹state_impl_inv (p1::int) (mfs1::int vec list) (dmu1::int mat) (state1::int vec list × int iarray iarray × int iarray × int iarray) ∧ (p1'::int) = p1 ∧ (g_idx1'::nat) = (g_idx1::nat)›*) have impl1: \"state_impl_inv p1 mfs1 dmu1 state1\" and id: \"p1' = p1\" \"g_idx1' = g_idx1\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ state_impl_inv p1 mfs1 dmu1 state1›\n 2. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ p1' = p1›\n 3. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ g_idx1' = g_idx1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n from LLL_initial_invariant_mod[OF init] (*‹∃(fs::int vec list) b::rat. LLL_invariant_mod fs (mfs1::int vec list) (dmu1::int mat) (p1::int) False b (0::nat)›*) obtain fs1 and b1 where inv1: \"LLL_invariant_mod fs1 mfs1 dmu1 p1 ?first b1 0\"\n (*goal: ‹(⋀fs1 b1. LLL_invariant_mod fs1 mfs1 dmu1 p1 False b1 0 ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain p2 and mfs2 and dmu2 where main: \"basis_reduction_mod_main p1 ?first mfs1 dmu1 g_idx1 0 0 = (p2, mfs2, dmu2)\"\n (*goal: ‹(⋀p2 mfs2 dmu2. basis_reduction_mod_main p1 False mfs1 dmu1 g_idx1 0 0 = (p2, mfs2, dmu2) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n from basis_reduction_mod_main[OF inv1 main] (*‹∃fs' b'. LLL_invariant_mod fs' mfs2 dmu2 p2 False b' m›*) obtain fs2 and b2 where inv2: \" LLL_invariant_mod fs2 mfs2 dmu2 p2 ?first b2 m\"\n (*goal: ‹(⋀fs2 b2. LLL_invariant_mod fs2 mfs2 dmu2 p2 False b2 m ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain p2' and state2 where main': \"LLL_main p1 ?first state1 g_idx1 0 0 = (p2', state2)\"\n (*goal: ‹(⋀(p2'::int) state2::int vec list × int iarray iarray × int iarray × int iarray. LLL_main (p1::int) False (state1::int vec list × int iarray iarray × int iarray × int iarray) (g_idx1::nat) (0::nat) (0::int) = (p2', state2) ⟹ thesis::bool) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀(x1::?'a) x2::?'b. (?y::?'a × ?'b) = (x1, x2) ⟹ ?P::bool) ⟹ ?P›*))\n from LLL_main[OF impl1 inv1 main, unfolded id, OF main'] (*‹state_impl_inv p2 mfs2 dmu2 state2 ∧ p2' = p2›*) have impl2: \"state_impl_inv p2 mfs2 dmu2 state2\" and p2: \"p2' = p2\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p2 mfs2 dmu2 state2 ∧ p2' = p2 ⟹ state_impl_inv p2 mfs2 dmu2 state2›\n 2. ‹state_impl_inv p2 mfs2 dmu2 state2 ∧ p2' = p2 ⟹ p2' = p2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n obtain mfs3 and dmu3 where outer: \"basis_reduction_mod_add_rows_outer_loop p2 mfs2 dmu2 (m - 1) = (mfs3, dmu3)\"\n (*goal: ‹(⋀(mfs3::int vec list) dmu3::int mat. basis_reduction_mod_add_rows_outer_loop (p2::int) (mfs2::int vec list) (dmu2::int mat) ((m::nat) - (1::nat)) = (mfs3, dmu3) ⟹ thesis::bool) ⟹ thesis›*)\n by force\n obtain mfsi3 and dmui3 and di3 and mods3 where outer': \"LLL_add_rows_outer_loop p2 state2 (m - 1) = (mfsi3, dmui3, di3, mods3)\"\n (*goal: ‹(⋀(mfsi3::int vec list) (dmui3::int iarray iarray) (di3::int iarray) mods3::int iarray. LLL_add_rows_outer_loop (p2::int) (state2::int vec list × int iarray iarray × int iarray × int iarray) ((m::nat) - (1::nat)) = (mfsi3, dmui3, di3, mods3) ⟹ thesis::bool) ⟹ thesis›*)\n by (metis prod_cases4 (*‹(⋀(a::?'a) (b::?'b) (c::?'c) d::?'d. (?y::?'a × ?'b × ?'c × ?'d) = (a, b, c, d) ⟹ ?thesis::bool) ⟹ ?thesis›*))\n from LLL_add_rows_outer_loop[OF impl2 inv2 outer outer' le_refl] (*‹state_impl_inv p2 mfs3 dmu3 (mfsi3, dmui3, di3, mods3)›*) have \"state_impl_inv p2 mfs3 dmu3 (mfsi3, dmui3, di3, mods3)\" .\n hence identity: \"mfs3 = mfsi3\"\n unfolding state_impl_inv.simps\n (*goal: ‹mfs3 = mfsi3›*)\n by auto\n note res = reduce_basis_mod_def[unfolded init main split Let_def outer] (*‹reduce_basis_mod = mfs3›*)\n note res' = LLL_reduce_basis_def[unfolded init' Let_def main' id split p2 outer' idm if_False] (*‹LLL_reduce_basis = mfsi3›*)\n show \"?thesis\"\n (*goal: ‹LLL_reduce_basis = reduce_basis_mod›*)\n unfolding res res' identity\n (*goal: ‹mfsi3 = mfsi3›*)\n by standard\nqed\n\nlemma LLL_reduce_basis_iso: \"LLL_reduce_basis_iso = reduce_basis_iso\" \nproof (cases \"m = 0\")\n (*goals:\n 1. ‹m = 0 ⟹ LLL_reduce_basis_iso = reduce_basis_iso›\n 2. ‹m ≠ 0 ⟹ LLL_reduce_basis_iso = reduce_basis_iso›*)\n case True (*‹m = 0›*)\n from LLL_invD[OF reduce_basis_iso_inv [ OF refl ]] (*‹gs.lin_indpt_list (map of_int_hom.vec_hom reduce_basis_iso)› ‹length (map of_int_hom.vec_hom reduce_basis_iso) = m› ‹set reduce_basis_iso ⊆ carrier_vec n› ‹?i < m ⟹ reduce_basis_iso ! ?i ∈ carrier_vec n› ‹?i < m ⟹ gram_schmidt_fs.gso n (map of_int_hom.vec_hom reduce_basis_iso) ?i ∈ Rn› ‹length reduce_basis_iso = m› ‹lattice_of reduce_basis_iso = L› ‹gram_schmidt_fs.weakly_reduced n (map of_int_hom.vec_hom reduce_basis_iso) α m› ‹m ≤ m› ‹gram_schmidt_fs.reduced n (map of_int_hom.vec_hom reduce_basis_iso) α m› ‹True ∨ μ_small reduce_basis_iso m›*) True (*‹m = 0›*) have \"reduce_basis_iso = []\"\n by auto\n thus \"?thesis\"\n (*goal: ‹LLL_reduce_basis_iso = reduce_basis_iso›*)\n using True (*‹m = 0›*) unfolding LLL_reduce_basis_iso_def\n (*goal: ‹(if m = 0 then [] else let first = False; (p0, state0, g_idx0) = LLL_initial first; (p, state) = LLL_iso_main p0 first state0 g_idx0 0; (mfs, uu_, uu_, uu_) = LLL_add_rows_outer_loop p state (m - 1) in mfs) = reduce_basis_iso›*)\n by auto\nnext\n (*goal: ‹m ≠ 0 ⟹ LLL_reduce_basis_iso = reduce_basis_iso›*)\n case False (*‹(m::nat) ≠ (0::nat)›*)\n hence idm: \"(m = 0) = False\"\n by auto\n let ?first = False\n obtain p1 and mfs1 and dmu1 and g_idx1 where init: \"compute_initial_state ?first = (p1, mfs1, dmu1, g_idx1)\"\n (*goal: ‹(⋀p1 mfs1 dmu1 g_idx1. compute_initial_state False = (p1, mfs1, dmu1, g_idx1) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n obtain p1' and state1 and g_idx1' where init': \"LLL_initial ?first = (p1', state1, g_idx1')\"\n (*goal: ‹(⋀p1' state1 g_idx1'. LLL_initial False = (p1', state1, g_idx1') ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n from LLL_initial[OF init init' False] (*‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1›*) have impl1: \"state_impl_inv p1 mfs1 dmu1 state1\" and id: \"p1' = p1\" \"g_idx1' = g_idx1\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ state_impl_inv p1 mfs1 dmu1 state1›\n 2. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ p1' = p1›\n 3. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ g_idx1' = g_idx1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n from LLL_initial_invariant_mod[OF init] (*‹∃(fs::int vec list) b::rat. LLL_invariant_mod fs (mfs1::int vec list) (dmu1::int mat) (p1::int) False b (0::nat)›*) obtain fs1 and b1 where inv1: \"LLL_invariant_mod_weak fs1 mfs1 dmu1 p1 ?first b1\"\n (*goal: ‹(⋀fs1 b1. LLL_invariant_mod_weak fs1 mfs1 dmu1 p1 False b1 ⟹ thesis) ⟹ thesis›*)\n by (auto simp: LLL_invariant_mod_weak_def (*‹LLL_invariant_mod_weak ?fs ?mfs ?dmu ?p ?first ?b = (length ?fs = m ∧ length ?mfs = m ∧ lattice_of ?fs = L ∧ gs.lin_indpt_list (map of_int_hom.vec_hom ?fs) ∧ map (map_vec (λx. x symmod ?p)) ?fs = ?mfs ∧ (∀i'<m. ∀j'<i'. ¦dμ ?fs i' j'¦ < ?p * d ?fs j' * d ?fs (Suc j')) ∧ (∀i'<m. ∀j'<m. dμ ?fs i' j' = ?dmu $$ (i', j')) ∧ 1 < ?p ∧ g_bnd_mode ?first ?b ?fs ∧ mod_invariant ?b ?p ?first)›*) LLL_invariant_mod_def (*‹LLL_invariant_mod ?fs ?mfs ?dmu ?p ?first ?b ?i = (length ?fs = m ∧ length ?mfs = m ∧ ?i ≤ m ∧ lattice_of ?fs = L ∧ gs.lin_indpt_list (map of_int_hom.vec_hom ?fs) ∧ gram_schmidt_fs.weakly_reduced n (map of_int_hom.vec_hom ?fs) α ?i ∧ map (map_vec (λx. x symmod ?p)) ?fs = ?mfs ∧ (∀i'<m. ∀j'<i'. ¦dμ ?fs i' j'¦ < ?p * d ?fs j' * d ?fs (Suc j')) ∧ (∀i'<m. ∀j'<m. dμ ?fs i' j' = ?dmu $$ (i', j')) ∧ 1 < ?p ∧ g_bnd_mode ?first ?b ?fs ∧ mod_invariant ?b ?p ?first)›*))\n obtain p2 and mfs2 and dmu2 where main: \"basis_reduction_iso_main p1 ?first mfs1 dmu1 g_idx1 0 = (p2, mfs2, dmu2)\"\n (*goal: ‹(⋀(p2::int) (mfs2::int vec list) dmu2::int mat. basis_reduction_iso_main (p1::int) False (mfs1::int vec list) (dmu1::int mat) (g_idx1::nat) (0::int) = (p2, mfs2, dmu2) ⟹ thesis::bool) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n from basis_reduction_iso_main[OF inv1 main] (*‹∃(fs'::int vec list) b'::rat. LLL_invariant_mod fs' (mfs2::int vec list) (dmu2::int mat) (p2::int) False b' (m::nat)›*) obtain fs2 and b2 where inv2: \" LLL_invariant_mod fs2 mfs2 dmu2 p2 ?first b2 m\"\n (*goal: ‹(⋀fs2 b2. LLL_invariant_mod fs2 mfs2 dmu2 p2 False b2 m ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain p2' and state2 where main': \"LLL_iso_main p1 ?first state1 g_idx1 0 = (p2', state2)\"\n (*goal: ‹(⋀p2' state2. LLL_iso_main p1 False state1 g_idx1 0 = (p2', state2) ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n from LLL_iso_main[OF impl1 inv1 main, unfolded id, OF main'] (*‹state_impl_inv p2 mfs2 dmu2 state2 ∧ p2' = p2›*) have impl2: \"state_impl_inv p2 mfs2 dmu2 state2\" and p2: \"p2' = p2\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p2 mfs2 dmu2 state2 ∧ p2' = p2 ⟹ state_impl_inv p2 mfs2 dmu2 state2›\n 2. ‹state_impl_inv p2 mfs2 dmu2 state2 ∧ p2' = p2 ⟹ p2' = p2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n obtain mfs3 and dmu3 where outer: \"basis_reduction_mod_add_rows_outer_loop p2 mfs2 dmu2 (m - 1) = (mfs3, dmu3)\"\n (*goal: ‹(⋀mfs3 dmu3. basis_reduction_mod_add_rows_outer_loop p2 mfs2 dmu2 (m - 1) = (mfs3, dmu3) ⟹ thesis) ⟹ thesis›*)\n by force\n obtain mfsi3 and dmui3 and di3 and mods3 where outer': \"LLL_add_rows_outer_loop p2 state2 (m - 1) = (mfsi3, dmui3, di3, mods3)\"\n (*goal: ‹(⋀mfsi3 dmui3 di3 mods3. LLL_add_rows_outer_loop p2 state2 (m - 1) = (mfsi3, dmui3, di3, mods3) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases4 (*‹(⋀a b c d. ?y = (a, b, c, d) ⟹ ?thesis) ⟹ ?thesis›*))\n from LLL_add_rows_outer_loop[OF impl2 inv2 outer outer' le_refl] (*‹state_impl_inv (p2::int) (mfs3::int vec list) (dmu3::int mat) (mfsi3::int vec list, dmui3::int iarray iarray, di3::int iarray, mods3::int iarray)›*) have \"state_impl_inv p2 mfs3 dmu3 (mfsi3, dmui3, di3, mods3)\" .\n hence identity: \"mfs3 = mfsi3\"\n unfolding state_impl_inv.simps\n (*goal: ‹(mfs3::int vec list) = (mfsi3::int vec list)›*)\n by auto\n note res = reduce_basis_iso_def[unfolded init main split Let_def outer] (*‹reduce_basis_iso = mfs3›*)\n note res' = LLL_reduce_basis_iso_def[unfolded init' Let_def main' id split p2 outer' idm if_False] (*‹LLL_reduce_basis_iso = mfsi3›*)\n show \"?thesis\"\n (*goal: ‹LLL_reduce_basis_iso = reduce_basis_iso›*)\n unfolding res res' identity\n (*goal: ‹mfsi3 = mfsi3›*)\n by standard\nqed\n\nlemma LLL_short_vector: assumes m: \"m ≠ 0\" \n shows \"LLL_short_vector = short_vector_mod\" \nproof (-)\n (*goal: ‹LLL_short_vector = short_vector_mod›*)\n let ?first = True\n obtain p1 and mfs1 and dmu1 and g_idx1 where init: \"compute_initial_state ?first = (p1, mfs1, dmu1,g_idx1)\"\n (*goal: ‹(⋀p1 mfs1 dmu1 g_idx1. compute_initial_state True = (p1, mfs1, dmu1, g_idx1) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n obtain p1' and state1 and g_idx1' where init': \"LLL_initial ?first = (p1', state1, g_idx1')\"\n (*goal: ‹(⋀p1' state1 g_idx1'. LLL_initial True = (p1', state1, g_idx1') ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n from LLL_initial[OF init init' m] (*‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1›*) have impl1: \"state_impl_inv p1 mfs1 dmu1 state1\" and id: \"p1' = p1\" \"g_idx1' = g_idx1\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ state_impl_inv p1 mfs1 dmu1 state1›\n 2. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ p1' = p1›\n 3. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ g_idx1' = g_idx1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n from LLL_initial_invariant_mod[OF init] (*‹∃(fs::int vec list) b::rat. LLL_invariant_mod fs (mfs1::int vec list) (dmu1::int mat) (p1::int) True b (0::nat)›*) obtain fs1 and b1 where inv1: \"LLL_invariant_mod fs1 mfs1 dmu1 p1 ?first b1 0\"\n (*goal: ‹(⋀fs1 b1. LLL_invariant_mod fs1 mfs1 dmu1 p1 True b1 0 ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain p2 and mfs2 and dmu2 where main: \"basis_reduction_mod_main p1 ?first mfs1 dmu1 g_idx1 0 0 = (p2, mfs2, dmu2)\"\n (*goal: ‹(⋀p2 mfs2 dmu2. basis_reduction_mod_main p1 True mfs1 dmu1 g_idx1 0 0 = (p2, mfs2, dmu2) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n from basis_reduction_mod_main[OF inv1 main] (*‹∃fs' b'. LLL_invariant_mod fs' mfs2 dmu2 p2 True b' m›*) obtain fs2 and b2 where inv2: \" LLL_invariant_mod fs2 mfs2 dmu2 p2 ?first b2 m\"\n (*goal: ‹(⋀fs2 b2. LLL_invariant_mod fs2 mfs2 dmu2 p2 True b2 m ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain p2' and mfsi2 and dmui2 and di2 and mods2 where main': \"LLL_main p1 ?first state1 g_idx1 0 0 = (p2', (mfsi2, dmui2, di2, mods2))\"\n (*goal: ‹(⋀p2' mfsi2 dmui2 di2 mods2. LLL_main p1 True state1 g_idx1 0 0 = (p2', mfsi2, dmui2, di2, mods2) ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n from LLL_main[OF impl1 inv1 main, unfolded id, OF main'] (*‹state_impl_inv (p2::int) (mfs2::int vec list) (dmu2::int mat) (mfsi2::int vec list, dmui2::int iarray iarray, di2::int iarray, mods2::int iarray) ∧ (p2'::int) = p2›*) have impl2: \"state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2)\" and p2: \"p2' = p2\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2) ∧ p2' = p2 ⟹ state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2)›\n 2. ‹state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2) ∧ p2' = p2 ⟹ p2' = p2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence identity: \"mfs2 = mfsi2\"\n unfolding state_impl_inv.simps\n (*goal: ‹mfs2 = mfsi2›*)\n by auto\n note res = short_vector_mod_def[unfolded init main split Let_def] (*‹short_vector_mod = hd (mfs2::int vec list)›*)\n note res' = LLL_short_vector_def[unfolded init' Let_def main' id split p2] (*‹LLL_short_vector = hd mfsi2›*)\n show \"?thesis\"\n (*goal: ‹LLL_short_vector = short_vector_mod›*)\n unfolding res res' identity\n (*goal: ‹hd mfsi2 = hd mfsi2›*)\n by standard\nqed\n\nlemma LLL_short_vector_iso: assumes m: \"m ≠ 0\" \n shows \"LLL_short_vector_iso = short_vector_iso\" \nproof (-)\n (*goal: ‹LLL_short_vector_iso = short_vector_iso›*)\n let ?first = True\n obtain p1 and mfs1 and dmu1 and g_idx1 where init: \"compute_initial_state ?first = (p1, mfs1, dmu1,g_idx1)\"\n (*goal: ‹(⋀p1 mfs1 dmu1 g_idx1. compute_initial_state True = (p1, mfs1, dmu1, g_idx1) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n obtain p1' and state1 and g_idx1' where init': \"LLL_initial ?first = (p1', state1, g_idx1')\"\n (*goal: ‹(⋀p1' state1 g_idx1'. LLL_initial True = (p1', state1, g_idx1') ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n from LLL_initial[OF init init' m] (*‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1›*) have impl1: \"state_impl_inv p1 mfs1 dmu1 state1\" and id: \"p1' = p1\" \"g_idx1' = g_idx1\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ state_impl_inv p1 mfs1 dmu1 state1›\n 2. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ p1' = p1›\n 3. ‹state_impl_inv p1 mfs1 dmu1 state1 ∧ p1' = p1 ∧ g_idx1' = g_idx1 ⟹ g_idx1' = g_idx1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n from LLL_initial_invariant_mod[OF init] (*‹∃fs b. LLL_invariant_mod fs mfs1 dmu1 p1 True b 0›*) obtain fs1 and b1 where inv1: \"LLL_invariant_mod_weak fs1 mfs1 dmu1 p1 ?first b1\"\n (*goal: ‹(⋀fs1 b1. LLL_invariant_mod_weak fs1 mfs1 dmu1 p1 True b1 ⟹ thesis) ⟹ thesis›*)\n by (auto simp: LLL_invariant_mod_weak_def (*‹LLL_invariant_mod_weak ?fs ?mfs ?dmu ?p ?first ?b = (length ?fs = m ∧ length ?mfs = m ∧ lattice_of ?fs = L ∧ gs.lin_indpt_list (map of_int_hom.vec_hom ?fs) ∧ map (map_vec (λx. x symmod ?p)) ?fs = ?mfs ∧ (∀i'<m. ∀j'<i'. ¦dμ ?fs i' j'¦ < ?p * d ?fs j' * d ?fs (Suc j')) ∧ (∀i'<m. ∀j'<m. dμ ?fs i' j' = ?dmu $$ (i', j')) ∧ 1 < ?p ∧ g_bnd_mode ?first ?b ?fs ∧ mod_invariant ?b ?p ?first)›*) LLL_invariant_mod_def (*‹LLL_invariant_mod ?fs ?mfs ?dmu ?p ?first ?b ?i = (length ?fs = m ∧ length ?mfs = m ∧ ?i ≤ m ∧ lattice_of ?fs = L ∧ gs.lin_indpt_list (map of_int_hom.vec_hom ?fs) ∧ gram_schmidt_fs.weakly_reduced n (map of_int_hom.vec_hom ?fs) α ?i ∧ map (map_vec (λx. x symmod ?p)) ?fs = ?mfs ∧ (∀i'<m. ∀j'<i'. ¦dμ ?fs i' j'¦ < ?p * d ?fs j' * d ?fs (Suc j')) ∧ (∀i'<m. ∀j'<m. dμ ?fs i' j' = ?dmu $$ (i', j')) ∧ 1 < ?p ∧ g_bnd_mode ?first ?b ?fs ∧ mod_invariant ?b ?p ?first)›*))\n obtain p2 and mfs2 and dmu2 where main: \"basis_reduction_iso_main p1 ?first mfs1 dmu1 g_idx1 0 = (p2, mfs2, dmu2)\"\n (*goal: ‹(⋀p2 mfs2 dmu2. basis_reduction_iso_main p1 True mfs1 dmu1 g_idx1 0 = (p2, mfs2, dmu2) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases3 (*‹(⋀a b c. ?y = (a, b, c) ⟹ ?thesis) ⟹ ?thesis›*))\n from basis_reduction_iso_main[OF inv1 main] (*‹∃fs' b'. LLL_invariant_mod fs' mfs2 dmu2 p2 True b' m›*) obtain fs2 and b2 where inv2: \" LLL_invariant_mod fs2 mfs2 dmu2 p2 ?first b2 m\"\n (*goal: ‹(⋀fs2 b2. LLL_invariant_mod fs2 mfs2 dmu2 p2 True b2 m ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain p2' and mfsi2 and dmui2 and di2 and mods2 where main': \"LLL_iso_main p1 ?first state1 g_idx1 0 = (p2', (mfsi2, dmui2, di2, mods2))\"\n (*goal: ‹(⋀p2' mfsi2 dmui2 di2 mods2. LLL_iso_main p1 True state1 g_idx1 0 = (p2', mfsi2, dmui2, di2, mods2) ⟹ thesis) ⟹ thesis›*)\n by (metis prod.exhaust (*‹(⋀x1 x2. ?y = (x1, x2) ⟹ ?P) ⟹ ?P›*))\n from LLL_iso_main[OF impl1 inv1 main, unfolded id, OF main'] (*‹state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2) ∧ p2' = p2›*) have impl2: \"state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2)\" and p2: \"p2' = p2\"\n apply -\n (*goals:\n 1. ‹state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2) ∧ p2' = p2 ⟹ state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2)›\n 2. ‹state_impl_inv p2 mfs2 dmu2 (mfsi2, dmui2, di2, mods2) ∧ p2' = p2 ⟹ p2' = p2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence identity: \"mfs2 = mfsi2\"\n unfolding state_impl_inv.simps\n (*goal: ‹mfs2 = mfsi2›*)\n by auto\n note res = short_vector_iso_def[unfolded init main split Let_def] (*‹short_vector_iso = hd (mfs2::int vec list)›*)\n note res' = LLL_short_vector_iso_def[unfolded init' Let_def main' id split p2] (*‹LLL_short_vector_iso = hd (mfsi2::int vec list)›*)\n show \"?thesis\"\n (*goal: ‹LLL_short_vector_iso = short_vector_iso›*)\n unfolding res res' identity\n (*goal: ‹hd mfsi2 = hd mfsi2›*)\n by standard\nqed\n\nend\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Modular_arithmetic_LLL_and_HNF_algorithms/Storjohann_Impl.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"be0a330414c18019c786e8357934f847f3e0cd86cdd6e377c8105ce523c0b771\"\n}"}}},{"rowIdx":161,"cells":{"text":{"kind":"string","value":"(* Author: Tobias Nipkow *)\n\nsubsection \"Verification Conditions for Total Correctness\"\n\nsubsubsection \"The Standard Approach\"\n\ntheory VCG_Total_EX\nimports Hoare_Total_EX\nbegin\n\ntext‹Annotated commands: commands where loops are annotated with\ninvariants.›\n\ndatatype acom =\n Askip (\"SKIP\") |\n Aassign vname aexp (\"(_ ::= _)\" [1000, 61] 61) |\n Aseq acom acom (\"_;;/ _\" [60, 61] 60) |\n Aif bexp acom acom (\"(IF _/ THEN _/ ELSE _)\" [0, 0, 61] 61) |\n Awhile \"nat ⇒ assn\" bexp acom\n (\"({_}/ WHILE _/ DO _)\" [0, 0, 61] 61)\n\nnotation com.SKIP (\"SKIP\")\n\ntext‹Strip annotations:›\n\nfun strip :: \"acom ⇒ com\" where\n\"strip SKIP = SKIP\" |\n\"strip (x ::= a) = (x ::= a)\" |\n\"strip (C₁;; C₂) = (strip C₁;; strip C₂)\" |\n\"strip (IF b THEN C₁ ELSE C₂) = (IF b THEN strip C₁ ELSE strip C₂)\" |\n\"strip ({_} WHILE b DO C) = (WHILE b DO strip C)\"\n\ntext‹Weakest precondition from annotated commands:›\n\nfun pre :: \"acom ⇒ assn ⇒ assn\" where\n\"pre SKIP Q = Q\" |\n\"pre (x ::= a) Q = (λs. Q(s(x := aval a s)))\" |\n\"pre (C₁;; C₂) Q = pre C₁ (pre C₂ Q)\" |\n\"pre (IF b THEN C₁ ELSE C₂) Q =\n (λs. if bval b s then pre C₁ Q s else pre C₂ Q s)\" |\n\"pre ({I} WHILE b DO C) Q = (λs. ∃n. I n s)\"\n\ntext‹Verification condition:›\n\nfun vc :: \"acom ⇒ assn ⇒ bool\" where\n\"vc SKIP Q = True\" |\n\"vc (x ::= a) Q = True\" |\n\"vc (C₁;; C₂) Q = (vc C₁ (pre C₂ Q) ∧ vc C₂ Q)\" |\n\"vc (IF b THEN C₁ ELSE C₂) Q = (vc C₁ Q ∧ vc C₂ Q)\" |\n\"vc ({I} WHILE b DO C) Q =\n (∀s n. (I (Suc n) s ⟶ pre C (I n) s) ∧\n (I (Suc n) s ⟶ bval b s) ∧\n (I 0 s ⟶ ¬ bval b s ∧ Q s) ∧\n vc C (I n))\"\n\nlemma vc_sound: \"vc C Q ⟹ ⊢⇩t {pre C Q} strip C {Q}\"\nproof (induction C arbitrary: Q)\n (*goals:\n 1. ‹⋀Q::(char list ⇒ int) ⇒ bool. vc SKIP Q ⟹ ⊢⇩t {pre SKIP Q} strip SKIP {Q}›\n 2. ‹⋀(x1::char list) (x2::aexp) Q::(char list ⇒ int) ⇒ bool. vc (x1 ::= x2) Q ⟹ ⊢⇩t {pre (x1 ::= x2) Q} strip (x1 ::= x2) {Q}›\n 3. ‹⋀(C1::acom) (C2::acom) Q::(char list ⇒ int) ⇒ bool. ⟦⋀Q::(char list ⇒ int) ⇒ bool. vc C1 Q ⟹ ⊢⇩t {pre C1 Q} strip C1 {Q}; ⋀Q::(char list ⇒ int) ⇒ bool. vc C2 Q ⟹ ⊢⇩t {pre C2 Q} strip C2 {Q}; vc (C1;; C2) Q⟧ ⟹ ⊢⇩t {pre (C1;; C2) Q} strip (C1;; C2) {Q}›\n 4. ‹⋀(x1::bexp) (C1::acom) (C2::acom) Q::(char list ⇒ int) ⇒ bool. ⟦⋀Q::(char list ⇒ int) ⇒ bool. vc C1 Q ⟹ ⊢⇩t {pre C1 Q} strip C1 {Q}; ⋀Q::(char list ⇒ int) ⇒ bool. vc C2 Q ⟹ ⊢⇩t {pre C2 Q} strip C2 {Q}; vc (IF x1 THEN C1 ELSE C2) Q⟧ ⟹ ⊢⇩t {pre (IF x1 THEN C1 ELSE C2) Q} strip (IF x1 THEN C1 ELSE C2) {Q}›\n 5. ‹⋀(x1::nat ⇒ (char list ⇒ int) ⇒ bool) (x2::bexp) (C::acom) Q::(char list ⇒ int) ⇒ bool. ⟦⋀Q::(char list ⇒ int) ⇒ bool. vc C Q ⟹ ⊢⇩t {pre C Q} strip C {Q}; vc ({x1} WHILE x2 DO C) Q⟧ ⟹ ⊢⇩t {pre ({x1} WHILE x2 DO C) Q} strip ({x1} WHILE x2 DO C) {Q}›*)\n case (Awhile I b C) (*‹vc C ?Q ⟹ ⊢⇩t {pre C ?Q} strip C {?Q}› ‹vc ({I} WHILE b DO C) Q›*)\n show \"?case\"\n (*goal: ‹⊢⇩t {pre ({I} WHILE b DO C) Q} strip ({I} WHILE b DO C) {Q}›*)\n proof (simp, rule conseq[OF _ While[of I]] (*‹⟦∀s. ?P' s ⟶ (∃n. I n s); ⋀n. ⊢⇩t {I (Suc n)} ?c1 {I n}; ∀n s. I (Suc n) s ⟶ bval ?b1 s; ∀s. I 0 s ⟶ ¬ bval ?b1 s; ∀s. I 0 s ⟶ ?Q' s⟧ ⟹ ⊢⇩t {?P'} WHILE ?b1 DO ?c1 {?Q'}›*), goal_cases)\n (*goals:\n 1. ‹∀s. (∃n. I n s) ⟶ (∃n. I n s)›\n 2. ‹⋀n. ⊢⇩t {I (Suc n)} strip C {I n}›\n 3. ‹∀n s. I (Suc n) s ⟶ bval b s›\n 4. ‹∀s. I 0 s ⟶ ¬ bval b s›\n 5. ‹∀s. I 0 s ⟶ Q s›*)\n case (2 n) (*no hyothesis introduced yet*)\n show \"?case\"\n (*goal: ‹⊢⇩t {I (Suc n)} strip C {I n}›*)\n using Awhile.IH[of \"I n\"] (*‹vc (C::acom) ((I::nat ⇒ (char list ⇒ int) ⇒ bool) (n::nat)) ⟹ ⊢⇩t {pre C (I n)} strip C {I n}›*) Awhile.prems (*‹vc ({I} WHILE b DO C) Q›*) by (auto intro: strengthen_pre (*‹⟦∀s::char list ⇒ int. (?P'::(char list ⇒ int) ⇒ bool) s ⟶ (?P::(char list ⇒ int) ⇒ bool) s; ⊢⇩t {?P} ?c::com {?Q::(char list ⇒ int) ⇒ bool}⟧ ⟹ ⊢⇩t {?P'} ?c {?Q}›*))\n qed (insert Awhile.prems (*‹vc ({I} WHILE b DO C) Q›*), auto)\n (*solves the remaining goals:\n 1. ‹∀s. (∃n. I n s) ⟶ (∃n. I n s)›\n 2. ‹∀n s. I (Suc n) s ⟶ bval b s›\n 3. ‹∀s. I 0 s ⟶ ¬ bval b s›\n 4. ‹∀s. I 0 s ⟶ Q s›*)\nqed (auto intro: conseq (*‹⟦∀s::char list ⇒ int. (?P'::(char list ⇒ int) ⇒ bool) s ⟶ (?P::(char list ⇒ int) ⇒ bool) s; ⊢⇩t {?P} ?c::com {?Q::(char list ⇒ int) ⇒ bool}; ∀s::char list ⇒ int. ?Q s ⟶ (?Q'::(char list ⇒ int) ⇒ bool) s⟧ ⟹ ⊢⇩t {?P'} ?c {?Q'}›*) Seq (*‹⟦⊢⇩t {?P₁::(char list ⇒ int) ⇒ bool} ?c₁::com {?P₂::(char list ⇒ int) ⇒ bool}; ⊢⇩t {?P₂} ?c₂::com {?P₃::(char list ⇒ int) ⇒ bool}⟧ ⟹ ⊢⇩t {?P₁} ?c₁;; ?c₂ {?P₃}›*) If (*‹⟦⊢⇩t {λs::char list ⇒ int. (?P::(char list ⇒ int) ⇒ bool) s ∧ bval (?b::bexp) s} ?c₁::com {?Q::(char list ⇒ int) ⇒ bool}; ⊢⇩t {λs::char list ⇒ int. ?P s ∧ ¬ bval ?b s} ?c₂::com {?Q}⟧ ⟹ ⊢⇩t {?P} IF ?b THEN ?c₁ ELSE ?c₂ {?Q}›*) simp: Skip (*‹⊢⇩t {?P::(char list ⇒ int) ⇒ bool} SKIP {?P}›*) Assign (*‹⊢⇩t {λs::char list ⇒ int. (?P::(char list ⇒ int) ⇒ bool) (s[?a::aexp/?x::char list])} ?x ::= ?a {?P}›*))\n(*solves the remaining goals:\n 1. ‹⋀Q. vc SKIP Q ⟹ ⊢⇩t {pre SKIP Q} strip SKIP {Q}›\n 2. ‹⋀x1 x2 Q. vc (x1 ::= x2) Q ⟹ ⊢⇩t {pre (x1 ::= x2) Q} strip (x1 ::= x2) {Q}›\n 3. ‹⋀C1 C2 Q. ⟦⋀Q. vc C1 Q ⟹ ⊢⇩t {pre C1 Q} strip C1 {Q}; ⋀Q. vc C2 Q ⟹ ⊢⇩t {pre C2 Q} strip C2 {Q}; vc (C1;; C2) Q⟧ ⟹ ⊢⇩t {pre (C1;; C2) Q} strip (C1;; C2) {Q}›\n 4. ‹⋀x1 C1 C2 Q. ⟦⋀Q. vc C1 Q ⟹ ⊢⇩t {pre C1 Q} strip C1 {Q}; ⋀Q. vc C2 Q ⟹ ⊢⇩t {pre C2 Q} strip C2 {Q}; vc (IF x1 THEN C1 ELSE C2) Q⟧ ⟹ ⊢⇩t {pre (IF x1 THEN C1 ELSE C2) Q} strip (IF x1 THEN C1 ELSE C2) {Q}›*)\n\ntext‹When trying to extend the completeness proof of the VCG for partial correctness\nto total correctness one runs into the following problem.\nIn the case of the while-rule, the universally quantified ‹n› in the first premise\nmeans that for that premise the induction hypothesis does not yield a single\nannotated command ‹C› but merely that for every ‹n› such a ‹C› exists.›\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/IMP/VCG_Total_EX.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"35cfbd01b401d7793fdebb4f1e2c0f954b6330d260c33f4246ad8c60ca55a1ea\"\n}"}}},{"rowIdx":162,"cells":{"text":{"kind":"string","value":"section‹The Axiom of Extensionality in $M[G]$›\ntheory Extensionality_Axiom\nimports\n Names\nbegin\n\ncontext forcing_data\nbegin\n \nlemma extensionality_in_MG : \"extensionality(##(M[G]))\"\nproof (-)\n (*goal: ‹extensionality(##M[G])›*)\n {\n fix x and y and z\n assume asms: \"x∈M[G]\" \"y∈M[G]\" \"(∀w∈M[G] . w ∈ x ⟷ w ∈ y)\" (*‹(x::i) ∈ M[G::i]› ‹(y::i) ∈ M[G::i]› ‹∀w::i∈M[G::i]. w ∈ (x::i) ⟷ w ∈ (y::i)›*)\n from ‹x∈M[G]› (*‹x ∈ M[G]›*) have \"z∈x ⟷ z∈M[G] ∧ z∈x\"\n using transitivity_MG (*‹⟦?y ∈ ?x; ?x ∈ M[?G]⟧ ⟹ ?y ∈ M[?G]›*) by auto\n also (*calculation: ‹z ∈ x ⟷ z ∈ M[G] ∧ z ∈ x›*) have \"... ⟷ z∈y\"\n using asms (*‹x ∈ M[G]› ‹(y::i) ∈ M[G::i]› ‹∀w∈M[G]. w ∈ x ⟷ w ∈ y›*) transitivity_MG (*‹⟦?y ∈ ?x; ?x ∈ M[?G]⟧ ⟹ ?y ∈ M[?G]›*) by auto\n finally (*calculation: ‹z ∈ x ⟷ z ∈ y›*) have \"z∈x ⟷ z∈y\" .\n }\n then have \"∀x∈M[G] . ∀y∈M[G] . (∀z∈M[G] . z ∈ x ⟷ z ∈ y) ⟶ x = y\"\n by blast\n then show \"?thesis\"\n (*goal: ‹extensionality(##M[G])›*)\n unfolding extensionality_def\n (*goal: ‹∀x::i[##M[G::i]]. ∀y::i[##M[G]]. (∀z::i[##M[G]]. z ∈ x ⟷ z ∈ y) ⟶ x = y›*)\n by simp\nqed\n \nend (* context forcing_data *)\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Forcing/Extensionality_Axiom.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"ffd787f00e433070124962757c017a12eab1d6e478259bf8b66dd5f8395aaa1b\"\n}"}}},{"rowIdx":163,"cells":{"text":{"kind":"string","value":"(* Title: OAWN_SOS.thy\n License: BSD 2-Clause. See LICENSE.\n Author: Timothy Bourke\n*)\n\nsection \"Open semantics of the Algebra of Wireless Networks\"\n\ntheory OAWN_SOS\nimports TransitionSystems AWN\nbegin\n\ntext ‹\n These are variants of the SOS rules that work against a mixed global/local context, where\n the global context is represented by a function @{term σ} mapping ip addresses to states.\n›\n\nsubsection \"Open structural operational semantics for sequential process expressions \"\n\ninductive_set\n oseqp_sos\n :: \"('s, 'm, 'p, 'l) seqp_env ⇒ ip\n ⇒ ((ip ⇒ 's) × ('s, 'm, 'p, 'l) seqp, 'm seq_action) transition set\"\n for Γ :: \"('s, 'm, 'p, 'l) seqp_env\"\n and i :: ip\nwhere\n obroadcastT: \"σ' i = σ i ⟹\n ((σ, {l}broadcast(s⇩m⇩s⇩g).p), broadcast (s⇩m⇩s⇩g (σ i)), (σ', p)) ∈ oseqp_sos Γ i\"\n | ogroupcastT: \"σ' i = σ i ⟹\n ((σ, {l}groupcast(s⇩i⇩p⇩s, s⇩m⇩s⇩g).p), groupcast (s⇩i⇩p⇩s (σ i)) (s⇩m⇩s⇩g (σ i)), (σ', p)) ∈ oseqp_sos Γ i\"\n | ounicastT: \"σ' i = σ i ⟹\n ((σ, {l}unicast(s⇩i⇩p, s⇩m⇩s⇩g).p ▹ q), unicast (s⇩i⇩p (σ i)) (s⇩m⇩s⇩g (σ i)), (σ', p)) ∈ oseqp_sos Γ i\"\n | onotunicastT:\"σ' i = σ i ⟹\n ((σ, {l}unicast(s⇩i⇩p, s⇩m⇩s⇩g).p ▹ q), ¬unicast (s⇩i⇩p (σ i)), (σ', q)) ∈ oseqp_sos Γ i\"\n | osendT: \"σ' i = σ i ⟹\n ((σ, {l}send(s⇩m⇩s⇩g).p), send (s⇩m⇩s⇩g (σ i)), (σ', p)) ∈ oseqp_sos Γ i\"\n | odeliverT: \"σ' i = σ i ⟹\n ((σ, {l}deliver(s⇩d⇩a⇩t⇩a).p), deliver (s⇩d⇩a⇩t⇩a (σ i)), (σ', p)) ∈ oseqp_sos Γ i\"\n | oreceiveT: \"σ' i = u⇩m⇩s⇩g msg (σ i) ⟹\n ((σ, {l}receive(u⇩m⇩s⇩g).p), receive msg, (σ', p)) ∈ oseqp_sos Γ i\"\n | oassignT: \"σ' i = u (σ i) ⟹\n ((σ, {l}⟦u⟧ p), τ, (σ', p)) ∈ oseqp_sos Γ i\"\n\n | ocallT: \"((σ, Γ pn), a, (σ', p')) ∈ oseqp_sos Γ i ⟹\n ((σ, call(pn)), a, (σ', p')) ∈ oseqp_sos Γ i\" (* TPB: quite different to Table 1 *)\n\n | ochoiceT1: \"((σ, p), a, (σ', p')) ∈ oseqp_sos Γ i ⟹\n ((σ, p ⊕ q), a, (σ', p')) ∈ oseqp_sos Γ i\"\n | ochoiceT2: \"((σ, q), a, (σ', q')) ∈ oseqp_sos Γ i ⟹\n ((σ, p ⊕ q), a, (σ', q')) ∈ oseqp_sos Γ i\"\n\n | oguardT: \"σ' i ∈ g (σ i) ⟹ ((σ, {l}⟨g⟩ p), τ, (σ', p)) ∈ oseqp_sos Γ i\"\n\ninductive_cases\n oseq_callTE [elim]: \"((σ, call(pn)), a, (σ', q)) ∈ oseqp_sos Γ i\"\n and oseq_choiceTE [elim]: \"((σ, p1 ⊕ p2), a, (σ', q)) ∈ oseqp_sos Γ i\"\n\nlemma oseq_broadcastTE [elim]:\n \"⟦((σ, {l}broadcast(s⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i;\n ⟦a = broadcast (s⇩m⇩s⇩g (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}broadcast(s⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i\")\n (*goal: ‹⟦((σ, {l}broadcast(s⇩m⇩s⇩g) .\n p), a, σ', q) ∈ oseqp_sos Γ i; ⟦a = broadcast (s⇩m⇩s⇩g (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P›*)\n by simp\n\nlemma oseq_groupcastTE [elim]:\n \"⟦((σ, {l}groupcast(s⇩i⇩p⇩s, s⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i;\n ⟦a = groupcast (s⇩i⇩p⇩s (σ i)) (s⇩m⇩s⇩g (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}groupcast(s⇩i⇩p⇩s, s⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i\")\n (*goal: ‹⟦((σ, {l}groupcast(s⇩i⇩p⇩s, s⇩m⇩s⇩g) .\n p), a, σ', q) ∈ oseqp_sos Γ i; ⟦a = groupcast (s⇩i⇩p⇩s (σ i)) (s⇩m⇩s⇩g (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P›*)\n by simp\n\nlemma oseq_unicastTE [elim]:\n \"⟦((σ, {l}unicast(s⇩i⇩p, s⇩m⇩s⇩g). p ▹ q), a, (σ', r)) ∈ oseqp_sos Γ i;\n ⟦a = unicast (s⇩i⇩p (σ i)) (s⇩m⇩s⇩g (σ i)); σ' i = σ i; r = p⟧ ⟹ P;\n ⟦a = ¬unicast (s⇩i⇩p (σ i)); σ' i = σ i; r = q⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}unicast(s⇩i⇩p, s⇩m⇩s⇩g). p ▹ q), a, (σ', r)) ∈ oseqp_sos Γ i\")\n (*goals:\n 1. ‹⟦⟦a = unicast (s⇩i⇩p (σ i)) (s⇩m⇩s⇩g (σ i)); σ' i = σ i; r = p⟧ ⟹ P; ⟦a = ¬unicast (s⇩i⇩p (σ i)); σ' i = σ i; r = q⟧ ⟹ P; a = unicast (s⇩i⇩p (σ i)) (s⇩m⇩s⇩g (σ i)); r = p; σ' i = σ i⟧ ⟹ P›\n 2. ‹⟦⟦a = unicast (s⇩i⇩p (σ i)) (s⇩m⇩s⇩g (σ i)); σ' i = σ i; r = p⟧ ⟹ P; ⟦a = ¬unicast (s⇩i⇩p (σ i)); σ' i = σ i; r = q⟧ ⟹ P; a = ¬unicast (s⇩i⇩p (σ i)); r = q; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma oseq_sendTE [elim]:\n \"⟦((σ, {l}send(s⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i;\n ⟦a = send (s⇩m⇩s⇩g (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}send(s⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i\")\n (*goal: ‹⟦((σ, {l}send(s⇩m⇩s⇩g) .\n p), a, σ', q) ∈ oseqp_sos Γ i; ⟦a = send (s⇩m⇩s⇩g (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P›*)\n by simp\n\nlemma oseq_deliverTE [elim]:\n \"⟦((σ, {l}deliver(s⇩d⇩a⇩t⇩a). p), a, (σ', q)) ∈ oseqp_sos Γ i;\n ⟦a = deliver (s⇩d⇩a⇩t⇩a (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}deliver(s⇩d⇩a⇩t⇩a). p), a, (σ', q)) ∈ oseqp_sos Γ i\")\n (*goal: ‹⟦((σ, {l}deliver(s⇩d⇩a⇩t⇩a) .\n p), a, σ', q) ∈ oseqp_sos Γ i; ⟦a = deliver (s⇩d⇩a⇩t⇩a (σ i)); σ' i = σ i; q = p⟧ ⟹ P⟧ ⟹ P›*)\n by simp\n\nlemma oseq_receiveTE [elim]:\n \"⟦((σ, {l}receive(u⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i;\n ⋀msg. ⟦a = receive msg; σ' i = u⇩m⇩s⇩g msg (σ i); q = p⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}receive(u⇩m⇩s⇩g). p), a, (σ', q)) ∈ oseqp_sos Γ i\")\n (*goal: ‹⟦((σ, {l}receive(u⇩m⇩s⇩g) .\n p), a, σ', q) ∈ oseqp_sos Γ i; ⋀msg. ⟦a = receive msg; σ' i = u⇩m⇩s⇩g msg (σ i); q = p⟧ ⟹ P⟧ ⟹ P›*)\n by simp\n\nlemma oseq_assignTE [elim]:\n \"⟦((σ, {l}⟦u⟧ p), a, (σ', q)) ∈ oseqp_sos Γ i; ⟦a = τ; σ' i = u (σ i); q = p⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}⟦u⟧ p), a, (σ', q)) ∈ oseqp_sos Γ i\")\n (*goal: ‹⟦((σ::nat ⇒ 'a::type, {l::'d::type}⟦u::'a::type ⇒ 'a::type⟧\n (p::('a::type, 'b::type, 'c::type, 'd::type) seqp)), a::'b::type seq_action, σ'::nat ⇒ 'a::type, q::('a::type, 'b::type, 'c::type, 'd::type) seqp) ∈ oseqp_sos (Γ::'c::type ⇒ ('a::type, 'b::type, 'c::type, 'd::type) seqp) (i::nat); ⟦a = τ; σ' i = u (σ i); q = p⟧ ⟹ P::bool⟧ ⟹ P›*)\n by simp\n\nlemma oseq_guardTE [elim]:\n \"⟦((σ, {l}⟨g⟩ p), a, (σ', q)) ∈ oseqp_sos Γ i; ⟦a = τ; σ' i ∈ g (σ i); q = p⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, {l}⟨g⟩ p), a, (σ', q)) ∈ oseqp_sos Γ i\")\n (*goal: ‹⟦((σ, {l}⟨g⟩\n p), a, σ', q) ∈ oseqp_sos Γ i; ⟦a = τ; σ' i ∈ g (σ i); q = p⟧ ⟹ P⟧ ⟹ P›*)\n by simp\n\nlemmas oseqpTEs =\n oseq_broadcastTE\n oseq_groupcastTE\n oseq_unicastTE\n oseq_sendTE\n oseq_deliverTE\n oseq_receiveTE\n oseq_assignTE\n oseq_callTE\n oseq_choiceTE\n oseq_guardTE\n\ndeclare oseqp_sos.intros [intro]\n\nsubsection \"Open structural operational semantics for parallel process expressions \"\n\ninductive_set\n oparp_sos :: \"ip\n ⇒ ((ip ⇒ 's) × 's1, 'm seq_action) transition set\n ⇒ ('s2, 'm seq_action) transition set\n ⇒ ((ip ⇒ 's) × ('s1 × 's2), 'm seq_action) transition set\"\n for i :: ip\n and S :: \"((ip ⇒ 's) × 's1, 'm seq_action) transition set\"\n and T :: \"('s2, 'm seq_action) transition set\"\nwhere\n oparleft: \"⟦ ((σ, s), a, (σ', s')) ∈ S; ⋀m. a ≠ receive m ⟧ ⟹\n ((σ, (s, t)), a, (σ', (s', t))) ∈ oparp_sos i S T\"\n | oparright: \"⟦ (t, a, t') ∈ T; ⋀m. a ≠ send m; σ' i = σ i ⟧ ⟹\n ((σ, (s, t)), a, (σ', (s, t'))) ∈ oparp_sos i S T\"\n | oparboth: \"⟦ ((σ, s), receive m, (σ', s')) ∈ S; (t, send m, t') ∈ T ⟧ ⟹\n ((σ, (s, t)), τ, (σ', (s', t'))) ∈ oparp_sos i S T\"\n\nlemma opar_broadcastTE [elim]:\n \"⟦((σ, (s, t)), broadcast m, (σ', (s', t'))) ∈ oparp_sos i S T;\n ⟦((σ, s), broadcast m, (σ', s')) ∈ S; t' = t⟧ ⟹ P;\n ⟦(t, broadcast m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, (s, t)), broadcast m, (σ', (s', t'))) ∈ oparp_sos i S T\")\n (*goals:\n 1. ‹⟦⟦((σ, s), broadcast m, σ', s') ∈ S; t' = t⟧ ⟹ P; ⟦(t, broadcast m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; t' = t; ((σ, s), broadcast m, σ', s') ∈ S; ⋀ma. broadcast m ≠ receive ma⟧ ⟹ P›\n 2. ‹⟦⟦((σ, s), broadcast m, σ', s') ∈ S; t' = t⟧ ⟹ P; ⟦(t, broadcast m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; s' = s; (t, broadcast m, t') ∈ T; ⋀ma. broadcast m ≠ send ma; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma opar_groupcastTE [elim]:\n \"⟦((σ, (s, t)), groupcast ips m, (σ', (s', t'))) ∈ oparp_sos i S T;\n ⟦((σ, s), groupcast ips m, (σ', s')) ∈ S; t' = t⟧ ⟹ P;\n ⟦(t, groupcast ips m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, (s, t)), groupcast ips m, (σ', (s', t'))) ∈ oparp_sos i S T\")\n (*goals:\n 1. ‹⟦⟦((σ, s), groupcast ips m, σ', s') ∈ S; t' = t⟧ ⟹ P; ⟦(t, groupcast ips m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; t' = t; ((σ, s), groupcast ips m, σ', s') ∈ S; ⋀ma. groupcast ips m ≠ receive ma⟧ ⟹ P›\n 2. ‹⟦⟦((σ, s), groupcast ips m, σ', s') ∈ S; t' = t⟧ ⟹ P; ⟦(t, groupcast ips m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; s' = s; (t, groupcast ips m, t') ∈ T; ⋀ma. groupcast ips m ≠ send ma; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma opar_unicastTE [elim]:\n \"⟦((σ, (s, t)), unicast i m, (σ', (s', t'))) ∈ oparp_sos i S T;\n ⟦((σ, s), unicast i m, (σ', s')) ∈ S; t' = t⟧ ⟹ P;\n ⟦(t, unicast i m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, (s, t)), unicast i m, (σ', (s', t'))) ∈ oparp_sos i S T\")\n (*goals:\n 1. ‹⟦⟦((σ::nat ⇒ 'a::type, s::'b::type), unicast (i::nat) (m::'d::type), σ'::nat ⇒ 'a::type, s'::'b::type) ∈ (S::(((nat ⇒ 'a::type) × 'b::type) × 'd::type seq_action × (nat ⇒ 'a::type) × 'b::type) set); (t'::'c::type) = (t::'c::type)⟧ ⟹ P::bool; ⟦(t, unicast i m, t') ∈ (T::('c::type × 'd::type seq_action × 'c::type) set); s' = s; σ' i = σ i⟧ ⟹ P; t' = t; ((σ, s), unicast i m, σ', s') ∈ S; ⋀ma::'d::type. unicast i m ≠ receive ma⟧ ⟹ P›\n 2. ‹⟦⟦((σ::nat ⇒ 'a::type, s::'b::type), unicast (i::nat) (m::'d::type), σ'::nat ⇒ 'a::type, s'::'b::type) ∈ (S::(((nat ⇒ 'a::type) × 'b::type) × 'd::type seq_action × (nat ⇒ 'a::type) × 'b::type) set); (t'::'c::type) = (t::'c::type)⟧ ⟹ P::bool; ⟦(t, unicast i m, t') ∈ (T::('c::type × 'd::type seq_action × 'c::type) set); s' = s; σ' i = σ i⟧ ⟹ P; s' = s; (t, unicast i m, t') ∈ T; ⋀ma::'d::type. unicast i m ≠ send ma; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma opar_notunicastTE [elim]:\n \"⟦((σ, (s, t)), notunicast i, (σ', (s', t'))) ∈ oparp_sos i S T;\n ⟦((σ, s), notunicast i, (σ', s')) ∈ S; t' = t⟧ ⟹ P;\n ⟦(t, notunicast i, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, (s, t)), notunicast i, (σ', (s', t'))) ∈ oparp_sos i S T\")\n (*goals:\n 1. ‹⟦⟦((σ, s), ¬unicast i, σ', s') ∈ S; t' = t⟧ ⟹ P; ⟦(t, ¬unicast i, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; t' = t; ((σ, s), ¬unicast i, σ', s') ∈ S; ⋀m. ¬unicast i ≠ receive m⟧ ⟹ P›\n 2. ‹⟦⟦((σ, s), ¬unicast i, σ', s') ∈ S; t' = t⟧ ⟹ P; ⟦(t, ¬unicast i, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; s' = s; (t, ¬unicast i, t') ∈ T; ⋀m. ¬unicast i ≠ send m; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma opar_sendTE [elim]:\n \"⟦((σ, (s, t)), send m, (σ', (s', t'))) ∈ oparp_sos i S T;\n ⟦((σ, s), send m, (σ', s')) ∈ S; t' = t⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, (s, t)), send m, (σ', (s', t'))) ∈ oparp_sos i S T\")\n (*goals:\n 1. ‹⟦⟦((σ, s), send m, σ', s') ∈ S; t' = t⟧ ⟹ P; t' = t; ((σ, s), send m, σ', s') ∈ S; ⋀ma. send m ≠ receive ma⟧ ⟹ P›\n 2. ‹⟦⟦((σ, s), send m, σ', s') ∈ S; t' = t⟧ ⟹ P; s' = s; (t, send m, t') ∈ T; ⋀ma. send m ≠ send ma; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma opar_deliverTE [elim]:\n \"⟦((σ, (s, t)), deliver d, (σ', (s', t'))) ∈ oparp_sos i S T;\n ⟦((σ, s), deliver d, (σ', s')) ∈ S; t' = t⟧ ⟹ P;\n ⟦(t, deliver d, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, (s, t)), deliver d, (σ', (s', t'))) ∈ oparp_sos i S T\")\n (*goals:\n 1. ‹⟦⟦((σ::nat ⇒ 'a, s::'b), deliver (d::nat), σ'::nat ⇒ 'a, s'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'd seq_action × (nat ⇒ 'a) × 'b) set); (t'::'c) = (t::'c)⟧ ⟹ P::bool; ⟦(t, deliver d, t') ∈ (T::('c × 'd seq_action × 'c) set); s' = s; σ' (i::nat) = σ i⟧ ⟹ P; t' = t; ((σ, s), deliver d, σ', s') ∈ S; ⋀m::'d. deliver d ≠ receive m⟧ ⟹ P›\n 2. ‹⟦⟦((σ::nat ⇒ 'a, s::'b), deliver (d::nat), σ'::nat ⇒ 'a, s'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'd seq_action × (nat ⇒ 'a) × 'b) set); (t'::'c) = (t::'c)⟧ ⟹ P::bool; ⟦(t, deliver d, t') ∈ (T::('c × 'd seq_action × 'c) set); s' = s; σ' (i::nat) = σ i⟧ ⟹ P; s' = s; (t, deliver d, t') ∈ T; ⋀m::'d. deliver d ≠ send m; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma opar_receiveTE [elim]:\n \"⟦((σ, (s, t)), receive m, (σ', (s', t'))) ∈ oparp_sos i S T;\n ⟦(t, receive m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P⟧ ⟹ P\"\n apply (ind_cases \"((σ, (s, t)), receive m, (σ', (s', t'))) ∈ oparp_sos i S T\")\n (*goals:\n 1. ‹⟦⟦(t, receive m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; t' = t; ((σ, s), receive m, σ', s') ∈ S; ⋀ma. receive m ≠ receive ma⟧ ⟹ P›\n 2. ‹⟦⟦(t, receive m, t') ∈ T; s' = s; σ' i = σ i⟧ ⟹ P; s' = s; (t, receive m, t') ∈ T; ⋀ma. receive m ≠ send ma; σ' i = σ i⟧ ⟹ P›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\ninductive_cases opar_tauTE: \"((σ, (s, t)), τ, (σ', (s', t'))) ∈ oparp_sos i S T\"\n\nlemmas oparpTEs =\n opar_broadcastTE\n opar_groupcastTE\n opar_unicastTE\n opar_notunicastTE\n opar_sendTE\n opar_deliverTE\n opar_receiveTE\n\nlemma oparp_sos_cases [elim]:\n assumes \"((σ, (s, t)), a, (σ', (s', t'))) ∈ oparp_sos i S T\"\n and \"⟦ ((σ, s), a, (σ', s')) ∈ S; ⋀m. a ≠ receive m; t' = t ⟧ ⟹ P\"\n and \"⟦ (t, a, t') ∈ T; ⋀m. a ≠ send m; s' = s; σ' i = σ i ⟧ ⟹ P\"\n and \"⋀m. ⟦ a = τ; ((σ, s), receive m, (σ', s')) ∈ S; (t, send m, t') ∈ T ⟧ ⟹ P\"\n shows \"P\"\n using assms (*‹((σ, s, t), a, σ', s', t') ∈ oparp_sos i S T› ‹⟦((σ::nat ⇒ 'a, s::'b), a::'d seq_action, σ'::nat ⇒ 'a, s'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'd seq_action × (nat ⇒ 'a) × 'b) set); ⋀m::'d. a ≠ receive m; (t'::'c) = (t::'c)⟧ ⟹ P::bool› ‹⟦(t, a, t') ∈ T; ⋀m. a ≠ send m; s' = s; σ' i = σ i⟧ ⟹ P› ‹⟦(a::'d::type seq_action) = τ; ((σ::nat ⇒ 'a::type, s::'b::type), receive (?m::'d::type), σ'::nat ⇒ 'a::type, s'::'b::type) ∈ (S::(((nat ⇒ 'a::type) × 'b::type) × 'd::type seq_action × (nat ⇒ 'a::type) × 'b::type) set); (t::'c::type, send ?m, t'::'c::type) ∈ (T::('c::type × 'd::type seq_action × 'c::type) set)⟧ ⟹ P::bool›*) apply cases\n (*goals:\n 1. ‹⟦⟦((σ, s), a, σ', s') ∈ S; ⋀m. a ≠ receive m; t' = t⟧ ⟹ P; ⟦(t, a, t') ∈ T; ⋀m. a ≠ send m; s' = s; σ' i = σ i⟧ ⟹ P; ⋀m. ⟦a = τ; ((σ, s), receive m, σ', s') ∈ S; (t, send m, t') ∈ T⟧ ⟹ P; t' = t; ((σ, s), a, σ', s') ∈ S; ⋀m. a ≠ receive m⟧ ⟹ P›\n 2. ‹⟦⟦((σ, s), a, σ', s') ∈ S; ⋀m. a ≠ receive m; t' = t⟧ ⟹ P; ⟦(t, a, t') ∈ T; ⋀m. a ≠ send m; s' = s; σ' i = σ i⟧ ⟹ P; ⋀m. ⟦a = τ; ((σ, s), receive m, σ', s') ∈ S; (t, send m, t') ∈ T⟧ ⟹ P; s' = s; (t, a, t') ∈ T; ⋀m. a ≠ send m; σ' i = σ i⟧ ⟹ P›\n 3. ‹⋀m. ⟦⟦((σ, s), a, σ', s') ∈ S; ⋀m. a ≠ receive m; t' = t⟧ ⟹ P; ⟦(t, a, t') ∈ T; ⋀m. a ≠ send m; s' = s; σ' i = σ i⟧ ⟹ P; ⋀m. ⟦a = τ; ((σ, s), receive m, σ', s') ∈ S; (t, send m, t') ∈ T⟧ ⟹ P; a = τ; ((σ, s), receive m, σ', s') ∈ S; (t, send m, t') ∈ T⟧ ⟹ P›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n\ndefinition extg :: \"('a × 'b) × 'c ⇒ 'a × 'b × 'c\"\nwhere \"extg ≡ λ((σ, l1), l2). (σ, (l1, l2))\"\n\nlemma extgsimp [simp]:\n \"extg ((σ, l1), l2) = (σ, (l1, l2))\"\n unfolding extg_def\n (*goal: ‹(case ((σ, l1), l2) of (x, xa) ⇒ (case x of (σ, l1) ⇒ λl2. (σ, l1, l2)) xa) = (σ, l1, l2)›*)\n by simp\n\nlemma extg_range_prod: \"extg ` (i1 × i2) = {(σ, (s1, s2))|σ s1 s2. (σ, s1) ∈ i1 ∧ s2 ∈ i2}\"\n unfolding image_def extg_def\n (*goal: ‹{y. ∃x∈i1 × i2. y = (case x of (x, xa) ⇒ (case x of (σ, l1) ⇒ λl2. (σ, l1, l2)) xa)} = {(σ, s1, s2) |σ s1 s2. (σ, s1) ∈ i1 ∧ s2 ∈ i2}›*)\n apply (rule Collect_cong (*‹(⋀x. ?P x = ?Q x) ⟹ {x. ?P x} = {x. ?Q x}›*))\n (*goal: ‹{y::'a::type × 'b::type × 'c::type. ∃x::('a::type × 'b::type) × 'c::type∈(i1::('a::type × 'b::type) set) × (i2::'c::type set). y = (case x of (x::'a::type × 'b::type, xa::'c::type) ⇒ (case x of (σ::'a::type, l1::'b::type) ⇒ λl2::'c::type. (σ, l1, l2)) xa)} = {(σ, s1, s2) |(σ::'a::type) (s1::'b::type) s2::'c::type. (σ, s1) ∈ i1 ∧ s2 ∈ i2}›*)\n by (auto split: prod.split (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))›*))\n\ndefinition\n opar_comp :: \"((ip ⇒ 's) × 's1, 'm seq_action) automaton\n ⇒ ip\n ⇒ ('s2, 'm seq_action) automaton\n ⇒ ((ip ⇒ 's) × 's1 × 's2, 'm seq_action) automaton\"\n (\"(_ ⟨⟨⇘_⇙ _)\" [102, 0, 103] 102)\nwhere\n \"s ⟨⟨⇘i⇙ t ≡ ⦇ init = extg ` (init s × init t), trans = oparp_sos i (trans s) (trans t) ⦈\"\n\nlemma opar_comp_def':\n \"s ⟨⟨⇘i⇙ t = ⦇ init = {(σ, (s⇩l, t⇩l))|σ s⇩l t⇩l. (σ, s⇩l) ∈ init s ∧ t⇩l ∈ init t},\n trans = oparp_sos i (trans s) (trans t) ⦈\"\n unfolding opar_comp_def extg_def image_def\n (*goal: ‹⦇init = {y. ∃x∈init s × init t. y = (case x of (x, xa) ⇒ (case x of (σ, l1) ⇒ λl2. (σ, l1, l2)) xa)}, trans = oparp_sos i (automaton.trans s) (automaton.trans t)⦈ = ⦇init = {(σ, s⇩l, t⇩l) |σ s⇩l t⇩l. (σ, s⇩l) ∈ init s ∧ t⇩l ∈ init t}, trans = oparp_sos i (automaton.trans s) (automaton.trans t)⦈›*)\n by (auto split: prod.split (*‹?P (case ?prod of (x, xa) ⇒ ?f x xa) = (∀x1 x2. ?prod = (x1, x2) ⟶ ?P (?f x1 x2))›*))\n\nlemma trans_opar_comp [simp]:\n \"trans (s ⟨⟨⇘i⇙ t) = oparp_sos i (trans s) (trans t)\"\n unfolding opar_comp_def\n (*goal: ‹automaton.trans ⦇init = extg ` (init s × init t), trans = oparp_sos i (automaton.trans s) (automaton.trans t)⦈ = oparp_sos i (automaton.trans s) (automaton.trans t)›*)\n by simp\n\nlemma init_opar_comp [simp]:\n \"init (s ⟨⟨⇘i⇙ t) = extg ` (init s × init t)\"\n unfolding opar_comp_def\n (*goal: ‹init ⦇init = extg ` (init s × init t), trans = oparp_sos i (automaton.trans s) (automaton.trans t)⦈ = extg ` (init s × init t)›*)\n by simp\n\nsubsection \"Open structural operational semantics for node expressions \"\n\ninductive_set\n onode_sos :: \"((ip ⇒ 's) × 'l, 'm seq_action) transition set\n ⇒ ((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\"\n for S :: \"((ip ⇒ 's) × 'l, 'm seq_action) transition set\"\nwhere\n onode_bcast:\n \"((σ, s), broadcast m, (σ', s')) ∈ S ⟹ ((σ, NodeS i s R), R:*cast(m), (σ', NodeS i s' R)) ∈ onode_sos S\"\n\n | onode_gcast:\n \"((σ, s), groupcast D m, (σ', s')) ∈ S ⟹ ((σ, NodeS i s R), (R∩D):*cast(m), (σ', NodeS i s' R)) ∈ onode_sos S\"\n\n | onode_ucast:\n \"⟦ ((σ, s), unicast d m, (σ', s')) ∈ S; d∈R ⟧ ⟹ ((σ, NodeS i s R), {d}:*cast(m), (σ', NodeS i s' R)) ∈ onode_sos S\"\n\n (* Such assumptions aid later proofs, but they must be justified when transferring results\n to closed systems. *)\n | onode_notucast: \"⟦ ((σ, s), ¬unicast d, (σ', s')) ∈ S; d∉R; ∀j. j≠i ⟶ σ' j = σ j ⟧\n ⟹ ((σ, NodeS i s R), τ, (σ', NodeS i s' R)) ∈ onode_sos S\"\n\n | onode_deliver: \"⟦ ((σ, s), deliver d, (σ', s')) ∈ S; ∀j. j≠i ⟶ σ' j = σ j ⟧\n ⟹ ((σ, NodeS i s R), i:deliver(d), (σ', NodeS i s' R)) ∈ onode_sos S\"\n\n | onode_tau: \"⟦ ((σ, s), τ, (σ', s')) ∈ S; ∀j. j≠i ⟶ σ' j = σ j ⟧\n ⟹ ((σ, NodeS i s R), τ, (σ', NodeS i s' R)) ∈ onode_sos S\"\n\n | onode_receive:\n \"((σ, s), receive m, (σ', s')) ∈ S ⟹ ((σ, NodeS i s R), {i}¬{}:arrive(m), (σ', NodeS i s' R)) ∈ onode_sos S\"\n\n | onode_arrive:\n \"σ' i = σ i ⟹ ((σ, NodeS i s R), {}¬{i}:arrive(m), (σ', NodeS i s R)) ∈ onode_sos S\"\n\n | onode_connect1:\n \"σ' i = σ i ⟹ ((σ, NodeS i s R), connect(i, i'), (σ', NodeS i s (R ∪ {i'}))) ∈ onode_sos S\"\n\n | onode_connect2:\n \"σ' i = σ i ⟹ ((σ, NodeS i s R), connect(i', i), (σ', NodeS i s (R ∪ {i'}))) ∈ onode_sos S\"\n\n | onode_disconnect1:\n \"σ' i = σ i ⟹ ((σ, NodeS i s R), disconnect(i, i'), (σ', NodeS i s (R - {i'}))) ∈ onode_sos S\"\n\n | onode_disconnect2:\n \"σ' i = σ i ⟹ ((σ, NodeS i s R), disconnect(i', i), (σ', NodeS i s (R - {i'}))) ∈ onode_sos S\"\n\n | onode_connect_other:\n \"⟦ i ≠ i'; i ≠ i''; σ' i = σ i ⟧ ⟹ ((σ, NodeS i s R), connect(i', i''), (σ', NodeS i s R)) ∈ onode_sos S\"\n\n | onode_disconnect_other:\n \"⟦ i ≠ i'; i ≠ i''; σ' i = σ i ⟧ ⟹ ((σ, NodeS i s R), disconnect(i', i''), (σ', NodeS i s R)) ∈ onode_sos S\"\n\ninductive_cases\n onode_arriveTE [elim]: \"((σ, NodeS i s R), ii¬ni:arrive(m), (σ', NodeS i' s' R')) ∈ onode_sos S\"\n and onode_castTE [elim]: \"((σ, NodeS i s R), RR:*cast(m), (σ', NodeS i' s' R')) ∈ onode_sos S\"\n and onode_deliverTE [elim]: \"((σ, NodeS i s R), ii:deliver(d), (σ', NodeS i' s' R')) ∈ onode_sos S\"\n and onode_connectTE [elim]: \"((σ, NodeS i s R), connect(ii, ii'), (σ', NodeS i' s' R')) ∈ onode_sos S\"\n and onode_disconnectTE [elim]: \"((σ, NodeS i s R), disconnect(ii, ii'),(σ', NodeS i' s' R')) ∈ onode_sos S\"\n and onode_newpktTE [elim]: \"((σ, NodeS i s R), ii:newpkt(d, di), (σ', NodeS i' s' R')) ∈ onode_sos S\"\n and onode_tauTE [elim]: \"((σ, NodeS i s R), τ, (σ', NodeS i' s' R')) ∈ onode_sos S\"\n\nlemma oarrives_or_not:\n assumes \"((σ, NodeS i s R), ii¬ni:arrive(m), (σ', NodeS i' s' R')) ∈ onode_sos S\"\n shows \"(ii = {i} ∧ ni = {}) ∨ (ii = {} ∧ ni = {i})\"\n using assms (*‹((σ, NodeS i s R), ii¬ni:arrive(m), σ', NodeS i' s' R') ∈ onode_sos S›*) apply rule\n (*goals:\n 1. ‹⟦((σ, s), receive m, σ', s') ∈ S; ii = {i}; ni = {}; i' = i; R' = R⟧ ⟹ ii = {i} ∧ ni = {} ∨ ii = {} ∧ ni = {i}›\n 2. ‹⟦σ' i = σ i; ii = {}; ni = {i}; i' = i; s' = s; R' = R⟧ ⟹ ii = {i} ∧ ni = {} ∨ ii = {} ∧ ni = {i}›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\ndefinition\n onode_comp :: \"ip\n ⇒ ((ip ⇒ 's) × 'l, 'm seq_action) automaton\n ⇒ ip set\n ⇒ ((ip ⇒ 's) × 'l net_state, 'm node_action) automaton\"\n (\"(⟨_ : (_) : _⟩⇩o)\" [0, 0, 0] 104)\nwhere\n \"⟨i : onp : R⇩i⟩⇩o ≡ ⦇ init = {(σ, NodeS i s R⇩i)|σ s. (σ, s) ∈ init onp},\n trans = onode_sos (trans onp) ⦈\"\n\nlemma trans_onode_comp:\n \"trans (⟨i : S : R⟩⇩o) = onode_sos (trans S)\"\n unfolding onode_comp_def\n (*goal: ‹automaton.trans ⦇init = {(σ, NodeS i s R) |σ s. (σ, s) ∈ init S}, trans = onode_sos (automaton.trans S)⦈ = onode_sos (automaton.trans S)›*)\n by simp\n\nlemma init_onode_comp:\n \"init (⟨i : S : R⟩⇩o) = {(σ, NodeS i s R)|σ s. (σ, s) ∈ init S}\"\n unfolding onode_comp_def\n (*goal: ‹init ⦇init = {(σ, NodeS i s R) |σ s. (σ, s) ∈ init S}, trans = onode_sos (automaton.trans S)⦈ = {(σ, NodeS i s R) |σ s. (σ, s) ∈ init S}›*)\n by simp\n\nlemmas onode_comps = trans_onode_comp init_onode_comp\n\nlemma fst_par_onode_comp [simp]:\n \"trans (⟨i : s ⟨⟨⇘I⇙ t : R⟩⇩o) = onode_sos (oparp_sos I (trans s) (trans t))\"\n unfolding onode_comp_def\n (*goal: ‹automaton.trans ⦇init = {(σ, NodeS (i::nat) sa (R::nat set)) |(σ::nat ⇒ 'a) sa::'b × 'c. (σ, sa) ∈ init ((s::((nat ⇒ 'a) × 'b, 'd seq_action) automaton) ⟨⟨⇘I::nat⇙ (t::('c, 'd seq_action) automaton))}, trans = onode_sos (automaton.trans (s ⟨⟨⇘I⇙ t))⦈ = onode_sos (oparp_sos I (automaton.trans s) (automaton.trans t))›*)\n by simp\n\nlemma init_par_onode_comp [simp]:\n \"init (⟨i : s ⟨⟨⇘I⇙ t : R⟩⇩o) = {(σ, NodeS i (s1, s2) R)|σ s1 s2. ((σ, s1), s2) ∈ init s × init t}\"\n unfolding onode_comp_def\n (*goal: ‹init ⦇init = {(σ, NodeS i sa R) |σ sa. (σ, sa) ∈ init (s ⟨⟨⇘I⇙ t)}, trans = onode_sos (automaton.trans (s ⟨⟨⇘I⇙ t))⦈ = {(σ, NodeS i (s1, s2) R) |σ s1 s2. ((σ, s1), s2) ∈ init s × init t}›*)\n by (simp add: extg_range_prod (*‹extg ` (?i1.0 × ?i2.0) = {(σ, s1, s2) |σ s1 s2. (σ, s1) ∈ ?i1.0 ∧ s2 ∈ ?i2.0}›*))\n\nlemma onode_sos_dest_is_net_state:\n assumes \"((σ, p), a, s') ∈ onode_sos S\"\n shows \"∃σ' i' ζ' R'. s' = (σ', NodeS i' ζ' R')\"\n using assms (*‹((σ, p), a, s') ∈ onode_sos S›*) proof (-)\n (*goal: ‹((σ, p), a, s') ∈ onode_sos S ⟹ ∃σ' i' ζ' R'. s' = (σ', NodeS i' ζ' R')›*)\n assume \"((σ, p), a, s') ∈ onode_sos S\" (*‹((σ::nat ⇒ 'a, p::'b net_state), a::'c node_action, s'::(nat ⇒ 'a) × 'b net_state) ∈ onode_sos (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)›*)\n then obtain σ' and i' and ζ' and R' where \"s' = (σ', NodeS i' ζ' R')\"\n (*goal: ‹(⋀σ' i' ζ' R'. s' = (σ', NodeS i' ζ' R') ⟹ thesis) ⟹ thesis›*)\n apply (cases s')\n (*goal: ‹(⋀σ' i' ζ' R'. s' = (σ', NodeS i' ζ' R') ⟹ thesis) ⟹ thesis›*)\n by (auto elim!: onode_sos.cases (*‹⟦((?a1.0, ?a2.0), ?a3.0, ?a4.0, ?a5.0) ∈ onode_sos ?S; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = R:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), broadcast m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s D m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = (R ∩ D):*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), groupcast D m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s d m σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {d}:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), unicast d m, σ', s') ∈ ?S; d ∈ R⟧ ⟹ ?P; ⋀σ s d σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), ¬unicast d, σ', s') ∈ ?S; d ∉ R; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s d σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = i:deliver(d); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), deliver d, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), τ, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {i}¬{}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), receive m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ' i σ s R m. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {}¬{i}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s R; σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P⟧ ⟹ ?P›*))\n thus \"?thesis\"\n (*goal: ‹∃σ' i' ζ' R'. s' = (σ', NodeS i' ζ' R')›*)\n by simp\n qed\n\nlemma onode_sos_dest_is_net_state':\n assumes \"((σ, NodeS i p R), a, s') ∈ onode_sos S\"\n shows \"∃σ' ζ' R'. s' = (σ', NodeS i ζ' R')\"\n using assms (*‹((σ, NodeS i p R), a, s') ∈ onode_sos S›*) proof (-)\n (*goal: ‹((σ, NodeS i p R), a, s') ∈ onode_sos S ⟹ ∃σ' ζ' R'. s' = (σ', NodeS i ζ' R')›*)\n assume \"((σ, NodeS i p R), a, s') ∈ onode_sos S\" (*‹((σ::nat ⇒ 'a, NodeS (i::nat) (p::'b) (R::nat set)), a::'c node_action, s'::(nat ⇒ 'a) × 'b net_state) ∈ onode_sos (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)›*)\n then obtain σ' and ζ' and R' where \"s' = (σ', NodeS i ζ' R')\"\n (*goal: ‹(⋀σ' ζ' R'. s' = (σ', NodeS i ζ' R') ⟹ thesis) ⟹ thesis›*)\n apply (cases s')\n (*goal: ‹(⋀σ' ζ' R'. s' = (σ', NodeS i ζ' R') ⟹ thesis) ⟹ thesis›*)\n by (auto elim!: onode_sos.cases (*‹⟦((?a1.0, ?a2.0), ?a3.0, ?a4.0, ?a5.0) ∈ onode_sos ?S; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = R:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), broadcast m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s D m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = (R ∩ D):*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), groupcast D m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s d m σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {d}:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), unicast d m, σ', s') ∈ ?S; d ∈ R⟧ ⟹ ?P; ⋀σ s d σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), ¬unicast d, σ', s') ∈ ?S; d ∉ R; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s d σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = i:deliver(d); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), deliver d, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), τ, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {i}¬{}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), receive m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ' i σ s R m. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {}¬{i}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s R; σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P⟧ ⟹ ?P›*))\n thus \"?thesis\"\n (*goal: ‹∃σ' ζ' R'. s' = (σ', NodeS i ζ' R')›*)\n by simp\n qed\n\nlemma onode_sos_dest_is_net_state'':\n assumes \"((σ, NodeS i p R), a, (σ', s')) ∈ onode_sos S\"\n shows \"∃ζ' R'. s' = NodeS i ζ' R'\"\n proof (-)\n (*goal: ‹∃ζ' R'. s' = NodeS i ζ' R'›*)\n define ns' where \"ns' = (σ', s')\"\n with assms (*‹((σ::nat ⇒ 'a::type, NodeS (i::nat) (p::'b::type) (R::nat set)), a::'c::type node_action, σ'::nat ⇒ 'a::type, s'::'b::type net_state) ∈ onode_sos (S::(((nat ⇒ 'a::type) × 'b::type) × 'c::type seq_action × (nat ⇒ 'a::type) × 'b::type) set)›*) have \"((σ, NodeS i p R), a, ns') ∈ onode_sos S\"\n by simp\n then obtain σ'' and ζ' and R' where \"ns' = (σ'', NodeS i ζ' R')\"\n (*goal: ‹(⋀σ'' ζ' R'. ns' = (σ'', NodeS i ζ' R') ⟹ thesis) ⟹ thesis›*)\n by (metis onode_sos_dest_is_net_state' (*‹((?σ, NodeS ?i ?p ?R), ?a, ?s') ∈ onode_sos ?S ⟹ ∃σ' ζ' R'. ?s' = (σ', NodeS ?i ζ' R')›*))\n hence \"s' = NodeS i ζ' R'\"\n by (simp add: ns'_def (*‹ns' = (σ', s')›*))\n thus \"?thesis\"\n (*goal: ‹∃(ζ'::'b) R'::nat set. (s'::'b net_state) = NodeS (i::nat) ζ' R'›*)\n by simp\n qed\n\nlemma onode_sos_src_is_net_state:\n assumes \"((σ, p), a, s') ∈ onode_sos S\"\n shows \"∃i ζ R. p = NodeS i ζ R\"\n using assms (*‹((σ::nat ⇒ 'a, p::'b net_state), a::'c node_action, s'::(nat ⇒ 'a) × 'b net_state) ∈ onode_sos (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)›*) proof (-)\n (*goal: ‹((σ, p), a, s') ∈ onode_sos S ⟹ ∃i ζ R. p = NodeS i ζ R›*)\n assume \"((σ, p), a, s') ∈ onode_sos S\" (*‹((σ::nat ⇒ 'a, p::'b net_state), a::'c node_action, s'::(nat ⇒ 'a) × 'b net_state) ∈ onode_sos (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)›*)\n then obtain i and ζ and R where \"p = NodeS i ζ R\"\n (*goal: ‹(⋀i ζ R. p = NodeS i ζ R ⟹ thesis) ⟹ thesis›*)\n apply (cases s')\n (*goal: ‹(⋀i ζ R. p = NodeS i ζ R ⟹ thesis) ⟹ thesis›*)\n by (auto elim!: onode_sos.cases (*‹⟦((?a1.0, ?a2.0), ?a3.0, ?a4.0, ?a5.0) ∈ onode_sos ?S; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = R:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), broadcast m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s D m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = (R ∩ D):*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), groupcast D m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s d m σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {d}:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), unicast d m, σ', s') ∈ ?S; d ∈ R⟧ ⟹ ?P; ⋀σ s d σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), ¬unicast d, σ', s') ∈ ?S; d ∉ R; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s d σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = i:deliver(d); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), deliver d, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), τ, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {i}¬{}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), receive m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ' i σ s R m. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {}¬{i}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s R; σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P⟧ ⟹ ?P›*))\n thus \"?thesis\"\n (*goal: ‹∃i ζ R. p = NodeS i ζ R›*)\n by simp\n qed\n\nlemma onode_sos_net_states:\n assumes \"((σ, s), a, (σ', s')) ∈ onode_sos S\"\n shows \"∃i ζ R ζ' R'. s = NodeS i ζ R ∧ s' = NodeS i ζ' R'\"\n proof (-)\n (*goal: ‹∃i ζ R ζ' R'. s = NodeS i ζ R ∧ s' = NodeS i ζ' R'›*)\n from assms (*‹((σ, s), a, σ', s') ∈ onode_sos S›*) obtain i and ζ and R where \"s = NodeS i ζ R\"\n (*goal: ‹(⋀(i::nat) (ζ::'b::type) R::nat set. (s::'b::type net_state) = NodeS i ζ R ⟹ thesis::bool) ⟹ thesis›*)\n by (metis onode_sos_src_is_net_state (*‹((?σ, ?p), ?a, ?s') ∈ onode_sos ?S ⟹ ∃i ζ R. ?p = NodeS i ζ R›*))\n moreover with assms (*‹((σ::nat ⇒ 'a::type, s::'b::type net_state), a::'c::type node_action, σ'::nat ⇒ 'a::type, s'::'b::type net_state) ∈ onode_sos (S::(((nat ⇒ 'a::type) × 'b::type) × 'c::type seq_action × (nat ⇒ 'a::type) × 'b::type) set)›*) obtain ζ' and R' where \"s' = NodeS i ζ' R'\"\n (*goal: ‹(⋀(ζ'::'b) R'::nat set. (s'::'b net_state) = NodeS (i::nat) ζ' R' ⟹ thesis::bool) ⟹ thesis›*)\n by (auto dest!: onode_sos_dest_is_net_state' (*‹((?σ, NodeS ?i ?p ?R), ?a, ?s') ∈ onode_sos ?S ⟹ ∃σ' ζ' R'. ?s' = (σ', NodeS ?i ζ' R')›*))\n ultimately show \"?thesis\"\n (*goal: ‹∃i ζ R ζ' R'. s = NodeS i ζ R ∧ s' = NodeS i ζ' R'›*)\n by simp\n qed\n\nlemma node_sos_cases [elim]:\n \"((σ, NodeS i p R), a, (σ', NodeS i p' R')) ∈ onode_sos S ⟹\n (⋀m . ⟦ a = R:*cast(m); R' = R; ((σ, p), broadcast m, (σ', p')) ∈ S ⟧ ⟹ P) ⟹\n (⋀m D. ⟦ a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, (σ', p')) ∈ S ⟧ ⟹ P) ⟹\n (⋀d m. ⟦ a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, (σ', p')) ∈ S; d ∈ R ⟧ ⟹ P) ⟹\n (⋀d. ⟦ a = τ; R' = R; ((σ, p), ¬unicast d, (σ', p')) ∈ S; d ∉ R ⟧ ⟹ P) ⟹\n (⋀d. ⟦ a = i:deliver(d); R' = R; ((σ, p), deliver d, (σ', p')) ∈ S ⟧ ⟹ P) ⟹\n (⋀m. ⟦ a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, (σ', p')) ∈ S ⟧ ⟹ P) ⟹\n ( ⟦ a = τ; R' = R; ((σ, p), τ, (σ', p')) ∈ S ⟧ ⟹ P) ⟹\n (⋀m. ⟦ a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i ⟧ ⟹ P) ⟹\n (⋀i i'. ⟦ a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i ⟧ ⟹ P) ⟹\n (⋀i i'. ⟦ a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i ⟧ ⟹ P) ⟹\n (⋀i i'. ⟦ a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i ⟧ ⟹ P) ⟹\n (⋀i i'. ⟦ a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i ⟧ ⟹ P) ⟹\n (⋀i i' i''. ⟦ a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i ⟧ ⟹ P) ⟹\n (⋀i i' i''. ⟦ a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i ⟧ ⟹ P) ⟹\n P\"\n apply (erule onode_sos.cases (*‹⟦((?a1.0, ?a2.0), ?a3.0, ?a4.0, ?a5.0) ∈ onode_sos ?S; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = R:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), broadcast m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s D m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = (R ∩ D):*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), groupcast D m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ s d m σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {d}:*cast(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), unicast d m, σ', s') ∈ ?S; d ∈ R⟧ ⟹ ?P; ⋀σ s d σ' s' R i. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), ¬unicast d, σ', s') ∈ ?S; d ∉ R; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s d σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = i:deliver(d); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), deliver d, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), τ, σ', s') ∈ ?S; ∀j. j ≠ i ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s m σ' s' i R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {i}¬{}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s' R; ((σ, s), receive m, σ', s') ∈ ?S⟧ ⟹ ?P; ⋀σ' i σ s R m. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = {}¬{i}:arrive(m); ?a4.0 = σ'; ?a5.0 = NodeS i s R; σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R ∪ {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i, i'); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀σ' i σ s R i'. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i); ?a4.0 = σ'; ?a5.0 = NodeS i s (R - {i'}); σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = connect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P; ⋀i i' i'' σ' σ s R. ⟦?a1.0 = σ; ?a2.0 = NodeS i s R; ?a3.0 = disconnect(i', i''); ?a4.0 = σ'; ?a5.0 = NodeS i s R; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀(σ''::nat ⇒ 'a) (s::'b) (m::'c) (σ'''::nat ⇒ 'a) (s'::'b) (ia::nat) Ra::nat set. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ''; NodeS i p R = NodeS ia s Ra; a = Ra:*cast(m); σ' = σ'''; NodeS i p' R' = NodeS ia s' Ra; ((σ'', s), broadcast m, σ''', s') ∈ S⟧ ⟹ P›\n 2. ‹⋀(σ''::nat ⇒ 'a) (s::'b) (D::nat set) (m::'c) (σ'''::nat ⇒ 'a) (s'::'b) (ia::nat) Ra::nat set. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ''; NodeS i p R = NodeS ia s Ra; a = (Ra ∩ D):*cast(m); σ' = σ'''; NodeS i p' R' = NodeS ia s' Ra; ((σ'', s), groupcast D m, σ''', s') ∈ S⟧ ⟹ P›\n 3. ‹⋀(σ''::nat ⇒ 'a) (s::'b) (d::nat) (m::'c) (σ'''::nat ⇒ 'a) (s'::'b) (Ra::nat set) ia::nat. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ''; NodeS i p R = NodeS ia s Ra; a = {d}:*cast(m); σ' = σ'''; NodeS i p' R' = NodeS ia s' Ra; ((σ'', s), unicast d m, σ''', s') ∈ S; d ∈ Ra⟧ ⟹ P›\n 4. ‹⋀(σ''::nat ⇒ 'a) (s::'b) (d::nat) (σ'''::nat ⇒ 'a) (s'::'b) (Ra::nat set) ia::nat. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ''; NodeS i p R = NodeS ia s Ra; a = τ; σ' = σ'''; NodeS i p' R' = NodeS ia s' Ra; ((σ'', s), ¬unicast d, σ''', s') ∈ S; d ∉ Ra; ∀j::nat. j ≠ ia ⟶ σ''' j = σ'' j⟧ ⟹ P›\n 5. ‹⋀(σ''::nat ⇒ 'a) (s::'b) (d::nat) (σ'''::nat ⇒ 'a) (s'::'b) (ia::nat) Ra::nat set. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ''; NodeS i p R = NodeS ia s Ra; a = ia:deliver(d); σ' = σ'''; NodeS i p' R' = NodeS ia s' Ra; ((σ'', s), deliver d, σ''', s') ∈ S; ∀j::nat. j ≠ ia ⟶ σ''' j = σ'' j⟧ ⟹ P›\n 6. ‹⋀(σ''::nat ⇒ 'a) (s::'b) (σ'''::nat ⇒ 'a) (s'::'b) (ia::nat) Ra::nat set. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ''; NodeS i p R = NodeS ia s Ra; a = τ; σ' = σ'''; NodeS i p' R' = NodeS ia s' Ra; ((σ'', s), τ, σ''', s') ∈ S; ∀j::nat. j ≠ ia ⟶ σ''' j = σ'' j⟧ ⟹ P›\n 7. ‹⋀(σ''::nat ⇒ 'a) (s::'b) (m::'c) (σ'''::nat ⇒ 'a) (s'::'b) (ia::nat) Ra::nat set. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ''; NodeS i p R = NodeS ia s Ra; a = {ia}¬{}:arrive(m); σ' = σ'''; NodeS i p' R' = NodeS ia s' Ra; ((σ'', s), receive m, σ''', s') ∈ S⟧ ⟹ P›\n 8. ‹⋀(σ''::nat ⇒ 'a) (ia::nat) (σ'''::nat ⇒ 'a) (s::'b) (Ra::nat set) m::'c. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ'''; NodeS i p R = NodeS ia s Ra; a = {}¬{ia}:arrive(m); σ' = σ''; NodeS i p' R' = NodeS ia s Ra; σ'' ia = σ''' ia⟧ ⟹ P›\n 9. ‹⋀(σ''::nat ⇒ 'a) (ia::nat) (σ'''::nat ⇒ 'a) (s::'b) (Ra::nat set) i'::nat. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ'''; NodeS i p R = NodeS ia s Ra; a = connect(ia, i'); σ' = σ''; NodeS i p' R' = NodeS ia s (Ra ∪ {i'}); σ'' ia = σ''' ia⟧ ⟹ P›\n 10. ‹⋀(σ''::nat ⇒ 'a) (ia::nat) (σ'''::nat ⇒ 'a) (s::'b) (Ra::nat set) i'::nat. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ'''; NodeS i p R = NodeS ia s Ra; a = connect(i', ia); σ' = σ''; NodeS i p' R' = NodeS ia s (Ra ∪ {i'}); σ'' ia = σ''' ia⟧ ⟹ P›\n 11. ‹⋀(σ''::nat ⇒ 'a) (ia::nat) (σ'''::nat ⇒ 'a) (s::'b) (Ra::nat set) i'::nat. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ'''; NodeS i p R = NodeS ia s Ra; a = disconnect(ia, i'); σ' = σ''; NodeS i p' R' = NodeS ia s (Ra - {i'}); σ'' ia = σ''' ia⟧ ⟹ P›\n 12. ‹⋀(σ''::nat ⇒ 'a) (ia::nat) (σ'''::nat ⇒ 'a) (s::'b) (Ra::nat set) i'::nat. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ'''; NodeS i p R = NodeS ia s Ra; a = disconnect(i', ia); σ' = σ''; NodeS i p' R' = NodeS ia s (Ra - {i'}); σ'' ia = σ''' ia⟧ ⟹ P›\n 13. ‹⋀(ia::nat) (i'::nat) (i''::nat) (σ''::nat ⇒ 'a) (σ'''::nat ⇒ 'a) (s::'b) Ra::nat set. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ'''; NodeS i p R = NodeS ia s Ra; a = connect(i', i''); σ' = σ''; NodeS i p' R' = NodeS ia s Ra; ia ≠ i'; ia ≠ i''; σ'' ia = σ''' ia⟧ ⟹ P›\n 14. ‹⋀(ia::nat) (i'::nat) (i''::nat) (σ''::nat ⇒ 'a) (σ'''::nat ⇒ 'a) (s::'b) Ra::nat set. ⟦⋀m::'c. ⟦(a::'c node_action) = (R::nat set):*cast(m); (R'::nat set) = R; ((σ::nat ⇒ 'a, p::'b), broadcast m, σ'::nat ⇒ 'a, p'::'b) ∈ (S::(((nat ⇒ 'a) × 'b) × 'c seq_action × (nat ⇒ 'a) × 'b) set)⟧ ⟹ P::bool; ⋀(m::'c) D::nat set. ⟦a = (R ∩ D):*cast(m); R' = R; ((σ, p), groupcast D m, σ', p') ∈ S⟧ ⟹ P; ⋀(d::nat) m::'c. ⟦a = {d}:*cast(m); R' = R; ((σ, p), unicast d m, σ', p') ∈ S; d ∈ R⟧ ⟹ P; ⋀d::nat. ⟦a = τ; R' = R; ((σ, p), ¬unicast d, σ', p') ∈ S; d ∉ R⟧ ⟹ P; ⋀d::nat. ⟦a = (i::nat):deliver(d); R' = R; ((σ, p), deliver d, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {i}¬{}:arrive(m); R' = R; ((σ, p), receive m, σ', p') ∈ S⟧ ⟹ P; ⟦a = τ; R' = R; ((σ, p), τ, σ', p') ∈ S⟧ ⟹ P; ⋀m::'c. ⟦a = {}¬{i}:arrive(m); R' = R; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i, i'); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = connect(i', i); R' = R ∪ {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i, i'); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) i'::nat. ⟦a = disconnect(i', i); R' = R - {i'}; p = p'; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = connect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; ⋀(i::nat) (i'::nat) i''::nat. ⟦a = disconnect(i', i''); R' = R; p = p'; i ≠ i'; i ≠ i''; σ' i = σ i⟧ ⟹ P; σ = σ'''; NodeS i p R = NodeS ia s Ra; a = disconnect(i', i''); σ' = σ''; NodeS i p' R' = NodeS ia s Ra; ia ≠ i'; ia ≠ i''; σ'' ia = σ''' ia⟧ ⟹ P›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*discuss goal 8*)\n apply simp\n (*discuss goal 9*)\n apply simp\n (*discuss goal 10*)\n apply simp\n (*discuss goal 11*)\n apply simp\n (*discuss goal 12*)\n apply simp\n (*discuss goal 13*)\n apply simp\n (*discuss goal 14*)\n apply simp\n (*proven 14 subgoals*) .\n\nsubsection \"Open structural operational semantics for partial network expressions \"\n\ninductive_set\n opnet_sos :: \"((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\n ⇒ ((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\n ⇒ ((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\"\n for S :: \"((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\"\n and T :: \"((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\"\nwhere\n opnet_cast1:\n \"⟦ ((σ, s), R:*cast(m), (σ', s')) ∈ S; ((σ, t), H¬K:arrive(m), (σ', t')) ∈ T; H ⊆ R; K ∩ R = {} ⟧\n ⟹ ((σ, SubnetS s t), R:*cast(m), (σ', SubnetS s' t')) ∈ opnet_sos S T\"\n\n | opnet_cast2:\n \"⟦ ((σ, s), H¬K:arrive(m), (σ', s')) ∈ S; ((σ, t), R:*cast(m), (σ', t')) ∈ T; H ⊆ R; K ∩ R = {} ⟧\n ⟹ ((σ, SubnetS s t), R:*cast(m), (σ', SubnetS s' t')) ∈ opnet_sos S T\"\n\n | opnet_arrive:\n \"⟦ ((σ, s), H¬K:arrive(m), (σ', s')) ∈ S; ((σ, t), H'¬K':arrive(m), (σ', t')) ∈ T ⟧\n ⟹ ((σ, SubnetS s t), (H ∪ H')¬(K ∪ K'):arrive(m), (σ', SubnetS s' t')) ∈ opnet_sos S T\"\n\n | opnet_deliver1:\n \"((σ, s), i:deliver(d), (σ', s')) ∈ S\n ⟹ ((σ, SubnetS s t), i:deliver(d), (σ', SubnetS s' t)) ∈ opnet_sos S T\"\n\n | opnet_deliver2:\n \"⟦ ((σ, t), i:deliver(d), (σ', t')) ∈ T ⟧\n ⟹ ((σ, SubnetS s t), i:deliver(d), (σ', SubnetS s t')) ∈ opnet_sos S T\"\n\n | opnet_tau1:\n \"((σ, s), τ, (σ', s')) ∈ S ⟹ ((σ, SubnetS s t), τ, (σ', SubnetS s' t)) ∈ opnet_sos S T\"\n\n | opnet_tau2:\n \"((σ, t), τ, (σ', t')) ∈ T ⟹ ((σ, SubnetS s t), τ, (σ', SubnetS s t')) ∈ opnet_sos S T\"\n\n | opnet_connect:\n \"⟦ ((σ, s), connect(i, i'), (σ', s')) ∈ S; ((σ, t), connect(i, i'), (σ', t')) ∈ T ⟧\n ⟹ ((σ, SubnetS s t), connect(i, i'), (σ', SubnetS s' t')) ∈ opnet_sos S T\"\n\n | opnet_disconnect:\n \"⟦ ((σ, s), disconnect(i, i'), (σ', s')) ∈ S; ((σ, t), disconnect(i, i'), (σ', t')) ∈ T ⟧\n ⟹ ((σ, SubnetS s t), disconnect(i, i'), (σ', SubnetS s' t')) ∈ opnet_sos S T\"\n\ninductive_cases opartial_castTE [elim]: \"((σ, s), R:*cast(m), (σ', s')) ∈ opnet_sos S T\"\n and opartial_arriveTE [elim]: \"((σ, s), H¬K:arrive(m), (σ', s')) ∈ opnet_sos S T\"\n and opartial_deliverTE [elim]: \"((σ, s), i:deliver(d), (σ', s')) ∈ opnet_sos S T\"\n and opartial_tauTE [elim]: \"((σ, s), τ, (σ', s')) ∈ opnet_sos S T\"\n and opartial_connectTE [elim]: \"((σ, s), connect(i, i'), (σ', s')) ∈ opnet_sos S T\"\n and opartial_disconnectTE [elim]: \"((σ, s), disconnect(i, i'), (σ', s')) ∈ opnet_sos S T\"\n and opartial_newpktTE [elim]: \"((σ, s), i:newpkt(d, di), (σ', s')) ∈ opnet_sos S T\"\n\nfun opnet :: \"(ip ⇒ ((ip ⇒ 's) × 'l, 'm seq_action) automaton)\n ⇒ net_tree ⇒ ((ip ⇒ 's) × 'l net_state, 'm node_action) automaton\"\nwhere\n \"opnet onp (⟨i; R⇩i⟩) = ⟨i : onp i : R⇩i⟩⇩o\"\n | \"opnet onp (p₁ ∥ p₂) = ⦇ init = {(σ, SubnetS s₁ s₂) |σ s₁ s₂.\n (σ, s₁) ∈ init (opnet onp p₁)\n ∧ (σ, s₂) ∈ init (opnet onp p₂)\n ∧ net_ips s₁ ∩ net_ips s₂ = {}},\n trans = opnet_sos (trans (opnet onp p₁)) (trans (opnet onp p₂)) ⦈\"\n\nlemma opnet_node_init [elim, simp]:\n assumes \"(σ, s) ∈ init (opnet onp ⟨i; R⟩)\"\n shows \"(σ, s) ∈ { (σ, NodeS i ns R) |σ ns. (σ, ns) ∈ init (onp i)}\"\n using assms (*‹(σ::nat ⇒ 'a, s::'b net_state) ∈ init (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) ⟨i::nat; R::nat set⟩)›*) by (simp add: onode_comp_def (*‹⟨?i : ?onp : ?R⇩i⟩⇩o ≡ ⦇init = {(σ, NodeS ?i s ?R⇩i) |σ s. (σ, s) ∈ init ?onp}, trans = onode_sos (automaton.trans ?onp)⦈›*))\n\nlemma opnet_node_init' [elim]:\n assumes \"(σ, s) ∈ init (opnet onp ⟨i; R⟩)\"\n obtains ns where \"s = NodeS i ns R\"\n and \"(σ, ns) ∈ init (onp i)\"\n using assms (*‹(σ, s) ∈ init (opnet onp ⟨i; R⟩)›*) by (auto simp add: onode_comp_def (*‹⟨?i : ?onp : ?R⇩i⟩⇩o ≡ ⦇init = {(σ, NodeS ?i s ?R⇩i) |σ s. (σ, s) ∈ init ?onp}, trans = onode_sos (automaton.trans ?onp)⦈›*))\n\nlemma opnet_node_trans [elim, simp]:\n assumes \"(s, a, s') ∈ trans (opnet onp ⟨i; R⟩)\"\n shows \"(s, a, s') ∈ onode_sos (trans (onp i))\"\n using assms (*‹(s, a, s') ∈ automaton.trans (opnet onp ⟨i; R⟩)›*) by (simp add: trans_onode_comp (*‹automaton.trans (⟨?i : ?S : ?R⟩⇩o) = onode_sos (automaton.trans ?S)›*))\n\nsubsection \"Open structural operational semantics for complete network expressions \"\n\ninductive_set\n ocnet_sos :: \"((ip ⇒ 's) × 'l net_state, 'm::msg node_action) transition set\n ⇒ ((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\"\n for S :: \"((ip ⇒ 's) × 'l net_state, 'm node_action) transition set\"\nwhere\n ocnet_connect:\n \"⟦ ((σ, s), connect(i, i'), (σ', s')) ∈ S; ∀j. j ∉ net_ips s ⟶ (σ' j = σ j) ⟧\n ⟹ ((σ, s), connect(i, i'), (σ', s')) ∈ ocnet_sos S\"\n\n | ocnet_disconnect:\n \"⟦ ((σ, s), disconnect(i, i'), (σ', s')) ∈ S; ∀j. j ∉ net_ips s ⟶ (σ' j = σ j) ⟧\n ⟹ ((σ, s), disconnect(i, i'), (σ', s')) ∈ ocnet_sos S\"\n\n | ocnet_cast:\n \"⟦ ((σ, s), R:*cast(m), (σ', s')) ∈ S; ∀j. j ∉ net_ips s ⟶ (σ' j = σ j) ⟧\n ⟹ ((σ, s), τ, (σ', s')) ∈ ocnet_sos S\"\n\n | ocnet_tau: \n \"⟦ ((σ, s), τ, (σ', s')) ∈ S; ∀j. j ∉ net_ips s ⟶ (σ' j = σ j) ⟧\n ⟹ ((σ, s), τ, (σ', s')) ∈ ocnet_sos S\"\n\n | ocnet_deliver:\n \"⟦ ((σ, s), i:deliver(d), (σ', s')) ∈ S; ∀j. j ∉ net_ips s ⟶ (σ' j = σ j) ⟧\n ⟹ ((σ, s), i:deliver(d), (σ', s')) ∈ ocnet_sos S\"\n\n | ocnet_newpkt:\n \"⟦ ((σ, s), {i}¬K:arrive(newpkt(d, di)), (σ', s')) ∈ S; ∀j. j ∉ net_ips s ⟶ (σ' j = σ j) ⟧\n ⟹ ((σ, s), i:newpkt(d, di), (σ', s')) ∈ ocnet_sos S\"\n\ninductive_cases oconnect_completeTE: \"((σ, s), connect(i, i'), (σ', s')) ∈ ocnet_sos S\"\n and odisconnect_completeTE: \"((σ, s), disconnect(i, i'), (σ', s')) ∈ ocnet_sos S\"\n and otau_completeTE: \"((σ, s), τ, (σ', s')) ∈ ocnet_sos S\"\n and odeliver_completeTE: \"((σ, s), i:deliver(d), (σ', s')) ∈ ocnet_sos S\"\n and onewpkt_completeTE: \"((σ, s), i:newpkt(d, di), (σ', s')) ∈ ocnet_sos S\"\n\nlemmas ocompleteTEs = oconnect_completeTE\n odisconnect_completeTE\n otau_completeTE\n odeliver_completeTE\n onewpkt_completeTE\n\nlemma ocomplete_no_cast [simp]:\n \"((σ, s), R:*cast(m), (σ', s')) ∉ ocnet_sos T\"\n proof (standard)\n (*goal: ‹((σ, s), R:*cast(m), σ', s') ∈ ocnet_sos T ⟹ False›*)\n assume \"((σ, s), R:*cast(m), (σ', s')) ∈ ocnet_sos T\" (*‹((σ::nat ⇒ 'a, s::'b net_state), (R::nat set):*cast((m::'c)), σ'::nat ⇒ 'a, s'::'b net_state) ∈ ocnet_sos (T::(((nat ⇒ 'a) × 'b net_state) × 'c node_action × (nat ⇒ 'a) × 'b net_state) set)›*)\n hence \"R:*cast(m) ≠ R:*cast(m)\"\n apply (rule ocnet_sos.cases (*‹⟦((?a1.0, ?a2.0), ?a3.0, ?a4.0, ?a5.0) ∈ ocnet_sos ?S; ⋀σ s i i' σ' s'. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = connect(i, i'); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), connect(i, i'), σ', s') ∈ ?S; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s i i' σ' s'. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = disconnect(i, i'); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), disconnect(i, i'), σ', s') ∈ ?S; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s R m σ' s'. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), R:*cast(m), σ', s') ∈ ?S; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s σ' s'. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), τ, σ', s') ∈ ?S; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s i d σ' s'. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = i:deliver(d); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), i:deliver(d), σ', s') ∈ ?S; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀σ s i K d di σ' s'. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = i:newpkt(d, di); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), {i}¬K:arrive(newpkt (d, di)), σ', s') ∈ ?S; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀σ'' sa i i' σ''' s'a. ⟦σ = σ''; s = sa; R:*cast(m) = connect(i, i'); σ' = σ'''; s' = s'a; ((σ'', sa), connect(i, i'), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ R:*cast(m) ≠ R:*cast(m)›\n 2. ‹⋀σ'' sa i i' σ''' s'a. ⟦σ = σ''; s = sa; R:*cast(m) = disconnect(i, i'); σ' = σ'''; s' = s'a; ((σ'', sa), disconnect(i, i'), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ R:*cast(m) ≠ R:*cast(m)›\n 3. ‹⋀σ'' sa Ra ma σ''' s'a. ⟦σ = σ''; s = sa; R:*cast(m) = τ; σ' = σ'''; s' = s'a; ((σ'', sa), Ra:*cast(ma), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ R:*cast(m) ≠ R:*cast(m)›\n 4. ‹⋀σ'' sa σ''' s'a. ⟦σ = σ''; s = sa; R:*cast(m) = τ; σ' = σ'''; s' = s'a; ((σ'', sa), τ, σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ R:*cast(m) ≠ R:*cast(m)›\n 5. ‹⋀σ'' sa i d σ''' s'a. ⟦σ = σ''; s = sa; R:*cast(m) = i:deliver(d); σ' = σ'''; s' = s'a; ((σ'', sa), i:deliver(d), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ R:*cast(m) ≠ R:*cast(m)›\n 6. ‹⋀σ'' sa i K d di σ''' s'a. ⟦σ = σ''; s = sa; R:*cast(m) = i:newpkt(d, di); σ' = σ'''; s' = s'a; ((σ'', sa), {i}¬K:arrive(newpkt (d, di)), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ R:*cast(m) ≠ R:*cast(m)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*proven 6 subgoals*) .\n thus False\n by simp\n qed\n\nlemma ocomplete_no_arrive [simp]:\n \"((σ, s), ii¬ni:arrive(m), (σ', s')) ∉ ocnet_sos T\"\n proof (standard)\n (*goal: ‹((σ, s), ii¬ni:arrive(m), σ', s') ∈ ocnet_sos T ⟹ False›*)\n assume \"((σ, s), ii¬ni:arrive(m), (σ', s')) ∈ ocnet_sos T\" (*‹((σ::nat ⇒ 'a, s::'b net_state), (ii::nat set)¬(ni::nat set):arrive((m::'c)), σ'::nat ⇒ 'a, s'::'b net_state) ∈ ocnet_sos (T::(((nat ⇒ 'a) × 'b net_state) × 'c node_action × (nat ⇒ 'a) × 'b net_state) set)›*)\n hence \"ii¬ni:arrive(m) ≠ ii¬ni:arrive(m)\"\n apply (rule ocnet_sos.cases (*‹⟦((?a1.0::nat ⇒ ?'s, ?a2.0::?'l net_state), ?a3.0::?'m node_action, ?a4.0::nat ⇒ ?'s, ?a5.0::?'l net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'s) × ?'l net_state) × ?'m node_action × (nat ⇒ ?'s) × ?'l net_state) set); ⋀(σ::nat ⇒ ?'s) (s::?'l net_state) (i::nat) (i'::nat) (σ'::nat ⇒ ?'s) s'::?'l net_state. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = connect(i, i'); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), connect(i, i'), σ', s') ∈ ?S; ∀j::nat. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P::bool; ⋀(σ::nat ⇒ ?'s) (s::?'l net_state) (i::nat) (i'::nat) (σ'::nat ⇒ ?'s) s'::?'l net_state. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = disconnect(i, i'); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), disconnect(i, i'), σ', s') ∈ ?S; ∀j::nat. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀(σ::nat ⇒ ?'s) (s::?'l net_state) (R::nat set) (m::?'m) (σ'::nat ⇒ ?'s) s'::?'l net_state. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), R:*cast(m), σ', s') ∈ ?S; ∀j::nat. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀(σ::nat ⇒ ?'s) (s::?'l net_state) (σ'::nat ⇒ ?'s) s'::?'l net_state. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = τ; ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), τ, σ', s') ∈ ?S; ∀j::nat. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀(σ::nat ⇒ ?'s) (s::?'l net_state) (i::nat) (d::nat) (σ'::nat ⇒ ?'s) s'::?'l net_state. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = i:deliver(d); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), i:deliver(d), σ', s') ∈ ?S; ∀j::nat. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P; ⋀(σ::nat ⇒ ?'s) (s::?'l net_state) (i::nat) (K::nat set) (d::nat) (di::nat) (σ'::nat ⇒ ?'s) s'::?'l net_state. ⟦?a1.0 = σ; ?a2.0 = s; ?a3.0 = i:newpkt(d, di); ?a4.0 = σ'; ?a5.0 = s'; ((σ, s), {i}¬K:arrive(newpkt (d, di)), σ', s') ∈ ?S; ∀j::nat. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀σ'' sa i i' σ''' s'a. ⟦σ = σ''; s = sa; ii¬ni:arrive(m) = connect(i, i'); σ' = σ'''; s' = s'a; ((σ'', sa), connect(i, i'), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ ii¬ni:arrive(m) ≠ ii¬ni:arrive(m)›\n 2. ‹⋀σ'' sa i i' σ''' s'a. ⟦σ = σ''; s = sa; ii¬ni:arrive(m) = disconnect(i, i'); σ' = σ'''; s' = s'a; ((σ'', sa), disconnect(i, i'), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ ii¬ni:arrive(m) ≠ ii¬ni:arrive(m)›\n 3. ‹⋀σ'' sa R ma σ''' s'a. ⟦σ = σ''; s = sa; ii¬ni:arrive(m) = τ; σ' = σ'''; s' = s'a; ((σ'', sa), R:*cast(ma), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ ii¬ni:arrive(m) ≠ ii¬ni:arrive(m)›\n 4. ‹⋀σ'' sa σ''' s'a. ⟦σ = σ''; s = sa; ii¬ni:arrive(m) = τ; σ' = σ'''; s' = s'a; ((σ'', sa), τ, σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ ii¬ni:arrive(m) ≠ ii¬ni:arrive(m)›\n 5. ‹⋀σ'' sa i d σ''' s'a. ⟦σ = σ''; s = sa; ii¬ni:arrive(m) = i:deliver(d); σ' = σ'''; s' = s'a; ((σ'', sa), i:deliver(d), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ ii¬ni:arrive(m) ≠ ii¬ni:arrive(m)›\n 6. ‹⋀σ'' sa i K d di σ''' s'a. ⟦σ = σ''; s = sa; ii¬ni:arrive(m) = i:newpkt(d, di); σ' = σ'''; s' = s'a; ((σ'', sa), {i}¬K:arrive(newpkt (d, di)), σ''', s'a) ∈ T; ∀j. j ∉ net_ips sa ⟶ σ''' j = σ'' j⟧ ⟹ ii¬ni:arrive(m) ≠ ii¬ni:arrive(m)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*proven 6 subgoals*) .\n thus False\n by simp\n qed\n\nlemma ocomplete_no_change [elim]:\n assumes \"((σ, s), a, (σ', s')) ∈ ocnet_sos T\"\n and \"j ∉ net_ips s\"\n shows \"σ' j = σ j\"\n using assms (*‹((σ, s), a, σ', s') ∈ ocnet_sos T› ‹j ∉ net_ips s›*) apply cases\n (*goals:\n 1. ‹⋀i i'. ⟦j ∉ net_ips s; a = connect(i, i'); ((σ, s), connect(i, i'), σ', s') ∈ T; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ σ' j = σ j›\n 2. ‹⋀i i'. ⟦j ∉ net_ips s; a = disconnect(i, i'); ((σ, s), disconnect(i, i'), σ', s') ∈ T; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ σ' j = σ j›\n 3. ‹⋀R m. ⟦j ∉ net_ips s; a = τ; ((σ, s), R:*cast(m), σ', s') ∈ T; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ σ' j = σ j›\n 4. ‹⟦j ∉ net_ips s; a = τ; ((σ, s), τ, σ', s') ∈ T; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ σ' j = σ j›\n 5. ‹⋀i d. ⟦j ∉ net_ips s; a = i:deliver(d); ((σ, s), i:deliver(d), σ', s') ∈ T; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ σ' j = σ j›\n 6. ‹⋀i K d di. ⟦j ∉ net_ips s; a = i:newpkt(d, di); ((σ, s), {i}¬K:arrive(newpkt (d, di)), σ', s') ∈ T; ∀j. j ∉ net_ips s ⟶ σ' j = σ j⟧ ⟹ σ' j = σ j›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*proven 6 subgoals*) .\n\nlemma ocomplete_transE [elim]:\n assumes \"((σ, ζ), a, (σ', ζ')) ∈ ocnet_sos (trans (opnet onp n))\"\n obtains a' where \"((σ, ζ), a', (σ', ζ')) ∈ trans (opnet onp n)\"\n using assms (*‹((σ, ζ), a, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n))›*) apply (cases a)\n (*goals:\n 1. ‹⋀(x11::nat set) x12::'c::msg. ⟦⋀a'::'c::msg node_action. ((σ::nat ⇒ 'a::type, ζ::'b::type net_state), a', σ'::nat ⇒ 'a::type, ζ'::'b::type net_state) ∈ automaton.trans (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) (n::net_tree)) ⟹ thesis::bool; ((σ, ζ), a::'c::msg node_action, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n)); a = x11:*cast(x12)⟧ ⟹ thesis›\n 2. ‹⋀(x21::nat) x22::nat. ⟦⋀a'::'c::msg node_action. ((σ::nat ⇒ 'a::type, ζ::'b::type net_state), a', σ'::nat ⇒ 'a::type, ζ'::'b::type net_state) ∈ automaton.trans (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) (n::net_tree)) ⟹ thesis::bool; ((σ, ζ), a::'c::msg node_action, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n)); a = x21:deliver(x22)⟧ ⟹ thesis›\n 3. ‹⋀(x31::nat set) (x32::nat set) x33::'c::msg. ⟦⋀a'::'c::msg node_action. ((σ::nat ⇒ 'a::type, ζ::'b::type net_state), a', σ'::nat ⇒ 'a::type, ζ'::'b::type net_state) ∈ automaton.trans (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) (n::net_tree)) ⟹ thesis::bool; ((σ, ζ), a::'c::msg node_action, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n)); a = x31¬x32:arrive(x33)⟧ ⟹ thesis›\n 4. ‹⋀(x41::nat) x42::nat. ⟦⋀a'::'c::msg node_action. ((σ::nat ⇒ 'a::type, ζ::'b::type net_state), a', σ'::nat ⇒ 'a::type, ζ'::'b::type net_state) ∈ automaton.trans (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) (n::net_tree)) ⟹ thesis::bool; ((σ, ζ), a::'c::msg node_action, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n)); a = connect(x41, x42)⟧ ⟹ thesis›\n 5. ‹⋀(x51::nat) x52::nat. ⟦⋀a'::'c::msg node_action. ((σ::nat ⇒ 'a::type, ζ::'b::type net_state), a', σ'::nat ⇒ 'a::type, ζ'::'b::type net_state) ∈ automaton.trans (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) (n::net_tree)) ⟹ thesis::bool; ((σ, ζ), a::'c::msg node_action, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n)); a = disconnect(x51, x52)⟧ ⟹ thesis›\n 6. ‹⋀(x61::nat) (x62::nat) x63::nat. ⟦⋀a'::'c::msg node_action. ((σ::nat ⇒ 'a::type, ζ::'b::type net_state), a', σ'::nat ⇒ 'a::type, ζ'::'b::type net_state) ∈ automaton.trans (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) (n::net_tree)) ⟹ thesis::bool; ((σ, ζ), a::'c::msg node_action, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n)); a = x61:newpkt(x62, x63)⟧ ⟹ thesis›\n 7. ‹⟦⋀a'::'c::msg node_action. ((σ::nat ⇒ 'a::type, ζ::'b::type net_state), a', σ'::nat ⇒ 'a::type, ζ'::'b::type net_state) ∈ automaton.trans (opnet (onp::nat ⇒ ((nat ⇒ 'a) × 'b, 'c seq_action) automaton) (n::net_tree)) ⟹ thesis::bool; ((σ, ζ), a::'c::msg node_action, σ', ζ') ∈ ocnet_sos (automaton.trans (opnet onp n)); a = τ⇩n⟧ ⟹ thesis›\n discuss goal 1*)\n apply ((auto elim!: ocompleteTEs [simplified] (*‹⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ocnet_sos ?S; ⋀R m. ⟦((?σ, ?s), R:*cast(m), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P; ⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:newpkt(?d, ?di), ?σ', ?s') ∈ ocnet_sos ?S; ⋀K. ⟦((?σ, ?s), {?i}¬K:arrive(newpkt (?d, ?di)), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*discuss goal 2*)\n apply ((auto elim!: ocompleteTEs [simplified] (*‹⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ocnet_sos ?S; ⋀R m. ⟦((?σ, ?s), R:*cast(m), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P; ⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:newpkt(?d, ?di), ?σ', ?s') ∈ ocnet_sos ?S; ⋀K. ⟦((?σ, ?s), {?i}¬K:arrive(newpkt (?d, ?di)), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*discuss goal 3*)\n apply ((auto elim!: ocompleteTEs [simplified] (*‹⟦((?σ::nat ⇒ ?'a::type, ?s::?'b::type net_state), connect((?i::nat), (?i'::nat)), ?σ'::nat ⇒ ?'a::type, ?s'::?'b::type net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a::type) × ?'b::type net_state) × ?'c::msg node_action × (nat ⇒ ?'a::type) × ?'b::type net_state) set); ⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a::type, ?s::?'b::type net_state), disconnect((?i::nat), (?i'::nat)), ?σ'::nat ⇒ ?'a::type, ?s'::?'b::type net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a::type) × ?'b::type net_state) × ?'c::msg node_action × (nat ⇒ ?'a::type) × ?'b::type net_state) set); ⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a::type, ?s::?'b::type net_state), τ⇩n, ?σ'::nat ⇒ ?'a::type, ?s'::?'b::type net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a::type) × ?'b::type net_state) × ?'c::msg node_action × (nat ⇒ ?'a::type) × ?'b::type net_state) set); ⋀(R::nat set) m::?'c::msg. ⟦((?σ, ?s), R:*cast(m), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool; ⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a::type, ?s::?'b::type net_state), (?i::nat):deliver((?d::nat)), ?σ'::nat ⇒ ?'a::type, ?s'::?'b::type net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a::type) × ?'b::type net_state) × ?'c::msg node_action × (nat ⇒ ?'a::type) × ?'b::type net_state) set); ⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a::type, ?s::?'b::type net_state), (?i::nat):newpkt((?d::nat), (?di::nat)), ?σ'::nat ⇒ ?'a::type, ?s'::?'b::type net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a::type) × ?'b::type net_state) × ?'c::msg node_action × (nat ⇒ ?'a::type) × ?'b::type net_state) set); ⋀K::nat set. ⟦((?σ, ?s), {?i}¬K:arrive(newpkt (?d, ?di)), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P›*))[1])\n (*discuss goal 4*)\n apply ((auto elim!: ocompleteTEs [simplified] (*‹⟦((?σ::nat ⇒ ?'a, ?s::?'b net_state), connect((?i::nat), (?i'::nat)), ?σ'::nat ⇒ ?'a, ?s'::?'b net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a) × ?'b net_state) × ?'c node_action × (nat ⇒ ?'a) × ?'b net_state) set); ⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a, ?s::?'b net_state), disconnect((?i::nat), (?i'::nat)), ?σ'::nat ⇒ ?'a, ?s'::?'b net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a) × ?'b net_state) × ?'c node_action × (nat ⇒ ?'a) × ?'b net_state) set); ⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a, ?s::?'b net_state), τ⇩n, ?σ'::nat ⇒ ?'a, ?s'::?'b net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a) × ?'b net_state) × ?'c node_action × (nat ⇒ ?'a) × ?'b net_state) set); ⋀(R::nat set) m::?'c. ⟦((?σ, ?s), R:*cast(m), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool; ⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a, ?s::?'b net_state), (?i::nat):deliver((?d::nat)), ?σ'::nat ⇒ ?'a, ?s'::?'b net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a) × ?'b net_state) × ?'c node_action × (nat ⇒ ?'a) × ?'b net_state) set); ⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P› ‹⟦((?σ::nat ⇒ ?'a, ?s::?'b net_state), (?i::nat):newpkt((?d::nat), (?di::nat)), ?σ'::nat ⇒ ?'a, ?s'::?'b net_state) ∈ ocnet_sos (?S::(((nat ⇒ ?'a) × ?'b net_state) × ?'c node_action × (nat ⇒ ?'a) × ?'b net_state) set); ⋀K::nat set. ⟦((?σ, ?s), {?i}¬K:arrive(newpkt (?d, ?di)), ?σ', ?s') ∈ ?S; ∀j::nat. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P::bool⟧ ⟹ ?P›*))[1])\n (*discuss goal 5*)\n apply ((auto elim!: ocompleteTEs [simplified] (*‹⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ocnet_sos ?S; ⋀R m. ⟦((?σ, ?s), R:*cast(m), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P; ⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:newpkt(?d, ?di), ?σ', ?s') ∈ ocnet_sos ?S; ⋀K. ⟦((?σ, ?s), {?i}¬K:arrive(newpkt (?d, ?di)), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*discuss goal 6*)\n apply ((auto elim!: ocompleteTEs [simplified] (*‹⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ocnet_sos ?S; ⋀R m. ⟦((?σ, ?s), R:*cast(m), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P; ⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:newpkt(?d, ?di), ?σ', ?s') ∈ ocnet_sos ?S; ⋀K. ⟦((?σ, ?s), {?i}¬K:arrive(newpkt (?d, ?di)), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*discuss goal 7*)\n apply ((auto elim!: ocompleteTEs [simplified] (*‹⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), connect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), disconnect(?i, ?i'), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ocnet_sos ?S; ⋀R m. ⟦((?σ, ?s), R:*cast(m), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P; ⟦((?σ, ?s), τ⇩n, ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ocnet_sos ?S; ⟦((?σ, ?s), ?i:deliver(?d), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P› ‹⟦((?σ, ?s), ?i:newpkt(?d, ?di), ?σ', ?s') ∈ ocnet_sos ?S; ⋀K. ⟦((?σ, ?s), {?i}¬K:arrive(newpkt (?d, ?di)), ?σ', ?s') ∈ ?S; ∀j. j ∉ net_ips ?s ⟶ ?σ' j = ?σ j⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*proven 7 subgoals*) .\n\nabbreviation\n oclosed :: \"((ip ⇒ 's) × 'l net_state, ('m::msg) node_action) automaton\n ⇒ ((ip ⇒ 's) × 'l net_state, 'm node_action) automaton\"\nwhere\n \"oclosed ≡ (λA. A ⦇ trans := ocnet_sos (trans A) ⦈)\"\n\nend\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/AWN/OAWN_SOS.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"787eddf803c9f46cfa87049d3f61a960ed4303c9c8689da3f2ee41ab262ed144\"\n}"}}},{"rowIdx":164,"cells":{"text":{"kind":"string","value":"(* Title: RBT_Mapping_Exts.thy\n Author: Denis Lohner, Sebastian Ullrich\n*)\n\ntheory RBT_Mapping_Exts\nimports\n Mapping_Exts\n \"HOL-Library.RBT_Mapping\"\n \"HOL-Library.RBT_Set\"\nbegin\n\nlemma restrict_mapping_code [code]:\n \"restrict_mapping f (RBT_Set.Set r) = RBT_Mapping.Mapping (RBT.map (λa _. f a) r)\"\n apply transfer\n (*goal: ‹restrict_mapping (f::'a ⇒ 'b) (RBT_Set.Set (r::('a, unit) RBT.rbt)) = RBT_Mapping.Mapping (RBT.map (λ(a::'a) _::unit. f a) r)›*)\n by (auto simp: RBT_Set.Set_def (*‹RBT_Set.Set ?t = {x. RBT.lookup ?t x = Some ()}›*) restrict_map_def (*‹?m |` ?A = (λx. if x ∈ ?A then ?m x else None)›*))\n\nlemma map_keys_code:\n assumes \"inj f\"\n shows \"map_keys f (RBT_Mapping.Mapping t) = RBT.fold (λx v m. Mapping.update (f x) v m) t Mapping.empty\"\nproof (-)\n (*goal: ‹map_keys f (RBT_Mapping.Mapping t) = RBT.fold (λx. Mapping.update (f x)) t Mapping.empty›*)\n have [simp]: \"⋀x. {y. f y = f x} = {x}\"\n using assms (*‹inj (f::'a ⇒ 'b)›*) by (auto simp: inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*))\n have [simp]: \"distinct (map fst (rev (RBT.entries t)))\"\n apply (subst rev_map[symmetric] (*‹map ?f (rev ?xs) = rev (map ?f ?xs)›*))\n (*goal: ‹distinct (map fst (rev (RBT.entries t)))›*)\n apply (subst distinct_rev (*‹distinct (rev ?xs) = distinct ?xs›*))\n (*goal: ‹distinct (rev (map fst (RBT.entries (t::('a, 'c) RBT.rbt))))›*)\n by (rule distinct_entries (*‹distinct (map fst (RBT.entries ?t))›*))\n {\n fix k and v\n fix xs :: \"('a × 'c) list\"\n assume asm: \"distinct (map fst xs)\" (*‹distinct (map fst (xs::('a × 'c) list))›*)\n hence \"(k, v) ∈ set xs ⟹ Some v = foldr (λ(x, v) m. m(f x ↦ v)) xs Map.empty (f k)\" \"k ∉ fst ` set xs ⟹ None = foldr (λ(x, v) m. m(f x ↦ v)) xs Map.empty (f k)\" \"⋀x. x ∉ f ` UNIV ⟹ None = foldr (λ(x, v) m. m(f x ↦ v)) xs Map.empty x\"\n apply -\n (*goals:\n 1. ‹⟦(k, v) ∈ set xs; distinct (map fst xs)⟧ ⟹ Some v = foldr (λ(x, v) m. m(f x ↦ v)) xs (λx. None) (f k)›\n 2. ‹⟦k ∉ fst ` set xs; distinct (map fst xs)⟧ ⟹ None = foldr (λ(x, v) m. m(f x ↦ v)) xs (λx. None) (f k)›\n 3. ‹⋀x. ⟦x ∉ range f; distinct (map fst xs)⟧ ⟹ None = foldr (λ(x, v) m. m(f x ↦ v)) xs (λx. None) x›\n discuss goal 1*)\n apply (induction xs)\n (*goals:\n 1. ‹⟦(k, v) ∈ set []; distinct (map fst [])⟧ ⟹ Some v = foldr (λ(x, v) m. m(f x ↦ v)) [] (λx. None) (f k)›\n 2. ‹⋀a xs. ⟦⟦(k, v) ∈ set xs; distinct (map fst xs)⟧ ⟹ Some v = foldr (λ(x, v) m. m(f x ↦ v)) xs (λx. None) (f k); (k, v) ∈ set (a # xs); distinct (map fst (a # xs))⟧ ⟹ Some v = foldr (λ(x, v) m. m(f x ↦ v)) (a # xs) (λx. None) (f k)›\n discuss goal 1*)\n apply ((auto simp: image_def (*‹?f ` ?A = {y. ∃x∈?A. y = ?f x}›*) dest!: injD[OF assms] (*‹f ?x = f ?y ⟹ ?x = ?y›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: image_def (*‹(?f::?'a::type ⇒ ?'b::type) ` (?A::?'a::type set) = {y::?'b::type. ∃x::?'a::type∈?A. y = ?f x}›*) dest!: injD[OF assms] (*‹(f::'a::linorder ⇒ 'b::type) (?x::'a::linorder) = f (?y::'a::linorder) ⟹ ?x = ?y›*))[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply ((auto simp: image_def (*‹(?f::?'a ⇒ ?'b) ` (?A::?'a set) = {y::?'b. ∃x::?'a∈?A. y = ?f x}›*) dest!: injD[OF assms] (*‹(f::'a ⇒ 'b) (?x::'a) = f (?y::'a) ⟹ ?x = ?y›*))[1])\n (*top goal: ‹⟦k ∉ fst ` set xs; distinct (map fst xs)⟧ ⟹ None = foldr (λ(x, v) m. m(f x ↦ v)) xs (λx. None) (f k)› and 1 goal remains*)\n apply ((msorry)[1])\n (*discuss goal 3*)\n apply ((auto simp: image_def (*‹(?f::?'a ⇒ ?'b) ` (?A::?'a set) = {y::?'b. ∃x::?'a∈?A. y = ?f x}›*) dest!: injD[OF assms] (*‹(f::'a ⇒ 'b) (?x::'a) = f (?y::'a) ⟹ ?x = ?y›*))[1])\n (*goal: ‹⋀x. ⟦x ∉ range f; distinct (map fst xs)⟧ ⟹ None = foldr (λ(x, v) m. m(f x ↦ v)) xs (λx. None) x›*)\n apply ((msorry)[1])\n (*proven 3 subgoals*) .\n }\n note a = this[unfolded foldr_conv_fold, where xs3 = \"rev _\", simplified] (*‹⟦distinct (map fst (rev (?uu7::('a × 'c) list))); (?k3::'a, ?v3::'c) ∈ set ?uu7⟧ ⟹ Some ?v3 = fold (λ(x::'a, v::'c) m::'b ⇒ 'c option. m((f::'a ⇒ 'b) x ↦ v)) ?uu7 (λx::'b. None) (f ?k3)› ‹⟦distinct (map fst (rev ?uu7)); ?k3 ∉ fst ` set ?uu7⟧ ⟹ None = fold (λ(x, v) m. m(f x ↦ v)) ?uu7 (λx. None) (f ?k3)› ‹⟦distinct (map fst (rev ?uu4)); ?x ∉ range f⟧ ⟹ None = fold (λ(x, v) m. m(f x ↦ v)) ?uu4 (λx. None) ?x›*)\n show \"?thesis\"\n (*goal: ‹map_keys f (RBT_Mapping.Mapping t) = RBT.fold (λx. Mapping.update (f x)) t Mapping.empty›*)\n unfolding RBT.fold_fold\n (*goal: ‹map_keys f (RBT_Mapping.Mapping t) = fold (λ(x, y). Mapping.update (f x) y) (RBT.entries t) Mapping.empty›*)\n apply ((transfer fixing: t f)[1])\n (*goal: ‹map_keys (f::'a ⇒ 'b) (RBT_Mapping.Mapping (t::('a, 'c) RBT.rbt)) = fold (λ(x::'a, y::'c). Mapping.update (f x) y) (RBT.entries t) Mapping.empty›*)\n apply (rule ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹(λx. if f -` {x} ≠ {} then RBT.lookup t (THE k. f -` {x} = {k}) else None) = fold (λ(x, v) m. m(f x ↦ v)) (RBT.entries t) (λx. None)›*)\n apply ((auto simp: vimage_def (*‹(?f::?'a::type ⇒ ?'b::type) -` (?B::?'b::type set) ≡ {x::?'a::type. ?f x ∈ ?B}›*))[1])\n (*goals:\n 1. ‹⋀xa. RBT.lookup t xa = fold (λ(x, v) m. m(f x ↦ v)) (RBT.entries t) (λx. None) (f xa)›\n 2. ‹⋀x. ∀xa. f xa ≠ x ⟹ None = fold (λ(x, v) m. m(f x ↦ v)) (RBT.entries t) (λx. None) x›\n discuss goal 1*)\n apply ((rename_tac x)[1])\n (*top goal: ‹⋀xa. RBT.lookup t xa = fold (λ(x, v) m. m(f x ↦ v)) (RBT.entries t) (λx. None) (f xa)› and 1 goal remains*)\n apply ((case_tac \"RBT.lookup t x\")[1])\n (*goals:\n 1. ‹⋀x::'a. RBT.lookup (t::('a, 'c) RBT.rbt) x = None ⟹ RBT.lookup t x = fold (λ(x::'a, v::'c) m::'b ⇒ 'c option. m((f::'a ⇒ 'b) x ↦ v)) (RBT.entries t) (λx::'b. None) (f x)›\n 2. ‹⋀(x::'a) a::'c. RBT.lookup (t::('a, 'c) RBT.rbt) x = Some a ⟹ RBT.lookup t x = fold (λ(x::'a, v::'c) m::'b ⇒ 'c option. m((f::'a ⇒ 'b) x ↦ v)) (RBT.entries t) (λx::'b. None) (f x)›\n discuss goal 1*)\n apply ((auto simp: lookup_in_tree[symmetric] (*‹((?k, ?v) ∈ set (RBT.entries ?t)) = (RBT.lookup ?t ?k = Some ?v)›*) intro!: a( (*‹⟦distinct (map fst (rev ?uu7)); ?k3 ∉ fst ` set ?uu7⟧ ⟹ None = fold (λ(x, v) m. m(f x ↦ v)) ?uu7 (λx. None) (f ?k3)›*) 2))[1])\n (*discuss goal 2*)\n apply ((auto dest!: lookup_in_tree[THEN iffD1] (*‹RBT.lookup ?t1 ?k1 = Some ?v1 ⟹ (?k1, ?v1) ∈ set (RBT.entries ?t1)›*) intro!: a( (*‹⟦distinct (map fst (rev ?uu7)); (?k3, ?v3) ∈ set ?uu7⟧ ⟹ Some ?v3 = fold (λ(x, v) m. m(f x ↦ v)) ?uu7 (λx. None) (f ?k3)›*) 1))[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (rule a( (*‹⟦distinct (map fst (rev (?uu4::('a × 'c) list))); (?x::'b) ∉ range (f::'a ⇒ 'b)⟧ ⟹ None = fold (λ(x::'a, v::'c) m::'b ⇒ 'c option. m(f x ↦ v)) ?uu4 (λx::'b. None) ?x›*) 3))\n (*goals:\n 1. ‹⋀x. ∀xa. f xa ≠ x ⟹ distinct (map fst (rev (RBT.entries t)))›\n 2. ‹⋀x. ∀xa. f xa ≠ x ⟹ x ∉ range f›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\nqed\n\nlemma map_values_code [code]:\n \"map_values f (RBT_Mapping.Mapping t) = RBT.fold (λx v m. case (f x v) of None ⇒ m | Some v' ⇒ Mapping.update x v' m) t Mapping.empty\"\nproof (-)\n (*goal: ‹map_values (f::'a ⇒ 'c ⇒ 'b option) (RBT_Mapping.Mapping (t::('a, 'c) RBT.rbt)) = RBT.fold (λ(x::'a) (v::'c) m::('a, 'b) mapping. case f x v of None ⇒ m | Some (v'::'b) ⇒ Mapping.update x v' m) t Mapping.empty›*)\n {\n fix xs and m\n assume \"distinct (map fst (xs::('a × 'c) list))\" (*‹distinct (map fst (xs::('a × 'c) list))›*)\n hence \"fold (λp m. case f (fst p) (snd p) of None ⇒ m | Some v' ⇒ m(fst p ↦ v')) xs m\n = m ++ (λx. Option.bind (map_of xs x) (f x))\"\n apply (induction xs arbitrary: m)\n (*goals:\n 1. ‹⋀m. distinct (map fst []) ⟹ fold (λp m. case f (fst p) (snd p) of None ⇒ m | Some v' ⇒ m(fst p ↦ v')) [] m = m ++ (λx. Option.bind (map_of [] x) (f x))›\n 2. ‹⋀a xs m. ⟦⋀m. distinct (map fst xs) ⟹ fold (λp m. case f (fst p) (snd p) of None ⇒ m | Some v' ⇒ m(fst p ↦ v')) xs m = m ++ (λx. Option.bind (map_of xs x) (f x)); distinct (map fst (a # xs))⟧ ⟹ fold (λp m. case f (fst p) (snd p) of None ⇒ m | Some v' ⇒ m(fst p ↦ v')) (a # xs) m = m ++ (λx. Option.bind (map_of (a # xs) x) (f x))›\n discuss goal 1*)\n apply ((auto intro: rev_image_eqI (*‹⟦(?x::?'a) ∈ (?A::?'a set); (?b::?'b) = (?f::?'a ⇒ ?'b) ?x⟧ ⟹ ?b ∈ ?f ` ?A›*) split: bind_split (*‹(?P::?'a option ⇒ bool) (Option.bind (?m::?'b option) (?f::?'b ⇒ ?'a option)) = ((?m = None ⟶ ?P None) ∧ (∀v::?'b. ?m = Some v ⟶ ?P (?f v)))›*) option.splits (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) simp: map_add_def (*‹(?m1.0::?'a ⇒ ?'b option) ++ (?m2.0::?'a ⇒ ?'b option) = (λx::?'a. case ?m2.0 x of None ⇒ ?m1.0 x | Some (y::?'b) ⇒ Some y)›*) fun_eq_iff (*‹((?f::?'a ⇒ ?'b) = (?g::?'a ⇒ ?'b)) = (∀x::?'a. ?f x = ?g x)›*))[1])\n (*discuss goal 2*)\n apply ((auto intro: rev_image_eqI (*‹⟦(?x::?'a) ∈ (?A::?'a set); (?b::?'b) = (?f::?'a ⇒ ?'b) ?x⟧ ⟹ ?b ∈ ?f ` ?A›*) split: bind_split (*‹(?P::?'a option ⇒ bool) (Option.bind (?m::?'b option) (?f::?'b ⇒ ?'a option)) = ((?m = None ⟶ ?P None) ∧ (∀v::?'b. ?m = Some v ⟶ ?P (?f v)))›*) option.splits (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) simp: map_add_def (*‹(?m1.0::?'a ⇒ ?'b option) ++ (?m2.0::?'a ⇒ ?'b option) = (λx::?'a. case ?m2.0 x of None ⇒ ?m1.0 x | Some (y::?'b) ⇒ Some y)›*) fun_eq_iff (*‹((?f::?'a ⇒ ?'b) = (?g::?'a ⇒ ?'b)) = (∀x::?'a. ?f x = ?g x)›*))[1])\n (*proven 2 subgoals*) .\n }\n note bind_map_of_fold = this (*‹distinct (map fst (?xs2::('a × 'c) list)) ⟹ fold (λ(p::'a × 'c) m::'a ⇒ 'b option. case (f::'a ⇒ 'c ⇒ 'b option) (fst p) (snd p) of None ⇒ m | Some (v'::'b) ⇒ m(fst p ↦ v')) ?xs2 (?m2::'a ⇒ 'b option) = ?m2 ++ (λx::'a. Option.bind (map_of ?xs2 x) (f x))›*)\n show \"?thesis\"\n (*goal: ‹map_values f (RBT_Mapping.Mapping t) = RBT.fold (λx v m. case f x v of None ⇒ m | Some v' ⇒ Mapping.update x v' m) t Mapping.empty›*)\n unfolding RBT.fold_fold\n (*goal: ‹map_values (f::'a ⇒ 'c ⇒ 'b option) (RBT_Mapping.Mapping (t::('a, 'c) RBT.rbt)) = fold (λ(x::'a, v::'c) m::('a, 'b) mapping. case f x v of None ⇒ m | Some (v'::'b) ⇒ Mapping.update x v' m) (RBT.entries t) Mapping.empty›*)\n apply (transfer fixing: t f)\n (*goal: ‹map_values (f::'a ⇒ 'c ⇒ 'b option) (RBT_Mapping.Mapping (t::('a, 'c) RBT.rbt)) = fold (λ(x::'a, v::'c) m::('a, 'b) mapping. case f x v of None ⇒ m | Some (v'::'b) ⇒ Mapping.update x v' m) (RBT.entries t) Mapping.empty›*)\n apply (simp add: split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*))\n (*goal: ‹(λx. Option.bind (RBT.lookup t x) (f x)) = fold (λ(x, v) m. case f x v of None ⇒ m | Some v' ⇒ m(x ↦ v')) (RBT.entries t) (λx. None)›*)\n apply (rule bind_map_of_fold [of \"RBT.entries t\" Map.empty, simplified, symmetric] (*‹distinct (map fst (RBT.entries t)) ⟹ (λx. Option.bind (RBT.lookup t x) (f x)) = fold (λp m. case f (fst p) (snd p) of None ⇒ m | Some v' ⇒ m(fst p ↦ v')) (RBT.entries t) (λx. None)›*))\n (*goal: ‹(λx. Option.bind (RBT.lookup t x) (f x)) = fold (λp m. case f (fst p) (snd p) of None ⇒ m | Some v' ⇒ m(fst p ↦ v')) (RBT.entries t) (λx. None)›*)\n using RBT.distinct_entries (*‹distinct (map fst (RBT.entries ?t))›*) distinct_map (*‹distinct (map ?f ?xs) = (distinct ?xs ∧ inj_on ?f (set ?xs))›*) by auto\nqed\n\nlemma [code_unfold]: \"set (RBT.keys t) = RBT_Set.Set (RBT.map (λ_ _. ()) t)\"\n by (auto simp: RBT_Set.Set_def (*‹RBT_Set.Set ?t = {x. RBT.lookup ?t x = Some ()}›*) RBT.keys_def_alt (*‹RBT.keys ?t = map fst (RBT.entries ?t)›*) RBT.lookup_in_tree (*‹(RBT.lookup ?t ?k = Some ?v) = ((?k, ?v) ∈ set (RBT.entries ?t))›*) elim: rev_image_eqI (*‹⟦?x ∈ ?A; ?b = ?f ?x⟧ ⟹ ?b ∈ ?f ` ?A›*))\n\nlemma mmap_rbt_code [code]: \"mmap f (RBT_Mapping.Mapping t) = RBT_Mapping.Mapping (RBT.map (λ_. f) t)\"\n unfolding mmap_def\n (*goal: ‹Mapping.map id f (RBT_Mapping.Mapping t) = RBT_Mapping.Mapping (RBT.map (λ_. f) t)›*)\n apply transfer\n (*goal: ‹Mapping.map id f (RBT_Mapping.Mapping t) = RBT_Mapping.Mapping (RBT.map (λ_. f) t)›*)\n by auto\n\nlemma mapping_add_code [code]: \"mapping_add (RBT_Mapping.Mapping t1) (RBT_Mapping.Mapping t2) = RBT_Mapping.Mapping (RBT.union t1 t2)\"\n apply transfer\n (*goal: ‹mapping_add (RBT_Mapping.Mapping t1) (RBT_Mapping.Mapping t2) = RBT_Mapping.Mapping (RBT.union t1 t2)›*)\n by (simp add: lookup_union (*‹RBT.lookup (RBT.union ?s ?t) = RBT.lookup ?s ++ RBT.lookup ?t›*))\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Formal_SSA/RBT_Mapping_Exts.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"46c7b144ae33b2806e873c368fa56f38ac8690ee726828b1d9bc1c8a7372ef05\"\n}"}}},{"rowIdx":165,"cells":{"text":{"kind":"string","value":"(* Title: The Optimized Graceful Recursive Path Order for Lambda-Free Higher-Order Terms\n Author: Jasmin Blanchette <jasmin.blanchette at inria.fr>, 2016\n Maintainer: Jasmin Blanchette <jasmin.blanchette at inria.fr>\n*)\n\nsection ‹The Optimized Graceful Recursive Path Order for Lambda-Free Higher-Order Terms›\n\ntheory Lambda_Free_RPO_Optim\nimports Lambda_Free_RPO_Std\nbegin\n\ntext ‹\nThis theory defines the optimized variant of the graceful recursive path order\n(RPO) for ‹λ›-free higher-order terms.\n›\n\n\nsubsection ‹Setup›\n\nlocale rpo_optim = rpo_basis _ _ arity_sym arity_var\n for\n arity_sym :: \"'s ⇒ enat\" and\n arity_var :: \"'v ⇒ enat\" +\n assumes extf_ext_snoc: \"ext_snoc (extf f)\"\nbegin\n\nlemmas extf_snoc = ext_snoc.snoc[OF extf_ext_snoc]\n\n\nsubsection ‹Definition of the Order›\n\ndefinition\n chkargs :: \"(('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool\"\nwhere\n [simp]: \"chkargs gt t s ⟷ (∀s' ∈ set (args s). gt t s')\"\n\nlemma chkargs_mono[mono]: \"gt ≤ gt' ⟹ chkargs gt ≤ chkargs gt'\"\n by force\n\ninductive gt :: \"('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool\" (infix \">⇩t\" 50) where\n gt_arg: \"ti ∈ set (args t) ⟹ ti >⇩t s ∨ ti = s ⟹ t >⇩t s\"\n| gt_diff: \"head t >⇩h⇩d head s ⟹ chkvar t s ⟹ chkargs (>⇩t) t s ⟹ t >⇩t s\"\n| gt_same: \"head t = head s ⟹ chkargs (>⇩t) t s ⟹\n (∀f ∈ ground_heads (head t). extf f (>⇩t) (args t) (args s)) ⟹ t >⇩t s\"\n\nabbreviation ge :: \"('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool\" (infix \"≥⇩t\" 50) where\n \"t ≥⇩t s ≡ t >⇩t s ∨ t = s\"\n\n\nsubsection ‹Transitivity›\n\nlemma gt_in_args_imp: \"ti ∈ set (args t) ⟹ ti >⇩t s ⟹ t >⇩t s\"\n apply (cases t)\n (*goals:\n 1. ‹⋀x1::('s::type, 'v::type) hd. ⟦(ti::('s::type, 'v::type) tm) ∈ set (args (t::('s::type, 'v::type) tm)); ti >⇩t (s::('s::type, 'v::type) tm); t = Hd x1⟧ ⟹ t >⇩t s›\n 2. ‹⋀(x21::('s::type, 'v::type) tm) x22::('s::type, 'v::type) tm. ⟦(ti::('s::type, 'v::type) tm) ∈ set (args (t::('s::type, 'v::type) tm)); ti >⇩t (s::('s::type, 'v::type) tm); t = App x21 x22⟧ ⟹ t >⇩t s›\n discuss goal 1*)\n apply ((auto intro: gt_arg (*‹⟦?ti ∈ set (args ?t); ?ti ≥⇩t ?s⟧ ⟹ ?t >⇩t ?s›*))[1])\n (*discuss goal 2*)\n apply ((auto intro: gt_arg (*‹⟦?ti ∈ set (args ?t); ?ti ≥⇩t ?s⟧ ⟹ ?t >⇩t ?s›*))[1])\n (*proven 2 subgoals*) .\n\nlemma gt_imp_vars: \"t >⇩t s ⟹ vars t ⊇ vars s\"\napply (simp only: atomize_imp (*‹(?A ⟹ ?B) ≡ ?A ⟶ ?B›*))\n(*goal: ‹(t::('s, 'v) tm) >⇩t (s::('s, 'v) tm) ⟹ vars s ⊆ vars t›*)\napply (rule measure_induct_rule[of \"λ(t, s). size t + size s\" \"λ(t, s). t >⇩t s ⟶ vars t ⊇ vars s\" \"(t, s)\", simplified prod.case] (*‹(⋀x::('s::type, 'v::type) tm × ('s::type, 'v::type) tm. (⋀y::('s::type, 'v::type) tm × ('s::type, 'v::type) tm. (case y of (t::('s::type, 'v::type) tm, s::('s::type, 'v::type) tm) ⇒ size t + size s) < (case x of (t::('s::type, 'v::type) tm, s::('s::type, 'v::type) tm) ⇒ size t + size s) ⟹ case y of (t::('s::type, 'v::type) tm, s::('s::type, 'v::type) tm) ⇒ t >⇩t s ⟶ vars s ⊆ vars t) ⟹ case x of (t::('s::type, 'v::type) tm, s::('s::type, 'v::type) tm) ⇒ t >⇩t s ⟶ vars s ⊆ vars t) ⟹ (t::('s::type, 'v::type) tm) >⇩t (s::('s::type, 'v::type) tm) ⟶ vars s ⊆ vars t›*))\n(*goal: ‹t >⇩t s ⟶ vars s ⊆ vars t›*)\nproof (simp only: split_paired_all (*‹(⋀x. PROP ?P x) ≡ (⋀a b. PROP ?P (a, b))›*) prod.case (*‹(case (?x1.0, ?x2.0) of (x, xa) ⇒ ?f x xa) = ?f ?x1.0 ?x2.0›*) atomize_imp[symmetric] (*‹?A ⟶ ?B ≡ (?A ⟹ ?B)›*))\n (*goal: ‹⋀a b. ⟦⋀aa ba. ⟦size aa + size ba < size a + size b; aa >⇩t ba⟧ ⟹ vars ba ⊆ vars aa; a >⇩t b⟧ ⟹ vars b ⊆ vars a›*)\n fix t and s\n assume ih: \"⋀ta sa. size ta + size sa < size t + size s ⟹ ta >⇩t sa ⟹ vars ta ⊇ vars sa\" and t_gt_s: \"t >⇩t s\" (*‹⟦size (?ta::('s, 'v) tm) + size (?sa::('s, 'v) tm) < size (t::('s, 'v) tm) + size (s::('s, 'v) tm); ?ta >⇩t ?sa⟧ ⟹ vars ?sa ⊆ vars ?ta› ‹(t::('s, 'v) tm) >⇩t (s::('s, 'v) tm)›*)\n show \"vars t ⊇ vars s\"\n using t_gt_s (*‹t >⇩t s›*) proof (cases)\n (*goals:\n 1. ‹⋀ti. ⟦ti ∈ set (args t); ti ≥⇩t s⟧ ⟹ vars s ⊆ vars t›\n 2. ‹⟦head t >⇩h⇩d head s; chkvar t s; chkargs (>⇩t) t s⟧ ⟹ vars s ⊆ vars t›\n 3. ‹⟦head t = head s; chkargs (>⇩t) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)⟧ ⟹ vars s ⊆ vars t›*)\n case (gt_arg ti) (*‹(ti::('s, 'v) tm) ∈ set (args (t::('s, 'v) tm))› ‹ti ≥⇩t s›*)\n thus \"?thesis\"\n (*goal: ‹vars s ⊆ vars t›*)\n using ih[of ti s] (*‹⟦size ti + size s < size t + size s; ti >⇩t s⟧ ⟹ vars s ⊆ vars ti›*) by (metis size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*) vars_args_subseteq (*‹?si ∈ set (args ?s) ⟹ vars ?si ⊆ vars ?s›*) add_mono_thms_linordered_field( (*‹?i < ?j ∧ ?k = ?l ⟹ ?i + ?k < ?j + ?l›*) 1) order_trans (*‹⟦?x ≤ ?y; ?y ≤ ?z⟧ ⟹ ?x ≤ ?z›*))\n next\n (*goals:\n 1. ‹⟦head (t::('s, 'v) tm) >⇩h⇩d head (s::('s, 'v) tm); chkvar t s; chkargs (>⇩t) t s⟧ ⟹ vars s ⊆ vars t›\n 2. ‹⟦head (t::('s, 'v) tm) = head (s::('s, 'v) tm); chkargs (>⇩t) t s; ∀f::'s∈local.ground_heads (head t). (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) (args t) (args s)⟧ ⟹ vars s ⊆ vars t›*)\n case gt_diff (*‹head t >⇩h⇩d head s› ‹chkvar (t::('s, 'v) tm) (s::('s, 'v) tm)› ‹chkargs (>⇩t) t s›*)\n show \"?thesis\"\n (*goal: ‹vars s ⊆ vars t›*)\n proof (cases s)\n (*goals:\n 1. ‹⋀x1. s = Hd x1 ⟹ vars s ⊆ vars t›\n 2. ‹⋀x21 x22. s = App x21 x22 ⟹ vars s ⊆ vars t›*)\n case Hd (*‹s = Hd x1_›*)\n thus \"?thesis\"\n (*goal: ‹vars s ⊆ vars t›*)\n using gt_diff(2) (*‹chkvar t s›*) by (auto elim: hd.set_cases( (*‹⟦?e ∈ vars_hd ?a; ?a = Var ?e ⟹ ?thesis⟧ ⟹ ?thesis›*) 2))\n next\n (*goal: ‹⋀x21 x22. s = App x21 x22 ⟹ vars s ⊆ vars t›*)\n case (App s1 s2) (*‹(s::('s, 'v) tm) = App (s1::('s, 'v) tm) (s2::('s, 'v) tm)›*)\n thus \"?thesis\"\n (*goal: ‹vars s ⊆ vars t›*)\n using gt_diff (*‹head (t::('s::type, 'v::type) tm) >⇩h⇩d head (s::('s::type, 'v::type) tm)› ‹chkvar (t::('s, 'v) tm) (s::('s, 'v) tm)› ‹chkargs (>⇩t) t s›*) ih (*‹⟦size ?ta + size ?sa < size t + size s; ?ta >⇩t ?sa⟧ ⟹ vars ?sa ⊆ vars ?ta›*) apply simp\n (*goal: ‹vars s ⊆ vars t›*)\n by (metis (no_types) add.assoc (*‹?a + ?b + ?c = ?a + (?b + ?c)›*) gt.simps[unfolded chkargs_def chkvar_def] (*‹(?a1.0 >⇩t ?a2.0) = ((∃ti t s. ?a1.0 = t ∧ ?a2.0 = s ∧ ti ∈ set (args t) ∧ ti ≥⇩t s) ∨ (∃t s. ?a1.0 = t ∧ ?a2.0 = s ∧ head t >⇩h⇩d head s ∧ vars_hd (head s) ⊆ vars t ∧ Ball (set (args s)) ((>⇩t) t)) ∨ (∃t s. ?a1.0 = t ∧ ?a2.0 = s ∧ head t = head s ∧ Ball (set (args s)) ((>⇩t) t) ∧ (∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s))))›*) less_add_Suc1 (*‹?i < Suc (?i + ?m)›*))\n qed\n next\n (*goal: ‹⟦head t = head s; chkargs (>⇩t) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)⟧ ⟹ vars s ⊆ vars t›*)\n case gt_same (*‹head t = head s› ‹chkargs (>⇩t) (t::('s::type, 'v::type) tm) (s::('s::type, 'v::type) tm)› ‹∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)›*)\n thus \"?thesis\"\n (*goal: ‹vars s ⊆ vars t›*)\n proof (cases s rule: tm_exhaust_apps (*‹(⋀ζ ss. ?s = apps (Hd ζ) ss ⟹ ?P) ⟹ ?P›*))\n (*goal: ‹⋀(ζ::('s, 'v) hd) ss::('s, 'v) tm list. ⟦head (t::('s, 'v) tm) = head (s::('s, 'v) tm); chkargs (>⇩t) t s; ∀f::'s∈local.ground_heads (head t). (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) (args t) (args s); s = apps (Hd ζ) ss⟧ ⟹ vars s ⊆ vars t›*)\n case s: (apps ζ ss) (*‹s = apps (Hd ζ) ss›*)\n thus \"?thesis\"\n (*goal: ‹vars (s::('s, 'v) tm) ⊆ vars (t::('s, 'v) tm)›*)\n using gt_same (*‹head t = head s› ‹chkargs (>⇩t) t s› ‹∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)›*) unfolding chkargs_def s\n (*goal: ‹vars (apps (Hd (ζ::('s, 'v) hd)) (ss::('s, 'v) tm list)) ⊆ vars (t::('s, 'v) tm)›*)\n apply (auto intro!: vars_head_subseteq (*‹vars_hd (head ?s) ⊆ vars ?s›*))\n (*goal: ‹vars (apps (Hd ζ) ss) ⊆ vars t›*)\n by (metis ih[of t] (*‹⟦size t + size ?sa < size t + size s; t >⇩t ?sa⟧ ⟹ vars ?sa ⊆ vars t›*) insert_absorb (*‹?a ∈ ?A ⟹ insert ?a ?A = ?A›*) insert_subset (*‹(insert ?x ?A ⊆ ?B) = (?x ∈ ?B ∧ ?A ⊆ ?B)›*) nat_add_left_cancel_less (*‹(?k + ?m < ?k + ?n) = (?m < ?n)›*) s (*‹s = apps (Hd ζ) ss›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*) tm_collapse_apps (*‹apps (Hd (head ?s)) (args ?s) = ?s›*) tm_inject_apps (*‹(apps (Hd ?ζ) ?ss = apps (Hd ?ξ) ?ts) = (?ζ = ?ξ ∧ ?ss = ?ts)›*))\n qed\n qed\nqed\n\nlemma gt_trans: \"u >⇩t t ⟹ t >⇩t s ⟹ u >⇩t s\"\napply (simp only: atomize_imp (*‹(?A::bool ⟹ ?B::bool) ≡ ?A ⟶ ?B›*))\n(*goal: ‹⟦u >⇩t t; t >⇩t s⟧ ⟹ u >⇩t s›*)\napply (rule measure_induct_rule[of \"λ(u, t, s). {#size u, size t, size s#}\" \"λ(u, t, s). u >⇩t t ⟶ t >⇩t s ⟶ u >⇩t s\" \"(u, t, s)\", simplified prod.case] (*‹(⋀x. (⋀y. (case y of (u, t, s) ⇒ {#size u, size t, size s#}) < (case x of (u, t, s) ⇒ {#size u, size t, size s#}) ⟹ case y of (u, t, s) ⇒ u >⇩t t ⟶ t >⇩t s ⟶ u >⇩t s) ⟹ case x of (u, t, s) ⇒ u >⇩t t ⟶ t >⇩t s ⟶ u >⇩t s) ⟹ u >⇩t t ⟶ t >⇩t s ⟶ u >⇩t s›*))\n(*goal: ‹u >⇩t t ⟶ t >⇩t s ⟶ u >⇩t s›*)\nproof (simp only: split_paired_all (*‹(⋀x::?'a × ?'b. PROP (?P::?'a × ?'b ⇒ prop) x) ≡ (⋀(a::?'a) b::?'b. PROP ?P (a, b))›*) prod.case (*‹(case (?x1.0::?'a, ?x2.0::?'b) of (x::?'a, xa::?'b) ⇒ (?f::?'a ⇒ ?'b ⇒ ?'c) x xa) = ?f ?x1.0 ?x2.0›*) atomize_imp[symmetric] (*‹(?A::bool) ⟶ (?B::bool) ≡ (?A ⟹ ?B)›*))\n (*goal: ‹⋀a aa b. ⟦⋀ab ac ba. ⟦{#size ab, size ac, size ba#} < {#size a, size aa, size b#}; ab >⇩t ac; ac >⇩t ba⟧ ⟹ ab >⇩t ba; a >⇩t aa; aa >⇩t b⟧ ⟹ a >⇩t b›*)\n fix u and t and s\n assume ih: \"⋀ua ta sa. {#size ua, size ta, size sa#} < {#size u, size t, size s#} ⟹\n ua >⇩t ta ⟹ ta >⇩t sa ⟹ ua >⇩t sa\" and u_gt_t: \"u >⇩t t\" and t_gt_s: \"t >⇩t s\" (*‹⟦{#size (?ua::('s, 'v) tm), size (?ta::('s, 'v) tm), size (?sa::('s, 'v) tm)#} < {#size (u::('s, 'v) tm), size (t::('s, 'v) tm), size (s::('s, 'v) tm)#}; ?ua >⇩t ?ta; ?ta >⇩t ?sa⟧ ⟹ ?ua >⇩t ?sa› ‹(u::('s, 'v) tm) >⇩t (t::('s, 'v) tm)› ‹(t::('s, 'v) tm) >⇩t (s::('s, 'v) tm)›*)\n have chkvar: \"chkvar u s\"\n apply clarsimp\n (*goal: ‹chkvar u s›*)\n by (meson u_gt_t (*‹u >⇩t t›*) t_gt_s (*‹t >⇩t s›*) gt_imp_vars (*‹?t >⇩t ?s ⟹ vars ?s ⊆ vars ?t›*) hd.set_sel( (*‹is_Var ?a ⟹ var ?a ∈ vars_hd ?a›*) 2) vars_head_subseteq (*‹vars_hd (head ?s) ⊆ vars ?s›*) subsetCE (*‹⟦?A ⊆ ?B; ?c ∉ ?A ⟹ ?P; ?c ∈ ?B ⟹ ?P⟧ ⟹ ?P›*))\n have chk_u_s_if: \"chkargs (>⇩t) u s\" if chk_t_s: \"chkargs (>⇩t) t s\"\n proof (clarsimp simp only: chkargs_def (*‹chkargs ?gt ?t ?s = (∀s'∈set (args ?s). ?gt ?t s')›*))\n (*goal: ‹⋀s'::('s::type, 'v::type) tm. s' ∈ set (args (s::('s::type, 'v::type) tm)) ⟹ (u::('s::type, 'v::type) tm) >⇩t s'›*)\n fix s'\n assume \"s' ∈ set (args s)\" (*‹(s'::('s, 'v) tm) ∈ set (args (s::('s, 'v) tm))›*)\n thus \"u >⇩t s'\"\n using chk_t_s (*‹chkargs (>⇩t) t s›*) by (auto intro!: ih[of _ _ s', OF _ u_gt_t] (*‹⟦{#size u, size t, size s'#} < {#size u, size t, size s#}; t >⇩t s'⟧ ⟹ u >⇩t s'›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*))\n qed\n have u_gt_s_if_ui: \"ui ≥⇩t t ⟹ u >⇩t s\" if ui_in: \"ui ∈ set (args u)\" for ui\n using ih[of ui t s, simplified, OF size_in_args [ OF ui_in ] _ t_gt_s] (*‹ui >⇩t t ⟹ ui >⇩t s›*) gt_in_args_imp[OF ui_in, of s] (*‹ui >⇩t s ⟹ u >⇩t s›*) t_gt_s (*‹t >⇩t s›*) by blast\n show \"u >⇩t s\"\n using t_gt_s (*‹t >⇩t s›*) proof (cases)\n (*goals:\n 1. ‹⋀ti::('s, 'v) tm. ⟦ti ∈ set (args (t::('s, 'v) tm)); ti ≥⇩t (s::('s, 'v) tm)⟧ ⟹ (u::('s, 'v) tm) >⇩t s›\n 2. ‹⟦head (t::('s, 'v) tm) >⇩h⇩d head (s::('s, 'v) tm); chkvar t s; chkargs (>⇩t) t s⟧ ⟹ (u::('s, 'v) tm) >⇩t s›\n 3. ‹⟦head (t::('s, 'v) tm) = head (s::('s, 'v) tm); chkargs (>⇩t) t s; ∀f::'s∈local.ground_heads (head t). (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) (args t) (args s)⟧ ⟹ (u::('s, 'v) tm) >⇩t s›*)\n case gt_arg_t_s: (gt_arg ti) (*‹ti ∈ set (args t)› ‹ti ≥⇩t s›*)\n have u_gt_s_if_chk_u_t: \"?thesis\" if chk_u_t: \"chkargs (>⇩t) u t\"\n using ih[of u ti s] (*‹⟦{#size u, size ti, size s#} < {#size u, size t, size s#}; u >⇩t ti; ti >⇩t s⟧ ⟹ u >⇩t s›*) gt_arg_t_s (*‹(ti::('s, 'v) tm) ∈ set (args (t::('s, 'v) tm))› ‹(ti::('s, 'v) tm) ≥⇩t (s::('s, 'v) tm)›*) chk_u_t (*‹chkargs (>⇩t) u t›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*) by force\n show \"?thesis\"\n (*goal: ‹(u::('s::type, 'v::type) tm) >⇩t (s::('s::type, 'v::type) tm)›*)\n using u_gt_t (*‹u >⇩t t›*) apply cases\n (*goals:\n 1. ‹⋀ti. ⟦ti ∈ set (args u); ti ≥⇩t t⟧ ⟹ u >⇩t s›\n 2. ‹⟦head u >⇩h⇩d head t; chkvar u t; chkargs (>⇩t) u t⟧ ⟹ u >⇩t s›\n 3. ‹⟦head u = head t; chkargs (>⇩t) u t; ∀f∈local.ground_heads (head u). extf f (>⇩t) (args u) (args t)⟧ ⟹ u >⇩t s›\n discuss goal 1*)\n apply ((auto intro: u_gt_s_if_ui (*‹⟦?ui ∈ set (args u); ?ui ≥⇩t t⟧ ⟹ u >⇩t s›*) u_gt_s_if_chk_u_t (*‹chkargs (>⇩t) u t ⟹ u >⇩t s›*))[1])\n (*discuss goal 2*)\n apply ((auto intro: u_gt_s_if_ui (*‹⟦?ui ∈ set (args u); ?ui ≥⇩t t⟧ ⟹ u >⇩t s›*) u_gt_s_if_chk_u_t (*‹chkargs (>⇩t) u t ⟹ u >⇩t s›*))[1])\n (*discuss goal 3*)\n apply ((auto intro: u_gt_s_if_ui (*‹⟦(?ui::('s::type, 'v::type) tm) ∈ set (args (u::('s::type, 'v::type) tm)); ?ui ≥⇩t (t::('s::type, 'v::type) tm)⟧ ⟹ u >⇩t (s::('s::type, 'v::type) tm)›*) u_gt_s_if_chk_u_t (*‹chkargs (>⇩t) (u::('s::type, 'v::type) tm) (t::('s::type, 'v::type) tm) ⟹ u >⇩t (s::('s::type, 'v::type) tm)›*))[1])\n (*proven 3 subgoals*) .\n next\n (*goals:\n 1. ‹⟦head t >⇩h⇩d head s; chkvar t s; chkargs (>⇩t) t s⟧ ⟹ u >⇩t s›\n 2. ‹⟦head t = head s; chkargs (>⇩t) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)⟧ ⟹ u >⇩t s›*)\n case gt_diff_t_s: gt_diff (*‹head t >⇩h⇩d head s› ‹chkvar t s› ‹chkargs (>⇩t) t s›*)\n show \"?thesis\"\n (*goal: ‹u >⇩t s›*)\n using u_gt_t (*‹u >⇩t t›*) proof (cases)\n (*goals:\n 1. ‹⋀ti. ⟦ti ∈ set (args u); ti ≥⇩t t⟧ ⟹ u >⇩t s›\n 2. ‹⟦head u >⇩h⇩d head t; chkvar u t; chkargs (>⇩t) u t⟧ ⟹ u >⇩t s›\n 3. ‹⟦head u = head t; chkargs (>⇩t) u t; ∀f∈local.ground_heads (head u). extf f (>⇩t) (args u) (args t)⟧ ⟹ u >⇩t s›*)\n case gt_diff_u_t: gt_diff (*‹head u >⇩h⇩d head t› ‹chkvar u t› ‹chkargs (>⇩t) (u::('s, 'v) tm) (t::('s, 'v) tm)›*)\n have \"head u >⇩h⇩d head s\"\n using gt_diff_u_t(1) (*‹head u >⇩h⇩d head t›*) gt_diff_t_s(1) (*‹head t >⇩h⇩d head s›*) by (auto intro: gt_hd_trans (*‹⟦?χ >⇩h⇩d ?ξ; ?ξ >⇩h⇩d ?ζ⟧ ⟹ ?χ >⇩h⇩d ?ζ›*))\n thus \"?thesis\"\n (*goal: ‹u >⇩t s›*)\n by (rule gt_diff[OF _ chkvar chk_u_s_if[OF gt_diff_t_s(3)]] (*‹head u >⇩h⇩d head s ⟹ u >⇩t s›*))\n next\n (*goals:\n 1. ‹⋀ti::('s::type, 'v::type) tm. ⟦ti ∈ set (args (u::('s::type, 'v::type) tm)); ti ≥⇩t (t::('s::type, 'v::type) tm)⟧ ⟹ u >⇩t (s::('s::type, 'v::type) tm)›\n 2. ‹⟦head (u::('s::type, 'v::type) tm) = head (t::('s::type, 'v::type) tm); chkargs (>⇩t) u t; ∀f::'s::type∈local.ground_heads (head u). (extf::'s::type ⇒ (('s::type, 'v::type) tm ⇒ ('s::type, 'v::type) tm ⇒ bool) ⇒ ('s::type, 'v::type) tm list ⇒ ('s::type, 'v::type) tm list ⇒ bool) f (>⇩t) (args u) (args t)⟧ ⟹ u >⇩t (s::('s::type, 'v::type) tm)›*)\n case gt_same_u_t: gt_same (*‹head u = head t› ‹chkargs (>⇩t) (u::('s, 'v) tm) (t::('s, 'v) tm)› ‹∀f∈local.ground_heads (head u). extf f (>⇩t) (args u) (args t)›*)\n have \"head u >⇩h⇩d head s\"\n using gt_diff_t_s(1) (*‹head t >⇩h⇩d head s›*) gt_same_u_t(1) (*‹head (u::('s, 'v) tm) = head (t::('s, 'v) tm)›*) by auto\n thus \"?thesis\"\n (*goal: ‹u >⇩t s›*)\n by (rule gt_diff[OF _ chkvar chk_u_s_if[OF gt_diff_t_s(3)]] (*‹head u >⇩h⇩d head s ⟹ u >⇩t s›*))\n qed (auto intro: u_gt_s_if_ui (*‹⟦?ui ∈ set (args u); ?ui ≥⇩t t⟧ ⟹ u >⇩t s›*))\n (*solved the remaining goal: ‹⋀ti. ⟦ti ∈ set (args u); ti ≥⇩t t⟧ ⟹ u >⇩t s›*)\n next\n (*goal: ‹⟦head t = head s; chkargs (>⇩t) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)⟧ ⟹ u >⇩t s›*)\n case gt_same_t_s: gt_same (*‹head t = head s› ‹chkargs (>⇩t) (t::('s, 'v) tm) (s::('s, 'v) tm)› ‹∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)›*)\n show \"?thesis\"\n (*goal: ‹u >⇩t s›*)\n using u_gt_t (*‹(u::('s, 'v) tm) >⇩t (t::('s, 'v) tm)›*) proof (cases)\n (*goals:\n 1. ‹⋀ti. ⟦ti ∈ set (args u); ti ≥⇩t t⟧ ⟹ u >⇩t s›\n 2. ‹⟦head u >⇩h⇩d head t; chkvar u t; chkargs (>⇩t) u t⟧ ⟹ u >⇩t s›\n 3. ‹⟦head u = head t; chkargs (>⇩t) u t; ∀f∈local.ground_heads (head u). extf f (>⇩t) (args u) (args t)⟧ ⟹ u >⇩t s›*)\n case gt_diff_u_t: gt_diff (*‹head u >⇩h⇩d head t› ‹chkvar u t› ‹chkargs (>⇩t) u t›*)\n have \"head u >⇩h⇩d head s\"\n using gt_diff_u_t(1) (*‹head u >⇩h⇩d head t›*) gt_same_t_s(1) (*‹head t = head s›*) by simp\n thus \"?thesis\"\n (*goal: ‹u >⇩t s›*)\n by (rule gt_diff[OF _ chkvar chk_u_s_if[OF gt_same_t_s(2)]] (*‹head u >⇩h⇩d head s ⟹ u >⇩t s›*))\n next\n (*goals:\n 1. ‹⋀ti::('s, 'v) tm. ⟦ti ∈ set (args (u::('s, 'v) tm)); ti ≥⇩t (t::('s, 'v) tm)⟧ ⟹ u >⇩t (s::('s, 'v) tm)›\n 2. ‹⟦head (u::('s, 'v) tm) = head (t::('s, 'v) tm); chkargs (>⇩t) u t; ∀f::'s∈local.ground_heads (head u). (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) (args u) (args t)⟧ ⟹ u >⇩t (s::('s, 'v) tm)›*)\n case gt_same_u_t: gt_same (*‹head u = head t› ‹chkargs (>⇩t) u t› ‹∀f∈local.ground_heads (head u). extf f (>⇩t) (args u) (args t)›*)\n have hd_u_s: \"head u = head s\"\n using gt_same_u_t(1) (*‹head u = head t›*) gt_same_t_s(1) (*‹head t = head s›*) by simp\n let ?S = \"set (args u) ∪ set (args t) ∪ set (args s)\"\n have gt_trans_args: \"∀ua ∈ ?S. ∀ta ∈ ?S. ∀sa ∈ ?S. ua >⇩t ta ⟶ ta >⇩t sa ⟶ ua >⇩t sa\"\n proof (clarify)\n (*goal: ‹⋀uaa taa saa. ⟦uaa ∈ set (args u) ∪ set (args t) ∪ set (args s); taa ∈ set (args u) ∪ set (args t) ∪ set (args s); saa ∈ set (args u) ∪ set (args t) ∪ set (args s); uaa >⇩t taa; taa >⇩t saa⟧ ⟹ uaa >⇩t saa›*)\n fix sa and ta and ua\n assume ua_in: \"ua ∈ ?S\" and ta_in: \"ta ∈ ?S\" and sa_in: \"sa ∈ ?S\" and ua_gt_ta: \"ua >⇩t ta\" and ta_gt_sa: \"ta >⇩t sa\" (*‹(ua::('s, 'v) tm) ∈ set (args (u::('s, 'v) tm)) ∪ set (args (t::('s, 'v) tm)) ∪ set (args (s::('s, 'v) tm))› ‹(ta::('s, 'v) tm) ∈ set (args (u::('s, 'v) tm)) ∪ set (args (t::('s, 'v) tm)) ∪ set (args (s::('s, 'v) tm))› ‹(sa::('s, 'v) tm) ∈ set (args (u::('s, 'v) tm)) ∪ set (args (t::('s, 'v) tm)) ∪ set (args (s::('s, 'v) tm))› ‹(ua::('s, 'v) tm) >⇩t (ta::('s, 'v) tm)› ‹(ta::('s, 'v) tm) >⇩t (sa::('s, 'v) tm)›*)\n show \"ua >⇩t sa\"\n apply (auto intro!: ih[OF Max_lt_imp_lt_mset ua_gt_ta ta_gt_sa] (*‹⟦{#size u, size t, size s#} ≠ {#}; Max_mset {#size ua, size ta, size sa#} < Max_mset {#size u, size t, size s#}⟧ ⟹ ua >⇩t sa›*))\n (*goals:\n 1. ‹size ua < max (size u) (max (size t) (size s))›\n 2. ‹size ta < max (size u) (max (size t) (size s))›\n 3. ‹size sa < max (size u) (max (size t) (size s))›\n discuss goal 1*)\n apply (meson ua_in (*‹ua ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) ta_in (*‹ta ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) sa_in (*‹sa ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) Un_iff (*‹(?c ∈ ?A ∪ ?B) = (?c ∈ ?A ∨ ?c ∈ ?B)›*) max.strict_coboundedI1 (*‹?c < ?a ⟹ ?c < max ?a ?b›*) max.strict_coboundedI2 (*‹?c < ?b ⟹ ?c < max ?a ?b›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*))\n (*discuss goal 2*)\n apply (meson ua_in (*‹ua ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) ta_in (*‹ta ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) sa_in (*‹sa ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) Un_iff (*‹(?c ∈ ?A ∪ ?B) = (?c ∈ ?A ∨ ?c ∈ ?B)›*) max.strict_coboundedI1 (*‹?c < ?a ⟹ ?c < max ?a ?b›*) max.strict_coboundedI2 (*‹?c < ?b ⟹ ?c < max ?a ?b›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*))\n (*discuss goal 3*)\n apply (meson ua_in (*‹ua ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) ta_in (*‹ta ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) sa_in (*‹sa ∈ set (args u) ∪ set (args t) ∪ set (args s)›*) Un_iff (*‹(?c ∈ ?A ∪ ?B) = (?c ∈ ?A ∨ ?c ∈ ?B)›*) max.strict_coboundedI1 (*‹?c < ?a ⟹ ?c < max ?a ?b›*) max.strict_coboundedI2 (*‹?c < ?b ⟹ ?c < max ?a ?b›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*))\n (*proven 3 subgoals*) .\n qed\n have \"∀f ∈ ground_heads (head u). extf f (>⇩t) (args u) (args s)\"\n proof (clarify, rule extf_trans[OF _ _ _ gt_trans_args] (*‹⟦(?zs::('s, 'v) tm list) ∈ lists (set (args (u::('s, 'v) tm)) ∪ set (args (t::('s, 'v) tm)) ∪ set (args (s::('s, 'v) tm))); (?ys::('s, 'v) tm list) ∈ lists (set (args u) ∪ set (args t) ∪ set (args s)); (?xs::('s, 'v) tm list) ∈ lists (set (args u) ∪ set (args t) ∪ set (args s)); (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) (?f::'s) (>⇩t) ?zs ?ys; extf ?f (>⇩t) ?ys ?xs⟧ ⟹ extf ?f (>⇩t) ?zs ?xs›*))\n (*goals:\n 1. ‹⋀f::'s. f ∈ local.ground_heads (head (u::('s, 'v) tm)) ⟹ args u ∈ lists (set (args u) ∪ set (args (t::('s, 'v) tm)) ∪ set (args (s::('s, 'v) tm)))›\n 2. ‹⋀f::'s. f ∈ local.ground_heads (head (u::('s, 'v) tm)) ⟹ (?ys1::'s ⇒ ('s, 'v) tm list) f ∈ lists (set (args u) ∪ set (args (t::('s, 'v) tm)) ∪ set (args (s::('s, 'v) tm)))›\n 3. ‹⋀f::'s. f ∈ local.ground_heads (head (u::('s, 'v) tm)) ⟹ args (s::('s, 'v) tm) ∈ lists (set (args u) ∪ set (args (t::('s, 'v) tm)) ∪ set (args s))›\n 4. ‹⋀f::'s. f ∈ local.ground_heads (head (u::('s, 'v) tm)) ⟹ (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) (args u) ((?ys1::'s ⇒ ('s, 'v) tm list) f)›\n 5. ‹⋀f::'s. f ∈ local.ground_heads (head (u::('s, 'v) tm)) ⟹ (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) ((?ys1::'s ⇒ ('s, 'v) tm list) f) (args (s::('s, 'v) tm))›*)\n fix f\n assume f_in_grounds: \"f ∈ ground_heads (head u)\" (*‹(f::'s) ∈ local.ground_heads (head (u::('s, 'v) tm))›*)\n show \"extf f (>⇩t) (args u) (args t)\"\n using f_in_grounds (*‹f ∈ local.ground_heads (head u)›*) gt_same_u_t(3) (*‹∀f∈local.ground_heads (head u). extf f (>⇩t) (args u) (args t)›*) by blast\n show \"extf f (>⇩t) (args t) (args s)\"\n using f_in_grounds (*‹f ∈ local.ground_heads (head u)›*) gt_same_t_s(3) (*‹∀f::'s::type∈local.ground_heads (head (t::('s::type, 'v::type) tm)). (extf::'s::type ⇒ (('s::type, 'v::type) tm ⇒ ('s::type, 'v::type) tm ⇒ bool) ⇒ ('s::type, 'v::type) tm list ⇒ ('s::type, 'v::type) tm list ⇒ bool) f (>⇩t) (args t) (args (s::('s::type, 'v::type) tm))›*) unfolding gt_same_u_t(1)\n (*goal: ‹extf f (>⇩t) (args t) (args s)›*)\n by blast\n qed (auto)\n (*solves the remaining goals:\n 1. ‹⋀f. f ∈ local.ground_heads (head u) ⟹ args u ∈ lists (set (args u) ∪ set (args t) ∪ set (args s))›\n 2. ‹⋀f. f ∈ local.ground_heads (head u) ⟹ args t ∈ lists (set (args u) ∪ set (args t) ∪ set (args s))›\n 3. ‹⋀f. f ∈ local.ground_heads (head u) ⟹ args s ∈ lists (set (args u) ∪ set (args t) ∪ set (args s))›*)\n thus \"?thesis\"\n (*goal: ‹u >⇩t s›*)\n by (rule gt_same[OF hd_u_s chk_u_s_if[OF gt_same_t_s(2)]] (*‹∀f::'s∈local.ground_heads (head (u::('s, 'v) tm)). (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) (args u) (args (s::('s, 'v) tm)) ⟹ u >⇩t s›*))\n qed (auto intro: u_gt_s_if_ui (*‹⟦?ui ∈ set (args u); ?ui ≥⇩t t⟧ ⟹ u >⇩t s›*))\n (*solved the remaining goal: ‹⋀ti. ⟦ti ∈ set (args u); ti ≥⇩t t⟧ ⟹ u >⇩t s›*)\n qed\nqed\n\nlemma gt_sub_fun: \"App s t >⇩t s\"\n apply (rule gt_same (*‹⟦head ?t = head ?s; chkargs (>⇩t) ?t ?s; ∀f∈local.ground_heads (head ?t). extf f (>⇩t) (args ?t) (args ?s)⟧ ⟹ ?t >⇩t ?s›*))\n (*goals:\n 1. ‹head (App s t) = head s›\n 2. ‹chkargs (>⇩t) (App s t) s›\n 3. ‹∀f∈local.ground_heads (head (App s t)). extf f (>⇩t) (args (App s t)) (args s)›\n discuss goal 1*)\n apply ((auto intro: extf_snoc (*‹extf ?f ?gt (?xs @ [?x]) ?xs›*) gt_arg[of _ \"App s t\"] (*‹⟦?ti ∈ set (args (App s t)); ?ti ≥⇩t ?s⟧ ⟹ App s t >⇩t ?s›*))[1])\n (*discuss goal 2*)\n apply ((auto intro: extf_snoc (*‹extf ?f ?gt (?xs @ [?x]) ?xs›*) gt_arg[of _ \"App s t\"] (*‹⟦?ti ∈ set (args (App s t)); ?ti ≥⇩t ?s⟧ ⟹ App s t >⇩t ?s›*))[1])\n (*discuss goal 3*)\n apply ((auto intro: extf_snoc (*‹extf ?f ?gt (?xs @ [?x]) ?xs›*) gt_arg[of _ \"App s t\"] (*‹⟦?ti ∈ set (args (App s t)); ?ti ≥⇩t ?s⟧ ⟹ App s t >⇩t ?s›*))[1])\n (*proven 3 subgoals*) .\n\nend\n\n\nsubsection ‹Conditional Equivalence with Unoptimized Version›\n\ncontext rpo\nbegin\n\ncontext\n assumes extf_ext_snoc: \"⋀f. ext_snoc (extf f)\"\nbegin\n\nlemma rpo_optim: \"rpo_optim ground_heads_var (>⇩s) extf arity_sym arity_var\"\n unfolding rpo_optim_def rpo_optim_axioms_def\n (*goal: ‹rpo_basis ground_heads_var (>⇩s) arity_sym arity_var extf ∧ (∀f. ext_snoc (extf f))›*)\n using rpo_basis_axioms (*‹rpo_basis ground_heads_var (>⇩s) arity_sym arity_var extf›*) extf_ext_snoc (*‹ext_snoc (extf ?f)›*) by auto\n\nabbreviation\n chkargs :: \"(('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool\"\nwhere\n \"chkargs ≡ rpo_optim.chkargs\"\n\nabbreviation gt_optim :: \"('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool\" (infix \">⇩t⇩o\" 50) where\n \"(>⇩t⇩o) ≡ rpo_optim.gt ground_heads_var (>⇩s) extf\"\n\nabbreviation ge_optim :: \"('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool\" (infix \"≥⇩t⇩o\" 50) where\n \"(≥⇩t⇩o) ≡ rpo_optim.ge ground_heads_var (>⇩s) extf\"\n\ntheorem gt_iff_optim: \"t >⇩t s ⟷ t >⇩t⇩o s\"\napply (rule measure_induct_rule[of \"λ(t, s). size t + size s\" \"λ(t, s). t >⇩t s ⟷ t >⇩t⇩o s\" \"(t, s)\", simplified prod.case] (*‹(⋀x. (⋀y. (case y of (t, s) ⇒ size t + size s) < (case x of (t, s) ⇒ size t + size s) ⟹ case y of (t, s) ⇒ (t >⇩t s) = (t >⇩t⇩o s)) ⟹ case x of (t, s) ⇒ (t >⇩t s) = (t >⇩t⇩o s)) ⟹ (t >⇩t s) = (t >⇩t⇩o s)›*))\n(*goal: ‹(t >⇩t s) = (t >⇩t⇩o s)›*)\nproof (simp only: split_paired_all (*‹(⋀x. PROP ?P x) ≡ (⋀a b. PROP ?P (a, b))›*) prod.case (*‹(case (?x1.0, ?x2.0) of (x, xa) ⇒ ?f x xa) = ?f ?x1.0 ?x2.0›*))\n (*goal: ‹⋀(a::('s, 'v) tm) b::('s, 'v) tm. (⋀(aa::('s, 'v) tm) ba::('s, 'v) tm. size aa + size ba < size a + size b ⟹ (aa >⇩t ba) = (aa >⇩t⇩o ba)) ⟹ (a >⇩t b) = (a >⇩t⇩o b)›*)\n fix t :: \"('s, 'v) tm\" and s :: \"('s, 'v) tm\"\n assume ih: \"⋀ta sa. size ta + size sa < size t + size s ⟹ ta >⇩t sa ⟷ ta >⇩t⇩o sa\" (*‹size (?ta::('s, 'v) tm) + size (?sa::('s, 'v) tm) < size (t::('s, 'v) tm) + size (s::('s, 'v) tm) ⟹ (?ta >⇩t ?sa) = (?ta >⇩t⇩o ?sa)›*)\n show \"t >⇩t s ⟷ t >⇩t⇩o s\"\n proof (standard)\n (*goals:\n 1. ‹t >⇩t s ⟹ t >⇩t⇩o s›\n 2. ‹t >⇩t⇩o s ⟹ t >⇩t s›*)\n assume t_gt_s: \"t >⇩t s\" (*‹(t::('s, 'v) tm) >⇩t (s::('s, 'v) tm)›*)\n have chkargs_if_chksubs: \"chkargs (>⇩t⇩o) t s\" if chksubs: \"chksubs (>⇩t) t s\"\n unfolding rpo_optim.chkargs_def[OF rpo_optim]\n (*goal: ‹Ball (set (args s)) ((>⇩t⇩o) t)›*)\n proof (cases s, simp_all, intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) ballI (*‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*))\n (*goals:\n 1. ‹⋀x21 x22. s = App x21 x22 ⟹ t >⇩t⇩o x22›\n 2. ‹⋀x21 x22 x. ⟦s = App x21 x22; x ∈ set (args x21)⟧ ⟹ t >⇩t⇩o x›*)\n fix s1 and s2\n assume s: \"s = App s1 s2\" (*‹(s::('s, 'v) tm) = App (s1::('s, 'v) tm) (s2::('s, 'v) tm)›*)\n have t_gt_s2: \"t >⇩t s2\"\n using chksubs (*‹chksubs (>⇩t) (t::('s, 'v) tm) (s::('s, 'v) tm)›*) s (*‹(s::('s, 'v) tm) = App (s1::('s, 'v) tm) (s2::('s, 'v) tm)›*) by simp\n show \"t >⇩t⇩o s2\"\n apply (rule ih[THEN iffD1, OF _ t_gt_s2] (*‹size t + size s2 < size t + size s ⟹ t >⇩t⇩o s2›*))\n (*goal: ‹t >⇩t⇩o s2›*)\n by (simp add: s (*‹s = App s1 s2›*))\n {\n fix s1i\n assume s1i_in: \"s1i ∈ set (args s1)\" (*‹(s1i::('s, 'v) tm) ∈ set (args (s1::('s, 'v) tm))›*)\n have \"t >⇩t s1\"\n using chksubs (*‹chksubs (>⇩t) t s›*) s (*‹s = App s1 s2›*) by simp\n moreover have \"s1 >⇩t s1i\"\n using s1i_in (*‹s1i ∈ set (args s1)›*) gt_proper_sub (*‹proper_sub ?s ?t ⟹ ?t >⇩t ?s›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*) sub_args (*‹(?s::(?'a, ?'b) tm) ∈ set (args (?t::(?'a, ?'b) tm)) ⟹ sub ?s ?t›*) by fastforce\n ultimately have t_gt_s1i: \"t >⇩t s1i\"\n by (rule gt_trans (*‹⟦?u >⇩t ?t; ?t >⇩t ?s⟧ ⟹ ?u >⇩t ?s›*))\n have sz_s1i: \"size s1i < size s\"\n using size_in_args[OF s1i_in] (*‹size s1i < size s1›*) s (*‹(s::('s, 'v) tm) = App (s1::('s, 'v) tm) (s2::('s, 'v) tm)›*) by simp\n show \"t >⇩t⇩o s1i\"\n apply (rule ih[THEN iffD1, OF _ t_gt_s1i] (*‹size (t::('s, 'v) tm) + size (s1i::('s, 'v) tm) < size t + size (s::('s, 'v) tm) ⟹ t >⇩t⇩o s1i›*))\n (*goal: ‹t >⇩t⇩o s1i›*)\n by (simp add: sz_s1i (*‹size s1i < size s›*))\n }\n qed\n show \"t >⇩t⇩o s\"\n using t_gt_s (*‹(t::('s, 'v) tm) >⇩t (s::('s, 'v) tm)›*) proof (cases)\n (*goals:\n 1. ‹⟦is_App (t::('s, 'v) tm); fun t ≥⇩t (s::('s, 'v) tm) ∨ arg t ≥⇩t s⟧ ⟹ t >⇩t⇩o s›\n 2. ‹⟦head (t::('s, 'v) tm) >⇩h⇩d head (s::('s, 'v) tm); chkvar t s; chksubs (>⇩t) t s⟧ ⟹ t >⇩t⇩o s›\n 3. ‹⟦head (t::('s, 'v) tm) = head (s::('s, 'v) tm); chksubs (>⇩t) t s; ∀f::'s∈local.ground_heads (head t). (extf::'s ⇒ (('s, 'v) tm ⇒ ('s, 'v) tm ⇒ bool) ⇒ ('s, 'v) tm list ⇒ ('s, 'v) tm list ⇒ bool) f (>⇩t) (args t) (args s)⟧ ⟹ t >⇩t⇩o s›*)\n case gt_sub (*‹is_App t› ‹fun t ≥⇩t s ∨ arg t ≥⇩t s›*)\n note t_app = this(1) (*‹is_App t›*) and ti_geo_s = this(2) (*‹fun t ≥⇩t s ∨ arg t ≥⇩t s›*)\n obtain t1 and t2 where t: \"t = App t1 t2\"\n (*goal: ‹(⋀t1 t2. t = App t1 t2 ⟹ thesis) ⟹ thesis›*)\n using t_app (*‹is_App t›*) by (metis tm.collapse( (*‹is_App (?tm::(?'s, ?'v) tm) ⟹ App (fun ?tm) (arg ?tm) = ?tm›*) 2))\n have t_gto_t1: \"t >⇩t⇩o t1\"\n unfolding t\n (*goal: ‹App t1 t2 >⇩t⇩o t1›*)\n by (rule rpo_optim.gt_sub_fun[OF rpo_optim] (*‹App ?s ?t >⇩t⇩o ?s›*))\n have t_gto_t2: \"t >⇩t⇩o t2\"\n unfolding t\n (*goal: ‹App t1 t2 >⇩t⇩o t2›*)\n apply (rule rpo_optim.gt_arg[OF rpo_optim, of t2] (*‹⟦t2 ∈ set (args ?t); t2 ≥⇩t⇩o ?s⟧ ⟹ ?t >⇩t⇩o ?s›*))\n (*goals:\n 1. ‹t2 ∈ set (args (App t1 t2))›\n 2. ‹t2 ≥⇩t⇩o t2›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n {\n assume t1_gt_s: \"t1 >⇩t s\" (*‹(t1::('s, 'v) tm) >⇩t (s::('s, 'v) tm)›*)\n have \"t1 >⇩t⇩o s\"\n apply (rule ih[THEN iffD1, OF _ t1_gt_s] (*‹size t1 + size s < size t + size s ⟹ t1 >⇩t⇩o s›*))\n (*goal: ‹t1 >⇩t⇩o s›*)\n by (simp add: t (*‹t = App t1 t2›*))\n hence \"?thesis\"\n by (rule rpo_optim.gt_trans[OF rpo_optim t_gto_t1] (*‹t1 >⇩t⇩o ?s ⟹ t >⇩t⇩o ?s›*))\n }\n moreover {\n assume t2_gt_s: \"t2 >⇩t s\" (*‹(t2::('s, 'v) tm) >⇩t (s::('s, 'v) tm)›*)\n have \"t2 >⇩t⇩o s\"\n apply (rule ih[THEN iffD1, OF _ t2_gt_s] (*‹size t2 + size s < size t + size s ⟹ t2 >⇩t⇩o s›*))\n (*goal: ‹t2 >⇩t⇩o s›*)\n by (simp add: t (*‹t = App t1 t2›*))\n hence \"?thesis\"\n by (rule rpo_optim.gt_trans[OF rpo_optim t_gto_t2] (*‹t2 >⇩t⇩o ?s ⟹ t >⇩t⇩o ?s›*))\n }\n ultimately show \"?thesis\"\n (*goal: ‹t >⇩t⇩o s›*)\n using t (*‹(t::('s, 'v) tm) = App (t1::('s, 'v) tm) (t2::('s, 'v) tm)›*) ti_geo_s (*‹fun (t::('s::type, 'v::type) tm) ≥⇩t (s::('s::type, 'v::type) tm) ∨ arg t ≥⇩t s›*) t_gto_t1 (*‹t >⇩t⇩o t1›*) t_gto_t2 (*‹(t::('s::type, 'v::type) tm) >⇩t⇩o (t2::('s::type, 'v::type) tm)›*) by auto\n next\n (*goals:\n 1. ‹⟦head (t::('s::type, 'v::type) tm) >⇩h⇩d head (s::('s::type, 'v::type) tm); chkvar t s; chksubs (>⇩t) t s⟧ ⟹ t >⇩t⇩o s›\n 2. ‹⟦head (t::('s::type, 'v::type) tm) = head (s::('s::type, 'v::type) tm); chksubs (>⇩t) t s; ∀f::'s::type∈local.ground_heads (head t). (extf::'s::type ⇒ (('s::type, 'v::type) tm ⇒ ('s::type, 'v::type) tm ⇒ bool) ⇒ ('s::type, 'v::type) tm list ⇒ ('s::type, 'v::type) tm list ⇒ bool) f (>⇩t) (args t) (args s)⟧ ⟹ t >⇩t⇩o s›*)\n case gt_diff (*‹head t >⇩h⇩d head s› ‹chkvar t s› ‹chksubs (>⇩t) t s›*)\n note hd_t_gt_s = this(1) (*‹head t >⇩h⇩d head s›*) and chkvar = this(2) (*‹chkvar t s›*) and chksubs = this(3) (*‹chksubs (>⇩t) t s›*)\n show \"?thesis\"\n (*goal: ‹(t::('s, 'v) tm) >⇩t⇩o (s::('s, 'v) tm)›*)\n by (rule rpo_optim.gt_diff[OF rpo_optim hd_t_gt_s chkvar chkargs_if_chksubs[OF chksubs]] (*‹t >⇩t⇩o s›*))\n next\n (*goal: ‹⟦head t = head s; chksubs (>⇩t) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)⟧ ⟹ t >⇩t⇩o s›*)\n case gt_same (*‹head (t::('s, 'v) tm) = head (s::('s, 'v) tm)› ‹chksubs (>⇩t) t s› ‹∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)›*)\n note hd_t_eq_s = this(1) (*‹head t = head s›*) and chksubs = this(2) (*‹chksubs (>⇩t) t s›*) and extf = this(3) (*‹∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)›*)\n have extf_gto: \"∀f ∈ ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)\"\n proof (rule ballI (*‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*), rule extf_mono_strong[of _ _ \"(>⇩t)\", rule_format] (*‹⟦⋀y x. ⟦y ∈ set ?ys; x ∈ set ?xs; y >⇩t x⟧ ⟹ ?gt' y x; extf ?f (>⇩t) ?ys ?xs⟧ ⟹ extf ?f ?gt' ?ys ?xs›*))\n (*goals:\n 1. ‹⋀f y x. ⟦f ∈ local.ground_heads (head t); y ∈ set (args t); x ∈ set (args s); y >⇩t x⟧ ⟹ y >⇩t⇩o x›\n 2. ‹⋀f. f ∈ local.ground_heads (head t) ⟹ extf f (>⇩t) (args t) (args s)›*)\n fix f\n assume f_in_ground: \"f ∈ ground_heads (head t)\" (*‹(f::'s) ∈ local.ground_heads (head (t::('s, 'v) tm))›*)\n {\n fix ta and sa\n assume ta_in: \"ta ∈ set (args t)\" and sa_in: \"sa ∈ set (args s)\" and ta_gt_sa: \"ta >⇩t sa\" (*‹(ta::('s, 'v) tm) ∈ set (args (t::('s, 'v) tm))› ‹(sa::('s, 'v) tm) ∈ set (args (s::('s, 'v) tm))› ‹(ta::('s, 'v) tm) >⇩t (sa::('s, 'v) tm)›*)\n show \"ta >⇩t⇩o sa\"\n apply (rule ih[THEN iffD1, OF _ ta_gt_sa] (*‹size ta + size sa < size t + size s ⟹ ta >⇩t⇩o sa›*))\n (*goal: ‹ta >⇩t⇩o sa›*)\n by (simp add: ta_in (*‹ta ∈ set (args t)›*) sa_in (*‹sa ∈ set (args s)›*) add_less_mono (*‹⟦?i < ?j; ?k < ?l⟧ ⟹ ?i + ?k < ?j + ?l›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*))\n }\n show \"extf f (>⇩t) (args t) (args s)\"\n using f_in_ground (*‹f ∈ local.ground_heads (head t)›*) extf (*‹∀f∈local.ground_heads (head t). extf f (>⇩t) (args t) (args s)›*) by simp\n qed\n show \"?thesis\"\n (*goal: ‹(t::('s::type, 'v::type) tm) >⇩t⇩o (s::('s::type, 'v::type) tm)›*)\n by (rule rpo_optim.gt_same[OF rpo_optim hd_t_eq_s chkargs_if_chksubs[OF chksubs] extf_gto] (*‹(t::('s, 'v) tm) >⇩t⇩o (s::('s, 'v) tm)›*))\n qed\n next\n (*goal: ‹t >⇩t⇩o s ⟹ t >⇩t s›*)\n assume t_gto_s: \"t >⇩t⇩o s\" (*‹(t::('s, 'v) tm) >⇩t⇩o (s::('s, 'v) tm)›*)\n have chksubs_if_chkargs: \"chksubs (>⇩t) t s\" if chkargs: \"chkargs (>⇩t⇩o) t s\"\n unfolding chksubs_def\n (*goal: ‹case s of Hd hd ⇒ True | App s1 s2 ⇒ t >⇩t s1 ∧ t >⇩t s2›*)\n proof (cases s, simp_all, rule conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹⋀x21 x22. s = App x21 x22 ⟹ t >⇩t x21›\n 2. ‹⋀x21 x22. s = App x21 x22 ⟹ t >⇩t x22›*)\n fix s1 and s2\n assume s: \"s = App s1 s2\" (*‹(s::('s, 'v) tm) = App (s1::('s, 'v) tm) (s2::('s, 'v) tm)›*)\n have \"s >⇩t⇩o s1\"\n unfolding s\n (*goal: ‹App s1 s2 >⇩t⇩o s1›*)\n by (rule rpo_optim.gt_sub_fun[OF rpo_optim] (*‹App ?s ?t >⇩t⇩o ?s›*))\n hence t_gto_s1: \"t >⇩t⇩o s1\"\n by (rule rpo_optim.gt_trans[OF rpo_optim t_gto_s] (*‹s >⇩t⇩o ?s ⟹ t >⇩t⇩o ?s›*))\n show \"t >⇩t s1\"\n apply (rule ih[THEN iffD2, OF _ t_gto_s1] (*‹size t + size s1 < size t + size s ⟹ t >⇩t s1›*))\n (*goal: ‹t >⇩t s1›*)\n by (simp add: s (*‹s = App s1 s2›*))\n have t_gto_s2: \"t >⇩t⇩o s2\"\n using chkargs (*‹chkargs (>⇩t⇩o) (t::('s, 'v) tm) (s::('s, 'v) tm)›*) unfolding rpo_optim.chkargs_def[OF rpo_optim] s\n (*goal: ‹(t::('s, 'v) tm) >⇩t⇩o (s2::('s, 'v) tm)›*)\n by simp\n show \"t >⇩t s2\"\n apply (rule ih[THEN iffD2, OF _ t_gto_s2] (*‹size t + size s2 < size t + size s ⟹ t >⇩t s2›*))\n (*goal: ‹t >⇩t s2›*)\n by (simp add: s (*‹s = App s1 s2›*))\n qed\n show \"t >⇩t s\"\n proof (cases rule: rpo_optim.gt.cases[OF rpo_optim t_gto_s, case_names gto_arg gto_diff gto_same] (*‹⟦⋀ti t s. ⟦t = t; s = s; ti ∈ set (args t); ti ≥⇩t⇩o s⟧ ⟹ ?P; ⋀t s. ⟦t = t; s = s; head t >⇩h⇩d head s; chkvar t s; chkargs (>⇩t⇩o) t s⟧ ⟹ ?P; ⋀t s. ⟦t = t; s = s; head t = head s; chkargs (>⇩t⇩o) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀ti t s. ⟦t = t; s = s; ti ∈ set (args t); ti ≥⇩t⇩o s⟧ ⟹ t >⇩t s›\n 2. ‹⋀t s. ⟦t = t; s = s; head t >⇩h⇩d head s; chkvar t s; chkargs (>⇩t⇩o) t s⟧ ⟹ t >⇩t s›\n 3. ‹⋀t s. ⟦t = t; s = s; head t = head s; chkargs (>⇩t⇩o) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)⟧ ⟹ t >⇩t s›*)\n case (gto_arg ti) (*‹(t::('s, 'v) tm) = (t_::('s, 'v) tm)› ‹s = s_› ‹ti ∈ set (args t_)› ‹ti ≥⇩t⇩o s_›*)\n hence ti_in: \"ti ∈ set (args t)\" and ti_geo_s: \"ti ≥⇩t⇩o s\"\n apply -\n (*goals:\n 1. ‹⟦t = t_; s = s_; ti ∈ set (args t_); ti ≥⇩t⇩o s_⟧ ⟹ ti ∈ set (args t)›\n 2. ‹⟦t = t_; s = s_; ti ∈ set (args t_); ti ≥⇩t⇩o s_⟧ ⟹ ti ≥⇩t⇩o s›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n obtain ζ and ts where t: \"t = apps (Hd ζ) ts\"\n (*goal: ‹(⋀ζ ts. t = apps (Hd ζ) ts ⟹ thesis) ⟹ thesis›*)\n by (fact tm_exhaust_apps (*‹(⋀ζ ss. ?s = apps (Hd ζ) ss ⟹ ?P) ⟹ ?P›*))\n {\n assume ti_gto_s: \"ti >⇩t⇩o s\" (*‹(ti::('s, 'v) tm) >⇩t⇩o (s::('s, 'v) tm)›*)\n hence ti_gt_s: \"ti >⇩t s\"\n using ih[of ti s] (*‹size ti + size s < size t + size s ⟹ (ti >⇩t s) = (ti >⇩t⇩o s)›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*) ti_in (*‹ti ∈ set (args t)›*) by auto\n moreover have \"t >⇩t ti\"\n using sub_args[OF ti_in] (*‹sub ti t›*) gt_proper_sub (*‹proper_sub ?s ?t ⟹ ?t >⇩t ?s›*) size_in_args[OF ti_in] (*‹size ti < size t›*) by blast\n ultimately have \"?thesis\"\n using gt_trans (*‹⟦?u >⇩t ?t; ?t >⇩t ?s⟧ ⟹ ?u >⇩t ?s›*) by blast\n }\n moreover {\n assume \"ti = s\" (*‹(ti::('s, 'v) tm) = (s::('s, 'v) tm)›*)\n hence \"?thesis\"\n using sub_args[OF ti_in] (*‹sub ti t›*) gt_proper_sub (*‹proper_sub (?s::('s, 'v) tm) (?t::('s, 'v) tm) ⟹ ?t >⇩t ?s›*) size_in_args[OF ti_in] (*‹size ti < size t›*) by blast\n }\n ultimately show \"?thesis\"\n (*goal: ‹t >⇩t s›*)\n using ti_geo_s (*‹ti ≥⇩t⇩o s›*) by blast\n next\n (*goals:\n 1. ‹⋀t s. ⟦t = t; s = s; head t >⇩h⇩d head s; chkvar t s; chkargs (>⇩t⇩o) t s⟧ ⟹ t >⇩t s›\n 2. ‹⋀t s. ⟦t = t; s = s; head t = head s; chkargs (>⇩t⇩o) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)⟧ ⟹ t >⇩t s›*)\n case gto_diff (*‹(t::('s, 'v) tm) = (t_::('s, 'v) tm)› ‹s = s_› ‹head t_ >⇩h⇩d head s_› ‹chkvar t_ s_› ‹chkargs (>⇩t⇩o) t_ s_›*)\n hence hd_t_gt_s: \"head t >⇩h⇩d head s\" and chkvar: \"chkvar t s\" and chkargs: \"chkargs (>⇩t⇩o) t s\"\n apply -\n (*goals:\n 1. ‹⟦t = t_; s = s_; head t_ >⇩h⇩d head s_; chkvar t_ s_; chkargs (>⇩t⇩o) t_ s_⟧ ⟹ head t >⇩h⇩d head s›\n 2. ‹⟦t = t_; s = s_; head t_ >⇩h⇩d head s_; chkvar t_ s_; chkargs (>⇩t⇩o) t_ s_⟧ ⟹ chkvar t s›\n 3. ‹⟦t = t_; s = s_; head t_ >⇩h⇩d head s_; chkvar t_ s_; chkargs (>⇩t⇩o) t_ s_⟧ ⟹ chkargs (>⇩t⇩o) t s›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*) .\n have \"chksubs (>⇩t) t s\"\n by (rule chksubs_if_chkargs[OF chkargs] (*‹chksubs (>⇩t) t s›*))\n thus \"?thesis\"\n (*goal: ‹(t::('s, 'v) tm) >⇩t (s::('s, 'v) tm)›*)\n by (rule gt_diff[OF hd_t_gt_s chkvar] (*‹chksubs (>⇩t) t s ⟹ t >⇩t s›*))\n next\n (*goal: ‹⋀t s. ⟦t = t; s = s; head t = head s; chkargs (>⇩t⇩o) t s; ∀f∈local.ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)⟧ ⟹ t >⇩t s›*)\n case gto_same (*‹(t::('s::type, 'v::type) tm) = (t_::('s::type, 'v::type) tm)› ‹s = s_› ‹head t_ = head s_› ‹chkargs (>⇩t⇩o) t_ s_› ‹∀f∈local.ground_heads (head t_). extf f (>⇩t⇩o) (args t_) (args s_)›*)\n hence hd_t_eq_s: \"head t = head s\" and chkargs: \"chkargs (>⇩t⇩o) t s\" and extf_gto: \"∀f ∈ ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)\"\n apply -\n (*goals:\n 1. ‹⟦t = t_; s = s_; head t_ = head s_; chkargs (>⇩t⇩o) t_ s_; ∀f∈local.ground_heads (head t_). extf f (>⇩t⇩o) (args t_) (args s_)⟧ ⟹ head t = head s›\n 2. ‹⟦t = t_; s = s_; head t_ = head s_; chkargs (>⇩t⇩o) t_ s_; ∀f∈local.ground_heads (head t_). extf f (>⇩t⇩o) (args t_) (args s_)⟧ ⟹ chkargs (>⇩t⇩o) t s›\n 3. ‹⟦t = t_; s = s_; head t_ = head s_; chkargs (>⇩t⇩o) t_ s_; ∀f∈local.ground_heads (head t_). extf f (>⇩t⇩o) (args t_) (args s_)⟧ ⟹ ∀f∈local.ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*) .\n have chksubs: \"chksubs (>⇩t) t s\"\n by (rule chksubs_if_chkargs[OF chkargs] (*‹chksubs (>⇩t) t s›*))\n have extf: \"∀f ∈ ground_heads (head t). extf f (>⇩t) (args t) (args s)\"\n proof (rule ballI (*‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*), rule extf_mono_strong[of _ _ \"(>⇩t⇩o)\", rule_format] (*‹⟦⋀y x. ⟦y ∈ set ?ys; x ∈ set ?xs; y >⇩t⇩o x⟧ ⟹ ?gt' y x; extf ?f (>⇩t⇩o) ?ys ?xs⟧ ⟹ extf ?f ?gt' ?ys ?xs›*))\n (*goals:\n 1. ‹⋀f y x. ⟦f ∈ local.ground_heads (head t); y ∈ set (args t); x ∈ set (args s); y >⇩t⇩o x⟧ ⟹ y >⇩t x›\n 2. ‹⋀f. f ∈ local.ground_heads (head t) ⟹ extf f (>⇩t⇩o) (args t) (args s)›*)\n fix f\n assume f_in_ground: \"f ∈ ground_heads (head t)\" (*‹(f::'s) ∈ local.ground_heads (head (t::('s, 'v) tm))›*)\n {\n fix ta and sa\n assume ta_in: \"ta ∈ set (args t)\" and sa_in: \"sa ∈ set (args s)\" and ta_gto_sa: \"ta >⇩t⇩o sa\" (*‹(ta::('s, 'v) tm) ∈ set (args (t::('s, 'v) tm))› ‹(sa::('s, 'v) tm) ∈ set (args (s::('s, 'v) tm))› ‹(ta::('s, 'v) tm) >⇩t⇩o (sa::('s, 'v) tm)›*)\n show \"ta >⇩t sa\"\n apply (rule ih[THEN iffD2, OF _ ta_gto_sa] (*‹size ta + size sa < size t + size s ⟹ ta >⇩t sa›*))\n (*goal: ‹ta >⇩t sa›*)\n by (simp add: ta_in (*‹ta ∈ set (args t)›*) sa_in (*‹sa ∈ set (args s)›*) add_less_mono (*‹⟦?i < ?j; ?k < ?l⟧ ⟹ ?i + ?k < ?j + ?l›*) size_in_args (*‹?s ∈ set (args ?t) ⟹ size ?s < size ?t›*))\n }\n show \"extf f (>⇩t⇩o) (args t) (args s)\"\n using f_in_ground (*‹f ∈ local.ground_heads (head t)›*) extf_gto (*‹∀f∈local.ground_heads (head t). extf f (>⇩t⇩o) (args t) (args s)›*) by simp\n qed\n show \"?thesis\"\n (*goal: ‹t >⇩t s›*)\n by (rule gt_same[OF hd_t_eq_s chksubs extf] (*‹(t::('s::type, 'v::type) tm) >⇩t (s::('s::type, 'v::type) tm)›*))\n qed\n qed\nqed\n\nend\n\nend\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Lambda_Free_RPOs/Lambda_Free_RPO_Optim.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"2b83bff277b2a507568849a83e05062f43f9d7cf85601d3f7a1806c347a88a66\"\n}"}}},{"rowIdx":166,"cells":{"text":{"kind":"string","value":"(* Title: HOL/UNITY/ProgressSets.thy\n Author: Lawrence C Paulson, Cambridge University Computer Laboratory\n Copyright 2003 University of Cambridge\n\nProgress Sets. From \n\n David Meier and Beverly Sanders,\n Composing Leads-to Properties\n Theoretical Computer Science 243:1-2 (2000), 339-361.\n\n David Meier,\n Progress Properties in Program Refinement and Parallel Composition\n Swiss Federal Institute of Technology Zurich (1997)\n*)\n\nsection‹Progress Sets›\n\ntheory ProgressSets imports Transformers begin\n\nsubsection ‹Complete Lattices and the Operator \\<^term>‹cl››\n\ndefinition lattice :: \"'a set set => bool\" where\n ― ‹Meier calls them closure sets, but they are just complete lattices›\n \"lattice L ==\n (∀M. M ⊆ L --> ⋂M ∈ L) & (∀M. M ⊆ L --> ⋃M ∈ L)\"\n\ndefinition cl :: \"['a set set, 'a set] => 'a set\" where\n ― ‹short for ``closure''›\n \"cl L r == ⋂{x. x∈L & r ⊆ x}\"\n\nlemma UNIV_in_lattice: \"lattice L ==> UNIV ∈ L\"\nby (force simp add: lattice_def (*‹lattice ?L ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*))\n\nlemma empty_in_lattice: \"lattice L ==> {} ∈ L\"\nby (force simp add: lattice_def (*‹lattice (?L::?'a set set) ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*))\n\nlemma Union_in_lattice: \"[|M ⊆ L; lattice L|] ==> ⋃M ∈ L\"\nby (simp add: lattice_def (*‹lattice ?L ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*))\n\nlemma Inter_in_lattice: \"[|M ⊆ L; lattice L|] ==> ⋂M ∈ L\"\nby (simp add: lattice_def (*‹lattice ?L ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*))\n\nlemma UN_in_lattice:\n \"[|lattice L; !!i. i∈I ==> r i ∈ L|] ==> (⋃i∈I. r i) ∈ L\"\nby (blast intro: Union_in_lattice (*‹⟦?M ⊆ ?L; lattice ?L⟧ ⟹ ⋃ ?M ∈ ?L›*))\n\nlemma INT_in_lattice:\n \"[|lattice L; !!i. i∈I ==> r i ∈ L|] ==> (⋂i∈I. r i) ∈ L\"\nby (blast intro: Inter_in_lattice (*‹⟦?M ⊆ ?L; lattice ?L⟧ ⟹ ⋂ ?M ∈ ?L›*))\n\nlemma Un_in_lattice: \"[|x∈L; y∈L; lattice L|] ==> x∪y ∈ L\"\n using Union_in_lattice[of \"{x, y}\" L] (*‹⟦{x, y} ⊆ L; lattice L⟧ ⟹ ⋃ {x, y} ∈ L›*) by simp\n\nlemma Int_in_lattice: \"[|x∈L; y∈L; lattice L|] ==> x∩y ∈ L\"\n using Inter_in_lattice[of \"{x, y}\" L] (*‹⟦{x::'a set, y::'a set} ⊆ (L::'a set set); lattice L⟧ ⟹ ⋂ {x, y} ∈ L›*) by simp\n\nlemma lattice_stable: \"lattice {X. F ∈ stable X}\"\napply (simp add: lattice_def (*‹lattice (?L::?'a set set) ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*) stable_def (*‹UNITY.stable (?A::?'a set) ≡ ?A co ?A›*) constrains_def (*‹(?A::?'a set) co (?B::?'a set) ≡ {F::?'a program. ∀act::(?'a × ?'a) set∈Acts F. act `` ?A ⊆ ?B}›*))\n(*goal: ‹lattice {X::'a set. (F::'a program) ∈ UNITY.stable X}›*)\nby blast\n\ntext‹The next three results state that \\<^term>‹cl L r› is the minimal\n element of \\<^term>‹L› that includes \\<^term>‹r›.›\nlemma cl_in_lattice: \"lattice L ==> cl L r ∈ L\"\n by (simp add: lattice_def (*‹lattice ?L ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*) cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n\nlemma cl_least: \"[|c∈L; r⊆c|] ==> cl L r ⊆ c\" \nby (force simp add: cl_def (*‹cl (?L::?'a::type set set) (?r::?'a::type set) ≡ ⋂ {x::?'a::type set ∈ ?L. ?r ⊆ x}›*))\n\ntext‹The next three lemmas constitute assertion (4.61)›\nlemma cl_mono: \"r ⊆ r' ==> cl L r ⊆ cl L r'\"\napply (simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n(*goal: ‹r ⊆ r' ⟹ cl L r ⊆ cl L r'›*)\nby blast\n\nlemma subset_cl: \"r ⊆ cl L r\"\nby (simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*) le_Inf_iff (*‹(?b ≤ Inf ?A) = (∀a∈?A. ?b ≤ a)›*))\n\ntext‹A reformulation of @{thm subset_cl}›\nlemma clI: \"x ∈ r ==> x ∈ cl L r\"\napply (simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n(*goal: ‹x ∈ r ⟹ x ∈ cl L r›*)\nby blast\n\ntext‹A reformulation of @{thm cl_least}›\nlemma clD: \"[|c ∈ cl L r; B ∈ L; r ⊆ B|] ==> c ∈ B\"\nby (force simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n\nlemma cl_UN_subset: \"(⋃i∈I. cl L (r i)) ⊆ cl L (⋃i∈I. r i)\"\napply (simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n(*goal: ‹(⋃i∈I. cl L (r i)) ⊆ cl L (⋃ (r ` I))›*)\nby blast\n\nlemma cl_Un: \"lattice L ==> cl L (r∪s) = cl L r ∪ cl L s\"\napply (rule equalityI (*‹⟦(?A::?'a set) ⊆ (?B::?'a set); ?B ⊆ ?A⟧ ⟹ ?A = ?B›*))\n(*goals:\n 1. ‹lattice L ⟹ cl L (r ∪ s) ⊆ cl L r ∪ cl L s›\n 2. ‹lattice L ⟹ cl L r ∪ cl L s ⊆ cl L (r ∪ s)›\ndiscuss goal 1*)\n apply (rule cl_least (*‹⟦?c ∈ ?L; ?r ⊆ ?c⟧ ⟹ cl ?L ?r ⊆ ?c›*))\n (*goals:\n 1. ‹lattice L ⟹ cl L r ∪ cl L s ∈ L›\n 2. ‹lattice L ⟹ r ∪ s ⊆ cl L r ∪ cl L s›\n discuss goal 1*)\n apply (blast intro: Un_in_lattice (*‹⟦(?x::?'a set) ∈ (?L::?'a set set); (?y::?'a set) ∈ ?L; lattice ?L⟧ ⟹ ?x ∪ ?y ∈ ?L›*) cl_in_lattice (*‹lattice (?L::?'a set set) ⟹ cl ?L (?r::?'a set) ∈ ?L›*))\n (*discuss goal 2*)\n apply (blast intro: subset_cl [THEN subsetD] (*‹?c ∈ ?A ⟹ ?c ∈ cl ?L1 ?A›*))\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply (simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n (*goal: ‹lattice L ⟹ cl L r ∪ cl L s ⊆ cl L (r ∪ s)›*)\n apply blast\n(*proven 2 subgoals*) .\n\nlemma cl_UN: \"lattice L ==> cl L (⋃i∈I. r i) = (⋃i∈I. cl L (r i))\"\napply (rule equalityI (*‹⟦?A ⊆ ?B; ?B ⊆ ?A⟧ ⟹ ?A = ?B›*))\n(*goals:\n 1. ‹lattice L ⟹ cl L (⋃ (r ` I)) ⊆ (⋃i∈I. cl L (r i))›\n 2. ‹lattice L ⟹ (⋃i∈I. cl L (r i)) ⊆ cl L (⋃ (r ` I))›\ndiscuss goal 1*)\n apply (rule cl_least (*‹⟦?c ∈ ?L; ?r ⊆ ?c⟧ ⟹ cl ?L ?r ⊆ ?c›*))\n (*goals:\n 1. ‹lattice (L::'a::type set set) ⟹ (⋃i::'b::type∈I::'b::type set. cl L ((r::'b::type ⇒ 'a::type set) i)) ∈ L›\n 2. ‹lattice (L::'a::type set set) ⟹ ⋃ ((r::'b::type ⇒ 'a::type set) ` (I::'b::type set)) ⊆ (⋃i::'b::type∈I. cl L (r i))›\n discuss goal 1*)\n apply (blast intro: UN_in_lattice (*‹⟦lattice ?L; ⋀i. i ∈ ?I ⟹ ?r i ∈ ?L⟧ ⟹ ⋃ (?r ` ?I) ∈ ?L›*) cl_in_lattice (*‹lattice ?L ⟹ cl ?L ?r ∈ ?L›*))\n (*discuss goal 2*)\n apply (blast intro: subset_cl [THEN subsetD] (*‹?c ∈ ?A ⟹ ?c ∈ cl ?L1 ?A›*))\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply (simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n (*goal: ‹lattice L ⟹ (⋃i∈I. cl L (r i)) ⊆ cl L (⋃ (r ` I))›*)\n apply blast\n(*proven 2 subgoals*) .\n\nlemma cl_Int_subset: \"cl L (r∩s) ⊆ cl L r ∩ cl L s\"\napply (simp add: cl_def (*‹cl (?L::?'a::type set set) (?r::?'a::type set) ≡ ⋂ {x::?'a::type set ∈ ?L. ?r ⊆ x}›*))\n(*goal: ‹cl L (r ∩ s) ⊆ cl L r ∩ cl L s›*)\nby blast\n\nlemma cl_idem [simp]: \"cl L (cl L r) = cl L r\"\napply (simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n(*goal: ‹cl (L::'a set set) (cl L (r::'a set)) = cl L r›*)\nby blast\n\nlemma cl_ident: \"r∈L ==> cl L r = r\" \nby (force simp add: cl_def (*‹cl ?L ?r ≡ ⋂ {x ∈ ?L. ?r ⊆ x}›*))\n\nlemma cl_empty [simp]: \"lattice L ==> cl L {} = {}\"\nby (simp add: cl_ident (*‹(?r::?'a set) ∈ (?L::?'a set set) ⟹ cl ?L ?r = ?r›*) empty_in_lattice (*‹lattice (?L::?'a set set) ⟹ {} ∈ ?L›*))\n\nlemma cl_UNIV [simp]: \"lattice L ==> cl L UNIV = UNIV\"\nby (simp add: cl_ident (*‹?r ∈ ?L ⟹ cl ?L ?r = ?r›*) UNIV_in_lattice (*‹lattice ?L ⟹ UNIV ∈ ?L›*))\n\ntext‹Assertion (4.62)›\nlemma cl_ident_iff: \"lattice L ==> (cl L r = r) = (r∈L)\" \napply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n(*goals:\n 1. ‹⟦lattice (L::'a::type set set); cl L (r::'a::type set) = r⟧ ⟹ r ∈ L›\n 2. ‹⟦lattice (L::'a::type set set); (r::'a::type set) ∈ L⟧ ⟹ cl L r = r›\ndiscuss goal 1*)\n apply (erule subst (*‹⟦(?s::?'a::type) = (?t::?'a::type); (?P::?'a::type ⇒ bool) ?s⟧ ⟹ ?P ?t›*))\n (*top goal: ‹⟦lattice L; cl L r = r⟧ ⟹ r ∈ L› and 1 goal remains*)\n apply (erule cl_in_lattice (*‹lattice ?L ⟹ cl ?L ?r ∈ ?L›*))\n(*discuss goal 2*)\n apply (erule cl_ident (*‹(?r::?'a::type set) ∈ (?L::?'a::type set set) ⟹ cl ?L ?r = ?r›*))\n(*proven 2 subgoals*) .\n\nlemma cl_subset_in_lattice: \"[|cl L r ⊆ r; lattice L|] ==> r∈L\" \nby (simp add: cl_ident_iff [symmetric] (*‹lattice ?L ⟹ (?r ∈ ?L) = (cl ?L ?r = ?r)›*) equalityI (*‹⟦?A ⊆ ?B; ?B ⊆ ?A⟧ ⟹ ?A = ?B›*) subset_cl (*‹?r ⊆ cl ?L ?r›*))\n\n\nsubsection ‹Progress Sets and the Main Lemma›\ntext‹A progress set satisfies certain closure conditions and is a \nsimple way of including the set \\<^term>‹wens_set F B›.›\n\ndefinition closed :: \"['a program, 'a set, 'a set, 'a set set] => bool\" where\n \"closed F T B L == ∀M. ∀act ∈ Acts F. B⊆M & T∩M ∈ L -->\n T ∩ (B ∪ wp act M) ∈ L\"\n\ndefinition progress_set :: \"['a program, 'a set, 'a set] => 'a set set set\" where\n \"progress_set F T B ==\n {L. lattice L & B ∈ L & T ∈ L & closed F T B L}\"\n\nlemma closedD:\n \"[|closed F T B L; act ∈ Acts F; B⊆M; T∩M ∈ L|] \n ==> T ∩ (B ∪ wp act M) ∈ L\" \nby (simp add: closed_def (*‹closed ?F ?T ?B ?L ≡ ∀M. ∀act∈Acts ?F. ?B ⊆ M ∧ ?T ∩ M ∈ ?L ⟶ ?T ∩ (?B ∪ wp act M) ∈ ?L›*)) \n\ntext‹Note: the formalization below replaces Meier's \\<^term>‹q› by \\<^term>‹B›\nand \\<^term>‹m› by \\<^term>‹X›.›\n\ntext‹Part of the proof of the claim at the bottom of page 97. It's\nproved separately because the argument requires a generalization over\nall \\<^term>‹act ∈ Acts F›.›\nlemma lattice_awp_lemma:\n assumes TXC: \"T∩X ∈ C\" ― ‹induction hypothesis in theorem below›\n and BsubX: \"B ⊆ X\" ― ‹holds in inductive step›\n and latt: \"lattice C\"\n and TC: \"T ∈ C\"\n and BC: \"B ∈ C\"\n and clos: \"closed F T B C\"\n shows \"T ∩ (B ∪ awp F (X ∪ cl C (T∩r))) ∈ C\"\napply (simp del: INT_simps (*‹(⋂x∈?C. ?A x ∩ ?B) = (if ?C = {} then UNIV else ⋂ (?A ` ?C) ∩ ?B)› ‹(⋂x∈?C. ?A ∩ ?B x) = (if ?C = {} then UNIV else ?A ∩ ⋂ (?B ` ?C))› ‹(⋂x∈?C. ?A x - ?B) = (if ?C = {} then UNIV else ⋂ (?A ` ?C) - ?B)› ‹(⋂x∈?C. ?A - ?B x) = (if ?C = {} then UNIV else ?A - ⋃ (?B ` ?C))› ‹(⋂x∈?C. insert ?a (?B x)) = insert ?a (⋂ (?B ` ?C))› ‹(⋂x∈?C. ?A x ∪ ?B) = ⋂ (?A ` ?C) ∪ ?B› ‹(⋂x∈?C. ?A ∪ ?B x) = ?A ∪ ⋂ (?B ` ?C)› ‹⋂ (?B ` ⋃ ?A) = (⋂y∈?A. ⋂ (?B ` y))› ‹⋂ (?C ` ⋃ (?B ` ?A)) = (⋂x∈?A. ⋂ (?C ` ?B x))› ‹⋂ (?B ` ?f ` ?A) = (⋂a∈?A. ?B (?f a))›*) add: awp_def (*‹awp ?F ?B ≡ ⋂act∈Acts ?F. wp act ?B›*) INT_extend_simps (*‹⋂ (?A ` ?C) ∩ ?B = (if ?C = {} then ?B else ⋂x∈?C. ?A x ∩ ?B)› ‹?A ∩ ⋂ (?B ` ?C) = (if ?C = {} then ?A else ⋂x∈?C. ?A ∩ ?B x)› ‹⋂ (?A ` ?C) - ?B = (if ?C = {} then UNIV - ?B else ⋂x∈?C. ?A x - ?B)› ‹?A - ⋃ (?B ` ?C) = (if ?C = {} then ?A else ⋂x∈?C. ?A - ?B x)› ‹insert ?a (⋂ (?B ` ?C)) = (⋂x∈?C. insert ?a (?B x))› ‹⋂ (?A ` ?C) ∪ ?B = (⋂x∈?C. ?A x ∪ ?B)› ‹?A ∪ ⋂ (?B ` ?C) = (⋂x∈?C. ?A ∪ ?B x)› ‹(⋂y∈?A. ⋂ (?B ` y)) = ⋂ (?B ` ⋃ ?A)› ‹(⋂x∈?A. ⋂ (?C ` ?B x)) = ⋂ (?C ` ⋃ (?B ` ?A))› ‹(⋂a∈?A. ?B (?f a)) = ⋂ (?B ` ?f ` ?A)›*))\n(*goal: ‹T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r))) ∈ C›*)\napply (rule INT_in_lattice [OF latt] (*‹(⋀i::?'b. i ∈ (?I::?'b set) ⟹ (?r::?'b ⇒ 'a set) i ∈ (C::'a set set)) ⟹ ⋂ (?r ` ?I) ∈ C›*))\n(*goal: ‹(⋂x∈Acts F. T ∩ (B ∪ wp x (X ∪ cl C (T ∩ r)))) ∈ C›*)\napply (erule closedD [OF clos] (*‹⟦?act ∈ Acts F; B ⊆ ?M; T ∩ ?M ∈ C⟧ ⟹ T ∩ (B ∪ wp ?act ?M) ∈ C›*))\n(*goals:\n 1. ‹⋀x. B ⊆ X ∪ cl C (T ∩ r)›\n 2. ‹⋀x. T ∩ (X ∪ cl C (T ∩ r)) ∈ C›\ndiscuss goal 1*)\n apply (simp add: subset_trans [OF BsubX Un_upper1] (*‹B ⊆ X ∪ ?B1›*))\n(*discuss goal 2*)\n apply (subgoal_tac \"T ∩ (X ∪ cl C (T∩r)) = (T∩X) ∪ cl C (T∩r)\")\n (*goals:\n 1. ‹⋀x. T ∩ (X ∪ cl C (T ∩ r)) = T ∩ X ∪ cl C (T ∩ r) ⟹ T ∩ (X ∪ cl C (T ∩ r)) ∈ C›\n 2. ‹⋀x. T ∩ (X ∪ cl C (T ∩ r)) = T ∩ X ∪ cl C (T ∩ r)›\n discuss goal 1*)\n apply (erule ssubst (*‹⟦?t = ?s; ?P ?s⟧ ⟹ ?P ?t›*))\n (*top goal: ‹⋀x. T ∩ (X ∪ cl C (T ∩ r)) = T ∩ X ∪ cl C (T ∩ r) ⟹ T ∩ (X ∪ cl C (T ∩ r)) ∈ C› and 1 goal remains*)\n apply (blast intro: Un_in_lattice (*‹⟦?x ∈ ?L; ?y ∈ ?L; lattice ?L⟧ ⟹ ?x ∪ ?y ∈ ?L›*) latt (*‹lattice C›*) cl_in_lattice (*‹lattice ?L ⟹ cl ?L ?r ∈ ?L›*) TXC (*‹T ∩ X ∈ C›*))\n (*discuss goal 2*)\n apply (blast intro: TC (*‹(T::'a set) ∈ (C::'a set set)›*) clD (*‹⟦(?c::?'a) ∈ cl (?L::?'a set set) (?r::?'a set); (?B::?'a set) ∈ ?L; ?r ⊆ ?B⟧ ⟹ ?c ∈ ?B›*))\n (*proven 2 subgoals*)\n(*proven 2 subgoals*) .\n\ntext‹Remainder of the proof of the claim at the bottom of page 97.›\nlemma lattice_lemma:\n assumes TXC: \"T∩X ∈ C\" ― ‹induction hypothesis in theorem below›\n and BsubX: \"B ⊆ X\" ― ‹holds in inductive step›\n and act: \"act ∈ Acts F\"\n and latt: \"lattice C\"\n and TC: \"T ∈ C\"\n and BC: \"B ∈ C\"\n and clos: \"closed F T B C\"\n shows \"T ∩ (wp act X ∩ awp F (X ∪ cl C (T∩r)) ∪ X) ∈ C\"\napply (subgoal_tac \"T ∩ (B ∪ wp act X) ∈ C\")\n(*goals:\n 1. ‹(T::'a::type set) ∩ ((B::'a::type set) ∪ wp (act::('a::type × 'a::type) set) (X::'a::type set)) ∈ (C::'a::type set set) ⟹ T ∩ (wp act X ∩ awp (F::'a::type program) (X ∪ cl C (T ∩ (r::'a::type set))) ∪ X) ∈ C›\n 2. ‹(T::'a::type set) ∩ ((B::'a::type set) ∪ wp (act::('a::type × 'a::type) set) (X::'a::type set)) ∈ (C::'a::type set set)›\ndiscuss goal 1*)\n apply (drule Int_in_lattice [OF _ lattice_awp_lemma [OF TXC BsubX latt TC BC clos, of r] latt] (*‹?x ∈ C ⟹ ?x ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) ∈ C›*))\n (*top goal: ‹T ∩ (B ∪ wp act X) ∈ C ⟹ T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ r)) ∪ X) ∈ C› and 1 goal remains*)\n apply (subgoal_tac \"T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T∩r)))) = \n T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T∩r)))\")\n (*goals:\n 1. ‹⟦T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) ∈ C; T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) = T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r)))⟧ ⟹ T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ r)) ∪ X) ∈ C›\n 2. ‹T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) ∈ C ⟹ T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) = T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r)))›\n discuss goal 1*)\n apply simp\n (*top goal: ‹⟦T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) ∈ C; T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) = T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r)))⟧ ⟹ T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ r)) ∪ X) ∈ C› and 2 goals remain*)\n apply (drule Un_in_lattice [OF _ TXC latt] (*‹(?x::'a::type set) ∈ (C::'a::type set set) ⟹ ?x ∪ (T::'a::type set) ∩ (X::'a::type set) ∈ C›*))\n (*top goal: ‹⟦T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r))) ∈ C; T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) = T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r)))⟧ ⟹ T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ r)) ∪ X) ∈ C› and 2 goals remain*)\n apply (subgoal_tac \"T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T∩r))) ∪ T∩X = \n T ∩ (wp act X ∩ awp F (X ∪ cl C (T∩r)) ∪ X)\")\n (*goals:\n 1. ‹⟦T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) = T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r))); T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r))) ∪ T ∩ X ∈ C; T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r))) ∪ T ∩ X = T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ r)) ∪ X)⟧ ⟹ T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ r)) ∪ X) ∈ C›\n 2. ‹⟦T ∩ (B ∪ wp act X) ∩ (T ∩ (B ∪ awp F (X ∪ cl C (T ∩ r)))) = T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r))); T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r))) ∪ T ∩ X ∈ C⟧ ⟹ T ∩ (B ∪ wp act X ∩ awp F (X ∪ cl C (T ∩ r))) ∪ T ∩ X = T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ r)) ∪ X)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply (blast intro: BsubX [THEN subsetD] (*‹?c ∈ B ⟹ ?c ∈ X›*))\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply (simp add: closedD [OF clos] (*‹⟦?act ∈ Acts F; B ⊆ ?M; T ∩ ?M ∈ C⟧ ⟹ T ∩ (B ∪ wp ?act ?M) ∈ C›*) act (*‹act ∈ Acts F›*) BsubX (*‹B ⊆ X›*) TXC (*‹T ∩ X ∈ C›*))\n(*proven 2 subgoals*) .\n\n\ntext‹Induction step for the main lemma›\nlemma progress_induction_step:\n assumes TXC: \"T∩X ∈ C\" ― ‹induction hypothesis in theorem below›\n and act: \"act ∈ Acts F\"\n and Xwens: \"X ∈ wens_set F B\"\n and latt: \"lattice C\"\n and TC: \"T ∈ C\"\n and BC: \"B ∈ C\"\n and clos: \"closed F T B C\"\n and Fstable: \"F ∈ stable T\"\n shows \"T ∩ wens F act X ∈ C\"\nproof (-)\n (*goal: ‹T ∩ wens F act X ∈ C›*)\n from Xwens (*‹X ∈ wens_set F B›*) have BsubX: \"B ⊆ X\"\n by (rule wens_set_imp_subset (*‹(?A::?'a set) ∈ wens_set (?F::?'a program) (?B::?'a set) ⟹ ?B ⊆ ?A›*))\n let ?r = \"wens F act X\"\n have \"?r ⊆ (wp act X ∩ awp F (X∪?r)) ∪ X\"\n by (simp add: wens_unfold [symmetric] (*‹wp ?act ?B ∩ awp ?F (?B ∪ wens ?F ?act ?B) ∪ ?B = wens ?F ?act ?B›*))\n then have \"T∩?r ⊆ T ∩ ((wp act X ∩ awp F (X∪?r)) ∪ X)\"\n by blast\n then have \"T∩?r ⊆ T ∩ ((wp act X ∩ awp F (T ∩ (X∪?r))) ∪ X)\"\n apply (simp add: awp_Int_eq (*‹awp (?F::?'a::type program) ((?A::?'a::type set) ∩ (?B::?'a::type set)) = awp ?F ?A ∩ awp ?F ?B›*) Fstable (*‹(F::'a::type program) ∈ UNITY.stable (T::'a::type set)›*) stable_imp_awp_ident (*‹(?F::?'a::type program) ∈ UNITY.stable (?A::?'a::type set) ⟹ awp ?F ?A = ?A›*))\n (*goal: ‹T ∩ wens F act X ⊆ T ∩ (wp act X ∩ awp F (T ∩ (X ∪ wens F act X)) ∪ X)›*)\n by blast\n then have \"T∩?r ⊆ T ∩ ((wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X)\"\n by (blast intro: awp_mono [THEN [2] rev_subsetD] (*‹⟦(?c::?'a) ∈ awp (?F1::?'a program) (?A1::?'a set); ?A1 ⊆ (?B1::?'a set)⟧ ⟹ ?c ∈ awp ?F1 ?B1›*) subset_cl [THEN subsetD] (*‹(?c::?'a) ∈ (?A::?'a set) ⟹ ?c ∈ cl (?L1::?'a set set) ?A›*))\n then have \"cl C (T∩?r) ⊆ \n cl C (T ∩ ((wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X))\"\n by (rule cl_mono (*‹?r ⊆ ?r' ⟹ cl ?L ?r ⊆ cl ?L ?r'›*))\n then have \"cl C (T∩?r) ⊆ \n T ∩ ((wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X)\"\n by (simp add: cl_ident (*‹?r ∈ ?L ⟹ cl ?L ?r = ?r›*) lattice_lemma [OF TXC BsubX act latt TC BC clos] (*‹T ∩ (wp act X ∩ awp F (X ∪ cl C (T ∩ ?r)) ∪ X) ∈ C›*))\n then have \"cl C (T∩?r) ⊆ (wp act X ∩ awp F (X ∪ cl C (T∩?r))) ∪ X\"\n by blast\n then have \"cl C (T∩?r) ⊆ ?r\"\n by (blast intro!: subset_wens (*‹?A - ?B ⊆ wp ?act ?B ∩ awp ?F (?B ∪ ?A) ⟹ ?A ⊆ wens ?F ?act ?B›*))\n then have cl_subset: \"cl C (T∩?r) ⊆ T∩?r\"\n by (simp add: cl_ident (*‹?r ∈ ?L ⟹ cl ?L ?r = ?r›*) TC (*‹T ∈ C›*) subset_trans [OF cl_mono [OF Int_lower1]] (*‹cl ?L1 ?r'1 ⊆ ?C ⟹ cl ?L1 (?r'1 ∩ ?B2) ⊆ ?C›*))\n show \"?thesis\"\n (*goal: ‹T ∩ wens F act X ∈ C›*)\n by (rule cl_subset_in_lattice [OF cl_subset latt] (*‹T ∩ wens F act X ∈ C›*))\nqed\n\ntext‹Proved on page 96 of Meier's thesis. The special case when\n \\<^term>‹T=UNIV› states that every progress set for the program \\<^term>‹F›\n and set \\<^term>‹B› includes the set \\<^term>‹wens_set F B›.›\nlemma progress_set_lemma:\n \"[|C ∈ progress_set F T B; r ∈ wens_set F B; F ∈ stable T|] ==> T∩r ∈ C\"\napply (simp add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*), clarify)\n(*goal: ‹⟦(C::'a set set) ∈ progress_set (F::'a program) (T::'a set) (B::'a set); (r::'a set) ∈ wens_set F B; F ∈ UNITY.stable T⟧ ⟹ T ∩ r ∈ C›*)\napply (erule wens_set.induct (*‹⟦(?x::?'a set) ∈ wens_set (?F::?'a program) (?B::?'a set); (?P::?'a set ⇒ bool) ?B; ⋀(X::?'a set) act::(?'a × ?'a) set. ⟦X ∈ wens_set ?F ?B; ?P X; act ∈ Acts ?F⟧ ⟹ ?P (wens ?F act X); ⋀W::?'a set set. ⟦W ≠ {}; ∀U::?'a set∈W. U ∈ wens_set ?F ?B ∧ ?P U⟧ ⟹ ?P (⋃ W)⟧ ⟹ ?P ?x›*))\n(*goal: ‹⟦r ∈ wens_set F B; F ∈ UNITY.stable T; lattice C; B ∈ C; T ∈ C; closed F T B C⟧ ⟹ T ∩ r ∈ C›*)\ntxt‹Base›\napply (simp add: Int_in_lattice (*‹⟦?x ∈ ?L; ?y ∈ ?L; lattice ?L⟧ ⟹ ?x ∩ ?y ∈ ?L›*))\n(*top goal: ‹⟦F ∈ UNITY.stable T; lattice C; B ∈ C; T ∈ C; closed F T B C⟧ ⟹ T ∩ B ∈ C› and 2 goals remain*)\ntxt‹The difficult \\<^term>‹wens› case›\napply (simp add: progress_induction_step (*‹⟦?T ∩ ?X ∈ ?C; ?act ∈ Acts ?F; ?X ∈ wens_set ?F ?B; lattice ?C; ?T ∈ ?C; ?B ∈ ?C; closed ?F ?T ?B ?C; ?F ∈ UNITY.stable ?T⟧ ⟹ ?T ∩ wens ?F ?act ?X ∈ ?C›*))\n(*top goal: ‹⋀(X::'a set) act::('a × 'a) set. ⟦(F::'a program) ∈ UNITY.stable (T::'a set); lattice (C::'a set set); (B::'a set) ∈ C; T ∈ C; closed F T B C; X ∈ wens_set F B; T ∩ X ∈ C; act ∈ Acts F⟧ ⟹ T ∩ wens F act X ∈ C› and 1 goal remains*)\ntxt‹Disjunctive case›\napply (subgoal_tac \"(⋃U∈W. T ∩ U) ∈ C\")\n(*goals:\n 1. ‹⋀W::'a set set. ⟦(F::'a program) ∈ UNITY.stable (T::'a set); lattice (C::'a set set); (B::'a set) ∈ C; T ∈ C; closed F T B C; W ≠ {}; ∀U::'a set∈W. U ∈ wens_set F B ∧ T ∩ U ∈ C; ⋃ ((∩) T ` W) ∈ C⟧ ⟹ T ∩ ⋃ W ∈ C›\n 2. ‹⋀W::'a set set. ⟦(F::'a program) ∈ UNITY.stable (T::'a set); lattice (C::'a set set); (B::'a set) ∈ C; T ∈ C; closed F T B C; W ≠ {}; ∀U::'a set∈W. U ∈ wens_set F B ∧ T ∩ U ∈ C⟧ ⟹ ⋃ ((∩) T ` W) ∈ C›\ndiscuss goal 1*)\n apply simp\n(*discuss goal 2*)\n apply (blast intro: UN_in_lattice (*‹⟦lattice (?L::?'a set set); ⋀i::?'b. i ∈ (?I::?'b set) ⟹ (?r::?'b ⇒ ?'a set) i ∈ ?L⟧ ⟹ ⋃ (?r ` ?I) ∈ ?L›*))\n(*proven 2 subgoals*) .\n\n\nsubsection ‹The Progress Set Union Theorem›\n\nlemma closed_mono:\n assumes BB': \"B ⊆ B'\"\n and TBwp: \"T ∩ (B ∪ wp act M) ∈ C\"\n and B'C: \"B' ∈ C\"\n and TC: \"T ∈ C\"\n and latt: \"lattice C\"\n shows \"T ∩ (B' ∪ wp act M) ∈ C\"\nproof (-)\n (*goal: ‹T ∩ (B' ∪ wp act M) ∈ C›*)\n from TBwp (*‹(T::'a::type set) ∩ ((B::'a::type set) ∪ wp (act::('a::type × 'a::type) set) (M::'a::type set)) ∈ (C::'a::type set set)›*) have \"(T∩B) ∪ (T ∩ wp act M) ∈ C\"\n by (simp add: Int_Un_distrib (*‹?A ∩ (?B ∪ ?C) = ?A ∩ ?B ∪ ?A ∩ ?C›*))\n then have TBBC: \"(T∩B') ∪ ((T∩B) ∪ (T ∩ wp act M)) ∈ C\"\n by (blast intro: Int_in_lattice (*‹⟦(?x::?'a set) ∈ (?L::?'a set set); (?y::?'a set) ∈ ?L; lattice ?L⟧ ⟹ ?x ∩ ?y ∈ ?L›*) Un_in_lattice (*‹⟦(?x::?'a set) ∈ (?L::?'a set set); (?y::?'a set) ∈ ?L; lattice ?L⟧ ⟹ ?x ∪ ?y ∈ ?L›*) TC (*‹(T::'a set) ∈ (C::'a set set)›*) B'C (*‹(B'::'a set) ∈ (C::'a set set)›*) latt (*‹lattice (C::'a set set)›*))\n show \"?thesis\"\n (*goal: ‹T ∩ (B' ∪ wp act M) ∈ C›*)\n apply (rule eqelem_imp_iff [THEN iffD1, OF _ TBBC] (*‹T ∩ B' ∪ (T ∩ B ∪ T ∩ wp act M) = ?y1 ⟹ ?y1 ∈ C›*))\n (*goal: ‹(T::'a::type set) ∩ ((B'::'a::type set) ∪ wp (act::('a::type × 'a::type) set) (M::'a::type set)) ∈ (C::'a::type set set)›*)\n by (blast intro: BB' [THEN subsetD] (*‹?c ∈ B ⟹ ?c ∈ B'›*))\nqed\n\n\nlemma progress_set_mono:\n assumes BB': \"B ⊆ B'\"\n shows\n \"[| B' ∈ C; C ∈ progress_set F T B|] \n ==> C ∈ progress_set F T B'\"\nby (simp add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*) closed_def (*‹closed ?F ?T ?B ?L ≡ ∀M. ∀act∈Acts ?F. ?B ⊆ M ∧ ?T ∩ M ∈ ?L ⟶ ?T ∩ (?B ∪ wp act M) ∈ ?L›*) closed_mono [OF BB'] (*‹⟦?T ∩ (B ∪ wp ?act ?M) ∈ ?C; B' ∈ ?C; ?T ∈ ?C; lattice ?C⟧ ⟹ ?T ∩ (B' ∪ wp ?act ?M) ∈ ?C›*) subset_trans [OF BB'] (*‹B' ⊆ ?C ⟹ B ⊆ ?C›*)) \n\ntheorem progress_set_Union:\n assumes leadsTo: \"F ∈ A leadsTo B'\"\n and prog: \"C ∈ progress_set F T B\"\n and Fstable: \"F ∈ stable T\"\n and BB': \"B ⊆ B'\"\n and B'C: \"B' ∈ C\"\n and Gco: \"!!X. X∈C ==> G ∈ X-B co X\"\n shows \"F⊔G ∈ T∩A leadsTo B'\"\napply (insert prog (*‹C ∈ progress_set F T B›*) Fstable (*‹F ∈ UNITY.stable T›*))\n(*goal: ‹F ⊔ G ∈ T ∩ A ⟼ B'›*)\napply (rule leadsTo_Join [OF leadsTo] (*‹⟦?T - B' ⊆ awp F ?T; ⋀X. X ∈ wens_set F B' ⟹ ?T ∩ X - B' ⊆ awp ?G (?T ∩ X)⟧ ⟹ F ⊔ ?G ∈ ?T ∩ A ⟼ B'›*))\n(*goals:\n 1. ‹⟦(C::'a set set) ∈ progress_set (F::'a program) (T::'a set) (B::'a set); F ∈ UNITY.stable T⟧ ⟹ T - (B'::'a set) ⊆ awp F T›\n 2. ‹⋀X::'a set. ⟦(C::'a set set) ∈ progress_set (F::'a program) (T::'a set) (B::'a set); F ∈ UNITY.stable T; X ∈ wens_set F (B'::'a set)⟧ ⟹ T ∩ X - B' ⊆ awp (G::'a program) (T ∩ X)›\ndiscuss goal 1*)\n apply (force simp add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*) awp_iff_stable [symmetric] (*‹(?F ∈ UNITY.stable ?A) = (?A ⊆ awp ?F ?A)›*))\n(*discuss goal 2*)\n apply (simp add: awp_iff_constrains (*‹(?A ⊆ awp ?F ?B) = (?F ∈ ?A co ?B)›*))\n (*goal: ‹⋀X. ⟦C ∈ progress_set F T B; F ∈ UNITY.stable T; X ∈ wens_set F B'⟧ ⟹ T ∩ X - B' ⊆ awp G (T ∩ X)›*)\n apply (drule progress_set_mono [OF BB' B'C] (*‹C ∈ progress_set ?F ?T B ⟹ C ∈ progress_set ?F ?T B'›*))\n (*goal: ‹⋀X. ⟦C ∈ progress_set F T B; F ∈ UNITY.stable T; X ∈ wens_set F B'⟧ ⟹ G ∈ T ∩ X - B' co T ∩ X›*)\n apply (blast intro: progress_set_lemma (*‹⟦?C ∈ progress_set ?F ?T ?B; ?r ∈ wens_set ?F ?B; ?F ∈ UNITY.stable ?T⟧ ⟹ ?T ∩ ?r ∈ ?C›*) Gco (*‹?X ∈ C ⟹ G ∈ ?X - B co ?X›*) constrains_weaken_L (*‹⟦?F ∈ ?A co ?A'; ?B ⊆ ?A⟧ ⟹ ?F ∈ ?B co ?A'›*) BB' [THEN subsetD] (*‹?c ∈ B ⟹ ?c ∈ B'›*))\n(*proven 2 subgoals*) .\n\n\nsubsection ‹Some Progress Sets›\n\nlemma UNIV_in_progress_set: \"UNIV ∈ progress_set F T B\"\nby (simp add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*) lattice_def (*‹lattice ?L ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*) closed_def (*‹closed ?F ?T ?B ?L ≡ ∀M. ∀act∈Acts ?F. ?B ⊆ M ∧ ?T ∩ M ∈ ?L ⟶ ?T ∩ (?B ∪ wp act M) ∈ ?L›*))\n\n\n\nsubsubsection ‹Lattices and Relations›\ntext‹From Meier's thesis, section 4.5.3›\n\ndefinition relcl :: \"'a set set => ('a * 'a) set\" where\n ― ‹Derived relation from a lattice›\n \"relcl L == {(x,y). y ∈ cl L {x}}\"\n \ndefinition latticeof :: \"('a * 'a) set => 'a set set\" where\n ― ‹Derived lattice from a relation: the set of upwards-closed sets›\n \"latticeof r == {X. ∀s t. s ∈ X & (s,t) ∈ r --> t ∈ X}\"\n\n\nlemma relcl_refl: \"(a,a) ∈ relcl L\"\nby (simp add: relcl_def (*‹relcl ?L ≡ {(x, y). y ∈ cl ?L {x}}›*) subset_cl [THEN subsetD] (*‹?c ∈ ?A ⟹ ?c ∈ cl ?L1 ?A›*))\n\nlemma relcl_trans:\n \"[| (a,b) ∈ relcl L; (b,c) ∈ relcl L; lattice L |] ==> (a,c) ∈ relcl L\"\napply (simp add: relcl_def (*‹relcl (?L::?'a set set) ≡ {(x::?'a, y::?'a). y ∈ cl ?L {x}}›*))\n(*goal: ‹⟦(a::'a::type, b::'a::type) ∈ relcl (L::'a::type set set); (b, c::'a::type) ∈ relcl L; lattice L⟧ ⟹ (a, c) ∈ relcl L›*)\nby (blast intro: clD (*‹⟦?c ∈ cl ?L ?r; ?B ∈ ?L; ?r ⊆ ?B⟧ ⟹ ?c ∈ ?B›*) cl_in_lattice (*‹lattice ?L ⟹ cl ?L ?r ∈ ?L›*))\n\nlemma refl_relcl: \"lattice L ==> refl (relcl L)\"\nby (simp add: refl_onI (*‹⟦?r ⊆ ?A × ?A; ⋀x. x ∈ ?A ⟹ (x, x) ∈ ?r⟧ ⟹ refl_on ?A ?r›*) relcl_def (*‹relcl ?L ≡ {(x, y). y ∈ cl ?L {x}}›*) subset_cl [THEN subsetD] (*‹?c ∈ ?A ⟹ ?c ∈ cl ?L1 ?A›*))\n\nlemma trans_relcl: \"lattice L ==> trans (relcl L)\"\nby (blast intro: relcl_trans (*‹⟦(?a, ?b) ∈ relcl ?L; (?b, ?c) ∈ relcl ?L; lattice ?L⟧ ⟹ (?a, ?c) ∈ relcl ?L›*) transI (*‹(⋀x y z. ⟦(x, y) ∈ ?r; (y, z) ∈ ?r⟧ ⟹ (x, z) ∈ ?r) ⟹ trans ?r›*))\n\nlemma lattice_latticeof: \"lattice (latticeof r)\"\nby (auto simp add: lattice_def (*‹lattice ?L ≡ (∀M⊆?L. ⋂ M ∈ ?L) ∧ (∀M⊆?L. ⋃ M ∈ ?L)›*) latticeof_def (*‹latticeof ?r ≡ {X. ∀s t. s ∈ X ∧ (s, t) ∈ ?r ⟶ t ∈ X}›*))\n\nlemma lattice_singletonI:\n \"[|lattice L; !!s. s ∈ X ==> {s} ∈ L|] ==> X ∈ L\"\napply (cut_tac UN_singleton [of X] (*‹(⋃x∈X. {x}) = X›*))\n(*goal: ‹⟦lattice L; ⋀s. s ∈ X ⟹ {s} ∈ L⟧ ⟹ X ∈ L›*)\napply (erule subst (*‹⟦(?s::?'a) = (?t::?'a); (?P::?'a ⇒ bool) ?s⟧ ⟹ ?P ?t›*))\n(*goal: ‹⟦lattice L; ⋀s. s ∈ X ⟹ {s} ∈ L; (⋃x∈X. {x}) = X⟧ ⟹ X ∈ L›*)\nby (simp only: UN_in_lattice (*‹⟦lattice (?L::?'a set set); ⋀i::?'b. i ∈ (?I::?'b set) ⟹ (?r::?'b ⇒ ?'a set) i ∈ ?L⟧ ⟹ ⋃ (?r ` ?I) ∈ ?L›*))\n\ntext‹Equation (4.71) of Meier's thesis. He gives no proof.›\nlemma cl_latticeof:\n \"[|refl r; trans r|] \n ==> cl (latticeof r) X = {t. ∃s. s∈X & (s,t) ∈ r}\" \napply (rule equalityI (*‹⟦(?A::?'a set) ⊆ (?B::?'a set); ?B ⊆ ?A⟧ ⟹ ?A = ?B›*))\n(*goals:\n 1. ‹⟦refl (r::('a::type × 'a::type) set); trans r⟧ ⟹ cl (latticeof r) (X::'a::type set) ⊆ {t::'a::type. ∃s::'a::type. s ∈ X ∧ (s, t) ∈ r}›\n 2. ‹⟦refl (r::('a::type × 'a::type) set); trans r⟧ ⟹ {t::'a::type. ∃s::'a::type. s ∈ (X::'a::type set) ∧ (s, t) ∈ r} ⊆ cl (latticeof r) X›\ndiscuss goal 1*)\n apply (rule cl_least (*‹⟦?c ∈ ?L; ?r ⊆ ?c⟧ ⟹ cl ?L ?r ⊆ ?c›*))\n (*goals:\n 1. ‹⟦refl r; trans r⟧ ⟹ {t. ∃s. s ∈ X ∧ (s, t) ∈ r} ∈ latticeof r›\n 2. ‹⟦refl r; trans r⟧ ⟹ X ⊆ {t. ∃s. s ∈ X ∧ (s, t) ∈ r}›\n discuss goal 1*)\n apply (simp (no_asm_use) add: latticeof_def (*‹latticeof ?r ≡ {X. ∀s t. s ∈ X ∧ (s, t) ∈ ?r ⟶ t ∈ X}›*) trans_def (*‹trans ?r = (∀x y z. (x, y) ∈ ?r ⟶ (y, z) ∈ ?r ⟶ (x, z) ∈ ?r)›*))\n (*top goal: ‹⟦refl r; trans r⟧ ⟹ {t. ∃s. s ∈ X ∧ (s, t) ∈ r} ∈ latticeof r› and 2 goals remain*)\n apply blast\n (*discuss goal 2*)\n apply (simp add: latticeof_def (*‹latticeof ?r ≡ {X. ∀s t. s ∈ X ∧ (s, t) ∈ ?r ⟶ t ∈ X}›*) refl_on_def (*‹refl_on ?A ?r = (?r ⊆ ?A × ?A ∧ (∀x∈?A. (x, x) ∈ ?r))›*))\n (*top goal: ‹⟦refl r; trans r⟧ ⟹ X ⊆ {t. ∃s. s ∈ X ∧ (s, t) ∈ r}› and 1 goal remains*)\n apply blast\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply (simp add: latticeof_def (*‹latticeof ?r ≡ {X. ∀s t. s ∈ X ∧ (s, t) ∈ ?r ⟶ t ∈ X}›*))\n (*goal: ‹⟦refl r; trans r⟧ ⟹ {t. ∃s. s ∈ X ∧ (s, t) ∈ r} ⊆ cl (latticeof r) X›*)\n apply clarify\n (*goal: ‹⟦refl r; trans r⟧ ⟹ {t. ∃s. s ∈ X ∧ (s, t) ∈ r} ⊆ cl {X. ∀s t. s ∈ X ∧ (s, t) ∈ r ⟶ t ∈ X} X›*)\n apply (unfold cl_def (*‹cl (?L::?'a set set) (?r::?'a set) ≡ ⋂ {x::?'a set ∈ ?L. ?r ⊆ x}›*))\n (*goal: ‹⋀x s. ⟦refl r; trans r; s ∈ X; (s, x) ∈ r⟧ ⟹ x ∈ cl {X. ∀s t. s ∈ X ∧ (s, t) ∈ r ⟶ t ∈ X} X›*)\n apply blast\n(*proven 2 subgoals*) .\n\ntext‹Related to (4.71).›\nlemma cl_eq_Collect_relcl:\n \"lattice L ==> cl L X = {t. ∃s. s∈X & (s,t) ∈ relcl L}\" \napply (cut_tac UN_singleton [of X] (*‹(⋃x∈X. {x}) = X›*))\n(*goal: ‹lattice L ⟹ cl L X = {t. ∃s. s ∈ X ∧ (s, t) ∈ relcl L}›*)\napply (erule subst (*‹⟦?s = ?t; ?P ?s⟧ ⟹ ?P ?t›*))\n(*goal: ‹⟦lattice L; (⋃x∈X. {x}) = X⟧ ⟹ cl L X = {t. ∃s. s ∈ X ∧ (s, t) ∈ relcl L}›*)\nby (force simp only: relcl_def (*‹relcl ?L ≡ {(x, y). y ∈ cl ?L {x}}›*) cl_UN (*‹lattice ?L ⟹ cl ?L (⋃ (?r ` ?I)) = (⋃i∈?I. cl ?L (?r i))›*))\n\ntext‹Meier's theorem of section 4.5.3›\ntheorem latticeof_relcl_eq: \"lattice L ==> latticeof (relcl L) = L\"\napply (rule equalityI (*‹⟦?A ⊆ ?B; ?B ⊆ ?A⟧ ⟹ ?A = ?B›*))\n(*goal: ‹lattice L ⟹ latticeof (relcl L) = L›*)\nprefer 2\n(*top goal: ‹lattice L ⟹ L ⊆ latticeof (relcl L)› and 1 goal remains*)\napply (force simp add: latticeof_def (*‹latticeof (?r::(?'a::type × ?'a::type) set) ≡ {X::?'a::type set. ∀(s::?'a::type) t::?'a::type. s ∈ X ∧ (s, t) ∈ ?r ⟶ t ∈ X}›*) relcl_def (*‹relcl (?L::?'a::type set set) ≡ {(x::?'a::type, y::?'a::type). y ∈ cl ?L {x}}›*) cl_def (*‹cl (?L::?'a::type set set) (?r::?'a::type set) ≡ ⋂ {x::?'a::type set ∈ ?L. ?r ⊆ x}›*), clarify)\n(*top goal: ‹lattice L ⟹ L ⊆ latticeof (relcl L)› and 1 goal remains*)\napply (rename_tac X)\n(*goal: ‹⋀x::'a set. ⟦lattice (L::'a set set); x ∈ latticeof (relcl L)⟧ ⟹ x ∈ L›*)\napply (rule cl_subset_in_lattice (*‹⟦cl ?L ?r ⊆ ?r; lattice ?L⟧ ⟹ ?r ∈ ?L›*))\n(*goal: ‹⋀X. ⟦lattice L; X ∈ latticeof (relcl L)⟧ ⟹ X ∈ L›*)\nprefer 2\n(*top goal: ‹⋀X. ⟦lattice L; X ∈ latticeof (relcl L)⟧ ⟹ lattice L› and 1 goal remains*)\napply assumption\n(*top goal: ‹⋀X. ⟦lattice L; X ∈ latticeof (relcl L)⟧ ⟹ lattice L› and 1 goal remains*)\napply (drule cl_ident_iff [OF lattice_latticeof, THEN iffD2] (*‹?r1 ∈ latticeof ?r2 ⟹ cl (latticeof ?r2) ?r1 = ?r1›*))\n(*goal: ‹⋀X. ⟦lattice L; X ∈ latticeof (relcl L)⟧ ⟹ cl L X ⊆ X›*)\napply (drule equalityD1 (*‹?A = ?B ⟹ ?A ⊆ ?B›*))\n(*goal: ‹⋀X::'a set. ⟦lattice (L::'a set set); cl (latticeof (relcl L)) X = X⟧ ⟹ cl L X ⊆ X›*)\napply (rule subset_trans (*‹⟦(?A::?'a::type set) ⊆ (?B::?'a::type set); ?B ⊆ (?C::?'a::type set)⟧ ⟹ ?A ⊆ ?C›*))\n(*goal: ‹⋀X. ⟦lattice L; cl (latticeof (relcl L)) X ⊆ X⟧ ⟹ cl L X ⊆ X›*)\nprefer 2\n(*top goal: ‹⋀X. ⟦lattice L; cl (latticeof (relcl L)) X ⊆ X⟧ ⟹ ?B16 X ⊆ X› and 1 goal remains*)\napply assumption\n(*top goal: ‹⋀X. ⟦lattice L; cl (latticeof (relcl L)) X ⊆ X⟧ ⟹ ?B16 X ⊆ X› and 1 goal remains*)\napply (thin_tac \"_ ⊆ X\")\n(*goal: ‹⋀X. ⟦lattice L; cl (latticeof (relcl L)) X ⊆ X⟧ ⟹ cl L X ⊆ cl (latticeof (relcl L)) X›*)\napply (cut_tac A=X in UN_singleton (*‹(⋃x∈?A. {x}) = ?A›*))\n(*goal: ‹⋀X. lattice L ⟹ cl L X ⊆ cl (latticeof (relcl L)) X›*)\napply (erule subst (*‹⟦?s = ?t; ?P ?s⟧ ⟹ ?P ?t›*))\n(*goal: ‹⋀X. ⟦lattice L; (⋃x∈X. {x}) = X⟧ ⟹ cl L X ⊆ cl (latticeof (relcl L)) X›*)\napply (simp only: cl_UN (*‹lattice ?L ⟹ cl ?L (⋃ (?r ` ?I)) = (⋃i∈?I. cl ?L (?r i))›*) lattice_latticeof (*‹lattice (latticeof ?r)›*) cl_latticeof [OF refl_relcl trans_relcl] (*‹⟦lattice ?L1; lattice ?L1⟧ ⟹ cl (latticeof (relcl ?L1)) ?X = {t. ∃s. s ∈ ?X ∧ (s, t) ∈ relcl ?L1}›*))\n(*goal: ‹⋀X. lattice L ⟹ cl L (⋃x∈X. {x}) ⊆ cl (latticeof (relcl L)) (⋃x∈X. {x})›*)\nby (simp add: relcl_def (*‹relcl ?L ≡ {(x, y). y ∈ cl ?L {x}}›*))\n\ntheorem relcl_latticeof_eq:\n \"[|refl r; trans r|] ==> relcl (latticeof r) = r\"\nby (simp add: relcl_def (*‹relcl ?L ≡ {(x, y). y ∈ cl ?L {x}}›*) cl_latticeof (*‹⟦refl ?r; trans ?r⟧ ⟹ cl (latticeof ?r) ?X = {t. ∃s. s ∈ ?X ∧ (s, t) ∈ ?r}›*))\n\n\nsubsubsection ‹Decoupling Theorems›\n\ndefinition decoupled :: \"['a program, 'a program] => bool\" where\n \"decoupled F G ==\n ∀act ∈ Acts F. ∀B. G ∈ stable B --> G ∈ stable (wp act B)\"\n\n\ntext‹Rao's Decoupling Theorem›\nlemma stableco: \"F ∈ stable A ==> F ∈ A-B co A\"\napply (simp add: stable_def (*‹UNITY.stable ?A ≡ ?A co ?A›*) constrains_def (*‹?A co ?B ≡ {F. ∀act∈Acts F. act `` ?A ⊆ ?B}›*))\n(*goal: ‹(F::'a::type program) ∈ UNITY.stable (A::'a::type set) ⟹ F ∈ A - (B::'a::type set) co A›*)\nby blast \n\ntheorem decoupling:\n assumes leadsTo: \"F ∈ A leadsTo B\"\n and Gstable: \"G ∈ stable B\"\n and dec: \"decoupled F G\"\n shows \"F⊔G ∈ A leadsTo B\"\nproof (-)\n (*goal: ‹F ⊔ G ∈ A ⟼ B›*)\n have prog: \"{X. G ∈ stable X} ∈ progress_set F UNIV B\"\n by (simp add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*) lattice_stable (*‹lattice {X. ?F ∈ UNITY.stable X}›*) Gstable (*‹G ∈ UNITY.stable B›*) closed_def (*‹closed ?F ?T ?B ?L ≡ ∀M. ∀act∈Acts ?F. ?B ⊆ M ∧ ?T ∩ M ∈ ?L ⟶ ?T ∩ (?B ∪ wp act M) ∈ ?L›*) stable_Un [OF Gstable] (*‹G ∈ UNITY.stable ?A' ⟹ G ∈ UNITY.stable (B ∪ ?A')›*) dec [unfolded decoupled_def] (*‹∀act∈Acts F. ∀B. G ∈ UNITY.stable B ⟶ G ∈ UNITY.stable (wp act B)›*))\n have \"F⊔G ∈ (UNIV∩A) leadsTo B\"\n apply (rule progress_set_Union [OF leadsTo prog] (*‹⟦F ∈ UNITY.stable UNIV; B ⊆ B; B ∈ {X. G ∈ UNITY.stable X}; ⋀X. X ∈ {X. G ∈ UNITY.stable X} ⟹ ?G ∈ X - B co X⟧ ⟹ F ⊔ ?G ∈ UNIV ∩ A ⟼ B›*))\n (*goals:\n 1. ‹F ∈ UNITY.stable UNIV›\n 2. ‹B ⊆ B›\n 3. ‹B ∈ {X. G ∈ UNITY.stable X}›\n 4. ‹⋀X. X ∈ {X. G ∈ UNITY.stable X} ⟹ G ∈ X - B co X›\n discuss goal 1*)\n apply (simp add: Gstable (*‹(G::'a::type program) ∈ UNITY.stable (B::'a::type set)›*) stableco (*‹(?F::?'a::type program) ∈ UNITY.stable (?A::?'a::type set) ⟹ ?F ∈ ?A - (?B::?'a::type set) co ?A›*))\n (*discuss goal 2*)\n apply (simp add: Gstable (*‹(G::'a program) ∈ UNITY.stable (B::'a set)›*) stableco (*‹(?F::?'a program) ∈ UNITY.stable (?A::?'a set) ⟹ ?F ∈ ?A - (?B::?'a set) co ?A›*))\n (*discuss goal 3*)\n apply (simp add: Gstable (*‹G ∈ UNITY.stable B›*) stableco (*‹?F ∈ UNITY.stable ?A ⟹ ?F ∈ ?A - ?B co ?A›*))\n (*discuss goal 4*)\n apply (simp add: Gstable (*‹G ∈ UNITY.stable B›*) stableco (*‹?F ∈ UNITY.stable ?A ⟹ ?F ∈ ?A - ?B co ?A›*))\n (*proven 4 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹(F::'a::type program) ⊔ (G::'a::type program) ∈ (A::'a::type set) ⟼ (B::'a::type set)›*)\n by simp\nqed\n\n\ntext‹Rao's Weak Decoupling Theorem›\ntheorem weak_decoupling:\n assumes leadsTo: \"F ∈ A leadsTo B\"\n and stable: \"F⊔G ∈ stable B\"\n and dec: \"decoupled F (F⊔G)\"\n shows \"F⊔G ∈ A leadsTo B\"\nproof (-)\n (*goal: ‹F ⊔ G ∈ A ⟼ B›*)\n have prog: \"{X. F⊔G ∈ stable X} ∈ progress_set F UNIV B\"\n by (simp del: Join_stable (*‹(?F ⊔ ?G ∈ UNITY.stable ?A) = (?F ∈ UNITY.stable ?A ∧ ?G ∈ UNITY.stable ?A)›*) add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*) lattice_stable (*‹lattice {X. ?F ∈ UNITY.stable X}›*) stable (*‹F ⊔ G ∈ UNITY.stable B›*) closed_def (*‹closed ?F ?T ?B ?L ≡ ∀M. ∀act∈Acts ?F. ?B ⊆ M ∧ ?T ∩ M ∈ ?L ⟶ ?T ∩ (?B ∪ wp act M) ∈ ?L›*) stable_Un [OF stable] (*‹F ⊔ G ∈ UNITY.stable ?A' ⟹ F ⊔ G ∈ UNITY.stable (B ∪ ?A')›*) dec [unfolded decoupled_def] (*‹∀act∈Acts F. ∀B. F ⊔ G ∈ UNITY.stable B ⟶ F ⊔ G ∈ UNITY.stable (wp act B)›*))\n have \"F⊔G ∈ (UNIV∩A) leadsTo B\"\n apply (rule progress_set_Union [OF leadsTo prog] (*‹⟦F ∈ UNITY.stable UNIV; B ⊆ B; B ∈ {X. F ⊔ G ∈ UNITY.stable X}; ⋀X. X ∈ {X. F ⊔ G ∈ UNITY.stable X} ⟹ ?G ∈ X - B co X⟧ ⟹ F ⊔ ?G ∈ UNIV ∩ A ⟼ B›*))\n (*goals:\n 1. ‹F ∈ UNITY.stable UNIV›\n 2. ‹B ⊆ B›\n 3. ‹B ∈ {X. F ⊔ G ∈ UNITY.stable X}›\n 4. ‹⋀X. X ∈ {X. F ⊔ G ∈ UNITY.stable X} ⟹ G ∈ X - B co X›\n discuss goal 1*)\n apply (simp del: Join_stable (*‹(?F ⊔ ?G ∈ UNITY.stable ?A) = (?F ∈ UNITY.stable ?A ∧ ?G ∈ UNITY.stable ?A)›*) add: stable (*‹F ⊔ G ∈ UNITY.stable B›*))\n (*discuss goal 2*)\n apply (simp del: Join_stable (*‹((?F::?'a program) ⊔ (?G::?'a program) ∈ UNITY.stable (?A::?'a set)) = (?F ∈ UNITY.stable ?A ∧ ?G ∈ UNITY.stable ?A)›*) add: stable (*‹(F::'a program) ⊔ (G::'a program) ∈ UNITY.stable (B::'a set)›*))\n (*discuss goal 3*)\n apply (simp del: Join_stable (*‹(?F ⊔ ?G ∈ UNITY.stable ?A) = (?F ∈ UNITY.stable ?A ∧ ?G ∈ UNITY.stable ?A)›*) add: stable (*‹F ⊔ G ∈ UNITY.stable B›*))\n (*discuss goal 4*)\n apply (simp del: Join_stable (*‹((?F::?'a program) ⊔ (?G::?'a program) ∈ UNITY.stable (?A::?'a set)) = (?F ∈ UNITY.stable ?A ∧ ?G ∈ UNITY.stable ?A)›*) add: stable (*‹(F::'a program) ⊔ (G::'a program) ∈ UNITY.stable (B::'a set)›*))\n (*goal: ‹⋀X. X ∈ {X. F ⊔ G ∈ UNITY.stable X} ⟹ G ∈ X - B co X›*)\n apply (simp add: stableco (*‹?F ∈ UNITY.stable ?A ⟹ ?F ∈ ?A - ?B co ?A›*))\n (*proven 4 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹F ⊔ G ∈ A ⟼ B›*)\n by simp\nqed\n\ntext‹The ``Decoupling via \\<^term>‹G'› Union Theorem''›\ntheorem decoupling_via_aux:\n assumes leadsTo: \"F ∈ A leadsTo B\"\n and prog: \"{X. G' ∈ stable X} ∈ progress_set F UNIV B\"\n and GG': \"G ≤ G'\" \n ― ‹Beware! This is the converse of the refinement relation!›\n shows \"F⊔G ∈ A leadsTo B\"\nproof (-)\n (*goal: ‹F ⊔ G ∈ A ⟼ B›*)\n from prog (*‹{X. G' ∈ UNITY.stable X} ∈ progress_set F UNIV B›*) have stable: \"G' ∈ stable B\"\n by (simp add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*))\n have \"F⊔G ∈ (UNIV∩A) leadsTo B\"\n apply (rule progress_set_Union [OF leadsTo prog] (*‹⟦F ∈ UNITY.stable UNIV; B ⊆ B; B ∈ {X. G' ∈ UNITY.stable X}; ⋀X. X ∈ {X. G' ∈ UNITY.stable X} ⟹ ?G ∈ X - B co X⟧ ⟹ F ⊔ ?G ∈ UNIV ∩ A ⟼ B›*))\n (*goals:\n 1. ‹F ∈ UNITY.stable UNIV›\n 2. ‹B ⊆ B›\n 3. ‹B ∈ {X. G' ∈ UNITY.stable X}›\n 4. ‹⋀X. X ∈ {X. G' ∈ UNITY.stable X} ⟹ G ∈ X - B co X›\n discuss goal 1*)\n apply (simp add: stable (*‹G' ∈ UNITY.stable B›*) stableco (*‹?F ∈ UNITY.stable ?A ⟹ ?F ∈ ?A - ?B co ?A›*) component_stable [OF GG'] (*‹G' ∈ UNITY.stable ?A ⟹ G ∈ UNITY.stable ?A›*))\n (*discuss goal 2*)\n apply (simp add: stable (*‹G' ∈ UNITY.stable B›*) stableco (*‹?F ∈ UNITY.stable ?A ⟹ ?F ∈ ?A - ?B co ?A›*) component_stable [OF GG'] (*‹G' ∈ UNITY.stable ?A ⟹ G ∈ UNITY.stable ?A›*))\n (*discuss goal 3*)\n apply (simp add: stable (*‹(G'::'a program) ∈ UNITY.stable (B::'a set)›*) stableco (*‹(?F::?'a program) ∈ UNITY.stable (?A::?'a set) ⟹ ?F ∈ ?A - (?B::?'a set) co ?A›*) component_stable [OF GG'] (*‹(G'::'a program) ∈ UNITY.stable (?A::'a set) ⟹ (G::'a program) ∈ UNITY.stable ?A›*))\n (*discuss goal 4*)\n apply (simp add: stable (*‹G' ∈ UNITY.stable B›*) stableco (*‹?F ∈ UNITY.stable ?A ⟹ ?F ∈ ?A - ?B co ?A›*) component_stable [OF GG'] (*‹G' ∈ UNITY.stable ?A ⟹ G ∈ UNITY.stable ?A›*))\n (*proven 4 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹(F::'a program) ⊔ (G::'a program) ∈ (A::'a set) ⟼ (B::'a set)›*)\n by simp\nqed\n\n\nsubsection‹Composition Theorems Based on Monotonicity and Commutativity›\n\nsubsubsection‹Commutativity of \\<^term>‹cl L› and assignment.›\ndefinition commutes :: \"['a program, 'a set, 'a set, 'a set set] => bool\" where\n \"commutes F T B L ==\n ∀M. ∀act ∈ Acts F. B ⊆ M --> \n cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T∩M)))\"\n\n\ntext‹From Meier's thesis, section 4.5.6›\nlemma commutativity1_lemma:\n assumes commutes: \"commutes F T B L\" \n and lattice: \"lattice L\"\n and BL: \"B ∈ L\"\n and TL: \"T ∈ L\"\n shows \"closed F T B L\"\napply (simp add: closed_def (*‹closed ?F ?T ?B ?L ≡ ∀M. ∀act∈Acts ?F. ?B ⊆ M ∧ ?T ∩ M ∈ ?L ⟶ ?T ∩ (?B ∪ wp act M) ∈ ?L›*))\n(*goal: ‹closed F T B L›*)\napply clarify\n(*goal: ‹∀M. B ⊆ M ∧ T ∩ M ∈ L ⟶ (∀act∈Acts F. T ∩ (B ∪ wp act M) ∈ L)›*)\napply (rule ProgressSets.cl_subset_in_lattice [OF _ lattice] (*‹cl L ?r ⊆ ?r ⟹ ?r ∈ L›*))\n(*goal: ‹⋀M act. ⟦act ∈ Acts F; B ⊆ M; T ∩ M ∈ L⟧ ⟹ T ∩ (B ∪ wp act M) ∈ L›*)\napply (simp add: Int_Un_distrib (*‹(?A::?'a set) ∩ ((?B::?'a set) ∪ (?C::?'a set)) = ?A ∩ ?B ∪ ?A ∩ ?C›*) cl_Un [OF lattice] (*‹cl (L::'a set set) ((?r::'a set) ∪ (?s::'a set)) = cl L ?r ∪ cl L ?s›*) cl_ident (*‹(?r::?'a set) ∈ (?L::?'a set set) ⟹ cl ?L ?r = ?r›*) Int_in_lattice [OF TL BL lattice] (*‹(T::'a set) ∩ (B::'a set) ∈ (L::'a set set)›*) Un_upper1 (*‹(?A::?'a set) ⊆ ?A ∪ (?B::?'a set)›*))\n(*goal: ‹⋀(M::'a::type set) act::('a::type × 'a::type) set. ⟦act ∈ Acts (F::'a::type program); (B::'a::type set) ⊆ M; (T::'a::type set) ∩ M ∈ (L::'a::type set set)⟧ ⟹ cl L (T ∩ (B ∪ wp act M)) ⊆ T ∩ (B ∪ wp act M)›*)\napply (subgoal_tac \"cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T ∩ M)))\")\n(*goals:\n 1. ‹⋀(M::'a::type set) act::('a::type × 'a::type) set. ⟦act ∈ Acts (F::'a::type program); (B::'a::type set) ⊆ M; (T::'a::type set) ∩ M ∈ (L::'a::type set set); cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T ∩ M)))⟧ ⟹ cl L (T ∩ wp act M) ⊆ T ∩ B ∪ T ∩ wp act M›\n 2. ‹⋀(M::'a::type set) act::('a::type × 'a::type) set. ⟦act ∈ Acts (F::'a::type program); (B::'a::type set) ⊆ M; (T::'a::type set) ∩ M ∈ (L::'a::type set set)⟧ ⟹ cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T ∩ M)))›\ndiscuss goal 1*)\n apply (erule subset_trans (*‹⟦?A ⊆ ?B; ?B ⊆ ?C⟧ ⟹ ?A ⊆ ?C›*))\n (*top goal: ‹⋀M act. ⟦act ∈ Acts F; B ⊆ M; T ∩ M ∈ L; cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T ∩ M)))⟧ ⟹ cl L (T ∩ wp act M) ⊆ T ∩ B ∪ T ∩ wp act M› and 1 goal remains*)\n apply (simp add: cl_ident (*‹?r ∈ ?L ⟹ cl ?L ?r = ?r›*))\n (*top goal: ‹⋀M act. ⟦act ∈ Acts F; B ⊆ M; T ∩ M ∈ L⟧ ⟹ T ∩ (B ∪ wp act (cl L (T ∩ M))) ⊆ T ∩ B ∪ T ∩ wp act M› and 1 goal remains*)\n apply (blast intro: rev_subsetD [OF _ wp_mono] (*‹⟦(?c::?'a) ∈ wp (?act1::(?'a × ?'a) set) (?A1::?'a set); ?A1 ⊆ (?B1::?'a set)⟧ ⟹ ?c ∈ wp ?act1 ?B1›*))\n(*discuss goal 2*)\n apply (cut_tac commutes (*‹commutes F T B L›*))\n (*goal: ‹⋀M act. ⟦act ∈ Acts F; B ⊆ M; T ∩ M ∈ L⟧ ⟹ cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T ∩ M)))›*)\n apply (simp add: commutes_def (*‹commutes (?F::?'a::type program) (?T::?'a::type set) (?B::?'a::type set) (?L::?'a::type set set) ≡ ∀M::?'a::type set. ∀act::(?'a::type × ?'a::type) set∈Acts ?F. ?B ⊆ M ⟶ cl ?L (?T ∩ wp act M) ⊆ ?T ∩ (?B ∪ wp act (cl ?L (?T ∩ M)))›*))\n(*proven 2 subgoals*) .\n\ntext‹Version packaged with @{thm progress_set_Union}›\nlemma commutativity1:\n assumes leadsTo: \"F ∈ A leadsTo B\"\n and lattice: \"lattice L\"\n and BL: \"B ∈ L\"\n and TL: \"T ∈ L\"\n and Fstable: \"F ∈ stable T\"\n and Gco: \"!!X. X∈L ==> G ∈ X-B co X\"\n and commutes: \"commutes F T B L\" \n shows \"F⊔G ∈ T∩A leadsTo B\"\napply (rule progress_set_Union [OF leadsTo _ Fstable subset_refl BL Gco] (*‹⟦L ∈ progress_set F T B; ⋀X. X ∈ L ⟹ X ∈ L⟧ ⟹ F ⊔ G ∈ T ∩ A ⟼ B›*))\n(*goals:\n 1. ‹(L::'a set set) ∈ progress_set (F::'a program) (T::'a set) (B::'a set)›\n 2. ‹⋀X::'a set. X ∈ (L::'a set set) ⟹ X ∈ L›\ndiscuss goal 1*)\n apply (simp add: progress_set_def (*‹progress_set ?F ?T ?B ≡ {L. lattice L ∧ ?B ∈ L ∧ ?T ∈ L ∧ closed ?F ?T ?B L}›*) commutativity1_lemma (*‹⟦commutes ?F ?T ?B ?L; lattice ?L; ?B ∈ ?L; ?T ∈ ?L⟧ ⟹ closed ?F ?T ?B ?L›*) commutes (*‹commutes F T B L›*) lattice (*‹lattice L›*) BL (*‹B ∈ L›*) TL (*‹T ∈ L›*))\n(*discuss goal 2*) .\n(*proven 2 subgoals*) \n\n\n\ntext‹Possibly move to Relation.thy, after \\<^term>‹single_valued››\ndefinition funof :: \"[('a*'b)set, 'a] => 'b\" where\n \"funof r == (λx. THE y. (x,y) ∈ r)\"\n\nlemma funof_eq: \"[|single_valued r; (x,y) ∈ r|] ==> funof r x = y\"\napply (simp add: funof_def (*‹funof (?r::(?'a × ?'b) set) ≡ λx::?'a. THE y::?'b. (x, y) ∈ ?r›*) single_valued_def (*‹single_valued (?r::(?'a × ?'b) set) = (∀(x::?'a) y::?'b. (x, y) ∈ ?r ⟶ (∀z::?'b. (x, z) ∈ ?r ⟶ y = z))›*))\n(*goal: ‹⟦single_valued r; (x, y) ∈ r⟧ ⟹ funof r x = y›*)\nby blast\n\nlemma funof_Pair_in:\n \"[|single_valued r; x ∈ Domain r|] ==> (x, funof r x) ∈ r\"\nby (force simp add: funof_eq (*‹⟦single_valued ?r; (?x, ?y) ∈ ?r⟧ ⟹ funof ?r ?x = ?y›*)) \n\nlemma funof_in:\n \"[|r``{x} ⊆ A; single_valued r; x ∈ Domain r|] ==> funof r x ∈ A\" \nby (force simp add: funof_eq (*‹⟦single_valued ?r; (?x, ?y) ∈ ?r⟧ ⟹ funof ?r ?x = ?y›*))\n \nlemma funof_imp_wp: \"[|funof act t ∈ A; single_valued act|] ==> t ∈ wp act A\"\nby (force simp add: in_wp_iff (*‹(?a ∈ wp ?act ?B) = (∀x. (?a, x) ∈ ?act ⟶ x ∈ ?B)›*) funof_eq (*‹⟦single_valued ?r; (?x, ?y) ∈ ?r⟧ ⟹ funof ?r ?x = ?y›*))\n\n\nsubsubsection‹Commutativity of Functions and Relation›\ntext‹Thesis, page 109›\n\n(*FIXME: this proof is still an ungodly mess*)\ntext‹From Meier's thesis, section 4.5.6›\nlemma commutativity2_lemma:\n assumes dcommutes: \n \"⋀act s t. act ∈ Acts F ⟹ s ∈ T ⟹ (s, t) ∈ relcl L ⟹\n s ∈ B | t ∈ B | (funof act s, funof act t) ∈ relcl L\"\n and determ: \"!!act. act ∈ Acts F ==> single_valued act\"\n and total: \"!!act. act ∈ Acts F ==> Domain act = UNIV\"\n and lattice: \"lattice L\"\n and BL: \"B ∈ L\"\n and TL: \"T ∈ L\"\n and Fstable: \"F ∈ stable T\"\n shows \"commutes F T B L\"\nproof (-)\n (*goal: ‹commutes (F::'a program) (T::'a set) (B::'a set) (L::'a set set)›*)\n {\n fix M and act and t\n assume 1: \"B ⊆ M\" \"act ∈ Acts F\" \"t ∈ cl L (T ∩ wp act M)\" (*‹(B::'a set) ⊆ (M::'a set)› ‹(act::('a × 'a) set) ∈ Acts (F::'a program)› ‹(t::'a) ∈ cl (L::'a set set) ((T::'a set) ∩ wp (act::('a × 'a) set) (M::'a set))›*)\n then have \"∃s. (s,t) ∈ relcl L ∧ s ∈ T ∩ wp act M\"\n by (force simp add: cl_eq_Collect_relcl [OF lattice] (*‹cl (L::'a set set) (?X::'a set) = {t::'a. ∃s::'a. s ∈ ?X ∧ (s, t) ∈ relcl L}›*))\n then obtain s where 2: \"(s, t) ∈ relcl L\" \"s ∈ T\" \"s ∈ wp act M\"\n (*goal: ‹(⋀s. ⟦(s, t) ∈ relcl L; s ∈ T; s ∈ wp act M⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n then have 3: \"∀u∈L. s ∈ u --> t ∈ u\"\n apply (intro ballI (*‹(⋀x::?'a::type. x ∈ (?A::?'a::type set) ⟹ (?P::?'a::type ⇒ bool) x) ⟹ ∀x::?'a::type∈?A. ?P x›*) impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹∀u∈L. s ∈ u ⟶ t ∈ u›*)\n apply (subst cl_ident [symmetric] (*‹?t ∈ ?L ⟹ ?t = cl ?L ?t›*))\n (*goals:\n 1. ‹⋀u. ⟦(s, t) ∈ relcl L; s ∈ T; s ∈ wp act M; u ∈ L; s ∈ u⟧ ⟹ u ∈ ?L5 u›\n 2. ‹⋀u. ⟦(s, t) ∈ relcl L; s ∈ T; s ∈ wp act M; u ∈ L; s ∈ u⟧ ⟹ t ∈ cl (?L5 u) u›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (simp add: relcl_def (*‹relcl ?L ≡ {(x, y). y ∈ cl ?L {x}}›*))\n (*goal: ‹⋀u. ⟦(s, t) ∈ relcl L; s ∈ T; s ∈ wp act M; u ∈ L; s ∈ u⟧ ⟹ t ∈ cl L u›*)\n apply (blast intro: cl_mono [THEN [2] rev_subsetD] (*‹⟦(?c::?'a) ∈ cl (?L1::?'a set set) (?r1::?'a set); ?r1 ⊆ (?r'1::?'a set)⟧ ⟹ ?c ∈ cl ?L1 ?r'1›*))\n (*proven 2 subgoals*) .\n with \"1\" (*‹B ⊆ M› ‹act ∈ Acts F› ‹(t::'a::type) ∈ cl (L::'a::type set set) ((T::'a::type set) ∩ wp (act::('a::type × 'a::type) set) (M::'a::type set))›*) \"2\" (*‹(s, t) ∈ relcl L› ‹s ∈ T› ‹s ∈ wp act M›*) Fstable (*‹F ∈ UNITY.stable T›*) have 4: \"funof act s ∈ T∩M\"\n by (force intro!: funof_in (*‹⟦?r `` {?x} ⊆ ?A; single_valued ?r; ?x ∈ Domain ?r⟧ ⟹ funof ?r ?x ∈ ?A›*) simp add: wp_def (*‹wp ?act ?B ≡ - ?act¯ `` (- ?B)›*) stable_def (*‹UNITY.stable ?A ≡ ?A co ?A›*) constrains_def (*‹?A co ?B ≡ {F. ∀act∈Acts F. act `` ?A ⊆ ?B}›*) determ (*‹?act ∈ Acts F ⟹ single_valued ?act›*) total (*‹?act ∈ Acts F ⟹ Domain ?act = UNIV›*))\n with \"1\" (*‹B ⊆ M› ‹act ∈ Acts F› ‹t ∈ cl L (T ∩ wp act M)›*) \"2\" (*‹(s, t) ∈ relcl L› ‹s ∈ T› ‹s ∈ wp act M›*) \"3\" (*‹∀u∈L. s ∈ u ⟶ t ∈ u›*) have 5: \"s ∈ B | t ∈ B | (funof act s, funof act t) ∈ relcl L\"\n apply (intro dcommutes (*‹⟦(?act::('a × 'a) set) ∈ Acts (F::'a program); (?s::'a) ∈ (T::'a set); (?s, ?t::'a) ∈ relcl (L::'a set set)⟧ ⟹ ?s ∈ (B::'a set) ∨ ?t ∈ B ∨ (funof ?act ?s, funof ?act ?t) ∈ relcl L›*))\n (*goals:\n 1. ‹⟦B ⊆ M; act ∈ Acts F; t ∈ cl L (T ∩ wp act M); (s, t) ∈ relcl L; s ∈ T; s ∈ wp act M; ∀u∈L. s ∈ u ⟶ t ∈ u; funof act s ∈ T ∩ M⟧ ⟹ act ∈ Acts F›\n 2. ‹⟦B ⊆ M; act ∈ Acts F; t ∈ cl L (T ∩ wp act M); (s, t) ∈ relcl L; s ∈ T; s ∈ wp act M; ∀u∈L. s ∈ u ⟶ t ∈ u; funof act s ∈ T ∩ M⟧ ⟹ s ∈ T›\n 3. ‹⟦B ⊆ M; act ∈ Acts F; t ∈ cl L (T ∩ wp act M); (s, t) ∈ relcl L; s ∈ T; s ∈ wp act M; ∀u∈L. s ∈ u ⟶ t ∈ u; funof act s ∈ T ∩ M⟧ ⟹ (s, t) ∈ relcl L›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply assumption\n (*proven 3 subgoals*) .\n with \"1\" (*‹B ⊆ M› ‹act ∈ Acts F› ‹t ∈ cl L (T ∩ wp act M)›*) \"2\" (*‹(s, t) ∈ relcl L› ‹s ∈ T› ‹s ∈ wp act M›*) \"3\" (*‹∀u::'a set∈L::'a set set. (s::'a) ∈ u ⟶ (t::'a) ∈ u›*) \"4\" (*‹funof act s ∈ T ∩ M›*) have \"t ∈ B | funof act t ∈ cl L (T∩M)\"\n apply (simp add: relcl_def (*‹relcl ?L ≡ {(x, y). y ∈ cl ?L {x}}›*))\n (*goal: ‹t ∈ B ∨ funof act t ∈ cl L (T ∩ M)›*)\n by (blast intro: BL (*‹B ∈ L›*) cl_mono [THEN [2] rev_subsetD] (*‹⟦?c ∈ cl ?L1 ?r1; ?r1 ⊆ ?r'1⟧ ⟹ ?c ∈ cl ?L1 ?r'1›*))\n with \"1\" (*‹B ⊆ M› ‹act ∈ Acts F› ‹t ∈ cl L (T ∩ wp act M)›*) \"2\" (*‹(s, t) ∈ relcl L› ‹s ∈ T› ‹s ∈ wp act M›*) \"3\" (*‹∀u∈L. s ∈ u ⟶ t ∈ u›*) \"4\" (*‹funof act s ∈ T ∩ M›*) \"5\" (*‹s ∈ B ∨ t ∈ B ∨ (funof act s, funof act t) ∈ relcl L›*) have \"t ∈ B | t ∈ wp act (cl L (T∩M))\"\n by (blast intro: funof_imp_wp (*‹⟦funof ?act ?t ∈ ?A; single_valued ?act⟧ ⟹ ?t ∈ wp ?act ?A›*) determ (*‹?act ∈ Acts F ⟹ single_valued ?act›*))\n with \"2\" (*‹(s, t) ∈ relcl L› ‹s ∈ T› ‹s ∈ wp act M›*) \"3\" (*‹∀u∈L. s ∈ u ⟶ t ∈ u›*) have \"t ∈ T ∧ (t ∈ B ∨ t ∈ wp act (cl L (T ∩ M)))\"\n by (blast intro: TL (*‹(T::'a::type set) ∈ (L::'a::type set set)›*) cl_mono [THEN [2] rev_subsetD] (*‹⟦(?c::?'a::type) ∈ cl (?L1::?'a::type set set) (?r1::?'a::type set); ?r1 ⊆ (?r'1::?'a::type set)⟧ ⟹ ?c ∈ cl ?L1 ?r'1›*))\n then have \"t ∈ T ∩ (B ∪ wp act (cl L (T ∩ M)))\"\n by simp\n }\n then show \"commutes F T B L\"\n unfolding commutes_def\n (*goal: ‹∀M. ∀act∈Acts F. B ⊆ M ⟶ cl L (T ∩ wp act M) ⊆ T ∩ (B ∪ wp act (cl L (T ∩ M)))›*)\n by clarify\nqed\n \ntext‹Version packaged with @{thm progress_set_Union}›\nlemma commutativity2:\n assumes leadsTo: \"F ∈ A leadsTo B\"\n and dcommutes: \n \"∀act ∈ Acts F. \n ∀s ∈ T. ∀t. (s,t) ∈ relcl L --> \n s ∈ B | t ∈ B | (funof act s, funof act t) ∈ relcl L\"\n and determ: \"!!act. act ∈ Acts F ==> single_valued act\"\n and total: \"!!act. act ∈ Acts F ==> Domain act = UNIV\"\n and lattice: \"lattice L\"\n and BL: \"B ∈ L\"\n and TL: \"T ∈ L\"\n and Fstable: \"F ∈ stable T\"\n and Gco: \"!!X. X∈L ==> G ∈ X-B co X\"\n shows \"F⊔G ∈ T∩A leadsTo B\"\napply (rule commutativity1 [OF leadsTo lattice] (*‹⟦B ∈ L; ?T ∈ L; F ∈ UNITY.stable ?T; ⋀X. X ∈ L ⟹ ?G ∈ X - B co X; commutes F ?T B L⟧ ⟹ F ⊔ ?G ∈ ?T ∩ A ⟼ B›*))\n(*goals:\n 1. ‹B ∈ L›\n 2. ‹T ∈ L›\n 3. ‹F ∈ UNITY.stable T›\n 4. ‹⋀X. X ∈ L ⟹ G ∈ X - B co X›\n 5. ‹commutes F T B L›\ndiscuss goal 1*)\n apply (simp add: Gco (*‹(?X::'a set) ∈ (L::'a set set) ⟹ (G::'a program) ∈ ?X - (B::'a set) co ?X›*) commutativity2_lemma (*‹⟦⋀(act::(?'a × ?'a) set) (s::?'a) t::?'a. ⟦act ∈ Acts (?F::?'a program); s ∈ (?T::?'a set); (s, t) ∈ relcl (?L::?'a set set)⟧ ⟹ s ∈ (?B::?'a set) ∨ t ∈ ?B ∨ (funof act s, funof act t) ∈ relcl ?L; ⋀act::(?'a × ?'a) set. act ∈ Acts ?F ⟹ single_valued act; ⋀act::(?'a × ?'a) set. act ∈ Acts ?F ⟹ Domain act = UNIV; lattice ?L; ?B ∈ ?L; ?T ∈ ?L; ?F ∈ UNITY.stable ?T⟧ ⟹ commutes ?F ?T ?B ?L›*) dcommutes (*‹∀act::('a × 'a) set∈Acts (F::'a program). ∀s::'a∈T::'a set. ∀t::'a. (s, t) ∈ relcl (L::'a set set) ⟶ s ∈ (B::'a set) ∨ t ∈ B ∨ (funof act s, funof act t) ∈ relcl L›*) determ (*‹(?act::('a × 'a) set) ∈ Acts (F::'a program) ⟹ single_valued ?act›*) total (*‹(?act::('a × 'a) set) ∈ Acts (F::'a program) ⟹ Domain ?act = UNIV›*) lattice (*‹lattice (L::'a set set)›*) BL (*‹(B::'a set) ∈ (L::'a set set)›*) TL (*‹(T::'a set) ∈ (L::'a set set)›*) Fstable (*‹(F::'a program) ∈ UNITY.stable (T::'a set)›*))\n(*discuss goal 2*)\n apply (simp add: Gco (*‹?X ∈ L ⟹ G ∈ ?X - B co ?X›*) commutativity2_lemma (*‹⟦⋀act s t. ⟦act ∈ Acts ?F; s ∈ ?T; (s, t) ∈ relcl ?L⟧ ⟹ s ∈ ?B ∨ t ∈ ?B ∨ (funof act s, funof act t) ∈ relcl ?L; ⋀act. act ∈ Acts ?F ⟹ single_valued act; ⋀act. act ∈ Acts ?F ⟹ Domain act = UNIV; lattice ?L; ?B ∈ ?L; ?T ∈ ?L; ?F ∈ UNITY.stable ?T⟧ ⟹ commutes ?F ?T ?B ?L›*) dcommutes (*‹∀act∈Acts F. ∀s∈T. ∀t. (s, t) ∈ relcl L ⟶ s ∈ B ∨ t ∈ B ∨ (funof act s, funof act t) ∈ relcl L›*) determ (*‹?act ∈ Acts F ⟹ single_valued ?act›*) total (*‹?act ∈ Acts F ⟹ Domain ?act = UNIV›*) lattice (*‹lattice L›*) BL (*‹B ∈ L›*) TL (*‹T ∈ L›*) Fstable (*‹F ∈ UNITY.stable T›*))\n(*discuss goal 3*)\n apply (simp add: Gco (*‹?X ∈ L ⟹ G ∈ ?X - B co ?X›*) commutativity2_lemma (*‹⟦⋀act s t. ⟦act ∈ Acts ?F; s ∈ ?T; (s, t) ∈ relcl ?L⟧ ⟹ s ∈ ?B ∨ t ∈ ?B ∨ (funof act s, funof act t) ∈ relcl ?L; ⋀act. act ∈ Acts ?F ⟹ single_valued act; ⋀act. act ∈ Acts ?F ⟹ Domain act = UNIV; lattice ?L; ?B ∈ ?L; ?T ∈ ?L; ?F ∈ UNITY.stable ?T⟧ ⟹ commutes ?F ?T ?B ?L›*) dcommutes (*‹∀act∈Acts F. ∀s∈T. ∀t. (s, t) ∈ relcl L ⟶ s ∈ B ∨ t ∈ B ∨ (funof act s, funof act t) ∈ relcl L›*) determ (*‹?act ∈ Acts F ⟹ single_valued ?act›*) total (*‹?act ∈ Acts F ⟹ Domain ?act = UNIV›*) lattice (*‹lattice L›*) BL (*‹B ∈ L›*) TL (*‹T ∈ L›*) Fstable (*‹F ∈ UNITY.stable T›*))\n(*discuss goal 4*)\n apply (simp add: Gco (*‹?X ∈ L ⟹ G ∈ ?X - B co ?X›*) commutativity2_lemma (*‹⟦⋀act s t. ⟦act ∈ Acts ?F; s ∈ ?T; (s, t) ∈ relcl ?L⟧ ⟹ s ∈ ?B ∨ t ∈ ?B ∨ (funof act s, funof act t) ∈ relcl ?L; ⋀act. act ∈ Acts ?F ⟹ single_valued act; ⋀act. act ∈ Acts ?F ⟹ Domain act = UNIV; lattice ?L; ?B ∈ ?L; ?T ∈ ?L; ?F ∈ UNITY.stable ?T⟧ ⟹ commutes ?F ?T ?B ?L›*) dcommutes (*‹∀act∈Acts F. ∀s∈T. ∀t. (s, t) ∈ relcl L ⟶ s ∈ B ∨ t ∈ B ∨ (funof act s, funof act t) ∈ relcl L›*) determ (*‹?act ∈ Acts F ⟹ single_valued ?act›*) total (*‹?act ∈ Acts F ⟹ Domain ?act = UNIV›*) lattice (*‹lattice L›*) BL (*‹B ∈ L›*) TL (*‹T ∈ L›*) Fstable (*‹F ∈ UNITY.stable T›*))\n(*discuss goal 5*)\n apply (simp add: Gco (*‹?X ∈ L ⟹ G ∈ ?X - B co ?X›*) commutativity2_lemma (*‹⟦⋀act s t. ⟦act ∈ Acts ?F; s ∈ ?T; (s, t) ∈ relcl ?L⟧ ⟹ s ∈ ?B ∨ t ∈ ?B ∨ (funof act s, funof act t) ∈ relcl ?L; ⋀act. act ∈ Acts ?F ⟹ single_valued act; ⋀act. act ∈ Acts ?F ⟹ Domain act = UNIV; lattice ?L; ?B ∈ ?L; ?T ∈ ?L; ?F ∈ UNITY.stable ?T⟧ ⟹ commutes ?F ?T ?B ?L›*) dcommutes (*‹∀act∈Acts F. ∀s∈T. ∀t. (s, t) ∈ relcl L ⟶ s ∈ B ∨ t ∈ B ∨ (funof act s, funof act t) ∈ relcl L›*) determ (*‹?act ∈ Acts F ⟹ single_valued ?act›*) total (*‹?act ∈ Acts F ⟹ Domain ?act = UNIV›*) lattice (*‹lattice L›*) BL (*‹B ∈ L›*) TL (*‹T ∈ L›*) Fstable (*‹F ∈ UNITY.stable T›*))\n(*proven 5 subgoals*) .\n\n\nsubsection ‹Monotonicity›\ntext‹From Meier's thesis, section 4.5.7, page 110›\n(*to be continued?*)\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/UNITY/ProgressSets.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"5d4f04e37f6e967b0f16b7c6d2056d04c7020784a38acbc97c72b6536562cfe1\"\n}"}}},{"rowIdx":167,"cells":{"text":{"kind":"string","value":"(* Author: Tobias Nipkow *)\n\nsection \"Tries (List Version)\"\n\ntheory Tries\nimports Trie\nbegin\n\ntext‹This is a specialization of tries where values are lists.›\n\ntype_synonym ('k,'v)tries = \"('k,'v list)trie\"\n\ndefinition lookup_tries :: \"('k,'v)tries ⇒ 'k list ⇒ 'v list\" where\n\"lookup_tries t ks = (case lookup_trie t ks of None ⇒ [] | Some vs ⇒ vs)\"\n\ndefinition update_with_tries ::\n \"'k list ⇒ ('v list ⇒ 'v list) ⇒ ('k, 'v) tries ⇒ ('k, 'v) tries\" where\n\"update_with_tries ks f =\n update_with_trie ks (λvo. case vo of None ⇒ f [] | Some vs ⇒ f vs)\"\n\ndefinition insert_tries :: \"'k list ⇒ 'v ⇒ ('k,'v)tries ⇒ ('k,'v)tries\" where\n\"insert_tries ks v =\n update_with_trie ks (λvo. case vo of None ⇒ [v] | Some vs ⇒ v#vs)\"\n\ndefinition inserts_tries :: \"('v ⇒ 'k list) ⇒ 'v list ⇒ ('k,'v)tries ⇒ ('k,'v)tries\" where\n\"inserts_tries key = fold (%v. insert_tries (key v) v)\"\n\ndefinition tries_of_list :: \"('v ⇒ 'k list) ⇒ 'v list ⇒ ('k,'v)tries\" where\n\"tries_of_list key vs = inserts_tries key vs (Trie None [])\"\n\ndefinition set_tries :: \"('k,'v)tries ⇒ 'v set\" where\n\"set_tries t = Union {gs. ∃a. gs = set(lookup_tries t a)}\"\n\ndefinition all_tries :: \"('v ⇒ bool) ⇒ ('k,'v)tries ⇒ bool\" where\n\"all_tries P = all_trie (list_all P)\"\n\n\nsubsection ‹@{const lookup_tries}›\n\nlemma lookup_Nil[simp]:\n \"lookup_tries (Trie vo ps) [] = (case vo of None ⇒ [] | Some vs ⇒ vs)\"\nby (simp add: lookup_tries_def (*‹lookup_tries ?t ?ks = (case lookup_trie ?t ?ks of None ⇒ [] | Some vs ⇒ vs)›*))\n\nlemma lookup_Cons[simp]: \"lookup_tries (Trie vo ps) (a#as) =\n (case map_of ps a of None ⇒ [] | Some at ⇒ lookup_tries at as)\"\nby (simp add: lookup_tries_def (*‹lookup_tries ?t ?ks = (case lookup_trie ?t ?ks of None ⇒ [] | Some vs ⇒ vs)›*) split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma lookup_empty[simp]: \"lookup_tries (Trie None []) as = []\"\napply (case_tac as)\n(*goals:\n 1. ‹as = [] ⟹ lookup_tries (Trie None []) as = []›\n 2. ‹⋀a list. as = a # list ⟹ lookup_tries (Trie None []) as = []›\ndiscuss goal 1*)\n apply (simp add: lookup_tries_def (*‹lookup_tries ?t ?ks = (case lookup_trie ?t ?ks of None ⇒ [] | Some vs ⇒ vs)›*))\n(*discuss goal 2*)\n apply (simp add: lookup_tries_def (*‹lookup_tries (?t::(?'k::type, ?'v::type list) trie) (?ks::?'k::type list) = (case lookup_trie ?t ?ks of None ⇒ [] | Some (vs::?'v::type list) ⇒ vs)›*))\n(*proven 2 subgoals*) .\n\ntheorem lookup_update:\n \"lookup_tries (update_trie as vs t) bs =\n (if as=bs then vs else lookup_tries t bs)\"\nby (auto simp add: lookup_tries_def (*‹lookup_tries ?t ?ks = (case lookup_trie ?t ?ks of None ⇒ [] | Some vs ⇒ vs)›*) lookup_update (*‹lookup_trie (update_trie ?ks ?v ?t) ?ks' = (if ?ks = ?ks' then Some ?v else lookup_trie ?t ?ks')›*))\n\ntheorem lookup_update_with:\n \"lookup_tries (update_with_tries as f t) bs =\n (if as=bs then f(lookup_tries t as) else lookup_tries t bs)\"\nby (auto simp add: lookup_tries_def (*‹lookup_tries ?t ?ks = (case lookup_trie ?t ?ks of None ⇒ [] | Some vs ⇒ vs)›*) update_with_tries_def (*‹update_with_tries ?ks ?f = update_with_trie ?ks (λvo. case vo of None ⇒ ?f [] | Some vs ⇒ ?f vs)›*) lookup_update_with_trie (*‹lookup_trie (update_with_trie ?ks ?f ?t) ?ks' = (if ?ks' = ?ks then Some (?f (lookup_trie ?t ?ks')) else lookup_trie ?t ?ks')›*) split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\n\nsubsection ‹@{const insert_tries}, @{const inserts_tries}, @{const tries_of_list}›\n\nlemma invar_insert_tries: \"invar_trie t ⟹ invar_trie(insert_tries as v t)\"\nby (simp add: insert_tries_def (*‹insert_tries ?ks ?v = update_with_trie ?ks (λvo. case vo of None ⇒ [?v] | Some vs ⇒ ?v # vs)›*) invar_update_with_trie (*‹invar_trie ?t ⟹ invar_trie (update_with_trie ?ks ?f ?t)›*) split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma invar_inserts_tries:\n \"invar_trie t ⟹ invar_trie (inserts_tries key xs t)\"\napply (induct xs arbitrary: t)\n(*goals:\n 1. ‹⋀t. invar_trie t ⟹ invar_trie (inserts_tries key [] t)›\n 2. ‹⋀a xs t. ⟦⋀t. invar_trie t ⟹ invar_trie (inserts_tries key xs t); invar_trie t⟧ ⟹ invar_trie (inserts_tries key (a # xs) t)›\ndiscuss goal 1*)\n apply ((auto simp: invar_insert_tries (*‹invar_trie ?t ⟹ invar_trie (insert_tries ?as ?v ?t)›*) inserts_tries_def (*‹inserts_tries ?key = fold (λv. insert_tries (?key v) v)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: invar_insert_tries (*‹invar_trie ?t ⟹ invar_trie (insert_tries ?as ?v ?t)›*) inserts_tries_def (*‹inserts_tries ?key = fold (λv. insert_tries (?key v) v)›*))[1])\n(*proven 2 subgoals*) .\n\nlemma invar_of_list: \"invar_trie (tries_of_list key xs)\"\nby (simp add: tries_of_list_def (*‹tries_of_list ?key ?vs = inserts_tries ?key ?vs (Trie None [])›*) invar_inserts_tries (*‹invar_trie ?t ⟹ invar_trie (inserts_tries ?key ?xs ?t)›*))\n\nlemma set_lookup_insert_tries: \"set (lookup_tries (insert_tries ks a t) ks') =\n (if ks' = ks then Set.insert a (set(lookup_tries t ks')) else set(lookup_tries t ks'))\"\nby (simp add: lookup_tries_def (*‹lookup_tries ?t ?ks = (case lookup_trie ?t ?ks of None ⇒ [] | Some vs ⇒ vs)›*) insert_tries_def (*‹insert_tries ?ks ?v = update_with_trie ?ks (λvo. case vo of None ⇒ [?v] | Some vs ⇒ ?v # vs)›*) lookup_update_with_trie (*‹lookup_trie (update_with_trie ?ks ?f ?t) ?ks' = (if ?ks' = ?ks then Some (?f (lookup_trie ?t ?ks')) else lookup_trie ?t ?ks')›*) set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*) split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma in_set_lookup_inserts_tries:\n \"(v ∈ set(lookup_tries (inserts_tries key vs t) (key v))) =\n (v ∈ set vs ∪ set(lookup_tries t (key v)))\"\napply (induct vs arbitrary: t)\n(*goals:\n 1. ‹⋀t::('b, 'a list) trie. ((v::'a) ∈ set (lookup_tries (inserts_tries (key::'a ⇒ 'b list) [] t) (key v))) = (v ∈ set [] ∪ set (lookup_tries t (key v)))›\n 2. ‹⋀(a::'a) (vs::'a list) t::('b, 'a list) trie. (⋀t::('b, 'a list) trie. ((v::'a) ∈ set (lookup_tries (inserts_tries (key::'a ⇒ 'b list) vs t) (key v))) = (v ∈ set vs ∪ set (lookup_tries t (key v)))) ⟹ (v ∈ set (lookup_tries (inserts_tries key (a # vs) t) (key v))) = (v ∈ set (a # vs) ∪ set (lookup_tries t (key v)))›\ndiscuss goal 1*)\n apply ((auto simp add: inserts_tries_def (*‹inserts_tries (?key::?'v::type ⇒ ?'k::type list) = fold (λv::?'v::type. insert_tries (?key v) v)›*) set_lookup_insert_tries (*‹set (lookup_tries (insert_tries (?ks::?'b::type list) (?a::?'a::type) (?t::(?'b::type, ?'a::type list) trie)) (?ks'::?'b::type list)) = (if ?ks' = ?ks then insert ?a (set (lookup_tries ?t ?ks')) else set (lookup_tries ?t ?ks'))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp add: inserts_tries_def (*‹inserts_tries ?key = fold (λv. insert_tries (?key v) v)›*) set_lookup_insert_tries (*‹set (lookup_tries (insert_tries ?ks ?a ?t) ?ks') = (if ?ks' = ?ks then insert ?a (set (lookup_tries ?t ?ks')) else set (lookup_tries ?t ?ks'))›*))[1])\n(*proven 2 subgoals*) .\n\nlemma in_set_lookup_of_list:\n \"v ∈ set(lookup_tries (tries_of_list key vs) (key v)) = (v ∈ set vs)\"\nby (simp add: tries_of_list_def (*‹tries_of_list ?key ?vs = inserts_tries ?key ?vs (Trie None [])›*) in_set_lookup_inserts_tries (*‹(?v ∈ set (lookup_tries (inserts_tries ?key ?vs ?t) (?key ?v))) = (?v ∈ set ?vs ∪ set (lookup_tries ?t (?key ?v)))›*))\n\nlemma in_set_lookup_inserts_triesD:\n \"v ∈ set(lookup_tries (inserts_tries key vs t) xs) ⟹\n v ∈ set vs ∪ set(lookup_tries t xs)\"\napply (induct vs arbitrary: t)\n(*goals:\n 1. ‹⋀t. v ∈ set (lookup_tries (inserts_tries key [] t) xs) ⟹ v ∈ set [] ∪ set (lookup_tries t xs)›\n 2. ‹⋀a vs t. ⟦⋀t. v ∈ set (lookup_tries (inserts_tries key vs t) xs) ⟹ v ∈ set vs ∪ set (lookup_tries t xs); v ∈ set (lookup_tries (inserts_tries key (a # vs) t) xs)⟧ ⟹ v ∈ set (a # vs) ∪ set (lookup_tries t xs)›\ndiscuss goal 1*)\n apply (simp add: inserts_tries_def (*‹inserts_tries ?key = fold (λv. insert_tries (?key v) v)›*))\n(*discuss goal 2*)\n apply (simp add: inserts_tries_def (*‹inserts_tries ?key = fold (λv. insert_tries (?key v) v)›*))\n (*goal: ‹⋀a vs t. ⟦⋀t. v ∈ set (lookup_tries (inserts_tries key vs t) xs) ⟹ v ∈ set vs ∪ set (lookup_tries t xs); v ∈ set (lookup_tries (inserts_tries key (a # vs) t) xs)⟧ ⟹ v ∈ set (a # vs) ∪ set (lookup_tries t xs)›*)\n apply (fastforce simp add: set_lookup_insert_tries (*‹set (lookup_tries (insert_tries ?ks ?a ?t) ?ks') = (if ?ks' = ?ks then insert ?a (set (lookup_tries ?t ?ks')) else set (lookup_tries ?t ?ks'))›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n(*proven 2 subgoals*) .\n \nlemma in_set_lookup_of_listD:\n \"v ∈ set(lookup_tries (tries_of_list f vs) xs) ⟹ v ∈ set vs\"\nsorry\n\n\nsubsection ‹@{const set_tries}›\n\nlemma set_tries_eq_ran: \"set_tries t = Union(set ` ran(lookup_trie t))\"\napply (auto simp add: set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*) set_tries_def (*‹set_tries ?t = ⋃ {gs. ∃a. gs = set (lookup_tries ?t a)}›*) lookup_tries_def (*‹lookup_tries ?t ?ks = (case lookup_trie ?t ?ks of None ⇒ [] | Some vs ⇒ vs)›*) ran_def (*‹ran ?m = {b. ∃a. ?m a = Some b}›*))\n(*goals:\n 1. ‹⋀x xa a uu_. ⟦∀x. (x ∈ xa) = (∃uu_. Some uu_ = lookup_trie t a ∧ x ∈ set uu_); Some uu_ = lookup_trie t a; x ∈ set uu_⟧ ⟹ ∃xa. (∃a. lookup_trie t a = Some xa) ∧ x ∈ set xa›\n 2. ‹⋀x xa a. ⟦x ∈ set xa; lookup_trie t a = Some xa⟧ ⟹ ∃xa. (∃a. ∀x. (x ∈ xa) = (∃uu_. Some uu_ = lookup_trie t a ∧ x ∈ set uu_)) ∧ x ∈ xa›\ndiscuss goal 1*)\n apply metis\n(*discuss goal 2*)\n apply (metis option.inject (*‹(Some ?x2.0 = Some ?y2.0) = (?x2.0 = ?y2.0)›*))\n(*proven 2 subgoals*) .\n\nlemma set_tries_empty[simp]: \"set_tries (Trie None []) = {}\"\nby (simp add: set_tries_def (*‹set_tries ?t = ⋃ {gs. ∃a. gs = set (lookup_tries ?t a)}›*))\n\nlemma set_tries_insert[simp]:\n \"set_tries (insert_tries a x t) = Set.insert x (set_tries t)\"\napply(auto simp: set_tries_def lookup_update set_lookup_insert_tries)\nby (metis insert_iff)\n\nlemma set_insert_tries:\n \"set_tries (inserts_tries key xs t) =\n set xs Un set_tries t\"\napply (induct xs arbitrary: t)\n(*goals:\n 1. ‹⋀t. set_tries (inserts_tries key [] t) = set [] ∪ set_tries t›\n 2. ‹⋀a xs t. (⋀t. set_tries (inserts_tries key xs t) = set xs ∪ set_tries t) ⟹ set_tries (inserts_tries key (a # xs) t) = set (a # xs) ∪ set_tries t›\ndiscuss goal 1*)\n apply ((auto simp: inserts_tries_def (*‹inserts_tries ?key = fold (λv. insert_tries (?key v) v)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: inserts_tries_def (*‹inserts_tries ?key = fold (λv. insert_tries (?key v) v)›*))[1])\n(*proven 2 subgoals*) .\n\nlemma set_tries_of_list[simp]:\n \"set_tries(tries_of_list key xs) = set xs\"\nby (simp add: tries_of_list_def (*‹tries_of_list ?key ?vs = inserts_tries ?key ?vs (Trie None [])›*) set_insert_tries (*‹set_tries (inserts_tries ?key ?xs ?t) = set ?xs ∪ set_tries ?t›*))\n\nlemma in_set_lookup_set_triesD:\n \"x∈set (lookup_tries t a) ⟹ x ∈ set_tries t\"\nby (auto simp: set_tries_def (*‹set_tries ?t = ⋃ {gs. ∃a. gs = set (lookup_tries ?t a)}›*))\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Trie/Tries.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"9d42301d5219b3a4ce584f64c749e7482d429206ffa61c512b6c3ce430d82d2a\"\n}"}}},{"rowIdx":168,"cells":{"text":{"kind":"string","value":"theory Fused_Resource imports\n Fold_Spmf\nbegin\n\n(* TODO: move these somewhere *)\ncontext includes ℐ.lifting begin\nlift_definition eℐ :: \"('a, 'b) ℐ ⇒ ('a, 'b × 'c) ℐ\" is \"λℐ x. ℐ x × UNIV\" .\n\nlemma outs_ℐ_eℐ[simp]: \"outs_ℐ (eℐ ℐ) = outs_ℐ ℐ\"\n apply transfer\n (*goal: ‹outs_ℐ (eℐ ℐ) = outs_ℐ ℐ›*)\n by simp\n\nlemma responses_ℐ_eℐ [simp]: \"responses_ℐ (eℐ ℐ) x = responses_ℐ ℐ x × UNIV\"\n apply transfer\n (*goal: ‹responses_ℐ (eℐ (ℐ::('c::type, 'a::type) ℐ)) (x::'c::type) = responses_ℐ ℐ x × UNIV›*)\n by simp\n\nlemma eℐ_map_ℐ: \"eℐ (map_ℐ f g ℐ) = map_ℐ f (apfst g) (eℐ ℐ)\"\n apply transfer\n (*goal: ‹eℐ (map_ℐ f g ℐ) = map_ℐ f (apfst g) (eℐ ℐ)›*)\n by (auto simp add: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) intro: rev_image_eqI (*‹⟦?x ∈ ?A; ?b = ?f ?x⟧ ⟹ ?b ∈ ?f ` ?A›*))\n\nlemma eℐ_inverse [simp]: \"map_ℐ id fst (eℐ ℐ) = ℐ\"\n apply transfer\n (*goal: ‹map_ℐ id fst (eℐ ℐ) = ℐ›*)\n by auto\nend\nlifting_update ℐ.lifting\nlifting_forget ℐ.lifting\n\n\nsection ‹Fused Resource›\n\nsubsection ‹Event Oracles -- they generate events›\n\ntype_synonym \n ('state, 'event, 'input, 'output) eoracle = \"('state, 'input, 'output × 'event list) oracle'\" \n\ndefinition\n parallel_eoracle :: \"\n ('s1, 'e1, 'i1, 'o1) eoracle ⇒ ('s2, 'e2, 'i2, 'o2) eoracle ⇒\n ('s1 × 's2, 'e1 + 'e2, 'i1 + 'i2, 'o1 + 'o2) eoracle\"\n where\n \"parallel_eoracle eoracle1 eoracle2 state ≡\n comp\n (map_spmf \n (map_prod \n (case_sum \n (map_prod Inl (map Inl)) \n (map_prod Inr (map Inr))) \n id))\n (parallel_oracle eoracle1 eoracle2 state)\"\n\ndefinition\n plus_eoracle :: \"\n ('s, 'e1, 'i1, 'o1) eoracle ⇒ ('s, 'e2, 'i2, 'o2) eoracle ⇒\n ('s, 'e1 + 'e2, 'i1 + 'i2, 'o1 + 'o2) eoracle\"\n where\n \"plus_eoracle eoracle1 eoracle2 state ≡\n comp\n (map_spmf \n (map_prod \n (case_sum \n (map_prod Inl (map Inl)) \n (map_prod Inr (map Inr))) \n id))\n (plus_oracle eoracle1 eoracle2 state)\"\n\ndefinition\n translate_eoracle :: \"\n ('s_event, 'e1, 'e2 list) oracle' ⇒ ('s_event × 's, 'e1, 'i, 'o) eoracle ⇒\n ('s_event × 's, 'e2, 'i, 'o) eoracle\"\n where\n \"translate_eoracle translator eoracle state inp ≡ \n bind_spmf\n (eoracle state inp)\n (λ((out, e_in), s).\n let conc = (λ(es, st) e. map_spmf (map_prod ((@) es) id) (translator st e)) in do {\n (e_out, s_event) ← foldl_spmf conc (return_spmf ([], fst s)) e_in;\n return_spmf ((out, e_out), s_event, snd s)\n })\"\n\n\nsubsection ‹Event Handlers -- they absorb (and silently handle) events›\n\ntype_synonym\n ('state, 'event) handler = \"'state ⇒ 'event ⇒ 'state spmf\"\n\nfun\n parallel_handler :: \"('s1, 'e1) handler ⇒ ('s2, 'e2) handler ⇒ ('s1 × 's2, 'e1 + 'e2) handler\"\n where\n \"parallel_handler left _ s (Inl e1) = map_spmf (λs1'. (s1', snd s)) (left (fst s) e1)\"\n | \"parallel_handler _ right s (Inr e2) = map_spmf (λs2'. (fst s, s2')) (right (snd s) e2)\"\n\ndefinition\n plus_handler :: \"('s, 'e1) handler ⇒ ('s, 'e2) handler ⇒ ('s, 'e1 + 'e2) handler\"\n where\n \"plus_handler left right s ≡ case_sum (left s) (right s)\"\n\nlemma parallel_handler_left:\n \"map_fun id (map_fun Inl id) (parallel_handler left right) = \n (λ(s_l, s_r) q. map_spmf (λs_l'. (s_l', s_r)) (left s_l q))\"\n by force\n\nlemma parallel_handler_right:\n \"map_fun id (map_fun Inr id) (parallel_handler left right) = \n (λ(s_l, s_r) q. map_spmf (λs_r'. (s_l, s_r')) (right s_r q))\"\n by force\n\nlemma in_set_spmf_parallel_handler:\n \"s' ∈ set_spmf (parallel_handler left right s x) ⟷\n (case x of Inl e ⇒ fst s' ∈ set_spmf (left (fst s) e) ∧ snd s' = snd s\n | Inr e ⇒ snd s' ∈ set_spmf (right (snd s) e) ∧ fst s' = fst s)\"\n apply (cases s)\n (*goal: ‹((s'::'a::type × 'b::type) ∈ set_spmf (parallel_handler (left::'a::type ⇒ 'c::type ⇒ 'a::type spmf) (right::'b::type ⇒ 'd::type ⇒ 'b::type spmf) (s::'a::type × 'b::type) (x::'c::type + 'd::type))) = (case x of Inl (e::'c::type) ⇒ fst s' ∈ set_spmf (left (fst s) e) ∧ snd s' = snd s | Inr (e::'d::type) ⇒ snd s' ∈ set_spmf (right (snd s) e) ∧ fst s' = fst s)›*)\n apply (cases s')\n (*goal: ‹⋀a b. s = (a, b) ⟹ (s' ∈ set_spmf (parallel_handler left right s x)) = (case x of Inl e ⇒ fst s' ∈ set_spmf (left (fst s) e) ∧ snd s' = snd s | Inr e ⇒ snd s' ∈ set_spmf (right (snd s) e) ∧ fst s' = fst s)›*)\n by (auto split: sum.split (*‹(?P::?'c::type ⇒ bool) (case ?sum::?'a::type + ?'b::type of Inl (x::?'a::type) ⇒ (?f1.0::?'a::type ⇒ ?'c::type) x | Inr (x::?'b::type) ⇒ (?f2.0::?'b::type ⇒ ?'c::type) x) = ((∀x1::?'a::type. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2::?'b::type. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))\n\nsubsection ‹Fused Resource Construction›\n\ncodatatype \n ('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core =\n Core\n (cpoke: \"('s_core, 'event) handler\")\n (cfunc_adv: \"('s_core, 'iadv_core, 'oadv_core) oracle'\")\n (cfunc_usr: \"('s_core, 'iusr_core, 'ousr_core) oracle'\")\n\ndeclare core.sel_transfer[transfer_rule del]\ndeclare core.ctr_transfer[transfer_rule del]\ndeclare core.case_transfer[transfer_rule del]\n\ncontext \n includes lifting_syntax \nbegin\n\ninductive \n rel_core':: \"\n ('s_core ⇒ 's_core' ⇒ bool) ⇒ \n ('event ⇒ 'event' ⇒ bool) ⇒ \n ('iadv_core ⇒ 'iadv_core' ⇒ bool) ⇒\n ('iusr_core ⇒ 'iusr_core' ⇒ bool) ⇒ \n ('oadv_core ⇒ 'oadv_core' ⇒ bool) ⇒ \n ('ousr_core ⇒ 'ousr_core' ⇒ bool) ⇒\n ('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core ⇒ \n ('s_core', 'event', 'iadv_core', 'iusr_core', 'oadv_core', 'ousr_core') core ⇒ bool\"\n for S E IA IU OA OU\n where \"rel_core' S E IA IU OA OU (Core cpoke cfunc_adv cfunc_usr) (Core cpoke' cfunc_adv' cfunc_usr')\"\n if \n \"(S ===> E ===> rel_spmf S) cpoke cpoke'\" and \n \"(S ===> IA ===> rel_spmf (rel_prod OA S)) cfunc_adv cfunc_adv'\" and \n \"(S ===> IU ===> rel_spmf (rel_prod OU S)) cfunc_usr cfunc_usr'\"\n for cpoke cfunc_adv cfunc_usr\n\ninductive_simps \n rel_core'_simps [simp]: \n \"rel_core' S E IA IU OA OU (Core cpoke' cfunc_adv' cfunc_usr') (Core cpoke'' cfunc_adv'' cfunc_usr'')\"\n\nlemma \n rel_core'_eq [relator_eq]:\n \"rel_core' (=) (=) (=) (=) (=) (=) = (=)\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹rel_core' (=) (=) (=) (=) (=) (=) = (=)›*)\n subgoal for x and y\n apply (cases x)\n (*goal: ‹rel_core' (=) (=) (=) (=) (=) (=) (x::('a::type, 'b::type, 'c::type, 'd::type, 'e::type, 'f::type) core) (y::('a::type, 'b::type, 'c::type, 'd::type, 'e::type, 'f::type) core) = (x = y)›*)\n apply (cases y)\n (*goal: ‹⋀x1 x2 x3. x = Core x1 x2 x3 ⟹ rel_core' (=) (=) (=) (=) (=) (=) x y = (x = y)›*)\n by (auto simp add: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) rel_fun_def (*‹?A ===> ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) relator_eq (*‹rel_exception (=) = (=)› ‹rel_converter (=) (=) (=) (=) = (=)› ‹rel_ℐ (=) (=) = (=)› ‹rel_gpv (=) (=) = (=)› ‹rel_generat (=) (=) (=) = (=)› ‹Resumption.resumption.rel_resumption (=) (=) = (=)› ‹rel_writerT (=) (=) (=) = (=)› ‹rel_stateT (=) (=) = (=)› ‹rel_nondetT (=) = (=)› ‹Monomorphic_Monad.rel_envT (=) (=) = (=)› ‹rel_optionT (=) = (=)› ‹rel_id (=) = (=)› and more 21 facts*)) .\n\nlemma \n rel_core'_mono [relator_mono]:\n \"rel_core' S E IA IU OA OU ≤ rel_core' S E' IA' IU' OA' OU'\"\n if \"E' ≤ E\" \"IA' ≤ IA\" \"IU' ≤ IU\" \"OA ≤ OA'\" \"OU ≤ OU'\"\n apply (rule predicate2I (*‹(⋀x y. ?P x y ⟹ ?Q x y) ⟹ ?P ≤ ?Q›*))\n (*goal: ‹rel_core' S E IA IU OA OU ≤ rel_core' S E' IA' IU' OA' OU'›*)\n subgoal for x and y\n apply (cases x)\n (*goal: ‹rel_core' (S::'k ⇒ 'l ⇒ bool) (E::'a ⇒ 'b ⇒ bool) (IA::'c ⇒ 'd ⇒ bool) (IU::'e ⇒ 'f ⇒ bool) (OA::'g ⇒ 'h ⇒ bool) (OU::'i ⇒ 'j ⇒ bool) (x::('k, 'a, 'c, 'e, 'g, 'i) core) (y::('l, 'b, 'd, 'f, 'h, 'j) core) ⟹ rel_core' S (E'::'a ⇒ 'b ⇒ bool) (IA'::'c ⇒ 'd ⇒ bool) (IU'::'e ⇒ 'f ⇒ bool) (OA'::'g ⇒ 'h ⇒ bool) (OU'::'i ⇒ 'j ⇒ bool) x y›*)\n apply (cases y)\n (*goal: ‹⋀x1 x2 x3. ⟦rel_core' S E IA IU OA OU x y; x = Core x1 x2 x3⟧ ⟹ rel_core' S E' IA' IU' OA' OU' x y›*)\n apply clarsimp\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦rel_core' S E IA IU OA OU x y; x = Core x1 x2 x3; y = Core x1a x2a x3a⟧ ⟹ rel_core' S E' IA' IU' OA' OU' x y›*)\n apply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹⋀(x1::'k::type ⇒ 'a::type ⇒ 'k::type spmf) (x2::'k::type ⇒ 'c::type ⇒ ('g::type × 'k::type) spmf) (x3::'k::type ⇒ 'e::type ⇒ ('i::type × 'k::type) spmf) (x1a::'l::type ⇒ 'b::type ⇒ 'l::type spmf) (x2a::'l::type ⇒ 'd::type ⇒ ('h::type × 'l::type) spmf) x3a::'l::type ⇒ 'f::type ⇒ ('j::type × 'l::type) spmf. ⟦(x::('k::type, 'a::type, 'c::type, 'e::type, 'g::type, 'i::type) core) = Core x1 x2 x3; (y::('l::type, 'b::type, 'd::type, 'f::type, 'h::type, 'j::type) core) = Core x1a x2a x3a; ((S::'k::type ⇒ 'l::type ⇒ bool) ===> (E::'a::type ⇒ 'b::type ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IA::'c::type ⇒ 'd::type ⇒ bool) ===> rel_spmf (rel_prod (OA::'g::type ⇒ 'h::type ⇒ bool) S)) x2 x2a; (S ===> (IU::'e::type ⇒ 'f::type ⇒ bool) ===> rel_spmf (rel_prod (OU::'i::type ⇒ 'j::type ⇒ bool) S)) x3 x3a⟧ ⟹ (S ===> (E'::'a::type ⇒ 'b::type ⇒ bool) ===> rel_spmf S) x1 x1a›\n 2. ‹⋀(x1::'k::type ⇒ 'a::type ⇒ 'k::type spmf) (x2::'k::type ⇒ 'c::type ⇒ ('g::type × 'k::type) spmf) (x3::'k::type ⇒ 'e::type ⇒ ('i::type × 'k::type) spmf) (x1a::'l::type ⇒ 'b::type ⇒ 'l::type spmf) (x2a::'l::type ⇒ 'd::type ⇒ ('h::type × 'l::type) spmf) x3a::'l::type ⇒ 'f::type ⇒ ('j::type × 'l::type) spmf. ⟦(x::('k::type, 'a::type, 'c::type, 'e::type, 'g::type, 'i::type) core) = Core x1 x2 x3; (y::('l::type, 'b::type, 'd::type, 'f::type, 'h::type, 'j::type) core) = Core x1a x2a x3a; ((S::'k::type ⇒ 'l::type ⇒ bool) ===> (E::'a::type ⇒ 'b::type ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IA::'c::type ⇒ 'd::type ⇒ bool) ===> rel_spmf (rel_prod (OA::'g::type ⇒ 'h::type ⇒ bool) S)) x2 x2a; (S ===> (IU::'e::type ⇒ 'f::type ⇒ bool) ===> rel_spmf (rel_prod (OU::'i::type ⇒ 'j::type ⇒ bool) S)) x3 x3a⟧ ⟹ (S ===> (IA'::'c::type ⇒ 'd::type ⇒ bool) ===> rel_spmf (rel_prod (OA'::'g::type ⇒ 'h::type ⇒ bool) S)) x2 x2a›\n 3. ‹⋀(x1::'k::type ⇒ 'a::type ⇒ 'k::type spmf) (x2::'k::type ⇒ 'c::type ⇒ ('g::type × 'k::type) spmf) (x3::'k::type ⇒ 'e::type ⇒ ('i::type × 'k::type) spmf) (x1a::'l::type ⇒ 'b::type ⇒ 'l::type spmf) (x2a::'l::type ⇒ 'd::type ⇒ ('h::type × 'l::type) spmf) x3a::'l::type ⇒ 'f::type ⇒ ('j::type × 'l::type) spmf. ⟦(x::('k::type, 'a::type, 'c::type, 'e::type, 'g::type, 'i::type) core) = Core x1 x2 x3; (y::('l::type, 'b::type, 'd::type, 'f::type, 'h::type, 'j::type) core) = Core x1a x2a x3a; ((S::'k::type ⇒ 'l::type ⇒ bool) ===> (E::'a::type ⇒ 'b::type ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IA::'c::type ⇒ 'd::type ⇒ bool) ===> rel_spmf (rel_prod (OA::'g::type ⇒ 'h::type ⇒ bool) S)) x2 x2a; (S ===> (IU::'e::type ⇒ 'f::type ⇒ bool) ===> rel_spmf (rel_prod (OU::'i::type ⇒ 'j::type ⇒ bool) S)) x3 x3a⟧ ⟹ (S ===> (IU'::'e::type ⇒ 'f::type ⇒ bool) ===> rel_spmf (rel_prod (OU'::'i::type ⇒ 'j::type ⇒ bool) S)) x3 x3a›\n discuss goal 1*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a⟧ ⟹ S xa ya ⟶ S xa ya›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a; (E ===> rel_spmf S) xa ya⟧ ⟹ (E' ===> rel_spmf S) xa ya›\n discuss goal 1*)\n apply (rule impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*) that (*‹(E'::'a ⇒ 'b ⇒ bool) ≤ (E::'a ⇒ 'b ⇒ bool)› ‹(IA'::'c ⇒ 'd ⇒ bool) ≤ (IA::'c ⇒ 'd ⇒ bool)› ‹(IU'::'e ⇒ 'f ⇒ bool) ≤ (IU::'e ⇒ 'f ⇒ bool)› ‹(OA::'g ⇒ 'h ⇒ bool) ≤ (OA'::'g ⇒ 'h ⇒ bool)› ‹(OU::'i ⇒ 'j ⇒ bool) ≤ (OU'::'i ⇒ 'j ⇒ bool)›*) order_refl (*‹(?x::?'a) ≤ ?x›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a⟧ ⟹ S xa ya ⟶ S xa ya› and 3 goals remain*)\n apply assumption\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a⟧ ⟹ E' xaa yaa ⟶ E xaa yaa›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a; rel_spmf S xaa yaa⟧ ⟹ rel_spmf S xaa yaa›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E' ≤ E› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'›*) order_refl (*‹?x ≤ ?x›*))\n (*top goal: ‹⋀(x1::'k ⇒ 'a ⇒ 'k spmf) (x2::'k ⇒ 'c ⇒ ('g × 'k) spmf) (x3::'k ⇒ 'e ⇒ ('i × 'k) spmf) (x1a::'l ⇒ 'b ⇒ 'l spmf) (x2a::'l ⇒ 'd ⇒ ('h × 'l) spmf) (x3a::'l ⇒ 'f ⇒ ('j × 'l) spmf) (xa::'a ⇒ 'k spmf) (ya::'b ⇒ 'l spmf) (xaa::'a) yaa::'b. ⟦(x::('k, 'a, 'c, 'e, 'g, 'i) core) = Core x1 x2 x3; (y::('l, 'b, 'd, 'f, 'h, 'j) core) = Core x1a x2a x3a; ((S::'k ⇒ 'l ⇒ bool) ===> (IA::'c ⇒ 'd ⇒ bool) ===> rel_spmf (rel_prod (OA::'g ⇒ 'h ⇒ bool) S)) x2 x2a; (S ===> (IU::'e ⇒ 'f ⇒ bool) ===> rel_spmf (rel_prod (OU::'i ⇒ 'j ⇒ bool) S)) x3 x3a⟧ ⟹ (E'::'a ⇒ 'b ⇒ bool) xaa yaa ⟶ (E::'a ⇒ 'b ⇒ bool) xaa yaa› and 3 goals remain*)\n apply (erule that[THEN predicate2D] (*‹(E'::'a ⇒ 'b ⇒ bool) (?x::'a) (?y::'b) ⟹ (E::'a ⇒ 'b ⇒ bool) ?x ?y› ‹(IA'::'c ⇒ 'd ⇒ bool) (?x::'c) (?y::'d) ⟹ (IA::'c ⇒ 'd ⇒ bool) ?x ?y› ‹(IU'::'e ⇒ 'f ⇒ bool) (?x::'e) (?y::'f) ⟹ (IU::'e ⇒ 'f ⇒ bool) ?x ?y› ‹(OA::'g ⇒ 'h ⇒ bool) (?x::'g) (?y::'h) ⟹ (OA'::'g ⇒ 'h ⇒ bool) ?x ?y› ‹(OU::'i ⇒ 'j ⇒ bool) (?x::'i) (?y::'j) ⟹ (OU'::'i ⇒ 'j ⇒ bool) ?x ?y›*))\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a; rel_spmf S xaa yaa⟧ ⟹ rel_spmf S xaa yaa› and 2 goals remain*)\n apply assumption\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a⟧ ⟹ S xa ya ⟶ S xa ya›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a; (IA ===> rel_spmf (rel_prod OA S)) xa ya⟧ ⟹ (IA' ===> rel_spmf (rel_prod OA' S)) xa ya›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E' ≤ E› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'›*) order_refl (*‹?x ≤ ?x›*))\n (*top goal: ‹⋀(x1::'k ⇒ 'a ⇒ 'k spmf) (x2::'k ⇒ 'c ⇒ ('g × 'k) spmf) (x3::'k ⇒ 'e ⇒ ('i × 'k) spmf) (x1a::'l ⇒ 'b ⇒ 'l spmf) (x2a::'l ⇒ 'd ⇒ ('h × 'l) spmf) (x3a::'l ⇒ 'f ⇒ ('j × 'l) spmf) (xa::'k) ya::'l. ⟦(x::('k, 'a, 'c, 'e, 'g, 'i) core) = Core x1 x2 x3; (y::('l, 'b, 'd, 'f, 'h, 'j) core) = Core x1a x2a x3a; ((S::'k ⇒ 'l ⇒ bool) ===> (E::'a ⇒ 'b ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IU::'e ⇒ 'f ⇒ bool) ===> rel_spmf (rel_prod (OU::'i ⇒ 'j ⇒ bool) S)) x3 x3a⟧ ⟹ S xa ya ⟶ S xa ya› and 2 goals remain*)\n apply assumption\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀(x1::'k ⇒ 'a ⇒ 'k spmf) (x2::'k ⇒ 'c ⇒ ('g × 'k) spmf) (x3::'k ⇒ 'e ⇒ ('i × 'k) spmf) (x1a::'l ⇒ 'b ⇒ 'l spmf) (x2a::'l ⇒ 'd ⇒ ('h × 'l) spmf) (x3a::'l ⇒ 'f ⇒ ('j × 'l) spmf) (xa::'c ⇒ ('g × 'k) spmf) (ya::'d ⇒ ('h × 'l) spmf) (xaa::'c) yaa::'d. ⟦(x::('k, 'a, 'c, 'e, 'g, 'i) core) = Core x1 x2 x3; (y::('l, 'b, 'd, 'f, 'h, 'j) core) = Core x1a x2a x3a; ((S::'k ⇒ 'l ⇒ bool) ===> (E::'a ⇒ 'b ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IU::'e ⇒ 'f ⇒ bool) ===> rel_spmf (rel_prod (OU::'i ⇒ 'j ⇒ bool) S)) x3 x3a⟧ ⟹ (IA'::'c ⇒ 'd ⇒ bool) xaa yaa ⟶ (IA::'c ⇒ 'd ⇒ bool) xaa yaa›\n 2. ‹⋀(x1::'k ⇒ 'a ⇒ 'k spmf) (x2::'k ⇒ 'c ⇒ ('g × 'k) spmf) (x3::'k ⇒ 'e ⇒ ('i × 'k) spmf) (x1a::'l ⇒ 'b ⇒ 'l spmf) (x2a::'l ⇒ 'd ⇒ ('h × 'l) spmf) (x3a::'l ⇒ 'f ⇒ ('j × 'l) spmf) (xa::'c ⇒ ('g × 'k) spmf) (ya::'d ⇒ ('h × 'l) spmf) (xaa::('g × 'k) spmf) yaa::('h × 'l) spmf. ⟦(x::('k, 'a, 'c, 'e, 'g, 'i) core) = Core x1 x2 x3; (y::('l, 'b, 'd, 'f, 'h, 'j) core) = Core x1a x2a x3a; ((S::'k ⇒ 'l ⇒ bool) ===> (E::'a ⇒ 'b ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IU::'e ⇒ 'f ⇒ bool) ===> rel_spmf (rel_prod (OU::'i ⇒ 'j ⇒ bool) S)) x3 x3a; rel_spmf (rel_prod (OA::'g ⇒ 'h ⇒ bool) S) xaa yaa⟧ ⟹ rel_spmf (rel_prod (OA'::'g ⇒ 'h ⇒ bool) S) xaa yaa›\n discuss goal 1*)\n apply (rule impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*) that (*‹(E'::'a ⇒ 'b ⇒ bool) ≤ (E::'a ⇒ 'b ⇒ bool)› ‹(IA'::'c ⇒ 'd ⇒ bool) ≤ (IA::'c ⇒ 'd ⇒ bool)› ‹(IU'::'e ⇒ 'f ⇒ bool) ≤ (IU::'e ⇒ 'f ⇒ bool)› ‹(OA::'g ⇒ 'h ⇒ bool) ≤ (OA'::'g ⇒ 'h ⇒ bool)› ‹(OU::'i ⇒ 'j ⇒ bool) ≤ (OU'::'i ⇒ 'j ⇒ bool)›*) order_refl (*‹(?x::?'a) ≤ ?x›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a⟧ ⟹ IA' xaa yaa ⟶ IA xaa yaa› and 2 goals remain*)\n apply (erule that[THEN predicate2D] (*‹E' ?x ?y ⟹ E ?x ?y› ‹IA' ?x ?y ⟹ IA ?x ?y› ‹IU' ?x ?y ⟹ IU ?x ?y› ‹OA ?x ?y ⟹ OA' ?x ?y› ‹OU ?x ?y ⟹ OU' ?x ?y›*))\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*top goal: ‹⋀(x1::'k ⇒ 'a ⇒ 'k spmf) (x2::'k ⇒ 'c ⇒ ('g × 'k) spmf) (x3::'k ⇒ 'e ⇒ ('i × 'k) spmf) (x1a::'l ⇒ 'b ⇒ 'l spmf) (x2a::'l ⇒ 'd ⇒ ('h × 'l) spmf) (x3a::'l ⇒ 'f ⇒ ('j × 'l) spmf) (xa::'c ⇒ ('g × 'k) spmf) (ya::'d ⇒ ('h × 'l) spmf) (xaa::('g × 'k) spmf) yaa::('h × 'l) spmf. ⟦(x::('k, 'a, 'c, 'e, 'g, 'i) core) = Core x1 x2 x3; (y::('l, 'b, 'd, 'f, 'h, 'j) core) = Core x1a x2a x3a; ((S::'k ⇒ 'l ⇒ bool) ===> (E::'a ⇒ 'b ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IU::'e ⇒ 'f ⇒ bool) ===> rel_spmf (rel_prod (OU::'i ⇒ 'j ⇒ bool) S)) x3 x3a; rel_spmf (rel_prod (OA::'g ⇒ 'h ⇒ bool) S) xaa yaa⟧ ⟹ rel_spmf (rel_prod (OA'::'g ⇒ 'h ⇒ bool) S) xaa yaa› and 1 goal remains*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a⟧ ⟹ OA ≤ OA'›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IU ===> rel_spmf (rel_prod OU S)) x3 x3a⟧ ⟹ S ≤ S›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E' ≤ E› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'›*) order_refl (*‹?x ≤ ?x›*))\n (*discuss goal 2*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E' ≤ E› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'›*) order_refl (*‹?x ≤ ?x›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*discuss goal 3*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a⟧ ⟹ S xa ya ⟶ S xa ya›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; (IU ===> rel_spmf (rel_prod OU S)) xa ya⟧ ⟹ (IU' ===> rel_spmf (rel_prod OU' S)) xa ya›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E' ≤ E› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'›*) order_refl (*‹?x ≤ ?x›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a⟧ ⟹ S xa ya ⟶ S xa ya› and 1 goal remains*)\n apply assumption\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a⟧ ⟹ IU' xaa yaa ⟶ IU xaa yaa›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; rel_spmf (rel_prod OU S) xaa yaa⟧ ⟹ rel_spmf (rel_prod OU' S) xaa yaa›\n discuss goal 1*)\n apply (rule impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*) that (*‹(E'::'a ⇒ 'b ⇒ bool) ≤ (E::'a ⇒ 'b ⇒ bool)› ‹(IA'::'c ⇒ 'd ⇒ bool) ≤ (IA::'c ⇒ 'd ⇒ bool)› ‹(IU'::'e ⇒ 'f ⇒ bool) ≤ (IU::'e ⇒ 'f ⇒ bool)› ‹(OA::'g ⇒ 'h ⇒ bool) ≤ (OA'::'g ⇒ 'h ⇒ bool)› ‹(OU::'i ⇒ 'j ⇒ bool) ≤ (OU'::'i ⇒ 'j ⇒ bool)›*) order_refl (*‹(?x::?'a) ≤ ?x›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a⟧ ⟹ IU' xaa yaa ⟶ IU xaa yaa› and 1 goal remains*)\n apply (erule that[THEN predicate2D] (*‹E' ?x ?y ⟹ E ?x ?y› ‹IA' ?x ?y ⟹ IA ?x ?y› ‹IU' ?x ?y ⟹ IU ?x ?y› ‹OA ?x ?y ⟹ OA' ?x ?y› ‹OU ?x ?y ⟹ OU' ?x ?y›*))\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a; (S ===> E ===> rel_spmf S) x1 x1a; (S ===> IA ===> rel_spmf (rel_prod OA S)) x2 x2a; rel_spmf (rel_prod OU S) xaa yaa⟧ ⟹ rel_spmf (rel_prod OU' S) xaa yaa›*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀(x1::'k::type ⇒ 'a::type ⇒ 'k::type spmf) (x2::'k::type ⇒ 'c::type ⇒ ('g::type × 'k::type) spmf) (x3::'k::type ⇒ 'e::type ⇒ ('i::type × 'k::type) spmf) (x1a::'l::type ⇒ 'b::type ⇒ 'l::type spmf) (x2a::'l::type ⇒ 'd::type ⇒ ('h::type × 'l::type) spmf) (x3a::'l::type ⇒ 'f::type ⇒ ('j::type × 'l::type) spmf) (xa::'e::type ⇒ ('i::type × 'k::type) spmf) (ya::'f::type ⇒ ('j::type × 'l::type) spmf) (xaa::('i::type × 'k::type) spmf) (yaa::('j::type × 'l::type) spmf) (xb::'i::type × 'k::type) yb::'j::type × 'l::type. ⟦(x::('k::type, 'a::type, 'c::type, 'e::type, 'g::type, 'i::type) core) = Core x1 x2 x3; (y::('l::type, 'b::type, 'd::type, 'f::type, 'h::type, 'j::type) core) = Core x1a x2a x3a; ((S::'k::type ⇒ 'l::type ⇒ bool) ===> (E::'a::type ⇒ 'b::type ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IA::'c::type ⇒ 'd::type ⇒ bool) ===> rel_spmf (rel_prod (OA::'g::type ⇒ 'h::type ⇒ bool) S)) x2 x2a⟧ ⟹ (OU::'i::type ⇒ 'j::type ⇒ bool) ≤ (OU'::'i::type ⇒ 'j::type ⇒ bool)›\n 2. ‹⋀(x1::'k::type ⇒ 'a::type ⇒ 'k::type spmf) (x2::'k::type ⇒ 'c::type ⇒ ('g::type × 'k::type) spmf) (x3::'k::type ⇒ 'e::type ⇒ ('i::type × 'k::type) spmf) (x1a::'l::type ⇒ 'b::type ⇒ 'l::type spmf) (x2a::'l::type ⇒ 'd::type ⇒ ('h::type × 'l::type) spmf) (x3a::'l::type ⇒ 'f::type ⇒ ('j::type × 'l::type) spmf) (xa::'e::type ⇒ ('i::type × 'k::type) spmf) (ya::'f::type ⇒ ('j::type × 'l::type) spmf) (xaa::('i::type × 'k::type) spmf) (yaa::('j::type × 'l::type) spmf) (xb::'i::type × 'k::type) yb::'j::type × 'l::type. ⟦(x::('k::type, 'a::type, 'c::type, 'e::type, 'g::type, 'i::type) core) = Core x1 x2 x3; (y::('l::type, 'b::type, 'd::type, 'f::type, 'h::type, 'j::type) core) = Core x1a x2a x3a; ((S::'k::type ⇒ 'l::type ⇒ bool) ===> (E::'a::type ⇒ 'b::type ⇒ bool) ===> rel_spmf S) x1 x1a; (S ===> (IA::'c::type ⇒ 'd::type ⇒ bool) ===> rel_spmf (rel_prod (OA::'g::type ⇒ 'h::type ⇒ bool) S)) x2 x2a⟧ ⟹ S ≤ S›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E' ≤ E› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'›*) order_refl (*‹?x ≤ ?x›*))\n (*discuss goal 2*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E' ≤ E› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'›*) order_refl (*‹?x ≤ ?x›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 3 subgoals*) . .\n\nlemma \n cpoke_parametric [transfer_rule]:\n \"(rel_core' S E IA IU OA OU ===> S ===> E ===> rel_spmf S) cpoke cpoke\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(rel_core' S E IA IU OA OU ===> S ===> E ===> rel_spmf S) cpoke cpoke›*)\n apply (erule rel_core'.cases (*‹⟦rel_core' ?S ?E ?IA ?IU ?OA ?OU ?a1.0 ?a2.0; ⋀cpoke cfunc_adv cfunc_usr cpoke' cfunc_adv' cfunc_usr'. ⟦?a1.0 = Core cpoke cfunc_adv cfunc_usr; ?a2.0 = Core cpoke' cfunc_adv' cfunc_usr'; (?S ===> ?E ===> rel_spmf ?S) cpoke cpoke'; (?S ===> ?IA ===> rel_spmf (rel_prod ?OA ?S)) cfunc_adv cfunc_adv'; (?S ===> ?IU ===> rel_spmf (rel_prod ?OU ?S)) cfunc_usr cfunc_usr'⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹⋀(x::('a, 'b, 'c, 'd, 'e, 'f) core) y::('g, 'h, 'i, 'j, 'k, 'l) core. rel_core' (S::'a ⇒ 'g ⇒ bool) (E::'b ⇒ 'h ⇒ bool) (IA::'c ⇒ 'i ⇒ bool) (IU::'d ⇒ 'j ⇒ bool) (OA::'e ⇒ 'k ⇒ bool) (OU::'f ⇒ 'l ⇒ bool) x y ⟹ (S ===> E ===> rel_spmf S) (cpoke x) (cpoke y)›*)\n by simp\n\nlemma \n cfunc_adv_parametric [transfer_rule]:\n \"(rel_core' S E IA IU OA OU ===> S ===> IA ===> rel_spmf (rel_prod OA S)) cfunc_adv cfunc_adv\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(rel_core' S E IA IU OA OU ===> S ===> IA ===> rel_spmf (rel_prod OA S)) cfunc_adv cfunc_adv›*)\n apply (erule rel_core'.cases (*‹⟦rel_core' ?S ?E ?IA ?IU ?OA ?OU ?a1.0 ?a2.0; ⋀cpoke cfunc_adv cfunc_usr cpoke' cfunc_adv' cfunc_usr'. ⟦?a1.0 = Core cpoke cfunc_adv cfunc_usr; ?a2.0 = Core cpoke' cfunc_adv' cfunc_usr'; (?S ===> ?E ===> rel_spmf ?S) cpoke cpoke'; (?S ===> ?IA ===> rel_spmf (rel_prod ?OA ?S)) cfunc_adv cfunc_adv'; (?S ===> ?IU ===> rel_spmf (rel_prod ?OU ?S)) cfunc_usr cfunc_usr'⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹⋀x y. rel_core' S E IA IU OA OU x y ⟹ (S ===> IA ===> rel_spmf (rel_prod OA S)) (cfunc_adv x) (cfunc_adv y)›*)\n by simp\n\nlemma \n cfunc_usr_parametric [transfer_rule]:\n \"(rel_core' S E IA IU OA OU ===> S ===> IU ===> rel_spmf (rel_prod OU S)) cfunc_usr cfunc_usr\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(rel_core' S E IA IU OA OU ===> S ===> IU ===> rel_spmf (rel_prod OU S)) cfunc_usr cfunc_usr›*)\n apply (erule rel_core'.cases (*‹⟦rel_core' ?S ?E ?IA ?IU ?OA ?OU ?a1.0 ?a2.0; ⋀cpoke cfunc_adv cfunc_usr cpoke' cfunc_adv' cfunc_usr'. ⟦?a1.0 = Core cpoke cfunc_adv cfunc_usr; ?a2.0 = Core cpoke' cfunc_adv' cfunc_usr'; (?S ===> ?E ===> rel_spmf ?S) cpoke cpoke'; (?S ===> ?IA ===> rel_spmf (rel_prod ?OA ?S)) cfunc_adv cfunc_adv'; (?S ===> ?IU ===> rel_spmf (rel_prod ?OU ?S)) cfunc_usr cfunc_usr'⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹⋀x y. rel_core' S E IA IU OA OU x y ⟹ (S ===> IU ===> rel_spmf (rel_prod OU S)) (cfunc_usr x) (cfunc_usr y)›*)\n by simp\n\nlemma \n Core_parametric [transfer_rule]:\n \"((S ===> E ===> rel_spmf S) ===> (S ===> IA ===> rel_spmf (rel_prod OA S)) ===> (S ===> IU ===> rel_spmf (rel_prod OU S))\n ===> rel_core' S E IA IU OA OU) Core Core\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹((S ===> E ===> rel_spmf S) ===> (S ===> IA ===> rel_spmf (rel_prod OA S)) ===> (S ===> IU ===> rel_spmf (rel_prod OU S)) ===> rel_core' S E IA IU OA OU) Core Core›*)\n apply (rule rel_funI (*‹(⋀(x::?'a) y::?'b. (?A::?'a ⇒ ?'b ⇒ bool) x y ⟹ (?B::?'c ⇒ ?'d ⇒ bool) ((?f::?'a ⇒ ?'c) x) ((?g::?'b ⇒ ?'d) y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y. (S ===> E ===> rel_spmf S) x y ⟹ ((S ===> IA ===> rel_spmf (rel_prod OA S)) ===> (S ===> IU ===> rel_spmf (rel_prod OU S)) ===> rel_core' S E IA IU OA OU) (Core x) (Core y)›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y xa ya. ⟦(S ===> E ===> rel_spmf S) x y; (S ===> IA ===> rel_spmf (rel_prod OA S)) xa ya⟧ ⟹ ((S ===> IU ===> rel_spmf (rel_prod OU S)) ===> rel_core' S E IA IU OA OU) (Core x xa) (Core y ya)›*)\n by simp\n\nlemma \n case_core_parametric [transfer_rule]:\n \"(((S ===> E ===> rel_spmf S) ===> \n (S ===> IA ===> rel_spmf (rel_prod OA S)) ===> \n (S ===> IU ===> rel_spmf (rel_prod OU S)) ===> X) ===> \n rel_core' S E IA IU OA OU ===> X) case_core case_core\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(((S ===> E ===> rel_spmf S) ===> (S ===> IA ===> rel_spmf (rel_prod OA S)) ===> (S ===> IU ===> rel_spmf (rel_prod OU S)) ===> X) ===> rel_core' S E IA IU OA OU ===> X) case_core case_core›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀(x::('a ⇒ 'b ⇒ 'a spmf) ⇒ ('a ⇒ 'c ⇒ ('d × 'a) spmf) ⇒ ('a ⇒ 'e ⇒ ('f × 'a) spmf) ⇒ 'g) y::('h ⇒ 'i ⇒ 'h spmf) ⇒ ('h ⇒ 'j ⇒ ('k × 'h) spmf) ⇒ ('h ⇒ 'l ⇒ ('m × 'h) spmf) ⇒ 'n. (((S::'a ⇒ 'h ⇒ bool) ===> (E::'b ⇒ 'i ⇒ bool) ===> rel_spmf S) ===> (S ===> (IA::'c ⇒ 'j ⇒ bool) ===> rel_spmf (rel_prod (OA::'d ⇒ 'k ⇒ bool) S)) ===> (S ===> (IU::'e ⇒ 'l ⇒ bool) ===> rel_spmf (rel_prod (OU::'f ⇒ 'm ⇒ bool) S)) ===> (X::'g ⇒ 'n ⇒ bool)) x y ⟹ (rel_core' S E IA IU OA OU ===> X) (case_core x) (case_core y)›*)\n by (auto 4 4 split: core.split (*‹?P (case ?core of Core x xa xb ⇒ ?f x xa xb) = (∀x1 x2 x3. ?core = Core x1 x2 x3 ⟶ ?P (?f x1 x2 x3))›*) dest: rel_funD (*‹⟦(?A ===> ?B) ?f ?g; ?A ?x ?y⟧ ⟹ ?B (?f ?x) (?g ?y)›*))\n\nlemma \n corec_core_parametric [transfer_rule]:\n \"((X ===> S ===> E ===> rel_spmf S) ===> \n (X ===> S ===> IA ===> rel_spmf (rel_prod OA S)) ===> \n (X ===> S ===> IU ===> rel_spmf (rel_prod OU S)) ===> \n X ===> rel_core' S E IA IU OA OU) corec_core corec_core\"\n apply (rule rel_funI (*‹(⋀(x::?'a::type) y::?'b::type. (?A::?'a::type ⇒ ?'b::type ⇒ bool) x y ⟹ (?B::?'c::type ⇒ ?'d::type ⇒ bool) ((?f::?'a::type ⇒ ?'c::type) x) ((?g::?'b::type ⇒ ?'d::type) y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(((X::'a ⇒ 'd ⇒ bool) ===> (S::'b ⇒ 'e ⇒ bool) ===> (E::'c ⇒ 'f ⇒ bool) ===> rel_spmf S) ===> (X ===> S ===> (IA::'g ⇒ 'k ⇒ bool) ===> rel_spmf (rel_prod (OA::'h ⇒ 'l ⇒ bool) S)) ===> (X ===> S ===> (IU::'i ⇒ 'm ⇒ bool) ===> rel_spmf (rel_prod (OU::'j ⇒ 'n ⇒ bool) S)) ===> X ===> rel_core' S E IA IU OA OU) corec_core corec_core›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y. (X ===> S ===> E ===> rel_spmf S) x y ⟹ ((X ===> S ===> IA ===> rel_spmf (rel_prod OA S)) ===> (X ===> S ===> IU ===> rel_spmf (rel_prod OU S)) ===> X ===> rel_core' S E IA IU OA OU) (corec_core x) (corec_core y)›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y xa ya. ⟦(X ===> S ===> E ===> rel_spmf S) x y; (X ===> S ===> IA ===> rel_spmf (rel_prod OA S)) xa ya⟧ ⟹ ((X ===> S ===> IU ===> rel_spmf (rel_prod OU S)) ===> X ===> rel_core' S E IA IU OA OU) (corec_core x xa) (corec_core y ya)›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀(x::'a ⇒ 'b ⇒ 'c ⇒ 'b spmf) (y::'d ⇒ 'e ⇒ 'f ⇒ 'e spmf) (xa::'a ⇒ 'b ⇒ 'g ⇒ ('h × 'b) spmf) (ya::'d ⇒ 'e ⇒ 'k ⇒ ('l × 'e) spmf) (xb::'a ⇒ 'b ⇒ 'i ⇒ ('j × 'b) spmf) yb::'d ⇒ 'e ⇒ 'm ⇒ ('n × 'e) spmf. ⟦((X::'a ⇒ 'd ⇒ bool) ===> (S::'b ⇒ 'e ⇒ bool) ===> (E::'c ⇒ 'f ⇒ bool) ===> rel_spmf S) x y; (X ===> S ===> (IA::'g ⇒ 'k ⇒ bool) ===> rel_spmf (rel_prod (OA::'h ⇒ 'l ⇒ bool) S)) xa ya; (X ===> S ===> (IU::'i ⇒ 'm ⇒ bool) ===> rel_spmf (rel_prod (OU::'j ⇒ 'n ⇒ bool) S)) xb yb⟧ ⟹ (X ===> rel_core' S E IA IU OA OU) (corec_core x xa xb) (corec_core y ya yb)›*)\n by (auto simp add: core.corec (*‹corec_core ?g1.0 ?g2.0 ?g3.0 ?a = Core (?g1.0 ?a) (?g2.0 ?a) (?g3.0 ?a)›*) dest: rel_funD (*‹⟦(?A ===> ?B) ?f ?g; ?A ?x ?y⟧ ⟹ ?B (?f ?x) (?g ?y)›*))\n\nprimcorec map_core' ::\n \"('event' ⇒ 'event) ⇒ \n ('iadv_core' ⇒ 'iadv_core) ⇒\n ('iusr_core' ⇒ 'iusr_core) ⇒ \n ('oadv_core ⇒ 'oadv_core') ⇒ \n ('ousr_core ⇒ 'ousr_core') ⇒\n ('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core ⇒ \n ('s_core, 'event', 'iadv_core', 'iusr_core', 'oadv_core', 'ousr_core') core\"\n where\n \"cpoke (map_core' e ia iu oa ou core) = (id ---> e ---> id) (cpoke core)\"\n| \"cfunc_adv (map_core' e ia iu oa ou core) = (id ---> ia ---> map_spmf (map_prod oa id)) (cfunc_adv core)\"\n| \"cfunc_usr (map_core' e ia iu oa ou core) = (id ---> iu ---> map_spmf (map_prod ou id)) (cfunc_usr core)\"\n\nlemmas map_core'_simps [simp] = map_core'.ctr[where core=\"Core _ _ _\", simplified]\n\nparametric_constant map_core'_parametric[transfer_rule]: map_core'_def\n\nlemma core'_rel_Grp:\n \"rel_core' (=) (BNF_Def.Grp UNIV e)¯¯ (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou)\n = BNF_Def.Grp UNIV (map_core' e ia iu oa ou)\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹rel_core' (=) (BNF_Def.Grp UNIV (e::'g::type ⇒ 'b::type))¯¯ (BNF_Def.Grp UNIV (ia::'h::type ⇒ 'c::type))¯¯ (BNF_Def.Grp UNIV (iu::'i::type ⇒ 'd::type))¯¯ (BNF_Def.Grp UNIV (oa::'e::type ⇒ 'j::type)) (BNF_Def.Grp UNIV (ou::'f::type ⇒ 'k::type)) = BNF_Def.Grp UNIV (map_core' e ia iu oa ou)›*)\n subgoal for x and y\n apply (cases x)\n (*goal: ‹rel_core' (=) (BNF_Def.Grp UNIV e)¯¯ (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) x y = BNF_Def.Grp UNIV (map_core' e ia iu oa ou) x y›*)\n apply (cases y)\n (*goal: ‹⋀x1 x2 x3. x = Core x1 x2 x3 ⟹ rel_core' (=) (BNF_Def.Grp UNIV e)¯¯ (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) x y = BNF_Def.Grp UNIV (map_core' e ia iu oa ou) x y›*)\n apply clarsimp\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a⟧ ⟹ rel_core' (=) (BNF_Def.Grp UNIV e)¯¯ (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) x y = BNF_Def.Grp UNIV (map_core' e ia iu oa ou) x y›*)\n apply (subst (2 4 6) eq_alt_conversep (*‹(=) = (BNF_Def.Grp UNIV id)¯¯›*))\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a⟧ ⟹ (((=) ===> (BNF_Def.Grp UNIV e)¯¯ ===> rel_spmf (=)) x1 x1a ∧ ((=) ===> (BNF_Def.Grp UNIV ia)¯¯ ===> rel_spmf (rel_prod (BNF_Def.Grp UNIV oa) (=))) x2 x2a ∧ ((=) ===> (BNF_Def.Grp UNIV iu)¯¯ ===> rel_spmf (rel_prod (BNF_Def.Grp UNIV ou) (=))) x3 x3a) = BNF_Def.Grp UNIV (map_core' e ia iu oa ou) (Core x1 x2 x3) (Core x1a x2a x3a)›*)\n apply (subst (2 3 4) eq_alt (*‹(=) = BNF_Def.Grp UNIV id›*))\n (*goal: ‹⋀(x1::'a ⇒ 'b ⇒ 'a spmf) (x2::'a ⇒ 'c ⇒ ('e × 'a) spmf) (x3::'a ⇒ 'd ⇒ ('f × 'a) spmf) (x1a::'a ⇒ 'g ⇒ 'a spmf) (x2a::'a ⇒ 'h ⇒ ('j × 'a) spmf) x3a::'a ⇒ 'i ⇒ ('k × 'a) spmf. ⟦(x::('a, 'b, 'c, 'd, 'e, 'f) core) = Core x1 x2 x3; (y::('a, 'g, 'h, 'i, 'j, 'k) core) = Core x1a x2a x3a⟧ ⟹ (((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV (e::'g ⇒ 'b))¯¯ ===> rel_spmf (=)) x1 x1a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV (ia::'h ⇒ 'c))¯¯ ===> rel_spmf (rel_prod (BNF_Def.Grp UNIV (oa::'e ⇒ 'j)) (=))) x2 x2a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV (iu::'i ⇒ 'd))¯¯ ===> rel_spmf (rel_prod (BNF_Def.Grp UNIV (ou::'f ⇒ 'k)) (=))) x3 x3a) = BNF_Def.Grp UNIV (map_core' e ia iu oa ou) (Core x1 x2 x3) (Core x1a x2a x3a)›*)\n apply (simp add: pmf.rel_Grp (*‹rel_pmf (BNF_Def.Grp ?A ?f) = BNF_Def.Grp {x. set_pmf x ⊆ ?A} (map_pmf ?f)›*) option.rel_Grp (*‹rel_option (BNF_Def.Grp ?A ?f) = BNF_Def.Grp {x. set_option x ⊆ ?A} (map_option ?f)›*) prod.rel_Grp (*‹rel_prod (BNF_Def.Grp ?A1.0 ?f1.0) (BNF_Def.Grp ?A2.0 ?f2.0) = BNF_Def.Grp {x. Basic_BNFs.fsts x ⊆ ?A1.0 ∧ Basic_BNFs.snds x ⊆ ?A2.0} (map_prod ?f1.0 ?f2.0)›*) rel_fun_conversep_grp_grp (*‹(BNF_Def.Grp UNIV ?f)¯¯ ===> BNF_Def.Grp ?B ?g = BNF_Def.Grp {x. range (x ∘ ?f) ⊆ ?B} (?f ---> ?g)›*))\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Core x1 x2 x3; y = Core x1a x2a x3a⟧ ⟹ (((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV e)¯¯ ===> rel_spmf (BNF_Def.Grp UNIV id)) x1 x1a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV ia)¯¯ ===> rel_spmf (rel_prod (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV id))) x2 x2a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV iu)¯¯ ===> rel_spmf (rel_prod (BNF_Def.Grp UNIV ou) (BNF_Def.Grp UNIV id))) x3 x3a) = BNF_Def.Grp UNIV (map_core' e ia iu oa ou) (Core x1 x2 x3) (Core x1a x2a x3a)›*)\n by (auto simp add: Grp_def (*‹BNF_Def.Grp ?A ?f = (λa b. b = ?f a ∧ a ∈ ?A)›*) spmf.map_id[abs_def] (*‹map_spmf id ≡ λp. p›*) id_def[symmetric] (*‹(λx. x) = id›*)) .\n\nend\n\ninductive WT_core :: \"('iadv, 'oadv) ℐ ⇒ ('iusr, 'ousr) ℐ ⇒ ('s ⇒ bool) ⇒ ('s, 'event, 'iadv, 'iusr, 'oadv, 'ousr) core ⇒ bool\"\n for ℐ_adv ℐ_usr I core where\n \"WT_core ℐ_adv ℐ_usr I core\" if\n \"⋀s e s'. ⟦ s' ∈ set_spmf (cpoke core s e); I s ⟧ ⟹ I s'\"\n \"⋀s x y s'. ⟦ (y, s') ∈ set_spmf (cfunc_adv core s x); x ∈ outs_ℐ ℐ_adv; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_adv x ∧ I s'\"\n \"⋀s x y s'. ⟦ (y, s') ∈ set_spmf (cfunc_usr core s x); x ∈ outs_ℐ ℐ_usr; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_usr x ∧ I s'\"\n\nlemma WT_coreD:\n assumes \"WT_core ℐ_adv ℐ_usr I core\"\n shows WT_coreD_cpoke: \"⋀s e s'. ⟦ s' ∈ set_spmf (cpoke core s e); I s ⟧ ⟹ I s'\"\n and WT_coreD_cfunc_adv: \"⋀s x y s'. ⟦ (y, s') ∈ set_spmf (cfunc_adv core s x); x ∈ outs_ℐ ℐ_adv; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_adv x ∧ I s'\"\n and WT_coreD_cfund_usr: \"⋀s x y s'. ⟦ (y, s') ∈ set_spmf (cfunc_usr core s x); x ∈ outs_ℐ ℐ_usr; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_usr x ∧ I s'\"\n using assms (*‹WT_core (ℐ_adv::('a, 'b) ℐ) (ℐ_usr::('c, 'd) ℐ) (I::'e ⇒ bool) (core::('e, 'f, 'a, 'c, 'b, 'd) core)›*) apply -\n (*goals:\n 1. ‹⋀s e s'. ⟦s' ∈ set_spmf (cpoke core s e); I s; WT_core ℐ_adv ℐ_usr I core⟧ ⟹ I s'›\n 2. ‹⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_adv core s x); x ∈ outs_ℐ ℐ_adv; I s; WT_core ℐ_adv ℐ_usr I core⟧ ⟹ y ∈ responses_ℐ ℐ_adv x ∧ I s'›\n 3. ‹⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_usr core s x); x ∈ outs_ℐ ℐ_usr; I s; WT_core ℐ_adv ℐ_usr I core⟧ ⟹ y ∈ responses_ℐ ℐ_usr x ∧ I s'›\n discuss goal 1*)\n apply ((auto elim!: WT_core.cases (*‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; ⟦⋀s e s'. ⟦s' ∈ set_spmf (cpoke ?core s e); ?I s⟧ ⟹ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_adv ?core s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_usr ?core s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*discuss goal 2*)\n apply ((auto elim!: WT_core.cases (*‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; ⟦⋀s e s'. ⟦s' ∈ set_spmf (cpoke ?core s e); ?I s⟧ ⟹ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_adv ?core s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_usr ?core s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*discuss goal 3*)\n apply ((auto elim!: WT_core.cases (*‹⟦WT_core (?ℐ_adv::(?'iadv, ?'oadv) ℐ) (?ℐ_usr::(?'iusr, ?'ousr) ℐ) (?I::?'s ⇒ bool) (?core::(?'s, ?'event, ?'iadv, ?'iusr, ?'oadv, ?'ousr) core); ⟦⋀(s::?'s) (e::?'event) s'::?'s. ⟦s' ∈ set_spmf (cpoke ?core s e); ?I s⟧ ⟹ ?I s'; ⋀(s::?'s) (x::?'iadv) (y::?'oadv) s'::?'s. ⟦(y, s') ∈ set_spmf (cfunc_adv ?core s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀(s::?'s) (x::?'iusr) (y::?'ousr) s'::?'s. ⟦(y, s') ∈ set_spmf (cfunc_usr ?core s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'⟧ ⟹ ?P::bool⟧ ⟹ ?P›*))[1])\n (*proven 3 subgoals*) .\n\nlemma WT_coreD_foldl_spmf_cpoke:\n assumes \"WT_core ℐ_adv ℐ_usr I core\"\n and \"s' ∈ set_spmf (foldl_spmf (cpoke core) p es)\"\n and \"∀s ∈ set_spmf p. I s\"\n shows \"I s'\"\n using assms(2,3) (*‹s' ∈ set_spmf (foldl_spmf (cpoke core) p es)› ‹∀s∈set_spmf p. I s›*) apply (induction es arbitrary: p)\n (*goals:\n 1. ‹⋀p. ⟦s' ∈ set_spmf (foldl_spmf (cpoke core) p []); Ball (set_spmf p) I⟧ ⟹ I s'›\n 2. ‹⋀a es p. ⟦⋀p. ⟦s' ∈ set_spmf (foldl_spmf (cpoke core) p es); Ball (set_spmf p) I⟧ ⟹ I s'; s' ∈ set_spmf (foldl_spmf (cpoke core) p (a # es)); Ball (set_spmf p) I⟧ ⟹ I s'›\n discuss goal 1*)\n apply (fastforce dest: WT_coreD_cpoke[OF assms(1)] (*‹⟦?s' ∈ set_spmf (cpoke core ?s ?e); I ?s⟧ ⟹ I ?s'›*) simp add: bind_UNION (*‹?A ⤜ ?f = ⋃ (?f ` ?A)›*))\n (*discuss goal 2*)\n apply (fastforce dest: WT_coreD_cpoke[OF assms(1)] (*‹⟦?s' ∈ set_spmf (cpoke core ?s ?e); I ?s⟧ ⟹ I ?s'›*) simp add: bind_UNION (*‹?A ⤜ ?f = ⋃ (?f ` ?A)›*))\n (*proven 2 subgoals*) .\n\nlemma WT_core_trivial:\n assumes adv: \"⋀s. ℐ_adv ⊢c cfunc_adv core s √\"\n and usr: \"⋀s. ℐ_usr ⊢c cfunc_usr core s √\"\n shows \"WT_core ℐ_adv ℐ_usr (λ_. True) core\"\n apply (rule WT_core.intros (*‹⟦⋀s e s'. ⟦s' ∈ set_spmf (cpoke ?core s e); ?I s⟧ ⟹ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_adv ?core s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_usr ?core s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'⟧ ⟹ WT_core ?ℐ_adv ?ℐ_usr ?I ?core›*))\n (*goals:\n 1. ‹⋀s e s'. ⟦s' ∈ set_spmf (cpoke core s e); True⟧ ⟹ True›\n 2. ‹⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_adv core s x); x ∈ outs_ℐ ℐ_adv; True⟧ ⟹ y ∈ responses_ℐ ℐ_adv x ∧ True›\n 3. ‹⋀s x y s'. ⟦(y, s') ∈ set_spmf (cfunc_usr core s x); x ∈ outs_ℐ ℐ_usr; True⟧ ⟹ y ∈ responses_ℐ ℐ_usr x ∧ True›\n discuss goal 1*)\n apply ((auto dest: assms[THEN WT_calleeD] (*‹⟦(?ret, ?s) ∈ set_spmf (cfunc_adv core ?s1 ?out); ?out ∈ outs_ℐ ℐ_adv⟧ ⟹ ?ret ∈ responses_ℐ ℐ_adv ?out› ‹⟦(?ret, ?s) ∈ set_spmf (cfunc_usr core ?s1 ?out); ?out ∈ outs_ℐ ℐ_usr⟧ ⟹ ?ret ∈ responses_ℐ ℐ_usr ?out›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: assms[THEN WT_calleeD] (*‹⟦(?ret, ?s) ∈ set_spmf (cfunc_adv core ?s1 ?out); ?out ∈ outs_ℐ ℐ_adv⟧ ⟹ ?ret ∈ responses_ℐ ℐ_adv ?out› ‹⟦(?ret, ?s) ∈ set_spmf (cfunc_usr core ?s1 ?out); ?out ∈ outs_ℐ ℐ_usr⟧ ⟹ ?ret ∈ responses_ℐ ℐ_usr ?out›*))[1])\n (*discuss goal 3*)\n apply ((auto dest: assms[THEN WT_calleeD] (*‹⟦(?ret, ?s) ∈ set_spmf (cfunc_adv core ?s1 ?out); ?out ∈ outs_ℐ ℐ_adv⟧ ⟹ ?ret ∈ responses_ℐ ℐ_adv ?out› ‹⟦(?ret, ?s) ∈ set_spmf (cfunc_usr core ?s1 ?out); ?out ∈ outs_ℐ ℐ_usr⟧ ⟹ ?ret ∈ responses_ℐ ℐ_usr ?out›*))[1])\n (*proven 3 subgoals*) .\n\ncodatatype \n ('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme =\n Rest\n (rinit: \"'more\")\n (rfunc_adv: \"('s_rest, 'event, 'iadv_rest, 'oadv_rest) eoracle\")\n (rfunc_usr: \"('s_rest, 'event, 'iusr_rest, 'ousr_rest) eoracle\")\n\ndeclare rest_scheme.sel_transfer[transfer_rule del]\ndeclare rest_scheme.ctr_transfer[transfer_rule del]\ndeclare rest_scheme.case_transfer[transfer_rule del]\n\ncontext \n includes lifting_syntax \nbegin\n\ninductive \n rel_rest':: \"\n ('s_rest ⇒ 's_rest' ⇒ bool) ⇒ \n ('event ⇒ 'event' ⇒ bool) ⇒ \n ('iadv_rest ⇒ 'iadv_rest' ⇒ bool) ⇒\n ('iusr_rest ⇒ 'iusr_rest' ⇒ bool) ⇒ \n ('oadv_rest ⇒ 'oadv_rest' ⇒ bool) ⇒ \n ('ousr_rest ⇒ 'ousr_rest' ⇒ bool) ⇒\n ('more ⇒ 'more' ⇒ bool) ⇒ \n ('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme ⇒ \n ('s_rest', 'event', 'iadv_rest', 'iusr_rest', 'oadv_rest', 'ousr_rest', 'more') rest_scheme ⇒ bool\"\n for S E IA IU OA OU M\n where \"rel_rest' S E IA IU OA OU M (Rest rinit rfunc_adv rfunc_usr) (Rest rinit' rfunc_adv' rfunc_usr')\"\n if \n \"M rinit rinit'\" and \n \"(S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) rfunc_adv rfunc_adv'\" and \n \"(S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) rfunc_usr rfunc_usr'\"\n for rinit rfunc_adv rfunc_usr\n\ninductive_simps \n rel_rest'_simps [simp]: \n \"rel_rest' S E IA IU OA OU M (Rest rinit' rfunc_adv' rfunc_usr') (Rest rinit'' rfunc_adv'' rfunc_usr'')\"\n\nlemma \n rel_rest'_eq [relator_eq]:\n \"rel_rest' (=) (=) (=) (=) (=) (=) (=) = (=)\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹rel_rest' (=) (=) (=) (=) (=) (=) (=) = (=)›*)\n subgoal for x and y\n apply (cases x)\n (*goal: ‹rel_rest' (=) (=) (=) (=) (=) (=) (=) (x::('a::type, 'b::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type) rest_scheme) (y::('a::type, 'b::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type) rest_scheme) = (x = y)›*)\n apply (cases y)\n (*goal: ‹⋀x1 x2 x3. x = Rest x1 x2 x3 ⟹ rel_rest' (=) (=) (=) (=) (=) (=) (=) x y = (x = y)›*)\n by (auto simp add: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) rel_fun_def (*‹?A ===> ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) relator_eq (*‹rel_core' (=) (=) (=) (=) (=) (=) = (=)› ‹rel_exception (=) = (=)› ‹rel_converter (=) (=) (=) (=) = (=)› ‹rel_ℐ (=) (=) = (=)› ‹rel_gpv (=) (=) = (=)› ‹rel_generat (=) (=) (=) = (=)› ‹Resumption.resumption.rel_resumption (=) (=) = (=)› ‹rel_writerT (=) (=) (=) = (=)› ‹rel_stateT (=) (=) = (=)› ‹rel_nondetT (=) = (=)› ‹Monomorphic_Monad.rel_envT (=) (=) = (=)› ‹rel_optionT (=) = (=)› and more 22 facts*)) .\n\nlemma \n rel_rest'_mono [relator_mono]:\n \"rel_rest' S E IA IU OA OU M ≤ rel_rest' S E' IA' IU' OA' OU' M'\"\n if \"E ≤ E'\" \"IA' ≤ IA\" \"IU' ≤ IU\" \"OA ≤ OA'\" \"OU ≤ OU'\" \"M ≤ M'\"\n apply (rule predicate2I (*‹(⋀x y. ?P x y ⟹ ?Q x y) ⟹ ?P ≤ ?Q›*))\n (*goal: ‹rel_rest' (S::'m ⇒ 'n ⇒ bool) (E::'a ⇒ 'b ⇒ bool) (IA::'c ⇒ 'd ⇒ bool) (IU::'e ⇒ 'f ⇒ bool) (OA::'g ⇒ 'h ⇒ bool) (OU::'i ⇒ 'j ⇒ bool) (M::'k ⇒ 'l ⇒ bool) ≤ rel_rest' S (E'::'a ⇒ 'b ⇒ bool) (IA'::'c ⇒ 'd ⇒ bool) (IU'::'e ⇒ 'f ⇒ bool) (OA'::'g ⇒ 'h ⇒ bool) (OU'::'i ⇒ 'j ⇒ bool) (M'::'k ⇒ 'l ⇒ bool)›*)\n subgoal for x and y\n apply (cases x)\n (*goal: ‹rel_rest' S E IA IU OA OU M x y ⟹ rel_rest' S E' IA' IU' OA' OU' M' x y›*)\n apply (cases y)\n (*goal: ‹⋀x1 x2 x3. ⟦rel_rest' S E IA IU OA OU M x y; x = Rest x1 x2 x3⟧ ⟹ rel_rest' S E' IA' IU' OA' OU' M' x y›*)\n apply clarsimp\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦rel_rest' S E IA IU OA OU M x y; x = Rest x1 x2 x3; y = Rest x1a x2a x3a⟧ ⟹ rel_rest' S E' IA' IU' OA' OU' M' x y›*)\n apply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ M' x1 x1a›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ (S ===> IA' ===> rel_spmf (rel_prod (rel_prod OA' (list_all2 E')) S)) x2 x2a›\n 3. ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ (S ===> IU' ===> rel_spmf (rel_prod (rel_prod OU' (list_all2 E')) S)) x3 x3a›\n discuss goal 1*)\n apply (erule that[THEN predicate2D] (*‹(E::'a::type ⇒ 'b::type ⇒ bool) (?x::'a::type) (?y::'b::type) ⟹ (E'::'a::type ⇒ 'b::type ⇒ bool) ?x ?y› ‹(IA'::'c::type ⇒ 'd::type ⇒ bool) (?x::'c::type) (?y::'d::type) ⟹ (IA::'c::type ⇒ 'd::type ⇒ bool) ?x ?y› ‹(IU'::'e::type ⇒ 'f::type ⇒ bool) (?x::'e::type) (?y::'f::type) ⟹ (IU::'e::type ⇒ 'f::type ⇒ bool) ?x ?y› ‹(OA::'g::type ⇒ 'h::type ⇒ bool) (?x::'g::type) (?y::'h::type) ⟹ (OA'::'g::type ⇒ 'h::type ⇒ bool) ?x ?y› ‹(OU::'i::type ⇒ 'j::type ⇒ bool) (?x::'i::type) (?y::'j::type) ⟹ (OU'::'i::type ⇒ 'j::type ⇒ bool) ?x ?y› ‹(M::'k::type ⇒ 'l::type ⇒ bool) (?x::'k::type) (?y::'l::type) ⟹ (M'::'k::type ⇒ 'l::type ⇒ bool) ?x ?y›*))\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ S xa ya ⟶ S xa ya›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a; (IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) xa ya⟧ ⟹ (IA' ===> rel_spmf (rel_prod (rel_prod OA' (list_all2 E')) S)) xa ya›\n discuss goal 1*)\n apply (rule impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*) that (*‹(E::'a ⇒ 'b ⇒ bool) ≤ (E'::'a ⇒ 'b ⇒ bool)› ‹(IA'::'c ⇒ 'd ⇒ bool) ≤ (IA::'c ⇒ 'd ⇒ bool)› ‹(IU'::'e ⇒ 'f ⇒ bool) ≤ (IU::'e ⇒ 'f ⇒ bool)› ‹(OA::'g ⇒ 'h ⇒ bool) ≤ (OA'::'g ⇒ 'h ⇒ bool)› ‹(OU::'i ⇒ 'j ⇒ bool) ≤ (OU'::'i ⇒ 'j ⇒ bool)› ‹(M::'k ⇒ 'l ⇒ bool) ≤ (M'::'k ⇒ 'l ⇒ bool)›*) order_refl (*‹(?x::?'a) ≤ ?x›*) prod.rel_mono (*‹⟦(?R1.0::?'a ⇒ ?'c ⇒ bool) ≤ (?R1a::?'a ⇒ ?'c ⇒ bool); (?R2.0::?'b ⇒ ?'d ⇒ bool) ≤ (?R2a::?'b ⇒ ?'d ⇒ bool)⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹(?R::?'a ⇒ ?'b ⇒ bool) ≤ (?Ra::?'a ⇒ ?'b ⇒ bool) ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*top goal: ‹⋀(x1::'k) (x2::'m ⇒ 'c ⇒ (('g × 'a list) × 'm) spmf) (x3::'m ⇒ 'e ⇒ (('i × 'a list) × 'm) spmf) (x1a::'l) (x2a::'n ⇒ 'd ⇒ (('h × 'b list) × 'n) spmf) (x3a::'n ⇒ 'f ⇒ (('j × 'b list) × 'n) spmf) (xa::'m) ya::'n. ⟦(x::('m, 'a, 'c, 'e, 'g, 'i, 'k) rest_scheme) = Rest x1 x2 x3; (y::('n, 'b, 'd, 'f, 'h, 'j, 'l) rest_scheme) = Rest x1a x2a x3a; (M::'k ⇒ 'l ⇒ bool) x1 x1a; ((S::'m ⇒ 'n ⇒ bool) ===> (IU::'e ⇒ 'f ⇒ bool) ===> rel_spmf (rel_prod (rel_prod (OU::'i ⇒ 'j ⇒ bool) (list_all2 (E::'a ⇒ 'b ⇒ bool))) S)) x3 x3a⟧ ⟹ S xa ya ⟶ S xa ya› and 2 goals remain*)\n apply assumption\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ IA' xaa yaa ⟶ IA xaa yaa›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a; rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S) xaa yaa⟧ ⟹ rel_spmf (rel_prod (rel_prod OA' (list_all2 E')) S) xaa yaa›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ IA' xaa yaa ⟶ IA xaa yaa› and 2 goals remain*)\n apply (erule that[THEN predicate2D] (*‹E ?x ?y ⟹ E' ?x ?y› ‹IA' ?x ?y ⟹ IA ?x ?y› ‹IU' ?x ?y ⟹ IU ?x ?y› ‹OA ?x ?y ⟹ OA' ?x ?y› ‹OU ?x ?y ⟹ OU' ?x ?y› ‹M ?x ?y ⟹ M' ?x ?y›*))\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a; rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S) xaa yaa⟧ ⟹ rel_spmf (rel_prod (rel_prod OA' (list_all2 E')) S) xaa yaa› and 1 goal remains*)\n apply (erule rel_fun_mono (*‹⟦((?X::?'a ⇒ ?'b ⇒ bool) ===> (?A::?'c ⇒ ?'d ⇒ bool)) (?f::?'a ⇒ ?'c) (?g::?'b ⇒ ?'d); ⋀(x::?'a) y::?'b. (?Y::?'a ⇒ ?'b ⇒ bool) x y ⟶ ?X x y; ⋀(x::?'c) y::?'d. ?A x y ⟹ (?B::?'c ⇒ ?'d ⇒ bool) x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf (?A::?'a ⇒ ?'b ⇒ bool) (?f::?'a spmf) (?g::?'b spmf); ⋀(x::?'a) y::?'b. ?A x y ⟹ (?B::?'a ⇒ ?'b ⇒ bool) x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod (?R1.1::?'a1 ⇒ ?'c1 ⇒ bool) (?R2.1::?'b1 ⇒ ?'d1 ⇒ bool) (?x::?'a1 × ?'b1) (?y::?'c1 × ?'d1); ?R1.1 ≤ (?R1a1::?'a1 ⇒ ?'c1 ⇒ bool); ?R2.1 ≤ (?R2a1::?'b1 ⇒ ?'d1 ⇒ bool)⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ rel_prod OA (list_all2 E) ≤ rel_prod OA' (list_all2 E')›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ S ≤ S›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ OA ≤ OA'›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ list_all2 E ≤ list_all2 E'›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*discuss goal 2*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) x3 x3a⟧ ⟹ list_all2 E ≤ list_all2 E'› and 2 goals remain*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*discuss goal 3*)\n apply (erule rel_fun_mono (*‹⟦((?X::?'a ⇒ ?'b ⇒ bool) ===> (?A::?'c ⇒ ?'d ⇒ bool)) (?f::?'a ⇒ ?'c) (?g::?'b ⇒ ?'d); ⋀(x::?'a) y::?'b. (?Y::?'a ⇒ ?'b ⇒ bool) x y ⟶ ?X x y; ⋀(x::?'c) y::?'d. ?A x y ⟹ (?B::?'c ⇒ ?'d ⇒ bool) x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf (?A::?'a ⇒ ?'b ⇒ bool) (?f::?'a spmf) (?g::?'b spmf); ⋀(x::?'a) y::?'b. ?A x y ⟹ (?B::?'a ⇒ ?'b ⇒ bool) x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod (?R1.1::?'a1 ⇒ ?'c1 ⇒ bool) (?R2.1::?'b1 ⇒ ?'d1 ⇒ bool) (?x::?'a1 × ?'b1) (?y::?'c1 × ?'d1); ?R1.1 ≤ (?R1a1::?'a1 ⇒ ?'c1 ⇒ bool); ?R2.1 ≤ (?R2a1::?'b1 ⇒ ?'d1 ⇒ bool)⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ S xa ya ⟶ S xa ya›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a; (IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) xa ya⟧ ⟹ (IU' ===> rel_spmf (rel_prod (rel_prod OU' (list_all2 E')) S)) xa ya›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ S xa ya ⟶ S xa ya› and 1 goal remains*)\n apply assumption\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ IU' xaa yaa ⟶ IU xaa yaa›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a; rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S) xaa yaa⟧ ⟹ rel_spmf (rel_prod (rel_prod OU' (list_all2 E')) S) xaa yaa›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ IU' xaa yaa ⟶ IU xaa yaa› and 1 goal remains*)\n apply (erule that[THEN predicate2D] (*‹E ?x ?y ⟹ E' ?x ?y› ‹IA' ?x ?y ⟹ IA ?x ?y› ‹IU' ?x ?y ⟹ IU ?x ?y› ‹OA ?x ?y ⟹ OA' ?x ?y› ‹OU ?x ?y ⟹ OU' ?x ?y› ‹M ?x ?y ⟹ M' ?x ?y›*))\n (*discuss goal 2*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a; rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S) xaa yaa⟧ ⟹ rel_spmf (rel_prod (rel_prod OU' (list_all2 E')) S) xaa yaa›*)\n apply (erule rel_fun_mono (*‹⟦(?X ===> ?A) ?f ?g; ⋀x y. ?Y x y ⟶ ?X x y; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ (?Y ===> ?B) ?f ?g›*) rel_spmf_mono (*‹⟦rel_spmf ?A ?f ?g; ⋀x y. ?A x y ⟹ ?B x y⟧ ⟹ rel_spmf ?B ?f ?g›*) prod.rel_mono[THEN predicate2D, rotated -1] (*‹⟦rel_prod ?R1.1 ?R2.1 ?x ?y; ?R1.1 ≤ ?R1a1; ?R2.1 ≤ ?R2a1⟧ ⟹ rel_prod ?R1a1 ?R2a1 ?x ?y›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ rel_prod OU (list_all2 E) ≤ rel_prod OU' (list_all2 E')›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ S ≤ S›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*goals:\n 1. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ OU ≤ OU'›\n 2. ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ list_all2 E ≤ list_all2 E'›\n discuss goal 1*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*discuss goal 2*)\n apply (rule impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*) that (*‹(E::'a ⇒ 'b ⇒ bool) ≤ (E'::'a ⇒ 'b ⇒ bool)› ‹(IA'::'c ⇒ 'd ⇒ bool) ≤ (IA::'c ⇒ 'd ⇒ bool)› ‹(IU'::'e ⇒ 'f ⇒ bool) ≤ (IU::'e ⇒ 'f ⇒ bool)› ‹(OA::'g ⇒ 'h ⇒ bool) ≤ (OA'::'g ⇒ 'h ⇒ bool)› ‹(OU::'i ⇒ 'j ⇒ bool) ≤ (OU'::'i ⇒ 'j ⇒ bool)› ‹(M::'k ⇒ 'l ⇒ bool) ≤ (M'::'k ⇒ 'l ⇒ bool)›*) order_refl (*‹(?x::?'a) ≤ ?x›*) prod.rel_mono (*‹⟦(?R1.0::?'a ⇒ ?'c ⇒ bool) ≤ (?R1a::?'a ⇒ ?'c ⇒ bool); (?R2.0::?'b ⇒ ?'d ⇒ bool) ≤ (?R2a::?'b ⇒ ?'d ⇒ bool)⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹(?R::?'a ⇒ ?'b ⇒ bool) ≤ (?Ra::?'a ⇒ ?'b ⇒ bool) ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*top goal: ‹⋀x1 x2 x3 x1a x2a x3a xa ya xaa yaa xb yb. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a; M x1 x1a; (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) x2 x2a⟧ ⟹ list_all2 E ≤ list_all2 E'› and 1 goal remains*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*) that (*‹E ≤ E'› ‹IA' ≤ IA› ‹IU' ≤ IU› ‹OA ≤ OA'› ‹OU ≤ OU'› ‹M ≤ M'›*) order_refl (*‹?x ≤ ?x›*) prod.rel_mono (*‹⟦?R1.0 ≤ ?R1a; ?R2.0 ≤ ?R2a⟧ ⟹ rel_prod ?R1.0 ?R2.0 ≤ rel_prod ?R1a ?R2a›*) list.rel_mono (*‹?R ≤ ?Ra ⟹ list_all2 ?R ≤ list_all2 ?Ra›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 3 subgoals*) . .\n\nlemma rel_rest'_sel: \"rel_rest' S E IA IU OA OU M rest1 rest2\"\n if \"M (rinit rest1) (rinit rest2)\"\n and \"(S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) (rfunc_adv rest1) (rfunc_adv rest2)\" \n and \"(S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) (rfunc_usr rest1) (rfunc_usr rest2)\"\n using that (*‹(M::'a ⇒ 'b ⇒ bool) (rinit (rest1::('c, 'd, 'e, 'f, 'g, 'h, 'a) rest_scheme)) (rinit (rest2::('i, 'j, 'k, 'l, 'm, 'n, 'b) rest_scheme))› ‹(S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) (rfunc_adv rest1) (rfunc_adv rest2)› ‹(S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) (rfunc_usr rest1) (rfunc_usr rest2)›*) apply (cases rest1)\n (*goal: ‹rel_rest' S E IA IU OA OU M rest1 rest2›*)\n apply (cases rest2)\n (*goal: ‹⋀x1 x2 x3. ⟦M (rinit rest1) (rinit rest2); (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) (rfunc_adv rest1) (rfunc_adv rest2); (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) (rfunc_usr rest1) (rfunc_usr rest2); rest1 = Rest x1 x2 x3⟧ ⟹ rel_rest' S E IA IU OA OU M rest1 rest2›*)\n by simp\n\nlemma rinit_parametric [transfer_rule]: \"(rel_rest' S E IA IU OA OU M ===> M) rinit rinit\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(rel_rest' S E IA IU OA OU M ===> M) rinit rinit›*)\n apply (erule rel_rest'.cases (*‹⟦rel_rest' ?S ?E ?IA ?IU ?OA ?OU ?M ?a1.0 ?a2.0; ⋀rinit rfunc_adv rfunc_usr rinit' rfunc_adv' rfunc_usr'. ⟦?a1.0 = Rest rinit rfunc_adv rfunc_usr; ?a2.0 = Rest rinit' rfunc_adv' rfunc_usr'; ?M rinit rinit'; (?S ===> ?IA ===> rel_spmf (rel_prod (rel_prod ?OA (list_all2 ?E)) ?S)) rfunc_adv rfunc_adv'; (?S ===> ?IU ===> rel_spmf (rel_prod (rel_prod ?OU (list_all2 ?E)) ?S)) rfunc_usr rfunc_usr'⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹⋀x y. rel_rest' S E IA IU OA OU M x y ⟹ M (rinit x) (rinit y)›*)\n by simp\n\nlemma rfunc_adv_parametric [transfer_rule]:\n \"(rel_rest' S E IA IU OA OU M ===> S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) rfunc_adv rfunc_adv\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(rel_rest' S E IA IU OA OU M ===> S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) rfunc_adv rfunc_adv›*)\n apply (erule rel_rest'.cases (*‹⟦rel_rest' ?S ?E ?IA ?IU ?OA ?OU ?M ?a1.0 ?a2.0; ⋀rinit rfunc_adv rfunc_usr rinit' rfunc_adv' rfunc_usr'. ⟦?a1.0 = Rest rinit rfunc_adv rfunc_usr; ?a2.0 = Rest rinit' rfunc_adv' rfunc_usr'; ?M rinit rinit'; (?S ===> ?IA ===> rel_spmf (rel_prod (rel_prod ?OA (list_all2 ?E)) ?S)) rfunc_adv rfunc_adv'; (?S ===> ?IU ===> rel_spmf (rel_prod (rel_prod ?OU (list_all2 ?E)) ?S)) rfunc_usr rfunc_usr'⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹⋀x y. rel_rest' S E IA IU OA OU M x y ⟹ (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) (rfunc_adv x) (rfunc_adv y)›*)\n by simp\n\nlemma rfunc_usr_parametric [transfer_rule]:\n \"(rel_rest' S E IA IU OA OU M ===> S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) rfunc_usr rfunc_usr\"\n apply (rule rel_funI (*‹(⋀(x::?'a::type) y::?'b::type. (?A::?'a::type ⇒ ?'b::type ⇒ bool) x y ⟹ (?B::?'c::type ⇒ ?'d::type ⇒ bool) ((?f::?'a::type ⇒ ?'c::type) x) ((?g::?'b::type ⇒ ?'d::type) y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(rel_rest' (S::'a::type ⇒ 'h::type ⇒ bool) (E::'b::type ⇒ 'i::type ⇒ bool) (IA::'c::type ⇒ 'j::type ⇒ bool) (IU::'d::type ⇒ 'k::type ⇒ bool) (OA::'e::type ⇒ 'l::type ⇒ bool) (OU::'f::type ⇒ 'm::type ⇒ bool) (M::'g::type ⇒ 'n::type ⇒ bool) ===> S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) rfunc_usr rfunc_usr›*)\n apply (erule rel_rest'.cases (*‹⟦rel_rest' ?S ?E ?IA ?IU ?OA ?OU ?M ?a1.0 ?a2.0; ⋀rinit rfunc_adv rfunc_usr rinit' rfunc_adv' rfunc_usr'. ⟦?a1.0 = Rest rinit rfunc_adv rfunc_usr; ?a2.0 = Rest rinit' rfunc_adv' rfunc_usr'; ?M rinit rinit'; (?S ===> ?IA ===> rel_spmf (rel_prod (rel_prod ?OA (list_all2 ?E)) ?S)) rfunc_adv rfunc_adv'; (?S ===> ?IU ===> rel_spmf (rel_prod (rel_prod ?OU (list_all2 ?E)) ?S)) rfunc_usr rfunc_usr'⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹⋀(x::('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme) y::('h, 'i, 'j, 'k, 'l, 'm, 'n) rest_scheme. rel_rest' (S::'a ⇒ 'h ⇒ bool) (E::'b ⇒ 'i ⇒ bool) (IA::'c ⇒ 'j ⇒ bool) (IU::'d ⇒ 'k ⇒ bool) (OA::'e ⇒ 'l ⇒ bool) (OU::'f ⇒ 'm ⇒ bool) (M::'g ⇒ 'n ⇒ bool) x y ⟹ (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) (rfunc_usr x) (rfunc_usr y)›*)\n by simp\n\nlemma Rest_parametric [transfer_rule]:\n \"(M ===> (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S))\n ===> (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S))\n ===> rel_rest' S E IA IU OA OU M) Rest Rest\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(M ===> (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) ===> (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) ===> rel_rest' S E IA IU OA OU M) Rest Rest›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y. M x y ⟹ ((S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) ===> (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) ===> rel_rest' S E IA IU OA OU M) (Rest x) (Rest y)›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀(x::'a) (y::'b) (xa::'c ⇒ 'd ⇒ (('e × 'f list) × 'c) spmf) ya::'i ⇒ 'j ⇒ (('k × 'l list) × 'i) spmf. ⟦(M::'a ⇒ 'b ⇒ bool) x y; ((S::'c ⇒ 'i ⇒ bool) ===> (IA::'d ⇒ 'j ⇒ bool) ===> rel_spmf (rel_prod (rel_prod (OA::'e ⇒ 'k ⇒ bool) (list_all2 (E::'f ⇒ 'l ⇒ bool))) S)) xa ya⟧ ⟹ ((S ===> (IU::'g ⇒ 'm ⇒ bool) ===> rel_spmf (rel_prod (rel_prod (OU::'h ⇒ 'n ⇒ bool) (list_all2 E)) S)) ===> rel_rest' S E IA IU OA OU M) (Rest x xa) (Rest y ya)›*)\n by simp\n\nlemma case_rest_scheme_parametric [transfer_rule]:\n \"((M ===> \n (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) ===> \n (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) ===> X) ===> \n rel_rest' S E IA IU OA OU M ===> X) case_rest_scheme case_rest_scheme\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹((M ===> (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) ===> (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) ===> X) ===> rel_rest' S E IA IU OA OU M ===> X) case_rest_scheme case_rest_scheme›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y. (M ===> (S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) ===> (S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) ===> X) x y ⟹ (rel_rest' S E IA IU OA OU M ===> X) (case_rest_scheme x) (case_rest_scheme y)›*)\n by (auto 4 4 split: rest_scheme.split (*‹?P (case ?rest_scheme of Rest x xa xb ⇒ ?f x xa xb) = (∀x1 x2 x3. ?rest_scheme = Rest x1 x2 x3 ⟶ ?P (?f x1 x2 x3))›*) dest: rel_funD (*‹⟦(?A ===> ?B) ?f ?g; ?A ?x ?y⟧ ⟹ ?B (?f ?x) (?g ?y)›*))\n\nlemma corec_rest_scheme_parametric [transfer_rule]:\n \"((X ===> M) ===> \n (X ===> S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) ===> \n (X ===> S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) ===> \n X ===> rel_rest' S E IA IU OA OU M) corec_rest_scheme corec_rest_scheme\"\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹(((X::'a::type ⇒ 'c::type ⇒ bool) ===> (M::'b::type ⇒ 'd::type ⇒ bool)) ===> (X ===> (S::'e::type ⇒ 'k::type ⇒ bool) ===> (IA::'f::type ⇒ 'l::type ⇒ bool) ===> rel_spmf (rel_prod (rel_prod (OA::'g::type ⇒ 'm::type ⇒ bool) (list_all2 (E::'h::type ⇒ 'n::type ⇒ bool))) S)) ===> (X ===> S ===> (IU::'i::type ⇒ 'o::type ⇒ bool) ===> rel_spmf (rel_prod (rel_prod (OU::'j::type ⇒ 'p::type ⇒ bool) (list_all2 E)) S)) ===> X ===> rel_rest' S E IA IU OA OU M) corec_rest_scheme corec_rest_scheme›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀(x::'a::type ⇒ 'b::type) y::'c::type ⇒ 'd::type. ((X::'a::type ⇒ 'c::type ⇒ bool) ===> (M::'b::type ⇒ 'd::type ⇒ bool)) x y ⟹ ((X ===> (S::'e::type ⇒ 'k::type ⇒ bool) ===> (IA::'f::type ⇒ 'l::type ⇒ bool) ===> rel_spmf (rel_prod (rel_prod (OA::'g::type ⇒ 'm::type ⇒ bool) (list_all2 (E::'h::type ⇒ 'n::type ⇒ bool))) S)) ===> (X ===> S ===> (IU::'i::type ⇒ 'o::type ⇒ bool) ===> rel_spmf (rel_prod (rel_prod (OU::'j::type ⇒ 'p::type ⇒ bool) (list_all2 E)) S)) ===> X ===> rel_rest' S E IA IU OA OU M) (corec_rest_scheme x) (corec_rest_scheme y)›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y xa ya. ⟦(X ===> M) x y; (X ===> S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) xa ya⟧ ⟹ ((X ===> S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) ===> X ===> rel_rest' S E IA IU OA OU M) (corec_rest_scheme x xa) (corec_rest_scheme y ya)›*)\n apply (rule rel_funI (*‹(⋀x y. ?A x y ⟹ ?B (?f x) (?g y)) ⟹ (?A ===> ?B) ?f ?g›*))\n (*goal: ‹⋀x y xa ya xb yb. ⟦(X ===> M) x y; (X ===> S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) xa ya; (X ===> S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) xb yb⟧ ⟹ (X ===> rel_rest' S E IA IU OA OU M) (corec_rest_scheme x xa xb) (corec_rest_scheme y ya yb)›*)\n by (auto simp add: rest_scheme.corec (*‹corec_rest_scheme ?g1.0 ?g2.0 ?g3.0 ?a = Rest (?g1.0 ?a) (?g2.0 ?a) (?g3.0 ?a)›*) dest: rel_funD (*‹⟦(?A ===> ?B) ?f ?g; ?A ?x ?y⟧ ⟹ ?B (?f ?x) (?g ?y)›*))\n\nprimcorec map_rest' ::\n \"('event ⇒ 'event') ⇒ \n ('iadv_rest' ⇒ 'iadv_rest) ⇒\n ('iusr_rest' ⇒ 'iusr_rest) ⇒ \n ('oadv_rest ⇒ 'oadv_rest') ⇒ \n ('ousr_rest ⇒ 'ousr_rest') ⇒\n ('more ⇒ 'more') ⇒\n ('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme ⇒ \n ('s_rest, 'event', 'iadv_rest', 'iusr_rest', 'oadv_rest', 'ousr_rest', 'more') rest_scheme\"\n where\n \"rinit (map_rest' e ia iu oa ou m rest) = m (rinit rest)\"\n| \"rfunc_adv (map_rest' e ia iu oa ou m rest) =\n (id ---> ia ---> map_spmf (map_prod (map_prod oa (map e)) id)) (rfunc_adv rest)\"\n| \"rfunc_usr (map_rest' e ia iu oa ou m rest) =\n (id ---> iu ---> map_spmf (map_prod (map_prod ou (map e)) id)) (rfunc_usr rest)\"\n\nlemmas map_rest'_simps [simp] = map_rest'.ctr[where rest=\"Rest _ _ _\", simplified]\n\nparametric_constant map_rest'_parametric[transfer_rule]: map_rest'_def\n\nlemma rest'_rel_Grp:\n \"rel_rest' (=) (BNF_Def.Grp UNIV e) (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) (BNF_Def.Grp UNIV m)\n = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m)\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹rel_rest' (=) (BNF_Def.Grp UNIV e) (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) (BNF_Def.Grp UNIV m) = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m)›*)\n subgoal for x and y\n apply (cases x)\n (*goal: ‹rel_rest' (=) (BNF_Def.Grp UNIV e) (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) (BNF_Def.Grp UNIV m) x y = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m) x y›*)\n apply (cases y)\n (*goal: ‹⋀x1 x2 x3. x = Rest x1 x2 x3 ⟹ rel_rest' (=) (BNF_Def.Grp UNIV e) (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) (BNF_Def.Grp UNIV m) x y = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m) x y›*)\n apply clarsimp\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a⟧ ⟹ rel_rest' (=) (BNF_Def.Grp UNIV e) (BNF_Def.Grp UNIV ia)¯¯ (BNF_Def.Grp UNIV iu)¯¯ (BNF_Def.Grp UNIV oa) (BNF_Def.Grp UNIV ou) (BNF_Def.Grp UNIV m) x y = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m) x y›*)\n apply (subst (2 4) eq_alt_conversep (*‹(=) = (BNF_Def.Grp UNIV id)¯¯›*))\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a⟧ ⟹ (BNF_Def.Grp UNIV m x1 x1a ∧ ((=) ===> (BNF_Def.Grp UNIV ia)¯¯ ===> rel_spmf (rel_prod (rel_prod (BNF_Def.Grp UNIV oa) (list_all2 (BNF_Def.Grp UNIV e))) (=))) x2 x2a ∧ ((=) ===> (BNF_Def.Grp UNIV iu)¯¯ ===> rel_spmf (rel_prod (rel_prod (BNF_Def.Grp UNIV ou) (list_all2 (BNF_Def.Grp UNIV e))) (=))) x3 x3a) = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m) (Rest x1 x2 x3) (Rest x1a x2a x3a)›*)\n apply (subst (2 3) eq_alt (*‹(=) = BNF_Def.Grp UNIV id›*))\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a⟧ ⟹ (BNF_Def.Grp UNIV m x1 x1a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV ia)¯¯ ===> rel_spmf (rel_prod (rel_prod (BNF_Def.Grp UNIV oa) (list_all2 (BNF_Def.Grp UNIV e))) (=))) x2 x2a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV iu)¯¯ ===> rel_spmf (rel_prod (rel_prod (BNF_Def.Grp UNIV ou) (list_all2 (BNF_Def.Grp UNIV e))) (=))) x3 x3a) = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m) (Rest x1 x2 x3) (Rest x1a x2a x3a)›*)\n apply (simp add: pmf.rel_Grp (*‹rel_pmf (BNF_Def.Grp ?A ?f) = BNF_Def.Grp {x. set_pmf x ⊆ ?A} (map_pmf ?f)›*) list.rel_Grp (*‹list_all2 (BNF_Def.Grp ?A ?f) = BNF_Def.Grp {x. set x ⊆ ?A} (map ?f)›*) option.rel_Grp (*‹rel_option (BNF_Def.Grp ?A ?f) = BNF_Def.Grp {x. set_option x ⊆ ?A} (map_option ?f)›*) prod.rel_Grp (*‹rel_prod (BNF_Def.Grp ?A1.0 ?f1.0) (BNF_Def.Grp ?A2.0 ?f2.0) = BNF_Def.Grp {x. Basic_BNFs.fsts x ⊆ ?A1.0 ∧ Basic_BNFs.snds x ⊆ ?A2.0} (map_prod ?f1.0 ?f2.0)›*) rel_fun_conversep_grp_grp (*‹(BNF_Def.Grp UNIV ?f)¯¯ ===> BNF_Def.Grp ?B ?g = BNF_Def.Grp {x. range (x ∘ ?f) ⊆ ?B} (?f ---> ?g)›*))\n (*goal: ‹⋀x1 x2 x3 x1a x2a x3a. ⟦x = Rest x1 x2 x3; y = Rest x1a x2a x3a⟧ ⟹ (BNF_Def.Grp UNIV m x1 x1a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV ia)¯¯ ===> rel_spmf (rel_prod (rel_prod (BNF_Def.Grp UNIV oa) (list_all2 (BNF_Def.Grp UNIV e))) (BNF_Def.Grp UNIV id))) x2 x2a ∧ ((BNF_Def.Grp UNIV id)¯¯ ===> (BNF_Def.Grp UNIV iu)¯¯ ===> rel_spmf (rel_prod (rel_prod (BNF_Def.Grp UNIV ou) (list_all2 (BNF_Def.Grp UNIV e))) (BNF_Def.Grp UNIV id))) x3 x3a) = BNF_Def.Grp UNIV (map_rest' e ia iu oa ou m) (Rest x1 x2 x3) (Rest x1a x2a x3a)›*)\n by (auto simp add: Grp_def (*‹BNF_Def.Grp ?A ?f = (λa b. b = ?f a ∧ a ∈ ?A)›*) spmf.map_id[abs_def] (*‹map_spmf id ≡ λp. p›*) id_def[symmetric] (*‹(λx. x) = id›*)) .\n\nend\n\ntype_synonym \n ('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest) rest_wstate =\n \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 's_rest) rest_scheme\"\n\ninductive WT_rest :: \"('iadv, 'oadv) ℐ ⇒ ('iusr, 'ousr) ℐ ⇒ ('s ⇒ bool) ⇒ ('s, 'event, 'iadv, 'iusr, 'oadv, 'ousr) rest_wstate ⇒ bool\"\n for ℐ_adv ℐ_usr I rest where\n \"WT_rest ℐ_adv ℐ_usr I rest\" if\n \"⋀s x y es s'. ⟦ ((y, es), s') ∈ set_spmf (rfunc_adv rest s x); x ∈ outs_ℐ ℐ_adv; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_adv x ∧ I s'\"\n \"⋀s x y es s'. ⟦ ((y, es), s') ∈ set_spmf (rfunc_usr rest s x); x ∈ outs_ℐ ℐ_usr; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_usr x ∧ I s'\"\n \"I (rinit rest)\"\n\nlemma WT_restD:\n assumes \"WT_rest ℐ_adv ℐ_usr I rest\"\n shows WT_restD_rfunc_adv: \"⋀s x y es s'. ⟦ ((y, es), s') ∈ set_spmf (rfunc_adv rest s x); x ∈ outs_ℐ ℐ_adv; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_adv x ∧ I s'\"\n and WT_restD_rfunc_usr: \"⋀s x y es s'. ⟦ ((y, es), s') ∈ set_spmf (rfunc_usr rest s x); x ∈ outs_ℐ ℐ_usr; I s ⟧ ⟹ y ∈ responses_ℐ ℐ_usr x ∧ I s'\"\n and WT_restD_rinit: \"I (rinit rest)\"\n using assms (*‹WT_rest ℐ_adv ℐ_usr I rest›*) apply -\n (*goals:\n 1. ‹⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_adv rest s x); x ∈ outs_ℐ ℐ_adv; I s; WT_rest ℐ_adv ℐ_usr I rest⟧ ⟹ y ∈ responses_ℐ ℐ_adv x ∧ I s'›\n 2. ‹⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_usr rest s x); x ∈ outs_ℐ ℐ_usr; I s; WT_rest ℐ_adv ℐ_usr I rest⟧ ⟹ y ∈ responses_ℐ ℐ_usr x ∧ I s'›\n 3. ‹WT_rest ℐ_adv ℐ_usr I rest ⟹ I (rinit rest)›\n discuss goal 1*)\n apply ((auto elim!: WT_rest.cases (*‹⟦WT_rest (?ℐ_adv::(?'iadv, ?'oadv) ℐ) (?ℐ_usr::(?'iusr, ?'ousr) ℐ) (?I::?'s ⇒ bool) (?rest::(?'s, ?'event, ?'iadv, ?'iusr, ?'oadv, ?'ousr, ?'s) rest_scheme); ⟦⋀(s::?'s) (x::?'iadv) (y::?'oadv) (es::?'event list) s'::?'s. ⟦((y, es), s') ∈ set_spmf (rfunc_adv ?rest s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀(s::?'s) (x::?'iusr) (y::?'ousr) (es::?'event list) s'::?'s. ⟦((y, es), s') ∈ set_spmf (rfunc_usr ?rest s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'; ?I (rinit ?rest)⟧ ⟹ ?P::bool⟧ ⟹ ?P›*))[1])\n (*discuss goal 2*)\n apply ((auto elim!: WT_rest.cases (*‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ⟦⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_adv ?rest s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_usr ?rest s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'; ?I (rinit ?rest)⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*discuss goal 3*)\n apply ((auto elim!: WT_rest.cases (*‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ⟦⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_adv ?rest s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_usr ?rest s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'; ?I (rinit ?rest)⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n (*proven 3 subgoals*) .\n\nabbreviation\n fuse_cfunc :: \"\n ('o ⇒ 'x) ⇒ ('s_core, 'i, 'o) oracle' ⇒ ('s_core × 's_rest, 'i , 'x) oracle'\"\n where\n \"fuse_cfunc redirect cfunc state inp ≡ do {\n let handle = map_prod redirect (prod.swap o Pair (snd state));\n (os_cfunc :: 'o × 's_core) ← cfunc (fst state) inp;\n return_spmf (handle os_cfunc)\n }\"\n\nabbreviation\n fuse_rfunc :: \"\n ('o ⇒ 'x) ⇒ ('s_rest, 'e, 'i, 'o) eoracle ⇒ ('s_core, 'e) handler ⇒ \n ('s_core × 's_rest, 'i , 'x) oracle'\"\n where\n \"fuse_rfunc redirect rfunc notify state inp ≡ \n bind_spmf \n (rfunc (snd state) inp)\n (λ((o_rfunc, e_lst), s_rfunc). \n bind_spmf \n (foldl_spmf notify (return_spmf (fst state)) e_lst)\n (λs_notify. return_spmf (redirect o_rfunc, s_notify, s_rfunc)))\n \"\n\n\nlocale fused_resource =\n fixes \n core :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\" and \n core_init :: 's_core\nbegin\n\nfun\n fuse :: \"\n ('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'm) rest_scheme ⇒\n ('s_core × 's_rest, \n ('iadv_core + 'iadv_rest) + ('iusr_core + 'iusr_rest), \n ('oadv_core + 'oadv_rest) + ('ousr_core + 'ousr_rest)) oracle'\" \n where\n \"fuse rest state (Inl (Inl iadv_core)) = \n fuse_cfunc (Inl o Inl) (cfunc_adv core) state iadv_core\"\n | \"fuse rest state (Inl (Inr iadv_rest)) = \n fuse_rfunc (Inl o Inr) (rfunc_adv rest) (cpoke core) state iadv_rest\"\n | \"fuse rest state (Inr (Inl iusr_core)) = \n fuse_cfunc (Inr o Inl) (cfunc_usr core) state iusr_core\"\n | \"fuse rest state (Inr (Inr iusr_rest)) = \n fuse_rfunc (Inr o Inr) (rfunc_usr rest) (cpoke core) state iusr_rest\"\n\ncase_of_simps fuse_case: fused_resource.fuse.simps\n\nlemma callee_invariant_on_fuse:\n assumes \"WT_core ℐ_adv_core ℐ_usr_core I_core core\" \n and \"WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest\"\n shows \"callee_invariant_on (fuse rest) (pred_prod I_core I_rest) ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest))\"\nproof (unfold_locales, goal_cases)\n (*goals:\n 1. ‹⋀s x y s'. ⟦(y, s') ∈ set_spmf (fuse rest s x); pred_prod I_core I_rest s; x ∈ outs_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest))⟧ ⟹ pred_prod I_core I_rest s'›\n 2. ‹⋀s. pred_prod I_core I_rest s ⟹ (ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢c fuse rest s √›*)\n case (1 s x y s') (*‹(y::('oadv_core + 'b) + 'ousr_core + 'd, s'::'s_core × 'e) ∈ set_spmf (fuse (rest::('e, 'event, 'a, 'c, 'b, 'd, 'e) rest_scheme) (s::'s_core × 'e) (x::('iadv_core + 'a) + 'iusr_core + 'c))› ‹pred_prod I_core I_rest s› ‹x ∈ outs_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest))›*)\n then show \"?case\"\n (*goal: ‹pred_prod I_core I_rest s'›*)\n using assms (*‹WT_core ℐ_adv_core ℐ_usr_core I_core core› ‹WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest›*) apply (cases s)\n (*goal: ‹pred_prod I_core I_rest s'›*)\n apply (cases s')\n (*goal: ‹⋀a b. ⟦(y, s') ∈ set_spmf (fuse rest s x); pred_prod I_core I_rest s; x ∈ outs_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)); WT_core ℐ_adv_core ℐ_usr_core I_core core; WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest; s = (a, b)⟧ ⟹ pred_prod I_core I_rest s'›*)\n by (auto 4 4 dest: WT_restD (*‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_adv ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_usr ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'› ‹WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest ⟹ ?I (rinit ?rest)›*) WT_coreD (*‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; ?s' ∈ set_spmf (cpoke ?core ?s ?e); ?I ?s⟧ ⟹ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_adv ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_usr ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'›*) WT_coreD_foldl_spmf_cpoke (*‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; ?s' ∈ set_spmf (foldl_spmf (cpoke ?core) ?p ?es); ∀s∈set_spmf ?p. ?I s⟧ ⟹ ?I ?s'›*))\nnext\n (*goal: ‹⋀s. pred_prod I_core I_rest s ⟹ (ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢c fuse rest s √›*)\n case (2 s) (*‹pred_prod I_core I_rest s›*)\n show \"?case\"\n (*goal: ‹(ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢c fuse rest s √›*)\n apply (rule WT_calleeI (*‹(⋀call ret s. ⟦call ∈ outs_ℐ ?ℐ; (ret, s) ∈ set_spmf (?callee call)⟧ ⟹ ret ∈ responses_ℐ ?ℐ call) ⟹ ?ℐ ⊢c ?callee √›*))\n (*goal: ‹((ℐ_adv_core::('iadv_core::type, 'oadv_core::type) ℐ) ⊕⇩ℐ (ℐ_adv_rest::('a::type, 'b::type) ℐ)) ⊕⇩ℐ ((ℐ_usr_core::('iusr_core::type, 'ousr_core::type) ℐ) ⊕⇩ℐ (ℐ_usr_rest::('c::type, 'd::type) ℐ)) ⊢c fuse (rest::('e::type, 'event::type, 'a::type, 'c::type, 'b::type, 'd::type, 'e::type) rest_scheme) (s::'s_core::type × 'e::type) √›*)\n subgoal for x and y and s'\n using \"2\" (*‹pred_prod I_core I_rest s›*) assms (*‹WT_core ℐ_adv_core ℐ_usr_core I_core core› ‹WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest›*) apply (cases \"(rest, s, x)\" rule: fuse.cases (*‹⟦⋀(rest::(?'s_rest::type, 'event::type, ?'iadv_rest::type, ?'iusr_rest::type, ?'oadv_rest::type, ?'ousr_rest::type, ?'m::type) rest_scheme) (state::'s_core::type × ?'s_rest::type) iadv_core::'iadv_core::type. (?x::(?'s_rest::type, 'event::type, ?'iadv_rest::type, ?'iusr_rest::type, ?'oadv_rest::type, ?'ousr_rest::type, ?'m::type) rest_scheme × ('s_core::type × ?'s_rest::type) × (('iadv_core::type + ?'iadv_rest::type) + 'iusr_core::type + ?'iusr_rest::type)) = (rest, state, Inl (Inl iadv_core)) ⟹ ?P::bool; ⋀(rest::(?'s_rest::type, 'event::type, ?'iadv_rest::type, ?'iusr_rest::type, ?'oadv_rest::type, ?'ousr_rest::type, ?'m::type) rest_scheme) (state::'s_core::type × ?'s_rest::type) iadv_rest::?'iadv_rest::type. ?x = (rest, state, Inl (Inr iadv_rest)) ⟹ ?P; ⋀(rest::(?'s_rest::type, 'event::type, ?'iadv_rest::type, ?'iusr_rest::type, ?'oadv_rest::type, ?'ousr_rest::type, ?'m::type) rest_scheme) (state::'s_core::type × ?'s_rest::type) iusr_core::'iusr_core::type. ?x = (rest, state, Inr (Inl iusr_core)) ⟹ ?P; ⋀(rest::(?'s_rest::type, 'event::type, ?'iadv_rest::type, ?'iusr_rest::type, ?'oadv_rest::type, ?'ousr_rest::type, ?'m::type) rest_scheme) (state::'s_core::type × ?'s_rest::type) iusr_rest::?'iusr_rest::type. ?x = (rest, state, Inr (Inr iusr_rest)) ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀resta state iadv_core. ⟦x ∈ outs_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)); (y, s') ∈ set_spmf (fuse rest s x); pred_prod I_core I_rest s; WT_core ℐ_adv_core ℐ_usr_core I_core core; WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest; (rest, s, x) = (resta, state, Inl (Inl iadv_core))⟧ ⟹ y ∈ responses_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)) x›\n 2. ‹⋀resta state iadv_rest. ⟦x ∈ outs_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)); (y, s') ∈ set_spmf (fuse rest s x); pred_prod I_core I_rest s; WT_core ℐ_adv_core ℐ_usr_core I_core core; WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest; (rest, s, x) = (resta, state, Inl (Inr iadv_rest))⟧ ⟹ y ∈ responses_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)) x›\n 3. ‹⋀resta state iusr_core. ⟦x ∈ outs_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)); (y, s') ∈ set_spmf (fuse rest s x); pred_prod I_core I_rest s; WT_core ℐ_adv_core ℐ_usr_core I_core core; WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest; (rest, s, x) = (resta, state, Inr (Inl iusr_core))⟧ ⟹ y ∈ responses_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)) x›\n 4. ‹⋀resta state iusr_rest. ⟦x ∈ outs_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)); (y, s') ∈ set_spmf (fuse rest s x); pred_prod I_core I_rest s; WT_core ℐ_adv_core ℐ_usr_core I_core core; WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest; (rest, s, x) = (resta, state, Inr (Inr iusr_rest))⟧ ⟹ y ∈ responses_ℐ ((ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)) x›\n discuss goal 1*)\n apply ((auto simp add: pred_prod_beta (*‹pred_prod ?P ?Q ?xy = (?P (fst ?xy) ∧ ?Q (snd ?xy))›*) dest: WT_restD (*‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_adv ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_usr ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'› ‹WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest ⟹ ?I (rinit ?rest)›*) WT_coreD (*‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; ?s' ∈ set_spmf (cpoke ?core ?s ?e); ?I ?s⟧ ⟹ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_adv ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_usr ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: pred_prod_beta (*‹pred_prod ?P ?Q ?xy = (?P (fst ?xy) ∧ ?Q (snd ?xy))›*) dest: WT_restD (*‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_adv ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_usr ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'› ‹WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest ⟹ ?I (rinit ?rest)›*) WT_coreD (*‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; ?s' ∈ set_spmf (cpoke ?core ?s ?e); ?I ?s⟧ ⟹ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_adv ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_usr ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: pred_prod_beta (*‹pred_prod (?P::?'a::type ⇒ bool) (?Q::?'b::type ⇒ bool) (?xy::?'a::type × ?'b::type) = (?P (fst ?xy) ∧ ?Q (snd ?xy))›*) dest: WT_restD (*‹⟦WT_rest (?ℐ_adv::(?'a::type, ?'b::type) ℐ) (?ℐ_usr::(?'c::type, ?'d::type) ℐ) (?I::?'e::type ⇒ bool) (?rest::(?'e::type, ?'f::type, ?'a::type, ?'c::type, ?'b::type, ?'d::type, ?'e::type) rest_scheme); ((?y::?'b::type, ?es::?'f::type list), ?s'::?'e::type) ∈ set_spmf (rfunc_adv ?rest (?s::?'e::type) (?x::?'a::type)); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_rest (?ℐ_adv::(?'a::type, ?'b::type) ℐ) (?ℐ_usr::(?'c::type, ?'d::type) ℐ) (?I::?'e::type ⇒ bool) (?rest::(?'e::type, ?'f::type, ?'a::type, ?'c::type, ?'b::type, ?'d::type, ?'e::type) rest_scheme); ((?y::?'d::type, ?es::?'f::type list), ?s'::?'e::type) ∈ set_spmf (rfunc_usr ?rest (?s::?'e::type) (?x::?'c::type)); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'› ‹WT_rest (?ℐ_adv::(?'a::type, ?'b::type) ℐ) (?ℐ_usr::(?'c::type, ?'d::type) ℐ) (?I::?'e::type ⇒ bool) (?rest::(?'e::type, ?'f::type, ?'a::type, ?'c::type, ?'b::type, ?'d::type, ?'e::type) rest_scheme) ⟹ ?I (rinit ?rest)›*) WT_coreD (*‹⟦WT_core (?ℐ_adv::(?'a::type, ?'b::type) ℐ) (?ℐ_usr::(?'c::type, ?'d::type) ℐ) (?I::?'e::type ⇒ bool) (?core::(?'e::type, ?'f::type, ?'a::type, ?'c::type, ?'b::type, ?'d::type) core); (?s'::?'e::type) ∈ set_spmf (cpoke ?core (?s::?'e::type) (?e::?'f::type)); ?I ?s⟧ ⟹ ?I ?s'› ‹⟦WT_core (?ℐ_adv::(?'a::type, ?'b::type) ℐ) (?ℐ_usr::(?'c::type, ?'d::type) ℐ) (?I::?'e::type ⇒ bool) (?core::(?'e::type, ?'f::type, ?'a::type, ?'c::type, ?'b::type, ?'d::type) core); (?y::?'b::type, ?s'::?'e::type) ∈ set_spmf (cfunc_adv ?core (?s::?'e::type) (?x::?'a::type)); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_core (?ℐ_adv::(?'a::type, ?'b::type) ℐ) (?ℐ_usr::(?'c::type, ?'d::type) ℐ) (?I::?'e::type ⇒ bool) (?core::(?'e::type, ?'f::type, ?'a::type, ?'c::type, ?'b::type, ?'d::type) core); (?y::?'d::type, ?s'::?'e::type) ∈ set_spmf (cfunc_usr ?core (?s::?'e::type) (?x::?'c::type)); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'›*))[1])\n (*discuss goal 4*)\n apply ((auto simp add: pred_prod_beta (*‹pred_prod ?P ?Q ?xy = (?P (fst ?xy) ∧ ?Q (snd ?xy))›*) dest: WT_restD (*‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_adv ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest; ((?y, ?es), ?s') ∈ set_spmf (rfunc_usr ?rest ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'› ‹WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest ⟹ ?I (rinit ?rest)›*) WT_coreD (*‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; ?s' ∈ set_spmf (cpoke ?core ?s ?e); ?I ?s⟧ ⟹ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_adv ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_adv; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_adv ?x ∧ ?I ?s'› ‹⟦WT_core ?ℐ_adv ?ℐ_usr ?I ?core; (?y, ?s') ∈ set_spmf (cfunc_usr ?core ?s ?x); ?x ∈ outs_ℐ ?ℐ_usr; ?I ?s⟧ ⟹ ?y ∈ responses_ℐ ?ℐ_usr ?x ∧ ?I ?s'›*))[1])\n (*proven 4 subgoals*) . .\nqed\n\ndefinition \n resource :: \"\n ('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest) rest_wstate ⇒\n (('iadv_core + 'iadv_rest) + ('iusr_core + 'iusr_rest),\n ('oadv_core + 'oadv_rest) + ('ousr_core + 'ousr_rest)) resource\" \n where\n \"resource rest = resource_of_oracle (fuse rest) (core_init, rinit rest)\"\n\nlemma WT_resource [WT_intro]:\n assumes \"WT_core ℐ_adv_core ℐ_usr_core I_core core\"\n and \"WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest\"\n and \"I_core core_init\"\n shows \"(ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢res resource rest √\"\nproof (-)\n (*goal: ‹(ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢res resource rest √›*)\n interpret callee_invariant_on \"fuse rest\" \"pred_prod I_core I_rest\" \"(ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest)\"\n using assms(1,2) (*‹WT_core ℐ_adv_core ℐ_usr_core I_core core› ‹WT_rest ℐ_adv_rest ℐ_usr_rest I_rest rest›*) by (rule callee_invariant_on_fuse (*‹⟦WT_core ?ℐ_adv_core ?ℐ_usr_core ?I_core core; WT_rest ?ℐ_adv_rest ?ℐ_usr_rest ?I_rest ?rest⟧ ⟹ callee_invariant_on (fuse ?rest) (pred_prod ?I_core ?I_rest) ((?ℐ_adv_core ⊕⇩ℐ ?ℐ_adv_rest) ⊕⇩ℐ (?ℐ_usr_core ⊕⇩ℐ ?ℐ_usr_rest))›*))\n show \"?thesis\"\n (*goal: ‹((ℐ_adv_core::('iadv_core, 'oadv_core) ℐ) ⊕⇩ℐ (ℐ_adv_rest::('a, 'b) ℐ)) ⊕⇩ℐ ((ℐ_usr_core::('iusr_core, 'ousr_core) ℐ) ⊕⇩ℐ (ℐ_usr_rest::('c, 'd) ℐ)) ⊢res resource (rest::('e, 'event, 'a, 'c, 'b, 'd, 'e) rest_scheme) √›*)\n unfolding resource_def\n (*goal: ‹(ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢res RES (fuse rest) (core_init, rinit rest) √›*)\n apply (rule WT_resource_of_oracle (*‹pred_prod I_core I_rest ?s ⟹ (ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢res RES (fuse rest) ?s √›*))\n (*goal: ‹(ℐ_adv_core ⊕⇩ℐ ℐ_adv_rest) ⊕⇩ℐ (ℐ_usr_core ⊕⇩ℐ ℐ_usr_rest) ⊢res RES (fuse rest) (core_init, rinit rest) √›*)\n by (simp add: assms( (*‹I_core core_init›*) 3) WT_restD_rinit[OF assms(2)] (*‹I_rest (rinit rest)›*))\nqed\n\nend\n\nparametric_constant \n fuse_parametric [transfer_rule]: fused_resource.fuse_case\n\nsubsection ‹More helpful construction functions›\n\ncontext\n fixes\n core1 :: \"('s_core1, 'event1, 'iadv_core1, 'iusr_core1, 'oadv_core1, 'ousr_core1) core\" and\n core2 :: \"('s_core2, 'event2, 'iadv_core2, 'iusr_core2, 'oadv_core2, 'ousr_core2) core\"\nbegin\n\nprimcorec parallel_core :: \"\n ('s_core1 × 's_core2, 'event1 + 'event2, \n 'iadv_core1 + 'iadv_core2, 'iusr_core1 + 'iusr_core2,\n 'oadv_core1 + 'oadv_core2, 'ousr_core1 + 'ousr_core2) core\" \n where\n \"cpoke parallel_core = parallel_handler (cpoke core1) (cpoke core2)\"\n | \"cfunc_adv parallel_core = parallel_oracle (cfunc_adv core1) (cfunc_adv core2)\"\n | \"cfunc_usr parallel_core = parallel_oracle (cfunc_usr core1) (cfunc_usr core2)\"\n\nend\n\n\ncontext\n fixes\n cnv_adv :: \"'s_adv ⇒ 'iadv ⇒ ('oadv × 's_adv, 'iadv_core, 'oadv_core) gpv\" and\n cnv_usr :: \"'s_usr ⇒ 'iusr ⇒ ('ousr × 's_usr, 'iusr_core, 'ousr_core) gpv\" and\n core :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\nbegin\n\nprimcorec \n attach_core :: \"(('s_adv × 's_usr) × 's_core, 'event, 'iadv, 'iusr, 'oadv, 'ousr) core\"\n where\n \"cpoke attach_core = (λ(s_advusr, s_core) event. \n map_spmf (λs_core'. (s_advusr, s_core')) (cpoke core s_core event))\"\n | \"cfunc_adv attach_core = (λ((s_adv, s_usr), s_core) iadv. \n map_spmf \n (λ((oadv, s_adv'), s_core'). (oadv, ((s_adv', s_usr), s_core'))) \n (exec_gpv (cfunc_adv core) (cnv_adv s_adv iadv) s_core))\"\n | \"cfunc_usr attach_core = (λ((s_adv, s_usr), s_core) iusr. \n map_spmf \n (λ((ousr, s_usr'), s_core'). (ousr, ((s_adv, s_usr'), s_core'))) \n (exec_gpv (cfunc_usr core) (cnv_usr s_usr iusr) s_core))\"\n\nend\n\n\nlemma \n attach_core_id_oracle_adv: \"cfunc_adv (attach_core 1⇩I cnv core) = \n (λ(s_cnv, s_core) q. map_spmf (λ(out, s_core'). (out, s_cnv, s_core')) (cfunc_adv core s_core q))\"\n by (simp add: id_oracle_def (*‹1⇩I ?s ?x = Pause ?x (λx. Done (x, ?s))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) map_spmf_conv_bind_spmf (*‹map_spmf ?f ?p = ?p ⤜ (λx. return_spmf (?f x))›*))\n\nlemma \n attach_core_id_oracle_usr: \"cfunc_usr (attach_core cnv 1⇩I core) = \n (λ(s_cnv, s_core) q. map_spmf (λ(out, s_core'). (out, s_cnv, s_core')) (cfunc_usr core s_core q))\"\n by (simp add: id_oracle_def (*‹1⇩I ?s ?x = Pause ?x (λx. Done (x, ?s))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) map_spmf_conv_bind_spmf (*‹map_spmf ?f ?p = ?p ⤜ (λx. return_spmf (?f x))›*))\n\n\ncontext\n fixes\n rest1 :: \"('s_rest1, 'event1, 'iadv_rest1, 'iusr_rest1, 'oadv_rest1, 'ousr_rest1, 'more1) rest_scheme\" and\n rest2 :: \"('s_rest2, 'event2, 'iadv_rest2, 'iusr_rest2, 'oadv_rest2, 'ousr_rest2, 'more2) rest_scheme\"\nbegin\n\nprimcorec parallel_rest :: \"\n ('s_rest1 × 's_rest2, 'event1 + 'event2, 'iadv_rest1 + 'iadv_rest2, 'iusr_rest1 + 'iusr_rest2, \n 'oadv_rest1 + 'oadv_rest2, 'ousr_rest1 + 'ousr_rest2, 'more1 × 'more2) rest_scheme\" \n where\n \"rinit parallel_rest = (rinit rest1, rinit rest2)\"\n | \"rfunc_adv parallel_rest = parallel_eoracle (rfunc_adv rest1) (rfunc_adv rest2)\"\n | \"rfunc_usr parallel_rest = parallel_eoracle (rfunc_usr rest1) (rfunc_usr rest2)\"\n\nend\n\nlemma WT_parallel_rest [WT_intro]:\n \"WT_rest (ℐ_adv1 ⊕⇩ℐ ℐ_adv2) (ℐ_usr1 ⊕⇩ℐ ℐ_usr2) (pred_prod I1 I2) (parallel_rest rest1 rest2)\"\n if \"WT_rest ℐ_adv1 ℐ_usr1 I1 rest1\"\n and \"WT_rest ℐ_adv2 ℐ_usr2 I2 rest2\"\n apply (rule WT_rest.intros (*‹⟦⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_adv ?rest s x); x ∈ outs_ℐ ?ℐ_adv; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_adv x ∧ ?I s'; ⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_usr ?rest s x); x ∈ outs_ℐ ?ℐ_usr; ?I s⟧ ⟹ y ∈ responses_ℐ ?ℐ_usr x ∧ ?I s'; ?I (rinit ?rest)⟧ ⟹ WT_rest ?ℐ_adv ?ℐ_usr ?I ?rest›*))\n (*goals:\n 1. ‹⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_adv (parallel_rest rest1 rest2) s x); x ∈ outs_ℐ (ℐ_adv1 ⊕⇩ℐ ℐ_adv2); pred_prod I1 I2 s⟧ ⟹ y ∈ responses_ℐ (ℐ_adv1 ⊕⇩ℐ ℐ_adv2) x ∧ pred_prod I1 I2 s'›\n 2. ‹⋀s x y es s'. ⟦((y, es), s') ∈ set_spmf (rfunc_usr (parallel_rest rest1 rest2) s x); x ∈ outs_ℐ (ℐ_usr1 ⊕⇩ℐ ℐ_usr2); pred_prod I1 I2 s⟧ ⟹ y ∈ responses_ℐ (ℐ_usr1 ⊕⇩ℐ ℐ_usr2) x ∧ pred_prod I1 I2 s'›\n 3. ‹pred_prod I1 I2 (rinit (parallel_rest rest1 rest2))›\n discuss goal 1*)\n apply ((auto 4 3 simp add: parallel_eoracle_def (*‹parallel_eoracle ?eoracle1.0 ?eoracle2.0 ?state ≡ map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) ∘ (?eoracle1.0 ‡⇩O ?eoracle2.0) ?state›*) simp add: that[THEN WT_restD_rinit] (*‹I1 (rinit rest1)› ‹I2 (rinit rest2)›*) dest: that[THEN WT_restD_rfunc_adv] (*‹⟦((?y, ?es), ?s') ∈ set_spmf (rfunc_adv rest1 ?s ?x); ?x ∈ outs_ℐ ℐ_adv1; I1 ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv1 ?x ∧ I1 ?s'› ‹⟦((?y, ?es), ?s') ∈ set_spmf (rfunc_adv rest2 ?s ?x); ?x ∈ outs_ℐ ℐ_adv2; I2 ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv2 ?x ∧ I2 ?s'›*) that[THEN WT_restD_rfunc_usr] (*‹⟦((?y, ?es), ?s') ∈ set_spmf (rfunc_usr rest1 ?s ?x); ?x ∈ outs_ℐ ℐ_usr1; I1 ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr1 ?x ∧ I1 ?s'› ‹⟦((?y, ?es), ?s') ∈ set_spmf (rfunc_usr rest2 ?s ?x); ?x ∈ outs_ℐ ℐ_usr2; I2 ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr2 ?x ∧ I2 ?s'›*))[1])\n (*discuss goal 2*)\n apply ((auto 4 3 simp add: parallel_eoracle_def (*‹parallel_eoracle (?eoracle1.0::?'s1.0 ⇒ ?'i1.0 ⇒ ((?'o1.0 × ?'e1.0 list) × ?'s1.0) spmf) (?eoracle2.0::?'s2.0 ⇒ ?'i2.0 ⇒ ((?'o2.0 × ?'e2.0 list) × ?'s2.0) spmf) (?state::?'s1.0 × ?'s2.0) ≡ map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) ∘ (?eoracle1.0 ‡⇩O ?eoracle2.0) ?state›*) simp add: that[THEN WT_restD_rinit] (*‹(I1::'e ⇒ bool) (rinit (rest1::('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme))› ‹(I2::'k ⇒ bool) (rinit (rest2::('k, 'l, 'g, 'i, 'h, 'j, 'k) rest_scheme))›*) dest: that[THEN WT_restD_rfunc_adv] (*‹⟦((?y::'b, ?es::'f list), ?s'::'e) ∈ set_spmf (rfunc_adv (rest1::('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme) (?s::'e) (?x::'a)); ?x ∈ outs_ℐ (ℐ_adv1::('a, 'b) ℐ); (I1::'e ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv1 ?x ∧ I1 ?s'› ‹⟦((?y::'h, ?es::'l list), ?s'::'k) ∈ set_spmf (rfunc_adv (rest2::('k, 'l, 'g, 'i, 'h, 'j, 'k) rest_scheme) (?s::'k) (?x::'g)); ?x ∈ outs_ℐ (ℐ_adv2::('g, 'h) ℐ); (I2::'k ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv2 ?x ∧ I2 ?s'›*) that[THEN WT_restD_rfunc_usr] (*‹⟦((?y::'d, ?es::'f list), ?s'::'e) ∈ set_spmf (rfunc_usr (rest1::('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme) (?s::'e) (?x::'c)); ?x ∈ outs_ℐ (ℐ_usr1::('c, 'd) ℐ); (I1::'e ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr1 ?x ∧ I1 ?s'› ‹⟦((?y::'j, ?es::'l list), ?s'::'k) ∈ set_spmf (rfunc_usr (rest2::('k, 'l, 'g, 'i, 'h, 'j, 'k) rest_scheme) (?s::'k) (?x::'i)); ?x ∈ outs_ℐ (ℐ_usr2::('i, 'j) ℐ); (I2::'k ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr2 ?x ∧ I2 ?s'›*))[1])\n (*discuss goal 3*)\n apply ((auto 4 3 simp add: parallel_eoracle_def (*‹parallel_eoracle (?eoracle1.0::?'s1.0 ⇒ ?'i1.0 ⇒ ((?'o1.0 × ?'e1.0 list) × ?'s1.0) spmf) (?eoracle2.0::?'s2.0 ⇒ ?'i2.0 ⇒ ((?'o2.0 × ?'e2.0 list) × ?'s2.0) spmf) (?state::?'s1.0 × ?'s2.0) ≡ map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) ∘ (?eoracle1.0 ‡⇩O ?eoracle2.0) ?state›*) simp add: that[THEN WT_restD_rinit] (*‹(I1::'e ⇒ bool) (rinit (rest1::('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme))› ‹(I2::'k ⇒ bool) (rinit (rest2::('k, 'l, 'g, 'i, 'h, 'j, 'k) rest_scheme))›*) dest: that[THEN WT_restD_rfunc_adv] (*‹⟦((?y::'b, ?es::'f list), ?s'::'e) ∈ set_spmf (rfunc_adv (rest1::('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme) (?s::'e) (?x::'a)); ?x ∈ outs_ℐ (ℐ_adv1::('a, 'b) ℐ); (I1::'e ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv1 ?x ∧ I1 ?s'› ‹⟦((?y::'h, ?es::'l list), ?s'::'k) ∈ set_spmf (rfunc_adv (rest2::('k, 'l, 'g, 'i, 'h, 'j, 'k) rest_scheme) (?s::'k) (?x::'g)); ?x ∈ outs_ℐ (ℐ_adv2::('g, 'h) ℐ); (I2::'k ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv2 ?x ∧ I2 ?s'›*) that[THEN WT_restD_rfunc_usr] (*‹⟦((?y::'d, ?es::'f list), ?s'::'e) ∈ set_spmf (rfunc_usr (rest1::('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme) (?s::'e) (?x::'c)); ?x ∈ outs_ℐ (ℐ_usr1::('c, 'd) ℐ); (I1::'e ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr1 ?x ∧ I1 ?s'› ‹⟦((?y::'j, ?es::'l list), ?s'::'k) ∈ set_spmf (rfunc_usr (rest2::('k, 'l, 'g, 'i, 'h, 'j, 'k) rest_scheme) (?s::'k) (?x::'i)); ?x ∈ outs_ℐ (ℐ_usr2::('i, 'j) ℐ); (I2::'k ⇒ bool) ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr2 ?x ∧ I2 ?s'›*))[1])\n (*proven 3 subgoals*) .\n\ncontext\n fixes\n cnv_adv :: \"'s_adv ⇒ 'iadv ⇒ ('oadv × 's_adv, 'iadv_rest, 'oadv_rest) gpv\" and\n cnv_usr :: \"'s_usr ⇒ 'iusr ⇒ ('ousr × 's_usr, 'iusr_rest, 'ousr_rest) gpv\" and\n f_init :: \"'more ⇒ 'more'\" and\n rest :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\"\nbegin\n\nprimcorec \n attach_rest :: \"\n (('s_adv × 's_usr) × 's_rest, 'event, 'iadv, 'iusr, 'oadv, 'ousr, 'more') rest_scheme\"\n where\n \"rinit attach_rest = f_init (rinit rest)\"\n | \"rfunc_adv attach_rest = (λ((s_adv, s_usr), s_rest) iadv.\n let orc_of = λorc (s, es) q. map_spmf (λ ((out, e), s'). (out, s', es @ e)) (orc s q) in\n let eorc_of = λ((oadv, s_adv'), (s_rest', es)). ((oadv, es), ((s_adv', s_usr), s_rest')) in\n map_spmf eorc_of (exec_gpv (orc_of (rfunc_adv rest)) (cnv_adv s_adv iadv) (s_rest, [])))\"\n | \"rfunc_usr attach_rest = (λ((s_adv, s_usr), s_rest) iusr. \n let orc_of = λorc (s, es) q. map_spmf (λ ((out, e), s'). (out, s', es @ e)) (orc s q) in\n let eorc_of = λ((ousr, s_usr'), (s_rest', es)). ((ousr, es), ((s_adv, s_usr'), s_rest')) in\n map_spmf eorc_of (exec_gpv (orc_of (rfunc_usr rest)) (cnv_usr s_usr iusr) (s_rest, [])))\"\n \nend\n\n\nlemma \n attach_rest_id_oracle_adv: \"rfunc_adv (attach_rest 1⇩I cnv f_init rest) = \n (λ(s_cnv, s_core) q. map_spmf (λ(out, s_core'). (out, s_cnv, s_core')) (rfunc_adv rest s_core q))\"\n by (simp add: id_oracle_def (*‹1⇩I ?s ?x = Pause ?x (λx. Done (x, ?s))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) map_spmf_conv_bind_spmf (*‹map_spmf ?f ?p = ?p ⤜ (λx. return_spmf (?f x))›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*)) \n\nlemma\n attach_rest_id_oracle_usr: \"rfunc_usr (attach_rest cnv 1⇩I f_init rest) = \n (λ(s_cnv, s_core) q. map_spmf (λ(out, s_core'). (out, s_cnv, s_core')) (rfunc_usr rest s_core q))\"\n by (simp add: id_oracle_def (*‹1⇩I ?s ?x = Pause ?x (λx. Done (x, ?s))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) map_spmf_conv_bind_spmf (*‹map_spmf ?f ?p = ?p ⤜ (λx. return_spmf (?f x))›*)) \n\n\n\nsection ‹Traces›\n\ntype_synonym ('event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) trace_core =\n \"('event + 'iadv_core × 'oadv_core + 'iusr_core × 'ousr_core) list \n ⇒ ('event ⇒ real)\n × ('iadv_core ⇒ 'oadv_core spmf) \n × ('iusr_core ⇒ 'ousr_core spmf)\"\n\ncontext \n fixes core :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\nbegin\n\nprimrec trace_core' :: \"'s_core spmf ⇒ ('event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) trace_core\" where\n \"trace_core' S [] = \n (λe. weight_spmf' (bind_spmf S (λs. cpoke core s e)),\n λia. bind_spmf S (λs. map_spmf fst (cfunc_adv core s ia)),\n λiu. bind_spmf S (λs. map_spmf fst (cfunc_usr core s iu)))\"\n| \"trace_core' S (obs # tr) = (case obs of\n Inl e ⇒ trace_core' (mk_lossless (bind_spmf S (λs. cpoke core s e))) tr\n | Inr (Inl (ia, oa)) ⇒ trace_core' (cond_spmf_fst (bind_spmf S (λs. cfunc_adv core s ia)) oa) tr\n | Inr (Inr (iu, ou)) ⇒ trace_core' (cond_spmf_fst (bind_spmf S (λs. cfunc_usr core s iu)) ou) tr\n )\"\n\nend\n\ndeclare trace_core'.simps [simp del]\ncase_of_simps trace_core'_unfold: trace_core'.simps[unfolded weight_spmf'_def]\nsimps_of_case trace_core'_simps [simp]: trace_core'_unfold\n\ncontext includes lifting_syntax begin\n\nlemma trace_core'_parametric [transfer_rule]:\n \"(rel_core' S E IA IU (=) (=) ===>\n rel_spmf S ===>\n list_all2 (rel_sum E (rel_sum (rel_prod IA (=)) (rel_prod IU (=)))) ===>\n rel_prod (E ===> (=)) (rel_prod (IA ===> (=)) (IU ===> (=))))\n trace_core' trace_core'\"\n unfolding trace_core'_def\n (*goal: ‹(rel_core' S E IA IU (=) (=) ===> rel_spmf S ===> list_all2 (rel_sum E (rel_sum (rel_prod IA (=)) (rel_prod IU (=)))) ===> rel_prod (E ===> (=)) (rel_prod (IA ===> (=)) (IU ===> (=)))) (λcore uu uua. rec_list (λS. (λe. weight_spmf' (S ⤜ (λs. cpoke core s e)), λia. S ⤜ (λs. map_spmf fst (cfunc_adv core s ia)), λiu. S ⤜ (λs. map_spmf fst (cfunc_usr core s iu)))) (λobs tr tra S. case obs of Inl e ⇒ tra (mk_lossless (S ⤜ (λs. cpoke core s e))) | Inr (Inl (ia, oa)) ⇒ tra (cond_spmf_fst (S ⤜ (λs. cfunc_adv core s ia)) oa) | Inr (Inr (iu, ou)) ⇒ tra (cond_spmf_fst (S ⤜ (λs. cfunc_usr core s iu)) ou)) uua uu) (λcore uu uua. rec_list (λS. (λe. weight_spmf' (S ⤜ (λs. cpoke core s e)), λia. S ⤜ (λs. map_spmf fst (cfunc_adv core s ia)), λiu. S ⤜ (λs. map_spmf fst (cfunc_usr core s iu)))) (λobs tr tra S. case obs of Inl e ⇒ tra (mk_lossless (S ⤜ (λs. cpoke core s e))) | Inr (Inl (ia, oa)) ⇒ tra (cond_spmf_fst (S ⤜ (λs. cfunc_adv core s ia)) oa) | Inr (Inr (iu, ou)) ⇒ tra (cond_spmf_fst (S ⤜ (λs. cfunc_usr core s iu)) ou)) uua uu)›*)\n by transfer_prover\n\ndefinition trace_core_eq \n :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\n ⇒ ('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\n ⇒ 'event set ⇒ 'iadv_core set ⇒ 'iusr_core set\n ⇒ 's_core spmf ⇒ 's_core' spmf ⇒ bool\" where\n \"trace_core_eq core1 core2 E IA IU p q ⟷\n (∀tr. set tr ⊆ E <+> (IA × UNIV) <+> (IU × UNIV) ⟶ \n rel_prod (eq_onp (λe. e ∈ E) ===> (=)) (rel_prod (eq_onp (λia. ia ∈ IA) ===> (=)) (eq_onp (λiu. iu ∈ IU) ===> (=)))\n (trace_core' core1 p tr) (trace_core' core2 q tr))\"\n\nend\n\nlemma trace_core_eqD:\n assumes \"trace_core_eq core1 core2 E IA IU p q\"\n and \"set tr ⊆ E <+> (IA × UNIV) <+> (IU × UNIV)\"\n shows trace_core_eqD_cpoke: \n \"e ∈ E ⟹ fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e\"\n and trace_core_eqD_cfunc_adv: \n \"ia ∈ IA ⟹ fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia\"\n and trace_core_eqD_cfunc_usr:\n \"iu ∈ IU ⟹ snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu\"\n using assms (*‹trace_core_eq core1 core2 E IA IU p q› ‹set tr ⊆ E <+> IA × UNIV <+> IU × UNIV›*) apply -\n (*goals:\n 1. ‹⟦e ∈ E; trace_core_eq core1 core2 E IA IU p q; set tr ⊆ E <+> IA × UNIV <+> IU × UNIV⟧ ⟹ fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e›\n 2. ‹⟦ia ∈ IA; trace_core_eq core1 core2 E IA IU p q; set tr ⊆ E <+> IA × UNIV <+> IU × UNIV⟧ ⟹ fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia›\n 3. ‹⟦iu ∈ IU; trace_core_eq core1 core2 E IA IU p q; set tr ⊆ E <+> IA × UNIV <+> IU × UNIV⟧ ⟹ snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu›\n discuss goal 1*)\n apply ((auto simp add: trace_core_eq_def (*‹trace_core_eq ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q = (∀tr. set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λe. e ∈ ?E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ ?IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ ?IU)) (=))) (trace_core' ?core1.0 ?p tr) (trace_core' ?core2.0 ?q tr))›*) rel_fun_def (*‹rel_fun ?A ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*) rel_prod_sel (*‹rel_prod ?R1.0 ?R2.0 ?p ?q = (?R1.0 (fst ?p) (fst ?q) ∧ ?R2.0 (snd ?p) (snd ?q))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: trace_core_eq_def (*‹trace_core_eq ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q = (∀tr. set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λe. e ∈ ?E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ ?IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ ?IU)) (=))) (trace_core' ?core1.0 ?p tr) (trace_core' ?core2.0 ?q tr))›*) rel_fun_def (*‹rel_fun ?A ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*) rel_prod_sel (*‹rel_prod ?R1.0 ?R2.0 ?p ?q = (?R1.0 (fst ?p) (fst ?q) ∧ ?R2.0 (snd ?p) (snd ?q))›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: trace_core_eq_def (*‹trace_core_eq ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q = (∀tr. set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λe. e ∈ ?E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ ?IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ ?IU)) (=))) (trace_core' ?core1.0 ?p tr) (trace_core' ?core2.0 ?q tr))›*) rel_fun_def (*‹rel_fun ?A ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*) rel_prod_sel (*‹rel_prod ?R1.0 ?R2.0 ?p ?q = (?R1.0 (fst ?p) (fst ?q) ∧ ?R2.0 (snd ?p) (snd ?q))›*))[1])\n (*proven 3 subgoals*) .\n\nlemma trace_core_eqI:\n assumes \"⋀tr e. ⟦ set tr ⊆ E <+> (IA × UNIV) <+> (IU × UNIV); e ∈ E ⟧ \n ⟹ fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e\"\n and \"⋀tr ia. ⟦ set tr ⊆ E <+> (IA × UNIV) <+> (IU × UNIV); ia ∈ IA ⟧\n ⟹ fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia\"\n and \"⋀tr iu. ⟦ set tr ⊆ E <+> (IA × UNIV) <+> (IU × UNIV); iu ∈ IU ⟧\n ⟹ snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu\"\n shows \"trace_core_eq core1 core2 E IA IU p q\"\n using assms (*‹⟦set ?tr ⊆ E <+> IA × UNIV <+> IU × UNIV; ?e ∈ E⟧ ⟹ fst (trace_core' core1 p ?tr) ?e = fst (trace_core' core2 q ?tr) ?e› ‹⟦set ?tr ⊆ E <+> IA × UNIV <+> IU × UNIV; ?ia ∈ IA⟧ ⟹ fst (snd (trace_core' core1 p ?tr)) ?ia = fst (snd (trace_core' core2 q ?tr)) ?ia› ‹⟦set ?tr ⊆ E <+> IA × UNIV <+> IU × UNIV; ?iu ∈ IU⟧ ⟹ snd (snd (trace_core' core1 p ?tr)) ?iu = snd (snd (trace_core' core2 q ?tr)) ?iu›*) by (auto simp add: trace_core_eq_def (*‹trace_core_eq ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q = (∀tr. set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λe. e ∈ ?E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ ?IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ ?IU)) (=))) (trace_core' ?core1.0 ?p tr) (trace_core' ?core2.0 ?q tr))›*) rel_fun_def (*‹rel_fun ?A ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*) rel_prod_sel (*‹rel_prod ?R1.0 ?R2.0 ?p ?q = (?R1.0 (fst ?p) (fst ?q) ∧ ?R2.0 (snd ?p) (snd ?q))›*))\n\nlemma trace_core_return_pmf_None [simp]:\n \"trace_core' core (return_pmf None) tr = (λ_. 0, λ_. return_pmf None, λ_. return_pmf None)\"\n apply (induction tr)\n (*goals:\n 1. ‹trace_core' core (return_pmf None) [] = (λ_. 0, λ_. return_pmf None, λ_. return_pmf None)›\n 2. ‹⋀a tr. trace_core' core (return_pmf None) tr = (λ_. 0, λ_. return_pmf None, λ_. return_pmf None) ⟹ trace_core' core (return_pmf None) (a # tr) = (λ_. 0, λ_. return_pmf None, λ_. return_pmf None)›\n discuss goal 1*)\n apply (simp add: trace_core'.simps (*‹trace_core' ?core ?S [] = (λe. weight_spmf' (?S ⤜ (λs. cpoke ?core s e)), λia. ?S ⤜ (λs. map_spmf fst (cfunc_adv ?core s ia)), λiu. ?S ⤜ (λs. map_spmf fst (cfunc_usr ?core s iu)))› ‹trace_core' ?core ?S (?obs # ?tr) = (case ?obs of Inl e ⇒ trace_core' ?core (mk_lossless (?S ⤜ (λs. cpoke ?core s e))) ?tr | Inr (Inl (ia, oa)) ⇒ trace_core' ?core (cond_spmf_fst (?S ⤜ (λs. cfunc_adv ?core s ia)) oa) ?tr | Inr (Inr (iu, ou)) ⇒ trace_core' ?core (cond_spmf_fst (?S ⤜ (λs. cfunc_usr ?core s iu)) ou) ?tr)›*) split: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))\n (*discuss goal 2*)\n apply (simp add: trace_core'.simps (*‹trace_core' (?core::(?'s_core, ?'event, ?'iadv_core, ?'iusr_core, ?'oadv_core, ?'ousr_core) core) (?S::?'s_core spmf) [] = (λe::?'event. weight_spmf' (?S ⤜ (λs::?'s_core. cpoke ?core s e)), λia::?'iadv_core. ?S ⤜ (λs::?'s_core. map_spmf fst (cfunc_adv ?core s ia)), λiu::?'iusr_core. ?S ⤜ (λs::?'s_core. map_spmf fst (cfunc_usr ?core s iu)))› ‹trace_core' (?core::(?'s_core, ?'event, ?'iadv_core, ?'iusr_core, ?'oadv_core, ?'ousr_core) core) (?S::?'s_core spmf) ((?obs::?'event + ?'iadv_core × ?'oadv_core + ?'iusr_core × ?'ousr_core) # (?tr::(?'event + ?'iadv_core × ?'oadv_core + ?'iusr_core × ?'ousr_core) list)) = (case ?obs of Inl (e::?'event) ⇒ trace_core' ?core (mk_lossless (?S ⤜ (λs::?'s_core. cpoke ?core s e))) ?tr | Inr (Inl (ia::?'iadv_core, oa::?'oadv_core)) ⇒ trace_core' ?core (cond_spmf_fst (?S ⤜ (λs::?'s_core. cfunc_adv ?core s ia)) oa) ?tr | Inr (Inr (iu::?'iusr_core, ou::?'ousr_core)) ⇒ trace_core' ?core (cond_spmf_fst (?S ⤜ (λs::?'s_core. cfunc_usr ?core s iu)) ou) ?tr)›*) split: sum.split (*‹(?P::?'c ⇒ bool) (case ?sum::?'a + ?'b of Inl (x::?'a) ⇒ (?f1.0::?'a ⇒ ?'c) x | Inr (x::?'b) ⇒ (?f2.0::?'b ⇒ ?'c) x) = ((∀x1::?'a. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2::?'b. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))\n (*proven 2 subgoals*) .\n\nlemma rel_core'_into_trace_core_eq: \"trace_core_eq core core' E IA IU p q\" \n if \"rel_core' S (eq_onp (λe. e ∈ E)) (eq_onp (λia. ia ∈ IA)) (eq_onp (λiu. iu ∈ IU)) (=) (=) core core'\"\n \"rel_spmf S p q\"\n using trace_core'_parametric[THEN rel_funD, THEN rel_funD, OF that] (*‹rel_fun (list_all2 (rel_sum (eq_onp (λe::'c::type. e ∈ (E::'c::type set))) (rel_sum (rel_prod (eq_onp (λia::'d::type. ia ∈ (IA::'d::type set))) (=)) (rel_prod (eq_onp (λiu::'e::type. iu ∈ (IU::'e::type set))) (=))))) (rel_prod (rel_fun (eq_onp (λe::'c::type. e ∈ E)) (=)) (rel_prod (rel_fun (eq_onp (λia::'d::type. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu::'e::type. iu ∈ IU)) (=)))) (trace_core' (core::('a::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type) core) (p::'a::type spmf)) (trace_core' (core'::('b::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type) core) (q::'b::type spmf))›*) unfolding trace_core_eq_def\n (*goal: ‹∀tr. set tr ⊆ E <+> IA × UNIV <+> IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λe. e ∈ E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ IU)) (=))) (trace_core' core p tr) (trace_core' core' q tr)›*)\n apply (intro strip (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q› ‹(⋀x. ?P x) ⟹ ∀x. ?P x› ‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*))\n (*goal: ‹∀tr. set tr ⊆ E <+> IA × UNIV <+> IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λe. e ∈ E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ IU)) (=))) (trace_core' core p tr) (trace_core' core' q tr)›*)\n subgoal for tr\n apply (simp add: eq_onp_True[symmetric] (*‹(=) = eq_onp (λ_. True)›*) prod.rel_eq_onp (*‹rel_prod (eq_onp ?P1.0) (eq_onp ?P2.0) = eq_onp (pred_prod ?P1.0 ?P2.0)›*) sum.rel_eq_onp (*‹rel_sum (eq_onp ?P1.0) (eq_onp ?P2.0) = eq_onp (pred_sum ?P1.0 ?P2.0)›*) list.rel_eq_onp (*‹list_all2 (eq_onp ?P) = eq_onp (list_all ?P)›*))\n (*goal: ‹⟦rel_fun (list_all2 (rel_sum (eq_onp (λe. e ∈ E)) (rel_sum (rel_prod (eq_onp (λia. ia ∈ IA)) (=)) (rel_prod (eq_onp (λiu. iu ∈ IU)) (=))))) (rel_prod (rel_fun (eq_onp (λe. e ∈ E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ IU)) (=)))) (trace_core' core p) (trace_core' core' q); set tr ⊆ E <+> IA × UNIV <+> IU × UNIV⟧ ⟹ rel_prod (rel_fun (eq_onp (λe. e ∈ E)) (=)) (rel_prod (rel_fun (eq_onp (λia. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ IU)) (=))) (trace_core' core p tr) (trace_core' core' q tr)›*)\n by (auto 4 3 simp add: eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*) list_all_iff (*‹list_all ?P ?x = Ball (set ?x) ?P›*) dest: rel_funD[where x=tr and y=tr] (*‹⟦rel_fun ?A ?B ?f ?g; ?A tr tr⟧ ⟹ ?B (?f tr) (?g tr)›*)) .\n\nlemma trace_core_eq_simI:\n fixes core1 :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and core2 :: \"('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and S :: \"'s_core spmf ⇒ 's_core' spmf ⇒ bool\"\n assumes start: \"S p q\"\n and step_cpoke: \"⋀p q e. ⟦ S p q; e ∈ E ⟧ ⟹ \n weight_spmf (bind_spmf p (λs. cpoke core1 s e)) = weight_spmf (bind_spmf q (λs. cpoke core2 s e))\"\n and sim_cpoke: \"⋀p q e. ⟦ S p q; e ∈ E ⟧ ⟹ \n S (mk_lossless (bind_spmf p (λs. cpoke core1 s e))) (mk_lossless (bind_spmf q (λs. cpoke core2 s e)))\"\n and step_cfunc_adv: \"⋀p q ia. ⟦ S p q; ia ∈ IA ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = bind_spmf q (λs2. map_spmf fst (cfunc_adv core2 s2 ia))\"\n and sim_cfunc_adv: \"⋀p q ia s1 s2 s1' s2' oa. ⟦ S p q; ia ∈ IA; \n s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv core1 s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia) ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (bind_spmf q (λs2. cfunc_adv core2 s2 ia)) oa)\"\n and step_cfunc_usr: \"⋀p q iu. ⟦ S p q; iu ∈ IU ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = bind_spmf q (λs2. map_spmf fst (cfunc_usr core2 s2 iu))\"\n and sim_cfunc_usr: \"⋀p q iu s1 s2 s1' s2' ou. ⟦ S p q; iu ∈ IU; \n s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr core1 s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu) ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (bind_spmf q (λs2. cfunc_usr core2 s2 iu)) ou)\"\n shows \"trace_core_eq core1 core2 E IA IU p q\"\nproof(rule trace_core_eqI)\n fix tr :: \" ('event + 'iadv_core × 'oadv_core + 'iusr_core × 'ousr_core) list\"\n assume \"set tr ⊆ E <+> IA × UNIV <+> IU × UNIV\"\n then have \"(∀e∈E. fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e) ∧\n (∀ia∈IA. fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia) ∧\n (∀iu∈IU. snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu)\"\n using start\n proof(induction tr arbitrary: p q)\n case Nil\n then show ?case by(simp add: step_cpoke step_cfunc_adv step_cfunc_usr)\n next\n case (Cons a tr)\n from Cons.prems(1) have tr: \"set tr ⊆ E <+> IA × UNIV <+> IU × UNIV\" by simp\n from Cons.prems(1)\n consider (cpoke) e where \"a = Inl e\" \"e ∈ E\"\n | (cfunc_adv) ia oa where \"a = Inr (Inl (ia, oa))\" \"ia ∈ IA\"\n | (cfunc_usr) iu ou where \"a = Inr (Inr (iu, ou))\" \"iu ∈ IU\" by auto\n then show ?case\n proof cases\n case cpoke\n then show ?thesis using tr Cons.prems(2) by(auto simp add: sim_cpoke intro!: Cons.IH)\n next\n case cfunc_adv\n let ?p = \"bind_spmf p (λs1. cfunc_adv core1 s1 ia)\"\n let ?q = \"bind_spmf q (λs2. cfunc_adv core2 s2 ia)\"\n show ?thesis\n proof(cases \"oa ∈ fst ` set_spmf ?p\")\n case True\n with step_cfunc_adv[OF Cons.prems(2) cfunc_adv(2), THEN arg_cong[where f=set_spmf]]\n have \"oa ∈ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def)\n then show ?thesis using True Cons.prems cfunc_adv\n by(clarsimp)(rule Cons.IH; blast intro: sim_cfunc_adv)\n next\n case False\n hence \"cond_spmf_fst ?p oa = return_pmf None\" by simp\n moreover\n from step_cfunc_adv[OF Cons.prems(2) cfunc_adv(2), THEN arg_cong[where f=set_spmf]] False\n have oa': \"oa ∉ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def) simp\n hence \"cond_spmf_fst ?q oa = return_pmf None\" by simp\n ultimately show ?thesis using cfunc_adv by(simp del: cond_spmf_fst_eq_return_None)\n qed\n next\n case cfunc_usr\n let ?p = \"bind_spmf p (λs1. cfunc_usr core1 s1 iu)\"\n let ?q = \"bind_spmf q (λs2. cfunc_usr core2 s2 iu)\"\n show ?thesis\n proof(cases \"ou ∈ fst ` set_spmf ?p\")\n case True\n with step_cfunc_usr[OF Cons.prems(2) cfunc_usr(2), THEN arg_cong[where f=set_spmf]]\n have \"ou ∈ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def)\n then show ?thesis using True Cons.prems cfunc_usr\n by(clarsimp)(rule Cons.IH; blast intro: sim_cfunc_usr)\n next\n case False\n hence \"cond_spmf_fst ?p ou = return_pmf None\" by simp\n moreover\n from step_cfunc_usr[OF Cons.prems(2) cfunc_usr(2), THEN arg_cong[where f=set_spmf]] False\n have oa': \"ou ∉ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def) simp\n hence \"cond_spmf_fst ?q ou = return_pmf None\" by simp\n ultimately show ?thesis using cfunc_usr by(simp del: cond_spmf_fst_eq_return_None)\n qed\n qed\n qed\n then show \"e ∈ E ⟹ fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e\"\n and \"ia ∈ IA ⟹ fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia\"\n and \"iu ∈ IU ⟹ snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu\"\n for e ia iu by blast+\nqed\n\ncontext \n fixes core :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\nbegin\n\nfun trace_core_aux \n :: \"'s_core spmf ⇒ ('event + 'iadv_core × 'oadv_core + 'iusr_core × 'ousr_core) list ⇒ 's_core spmf\" where\n \"trace_core_aux p [] = p\"\n| \"trace_core_aux p (Inl e # tr) = trace_core_aux (mk_lossless (bind_spmf p (λs. cpoke core s e))) tr\"\n| \"trace_core_aux p (Inr (Inl (ia, oa)) # tr) = trace_core_aux (cond_spmf_fst (bind_spmf p (λs. cfunc_adv core s ia)) oa) tr\"\n| \"trace_core_aux p (Inr (Inr (iu, ou)) # tr) = trace_core_aux (cond_spmf_fst (bind_spmf p (λs. cfunc_usr core s iu)) ou) tr\"\n\nend\n\nlemma trace_core_conv_trace_core_aux:\n \"trace_core' core p tr = \n (λe. weight_spmf (bind_spmf (trace_core_aux core p tr) (λs. cpoke core s e)),\n λia. bind_spmf (trace_core_aux core p tr) (λs. map_spmf fst (cfunc_adv core s ia)),\n λiu. bind_spmf (trace_core_aux core p tr) (λs. map_spmf fst (cfunc_usr core s iu)))\"\n apply (induction p tr rule: trace_core_aux.induct (*‹⟦⋀p. ?P p []; ⋀p e tr. ?P (mk_lossless (p ⤜ (λs. cpoke ?core s e))) tr ⟹ ?P p (Inl e # tr); ⋀p ia oa tr. ?P (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) tr ⟹ ?P p (Inr (Inl (ia, oa)) # tr); ⋀p iu ou tr. ?P (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) tr ⟹ ?P p (Inr (Inr (iu, ou)) # tr)⟧ ⟹ ?P ?a0.0 ?a1.0›*))\n (*goals:\n 1. ‹⋀p. trace_core' core p [] = (λe. weight_spmf (trace_core_aux core p [] ⤜ (λs. cpoke core s e)), λia. trace_core_aux core p [] ⤜ (λs. map_spmf fst (cfunc_adv core s ia)), λiu. trace_core_aux core p [] ⤜ (λs. map_spmf fst (cfunc_usr core s iu)))›\n 2. ‹⋀p e tr. trace_core' core (mk_lossless (p ⤜ (λs. cpoke ?core s e))) tr = (λea. weight_spmf (trace_core_aux core (mk_lossless (p ⤜ (λs. cpoke ?core s e))) tr ⤜ (λs. cpoke core s ea)), λia. trace_core_aux core (mk_lossless (p ⤜ (λs. cpoke ?core s e))) tr ⤜ (λs. map_spmf fst (cfunc_adv core s ia)), λiu. trace_core_aux core (mk_lossless (p ⤜ (λs. cpoke ?core s e))) tr ⤜ (λs. map_spmf fst (cfunc_usr core s iu))) ⟹ trace_core' core p (Inl e # tr) = (λea. weight_spmf (trace_core_aux core p (Inl e # tr) ⤜ (λs. cpoke core s ea)), λia. trace_core_aux core p (Inl e # tr) ⤜ (λs. map_spmf fst (cfunc_adv core s ia)), λiu. trace_core_aux core p (Inl e # tr) ⤜ (λs. map_spmf fst (cfunc_usr core s iu)))›\n 3. ‹⋀p ia oa tr. trace_core' core (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) tr = (λe. weight_spmf (trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) tr ⤜ (λs. cpoke core s e)), λiaa. trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) tr ⤜ (λs. map_spmf fst (cfunc_adv core s iaa)), λiu. trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) tr ⤜ (λs. map_spmf fst (cfunc_usr core s iu))) ⟹ trace_core' core p (Inr (Inl (ia, oa)) # tr) = (λe. weight_spmf (trace_core_aux core p (Inr (Inl (ia, oa)) # tr) ⤜ (λs. cpoke core s e)), λiaa. trace_core_aux core p (Inr (Inl (ia, oa)) # tr) ⤜ (λs. map_spmf fst (cfunc_adv core s iaa)), λiu. trace_core_aux core p (Inr (Inl (ia, oa)) # tr) ⤜ (λs. map_spmf fst (cfunc_usr core s iu)))›\n 4. ‹⋀p iu ou tr. trace_core' core (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) tr = (λe. weight_spmf (trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) tr ⤜ (λs. cpoke core s e)), λia. trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) tr ⤜ (λs. map_spmf fst (cfunc_adv core s ia)), λiua. trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) tr ⤜ (λs. map_spmf fst (cfunc_usr core s iua))) ⟹ trace_core' core p (Inr (Inr (iu, ou)) # tr) = (λe. weight_spmf (trace_core_aux core p (Inr (Inr (iu, ou)) # tr) ⤜ (λs. cpoke core s e)), λia. trace_core_aux core p (Inr (Inr (iu, ou)) # tr) ⤜ (λs. map_spmf fst (cfunc_adv core s ia)), λiua. trace_core_aux core p (Inr (Inr (iu, ou)) # tr) ⤜ (λs. map_spmf fst (cfunc_usr core s iua)))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*proven 4 subgoals*) .\n\nlemma trace_core_aux_append:\n \"trace_core_aux core p (tr @ tr') = trace_core_aux core (trace_core_aux core p tr) tr'\"\n apply (induction p tr rule: trace_core_aux.induct (*‹⟦⋀p. ?P p []; ⋀p e tr. ?P (mk_lossless (p ⤜ (λs. cpoke ?core s e))) tr ⟹ ?P p (Inl e # tr); ⋀p ia oa tr. ?P (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) tr ⟹ ?P p (Inr (Inl (ia, oa)) # tr); ⋀p iu ou tr. ?P (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) tr ⟹ ?P p (Inr (Inr (iu, ou)) # tr)⟧ ⟹ ?P ?a0.0 ?a1.0›*))\n (*goals:\n 1. ‹⋀p. trace_core_aux core p ([] @ tr') = trace_core_aux core (trace_core_aux core p []) tr'›\n 2. ‹⋀p e tr. trace_core_aux core (mk_lossless (p ⤜ (λs. cpoke ?core s e))) (tr @ tr') = trace_core_aux core (trace_core_aux core (mk_lossless (p ⤜ (λs. cpoke ?core s e))) tr) tr' ⟹ trace_core_aux core p ((Inl e # tr) @ tr') = trace_core_aux core (trace_core_aux core p (Inl e # tr)) tr'›\n 3. ‹⋀p ia oa tr. trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) (tr @ tr') = trace_core_aux core (trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_adv ?core s ia)) oa) tr) tr' ⟹ trace_core_aux core p ((Inr (Inl (ia, oa)) # tr) @ tr') = trace_core_aux core (trace_core_aux core p (Inr (Inl (ia, oa)) # tr)) tr'›\n 4. ‹⋀p iu ou tr. trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) (tr @ tr') = trace_core_aux core (trace_core_aux core (cond_spmf_fst (p ⤜ (λs. cfunc_usr ?core s iu)) ou) tr) tr' ⟹ trace_core_aux core p ((Inr (Inr (iu, ou)) # tr) @ tr') = trace_core_aux core (trace_core_aux core p (Inr (Inr (iu, ou)) # tr)) tr'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n\ninductive trace_core_closure \n :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\n ⇒ ('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\n ⇒ 'event set ⇒ 'iadv_core set ⇒ 'iusr_core set\n ⇒ 's_core spmf ⇒ 's_core' spmf ⇒ 's_core spmf ⇒ 's_core' spmf ⇒ bool\"\n for core1 core2 E IA IU p q where\n \"trace_core_closure core1 core2 E IA IU p q (trace_core_aux core1 p tr) (trace_core_aux core2 q tr)\"\n if \"set tr ⊆ E <+> IA × UNIV <+> IU × UNIV\"\n\nlemma trace_core_closure_start: \"trace_core_closure core1 core2 E IA IU p q p q\"\n by (simp add: trace_core_closure.simps (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?a1.0 ?a2.0 = (∃tr. ?a1.0 = trace_core_aux ?core1.0 ?p tr ∧ ?a2.0 = trace_core_aux ?core2.0 ?q tr ∧ set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV)›*) exI[where x=\"[]\"] (*‹?P [] ⟹ ∃x. ?P x›*))\n\nlemma trace_core_closure_step:\n assumes \"trace_core_eq core1 core2 E IA IU p q\"\n and \"trace_core_closure core1 core2 E IA IU p q p' q'\"\n shows trace_core_closure_step_cpoke:\n \"e ∈ E ⟹ weight_spmf (bind_spmf p' (λs. cpoke core1 s e)) = weight_spmf (bind_spmf q' (λs. cpoke core2 s e))\"\n (is \"PROP ?thesis1\")\n and trace_core_closure_step_cfunc_adv: \n \"ia ∈ IA ⟹ bind_spmf p' (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = bind_spmf q' (λs2. map_spmf fst (cfunc_adv core2 s2 ia))\"\n (is \"PROP ?thesis2\")\n and trace_core_closure_step_cfunc_usr:\n \"iu ∈ IU ⟹ bind_spmf p' (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = bind_spmf q' (λs2. map_spmf fst (cfunc_usr core2 s2 iu))\"\n (is \"PROP ?thesis3\")\nproof (-)\n (*goals:\n 1. ‹e ∈ E ⟹ weight_spmf (p' ⤜ (λs. cpoke core1 s e)) = weight_spmf (q' ⤜ (λs. cpoke core2 s e))›\n 2. ‹ia ∈ IA ⟹ p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›\n 3. ‹iu ∈ IU ⟹ p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›*)\n from assms(2) (*‹trace_core_closure core1 core2 E IA IU p q p' q'›*) obtain tr where p: \"p' = trace_core_aux core1 p tr\" and q: \"q' = trace_core_aux core2 q tr\" and tr: \"set tr ⊆ E <+> IA × UNIV <+> IU × UNIV\"\n (*goal: ‹(⋀tr. ⟦p' = trace_core_aux core1 p tr; q' = trace_core_aux core2 q tr; set tr ⊆ E <+> IA × UNIV <+> IU × UNIV⟧ ⟹ thesis) ⟹ thesis›*)\n by cases\n from trace_core_eqD[OF assms ( 1 ) tr] (*‹?e ∈ E ⟹ fst (trace_core' core1 p tr) ?e = fst (trace_core' core2 q tr) ?e› ‹?ia ∈ IA ⟹ fst (snd (trace_core' core1 p tr)) ?ia = fst (snd (trace_core' core2 q tr)) ?ia› ‹?iu ∈ IU ⟹ snd (snd (trace_core' core1 p tr)) ?iu = snd (snd (trace_core' core2 q tr)) ?iu›*) p (*‹p' = trace_core_aux core1 p tr›*) q (*‹(q'::'g spmf) = trace_core_aux (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (q::'g spmf) (tr::('b + 'c × 'e + 'd × 'f) list)›*) show \"PROP ?thesis1\" and \"PROP ?thesis2\" \"PROP ?thesis3\"\n (*goals:\n 1. ‹e ∈ E ⟹ weight_spmf (p' ⤜ (λs. cpoke core1 s e)) = weight_spmf (q' ⤜ (λs. cpoke core2 s e))›\n 2. ‹ia ∈ IA ⟹ p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›\n 3. ‹iu ∈ IU ⟹ p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›*)\n apply -\n (*goals:\n 1. ‹⟦e ∈ E; ⋀e. e ∈ E ⟹ fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e; ⋀ia. ia ∈ IA ⟹ fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia; ⋀iu. iu ∈ IU ⟹ snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu; p' = trace_core_aux core1 p tr; q' = trace_core_aux core2 q tr⟧ ⟹ weight_spmf (p' ⤜ (λs. cpoke core1 s e)) = weight_spmf (q' ⤜ (λs. cpoke core2 s e))›\n 2. ‹⟦ia ∈ IA; ⋀e. e ∈ E ⟹ fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e; ⋀ia. ia ∈ IA ⟹ fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia; ⋀iu. iu ∈ IU ⟹ snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu; p' = trace_core_aux core1 p tr; q' = trace_core_aux core2 q tr⟧ ⟹ p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›\n 3. ‹⟦iu ∈ IU; ⋀e. e ∈ E ⟹ fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e; ⋀ia. ia ∈ IA ⟹ fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia; ⋀iu. iu ∈ IU ⟹ snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu; p' = trace_core_aux core1 p tr; q' = trace_core_aux core2 q tr⟧ ⟹ p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›\n discuss goal 1*)\n apply (simp add: trace_core_conv_trace_core_aux (*‹trace_core' ?core ?p ?tr = (λe. weight_spmf (trace_core_aux ?core ?p ?tr ⤜ (λs. cpoke ?core s e)), λia. trace_core_aux ?core ?p ?tr ⤜ (λs. map_spmf fst (cfunc_adv ?core s ia)), λiu. trace_core_aux ?core ?p ?tr ⤜ (λs. map_spmf fst (cfunc_usr ?core s iu)))›*))\n (*discuss goal 2*)\n apply (simp add: trace_core_conv_trace_core_aux (*‹trace_core' ?core ?p ?tr = (λe. weight_spmf (trace_core_aux ?core ?p ?tr ⤜ (λs. cpoke ?core s e)), λia. trace_core_aux ?core ?p ?tr ⤜ (λs. map_spmf fst (cfunc_adv ?core s ia)), λiu. trace_core_aux ?core ?p ?tr ⤜ (λs. map_spmf fst (cfunc_usr ?core s iu)))›*))\n (*discuss goal 3*)\n apply (simp add: trace_core_conv_trace_core_aux (*‹trace_core' (?core::(?'f::type, ?'a::type, ?'b::type, ?'d::type, ?'c::type, ?'e::type) core) (?p::?'f::type spmf) (?tr::(?'a::type + ?'b::type × ?'c::type + ?'d::type × ?'e::type) list) = (λe::?'a::type. weight_spmf (trace_core_aux ?core ?p ?tr ⤜ (λs::?'f::type. cpoke ?core s e)), λia::?'b::type. trace_core_aux ?core ?p ?tr ⤜ (λs::?'f::type. map_spmf fst (cfunc_adv ?core s ia)), λiu::?'d::type. trace_core_aux ?core ?p ?tr ⤜ (λs::?'f::type. map_spmf fst (cfunc_usr ?core s iu)))›*))\n (*proven 3 subgoals*) .\nqed\n\nlemma trace_core_closure_sim:\n fixes core1 core2 E IA IU p q\n defines \"S ≡ trace_core_closure core1 core2 E IA IU p q\"\n assumes \"S p' q'\"\n shows trace_core_closure_sim_cpoke:\n \"e ∈ E ⟹ S (mk_lossless (bind_spmf p' (λs. cpoke core1 s e))) (mk_lossless (bind_spmf q' (λs. cpoke core2 s e)))\"\n (is \"PROP ?thesis1\")\n and trace_core_closure_sim_cfunc_adv: \"ia ∈ IA \n ⟹ S (cond_spmf_fst (bind_spmf p' (λs1. cfunc_adv core1 s1 ia)) oa) \n (cond_spmf_fst (bind_spmf q' (λs2. cfunc_adv core2 s2 ia)) oa)\"\n (is \"PROP ?thesis2\")\n and trace_core_closure_sim_cfunc_usr: \"iu ∈ IU \n ⟹ S (cond_spmf_fst (bind_spmf p' (λs1. cfunc_usr core1 s1 iu)) ou)\n (cond_spmf_fst (bind_spmf q' (λs2. cfunc_usr core2 s2 iu)) ou)\"\n (is \"PROP ?thesis3\")\nproof (-)\n (*goals:\n 1. ‹e ∈ E ⟹ S (mk_lossless (p' ⤜ (λs. cpoke core1 s e))) (mk_lossless (q' ⤜ (λs. cpoke core2 s e)))›\n 2. ‹ia ∈ IA ⟹ S (cond_spmf_fst (p' ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q' ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›\n 3. ‹iu ∈ IU ⟹ S (cond_spmf_fst (p' ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q' ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n from assms(2) (*‹(S::'a spmf ⇒ 'b spmf ⇒ bool) (p'::'a spmf) (q'::'b spmf)›*) obtain tr where p: \"p' = trace_core_aux core1 p tr\" and q: \"q' = trace_core_aux core2 q tr\" and tr: \"set tr ⊆ E <+> IA × UNIV <+> IU × UNIV\"\n (*goal: ‹(⋀tr. ⟦p' = trace_core_aux core1 p tr; q' = trace_core_aux core2 q tr; set tr ⊆ E <+> IA × UNIV <+> IU × UNIV⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding S_def\n (*goal: ‹(⋀tr. ⟦p' = trace_core_aux core1 p tr; q' = trace_core_aux core2 q tr; set tr ⊆ E <+> IA × UNIV <+> IU × UNIV⟧ ⟹ thesis) ⟹ thesis›*)\n by cases\n show \"PROP ?thesis1\"\n (*goal: ‹e ∈ E ⟹ S (mk_lossless (p' ⤜ (λs. cpoke core1 s e))) (mk_lossless (q' ⤜ (λs. cpoke core2 s e)))›*)\n using p (*‹p' = trace_core_aux core1 p tr›*) q (*‹q' = trace_core_aux core2 q tr›*) tr (*‹set tr ⊆ E <+> IA × UNIV <+> IU × UNIV›*) by (auto simp add: S_def (*‹S::'a::type spmf ⇒ 'b::type spmf ⇒ bool ≡ trace_core_closure (core1::('a::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type) core) (core2::('b::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type) core) (E::'c::type set) (IA::'d::type set) (IU::'e::type set) (p::'a::type spmf) (q::'b::type spmf)›*) trace_core_closure.simps (*‹trace_core_closure (?core1.0::(?'s_core::type, ?'event::type, ?'iadv_core::type, ?'iusr_core::type, ?'oadv_core::type, ?'ousr_core::type) core) (?core2.0::(?'s_core'::type, ?'event::type, ?'iadv_core::type, ?'iusr_core::type, ?'oadv_core::type, ?'ousr_core::type) core) (?E::?'event::type set) (?IA::?'iadv_core::type set) (?IU::?'iusr_core::type set) (?p::?'s_core::type spmf) (?q::?'s_core'::type spmf) (?a1.0::?'s_core::type spmf) (?a2.0::?'s_core'::type spmf) = (∃tr::(?'event::type + ?'iadv_core::type × ?'oadv_core::type + ?'iusr_core::type × ?'ousr_core::type) list. ?a1.0 = trace_core_aux ?core1.0 ?p tr ∧ ?a2.0 = trace_core_aux ?core2.0 ?q tr ∧ set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV)›*) trace_core_aux_append (*‹trace_core_aux (?core::(?'a::type, ?'b::type, ?'c::type, ?'d::type, ?'e::type, ?'f::type) core) (?p::?'a::type spmf) ((?tr::(?'b::type + ?'c::type × ?'e::type + ?'d::type × ?'f::type) list) @ (?tr'::(?'b::type + ?'c::type × ?'e::type + ?'d::type × ?'f::type) list)) = trace_core_aux ?core (trace_core_aux ?core ?p ?tr) ?tr'›*) intro!: exI[where x=\"tr @ [Inl _]\"] (*‹(?P::('c::type + 'd::type × 'f::type + 'e::type × 'g::type) list ⇒ bool) ((tr::('c::type + 'd::type × 'f::type + 'e::type × 'g::type) list) @ [Inl (?uu4::'c::type)]) ⟹ ∃x::('c::type + 'd::type × 'f::type + 'e::type × 'g::type) list. ?P x›*))\n show \"PROP ?thesis2\"\n (*goal: ‹ia ∈ IA ⟹ S (cond_spmf_fst (p' ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q' ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›*)\n using p (*‹p' = trace_core_aux core1 p tr›*) q (*‹q' = trace_core_aux core2 q tr›*) tr (*‹set tr ⊆ E <+> IA × UNIV <+> IU × UNIV›*) by (auto simp add: S_def (*‹S ≡ trace_core_closure core1 core2 E IA IU p q›*) trace_core_closure.simps (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?a1.0 ?a2.0 = (∃tr. ?a1.0 = trace_core_aux ?core1.0 ?p tr ∧ ?a2.0 = trace_core_aux ?core2.0 ?q tr ∧ set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV)›*) trace_core_aux_append (*‹trace_core_aux ?core ?p (?tr @ ?tr') = trace_core_aux ?core (trace_core_aux ?core ?p ?tr) ?tr'›*) intro!: exI[where x=\"tr @ [Inr (Inl (_, _))]\"] (*‹?P (tr @ [Inr (Inl (?uu4, ?uua4))]) ⟹ ∃x. ?P x›*))\n show \"PROP ?thesis3\"\n (*goal: ‹iu ∈ IU ⟹ S (cond_spmf_fst (p' ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q' ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n using p (*‹(p'::'a spmf) = trace_core_aux (core1::('a, 'c, 'd, 'e, 'f, 'g) core) (p::'a spmf) (tr::('c + 'd × 'f + 'e × 'g) list)›*) q (*‹q' = trace_core_aux core2 q tr›*) tr (*‹set (tr::('c::type + 'd::type × 'f::type + 'e::type × 'g::type) list) ⊆ (E::'c::type set) <+> (IA::'d::type set) × UNIV <+> (IU::'e::type set) × UNIV›*) by (auto simp add: S_def (*‹S ≡ trace_core_closure core1 core2 E IA IU p q›*) trace_core_closure.simps (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?a1.0 ?a2.0 = (∃tr. ?a1.0 = trace_core_aux ?core1.0 ?p tr ∧ ?a2.0 = trace_core_aux ?core2.0 ?q tr ∧ set tr ⊆ ?E <+> ?IA × UNIV <+> ?IU × UNIV)›*) trace_core_aux_append (*‹trace_core_aux ?core ?p (?tr @ ?tr') = trace_core_aux ?core (trace_core_aux ?core ?p ?tr) ?tr'›*) intro!: exI[where x=\"tr @ [Inr (Inr (_, _))]\"] (*‹?P (tr @ [Inr (Inr (?uu4, ?uua4))]) ⟹ ∃x. ?P x›*))\nqed\n\nproposition trace_core_eq_complete:\n assumes \"trace_core_eq core1 core2 E IA IU p q\"\n obtains S\n where \"S p q\"\n and \"⋀p q e. ⟦ S p q; e ∈ E ⟧ ⟹ \n weight_spmf (bind_spmf p (λs. cpoke core1 s e)) = weight_spmf (bind_spmf q (λs. cpoke core2 s e))\"\n and \"⋀p q e. ⟦ S p q; e ∈ E ⟧ ⟹ \n S (mk_lossless (bind_spmf p (λs. cpoke core1 s e))) (mk_lossless (bind_spmf q (λs. cpoke core2 s e)))\"\n and \"⋀p q ia. ⟦ S p q; ia ∈ IA ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = bind_spmf q (λs2. map_spmf fst (cfunc_adv core2 s2 ia))\"\n and \"⋀p q ia oa. ⟦ S p q; ia ∈ IA ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (bind_spmf q (λs2. cfunc_adv core2 s2 ia)) oa)\"\n and \"⋀p q iu. ⟦ S p q; iu ∈ IU ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = bind_spmf q (λs2. map_spmf fst (cfunc_usr core2 s2 iu))\"\n and \"⋀p q iu ou. ⟦ S p q; iu ∈ IU ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (bind_spmf q (λs2. cfunc_usr core2 s2 iu)) ou)\"\nproof (-)\n (*goal: ‹(⋀S. ⟦S p q; ⋀p q e. ⟦S p q; e ∈ E⟧ ⟹ weight_spmf (p ⤜ (λs. cpoke core1 s e)) = weight_spmf (q ⤜ (λs. cpoke core2 s e)); ⋀p q e. ⟦S p q; e ∈ E⟧ ⟹ S (mk_lossless (p ⤜ (λs. cpoke core1 s e))) (mk_lossless (q ⤜ (λs. cpoke core2 s e))); ⋀p q ia. ⟦S p q; ia ∈ IA⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia)); ⋀p q ia oa. ⟦S p q; ia ∈ IA⟧ ⟹ S (cond_spmf_fst (p ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. cfunc_adv core2 s2 ia)) oa); ⋀p q iu. ⟦S p q; iu ∈ IU⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu)); ⋀p q iu ou. ⟦S p q; iu ∈ IU⟧ ⟹ S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)⟧ ⟹ thesis) ⟹ thesis›*)\n show thesis\n apply (rule that[where S=\"trace_core_closure core1 core2 E IA IU p q\"] (*‹⟦trace_core_closure core1 core2 E IA IU p q p q; ⋀pa qa e. ⟦trace_core_closure core1 core2 E IA IU p q pa qa; e ∈ E⟧ ⟹ weight_spmf (pa ⤜ (λs. cpoke core1 s e)) = weight_spmf (qa ⤜ (λs. cpoke core2 s e)); ⋀pa qa e. ⟦trace_core_closure core1 core2 E IA IU p q pa qa; e ∈ E⟧ ⟹ trace_core_closure core1 core2 E IA IU p q (mk_lossless (pa ⤜ (λs. cpoke core1 s e))) (mk_lossless (qa ⤜ (λs. cpoke core2 s e))); ⋀pa qa ia. ⟦trace_core_closure core1 core2 E IA IU p q pa qa; ia ∈ IA⟧ ⟹ pa ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = qa ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia)); ⋀pa qa ia oa. ⟦trace_core_closure core1 core2 E IA IU p q pa qa; ia ∈ IA⟧ ⟹ trace_core_closure core1 core2 E IA IU p q (cond_spmf_fst (pa ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (qa ⤜ (λs2. cfunc_adv core2 s2 ia)) oa); ⋀pa qa iu. ⟦trace_core_closure core1 core2 E IA IU p q pa qa; iu ∈ IU⟧ ⟹ pa ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = qa ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu)); ⋀pa qa iu ou. ⟦trace_core_closure core1 core2 E IA IU p q pa qa; iu ∈ IU⟧ ⟹ trace_core_closure core1 core2 E IA IU p q (cond_spmf_fst (pa ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (qa ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)⟧ ⟹ thesis›*))\n (*goals:\n 1. ‹trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) p q›\n 2. ‹⋀(pa::'a spmf) (qa::'g spmf) e::'b. ⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) pa qa; e ∈ E⟧ ⟹ weight_spmf (pa ⤜ (λs::'a. cpoke core1 s e)) = weight_spmf (qa ⤜ (λs::'g. cpoke core2 s e))›\n 3. ‹⋀(pa::'a spmf) (qa::'g spmf) e::'b. ⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) pa qa; e ∈ E⟧ ⟹ trace_core_closure core1 core2 E IA IU p q (mk_lossless (pa ⤜ (λs::'a. cpoke core1 s e))) (mk_lossless (qa ⤜ (λs::'g. cpoke core2 s e)))›\n 4. ‹⋀(pa::'a spmf) (qa::'g spmf) ia::'c. ⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) pa qa; ia ∈ IA⟧ ⟹ pa ⤜ (λs1::'a. map_spmf fst (cfunc_adv core1 s1 ia)) = qa ⤜ (λs2::'g. map_spmf fst (cfunc_adv core2 s2 ia))›\n 5. ‹⋀(pa::'a spmf) (qa::'g spmf) (ia::'c) oa::'e. ⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) pa qa; ia ∈ IA⟧ ⟹ trace_core_closure core1 core2 E IA IU p q (cond_spmf_fst (pa ⤜ (λs1::'a. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (qa ⤜ (λs2::'g. cfunc_adv core2 s2 ia)) oa)›\n 6. ‹⋀(pa::'a spmf) (qa::'g spmf) iu::'d. ⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) pa qa; iu ∈ IU⟧ ⟹ pa ⤜ (λs1::'a. map_spmf fst (cfunc_usr core1 s1 iu)) = qa ⤜ (λs2::'g. map_spmf fst (cfunc_usr core2 s2 iu))›\n 7. ‹⋀(pa::'a spmf) (qa::'g spmf) (iu::'d) ou::'f. ⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) pa qa; iu ∈ IU⟧ ⟹ trace_core_closure core1 core2 E IA IU p q (cond_spmf_fst (pa ⤜ (λs1::'a. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (qa ⤜ (λs2::'g. cfunc_usr core2 s2 iu)) ou)›\n discuss goal 1*)\n apply ((auto intro: trace_core_closure_start (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p ?q›*) trace_core_closure_step[OF assms] (*‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?e ∈ E⟧ ⟹ weight_spmf (?p' ⤜ (λs. cpoke core1 s ?e)) = weight_spmf (?q' ⤜ (λs. cpoke core2 s ?e))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) trace_core_closure_sim (*‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?e ∈ ?E⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (mk_lossless (?p' ⤜ (λs. cpoke ?core1.0 s ?e))) (mk_lossless (?q' ⤜ (λs. cpoke ?core2.0 s ?e)))› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_adv ?core1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_adv ?core2.0 s2 ?ia)) ?oa)› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_usr ?core1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_usr ?core2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 2*)\n apply ((auto intro: trace_core_closure_start (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p ?q›*) trace_core_closure_step[OF assms] (*‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?e ∈ E⟧ ⟹ weight_spmf (?p' ⤜ (λs. cpoke core1 s ?e)) = weight_spmf (?q' ⤜ (λs. cpoke core2 s ?e))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) trace_core_closure_sim (*‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?e ∈ ?E⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (mk_lossless (?p' ⤜ (λs. cpoke ?core1.0 s ?e))) (mk_lossless (?q' ⤜ (λs. cpoke ?core2.0 s ?e)))› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_adv ?core1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_adv ?core2.0 s2 ?ia)) ?oa)› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_usr ?core1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_usr ?core2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 3*)\n apply ((auto intro: trace_core_closure_start (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p ?q›*) trace_core_closure_step[OF assms] (*‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?e ∈ E⟧ ⟹ weight_spmf (?p' ⤜ (λs. cpoke core1 s ?e)) = weight_spmf (?q' ⤜ (λs. cpoke core2 s ?e))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) trace_core_closure_sim (*‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?e ∈ ?E⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (mk_lossless (?p' ⤜ (λs. cpoke ?core1.0 s ?e))) (mk_lossless (?q' ⤜ (λs. cpoke ?core2.0 s ?e)))› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_adv ?core1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_adv ?core2.0 s2 ?ia)) ?oa)› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_usr ?core1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_usr ?core2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 4*)\n apply ((auto intro: trace_core_closure_start (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p ?q›*) trace_core_closure_step[OF assms] (*‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?e ∈ E⟧ ⟹ weight_spmf (?p' ⤜ (λs. cpoke core1 s ?e)) = weight_spmf (?q' ⤜ (λs. cpoke core2 s ?e))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) trace_core_closure_sim (*‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?e ∈ ?E⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (mk_lossless (?p' ⤜ (λs. cpoke ?core1.0 s ?e))) (mk_lossless (?q' ⤜ (λs. cpoke ?core2.0 s ?e)))› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_adv ?core1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_adv ?core2.0 s2 ?ia)) ?oa)› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_usr ?core1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_usr ?core2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 5*)\n apply ((auto intro: trace_core_closure_start (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p ?q›*) trace_core_closure_step[OF assms] (*‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?e ∈ E⟧ ⟹ weight_spmf (?p' ⤜ (λs. cpoke core1 s ?e)) = weight_spmf (?q' ⤜ (λs. cpoke core2 s ?e))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) trace_core_closure_sim (*‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?e ∈ ?E⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (mk_lossless (?p' ⤜ (λs. cpoke ?core1.0 s ?e))) (mk_lossless (?q' ⤜ (λs. cpoke ?core2.0 s ?e)))› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_adv ?core1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_adv ?core2.0 s2 ?ia)) ?oa)› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_usr ?core1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_usr ?core2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 6*)\n apply ((auto intro: trace_core_closure_start (*‹trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p ?q›*) trace_core_closure_step[OF assms] (*‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?e ∈ E⟧ ⟹ weight_spmf (?p' ⤜ (λs. cpoke core1 s ?e)) = weight_spmf (?q' ⤜ (λs. cpoke core2 s ?e))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia))› ‹⟦trace_core_closure core1 core2 E IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) trace_core_closure_sim (*‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?e ∈ ?E⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (mk_lossless (?p' ⤜ (λs. cpoke ?core1.0 s ?e))) (mk_lossless (?q' ⤜ (λs. cpoke ?core2.0 s ?e)))› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_adv ?core1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_adv ?core2.0 s2 ?ia)) ?oa)› ‹⟦trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. cfunc_usr ?core1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. cfunc_usr ?core2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 7*)\n apply ((auto intro: trace_core_closure_start (*‹trace_core_closure (?core1.0::(?'a, ?'b, ?'c, ?'d, ?'e, ?'f) core) (?core2.0::(?'g, ?'b, ?'c, ?'d, ?'e, ?'f) core) (?E::?'b set) (?IA::?'c set) (?IU::?'d set) (?p::?'a spmf) (?q::?'g spmf) ?p ?q›*) trace_core_closure_step[OF assms] (*‹⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) (?p'::'a spmf) (?q'::'g spmf); (?e::'b) ∈ E⟧ ⟹ weight_spmf (?p' ⤜ (λs::'a. cpoke core1 s ?e)) = weight_spmf (?q' ⤜ (λs::'g. cpoke core2 s ?e))› ‹⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) (?p'::'a spmf) (?q'::'g spmf); (?ia::'c) ∈ IA⟧ ⟹ ?p' ⤜ (λs1::'a. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q' ⤜ (λs2::'g. map_spmf fst (cfunc_adv core2 s2 ?ia))› ‹⟦trace_core_closure (core1::('a, 'b, 'c, 'd, 'e, 'f) core) (core2::('g, 'b, 'c, 'd, 'e, 'f) core) (E::'b set) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'g spmf) (?p'::'a spmf) (?q'::'g spmf); (?iu::'d) ∈ IU⟧ ⟹ ?p' ⤜ (λs1::'a. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q' ⤜ (λs2::'g. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) trace_core_closure_sim (*‹⟦trace_core_closure (?core1.0::(?'a, ?'c, ?'d, ?'e, ?'f, ?'g) core) (?core2.0::(?'b, ?'c, ?'d, ?'e, ?'f, ?'g) core) (?E::?'c set) (?IA::?'d set) (?IU::?'e set) (?p::?'a spmf) (?q::?'b spmf) (?p'::?'a spmf) (?q'::?'b spmf); (?e::?'c) ∈ ?E⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (mk_lossless (?p' ⤜ (λs::?'a. cpoke ?core1.0 s ?e))) (mk_lossless (?q' ⤜ (λs::?'b. cpoke ?core2.0 s ?e)))› ‹⟦trace_core_closure (?core1.0::(?'a, ?'c, ?'d, ?'e, ?'f, ?'g) core) (?core2.0::(?'b, ?'c, ?'d, ?'e, ?'f, ?'g) core) (?E::?'c set) (?IA::?'d set) (?IU::?'e set) (?p::?'a spmf) (?q::?'b spmf) (?p'::?'a spmf) (?q'::?'b spmf); (?ia::?'d) ∈ ?IA⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1::?'a. cfunc_adv ?core1.0 s1 ?ia)) (?oa::?'f)) (cond_spmf_fst (?q' ⤜ (λs2::?'b. cfunc_adv ?core2.0 s2 ?ia)) ?oa)› ‹⟦trace_core_closure (?core1.0::(?'a, ?'c, ?'d, ?'e, ?'f, ?'g) core) (?core2.0::(?'b, ?'c, ?'d, ?'e, ?'f, ?'g) core) (?E::?'c set) (?IA::?'d set) (?IU::?'e set) (?p::?'a spmf) (?q::?'b spmf) (?p'::?'a spmf) (?q'::?'b spmf); (?iu::?'e) ∈ ?IU⟧ ⟹ trace_core_closure ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1::?'a. cfunc_usr ?core1.0 s1 ?iu)) (?ou::?'g)) (cond_spmf_fst (?q' ⤜ (λs2::?'b. cfunc_usr ?core2.0 s2 ?iu)) ?ou)›*))[1])\n (*proven 7 subgoals*) .\nqed\n\n\n\ntype_synonym ('event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest) trace_rest =\n \"('iadv_rest × 'oadv_rest × 'event list + 'iusr_rest × 'ousr_rest × 'event list) list\n ⇒ ('iadv_rest ⇒ ('oadv_rest × 'event list) spmf) \n × ('iusr_rest ⇒ ('ousr_rest × 'event list) spmf)\"\n\ncontext\n fixes rest :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\"\nbegin\n\nprimrec trace_rest' :: \"'s_rest spmf ⇒ ('event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest) trace_rest\" where\n \"trace_rest' S [] =\n (λia. bind_spmf S (λs. map_spmf fst (rfunc_adv rest s ia)),\n λiu. bind_spmf S (λs. map_spmf fst (rfunc_usr rest s iu)))\"\n| \"trace_rest' S (obs # tr) = (case obs of\n Inl (ia, oa) ⇒ trace_rest' (cond_spmf_fst (bind_spmf S (λs. rfunc_adv rest s ia)) oa) tr\n | Inr (iu, ou) ⇒ trace_rest' (cond_spmf_fst (bind_spmf S (λs. rfunc_usr rest s iu)) ou) tr)\"\n\nend\n\ndeclare trace_rest'.simps [simp del]\ncase_of_simps trace_rest'_unfold: trace_rest'.simps\nsimps_of_case trace_rest'_simps [simp]: trace_rest'_unfold\n\ncontext includes lifting_syntax begin\n\nlemma trace_rest'_parametric [transfer_rule]:\n \"(rel_rest' S (=) IA IU (=) (=) M ===> rel_spmf S ===>\n list_all2 (rel_sum (rel_prod IA (=)) (rel_prod IU (=))) ===>\n rel_prod (IA ===> (=)) (IU ===> (=)))\n trace_rest' trace_rest'\"\n unfolding trace_rest'_def\n (*goal: ‹(rel_rest' S (=) IA IU (=) (=) M ===> rel_spmf S ===> list_all2 (rel_sum (rel_prod IA (=)) (rel_prod IU (=))) ===> rel_prod (IA ===> (=)) (IU ===> (=))) (λrest uu uua. rec_list (λS. (λia. S ⤜ (λs. map_spmf fst (rfunc_adv rest s ia)), λiu. S ⤜ (λs. map_spmf fst (rfunc_usr rest s iu)))) (λobs tr tra S. case obs of Inl (ia, oa) ⇒ tra (cond_spmf_fst (S ⤜ (λs. rfunc_adv rest s ia)) oa) | Inr (iu, ou) ⇒ tra (cond_spmf_fst (S ⤜ (λs. rfunc_usr rest s iu)) ou)) uua uu) (λrest uu uua. rec_list (λS. (λia. S ⤜ (λs. map_spmf fst (rfunc_adv rest s ia)), λiu. S ⤜ (λs. map_spmf fst (rfunc_usr rest s iu)))) (λobs tr tra S. case obs of Inl (ia, oa) ⇒ tra (cond_spmf_fst (S ⤜ (λs. rfunc_adv rest s ia)) oa) | Inr (iu, ou) ⇒ tra (cond_spmf_fst (S ⤜ (λs. rfunc_usr rest s iu)) ou)) uua uu)›*)\n by transfer_prover\n\ndefinition trace_rest_eq\n :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more1) rest_scheme\n ⇒ ('s_rest', 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more2) rest_scheme\n ⇒ 'iadv_rest set ⇒ 'iusr_rest set\n ⇒ 's_rest spmf ⇒ 's_rest' spmf ⇒ bool\" where\n \"trace_rest_eq rest1 rest2 IA IU p q ⟷\n (∀tr. set tr ⊆ (IA × UNIV) <+> (IU × UNIV) ⟶\n rel_prod (eq_onp (λia. ia ∈ IA) ===> (=)) (eq_onp (λiu. iu ∈ IU) ===> (=))\n (trace_rest' rest1 p tr) (trace_rest' rest2 q tr))\"\n\nend\n\nlemma trace_rest_eqD:\n assumes \"trace_rest_eq rest1 rest2 IA IU p q\"\n and \"set tr ⊆ (IA × UNIV) <+> (IU × UNIV)\"\n shows trace_rest_eqD_rfunc_adv:\n \"ia ∈ IA ⟹ fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia\"\n and trace_rest_eqD_rfunc_usr:\n \"iu ∈ IU ⟹ snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu\"\n using assms (*‹trace_rest_eq (rest1::('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme) (rest2::('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'h spmf)› ‹set (tr::('c × 'e × 'b list + 'd × 'f × 'b list) list) ⊆ (IA::'c set) × UNIV <+> (IU::'d set) × UNIV›*) apply -\n (*goals:\n 1. ‹⟦ia ∈ IA; trace_rest_eq rest1 rest2 IA IU p q; set tr ⊆ IA × UNIV <+> IU × UNIV⟧ ⟹ fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia›\n 2. ‹⟦iu ∈ IU; trace_rest_eq rest1 rest2 IA IU p q; set tr ⊆ IA × UNIV <+> IU × UNIV⟧ ⟹ snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu›\n discuss goal 1*)\n apply ((auto simp add: trace_rest_eq_def (*‹trace_rest_eq ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q = (∀tr. set tr ⊆ ?IA × UNIV <+> ?IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λia. ia ∈ ?IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ ?IU)) (=)) (trace_rest' ?rest1.0 ?p tr) (trace_rest' ?rest2.0 ?q tr))›*) rel_fun_def (*‹rel_fun ?A ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) rel_prod_sel (*‹rel_prod ?R1.0 ?R2.0 ?p ?q = (?R1.0 (fst ?p) (fst ?q) ∧ ?R2.0 (snd ?p) (snd ?q))›*) eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: trace_rest_eq_def (*‹trace_rest_eq ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q = (∀tr. set tr ⊆ ?IA × UNIV <+> ?IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λia. ia ∈ ?IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ ?IU)) (=)) (trace_rest' ?rest1.0 ?p tr) (trace_rest' ?rest2.0 ?q tr))›*) rel_fun_def (*‹rel_fun ?A ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) rel_prod_sel (*‹rel_prod ?R1.0 ?R2.0 ?p ?q = (?R1.0 (fst ?p) (fst ?q) ∧ ?R2.0 (snd ?p) (snd ?q))›*) eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*))[1])\n (*proven 2 subgoals*) .\n\nlemma trace_rest_eqI:\n assumes \"⋀tr ia. ⟦ set tr ⊆ (IA × UNIV) <+> (IU × UNIV); ia ∈ IA ⟧\n ⟹ fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia\"\n and \"⋀tr iu. ⟦ set tr ⊆ (IA × UNIV) <+> (IU × UNIV); iu ∈ IU ⟧\n ⟹ snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu\"\n shows \"trace_rest_eq rest1 rest2 IA IU p q\"\n using assms (*‹⟦set ?tr ⊆ IA × UNIV <+> IU × UNIV; ?ia ∈ IA⟧ ⟹ fst (trace_rest' rest1 p ?tr) ?ia = fst (trace_rest' rest2 q ?tr) ?ia› ‹⟦set ?tr ⊆ IA × UNIV <+> IU × UNIV; ?iu ∈ IU⟧ ⟹ snd (trace_rest' rest1 p ?tr) ?iu = snd (trace_rest' rest2 q ?tr) ?iu›*) by (auto simp add: trace_rest_eq_def (*‹trace_rest_eq ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q = (∀tr. set tr ⊆ ?IA × UNIV <+> ?IU × UNIV ⟶ rel_prod (rel_fun (eq_onp (λia. ia ∈ ?IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ ?IU)) (=)) (trace_rest' ?rest1.0 ?p tr) (trace_rest' ?rest2.0 ?q tr))›*) rel_fun_def (*‹rel_fun ?A ?B = (λf g. ∀x y. ?A x y ⟶ ?B (f x) (g y))›*) eq_onp_def (*‹eq_onp ?R = (λx y. ?R x ∧ x = y)›*) rel_prod_sel (*‹rel_prod ?R1.0 ?R2.0 ?p ?q = (?R1.0 (fst ?p) (fst ?q) ∧ ?R2.0 (snd ?p) (snd ?q))›*))\n\nlemma trace_rest_return_pmf_None [simp]:\n \"trace_rest' rest (return_pmf None) tr = (λ_. return_pmf None, λ_. return_pmf None)\"\n apply (induction tr)\n (*goals:\n 1. ‹trace_rest' (rest::('f, 'c, 'a, 'd, 'b, 'e, 'g) rest_scheme) (return_pmf None) [] = (λ_::'a. return_pmf None, λ_::'d. return_pmf None)›\n 2. ‹⋀(a::'a × 'b × 'c list + 'd × 'e × 'c list) tr::('a × 'b × 'c list + 'd × 'e × 'c list) list. trace_rest' (rest::('f, 'c, 'a, 'd, 'b, 'e, 'g) rest_scheme) (return_pmf None) tr = (λ_::'a. return_pmf None, λ_::'d. return_pmf None) ⟹ trace_rest' rest (return_pmf None) (a # tr) = (λ_::'a. return_pmf None, λ_::'d. return_pmf None)›\n discuss goal 1*)\n apply (simp add: trace_rest'.simps (*‹trace_rest' ?rest ?S [] = (λia. ?S ⤜ (λs. map_spmf fst (rfunc_adv ?rest s ia)), λiu. ?S ⤜ (λs. map_spmf fst (rfunc_usr ?rest s iu)))› ‹trace_rest' ?rest ?S (?obs # ?tr) = (case ?obs of Inl (ia, oa) ⇒ trace_rest' ?rest (cond_spmf_fst (?S ⤜ (λs. rfunc_adv ?rest s ia)) oa) ?tr | Inr (iu, ou) ⇒ trace_rest' ?rest (cond_spmf_fst (?S ⤜ (λs. rfunc_usr ?rest s iu)) ou) ?tr)›*) split: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))\n (*discuss goal 2*)\n apply (simp add: trace_rest'.simps (*‹trace_rest' ?rest ?S [] = (λia. ?S ⤜ (λs. map_spmf fst (rfunc_adv ?rest s ia)), λiu. ?S ⤜ (λs. map_spmf fst (rfunc_usr ?rest s iu)))› ‹trace_rest' ?rest ?S (?obs # ?tr) = (case ?obs of Inl (ia, oa) ⇒ trace_rest' ?rest (cond_spmf_fst (?S ⤜ (λs. rfunc_adv ?rest s ia)) oa) ?tr | Inr (iu, ou) ⇒ trace_rest' ?rest (cond_spmf_fst (?S ⤜ (λs. rfunc_usr ?rest s iu)) ou) ?tr)›*) split: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))\n (*proven 2 subgoals*) .\n\nlemma rel_rest'_into_trace_rest_eq: \"trace_rest_eq rest rest' IA IU p q\" \n if \"rel_rest' S (=) (eq_onp (λia. ia ∈ IA)) (eq_onp (λiu. iu ∈ IU)) (=) (=) M rest rest'\"\n \"rel_spmf S p q\"\n using trace_rest'_parametric[THEN rel_funD, THEN rel_funD, OF that] (*‹rel_fun (list_all2 (rel_sum (rel_prod (eq_onp (λia::'d. ia ∈ (IA::'d set))) (=)) (rel_prod (eq_onp (λiu::'e. iu ∈ (IU::'e set))) (=)))) (rel_prod (rel_fun (eq_onp (λia::'d. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu::'e. iu ∈ IU)) (=))) (trace_rest' (rest::('a, 'c, 'd, 'e, 'f, 'g, 'h) rest_scheme) (p::'a spmf)) (trace_rest' (rest'::('b, 'c, 'd, 'e, 'f, 'g, 'i) rest_scheme) (q::'b spmf))›*) unfolding trace_rest_eq_def\n (*goal: ‹∀tr::('d::type × 'f::type × 'c::type list + 'e::type × 'g::type × 'c::type list) list. set tr ⊆ (IA::'d::type set) × UNIV <+> (IU::'e::type set) × UNIV ⟶ rel_prod (rel_fun (eq_onp (λia::'d::type. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu::'e::type. iu ∈ IU)) (=)) (trace_rest' (rest::('a::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type, 'h::type) rest_scheme) (p::'a::type spmf) tr) (trace_rest' (rest'::('b::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type, 'i::type) rest_scheme) (q::'b::type spmf) tr)›*)\n apply (intro strip (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q› ‹(⋀x. ?P x) ⟹ ∀x. ?P x› ‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*))\n (*goal: ‹∀tr::('d × 'f × 'c list + 'e × 'g × 'c list) list. set tr ⊆ (IA::'d set) × UNIV <+> (IU::'e set) × UNIV ⟶ rel_prod (rel_fun (eq_onp (λia::'d. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu::'e. iu ∈ IU)) (=)) (trace_rest' (rest::('a, 'c, 'd, 'e, 'f, 'g, 'h) rest_scheme) (p::'a spmf) tr) (trace_rest' (rest'::('b, 'c, 'd, 'e, 'f, 'g, 'i) rest_scheme) (q::'b spmf) tr)›*)\n subgoal for tr\n apply (simp add: eq_onp_True[symmetric] (*‹(=) = eq_onp (λ_. True)›*) prod.rel_eq_onp (*‹rel_prod (eq_onp ?P1.0) (eq_onp ?P2.0) = eq_onp (pred_prod ?P1.0 ?P2.0)›*) sum.rel_eq_onp (*‹rel_sum (eq_onp ?P1.0) (eq_onp ?P2.0) = eq_onp (pred_sum ?P1.0 ?P2.0)›*) list.rel_eq_onp (*‹list_all2 (eq_onp ?P) = eq_onp (list_all ?P)›*))\n (*goal: ‹⟦rel_fun (list_all2 (rel_sum (rel_prod (eq_onp (λia. ia ∈ IA)) (=)) (rel_prod (eq_onp (λiu. iu ∈ IU)) (=)))) (rel_prod (rel_fun (eq_onp (λia. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ IU)) (=))) (trace_rest' rest p) (trace_rest' rest' q); set tr ⊆ IA × UNIV <+> IU × UNIV⟧ ⟹ rel_prod (rel_fun (eq_onp (λia. ia ∈ IA)) (=)) (rel_fun (eq_onp (λiu. iu ∈ IU)) (=)) (trace_rest' rest p tr) (trace_rest' rest' q tr)›*)\n by (auto 4 3 simp add: eq_onp_def (*‹eq_onp (?R::?'a::type ⇒ bool) = (λ(x::?'a::type) y::?'a::type. ?R x ∧ x = y)›*) list_all_iff (*‹list_all (?P::?'a::type ⇒ bool) (?x::?'a::type list) = Ball (set ?x) ?P›*) dest: rel_funD[where x=tr and y=tr] (*‹⟦rel_fun (?A::('d::type × 'f::type × 'c::type list + 'e::type × 'g::type × 'c::type list) list ⇒ ('d::type × 'f::type × 'c::type list + 'e::type × 'g::type × 'c::type list) list ⇒ bool) (?B::?'c::type ⇒ ?'d::type ⇒ bool) (?f::('d::type × 'f::type × 'c::type list + 'e::type × 'g::type × 'c::type list) list ⇒ ?'c::type) (?g::('d::type × 'f::type × 'c::type list + 'e::type × 'g::type × 'c::type list) list ⇒ ?'d::type); ?A (tr::('d::type × 'f::type × 'c::type list + 'e::type × 'g::type × 'c::type list) list) tr⟧ ⟹ ?B (?f tr) (?g tr)›*)) .\n\nlemma trace_rest_eq_simI:\n fixes rest1 :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\"\n and rest2 :: \"('s_rest', 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\"\n and S :: \"'s_rest spmf ⇒ 's_rest' spmf ⇒ bool\"\n assumes start: \"S p q\"\n and step_rfunc_adv: \"⋀p q ia. ⟦ S p q; ia ∈ IA ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = bind_spmf q (λs2. map_spmf fst (rfunc_adv rest2 s2 ia))\"\n and sim_rfunc_adv: \"⋀p q ia s1 s2 s1' s2' oa. ⟦ S p q; ia ∈ IA; \n s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (rfunc_adv rest1 s1 ia); (oa, s2') ∈ set_spmf (rfunc_adv rest2 s2 ia) ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (bind_spmf q (λs2. rfunc_adv rest2 s2 ia)) oa)\"\n and step_rfunc_usr: \"⋀p q iu. ⟦ S p q; iu ∈ IU ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = bind_spmf q (λs2. map_spmf fst (rfunc_usr rest2 s2 iu))\"\n and sim_rfunc_usr: \"⋀p q iu s1 s2 s1' s2' ou. ⟦ S p q; iu ∈ IU; \n s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (rfunc_usr rest1 s1 iu); (ou, s2') ∈ set_spmf (rfunc_usr rest2 s2 iu) ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (bind_spmf q (λs2. rfunc_usr rest2 s2 iu)) ou)\"\n shows \"trace_rest_eq rest1 rest2 IA IU p q\"\nproof(rule trace_rest_eqI)\n fix tr :: \"('iadv_rest × 'oadv_rest × 'event list + 'iusr_rest × 'ousr_rest × 'event list) list\"\n assume \"set tr ⊆ IA × UNIV <+> IU × UNIV\"\n then have \"(∀ia∈IA. fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia) ∧\n (∀iu∈IU. snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu)\"\n using start\n proof(induction tr arbitrary: p q)\n case Nil\n then show ?case by(simp add: step_rfunc_adv step_rfunc_usr)\n next\n case (Cons a tr)\n from Cons.prems(1) have tr: \"set tr ⊆ IA × UNIV <+> IU × UNIV\" by simp\n from Cons.prems(1)\n consider (rfunc_adv) ia oa where \"a = Inl (ia, oa)\" \"ia ∈ IA\"\n | (rfunc_usr) iu ou where \"a = Inr (iu, ou)\" \"iu ∈ IU\" by auto\n then show ?case\n proof cases\n case rfunc_adv\n let ?p = \"bind_spmf p (λs1. rfunc_adv rest1 s1 ia)\"\n let ?q = \"bind_spmf q (λs2. rfunc_adv rest2 s2 ia)\"\n show ?thesis\n proof(cases \"oa ∈ fst ` set_spmf ?p\")\n case True\n with step_rfunc_adv[OF Cons.prems(2) rfunc_adv(2), THEN arg_cong[where f=set_spmf]]\n have \"oa ∈ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def)\n then show ?thesis using True Cons.prems rfunc_adv\n by(clarsimp)(rule Cons.IH; blast intro: sim_rfunc_adv)\n next\n case False\n hence \"cond_spmf_fst ?p oa = return_pmf None\" by simp\n moreover\n from step_rfunc_adv[OF Cons.prems(2) rfunc_adv(2), THEN arg_cong[where f=set_spmf]] False\n have oa': \"oa ∉ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def) simp\n hence \"cond_spmf_fst ?q oa = return_pmf None\" by simp\n ultimately show ?thesis using rfunc_adv by(simp del: cond_spmf_fst_eq_return_None)\n qed\n next\n case rfunc_usr\n let ?p = \"bind_spmf p (λs1. rfunc_usr rest1 s1 iu)\"\n let ?q = \"bind_spmf q (λs2. rfunc_usr rest2 s2 iu)\"\n show ?thesis\n proof(cases \"ou ∈ fst ` set_spmf ?p\")\n case True\n with step_rfunc_usr[OF Cons.prems(2) rfunc_usr(2), THEN arg_cong[where f=set_spmf]]\n have \"ou ∈ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def)\n then show ?thesis using True Cons.prems rfunc_usr\n by(clarsimp)(rule Cons.IH; blast intro: sim_rfunc_usr)\n next\n case False\n hence \"cond_spmf_fst ?p ou = return_pmf None\" by simp\n moreover\n from step_rfunc_usr[OF Cons.prems(2) rfunc_usr(2), THEN arg_cong[where f=set_spmf]] False\n have oa': \"ou ∉ fst ` set_spmf ?q\"\n unfolding set_map_spmf[symmetric] by(simp only: map_bind_spmf o_def) simp\n hence \"cond_spmf_fst ?q ou = return_pmf None\" by simp\n ultimately show ?thesis using rfunc_usr by(simp del: cond_spmf_fst_eq_return_None)\n qed\n qed\n qed\n then show \"ia ∈ IA ⟹ fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia\"\n and \"iu ∈ IU ⟹ snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu\"\n for ia iu by blast+\nqed\n\ncontext \n fixes rest :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\"\nbegin\n\nfun trace_rest_aux \n :: \"'s_rest spmf ⇒ ('iadv_rest × 'oadv_rest × 'event list + 'iusr_rest × 'ousr_rest × 'event list) list ⇒ 's_rest spmf\" where\n \"trace_rest_aux p [] = p\"\n| \"trace_rest_aux p (Inl (ia, oaes) # tr) = trace_rest_aux (cond_spmf_fst (bind_spmf p (λs. rfunc_adv rest s ia)) oaes) tr\"\n| \"trace_rest_aux p (Inr (iu, oues) # tr) = trace_rest_aux (cond_spmf_fst (bind_spmf p (λs. rfunc_usr rest s iu)) oues) tr\"\n\nend\n\nlemma trace_rest_conv_trace_rest_aux:\n \"trace_rest' rest p tr = \n (λia. bind_spmf (trace_rest_aux rest p tr) (λs. map_spmf fst (rfunc_adv rest s ia)),\n λiu. bind_spmf (trace_rest_aux rest p tr) (λs. map_spmf fst (rfunc_usr rest s iu)))\"\n apply (induction p tr rule: trace_rest_aux.induct (*‹⟦⋀p. ?P p []; ⋀p ia oaes tr. ?P (cond_spmf_fst (p ⤜ (λs. rfunc_adv ?rest s ia)) oaes) tr ⟹ ?P p (Inl (ia, oaes) # tr); ⋀p iu oues tr. ?P (cond_spmf_fst (p ⤜ (λs. rfunc_usr ?rest s iu)) oues) tr ⟹ ?P p (Inr (iu, oues) # tr)⟧ ⟹ ?P ?a0.0 ?a1.0›*))\n (*goals:\n 1. ‹⋀p. trace_rest' rest p [] = (λia. trace_rest_aux rest p [] ⤜ (λs. map_spmf fst (rfunc_adv rest s ia)), λiu. trace_rest_aux rest p [] ⤜ (λs. map_spmf fst (rfunc_usr rest s iu)))›\n 2. ‹⋀p ia oaes tr. trace_rest' rest (cond_spmf_fst (p ⤜ (λs. rfunc_adv ?rest s ia)) oaes) tr = (λiaa. trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_adv ?rest s ia)) oaes) tr ⤜ (λs. map_spmf fst (rfunc_adv rest s iaa)), λiu. trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_adv ?rest s ia)) oaes) tr ⤜ (λs. map_spmf fst (rfunc_usr rest s iu))) ⟹ trace_rest' rest p (Inl (ia, oaes) # tr) = (λiaa. trace_rest_aux rest p (Inl (ia, oaes) # tr) ⤜ (λs. map_spmf fst (rfunc_adv rest s iaa)), λiu. trace_rest_aux rest p (Inl (ia, oaes) # tr) ⤜ (λs. map_spmf fst (rfunc_usr rest s iu)))›\n 3. ‹⋀p iu oues tr. trace_rest' rest (cond_spmf_fst (p ⤜ (λs. rfunc_usr ?rest s iu)) oues) tr = (λia. trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_usr ?rest s iu)) oues) tr ⤜ (λs. map_spmf fst (rfunc_adv rest s ia)), λiua. trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_usr ?rest s iu)) oues) tr ⤜ (λs. map_spmf fst (rfunc_usr rest s iua))) ⟹ trace_rest' rest p (Inr (iu, oues) # tr) = (λia. trace_rest_aux rest p (Inr (iu, oues) # tr) ⤜ (λs. map_spmf fst (rfunc_adv rest s ia)), λiua. trace_rest_aux rest p (Inr (iu, oues) # tr) ⤜ (λs. map_spmf fst (rfunc_usr rest s iua)))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*proven 3 subgoals*) .\n\nlemma trace_rest_aux_append:\n \"trace_rest_aux rest p (tr @ tr') = trace_rest_aux rest (trace_rest_aux rest p tr) tr'\"\n apply (induction p tr rule: trace_rest_aux.induct (*‹⟦⋀p. ?P p []; ⋀p ia oaes tr. ?P (cond_spmf_fst (p ⤜ (λs. rfunc_adv ?rest s ia)) oaes) tr ⟹ ?P p (Inl (ia, oaes) # tr); ⋀p iu oues tr. ?P (cond_spmf_fst (p ⤜ (λs. rfunc_usr ?rest s iu)) oues) tr ⟹ ?P p (Inr (iu, oues) # tr)⟧ ⟹ ?P ?a0.0 ?a1.0›*))\n (*goals:\n 1. ‹⋀p. trace_rest_aux rest p ([] @ tr') = trace_rest_aux rest (trace_rest_aux rest p []) tr'›\n 2. ‹⋀p ia oaes tr. trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_adv ?rest s ia)) oaes) (tr @ tr') = trace_rest_aux rest (trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_adv ?rest s ia)) oaes) tr) tr' ⟹ trace_rest_aux rest p ((Inl (ia, oaes) # tr) @ tr') = trace_rest_aux rest (trace_rest_aux rest p (Inl (ia, oaes) # tr)) tr'›\n 3. ‹⋀p iu oues tr. trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_usr ?rest s iu)) oues) (tr @ tr') = trace_rest_aux rest (trace_rest_aux rest (cond_spmf_fst (p ⤜ (λs. rfunc_usr ?rest s iu)) oues) tr) tr' ⟹ trace_rest_aux rest p ((Inr (iu, oues) # tr) @ tr') = trace_rest_aux rest (trace_rest_aux rest p (Inr (iu, oues) # tr)) tr'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n\ninductive trace_rest_closure \n :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\n ⇒ ('s_rest', 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more') rest_scheme\n ⇒ 'iadv_rest set ⇒ 'iusr_rest set\n ⇒ 's_rest spmf ⇒ 's_rest' spmf ⇒ 's_rest spmf ⇒ 's_rest' spmf ⇒ bool\"\n for rest1 rest2 IA IU p q where\n \"trace_rest_closure rest1 rest2 IA IU p q (trace_rest_aux rest1 p tr) (trace_rest_aux rest2 q tr)\"\n if \"set tr ⊆ IA × UNIV <+> IU × UNIV\"\n\nlemma trace_rest_closure_start: \"trace_rest_closure rest1 rest2 IA IU p q p q\"\n by (simp add: trace_rest_closure.simps (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?a1.0 ?a2.0 = (∃tr. ?a1.0 = trace_rest_aux ?rest1.0 ?p tr ∧ ?a2.0 = trace_rest_aux ?rest2.0 ?q tr ∧ set tr ⊆ ?IA × UNIV <+> ?IU × UNIV)›*) exI[where x=\"[]\"] (*‹?P [] ⟹ ∃x. ?P x›*))\n\nlemma trace_rest_closure_step:\n assumes \"trace_rest_eq rest1 rest2 IA IU p q\"\n and \"trace_rest_closure rest1 rest2 IA IU p q p' q'\"\n shows trace_rest_closure_step_rfunc_adv: \n \"ia ∈ IA ⟹ bind_spmf p' (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = bind_spmf q' (λs2. map_spmf fst (rfunc_adv rest2 s2 ia))\"\n (is \"PROP ?thesis1\")\n and trace_rest_closure_step_rfunc_usr:\n \"iu ∈ IU ⟹ bind_spmf p' (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = bind_spmf q' (λs2. map_spmf fst (rfunc_usr rest2 s2 iu))\"\n (is \"PROP ?thesis2\")\nproof (-)\n (*goals:\n 1. ‹ia ∈ IA ⟹ p' ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = q' ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ia))›\n 2. ‹iu ∈ IU ⟹ p' ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = q' ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 iu))›*)\n from assms(2) (*‹trace_rest_closure rest1 rest2 IA IU p q p' q'›*) obtain tr where p: \"p' = trace_rest_aux rest1 p tr\" and q: \"q' = trace_rest_aux rest2 q tr\" and tr: \"set tr ⊆ IA × UNIV <+> IU × UNIV\"\n (*goal: ‹(⋀tr. ⟦p' = trace_rest_aux rest1 p tr; q' = trace_rest_aux rest2 q tr; set tr ⊆ IA × UNIV <+> IU × UNIV⟧ ⟹ thesis) ⟹ thesis›*)\n by cases\n from trace_rest_eqD[OF assms ( 1 ) tr] (*‹?ia ∈ IA ⟹ fst (trace_rest' rest1 p tr) ?ia = fst (trace_rest' rest2 q tr) ?ia› ‹?iu ∈ IU ⟹ snd (trace_rest' rest1 p tr) ?iu = snd (trace_rest' rest2 q tr) ?iu›*) p (*‹p' = trace_rest_aux rest1 p tr›*) q (*‹q' = trace_rest_aux rest2 q tr›*) show \"PROP ?thesis1\" and \"PROP ?thesis2\"\n (*goals:\n 1. ‹(ia::'c) ∈ (IA::'c set) ⟹ (p'::'a spmf) ⤜ (λs1::'a. map_spmf fst (rfunc_adv (rest1::('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme) s1 ia)) = (q'::'h spmf) ⤜ (λs2::'h. map_spmf fst (rfunc_adv (rest2::('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme) s2 ia))›\n 2. ‹(iu::'d) ∈ (IU::'d set) ⟹ (p'::'a spmf) ⤜ (λs1::'a. map_spmf fst (rfunc_usr (rest1::('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme) s1 iu)) = (q'::'h spmf) ⤜ (λs2::'h. map_spmf fst (rfunc_usr (rest2::('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme) s2 iu))›*)\n apply -\n (*goals:\n 1. ‹⟦ia ∈ IA; ⋀ia. ia ∈ IA ⟹ fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia; ⋀iu. iu ∈ IU ⟹ snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu; p' = trace_rest_aux rest1 p tr; q' = trace_rest_aux rest2 q tr⟧ ⟹ p' ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = q' ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ia))›\n 2. ‹⟦iu ∈ IU; ⋀ia. ia ∈ IA ⟹ fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia; ⋀iu. iu ∈ IU ⟹ snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu; p' = trace_rest_aux rest1 p tr; q' = trace_rest_aux rest2 q tr⟧ ⟹ p' ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = q' ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 iu))›\n discuss goal 1*)\n apply (simp add: trace_rest_conv_trace_rest_aux (*‹trace_rest' ?rest ?p ?tr = (λia. trace_rest_aux ?rest ?p ?tr ⤜ (λs. map_spmf fst (rfunc_adv ?rest s ia)), λiu. trace_rest_aux ?rest ?p ?tr ⤜ (λs. map_spmf fst (rfunc_usr ?rest s iu)))›*))\n (*discuss goal 2*)\n apply (simp add: trace_rest_conv_trace_rest_aux (*‹trace_rest' (?rest::(?'f, ?'c, ?'a, ?'d, ?'b, ?'e, ?'g) rest_scheme) (?p::?'f spmf) (?tr::(?'a × ?'b × ?'c list + ?'d × ?'e × ?'c list) list) = (λia::?'a. trace_rest_aux ?rest ?p ?tr ⤜ (λs::?'f. map_spmf fst (rfunc_adv ?rest s ia)), λiu::?'d. trace_rest_aux ?rest ?p ?tr ⤜ (λs::?'f. map_spmf fst (rfunc_usr ?rest s iu)))›*))\n (*proven 2 subgoals*) .\nqed\n\nlemma trace_rest_closure_sim:\n fixes rest1 rest2 IA IU p q\n defines \"S ≡ trace_rest_closure rest1 rest2 IA IU p q\"\n assumes \"S p' q'\"\n shows trace_rest_closure_sim_rfunc_adv: \"ia ∈ IA \n ⟹ S (cond_spmf_fst (bind_spmf p' (λs1. rfunc_adv rest1 s1 ia)) oa) \n (cond_spmf_fst (bind_spmf q' (λs2. rfunc_adv rest2 s2 ia)) oa)\"\n (is \"PROP ?thesis1\")\n and trace_rest_closure_sim_rfunc_usr: \"iu ∈ IU \n ⟹ S (cond_spmf_fst (bind_spmf p' (λs1. rfunc_usr rest1 s1 iu)) ou)\n (cond_spmf_fst (bind_spmf q' (λs2. rfunc_usr rest2 s2 iu)) ou)\"\n (is \"PROP ?thesis2\")\nproof (-)\n (*goals:\n 1. ‹ia ∈ IA ⟹ S (cond_spmf_fst (p' ⤜ (λs1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (q' ⤜ (λs2. rfunc_adv rest2 s2 ia)) oa)›\n 2. ‹iu ∈ IU ⟹ S (cond_spmf_fst (p' ⤜ (λs1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (q' ⤜ (λs2. rfunc_usr rest2 s2 iu)) ou)›*)\n from assms(2) (*‹S p' q'›*) obtain tr where p: \"p' = trace_rest_aux rest1 p tr\" and q: \"q' = trace_rest_aux rest2 q tr\" and tr: \"set tr ⊆ IA × UNIV <+> IU × UNIV\"\n (*goal: ‹(⋀tr. ⟦p' = trace_rest_aux rest1 p tr; q' = trace_rest_aux rest2 q tr; set tr ⊆ IA × UNIV <+> IU × UNIV⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding S_def\n (*goal: ‹(⋀tr::('d × 'f × 'c list + 'e × 'g × 'c list) list. ⟦(p'::'a spmf) = trace_rest_aux (rest1::('a, 'c, 'd, 'e, 'f, 'g, 'h) rest_scheme) (p::'a spmf) tr; (q'::'b spmf) = trace_rest_aux (rest2::('b, 'c, 'd, 'e, 'f, 'g, 'i) rest_scheme) (q::'b spmf) tr; set tr ⊆ (IA::'d set) × UNIV <+> (IU::'e set) × UNIV⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by cases\n show \"PROP ?thesis1\"\n (*goal: ‹ia ∈ IA ⟹ S (cond_spmf_fst (p' ⤜ (λs1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (q' ⤜ (λs2. rfunc_adv rest2 s2 ia)) oa)›*)\n using p (*‹p' = trace_rest_aux rest1 p tr›*) q (*‹q' = trace_rest_aux rest2 q tr›*) tr (*‹set tr ⊆ IA × UNIV <+> IU × UNIV›*) by (auto simp add: S_def (*‹S ≡ trace_rest_closure rest1 rest2 IA IU p q›*) trace_rest_closure.simps (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?a1.0 ?a2.0 = (∃tr. ?a1.0 = trace_rest_aux ?rest1.0 ?p tr ∧ ?a2.0 = trace_rest_aux ?rest2.0 ?q tr ∧ set tr ⊆ ?IA × UNIV <+> ?IU × UNIV)›*) trace_rest_aux_append (*‹trace_rest_aux ?rest ?p (?tr @ ?tr') = trace_rest_aux ?rest (trace_rest_aux ?rest ?p ?tr) ?tr'›*) intro!: exI[where x=\"tr @ [Inl (_, _)]\"] (*‹?P (tr @ [Inl (?uu4, ?uua4)]) ⟹ ∃x. ?P x›*))\n show \"PROP ?thesis2\"\n (*goal: ‹iu ∈ IU ⟹ S (cond_spmf_fst (p' ⤜ (λs1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (q' ⤜ (λs2. rfunc_usr rest2 s2 iu)) ou)›*)\n using p (*‹p' = trace_rest_aux rest1 p tr›*) q (*‹(q'::'b::type spmf) = trace_rest_aux (rest2::('b::type, 'c::type, 'd::type, 'e::type, 'f::type, 'g::type, 'i::type) rest_scheme) (q::'b::type spmf) (tr::('d::type × 'f::type × 'c::type list + 'e::type × 'g::type × 'c::type list) list)›*) tr (*‹set tr ⊆ IA × UNIV <+> IU × UNIV›*) by (auto simp add: S_def (*‹S ≡ trace_rest_closure rest1 rest2 IA IU p q›*) trace_rest_closure.simps (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?a1.0 ?a2.0 = (∃tr. ?a1.0 = trace_rest_aux ?rest1.0 ?p tr ∧ ?a2.0 = trace_rest_aux ?rest2.0 ?q tr ∧ set tr ⊆ ?IA × UNIV <+> ?IU × UNIV)›*) trace_rest_aux_append (*‹trace_rest_aux ?rest ?p (?tr @ ?tr') = trace_rest_aux ?rest (trace_rest_aux ?rest ?p ?tr) ?tr'›*) intro!: exI[where x=\"tr @ [Inr (_, _)]\"] (*‹?P (tr @ [Inr (?uu4, ?uua4)]) ⟹ ∃x. ?P x›*))\nqed\n\nproposition trace_rest_eq_complete:\n assumes \"trace_rest_eq rest1 rest2 IA IU p q\"\n obtains S\n where \"S p q\"\n and \"⋀p q ia. ⟦ S p q; ia ∈ IA ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = bind_spmf q (λs2. map_spmf fst (rfunc_adv rest2 s2 ia))\"\n and \"⋀p q ia oa. ⟦ S p q; ia ∈ IA ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (bind_spmf q (λs2. rfunc_adv rest2 s2 ia)) oa)\"\n and \"⋀p q iu. ⟦ S p q; iu ∈ IU ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = bind_spmf q (λs2. map_spmf fst (rfunc_usr rest2 s2 iu))\"\n and \"⋀p q iu ou. ⟦ S p q; iu ∈ IU ⟧\n ⟹ S (cond_spmf_fst (bind_spmf p (λs1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (bind_spmf q (λs2. rfunc_usr rest2 s2 iu)) ou)\"\nproof (-)\n (*goal: ‹(⋀S. ⟦S p q; ⋀p q ia. ⟦S p q; ia ∈ IA⟧ ⟹ p ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = q ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ia)); ⋀p q ia oa. ⟦S p q; ia ∈ IA⟧ ⟹ S (cond_spmf_fst (p ⤜ (λs1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. rfunc_adv rest2 s2 ia)) oa); ⋀p q iu. ⟦S p q; iu ∈ IU⟧ ⟹ p ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = q ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 iu)); ⋀p q iu ou. ⟦S p q; iu ∈ IU⟧ ⟹ S (cond_spmf_fst (p ⤜ (λs1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. rfunc_usr rest2 s2 iu)) ou)⟧ ⟹ thesis) ⟹ thesis›*)\n show thesis\n apply (rule that[where S=\"trace_rest_closure rest1 rest2 IA IU p q\"] (*‹⟦trace_rest_closure (rest1::('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme) (rest2::('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme) (IA::'c set) (IU::'d set) (p::'a spmf) (q::'h spmf) p q; ⋀(pa::'a spmf) (qa::'h spmf) ia::'c. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; ia ∈ IA⟧ ⟹ pa ⤜ (λs1::'a. map_spmf fst (rfunc_adv rest1 s1 ia)) = qa ⤜ (λs2::'h. map_spmf fst (rfunc_adv rest2 s2 ia)); ⋀(pa::'a spmf) (qa::'h spmf) (ia::'c) oa::'e × 'b list. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; ia ∈ IA⟧ ⟹ trace_rest_closure rest1 rest2 IA IU p q (cond_spmf_fst (pa ⤜ (λs1::'a. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (qa ⤜ (λs2::'h. rfunc_adv rest2 s2 ia)) oa); ⋀(pa::'a spmf) (qa::'h spmf) iu::'d. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; iu ∈ IU⟧ ⟹ pa ⤜ (λs1::'a. map_spmf fst (rfunc_usr rest1 s1 iu)) = qa ⤜ (λs2::'h. map_spmf fst (rfunc_usr rest2 s2 iu)); ⋀(pa::'a spmf) (qa::'h spmf) (iu::'d) ou::'f × 'b list. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; iu ∈ IU⟧ ⟹ trace_rest_closure rest1 rest2 IA IU p q (cond_spmf_fst (pa ⤜ (λs1::'a. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (qa ⤜ (λs2::'h. rfunc_usr rest2 s2 iu)) ou)⟧ ⟹ thesis::bool›*))\n (*goals:\n 1. ‹trace_rest_closure rest1 rest2 IA IU p q p q›\n 2. ‹⋀pa qa ia. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; ia ∈ IA⟧ ⟹ pa ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = qa ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ia))›\n 3. ‹⋀pa qa ia oa. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; ia ∈ IA⟧ ⟹ trace_rest_closure rest1 rest2 IA IU p q (cond_spmf_fst (pa ⤜ (λs1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (qa ⤜ (λs2. rfunc_adv rest2 s2 ia)) oa)›\n 4. ‹⋀pa qa iu. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; iu ∈ IU⟧ ⟹ pa ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = qa ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 iu))›\n 5. ‹⋀pa qa iu ou. ⟦trace_rest_closure rest1 rest2 IA IU p q pa qa; iu ∈ IU⟧ ⟹ trace_rest_closure rest1 rest2 IA IU p q (cond_spmf_fst (pa ⤜ (λs1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (qa ⤜ (λs2. rfunc_usr rest2 s2 iu)) ou)›\n discuss goal 1*)\n apply ((auto intro: trace_rest_closure_start (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p ?q›*) trace_rest_closure_step[OF assms] (*‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ?ia))› ‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 ?iu))›*) trace_rest_closure_sim (*‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_adv ?rest1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_adv ?rest2.0 s2 ?ia)) ?oa)› ‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_usr ?rest1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_usr ?rest2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 2*)\n apply ((auto intro: trace_rest_closure_start (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p ?q›*) trace_rest_closure_step[OF assms] (*‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ?ia))› ‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 ?iu))›*) trace_rest_closure_sim (*‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_adv ?rest1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_adv ?rest2.0 s2 ?ia)) ?oa)› ‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_usr ?rest1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_usr ?rest2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 3*)\n apply ((auto intro: trace_rest_closure_start (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p ?q›*) trace_rest_closure_step[OF assms] (*‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ?ia))› ‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 ?iu))›*) trace_rest_closure_sim (*‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_adv ?rest1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_adv ?rest2.0 s2 ?ia)) ?oa)› ‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_usr ?rest1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_usr ?rest2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 4*)\n apply ((auto intro: trace_rest_closure_start (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p ?q›*) trace_rest_closure_step[OF assms] (*‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ?ia))› ‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 ?iu))›*) trace_rest_closure_sim (*‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_adv ?rest1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_adv ?rest2.0 s2 ?ia)) ?oa)› ‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_usr ?rest1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_usr ?rest2.0 s2 ?iu)) ?ou)›*))[1])\n (*discuss goal 5*)\n apply ((auto intro: trace_rest_closure_start (*‹trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p ?q›*) trace_rest_closure_step[OF assms] (*‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?ia ∈ IA⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_adv rest1 s1 ?ia)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_adv rest2 s2 ?ia))› ‹⟦trace_rest_closure rest1 rest2 IA IU p q ?p' ?q'; ?iu ∈ IU⟧ ⟹ ?p' ⤜ (λs1. map_spmf fst (rfunc_usr rest1 s1 ?iu)) = ?q' ⤜ (λs2. map_spmf fst (rfunc_usr rest2 s2 ?iu))›*) trace_rest_closure_sim (*‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?ia ∈ ?IA⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_adv ?rest1.0 s1 ?ia)) ?oa) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_adv ?rest2.0 s2 ?ia)) ?oa)› ‹⟦trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ?p' ?q'; ?iu ∈ ?IU⟧ ⟹ trace_rest_closure ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q (cond_spmf_fst (?p' ⤜ (λs1. rfunc_usr ?rest1.0 s1 ?iu)) ?ou) (cond_spmf_fst (?q' ⤜ (λs2. rfunc_usr ?rest2.0 s2 ?iu)) ?ou)›*))[1])\n (*proven 5 subgoals*) .\nqed\n\ndefinition callee_of_core\n :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\n ⇒ ('s_core, 'event + 'iadv_core + 'iusr_core, unit + 'oadv_core + 'ousr_core) oracle'\" where\n \"callee_of_core core =\n map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke core) ⊕⇩O cfunc_adv core ⊕⇩O cfunc_usr core\"\n\nlemma callee_of_core_simps [simp]:\n \"callee_of_core core s (Inl e) = map_spmf (Pair (Inl ())) (cpoke core s e)\"\n \"callee_of_core core s (Inr (Inl iadv_core)) = map_spmf (apfst (Inr ∘ Inl)) (cfunc_adv core s iadv_core)\"\n \"callee_of_core core s (Inr (Inr iusr_core)) = map_spmf (apfst (Inr ∘ Inr)) (cfunc_usr core s iusr_core)\"\n \n (*goals:\n 1. ‹callee_of_core core s (Inl e) = map_spmf (Pair (Inl ())) (cpoke core s e)›\n 2. ‹callee_of_core core s (Inr (Inl iadv_core)) = map_spmf (apfst (Inr ∘ Inl)) (cfunc_adv core s iadv_core)›\n 3. ‹callee_of_core core s (Inr (Inr iusr_core)) = map_spmf (apfst (Inr ∘ Inr)) (cfunc_usr core s iusr_core)›\n discuss goal 1*)\n apply (simp add: callee_of_core_def (*‹callee_of_core (?core::(?'s_core, ?'event, ?'iadv_core, ?'iusr_core, ?'oadv_core, ?'ousr_core) core) = map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke ?core) ⊕⇩O cfunc_adv ?core ⊕⇩O cfunc_usr ?core›*) spmf.map_comp (*‹map_spmf (?f::?'b ⇒ ?'a) (map_spmf (?g::?'c ⇒ ?'b) (?p::?'c spmf)) = map_spmf (?f ∘ ?g) ?p›*) o_def (*‹(?f::?'b ⇒ ?'c) ∘ (?g::?'a ⇒ ?'b) = (λx::?'a. ?f (?g x))›*) apfst_def (*‹apfst (?f::?'a ⇒ ?'c) = map_prod ?f id›*) prod.map_comp (*‹map_prod (?g1.0::?'c ⇒ ?'e) (?g2.0::?'d ⇒ ?'f) (map_prod (?f1.0::?'a ⇒ ?'c) (?f2.0::?'b ⇒ ?'d) (?v::?'a × ?'b)) = map_prod (?g1.0 ∘ ?f1.0) (?g2.0 ∘ ?f2.0) ?v›*) id_def (*‹id = (λx::?'a. x)›*))\n (*discuss goal 2*)\n apply (simp add: callee_of_core_def (*‹callee_of_core (?core::(?'s_core::type, ?'event::type, ?'iadv_core::type, ?'iusr_core::type, ?'oadv_core::type, ?'ousr_core::type) core) = map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke ?core) ⊕⇩O cfunc_adv ?core ⊕⇩O cfunc_usr ?core›*) spmf.map_comp (*‹map_spmf (?f::?'b::type ⇒ ?'a::type) (map_spmf (?g::?'c::type ⇒ ?'b::type) (?p::?'c::type spmf)) = map_spmf (?f ∘ ?g) ?p›*) o_def (*‹(?f::?'b::type ⇒ ?'c::type) ∘ (?g::?'a::type ⇒ ?'b::type) = (λx::?'a::type. ?f (?g x))›*) apfst_def (*‹apfst (?f::?'a::type ⇒ ?'c::type) = map_prod ?f id›*) prod.map_comp (*‹map_prod (?g1.0::?'c::type ⇒ ?'e::type) (?g2.0::?'d::type ⇒ ?'f::type) (map_prod (?f1.0::?'a::type ⇒ ?'c::type) (?f2.0::?'b::type ⇒ ?'d::type) (?v::?'a::type × ?'b::type)) = map_prod (?g1.0 ∘ ?f1.0) (?g2.0 ∘ ?f2.0) ?v›*) id_def (*‹id = (λx::?'a::type. x)›*))\n (*discuss goal 3*)\n apply (simp add: callee_of_core_def (*‹callee_of_core ?core = map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke ?core) ⊕⇩O cfunc_adv ?core ⊕⇩O cfunc_usr ?core›*) spmf.map_comp (*‹map_spmf ?f (map_spmf ?g ?p) = map_spmf (?f ∘ ?g) ?p›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) apfst_def (*‹apfst ?f = map_prod ?f id›*) prod.map_comp (*‹map_prod ?g1.0 ?g2.0 (map_prod ?f1.0 ?f2.0 ?v) = map_prod (?g1.0 ∘ ?f1.0) (?g2.0 ∘ ?f2.0) ?v›*) id_def (*‹id = (λx. x)›*))\n (*proven 3 subgoals*) .\n\nlemma WT_callee_of_core [WT_intro]:\n assumes WT: \"WT_core ℐ_adv ℐ_usr I core\"\n and I: \"I s\"\n shows \"ℐ_full ⊕⇩ℐ (ℐ_adv ⊕⇩ℐ ℐ_usr) ⊢c callee_of_core core s √\"\n apply (rule WT_calleeI (*‹(⋀(call::?'call::type) (ret::?'ret::type) s::?'s::type. ⟦call ∈ outs_ℐ (?ℐ::(?'call::type, ?'ret::type) ℐ); (ret, s) ∈ set_spmf ((?callee::?'call::type ⇒ (?'ret::type × ?'s::type) spmf) call)⟧ ⟹ ret ∈ responses_ℐ ?ℐ call) ⟹ ?ℐ ⊢c ?callee √›*))\n (*goal: ‹ℐ_full ⊕⇩ℐ (ℐ_adv ⊕⇩ℐ ℐ_usr) ⊢c callee_of_core core s √›*)\n subgoal for x and y and s'\n using I (*‹(I::'e ⇒ bool) (s::'e)›*) WT_coreD[OF WT] (*‹⟦?s' ∈ set_spmf (cpoke core ?s ?e); I ?s⟧ ⟹ I ?s'› ‹⟦(?y, ?s') ∈ set_spmf (cfunc_adv core ?s ?x); ?x ∈ outs_ℐ ℐ_adv; I ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv ?x ∧ I ?s'› ‹⟦(?y, ?s') ∈ set_spmf (cfunc_usr core ?s ?x); ?x ∈ outs_ℐ ℐ_usr; I ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr ?x ∧ I ?s'›*) by (auto simp add: callee_of_core_def (*‹callee_of_core (?core::(?'s_core::type, ?'event::type, ?'iadv_core::type, ?'iusr_core::type, ?'oadv_core::type, ?'ousr_core::type) core) = map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke ?core) ⊕⇩O cfunc_adv ?core ⊕⇩O cfunc_usr ?core›*) plus_oracle_def (*‹((?left::?'s::type ⇒ ?'a::type ⇒ (?'b::type × ?'s::type) spmf) ⊕⇩O (?right::?'s::type ⇒ ?'c::type ⇒ (?'d::type × ?'s::type) spmf)) (?s::?'s::type) ≡ λuu::?'a::type + ?'c::type. case uu of Inl (a::?'a::type) ⇒ map_spmf (apfst Inl) (?left ?s a) | Inr (b::?'c::type) ⇒ map_spmf (apfst Inr) (?right ?s b)›*) split!: sum.splits (*‹(?P::?'c::type ⇒ bool) (case ?sum::?'a::type + ?'b::type of Inl (x::?'a::type) ⇒ (?f1.0::?'a::type ⇒ ?'c::type) x | Inr (x::?'b::type) ⇒ (?f2.0::?'b::type ⇒ ?'c::type) x) = ((∀x1::?'a::type. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2::?'b::type. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'c::type ⇒ bool) (case ?sum::?'a::type + ?'b::type of Inl (x::?'a::type) ⇒ (?f1.0::?'a::type ⇒ ?'c::type) x | Inr (x::?'b::type) ⇒ (?f2.0::?'b::type ⇒ ?'c::type) x) = (¬ ((∃x1::?'a::type. ?sum = Inl x1 ∧ ¬ ?P (?f1.0 x1)) ∨ (∃x2::?'b::type. ?sum = Inr x2 ∧ ¬ ?P (?f2.0 x2))))›*)) .\n\nlemma WT_core_callee_invariant_on [WT_intro]:\n assumes WT: \"WT_core ℐ_adv ℐ_usr I core\"\n shows \"callee_invariant_on (callee_of_core core) I (ℐ_full ⊕⇩ℐ (ℐ_adv ⊕⇩ℐ ℐ_usr))\"\n apply unfold_locales\n (*goal: ‹callee_invariant_on (callee_of_core core) I (ℐ_full ⊕⇩ℐ (ℐ_adv ⊕⇩ℐ ℐ_usr))›*)\n subgoal for s and x and y and s'\n by (auto simp add: callee_of_core_def (*‹callee_of_core ?core = map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke ?core) ⊕⇩O cfunc_adv ?core ⊕⇩O cfunc_usr ?core›*) plus_oracle_def (*‹(?left ⊕⇩O ?right) ?s ≡ λuu. case uu of Inl a ⇒ map_spmf (apfst Inl) (?left ?s a) | Inr b ⇒ map_spmf (apfst Inr) (?right ?s b)›*) split!: sum.splits (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = (¬ ((∃x1. ?sum = Inl x1 ∧ ¬ ?P (?f1.0 x1)) ∨ (∃x2. ?sum = Inr x2 ∧ ¬ ?P (?f2.0 x2))))›*) dest: WT_coreD[OF assms] (*‹⟦?s' ∈ set_spmf (cpoke core ?s ?e); I ?s⟧ ⟹ I ?s'› ‹⟦(?y, ?s') ∈ set_spmf (cfunc_adv core ?s ?x); ?x ∈ outs_ℐ ℐ_adv; I ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv ?x ∧ I ?s'› ‹⟦(?y, ?s') ∈ set_spmf (cfunc_usr core ?s ?x); ?x ∈ outs_ℐ ℐ_usr; I ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr ?x ∧ I ?s'›*))\n subgoal for \n by (rule WT_callee_of_core[OF WT] (*‹I ?s ⟹ ℐ_full ⊕⇩ℐ (ℐ_adv ⊕⇩ℐ ℐ_usr) ⊢c callee_of_core core ?s √›*)) .\n\ndefinition callee_of_rest\n :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\n ⇒ ('s_rest, 'iadv_rest + 'iusr_rest, 'oadv_rest × 'event list + 'ousr_rest × 'event list) oracle'\" where\n \"callee_of_rest rest = rfunc_adv rest ⊕⇩O rfunc_usr rest\"\n\nlemma callee_of_rest_simps [simp]:\n \"callee_of_rest rest s (Inl iadv_rest) = map_spmf (apfst Inl) (rfunc_adv rest s iadv_rest)\"\n \"callee_of_rest rest s (Inr iusr_rest) = map_spmf (apfst Inr) (rfunc_usr rest s iusr_rest)\"\n \n (*goals:\n 1. ‹callee_of_rest rest s (Inl iadv_rest) = map_spmf (apfst Inl) (rfunc_adv rest s iadv_rest)›\n 2. ‹callee_of_rest rest s (Inr iusr_rest) = map_spmf (apfst Inr) (rfunc_usr rest s iusr_rest)›\n discuss goal 1*)\n apply (simp add: callee_of_rest_def (*‹callee_of_rest ?rest = rfunc_adv ?rest ⊕⇩O rfunc_usr ?rest›*))\n (*discuss goal 2*)\n apply (simp add: callee_of_rest_def (*‹callee_of_rest ?rest = rfunc_adv ?rest ⊕⇩O rfunc_usr ?rest›*))\n (*proven 2 subgoals*) .\n\nlemma WT_callee_of_rest [WT_intro]:\n assumes WT: \"WT_rest ℐ_adv ℐ_usr I rest\"\n and I: \"I s\"\n shows \"eℐ ℐ_adv ⊕⇩ℐ eℐ ℐ_usr ⊢c callee_of_rest rest s √\"\n apply (rule WT_calleeI (*‹(⋀call ret s. ⟦call ∈ outs_ℐ ?ℐ; (ret, s) ∈ set_spmf (?callee call)⟧ ⟹ ret ∈ responses_ℐ ?ℐ call) ⟹ ?ℐ ⊢c ?callee √›*))\n (*goal: ‹eℐ ℐ_adv ⊕⇩ℐ eℐ ℐ_usr ⊢c callee_of_rest rest s √›*)\n subgoal for x and y and s'\n using I (*‹I s›*) WT_restD[OF WT] (*‹⟦((?y, ?es), ?s') ∈ set_spmf (rfunc_adv rest ?s ?x); ?x ∈ outs_ℐ ℐ_adv; I ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_adv ?x ∧ I ?s'› ‹⟦((?y, ?es), ?s') ∈ set_spmf (rfunc_usr rest ?s ?x); ?x ∈ outs_ℐ ℐ_usr; I ?s⟧ ⟹ ?y ∈ responses_ℐ ℐ_usr ?x ∧ I ?s'› ‹I (rinit rest)›*) by (auto simp add: callee_of_core_def (*‹callee_of_core ?core = map_fun id (map_fun id (map_spmf (Pair ()))) (cpoke ?core) ⊕⇩O cfunc_adv ?core ⊕⇩O cfunc_usr ?core›*) plus_oracle_def (*‹(?left ⊕⇩O ?right) ?s ≡ λuu. case uu of Inl a ⇒ map_spmf (apfst Inl) (?left ?s a) | Inr b ⇒ map_spmf (apfst Inr) (?right ?s b)›*) split!: sum.splits (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = (¬ ((∃x1. ?sum = Inl x1 ∧ ¬ ?P (?f1.0 x1)) ∨ (∃x2. ?sum = Inr x2 ∧ ¬ ?P (?f2.0 x2))))›*)) .\n\nfun fuse_callee \n :: \"('iadv_core + 'iadv_rest) + ('iusr_core + 'iusr_rest) ⇒\n (('oadv_core + 'oadv_rest) + ('ousr_core + 'ousr_rest),\n ('event + 'iadv_core + 'iusr_core) + ('iadv_rest + 'iusr_rest),\n (unit + 'oadv_core + 'ousr_core) + ('oadv_rest × 'event list + 'ousr_rest × 'event list)) gpv\"\n where\n \"fuse_callee (Inl (Inl iadv_core)) = Pause (Inl (Inr (Inl iadv_core))) (λx. case x of\n Inl (Inr (Inl oadv_core)) ⇒ Done (Inl (Inl oadv_core))\n | _ ⇒ Fail)\"\n| \"fuse_callee (Inl (Inr iadv_rest)) = Pause (Inr (Inl iadv_rest)) (λx. case x of\n Inr (Inl (oadv_rest, es)) ⇒ bind_gpv (pauses (map (Inl ∘ Inl) es)) (λ_. Done (Inl (Inr oadv_rest)))\n | _ ⇒ Fail)\"\n| \"fuse_callee (Inr (Inl iusr_core)) = Pause (Inl (Inr (Inr iusr_core))) (λx. case x of\n Inl (Inr (Inr oadv_core)) ⇒ Done (Inr (Inl oadv_core)))\"\n| \"fuse_callee (Inr (Inr iusr_rest)) = Pause (Inr (Inr iusr_rest)) (λx. case x of\n Inr (Inr (ousr_rest, es)) ⇒ bind_gpv (pauses (map (Inl ∘ Inl) es)) (λ_. Done (Inr (Inr ousr_rest))))\"\n\ncase_of_simps fuse_callee_case: fuse_callee.simps\n\n(* parametric_constant fuse_callee_parametric [transfer_rule]: fuse_callee_case *)\n\ndefinition fuse_converter \n :: \"(('iadv_core + 'iadv_rest) + ('iusr_core + 'iusr_rest), \n ('oadv_core + 'oadv_rest) + ('ousr_core + 'ousr_rest),\n ('event + 'iadv_core + 'iusr_core) + ('iadv_rest + 'iusr_rest),\n (unit + 'oadv_core + 'ousr_core) + ('oadv_rest × 'event list + 'ousr_rest × 'event list)) converter\"\n where\n \"fuse_converter = converter_of_callee (stateless_callee fuse_callee) ()\"\n\nlemma fuse_converter:\n \"resource_of_oracle (fused_resource.fuse core rest) (s_core, s_rest) = \n fuse_converter ⊳ (resource_of_oracle (callee_of_core core) s_core ∥ resource_of_oracle (callee_of_rest rest) s_rest)\"\n unfolding fuse_converter_def resource_of_parallel_oracle[symmetric] attach_CNV_RES attach_stateless_callee resource_of_oracle_extend_state_oracle\n (*goal: ‹RES (fused_resource.fuse (core::('i, 'k, 'a, 'c, 'e, 'g) core) (rest::('j, 'k, 'b, 'd, 'f, 'h, 'l) rest_scheme)) (s_core::'i, s_rest::'j) = RES (λ(s::'i × 'j) q::('a + 'b) + 'c + 'd. exec_gpv (callee_of_core core ‡⇩O callee_of_rest rest) (fuse_callee q) s) (s_core, s_rest)›*)\n proof (rule arg_cong2[where f=resource_of_oracle] (*‹⟦?a = ?b; ?c = ?d⟧ ⟹ RES ?a ?c = RES ?b ?d›*); clarsimp simp add: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*))\n (*goal: ‹⋀(a::'i) (b::'j) x::('a + 'b) + 'c + 'd. fused_resource.fuse (core::('i, 'k, 'a, 'c, 'e, 'g) core) (rest::('j, 'k, 'b, 'd, 'f, 'h, 'l) rest_scheme) (a, b) x = exec_gpv (callee_of_core core ‡⇩O callee_of_rest rest) (fuse_callee x) (a, b)›*)\n interpret fused_resource core core_init for core_init .\n have \"foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) xs\n = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) xs)\" for s_core and s_rest and xs\n apply (induction xs arbitrary: s_core)\n (*goals:\n 1. ‹⋀s_core. foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) [] = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) [])›\n 2. ‹⋀a xs s_core. (⋀s_core. foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) xs = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) xs)) ⟹ foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) (a # xs) = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) (a # xs))›\n discuss goal 1*)\n apply (simp add: spmf.map_comp (*‹map_spmf ?f (map_spmf ?g ?p) = map_spmf (?f ∘ ?g) ?p›*) foldl_spmf_Cons' (*‹foldl_spmf ?f (return_spmf ?a) (?x # ?xs) = ?f ?a ?x ⤜ (λa'. foldl_spmf ?f (return_spmf a') ?xs)›*) map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) del: foldl_spmf_Cons (*‹foldl_spmf ?f ?p (?x # ?xs) = foldl_spmf ?f (?p ⤜ (λa. ?f a ?x)) ?xs›*))\n (*discuss goal 2*)\n apply (simp add: spmf.map_comp (*‹map_spmf (?f::?'b ⇒ ?'a) (map_spmf (?g::?'c ⇒ ?'b) (?p::?'c spmf)) = map_spmf (?f ∘ ?g) ?p›*) foldl_spmf_Cons' (*‹foldl_spmf (?f::?'a ⇒ ?'b ⇒ ?'a spmf) (return_spmf (?a::?'a)) ((?x::?'b) # (?xs::?'b list)) = ?f ?a ?x ⤜ (λa'::?'a. foldl_spmf ?f (return_spmf a') ?xs)›*) map_bind_spmf (*‹map_spmf (?f::?'b ⇒ ?'a) ((?p::?'c spmf) ⤜ (?g::?'c ⇒ ?'b spmf)) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf (?f::?'c ⇒ ?'b) (?p::?'c spmf) ⤜ (?g::?'b ⇒ ?'a spmf) = ?p ⤜ ?g ∘ ?f›*) o_def (*‹(?f::?'b ⇒ ?'c) ∘ (?g::?'a ⇒ ?'b) = (λx::?'a. ?f (?g x))›*) del: foldl_spmf_Cons (*‹foldl_spmf (?f::?'b ⇒ ?'a ⇒ ?'b spmf) (?p::?'b spmf) ((?x::?'a) # (?xs::?'a list)) = foldl_spmf ?f (?p ⤜ (λa::?'b. ?f a ?x)) ?xs›*))\n (*proven 2 subgoals*) .\n then show \"fuse rest (s_core, s_rest) q = exec_gpv (callee_of_core core ‡⇩O callee_of_rest rest) (fuse_callee q) (s_core, s_rest)\" for s_core and s_rest and q\n apply (cases q rule: fuse_callee.cases (*‹⟦⋀iadv_core. ?x = Inl (Inl iadv_core) ⟹ ?P; ⋀iadv_rest. ?x = Inl (Inr iadv_rest) ⟹ ?P; ⋀iusr_core. ?x = Inr (Inl iusr_core) ⟹ ?P; ⋀iusr_rest. ?x = Inr (Inr iusr_rest) ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀iadv_core. ⟦⋀s_core s_rest xs. foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) xs = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) xs); q = Inl (Inl iadv_core)⟧ ⟹ fuse rest (s_core, s_rest) q = exec_gpv (callee_of_core core ‡⇩O callee_of_rest rest) (fuse_callee q) (s_core, s_rest)›\n 2. ‹⋀iadv_rest. ⟦⋀s_core s_rest xs. foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) xs = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) xs); q = Inl (Inr iadv_rest)⟧ ⟹ fuse rest (s_core, s_rest) q = exec_gpv (callee_of_core core ‡⇩O callee_of_rest rest) (fuse_callee q) (s_core, s_rest)›\n 3. ‹⋀iusr_core. ⟦⋀s_core s_rest xs. foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) xs = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) xs); q = Inr (Inl iusr_core)⟧ ⟹ fuse rest (s_core, s_rest) q = exec_gpv (callee_of_core core ‡⇩O callee_of_rest rest) (fuse_callee q) (s_core, s_rest)›\n 4. ‹⋀iusr_rest. ⟦⋀s_core s_rest xs. foldl_spmf (map_fun id (map_fun (Inl ∘ Inl) id) (map_fun id (map_fun id (map_spmf snd)) (callee_of_core core ‡⇩O callee_of_rest rest))) (return_spmf (s_core, s_rest)) xs = map_spmf (λs_core. (s_core, s_rest)) (foldl_spmf (cpoke core) (return_spmf s_core) xs); q = Inr (Inr iusr_rest)⟧ ⟹ fuse rest (s_core, s_rest) q = exec_gpv (callee_of_core core ‡⇩O callee_of_rest rest) (fuse_callee q) (s_core, s_rest)›\n discuss goal 1*)\n apply (clarsimp simp add: map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) exec_gpv_bind (*‹exec_gpv ?callee (?c ⤜ ?f) ?s = exec_gpv ?callee ?c ?s ⤜ (λxa. case xa of (x, s') ⇒ exec_gpv ?callee (?f x) s')›*) exec_gpv_pauses (*‹exec_gpv ?callee (pauses ?xs) ?s = map_spmf (Pair ()) (foldl_spmf (map_fun id (map_fun id (map_spmf snd)) ?callee) (return_spmf ?s) ?xs)›*) intro!: bind_spmf_cong[OF refl] (*‹(⋀x. x ∈ set_spmf ?p ⟹ ?f x = ?g x) ⟹ ?p ⤜ ?f = ?p ⤜ ?g›*))\n (*discuss goal 2*)\n apply (clarsimp simp add: map_bind_spmf (*‹map_spmf (?f::?'b::type ⇒ ?'a::type) ((?p::?'c::type spmf) ⤜ (?g::?'c::type ⇒ ?'b::type spmf)) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf (?f::?'c::type ⇒ ?'b::type) (?p::?'c::type spmf) ⤜ (?g::?'b::type ⇒ ?'a::type spmf) = ?p ⤜ ?g ∘ ?f›*) exec_gpv_bind (*‹exec_gpv (?callee::?'b::type ⇒ ?'c::type ⇒ (?'d::type × ?'b::type) spmf) ((?c::(?'e::type, ?'c::type, ?'d::type) gpv) ⤜ (?f::?'e::type ⇒ (?'a::type, ?'c::type, ?'d::type) gpv)) (?s::?'b::type) = exec_gpv ?callee ?c ?s ⤜ (λxa::?'e::type × ?'b::type. case xa of (x::?'e::type, s'::?'b::type) ⇒ exec_gpv ?callee (?f x) s')›*) exec_gpv_pauses (*‹exec_gpv (?callee::?'a::type ⇒ ?'b::type ⇒ (?'c::type × ?'a::type) spmf) (pauses (?xs::?'b::type list)) (?s::?'a::type) = map_spmf (Pair ()) (foldl_spmf (map_fun id (map_fun id (map_spmf snd)) ?callee) (return_spmf ?s) ?xs)›*) intro!: bind_spmf_cong[OF refl] (*‹(⋀x::?'a::type. x ∈ set_spmf (?p::?'a::type spmf) ⟹ (?f::?'a::type ⇒ ?'b::type spmf) x = (?g::?'a::type ⇒ ?'b::type spmf) x) ⟹ ?p ⤜ ?f = ?p ⤜ ?g›*))\n (*discuss goal 3*)\n apply (clarsimp simp add: map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) exec_gpv_bind (*‹exec_gpv ?callee (?c ⤜ ?f) ?s = exec_gpv ?callee ?c ?s ⤜ (λxa. case xa of (x, s') ⇒ exec_gpv ?callee (?f x) s')›*) exec_gpv_pauses (*‹exec_gpv ?callee (pauses ?xs) ?s = map_spmf (Pair ()) (foldl_spmf (map_fun id (map_fun id (map_spmf snd)) ?callee) (return_spmf ?s) ?xs)›*) intro!: bind_spmf_cong[OF refl] (*‹(⋀x. x ∈ set_spmf ?p ⟹ ?f x = ?g x) ⟹ ?p ⤜ ?f = ?p ⤜ ?g›*))\n (*discuss goal 4*)\n apply (clarsimp simp add: map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) exec_gpv_bind (*‹exec_gpv ?callee (?c ⤜ ?f) ?s = exec_gpv ?callee ?c ?s ⤜ (λxa. case xa of (x, s') ⇒ exec_gpv ?callee (?f x) s')›*) exec_gpv_pauses (*‹exec_gpv ?callee (pauses ?xs) ?s = map_spmf (Pair ()) (foldl_spmf (map_fun id (map_fun id (map_spmf snd)) ?callee) (return_spmf ?s) ?xs)›*) intro!: bind_spmf_cong[OF refl] (*‹(⋀x. x ∈ set_spmf ?p ⟹ ?f x = ?g x) ⟹ ?p ⤜ ?f = ?p ⤜ ?g›*))\n (*proven 4 subgoals*) .\n qed\n\nlemma trace_eq_callee_of_coreI:\n \"trace_callee_eq (callee_of_core core1) (callee_of_core core2) (E <+> IA <+> IU) p q\"\n if \"trace_core_eq core1 core2 E IA IU p q\"\nproof -\n from that obtain S_core \n where core_start: \"S_core p q\"\n and step_cpoke: \"⋀p q e. S_core p q ⟹ e ∈ E\n ⟹ weight_spmf (bind_spmf p (λs. cpoke core1 s e)) = weight_spmf (bind_spmf q (λs. cpoke core2 s e))\"\n and sim_cpoke: \"⋀p q e. S_core p q ⟹ e ∈ E \n ⟹ S_core (mk_lossless (bind_spmf p (λs. cpoke core1 s e))) (mk_lossless (bind_spmf q (λs. cpoke core2 s e)))\"\n and step_cfunc_adv: \"⋀p q ia. ⟦ S_core p q; ia ∈ IA ⟧\n ⟹ bind_spmf p (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = bind_spmf q (λs2. map_spmf fst (cfunc_adv core2 s2 ia))\"\n and sim_cfunc_adv: \"⋀p q ia oa. ⟦ S_core p q; ia ∈ IA ⟧ ⟹\n S_core (cond_spmf_fst (bind_spmf p (λs1. cfunc_adv core1 s1 ia)) oa)\n (cond_spmf_fst (bind_spmf q (λs2. cfunc_adv core2 s2 ia)) oa)\"\n and step_cfunc_usr: \"⋀p q iu. ⟦ S_core p q; iu ∈ IU ⟧\n ⟹ bind_spmf p (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = bind_spmf q (λs2. map_spmf fst (cfunc_usr core2 s2 iu))\"\n and sim_cfunc_usr: \"⋀p q iu ou. ⟦ S_core p q; iu ∈ IU ⟧ ⟹\n S_core (cond_spmf_fst (bind_spmf p (λs1. cfunc_usr core1 s1 iu)) ou)\n (cond_spmf_fst (bind_spmf q (λs2. cfunc_usr core2 s2 iu)) ou)\"\n by(rule trace_core_eq_complete) blast\n\n show ?thesis using core_start\n proof(coinduct rule: trace'_eqI_sim[consumes 1, case_names step sim])\n case (step p q a)\n then consider (cpoke) e where \"a = Inl e\" \"e ∈ E\"\n | (cfunc_adv) ia where \"a = Inr (Inl ia)\" \"ia ∈ IA\"\n | (cfunc_usr) iu where \"a = Inr (Inr iu)\" \"iu ∈ IU\" by auto\n then show ?case\n proof cases\n case cpoke\n with step_cpoke[OF step(1), of e] show ?thesis \n by(simp add: spmf.map_comp o_def map_spmf_const weight_bind_spmf)\n (auto intro!: spmf_eqI simp add: spmf_bind spmf_scale_spmf max_def min_absorb2 weight_spmf_le_1)\n next\n case cfunc_adv\n with step_cfunc_adv[OF step(1) cfunc_adv(2)] show ?thesis\n by(simp add: spmf.map_comp)(simp add: spmf.map_comp[symmetric] map_bind_spmf[unfolded o_def, symmetric])\n next\n case cfunc_usr\n with step_cfunc_usr[OF step(1) cfunc_usr(2)] show ?thesis\n by(simp add: spmf.map_comp)(simp add: spmf.map_comp[symmetric] map_bind_spmf[unfolded o_def, symmetric])\n qed \n next\n case (sim p q a res b s')\n then consider (cpoke) e where \"a = Inl e\" \"e ∈ E\"\n | (cfunc_adv) ia where \"a = Inr (Inl ia)\" \"ia ∈ IA\"\n | (cfunc_usr) iu where \"a = Inr (Inr iu)\" \"iu ∈ IU\" by auto\n then show ?case\n proof cases\n case cpoke\n with sim_cpoke[OF sim(1) , of e] sim show ?thesis \n by(clarsimp simp add: map_bind_spmf[unfolded o_def, symmetric])\n next\n case cfunc_adv\n with sim_cfunc_adv[OF sim(1) cfunc_adv(2)] sim show ?thesis\n apply(clarsimp simp add: map_bind_spmf[unfolded o_def, symmetric] apfst_def map_prod_def)\n apply(subst (1 2) cond_spmf_fst_map_prod_inj)\n apply(simp_all add: o_def[symmetric] inj_compose del: o_apply)\n done\n next\n case cfunc_usr\n with sim_cfunc_usr[OF sim(1) cfunc_usr(2)] sim show ?thesis\n apply(clarsimp simp add: map_bind_spmf[unfolded o_def, symmetric] apfst_def map_prod_def)\n apply(subst (1 2) cond_spmf_fst_map_prod_inj)\n apply(simp_all add: o_def[symmetric] inj_compose del: o_apply)\n done\n qed\n qed\nqed\n\nlemma trace_eq_callee_of_restI:\n \"trace_callee_eq (callee_of_rest rest1) (callee_of_rest rest2) (IA <+> IU) p q\"\n if \"trace_rest_eq rest1 rest2 IA IU p q\"\nproof -\n from that obtain S_rest \n where rest_start: \"S_rest p q\"\n and step_rfunc_adv: \"⋀p q ia. ⟦ S_rest p q; ia ∈ IA ⟧\n ⟹ bind_spmf p (λs1. map_spmf fst (rfunc_adv rest1 s1 ia)) = bind_spmf q (λs2. map_spmf fst (rfunc_adv rest2 s2 ia))\"\n and sim_rfunc_adv: \"⋀p q ia oa. ⟦ S_rest p q; ia ∈ IA ⟧ ⟹\n S_rest (cond_spmf_fst (bind_spmf p (λs1. rfunc_adv rest1 s1 ia)) oa)\n (cond_spmf_fst (bind_spmf q (λs2. rfunc_adv rest2 s2 ia)) oa)\"\n and step_rfunc_usr: \"⋀p q iu. ⟦ S_rest p q; iu ∈ IU ⟧\n ⟹ bind_spmf p (λs1. map_spmf fst (rfunc_usr rest1 s1 iu)) = bind_spmf q (λs2. map_spmf fst (rfunc_usr rest2 s2 iu))\"\n and sim_rfunc_usr: \"⋀p q iu ou. ⟦ S_rest p q; iu ∈ IU ⟧ ⟹\n S_rest (cond_spmf_fst (bind_spmf p (λs1. rfunc_usr rest1 s1 iu)) ou)\n (cond_spmf_fst (bind_spmf q (λs2. rfunc_usr rest2 s2 iu)) ou)\"\n by(rule trace_rest_eq_complete) blast\n\n show ?thesis using rest_start\n proof(coinduct rule: trace'_eqI_sim[consumes 1, case_names step sim])\n case (step p q a)\n then consider (rfunc_adv) ia where \"a = Inl ia\" \"ia ∈ IA\"\n | (rfunc_usr) iu where \"a = Inr iu\" \"iu ∈ IU\" by auto\n then show ?case\n proof cases\n case rfunc_adv\n with step_rfunc_adv[OF step(1) rfunc_adv(2)] show ?thesis\n by(simp add: spmf.map_comp)(simp add: spmf.map_comp[symmetric] map_bind_spmf[unfolded o_def, symmetric])\n next\n case rfunc_usr\n with step_rfunc_usr[OF step(1) rfunc_usr(2)] show ?thesis\n by(simp add: spmf.map_comp)(simp add: spmf.map_comp[symmetric] map_bind_spmf[unfolded o_def, symmetric])\n qed \n next\n case (sim p q a res b s')\n then consider (rfunc_adv) ia where \"a = Inl ia\" \"ia ∈ IA\"\n | (rfunc_usr) iu where \"a = Inr iu\" \"iu ∈ IU\" by auto\n then show ?case\n proof cases\n case rfunc_adv\n with sim_rfunc_adv[OF sim(1) rfunc_adv(2)] sim show ?thesis\n by(clarsimp simp add: map_bind_spmf[unfolded o_def, symmetric] apfst_def map_prod_def)\n (subst (1 2) cond_spmf_fst_map_prod_inj; simp)\n next\n case rfunc_usr\n with sim_rfunc_usr[OF sim(1) rfunc_usr(2)] sim show ?thesis\n by(clarsimp simp add: map_bind_spmf[unfolded o_def, symmetric] apfst_def map_prod_def)\n (subst (1 2) cond_spmf_fst_map_prod_inj; simp)\n qed\n qed\nqed\n\nlemma trace_callee_resource_of_oracle:\n \"trace_callee run_resource (map_spmf (resource_of_oracle callee) p) = trace_callee callee p\"\n (is \"?lhs = ?rhs\")\nproof (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹⋀x xa. trace_callee run_resource (map_spmf (RES callee) p) x xa = trace_callee callee p x xa›*)\n show \"?lhs tr x = ?rhs tr x\" for tr and x\n proof (induction tr arbitrary: p)\n (*goals:\n 1. ‹⋀p::'c spmf. trace_callee run_resource (map_spmf (RES (callee::'c ⇒ 'a ⇒ ('b × 'c) spmf)) p) [] (x::'a) = trace_callee callee p [] x›\n 2. ‹⋀(a::'a × 'b) (tr::('a × 'b) list) p::'c spmf. (⋀p::'c spmf. trace_callee run_resource (map_spmf (RES (callee::'c ⇒ 'a ⇒ ('b × 'c) spmf)) p) tr (x::'a) = trace_callee callee p tr x) ⟹ trace_callee run_resource (map_spmf (RES callee) p) (a # tr) x = trace_callee callee p (a # tr) x›*)\n case Nil (*no hyothesis introduced yet*)\n show \"?case\"\n (*goal: ‹trace_callee run_resource (map_spmf (RES callee) p) [] x = trace_callee callee p [] x›*)\n by (simp add: bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) spmf.map_comp (*‹map_spmf ?f (map_spmf ?g ?p) = map_spmf (?f ∘ ?g) ?p›*))\n next\n (*goal: ‹⋀a tr p. (⋀p. trace_callee run_resource (map_spmf (RES callee) p) tr x = trace_callee callee p tr x) ⟹ trace_callee run_resource (map_spmf (RES callee) p) (a # tr) x = trace_callee callee p (a # tr) x›*)\n case (Cons a tr) (*‹trace_callee run_resource (map_spmf (RES callee) ?p) tr x = trace_callee callee ?p tr x›*)\n obtain y and z where a[simp]: \"a = (y, z)\"\n (*goal: ‹(⋀y z. a = (y, z) ⟹ thesis) ⟹ thesis›*)\n by (cases a)\n have \"trace_callee run_resource (map_spmf (RES callee) p) (a # tr) x =\n trace_callee run_resource (cond_spmf_fst (map_spmf (λ(x, y). (x, RES callee y)) (p ⤜ (λx. (callee x y)))) z) tr x\"\n by (clarsimp simp add: bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) map_prod_def (*‹map_prod ?f ?g = (λ(x, y). (?f x, ?g y))›*) map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*))\n also (*calculation: ‹trace_callee run_resource (map_spmf (RES (callee::'c ⇒ 'a ⇒ ('b × 'c) spmf)) (p::'c spmf)) ((a::'a × 'b) # (tr::('a × 'b) list)) (x::'a) = trace_callee run_resource (cond_spmf_fst (map_spmf (λ(x::'b, y::'c). (x, RES callee y)) (p ⤜ (λx::'c. callee x (y::'a)))) (z::'b)) tr x›*) have \"… = trace_callee run_resource (map_spmf (RES callee) (cond_spmf_fst (p ⤜ (λx. (callee x y))) z)) tr x\"\n apply (subst cond_spmf_fst_map_prod_inj (*‹inj (?f::?'a ⇒ ?'b) ⟹ cond_spmf_fst (map_spmf (λ(x::?'a, y::?'d). (?f x, (?g::?'a ⇒ ?'d ⇒ ?'c) x y)) (?p::(?'a × ?'d) spmf)) (?f (?x::?'a)) = map_spmf (?g ?x) (cond_spmf_fst ?p ?x)›*))\n (*goals:\n 1. ‹inj (λx::'b. x)›\n 2. ‹trace_callee run_resource (map_spmf (RES (callee::'c ⇒ 'a ⇒ ('b × 'c) spmf)) (cond_spmf_fst ((p::'c spmf) ⤜ (λx::'c. callee x (y::'a))) (z::'b))) (tr::('a × 'b) list) (x::'a) = trace_callee run_resource (map_spmf (RES callee) (cond_spmf_fst (p ⤜ (λx::'c. callee x y)) z)) tr x›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n finally (*calculation: ‹trace_callee run_resource (map_spmf (RES (callee::'c::type ⇒ 'a::type ⇒ ('b::type × 'c::type) spmf)) (p::'c::type spmf)) ((a::'a::type × 'b::type) # (tr::('a::type × 'b::type) list)) (x::'a::type) = trace_callee run_resource (map_spmf (RES callee) (cond_spmf_fst (p ⤜ (λx::'c::type. callee x (y::'a::type))) (z::'b::type))) tr x›*) show \"?case\"\n (*goal: ‹trace_callee run_resource (map_spmf (RES callee) p) (a # tr) x = trace_callee callee p (a # tr) x›*)\n using Cons.IH (*‹trace_callee run_resource (map_spmf (RES callee) ?p) tr x = trace_callee callee ?p tr x›*) by simp\n qed\nqed\n\nlemma trace_callee_resource_of_oracle':\n \"trace_callee run_resource (return_spmf (resource_of_oracle callee s)) = trace_callee callee (return_spmf s)\"\n using trace_callee_resource_of_oracle[where p = \"return_spmf s\"] (*‹trace_callee run_resource (map_spmf (RES ?callee) (return_spmf s)) = trace_callee ?callee (return_spmf s)›*) by simp\n\nlemma trace_eq_resource_of_oracle:\n \"trace_eq A (map_spmf (resource_of_oracle callee1) p) (map_spmf (resource_of_oracle callee2) q) =\n trace_callee_eq callee1 callee2 A p q\"\n unfolding trace_callee_eq_def trace_callee_resource_of_oracle\n (*goal: ‹(∀xs. set xs ⊆ A × UNIV ⟶ (∀x∈A. trace_callee callee1 p xs x = trace_callee callee2 q xs x)) = (∀xs. set xs ⊆ A × UNIV ⟶ (∀x∈A. trace_callee callee1 p xs x = trace_callee callee2 q xs x))›*)\n by (rule refl (*‹?t = ?t›*))\n\nlemma WT_fuse_converter [WT_intro]:\n \"(ℐAC ⊕⇩ℐ map_ℐ id fst ℐAR) ⊕⇩ℐ (ℐUC ⊕⇩ℐ map_ℐ id fst ℐUR), (ℐE ⊕⇩ℐ (ℐAC ⊕⇩ℐ ℐUC)) ⊕⇩ℐ (ℐAR ⊕⇩ℐ ℐUR) ⊢⇩C fuse_converter √\"\n if \"∀x. ∀(y, es)∈responses_ℐ ℐAR x. set es ⊆ outs_ℐ ℐE\" \"∀x. ∀(y, es)∈responses_ℐ ℐUR x. set es ⊆ outs_ℐ ℐE\"\n unfolding fuse_converter_def\n (*goal: ‹(ℐAC ⊕⇩ℐ map_ℐ id fst ℐAR) ⊕⇩ℐ (ℐUC ⊕⇩ℐ map_ℐ id fst ℐUR), (ℐE ⊕⇩ℐ (ℐAC ⊕⇩ℐ ℐUC)) ⊕⇩ℐ (ℐAR ⊕⇩ℐ ℐUR) ⊢⇩C CNV (stateless_callee fuse_callee) () √›*)\n using that (*‹∀x. ∀(y, es)∈responses_ℐ ℐAR x. set es ⊆ outs_ℐ ℐE› ‹∀x. ∀(y, es)∈responses_ℐ ℐUR x. set es ⊆ outs_ℐ ℐE›*) apply (intro WT_converter_of_callee (*‹⟦⋀s q. q ∈ outs_ℐ ?ℐ ⟹ ?ℐ' ⊢g ?callee s q √; ⋀s q r s'. ⟦q ∈ outs_ℐ ?ℐ; (r, s') ∈ results_gpv ?ℐ' (?callee s q)⟧ ⟹ r ∈ responses_ℐ ?ℐ q⟧ ⟹ ?ℐ, ?ℐ' ⊢⇩C CNV ?callee ?s √›*))\n (*goals:\n 1. ‹⋀s q. ⟦∀x. ∀(y, es)∈responses_ℐ ℐAR x. set es ⊆ outs_ℐ ℐE; ∀x. ∀(y, es)∈responses_ℐ ℐUR x. set es ⊆ outs_ℐ ℐE; q ∈ outs_ℐ ((ℐAC ⊕⇩ℐ map_ℐ id fst ℐAR) ⊕⇩ℐ (ℐUC ⊕⇩ℐ map_ℐ id fst ℐUR))⟧ ⟹ (ℐE ⊕⇩ℐ (ℐAC ⊕⇩ℐ ℐUC)) ⊕⇩ℐ (ℐAR ⊕⇩ℐ ℐUR) ⊢g stateless_callee fuse_callee s q √›\n 2. ‹⋀s q r s'. ⟦∀x. ∀(y, es)∈responses_ℐ ℐAR x. set es ⊆ outs_ℐ ℐE; ∀x. ∀(y, es)∈responses_ℐ ℐUR x. set es ⊆ outs_ℐ ℐE; q ∈ outs_ℐ ((ℐAC ⊕⇩ℐ map_ℐ id fst ℐAR) ⊕⇩ℐ (ℐUC ⊕⇩ℐ map_ℐ id fst ℐUR)); (r, s') ∈ results_gpv ((ℐE ⊕⇩ℐ (ℐAC ⊕⇩ℐ ℐUC)) ⊕⇩ℐ (ℐAR ⊕⇩ℐ ℐUR)) (stateless_callee fuse_callee s q)⟧ ⟹ r ∈ responses_ℐ ((ℐAC ⊕⇩ℐ map_ℐ id fst ℐAR) ⊕⇩ℐ (ℐUC ⊕⇩ℐ map_ℐ id fst ℐUR)) q›\n discuss goal 1*)\n apply (fastforce simp add: stateless_callee_def (*‹stateless_callee ?callee ?s = map_gpv (λb. (b, ?s)) id ∘ ?callee›*) image_image (*‹?f ` ?g ` ?A = (λx. ?f (?g x)) ` ?A›*) intro: rev_image_eqI (*‹⟦?x ∈ ?A; ?b = ?f ?x⟧ ⟹ ?b ∈ ?f ` ?A›*) intro!: WT_gpv_pauses (*‹set ?xs ⊆ outs_ℐ ?ℐ ⟹ ?ℐ ⊢g pauses ?xs √›*) split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 2*)\n apply (fastforce simp add: stateless_callee_def (*‹stateless_callee ?callee ?s = map_gpv (λb. (b, ?s)) id ∘ ?callee›*) image_image (*‹?f ` ?g ` ?A = (λx. ?f (?g x)) ` ?A›*) intro: rev_image_eqI (*‹⟦?x ∈ ?A; ?b = ?f ?x⟧ ⟹ ?b ∈ ?f ` ?A›*) intro!: WT_gpv_pauses (*‹set ?xs ⊆ outs_ℐ ?ℐ ⟹ ?ℐ ⊢g pauses ?xs √›*) split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*proven 2 subgoals*) .\n\ntheorem fuse_trace_eq:\n fixes core1 :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and core2 :: \"('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and rest1 :: \"('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more1) rest_scheme\"\n and rest2 :: \"('s_rest', 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more2) rest_scheme\"\n assumes core: \"trace_core_eq core1 core2 (outs_ℐ ℐE) (outs_ℐ ℐCA) (outs_ℐ ℐCU) (return_spmf s_core) (return_spmf s_core')\"\n and rest: \"trace_rest_eq rest1 rest2 (outs_ℐ ℐRA) (outs_ℐ ℐRU) (return_spmf s_rest) (return_spmf s_rest')\"\n and IC1: \"callee_invariant_on (callee_of_core core1) IC1 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))\" \"IC1 s_core\"\n and IC2: \"callee_invariant_on (callee_of_core core2) IC2 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))\" \"IC2 s_core'\"\n and IR1: \"callee_invariant_on (callee_of_rest rest1) IR1 (ℐRA ⊕⇩ℐ ℐRU)\" \"IR1 s_rest\"\n and IR2: \"callee_invariant_on (callee_of_rest rest2) IR2 (ℐRA ⊕⇩ℐ ℐRU)\" \"IR2 s_rest'\"\n and E1 [WT_intro]: \"∀x. ∀(y, es)∈responses_ℐ ℐRA x. set es ⊆ outs_ℐ ℐE\"\n and E2 [WT_intro]: \"∀x. ∀(y, es)∈responses_ℐ ℐRU x. set es ⊆ outs_ℐ ℐE\"\n shows \"trace_callee_eq (fused_resource.fuse core1 rest1) (fused_resource.fuse core2 rest2)\n ((outs_ℐ ℐCA <+> outs_ℐ ℐRA) <+> (outs_ℐ ℐCU <+> outs_ℐ ℐRU)) (return_spmf (s_core, s_rest)) (return_spmf (s_core', s_rest'))\"\nproof (-)\n (*goal: ‹((outs_ℐ (ℐCA::('iadv_core, 'oadv_core) ℐ) <+> outs_ℐ (ℐRA::('iadv_rest, 'oadv_rest × 'event list) ℐ)) <+> outs_ℐ (ℐCU::('iusr_core, 'ousr_core) ℐ) <+> outs_ℐ (ℐRU::('iusr_rest, 'ousr_rest × 'event list) ℐ)) ⊢⇩C fused_resource.fuse (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) (rest1::('s_rest, 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more1) rest_scheme)((s_core::'s_core, s_rest::'s_rest)) ≈ fused_resource.fuse (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) (rest2::('s_rest', 'event, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more2) rest_scheme)((s_core'::'s_core', s_rest'::'s_rest'))›*)\n let ?ℐC = \"ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU)\"\n let ?ℐR = \"ℐRA ⊕⇩ℐ ℐRU\"\n let ?ℐ' = \"?ℐC ⊕⇩ℐ ?ℐR\"\n let ?ℐ = \"(ℐCA ⊕⇩ℐ map_ℐ id fst ℐRA) ⊕⇩ℐ (ℐCU ⊕⇩ℐ map_ℐ id fst ℐRU)\"\n interpret fuse1: fused_resource core1 s1 for s1 .\n interpret fuse2: fused_resource core2 s2 for s2 .\n interpret IC1: callee_invariant_on \"callee_of_core core1\" IC1 \"?ℐC\"\n by fact\n interpret IC2: callee_invariant_on \"callee_of_core core2\" IC2 \"?ℐC\"\n by fact\n interpret IR1: callee_invariant_on \"callee_of_rest rest1\" IR1 \"?ℐR\"\n by fact\n interpret IR2: callee_invariant_on \"callee_of_rest rest2\" IR2 \"?ℐR\"\n by fact\n from core (*‹trace_core_eq core1 core2 (outs_ℐ ℐE) (outs_ℐ ℐCA) (outs_ℐ ℐCU) (return_spmf s_core) (return_spmf s_core')›*) have \"outs_ℐ ?ℐC ⊢⇩C callee_of_core core1(s_core) ≈ callee_of_core core2(s_core')\"\n by (simp add: trace_eq_callee_of_coreI (*‹trace_core_eq ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q ⟹ trace_callee_eq (callee_of_core ?core1.0) (callee_of_core ?core2.0) (?E <+> ?IA <+> ?IU) ?p ?q›*))\n hence \"outs_ℐ ?ℐC ⊢⇩R RES (callee_of_core core1) s_core ≈ RES (callee_of_core core2) s_core'\"\n by simp\n moreover have \"outs_ℐ ?ℐR ⊢⇩C callee_of_rest rest1(s_rest) ≈ callee_of_rest rest2(s_rest')\"\n using rest (*‹trace_rest_eq rest1 rest2 (outs_ℐ ℐRA) (outs_ℐ ℐRU) (return_spmf s_rest) (return_spmf s_rest')›*) by (simp add: trace_eq_callee_of_restI (*‹trace_rest_eq ?rest1.0 ?rest2.0 ?IA ?IU ?p ?q ⟹ trace_callee_eq (callee_of_rest ?rest1.0) (callee_of_rest ?rest2.0) (?IA <+> ?IU) ?p ?q›*))\n hence \"outs_ℐ ?ℐR ⊢⇩R RES (callee_of_rest rest1) s_rest ≈ RES (callee_of_rest rest2) s_rest'\"\n by simp\n ultimately have \"outs_ℐ ?ℐ' ⊢⇩R\n RES (callee_of_core core1) s_core ∥ RES (callee_of_rest rest1) s_rest ≈\n RES (callee_of_core core2) s_core' ∥ RES (callee_of_rest rest2) s_rest'\"\n by (simp add: trace_eq'_parallel_resource (*‹⟦?A ⊢⇩R ?res1.0 ≈ ?res1'; ?B ⊢⇩R ?res2.0 ≈ ?res2'⟧ ⟹ (?A <+> ?B) ⊢⇩R ?res1.0 ∥ ?res2.0 ≈ ?res1' ∥ ?res2'›*))\n hence \"outs_ℐ ?ℐ ⊢⇩R fuse_converter ⊳ (RES (callee_of_core core1) s_core ∥ RES (callee_of_rest rest1) s_rest) ≈\n fuse_converter ⊳ (RES (callee_of_core core2) s_core' ∥ RES (callee_of_rest rest2) s_rest')\"\n apply (rule attach_trace_eq' (*‹⟦outs_ℐ (?ℐ::(?'a, ?'b) ℐ) ⊢⇩R (?res1.0::(?'a, ?'b) resource) ≈ (?res2.0::(?'a, ?'b) resource); ?ℐ ⊢res ?res1.0 √; ?ℐ ⊢res ?res2.0 √; (?ℐ'::(?'c, ?'d) ℐ), ?ℐ ⊢⇩C ?conv::(?'c, ?'d, ?'a, ?'b) converter √⟧ ⟹ outs_ℐ ?ℐ' ⊢⇩R ?conv ⊳ ?res1.0 ≈ ?conv ⊳ ?res2.0›*))\n (*goals:\n 1. ‹(ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU)) ⊕⇩ℐ (ℐRA ⊕⇩ℐ ℐRU) ⊢res RES (callee_of_core core1) s_core ∥ RES (callee_of_rest rest1) s_rest √›\n 2. ‹(ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU)) ⊕⇩ℐ (ℐRA ⊕⇩ℐ ℐRU) ⊢res RES (callee_of_core core2) s_core' ∥ RES (callee_of_rest rest2) s_rest' √›\n 3. ‹(ℐCA ⊕⇩ℐ map_ℐ id fst ℐRA) ⊕⇩ℐ (ℐCU ⊕⇩ℐ map_ℐ id fst ℐRU), (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU)) ⊕⇩ℐ (ℐRA ⊕⇩ℐ ℐRU) ⊢⇩C fuse_converter √›\n discuss goal 1*)\n apply (intro WT_intro (*‹map_ℐ ?g ?h ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv' ?f ?g ?h ?gpv √› ‹map_ℐ ?g id ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv ?f ?g ?gpv √› ‹⟦?ℐ1.0 ⊢res ?res1.0 √; ?ℐ2.0 ⊢res ?res2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0 ⊢res ?res1.0 ∥ ?res2.0 √› ‹⟦map_ℐ (inv ?f) (inv ?g) ?ℐ, map_ℐ ?f' ?g' ?ℐ' ⊢⇩C ?conv √; inj ?f; surj ?g⟧ ⟹ ?ℐ, ?ℐ' ⊢⇩C map_converter ?f ?g ?f' ?g' ?conv √› ‹?ℐ, ?ℐ ⊢⇩C 1⇩C √› ‹?ℐ, ?ℐ' ⊢⇩C ⊥⇩C √› ‹⟦?ℐ1.0, ?ℐ2.0 ⊢⇩C ?conv1.0 √; ?ℐ1', ?ℐ2' ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ1', ?ℐ2.0 ⊕⇩ℐ ?ℐ2' ⊢⇩C ?conv1.0 |₌ ?conv2.0 √› ‹⟦?ℐ1.0, ?ℐ ⊢⇩C ?conv1.0 √; ?ℐ2.0, ?ℐ ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0, ?ℐ ⊢⇩C ?conv1.0 |⇩∝ ?conv2.0 √› ‹?ℐ ⊢res ?res √ ⟹ ?ℐ, ?ℐ' ⊢⇩C converter_of_resource ?res √› ‹?ℐ, ?ℐ' ⊢⇩C ?cnv √ ⟹ ?ℐ, ?ℐ' ⊢⇩C restrict_converter ?A ?ℐ' ?cnv √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ' ⊢res ?res √⟧ ⟹ ?ℐ ⊢res ?conv ⊳ ?res √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ', ?ℐ'' ⊢⇩C ?conv' √⟧ ⟹ ?ℐ, ?ℐ'' ⊢⇩C ?conv ⊙ ?conv' √› and more 23 facts*) IC1.WT_resource_of_oracle (*‹IC1 ?s ⟹ ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU) ⊢res RES (callee_of_core core1) ?s √›*) IC1 (*‹callee_invariant_on (callee_of_core core1) IC1 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))› ‹IC1 s_core›*) IC2.WT_resource_of_oracle (*‹IC2 ?s ⟹ ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU) ⊢res RES (callee_of_core core2) ?s √›*) IC2 (*‹callee_invariant_on (callee_of_core core2) IC2 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))› ‹IC2 s_core'›*) IR1.WT_resource_of_oracle (*‹IR1 ?s ⟹ ℐRA ⊕⇩ℐ ℐRU ⊢res RES (callee_of_rest rest1) ?s √›*) IR1 (*‹callee_invariant_on (callee_of_rest rest1) IR1 (ℐRA ⊕⇩ℐ ℐRU)› ‹IR1 s_rest›*) IR2.WT_resource_of_oracle (*‹IR2 ?s ⟹ ℐRA ⊕⇩ℐ ℐRU ⊢res RES (callee_of_rest rest2) ?s √›*) IR2 (*‹callee_invariant_on (callee_of_rest rest2) IR2 (ℐRA ⊕⇩ℐ ℐRU)› ‹IR2 s_rest'›*))\n (*discuss goal 2*)\n apply (intro WT_intro (*‹map_ℐ ?g ?h ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv' ?f ?g ?h ?gpv √› ‹map_ℐ ?g id ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv ?f ?g ?gpv √› ‹⟦?ℐ1.0 ⊢res ?res1.0 √; ?ℐ2.0 ⊢res ?res2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0 ⊢res ?res1.0 ∥ ?res2.0 √› ‹⟦map_ℐ (inv ?f) (inv ?g) ?ℐ, map_ℐ ?f' ?g' ?ℐ' ⊢⇩C ?conv √; inj ?f; surj ?g⟧ ⟹ ?ℐ, ?ℐ' ⊢⇩C map_converter ?f ?g ?f' ?g' ?conv √› ‹?ℐ, ?ℐ ⊢⇩C 1⇩C √› ‹?ℐ, ?ℐ' ⊢⇩C ⊥⇩C √› ‹⟦?ℐ1.0, ?ℐ2.0 ⊢⇩C ?conv1.0 √; ?ℐ1', ?ℐ2' ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ1', ?ℐ2.0 ⊕⇩ℐ ?ℐ2' ⊢⇩C ?conv1.0 |₌ ?conv2.0 √› ‹⟦?ℐ1.0, ?ℐ ⊢⇩C ?conv1.0 √; ?ℐ2.0, ?ℐ ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0, ?ℐ ⊢⇩C ?conv1.0 |⇩∝ ?conv2.0 √› ‹?ℐ ⊢res ?res √ ⟹ ?ℐ, ?ℐ' ⊢⇩C converter_of_resource ?res √› ‹?ℐ, ?ℐ' ⊢⇩C ?cnv √ ⟹ ?ℐ, ?ℐ' ⊢⇩C restrict_converter ?A ?ℐ' ?cnv √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ' ⊢res ?res √⟧ ⟹ ?ℐ ⊢res ?conv ⊳ ?res √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ', ?ℐ'' ⊢⇩C ?conv' √⟧ ⟹ ?ℐ, ?ℐ'' ⊢⇩C ?conv ⊙ ?conv' √› and more 23 facts*) IC1.WT_resource_of_oracle (*‹IC1 ?s ⟹ ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU) ⊢res RES (callee_of_core core1) ?s √›*) IC1 (*‹callee_invariant_on (callee_of_core core1) IC1 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))› ‹IC1 s_core›*) IC2.WT_resource_of_oracle (*‹IC2 ?s ⟹ ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU) ⊢res RES (callee_of_core core2) ?s √›*) IC2 (*‹callee_invariant_on (callee_of_core core2) IC2 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))› ‹IC2 s_core'›*) IR1.WT_resource_of_oracle (*‹IR1 ?s ⟹ ℐRA ⊕⇩ℐ ℐRU ⊢res RES (callee_of_rest rest1) ?s √›*) IR1 (*‹callee_invariant_on (callee_of_rest rest1) IR1 (ℐRA ⊕⇩ℐ ℐRU)› ‹IR1 s_rest›*) IR2.WT_resource_of_oracle (*‹IR2 ?s ⟹ ℐRA ⊕⇩ℐ ℐRU ⊢res RES (callee_of_rest rest2) ?s √›*) IR2 (*‹callee_invariant_on (callee_of_rest rest2) IR2 (ℐRA ⊕⇩ℐ ℐRU)› ‹IR2 s_rest'›*))\n (*discuss goal 3*)\n apply (intro WT_intro (*‹map_ℐ ?g ?h ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv' ?f ?g ?h ?gpv √› ‹map_ℐ ?g id ?ℐ ⊢g ?gpv √ ⟹ ?ℐ ⊢g map_gpv ?f ?g ?gpv √› ‹⟦?ℐ1.0 ⊢res ?res1.0 √; ?ℐ2.0 ⊢res ?res2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0 ⊢res ?res1.0 ∥ ?res2.0 √› ‹⟦map_ℐ (inv ?f) (inv ?g) ?ℐ, map_ℐ ?f' ?g' ?ℐ' ⊢⇩C ?conv √; inj ?f; surj ?g⟧ ⟹ ?ℐ, ?ℐ' ⊢⇩C map_converter ?f ?g ?f' ?g' ?conv √› ‹?ℐ, ?ℐ ⊢⇩C 1⇩C √› ‹?ℐ, ?ℐ' ⊢⇩C ⊥⇩C √› ‹⟦?ℐ1.0, ?ℐ2.0 ⊢⇩C ?conv1.0 √; ?ℐ1', ?ℐ2' ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ1', ?ℐ2.0 ⊕⇩ℐ ?ℐ2' ⊢⇩C ?conv1.0 |₌ ?conv2.0 √› ‹⟦?ℐ1.0, ?ℐ ⊢⇩C ?conv1.0 √; ?ℐ2.0, ?ℐ ⊢⇩C ?conv2.0 √⟧ ⟹ ?ℐ1.0 ⊕⇩ℐ ?ℐ2.0, ?ℐ ⊢⇩C ?conv1.0 |⇩∝ ?conv2.0 √› ‹?ℐ ⊢res ?res √ ⟹ ?ℐ, ?ℐ' ⊢⇩C converter_of_resource ?res √› ‹?ℐ, ?ℐ' ⊢⇩C ?cnv √ ⟹ ?ℐ, ?ℐ' ⊢⇩C restrict_converter ?A ?ℐ' ?cnv √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ' ⊢res ?res √⟧ ⟹ ?ℐ ⊢res ?conv ⊳ ?res √› ‹⟦?ℐ, ?ℐ' ⊢⇩C ?conv √; ?ℐ', ?ℐ'' ⊢⇩C ?conv' √⟧ ⟹ ?ℐ, ?ℐ'' ⊢⇩C ?conv ⊙ ?conv' √› and more 23 facts*) IC1.WT_resource_of_oracle (*‹IC1 ?s ⟹ ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU) ⊢res RES (callee_of_core core1) ?s √›*) IC1 (*‹callee_invariant_on (callee_of_core core1) IC1 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))› ‹IC1 s_core›*) IC2.WT_resource_of_oracle (*‹IC2 ?s ⟹ ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU) ⊢res RES (callee_of_core core2) ?s √›*) IC2 (*‹callee_invariant_on (callee_of_core core2) IC2 (ℐE ⊕⇩ℐ (ℐCA ⊕⇩ℐ ℐCU))› ‹IC2 s_core'›*) IR1.WT_resource_of_oracle (*‹IR1 ?s ⟹ ℐRA ⊕⇩ℐ ℐRU ⊢res RES (callee_of_rest rest1) ?s √›*) IR1 (*‹callee_invariant_on (callee_of_rest rest1) IR1 (ℐRA ⊕⇩ℐ ℐRU)› ‹IR1 s_rest›*) IR2.WT_resource_of_oracle (*‹IR2 ?s ⟹ ℐRA ⊕⇩ℐ ℐRU ⊢res RES (callee_of_rest rest2) ?s √›*) IR2 (*‹callee_invariant_on (callee_of_rest rest2) IR2 (ℐRA ⊕⇩ℐ ℐRU)› ‹IR2 s_rest'›*))\n (*proven 3 subgoals*) .\n hence \"trace_eq' (outs_ℐ ?ℐ) (resource_of_oracle (fuse1.fuse rest1) (s_core, s_rest)) (resource_of_oracle (fuse2.fuse rest2) (s_core', s_rest'))\"\n unfolding fuse_converter\n (*goal: ‹outs_ℐ ((ℐCA ⊕⇩ℐ map_ℐ id fst ℐRA) ⊕⇩ℐ (ℐCU ⊕⇩ℐ map_ℐ id fst ℐRU)) ⊢⇩R fuse_converter ⊳ RES (callee_of_core core1) s_core ∥ RES (callee_of_rest rest1) s_rest ≈ fuse_converter ⊳ RES (callee_of_core core2) s_core' ∥ RES (callee_of_rest rest2) s_rest'›*)\n by simp\n then show \"?thesis\"\n (*goal: ‹((outs_ℐ (ℐCA::('iadv_core::type, 'oadv_core::type) ℐ) <+> outs_ℐ (ℐRA::('iadv_rest::type, 'oadv_rest::type × 'event::type list) ℐ)) <+> outs_ℐ (ℐCU::('iusr_core::type, 'ousr_core::type) ℐ) <+> outs_ℐ (ℐRU::('iusr_rest::type, 'ousr_rest::type × 'event::type list) ℐ)) ⊢⇩C fuse1.fuse (rest1::('s_rest::type, 'event::type, 'iadv_rest::type, 'iusr_rest::type, 'oadv_rest::type, 'ousr_rest::type, 'more1::type) rest_scheme)((s_core::'s_core::type, s_rest::'s_rest::type)) ≈ fuse2.fuse (rest2::('s_rest'::type, 'event::type, 'iadv_rest::type, 'iusr_rest::type, 'oadv_rest::type, 'ousr_rest::type, 'more2::type) rest_scheme)((s_core'::'s_core'::type, s_rest'::'s_rest'::type))›*)\n by simp\nqed\n\n\ninductive trace_eq_simcl :: \"('s1 spmf ⇒ 's2 spmf ⇒ bool) ⇒ 's1 spmf ⇒ 's2 spmf ⇒ bool\"\n for S where\n base: \"trace_eq_simcl S p q\" if \"S p q\" for p q\n| bind_nat: \"trace_eq_simcl S (bind_spmf p f) (bind_spmf p g)\" \nif \"⋀x :: nat. x ∈ set_spmf p ⟹ S (f x) (g x)\"\n\nlemma trace_eq_simcl_bindI [intro?]: \"trace_eq_simcl S (bind_spmf p f) (bind_spmf p g)\"\n if \"⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)\"\n apply (subst (1 2) bind_spmf_to_nat_on[symmetric] (*‹?p ⤜ ?f = map_spmf (to_nat_on (set_spmf ?p)) ?p ⤜ (λn. ?f (from_nat_into (set_spmf ?p) n))›*))\n (*goal: ‹trace_eq_simcl S (p ⤜ f) (p ⤜ g)›*)\n by (auto intro!: trace_eq_simcl.bind_nat (*‹(⋀x. x ∈ set_spmf ?p ⟹ ?S (?f x) (?g x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*) simp add: that (*‹?x ∈ set_spmf p ⟹ S (f ?x) (g ?x)›*))\n\nlemma trace_eq_simcl_bind: \"trace_eq_simcl S (bind_spmf p f) (bind_spmf p g)\"\n if *: \"⋀x :: 'a. x ∈ set_spmf p ⟹ trace_eq_simcl S (f x) (g x)\"\nproof (-)\n (*goal: ‹trace_eq_simcl (S::'b spmf ⇒ 'c spmf ⇒ bool) ((p::'a spmf) ⤜ (f::'a ⇒ 'b spmf)) (p ⤜ (g::'a ⇒ 'c spmf))›*)\n obtain P :: \"'a ⇒ nat spmf\" and F and G where \"**\": \"⋀x. x ∈ set_spmf p ⟹ f x = bind_spmf (P x) (F x) ∧ g x = bind_spmf (P x) (G x) ∧ (∀y∈set_spmf (P x). S (F x y) (G x y))\"\n (*goal: ‹(⋀P F G. (⋀x. x ∈ set_spmf p ⟹ f x = P x ⤜ F x ∧ g x = P x ⤜ G x ∧ (∀y∈set_spmf (P x). S (F x y) (G x y))) ⟹ thesis) ⟹ thesis›*)\n apply atomize_elim\n (*goal: ‹(⋀P F G. (⋀x. x ∈ set_spmf p ⟹ f x = P x ⤜ F x ∧ g x = P x ⤜ G x ∧ (∀y∈set_spmf (P x). S (F x y) (G x y))) ⟹ thesis) ⟹ thesis›*)\n apply (subst choice_iff[symmetric] (*‹(∃f. ∀x. ?Q x (f x)) = (∀x. ∃y. ?Q x y)›*))\n (*goal: ‹∃P F G. ∀x. x ∈ set_spmf p ⟶ f x = P x ⤜ F x ∧ g x = P x ⤜ G x ∧ (∀y∈set_spmf (P x). S (F x y) (G x y))›*)\n apply (subst choice_iff[symmetric] (*‹(∃f. ∀x. ?Q x (f x)) = (∀x. ∃y. ?Q x y)›*))\n (*goal: ‹∃P F. ∀x. ∃y. x ∈ set_spmf p ⟶ f x = P x ⤜ F x ∧ g x = P x ⤜ y ∧ (∀ya∈set_spmf (P x). S (F x ya) (y ya))›*)\n apply (subst choice_iff[symmetric] (*‹(∃f. ∀x. ?Q x (f x)) = (∀x. ∃y. ?Q x y)›*))\n (*goal: ‹∃P::'a::type ⇒ nat spmf. ∀x::'a::type. ∃(y::nat ⇒ 'b::type spmf) ya::nat ⇒ 'c::type spmf. x ∈ set_spmf (p::'a::type spmf) ⟶ (f::'a::type ⇒ 'b::type spmf) x = P x ⤜ y ∧ (g::'a::type ⇒ 'c::type spmf) x = P x ⤜ ya ∧ (∀yb::nat∈set_spmf (P x). (S::'b::type spmf ⇒ 'c::type spmf ⇒ bool) (y yb) (ya yb))›*)\n by (fastforce dest!: * (*‹(?x::'a) ∈ set_spmf (p::'a spmf) ⟹ trace_eq_simcl (S::'b spmf ⇒ 'c spmf ⇒ bool) ((f::'a ⇒ 'b spmf) ?x) ((g::'a ⇒ 'c spmf) ?x)›*) elim!: trace_eq_simcl.cases (*‹⟦trace_eq_simcl (?S::?'s1.0 spmf ⇒ ?'s2.0 spmf ⇒ bool) (?a1.0::?'s1.0 spmf) (?a2.0::?'s2.0 spmf); ⋀(p::?'s1.0 spmf) q::?'s2.0 spmf. ⟦?a1.0 = p; ?a2.0 = q; ?S p q⟧ ⟹ ?P::bool; ⋀(p::nat spmf) (f::nat ⇒ ?'s1.0 spmf) g::nat ⇒ ?'s2.0 spmf. ⟦?a1.0 = p ⤜ f; ?a2.0 = p ⤜ g; ⋀x::nat. x ∈ set_spmf p ⟹ ?S (f x) (g x)⟧ ⟹ ?P⟧ ⟹ ?P›*) intro: exI[where x=\"return_spmf _\"] (*‹(?P::?'d1 spmf ⇒ bool) (return_spmf (?uu5::?'d1)) ⟹ ∃x::?'d1 spmf. ?P x›*))\n have \"bind_spmf p f = bind_spmf (bind_spmf p (λx. map_spmf (Pair x) (P x))) (λ(x, y). F x y)\"\n by (simp add: bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) ** (*‹?x ∈ set_spmf p ⟹ f ?x = P ?x ⤜ F ?x ∧ g ?x = P ?x ⤜ G ?x ∧ (∀y∈set_spmf (P ?x). S (F ?x y) (G ?x y))›*) cong: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))\n moreover have \"bind_spmf p g = bind_spmf (bind_spmf p (λx. map_spmf (Pair x) (P x))) (λ(x, y). G x y)\"\n by (simp add: bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) ** (*‹?x ∈ set_spmf p ⟹ f ?x = P ?x ⤜ F ?x ∧ g ?x = P ?x ⤜ G ?x ∧ (∀y∈set_spmf (P ?x). S (F ?x y) (G ?x y))›*) cong: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))\n ultimately show \"?thesis\"\n (*goal: ‹trace_eq_simcl S (p ⤜ f) (p ⤜ g)›*)\n apply (simp only:)\n (*goal: ‹trace_eq_simcl S (p ⤜ f) (p ⤜ g)›*)\n apply (rule trace_eq_simcl_bindI (*‹(⋀x::?'a. x ∈ set_spmf (?p::?'a spmf) ⟹ (?S::?'b spmf ⇒ ?'c spmf ⇒ bool) ((?f::?'a ⇒ ?'b spmf) x) ((?g::?'a ⇒ ?'c spmf) x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹⟦(p::'a spmf) ⤜ (f::'a ⇒ 'b spmf) = p ⤜ (λx::'a. map_spmf (Pair x) ((P::'a ⇒ nat spmf) x)) ⤜ (λ(x::'a, y::nat). (F::'a ⇒ nat ⇒ 'b spmf) x y); p ⤜ (g::'a ⇒ 'c spmf) = p ⤜ (λx::'a. map_spmf (Pair x) (P x)) ⤜ (λ(x::'a, y::nat). (G::'a ⇒ nat ⇒ 'c spmf) x y)⟧ ⟹ trace_eq_simcl (S::'b spmf ⇒ 'c spmf ⇒ bool) (p ⤜ (λx::'a. map_spmf (Pair x) (P x)) ⤜ (λ(x::'a, y::nat). F x y)) (p ⤜ (λx::'a. map_spmf (Pair x) (P x)) ⤜ (λ(x::'a, y::nat). G x y))›*)\n by (clarsimp simp add: ** (*‹(?x::'a) ∈ set_spmf (p::'a spmf) ⟹ (f::'a ⇒ 'b spmf) ?x = (P::'a ⇒ nat spmf) ?x ⤜ (F::'a ⇒ nat ⇒ 'b spmf) ?x ∧ (g::'a ⇒ 'c spmf) ?x = P ?x ⤜ (G::'a ⇒ nat ⇒ 'c spmf) ?x ∧ (∀y::nat∈set_spmf (P ?x). (S::'b spmf ⇒ 'c spmf ⇒ bool) (F ?x y) (G ?x y))›*))\nqed\n\nlemma trace_eq_simcl_bind1_scale: \"trace_eq_simcl S (bind_spmf p f) (scale_spmf (weight_spmf p) q)\"\n if \"∀x∈set_spmf p. trace_eq_simcl S (f x) q\"\nproof (-)\n (*goal: ‹trace_eq_simcl S (p ⤜ f) (scale_spmf (weight_spmf p) q)›*)\n have \"trace_eq_simcl S (bind_spmf p f) (bind_spmf p (λ_. q))\"\n apply (rule trace_eq_simcl_bind (*‹(⋀x. x ∈ set_spmf ?p ⟹ trace_eq_simcl ?S (?f x) (?g x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹trace_eq_simcl (S::'b spmf ⇒ 'c spmf ⇒ bool) ((p::'a spmf) ⤜ (f::'a ⇒ 'b spmf)) (p ⤜ (λ_::'a. q::'c spmf))›*)\n by (simp add: that (*‹∀x∈set_spmf p. trace_eq_simcl S (f x) q›*))\n thus \"?thesis\"\n (*goal: ‹trace_eq_simcl (S::'b spmf ⇒ 'c spmf ⇒ bool) ((p::'a spmf) ⤜ (f::'a ⇒ 'b spmf)) (scale_spmf (weight_spmf p) (q::'c spmf))›*)\n by (simp add: bind_spmf_const (*‹?p ⤜ (λx. ?q) = scale_spmf (weight_spmf ?p) ?q›*))\nqed\n\nlemma trace_eq_simcl_bind1: \"trace_eq_simcl S (bind_spmf p f) q\"\n if \"∀x∈set_spmf p. trace_eq_simcl S (f x) q\" \"lossless_spmf p\"\n using trace_eq_simcl_bind1_scale[OF that ( 1 )] (*‹trace_eq_simcl S (p ⤜ f) (scale_spmf (weight_spmf p) q)›*) that(2) (*‹lossless_spmf p›*) by (simp add: lossless_weight_spmfD (*‹lossless_spmf (?p::?'a spmf) ⟹ weight_spmf ?p = (1::real)›*))\n\nlemma trace_eq_simcl_bind2_scale: \"trace_eq_simcl S (scale_spmf (weight_spmf q) p) (bind_spmf q f)\"\n if \"∀x∈set_spmf q. trace_eq_simcl S p (f x)\"\nproof (-)\n (*goal: ‹trace_eq_simcl S (scale_spmf (weight_spmf q) p) (q ⤜ f)›*)\n have \"trace_eq_simcl S (bind_spmf q (λ_. p)) (bind_spmf q f)\"\n apply (rule trace_eq_simcl_bind (*‹(⋀x. x ∈ set_spmf ?p ⟹ trace_eq_simcl ?S (?f x) (?g x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹trace_eq_simcl S (q ⤜ (λ_. p)) (q ⤜ f)›*)\n by (simp add: that (*‹∀x∈set_spmf q. trace_eq_simcl S p (f x)›*))\n thus \"?thesis\"\n (*goal: ‹trace_eq_simcl S (scale_spmf (weight_spmf q) p) (q ⤜ f)›*)\n by (simp add: bind_spmf_const (*‹?p ⤜ (λx. ?q) = scale_spmf (weight_spmf ?p) ?q›*))\nqed\n\nlemma trace_eq_simcl_bind2: \"trace_eq_simcl S p (bind_spmf q f)\"\n if \"∀x∈set_spmf q. trace_eq_simcl S p (f x)\" \"lossless_spmf q\"\n using trace_eq_simcl_bind2_scale[OF that ( 1 )] (*‹trace_eq_simcl S (scale_spmf (weight_spmf q) p) (q ⤜ f)›*) that(2) (*‹lossless_spmf q›*) by (simp add: lossless_weight_spmfD (*‹lossless_spmf ?p ⟹ weight_spmf ?p = 1›*))\n\nlemma trace_eq_simcl_return_pmf_None [simp, intro!]: \"trace_eq_simcl S (return_pmf None) (return_pmf None)\" \n for S :: \"'s1 spmf ⇒ 's2 spmf ⇒ bool\"\nproof (-)\n (*goal: ‹trace_eq_simcl S (return_pmf None) (return_pmf None)›*)\n have \"trace_eq_simcl S (bind_spmf (return_pmf None) (undefined :: nat ⇒ 's1 spmf)) (bind_spmf (return_pmf None) (undefined :: nat ⇒ 's2 spmf))\"\n apply (rule trace_eq_simcl_bindI (*‹(⋀x. x ∈ set_spmf ?p ⟹ ?S (?f x) (?g x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹trace_eq_simcl S (return_pmf None ⤜ undefined) (return_pmf None ⤜ undefined)›*)\n by simp\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl S (return_pmf None) (return_pmf None)›*)\n by simp\nqed\n\nlemma trace_eq_simcl_map: \"trace_eq_simcl S (map_spmf f p) (map_spmf g p)\"\n if \"∀x∈set_spmf p. S (return_spmf (f x)) (return_spmf (g x))\"\n unfolding map_spmf_conv_bind_spmf\n (*goal: ‹trace_eq_simcl S (p ⤜ (λx. return_spmf (f x))) (p ⤜ (λx. return_spmf (g x)))›*)\n apply (rule trace_eq_simcl_bindI (*‹(⋀x. x ∈ set_spmf ?p ⟹ ?S (?f x) (?g x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹trace_eq_simcl S (p ⤜ (λx. return_spmf (f x))) (p ⤜ (λx. return_spmf (g x)))›*)\n by (simp add: that (*‹∀x∈set_spmf p. S (return_spmf (f x)) (return_spmf (g x))›*))\n\nlemma trace_eq_simcl_map1: \"trace_eq_simcl S (map_spmf f p) q\"\n if \"∀x∈set_spmf p. trace_eq_simcl S (return_spmf (f x)) q\" \"lossless_spmf p\"\n unfolding map_spmf_conv_bind_spmf\n (*goal: ‹trace_eq_simcl S (p ⤜ (λx. return_spmf (f x))) q›*)\n apply (rule trace_eq_simcl_bind1 (*‹⟦∀x∈set_spmf ?p. trace_eq_simcl ?S (?f x) ?q; lossless_spmf ?p⟧ ⟹ trace_eq_simcl ?S (?p ⤜ ?f) ?q›*))\n (*goals:\n 1. ‹∀x∈set_spmf p. trace_eq_simcl S (return_spmf (f x)) q›\n 2. ‹lossless_spmf p›\n discuss goal 1*)\n apply (simp add: that (*‹∀x∈set_spmf p. trace_eq_simcl S (return_spmf (f x)) q› ‹lossless_spmf p›*))\n (*discuss goal 2*)\n apply (simp add: that (*‹∀x∈set_spmf p. trace_eq_simcl S (return_spmf (f x)) q› ‹lossless_spmf p›*))\n (*proven 2 subgoals*) .\n\nlemma trace_eq_simcl_map2: \"trace_eq_simcl S p (map_spmf f q)\"\n if \"∀x∈set_spmf q. trace_eq_simcl S p (return_spmf (f x))\" \"lossless_spmf q\"\n unfolding map_spmf_conv_bind_spmf\n (*goal: ‹trace_eq_simcl S p (q ⤜ (λx. return_spmf (f x)))›*)\n apply (rule trace_eq_simcl_bind2 (*‹⟦∀x::?'a::type∈set_spmf (?q::?'a::type spmf). trace_eq_simcl (?S::?'b::type spmf ⇒ ?'c::type spmf ⇒ bool) (?p::?'b::type spmf) ((?f::?'a::type ⇒ ?'c::type spmf) x); lossless_spmf ?q⟧ ⟹ trace_eq_simcl ?S ?p (?q ⤜ ?f)›*))\n (*goals:\n 1. ‹∀x∈set_spmf q. trace_eq_simcl S p (return_spmf (f x))›\n 2. ‹lossless_spmf q›\n discuss goal 1*)\n apply (simp add: that (*‹∀x∈set_spmf q. trace_eq_simcl S p (return_spmf (f x))› ‹lossless_spmf q›*))\n (*discuss goal 2*)\n apply (simp add: that (*‹∀x::'a∈set_spmf (q::'a spmf). trace_eq_simcl (S::'b spmf ⇒ 'c spmf ⇒ bool) (p::'b spmf) (return_spmf ((f::'a ⇒ 'c) x))› ‹lossless_spmf (q::'a spmf)›*))\n (*proven 2 subgoals*) .\n\nlemma trace_eq_simcl_return_spmf [simp]: \"trace_eq_simcl S (return_spmf x) (return_spmf y) ⟷ S (return_spmf x) (return_spmf y)\"\n apply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goal: ‹trace_eq_simcl S (return_spmf x) (return_spmf y) = S (return_spmf x) (return_spmf y)›*)\n subgoal for \n apply (erule trace_eq_simcl.cases (*‹⟦trace_eq_simcl ?S ?a1.0 ?a2.0; ⋀p q. ⟦?a1.0 = p; ?a2.0 = q; ?S p q⟧ ⟹ ?P; ⋀p f g. ⟦?a1.0 = p ⤜ f; ?a2.0 = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ ?S (f x) (g x)⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀p q. ⟦return_spmf x = p; return_spmf y = q; S p q⟧ ⟹ S (return_spmf x) (return_spmf y)›\n 2. ‹⋀p f g. ⟦return_spmf x = p ⤜ f; return_spmf y = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ S (return_spmf x) (return_spmf y)›\n discuss goal 1*)\n apply (clarsimp dest!: sym[where s=\"return_spmf _\"] (*‹return_spmf ?uu5 = ?t ⟹ ?t = return_spmf ?uu5›*))\n (*discuss goal 2*)\n apply (clarsimp dest!: sym[where s=\"return_spmf _\"] (*‹return_spmf ?uu5 = ?t ⟹ ?t = return_spmf ?uu5›*))\n (*goal: ‹⋀(p::nat spmf) (f::nat ⇒ 'a spmf) g::nat ⇒ 'b spmf. ⟦return_spmf (x::'a) = p ⤜ f; return_spmf (y::'b) = p ⤜ g; ⋀x::nat. x ∈ set_spmf p ⟹ (S::'a spmf ⇒ 'b spmf ⇒ bool) (f x) (g x)⟧ ⟹ S (return_spmf x) (return_spmf y)›*)\n apply (auto 4 4 simp add: bind_eq_return_spmf (*‹(?p ⤜ ?f = return_spmf ?x) = ((∀y∈set_spmf ?p. ?f y = return_spmf ?x) ∧ lossless_spmf ?p)›*) dest!: lossless_spmfD_set_spmf_nonempty (*‹lossless_spmf ?p ⟹ set_spmf ?p ≠ {}›*))\n (*proven 2 subgoals*) .\n by (simp add: trace_eq_simcl.base (*‹?S ?p ?q ⟹ trace_eq_simcl ?S ?p ?q›*))\n\nlemma trace_eq_simcl_callee:\n fixes callee1 :: \"('a, 'b, 's1) callee\" and callee2 :: \"('a, 'b, 's2) callee\"\n assumes step: \"⋀p q a. ⟦ S p q; a ∈ A ⟧ ⟹\n bind_spmf p (λs. map_spmf fst (callee1 s a)) = bind_spmf q (λs. map_spmf fst (callee2 s a))\"\n and sim: \"⋀p q a res b s'. ⟦ S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a) ⟧\n ⟹ trace_eq_simcl S (cond_spmf_fst (bind_spmf p (λs. callee1 s a)) b)\n (cond_spmf_fst (bind_spmf q (λs. callee2 s a)) b)\"\n and start: \"trace_eq_simcl S p q\" and a: \"a ∈ A\"\n shows trace_eq_simcl_callee_step: \"bind_spmf p (λs. map_spmf fst (callee1 s a)) = bind_spmf q (λs. map_spmf fst (callee2 s a))\" (is \"?step\")\n and trace_eq_simcl_callee_sim: \"⋀res b s'. ⟦ res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a) ⟧\n ⟹ trace_eq_simcl S (cond_spmf_fst (bind_spmf p (λs. callee1 s a)) b)\n (cond_spmf_fst (bind_spmf q (λs. callee2 s a)) b)\" (is \"⋀res b s'. ⟦ ?res res; ?b res b s' ⟧ ⟹ ?sim res b s'\")\nproof (-)\n (*goals:\n 1. ‹p ⤜ (λs. map_spmf fst (callee1 s a)) = q ⤜ (λs. map_spmf fst (callee2 s a))›\n 2. ‹⋀res b s'. ⟦res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs. callee1 s a)) b) (cond_spmf_fst (q ⤜ (λs. callee2 s a)) b)›*)\n show eq: \"?step\"\n using start (*‹trace_eq_simcl S p q›*) a (*‹a ∈ A›*) apply cases\n (*goals:\n 1. ‹⟦a ∈ A; S p q⟧ ⟹ p ⤜ (λs. map_spmf fst (callee1 s a)) = q ⤜ (λs. map_spmf fst (callee2 s a))›\n 2. ‹⋀pa f g. ⟦a ∈ A; p = pa ⤜ f; q = pa ⤜ g; ⋀x. x ∈ set_spmf pa ⟹ S (f x) (g x)⟧ ⟹ p ⤜ (λs. map_spmf fst (callee1 s a)) = q ⤜ (λs. map_spmf fst (callee2 s a))›\n discuss goal 1*)\n apply ((auto intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*) intro: step (*‹⟦S ?p ?q; ?a ∈ A⟧ ⟹ ?p ⤜ (λs. map_spmf fst (callee1 s ?a)) = ?q ⤜ (λs. map_spmf fst (callee2 s ?a))›*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*) intro: step (*‹⟦S ?p ?q; ?a ∈ A⟧ ⟹ ?p ⤜ (λs. map_spmf fst (callee1 s ?a)) = ?q ⤜ (λs. map_spmf fst (callee2 s ?a))›*))[1])\n (*proven 2 subgoals*) .\n show \"?sim res b s'\" if \"?res res\" \"?b res b s'\" for res and b and s'\n using start (*‹trace_eq_simcl S p q›*) proof (cases)\n (*goals:\n 1. ‹S p q ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs. callee1 s a)) b) (cond_spmf_fst (q ⤜ (λs. callee2 s a)) b)›\n 2. ‹⋀pa f g. ⟦p = pa ⤜ f; q = pa ⤜ g; ⋀x. x ∈ set_spmf pa ⟹ S (f x) (g x)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs. callee1 s a)) b) (cond_spmf_fst (q ⤜ (λs. callee2 s a)) b)›*)\n case base (*‹S p q›*)\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs. callee1 s a)) b) (cond_spmf_fst (q ⤜ (λs. callee2 s a)) b)›*)\n using a (*‹a ∈ A›*) that (*‹(res::'s2) ∈ set_spmf (q::'s2 spmf)› ‹(b, s') ∈ set_spmf (callee2 res a)›*) by (rule sim (*‹⟦(S::'s1 spmf ⇒ 's2 spmf ⇒ bool) (?p::'s1 spmf) (?q::'s2 spmf); (?a::'a) ∈ (A::'a set); (?res::'s2) ∈ set_spmf ?q; (?b::'b, ?s'::'s2) ∈ set_spmf ((callee2::'s2 ⇒ 'a ⇒ ('b × 's2) spmf) ?res ?a)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs::'s1. (callee1::'s1 ⇒ 'a ⇒ ('b × 's1) spmf) s ?a)) ?b) (cond_spmf_fst (?q ⤜ (λs::'s2. callee2 s ?a)) ?b)›*))\n next\n (*goal: ‹⋀pa f g. ⟦p = pa ⤜ f; q = pa ⤜ g; ⋀x. x ∈ set_spmf pa ⟹ S (f x) (g x)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs. callee1 s a)) b) (cond_spmf_fst (q ⤜ (λs. callee2 s a)) b)›*)\n case (bind_nat X f g) (*‹p = X ⤜ f› ‹q = X ⤜ g› ‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*)\n let ?Y = \"cond_bind_spmf_fst X (λy. map_spmf fst (bind_spmf (f y) (λs. callee1 s a))) b\"\n let ?Y' = \"cond_bind_spmf_fst X (λy. map_spmf fst (bind_spmf (g y) (λs. callee2 s a))) b\"\n have \"cond_spmf_fst (bind_spmf p (λs. callee1 s a)) b = bind_spmf ?Y (λx. cond_spmf_fst (bind_spmf (f x) (λs. callee1 s a)) b)\"\n unfolding bind_nat\n (*goal: ‹cond_spmf_fst (X ⤜ f ⤜ (λs. callee1 s a)) b = cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs. callee1 s a))) b ⤜ (λx. cond_spmf_fst (f x ⤜ (λs. callee1 s a)) b)›*)\n by (simp add: cond_spmf_fst_bind (*‹cond_spmf_fst (?p ⤜ ?f) ?x = cond_bind_spmf_fst ?p (map_spmf fst ∘ ?f) ?x ⤜ (λy. cond_spmf_fst (?f y) ?x)›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n moreover have \"cond_spmf_fst (bind_spmf q (λs. callee2 s a)) b = bind_spmf ?Y' (λx. cond_spmf_fst (bind_spmf (g x) (λs. callee2 s a)) b)\"\n unfolding bind_nat\n (*goal: ‹cond_spmf_fst (X ⤜ g ⤜ (λs. callee2 s a)) b = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs. callee2 s a))) b ⤜ (λx. cond_spmf_fst (g x ⤜ (λs. callee2 s a)) b)›*)\n by (simp add: cond_spmf_fst_bind (*‹cond_spmf_fst (?p ⤜ ?f) ?x = cond_bind_spmf_fst ?p (map_spmf fst ∘ ?f) ?x ⤜ (λy. cond_spmf_fst (?f y) ?x)›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n moreover have \"?Y = ?Y'\"\n using bind_nat (*‹p = X ⤜ f› ‹(q::'s2 spmf) = (X::nat spmf) ⤜ (g::nat ⇒ 's2 spmf)› ‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*) eq (*‹p ⤜ (λs. map_spmf fst (callee1 s a)) = q ⤜ (λs. map_spmf fst (callee2 s a))›*) apply (intro spmf_eqI (*‹(⋀i. spmf ?p i = spmf ?q i) ⟹ ?p = ?q›*))\n (*goal: ‹cond_bind_spmf_fst (X::nat spmf) (λy::nat. map_spmf fst ((f::nat ⇒ 's1 spmf) y ⤜ (λs::'s1. (callee1::'s1 ⇒ 'a ⇒ ('b × 's1) spmf) s (a::'a)))) (b::'b) = cond_bind_spmf_fst X (λy::nat. map_spmf fst ((g::nat ⇒ 's2 spmf) y ⤜ (λs::'s2. (callee2::'s2 ⇒ 'a ⇒ ('b × 's2) spmf) s a))) b›*)\n by (fastforce simp add: map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) spmf_eq_0_set_spmf (*‹(spmf ?p ?x = 0) = (?x ∉ set_spmf ?p)›*) dest: step[OF _ a] (*‹S ?p ?q ⟹ ?p ⤜ (λs. map_spmf fst (callee1 s a)) = ?q ⤜ (λs. map_spmf fst (callee2 s a))›*))\n ultimately show \"trace_eq_simcl S (cond_spmf_fst (bind_spmf p (λs. callee1 s a)) b)\n (cond_spmf_fst (bind_spmf q (λs. callee2 s a)) b)\"\n using bind_nat (*‹p = X ⤜ f› ‹(q::'s2::type spmf) = (X::nat spmf) ⤜ (g::nat ⇒ 's2::type spmf)› ‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*) a (*‹(a::'a) ∈ (A::'a set)›*) apply simp\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs. callee1 s a)) b) (cond_spmf_fst (q ⤜ (λs. callee2 s a)) b)›*)\n apply (rule trace_eq_simcl_bind (*‹(⋀x::?'a. x ∈ set_spmf (?p::?'a spmf) ⟹ trace_eq_simcl (?S::?'b spmf ⇒ ?'c spmf ⇒ bool) ((?f::?'a ⇒ ?'b spmf) x) ((?g::?'a ⇒ ?'c spmf) x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹⟦cond_spmf_fst (X ⤜ (λy. f y ⤜ (λs. callee1 s a))) b = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs. callee2 s a))) b ⤜ (λx. cond_spmf_fst (f x ⤜ (λs. callee1 s a)) b); cond_spmf_fst (X ⤜ (λy. g y ⤜ (λs. callee2 s a))) b = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs. callee2 s a))) b ⤜ (λx. cond_spmf_fst (g x ⤜ (λs. callee2 s a)) b); cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs. callee1 s a))) b = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs. callee2 s a))) b; p = X ⤜ f; q = X ⤜ g; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); a ∈ A⟧ ⟹ trace_eq_simcl S (cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs. callee2 s a))) b ⤜ (λx. cond_spmf_fst (f x ⤜ (λs. callee1 s a)) b)) (cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs. callee2 s a))) b ⤜ (λx. cond_spmf_fst (g x ⤜ (λs. callee2 s a)) b))›*)\n by (auto intro!: sim (*‹⟦S ?p ?q; ?a ∈ A; ?res ∈ set_spmf ?q; (?b, ?s') ∈ set_spmf (callee2 ?res ?a)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs. callee1 s ?a)) ?b) (cond_spmf_fst (?q ⤜ (λs. callee2 s ?a)) ?b)›*) simp add: bind_UNION (*‹?A ⤜ ?f = ⋃ (?f ` ?A)›*))\n qed\nqed\n\nproposition trace'_eqI_sim_upto:\n fixes callee1 :: \"('a, 'b, 's1) callee\" and callee2 :: \"('a, 'b, 's2) callee\"\n assumes start: \"S p q\"\n and step: \"⋀p q a. ⟦ S p q; a ∈ A ⟧ ⟹\n bind_spmf p (λs. map_spmf fst (callee1 s a)) = bind_spmf q (λs. map_spmf fst (callee2 s a))\"\n and sim: \"⋀p q a res b s'. ⟦ S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a) ⟧\n ⟹ trace_eq_simcl S (cond_spmf_fst (bind_spmf p (λs. callee1 s a)) b)\n (cond_spmf_fst (bind_spmf q (λs. callee2 s a)) b)\"\n shows \"trace_callee_eq callee1 callee2 A p q\"\nproof (-)\n (*goal: ‹trace_callee_eq callee1 callee2 A p q›*)\n let ?S = \"trace_eq_simcl S\"\n from start (*‹S p q›*) have \"?S p q\"\n by (rule trace_eq_simcl.base (*‹?S ?p ?q ⟹ trace_eq_simcl ?S ?p ?q›*))\n then show \"?thesis\"\n (*goal: ‹trace_callee_eq callee1 callee2 A p q›*)\n apply (rule trace'_eqI_sim (*‹⟦?S ?p ?q; ⋀p q a. ⟦?S p q; a ∈ ?A⟧ ⟹ p ⤜ (λs. map_spmf fst (?callee1.0 s a)) = q ⤜ (λs. map_spmf fst (?callee2.0 s a)); ⋀p q a res b s'. ⟦?S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (?callee2.0 res a)⟧ ⟹ ?S (cond_spmf_fst (p ⤜ (λs. ?callee1.0 s a)) b) (cond_spmf_fst (q ⤜ (λs. ?callee2.0 s a)) b)⟧ ⟹ trace_callee_eq ?callee1.0 ?callee2.0 ?A ?p ?q›*))\n (*goals:\n 1. ‹⋀p q a. ⟦trace_eq_simcl S p q; a ∈ A⟧ ⟹ p ⤜ (λs. map_spmf fst (callee1 s a)) = q ⤜ (λs. map_spmf fst (callee2 s a))›\n 2. ‹⋀p q a res b s'. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs. callee1 s a)) b) (cond_spmf_fst (q ⤜ (λs. callee2 s a)) b)›\n discuss goal 1*)\n apply (rule trace_eq_simcl_callee[OF step sim] (*‹⟦⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ S p q; ⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ S p q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?res1 p q a res b s' ∈ set_spmf q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ (b, ?s'1 p q a res b s') ∈ set_spmf (callee2 (?res1 p q a res b s') (?a3 a)); trace_eq_simcl S ?p ?q; ?a ∈ ?A⟧ ⟹ ?p ⤜ (λs. map_spmf fst (callee1 s (?a3 ?a))) = ?q ⤜ (λs. map_spmf fst (callee2 s (?a3 ?a)))› ‹⟦⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ S p q; ⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ S p q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?res1 p q a res b s' ∈ set_spmf q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ (b, ?s'1 p q a res b s') ∈ set_spmf (callee2 (?res1 p q a res b s') (?a3 a)); trace_eq_simcl S ?p ?q; ?a ∈ ?A; ?res ∈ set_spmf ?q; (?b, ?s') ∈ set_spmf (callee2 ?res (?a3 ?a))⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs. callee1 s (?a3 ?a))) ?b) (cond_spmf_fst (?q ⤜ (λs. callee2 s (?a3 ?a))) ?b)›*))\n (*goals:\n 1. ‹⋀(p::'s1 spmf) (q::'s2 spmf) (a::'a) (pa::'s1 spmf) (qa::'s2 spmf) aa::'a. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set); S pa qa; aa ∈ (?A3::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 'a set) p q a⟧ ⟹ S pa qa›\n 2. ‹⋀(p::'s1 spmf) (q::'s2 spmf) (a::'a) (pa::'s1 spmf) (qa::'s2 spmf) aa::'a. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set); S pa qa; aa ∈ (?A3::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 'a set) p q a⟧ ⟹ aa ∈ A›\n 3. ‹⋀(p::'s1 spmf) (q::'s2 spmf) (a::'a) (pa::'s1 spmf) (qa::'s2 spmf) (aa::'a) (res::'s2) (b::'b) s'::'s2. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set); S pa qa; aa ∈ (?A3::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 'a set) p q a; res ∈ set_spmf qa; (b, s') ∈ set_spmf ((callee2::'s2 ⇒ 'a ⇒ ('b × 's2) spmf) res aa)⟧ ⟹ S pa qa›\n 4. ‹⋀(p::'s1 spmf) (q::'s2 spmf) (a::'a) (pa::'s1 spmf) (qa::'s2 spmf) (aa::'a) (res::'s2) (b::'b) s'::'s2. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set); S pa qa; aa ∈ (?A3::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 'a set) p q a; res ∈ set_spmf qa; (b, s') ∈ set_spmf ((callee2::'s2 ⇒ 'a ⇒ ('b × 's2) spmf) res aa)⟧ ⟹ aa ∈ A›\n 5. ‹⋀(p::'s1 spmf) (q::'s2 spmf) (a::'a) (pa::'s1 spmf) (qa::'s2 spmf) (aa::'a) (res::'s2) (b::'b) s'::'s2. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set); S pa qa; aa ∈ (?A3::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 'a set) p q a; res ∈ set_spmf qa; (b, s') ∈ set_spmf ((callee2::'s2 ⇒ 'a ⇒ ('b × 's2) spmf) res aa)⟧ ⟹ (?res4::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 's1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 's2 ⇒ 'b ⇒ 's2 ⇒ 's2) p q a pa qa aa res b s' ∈ set_spmf qa›\n 6. ‹⋀(p::'s1 spmf) (q::'s2 spmf) (a::'a) (pa::'s1 spmf) (qa::'s2 spmf) (aa::'a) (res::'s2) (b::'b) s'::'s2. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set); S pa qa; aa ∈ (?A3::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 'a set) p q a; res ∈ set_spmf qa; (b, s') ∈ set_spmf ((callee2::'s2 ⇒ 'a ⇒ ('b × 's2) spmf) res aa)⟧ ⟹ (b, (?s'4::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 's1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 's2 ⇒ 'b ⇒ 's2 ⇒ 's2) p q a pa qa aa res b s') ∈ set_spmf (callee2 ((?res4::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 's1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 's2 ⇒ 'b ⇒ 's2 ⇒ 's2) p q a pa qa aa res b s') aa)›\n 7. ‹⋀(p::'s1 spmf) (q::'s2 spmf) a::'a. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set)⟧ ⟹ trace_eq_simcl S p q›\n 8. ‹⋀(p::'s1 spmf) (q::'s2 spmf) a::'a. ⟦trace_eq_simcl (S::'s1 spmf ⇒ 's2 spmf ⇒ bool) p q; a ∈ (A::'a set)⟧ ⟹ a ∈ (?A3::'s1 spmf ⇒ 's2 spmf ⇒ 'a ⇒ 'a set) p q a›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply assumption\n (*discuss goal 4*)\n apply assumption\n (*discuss goal 5*)\n apply assumption\n (*discuss goal 6*)\n apply assumption\n (*discuss goal 7*)\n apply assumption\n (*discuss goal 8*)\n apply assumption\n (*proven 8 subgoals*)\n (*discuss goal 2*)\n apply (rule trace_eq_simcl_callee[OF step sim] (*‹⟦⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ S p q; ⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ S p q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?res1 p q a res b s' ∈ set_spmf q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ (b, ?s'1 p q a res b s') ∈ set_spmf (callee2 (?res1 p q a res b s') (?a3 a)); trace_eq_simcl S ?p ?q; ?a ∈ ?A⟧ ⟹ ?p ⤜ (λs. map_spmf fst (callee1 s (?a3 ?a))) = ?q ⤜ (λs. map_spmf fst (callee2 s (?a3 ?a)))› ‹⟦⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ S p q; ⋀p q a. ⟦S p q; a ∈ ?A⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ S p q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?a3 a ∈ A; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ ?res1 p q a res b s' ∈ set_spmf q; ⋀p q a res b s'. ⟦S p q; a ∈ ?A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res (?a3 a))⟧ ⟹ (b, ?s'1 p q a res b s') ∈ set_spmf (callee2 (?res1 p q a res b s') (?a3 a)); trace_eq_simcl S ?p ?q; ?a ∈ ?A; ?res ∈ set_spmf ?q; (?b, ?s') ∈ set_spmf (callee2 ?res (?a3 ?a))⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs. callee1 s (?a3 ?a))) ?b) (cond_spmf_fst (?q ⤜ (λs. callee2 s (?a3 ?a))) ?b)›*))\n (*goals:\n 1. ‹⋀p q a res b s' pa qa aa. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a); S pa qa; aa ∈ ?A16 p q a res b s'⟧ ⟹ S pa qa›\n 2. ‹⋀p q a res b s' pa qa aa. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a); S pa qa; aa ∈ ?A16 p q a res b s'⟧ ⟹ aa ∈ A›\n 3. ‹⋀p q a res b s' pa qa aa resa ba s'a. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a); S pa qa; aa ∈ ?A16 p q a res b s'; resa ∈ set_spmf qa; (ba, s'a) ∈ set_spmf (callee2 resa aa)⟧ ⟹ S pa qa›\n 4. ‹⋀p q a res b s' pa qa aa resa ba s'a. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a); S pa qa; aa ∈ ?A16 p q a res b s'; resa ∈ set_spmf qa; (ba, s'a) ∈ set_spmf (callee2 resa aa)⟧ ⟹ aa ∈ A›\n 5. ‹⋀p q a res b s' pa qa aa resa ba s'a. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a); S pa qa; aa ∈ ?A16 p q a res b s'; resa ∈ set_spmf qa; (ba, s'a) ∈ set_spmf (callee2 resa aa)⟧ ⟹ ?res17 p q a res b s' pa qa aa resa ba s'a ∈ set_spmf qa›\n 6. ‹⋀p q a res b s' pa qa aa resa ba s'a. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a); S pa qa; aa ∈ ?A16 p q a res b s'; resa ∈ set_spmf qa; (ba, s'a) ∈ set_spmf (callee2 resa aa)⟧ ⟹ (ba, ?s'17 p q a res b s' pa qa aa resa ba s'a) ∈ set_spmf (callee2 (?res17 p q a res b s' pa qa aa resa ba s'a) aa)›\n 7. ‹⋀p q a res b s'. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a)⟧ ⟹ trace_eq_simcl S p q›\n 8. ‹⋀p q a res b s'. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a)⟧ ⟹ a ∈ ?A16 p q a res b s'›\n 9. ‹⋀p q a res b s'. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a)⟧ ⟹ ?res16 p q a res b s' ∈ set_spmf q›\n 10. ‹⋀p q a res b s'. ⟦trace_eq_simcl S p q; a ∈ A; res ∈ set_spmf q; (b, s') ∈ set_spmf (callee2 res a)⟧ ⟹ (b, ?s'16 p q a res b s') ∈ set_spmf (callee2 (?res16 p q a res b s') a)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply assumption\n (*discuss goal 4*)\n apply assumption\n (*discuss goal 5*)\n apply assumption\n (*discuss goal 6*)\n apply assumption\n (*discuss goal 7*)\n apply assumption\n (*discuss goal 8*)\n apply assumption\n (*discuss goal 9*)\n apply assumption\n (*discuss goal 10*)\n apply assumption\n (*proven 10 subgoals*)\n (*proven 2 subgoals*) .\nqed\n\nlemma trace_core_eq_simI_upto:\n fixes core1 :: \"('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and core2 :: \"('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and S :: \"'s_core spmf ⇒ 's_core' spmf ⇒ bool\"\n assumes start: \"S p q\"\n and step_cpoke: \"⋀p q e. ⟦ S p q; e ∈ E ⟧ ⟹ \n weight_spmf (bind_spmf p (λs. cpoke core1 s e)) = weight_spmf (bind_spmf q (λs. cpoke core2 s e))\"\n and sim_cpoke: \"⋀p q e. ⟦ S p q; e ∈ E ⟧ ⟹ \n trace_eq_simcl S (mk_lossless (bind_spmf p (λs. cpoke core1 s e))) (mk_lossless (bind_spmf q (λs. cpoke core2 s e)))\"\n and step_cfunc_adv: \"⋀p q ia. ⟦ S p q; ia ∈ IA ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = bind_spmf q (λs2. map_spmf fst (cfunc_adv core2 s2 ia))\"\n and sim_cfunc_adv: \"⋀p q ia s1 s2 s1' s2' oa. ⟦ S p q; ia ∈ IA; \n s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv core1 s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia) ⟧\n ⟹ trace_eq_simcl S (cond_spmf_fst (bind_spmf p (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (bind_spmf q (λs2. cfunc_adv core2 s2 ia)) oa)\"\n and step_cfunc_usr: \"⋀p q iu. ⟦ S p q; iu ∈ IU ⟧ ⟹ \n bind_spmf p (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = bind_spmf q (λs2. map_spmf fst (cfunc_usr core2 s2 iu))\"\n and sim_cfunc_usr: \"⋀p q iu s1 s2 s1' s2' ou. ⟦ S p q; iu ∈ IU; \n s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr core1 s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu) ⟧\n ⟹ trace_eq_simcl S (cond_spmf_fst (bind_spmf p (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (bind_spmf q (λs2. cfunc_usr core2 s2 iu)) ou)\"\n shows \"trace_core_eq core1 core2 E IA IU p q\"\nproof (-)\n (*goal: ‹trace_core_eq core1 core2 E IA IU p q›*)\n let ?S = \"trace_eq_simcl S\"\n from start (*‹S p q›*) have \"?S p q\"\n by (rule trace_eq_simcl.base (*‹?S ?p ?q ⟹ trace_eq_simcl ?S ?p ?q›*))\n then show \"?thesis\"\n (*goal: ‹trace_core_eq core1 core2 E IA IU p q›*)\n proof (rule trace_core_eq_simI (*‹⟦?S ?p ?q; ⋀p q e. ⟦?S p q; e ∈ ?E⟧ ⟹ weight_spmf (p ⤜ (λs. cpoke ?core1.0 s e)) = weight_spmf (q ⤜ (λs. cpoke ?core2.0 s e)); ⋀p q e. ⟦?S p q; e ∈ ?E⟧ ⟹ ?S (mk_lossless (p ⤜ (λs. cpoke ?core1.0 s e))) (mk_lossless (q ⤜ (λs. cpoke ?core2.0 s e))); ⋀p q ia. ⟦?S p q; ia ∈ ?IA⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_adv ?core1.0 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv ?core2.0 s2 ia)); ⋀p q ia s1 s2 s1' s2' oa. ⟦?S p q; ia ∈ ?IA; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv ?core1.0 s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv ?core2.0 s2 ia)⟧ ⟹ ?S (cond_spmf_fst (p ⤜ (λs1. cfunc_adv ?core1.0 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. cfunc_adv ?core2.0 s2 ia)) oa); ⋀p q iu. ⟦?S p q; iu ∈ ?IU⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_usr ?core1.0 s1 iu)) = q ⤜ (λs2. map_spmf fst (cfunc_usr ?core2.0 s2 iu)); ⋀p q iu s1 s2 s1' s2' ou. ⟦?S p q; iu ∈ ?IU; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr ?core1.0 s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr ?core2.0 s2 iu)⟧ ⟹ ?S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr ?core1.0 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr ?core2.0 s2 iu)) ou)⟧ ⟹ trace_core_eq ?core1.0 ?core2.0 ?E ?IA ?IU ?p ?q›*), goal_cases Step_cpoke Sim_cpoke Step_cfunc_adv Sim_cfunc_adv Step_cfunc_usr Sim_cfunc_usr)\n (*goals:\n 1. ‹⋀p q e. ⟦trace_eq_simcl S p q; e ∈ E⟧ ⟹ weight_spmf (p ⤜ (λs. cpoke core1 s e)) = weight_spmf (q ⤜ (λs. cpoke core2 s e))›\n 2. ‹⋀p q e. ⟦trace_eq_simcl S p q; e ∈ E⟧ ⟹ trace_eq_simcl S (mk_lossless (p ⤜ (λs. cpoke core1 s e))) (mk_lossless (q ⤜ (λs. cpoke core2 s e)))›\n 3. ‹⋀p q ia. ⟦trace_eq_simcl S p q; ia ∈ IA⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›\n 4. ‹⋀p q ia s1 s2 s1' s2' oa. ⟦trace_eq_simcl S p q; ia ∈ IA; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv core1 s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›\n 5. ‹⋀p q iu. ⟦trace_eq_simcl S p q; iu ∈ IU⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›\n 6. ‹⋀p q iu s1 s2 s1' s2' ou. ⟦trace_eq_simcl S p q; iu ∈ IU; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr core1 s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n {\n case (Step_cpoke p q e) (*‹trace_eq_simcl S p q› ‹e ∈ E›*)\n then show \"?case\"\n (*goal: ‹weight_spmf (p ⤜ (λs. cpoke core1 s e)) = weight_spmf (q ⤜ (λs. cpoke core2 s e))›*)\n using step_cpoke (*‹⟦S ?p ?q; ?e ∈ E⟧ ⟹ weight_spmf (?p ⤜ (λs. cpoke core1 s ?e)) = weight_spmf (?q ⤜ (λs. cpoke core2 s ?e))›*) apply cases\n (*goals:\n 1. ‹⟦(e::'event::type) ∈ (E::'event::type set); ⋀(p::'s_core::type spmf) (q::'s_core'::type spmf) e::'event::type. ⟦(S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) p q; e ∈ E⟧ ⟹ weight_spmf (p ⤜ (λs::'s_core::type. cpoke (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s e)) = weight_spmf (q ⤜ (λs::'s_core'::type. cpoke (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s e)); S (p::'s_core::type spmf) (q::'s_core'::type spmf)⟧ ⟹ weight_spmf (p ⤜ (λs::'s_core::type. cpoke core1 s e)) = weight_spmf (q ⤜ (λs::'s_core'::type. cpoke core2 s e))›\n 2. ‹⋀(p::nat spmf) (f::nat ⇒ 's_core::type spmf) g::nat ⇒ 's_core'::type spmf. ⟦(e::'event::type) ∈ (E::'event::type set); ⋀(p::'s_core::type spmf) (q::'s_core'::type spmf) e::'event::type. ⟦(S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) p q; e ∈ E⟧ ⟹ weight_spmf (p ⤜ (λs::'s_core::type. cpoke (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s e)) = weight_spmf (q ⤜ (λs::'s_core'::type. cpoke (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s e)); (p::'s_core::type spmf) = p ⤜ f; (q::'s_core'::type spmf) = p ⤜ g; ⋀x::nat. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ weight_spmf (p ⤜ (λs::'s_core::type. cpoke core1 s e)) = weight_spmf (q ⤜ (λs::'s_core'::type. cpoke core2 s e))›\n discuss goal 1*)\n apply ((auto simp add: weight_bind_spmf (*‹weight_spmf (?x ⤜ ?f) = integral⇧L (measure_spmf ?x) (weight_spmf ∘ ?f)›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) intro!: Bochner_Integration.integral_cong_AE (*‹⟦?f ∈ borel_measurable ?M; ?g ∈ borel_measurable ?M; AE x in ?M. ?f x = ?g x⟧ ⟹ integral⇧L ?M ?f = integral⇧L ?M ?g›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: weight_bind_spmf (*‹weight_spmf (?x ⤜ ?f) = integral⇧L (measure_spmf ?x) (weight_spmf ∘ ?f)›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) intro!: Bochner_Integration.integral_cong_AE (*‹⟦?f ∈ borel_measurable ?M; ?g ∈ borel_measurable ?M; AE x in ?M. ?f x = ?g x⟧ ⟹ integral⇧L ?M ?f = integral⇧L ?M ?g›*))[1])\n (*proven 2 subgoals*) .\n }\n note eq = this (*‹⟦trace_eq_simcl S ?pa2 ?qa2; ?e2 ∈ E⟧ ⟹ weight_spmf (?pa2 ⤜ (λs. cpoke core1 s ?e2)) = weight_spmf (?qa2 ⤜ (λs. cpoke core2 s ?e2))›*)\n case (Sim_cpoke p q e) (*‹trace_eq_simcl S p q› ‹e ∈ E›*)\n then show \"?case\"\n (*goal: ‹trace_eq_simcl (S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (mk_lossless ((p::'s_core::type spmf) ⤜ (λs::'s_core::type. cpoke (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s (e::'event::type)))) (mk_lossless ((q::'s_core'::type spmf) ⤜ (λs::'s_core'::type. cpoke (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s e)))›*)\n proof (cases)\n (*goals:\n 1. ‹⟦e ∈ E; S p q⟧ ⟹ trace_eq_simcl S (mk_lossless (p ⤜ (λs. cpoke core1 s e))) (mk_lossless (q ⤜ (λs. cpoke core2 s e)))›\n 2. ‹⋀p f g. ⟦e ∈ E; p = p ⤜ f; q = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ trace_eq_simcl S (mk_lossless (p ⤜ (λs. cpoke core1 s e))) (mk_lossless (q ⤜ (λs. cpoke core2 s e)))›*)\n case base (*‹S p q›*)\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl S (mk_lossless (p ⤜ (λs. cpoke core1 s e))) (mk_lossless (q ⤜ (λs. cpoke core2 s e)))›*)\n using Sim_cpoke(2) (*‹(e::'event) ∈ (E::'event set)›*) by (rule sim_cpoke (*‹⟦S ?p ?q; ?e ∈ E⟧ ⟹ trace_eq_simcl S (mk_lossless (?p ⤜ (λs. cpoke core1 s ?e))) (mk_lossless (?q ⤜ (λs. cpoke core2 s ?e)))›*))\n next\n (*goal: ‹⋀p f g. ⟦e ∈ E; p = p ⤜ f; q = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ trace_eq_simcl S (mk_lossless (p ⤜ (λs. cpoke core1 s e))) (mk_lossless (q ⤜ (λs. cpoke core2 s e)))›*)\n case (bind_nat X f g) (*‹p = X ⤜ f› ‹q = X ⤜ g› ‹(?x::nat) ∈ set_spmf (X::nat spmf) ⟹ (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) ((f::nat ⇒ 's_core spmf) ?x) ((g::nat ⇒ 's_core' spmf) ?x)›*)\n then have \"cond_bind_spmf X (λy. f y ⤜ (λs. cpoke core1 s e)) UNIV = cond_bind_spmf X (λy. g y ⤜ (λs. cpoke core2 s e)) UNIV\"\n using eq[OF Sim_cpoke] (*‹weight_spmf (p ⤜ (λs. cpoke core1 s e)) = weight_spmf (q ⤜ (λs. cpoke core2 s e))›*) step_cpoke (*‹⟦(S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (?p::'s_core spmf) (?q::'s_core' spmf); (?e::'event) ∈ (E::'event set)⟧ ⟹ weight_spmf (?p ⤜ (λs::'s_core. cpoke (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s ?e)) = weight_spmf (?q ⤜ (λs::'s_core'. cpoke (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s ?e))›*) Sim_cpoke (*‹trace_eq_simcl S p q› ‹e ∈ E›*) apply (intro spmf_eqI (*‹(⋀i. spmf ?p i = spmf ?q i) ⟹ ?p = ?q›*))\n (*goal: ‹cond_bind_spmf X (λy. f y ⤜ (λs. cpoke core1 s e)) UNIV = cond_bind_spmf X (λy. g y ⤜ (λs. cpoke core2 s e)) UNIV›*)\n by (simp add: weight_spmf_def (*‹weight_spmf ?p = Sigma_Algebra.measure (measure_spmf ?p) UNIV›*) measure_spmf_zero_iff (*‹(Sigma_Algebra.measure (measure_spmf ?p) ?A = 0) = (set_spmf ?p ∩ ?A = {})›*) bind_UNION (*‹?A ⤜ ?f = ⋃ (?f ` ?A)›*) spmf_eq_0_set_spmf (*‹(spmf ?p ?x = 0) = (?x ∉ set_spmf ?p)›*))\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl S (mk_lossless (p ⤜ (λs. cpoke core1 s e))) (mk_lossless (q ⤜ (λs. cpoke core2 s e)))›*)\n using bind_nat (*‹p = X ⤜ f› ‹q = X ⤜ g› ‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*) Sim_cpoke (*‹trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (p::'s_core spmf) (q::'s_core' spmf)› ‹(e::'event) ∈ (E::'event set)›*) sim_cpoke (*‹⟦S ?p ?q; ?e ∈ E⟧ ⟹ trace_eq_simcl S (mk_lossless (?p ⤜ (λs. cpoke core1 s ?e))) (mk_lossless (?q ⤜ (λs. cpoke core2 s ?e)))›*) by (auto simp add: cond_bind_spmf (*‹cond_spmf (?p ⤜ ?f) ?A = cond_bind_spmf ?p ?f ?A ⤜ (λx. cond_spmf (?f x) ?A)›*) cond_spmf_UNIV[symmetric] (*‹mk_lossless ?p = cond_spmf ?p UNIV›*) simp del: cond_spmf_UNIV (*‹cond_spmf ?p UNIV = mk_lossless ?p›*) intro: trace_eq_simcl_bind (*‹(⋀x. x ∈ set_spmf ?p ⟹ trace_eq_simcl ?S (?f x) (?g x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n qed\n next\n (*goals:\n 1. ‹⋀(p::'s_core spmf) (q::'s_core' spmf) ia::'iadv_core. ⟦trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) p q; ia ∈ (IA::'iadv_core set)⟧ ⟹ p ⤜ (λs1::'s_core. map_spmf fst (cfunc_adv (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 ia)) = q ⤜ (λs2::'s_core'. map_spmf fst (cfunc_adv (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 ia))›\n 2. ‹⋀(p::'s_core spmf) (q::'s_core' spmf) (ia::'iadv_core) (s1::'s_core) (s2::'s_core') (s1'::'s_core) (s2'::'s_core') oa::'oadv_core. ⟦trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) p q; ia ∈ (IA::'iadv_core set); s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 ia)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1::'s_core. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2::'s_core'. cfunc_adv core2 s2 ia)) oa)›\n 3. ‹⋀(p::'s_core spmf) (q::'s_core' spmf) iu::'iusr_core. ⟦trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) p q; iu ∈ (IU::'iusr_core set)⟧ ⟹ p ⤜ (λs1::'s_core. map_spmf fst (cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 iu)) = q ⤜ (λs2::'s_core'. map_spmf fst (cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 iu))›\n 4. ‹⋀(p::'s_core spmf) (q::'s_core' spmf) (iu::'iusr_core) (s1::'s_core) (s2::'s_core') (s1'::'s_core) (s2'::'s_core') ou::'ousr_core. ⟦trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) p q; iu ∈ (IU::'iusr_core set); s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 iu)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1::'s_core. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2::'s_core'. cfunc_usr core2 s2 iu)) ou)›*)\n {\n case (Step_cfunc_adv p q ia) (*‹trace_eq_simcl S p q› ‹ia ∈ IA›*)\n then show \"?case\"\n (*goal: ‹p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›*)\n using step_cfunc_adv (*‹⟦S ?p ?q; ?ia ∈ IA⟧ ⟹ ?p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia)) = ?q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia))›*) apply cases\n (*goals:\n 1. ‹⟦ia ∈ IA; ⋀p q ia. ⟦S p q; ia ∈ IA⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia)); S p q⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›\n 2. ‹⋀p f g. ⟦ia ∈ IA; ⋀p q ia. ⟦S p q; ia ∈ IA⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia)); p = p ⤜ f; q = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›\n discuss goal 1*)\n apply ((auto intro!: bind_spmf_cong (*‹⟦(?p::?'a spmf) = (?q::?'a spmf); ⋀x::?'a. x ∈ set_spmf ?q ⟹ (?f::?'a ⇒ ?'b spmf) x = (?g::?'a ⇒ ?'b spmf) x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*proven 2 subgoals*) .\n }\n note eq = this (*‹⟦trace_eq_simcl S ?pa2 ?qa2; ?ia2 ∈ IA⟧ ⟹ ?pa2 ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ?ia2)) = ?qa2 ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ?ia2))›*)\n case (Sim_cfunc_adv p q ia s1 s2 s1' s2' oa) (*‹trace_eq_simcl S p q› ‹ia ∈ IA› ‹(s1::'s_core) ∈ set_spmf (p::'s_core spmf)› ‹s2 ∈ set_spmf q› ‹(oa::'oadv_core, s1'::'s_core) ∈ set_spmf (cfunc_adv (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) (s1::'s_core) (ia::'iadv_core))› ‹(oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia)›*)\n then show \"?case\"\n (*goal: ‹trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (cond_spmf_fst ((p::'s_core spmf) ⤜ (λs1::'s_core. cfunc_adv (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 (ia::'iadv_core))) (oa::'oadv_core)) (cond_spmf_fst ((q::'s_core' spmf) ⤜ (λs2::'s_core'. cfunc_adv (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 ia)) oa)›*)\n proof (cases)\n (*goals:\n 1. ‹⟦ia ∈ IA; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv core1 s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia); S p q⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›\n 2. ‹⋀p f g. ⟦ia ∈ IA; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv core1 s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia); p = p ⤜ f; q = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›*)\n case base (*‹S p q›*)\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl (S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (cond_spmf_fst ((p::'s_core::type spmf) ⤜ (λs1::'s_core::type. cfunc_adv (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s1 (ia::'iadv_core::type))) (oa::'oadv_core::type)) (cond_spmf_fst ((q::'s_core'::type spmf) ⤜ (λs2::'s_core'::type. cfunc_adv (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s2 ia)) oa)›*)\n using Sim_cfunc_adv(2-) (*‹ia ∈ IA› ‹(s1::'s_core) ∈ set_spmf (p::'s_core spmf)› ‹s2 ∈ set_spmf q› ‹(oa, s1') ∈ set_spmf (cfunc_adv core1 s1 ia)› ‹(oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia)›*) by (rule sim_cfunc_adv (*‹⟦(S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (?p::'s_core spmf) (?q::'s_core' spmf); (?ia::'iadv_core) ∈ (IA::'iadv_core set); (?s1.0::'s_core) ∈ set_spmf ?p; (?s2.0::'s_core') ∈ set_spmf ?q; (?oa::'oadv_core, ?s1'::'s_core) ∈ set_spmf (cfunc_adv (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) ?s1.0 ?ia); (?oa, ?s2'::'s_core') ∈ set_spmf (cfunc_adv (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) ?s2.0 ?ia)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs1::'s_core. cfunc_adv core1 s1 ?ia)) ?oa) (cond_spmf_fst (?q ⤜ (λs2::'s_core'. cfunc_adv core2 s2 ?ia)) ?oa)›*))\n next\n (*goal: ‹⋀p f g. ⟦ia ∈ IA; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (oa, s1') ∈ set_spmf (cfunc_adv core1 s1 ia); (oa, s2') ∈ set_spmf (cfunc_adv core2 s2 ia); p = p ⤜ f; q = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›*)\n case (bind_nat X f g) (*‹p = X ⤜ f› ‹q = X ⤜ g› ‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*)\n then have \"cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa =\n cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa\"\n using eq[OF Sim_cfunc_adv ( 1 , 2 )] (*‹p ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)) = q ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))›*) apply (intro spmf_eqI (*‹(⋀i. spmf ?p i = spmf ?q i) ⟹ ?p = ?q›*))\n (*goal: ‹cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa›*)\n by (fastforce simp add: map_bind_spmf (*‹map_spmf (?f::?'b ⇒ ?'a) ((?p::?'c spmf) ⤜ (?g::?'c ⇒ ?'b spmf)) = ?p ⤜ map_spmf ?f ∘ ?g›*) o_def (*‹(?f::?'b ⇒ ?'c) ∘ (?g::?'a ⇒ ?'b) = (λx::?'a. ?f (?g x))›*) spmf_eq_0_set_spmf (*‹(spmf (?p::?'a spmf) (?x::?'a) = (0::real)) = (?x ∉ set_spmf ?p)›*) dest: step_cfunc_adv[OF _ Sim_cfunc_adv(2)] (*‹(S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (?p::'s_core spmf) (?q::'s_core' spmf) ⟹ ?p ⤜ (λs1::'s_core. map_spmf fst (cfunc_adv (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 (ia::'iadv_core))) = ?q ⤜ (λs2::'s_core'. map_spmf fst (cfunc_adv (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 ia))›*))\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›*)\n using bind_nat(3-) (*‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*) Sim_cfunc_adv(1-2) (*‹trace_eq_simcl S p q› ‹ia ∈ IA›*) unfolding bind_nat(1,2) bind_spmf_assoc\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (X ⤜ (λy. f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa) (cond_spmf_fst (X ⤜ (λy. g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa)›*)\n apply (subst (1 2) cond_spmf_fst_bind (*‹cond_spmf_fst (?p ⤜ ?f) ?x = cond_bind_spmf_fst ?p (map_spmf fst ∘ ?f) ?x ⤜ (λy. cond_spmf_fst (?f y) ?x)›*))\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (X ⤜ (λy. f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa) (cond_spmf_fst (X ⤜ (λy. g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa)›*)\n apply (simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n (*goal: ‹⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); ia ∈ IA⟧ ⟹ trace_eq_simcl S (cond_bind_spmf_fst X (map_spmf fst ∘ (λy. f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa ⤜ (λy. cond_spmf_fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia)) oa)) (cond_bind_spmf_fst X (map_spmf fst ∘ (λy. g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa ⤜ (λy. cond_spmf_fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia)) oa))›*)\n apply (rule trace_eq_simcl_bind (*‹(⋀x::?'a. x ∈ set_spmf (?p::?'a spmf) ⟹ trace_eq_simcl (?S::?'b spmf ⇒ ?'c spmf ⇒ bool) ((?f::?'a ⇒ ?'b spmf) x) ((?g::?'a ⇒ ?'c spmf) x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); ia ∈ IA⟧ ⟹ trace_eq_simcl S (cond_bind_spmf_fst X (λx. map_spmf fst (g x ⤜ (λs2. cfunc_adv core2 s2 ia))) oa ⤜ (λy. cond_spmf_fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia)) oa)) (cond_bind_spmf_fst X (λx. map_spmf fst (g x ⤜ (λs2. cfunc_adv core2 s2 ia))) oa ⤜ (λy. cond_spmf_fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia)) oa))›*)\n apply clarsimp\n (*goal: ‹⋀x. ⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); ia ∈ IA; x ∈ set_spmf (cond_bind_spmf_fst X (λx. map_spmf fst (g x ⤜ (λs2. cfunc_adv core2 s2 ia))) oa)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (g x ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›*)\n apply (frule step_cfunc_adv[OF bind_nat(3) Sim_cfunc_adv(2), THEN arg_cong[where f=\"set_spmf\"], THEN equalityD2] (*‹(?x4::nat) ∈ set_spmf (X::nat spmf) ⟹ set_spmf ((g::nat ⇒ 's_core'::type spmf) ?x4 ⤜ (λs2::'s_core'::type. map_spmf fst (cfunc_adv (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s2 (ia::'iadv_core::type)))) ⊆ set_spmf ((f::nat ⇒ 's_core::type spmf) ?x4 ⤜ (λs1::'s_core::type. map_spmf fst (cfunc_adv (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s1 ia)))›*))\n (*goal: ‹⋀x b xa. ⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); ia ∈ IA; x ∈ set_spmf X; xa ∈ set_spmf (g x); (oa, b) ∈ set_spmf (cfunc_adv core2 xa ia)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (g x ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›*)\n apply (clarsimp simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) bind_UNION (*‹?A ⤜ ?f = ⋃ (?f ` ?A)›*))\n (*goal: ‹⋀x b xa. ⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_adv core1 s1 ia))) oa = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_adv core2 s2 ia))) oa; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); ia ∈ IA; x ∈ set_spmf X; xa ∈ set_spmf (g x); (oa, b) ∈ set_spmf (cfunc_adv core2 xa ia); set_spmf (g x ⤜ (λs2. map_spmf fst (cfunc_adv core2 s2 ia))) ⊆ set_spmf (f x ⤜ (λs1. map_spmf fst (cfunc_adv core1 s1 ia)))⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (g x ⤜ (λs2. cfunc_adv core2 s2 ia)) oa)›*)\n apply (drule subsetD (*‹⟦?A ⊆ ?B; ?c ∈ ?A⟧ ⟹ ?c ∈ ?B›*))\n (*goals:\n 1. ‹⋀(x::nat) (b::'s_core'::type) xa::'s_core'::type. ⟦cond_bind_spmf_fst (X::nat spmf) (λy::nat. map_spmf fst ((f::nat ⇒ 's_core::type spmf) y ⤜ (λs1::'s_core::type. cfunc_adv (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s1 (ia::'iadv_core::type)))) (oa::'oadv_core::type) = cond_bind_spmf_fst X (λy::nat. map_spmf fst ((g::nat ⇒ 's_core'::type spmf) y ⤜ (λs2::'s_core'::type. cfunc_adv (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s2 ia))) oa; ⋀x::nat. x ∈ set_spmf X ⟹ (S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); ia ∈ (IA::'iadv_core::type set); x ∈ set_spmf X; xa ∈ set_spmf (g x); (oa, b) ∈ set_spmf (cfunc_adv core2 xa ia)⟧ ⟹ (?c26::nat ⇒ 's_core'::type ⇒ 's_core'::type ⇒ 'oadv_core::type) x b xa ∈ (⋃x::'s_core'::type∈set_spmf (g x). fst ` set_spmf (cfunc_adv core2 x ia))›\n 2. ‹⋀(x::nat) (b::'s_core'::type) xa::'s_core'::type. ⟦cond_bind_spmf_fst (X::nat spmf) (λy::nat. map_spmf fst ((f::nat ⇒ 's_core::type spmf) y ⤜ (λs1::'s_core::type. cfunc_adv (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s1 (ia::'iadv_core::type)))) (oa::'oadv_core::type) = cond_bind_spmf_fst X (λy::nat. map_spmf fst ((g::nat ⇒ 's_core'::type spmf) y ⤜ (λs2::'s_core'::type. cfunc_adv (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s2 ia))) oa; ⋀x::nat. x ∈ set_spmf X ⟹ (S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); ia ∈ (IA::'iadv_core::type set); x ∈ set_spmf X; xa ∈ set_spmf (g x); (oa, b) ∈ set_spmf (cfunc_adv core2 xa ia); (?c26::nat ⇒ 's_core'::type ⇒ 's_core'::type ⇒ 'oadv_core::type) x b xa ∈ (⋃x::'s_core::type∈set_spmf (f x). fst ` set_spmf (cfunc_adv core1 x ia))⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1::'s_core::type. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (g x ⤜ (λs2::'s_core'::type. cfunc_adv core2 s2 ia)) oa)›\n discuss goal 1*)\n apply fastforce\n (*discuss goal 2*)\n apply (auto intro: sim_cfunc_adv (*‹⟦(S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (?p::'s_core::type spmf) (?q::'s_core'::type spmf); (?ia::'iadv_core::type) ∈ (IA::'iadv_core::type set); (?s1.0::'s_core::type) ∈ set_spmf ?p; (?s2.0::'s_core'::type) ∈ set_spmf ?q; (?oa::'oadv_core::type, ?s1'::'s_core::type) ∈ set_spmf (cfunc_adv (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) ?s1.0 ?ia); (?oa, ?s2'::'s_core'::type) ∈ set_spmf (cfunc_adv (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) ?s2.0 ?ia)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs1::'s_core::type. cfunc_adv core1 s1 ?ia)) ?oa) (cond_spmf_fst (?q ⤜ (λs2::'s_core'::type. cfunc_adv core2 s2 ?ia)) ?oa)›*))\n (*proven 2 subgoals*) .\n qed\n next\n (*goals:\n 1. ‹⋀p q iu. ⟦trace_eq_simcl S p q; iu ∈ IU⟧ ⟹ p ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›\n 2. ‹⋀p q iu s1 s2 s1' s2' ou. ⟦trace_eq_simcl S p q; iu ∈ IU; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr core1 s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n {\n case (Step_cfunc_usr p q iu) (*‹trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (p::'s_core spmf) (q::'s_core' spmf)› ‹(iu::'iusr_core) ∈ (IU::'iusr_core set)›*)\n then show \"?case\"\n (*goal: ‹p ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›*)\n using step_cfunc_usr (*‹⟦S ?p ?q; ?iu ∈ IU⟧ ⟹ ?p ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu)) = ?q ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu))›*) apply cases\n (*goals:\n 1. ‹⟦(iu::'iusr_core) ∈ (IU::'iusr_core set); ⋀(p::'s_core spmf) (q::'s_core' spmf) iu::'iusr_core. ⟦(S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) p q; iu ∈ IU⟧ ⟹ p ⤜ (λs1::'s_core. map_spmf fst (cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 iu)) = q ⤜ (λs2::'s_core'. map_spmf fst (cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 iu)); S (p::'s_core spmf) (q::'s_core' spmf)⟧ ⟹ p ⤜ (λs1::'s_core. map_spmf fst (cfunc_usr core1 s1 iu)) = q ⤜ (λs2::'s_core'. map_spmf fst (cfunc_usr core2 s2 iu))›\n 2. ‹⋀(p::nat spmf) (f::nat ⇒ 's_core spmf) g::nat ⇒ 's_core' spmf. ⟦(iu::'iusr_core) ∈ (IU::'iusr_core set); ⋀(p::'s_core spmf) (q::'s_core' spmf) iu::'iusr_core. ⟦(S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) p q; iu ∈ IU⟧ ⟹ p ⤜ (λs1::'s_core. map_spmf fst (cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 iu)) = q ⤜ (λs2::'s_core'. map_spmf fst (cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 iu)); (p::'s_core spmf) = p ⤜ f; (q::'s_core' spmf) = p ⤜ g; ⋀x::nat. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ p ⤜ (λs1::'s_core. map_spmf fst (cfunc_usr core1 s1 iu)) = q ⤜ (λs2::'s_core'. map_spmf fst (cfunc_usr core2 s2 iu))›\n discuss goal 1*)\n apply ((auto intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*proven 2 subgoals*) .\n }\n note eq = this (*‹⟦trace_eq_simcl S ?pa2 ?qa2; ?iu2 ∈ IU⟧ ⟹ ?pa2 ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 ?iu2)) = ?qa2 ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 ?iu2))›*)\n case (Sim_cfunc_usr p q iu s1 s2 s1' s2' ou) (*‹trace_eq_simcl S p q› ‹(iu::'iusr_core) ∈ (IU::'iusr_core set)› ‹s1 ∈ set_spmf p› ‹s2 ∈ set_spmf q› ‹(ou, s1') ∈ set_spmf (cfunc_usr core1 s1 iu)› ‹(ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu)›*)\n then show \"?case\"\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n proof (cases)\n (*goals:\n 1. ‹⟦iu ∈ IU; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr core1 s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu); S p q⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›\n 2. ‹⋀p f g. ⟦iu ∈ IU; s1 ∈ set_spmf p; s2 ∈ set_spmf q; (ou, s1') ∈ set_spmf (cfunc_usr core1 s1 iu); (ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu); p = p ⤜ f; q = p ⤜ g; ⋀x. x ∈ set_spmf p ⟹ S (f x) (g x)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n case base (*‹S p q›*)\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (cond_spmf_fst ((p::'s_core spmf) ⤜ (λs1::'s_core. cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 (iu::'iusr_core))) (ou::'ousr_core)) (cond_spmf_fst ((q::'s_core' spmf) ⤜ (λs2::'s_core'. cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 iu)) ou)›*)\n using Sim_cfunc_usr(2-) (*‹iu ∈ IU› ‹s1 ∈ set_spmf p› ‹s2 ∈ set_spmf q› ‹(ou::'ousr_core::type, s1'::'s_core::type) ∈ set_spmf (cfunc_usr (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) (s1::'s_core::type) (iu::'iusr_core::type))› ‹(ou, s2') ∈ set_spmf (cfunc_usr core2 s2 iu)›*) by (rule sim_cfunc_usr (*‹⟦(S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (?p::'s_core spmf) (?q::'s_core' spmf); (?iu::'iusr_core) ∈ (IU::'iusr_core set); (?s1.0::'s_core) ∈ set_spmf ?p; (?s2.0::'s_core') ∈ set_spmf ?q; (?ou::'ousr_core, ?s1'::'s_core) ∈ set_spmf (cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) ?s1.0 ?iu); (?ou, ?s2'::'s_core') ∈ set_spmf (cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) ?s2.0 ?iu)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs1::'s_core. cfunc_usr core1 s1 ?iu)) ?ou) (cond_spmf_fst (?q ⤜ (λs2::'s_core'. cfunc_usr core2 s2 ?iu)) ?ou)›*))\n next\n (*goal: ‹⋀(p::nat spmf) (f::nat ⇒ 's_core::type spmf) g::nat ⇒ 's_core'::type spmf. ⟦(iu::'iusr_core::type) ∈ (IU::'iusr_core::type set); (s1::'s_core::type) ∈ set_spmf (p::'s_core::type spmf); (s2::'s_core'::type) ∈ set_spmf (q::'s_core'::type spmf); (ou::'ousr_core::type, s1'::'s_core::type) ∈ set_spmf (cfunc_usr (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s1 iu); (ou, s2'::'s_core'::type) ∈ set_spmf (cfunc_usr (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s2 iu); p = p ⤜ f; q = p ⤜ g; ⋀x::nat. x ∈ set_spmf p ⟹ (S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (f x) (g x)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1::'s_core::type. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2::'s_core'::type. cfunc_usr core2 s2 iu)) ou)›*)\n case (bind_nat X f g) (*‹p = X ⤜ f› ‹q = X ⤜ g› ‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*)\n then have \"cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou =\n cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou\"\n using eq[OF Sim_cfunc_usr ( 1 , 2 )] (*‹p ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = q ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›*) apply (intro spmf_eqI (*‹(⋀i. spmf ?p i = spmf ?q i) ⟹ ?p = ?q›*))\n (*goal: ‹cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou›*)\n by (fastforce simp add: map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) spmf_eq_0_set_spmf (*‹(spmf ?p ?x = 0) = (?x ∉ set_spmf ?p)›*) dest: step_cfunc_usr[OF _ Sim_cfunc_usr(2)] (*‹S ?p ?q ⟹ ?p ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)) = ?q ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))›*))\n then show \"?thesis\"\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (p ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n using bind_nat(3-) (*‹?x ∈ set_spmf X ⟹ S (f ?x) (g ?x)›*) Sim_cfunc_usr(1-2) (*‹trace_eq_simcl (S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (p::'s_core::type spmf) (q::'s_core'::type spmf)› ‹(iu::'iusr_core) ∈ (IU::'iusr_core set)›*) unfolding bind_nat(1,2) bind_spmf_assoc\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (X ⤜ (λy. f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou) (cond_spmf_fst (X ⤜ (λy. g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou)›*)\n apply (subst (1 2) cond_spmf_fst_bind (*‹cond_spmf_fst (?p ⤜ ?f) ?x = cond_bind_spmf_fst ?p (map_spmf fst ∘ ?f) ?x ⤜ (λy. cond_spmf_fst (?f y) ?x)›*))\n (*goal: ‹trace_eq_simcl S (cond_spmf_fst (X ⤜ (λy. f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou) (cond_spmf_fst (X ⤜ (λy. g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou)›*)\n apply (simp add: o_def (*‹(?f::?'b ⇒ ?'c) ∘ (?g::?'a ⇒ ?'b) = (λx::?'a. ?f (?g x))›*))\n (*goal: ‹⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); iu ∈ IU⟧ ⟹ trace_eq_simcl S (cond_bind_spmf_fst X (map_spmf fst ∘ (λy. f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou ⤜ (λy. cond_spmf_fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu)) ou)) (cond_bind_spmf_fst X (map_spmf fst ∘ (λy. g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou ⤜ (λy. cond_spmf_fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu)) ou))›*)\n apply (rule trace_eq_simcl_bind (*‹(⋀x. x ∈ set_spmf ?p ⟹ trace_eq_simcl ?S (?f x) (?g x)) ⟹ trace_eq_simcl ?S (?p ⤜ ?f) (?p ⤜ ?g)›*))\n (*goal: ‹⟦cond_bind_spmf_fst (X::nat spmf) (λy::nat. map_spmf fst ((f::nat ⇒ 's_core::type spmf) y ⤜ (λs1::'s_core::type. cfunc_usr (core1::('s_core::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s1 (iu::'iusr_core::type)))) (ou::'ousr_core::type) = cond_bind_spmf_fst X (λy::nat. map_spmf fst ((g::nat ⇒ 's_core'::type spmf) y ⤜ (λs2::'s_core'::type. cfunc_usr (core2::('s_core'::type, 'event::type, 'iadv_core::type, 'iusr_core::type, 'oadv_core::type, 'ousr_core::type) core) s2 iu))) ou; ⋀x::nat. x ∈ set_spmf X ⟹ (S::'s_core::type spmf ⇒ 's_core'::type spmf ⇒ bool) (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); iu ∈ (IU::'iusr_core::type set)⟧ ⟹ trace_eq_simcl S (cond_bind_spmf_fst X (λx::nat. map_spmf fst (g x ⤜ (λs2::'s_core'::type. cfunc_usr core2 s2 iu))) ou ⤜ (λy::nat. cond_spmf_fst (f y ⤜ (λs1::'s_core::type. cfunc_usr core1 s1 iu)) ou)) (cond_bind_spmf_fst X (λx::nat. map_spmf fst (g x ⤜ (λs2::'s_core'::type. cfunc_usr core2 s2 iu))) ou ⤜ (λy::nat. cond_spmf_fst (g y ⤜ (λs2::'s_core'::type. cfunc_usr core2 s2 iu)) ou))›*)\n apply clarsimp\n (*goal: ‹⋀x. ⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); iu ∈ IU; x ∈ set_spmf (cond_bind_spmf_fst X (λx. map_spmf fst (g x ⤜ (λs2. cfunc_usr core2 s2 iu))) ou)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (g x ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n apply (frule step_cfunc_usr[OF bind_nat(3) Sim_cfunc_usr(2), THEN arg_cong[where f=\"set_spmf\"], THEN equalityD2] (*‹?x4 ∈ set_spmf X ⟹ set_spmf (g ?x4 ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))) ⊆ set_spmf (f ?x4 ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)))›*))\n (*goal: ‹⋀(x::nat) (b::'s_core') xa::'s_core'. ⟦cond_bind_spmf_fst (X::nat spmf) (λy::nat. map_spmf fst ((f::nat ⇒ 's_core spmf) y ⤜ (λs1::'s_core. cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s1 (iu::'iusr_core)))) (ou::'ousr_core) = cond_bind_spmf_fst X (λy::nat. map_spmf fst ((g::nat ⇒ 's_core' spmf) y ⤜ (λs2::'s_core'. cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) s2 iu))) ou; ⋀x::nat. x ∈ set_spmf X ⟹ (S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); iu ∈ (IU::'iusr_core set); x ∈ set_spmf X; xa ∈ set_spmf (g x); (ou, b) ∈ set_spmf (cfunc_usr core2 xa iu)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1::'s_core. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (g x ⤜ (λs2::'s_core'. cfunc_usr core2 s2 iu)) ou)›*)\n apply (clarsimp simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) bind_UNION (*‹?A ⤜ ?f = ⋃ (?f ` ?A)›*))\n (*goal: ‹⋀x b xa. ⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); iu ∈ IU; x ∈ set_spmf X; xa ∈ set_spmf (g x); (ou, b) ∈ set_spmf (cfunc_usr core2 xa iu); set_spmf (g x ⤜ (λs2. map_spmf fst (cfunc_usr core2 s2 iu))) ⊆ set_spmf (f x ⤜ (λs1. map_spmf fst (cfunc_usr core1 s1 iu)))⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (g x ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›*)\n apply (drule subsetD (*‹⟦?A ⊆ ?B; ?c ∈ ?A⟧ ⟹ ?c ∈ ?B›*))\n (*goals:\n 1. ‹⋀x b xa. ⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); iu ∈ IU; x ∈ set_spmf X; xa ∈ set_spmf (g x); (ou, b) ∈ set_spmf (cfunc_usr core2 xa iu)⟧ ⟹ ?c26 x b xa ∈ (⋃x∈set_spmf (g x). fst ` set_spmf (cfunc_usr core2 x iu))›\n 2. ‹⋀x b xa. ⟦cond_bind_spmf_fst X (λy. map_spmf fst (f y ⤜ (λs1. cfunc_usr core1 s1 iu))) ou = cond_bind_spmf_fst X (λy. map_spmf fst (g y ⤜ (λs2. cfunc_usr core2 s2 iu))) ou; ⋀x. x ∈ set_spmf X ⟹ S (f x) (g x); trace_eq_simcl S (X ⤜ f) (X ⤜ g); iu ∈ IU; x ∈ set_spmf X; xa ∈ set_spmf (g x); (ou, b) ∈ set_spmf (cfunc_usr core2 xa iu); ?c26 x b xa ∈ (⋃x∈set_spmf (f x). fst ` set_spmf (cfunc_usr core1 x iu))⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (f x ⤜ (λs1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (g x ⤜ (λs2. cfunc_usr core2 s2 iu)) ou)›\n discuss goal 1*)\n apply fastforce\n (*discuss goal 2*)\n apply (auto intro: sim_cfunc_usr (*‹⟦(S::'s_core spmf ⇒ 's_core' spmf ⇒ bool) (?p::'s_core spmf) (?q::'s_core' spmf); (?iu::'iusr_core) ∈ (IU::'iusr_core set); (?s1.0::'s_core) ∈ set_spmf ?p; (?s2.0::'s_core') ∈ set_spmf ?q; (?ou::'ousr_core, ?s1'::'s_core) ∈ set_spmf (cfunc_usr (core1::('s_core, 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) ?s1.0 ?iu); (?ou, ?s2'::'s_core') ∈ set_spmf (cfunc_usr (core2::('s_core', 'event, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core) ?s2.0 ?iu)⟧ ⟹ trace_eq_simcl S (cond_spmf_fst (?p ⤜ (λs1::'s_core. cfunc_usr core1 s1 ?iu)) ?ou) (cond_spmf_fst (?q ⤜ (λs2::'s_core'. cfunc_usr core2 s2 ?iu)) ?ou)›*))\n (*proven 2 subgoals*) .\n qed\n qed\nqed\n\n\n\ncontext\n fixes core :: \"('s_core, 'event1 + 'event2, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and rest :: \"('s_rest, 'event2, 'iadv_rest, 'iusr_rest, 'oadv_rest, 'ousr_rest, 'more) rest_scheme\" \nbegin\n\nprimcorec core_with_rest :: \n \"('s_core × 's_rest, 'event1, 'iadv_core + 'iadv_rest, 'iusr_core + 'iusr_rest, 'oadv_core + 'oadv_rest, 'ousr_core + 'ousr_rest) core\" \n where\n \"cpoke core_with_rest = (λ(s_core, s_rest) e. map_spmf (λs_core'. (s_core', s_rest)) (cpoke core s_core (Inl e)))\"\n | \"cfunc_adv core_with_rest = (λ(s_core, s_rest) iadv. case iadv of\n Inl iadv_core ⇒ map_spmf (λ(oadv_core, s_core'). (Inl oadv_core, (s_core', s_rest))) (cfunc_adv core s_core iadv_core)\n | Inr iadv_rest ⇒ \n bind_spmf (rfunc_adv rest s_rest iadv_rest) (λ((oadv_rest, es), s_rest'). \n map_spmf (λs_core'. (Inr oadv_rest, (s_core', s_rest'))) (foldl_spmf (cpoke core) (return_spmf s_core) (map Inr es))))\"\n | \"cfunc_usr core_with_rest = (λ(s_core, s_rest) iusr. case iusr of\n Inl iusr_core ⇒ map_spmf (λ(ousr_core, s_core'). (Inl ousr_core, (s_core', s_rest))) (cfunc_usr core s_core iusr_core)\n | Inr iusr_rest ⇒\n bind_spmf (rfunc_usr rest s_rest iusr_rest) (λ((ousr_rest, es), s_rest').\n map_spmf (λs_core'. (Inr ousr_rest, (s_core', s_rest'))) (foldl_spmf (cpoke core) (return_spmf s_core) (map Inr es))))\"\n\nend\n\nlemma fuse_core_with_rest:\n fixes core :: \"('s_core, 'event1 + 'event2, 'iadv_core, 'iusr_core, 'oadv_core, 'ousr_core) core\"\n and rest1 :: \"('s_rest1, 'event1, 'iadv_rest1, 'iusr_rest1, 'oadv_rest1, 'ousr_rest1, 'more1) rest_scheme\"\n and rest2 :: \"('s_rest2, 'event2, 'iadv_rest2, 'iusr_rest2, 'oadv_rest2, 'ousr_rest2, 'more2) rest_scheme\"\n shows\n \"fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, (s_rest1, s_rest2)) = \n map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl)))\n (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1))\"\n apply (rule ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, s_rest1, s_rest2) = map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl))) (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1))›*)\n subgoal for x\n apply (cases \"(parallel_rest rest1 rest2, (s_core, (s_rest1, s_rest2)), x)\" rule: fused_resource.fuse.cases (*‹⟦⋀(rest::(?'s_rest, ?'event, ?'iadv_rest, ?'iusr_rest, ?'oadv_rest, ?'ousr_rest, ?'m) rest_scheme) (state::?'s_core × ?'s_rest) iadv_core::?'iadv_core. (?x::(?'s_rest, ?'event, ?'iadv_rest, ?'iusr_rest, ?'oadv_rest, ?'ousr_rest, ?'m) rest_scheme × (?'s_core × ?'s_rest) × ((?'iadv_core + ?'iadv_rest) + ?'iusr_core + ?'iusr_rest)) = (rest, state, Inl (Inl iadv_core)) ⟹ ?P::bool; ⋀(rest::(?'s_rest, ?'event, ?'iadv_rest, ?'iusr_rest, ?'oadv_rest, ?'ousr_rest, ?'m) rest_scheme) (state::?'s_core × ?'s_rest) iadv_rest::?'iadv_rest. ?x = (rest, state, Inl (Inr iadv_rest)) ⟹ ?P; ⋀(rest::(?'s_rest, ?'event, ?'iadv_rest, ?'iusr_rest, ?'oadv_rest, ?'ousr_rest, ?'m) rest_scheme) (state::?'s_core × ?'s_rest) iusr_core::?'iusr_core. ?x = (rest, state, Inr (Inl iusr_core)) ⟹ ?P; ⋀(rest::(?'s_rest, ?'event, ?'iadv_rest, ?'iusr_rest, ?'oadv_rest, ?'ousr_rest, ?'m) rest_scheme) (state::?'s_core × ?'s_rest) iusr_rest::?'iusr_rest. ?x = (rest, state, Inr (Inr iusr_rest)) ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀rest state iadv_core. (parallel_rest rest1 rest2, (s_core, s_rest1, s_rest2), x) = (rest, state, Inl (Inl iadv_core)) ⟹ fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, s_rest1, s_rest2) x = map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl))) (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1)) x›\n 2. ‹⋀rest state iadv_rest. (parallel_rest rest1 rest2, (s_core, s_rest1, s_rest2), x) = (rest, state, Inl (Inr iadv_rest)) ⟹ fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, s_rest1, s_rest2) x = map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl))) (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1)) x›\n 3. ‹⋀rest state iusr_core. (parallel_rest rest1 rest2, (s_core, s_rest1, s_rest2), x) = (rest, state, Inr (Inl iusr_core)) ⟹ fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, s_rest1, s_rest2) x = map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl))) (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1)) x›\n 4. ‹⋀rest state iusr_rest. (parallel_rest rest1 rest2, (s_core, s_rest1, s_rest2), x) = (rest, state, Inr (Inr iusr_rest)) ⟹ fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, s_rest1, s_rest2) x = map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl))) (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1)) x›\n discuss goal 1*)\n apply ((auto simp add: fused_resource.fuse.simps (*‹fused_resource.fuse ?core ?rest ?state (Inl (Inl ?iadv_core)) = (let handle = map_prod (Inl ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_adv ?core (fst ?state) ?iadv_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inl (Inr ?iadv_rest)) = rfunc_adv ?rest (snd ?state) ?iadv_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inl ∘ Inr) o_rfunc, s_notify, s_rfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inl ?iusr_core)) = (let handle = map_prod (Inr ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_usr ?core (fst ?state) ?iusr_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inr ?iusr_rest)) = rfunc_usr ?rest (snd ?state) ?iusr_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inr ∘ Inr) o_rfunc, s_notify, s_rfunc)))›*) map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) map_prod_def (*‹map_prod ?f ?g = (λ(x, y). (?f x, ?g y))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) parallel_eoracle_def (*‹parallel_eoracle ?eoracle1.0 ?eoracle2.0 ?state ≡ map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) ∘ (?eoracle1.0 ‡⇩O ?eoracle2.0) ?state›*) parallel_oracle_def (*‹?left ‡⇩O ?right = (λ(s1, s2). case_sum (λa. map_spmf (map_prod Inl (λs1'. (s1', s2))) (?left s1 a)) (λb. map_spmf (map_prod Inr (Pair s1)) (?right s2 b)))›*) split!: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*) intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: fused_resource.fuse.simps (*‹fused_resource.fuse ?core ?rest ?state (Inl (Inl ?iadv_core)) = (let handle = map_prod (Inl ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_adv ?core (fst ?state) ?iadv_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inl (Inr ?iadv_rest)) = rfunc_adv ?rest (snd ?state) ?iadv_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inl ∘ Inr) o_rfunc, s_notify, s_rfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inl ?iusr_core)) = (let handle = map_prod (Inr ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_usr ?core (fst ?state) ?iusr_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inr ?iusr_rest)) = rfunc_usr ?rest (snd ?state) ?iusr_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inr ∘ Inr) o_rfunc, s_notify, s_rfunc)))›*) map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) map_prod_def (*‹map_prod ?f ?g = (λ(x, y). (?f x, ?g y))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) parallel_eoracle_def (*‹parallel_eoracle ?eoracle1.0 ?eoracle2.0 ?state ≡ map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) ∘ (?eoracle1.0 ‡⇩O ?eoracle2.0) ?state›*) parallel_oracle_def (*‹?left ‡⇩O ?right = (λ(s1, s2). case_sum (λa. map_spmf (map_prod Inl (λs1'. (s1', s2))) (?left s1 a)) (λb. map_spmf (map_prod Inr (Pair s1)) (?right s2 b)))›*) split!: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*) intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*top goal: ‹⋀rest state iadv_rest. (parallel_rest rest1 rest2, (s_core, s_rest1, s_rest2), x) = (rest, state, Inl (Inr iadv_rest)) ⟹ fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, s_rest1, s_rest2) x = map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl))) (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1)) x› and 2 goals remain*)\n apply (subst foldl_spmf_pair_left[simplified split_def] (*‹foldl_spmf (λ(p::?'a × ?'b) e::?'c. map_spmf (λl'::?'a. (l', snd p)) ((?f::?'a ⇒ ?'c ⇒ ?'a spmf) (fst p) e)) (return_spmf (?l::?'a, ?r::?'b)) (?es::?'c list) = map_spmf (λl'::?'a. (l', ?r)) (foldl_spmf ?f (return_spmf ?l) ?es)›*))\n (*top goal: ‹⋀x1 a b ba. ⟦x = Inl (Inr (Inl x1)); ((a, b), ba) ∈ set_spmf (rfunc_adv rest1 s_rest1 x1)⟧ ⟹ foldl_spmf (map_fun id (map_fun Inl id) (cpoke core)) (return_spmf s_core) b ⤜ (λs_notify. return_spmf (Inl (Inr (Inl a)), s_notify, ba, s_rest2)) = foldl_spmf (λp e. map_spmf (λs_core'. (s_core', snd p)) (cpoke core (fst p) (Inl e))) (return_spmf (s_core, s_rest2)) b ⤜ (λx. return_spmf (Inl (Inr (Inl a)), fst (rprodl (x, ba)), prod.swap (snd (rprodl (x, ba)))))› and 2 goals remain*)\n apply (simp add: map_fun_def (*‹map_fun ?f ?g ?h = ?g ∘ ?h ∘ ?f›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*))\n (*discuss goal 3*)\n apply ((auto simp add: fused_resource.fuse.simps (*‹fused_resource.fuse ?core ?rest ?state (Inl (Inl ?iadv_core)) = (let handle = map_prod (Inl ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_adv ?core (fst ?state) ?iadv_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inl (Inr ?iadv_rest)) = rfunc_adv ?rest (snd ?state) ?iadv_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inl ∘ Inr) o_rfunc, s_notify, s_rfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inl ?iusr_core)) = (let handle = map_prod (Inr ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_usr ?core (fst ?state) ?iusr_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inr ?iusr_rest)) = rfunc_usr ?rest (snd ?state) ?iusr_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inr ∘ Inr) o_rfunc, s_notify, s_rfunc)))›*) map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) map_prod_def (*‹map_prod ?f ?g = (λ(x, y). (?f x, ?g y))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) parallel_eoracle_def (*‹parallel_eoracle ?eoracle1.0 ?eoracle2.0 ?state ≡ map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) ∘ (?eoracle1.0 ‡⇩O ?eoracle2.0) ?state›*) parallel_oracle_def (*‹?left ‡⇩O ?right = (λ(s1, s2). case_sum (λa. map_spmf (map_prod Inl (λs1'. (s1', s2))) (?left s1 a)) (λb. map_spmf (map_prod Inr (Pair s1)) (?right s2 b)))›*) split!: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*) intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*discuss goal 4*)\n apply ((auto simp add: fused_resource.fuse.simps (*‹fused_resource.fuse ?core ?rest ?state (Inl (Inl ?iadv_core)) = (let handle = map_prod (Inl ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_adv ?core (fst ?state) ?iadv_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inl (Inr ?iadv_rest)) = rfunc_adv ?rest (snd ?state) ?iadv_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inl ∘ Inr) o_rfunc, s_notify, s_rfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inl ?iusr_core)) = (let handle = map_prod (Inr ∘ Inl) (prod.swap ∘ Pair (snd ?state)) in cfunc_usr ?core (fst ?state) ?iusr_core ⤜ (λos_cfunc. return_spmf (handle os_cfunc)))› ‹fused_resource.fuse ?core ?rest ?state (Inr (Inr ?iusr_rest)) = rfunc_usr ?rest (snd ?state) ?iusr_rest ⤜ (λ((o_rfunc, e_lst), s_rfunc). foldl_spmf (cpoke ?core) (return_spmf (fst ?state)) e_lst ⤜ (λs_notify. return_spmf ((Inr ∘ Inr) o_rfunc, s_notify, s_rfunc)))›*) map_bind_spmf (*‹map_spmf ?f (?p ⤜ ?g) = ?p ⤜ map_spmf ?f ∘ ?g›*) bind_map_spmf (*‹map_spmf ?f ?p ⤜ ?g = ?p ⤜ ?g ∘ ?f›*) map_prod_def (*‹map_prod ?f ?g = (λ(x, y). (?f x, ?g y))›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) parallel_eoracle_def (*‹parallel_eoracle ?eoracle1.0 ?eoracle2.0 ?state ≡ map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) ∘ (?eoracle1.0 ‡⇩O ?eoracle2.0) ?state›*) parallel_oracle_def (*‹?left ‡⇩O ?right = (λ(s1, s2). case_sum (λa. map_spmf (map_prod Inl (λs1'. (s1', s2))) (?left s1 a)) (λb. map_spmf (map_prod Inr (Pair s1)) (?right s2 b)))›*) split!: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*) intro!: bind_spmf_cong (*‹⟦?p = ?q; ⋀x. x ∈ set_spmf ?q ⟹ ?f x = ?g x⟧ ⟹ ?p ⤜ ?f = ?q ⤜ ?g›*))[1])\n (*goal: ‹⋀rest state iusr_rest. (parallel_rest rest1 rest2, (s_core, s_rest1, s_rest2), x) = (rest, state, Inr (Inr iusr_rest)) ⟹ fused_resource.fuse core (parallel_rest rest1 rest2) (s_core, s_rest1, s_rest2) x = map_fun (map_sum (lsumr ∘ map_sum id swap_sum) (lsumr ∘ map_sum id swap_sum)) (map_spmf (map_prod (map_sum (map_sum id swap_sum ∘ rsuml) (map_sum id swap_sum ∘ rsuml)) (map_prod id prod.swap ∘ rprodl))) (fused_resource.fuse (core_with_rest core rest2) rest1 ((s_core, s_rest2), s_rest1)) x›*)\n apply (subst foldl_spmf_pair_left[simplified split_def] (*‹foldl_spmf (λ(p::?'a::type × ?'b::type) e::?'c::type. map_spmf (λl'::?'a::type. (l', snd p)) ((?f::?'a::type ⇒ ?'c::type ⇒ ?'a::type spmf) (fst p) e)) (return_spmf (?l::?'a::type, ?r::?'b::type)) (?es::?'c::type list) = map_spmf (λl'::?'a::type. (l', ?r)) (foldl_spmf ?f (return_spmf ?l) ?es)›*))\n (*goal: ‹⋀x1 a b ba. ⟦x = Inr (Inr (Inl x1)); ((a, b), ba) ∈ set_spmf (rfunc_usr rest1 s_rest1 x1)⟧ ⟹ foldl_spmf (map_fun id (map_fun Inl id) (cpoke core)) (return_spmf s_core) b ⤜ (λs_notify. return_spmf (Inr (Inr (Inl a)), s_notify, ba, s_rest2)) = foldl_spmf (λp e. map_spmf (λs_core'. (s_core', snd p)) (cpoke core (fst p) (Inl e))) (return_spmf (s_core, s_rest2)) b ⤜ (λx. return_spmf (Inr (Inr (Inl a)), fst (rprodl (x, ba)), prod.swap (snd (rprodl (x, ba)))))›*)\n apply (simp add: map_fun_def (*‹map_fun (?f::?'c::type ⇒ ?'a::type) (?g::?'b::type ⇒ ?'d::type) (?h::?'a::type ⇒ ?'b::type) = ?g ∘ ?h ∘ ?f›*) o_def (*‹(?f::?'b::type ⇒ ?'c::type) ∘ (?g::?'a::type ⇒ ?'b::type) = (λx::?'a::type. ?f (?g x))›*) bind_map_spmf (*‹map_spmf (?f::?'c::type ⇒ ?'b::type) (?p::?'c::type spmf) ⤜ (?g::?'b::type ⇒ ?'a::type spmf) = ?p ⤜ ?g ∘ ?f›*))\n (*proven 4 subgoals*) . .\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Constructive_Cryptography_CM/Fused_Resource.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"b3c7ebc96745729e90a16276e18f756f8e3c7e01d1a872c0f9a971d9c6523926\"\n}"}}},{"rowIdx":169,"cells":{"text":{"kind":"string","value":"(* Title: HOL/TLA/TLA.thy\n Author: Stephan Merz\n Copyright: 1998 University of Munich\n*)\n\nsection ‹The temporal level of TLA›\n\ntheory TLA\nimports Init\nbegin\n\nconsts\n (** abstract syntax **)\n Box :: \"('w::world) form ⇒ temporal\"\n Dmd :: \"('w::world) form ⇒ temporal\"\n leadsto :: \"['w::world form, 'v::world form] ⇒ temporal\"\n Stable :: \"stpred ⇒ temporal\"\n WF :: \"[action, 'a stfun] ⇒ temporal\"\n SF :: \"[action, 'a stfun] ⇒ temporal\"\n\n (* Quantification over (flexible) state variables *)\n EEx :: \"('a stfun ⇒ temporal) ⇒ temporal\" (binder \"Eex \" 10)\n AAll :: \"('a stfun ⇒ temporal) ⇒ temporal\" (binder \"Aall \" 10)\n\n (** concrete syntax **)\nsyntax\n \"_Box\" :: \"lift ⇒ lift\" (\"(□_)\" [40] 40)\n \"_Dmd\" :: \"lift ⇒ lift\" (\"(◇_)\" [40] 40)\n \"_leadsto\" :: \"[lift,lift] ⇒ lift\" (\"(_ ↝ _)\" [23,22] 22)\n \"_stable\" :: \"lift ⇒ lift\" (\"(stable/ _)\")\n \"_WF\" :: \"[lift,lift] ⇒ lift\" (\"(WF'(_')'_(_))\" [0,60] 55)\n \"_SF\" :: \"[lift,lift] ⇒ lift\" (\"(SF'(_')'_(_))\" [0,60] 55)\n\n \"_EEx\" :: \"[idts, lift] ⇒ lift\" (\"(3∃∃ _./ _)\" [0,10] 10)\n \"_AAll\" :: \"[idts, lift] ⇒ lift\" (\"(3∀∀ _./ _)\" [0,10] 10)\n\ntranslations\n \"_Box\" == \"CONST Box\"\n \"_Dmd\" == \"CONST Dmd\"\n \"_leadsto\" == \"CONST leadsto\"\n \"_stable\" == \"CONST Stable\"\n \"_WF\" == \"CONST WF\"\n \"_SF\" == \"CONST SF\"\n \"_EEx v A\" == \"Eex v. A\"\n \"_AAll v A\" == \"Aall v. A\"\n\n \"sigma ⊨ □F\" <= \"_Box F sigma\"\n \"sigma ⊨ ◇F\" <= \"_Dmd F sigma\"\n \"sigma ⊨ F ↝ G\" <= \"_leadsto F G sigma\"\n \"sigma ⊨ stable P\" <= \"_stable P sigma\"\n \"sigma ⊨ WF(A)_v\" <= \"_WF A v sigma\"\n \"sigma ⊨ SF(A)_v\" <= \"_SF A v sigma\"\n \"sigma ⊨ ∃∃x. F\" <= \"_EEx x F sigma\"\n \"sigma ⊨ ∀∀x. F\" <= \"_AAll x F sigma\"\n\naxiomatization where\n (* Definitions of derived operators *)\n dmd_def: \"⋀F. TEMP ◇F == TEMP ¬□¬F\"\n\naxiomatization where\n boxInit: \"⋀F. TEMP □F == TEMP □Init F\" and\n leadsto_def: \"⋀F G. TEMP F ↝ G == TEMP □(Init F ⟶ ◇G)\" and\n stable_def: \"⋀P. TEMP stable P == TEMP □($P ⟶ P$)\" and\n WF_def: \"TEMP WF(A)_v == TEMP ◇□ Enabled(<A>_v) ⟶ □◇<A>_v\" and\n SF_def: \"TEMP SF(A)_v == TEMP □◇ Enabled(<A>_v) ⟶ □◇<A>_v\" and\n aall_def: \"TEMP (∀∀x. F x) == TEMP ¬ (∃∃x. ¬ F x)\"\n\naxiomatization where\n(* Base axioms for raw TLA. *)\n normalT: \"⋀F G. ⊢ □(F ⟶ G) ⟶ (□F ⟶ □G)\" and (* polymorphic *)\n reflT: \"⋀F. ⊢ □F ⟶ F\" and (* F::temporal *)\n transT: \"⋀F. ⊢ □F ⟶ □□F\" and (* polymorphic *)\n linT: \"⋀F G. ⊢ ◇F ∧ ◇G ⟶ (◇(F ∧ ◇G)) ∨ (◇(G ∧ ◇F))\" and\n discT: \"⋀F. ⊢ □(F ⟶ ◇(¬F ∧ ◇F)) ⟶ (F ⟶ □◇F)\" and\n primeI: \"⋀P. ⊢ □P ⟶ Init P`\" and\n primeE: \"⋀P F. ⊢ □(Init P ⟶ □F) ⟶ Init P` ⟶ (F ⟶ □F)\" and\n indT: \"⋀P F. ⊢ □(Init P ∧ ¬□F ⟶ Init P` ∧ F) ⟶ Init P ⟶ □F\" and\n allT: \"⋀F. ⊢ (∀x. □(F x)) = (□(∀ x. F x))\"\n\naxiomatization where\n necT: \"⋀F. ⊢ F ⟹ ⊢ □F\" (* polymorphic *)\n\naxiomatization where\n(* Flexible quantification: refinement mappings, history variables *)\n eexI: \"⊢ F x ⟶ (∃∃x. F x)\" and\n eexE: \"⟦ sigma ⊨ (∃∃x. F x); basevars vs;\n (⋀x. ⟦ basevars (x, vs); sigma ⊨ F x ⟧ ⟹ (G sigma)::bool)\n ⟧ ⟹ G sigma\" and\n history: \"⊢ ∃∃h. Init(h = ha) ∧ □(∀x. $h = #x ⟶ h` = hb x)\"\n\n\n(* Specialize intensional introduction/elimination rules for temporal formulas *)\n\nlemma tempI [intro!]: \"(⋀sigma. sigma ⊨ (F::temporal)) ⟹ ⊢ F\"\n apply (rule intI (*‹(⋀w. ?A w) ⟹ ⊢ ?A›*))\n (*goal: ‹(⋀sigma. F sigma) ⟹ ⊢ F›*)\n by (erule meta_spec (*‹(⋀x::?'a. PROP (?P::?'a ⇒ prop) x) ⟹ PROP ?P (?x::?'a)›*))\n\nlemma tempD [dest]: \"⊢ (F::temporal) ⟹ sigma ⊨ F\"\n by (erule intD (*‹⊢ ?A ⟹ ?A ?w›*))\n\n\n(* ======== Functions to \"unlift\" temporal theorems ====== *)\n\nML ‹\n(* The following functions are specialized versions of the corresponding\n functions defined in theory Intensional in that they introduce a\n \"world\" parameter of type \"behavior\".\n*)\nfun temp_unlift ctxt th =\n (rewrite_rule ctxt @{thms action_rews} (th RS @{thm tempD}))\n handle THM _ => action_unlift ctxt th;\n\n(* Turn ⊢ F = G into meta-level rewrite rule F == G *)\nval temp_rewrite = int_rewrite\n\nfun temp_use ctxt th =\n case Thm.concl_of th of\n Const _ $ (Const (\\<^const_name>‹Intensional.Valid›, _) $ _) =>\n ((flatten (temp_unlift ctxt th)) handle THM _ => th)\n | _ => th;\n\nfun try_rewrite ctxt th = temp_rewrite ctxt th handle THM _ => temp_use ctxt th;\n›\n\nattribute_setup temp_unlift =\n ‹Scan.succeed (Thm.rule_attribute [] (temp_unlift o Context.proof_of))›\nattribute_setup temp_rewrite =\n ‹Scan.succeed (Thm.rule_attribute [] (temp_rewrite o Context.proof_of))›\nattribute_setup temp_use =\n ‹Scan.succeed (Thm.rule_attribute [] (temp_use o Context.proof_of))›\nattribute_setup try_rewrite =\n ‹Scan.succeed (Thm.rule_attribute [] (try_rewrite o Context.proof_of))›\n\n\n(* ------------------------------------------------------------------------- *)\n(*** \"Simple temporal logic\": only □ and ◇ ***)\n(* ------------------------------------------------------------------------- *)\nsection \"Simple temporal logic\"\n\n(* □¬F == □¬Init F *)\nlemmas boxNotInit = boxInit [of \"LIFT ¬F\", unfolded Init_simps] for F\n\nlemma dmdInit: \"TEMP ◇F == TEMP ◇ Init F\"\n apply (unfold dmd_def (*‹◇(?F::?'a ⇒ bool) ≡ ¬ □¬ ?F›*))\n (*goal: ‹◇F ≡ ◇Init F›*)\n apply (unfold boxInit [of \"LIFT ¬F\"] (*‹□¬ F ≡ □Init ¬ F›*))\n (*goal: ‹¬ □¬ F ≡ ¬ □¬ Init F›*)\n by (simp (no_asm) add: Init_simps (*‹Init (?y::behavior ⇒ bool) ≡ ?y› ‹Init $(?P::state ⇒ bool) ≡ Init ?P› ‹Init ¬ (?F::?'a ⇒ bool) ≡ ¬ Init ?F› ‹Init ((?P::?'b ⇒ bool) ⟶ (?Q::?'b ⇒ bool)) ≡ Init ?P ⟶ Init ?Q› ‹Init ((?P::?'b ⇒ bool) ∧ (?Q::?'b ⇒ bool)) ≡ Init ?P ∧ Init ?Q› ‹Init ((?P::?'b ⇒ bool) ∨ (?Q::?'b ⇒ bool)) ≡ Init ?P ∨ Init ?Q› ‹Init (?P::?'b ⇒ bool) = (?Q::?'b ⇒ bool) ≡ Init ?P = (Init ?Q)› ‹Init (∀x::?'d. (?F::?'d ⇒ ?'c ⇒ bool) x) ≡ ∀x::?'d. Init ?F x› ‹Init (∃x::?'d. (?F::?'d ⇒ ?'c ⇒ bool) x) ≡ ∃x::?'d. Init ?F x› ‹Init (∃!x::?'d. (?F::?'d ⇒ ?'c ⇒ bool) x) ≡ ∃!x::?'d. Init ?F x›*))\n\nlemmas dmdNotInit = dmdInit [of \"LIFT ¬F\", unfolded Init_simps] for F\n\n(* boxInit and dmdInit cannot be used as rewrites, because they loop.\n Non-looping instances for state predicates and actions are occasionally useful.\n*)\nlemmas boxInit_stp = boxInit [where 'a = state]\nlemmas boxInit_act = boxInit [where 'a = \"state * state\"]\nlemmas dmdInit_stp = dmdInit [where 'a = state]\nlemmas dmdInit_act = dmdInit [where 'a = \"state * state\"]\n\n(* The symmetric equations can be used to get rid of Init *)\nlemmas boxInitD = boxInit [symmetric]\nlemmas dmdInitD = dmdInit [symmetric]\nlemmas boxNotInitD = boxNotInit [symmetric]\nlemmas dmdNotInitD = dmdNotInit [symmetric]\n\nlemmas Init_simps = Init_simps boxInitD dmdInitD boxNotInitD dmdNotInitD\n\n(* ------------------------ STL2 ------------------------------------------- *)\nlemmas STL2 = reflT\n\n(* The \"polymorphic\" (generic) variant *)\nlemma STL2_gen: \"⊢ □F ⟶ Init F\"\n apply (unfold boxInit [of F] (*‹□(F::'a ⇒ bool) ≡ □Init F›*))\n (*goal: ‹⊢ □F ⟶ Init F›*)\n by (rule STL2 (*‹⊢ □?F ⟶ ?F›*))\n\n(* see also STL2_pr below: \"⊢ □P ⟶ Init P & Init (P`)\" *)\n\n\n(* Dual versions for ◇ *)\nlemma InitDmd: \"⊢ F ⟶ ◇ F\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ F ⟶ ◇F›*)\n by (auto dest!: STL2 [temp_use] (*‹?sigma ⊨ □?F ⟹ ?F ?sigma›*))\n\nlemma InitDmd_gen: \"⊢ Init F ⟶ ◇F\"\n apply clarsimp\n (*goal: ‹⊢ Init (F::'a ⇒ bool) ⟶ ◇F›*)\n apply (drule InitDmd [temp_use] (*‹?F ?sigma ⟹ ?sigma ⊨ ◇?F›*))\n (*goal: ‹⋀sigma. sigma ⊨ Init F ⟹ sigma ⊨ ◇F›*)\n by (simp add: dmdInitD (*‹◇Init ?F ≡ ◇?F›*))\n\n\n(* ------------------------ STL3 ------------------------------------------- *)\nlemma STL3: \"⊢ (□□F) = (□F)\"\n by (auto elim: transT [temp_use] (*‹?sigma ⊨ □?F ⟹ ?sigma ⊨ □□?F›*) STL2 [temp_use] (*‹?sigma ⊨ □?F ⟹ ?F ?sigma›*))\n\n(* corresponding elimination rule introduces double boxes:\n ⟦ (sigma ⊨ □F); (sigma ⊨ □□F) ⟹ PROP W ⟧ ⟹ PROP W\n*)\nlemmas dup_boxE = STL3 [temp_unlift, THEN iffD2, elim_format]\nlemmas dup_boxD = STL3 [temp_unlift, THEN iffD1]\n\n(* dual versions for ◇ *)\nlemma DmdDmd: \"⊢ (◇◇F) = (◇F)\"\n by (auto simp add: dmd_def [try_rewrite] (*‹◇?F ≡ ¬ □¬ ?F›*) STL3 [try_rewrite] (*‹□□?F ≡ □?F›*))\n\nlemmas dup_dmdE = DmdDmd [temp_unlift, THEN iffD2, elim_format]\nlemmas dup_dmdD = DmdDmd [temp_unlift, THEN iffD1]\n\n\n(* ------------------------ STL4 ------------------------------------------- *)\nlemma STL4:\n assumes \"⊢ F ⟶ G\"\n shows \"⊢ □F ⟶ □G\"\n apply clarsimp\n (*goal: ‹⊢ □F ⟶ □G›*)\n apply (rule normalT [temp_use] (*‹⟦?sigma ⊨ □(?F ⟶ ?G); ?sigma ⊨ □?F⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ □F ⟹ sigma ⊨ □(?F2 sigma ⟶ G)›\n 2. ‹⋀sigma. sigma ⊨ □F ⟹ sigma ⊨ □?F2 sigma›\n discuss goal 1*)\n apply (rule assms [THEN necT, temp_use] (*‹?sigma ⊨ □(F ⟶ G)›*))\n (*discuss goal 2*)\n apply assumption\n (*proven 2 subgoals*) .\n\n(* Unlifted version as an elimination rule *)\nlemma STL4E: \"⟦ sigma ⊨ □F; ⊢ F ⟶ G ⟧ ⟹ sigma ⊨ □G\"\n by (erule (1) STL4 [temp_use] (*‹⟦⊢ ?F ⟶ ?G; ?sigma ⊨ □?F⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma STL4_gen: \"⊢ Init F ⟶ Init G ⟹ ⊢ □F ⟶ □G\"\n apply (drule STL4 (*‹⊢ ?F ⟶ ?G ⟹ ⊢ □?F ⟶ □?G›*))\n (*goal: ‹⊢ Init F ⟶ Init G ⟹ ⊢ □F ⟶ □G›*)\n by (simp add: boxInitD (*‹□Init ?F ≡ □?F›*))\n\nlemma STL4E_gen: \"⟦ sigma ⊨ □F; ⊢ Init F ⟶ Init G ⟧ ⟹ sigma ⊨ □G\"\n by (erule (1) STL4_gen [temp_use] (*‹⟦⊢ Init (?F::?'a ⇒ bool) ⟶ Init (?G::?'b ⇒ bool); (?sigma::behavior) ⊨ □?F⟧ ⟹ ?sigma ⊨ □?G›*))\n\n(* see also STL4Edup below, which allows an auxiliary boxed formula:\n □A /\\ F => G\n -----------------\n □A /\\ □F => □G\n*)\n\n(* The dual versions for ◇ *)\nlemma DmdImpl:\n assumes prem: \"⊢ F ⟶ G\"\n shows \"⊢ ◇F ⟶ ◇G\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ ◇F ⟶ ◇G›*)\n by (fastforce intro!: prem [temp_use] (*‹F ?w ⟹ G ?w›*) elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma DmdImplE: \"⟦ sigma ⊨ ◇F; ⊢ F ⟶ G ⟧ ⟹ sigma ⊨ ◇G\"\n by (erule (1) DmdImpl [temp_use] (*‹⟦⊢ (?F::?'a ⇒ bool) ⟶ (?G::?'a ⇒ bool); (?sigma::behavior) ⊨ ◇?F⟧ ⟹ ?sigma ⊨ ◇?G›*))\n\n(* ------------------------ STL5 ------------------------------------------- *)\nlemma STL5: \"⊢ (□F ∧ □G) = (□(F ∧ G))\"\n apply auto\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ □F; sigma ⊨ □G⟧ ⟹ sigma ⊨ □(F ∧ G)›\n 2. ‹⋀sigma. sigma ⊨ □(F ∧ G) ⟹ sigma ⊨ □F›\n 3. ‹⋀sigma. sigma ⊨ □(F ∧ G) ⟹ sigma ⊨ □G›\n discuss goal 1*)\n apply (subgoal_tac \"sigma ⊨ □ (G ⟶ (F ∧ G))\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ □F; sigma ⊨ □G; sigma ⊨ □(G ⟶ F ∧ G)⟧ ⟹ sigma ⊨ □(F ∧ G)›\n 2. ‹⋀sigma. ⟦sigma ⊨ □F; sigma ⊨ □G⟧ ⟹ sigma ⊨ □(G ⟶ F ∧ G)›\n discuss goal 1*)\n apply (erule normalT [temp_use] (*‹⟦?sigma ⊨ □(?F ⟶ ?G); ?sigma ⊨ □?F⟧ ⟹ ?sigma ⊨ □?G›*))\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ □F; sigma ⊨ □G; sigma ⊨ □(G ⟶ F ∧ G)⟧ ⟹ sigma ⊨ □(F ∧ G)› and 3 goals remain*)\n apply (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*discuss goal 2*)\n apply (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (fastforce elim!: STL4E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*))\n (*discuss goal 3*)\n apply (fastforce elim!: STL4E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*))\n (*proven 3 subgoals*) .\n\n(* rewrite rule to split conjunctions under boxes *)\nlemmas split_box_conj = STL5 [temp_unlift, symmetric]\n\n\n(* the corresponding elimination rule allows to combine boxes in the hypotheses\n (NB: F and G must have the same type, i.e., both actions or temporals.)\n Use \"addSE2\" etc. if you want to add this to a claset, otherwise it will loop!\n*)\nlemma box_conjE:\n assumes \"sigma ⊨ □F\"\n and \"sigma ⊨ □G\"\n and \"sigma ⊨ □(F∧G) ⟹ PROP R\"\n shows \"PROP R\"\n apply (rule assms (*‹sigma ⊨ □F› ‹sigma ⊨ □G› ‹sigma ⊨ □(F ∧ G) ⟹ PROP R›*) STL5 [temp_unlift, THEN iffD1] (*‹(?sigma1 ⊨ □?F2) ∧ (?sigma1 ⊨ □?G2) ⟹ ?sigma1 ⊨ □(?F2 ∧ ?G2)›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goal: ‹PROP R›*)\n apply (rule assms (*‹sigma ⊨ □F› ‹sigma ⊨ □G› ‹sigma ⊨ □(F ∧ G) ⟹ PROP R›*) STL5 [temp_unlift, THEN iffD1] (*‹(?sigma1 ⊨ □?F2) ∧ (?sigma1 ⊨ □?G2) ⟹ ?sigma1 ⊨ □(?F2 ∧ ?G2)›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goal: ‹sigma ⊨ □(F ∧ G)›*)\n apply (rule assms (*‹sigma ⊨ □F› ‹sigma ⊨ □G› ‹sigma ⊨ □(F ∧ G) ⟹ PROP R›*) STL5 [temp_unlift, THEN iffD1] (*‹(?sigma1 ⊨ □?F2) ∧ (?sigma1 ⊨ □?G2) ⟹ ?sigma1 ⊨ □(?F2 ∧ ?G2)›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹sigma ⊨ □F›\n 2. ‹sigma ⊨ □G›\n discuss goal 1*)\n apply (rule assms (*‹sigma ⊨ □F› ‹sigma ⊨ □G› ‹sigma ⊨ □(F ∧ G) ⟹ PROP R›*) STL5 [temp_unlift, THEN iffD1] (*‹(?sigma1 ⊨ □?F2) ∧ (?sigma1 ⊨ □?G2) ⟹ ?sigma1 ⊨ □(?F2 ∧ ?G2)›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*discuss goal 2*)\n apply (rule assms (*‹sigma ⊨ □F› ‹sigma ⊨ □G› ‹sigma ⊨ □(F ∧ G) ⟹ PROP R›*) STL5 [temp_unlift, THEN iffD1] (*‹(?sigma1 ⊨ □?F2) ∧ (?sigma1 ⊨ □?G2) ⟹ ?sigma1 ⊨ □(?F2 ∧ ?G2)›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*proven 2 subgoals*) .\n\n(* Instances of box_conjE for state predicates, actions, and temporals\n in case the general rule is \"too polymorphic\".\n*)\nlemmas box_conjE_temp = box_conjE [where 'a = behavior]\nlemmas box_conjE_stp = box_conjE [where 'a = state]\nlemmas box_conjE_act = box_conjE [where 'a = \"state * state\"]\n\n(* Define a tactic that tries to merge all boxes in an antecedent. The definition is\n a bit kludgy in order to simulate \"double elim-resolution\".\n*)\n\nlemma box_thin: \"⟦ sigma ⊨ □F; PROP W ⟧ ⟹ PROP W\" .\n\nML ‹\nfun merge_box_tac ctxt i =\n REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE} i, assume_tac ctxt i,\n eresolve_tac ctxt @{thms box_thin} i])\n\nfun merge_temp_box_tac ctxt i =\n REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE_temp} i, assume_tac ctxt i,\n Rule_Insts.eres_inst_tac ctxt [(((\"'a\", 0), Position.none), \"behavior\")] [] @{thm box_thin} i])\n\nfun merge_stp_box_tac ctxt i =\n REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE_stp} i, assume_tac ctxt i,\n Rule_Insts.eres_inst_tac ctxt [(((\"'a\", 0), Position.none), \"state\")] [] @{thm box_thin} i])\n\nfun merge_act_box_tac ctxt i =\n REPEAT_DETERM (EVERY [eresolve_tac ctxt @{thms box_conjE_act} i, assume_tac ctxt i,\n Rule_Insts.eres_inst_tac ctxt [(((\"'a\", 0), Position.none), \"state * state\")] [] @{thm box_thin} i])\n›\n\nmethod_setup merge_box = ‹Scan.succeed (SIMPLE_METHOD' o merge_box_tac)›\nmethod_setup merge_temp_box = ‹Scan.succeed (SIMPLE_METHOD' o merge_temp_box_tac)›\nmethod_setup merge_stp_box = ‹Scan.succeed (SIMPLE_METHOD' o merge_stp_box_tac)›\nmethod_setup merge_act_box = ‹Scan.succeed (SIMPLE_METHOD' o merge_act_box_tac)›\n\n(* rewrite rule to push universal quantification through box:\n (sigma ⊨ □(∀x. F x)) = (∀x. (sigma ⊨ □F x))\n*)\nlemmas all_box = allT [temp_unlift, symmetric]\n\nlemma DmdOr: \"⊢ (◇(F ∨ G)) = (◇F ∨ ◇G)\"\n apply (auto simp add: dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*) split_box_conj [try_rewrite] (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*))\n (*goals:\n 1. ‹⋀sigma. ⟦¬ (sigma ⊨ □¬ (F ∨ G)); sigma ⊨ □¬ F; sigma ⊨ □¬ G⟧ ⟹ False›\n 2. ‹⋀sigma. ⟦sigma ⊨ □¬ (F ∨ G); ¬ (sigma ⊨ □¬ F)⟧ ⟹ False›\n 3. ‹⋀sigma. ⟦sigma ⊨ □¬ (F ∨ G); ¬ (sigma ⊨ □¬ G)⟧ ⟹ False›\n discuss goal 1*)\n apply (erule contrapos_np (*‹⟦¬ ?Q; ¬ ?P ⟹ ?Q⟧ ⟹ ?P›*))\n (*top goal: ‹⋀sigma. ⟦¬ (sigma ⊨ □¬ (F ∨ G)); sigma ⊨ □¬ F; sigma ⊨ □¬ G⟧ ⟹ False› and 2 goals remain*)\n apply merge_box\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ □¬ F; sigma ⊨ □¬ G; ¬ False⟧ ⟹ sigma ⊨ □¬ (F ∨ G)› and 2 goals remain*)\n apply (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*discuss goal 2*)\n apply (erule contrapos_np (*‹⟦¬ ?Q; ¬ ?P ⟹ ?Q⟧ ⟹ ?P›*))\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ □¬ (F ∨ G); ¬ (sigma ⊨ □¬ F)⟧ ⟹ False› and 1 goal remains*)\n apply merge_box\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ □¬ (F ∨ G); ¬ False⟧ ⟹ sigma ⊨ □¬ F› and 1 goal remains*)\n apply (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*discuss goal 3*)\n apply (erule contrapos_np (*‹⟦¬ (?Q::bool); ¬ (?P::bool) ⟹ ?Q⟧ ⟹ ?P›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □¬ (F ∨ G); ¬ (sigma ⊨ □¬ G)⟧ ⟹ False›*)\n apply merge_box\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □¬ (F ∨ G); ¬ False⟧ ⟹ sigma ⊨ □¬ G›*)\n apply (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*proven 3 subgoals*) .\n\nlemma exT: \"⊢ (∃x. ◇(F x)) = (◇(∃x. F x))\"\n by (auto simp: dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*) Not_Rex [try_rewrite] (*‹¬ (∃x. ?F x) ≡ ∀x. ¬ ?F x›*) all_box [try_rewrite] (*‹(?sigma ⊨ □RAll ?F) = (∀x. ?sigma ⊨ □?F x)›*))\n\nlemmas ex_dmd = exT [temp_unlift, symmetric]\n\nlemma STL4Edup: \"⋀sigma. ⟦ sigma ⊨ □A; sigma ⊨ □F; ⊢ F ∧ □A ⟶ G ⟧ ⟹ sigma ⊨ □G\"\n apply (erule dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □A; sigma ⊨ □F; ⊢ F ∧ □A ⟶ G⟧ ⟹ sigma ⊨ □G›*)\n apply merge_box\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □F; ⊢ F ∧ □A ⟶ G; sigma ⊨ □□A⟧ ⟹ sigma ⊨ □G›*)\n apply (erule STL4E (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma. ⟦⊢ F ∧ □A ⟶ G; sigma ⊨ □(F ∧ □A)⟧ ⟹ sigma ⊨ □G›*)\n by assumption\n\nlemma DmdImpl2:\n \"⋀sigma. ⟦ sigma ⊨ ◇F; sigma ⊨ □(F ⟶ G) ⟧ ⟹ sigma ⊨ ◇G\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ ◇F; sigma ⊨ □(F ⟶ G)⟧ ⟹ sigma ⊨ ◇G›*)\n apply auto\n (*goal: ‹⋀sigma. ⟦sigma ⊨ ¬ □¬ F; sigma ⊨ □(F ⟶ G)⟧ ⟹ sigma ⊨ ¬ □¬ G›*)\n apply (erule notE (*‹⟦¬ ?P; ?P⟧ ⟹ ?R›*))\n (*goal: ‹⋀sigma. ⟦¬ (sigma ⊨ □¬ F); sigma ⊨ □(F ⟶ G); sigma ⊨ □¬ G⟧ ⟹ False›*)\n apply merge_box\n (*goal: ‹⋀sigma::behavior. ⟦sigma ⊨ □((F::'a ⇒ bool) ⟶ (G::'a ⇒ bool)); sigma ⊨ □¬ G⟧ ⟹ sigma ⊨ □¬ F›*)\n by (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma InfImpl:\n assumes 1: \"sigma ⊨ □◇F\"\n and 2: \"sigma ⊨ □G\"\n and 3: \"⊢ F ∧ G ⟶ H\"\n shows \"sigma ⊨ □◇H\"\n apply (insert 1 (*‹(sigma::behavior) ⊨ □◇(F::'a ⇒ bool)›*) 2 (*‹(sigma::behavior) ⊨ □(G::'a ⇒ bool)›*))\n (*goal: ‹sigma ⊨ □◇H›*)\n apply (erule_tac F = G in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⟦sigma ⊨ □◇F; sigma ⊨ □G⟧ ⟹ sigma ⊨ □◇H›*)\n apply merge_box\n (*goal: ‹⟦sigma ⊨ □◇F; sigma ⊨ □□G⟧ ⟹ sigma ⊨ □◇H›*)\n by (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*) DmdImpl2 [temp_use] (*‹⟦?sigma ⊨ ◇?F; ?sigma ⊨ □(?F ⟶ ?G)⟧ ⟹ ?sigma ⊨ ◇?G›*) intro!: 3 [temp_use] (*‹⟦F ?w; G ?w⟧ ⟹ H ?w›*))\n\n(* ------------------------ STL6 ------------------------------------------- *)\n(* Used in the proof of STL6, but useful in itself. *)\nlemma BoxDmd: \"⊢ □F ∧ ◇G ⟶ ◇(□F ∧ G)\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ □F ∧ ◇G ⟶ ◇(□F ∧ G)›*)\n apply clarsimp\n (*goal: ‹⊢ □F ∧ ¬ □¬ G ⟶ ¬ □¬ (□F ∧ G)›*)\n apply (erule dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □F; ¬ (sigma ⊨ □¬ G); sigma ⊨ □¬ (□F ∧ G)⟧ ⟹ False›*)\n apply merge_box\n (*goal: ‹⋀sigma::behavior. ⟦¬ (sigma ⊨ □¬ (G::behavior ⇒ bool)); sigma ⊨ □¬ (□(F::'a ⇒ bool) ∧ G); sigma ⊨ □□F⟧ ⟹ False›*)\n apply (erule contrapos_np (*‹⟦¬ (?Q::bool); ¬ (?P::bool) ⟹ ?Q⟧ ⟹ ?P›*))\n (*goal: ‹⋀sigma. ⟦¬ (sigma ⊨ □¬ G); sigma ⊨ □(¬ (□F ∧ G) ∧ □F)⟧ ⟹ False›*)\n by (fastforce elim!: STL4E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*))\n\n(* weaker than BoxDmd, but more polymorphic (and often just right) *)\nlemma BoxDmd_simple: \"⊢ □F ∧ ◇G ⟶ ◇(F ∧ G)\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ □(F::'a ⇒ bool) ∧ ◇(G::'a ⇒ bool) ⟶ ◇(F ∧ G)›*)\n apply clarsimp\n (*goal: ‹⊢ □F ∧ ¬ □¬ G ⟶ ¬ □¬ (F ∧ G)›*)\n apply merge_box\n (*goal: ‹⋀sigma::behavior. ⟦sigma ⊨ □(F::'a ⇒ bool); ¬ (sigma ⊨ □¬ (G::'a ⇒ bool)); sigma ⊨ □¬ (F ∧ G)⟧ ⟹ False›*)\n by (fastforce elim!: notE (*‹⟦¬ ?P; ?P⟧ ⟹ ?R›*) STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma BoxDmd2_simple: \"⊢ □F ∧ ◇G ⟶ ◇(G ∧ F)\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ □F ∧ ◇G ⟶ ◇(G ∧ F)›*)\n apply clarsimp\n (*goal: ‹⊢ □F ∧ ¬ □¬ G ⟶ ¬ □¬ (G ∧ F)›*)\n apply merge_box\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □F; ¬ (sigma ⊨ □¬ G); sigma ⊨ □¬ (G ∧ F)⟧ ⟹ False›*)\n by (fastforce elim!: notE (*‹⟦¬ ?P; ?P⟧ ⟹ ?R›*) STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma DmdImpldup:\n assumes 1: \"sigma ⊨ □A\"\n and 2: \"sigma ⊨ ◇F\"\n and 3: \"⊢ □A ∧ F ⟶ G\"\n shows \"sigma ⊨ ◇G\"\n apply (rule 2 [THEN 1 [THEN BoxDmd [temp_use]], THEN DmdImplE] (*‹⊢ □A ∧ F ⟶ ?G ⟹ sigma ⊨ ◇?G›*))\n (*goal: ‹sigma ⊨ ◇G›*)\n by (rule 3 (*‹⊢ □A ∧ F ⟶ G›*))\n\nlemma STL6: \"⊢ ◇□F ∧ ◇□G ⟶ ◇□(F ∧ G)\"\n apply (auto simp: STL5 [temp_rewrite, symmetric] (*‹□(?F ∧ ?G) ≡ □?F ∧ □?G›*))\n (*goal: ‹⊢ ◇□F ∧ ◇□G ⟶ ◇□(F ∧ G)›*)\n apply (drule linT [temp_use] (*‹⟦?sigma ⊨ ◇?F; ?sigma ⊨ ◇?G⟧ ⟹ (?sigma ⊨ ◇(?F ∧ ◇?G)) ∨ (?sigma ⊨ ◇(?G ∧ ◇?F))›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ ◇□G ⟹ sigma ⊨ ◇?G5 sigma›\n 2. ‹⋀sigma. ⟦sigma ⊨ ◇□G; (sigma ⊨ ◇(□F ∧ ◇?G5 sigma)) ∨ (sigma ⊨ ◇(?G5 sigma ∧ ◇□F))⟧ ⟹ sigma ⊨ ◇(□F ∧ □G)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (erule thin_rl (*‹⟦PROP ?V::prop; PROP ?W::prop⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ ◇□G; (sigma ⊨ ◇(□F ∧ ◇□G)) ∨ (sigma ⊨ ◇(□G ∧ ◇□F))⟧ ⟹ sigma ⊨ ◇(□F ∧ □G)›*)\n apply (rule DmdDmd [temp_unlift, THEN iffD1] (*‹?sigma1 ⊨ ◇◇?F2 ⟹ ?sigma1 ⊨ ◇?F2›*))\n (*goal: ‹⋀sigma::behavior. (sigma ⊨ ◇(□(F::'a ⇒ bool) ∧ ◇□(G::'a ⇒ bool))) ∨ (sigma ⊨ ◇(□G ∧ ◇□F)) ⟹ sigma ⊨ ◇(□F ∧ □G)›*)\n apply (erule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ ◇(□F ∧ ◇□G) ⟹ sigma ⊨ ◇◇(□F ∧ □G)›\n 2. ‹⋀sigma. sigma ⊨ ◇(□G ∧ ◇□F) ⟹ sigma ⊨ ◇◇(□F ∧ □G)›\n discuss goal 1*)\n apply (erule DmdImplE (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*top goal: ‹⋀sigma::behavior. sigma ⊨ ◇(□(F::'a ⇒ bool) ∧ ◇□(G::'a ⇒ bool)) ⟹ sigma ⊨ ◇◇(□F ∧ □G)› and 1 goal remains*)\n apply (rule BoxDmd (*‹⊢ □?F ∧ ◇?G ⟶ ◇(□?F ∧ ?G)›*))\n (*discuss goal 2*)\n apply (erule DmdImplE (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*goal: ‹⋀sigma. sigma ⊨ ◇(□G ∧ ◇□F) ⟹ sigma ⊨ ◇◇(□F ∧ □G)›*)\n apply auto\n (*goal: ‹⋀sigma::behavior. ⊢ □(G::'a ⇒ bool) ∧ ◇□(F::'a ⇒ bool) ⟶ ◇(□F ∧ □G)›*)\n apply (drule BoxDmd [temp_use] (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ ◇?G⟧ ⟹ ?sigma ⊨ ◇(□?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ ◇□F ⟹ sigma ⊨ ◇?G25 sigma›\n 2. ‹⋀sigma. ⟦sigma ⊨ ◇□F; sigma ⊨ ◇(□G ∧ ?G25 sigma)⟧ ⟹ sigma ⊨ ◇(□F ∧ □G)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (erule thin_rl (*‹⟦PROP ?V; PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma::behavior. ⟦sigma ⊨ ◇□(F::'a ⇒ bool); sigma ⊨ ◇(□(G::'a ⇒ bool) ∧ □F)⟧ ⟹ sigma ⊨ ◇(□F ∧ □G)›*)\n apply (fastforce elim!: DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n\n(* ------------------------ True / False ----------------------------------------- *)\nsection \"Simplification of constants\"\n\nlemma BoxConst: \"⊢ (□#P) = #P\"\n apply (rule tempI (*‹(⋀sigma. ?F sigma) ⟹ ⊢ ?F›*))\n (*goal: ‹⊢ (□#P) = #P›*)\n apply (cases P)\n (*goals:\n 1. ‹⋀sigma. P ⟹ sigma ⊨ (□#P) = #P›\n 2. ‹⋀sigma. ¬ P ⟹ sigma ⊨ (□#P) = #P›\n discuss goal 1*)\n apply ((auto intro!: necT [temp_use] (*‹⊢ ?F::?'a::world ⇒ bool ⟹ (?sigma::behavior) ⊨ □?F›*) dest: STL2_gen [temp_use] (*‹(?sigma::behavior) ⊨ □(?F::?'a::world ⇒ bool) ⟹ ?sigma ⊨ Init ?F›*) simp: Init_simps (*‹Init (?y::behavior ⇒ bool) ≡ ?y› ‹Init $(?P::state ⇒ bool) ≡ Init ?P› ‹Init ¬ (?F::?'a::world ⇒ bool) ≡ ¬ Init ?F› ‹Init ((?P::?'b::world ⇒ bool) ⟶ (?Q::?'b::world ⇒ bool)) ≡ Init ?P ⟶ Init ?Q› ‹Init ((?P::?'b::world ⇒ bool) ∧ (?Q::?'b::world ⇒ bool)) ≡ Init ?P ∧ Init ?Q› ‹Init ((?P::?'b::world ⇒ bool) ∨ (?Q::?'b::world ⇒ bool)) ≡ Init ?P ∨ Init ?Q› ‹Init (?P::?'b::world ⇒ bool) = (?Q::?'b::world ⇒ bool) ≡ Init ?P = (Init ?Q)› ‹Init (∀x::?'d::type. (?F::?'d::type ⇒ ?'c::world ⇒ bool) x) ≡ ∀x::?'d::type. Init ?F x› ‹Init (∃x::?'d::type. (?F::?'d::type ⇒ ?'c::world ⇒ bool) x) ≡ ∃x::?'d::type. Init ?F x› ‹Init (∃!x::?'d::type. (?F::?'d::type ⇒ ?'c::world ⇒ bool) x) ≡ ∃!x::?'d::type. Init ?F x› ‹□Init (?F::?'a::world ⇒ bool) ≡ □?F› ‹◇Init (?F::?'a::world ⇒ bool) ≡ ◇?F› and more 2 facts*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*) dest: STL2_gen [temp_use] (*‹?sigma ⊨ □?F ⟹ ?sigma ⊨ Init ?F›*) simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*))[1])\n (*proven 2 subgoals*) .\n\nlemma DmdConst: \"⊢ (◇#P) = #P\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ (◇#P) = #P›*)\n apply (cases P)\n (*goals:\n 1. ‹P ⟹ ⊢ (¬ □¬ #P) = #P›\n 2. ‹¬ P ⟹ ⊢ (¬ □¬ #P) = #P›\n discuss goal 1*)\n apply (simp add: BoxConst [try_rewrite] (*‹□#(?P::bool) ≡ #?P›*))\n (*discuss goal 2*)\n apply (simp add: BoxConst [try_rewrite] (*‹□#?P ≡ #?P›*))\n (*proven 2 subgoals*) .\n\nlemmas temp_simps [temp_rewrite, simp] = BoxConst DmdConst\n\n\n(* ------------------------ Further rewrites ----------------------------------------- *)\nsection \"Further rewrites\"\n\nlemma NotBox: \"⊢ (¬□F) = (◇¬F)\"\n by (simp add: dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n\nlemma NotDmd: \"⊢ (¬◇F) = (□¬F)\"\n by (simp add: dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n\n(* These are not declared by default, because they could be harmful,\n e.g. □F & ¬□F becomes □F & ◇¬F !! *)\nlemmas more_temp_simps1 =\n STL3 [temp_rewrite] DmdDmd [temp_rewrite] NotBox [temp_rewrite] NotDmd [temp_rewrite]\n NotBox [temp_unlift, THEN eq_reflection]\n NotDmd [temp_unlift, THEN eq_reflection]\n\nlemma BoxDmdBox: \"⊢ (□◇□F) = (◇□F)\"\n apply (auto dest!: STL2 [temp_use] (*‹?sigma ⊨ □?F ⟹ ?F ?sigma›*))\n (*goal: ‹⊢ (□◇□F) = (◇□F)›*)\n apply (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹⋀sigma. sigma ⊨ ◇□F ⟹ sigma ⊨ □◇□F›*)\n apply (subgoal_tac \"sigma ⊨ ◇□□F ∧ ◇□¬□F\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ ◇□F; ¬ (sigma ⊨ □◇□F); sigma ⊨ ◇□□F ∧ ◇□¬ □F⟧ ⟹ False›\n 2. ‹⋀sigma. ⟦sigma ⊨ ◇□F; ¬ (sigma ⊨ □◇□F)⟧ ⟹ sigma ⊨ ◇□□F ∧ ◇□¬ □F›\n discuss goal 1*)\n apply (erule thin_rl (*‹⟦PROP ?V; PROP ?W⟧ ⟹ PROP ?W›*))\n (*top goal: ‹⋀sigma::behavior. ⟦sigma ⊨ ◇□(F::'a ⇒ bool); ¬ (sigma ⊨ □◇□F); sigma ⊨ ◇□□F ∧ ◇□¬ □F⟧ ⟹ False› and 1 goal remains*)\n apply ((auto)[1])\n (*top goal: ‹⋀sigma. ⟦¬ (sigma ⊨ □◇□F); sigma ⊨ ◇□□F ∧ ◇□¬ □F⟧ ⟹ False› and 1 goal remains*)\n apply (drule STL6 [temp_use] (*‹⟦?sigma ⊨ ◇□?F; ?sigma ⊨ ◇□?G⟧ ⟹ ?sigma ⊨ ◇□(?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma. ⟦¬ (sigma ⊨ □◇□F); sigma ⊨ ◇□¬ □F⟧ ⟹ sigma ⊨ ◇□?G11 sigma›\n 2. ‹⋀sigma. ⟦¬ (sigma ⊨ □◇□F); sigma ⊨ ◇□¬ □F; sigma ⊨ ◇□(□F ∧ ?G11 sigma)⟧ ⟹ False›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply ((auto)[1])\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ ◇□F; ¬ (sigma ⊨ □◇□F)⟧ ⟹ sigma ⊨ ◇□□F›\n 2. ‹⋀sigma. ⟦sigma ⊨ ◇□F; ¬ (sigma ⊨ □◇□F)⟧ ⟹ sigma ⊨ ◇□¬ □F›\n discuss goal 1*)\n apply (simp add: more_temp_simps1 (*‹□□?F ≡ □?F› ‹◇◇?F ≡ ◇?F› ‹¬ □?F ≡ ◇¬ ?F› ‹¬ ◇?F ≡ □¬ ?F› ‹¬ (?sigma ⊨ □?F) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ (?sigma ⊨ ◇?F) ≡ ?sigma ⊨ □¬ ?F›*))\n (*discuss goal 2*)\n apply (simp add: more_temp_simps1 (*‹□□(?F::?'a ⇒ bool) ≡ □?F› ‹◇◇(?F::?'a ⇒ bool) ≡ ◇?F› ‹¬ □(?F::?'a ⇒ bool) ≡ ◇¬ ?F› ‹¬ ◇(?F::?'a ⇒ bool) ≡ □¬ ?F› ‹¬ ((?sigma::behavior) ⊨ □(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ ((?sigma::behavior) ⊨ ◇(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ □¬ ?F›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\nlemma DmdBoxDmd: \"⊢ (◇□◇F) = (□◇F)\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ (◇□◇F) = (□◇F)›*)\n by (auto simp: BoxDmdBox [unfolded dmd_def, try_rewrite] (*‹□¬ □¬ □(?F::?'a::world ⇒ bool) ≡ ¬ □¬ □?F›*))\n\nlemmas more_temp_simps2 = more_temp_simps1 BoxDmdBox [temp_rewrite] DmdBoxDmd [temp_rewrite]\n\n\n(* ------------------------ Miscellaneous ----------------------------------- *)\n\nlemma BoxOr: \"⋀sigma. ⟦ sigma ⊨ □F ∨ □G ⟧ ⟹ sigma ⊨ □(F ∨ G)\"\n by (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\n(* \"persistently implies infinitely often\" *)\nlemma DBImplBD: \"⊢ ◇□F ⟶ □◇F\"\n apply clarsimp\n (*goal: ‹⊢ ◇□F ⟶ □◇F›*)\n apply (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹⋀sigma::behavior. sigma ⊨ ◇□(F::'a ⇒ bool) ⟹ sigma ⊨ □◇F›*)\n apply (simp add: more_temp_simps2 (*‹□□?F ≡ □?F› ‹◇◇?F ≡ ◇?F› ‹¬ □?F ≡ ◇¬ ?F› ‹¬ ◇?F ≡ □¬ ?F› ‹¬ (?sigma ⊨ □?F) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ (?sigma ⊨ ◇?F) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□?F ≡ ◇□?F› ‹◇□◇?F ≡ □◇?F›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ ◇□F; ¬ (sigma ⊨ □◇F)⟧ ⟹ False›*)\n apply (drule STL6 [temp_use] (*‹⟦(?sigma::behavior) ⊨ ◇□(?F::?'a ⇒ bool); ?sigma ⊨ ◇□(?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ ◇□(?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ ◇□¬ F ⟹ sigma ⊨ ◇□?G4 sigma›\n 2. ‹⋀sigma. ⟦sigma ⊨ ◇□¬ F; sigma ⊨ ◇□(F ∧ ?G4 sigma)⟧ ⟹ False›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\nlemma BoxDmdDmdBox: \"⊢ □◇F ∧ ◇□G ⟶ □◇(F ∧ G)\"\n apply clarsimp\n (*goal: ‹⊢ □◇F ∧ ◇□G ⟶ □◇(F ∧ G)›*)\n apply (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹⋀sigma::behavior. ⟦sigma ⊨ □◇(F::'a ⇒ bool); sigma ⊨ ◇□(G::'a ⇒ bool)⟧ ⟹ sigma ⊨ □◇(F ∧ G)›*)\n apply (unfold more_temp_simps2 (*‹□□?F ≡ □?F› ‹◇◇?F ≡ ◇?F› ‹¬ □?F ≡ ◇¬ ?F› ‹¬ ◇?F ≡ □¬ ?F› ‹¬ (?sigma ⊨ □?F) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ (?sigma ⊨ ◇?F) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□?F ≡ ◇□?F› ‹◇□◇?F ≡ □◇?F›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □◇F; sigma ⊨ ◇□G; ¬ (sigma ⊨ □◇(F ∧ G))⟧ ⟹ False›*)\n apply (drule STL6 [temp_use] (*‹⟦?sigma ⊨ ◇□?F; ?sigma ⊨ ◇□?G⟧ ⟹ ?sigma ⊨ ◇□(?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ □◇F; sigma ⊨ ◇□¬ (F ∧ G)⟧ ⟹ sigma ⊨ ◇□?G7 sigma›\n 2. ‹⋀sigma. ⟦sigma ⊨ □◇F; sigma ⊨ ◇□¬ (F ∧ G); sigma ⊨ ◇□(G ∧ ?G7 sigma)⟧ ⟹ False›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (subgoal_tac \"sigma ⊨ ◇□¬F\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ □◇F; sigma ⊨ ◇□¬ (F ∧ G); sigma ⊨ ◇□(G ∧ ¬ (F ∧ G)); sigma ⊨ ◇□¬ F⟧ ⟹ False›\n 2. ‹⋀sigma. ⟦sigma ⊨ □◇F; sigma ⊨ ◇□¬ (F ∧ G); sigma ⊨ ◇□(G ∧ ¬ (F ∧ G))⟧ ⟹ sigma ⊨ ◇□¬ F›\n discuss goal 1*)\n apply (force simp: dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*discuss goal 2*)\n apply (fastforce elim: DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*) STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n\n(* ------------------------------------------------------------------------- *)\n(*** TLA-specific theorems: primed formulas ***)\n(* ------------------------------------------------------------------------- *)\nsection \"priming\"\n\n(* ------------------------ TLA2 ------------------------------------------- *)\nlemma STL2_pr: \"⊢ □P ⟶ Init P ∧ Init P`\"\n by (fastforce intro!: STL2_gen [temp_use] (*‹?sigma ⊨ □?F ⟹ ?sigma ⊨ Init ?F›*) primeI [temp_use] (*‹?sigma ⊨ □?P ⟹ ?sigma ⊨ Init ?P$›*))\n\n(* Auxiliary lemma allows priming of boxed actions *)\nlemma BoxPrime: \"⊢ □P ⟶ □($P ∧ P$)\"\n apply clarsimp\n (*goal: ‹⊢ □P ⟶ □($P ∧ P$)›*)\n apply (erule dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. sigma ⊨ □P ⟹ sigma ⊨ □($P ∧ P$)›*)\n apply (unfold boxInit_act (*‹□(?F::state × state ⇒ bool) ≡ □Init ?F›*))\n (*goal: ‹⋀sigma. sigma ⊨ □□P ⟹ sigma ⊨ □($P ∧ P$)›*)\n apply (erule STL4E (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma::behavior. sigma ⊨ □□(P::state ⇒ bool) ⟹ sigma ⊨ □Init ($P ∧ P$)›*)\n by (auto simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) dest!: STL2_pr [temp_use] (*‹?sigma ⊨ □?P ⟹ (?sigma ⊨ Init ?P) ∧ (?sigma ⊨ Init ?P$)›*))\n\nlemma TLA2:\n assumes \"⊢ $P ∧ P$ ⟶ A\"\n shows \"⊢ □P ⟶ □A\"\n apply clarsimp\n (*goal: ‹⊢ □(P::state ⇒ bool) ⟶ □(A::state × state ⇒ bool)›*)\n apply (drule BoxPrime [temp_use] (*‹?sigma ⊨ □?P ⟹ ?sigma ⊨ □($?P ∧ ?P$)›*))\n (*goal: ‹⋀sigma::behavior. sigma ⊨ □(P::state ⇒ bool) ⟹ sigma ⊨ □(A::state × state ⇒ bool)›*)\n by (auto simp: Init_stp_act_rev [try_rewrite] (*‹Init ?P ≡ Init $?P›*) intro!: assms [temp_use] (*‹⟦P ?s; P ?t⟧ ⟹ A (?s, ?t)›*) elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma TLA2E: \"⟦ sigma ⊨ □P; ⊢ $P ∧ P$ ⟶ A ⟧ ⟹ sigma ⊨ □A\"\n by (erule (1) TLA2 [temp_use] (*‹⟦⊢ $?P ∧ ?P$ ⟶ ?A; ?sigma ⊨ □?P⟧ ⟹ ?sigma ⊨ □?A›*))\n\nlemma DmdPrime: \"⊢ (◇P`) ⟶ (◇P)\"\n apply (unfold dmd_def (*‹◇(?F::?'a ⇒ bool) ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ ◇P$ ⟶ ◇P›*)\n by (fastforce elim!: TLA2E [temp_use] (*‹⟦?sigma ⊨ □?P; ⊢ $?P ∧ ?P$ ⟶ ?A⟧ ⟹ ?sigma ⊨ □?A›*))\n\nlemmas PrimeDmd = InitDmd_gen [temp_use, THEN DmdPrime [temp_use]]\n\n(* ------------------------ INV1, stable --------------------------------------- *)\nsection \"stable, invariant\"\n\nlemma ind_rule:\n \"⟦ sigma ⊨ □H; sigma ⊨ Init P; ⊢ H ⟶ (Init P ∧ ¬□F ⟶ Init(P`) ∧ F) ⟧\n ⟹ sigma ⊨ □F\"\n apply (rule indT [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(Init (?P::state ⇒ bool) ∧ ¬ □(?F::behavior ⇒ bool) ⟶ Init ?P$ ∧ ?F); ?sigma ⊨ Init ?P⟧ ⟹ ?sigma ⊨ □?F›*))\n (*goal: ‹⟦sigma ⊨ □H; sigma ⊨ Init P; ⊢ H ⟶ Init P ∧ ¬ □F ⟶ Init P$ ∧ F⟧ ⟹ sigma ⊨ □F›*)\n by (erule (2) STL4E (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma box_stp_act: \"⊢ (□$P) = (□P)\"\n by (simp add: boxInit_act (*‹□?F ≡ □Init ?F›*) Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*))\n\nlemmas box_stp_actI = box_stp_act [temp_use, THEN iffD2]\nlemmas box_stp_actD = box_stp_act [temp_use, THEN iffD1]\n\nlemmas more_temp_simps3 = box_stp_act [temp_rewrite] more_temp_simps2\n\nlemma INV1:\n \"⊢ (Init P) ⟶ (stable P) ⟶ □P\"\n apply (unfold stable_def (*‹stable ?P ≡ □($?P ⟶ ?P$)›*) boxInit_stp (*‹□?F ≡ □Init ?F›*) boxInit_act (*‹□?F ≡ □Init ?F›*))\n (*goal: ‹⊢ Init P ⟶ stable P ⟶ □P›*)\n apply clarsimp\n (*goal: ‹⊢ Init P ⟶ □Init ($P ⟶ P$) ⟶ □Init P›*)\n apply (erule ind_rule (*‹⟦?sigma ⊨ □?H; ?sigma ⊨ Init ?P; ⊢ ?H ⟶ Init ?P ∧ ¬ □?F ⟶ Init ?P$ ∧ ?F⟧ ⟹ ?sigma ⊨ □?F›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ Init P ⟹ sigma ⊨ Init ?P5 sigma›\n 2. ‹⋀sigma. sigma ⊨ Init P ⟹ ⊢ Init ($P ⟶ P$) ⟶ Init ?P5 sigma ∧ ¬ □Init P ⟶ Init ?P5 sigma$ ∧ Init P›\n discuss goal 1*)\n apply ((auto simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) elim: ind_rule (*‹⟦?sigma ⊨ □?H; ?sigma ⊨ Init ?P; ⊢ ?H ⟶ Init ?P ∧ ¬ □?F ⟶ Init ?P$ ∧ ?F⟧ ⟹ ?sigma ⊨ □?F›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) elim: ind_rule (*‹⟦?sigma ⊨ □?H; ?sigma ⊨ Init ?P; ⊢ ?H ⟶ Init ?P ∧ ¬ □?F ⟶ Init ?P$ ∧ ?F⟧ ⟹ ?sigma ⊨ □?F›*))[1])\n (*proven 2 subgoals*) .\n\nlemma StableT:\n \"⋀P. ⊢ $P ∧ A ⟶ P` ⟹ ⊢ □A ⟶ stable P\"\n apply (unfold stable_def (*‹stable ?P ≡ □($?P ⟶ ?P$)›*))\n (*goal: ‹⋀P. ⊢ $P ∧ A ⟶ P$ ⟹ ⊢ □A ⟶ stable P›*)\n by (fastforce elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma Stable: \"⟦ sigma ⊨ □A; ⊢ $P ∧ A ⟶ P` ⟧ ⟹ sigma ⊨ stable P\"\n by (erule (1) StableT [temp_use] (*‹⟦⊢ $?P ∧ ?A ⟶ ?P$; ?sigma ⊨ □?A⟧ ⟹ ?sigma ⊨ stable ?P›*))\n\n(* Generalization of INV1 *)\nlemma StableBox: \"⊢ (stable P) ⟶ □(Init P ⟶ □P)\"\n apply (unfold stable_def (*‹stable ?P ≡ □($?P ⟶ ?P$)›*))\n (*goal: ‹⊢ stable P::state ⇒ bool ⟶ □(Init P ⟶ □P)›*)\n apply clarsimp\n (*goal: ‹⊢ □($P ⟶ P$) ⟶ □(Init P ⟶ □P)›*)\n apply (erule dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. sigma ⊨ □($P ⟶ P$) ⟹ sigma ⊨ □(Init P ⟶ □P)›*)\n by (force simp: stable_def (*‹stable ?P ≡ □($?P ⟶ ?P$)›*) elim: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*) INV1 [temp_use] (*‹⟦?sigma ⊨ Init ?P; ?sigma ⊨ stable ?P⟧ ⟹ ?sigma ⊨ □?P›*))\n\nlemma DmdStable: \"⊢ (stable P) ∧ ◇P ⟶ ◇□P\"\n apply clarsimp\n (*goal: ‹⊢ stable P ∧ ◇P ⟶ ◇□P›*)\n apply (rule DmdImpl2 (*‹⟦(?sigma::behavior) ⊨ ◇(?F::?'a::world ⇒ bool); ?sigma ⊨ □(?F ⟶ (?G::?'a::world ⇒ bool))⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ stable P; sigma ⊨ ◇P⟧ ⟹ sigma ⊨ ◇□P›*)\n prefer 2\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ stable P; sigma ⊨ ◇P⟧ ⟹ sigma ⊨ □(?F5 sigma ⟶ □P)› and 1 goal remains*)\n apply (erule StableBox [temp_use] (*‹?sigma ⊨ stable ?P ⟹ ?sigma ⊨ □(Init ?P ⟶ □?P)›*))\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ stable P; sigma ⊨ ◇P⟧ ⟹ sigma ⊨ □(?F5 sigma ⟶ □P)› and 1 goal remains*)\n by (simp add: dmdInitD (*‹◇Init ?F ≡ ◇?F›*))\n\n(* ---------------- (Semi-)automatic invariant tactics ---------------------- *)\n\nML ‹\n(* inv_tac reduces goals of the form ... ⟹ sigma ⊨ □P *)\nfun inv_tac ctxt =\n SELECT_GOAL\n (EVERY\n [auto_tac ctxt,\n TRY (merge_box_tac ctxt 1),\n resolve_tac ctxt [temp_use ctxt @{thm INV1}] 1, (* fail if the goal is not a box *)\n TRYALL (eresolve_tac ctxt @{thms Stable})]);\n\n(* auto_inv_tac applies inv_tac and then tries to attack the subgoals\n in simple cases it may be able to handle goals like ⊢ MyProg ⟶ □Inv.\n In these simple cases the simplifier seems to be more useful than the\n auto-tactic, which applies too much propositional logic and simplifies\n too late.\n*)\nfun auto_inv_tac ctxt =\n SELECT_GOAL\n (inv_tac ctxt 1 THEN\n (TRYALL (action_simp_tac\n (ctxt addsimps [@{thm Init_stp}, @{thm Init_act}]) [] [@{thm squareE}])));\n›\n\nmethod_setup invariant = ‹\n Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o inv_tac))\n›\n\nmethod_setup auto_invariant = ‹\n Method.sections Clasimp.clasimp_modifiers >> (K (SIMPLE_METHOD' o auto_inv_tac))\n›\n\nlemma unless: \"⊢ □($P ⟶ P` ∨ Q`) ⟶ (stable P) ∨ ◇Q\"\n apply (unfold dmd_def (*‹◇(?F::?'a ⇒ bool) ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ □($P ⟶ P$ ∨ Q$) ⟶ stable P ∨ ◇Q›*)\n apply (clarsimp dest!: BoxPrime [temp_use] (*‹?sigma ⊨ □?P ⟹ ?sigma ⊨ □($?P ∧ ?P$)›*))\n (*goal: ‹⊢ □($P ⟶ P$ ∨ Q$) ⟶ stable P ∨ ¬ □¬ Q›*)\n apply merge_box\n (*goal: ‹⋀sigma::behavior. ⟦sigma ⊨ □($(P::state ⇒ bool) ⟶ P$ ∨ (Q::state ⇒ bool)$); ¬ (sigma ⊨ stable P); sigma ⊨ □($(¬ Q) ∧ ¬ Q$)⟧ ⟹ False›*)\n apply (erule contrapos_np (*‹⟦¬ ?Q; ¬ ?P ⟹ ?Q⟧ ⟹ ?P›*))\n (*goal: ‹⋀sigma. ⟦¬ (sigma ⊨ stable P); sigma ⊨ □(($P ⟶ P$ ∨ Q$) ∧ $(¬ Q) ∧ ¬ Q$)⟧ ⟹ False›*)\n by (fastforce elim!: Stable [temp_use] (*‹⟦?sigma ⊨ □?A; ⊢ $?P ∧ ?A ⟶ ?P$⟧ ⟹ ?sigma ⊨ stable ?P›*))\n\n\n(* --------------------- Recursive expansions --------------------------------------- *)\nsection \"recursive expansions\"\n\n(* Recursive expansions of □ and ◇ for state predicates *)\nlemma BoxRec: \"⊢ (□P) = (Init P ∧ □P`)\"\n apply (auto intro!: STL2_gen [temp_use] (*‹?sigma ⊨ □?F ⟹ ?sigma ⊨ Init ?F›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ □P ⟹ sigma ⊨ □P$›\n 2. ‹⋀sigma. ⟦sigma ⊨ Init P; sigma ⊨ □P$⟧ ⟹ sigma ⊨ □P›\n discuss goal 1*)\n apply (fastforce elim!: TLA2E [temp_use] (*‹⟦?sigma ⊨ □?P; ⊢ $?P ∧ ?P$ ⟶ ?A⟧ ⟹ ?sigma ⊨ □?A›*))\n (*discuss goal 2*)\n apply (auto simp: stable_def (*‹stable ?P::state ⇒ bool ≡ □($?P ⟶ ?P$)›*) elim!: INV1 [temp_use] (*‹⟦(?sigma::behavior) ⊨ Init (?P::state ⇒ bool); ?sigma ⊨ stable ?P⟧ ⟹ ?sigma ⊨ □?P›*) STL4E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*))\n (*proven 2 subgoals*) .\n\nlemma DmdRec: \"⊢ (◇P) = (Init P ∨ ◇P`)\"\n apply (unfold dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*) BoxRec [temp_rewrite] (*‹□?P ≡ Init ?P ∧ □?P$›*))\n (*goal: ‹⊢ (◇P) = (Init P ∨ ◇P$)›*)\n by (auto simp: Init_simps (*‹Init (?y::behavior ⇒ bool) ≡ ?y› ‹Init $(?P::state ⇒ bool) ≡ Init ?P› ‹Init ¬ (?F::?'a ⇒ bool) ≡ ¬ Init ?F› ‹Init ((?P::?'b ⇒ bool) ⟶ (?Q::?'b ⇒ bool)) ≡ Init ?P ⟶ Init ?Q› ‹Init ((?P::?'b ⇒ bool) ∧ (?Q::?'b ⇒ bool)) ≡ Init ?P ∧ Init ?Q› ‹Init ((?P::?'b ⇒ bool) ∨ (?Q::?'b ⇒ bool)) ≡ Init ?P ∨ Init ?Q› ‹Init (?P::?'b ⇒ bool) = (?Q::?'b ⇒ bool) ≡ Init ?P = (Init ?Q)› ‹Init (∀x::?'d. (?F::?'d ⇒ ?'c ⇒ bool) x) ≡ ∀x::?'d. Init ?F x› ‹Init (∃x::?'d. (?F::?'d ⇒ ?'c ⇒ bool) x) ≡ ∃x::?'d. Init ?F x› ‹Init (∃!x::?'d. (?F::?'d ⇒ ?'c ⇒ bool) x) ≡ ∃!x::?'d. Init ?F x› ‹□Init (?F::?'a ⇒ bool) ≡ □?F› ‹◇Init (?F::?'a ⇒ bool) ≡ ◇?F› and more 2 facts*))\n\nlemma DmdRec2: \"⋀sigma. ⟦ sigma ⊨ ◇P; sigma ⊨ □¬P` ⟧ ⟹ sigma ⊨ Init P\"\n by (force simp: DmdRec [temp_rewrite] (*‹◇(?P::state ⇒ bool) ≡ Init ?P ∨ ◇?P$›*) dmd_def (*‹◇(?F::?'a ⇒ bool) ≡ ¬ □¬ ?F›*))\n\nlemma InfinitePrime: \"⊢ (□◇P) = (□◇P`)\"\n apply auto\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ □◇P ⟹ sigma ⊨ □◇P$›\n 2. ‹⋀sigma. sigma ⊨ □◇P$ ⟹ sigma ⊨ □◇P›\n discuss goal 1*)\n apply (rule classical (*‹(¬ (?P::bool) ⟹ ?P) ⟹ ?P›*))\n (*top goal: ‹⋀sigma::behavior. sigma ⊨ □◇(P::state ⇒ bool) ⟹ sigma ⊨ □◇P$› and 1 goal remains*)\n apply (rule DBImplBD [temp_use] (*‹?sigma ⊨ ◇□?F ⟹ ?sigma ⊨ □◇?F›*))\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ □◇P; ¬ (sigma ⊨ □◇P$)⟧ ⟹ sigma ⊨ □◇P$› and 1 goal remains*)\n apply (subgoal_tac \"sigma ⊨ ◇□P\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ □◇P; ¬ (sigma ⊨ □◇P$); sigma ⊨ ◇□P⟧ ⟹ sigma ⊨ ◇□P$›\n 2. ‹⋀sigma. ⟦sigma ⊨ □◇P; ¬ (sigma ⊨ □◇P$)⟧ ⟹ sigma ⊨ ◇□P›\n discuss goal 1*)\n apply (fastforce elim!: DmdImplE [temp_use] (*‹⟦(?sigma::behavior) ⊨ ◇(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ ◇?G›*) TLA2E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?P::state ⇒ bool); ⊢ $?P ∧ ?P$ ⟶ (?A::state × state ⇒ bool)⟧ ⟹ ?sigma ⊨ □?A›*))\n (*discuss goal 2*)\n apply (subgoal_tac \"sigma ⊨ ◇□ (◇P ∧ □¬P`)\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ □◇P; ¬ (sigma ⊨ □◇P$); sigma ⊨ ◇□(◇P ∧ □¬ P$)⟧ ⟹ sigma ⊨ ◇□P›\n 2. ‹⋀sigma. ⟦sigma ⊨ □◇P; ¬ (sigma ⊨ □◇P$)⟧ ⟹ sigma ⊨ ◇□(◇P ∧ □¬ P$)›\n discuss goal 1*)\n apply (force simp: boxInit_stp [temp_use] (*‹□?F ≡ □Init ?F›*) elim!: DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*) STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*) DmdRec2 [temp_use] (*‹⟦?sigma ⊨ ◇?P; ?sigma ⊨ □¬ ?P$⟧ ⟹ ?sigma ⊨ Init ?P›*))\n (*discuss goal 2*)\n apply (force intro!: STL6 [temp_use] (*‹⟦(?sigma::behavior) ⊨ ◇□(?F::?'a ⇒ bool); ?sigma ⊨ ◇□(?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ ◇□(?F ∧ ?G)›*) simp: more_temp_simps3 (*‹□$(?P::state ⇒ bool) ≡ □?P› ‹□□(?F::?'a ⇒ bool) ≡ □?F› ‹◇◇(?F::?'a ⇒ bool) ≡ ◇?F› ‹¬ □(?F::?'a ⇒ bool) ≡ ◇¬ ?F› ‹¬ ◇(?F::?'a ⇒ bool) ≡ □¬ ?F› ‹¬ ((?sigma::behavior) ⊨ □(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ ((?sigma::behavior) ⊨ ◇(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□(?F::?'a ⇒ bool) ≡ ◇□?F› ‹◇□◇(?F::?'a ⇒ bool) ≡ □◇?F›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (fastforce intro: DmdPrime [temp_use] (*‹?sigma ⊨ ◇?P$ ⟹ ?sigma ⊨ ◇?P›*) elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*proven 2 subgoals*) .\n\nlemma InfiniteEnsures:\n \"⟦ sigma ⊨ □N; sigma ⊨ □◇A; ⊢ A ∧ N ⟶ P` ⟧ ⟹ sigma ⊨ □◇P\"\n apply (unfold InfinitePrime [temp_rewrite] (*‹□◇?P ≡ □◇?P$›*))\n (*goal: ‹⟦sigma ⊨ □N; sigma ⊨ □◇A; ⊢ A ∧ N ⟶ P$⟧ ⟹ sigma ⊨ □◇P›*)\n apply (rule InfImpl (*‹⟦?sigma ⊨ □◇?F; ?sigma ⊨ □?G; ⊢ ?F ∧ ?G ⟶ ?H⟧ ⟹ ?sigma ⊨ □◇?H›*))\n (*goals:\n 1. ‹⟦sigma ⊨ □N; sigma ⊨ □◇A; ⊢ A ∧ N ⟶ P$⟧ ⟹ sigma ⊨ □◇?F›\n 2. ‹⟦sigma ⊨ □N; sigma ⊨ □◇A; ⊢ A ∧ N ⟶ P$⟧ ⟹ sigma ⊨ □?G›\n 3. ‹⟦sigma ⊨ □N; sigma ⊨ □◇A; ⊢ A ∧ N ⟶ P$⟧ ⟹ ⊢ ?F ∧ ?G ⟶ P$›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply assumption\n (*proven 3 subgoals*) .\n\n(* ------------------------ fairness ------------------------------------------- *)\nsection \"fairness\"\n\n(* alternative definitions of fairness *)\nlemma WF_alt: \"⊢ WF(A)_v = (□◇¬Enabled(<A>_v) ∨ □◇<A>_v)\"\n apply (unfold WF_def (*‹WF(?A)_?v ≡ ◇□Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*) dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ WF(A)_v = (□◇¬ Enabled (<A>_v) ∨ □◇<A>_v)›*)\n by fastforce\n\nlemma SF_alt: \"⊢ SF(A)_v = (◇□¬Enabled(<A>_v) ∨ □◇<A>_v)\"\n apply (unfold SF_def (*‹SF(?A)_?v ≡ □◇Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*) dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ SF(A)_v = (◇□¬ Enabled (<A>_v) ∨ □◇<A>_v)›*)\n by fastforce\n\n(* theorems to \"box\" fairness conditions *)\nlemma BoxWFI: \"⊢ WF(A)_v ⟶ □WF(A)_v\"\n by (auto simp: WF_alt [try_rewrite] (*‹WF(?A)_?v ≡ □◇¬ Enabled (<?A>_?v) ∨ □◇<?A>_?v›*) more_temp_simps3 (*‹□$?P ≡ □?P› ‹□□?F ≡ □?F› ‹◇◇?F ≡ ◇?F› ‹¬ □?F ≡ ◇¬ ?F› ‹¬ ◇?F ≡ □¬ ?F› ‹¬ (?sigma ⊨ □?F) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ (?sigma ⊨ ◇?F) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□?F ≡ ◇□?F› ‹◇□◇?F ≡ □◇?F›*) intro!: BoxOr [temp_use] (*‹?sigma ⊨ □?F ∨ □?G ⟹ ?sigma ⊨ □(?F ∨ ?G)›*))\n\nlemma WF_Box: \"⊢ (□WF(A)_v) = WF(A)_v\"\n by (fastforce intro!: BoxWFI [temp_use] (*‹?sigma ⊨ WF(?A)_?v ⟹ ?sigma ⊨ □WF(?A)_?v›*) dest!: STL2 [temp_use] (*‹?sigma ⊨ □?F ⟹ ?F ?sigma›*))\n\nlemma BoxSFI: \"⊢ SF(A)_v ⟶ □SF(A)_v\"\n by (auto simp: SF_alt [try_rewrite] (*‹SF(?A)_?v ≡ ◇□¬ Enabled (<?A>_?v) ∨ □◇<?A>_?v›*) more_temp_simps3 (*‹□$?P ≡ □?P› ‹□□?F ≡ □?F› ‹◇◇?F ≡ ◇?F› ‹¬ □?F ≡ ◇¬ ?F› ‹¬ ◇?F ≡ □¬ ?F› ‹¬ (?sigma ⊨ □?F) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ (?sigma ⊨ ◇?F) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□?F ≡ ◇□?F› ‹◇□◇?F ≡ □◇?F›*) intro!: BoxOr [temp_use] (*‹?sigma ⊨ □?F ∨ □?G ⟹ ?sigma ⊨ □(?F ∨ ?G)›*))\n\nlemma SF_Box: \"⊢ (□SF(A)_v) = SF(A)_v\"\n by (fastforce intro!: BoxSFI [temp_use] (*‹(?sigma::behavior) ⊨ SF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a) ⟹ ?sigma ⊨ □SF(?A)_?v›*) dest!: STL2 [temp_use] (*‹(?sigma::behavior) ⊨ □(?F::behavior ⇒ bool) ⟹ ?F ?sigma›*))\n\nlemmas more_temp_simps = more_temp_simps3 WF_Box [temp_rewrite] SF_Box [temp_rewrite]\n\nlemma SFImplWF: \"⊢ SF(A)_v ⟶ WF(A)_v\"\n apply (unfold SF_def (*‹SF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'d::type) ≡ □◇Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*) WF_def (*‹WF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'d::type) ≡ ◇□Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*))\n (*goal: ‹⊢ SF(A)_v ⟶ WF(A)_v›*)\n by (fastforce dest!: DBImplBD [temp_use] (*‹?sigma ⊨ ◇□?F ⟹ ?sigma ⊨ □◇?F›*))\n\n(* A tactic that \"boxes\" all fairness conditions. Apply more_temp_simps to \"unbox\". *)\nML ‹\nfun box_fair_tac ctxt =\n SELECT_GOAL (REPEAT (dresolve_tac ctxt [@{thm BoxWFI}, @{thm BoxSFI}] 1))\n›\n\n\n(* ------------------------------ leads-to ------------------------------ *)\n\nsection \"↝\"\n\nlemma leadsto_init: \"⊢ (Init F) ∧ (F ↝ G) ⟶ ◇G\"\n apply (unfold leadsto_def (*‹(?F::?'b::world ⇒ bool) ↝ (?G::?'c::world ⇒ bool) ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ Init F ∧ (F ↝ G) ⟶ ◇G›*)\n by (auto dest!: STL2 [temp_use] (*‹?sigma ⊨ □?F ⟹ ?F ?sigma›*))\n\n(* ⊢ F & (F ↝ G) ⟶ ◇G *)\nlemmas leadsto_init_temp = leadsto_init [where 'a = behavior, unfolded Init_simps]\n\nlemma streett_leadsto: \"⊢ (□◇Init F ⟶ □◇G) = (◇(F ↝ G))\"\n apply (unfold leadsto_def (*‹(?F::?'b ⇒ bool) ↝ (?G::?'c ⇒ bool) ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ (□◇Init F ⟶ □◇G) = (◇(F ↝ G))›*)\n apply auto\n (*goals:\n 1. ‹⋀sigma. ¬ (sigma ⊨ □◇Init F) ⟹ sigma ⊨ ◇□(Init F ⟶ ◇G)›\n 2. ‹⋀sigma. sigma ⊨ □◇G ⟹ sigma ⊨ ◇□(Init F ⟶ ◇G)›\n 3. ‹⋀sigma. ⟦sigma ⊨ ◇□(Init F ⟶ ◇G); sigma ⊨ □◇Init F⟧ ⟹ sigma ⊨ □◇G›\n discuss goal 1*)\n apply (simp add: more_temp_simps (*‹□$?P ≡ □?P› ‹□□?F ≡ □?F› ‹◇◇?F ≡ ◇?F› ‹¬ □?F ≡ ◇¬ ?F› ‹¬ ◇?F ≡ □¬ ?F› ‹¬ (?sigma ⊨ □?F) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ (?sigma ⊨ ◇?F) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□?F ≡ ◇□?F› ‹◇□◇?F ≡ □◇?F› ‹□WF(?A)_?v ≡ WF(?A)_?v› ‹□SF(?A)_?v ≡ SF(?A)_?v›*))\n (*top goal: ‹⋀sigma. ¬ (sigma ⊨ □◇Init F) ⟹ sigma ⊨ ◇□(Init F ⟶ ◇G)› and 2 goals remain*)\n apply (fastforce elim!: DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*) STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*discuss goal 2*)\n apply (fastforce intro!: InitDmd [temp_use] (*‹(?F::behavior ⇒ bool) (?sigma::behavior) ⟹ ?sigma ⊨ ◇?F›*) elim!: STL4E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*))\n (*discuss goal 3*)\n apply (subgoal_tac \"sigma ⊨ □◇◇G\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ ◇□(Init F ⟶ ◇G); sigma ⊨ □◇Init F; sigma ⊨ □◇◇G⟧ ⟹ sigma ⊨ □◇G›\n 2. ‹⋀sigma. ⟦sigma ⊨ ◇□(Init F ⟶ ◇G); sigma ⊨ □◇Init F⟧ ⟹ sigma ⊨ □◇◇G›\n discuss goal 1*)\n apply (simp add: more_temp_simps (*‹□$(?P::state ⇒ bool) ≡ □?P› ‹□□(?F::?'a ⇒ bool) ≡ □?F› ‹◇◇(?F::?'a ⇒ bool) ≡ ◇?F› ‹¬ □(?F::?'a ⇒ bool) ≡ ◇¬ ?F› ‹¬ ◇(?F::?'a ⇒ bool) ≡ □¬ ?F› ‹¬ ((?sigma::behavior) ⊨ □(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ ((?sigma::behavior) ⊨ ◇(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□(?F::?'a ⇒ bool) ≡ ◇□?F› ‹◇□◇(?F::?'a ⇒ bool) ≡ □◇?F› ‹□WF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a) ≡ WF(?A)_?v› ‹□SF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a) ≡ SF(?A)_?v›*))\n (*discuss goal 2*)\n apply (drule BoxDmdDmdBox [temp_use] (*‹⟦?sigma ⊨ □◇?F; ?sigma ⊨ ◇□?G⟧ ⟹ ?sigma ⊨ □◇(?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma::behavior. sigma ⊨ ◇□(Init (F::'a ⇒ bool) ⟶ ◇(G::'b ⇒ bool)) ⟹ sigma ⊨ ◇□(?G17::behavior ⇒ behavior ⇒ bool) sigma›\n 2. ‹⋀sigma::behavior. ⟦sigma ⊨ ◇□(Init (F::'a ⇒ bool) ⟶ ◇(G::'b ⇒ bool)); sigma ⊨ □◇(Init F ∧ (?G17::behavior ⇒ behavior ⇒ bool) sigma)⟧ ⟹ sigma ⊨ □◇◇G›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (fastforce elim!: DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*) STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 3 subgoals*) .\n\nlemma leadsto_infinite: \"⊢ □◇F ∧ (F ↝ G) ⟶ □◇G\"\n apply clarsimp\n (*goal: ‹⊢ □◇F ∧ (F ↝ G) ⟶ □◇G›*)\n apply (erule InitDmd [temp_use, THEN streett_leadsto [temp_unlift, THEN iffD2, THEN mp]] (*‹⟦(?sigma2::behavior) ⊨ (?F3::?'a3 ⇒ bool) ↝ (?G3::?'b3 ⇒ bool); ?sigma2 ⊨ □◇Init ?F3⟧ ⟹ ?sigma2 ⊨ □◇?G3›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □◇F; sigma ⊨ F ↝ G⟧ ⟹ sigma ⊨ □◇G›*)\n by (simp add: dmdInitD (*‹◇Init (?F::?'a ⇒ bool) ≡ ◇?F›*))\n\n(* In particular, strong fairness is a Streett condition. The following\n rules are sometimes easier to use than WF2 or SF2 below.\n*)\nlemma leadsto_SF: \"⊢ (Enabled(<A>_v) ↝ <A>_v) ⟶ SF(A)_v\"\n apply (unfold SF_def (*‹SF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'d) ≡ □◇Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*))\n (*goal: ‹⊢ (Enabled (<A>_v) ↝ <A>_v) ⟶ SF(A)_v›*)\n by (clarsimp elim!: leadsto_infinite [temp_use] (*‹⟦?sigma ⊨ □◇?F; ?sigma ⊨ ?F ↝ ?G⟧ ⟹ ?sigma ⊨ □◇?G›*))\n\nlemma leadsto_WF: \"⊢ (Enabled(<A>_v) ↝ <A>_v) ⟶ WF(A)_v\"\n by (clarsimp intro!: SFImplWF [temp_use] (*‹(?sigma::behavior) ⊨ SF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a::type) ⟹ ?sigma ⊨ WF(?A)_?v›*) leadsto_SF [temp_use] (*‹(?sigma::behavior) ⊨ Enabled (<?A::state × state ⇒ bool>_(?v::state ⇒ ?'a::type)) ↝ <?A>_?v ⟹ ?sigma ⊨ SF(?A)_?v›*))\n\n(* introduce an invariant into the proof of a leadsto assertion.\n □I ⟶ ((P ↝ Q) = (P /\\ I ↝ Q))\n*)\nlemma INV_leadsto: \"⊢ □I ∧ (P ∧ I ↝ Q) ⟶ (P ↝ Q)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ □I ∧ (P ∧ I ↝ Q) ⟶ (P ↝ Q)›*)\n apply clarsimp\n (*goal: ‹⊢ □I ∧ □(Init (P ∧ I) ⟶ ◇Q) ⟶ □(Init P ⟶ ◇Q)›*)\n apply (erule STL4Edup (*‹⟦?sigma ⊨ □?A; ?sigma ⊨ □?F; ⊢ ?F ∧ □?A ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ □(Init (P ∧ I) ⟶ ◇Q) ⟹ sigma ⊨ □?F5 sigma›\n 2. ‹⋀sigma. sigma ⊨ □(Init (P ∧ I) ⟶ ◇Q) ⟹ ⊢ ?F5 sigma ∧ □I ⟶ Init P ⟶ ◇Q›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (auto simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) dest!: STL2_gen [temp_use] (*‹?sigma ⊨ □?F ⟹ ?sigma ⊨ Init ?F›*))\n (*proven 2 subgoals*) .\n\nlemma leadsto_classical: \"⊢ (Init F ∧ □¬G ↝ G) ⟶ (F ↝ G)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*) dmd_def (*‹◇?F ≡ ¬ □¬ ?F›*))\n (*goal: ‹⊢ (Init F ∧ □¬ G ↝ G) ⟶ (F ↝ G)›*)\n by (force simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma leadsto_false: \"⊢ (F ↝ #False) = (□¬F)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ (F ↝ #False) = (□¬ F)›*)\n by (simp add: boxNotInitD (*‹□¬ Init (?F::?'a ⇒ bool) ≡ □¬ ?F›*))\n\nlemma leadsto_exists: \"⊢ ((∃x. F x) ↝ G) = (∀x. (F x ↝ G))\"\n apply (unfold leadsto_def (*‹(?F::?'b::world ⇒ bool) ↝ (?G::?'c::world ⇒ bool) ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ ((∃x. F x) ↝ G) = (∀x. F x ↝ G)›*)\n by (auto simp: allT [try_rewrite] (*‹∀x. □?F x ≡ □(∀x. ?F x)›*) Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\n(* basic leadsto properties, cf. Unity *)\n\nlemma ImplLeadsto_gen: \"⊢ □(Init F ⟶ Init G) ⟶ (F ↝ G)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ □(Init F ⟶ Init G) ⟶ (F ↝ G)›*)\n by (auto intro!: InitDmd_gen [temp_use] (*‹?sigma ⊨ Init ?F ⟹ ?sigma ⊨ ◇?F›*) elim!: STL4E_gen [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ Init ?F ⟶ Init ?G⟧ ⟹ ?sigma ⊨ □?G›*) simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*))\n\nlemmas ImplLeadsto =\n ImplLeadsto_gen [where 'a = behavior and 'b = behavior, unfolded Init_simps]\n\nlemma ImplLeadsto_simple: \"⋀F G. ⊢ F ⟶ G ⟹ ⊢ F ↝ G\"\n by (auto simp: Init_def (*‹(?sigma ⊨ Init ?F) = ?F (first_world ?sigma)›*) intro!: ImplLeadsto_gen [temp_use] (*‹?sigma ⊨ □(Init ?F ⟶ Init ?G) ⟹ ?sigma ⊨ ?F ↝ ?G›*) necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*))\n\nlemma EnsuresLeadsto:\n assumes \"⊢ A ∧ $P ⟶ Q`\"\n shows \"⊢ □A ⟶ (P ↝ Q)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ □A ⟶ (P ↝ Q)›*)\n apply (clarsimp elim!: INV_leadsto [temp_use] (*‹⟦?sigma ⊨ □?I; ?sigma ⊨ ?P ∧ ?I ↝ ?Q⟧ ⟹ ?sigma ⊨ ?P ↝ ?Q›*))\n (*goal: ‹⊢ □A ⟶ □(Init P ⟶ ◇Q)›*)\n apply (erule STL4E_gen (*‹⟦?sigma ⊨ □?F; ⊢ Init ?F ⟶ Init ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma. sigma ⊨ □A ⟹ sigma ⊨ □(Init P ⟶ ◇Q)›*)\n by (auto simp: Init_defs (*‹(?sigma ⊨ Init ?P) = ?P (st1 ?sigma)› ‹(?sigma ⊨ Init ?A) = ?A (st1 ?sigma, st2 ?sigma)› ‹(?w ⊨ Init ?F) = ?F ?w›*) intro!: PrimeDmd [temp_use] (*‹?sigma ⊨ Init ?P$ ⟹ ?sigma ⊨ ◇?P›*) assms [temp_use] (*‹⟦A (?s, ?t); P ?s⟧ ⟹ Q ?t›*))\n\nlemma EnsuresLeadsto2: \"⊢ □($P ⟶ Q`) ⟶ (P ↝ Q)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ □($P ⟶ Q$) ⟶ (P ↝ Q)›*)\n apply clarsimp\n (*goal: ‹⊢ □($P ⟶ Q$) ⟶ □(Init P ⟶ ◇Q)›*)\n apply (erule STL4E_gen (*‹⟦?sigma ⊨ □?F; ⊢ Init ?F ⟶ Init ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma. sigma ⊨ □($P ⟶ Q$) ⟹ sigma ⊨ □(Init P ⟶ ◇Q)›*)\n by (auto simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) intro!: PrimeDmd [temp_use] (*‹?sigma ⊨ Init ?P$ ⟹ ?sigma ⊨ ◇?P›*))\n\nlemma ensures:\n assumes 1: \"⊢ $P ∧ N ⟶ P` ∨ Q`\"\n and 2: \"⊢ ($P ∧ N) ∧ A ⟶ Q`\"\n shows \"⊢ □N ∧ □(□P ⟶ ◇A) ⟶ (P ↝ Q)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ □(N::state × state ⇒ bool) ∧ □(□(P::state ⇒ bool) ⟶ ◇(A::state × state ⇒ bool)) ⟶ (P ↝ (Q::state ⇒ bool))›*)\n apply clarsimp\n (*goal: ‹⊢ □N ∧ □(□P ⟶ ◇A) ⟶ □(Init P ⟶ ◇Q)›*)\n apply (erule STL4Edup (*‹⟦?sigma ⊨ □?A; ?sigma ⊨ □?F; ⊢ ?F ∧ □?A ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ □(□P ⟶ ◇A) ⟹ sigma ⊨ □?F5 sigma›\n 2. ‹⋀sigma. sigma ⊨ □(□P ⟶ ◇A) ⟹ ⊢ ?F5 sigma ∧ □N ⟶ Init P ⟶ ◇Q›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply clarsimp\n (*goal: ‹⋀sigma. sigma ⊨ □(□P ⟶ ◇A) ⟹ ⊢ (□P ⟶ ◇A) ∧ □N ⟶ Init P ⟶ ◇Q›*)\n apply (subgoal_tac \"sigmaa ⊨ □($P ⟶ P` ∨ Q`) \")\n (*goals:\n 1. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□P ⟶ ◇A); (sigmaa ⊨ □P) ⟶ (sigmaa ⊨ ◇A); sigmaa ⊨ □N; sigmaa ⊨ Init P; sigmaa ⊨ □($P ⟶ P$ ∨ Q$)⟧ ⟹ sigmaa ⊨ ◇Q›\n 2. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□P ⟶ ◇A); (sigmaa ⊨ □P) ⟶ (sigmaa ⊨ ◇A); sigmaa ⊨ □N; sigmaa ⊨ Init P⟧ ⟹ sigmaa ⊨ □($P ⟶ P$ ∨ Q$)›\n discuss goal 1*)\n apply (drule unless [temp_use] (*‹?sigma ⊨ □($?P ⟶ ?P$ ∨ ?Q$) ⟹ (?sigma ⊨ stable ?P) ∨ (?sigma ⊨ ◇?Q)›*))\n (*top goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□P ⟶ ◇A); (sigmaa ⊨ □P) ⟶ (sigmaa ⊨ ◇A); sigmaa ⊨ □N; sigmaa ⊨ Init P; sigmaa ⊨ □($P ⟶ P$ ∨ Q$)⟧ ⟹ sigmaa ⊨ ◇Q› and 1 goal remains*)\n apply (clarsimp dest!: INV1 [temp_use] (*‹⟦?sigma ⊨ Init ?P; ?sigma ⊨ stable ?P⟧ ⟹ ?sigma ⊨ □?P›*))\n (*top goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□P ⟶ ◇A); (sigmaa ⊨ □P) ⟶ (sigmaa ⊨ ◇A); sigmaa ⊨ □N; sigmaa ⊨ Init P; (sigmaa ⊨ stable P) ∨ (sigmaa ⊨ ◇Q)⟧ ⟹ sigmaa ⊨ ◇Q› and 1 goal remains*)\n apply (rule 2 [THEN DmdImpl, temp_use, THEN DmdPrime [temp_use]] (*‹(?sigma::behavior) ⊨ ◇(($(P::state ⇒ bool) ∧ (N::state × state ⇒ bool)) ∧ (A::state × state ⇒ bool)) ⟹ ?sigma ⊨ ◇(Q::state ⇒ bool)›*))\n (*top goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□P ⟶ ◇A); sigmaa ⊨ □N; sigmaa ⊨ stable P; sigmaa ⊨ □P; sigmaa ⊨ ◇A⟧ ⟹ sigmaa ⊨ ◇Q› and 1 goal remains*)\n apply (force intro!: BoxDmd_simple [temp_use] (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ ◇?G⟧ ⟹ ?sigma ⊨ ◇(?F ∧ ?G)›*) simp: split_box_conj [try_rewrite] (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) box_stp_act [try_rewrite] (*‹□$?P ≡ □?P›*))\n (*discuss goal 2*)\n apply (force elim: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*) dest: 1 [temp_use] (*‹⟦P ?s; N (?s, ?t)⟧ ⟹ P ?t ∨ Q ?t›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\nlemma ensures_simple:\n \"⟦ ⊢ $P ∧ N ⟶ P` ∨ Q`;\n ⊢ ($P ∧ N) ∧ A ⟶ Q`\n ⟧ ⟹ ⊢ □N ∧ □◇A ⟶ (P ↝ Q)\"\n apply clarsimp\n (*goal: ‹⟦⊢ $P ∧ N ⟶ P$ ∨ Q$; ⊢ ($P ∧ N) ∧ A ⟶ Q$⟧ ⟹ ⊢ □N ∧ □◇A ⟶ (P ↝ Q)›*)\n apply (erule (2) ensures [temp_use] (*‹⟦⊢ $(?P::state ⇒ bool) ∧ (?N::state × state ⇒ bool) ⟶ ?P$ ∨ (?Q::state ⇒ bool)$; ⊢ ($?P ∧ ?N) ∧ (?A::state × state ⇒ bool) ⟶ ?Q$; (?sigma::behavior) ⊨ □?N; ?sigma ⊨ □(□?P ⟶ ◇?A)⟧ ⟹ ?sigma ⊨ ?P ↝ ?Q›*))\n (*goal: ‹⋀sigma. ⟦⊢ $P ∧ N ⟶ P$ ∨ Q$; ⊢ ($P ∧ N) ∧ A ⟶ Q$; sigma ⊨ □N; sigma ⊨ □◇A⟧ ⟹ sigma ⊨ P ↝ Q›*)\n by (force elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma EnsuresInfinite:\n \"⟦ sigma ⊨ □◇P; sigma ⊨ □A; ⊢ A ∧ $P ⟶ Q` ⟧ ⟹ sigma ⊨ □◇Q\"\n apply (erule leadsto_infinite [temp_use] (*‹⟦?sigma ⊨ □◇?F; ?sigma ⊨ ?F ↝ ?G⟧ ⟹ ?sigma ⊨ □◇?G›*))\n (*goal: ‹⟦sigma ⊨ □◇P; sigma ⊨ □A; ⊢ A ∧ $P ⟶ Q$⟧ ⟹ sigma ⊨ □◇Q›*)\n apply (erule EnsuresLeadsto [temp_use] (*‹⟦⊢ (?A::state × state ⇒ bool) ∧ $(?P::state ⇒ bool) ⟶ (?Q::state ⇒ bool)$; (?sigma::behavior) ⊨ □?A⟧ ⟹ ?sigma ⊨ ?P ↝ ?Q›*))\n (*goal: ‹⟦sigma ⊨ □A; ⊢ A ∧ $P ⟶ Q$⟧ ⟹ sigma ⊨ P ↝ Q›*)\n by assumption\n\n\n(*** Gronning's lattice rules (taken from TLP) ***)\nsection \"Lattice rules\"\n\nlemma LatticeReflexivity: \"⊢ F ↝ F\"\n apply (unfold leadsto_def (*‹(?F::?'b::world ⇒ bool) ↝ (?G::?'c::world ⇒ bool) ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ F ↝ F›*)\n apply (rule necT (*‹⊢ ?F ⟹ ⊢ □?F›*) InitDmd_gen (*‹⊢ Init ?F ⟶ ◇?F›*))\n (*goal: ‹⊢ □(Init F ⟶ ◇F)›*)\n by (rule necT (*‹⊢ ?F ⟹ ⊢ □?F›*) InitDmd_gen (*‹⊢ Init ?F ⟶ ◇?F›*))\n\nlemma LatticeTransitivity: \"⊢ (G ↝ H) ∧ (F ↝ G) ⟶ (F ↝ H)\"\n apply (unfold leadsto_def (*‹(?F::?'b ⇒ bool) ↝ (?G::?'c ⇒ bool) ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ (G ↝ H) ∧ (F ↝ G) ⟶ (F ↝ H)›*)\n apply clarsimp\n (*goal: ‹⊢ □(Init (G::'a::world ⇒ bool) ⟶ ◇(H::'b::world ⇒ bool)) ∧ □(Init (F::'c::world ⇒ bool) ⟶ ◇G) ⟶ □(Init F ⟶ ◇H)›*)\n apply (erule dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □(Init G ⟶ ◇H); sigma ⊨ □(Init F ⟶ ◇G)⟧ ⟹ sigma ⊨ □(Init F ⟶ ◇H)›*)\n apply merge_box\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □(Init F ⟶ ◇G); sigma ⊨ □□(Init G ⟶ ◇H)⟧ ⟹ sigma ⊨ □(Init F ⟶ ◇H)›*)\n apply (clarsimp elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma::behavior. sigma ⊨ □((Init (F::'c ⇒ bool) ⟶ ◇(G::'a ⇒ bool)) ∧ □(Init G ⟶ ◇(H::'b ⇒ bool))) ⟹ sigma ⊨ □(Init F ⟶ ◇H)›*)\n apply (rule dup_dmdD (*‹?sigma ⊨ ◇◇?F ⟹ ?sigma ⊨ ◇?F›*))\n (*goal: ‹⋀sigmaa. ⟦sigmaa ⊨ □(Init G ⟶ ◇H); sigmaa ⊨ Init F; sigmaa ⊨ ◇G⟧ ⟹ sigmaa ⊨ ◇H›*)\n apply (subgoal_tac \"sigmaa ⊨ ◇Init G\")\n (*goals:\n 1. ‹⋀sigmaa::behavior. ⟦sigmaa ⊨ □(Init (G::'a ⇒ bool) ⟶ ◇(H::'b ⇒ bool)); sigmaa ⊨ Init (F::'c ⇒ bool); sigmaa ⊨ ◇G; sigmaa ⊨ ◇Init G⟧ ⟹ sigmaa ⊨ ◇◇H›\n 2. ‹⋀sigmaa::behavior. ⟦sigmaa ⊨ □(Init (G::'a ⇒ bool) ⟶ ◇(H::'b ⇒ bool)); sigmaa ⊨ Init (F::'c ⇒ bool); sigmaa ⊨ ◇G⟧ ⟹ sigmaa ⊨ ◇Init G›\n discuss goal 1*)\n apply (erule DmdImpl2 (*‹⟦(?sigma::behavior) ⊨ ◇(?F::?'a::world ⇒ bool); ?sigma ⊨ □(?F ⟶ (?G::?'a::world ⇒ bool))⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*top goal: ‹⋀sigmaa. ⟦sigmaa ⊨ □(Init G ⟶ ◇H); sigmaa ⊨ Init F; sigmaa ⊨ ◇G; sigmaa ⊨ ◇Init G⟧ ⟹ sigmaa ⊨ ◇◇H› and 1 goal remains*)\n apply assumption\n (*discuss goal 2*)\n apply (simp add: dmdInitD (*‹◇Init ?F ≡ ◇?F›*))\n (*proven 2 subgoals*) .\n\nlemma LatticeDisjunctionElim1: \"⊢ (F ∨ G ↝ H) ⟶ (F ↝ H)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ (F ∨ G ↝ H) ⟶ (F ↝ H)›*)\n by (auto simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma LatticeDisjunctionElim2: \"⊢ (F ∨ G ↝ H) ⟶ (G ↝ H)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ ((F::'a ⇒ bool) ∨ (G::'a ⇒ bool) ↝ (H::'b ⇒ bool)) ⟶ (G ↝ H)›*)\n by (auto simp: Init_simps (*‹Init (?y::behavior ⇒ bool) ≡ ?y› ‹Init $(?P::state ⇒ bool) ≡ Init ?P› ‹Init ¬ (?F::?'a::world ⇒ bool) ≡ ¬ Init ?F› ‹Init ((?P::?'b::world ⇒ bool) ⟶ (?Q::?'b::world ⇒ bool)) ≡ Init ?P ⟶ Init ?Q› ‹Init ((?P::?'b::world ⇒ bool) ∧ (?Q::?'b::world ⇒ bool)) ≡ Init ?P ∧ Init ?Q› ‹Init ((?P::?'b::world ⇒ bool) ∨ (?Q::?'b::world ⇒ bool)) ≡ Init ?P ∨ Init ?Q› ‹Init (?P::?'b::world ⇒ bool) = (?Q::?'b::world ⇒ bool) ≡ Init ?P = (Init ?Q)› ‹Init (∀x::?'d::type. (?F::?'d::type ⇒ ?'c::world ⇒ bool) x) ≡ ∀x::?'d::type. Init ?F x› ‹Init (∃x::?'d::type. (?F::?'d::type ⇒ ?'c::world ⇒ bool) x) ≡ ∃x::?'d::type. Init ?F x› ‹Init (∃!x::?'d::type. (?F::?'d::type ⇒ ?'c::world ⇒ bool) x) ≡ ∃!x::?'d::type. Init ?F x› ‹□Init (?F::?'a::world ⇒ bool) ≡ □?F› ‹◇Init (?F::?'a::world ⇒ bool) ≡ ◇?F› and more 2 facts*) elim!: STL4E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?F::?'a::world ⇒ bool); ⊢ ?F ⟶ (?G::?'a::world ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma LatticeDisjunctionIntro: \"⊢ (F ↝ H) ∧ (G ↝ H) ⟶ (F ∨ G ↝ H)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ ((F::'a ⇒ bool) ↝ (H::'b ⇒ bool)) ∧ ((G::'a ⇒ bool) ↝ H) ⟶ (F ∨ G ↝ H)›*)\n apply clarsimp\n (*goal: ‹⊢ □(Init F ⟶ ◇H) ∧ □(Init G ⟶ ◇H) ⟶ □(Init (F ∨ G) ⟶ ◇H)›*)\n apply merge_box\n (*goal: ‹⋀sigma::behavior. ⟦sigma ⊨ □(Init (F::'a ⇒ bool) ⟶ ◇(H::'b ⇒ bool)); sigma ⊨ □(Init (G::'a ⇒ bool) ⟶ ◇H)⟧ ⟹ sigma ⊨ □(Init (F ∨ G) ⟶ ◇H)›*)\n by (auto simp: Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n\nlemma LatticeDisjunction: \"⊢ (F ∨ G ↝ H) = ((F ↝ H) ∧ (G ↝ H))\"\n by (auto intro: LatticeDisjunctionIntro [temp_use] (*‹⟦?sigma ⊨ ?F ↝ ?H; ?sigma ⊨ ?G ↝ ?H⟧ ⟹ ?sigma ⊨ ?F ∨ ?G ↝ ?H›*) LatticeDisjunctionElim1 [temp_use] (*‹?sigma ⊨ ?F ∨ ?G ↝ ?H ⟹ ?sigma ⊨ ?F ↝ ?H›*) LatticeDisjunctionElim2 [temp_use] (*‹?sigma ⊨ ?F ∨ ?G ↝ ?H ⟹ ?sigma ⊨ ?G ↝ ?H›*))\n\nlemma LatticeDiamond: \"⊢ (A ↝ B ∨ C) ∧ (B ↝ D) ∧ (C ↝ D) ⟶ (A ↝ D)\"\n apply clarsimp\n (*goal: ‹⊢ (A ↝ B ∨ C) ∧ (B ↝ D) ∧ (C ↝ D) ⟶ (A ↝ D)›*)\n apply (subgoal_tac \"sigma ⊨ (B ∨ C) ↝ D\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ A ↝ B ∨ C; sigma ⊨ B ↝ D; sigma ⊨ C ↝ D; sigma ⊨ B ∨ C ↝ D⟧ ⟹ sigma ⊨ A ↝ D›\n 2. ‹⋀sigma. ⟦sigma ⊨ A ↝ B ∨ C; sigma ⊨ B ↝ D; sigma ⊨ C ↝ D⟧ ⟹ sigma ⊨ B ∨ C ↝ D›\n discuss goal 1*)\n apply (erule_tac G = \"LIFT (B ∨ C)\" in LatticeTransitivity [temp_use] (*‹⟦?sigma ⊨ ?G ↝ ?H; ?sigma ⊨ ?F ↝ ?G⟧ ⟹ ?sigma ⊨ ?F ↝ ?H›*))\n (*top goal: ‹⋀sigma::behavior. ⟦sigma ⊨ (A::'a::world ⇒ bool) ↝ (B::'b::world ⇒ bool) ∨ (C::'b::world ⇒ bool); sigma ⊨ B ↝ (D::'c::world ⇒ bool); sigma ⊨ C ↝ D; sigma ⊨ B ∨ C ↝ D⟧ ⟹ sigma ⊨ A ↝ D› and 1 goal remains*)\n apply (fastforce intro!: LatticeDisjunctionIntro [temp_use] (*‹⟦?sigma ⊨ ?F ↝ ?H; ?sigma ⊨ ?G ↝ ?H⟧ ⟹ ?sigma ⊨ ?F ∨ ?G ↝ ?H›*))\n (*discuss goal 2*)\n apply (fastforce intro!: LatticeDisjunctionIntro [temp_use] (*‹⟦?sigma ⊨ ?F ↝ ?H; ?sigma ⊨ ?G ↝ ?H⟧ ⟹ ?sigma ⊨ ?F ∨ ?G ↝ ?H›*))\n (*proven 2 subgoals*) .\n\nlemma LatticeTriangle: \"⊢ (A ↝ D ∨ B) ∧ (B ↝ D) ⟶ (A ↝ D)\"\n apply clarsimp\n (*goal: ‹⊢ ((A::'a ⇒ bool) ↝ (D::'b ⇒ bool) ∨ (B::'b ⇒ bool)) ∧ (B ↝ D) ⟶ (A ↝ D)›*)\n apply (subgoal_tac \"sigma ⊨ (D ∨ B) ↝ D\")\n (*goals:\n 1. ‹⋀sigma. ⟦sigma ⊨ A ↝ D ∨ B; sigma ⊨ B ↝ D; sigma ⊨ D ∨ B ↝ D⟧ ⟹ sigma ⊨ A ↝ D›\n 2. ‹⋀sigma. ⟦sigma ⊨ A ↝ D ∨ B; sigma ⊨ B ↝ D⟧ ⟹ sigma ⊨ D ∨ B ↝ D›\n discuss goal 1*)\n apply (erule_tac G = \"LIFT (D ∨ B)\" in LatticeTransitivity [temp_use] (*‹⟦(?sigma::behavior) ⊨ (?G::?'a::world ⇒ bool) ↝ (?H::?'b::world ⇒ bool); ?sigma ⊨ (?F::?'c::world ⇒ bool) ↝ ?G⟧ ⟹ ?sigma ⊨ ?F ↝ ?H›*))\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ A ↝ D ∨ B; sigma ⊨ B ↝ D; sigma ⊨ D ∨ B ↝ D⟧ ⟹ sigma ⊨ A ↝ D› and 1 goal remains*)\n apply assumption\n (*discuss goal 2*)\n apply (auto intro: LatticeDisjunctionIntro [temp_use] (*‹⟦?sigma ⊨ ?F ↝ ?H; ?sigma ⊨ ?G ↝ ?H⟧ ⟹ ?sigma ⊨ ?F ∨ ?G ↝ ?H›*) LatticeReflexivity [temp_use] (*‹?sigma ⊨ ?F ↝ ?F›*))\n (*proven 2 subgoals*) .\n\nlemma LatticeTriangle2: \"⊢ (A ↝ B ∨ D) ∧ (B ↝ D) ⟶ (A ↝ D)\"\n apply clarsimp\n (*goal: ‹⊢ ((A::'a ⇒ bool) ↝ (B::'b ⇒ bool) ∨ (D::'b ⇒ bool)) ∧ (B ↝ D) ⟶ (A ↝ D)›*)\n apply (subgoal_tac \"sigma ⊨ B ∨ D ↝ D\")\n (*goals:\n 1. ‹⋀sigma::behavior. ⟦sigma ⊨ (A::'a ⇒ bool) ↝ (B::'b ⇒ bool) ∨ (D::'b ⇒ bool); sigma ⊨ B ↝ D; sigma ⊨ B ∨ D ↝ D⟧ ⟹ sigma ⊨ A ↝ D›\n 2. ‹⋀sigma::behavior. ⟦sigma ⊨ (A::'a ⇒ bool) ↝ (B::'b ⇒ bool) ∨ (D::'b ⇒ bool); sigma ⊨ B ↝ D⟧ ⟹ sigma ⊨ B ∨ D ↝ D›\n discuss goal 1*)\n apply (erule_tac G = \"LIFT (B ∨ D)\" in LatticeTransitivity [temp_use] (*‹⟦?sigma ⊨ ?G ↝ ?H; ?sigma ⊨ ?F ↝ ?G⟧ ⟹ ?sigma ⊨ ?F ↝ ?H›*))\n (*top goal: ‹⋀sigma. ⟦sigma ⊨ A ↝ B ∨ D; sigma ⊨ B ↝ D; sigma ⊨ B ∨ D ↝ D⟧ ⟹ sigma ⊨ A ↝ D› and 1 goal remains*)\n apply assumption\n (*discuss goal 2*)\n apply (auto intro: LatticeDisjunctionIntro [temp_use] (*‹⟦(?sigma::behavior) ⊨ (?F::?'a ⇒ bool) ↝ (?H::?'b ⇒ bool); ?sigma ⊨ (?G::?'a ⇒ bool) ↝ ?H⟧ ⟹ ?sigma ⊨ ?F ∨ ?G ↝ ?H›*) LatticeReflexivity [temp_use] (*‹(?sigma::behavior) ⊨ (?F::?'a ⇒ bool) ↝ ?F›*))\n (*proven 2 subgoals*) .\n\n(*** Lamport's fairness rules ***)\nsection \"Fairness rules\"\n\nlemma WF1:\n \"⟦ ⊢ $P ∧ N ⟶ P` ∨ Q`;\n ⊢ ($P ∧ N) ∧ <A>_v ⟶ Q`;\n ⊢ $P ∧ N ⟶ $(Enabled(<A>_v)) ⟧\n ⟹ ⊢ □N ∧ WF(A)_v ⟶ (P ↝ Q)\"\n apply (clarsimp dest!: BoxWFI [temp_use] (*‹?sigma ⊨ WF(?A)_?v ⟹ ?sigma ⊨ □WF(?A)_?v›*))\n (*goal: ‹⟦⊢ $P ∧ N ⟶ P$ ∨ Q$; ⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ $P ∧ N ⟶ $Enabled (<A>_v)⟧ ⟹ ⊢ □N ∧ WF(A)_v ⟶ (P ↝ Q)›*)\n apply (erule (2) ensures [temp_use] (*‹⟦⊢ $?P ∧ ?N ⟶ ?P$ ∨ ?Q$; ⊢ ($?P ∧ ?N) ∧ ?A ⟶ ?Q$; ?sigma ⊨ □?N; ?sigma ⊨ □(□?P ⟶ ◇?A)⟧ ⟹ ?sigma ⊨ ?P ↝ ?Q›*))\n (*goal: ‹⋀sigma. ⟦⊢ $P ∧ N ⟶ P$ ∨ Q$; ⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ $P ∧ N ⟶ $Enabled (<A>_v); sigma ⊨ □N; sigma ⊨ □WF(A)_v⟧ ⟹ sigma ⊨ P ↝ Q›*)\n apply (erule (1) STL4Edup (*‹⟦?sigma ⊨ □?A; ?sigma ⊨ □?F; ⊢ ?F ∧ □?A ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ $P ∧ N ⟶ $Enabled (<A>_v); sigma ⊨ □N; sigma ⊨ □WF(A)_v⟧ ⟹ sigma ⊨ □(□P ⟶ ◇<A>_v)›*)\n apply (clarsimp simp: WF_def (*‹WF(?A)_?v ≡ ◇□Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*))\n (*goal: ‹⋀sigma. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ $P ∧ N ⟶ $Enabled (<A>_v); sigma ⊨ □WF(A)_v⟧ ⟹ ⊢ WF(A)_v ∧ □N ⟶ □P ⟶ ◇<A>_v›*)\n apply (rule STL2 [temp_use] (*‹?sigma ⊨ □?F ⟹ ?F ?sigma›*))\n (*goal: ‹⋀sigma sigmaa. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ $P ∧ N ⟶ $Enabled (<A>_v); sigma ⊨ □(◇□Enabled (<A>_v) ⟶ □◇<A>_v); (sigmaa ⊨ ◇□Enabled (<A>_v)) ⟶ (sigmaa ⊨ □◇<A>_v); sigmaa ⊨ □N; sigmaa ⊨ □P⟧ ⟹ sigmaa ⊨ ◇<A>_v›*)\n apply (clarsimp elim!: mp (*‹⟦?P ⟶ ?Q; ?P⟧ ⟹ ?Q›*) intro!: InitDmd [temp_use] (*‹?F ?sigma ⟹ ?sigma ⊨ ◇?F›*))\n (*goal: ‹⋀sigma sigmaa. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ $P ∧ N ⟶ $Enabled (<A>_v); sigma ⊨ □(◇□Enabled (<A>_v) ⟶ □◇<A>_v); (sigmaa ⊨ ◇□Enabled (<A>_v)) ⟶ (sigmaa ⊨ □◇<A>_v); sigmaa ⊨ □N; sigmaa ⊨ □P⟧ ⟹ sigmaa ⊨ □◇<A>_v›*)\n apply (erule STL4 [temp_use, THEN box_stp_actD [temp_use]] (*‹⟦⊢ ?F1 ⟶ $?P; ?sigma ⊨ □?F1⟧ ⟹ ?sigma ⊨ □?P›*))\n (*goal: ‹⋀sigma sigmaa. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ $P ∧ N ⟶ $Enabled (<A>_v); sigma ⊨ □(◇□Enabled (<A>_v) ⟶ □◇<A>_v); sigmaa ⊨ □N; sigmaa ⊨ □P⟧ ⟹ sigmaa ⊨ □Enabled (<A>_v)›*)\n by (simp add: split_box_conj (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) box_stp_actI (*‹?sigma ⊨ □?P ⟹ ?sigma ⊨ □$?P›*))\n\n(* Sometimes easier to use; designed for action B rather than state predicate Q *)\nlemma WF_leadsto:\n assumes 1: \"⊢ N ∧ $P ⟶ $Enabled (<A>_v)\"\n and 2: \"⊢ N ∧ <A>_v ⟶ B\"\n and 3: \"⊢ □(N ∧ [¬A]_v) ⟶ stable P\"\n shows \"⊢ □N ∧ WF(A)_v ⟶ (P ↝ B)\"\n apply (unfold leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*))\n (*goal: ‹⊢ □N ∧ WF(A)_v ⟶ (P ↝ B)›*)\n apply (clarsimp dest!: BoxWFI [temp_use] (*‹(?sigma::behavior) ⊨ WF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a) ⟹ ?sigma ⊨ □WF(?A)_?v›*))\n (*goal: ‹⊢ □N ∧ WF(A)_v ⟶ □(Init P ⟶ ◇B)›*)\n apply (erule (1) STL4Edup (*‹⟦(?sigma::behavior) ⊨ □(?A::?'a::world ⇒ bool); ?sigma ⊨ □(?F::behavior ⇒ bool); ⊢ ?F ∧ □?A ⟶ (?G::behavior ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □N; sigma ⊨ □WF(A)_v⟧ ⟹ sigma ⊨ □(Init P ⟶ ◇B)›*)\n apply clarsimp\n (*goal: ‹⋀sigma::behavior. sigma ⊨ □WF(A::state × state ⇒ bool)_(v::state ⇒ 'a) ⟹ ⊢ WF(A)_v ∧ □(N::state × state ⇒ bool) ⟶ Init (P::state ⇒ bool) ⟶ ◇(B::state × state ⇒ bool)›*)\n apply (rule 2 [THEN DmdImpl, temp_use] (*‹(?sigma::behavior) ⊨ ◇((N::state × state ⇒ bool) ∧ <A::state × state ⇒ bool>_(v::state ⇒ 'a::type)) ⟹ ?sigma ⊨ ◇(B::state × state ⇒ bool)›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □WF(A)_v; sigmaa ⊨ WF(A)_v; sigmaa ⊨ □N; sigmaa ⊨ Init P⟧ ⟹ sigmaa ⊨ ◇B›*)\n apply (rule BoxDmd_simple [temp_use] (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ ◇?G⟧ ⟹ ?sigma ⊨ ◇(?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma sigmaa. ⟦sigma ⊨ □WF(A)_v; sigmaa ⊨ WF(A)_v; sigmaa ⊨ □N; sigmaa ⊨ Init P⟧ ⟹ sigmaa ⊨ □N›\n 2. ‹⋀sigma sigmaa. ⟦sigma ⊨ □WF(A)_v; sigmaa ⊨ WF(A)_v; sigmaa ⊨ □N; sigmaa ⊨ Init P⟧ ⟹ sigmaa ⊨ ◇<A>_v›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (rule classical (*‹(¬ ?P ⟹ ?P) ⟹ ?P›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □WF(A)_v; sigmaa ⊨ WF(A)_v; sigmaa ⊨ □N; sigmaa ⊨ Init P⟧ ⟹ sigmaa ⊨ ◇<A>_v›*)\n apply (rule STL2 [temp_use] (*‹(?sigma::behavior) ⊨ □(?F::behavior ⇒ bool) ⟹ ?F ?sigma›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □WF(A)_v; sigmaa ⊨ WF(A)_v; sigmaa ⊨ □N; sigmaa ⊨ Init P; ¬ (sigmaa ⊨ ◇<A>_v)⟧ ⟹ sigmaa ⊨ ◇<A>_v›*)\n apply (clarsimp simp: WF_def (*‹WF(?A)_?v ≡ ◇□Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*) elim!: mp (*‹⟦?P ⟶ ?Q; ?P⟧ ⟹ ?Q›*) intro!: InitDmd [temp_use] (*‹?F ?sigma ⟹ ?sigma ⊨ ◇?F›*))\n (*goal: ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □WF(A::state × state ⇒ bool)_(v::state ⇒ 'a::type); sigmaa ⊨ WF(A)_v; sigmaa ⊨ □(N::state × state ⇒ bool); sigmaa ⊨ Init (P::state ⇒ bool); ¬ (sigmaa ⊨ ◇<A>_v)⟧ ⟹ sigmaa ⊨ □◇<A>_v›*)\n apply (rule 1 [THEN STL4, temp_use, THEN box_stp_actD] (*‹(?sigma::behavior) ⊨ □((N::state × state ⇒ bool) ∧ $(P::state ⇒ bool)) ⟹ ?sigma ⊨ □Enabled (<A::state × state ⇒ bool>_(v::state ⇒ 'a::type))›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(◇□Enabled (<A>_v) ⟶ □◇<A>_v); sigmaa ⊨ □N; sigmaa ⊨ Init P; ¬ (sigmaa ⊨ ◇<A>_v)⟧ ⟹ sigmaa ⊨ □Enabled (<A>_v)›*)\n apply (simp (no_asm_simp) add: split_box_conj [try_rewrite] (*‹((?sigma::behavior) ⊨ □((?F::?'a::world ⇒ bool) ∧ (?G::?'a::world ⇒ bool))) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) box_stp_act [try_rewrite] (*‹□$(?P::state ⇒ bool) ≡ □?P›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(◇□Enabled (<A>_v) ⟶ □◇<A>_v); sigmaa ⊨ □N; sigmaa ⊨ Init P; ¬ (sigmaa ⊨ ◇<A>_v)⟧ ⟹ sigmaa ⊨ □(N ∧ $P)›*)\n apply (erule INV1 [temp_use] (*‹⟦?sigma ⊨ Init ?P; ?sigma ⊨ stable ?P⟧ ⟹ ?sigma ⊨ □?P›*))\n (*goal: ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(◇□Enabled (<A::state × state ⇒ bool>_(v::state ⇒ 'a::type)) ⟶ □◇<A>_v); sigmaa ⊨ □(N::state × state ⇒ bool); sigmaa ⊨ Init (P::state ⇒ bool); ¬ (sigmaa ⊨ ◇<A>_v)⟧ ⟹ sigmaa ⊨ □P›*)\n apply (rule 3 [temp_use] (*‹?sigma ⊨ □(N ∧ [¬ A]_v) ⟹ ?sigma ⊨ stable P›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(◇□Enabled (<A>_v) ⟶ □◇<A>_v); sigmaa ⊨ □N; ¬ (sigmaa ⊨ ◇<A>_v)⟧ ⟹ sigmaa ⊨ stable P›*)\n apply (simp add: split_box_conj [try_rewrite] (*‹((?sigma::behavior) ⊨ □((?F::?'a ⇒ bool) ∧ (?G::?'a ⇒ bool))) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) NotDmd [temp_use] (*‹(¬ ((?sigma::behavior) ⊨ ◇(?F::?'a ⇒ bool))) = (?sigma ⊨ □¬ ?F)›*) not_angle [try_rewrite] (*‹¬ <?A::state × state ⇒ bool>_(?v::state ⇒ ?'a) ≡ [¬ ?A]_?v›*))\n (*proven 2 subgoals*) .\n\nlemma SF1:\n \"⟦ ⊢ $P ∧ N ⟶ P` ∨ Q`;\n ⊢ ($P ∧ N) ∧ <A>_v ⟶ Q`;\n ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled(<A>_v) ⟧\n ⟹ ⊢ □N ∧ SF(A)_v ∧ □F ⟶ (P ↝ Q)\"\n apply (clarsimp dest!: BoxSFI [temp_use] (*‹?sigma ⊨ SF(?A)_?v ⟹ ?sigma ⊨ □SF(?A)_?v›*))\n (*goal: ‹⟦⊢ $P ∧ N ⟶ P$ ∨ Q$; ⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v)⟧ ⟹ ⊢ □N ∧ SF(A)_v ∧ □F ⟶ (P ↝ Q)›*)\n apply (erule (2) ensures [temp_use] (*‹⟦⊢ $?P ∧ ?N ⟶ ?P$ ∨ ?Q$; ⊢ ($?P ∧ ?N) ∧ ?A ⟶ ?Q$; ?sigma ⊨ □?N; ?sigma ⊨ □(□?P ⟶ ◇?A)⟧ ⟹ ?sigma ⊨ ?P ↝ ?Q›*))\n (*goal: ‹⋀sigma. ⟦⊢ $P ∧ N ⟶ P$ ∨ Q$; ⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □N; sigma ⊨ □F; sigma ⊨ □SF(A)_v⟧ ⟹ sigma ⊨ P ↝ Q›*)\n apply (erule_tac F = F in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □N; sigma ⊨ □F; sigma ⊨ □SF(A)_v⟧ ⟹ sigma ⊨ □(□P ⟶ ◇<A>_v)›*)\n apply merge_temp_box\n (*goal: ‹⋀sigma. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □N; sigma ⊨ □SF(A)_v; sigma ⊨ □□F⟧ ⟹ sigma ⊨ □(□P ⟶ ◇<A>_v)›*)\n apply (erule STL4Edup (*‹⟦?sigma ⊨ □?A; ?sigma ⊨ □?F; ⊢ ?F ∧ □?A ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goals:\n 1. ‹⋀sigma. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □(SF(A)_v ∧ □F)⟧ ⟹ sigma ⊨ □?F14 sigma›\n 2. ‹⋀sigma. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □(SF(A)_v ∧ □F)⟧ ⟹ ⊢ ?F14 sigma ∧ □N ⟶ □P ⟶ ◇<A>_v›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (clarsimp simp: SF_def (*‹SF(?A)_?v ≡ □◇Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*))\n (*goal: ‹⋀sigma. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □(SF(A)_v ∧ □F)⟧ ⟹ ⊢ (SF(A)_v ∧ □F) ∧ □N ⟶ □P ⟶ ◇<A>_v›*)\n apply (rule STL2 [temp_use] (*‹(?sigma::behavior) ⊨ □(?F::behavior ⇒ bool) ⟹ ?F ?sigma›*))\n (*goal: ‹⋀sigma sigmaa. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □((□◇Enabled (<A>_v) ⟶ □◇<A>_v) ∧ □F); (sigmaa ⊨ □◇Enabled (<A>_v)) ⟶ (sigmaa ⊨ □◇<A>_v); sigmaa ⊨ □F; sigmaa ⊨ □N; sigmaa ⊨ □P⟧ ⟹ sigmaa ⊨ ◇<A>_v›*)\n apply (erule mp (*‹⟦(?P::bool) ⟶ (?Q::bool); ?P⟧ ⟹ ?Q›*))\n (*goal: ‹⋀sigma sigmaa. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □((□◇Enabled (<A>_v) ⟶ □◇<A>_v) ∧ □F); (sigmaa ⊨ □◇Enabled (<A>_v)) ⟶ (sigmaa ⊨ □◇<A>_v); sigmaa ⊨ □F; sigmaa ⊨ □N; sigmaa ⊨ □P⟧ ⟹ sigmaa ⊨ □◇<A>_v›*)\n apply (erule STL4 [temp_use] (*‹⟦⊢ (?F::?'a::world ⇒ bool) ⟶ (?G::?'a::world ⇒ bool); (?sigma::behavior) ⊨ □?F⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma sigmaa. ⟦⊢ ($P ∧ N) ∧ <A>_v ⟶ Q$; ⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled (<A>_v); sigma ⊨ □((□◇Enabled (<A>_v) ⟶ □◇<A>_v) ∧ □F); sigmaa ⊨ □F; sigmaa ⊨ □N; sigmaa ⊨ □P⟧ ⟹ sigmaa ⊨ □◇Enabled (<A>_v)›*)\n apply (simp add: split_box_conj [try_rewrite] (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) STL3 [try_rewrite] (*‹□□?F ≡ □?F›*))\n (*proven 2 subgoals*) .\n\nlemma WF2:\n assumes 1: \"⊢ N ∧ <B>_f ⟶ <M>_g\"\n and 2: \"⊢ $P ∧ P` ∧ <N ∧ A>_f ⟶ B\"\n and 3: \"⊢ P ∧ Enabled(<M>_g) ⟶ Enabled(<A>_f)\"\n and 4: \"⊢ □(N ∧ [¬B]_f) ∧ WF(A)_f ∧ □F ∧ ◇□Enabled(<M>_g) ⟶ ◇□P\"\n shows \"⊢ □N ∧ WF(A)_f ∧ □F ⟶ WF(M)_g\"\n apply (clarsimp dest!: BoxWFI [temp_use] (*‹(?sigma::behavior) ⊨ WF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a::type) ⟹ ?sigma ⊨ □WF(?A)_?v›*) BoxDmdBox [temp_use, THEN iffD2] (*‹(?sigma1::behavior) ⊨ ◇□(?F1::?'a1::world ⇒ bool) ⟹ ?sigma1 ⊨ □◇□?F1›*) simp: WF_def [where A = M] (*‹WF(M::state × state ⇒ bool)_(?v::state ⇒ ?'d::type) ≡ ◇□Enabled (<M>_?v) ⟶ □◇<M>_?v›*))\n (*goal: ‹⊢ □(N::state × state ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ □(F::'c ⇒ bool) ⟶ WF(M::state × state ⇒ bool)_(g::state ⇒ 'b)›*)\n apply (erule_tac F = F in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma::behavior. ⟦sigma ⊨ □(N::state × state ⇒ bool); sigma ⊨ □(F::'c::world ⇒ bool); sigma ⊨ □WF(A::state × state ⇒ bool)_(f::state ⇒ 'a::type); sigma ⊨ □◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b::type))⟧ ⟹ sigma ⊨ □◇<M>_g›*)\n apply merge_temp_box\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □N; sigma ⊨ □WF(A)_f; sigma ⊨ □◇□Enabled (<M>_g); sigma ⊨ □□F⟧ ⟹ sigma ⊨ □◇<M>_g›*)\n apply (erule STL4Edup (*‹⟦?sigma ⊨ □?A; ?sigma ⊨ □?F; ⊢ ?F ∧ □?A ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)) ⟹ sigma ⊨ □?F14 sigma›\n 2. ‹⋀sigma. sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)) ⟹ ⊢ ?F14 sigma ∧ □N ⟶ ◇<M>_g›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]] (*‹⟦?sigma ⊨ □N; ?sigma ⊨ ◇<B>_f⟧ ⟹ ?sigma ⊨ ◇<M>_g›*))\n (*goal: ‹⋀sigma. sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)) ⟹ ⊢ (□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)) ∧ □N ⟶ ◇<M>_g›*)\n apply (rule classical (*‹(¬ ?P ⟹ ?P) ⟹ ?P›*))\n (*goal: ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □(N::state × state ⇒ bool)⟧ ⟹ sigmaa ⊨ ◇<B::state × state ⇒ bool>_f›*)\n apply (subgoal_tac \"sigmaa ⊨ ◇ (($P ∧ P` ∧ N) ∧ <A>_f)\")\n (*goals:\n 1. ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c::world ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a::type) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b::type))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □(N::state × state ⇒ bool); ¬ (sigmaa ⊨ ◇<B::state × state ⇒ bool>_f); sigmaa ⊨ ◇(($(P::state ⇒ bool) ∧ P$ ∧ N) ∧ <A>_f)⟧ ⟹ sigmaa ⊨ ◇<B>_f›\n 2. ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c::world ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a::type) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b::type))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □(N::state × state ⇒ bool); ¬ (sigmaa ⊨ ◇<B::state × state ⇒ bool>_f)⟧ ⟹ sigmaa ⊨ ◇(($(P::state ⇒ bool) ∧ P$ ∧ N) ∧ <A>_f)›\n discuss goal 1*)\n apply (force simp: angle_def (*‹<?A>_?v ≡ ?A ∧ ¬ unchanged ?v›*) intro!: 2 [temp_use] (*‹⟦P ?s; P ?t; (?s, ?t) ⊨ <N ∧ A>_f⟧ ⟹ B (?s, ?t)›*) elim!: DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*discuss goal 2*)\n apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use] (*‹?sigma ⊨ ◇(□?F ∧ ◇?G) ⟹ ?sigma ⊨ ◇(?F ∧ ?G)›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □N; ¬ (sigmaa ⊨ ◇<B>_f)⟧ ⟹ sigmaa ⊨ ◇(($P ∧ P$ ∧ N) ∧ <A>_f)›*)\n apply (simp add: NotDmd [temp_use] (*‹(¬ (?sigma ⊨ ◇?F)) = (?sigma ⊨ □¬ ?F)›*) not_angle [try_rewrite] (*‹¬ <?A>_?v ≡ [¬ ?A]_?v›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □N; ¬ (sigmaa ⊨ ◇<B>_f)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply merge_act_box\n (*goal: ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □(N::state × state ⇒ bool); sigmaa ⊨ □[¬ (B::state × state ⇒ bool)]_f⟧ ⟹ sigmaa ⊨ ◇(□($(P::state ⇒ bool) ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (frule 4 [temp_use] (*‹⟦(?sigma::behavior) ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_(f::state ⇒ 'a)); ?sigma ⊨ WF(A::state × state ⇒ bool)_f; ?sigma ⊨ □(F::'c ⇒ bool); ?sigma ⊨ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))⟧ ⟹ ?sigma ⊨ ◇□(P::state ⇒ bool)›*))\n (*goals:\n 1. ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_f)⟧ ⟹ sigmaa ⊨ WF(A)_f›\n 2. ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_f)⟧ ⟹ sigmaa ⊨ □F›\n 3. ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_f)⟧ ⟹ sigmaa ⊨ ◇□Enabled (<M>_g)›\n 4. ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ ◇□Enabled (<M>_g); sigmaa ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_f); sigmaa ⊨ ◇□(P::state ⇒ bool)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply assumption\n (*discuss goal 4*)\n apply (drule STL6 [temp_use] (*‹⟦?sigma ⊨ ◇□?F; ?sigma ⊨ ◇□?G⟧ ⟹ ?sigma ⊨ ◇□(?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□P⟧ ⟹ sigmaa ⊨ ◇□?G46 sigma sigmaa›\n 2. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□P; sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ ?G46 sigma sigmaa)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (erule_tac V = \"sigmaa ⊨ ◇□P\" in thin_rl (*‹⟦PROP ?V; PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_f); sigmaa ⊨ ◇□(P::state ⇒ bool); sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (erule_tac V = \"sigmaa ⊨ □F\" in thin_rl (*‹⟦PROP ?V; PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)); sigmaa ⊨ □F; sigmaa ⊨ WF(A)_f; sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (drule BoxWFI [temp_use] (*‹?sigma ⊨ WF(?A)_?v ⟹ ?sigma ⊨ □WF(?A)_?v›*))\n (*goal: ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□(F::'c ⇒ bool) ∧ WF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ ◇□Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))); sigmaa ⊨ WF(A)_f; sigmaa ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_f); sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ (P::state ⇒ bool))⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (erule_tac F = \"ACT N ∧ [¬B]_f\" in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)); sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P); sigmaa ⊨ □WF(A)_f⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply merge_temp_box\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□F ∧ WF(A)_f ∧ ◇□Enabled (<M>_g)); sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P); sigmaa ⊨ □WF(A)_f; sigmaa ⊨ □□(N ∧ [¬ B]_f)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (erule DmdImpldup (*‹⟦(?sigma::behavior) ⊨ □(?A::?'a::world ⇒ bool); ?sigma ⊨ ◇(?F::behavior ⇒ bool); ⊢ □?A ∧ ?F ⟶ (?G::behavior ⇒ bool)⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*goals:\n 1. ‹⋀sigma sigmaa. sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P) ⟹ sigmaa ⊨ ◇?F58 sigma sigmaa›\n 2. ‹⋀sigma sigmaa. sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P) ⟹ ⊢ □(□(N ∧ [¬ B]_f) ∧ WF(A)_f ∧ □(N ∧ [¬ B]_f)) ∧ ?F58 sigma sigmaa ⟶ □($P ∧ P$ ∧ N) ∧ ◇<A>_f›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (auto simp: split_box_conj [try_rewrite] (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) STL3 [try_rewrite] (*‹□□?F ≡ □?F›*) WF_Box [try_rewrite] (*‹□WF(?A)_?v ≡ WF(?A)_?v›*) box_stp_act [try_rewrite] (*‹□$?P ≡ □?P›*))\n (*goals:\n 1. ‹⋀sigmaa sigma. ⟦sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P); sigma ⊨ WF(A)_f; sigma ⊨ □N; sigma ⊨ □[¬ B]_f; sigma ⊨ □Enabled (<M>_g); sigma ⊨ □P⟧ ⟹ sigma ⊨ □P$›\n 2. ‹⋀sigmaa sigma. ⟦sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P); sigma ⊨ WF(A)_f; sigma ⊨ □N; sigma ⊨ □[¬ B]_f; sigma ⊨ □Enabled (<M>_g); sigma ⊨ □P⟧ ⟹ sigma ⊨ ◇<A>_f›\n discuss goal 1*)\n apply (force elim!: TLA2E [where P = P, temp_use] (*‹⟦?sigma ⊨ □P; ⊢ $P ∧ P$ ⟶ ?A⟧ ⟹ ?sigma ⊨ □?A›*))\n (*discuss goal 2*)\n apply (rule STL2 [temp_use] (*‹(?sigma::behavior) ⊨ □(?F::behavior ⇒ bool) ⟹ ?F ?sigma›*))\n (*goal: ‹⋀sigmaa sigma. ⟦sigmaa ⊨ ◇□(Enabled (<M>_g) ∧ P); sigma ⊨ WF(A)_f; sigma ⊨ □N; sigma ⊨ □[¬ B]_f; sigma ⊨ □Enabled (<M>_g); sigma ⊨ □P⟧ ⟹ sigma ⊨ ◇<A>_f›*)\n apply (force simp: WF_def (*‹WF(?A)_?v ≡ ◇□Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*) split_box_conj [try_rewrite] (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) elim!: mp (*‹⟦?P ⟶ ?Q; ?P⟧ ⟹ ?Q›*) intro!: InitDmd [temp_use] (*‹?F ?sigma ⟹ ?sigma ⊨ ◇?F›*) 3 [THEN STL4, temp_use] (*‹?sigma ⊨ □(P ∧ Enabled (<M>_g)) ⟹ ?sigma ⊨ □Enabled (<A>_f)›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 4 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\nlemma SF2:\n assumes 1: \"⊢ N ∧ <B>_f ⟶ <M>_g\"\n and 2: \"⊢ $P ∧ P` ∧ <N ∧ A>_f ⟶ B\"\n and 3: \"⊢ P ∧ Enabled(<M>_g) ⟶ Enabled(<A>_f)\"\n and 4: \"⊢ □(N ∧ [¬B]_f) ∧ SF(A)_f ∧ □F ∧ □◇Enabled(<M>_g) ⟶ ◇□P\"\n shows \"⊢ □N ∧ SF(A)_f ∧ □F ⟶ SF(M)_g\"\n apply (clarsimp dest!: BoxSFI [temp_use] (*‹?sigma ⊨ SF(?A)_?v ⟹ ?sigma ⊨ □SF(?A)_?v›*) simp: 2 [try_rewrite] (*‹⟦P ?s; P ?t; (?s, ?t) ⊨ <N ∧ A>_f⟧ ⟹ B (?s, ?t)›*) SF_def [where A = M] (*‹SF(M)_?v ≡ □◇Enabled (<M>_?v) ⟶ □◇<M>_?v›*))\n (*goal: ‹⊢ □N ∧ SF(A)_f ∧ □F ⟶ SF(M)_g›*)\n apply (erule_tac F = F in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □N; sigma ⊨ □F; sigma ⊨ □SF(A)_f; sigma ⊨ □◇Enabled (<M>_g)⟧ ⟹ sigma ⊨ □◇<M>_g›*)\n apply (erule_tac F = \"TEMP ◇Enabled (<M>_g) \" in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □N; sigma ⊨ □SF(A)_f; sigma ⊨ □◇Enabled (<M>_g); sigma ⊨ □□F⟧ ⟹ sigma ⊨ □◇<M>_g›*)\n apply merge_temp_box\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □N; sigma ⊨ □SF(A)_f; sigma ⊨ □□F; sigma ⊨ □□◇Enabled (<M>_g)⟧ ⟹ sigma ⊨ □◇<M>_g›*)\n apply (erule STL4Edup (*‹⟦?sigma ⊨ □?A; ?sigma ⊨ □?F; ⊢ ?F ∧ □?A ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F) ⟹ sigma ⊨ □?F14 sigma›\n 2. ‹⋀sigma. sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F) ⟹ ⊢ ?F14 sigma ∧ □N ⟶ ◇<M>_g›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (clarsimp intro!: BoxDmd_simple [temp_use, THEN 1 [THEN DmdImpl, temp_use]] (*‹⟦?sigma ⊨ □N; ?sigma ⊨ ◇<B>_f⟧ ⟹ ?sigma ⊨ ◇<M>_g›*))\n (*goal: ‹⋀sigma. sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F) ⟹ ⊢ (□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F) ∧ □N ⟶ ◇<M>_g›*)\n apply (rule classical (*‹(¬ (?P::bool) ⟹ ?P) ⟹ ?P›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □N⟧ ⟹ sigmaa ⊨ ◇<B>_f›*)\n apply (subgoal_tac \"sigmaa ⊨ ◇ (($P ∧ P` ∧ N) ∧ <A>_f)\")\n (*goals:\n 1. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □N; ¬ (sigmaa ⊨ ◇<B>_f); sigmaa ⊨ ◇(($P ∧ P$ ∧ N) ∧ <A>_f)⟧ ⟹ sigmaa ⊨ ◇<B>_f›\n 2. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □N; ¬ (sigmaa ⊨ ◇<B>_f)⟧ ⟹ sigmaa ⊨ ◇(($P ∧ P$ ∧ N) ∧ <A>_f)›\n discuss goal 1*)\n apply (force simp: angle_def (*‹<?A>_?v ≡ ?A ∧ ¬ unchanged ?v›*) intro!: 2 [temp_use] (*‹⟦P ?s; P ?t; (?s, ?t) ⊨ <N ∧ A>_f⟧ ⟹ B (?s, ?t)›*) elim!: DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*discuss goal 2*)\n apply (rule BoxDmd_simple [THEN DmdImpl, unfolded DmdDmd [temp_rewrite], temp_use] (*‹(?sigma::behavior) ⊨ ◇(□(?F::?'a ⇒ bool) ∧ ◇(?G::?'a ⇒ bool)) ⟹ ?sigma ⊨ ◇(?F ∧ ?G)›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □N; ¬ (sigmaa ⊨ ◇<B>_f)⟧ ⟹ sigmaa ⊨ ◇(($P ∧ P$ ∧ N) ∧ <A>_f)›*)\n apply (simp add: NotDmd [temp_use] (*‹(¬ (?sigma ⊨ ◇?F)) = (?sigma ⊨ □¬ ?F)›*) not_angle [try_rewrite] (*‹¬ <?A>_?v ≡ [¬ ?A]_?v›*))\n (*goal: ‹⋀(sigma::behavior) sigmaa::behavior. ⟦sigma ⊨ □(□◇Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b)) ∧ SF(A::state × state ⇒ bool)_(f::state ⇒ 'a) ∧ □(F::'c ⇒ bool)); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □(N::state × state ⇒ bool); ¬ (sigmaa ⊨ ◇<B::state × state ⇒ bool>_f)⟧ ⟹ sigmaa ⊨ ◇(□($(P::state ⇒ bool) ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply merge_act_box\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □N; sigmaa ⊨ □[¬ B]_f⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (frule 4 [temp_use] (*‹⟦(?sigma::behavior) ⊨ □((N::state × state ⇒ bool) ∧ [¬ (B::state × state ⇒ bool)]_(f::state ⇒ 'a)); ?sigma ⊨ SF(A::state × state ⇒ bool)_f; ?sigma ⊨ □(F::'c ⇒ bool); ?sigma ⊨ □◇Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b))⟧ ⟹ ?sigma ⊨ ◇□(P::state ⇒ bool)›*))\n (*goals:\n 1. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □(N ∧ [¬ B]_f)⟧ ⟹ sigmaa ⊨ SF(A)_f›\n 2. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □(N ∧ [¬ B]_f)⟧ ⟹ sigmaa ⊨ □F›\n 3. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □(N ∧ [¬ B]_f)⟧ ⟹ sigmaa ⊨ □◇Enabled (<M>_g)›\n 4. ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□P⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply assumption\n (*discuss goal 4*)\n apply (erule_tac V = \"sigmaa ⊨ □F\" in thin_rl (*‹⟦PROP ?V; PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □F; sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□P⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (drule BoxSFI [temp_use] (*‹(?sigma::behavior) ⊨ SF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a) ⟹ ?sigma ⊨ □SF(?A)_?v›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ SF(A)_f; sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□P⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (erule_tac F = \"TEMP ◇Enabled (<M>_g)\" in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □◇Enabled (<M>_g); sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□P; sigmaa ⊨ □SF(A)_f⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (erule_tac F = \"ACT N ∧ [¬B]_f\" in dup_boxE (*‹⟦?sigma ⊨ □?F; ?sigma ⊨ □□?F ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ □(N ∧ [¬ B]_f); sigmaa ⊨ ◇□P; sigmaa ⊨ □SF(A)_f; sigmaa ⊨ □□◇Enabled (<M>_g)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply merge_temp_box\n (*goal: ‹⋀sigma sigmaa. ⟦sigma ⊨ □(□◇Enabled (<M>_g) ∧ SF(A)_f ∧ □F); sigmaa ⊨ ◇□P; sigmaa ⊨ □SF(A)_f; sigmaa ⊨ □□◇Enabled (<M>_g); sigmaa ⊨ □□(N ∧ [¬ B]_f)⟧ ⟹ sigmaa ⊨ ◇(□($P ∧ P$ ∧ N) ∧ ◇<A>_f)›*)\n apply (erule DmdImpldup (*‹⟦?sigma ⊨ □?A; ?sigma ⊨ ◇?F; ⊢ □?A ∧ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*))\n (*goals:\n 1. ‹⋀sigma sigmaa. sigmaa ⊨ ◇□P ⟹ sigmaa ⊨ ◇?F57 sigma sigmaa›\n 2. ‹⋀sigma sigmaa. sigmaa ⊨ ◇□P ⟹ ⊢ □((SF(A)_f ∧ □◇Enabled (<M>_g)) ∧ □◇Enabled (<M>_g) ∧ □(N ∧ [¬ B]_f)) ∧ ?F57 sigma sigmaa ⟶ □($P ∧ P$ ∧ N) ∧ ◇<A>_f›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (auto simp: split_box_conj [try_rewrite] (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) STL3 [try_rewrite] (*‹□□?F ≡ □?F›*) SF_Box [try_rewrite] (*‹□SF(?A)_?v ≡ SF(?A)_?v›*) box_stp_act [try_rewrite] (*‹□$?P ≡ □?P›*))\n (*goals:\n 1. ‹⋀(sigmaa::behavior) sigma::behavior. ⟦sigmaa ⊨ ◇□(P::state ⇒ bool); sigma ⊨ SF(A::state × state ⇒ bool)_(f::state ⇒ 'a); sigma ⊨ □◇Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b)); sigma ⊨ □(N::state × state ⇒ bool); sigma ⊨ □[¬ (B::state × state ⇒ bool)]_f; sigma ⊨ □P⟧ ⟹ sigma ⊨ □P$›\n 2. ‹⋀(sigmaa::behavior) sigma::behavior. ⟦sigmaa ⊨ ◇□(P::state ⇒ bool); sigma ⊨ SF(A::state × state ⇒ bool)_(f::state ⇒ 'a); sigma ⊨ □◇Enabled (<M::state × state ⇒ bool>_(g::state ⇒ 'b)); sigma ⊨ □(N::state × state ⇒ bool); sigma ⊨ □[¬ (B::state × state ⇒ bool)]_f; sigma ⊨ □P⟧ ⟹ sigma ⊨ ◇<A>_f›\n discuss goal 1*)\n apply (force elim!: TLA2E [where P = P, temp_use] (*‹⟦?sigma ⊨ □P; ⊢ $P ∧ P$ ⟶ ?A⟧ ⟹ ?sigma ⊨ □?A›*))\n (*discuss goal 2*)\n apply (rule STL2 [temp_use] (*‹?sigma ⊨ □?F ⟹ ?F ?sigma›*))\n (*goal: ‹⋀sigmaa sigma. ⟦sigmaa ⊨ ◇□P; sigma ⊨ SF(A)_f; sigma ⊨ □◇Enabled (<M>_g); sigma ⊨ □N; sigma ⊨ □[¬ B]_f; sigma ⊨ □P⟧ ⟹ sigma ⊨ ◇<A>_f›*)\n apply (force simp: SF_def (*‹SF(?A)_?v ≡ □◇Enabled (<?A>_?v) ⟶ □◇<?A>_?v›*) split_box_conj [try_rewrite] (*‹(?sigma ⊨ □(?F ∧ ?G)) = ((?sigma ⊨ □?F) ∧ (?sigma ⊨ □?G))›*) elim!: mp (*‹⟦?P ⟶ ?Q; ?P⟧ ⟹ ?Q›*) InfImpl [temp_use] (*‹⟦?sigma ⊨ □◇?F; ?sigma ⊨ □?G; ⊢ ?F ∧ ?G ⟶ ?H⟧ ⟹ ?sigma ⊨ □◇?H›*) intro!: 3 [temp_use] (*‹⟦P ?w; ?w ⊨ Enabled (<M>_g)⟧ ⟹ ?w ⊨ Enabled (<A>_f)›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 4 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n(* ------------------------------------------------------------------------- *)\n(*** Liveness proofs by well-founded orderings ***)\n(* ------------------------------------------------------------------------- *)\nsection \"Well-founded orderings\"\n\nlemma wf_leadsto:\n assumes 1: \"wf r\"\n and 2: \"⋀x. sigma ⊨ F x ↝ (G ∨ (∃y. #((y,x)∈r) ∧ F y)) \"\n shows \"sigma ⊨ F x ↝ G\"\n apply (rule 1 [THEN wf_induct] (*‹(⋀x. ∀y. (y, x) ∈ r ⟶ ?P y ⟹ ?P x) ⟹ ?P ?a›*))\n (*goal: ‹sigma ⊨ F x ↝ G›*)\n apply (rule LatticeTriangle [temp_use] (*‹⟦?sigma ⊨ ?A ↝ ?D ∨ ?B; ?sigma ⊨ ?B ↝ ?D⟧ ⟹ ?sigma ⊨ ?A ↝ ?D›*))\n (*goals:\n 1. ‹⋀x. ∀y. (y, x) ∈ r ⟶ (sigma ⊨ F y ↝ G) ⟹ sigma ⊨ F x ↝ G ∨ ?B2 x›\n 2. ‹⋀x. ∀y. (y, x) ∈ r ⟶ (sigma ⊨ F y ↝ G) ⟹ sigma ⊨ ?B2 x ↝ G›\n discuss goal 1*)\n apply (rule 2 (*‹sigma ⊨ F ?x ↝ G ∨ (∃y. #((y, ?x) ∈ r) ∧ F y)›*))\n (*discuss goal 2*)\n apply (auto simp: leadsto_exists [try_rewrite] (*‹(∃x. ?F x) ↝ ?G ≡ ∀x. ?F x ↝ ?G›*))\n (*goal: ‹⋀x. ∀y. (y, x) ∈ r ⟶ (sigma ⊨ F y ↝ G) ⟹ sigma ⊨ (∃y. #((y, x) ∈ r) ∧ F y) ↝ G›*)\n apply (case_tac \"(y,x) ∈ r\")\n (*goals:\n 1. ‹⋀x y. ⟦∀y. (y, x) ∈ r ⟶ (sigma ⊨ F y ↝ G); (y, x) ∈ r⟧ ⟹ sigma ⊨ #((y, x) ∈ r) ∧ F y ↝ G›\n 2. ‹⋀x y. ⟦∀y. (y, x) ∈ r ⟶ (sigma ⊨ F y ↝ G); (y, x) ∉ r⟧ ⟹ sigma ⊨ #((y, x) ∈ r) ∧ F y ↝ G›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply (force simp: leadsto_def (*‹?F ↝ ?G ≡ □(Init ?F ⟶ ◇?G)›*) Init_simps (*‹Init ?y ≡ ?y› ‹Init $?P ≡ Init ?P› ‹Init ¬ ?F ≡ ¬ Init ?F› ‹Init (?P ⟶ ?Q) ≡ Init ?P ⟶ Init ?Q› ‹Init (?P ∧ ?Q) ≡ Init ?P ∧ Init ?Q› ‹Init (?P ∨ ?Q) ≡ Init ?P ∨ Init ?Q› ‹Init ?P = ?Q ≡ Init ?P = (Init ?Q)› ‹Init (∀x. ?F x) ≡ ∀x. Init ?F x› ‹Init (∃x. ?F x) ≡ ∃x. Init ?F x› ‹Init (∃!x. ?F x) ≡ ∃!x. Init ?F x› ‹□Init ?F ≡ □?F› ‹◇Init ?F ≡ ◇?F› and more 2 facts*) intro!: necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n(* If r is well-founded, state function v cannot decrease forever *)\nlemma wf_not_box_decrease: \"⋀r. wf r ⟹ ⊢ □[ (v`, $v) ∈ #r ]_v ⟶ ◇□[#False]_v\"\n apply clarsimp\n (*goal: ‹⋀r. wf r ⟹ ⊢ □[(v$, $v) ∈ #r]_v ⟶ ◇□[#False]_v›*)\n apply (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹⋀(r::('a × 'a) set) sigma::behavior. ⟦wf r; sigma ⊨ □[((v::state ⇒ 'a)$, $v) ∈ #r]_v⟧ ⟹ sigma ⊨ ◇□[#False]_v›*)\n apply (subgoal_tac \"sigma ⊨ (∃x. v=#x) ↝ #False\")\n (*goal: ‹⋀r sigma. ⟦wf r; sigma ⊨ □[(v$, $v) ∈ #r]_v; ¬ (sigma ⊨ ◇□[#False]_v)⟧ ⟹ False›*)\n apply (drule leadsto_false [temp_use, THEN iffD1, THEN STL2_gen [temp_use]] (*‹(?sigma::behavior) ⊨ (?F2::?'a ⇒ bool) ↝ #False ⟹ ?sigma ⊨ Init ¬ ?F2›*))\n (*top goal: ‹⋀r sigma. ⟦wf r; sigma ⊨ □[(v$, $v) ∈ #r]_v; ¬ (sigma ⊨ ◇□[#False]_v); sigma ⊨ (∃x. v = #x) ↝ #False⟧ ⟹ False› and 1 goal remains*)\n apply (force simp: Init_defs (*‹((?sigma::behavior) ⊨ Init (?P::state ⇒ bool)) = ?P (st1 ?sigma)› ‹((?sigma::behavior) ⊨ Init (?A::state × state ⇒ bool)) = ?A (st1 ?sigma, st2 ?sigma)› ‹((?w::behavior) ⊨ Init (?F::behavior ⇒ bool)) = ?F ?w›*))\n (*top goal: ‹⋀r sigma. ⟦wf r; sigma ⊨ □[(v$, $v) ∈ #r]_v; ¬ (sigma ⊨ ◇□[#False]_v); sigma ⊨ Init ¬ (∃x. v = #x)⟧ ⟹ False› and 1 goal remains*)\n apply (clarsimp simp: leadsto_exists [try_rewrite] (*‹(∃x::?'c. (?F::?'c ⇒ ?'a ⇒ bool) x) ↝ (?G::?'b ⇒ bool) ≡ ∀x::?'c. ?F x ↝ ?G›*) not_square [try_rewrite] (*‹¬ [?A::state × state ⇒ bool]_(?v::state ⇒ ?'a) ≡ <¬ ?A>_?v›*) more_temp_simps (*‹□$(?P::state ⇒ bool) ≡ □?P› ‹□□(?F::?'a ⇒ bool) ≡ □?F› ‹◇◇(?F::?'a ⇒ bool) ≡ ◇?F› ‹¬ □(?F::?'a ⇒ bool) ≡ ◇¬ ?F› ‹¬ ◇(?F::?'a ⇒ bool) ≡ □¬ ?F› ‹¬ ((?sigma::behavior) ⊨ □(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ ((?sigma::behavior) ⊨ ◇(?F::?'a ⇒ bool)) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□(?F::?'a ⇒ bool) ≡ ◇□?F› ‹◇□◇(?F::?'a ⇒ bool) ≡ □◇?F› ‹□WF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a) ≡ WF(?A)_?v› ‹□SF(?A::state × state ⇒ bool)_(?v::state ⇒ ?'a) ≡ SF(?A)_?v›*))\n (*goal: ‹⋀r sigma. ⟦wf r; sigma ⊨ □[(v$, $v) ∈ #r]_v; ¬ (sigma ⊨ ◇□[#False]_v)⟧ ⟹ sigma ⊨ (∃x. v = #x) ↝ #False›*)\n apply (erule wf_leadsto (*‹⟦wf ?r; ⋀x. ?sigma ⊨ ?F x ↝ ?G ∨ (∃y. #((y, x) ∈ ?r) ∧ ?F y)⟧ ⟹ ?sigma ⊨ ?F ?x ↝ ?G›*))\n (*goal: ‹⋀r sigma x. ⟦wf r; sigma ⊨ □[(v$, $v) ∈ #r]_v; sigma ⊨ □◇<#True>_v⟧ ⟹ sigma ⊨ v = #x ↝ #False›*)\n apply (rule ensures_simple [temp_use] (*‹⟦⊢ $?P ∧ ?N ⟶ ?P$ ∨ ?Q$; ⊢ ($?P ∧ ?N) ∧ ?A ⟶ ?Q$; ?sigma ⊨ □?N; ?sigma ⊨ □◇?A⟧ ⟹ ?sigma ⊨ ?P ↝ ?Q›*))\n (*goal: ‹⋀r sigma x xa. ⟦sigma ⊨ □[(v$, $v) ∈ #r]_v; sigma ⊨ □◇<#True>_v⟧ ⟹ sigma ⊨ v = #xa ↝ #False ∨ (∃y. #((y, xa) ∈ r) ∧ v = #y)›*)\n by (auto simp: square_def (*‹[?A]_?v ≡ ?A ∨ unchanged ?v›*) angle_def (*‹<?A>_?v ≡ ?A ∧ ¬ unchanged ?v›*))\n\n(* \"wf r ⟹ ⊢ ◇□[ (v`, $v) : #r ]_v ⟶ ◇□[#False]_v\" *)\nlemmas wf_not_dmd_box_decrease =\n wf_not_box_decrease [THEN DmdImpl, unfolded more_temp_simps]\n\n(* If there are infinitely many steps where v decreases, then there\n have to be infinitely many non-stuttering steps where v doesn't decrease.\n*)\nlemma wf_box_dmd_decrease:\n assumes 1: \"wf r\"\n shows \"⊢ □◇((v`, $v) ∈ #r) ⟶ □◇<(v`, $v) ∉ #r>_v\"\n apply clarsimp\n (*goal: ‹⊢ □◇(v$, $v) ∈ #r ⟶ □◇<(v$, $v) ∉ #r>_v›*)\n apply (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹⋀sigma::behavior. sigma ⊨ □◇((v::state ⇒ 'a::type)$, $v) ∈ #(r::('a::type × 'a::type) set) ⟹ sigma ⊨ □◇<(v$, $v) ∉ #r>_v›*)\n apply (simp add: not_angle [try_rewrite] (*‹¬ <?A>_?v ≡ [¬ ?A]_?v›*) more_temp_simps (*‹□$?P ≡ □?P› ‹□□?F ≡ □?F› ‹◇◇?F ≡ ◇?F› ‹¬ □?F ≡ ◇¬ ?F› ‹¬ ◇?F ≡ □¬ ?F› ‹¬ (?sigma ⊨ □?F) ≡ ?sigma ⊨ ◇¬ ?F› ‹¬ (?sigma ⊨ ◇?F) ≡ ?sigma ⊨ □¬ ?F› ‹□◇□?F ≡ ◇□?F› ‹◇□◇?F ≡ □◇?F› ‹□WF(?A)_?v ≡ WF(?A)_?v› ‹□SF(?A)_?v ≡ SF(?A)_?v›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □◇(v$, $v) ∈ #r; ¬ (sigma ⊨ □◇<(v$, $v) ∉ #r>_v)⟧ ⟹ False›*)\n apply (drule 1 [THEN wf_not_dmd_box_decrease [temp_use]] (*‹?sigma ⊨ ◇□[(?v$, $?v) ∈ #r]_?v ⟹ ?sigma ⊨ ◇□[#False]_?v›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ □◇(v$, $v) ∈ #r; sigma ⊨ ◇□[(v$, $v) ∈ #r]_v⟧ ⟹ False›*)\n apply (drule BoxDmdDmdBox [temp_use] (*‹⟦?sigma ⊨ □◇?F; ?sigma ⊨ ◇□?G⟧ ⟹ ?sigma ⊨ □◇(?F ∧ ?G)›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ ◇□[#False]_v ⟹ sigma ⊨ ◇□?G6 sigma›\n 2. ‹⋀sigma. ⟦sigma ⊨ ◇□[#False]_v; sigma ⊨ □◇((v$, $v) ∈ #r ∧ ?G6 sigma)⟧ ⟹ False›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (subgoal_tac \"sigma ⊨ □◇ ((#False) ::action)\")\n (*goals:\n 1. ‹⋀sigma::behavior. ⟦sigma ⊨ ◇□[#False]_(v::state ⇒ 'a); sigma ⊨ □◇((v$, $v) ∈ #(r::('a × 'a) set) ∧ [#False]_v); sigma ⊨ □◇#False⟧ ⟹ False›\n 2. ‹⋀sigma::behavior. ⟦sigma ⊨ ◇□[#False]_(v::state ⇒ 'a); sigma ⊨ □◇((v$, $v) ∈ #(r::('a × 'a) set) ∧ [#False]_v)⟧ ⟹ sigma ⊨ □◇#False›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply (erule STL4E (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*goal: ‹⋀sigma. ⟦sigma ⊨ ◇□[#False]_v; sigma ⊨ □◇((v$, $v) ∈ #r ∧ [#False]_v)⟧ ⟹ sigma ⊨ □◇#False›*)\n apply (rule DmdImpl (*‹⊢ ?F ⟶ ?G ⟹ ⊢ ◇?F ⟶ ◇?G›*))\n (*goal: ‹⋀sigma. sigma ⊨ ◇□[#False]_v ⟹ ⊢ ◇((v$, $v) ∈ #r ∧ [#False]_v) ⟶ ◇#False›*)\n apply (force intro: 1 [THEN wf_irrefl, temp_use] (*‹((?a::'a::type, ?a) ∉ (r::('a::type × 'a::type) set) ⟹ ?thesis::bool) ⟹ ?thesis›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n(* In particular, for natural numbers, if n decreases infinitely often\n then it has to increase infinitely often.\n*)\nlemma nat_box_dmd_decrease: \"⋀n::nat stfun. ⊢ □◇(n` < $n) ⟶ □◇($n < n`)\"\n apply clarsimp\n (*goal: ‹⋀n::state ⇒ nat. ⊢ □◇n$ < $n ⟶ □◇$n < n$›*)\n apply (subgoal_tac \"sigma ⊨ □◇<¬ ((n`,$n) ∈ #less_than)>_n\")\n (*goals:\n 1. ‹⋀(n::state ⇒ nat) sigma::behavior. ⟦sigma ⊨ □◇n$ < $n; sigma ⊨ □◇<(n$, $n) ∉ #less_than>_n⟧ ⟹ sigma ⊨ □◇$n < n$›\n 2. ‹⋀(n::state ⇒ nat) sigma::behavior. sigma ⊨ □◇n$ < $n ⟹ sigma ⊨ □◇<(n$, $n) ∉ #less_than>_n›\n discuss goal 1*)\n apply (erule thin_rl (*‹⟦PROP ?V; PROP ?W⟧ ⟹ PROP ?W›*))\n (*top goal: ‹⋀n sigma. ⟦sigma ⊨ □◇n$ < $n; sigma ⊨ □◇<(n$, $n) ∉ #less_than>_n⟧ ⟹ sigma ⊨ □◇$n < n$› and 1 goal remains*)\n apply (erule STL4E (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*))\n (*top goal: ‹⋀n sigma. sigma ⊨ □◇<(n$, $n) ∉ #less_than>_n ⟹ sigma ⊨ □◇$n < n$› and 1 goal remains*)\n apply (rule DmdImpl (*‹⊢ ?F ⟶ ?G ⟹ ⊢ ◇?F ⟶ ◇?G›*))\n (*top goal: ‹⋀n sigma. ⊢ ◇<(n$, $n) ∉ #less_than>_n ⟶ ◇$n < n$› and 1 goal remains*)\n apply (clarsimp simp: angle_def [try_rewrite] (*‹<?A>_?v ≡ ?A ∧ ¬ unchanged ?v›*))\n (*discuss goal 2*)\n apply (rule wf_box_dmd_decrease [temp_use] (*‹⟦wf ?r; ?sigma ⊨ □◇(?v$, $?v) ∈ #?r⟧ ⟹ ?sigma ⊨ □◇<(?v$, $?v) ∉ #?r>_?v›*))\n (*goals:\n 1. ‹⋀n sigma. sigma ⊨ □◇n$ < $n ⟹ wf less_than›\n 2. ‹⋀n sigma. sigma ⊨ □◇n$ < $n ⟹ sigma ⊨ □◇(n$, $n) ∈ #less_than›\n discuss goal 1*)\n apply ((auto elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*) DmdImplE [temp_use] (*‹⟦?sigma ⊨ ◇?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ ◇?G›*))[1])\n (*discuss goal 2*)\n apply ((auto elim!: STL4E [temp_use] (*‹⟦(?sigma::behavior) ⊨ □(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ □?G›*) DmdImplE [temp_use] (*‹⟦(?sigma::behavior) ⊨ ◇(?F::?'a ⇒ bool); ⊢ ?F ⟶ (?G::?'a ⇒ bool)⟧ ⟹ ?sigma ⊨ ◇?G›*))[1])\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n\n(* ------------------------------------------------------------------------- *)\n(*** Flexible quantification over state variables ***)\n(* ------------------------------------------------------------------------- *)\nsection \"Flexible quantification\"\n\nlemma aallI:\n assumes 1: \"basevars vs\"\n and 2: \"(⋀x. basevars (x,vs) ⟹ sigma ⊨ F x)\"\n shows \"sigma ⊨ (∀∀x. F x)\"\n by (auto simp: aall_def (*‹∀∀ x::state ⇒ ?'e. (?F::(state ⇒ ?'e) ⇒ behavior ⇒ bool) x ≡ ¬ (∃∃ x::state ⇒ ?'e. ¬ ?F x)›*) elim!: eexE [temp_use] (*‹⟦(∃∃ x::state ⇒ ?'a. (?F::(state ⇒ ?'a) ⇒ behavior ⇒ bool) x) (?sigma::behavior); basevars ?vs::state ⇒ ?'b; ⋀x::state ⇒ ?'a. ⟦basevars (x, ?vs); ?F x ?sigma⟧ ⟹ (?G::behavior ⇒ bool) ?sigma⟧ ⟹ ?G ?sigma›*) intro!: 1 (*‹basevars vs::state ⇒ 'a›*) dest!: 2 [temp_use] (*‹basevars (?x::state ⇒ 'b, vs::state ⇒ 'a) ⟹ (F::(state ⇒ 'b) ⇒ behavior ⇒ bool) ?x (sigma::behavior)›*))\n\nlemma aallE: \"⊢ (∀∀x. F x) ⟶ F x\"\n apply (unfold aall_def (*‹∀∀ x. ?F x ≡ ¬ (∃∃ x. ¬ ?F x)›*))\n (*goal: ‹⊢ (∀∀ x. F x) ⟶ F x›*)\n apply clarsimp\n (*goal: ‹⊢ ¬ (∃∃ x. ¬ F x) ⟶ F x›*)\n apply (erule contrapos_np (*‹⟦¬ (?Q::bool); ¬ (?P::bool) ⟹ ?Q⟧ ⟹ ?P›*))\n (*goal: ‹⋀sigma::behavior. ¬ (∃∃ x::state ⇒ 'a::type. ¬ (F::(state ⇒ 'a::type) ⇒ behavior ⇒ bool) x) sigma ⟹ F (x::state ⇒ 'a::type) sigma›*)\n by (force intro!: eexI [temp_use] (*‹?F ?x ?sigma ⟹ EEx ?F ?sigma›*))\n\n(* monotonicity of quantification *)\nlemma eex_mono:\n assumes 1: \"sigma ⊨ ∃∃x. F x\"\n and 2: \"⋀x. sigma ⊨ F x ⟶ G x\"\n shows \"sigma ⊨ ∃∃x. G x\"\n apply (rule unit_base [THEN 1 [THEN eexE]] (*‹(⋀x. ⟦basevars (x, ?vs); F x sigma⟧ ⟹ ?G sigma) ⟹ ?G sigma›*))\n (*goal: ‹(∃∃ x::state ⇒ 'a. (G::(state ⇒ 'a) ⇒ behavior ⇒ bool) x) (sigma::behavior)›*)\n apply (rule eexI [temp_use] (*‹?F ?x ?sigma ⟹ EEx ?F ?sigma›*))\n (*goal: ‹⋀x. ⟦basevars (x, ?vs); F x sigma⟧ ⟹ (∃∃ x. G x) sigma›*)\n by (erule 2 [unfolded intensional_rews, THEN mp] (*‹F ?x1 sigma ⟹ G ?x1 sigma›*))\n\nlemma aall_mono:\n assumes 1: \"sigma ⊨ ∀∀x. F(x)\"\n and 2: \"⋀x. sigma ⊨ F(x) ⟶ G(x)\"\n shows \"sigma ⊨ ∀∀x. G(x)\"\n apply (rule unit_base [THEN aallI] (*‹(⋀x. basevars (x, ?vs) ⟹ ?F x ?sigma) ⟹ (∀∀ x. ?F x) ?sigma›*))\n (*goal: ‹(∀∀ x. G x) sigma›*)\n apply (rule 2 [unfolded intensional_rews, THEN mp] (*‹F ?x1 sigma ⟹ G ?x1 sigma›*))\n (*goal: ‹⋀x. basevars (x, ?vs) ⟹ G x sigma›*)\n by (rule 1 [THEN aallE [temp_use]] (*‹F ?x sigma›*))\n\n(* Derived history introduction rule *)\nlemma historyI:\n assumes 1: \"sigma ⊨ Init I\"\n and 2: \"sigma ⊨ □N\"\n and 3: \"basevars vs\"\n and 4: \"⋀h. basevars(h,vs) ⟹ ⊢ I ∧ h = ha ⟶ HI h\"\n and 5: \"⋀h s t. ⟦ basevars(h,vs); N (s,t); h t = hb (h s) (s,t) ⟧ ⟹ HN h (s,t)\"\n shows \"sigma ⊨ ∃∃h. Init (HI h) ∧ □(HN h)\"\n apply (rule history [temp_use, THEN eexE] (*‹⟦basevars ?vs; ⋀x. ⟦basevars (x, ?vs); ?sigma ⊨ Init x = ?ha1 ∧ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa)⟧ ⟹ ?G ?sigma⟧ ⟹ ?G ?sigma›*))\n (*goals:\n 1. ‹basevars ?vs›\n 2. ‹⋀x. ⟦basevars (x, ?vs); sigma ⊨ Init x = ?ha1 ∧ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa)⟧ ⟹ (∃∃ h. Init HI h ∧ □HN h) sigma›\n discuss goal 1*)\n apply (rule 3 (*‹basevars vs::state ⇒ 'a›*))\n (*discuss goal 2*)\n apply (rule eexI [temp_use] (*‹?F ?x ?sigma ⟹ EEx ?F ?sigma›*))\n (*goal: ‹⋀x::state ⇒ ?'a::type. ⟦basevars (x, vs::state ⇒ 'a::type); (sigma::behavior) ⊨ Init x = (?ha1::state ⇒ ?'a::type) ∧ □(∀xa::?'a::type. $x = #xa ⟶ x$ = (?hb1::?'a::type ⇒ state × state ⇒ ?'a::type) xa)⟧ ⟹ (∃∃ h::state ⇒ 'b::type. Init (HI::(state ⇒ 'b::type) ⇒ state ⇒ bool) h ∧ □(HN::(state ⇒ 'b::type) ⇒ state × state ⇒ bool) h) sigma›*)\n apply clarsimp\n (*goal: ‹⋀x. ⟦basevars (x, vs); sigma ⊨ Init x = ?ha1 ∧ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa)⟧ ⟹ sigma ⊨ Init HI (?x6 x) ∧ □HN (?x6 x)›*)\n apply (rule conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹⋀x. ⟦basevars (x, vs); sigma ⊨ Init x = ?ha1; sigma ⊨ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa)⟧ ⟹ sigma ⊨ Init HI (?x6 x)›\n 2. ‹⋀x. ⟦basevars (x, vs); sigma ⊨ Init x = ?ha1; sigma ⊨ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa)⟧ ⟹ sigma ⊨ □HN (?x6 x)›\n discuss goal 1*)\n apply ((insert 2 (*‹sigma ⊨ □N›*))[1])\n (*top goal: ‹⋀x. ⟦basevars (x, vs); sigma ⊨ Init x = ?ha1; sigma ⊨ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa)⟧ ⟹ sigma ⊨ Init HI (?x6 x)› and 1 goal remains*)\n apply (insert 1 (*‹sigma ⊨ Init I›*))\n (*top goal: ‹⋀x. ⟦basevars (x, vs); sigma ⊨ Init x = ?ha1; sigma ⊨ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa); sigma ⊨ □N⟧ ⟹ sigma ⊨ Init HI (?x6 x)› and 1 goal remains*)\n apply (force simp: Init_defs (*‹(?sigma ⊨ Init ?P) = ?P (st1 ?sigma)› ‹(?sigma ⊨ Init ?A) = ?A (st1 ?sigma, st2 ?sigma)› ‹(?w ⊨ Init ?F) = ?F ?w›*) elim!: 4 [temp_use] (*‹⟦basevars (?h, vs); I ?w; ?h ?w = ha ?w⟧ ⟹ HI ?h ?w›*))\n (*discuss goal 2*)\n apply ((insert 2 (*‹sigma ⊨ □N›*))[1])\n (*goal: ‹⋀x. ⟦basevars (x, vs); sigma ⊨ Init x = ha; sigma ⊨ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa); sigma ⊨ Init I⟧ ⟹ sigma ⊨ □HN x›*)\n apply merge_box\n (*goal: ‹⋀x. ⟦basevars (x, vs); sigma ⊨ Init x = ha; sigma ⊨ □(∀xa. $x = #xa ⟶ x$ = ?hb1 xa); sigma ⊨ Init I; sigma ⊨ □N⟧ ⟹ sigma ⊨ □HN x›*)\n apply (force elim!: STL4E [temp_use] (*‹⟦?sigma ⊨ □?F; ⊢ ?F ⟶ ?G⟧ ⟹ ?sigma ⊨ □?G›*) 5 [temp_use] (*‹⟦basevars (?h, vs); N (?s, ?t); ?h ?t = hb (?h ?s) (?s, ?t)⟧ ⟹ HN ?h (?s, ?t)›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\n(* ----------------------------------------------------------------------\n example of a history variable: existence of a clock\n*)\n\nlemma \"⊢ ∃∃h. Init(h = #True) ∧ □(h` = (¬$h))\"\n apply (rule tempI (*‹(⋀sigma. ?F sigma) ⟹ ⊢ ?F›*))\n (*goal: ‹⊢ ∃∃ h. Init h = #True ∧ □h$ = (¬ $h)›*)\n apply (rule historyI (*‹⟦(?sigma::behavior) ⊨ Init (?I::state ⇒ bool); ?sigma ⊨ □(?N::state × state ⇒ bool); basevars ?vs::state ⇒ ?'a; ⋀h::state ⇒ ?'b. basevars (h, ?vs) ⟹ ⊢ ?I ∧ h = (?ha::state ⇒ ?'b) ⟶ (?HI::(state ⇒ ?'b) ⇒ state ⇒ bool) h; ⋀(h::state ⇒ ?'b) (s::state) t::state. ⟦basevars (h, ?vs); ?N (s, t); h t = (?hb::?'b ⇒ state × state ⇒ ?'b) (h s) (s, t)⟧ ⟹ (?HN::(state ⇒ ?'b) ⇒ state × state ⇒ bool) h (s, t)⟧ ⟹ (∃∃ h::state ⇒ ?'b. Init ?HI h ∧ □?HN h) ?sigma›*))\n (*goals:\n 1. ‹⋀sigma. sigma ⊨ Init ?I2 sigma›\n 2. ‹⋀sigma. sigma ⊨ □?N2 sigma›\n 3. ‹⋀sigma. basevars ?vs2 sigma›\n 4. ‹⋀sigma h. basevars (h, ?vs2 sigma) ⟹ ⊢ ?I2 sigma ∧ h = ?ha2 sigma ⟶ h = #True›\n 5. ‹⋀sigma h s t. ⟦basevars (h, ?vs2 sigma); ?N2 sigma (s, t); h t = ?hb2 sigma (h s) (s, t)⟧ ⟹ (s, t) ⊨ h$ = (¬ $h)›\n discuss goal 1*)\n apply (force simp: Init_defs (*‹(?sigma ⊨ Init ?P) = ?P (st1 ?sigma)› ‹(?sigma ⊨ Init ?A) = ?A (st1 ?sigma, st2 ?sigma)› ‹(?w ⊨ Init ?F) = ?F ?w›*) intro!: unit_base [temp_use] (*‹basevars ?v›*) necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*))\n (*discuss goal 2*)\n apply (force simp: Init_defs (*‹(?sigma ⊨ Init ?P) = ?P (st1 ?sigma)› ‹(?sigma ⊨ Init ?A) = ?A (st1 ?sigma, st2 ?sigma)› ‹(?w ⊨ Init ?F) = ?F ?w›*) intro!: unit_base [temp_use] (*‹basevars ?v›*) necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*))\n (*discuss goal 3*)\n apply (force simp: Init_defs (*‹(?sigma ⊨ Init ?P) = ?P (st1 ?sigma)› ‹(?sigma ⊨ Init ?A) = ?A (st1 ?sigma, st2 ?sigma)› ‹(?w ⊨ Init ?F) = ?F ?w›*) intro!: unit_base [temp_use] (*‹basevars ?v›*) necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*))\n (*discuss goal 4*)\n apply (force simp: Init_defs (*‹(?sigma ⊨ Init ?P) = ?P (st1 ?sigma)› ‹(?sigma ⊨ Init ?A) = ?A (st1 ?sigma, st2 ?sigma)› ‹(?w ⊨ Init ?F) = ?F ?w›*) intro!: unit_base [temp_use] (*‹basevars ?v›*) necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*))\n (*discuss goal 5*)\n apply (force simp: Init_defs (*‹(?sigma ⊨ Init ?P) = ?P (st1 ?sigma)› ‹(?sigma ⊨ Init ?A) = ?A (st1 ?sigma, st2 ?sigma)› ‹(?w ⊨ Init ?F) = ?F ?w›*) intro!: unit_base [temp_use] (*‹basevars ?v›*) necT [temp_use] (*‹⊢ ?F ⟹ ?sigma ⊨ □?F›*))\n (*proven 5 subgoals*) .\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/TLA/TLA.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"c9be2e8ab3779532f7f7bbda3f76ad9a5bd2a5a839e70af5d7b5bb5863001810\"\n}"}}},{"rowIdx":170,"cells":{"text":{"kind":"string","value":"section ‹ Key algorithms for WEST ›\n\ntheory WEST_Algorithms\n\nimports Mission_Time_LTL.MLTL_Properties\n\nbegin\n\nsubsection ‹Custom Types›\n\ndatatype WEST_bit = Zero | One | S\ntype_synonym state = \"nat set\"\ntype_synonym trace = \"nat set list\"\ntype_synonym state_regex = \"WEST_bit list\"\ntype_synonym trace_regex = \"WEST_bit list list\"\ntype_synonym WEST_regex = \"WEST_bit list list list\"\n\nsubsection ‹Trace Regular Expressions›\n\nfun WEST_get_bit:: \"trace_regex ⇒ nat ⇒ nat ⇒ WEST_bit\"\n where \"WEST_get_bit regex timestep var = (\n if timestep ≥ length regex then S\n else let regex_index = regex ! timestep in\n if var ≥ length regex_index then S\n else regex_index ! var\n )\"\n\ntext ‹ Returns the state at time i, list of variable states ›\nfun WEST_get_state:: \"trace_regex ⇒ nat ⇒ nat ⇒ state_regex\"\n where \"WEST_get_state regex time num_vars = (\n if time ≥ length regex then (map (λ k. S) [0 ..< num_vars])\n else regex ! time\n )\"\n\n\ntext ‹ Checks if one state of a trace matches one timeslice of a WEST regex ›\ndefinition match_timestep:: \"nat set ⇒ state_regex ⇒ bool\"\n where \"match_timestep state regex_state = (∀ x::nat. x < length regex_state ⟶ (\n ((regex_state ! x = One) ⟶ x ∈ state) ∧\n ((regex_state ! x = Zero) ⟶ x ∉ state)))\"\n\nfun trim_reversed_regex:: \"trace_regex ⇒ trace_regex\"\n where \"trim_reversed_regex [] = []\"\n | \"trim_reversed_regex (h#t) = (if (∀i<length h. (h!i) = S)\n then (trim_reversed_regex t) else (h#t))\"\n\nfun trim_regex:: \"trace_regex ⇒ trace_regex\"\n where \"trim_regex regex = rev (trim_reversed_regex (rev regex))\"\n\ndefinition match_regex:: \"nat set list ⇒ trace_regex ⇒ bool\"\n where \"match_regex trace regex = ((∀ time<length regex.\n (match_timestep (trace ! time) (regex ! time)))\n ∧(length trace ≥ length regex))\"\n\ndefinition match:: \"nat set list ⇒ WEST_regex ⇒ bool\"\n where \"match trace regex_list = (∃ i. i < length regex_list ∧\n (match_regex trace (regex_list ! i)))\"\n\nlemma match_example:\n shows \"match [{0::nat,1}, {1}, {0}]\n [\n [[Zero,Zero]],\n [[S,S], [S,One]]\n ] = True\"\nproof (-)\n (*goal: ‹match [{0::nat, 1::nat}, {1::nat}, {0::nat}] [[[Zero, Zero]], [[S, S], [S, One]]] = True›*)\n let ?regexList = \"[[[Zero,Zero]],[[S,S], [S,One]]]\"\n let ?trace = \"[{0::nat,1}, {1}, {0}]\"\n have \"(match_regex ?trace (?regexList!1))\"\n unfolding match_regex_def\n (*goal: ‹(∀time<length ([[[Zero, Zero]], [[S, S], [S, One]]] ! 1). match_timestep ([{0, 1}, {1}, {0}] ! time) ([[[Zero, Zero]], [[S, S], [S, One]]] ! 1 ! time)) ∧ length ([[[Zero, Zero]], [[S, S], [S, One]]] ! 1) ≤ length [{0, 1}, {1}, {0}]›*)\n by (simp add: match_timestep_def (*‹match_timestep (?state::nat set) (?regex_state::WEST_bit list) = (∀x<length ?regex_state. (?regex_state ! x = One ⟶ x ∈ ?state) ∧ (?regex_state ! x = Zero ⟶ x ∉ ?state))›*) nth_Cons' (*‹((?x::?'a) # (?xs::?'a list)) ! (?n::nat) = (if ?n = (0::nat) then ?x else ?xs ! (?n - (1::nat)))›*))\n then show \"?thesis\"\n (*goal: ‹match [{0, 1}, {1}, {0}] [[[Zero, Zero]], [[S, S], [S, One]]] = True›*)\n by (metis One_nat_def (*‹1 = Suc 0›*) add.commute (*‹?a + ?b = ?b + ?a›*) le_imp_less_Suc (*‹?m ≤ ?n ⟹ ?m < Suc ?n›*) le_numeral_extra( (*‹1 ≤ 1›*) 4) list.size( (*‹length [] = 0›*) 3) list.size( (*‹length (?x21.0 # ?x22.0) = length ?x22.0 + Suc 0›*) 4) match_def (*‹match ?trace ?regex_list = (∃i<length ?regex_list. match_regex ?trace (?regex_list ! i))›*) plus_1_eq_Suc (*‹(+) 1 = Suc›*))\nqed\n\n\ndefinition regex_equiv:: \"WEST_regex ⇒ WEST_regex ⇒ bool\"\n where \"regex_equiv rl1 rl2 = (\n ∀ π::nat set list. (match π rl1) ⟷ (match π rl2))\"\n\n\nlemma \"(regex_equiv [[[S,S]]]\n [[[S,One]],\n [[One,S]],\n [[Zero,Zero]]]) = True\"\nproof (-)\n (*goal: ‹regex_equiv [[[S, S]]] [[[S, One]], [[One, S]], [[Zero, Zero]]] = True›*)\n have d1: \"match π [[[S, One]], [[One, S]], [[Zero, Zero]]]\" if match: \"match π [[[S, S]]]\" for π\n proof (-)\n (*goal: ‹match (π::nat set list) [[[S, One]], [[One, S]], [[Zero, Zero]]]›*)\n have match_ss: \"match_regex π [[S, S]]\"\n using match (*‹match π [[[S, S]]]›*) unfolding match_def\n (*goal: ‹match_regex π [[S, S]]›*)\n by (metis One_nat_def (*‹1 = Suc 0›*) length_Cons (*‹length (?x # ?xs) = Suc (length ?xs)›*) less_one (*‹(?n < 1) = (?n = 0)›*) list.size( (*‹length [] = 0›*) 3) nth_Cons_0 (*‹(?x # ?xs) ! 0 = ?x›*))\n {\n assume \"*\": \"¬ (match_regex π [[S, One]]) ∧ ¬ (match_regex π [[One, S]])\" (*‹¬ match_regex (π::nat set list) [[S, One]] ∧ ¬ match_regex π [[One, S]]›*)\n have \"match_regex π [[Zero, Zero]]\"\n using match_ss (*‹match_regex π [[S, S]]›*) unfolding match_regex_def\n (*goal: ‹(∀time<length [[Zero, Zero]]. match_timestep (π ! time) ([[Zero, Zero]] ! time)) ∧ length [[Zero, Zero]] ≤ length π›*)\n by (smt (verit) \"*\" (*‹¬ match_regex π [[S, One]] ∧ ¬ match_regex π [[One, S]]›*) One_nat_def (*‹1 = Suc 0›*) WEST_bit.simps( (*‹One ≠ Zero›*) 2) length_Cons (*‹length (?x # ?xs) = Suc (length ?xs)›*) less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*) less_one (*‹(?n < 1) = (?n = 0)›*) list.size( (*‹length [] = 0›*) 3) match_regex_def (*‹match_regex ?trace ?regex = ((∀time<length ?regex. match_timestep (?trace ! time) (?regex ! time)) ∧ length ?regex ≤ length ?trace)›*) match_timestep_def (*‹match_timestep ?state ?regex_state = (∀x<length ?regex_state. (?regex_state ! x = One ⟶ x ∈ ?state) ∧ (?regex_state ! x = Zero ⟶ x ∉ ?state))›*) nth_Cons_0 (*‹(?x # ?xs) ! 0 = ?x›*) nth_Cons_Suc (*‹(?x # ?xs) ! Suc ?n = ?xs ! ?n›*) numeral_2_eq_2 (*‹2 = Suc (Suc 0)›*))\n }\n then show \"?thesis\"\n (*goal: ‹match π [[[S, One]], [[One, S]], [[Zero, Zero]]]›*)\n unfolding match_def\n (*goal: ‹∃i<length [[[S, One]], [[One, S]], [[Zero, Zero]]]. match_regex π ([[[S, One]], [[One, S]], [[Zero, Zero]]] ! i)›*)\n by (metis length_Cons (*‹length (?x # ?xs) = Suc (length ?xs)›*) less_Suc_eq_0_disj (*‹(?m < Suc ?n) = (?m = 0 ∨ (∃j. ?m = Suc j ∧ j < ?n))›*) nth_Cons_0 (*‹(?x # ?xs) ! 0 = ?x›*) nth_Cons_Suc (*‹(?x # ?xs) ! Suc ?n = ?xs ! ?n›*))\n qed\n have d2: \"match π [[[S, S]]]\" if match: \"match π [[[S, One]], [[One, S]], [[Zero, Zero]]]\" for π\n proof (-)\n (*goal: ‹match π [[[S, S]]]›*)\n {\n assume \"*\": \"match_regex π [[S, One]]\" (*‹match_regex (π::nat set list) [[S, One]]›*)\n then have \"match_regex π [[S, S]]\"\n unfolding match_regex_def\n (*goal: ‹(∀time<length [[S, S]]. match_timestep (π ! time) ([[S, S]] ! time)) ∧ length [[S, S]] ≤ length π›*)\n by (smt (verit, ccfv_SIG) One_nat_def (*‹(1::nat) = Suc (0::nat)›*) WEST_bit.simps( (*‹S ≠ Zero›*) 4) length_Cons (*‹length ((?x::?'a) # (?xs::?'a list)) = Suc (length ?xs)›*) less_2_cases (*‹(?n::nat) < (2::nat) ⟹ ?n = (0::nat) ∨ ?n = Suc (0::nat)›*) less_one (*‹((?n::nat) < (1::nat)) = (?n = (0::nat))›*) list.size( (*‹length [] = (0::nat)›*) 3) match_timestep_def (*‹match_timestep (?state::nat set) (?regex_state::WEST_bit list) = (∀x<length ?regex_state. (?regex_state ! x = One ⟶ x ∈ ?state) ∧ (?regex_state ! x = Zero ⟶ x ∉ ?state))›*) nth_Cons_0 (*‹((?x::?'a) # (?xs::?'a list)) ! (0::nat) = ?x›*) nth_Cons_Suc (*‹((?x::?'a) # (?xs::?'a list)) ! Suc (?n::nat) = ?xs ! ?n›*) numeral_2_eq_2 (*‹(2::nat) = Suc (Suc (0::nat))›*))\n then have \"match π [[[S, S]]]\"\n unfolding match_def\n (*goal: ‹∃i<length [[[S, S]]]. match_regex π ([[[S, S]]] ! i)›*)\n by simp\n }\n moreover {\n assume \"*\": \"match_regex π [[One, S]]\" (*‹match_regex (π::nat set list) [[One, S]]›*)\n then have \"match_regex π [[S, S]]\"\n unfolding match_regex_def\n (*goal: ‹(∀time<length [[S, S]]. match_timestep (π ! time) ([[S, S]] ! time)) ∧ length [[S, S]] ≤ length π›*)\n by (smt (verit, ccfv_SIG) One_nat_def (*‹1 = Suc 0›*) WEST_bit.simps( (*‹S ≠ Zero›*) 4) length_Cons (*‹length (?x # ?xs) = Suc (length ?xs)›*) less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*) less_one (*‹(?n < 1) = (?n = 0)›*) list.size( (*‹length [] = 0›*) 3) match_timestep_def (*‹match_timestep ?state ?regex_state = (∀x<length ?regex_state. (?regex_state ! x = One ⟶ x ∈ ?state) ∧ (?regex_state ! x = Zero ⟶ x ∉ ?state))›*) nth_Cons_0 (*‹(?x # ?xs) ! 0 = ?x›*) nth_Cons_Suc (*‹(?x # ?xs) ! Suc ?n = ?xs ! ?n›*) numeral_2_eq_2 (*‹2 = Suc (Suc 0)›*))\n then have \"match π [[[S, S]]]\"\n unfolding match_def\n (*goal: ‹∃i<length [[[S, S]]]. match_regex π ([[[S, S]]] ! i)›*)\n by simp\n }\n moreover {\n assume \"*\": \"match_regex π [[Zero, Zero]]\" (*‹match_regex (π::nat set list) [[Zero, Zero]]›*)\n then have \"match_regex π [[S, S]]\"\n unfolding match_regex_def\n (*goal: ‹(∀time<length [[S, S]]. match_timestep ((π::nat set list) ! time) ([[S, S]] ! time)) ∧ length [[S, S]] ≤ length π›*)\n by (smt (verit) One_nat_def (*‹1 = Suc 0›*) WEST_bit.distinct( (*‹One ≠ S›*) 5) length_Cons (*‹length (?x # ?xs) = Suc (length ?xs)›*) less_2_cases_iff (*‹(?n < 2) = (?n = 0 ∨ ?n = Suc 0)›*) less_one (*‹(?n < 1) = (?n = 0)›*) list.size( (*‹length [] = 0›*) 3) match_timestep_def (*‹match_timestep ?state ?regex_state = (∀x<length ?regex_state. (?regex_state ! x = One ⟶ x ∈ ?state) ∧ (?regex_state ! x = Zero ⟶ x ∉ ?state))›*) nth_Cons_0 (*‹(?x # ?xs) ! 0 = ?x›*) nth_Cons_Suc (*‹(?x # ?xs) ! Suc ?n = ?xs ! ?n›*) numeral_2_eq_2 (*‹2 = Suc (Suc 0)›*))\n then have \"match π [[[S, S]]]\"\n unfolding match_def\n (*goal: ‹∃i<length [[[S, S]]]. match_regex π ([[[S, S]]] ! i)›*)\n by simp\n }\n ultimately show \"?thesis\"\n (*goal: ‹match (π::nat set list) [[[S, S]]]›*)\n using match (*‹match π [[[S, One]], [[One, S]], [[Zero, Zero]]]›*) unfolding regex_equiv_def\n (*goal: ‹match (π::nat set list) [[[S, S]]]›*)\n by (smt (verit, del_insts) length_Cons (*‹length ((?x::?'a::type) # (?xs::?'a::type list)) = Suc (length ?xs)›*) less_Suc_eq_0_disj (*‹((?m::nat) < Suc (?n::nat)) = (?m = (0::nat) ∨ (∃j::nat. ?m = Suc j ∧ j < ?n))›*) match_def (*‹match (?trace::nat set list) (?regex_list::WEST_bit list list list) = (∃i<length ?regex_list. match_regex ?trace (?regex_list ! i))›*) nth_Cons_0 (*‹((?x::?'a::type) # (?xs::?'a::type list)) ! (0::nat) = ?x›*) nth_Cons_Suc (*‹((?x::?'a::type) # (?xs::?'a::type list)) ! Suc (?n::nat) = ?xs ! ?n›*))\n qed\n show \"?thesis\"\n (*goal: ‹regex_equiv [[[S, S]]] [[[S, One]], [[One, S]], [[Zero, Zero]]] = True›*)\n using d1 (*‹match (?π::nat set list) [[[S, S]]] ⟹ match ?π [[[S, One]], [[One, S]], [[Zero, Zero]]]›*) d2 (*‹match ?π [[[S, One]], [[One, S]], [[Zero, Zero]]] ⟹ match ?π [[[S, S]]]›*) unfolding regex_equiv_def\n (*goal: ‹(∀π. match π [[[S, S]]] = match π [[[S, One]], [[One, S]], [[Zero, Zero]]]) = True›*)\n by metis\nqed\n\n\nsubsection ‹ WEST Operations ›\n\nsubsubsection ‹ AND ›\n\n\nfun WEST_and_bitwise::\"WEST_bit ⇒\n WEST_bit ⇒\n WEST_bit option\"\n where \"WEST_and_bitwise b One = (if b = Zero then None else Some One)\"\n | \"WEST_and_bitwise b Zero = (if b = One then None else Some Zero)\"\n | \"WEST_and_bitwise b S = Some b\"\n\n\n\nfun WEST_and_state:: \"state_regex ⇒ state_regex ⇒ state_regex option\"\n where \"WEST_and_state [] [] = Some []\"\n | \"WEST_and_state (h1#t1) (h2#t2) =\n (case WEST_and_bitwise h1 h2 of\n None ⇒ None\n | Some b ⇒ (case WEST_and_state t1 t2 of\n None ⇒ None\n | Some L ⇒ Some (b#L)))\"\n | \"WEST_and_state _ _ = None\"\n\n\n\nfun WEST_and_trace:: \"trace_regex ⇒ trace_regex ⇒ trace_regex option\"\n where \"WEST_and_trace trace [] = Some trace\"\n | \"WEST_and_trace [] trace = Some trace\"\n | \"WEST_and_trace (h1#t1) (h2#t2) =\n (case WEST_and_state h1 h2 of\n None ⇒ None\n | Some state ⇒ (case WEST_and_trace t1 t2 of\n None ⇒ None\n | Some trace ⇒ Some (state#trace)))\"\n\n\nfun WEST_and_helper:: \"trace_regex ⇒ WEST_regex ⇒ WEST_regex\"\n where \"WEST_and_helper trace [] = []\"\n | \"WEST_and_helper trace (t#traces) =\n (case WEST_and_trace trace t of\n None ⇒ WEST_and_helper trace traces\n | Some res ⇒ res#(WEST_and_helper trace traces))\"\n\n\nfun WEST_and:: \"WEST_regex ⇒ WEST_regex ⇒ WEST_regex\"\n where \"WEST_and traceList [] = []\"\n | \"WEST_and [] traceList = []\"\n | \"WEST_and (trace#traceList1) traceList2 =\n (case WEST_and_helper trace traceList2 of\n [] ⇒ WEST_and traceList1 traceList2\n | traceList ⇒ traceList@(WEST_and traceList1 traceList2))\"\n\nsubsubsection ‹Simp›\n\nparagraph ‹Bitwise simplification operation›\n\nfun WEST_simp_bitwise:: \"WEST_bit ⇒ WEST_bit ⇒ WEST_bit\"\n where \"WEST_simp_bitwise b S = S\"\n | \"WEST_simp_bitwise b Zero = (if b = Zero then Zero else S)\"\n | \"WEST_simp_bitwise b One = (if b = One then One else S)\"\n\nfun WEST_simp_state:: \"state_regex ⇒ state_regex ⇒ state_regex\"\n where \"WEST_simp_state s1 s2 = (\n map (λ k. WEST_simp_bitwise (s1 ! k) (s2 ! k)) [0 ..< (length s1)])\"\n\n\nfun WEST_simp_trace:: \"trace_regex ⇒ trace_regex ⇒ nat => trace_regex\"\n where \"WEST_simp_trace trace1 trace2 num_vars = (\n map (λ k. (WEST_simp_state (WEST_get_state trace1 k num_vars) (WEST_get_state trace2 k num_vars)))\n [0 ..< (Max {(length trace1), (length trace2)})])\"\n\n\nparagraph ‹ Helper functions for defining WEST-simp ›\n\nfun count_nonS_trace:: \"state_regex ⇒ nat\"\n where \"count_nonS_trace [] = 0\"\n | \"count_nonS_trace (h#t) = (if (h ≠ S) then (1 + (count_nonS_trace t)) else (count_nonS_trace t))\"\n\nfun count_diff_state:: \"state_regex ⇒ state_regex ⇒ nat\"\n where \"count_diff_state [] [] = 0\"\n | \"count_diff_state trace [] = count_nonS_trace trace\"\n | \"count_diff_state [] trace = count_nonS_trace trace\"\n | \"count_diff_state (h1#t1) (h2#t2) = (if (h1 = h2) then (count_diff_state t1 t2) else (1 + (count_diff_state t1 t2)))\"\n\nfun count_diff:: \"trace_regex ⇒ trace_regex ⇒ nat\"\n where \"count_diff [] [] = 0\"\n | \"count_diff [] (h#t) = (count_diff_state [] h) + (count_diff [] t)\"\n | \"count_diff (h#t) [] = (count_diff_state [] h) + (count_diff [] t)\"\n | \"count_diff (h1#t1) (h2#t2) = (count_diff_state h1 h2) + (count_diff t1 t2)\"\n\nfun check_simp:: \"trace_regex ⇒ trace_regex ⇒ bool\"\n where \"check_simp trace1 trace2 = ((count_diff trace1 trace2) ≤ 1 ∧ length trace1 = length trace2)\"\n\nfun enumerate_pairs :: \"nat list ⇒ (nat * nat) list\" where\n \"enumerate_pairs [] = []\" |\n \"enumerate_pairs (x#xs) = map (λy. (x, y)) xs @ enumerate_pairs xs\"\n\nfun enum_pairs:: \"'a list ⇒ (nat * nat) list\"\n where \"enum_pairs L = enumerate_pairs [0 ..< length L]\"\n\nfun remove_element_at_index:: \"nat ⇒ 'a list ⇒ 'a list\"\n where \"remove_element_at_index n L = (take n L)@(drop (n+1) L)\"\n\ntext ‹ This assumes (fst h) < (snd h) ›\nfun update_L:: \"WEST_regex ⇒ (nat × nat) ⇒ nat ⇒ WEST_regex\"\n where \"update_L L h num_vars =\n(remove_element_at_index (fst h) (remove_element_at_index (snd h) L))@[WEST_simp_trace (L!(fst h)) (L!(snd h)) num_vars]\"\n\nparagraph ‹ Defining and Proving Termination of WEST-simp ›\n\nlemma length_enumerate_pairs:\n shows \"length (enumerate_pairs L) ≤ (length L)^2\"\nproof (induction L)\n (*goals:\n 1. ‹length (enumerate_pairs []) ≤ (length [])²›\n 2. ‹⋀(a::nat) L::nat list. length (enumerate_pairs L) ≤ (length L)² ⟹ length (enumerate_pairs (a # L)) ≤ (length (a # L))²›*)\n case Nil (*no hyothesis introduced yet*)\n then show \"?case\"\n (*goal: ‹length (enumerate_pairs []) ≤ (length [])²›*)\n by auto\nnext\n (*goal: ‹⋀a L. length (enumerate_pairs L) ≤ (length L)² ⟹ length (enumerate_pairs (a # L)) ≤ (length (a # L))²›*)\n case (Cons a L) (*‹length (enumerate_pairs (L::nat list)) ≤ (length L)²›*)\n have length_L: \"(length (a # L))² = (1 + (length L))^2\"\n by auto\n then have length_L: \"(length (a # L))² = 1 + 2*(length L) + (length L)^2\"\n by algebra\n have \"length (map (Pair a) L) ≤ length L\"\n by simp\n then show \"?case\"\n (*goal: ‹length (enumerate_pairs (a # L)) ≤ (length (a # L))²›*)\n unfolding enumerate_pairs.simps\n (*goal: ‹length (map (Pair a) L @ enumerate_pairs L) ≤ (length (a # L))²›*)\n using Cons (*‹length (enumerate_pairs (L::nat list)) ≤ (length L)²›*) length_L (*‹(length (a # L))² = 1 + 2 * length L + (length L)²›*) by simp\nqed\n\nlemma length_enum_pairs:\n shows \"length (enum_pairs L) ≤ (length L)^2\"\nproof (-)\n (*goal: ‹length (enum_pairs L) ≤ (length L)²›*)\n show \"?thesis\"\n (*goal: ‹length (enum_pairs L) ≤ (length L)²›*)\n unfolding enum_pairs.simps\n (*goal: ‹length (enumerate_pairs [0..<length L]) ≤ (length L)²›*)\n using length_enumerate_pairs (*‹length (enumerate_pairs ?L) ≤ (length ?L)²›*) by (metis length_upt (*‹length [?i..<?j] = ?j - ?i›*) minus_nat.diff_0 (*‹?m - 0 = ?m›*))\nqed\n\nlemma enumerate_pairs_fact:\n assumes \"∀ i j. (i < j ∧ i < length L ∧ j < length L) ⟶ (L!i) < (L!j)\"\n shows \"∀ pair ∈ set (enumerate_pairs L). (fst pair) < (snd pair)\"\n using assms (*‹∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j›*) proof (induct \"length L\" arbitrary:L)\n (*goals:\n 1. ‹⋀L. ⟦0 = length L; ∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j⟧ ⟹ ∀pair∈set (enumerate_pairs L). fst pair < snd pair›\n 2. ‹⋀x L. ⟦⋀L. ⟦x = length L; ∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j⟧ ⟹ ∀pair∈set (enumerate_pairs L). fst pair < snd pair; Suc x = length L; ∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j⟧ ⟹ ∀pair∈set (enumerate_pairs L). fst pair < snd pair›*)\n case 0 (*‹0 = length L› ‹∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j›*)\n then show \"?case\"\n (*goal: ‹∀pair∈set (enumerate_pairs L). fst pair < snd pair›*)\n by auto\n next\n (*goal: ‹⋀(x::nat) L::nat list. ⟦⋀L::nat list. ⟦x = length L; ∀(i::nat) j::nat. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j⟧ ⟹ ∀pair::nat × nat∈set (enumerate_pairs L). fst pair < snd pair; Suc x = length L; ∀(i::nat) j::nat. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j⟧ ⟹ ∀pair::nat × nat∈set (enumerate_pairs L). fst pair < snd pair›*)\n case (Suc x) (*‹⟦(x::nat) = length (?L::nat list); ∀(i::nat) j::nat. i < j ∧ i < length ?L ∧ j < length ?L ⟶ ?L ! i < ?L ! j⟧ ⟹ ∀pair::nat × nat∈set (enumerate_pairs ?L). fst pair < snd pair› ‹Suc x = length L› ‹∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j›*)\n then obtain h and T where obt_hT: \"L = h#T\"\n (*goal: ‹(⋀h T. L = h # T ⟹ thesis) ⟹ thesis›*)\n by (metis length_Suc_conv (*‹(length ?xs = Suc ?n) = (∃y ys. ?xs = y # ys ∧ length ys = ?n)›*))\n then have enum_L: \"enumerate_pairs L = map (Pair h) T @ enumerate_pairs T\"\n using enumerate_pairs.simps (*‹enumerate_pairs [] = []› ‹enumerate_pairs (?x # ?xs) = map (Pair ?x) ?xs @ enumerate_pairs ?xs›*) obt_hT (*‹L = h # T›*) by blast\n then have \"⋀ pair. pair ∈ set (enumerate_pairs L) ⟹ fst pair < snd pair\"\n proof (-)\n (*goal: ‹⋀pair. ⟦pair ∈ set (enumerate_pairs L); enumerate_pairs L = map (Pair h) T @ enumerate_pairs T⟧ ⟹ fst pair < snd pair›*)\n fix pair\n assume \"pair ∈ set (enumerate_pairs L)\" (*‹(pair::nat × nat) ∈ set (enumerate_pairs (L::nat list))›*)\n then have \"pair ∈ set (map (Pair h) T @ enumerate_pairs T)\"\n using enum_L (*‹enumerate_pairs L = map (Pair h) T @ enumerate_pairs T›*) by auto\n then have pair_or: \"pair ∈ set (map (Pair h) T) ∨ pair ∈ set(enumerate_pairs T)\"\n by auto\n {\n assume in_base: \"pair ∈ set (map (Pair h) T)\" (*‹(pair::nat × nat) ∈ set (map (Pair (h::nat)) (T::nat list))›*)\n have \"∀j. 0 < j ∧ j < length L ⟶ h < L ! j\"\n using Suc.prems (*‹∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j›*) obt_hT (*‹L = h # T›*) by force\n then have \"∀j < length T. h < T!j\"\n using obt_hT (*‹(L::nat list) = (h::nat) # (T::nat list)›*) by force\n then have \"∀t ∈ set T. h < t\"\n using obt_hT (*‹(L::nat list) = (h::nat) # (T::nat list)›*) by (metis in_set_conv_nth (*‹((?x::?'a) ∈ set (?xs::?'a list)) = (∃i<length ?xs. ?xs ! i = ?x)›*))\n then have \"fst pair < snd pair\"\n using in_base (*‹pair ∈ set (map (Pair h) T)›*) by auto\n }\n moreover {\n assume in_rec: \"pair ∈ set(enumerate_pairs T)\" (*‹(pair::nat × nat) ∈ set (enumerate_pairs (T::nat list))›*)\n have \"fst pair < snd pair\"\n using Suc.hyps(1)[of T] (*‹⟦(x::nat) = length (T::nat list); ∀(i::nat) j::nat. i < j ∧ i < length T ∧ j < length T ⟶ T ! i < T ! j⟧ ⟹ ∀pair::nat × nat∈set (enumerate_pairs T). fst pair < snd pair›*) Suc.prems (*‹∀i j. i < j ∧ i < length L ∧ j < length L ⟶ L ! i < L ! j›*) obt_hT (*‹L = h # T›*) in_rec (*‹pair ∈ set (enumerate_pairs T)›*) by (smt (verit, ccfv_SIG) Ex_less_Suc (*‹(∃i<Suc ?n. ?P i) = (?P ?n ∨ (∃i<?n. ?P i))›*) Suc.hyps( (*‹⟦x = length ?L; ∀i j. i < j ∧ i < length ?L ∧ j < length ?L ⟶ ?L ! i < ?L ! j⟧ ⟹ ∀pair∈set (enumerate_pairs ?L). fst pair < snd pair›*) 1) Suc.hyps( (*‹Suc x = length L›*) 2) length_Cons (*‹length (?x # ?xs) = Suc (length ?xs)›*) less_trans_Suc (*‹⟦?i < ?j; ?j < ?k⟧ ⟹ Suc ?i < ?k›*) nat.inject (*‹(Suc ?x2.0 = Suc ?y2.0) = (?x2.0 = ?y2.0)›*) nth_Cons_Suc (*‹(?x # ?xs) ! Suc ?n = ?xs ! ?n›*))\n }\n ultimately show \"fst pair < snd pair\"\n using enum_L (*‹enumerate_pairs L = map (Pair h) T @ enumerate_pairs T›*) obt_hT (*‹L = h # T›*) pair_or (*‹pair ∈ set (map (Pair h) T) ∨ pair ∈ set (enumerate_pairs T)›*) by blast\n qed\n then show \"?case\"\n (*goal: ‹∀pair∈set (enumerate_pairs L). fst pair < snd pair›*)\n by blast\n qed\n\nlemma enum_pairs_fact:\n shows \"∀ pair ∈ set (enum_pairs L). (fst pair) < (snd pair)\"\n unfolding enum_pairs.simps\n (*goal: ‹∀pair∈set (enumerate_pairs [0..<length L]). fst pair < snd pair›*)\n using enumerate_pairs_fact[of \"[0..<length L]\"] (*‹∀i j. i < j ∧ i < length [0..<length L] ∧ j < length [0..<length L] ⟶ [0..<length L] ! i < [0..<length L] ! j ⟹ ∀pair∈set (enumerate_pairs [0..<length L]). fst pair < snd pair›*) by simp\n\nlemma enum_pairs_bound_snd:\n assumes \" pair ∈ set (enumerate_pairs L)\"\n shows \"(snd pair) ∈ set L\"\n using assms (*‹pair ∈ set (enumerate_pairs L)›*) proof (induct \"length L\" arbitrary: L)\n (*goals:\n 1. ‹⋀L. ⟦0 = length L; pair ∈ set (enumerate_pairs L)⟧ ⟹ snd pair ∈ set L›\n 2. ‹⋀x L. ⟦⋀L. ⟦x = length L; pair ∈ set (enumerate_pairs L)⟧ ⟹ snd pair ∈ set L; Suc x = length L; pair ∈ set (enumerate_pairs L)⟧ ⟹ snd pair ∈ set L›*)\n case 0 (*‹(0::nat) = length (L::nat list)› ‹pair ∈ set (enumerate_pairs L)›*)\n then show \"?case\"\n (*goal: ‹snd pair ∈ set L›*)\n by auto\n next\n (*goal: ‹⋀x L. ⟦⋀L. ⟦x = length L; pair ∈ set (enumerate_pairs L)⟧ ⟹ snd pair ∈ set L; Suc x = length L; pair ∈ set (enumerate_pairs L)⟧ ⟹ snd pair ∈ set L›*)\n case (Suc x) (*‹⟦x = length ?L; pair ∈ set (enumerate_pairs ?L)⟧ ⟹ snd pair ∈ set ?L› ‹Suc x = length L› ‹pair ∈ set (enumerate_pairs L)›*)\n then obtain h and T where ht: \"L = h#T\"\n (*goal: ‹(⋀h T. L = h # T ⟹ thesis) ⟹ thesis›*)\n by (metis enumerate_pairs.cases (*‹⟦?x = [] ⟹ ?P; ⋀x xs. ?x = x # xs ⟹ ?P⟧ ⟹ ?P›*) enumerate_pairs.simps( (*‹enumerate_pairs [] = []›*) 1) in_set_member (*‹(?x ∈ set ?xs) = List.member ?xs ?x›*) member_rec( (*‹List.member [] ?y = False›*) 2))\n then have eo: \"pair ∈ set (map (Pair h) T) ∨ pair ∈ set (enumerate_pairs T)\"\n using Suc (*‹⟦(x::nat) = length (?L::nat list); (pair::nat × nat) ∈ set (enumerate_pairs ?L)⟧ ⟹ snd pair ∈ set ?L› ‹Suc (x::nat) = length (L::nat list)› ‹(pair::nat × nat) ∈ set (enumerate_pairs (L::nat list))›*) by simp\n {\n assume \"*\": \"pair ∈ set (map (Pair h) T)\" (*‹(pair::nat × nat) ∈ set (map (Pair (h::nat)) (T::nat list))›*)\n then have \"?case\"\n using ht (*‹L = h # T›*) using imageE (*‹⟦?b ∈ ?f ` ?A; ⋀x. ⟦?b = ?f x; x ∈ ?A⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) by auto\n }\n moreover {\n assume \"*\": \"pair ∈ set (enumerate_pairs T)\" (*‹(pair::nat × nat) ∈ set (enumerate_pairs (T::nat list))›*)\n then have \"snd pair ∈ set T\"\n using Suc(1)[of T] (*‹⟦(x::nat) = length (T::nat list); (pair::nat × nat) ∈ set (enumerate_pairs T)⟧ ⟹ snd pair ∈ set T›*) ht (*‹L = h # T›*) using Suc.hyps(2) (*‹Suc x = length L›*) by fastforce\n then have \"?case\"\n using ht (*‹L = h # T›*) by simp\n }\n ultimately show \"?case\"\n (*goal: ‹snd pair ∈ set L›*)\n using eo (*‹pair ∈ set (map (Pair h) T) ∨ pair ∈ set (enumerate_pairs T)›*) by blast\n qed\n\n\nlemma enum_pairs_bound:\n shows \"∀ pair ∈ set (enum_pairs L). (snd pair) < length L\"\n unfolding enum_pairs.simps enumerate_pairs.simps\n (*goal: ‹∀pair∈set (enumerate_pairs [0..<length L]). snd pair < length L›*)\n proof (induct \"length L\" arbitrary: L)\n (*goals:\n 1. ‹⋀L::'a list. (0::nat) = length L ⟹ ∀pair::nat × nat∈set (enumerate_pairs [0::nat..<length L]). snd pair < length L›\n 2. ‹⋀(x::nat) L::'a list. ⟦⋀L::'a list. x = length L ⟹ ∀pair::nat × nat∈set (enumerate_pairs [0::nat..<length L]). snd pair < length L; Suc x = length L⟧ ⟹ ∀pair::nat × nat∈set (enumerate_pairs [0::nat..<length L]). snd pair < length L›*)\n case 0 (*‹0 = length L›*)\n then show \"?case\"\n (*goal: ‹∀pair∈set (enumerate_pairs [0..<length L]). snd pair < length L›*)\n by simp\n next\n (*goal: ‹⋀x L. ⟦⋀L. x = length L ⟹ ∀pair∈set (enumerate_pairs [0..<length L]). snd pair < length L; Suc x = length L⟧ ⟹ ∀pair∈set (enumerate_pairs [0..<length L]). snd pair < length L›*)\n case (Suc x) (*‹x = length ?L ⟹ ∀pair∈set (enumerate_pairs [0..<length ?L]). snd pair < length ?L› ‹Suc x = length L›*)\n then have enum_L: \"enumerate_pairs ([0..<length L]) =\n map (Pair 0) [1..<length L] @ enumerate_pairs [1..<length L]\"\n using enumerate_pairs.simps(2)[of 0 \"[1 ..< length L]\"] (*‹enumerate_pairs (0 # [1..<length L]) = map (Pair 0) [1..<length L] @ enumerate_pairs [1..<length L]›*) by (metis One_nat_def (*‹1 = Suc 0›*) upt_conv_Cons (*‹?i < ?j ⟹ [?i..<?j] = ?i # [Suc ?i..<?j]›*) zero_less_Suc (*‹0 < Suc ?n›*))\n then have \"pair∈set (enumerate_pairs [0..<length L]) ⟹ snd pair < length L\" for pair\n using enum_pairs_bound_snd[of pair \"[0..<length L]\"] (*‹pair ∈ set (enumerate_pairs [0..<length L]) ⟹ snd pair ∈ set [0..<length L]›*) by auto\n then show \"?case\"\n (*goal: ‹∀pair∈set (enumerate_pairs [0..<length L]). snd pair < length L›*)\n unfolding enum_pairs.simps\n (*goal: ‹∀pair∈set (enumerate_pairs [0..<length L]). snd pair < length L›*)\n by blast\n qed\n\nlemma WEST_simp_termination1_bound:\n fixes a::\"nat\"\n shows \"a^3+a^2 < (a+1)^3\"\nproof (-)\n (*goal: ‹(a::nat) ^ (3::nat) + a² < (a + (1::nat)) ^ (3::nat)›*)\n have cubed: \"(a+1)^3 = a^3 + 3*a^2 + 3*a + 1\"\n proof (-)\n (*goal: ‹(a + 1) ^ 3 = a ^ 3 + 3 * a² + 3 * a + 1›*)\n have \"(a+1)^3 = (a+1)*(a+1)*(a+1)\"\n by algebra\n then show \"?thesis\"\n (*goal: ‹((a::nat) + (1::nat)) ^ (3::nat) = a ^ (3::nat) + (3::nat) * a² + (3::nat) * a + (1::nat)›*)\n by (simp add: add.commute (*‹?a + ?b = ?b + ?a›*) add_mult_distrib2 (*‹?k * (?m + ?n) = ?k * ?m + ?k * ?n›*) mult.commute (*‹?a * ?b = ?b * ?a›*) power2_eq_square (*‹?a² = ?a * ?a›*) power3_eq_cube (*‹?a ^ 3 = ?a * ?a * ?a›*))\n qed\n have \"0 < 2*a^2 + 2*a + 1\"\n by simp\n then have \"a^3 + a^2 < a^3 + 3*a^2 + 3*a + 1\"\n by simp\n then show \"?thesis\"\n (*goal: ‹a ^ 3 + a² < (a + 1) ^ 3›*)\n using cubed (*‹(a + 1) ^ 3 = a ^ 3 + 3 * a² + 3 * a + 1›*) by simp\nqed\n\nlemma WEST_simp_termination1:\n fixes L::\"WEST_regex\"\n assumes \"¬ (idx_pairs ≠ enum_pairs L ∨ length idx_pairs ≤ i)\"\n assumes \"check_simp (L ! fst (idx_pairs ! i)) (L ! snd (idx_pairs ! i))\"\n assumes \"x = update_L L (idx_pairs ! i) num_vars\"\n shows \"((x, enum_pairs x, 0, num_vars), L, idx_pairs, i, num_vars)\n ∈ measure (λ(L, idx_list, i, num_vars). length L ^ 3 + length idx_list - i)\"\nproof (-)\n (*goal: ‹((x, enum_pairs x, 0, num_vars), L, idx_pairs, i, num_vars) ∈ measure (λ(L, idx_list, i, num_vars). length L ^ 3 + length idx_list - i)›*)\n let ?i = \"fst (idx_pairs ! i)\"\n let ?j = \"snd (idx_pairs ! i)\"\n have i_le_j: \"?i < ?j\"\n using enum_pairs_fact (*‹∀pair∈set (enum_pairs ?L). fst pair < snd pair›*) assms (*‹¬ (idx_pairs ≠ enum_pairs L ∨ length idx_pairs ≤ i)› ‹check_simp (L ! fst (idx_pairs ! i)) (L ! snd (idx_pairs ! i))› ‹x = update_L L (idx_pairs ! i) num_vars›*) by (metis linorder_le_less_linear (*‹?x ≤ ?y ∨ ?y < ?x›*) nth_mem (*‹?n < length ?xs ⟹ ?xs ! ?n ∈ set ?xs›*))\n have j_bound: \"?j < length L\"\n using assms(1) (*‹¬ (idx_pairs ≠ enum_pairs L ∨ length idx_pairs ≤ i)›*) enum_pairs_bound[of L] (*‹∀pair::nat × nat∈set (enum_pairs (L::WEST_bit list list list)). snd pair < length L›*) by simp\n then have i_bound: \"?i < (length L)-1\"\n using i_le_j (*‹fst (idx_pairs ! i) < snd (idx_pairs ! i)›*) by auto\n have len_orsimp: \"length [WEST_simp_trace (L ! ?i) (L ! ?j) num_vars] = 1\"\n by simp\n have \"length (remove_element_at_index ?j L) = length L - 1\"\n using assms(3) (*‹(x::WEST_bit list list list) = update_L (L::WEST_bit list list list) ((idx_pairs::(nat × nat) list) ! (i::nat)) (num_vars::nat)›*) j_bound (*‹snd (idx_pairs ! i) < length L›*) by auto\n then have \"length (remove_element_at_index ?i (remove_element_at_index ?j L)) = length L - 2\"\n using assms(3) (*‹x = update_L L (idx_pairs ! i) num_vars›*) i_bound (*‹fst (idx_pairs ! i) < length L - 1›*) j_bound (*‹snd (idx_pairs ! i) < length L›*) by simp\n then have length_x: \"length x = (length L) - 1\"\n using assms(3) (*‹x = update_L L (idx_pairs ! i) num_vars›*) len_orsimp (*‹length [WEST_simp_trace (L ! fst (idx_pairs ! i)) (L ! snd (idx_pairs ! i)) num_vars] = 1›*) unfolding update_L.simps[of L \"idx_pairs ! i\" num_vars]\n (*goal: ‹length x = length L - 1›*)\n by (metis (no_types, lifting) add.commute (*‹?a + ?b = ?b + ?a›*) add_diff_inverse_nat (*‹¬ ?m < ?n ⟹ ?n + (?m - ?n) = ?m›*) diff_diff_left (*‹?i - ?j - ?k = ?i - (?j + ?k)›*) gr_implies_not0 (*‹?m < ?n ⟹ ?n ≠ 0›*) i_bound (*‹fst (idx_pairs ! i) < length L - 1›*) length_append (*‹length (?xs @ ?ys) = length ?xs + length ?ys›*) less_one (*‹(?n < 1) = (?n = 0)›*) nat_1_add_1 (*‹1 + 1 = 2›*))\n have i_bound: \"i < length idx_pairs\"\n using assms (*‹¬ (idx_pairs ≠ enum_pairs L ∨ length idx_pairs ≤ i)› ‹check_simp (L ! fst (idx_pairs ! i)) (L ! snd (idx_pairs ! i))› ‹x = update_L L (idx_pairs ! i) num_vars›*) by force\n {\n assume short_L: \"length L = 0\" (*‹length (L::WEST_bit list list list) = (0::nat)›*)\n then have \"?thesis\"\n using assms (*‹¬ (idx_pairs ≠ enum_pairs L ∨ length idx_pairs ≤ i)› ‹check_simp (L ! fst (idx_pairs ! i)) (L ! snd (idx_pairs ! i))› ‹x = update_L L (idx_pairs ! i) num_vars›*) using j_bound (*‹snd (idx_pairs ! i) < length L›*) by linarith\n }\n moreover {\n assume long_L: \"length L ≥ 1\" (*‹(1::nat) ≤ length (L::WEST_bit list list list)›*)\n then have \"length L - 1 ≥ 0\"\n by blast\n then have \"(length L - 1) ^ 3 + (length L - 1) ^ 2 < length L ^ 3\"\n using WEST_simp_termination1_bound[of \"length L-1\"] (*‹(length L - 1) ^ 3 + (length L - 1)² < (length L - 1 + 1) ^ 3›*) by (metis long_L (*‹1 ≤ length L›*) ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add (*‹⟦?a ≤ ?b; ?a ≤ ?b⟧ ⟹ (?b - ?a = ?c) = (?b = ?c + ?a)›*))\n then have \"(length L - 1) ^ 3 + length (enumerate_pairs [0..<length x]) < length L ^ 3\"\n using length_enumerate_pairs[of \"[0..<length x]\"] (*‹length (enumerate_pairs [0..<length x]) ≤ (length [0..<length x])²›*) length_x (*‹length x = length L - 1›*) by auto\n then have \"length x ^ 3 + length (enumerate_pairs [0..<length x])\n < length L ^ 3 + length idx_pairs - i\"\n using i_bound (*‹i < length idx_pairs›*) length_x (*‹length x = length L - 1›*) by simp\n then have \"?thesis\"\n by simp\n }\n ultimately show \"?thesis\"\n (*goal: ‹((x, enum_pairs x, 0, num_vars), L, idx_pairs, i, num_vars) ∈ measure (λ(L, idx_list, i, num_vars). length L ^ 3 + length idx_list - i)›*)\n by linarith\nqed\n\n\nfunction WEST_simp_helper:: \"WEST_regex ⇒ (nat × nat) list ⇒ nat ⇒ nat ⇒ WEST_regex\"\n where \"WEST_simp_helper L idx_pairs i num_vars =\n (if (idx_pairs ≠ enum_pairs L ∨ i ≥ length idx_pairs) then L else\n (if (check_simp (L!(fst (idx_pairs!i))) (L!(snd (idx_pairs!i)))) then\n (let newL = update_L L (idx_pairs!i) num_vars in\n WEST_simp_helper newL (enum_pairs newL) 0 num_vars)\n else WEST_simp_helper L idx_pairs (i+1) num_vars))\"\n apply fast by blast\ntermination\n apply (relation \"measure (λ(L , idx_list, i, num_vars). (length L^3 + length idx_list - i))\")\n apply simp using WEST_simp_termination1 apply blast by auto\n\ndeclare WEST_simp_helper.simps[simp del]\n\nfun WEST_simp:: \"WEST_regex ⇒ nat ⇒ WEST_regex\"\n where \"WEST_simp L num_vars =\n WEST_simp_helper L (enum_pairs L) 0 num_vars\"\n\nvalue \"WEST_simp [[[S, S, One]],[[S, One, S]], [[S, S, Zero]]] 3\"\nvalue \"WEST_simp [[[S, One]],[[One, S]], [[Zero, Zero]]] 2\"\nvalue \"WEST_simp [[[One, One]],[[Zero, Zero]], [[One, Zero]], [[Zero, One]]] 2\"\n\n\nsubsubsection ‹ AND and OR operations with WEST-simp ›\n\nfun WEST_and_simp:: \"WEST_regex ⇒ WEST_regex ⇒ nat ⇒ WEST_regex\"\n where \"WEST_and_simp L1 L2 num_vars = WEST_simp (WEST_and L1 L2) num_vars\"\n\nfun WEST_or_simp:: \"WEST_regex ⇒ WEST_regex ⇒ nat ⇒ WEST_regex\"\n where \"WEST_or_simp L1 L2 num_vars = WEST_simp (L1@L2) num_vars\"\n\nsubsubsection ‹ Useful Helper Functions ›\n\nfun arbitrary_state::\"nat ⇒ state_regex\"\n where \"arbitrary_state num_vars = map (λ k. S) [0 ..< num_vars]\"\n\nfun arbitrary_trace::\"nat ⇒ nat ⇒ trace_regex\"\n where \"arbitrary_trace num_vars num_pad = map (λ k. (arbitrary_state num_vars)) [0 ..< num_pad]\"\n\nfun shift:: \"WEST_regex ⇒ nat ⇒ nat ⇒ WEST_regex\"\n where \"shift traceList num_vars num_pad = map (λ trace. (arbitrary_trace num_vars num_pad)@trace) traceList\"\n\nfun pad:: \"trace_regex ⇒ nat ⇒ nat ⇒ trace_regex\"\n where \"pad trace num_vars num_pad = trace@(arbitrary_trace num_vars num_pad)\"\n\n\nsubsubsection ‹ WEST Temporal Operations ›\n\nfun WEST_global:: \"WEST_regex ⇒ nat ⇒ nat ⇒ nat ⇒ WEST_regex\"\nwhere \"WEST_global L a b num_vars = (\nif (a = b) then (shift L num_vars a)\n else ( if (a < b) then (WEST_and_simp (shift L num_vars b)\n (WEST_global L a (b-1) num_vars) num_vars)\n else []))\"\n\n\nfun WEST_future:: \"WEST_regex ⇒ nat ⇒ nat ⇒ nat ⇒ WEST_regex\"\n where \"WEST_future L a b num_vars = (\n if (a = b)\n then (shift L num_vars a)\n else (\n if (a < b)\n then WEST_or_simp (shift L num_vars b) (WEST_future L a (b-1) num_vars) num_vars\n else []))\"\n\n\nfun WEST_until:: \"WEST_regex ⇒ WEST_regex ⇒ nat ⇒\n nat ⇒ nat ⇒ WEST_regex\"\n where \"WEST_until L_φ L_ψ a b num_vars = (\n if (a=b)\n then (WEST_global L_ψ a a num_vars)\n else (\n if (a < b)\n then WEST_or_simp (WEST_until L_φ L_ψ a (b-1) num_vars)\n (WEST_and_simp (WEST_global L_φ a (b-1) num_vars)\n (WEST_global L_ψ b b num_vars) num_vars) num_vars\n else []))\"\n\n\nfun WEST_release_helper:: \"WEST_regex ⇒ WEST_regex ⇒\n nat ⇒ nat ⇒ nat ⇒ WEST_regex\"\n where \"WEST_release_helper L_φ L_ψ a ub num_vars = (\n if (a=ub)\n then (WEST_and_simp (WEST_global L_φ a a num_vars) (WEST_global L_ψ a a num_vars) num_vars)\n else (\n if (a < ub)\n then WEST_or_simp (WEST_release_helper L_φ L_ψ a (ub-1) num_vars)\n (WEST_and_simp (WEST_global L_ψ a ub num_vars)\n (WEST_global L_φ ub ub num_vars) num_vars) num_vars\n else []))\"\n\nfun WEST_release:: \"WEST_regex ⇒ WEST_regex ⇒ nat\n ⇒ nat ⇒ nat ⇒ WEST_regex\"\n where \"WEST_release L_φ L_ψ a b num_vars = (\n if (b > a)\n then (WEST_or_simp (WEST_global L_ψ a b num_vars) (WEST_release_helper L_φ L_ψ a (b-1) num_vars) num_vars)\n else (WEST_global L_ψ a b num_vars))\"\n\n\nsubsubsection ‹ WEST recursive reg Function ›\nlemma exhaustive:\n shows \"⋀x:: nat mltl × nat. ⋀P::bool. (⋀num_vars::nat. x = (True_mltl, num_vars) ⟹ P) ⟹\n (⋀num_vars::nat. x = (False_mltl, num_vars) ⟹ P) ⟹\n (⋀p num_vars::nat. x = (Prop_mltl p, num_vars) ⟹ P) ⟹\n (⋀p num_vars::nat. x = (Not_mltl (Prop_mltl p), num_vars) ⟹ P) ⟹\n (⋀φ ψ num_vars. x = (Or_mltl φ ψ, num_vars) ⟹ P) ⟹\n (⋀φ ψ num_vars. x = (And_mltl φ ψ, num_vars) ⟹ P) ⟹\n (⋀φ a b num_vars. x = (Future_mltl φ a b, num_vars) ⟹ P) ⟹\n (⋀φ a b num_vars. x = (Global_mltl φ a b, num_vars) ⟹ P) ⟹\n (⋀φ ψ a b num_vars. x = (Until_mltl φ ψ a b, num_vars) ⟹ P) ⟹\n (⋀φ ψ a b num_vars. x = (Release_mltl φ ψ a b, num_vars) ⟹ P) ⟹\n (⋀num_vars. x = (Not_mltl True_mltl, num_vars) ⟹ P) ⟹\n (⋀num_vars. x = (Not_mltl False_mltl, num_vars) ⟹ P) ⟹\n (⋀φ ψ num_vars. x = (Not_mltl (And_mltl φ ψ), num_vars) ⟹ P) ⟹\n (⋀φ ψ num_vars. x = (Not_mltl (Or_mltl φ ψ), num_vars) ⟹ P) ⟹\n (⋀φ a b num_vars. x = (Not_mltl (Future_mltl φ a b), num_vars) ⟹ P) ⟹\n (⋀φ a b num_vars. x = (Not_mltl (Global_mltl φ a b), num_vars) ⟹ P) ⟹\n (⋀φ ψ a b num_vars. x = (Not_mltl (Until_mltl φ ψ a b), num_vars) ⟹ P) ⟹\n (⋀φ ψ a b num_vars. x = (Not_mltl (Release_mltl φ ψ a b), num_vars) ⟹ P) ⟹\n (⋀φ num_vars. x = (Not_mltl (Not_mltl φ), num_vars) ⟹ P) ⟹ P\"\nproof (-)\n (*goal: ‹⋀x P. ⟦⋀num_vars. x = (True⇩m, num_vars) ⟹ P; ⋀num_vars. x = (False⇩m, num_vars) ⟹ P; ⋀p num_vars. x = (Prop⇩m (p), num_vars) ⟹ P; ⋀p num_vars. x = (Not⇩m Prop⇩m (p), num_vars) ⟹ P; ⋀φ ψ num_vars. x = (φ Or⇩m ψ, num_vars) ⟹ P; ⋀φ ψ num_vars. x = (φ And⇩m ψ, num_vars) ⟹ P; ⋀φ a b num_vars. x = (F⇩m [φ,a] b, num_vars) ⟹ P; ⋀φ a b num_vars. x = (G⇩m [φ,a] b, num_vars) ⟹ P; ⋀φ ψ a b num_vars. x = (φ U⇩m [ψ,a] b, num_vars) ⟹ P; ⋀φ ψ a b num_vars. x = (φ R⇩m [ψ,a] b, num_vars) ⟹ P; ⋀num_vars. x = (Not⇩m True⇩m, num_vars) ⟹ P; ⋀num_vars. x = (Not⇩m False⇩m, num_vars) ⟹ P; ⋀φ ψ num_vars. x = (Not⇩m (φ And⇩m ψ), num_vars) ⟹ P; ⋀φ ψ num_vars. x = (Not⇩m (φ Or⇩m ψ), num_vars) ⟹ P; ⋀φ a b num_vars. x = (Not⇩m F⇩m [φ,a] b, num_vars) ⟹ P; ⋀φ a b num_vars. x = (Not⇩m G⇩m [φ,a] b, num_vars) ⟹ P; ⋀φ ψ a b num_vars. x = (Not⇩m (φ U⇩m [ψ,a] b), num_vars) ⟹ P; ⋀φ ψ a b num_vars. x = (Not⇩m (φ R⇩m [ψ,a] b), num_vars) ⟹ P; ⋀φ num_vars. x = (Not⇩m Not⇩m φ, num_vars) ⟹ P⟧ ⟹ P›*)\n fix x :: \"nat mltl × nat\"\n fix P :: bool\n assume t: \"(⋀num_vars::nat. x = (True_mltl, num_vars) ⟹ P)\" (*‹(x::nat mltl × nat) = (True⇩m, ?num_vars::nat) ⟹ P::bool›*)\n assume fa: \"(⋀num_vars::nat. x = (False_mltl, num_vars) ⟹ P)\" (*‹(x::nat mltl × nat) = (False⇩m, ?num_vars::nat) ⟹ P::bool›*)\n assume p: \"(⋀p num_vars::nat. x = (Prop_mltl p, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Prop⇩m (?p::nat), ?num_vars::nat) ⟹ P::bool›*)\n assume n1: \"(⋀p num_vars::nat. x = (Not_mltl (Prop_mltl p), num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m Prop⇩m (?p::nat), ?num_vars::nat) ⟹ P::bool›*)\n assume o: \"(⋀φ ψ num_vars. x = (Or_mltl φ ψ, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = ((?φ::nat mltl) Or⇩m (?ψ::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume a: \"(⋀φ ψ num_vars. x = (And_mltl φ ψ, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = ((?φ::nat mltl) And⇩m (?ψ::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume f: \"(⋀φ a b num_vars. x = (Future_mltl φ a b, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (F⇩m [(?φ::nat),(?a::nat)] (?b::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume g: \"(⋀φ a b num_vars. x = (Global_mltl φ a b, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (G⇩m [(?φ::nat),(?a::nat)] (?b::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume u: \"(⋀φ ψ a b num_vars. x = (Until_mltl φ ψ a b, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = ((?φ::nat mltl) U⇩m [(?ψ::nat),(?a::nat)] (?b::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume r: \"(⋀φ ψ a b num_vars. x = (Release_mltl φ ψ a b, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = ((?φ::nat mltl) R⇩m [(?ψ::nat),(?a::nat)] (?b::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume n2: \"(⋀num_vars. x = (Not_mltl True_mltl, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m True⇩m, ?num_vars::nat) ⟹ P::bool›*)\n assume n3: \"(⋀num_vars. x = (Not_mltl False_mltl, num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m False⇩m, ?num_vars::nat) ⟹ P::bool›*)\n assume n4: \"(⋀φ ψ num_vars. x = (Not_mltl (And_mltl φ ψ), num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m ((?φ::nat mltl) And⇩m (?ψ::nat mltl)), ?num_vars::nat) ⟹ P::bool›*)\n assume n5: \"(⋀φ ψ num_vars. x = (Not_mltl (Or_mltl φ ψ), num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m ((?φ::nat mltl) Or⇩m (?ψ::nat mltl)), ?num_vars::nat) ⟹ P::bool›*)\n assume n6: \"(⋀φ a b num_vars. x = (Not_mltl (Future_mltl φ a b), num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m F⇩m [(?φ::nat),(?a::nat)] (?b::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume n7: \"(⋀φ a b num_vars. x = (Not_mltl (Global_mltl φ a b), num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m G⇩m [(?φ::nat),(?a::nat)] (?b::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n assume n8: \"(⋀φ ψ a b num_vars. x = (Not_mltl (Until_mltl φ ψ a b), num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m ((?φ::nat mltl) U⇩m [(?ψ::nat),(?a::nat)] (?b::nat mltl)), ?num_vars::nat) ⟹ P::bool›*)\n assume n9: \"(⋀φ ψ a b num_vars. x = (Not_mltl (Release_mltl φ ψ a b), num_vars) ⟹ P) \" (*‹(x::nat mltl × nat) = (Not⇩m ((?φ::nat mltl) R⇩m [(?ψ::nat),(?a::nat)] (?b::nat mltl)), ?num_vars::nat) ⟹ P::bool›*)\n assume n10: \"(⋀φ num_vars. x = (Not_mltl (Not_mltl φ), num_vars) ⟹ P)\" (*‹(x::nat mltl × nat) = (Not⇩m Not⇩m (?φ::nat mltl), ?num_vars::nat) ⟹ P::bool›*)\n show P\n proof (cases \"fst x\")\n (*goals:\n 1. ‹fst x = True⇩m ⟹ P›\n 2. ‹fst x = False⇩m ⟹ P›\n 3. ‹⋀x3. fst x = Prop⇩m (x3) ⟹ P›\n 4. ‹⋀x4. fst x = Not⇩m x4 ⟹ P›\n 5. ‹⋀x51 x52. fst x = x51 And⇩m x52 ⟹ P›\n 6. ‹⋀x61 x62. fst x = x61 Or⇩m x62 ⟹ P›\n 7. ‹⋀x71 x72 x73. fst x = F⇩m [x71,x72] x73 ⟹ P›\n 8. ‹⋀x81 x82 x83. fst x = G⇩m [x81,x82] x83 ⟹ P›\n 9. ‹⋀x91 x92 x93 x94. fst x = x91 U⇩m [x92,x93] x94 ⟹ P›\n 10. ‹⋀x101 x102 x103 x104. fst x = x101 R⇩m [x102,x103] x104 ⟹ P›*)\n case True_mltl (*‹fst x = True⇩m›*)\n then show \"?thesis\"\n (*goal: ‹P::bool›*)\n using t (*‹x = (True⇩m, ?num_vars) ⟹ P›*) by (metis eq_fst_iff (*‹(?a = fst ?p) = (∃b. ?p = (?a, b))›*))\n next\n (*goals:\n 1. ‹fst x = False⇩m ⟹ P›\n 2. ‹⋀x3. fst x = Prop⇩m (x3) ⟹ P›\n 3. ‹⋀x4. fst x = Not⇩m x4 ⟹ P›\n 4. ‹⋀x51 x52. fst x = x51 And⇩m x52 ⟹ P›\n 5. ‹⋀x61 x62. fst x = x61 Or⇩m x62 ⟹ P›\n 6. ‹⋀x71 x72 x73. fst x = F⇩m [x71,x72] x73 ⟹ P›\n 7. ‹⋀x81 x82 x83. fst x = G⇩m [x81,x82] x83 ⟹ P›\n 8. ‹⋀x91 x92 x93 x94. fst x = x91 U⇩m [x92,x93] x94 ⟹ P›\n 9. ‹⋀x101 x102 x103 x104. fst x = x101 R⇩m [x102,x103] x104 ⟹ P›*)\n case False_mltl (*‹fst x = False⇩m›*)\n then show \"?thesis\"\n (*goal: ‹P::bool›*)\n using fa (*‹x = (False⇩m, ?num_vars) ⟹ P›*) eq_fst_iff (*‹(?a = fst ?p) = (∃b. ?p = (?a, b))›*) by metis\n next\n (*goals:\n 1. ‹⋀x3::nat. fst (x::nat mltl × nat) = Prop⇩m (x3) ⟹ P::bool›\n 2. ‹⋀x4::nat mltl. fst (x::nat mltl × nat) = Not⇩m x4 ⟹ P::bool›\n 3. ‹⋀(x51::nat mltl) x52::nat mltl. fst (x::nat mltl × nat) = x51 And⇩m x52 ⟹ P::bool›\n 4. ‹⋀(x61::nat mltl) x62::nat mltl. fst (x::nat mltl × nat) = x61 Or⇩m x62 ⟹ P::bool›\n 5. ‹⋀(x71::nat) (x72::nat) x73::nat mltl. fst (x::nat mltl × nat) = F⇩m [x71,x72] x73 ⟹ P::bool›\n 6. ‹⋀(x81::nat) (x82::nat) x83::nat mltl. fst (x::nat mltl × nat) = G⇩m [x81,x82] x83 ⟹ P::bool›\n 7. ‹⋀(x91::nat mltl) (x92::nat) (x93::nat) x94::nat mltl. fst (x::nat mltl × nat) = x91 U⇩m [x92,x93] x94 ⟹ P::bool›\n 8. ‹⋀(x101::nat mltl) (x102::nat) (x103::nat) x104::nat mltl. fst (x::nat mltl × nat) = x101 R⇩m [x102,x103] x104 ⟹ P::bool›*)\n case (Prop_mltl p) (*‹fst x = Prop⇩m (p)›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using p (*‹(x::nat mltl × nat) = (Prop⇩m (?p::nat), ?num_vars::nat) ⟹ P::bool›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀x4. fst x = Not⇩m x4 ⟹ P›\n 2. ‹⋀x51 x52. fst x = x51 And⇩m x52 ⟹ P›\n 3. ‹⋀x61 x62. fst x = x61 Or⇩m x62 ⟹ P›\n 4. ‹⋀x71 x72 x73. fst x = F⇩m [x71,x72] x73 ⟹ P›\n 5. ‹⋀x81 x82 x83. fst x = G⇩m [x81,x82] x83 ⟹ P›\n 6. ‹⋀x91 x92 x93 x94. fst x = x91 U⇩m [x92,x93] x94 ⟹ P›\n 7. ‹⋀x101 x102 x103 x104. fst x = x101 R⇩m [x102,x103] x104 ⟹ P›*)\n case (Not_mltl φ) (*‹fst x = Not⇩m φ›*)\n then have fst_x: \"fst x = Not_mltl φ\"\n by auto\n then show \"?thesis\"\n (*goal: ‹P›*)\n proof (cases φ)\n (*goals:\n 1. ‹⟦fst x = Not⇩m φ; φ = True⇩m⟧ ⟹ P›\n 2. ‹⟦fst x = Not⇩m φ; φ = False⇩m⟧ ⟹ P›\n 3. ‹⋀x3. ⟦fst x = Not⇩m φ; φ = Prop⇩m (x3)⟧ ⟹ P›\n 4. ‹⋀x4. ⟦fst x = Not⇩m φ; φ = Not⇩m x4⟧ ⟹ P›\n 5. ‹⋀x51 x52. ⟦fst x = Not⇩m φ; φ = x51 And⇩m x52⟧ ⟹ P›\n 6. ‹⋀x61 x62. ⟦fst x = Not⇩m φ; φ = x61 Or⇩m x62⟧ ⟹ P›\n 7. ‹⋀x71 x72 x73. ⟦fst x = Not⇩m φ; φ = F⇩m [x71,x72] x73⟧ ⟹ P›\n 8. ‹⋀x81 x82 x83. ⟦fst x = Not⇩m φ; φ = G⇩m [x81,x82] x83⟧ ⟹ P›\n 9. ‹⋀x91 x92 x93 x94. ⟦fst x = Not⇩m φ; φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P›\n 10. ‹⋀x101 x102 x103 x104. ⟦fst x = Not⇩m φ; φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P›*)\n case True_mltl (*‹φ = True⇩m›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using n2 (*‹x = (Not⇩m True⇩m, ?num_vars) ⟹ P›*) by (metis Not_mltl (*‹fst x = Not⇩m φ›*) split_pairs (*‹((?A, ?B) = ?X) = (fst ?X = ?A ∧ snd ?X = ?B)›*))\n next\n (*goals:\n 1. ‹⟦fst x = Not⇩m φ; φ = False⇩m⟧ ⟹ P›\n 2. ‹⋀x3. ⟦fst x = Not⇩m φ; φ = Prop⇩m (x3)⟧ ⟹ P›\n 3. ‹⋀x4. ⟦fst x = Not⇩m φ; φ = Not⇩m x4⟧ ⟹ P›\n 4. ‹⋀x51 x52. ⟦fst x = Not⇩m φ; φ = x51 And⇩m x52⟧ ⟹ P›\n 5. ‹⋀x61 x62. ⟦fst x = Not⇩m φ; φ = x61 Or⇩m x62⟧ ⟹ P›\n 6. ‹⋀x71 x72 x73. ⟦fst x = Not⇩m φ; φ = F⇩m [x71,x72] x73⟧ ⟹ P›\n 7. ‹⋀x81 x82 x83. ⟦fst x = Not⇩m φ; φ = G⇩m [x81,x82] x83⟧ ⟹ P›\n 8. ‹⋀x91 x92 x93 x94. ⟦fst x = Not⇩m φ; φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P›\n 9. ‹⋀x101 x102 x103 x104. ⟦fst x = Not⇩m φ; φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P›*)\n case False_mltl (*‹φ = False⇩m›*)\n then show \"?thesis\"\n (*goal: ‹P::bool›*)\n using n3 (*‹x = (Not⇩m False⇩m, ?num_vars) ⟹ P›*) by (metis Not_mltl (*‹fst x = Not⇩m φ›*) prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀x3::nat. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = Prop⇩m (x3)⟧ ⟹ P::bool›\n 2. ‹⋀x4::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = Not⇩m x4⟧ ⟹ P::bool›\n 3. ‹⋀(x51::nat mltl) x52::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x51 And⇩m x52⟧ ⟹ P::bool›\n 4. ‹⋀(x61::nat mltl) x62::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x61 Or⇩m x62⟧ ⟹ P::bool›\n 5. ‹⋀(x71::nat) (x72::nat) x73::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = F⇩m [x71,x72] x73⟧ ⟹ P::bool›\n 6. ‹⋀(x81::nat) (x82::nat) x83::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = G⇩m [x81,x82] x83⟧ ⟹ P::bool›\n 7. ‹⋀(x91::nat mltl) (x92::nat) (x93::nat) x94::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P::bool›\n 8. ‹⋀(x101::nat mltl) (x102::nat) (x103::nat) x104::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P::bool›*)\n case (Prop_mltl p) (*‹φ = Prop⇩m (p)›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using n1 (*‹x = (Not⇩m Prop⇩m (?p), ?num_vars) ⟹ P›*) by (metis Not_mltl (*‹fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl)›*) split_pairs (*‹((?A::?'a, ?B::?'b) = (?X::?'a × ?'b)) = (fst ?X = ?A ∧ snd ?X = ?B)›*))\n next\n (*goals:\n 1. ‹⋀x4. ⟦fst x = Not⇩m φ; φ = Not⇩m x4⟧ ⟹ P›\n 2. ‹⋀x51 x52. ⟦fst x = Not⇩m φ; φ = x51 And⇩m x52⟧ ⟹ P›\n 3. ‹⋀x61 x62. ⟦fst x = Not⇩m φ; φ = x61 Or⇩m x62⟧ ⟹ P›\n 4. ‹⋀x71 x72 x73. ⟦fst x = Not⇩m φ; φ = F⇩m [x71,x72] x73⟧ ⟹ P›\n 5. ‹⋀x81 x82 x83. ⟦fst x = Not⇩m φ; φ = G⇩m [x81,x82] x83⟧ ⟹ P›\n 6. ‹⋀x91 x92 x93 x94. ⟦fst x = Not⇩m φ; φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P›\n 7. ‹⋀x101 x102 x103 x104. ⟦fst x = Not⇩m φ; φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P›*)\n case (Not_mltl φ1) (*‹φ = Not⇩m φ1›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using n10 (*‹x = (Not⇩m Not⇩m ?φ, ?num_vars) ⟹ P›*) fst_x (*‹fst x = Not⇩m φ›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀(x51::nat mltl) x52::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x51 And⇩m x52⟧ ⟹ P::bool›\n 2. ‹⋀(x61::nat mltl) x62::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x61 Or⇩m x62⟧ ⟹ P::bool›\n 3. ‹⋀(x71::nat) (x72::nat) x73::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = F⇩m [x71,x72] x73⟧ ⟹ P::bool›\n 4. ‹⋀(x81::nat) (x82::nat) x83::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = G⇩m [x81,x82] x83⟧ ⟹ P::bool›\n 5. ‹⋀(x91::nat mltl) (x92::nat) (x93::nat) x94::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P::bool›\n 6. ‹⋀(x101::nat mltl) (x102::nat) (x103::nat) x104::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P::bool›*)\n case (And_mltl φ1 φ2) (*‹(φ::nat mltl) = (φ1::nat mltl) And⇩m (φ2::nat mltl)›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n by (metis Not_mltl (*‹fst x = Not⇩m φ›*) n4 (*‹x = (Not⇩m (?φ And⇩m ?ψ), ?num_vars) ⟹ P›*) prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀x61 x62. ⟦fst x = Not⇩m φ; φ = x61 Or⇩m x62⟧ ⟹ P›\n 2. ‹⋀x71 x72 x73. ⟦fst x = Not⇩m φ; φ = F⇩m [x71,x72] x73⟧ ⟹ P›\n 3. ‹⋀x81 x82 x83. ⟦fst x = Not⇩m φ; φ = G⇩m [x81,x82] x83⟧ ⟹ P›\n 4. ‹⋀x91 x92 x93 x94. ⟦fst x = Not⇩m φ; φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P›\n 5. ‹⋀x101 x102 x103 x104. ⟦fst x = Not⇩m φ; φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P›*)\n case (Or_mltl φ1 φ2) (*‹φ = φ1 Or⇩m φ2›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using n5 (*‹x = (Not⇩m (?φ Or⇩m ?ψ), ?num_vars) ⟹ P›*) Not_mltl (*‹fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl)›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀x71 x72 x73. ⟦fst x = Not⇩m φ; φ = F⇩m [x71,x72] x73⟧ ⟹ P›\n 2. ‹⋀x81 x82 x83. ⟦fst x = Not⇩m φ; φ = G⇩m [x81,x82] x83⟧ ⟹ P›\n 3. ‹⋀x91 x92 x93 x94. ⟦fst x = Not⇩m φ; φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P›\n 4. ‹⋀x101 x102 x103 x104. ⟦fst x = Not⇩m φ; φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P›*)\n case (Future_mltl a b φ1) (*‹φ = F⇩m [a,b] φ1›*)\n then show \"?thesis\"\n (*goal: ‹P::bool›*)\n using n6 (*‹x = (Not⇩m F⇩m [?φ,?a] ?b, ?num_vars) ⟹ P›*) Not_mltl (*‹fst x = Not⇩m φ›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀(x81::nat) (x82::nat) x83::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = G⇩m [x81,x82] x83⟧ ⟹ P::bool›\n 2. ‹⋀(x91::nat mltl) (x92::nat) (x93::nat) x94::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P::bool›\n 3. ‹⋀(x101::nat mltl) (x102::nat) (x103::nat) x104::nat mltl. ⟦fst (x::nat mltl × nat) = Not⇩m (φ::nat mltl); φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P::bool›*)\n case (Global_mltl a b φ1) (*‹φ = G⇩m [a,b] φ1›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using n7 (*‹x = (Not⇩m G⇩m [?φ,?a] ?b, ?num_vars) ⟹ P›*) Not_mltl (*‹fst x = Not⇩m φ›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀x91 x92 x93 x94. ⟦fst x = Not⇩m φ; φ = x91 U⇩m [x92,x93] x94⟧ ⟹ P›\n 2. ‹⋀x101 x102 x103 x104. ⟦fst x = Not⇩m φ; φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P›*)\n case (Until_mltl φ1 a b φ2) (*‹φ = φ1 U⇩m [a,b] φ2›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using n8 (*‹x = (Not⇩m (?φ U⇩m [?ψ,?a] ?b), ?num_vars) ⟹ P›*) Not_mltl (*‹fst x = Not⇩m φ›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goal: ‹⋀x101 x102 x103 x104. ⟦fst x = Not⇩m φ; φ = x101 R⇩m [x102,x103] x104⟧ ⟹ P›*)\n case (Release_mltl φ1 a b φ2) (*‹φ = φ1 R⇩m [a,b] φ2›*)\n then show \"?thesis\"\n (*goal: ‹P::bool›*)\n using n9 (*‹x = (Not⇩m (?φ R⇩m [?ψ,?a] ?b), ?num_vars) ⟹ P›*) Not_mltl (*‹fst x = Not⇩m φ›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n qed\n next\n (*goals:\n 1. ‹⋀x51 x52. fst x = x51 And⇩m x52 ⟹ P›\n 2. ‹⋀x61 x62. fst x = x61 Or⇩m x62 ⟹ P›\n 3. ‹⋀x71 x72 x73. fst x = F⇩m [x71,x72] x73 ⟹ P›\n 4. ‹⋀x81 x82 x83. fst x = G⇩m [x81,x82] x83 ⟹ P›\n 5. ‹⋀x91 x92 x93 x94. fst x = x91 U⇩m [x92,x93] x94 ⟹ P›\n 6. ‹⋀x101 x102 x103 x104. fst x = x101 R⇩m [x102,x103] x104 ⟹ P›*)\n case (And_mltl φ1 φ2) (*‹fst x = φ1 And⇩m φ2›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using a (*‹x = (?φ And⇩m ?ψ, ?num_vars) ⟹ P›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀x61 x62. fst x = x61 Or⇩m x62 ⟹ P›\n 2. ‹⋀x71 x72 x73. fst x = F⇩m [x71,x72] x73 ⟹ P›\n 3. ‹⋀x81 x82 x83. fst x = G⇩m [x81,x82] x83 ⟹ P›\n 4. ‹⋀x91 x92 x93 x94. fst x = x91 U⇩m [x92,x93] x94 ⟹ P›\n 5. ‹⋀x101 x102 x103 x104. fst x = x101 R⇩m [x102,x103] x104 ⟹ P›*)\n case (Or_mltl φ1 φ2) (*‹fst x = φ1 Or⇩m φ2›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using o (*‹x = (?φ Or⇩m ?ψ, ?num_vars) ⟹ P›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀x71 x72 x73. fst x = F⇩m [x71,x72] x73 ⟹ P›\n 2. ‹⋀x81 x82 x83. fst x = G⇩m [x81,x82] x83 ⟹ P›\n 3. ‹⋀x91 x92 x93 x94. fst x = x91 U⇩m [x92,x93] x94 ⟹ P›\n 4. ‹⋀x101 x102 x103 x104. fst x = x101 R⇩m [x102,x103] x104 ⟹ P›*)\n case (Future_mltl a b φ1) (*‹fst x = F⇩m [a,b] φ1›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using f (*‹(x::nat mltl × nat) = (F⇩m [(?φ::nat),(?a::nat)] (?b::nat mltl), ?num_vars::nat) ⟹ P::bool›*) by (metis split_pairs (*‹((?A::?'a, ?B::?'b) = (?X::?'a × ?'b)) = (fst ?X = ?A ∧ snd ?X = ?B)›*))\n next\n (*goals:\n 1. ‹⋀(x81::nat) (x82::nat) x83::nat mltl. fst (x::nat mltl × nat) = G⇩m [x81,x82] x83 ⟹ P::bool›\n 2. ‹⋀(x91::nat mltl) (x92::nat) (x93::nat) x94::nat mltl. fst (x::nat mltl × nat) = x91 U⇩m [x92,x93] x94 ⟹ P::bool›\n 3. ‹⋀(x101::nat mltl) (x102::nat) (x103::nat) x104::nat mltl. fst (x::nat mltl × nat) = x101 R⇩m [x102,x103] x104 ⟹ P::bool›*)\n case (Global_mltl a b φ1) (*‹fst (x::nat mltl × nat) = G⇩m [(a::nat),(b::nat)] (φ1::nat mltl)›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using g (*‹x = (G⇩m [?φ,?a] ?b, ?num_vars) ⟹ P›*) by (metis prod.collapse (*‹(fst ?prod, snd ?prod) = ?prod›*))\n next\n (*goals:\n 1. ‹⋀(x91::nat mltl) (x92::nat) (x93::nat) x94::nat mltl. fst (x::nat mltl × nat) = x91 U⇩m [x92,x93] x94 ⟹ P::bool›\n 2. ‹⋀(x101::nat mltl) (x102::nat) (x103::nat) x104::nat mltl. fst (x::nat mltl × nat) = x101 R⇩m [x102,x103] x104 ⟹ P::bool›*)\n case (Until_mltl φ1 a b φ2) (*‹fst x = φ1 U⇩m [a,b] φ2›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using u (*‹x = (?φ U⇩m [?ψ,?a] ?b, ?num_vars) ⟹ P›*) by (metis split_pairs (*‹((?A, ?B) = ?X) = (fst ?X = ?A ∧ snd ?X = ?B)›*))\n next\n (*goal: ‹⋀x101 x102 x103 x104. fst x = x101 R⇩m [x102,x103] x104 ⟹ P›*)\n case (Release_mltl φ1 a b φ2) (*‹fst x = φ1 R⇩m [a,b] φ2›*)\n then show \"?thesis\"\n (*goal: ‹P›*)\n using r (*‹x = (?φ R⇩m [?ψ,?a] ?b, ?num_vars) ⟹ P›*) by (metis split_pairs (*‹((?A, ?B) = ?X) = (fst ?X = ?A ∧ snd ?X = ?B)›*))\n qed\nqed\n\n\nfun WEST_termination_measure:: \"(nat) mltl ⇒ nat\"\n where \"WEST_termination_measure True⇩m = 1\"\n | \"WEST_termination_measure (Not⇩m True⇩m) = 4\"\n | \"WEST_termination_measure False⇩m = 1\"\n | \"WEST_termination_measure (Not⇩m False⇩m) = 4\"\n | \"WEST_termination_measure (Prop⇩m (p)) = 1\"\n | \"WEST_termination_measure (Not⇩m (Prop⇩m (p))) = 4\"\n | \"WEST_termination_measure (φ Or⇩m ψ) = 1 + (WEST_termination_measure φ) + (WEST_termination_measure ψ)\"\n | \"WEST_termination_measure (φ And⇩m ψ) = 1 + (WEST_termination_measure φ) + (WEST_termination_measure ψ)\"\n | \"WEST_termination_measure (F⇩m [a,b] φ) = 1 + (WEST_termination_measure φ)\"\n | \"WEST_termination_measure (G⇩m [a,b] φ) = 1 + (WEST_termination_measure φ)\"\n | \"WEST_termination_measure (φ U⇩m[a,b] ψ) = 1 + (WEST_termination_measure φ) + (WEST_termination_measure ψ)\"\n | \"WEST_termination_measure (φ R⇩m[a,b] ψ) = 1 + (WEST_termination_measure φ) + (WEST_termination_measure ψ)\"\n | \"WEST_termination_measure (Not⇩m (φ Or⇩m ψ)) = 1 + 3 * (WEST_termination_measure (φ Or⇩m ψ))\"\n | \"WEST_termination_measure (Not⇩m (φ And⇩m ψ)) = 1 + 3 * (WEST_termination_measure (φ And⇩m ψ))\"\n | \"WEST_termination_measure (Not⇩m (F⇩m[a,b] φ)) = 1 + 3 * (WEST_termination_measure (F⇩m[a,b] φ))\"\n | \"WEST_termination_measure (Not⇩m (G⇩m[a,b] φ)) = 1 + 3 * (WEST_termination_measure (G⇩m[a,b] φ))\"\n | \"WEST_termination_measure (Not⇩m (φ U⇩m[a,b] ψ)) = 1 + 3 * (WEST_termination_measure (φ U⇩m[a,b] ψ))\"\n | \"WEST_termination_measure (Not⇩m (φ R⇩m[a,b] ψ)) = 1 + 3 * (WEST_termination_measure (φ R⇩m[a,b] ψ))\"\n | \"WEST_termination_measure (Not⇩m (Not⇩m φ)) = 1 + 3 * (WEST_termination_measure (Not⇩m φ))\"\n\nlemma WEST_termination_measure_not:\n fixes φ::\"(nat) mltl\"\n shows \"WEST_termination_measure (Not_mltl φ) = 1 + 3 * (WEST_termination_measure φ)\"\n apply (induction φ)\n (*goals:\n 1. ‹WEST_termination_measure (Not⇩m True⇩m) = (1::nat) + (3::nat) * WEST_termination_measure True⇩m›\n 2. ‹WEST_termination_measure (Not⇩m False⇩m) = (1::nat) + (3::nat) * WEST_termination_measure False⇩m›\n 3. ‹⋀x::nat. WEST_termination_measure (Not⇩m Prop⇩m (x)) = (1::nat) + (3::nat) * WEST_termination_measure Prop⇩m (x)›\n 4. ‹⋀φ::nat mltl. WEST_termination_measure (Not⇩m φ) = (1::nat) + (3::nat) * WEST_termination_measure φ ⟹ WEST_termination_measure (Not⇩m Not⇩m φ) = (1::nat) + (3::nat) * WEST_termination_measure (Not⇩m φ)›\n 5. ‹⋀(φ1::nat mltl) φ2::nat mltl. ⟦WEST_termination_measure (Not⇩m φ1) = (1::nat) + (3::nat) * WEST_termination_measure φ1; WEST_termination_measure (Not⇩m φ2) = (1::nat) + (3::nat) * WEST_termination_measure φ2⟧ ⟹ WEST_termination_measure (Not⇩m (φ1 And⇩m φ2)) = (1::nat) + (3::nat) * WEST_termination_measure (φ1 And⇩m φ2)›\n 6. ‹⋀(φ1::nat mltl) φ2::nat mltl. ⟦WEST_termination_measure (Not⇩m φ1) = (1::nat) + (3::nat) * WEST_termination_measure φ1; WEST_termination_measure (Not⇩m φ2) = (1::nat) + (3::nat) * WEST_termination_measure φ2⟧ ⟹ WEST_termination_measure (Not⇩m (φ1 Or⇩m φ2)) = (1::nat) + (3::nat) * WEST_termination_measure (φ1 Or⇩m φ2)›\n 7. ‹⋀(x1::nat) (x2::nat) φ::nat mltl. WEST_termination_measure (Not⇩m φ) = (1::nat) + (3::nat) * WEST_termination_measure φ ⟹ WEST_termination_measure (Not⇩m F⇩m [x1,x2] φ) = (1::nat) + (3::nat) * WEST_termination_measure (F⇩m [x1,x2] φ)›\n 8. ‹⋀(x1::nat) (x2::nat) φ::nat mltl. WEST_termination_measure (Not⇩m φ) = (1::nat) + (3::nat) * WEST_termination_measure φ ⟹ WEST_termination_measure (Not⇩m G⇩m [x1,x2] φ) = (1::nat) + (3::nat) * WEST_termination_measure (G⇩m [x1,x2] φ)›\n 9. ‹⋀(φ1::nat mltl) (x2::nat) (x3a::nat) φ2::nat mltl. ⟦WEST_termination_measure (Not⇩m φ1) = (1::nat) + (3::nat) * WEST_termination_measure φ1; WEST_termination_measure (Not⇩m φ2) = (1::nat) + (3::nat) * WEST_termination_measure φ2⟧ ⟹ WEST_termination_measure (Not⇩m (φ1 U⇩m [x2,x3a] φ2)) = (1::nat) + (3::nat) * WEST_termination_measure (φ1 U⇩m [x2,x3a] φ2)›\n 10. ‹⋀(φ1::nat mltl) (x2::nat) (x3a::nat) φ2::nat mltl. ⟦WEST_termination_measure (Not⇩m φ1) = (1::nat) + (3::nat) * WEST_termination_measure φ1; WEST_termination_measure (Not⇩m φ2) = (1::nat) + (3::nat) * WEST_termination_measure φ2⟧ ⟹ WEST_termination_measure (Not⇩m (φ1 R⇩m [x2,x3a] φ2)) = (1::nat) + (3::nat) * WEST_termination_measure (φ1 R⇩m [x2,x3a] φ2)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*discuss goal 8*)\n apply simp\n (*discuss goal 9*)\n apply simp\n (*discuss goal 10*)\n apply simp\n (*proven 10 subgoals*) .\n\n\nfunction WEST_reg_aux:: \"(nat) mltl ⇒ nat ⇒ WEST_regex\"\n where \"WEST_reg_aux True⇩m num_vars = [[(map (λ j. S) [0 ..< num_vars])]]\"\n | \"WEST_reg_aux False⇩m num_vars = []\"\n | \"WEST_reg_aux (Prop⇩m (p)) num_vars = [[(map (λ j. (if (p=j) then One else S)) [0 ..< num_vars])]]\"\n | \"WEST_reg_aux (Not⇩m (Prop⇩m (p))) num_vars = [[(map (λ j. (if (p=j) then Zero else S)) [0 ..< num_vars])]]\"\n | \"WEST_reg_aux (φ Or⇩m ψ) num_vars = WEST_or_simp (WEST_reg_aux φ num_vars) (WEST_reg_aux ψ num_vars) num_vars\"\n | \"WEST_reg_aux (φ And⇩m ψ) num_vars = (WEST_and_simp (WEST_reg_aux φ num_vars) (WEST_reg_aux ψ num_vars) num_vars)\"\n | \"WEST_reg_aux (F⇩m[a,b] φ) num_vars = (WEST_future (WEST_reg_aux φ num_vars) a b num_vars)\"\n | \"WEST_reg_aux (G⇩m[a,b] φ) num_vars = (WEST_global (WEST_reg_aux φ num_vars) a b num_vars)\"\n | \"WEST_reg_aux (φ U⇩m[a,b] ψ) num_vars = (WEST_until (WEST_reg_aux φ num_vars) (WEST_reg_aux ψ num_vars) a b num_vars)\"\n | \"WEST_reg_aux (φ R⇩m[a,b] ψ) num_vars = WEST_release (WEST_reg_aux φ num_vars) (WEST_reg_aux ψ num_vars) a b num_vars\"\n | \"WEST_reg_aux (Not⇩m True⇩m) num_vars = WEST_reg_aux False⇩m num_vars\"\n | \"WEST_reg_aux (Not⇩m False⇩m) num_vars = WEST_reg_aux True⇩m num_vars\"\n | \"WEST_reg_aux (Not⇩m (φ And⇩m ψ)) num_vars = WEST_reg_aux ((Not⇩m φ) Or⇩m (Not⇩m ψ)) num_vars\"\n | \"WEST_reg_aux (Not⇩m (φ Or⇩m ψ)) num_vars = WEST_reg_aux ((Not⇩m φ) And⇩m (Not⇩m ψ)) num_vars\"\n | \"WEST_reg_aux (Not⇩m (F⇩m[a,b] φ)) num_vars = WEST_reg_aux (G⇩m[a,b] (Not⇩m φ)) num_vars\"\n | \"WEST_reg_aux (Not⇩m (G⇩m[a,b] φ)) num_vars = WEST_reg_aux (F⇩m[a,b] (Not⇩m φ)) num_vars\"\n | \"WEST_reg_aux (Not⇩m (φ U⇩m[a,b] ψ)) num_vars = WEST_reg_aux ((Not⇩m φ) R⇩m[a,b] (Not⇩m ψ)) num_vars\"\n | \"WEST_reg_aux (Not⇩m (φ R⇩m[a,b] ψ)) num_vars = WEST_reg_aux ((Not⇩m φ) U⇩m[a,b] (Not⇩m ψ)) num_vars\"\n | \"WEST_reg_aux (Not⇩m (Not⇩m φ)) num_vars = WEST_reg_aux φ num_vars\"\n using exhaustive convert_nnf.cases using exhaustive apply (smt (z3)) (* May take a second to load *)\n defer apply blast apply simp_all .\ntermination\n apply (relation \"measure (λ(F,num_vars). (WEST_termination_measure F))\")\n using WEST_termination_measure_not by simp_all\n\n\nfun WEST_num_vars:: \"(nat) mltl ⇒ nat\"\n where \"WEST_num_vars True⇩m = 1\"\n | \"WEST_num_vars False⇩m = 1\"\n | \"WEST_num_vars (Prop⇩m (p)) = p+1\"\n | \"WEST_num_vars (Not⇩m φ) = WEST_num_vars φ\"\n | \"WEST_num_vars (φ And⇩m ψ) = Max {(WEST_num_vars φ), (WEST_num_vars ψ)}\"\n | \"WEST_num_vars (φ Or⇩m ψ) = Max {(WEST_num_vars φ), (WEST_num_vars ψ)}\"\n | \"WEST_num_vars (F⇩m[a,b] φ) = WEST_num_vars φ\"\n | \"WEST_num_vars (G⇩m[a,b] φ) = WEST_num_vars φ\"\n | \"WEST_num_vars (φ U⇩m[a,b] ψ) = Max {(WEST_num_vars φ), (WEST_num_vars ψ)}\"\n | \"WEST_num_vars (φ R⇩m[a,b] ψ) = Max {(WEST_num_vars φ), (WEST_num_vars ψ)}\"\n\n\nfun WEST_reg:: \"(nat) mltl ⇒ WEST_regex\"\n where \"WEST_reg F = (let nnf_F = convert_nnf F in WEST_reg_aux nnf_F (WEST_num_vars F))\"\n\nsubsubsection ‹ Adding padding ›\nfun pad_WEST_reg:: \"nat mltl ⇒ WEST_regex\"\n where \"pad_WEST_reg φ = (let unpadded = WEST_reg φ in\n (let complen = complen_mltl φ in\n (let num_vars = WEST_num_vars φ in\n (map (λ L. (if (length L < complen)then (pad L num_vars (complen-(length L))) else L))) unpadded)))\"\n\nfun simp_pad_WEST_reg:: \"nat mltl ⇒ WEST_regex\"\n where \"simp_pad_WEST_reg φ = WEST_simp (pad_WEST_reg φ) (WEST_num_vars φ)\"\n\n\nsection ‹ Some examples and Code Export ›\n\ntext ‹ Base cases ›\nvalue \"WEST_reg True⇩m\"\nvalue \"WEST_reg False⇩m\"\nvalue \"WEST_reg (Prop⇩m (1))\"\nvalue \"WEST_reg (Not⇩m (Prop⇩m (0)))\"\n\ntext ‹ Test cases for recursion ›\nvalue \"WEST_reg ((Not⇩m (Prop⇩m (0))) And⇩m (Prop⇩m (1)))\"\nvalue \"WEST_reg (F⇩m[0,2] (Prop⇩m (1)))\"\nvalue \"WEST_reg ((Not⇩m (Prop⇩m (0))) Or⇩m (Prop⇩m (0)))\"\n\nvalue \"pad_WEST_reg ((Prop⇩m (0)) U⇩m[0,2] (Prop⇩m (0)))\"\nvalue \"simp_pad_WEST_reg ((Prop_mltl 0) U⇩m[0,2] (Prop_mltl 0))\"\n\nexport_code WEST_reg in Haskell module_name WEST\nexport_code simp_pad_WEST_reg in Haskell module_name WEST_simp_pad\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Mission_Time_LTL_to_Regular_Expression/WEST_Algorithms.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"bc00eb032d5931517ae0922b03d5ce3fef1593da049f1a4817d803e079a38bcc\"\n}"}}},{"rowIdx":171,"cells":{"text":{"kind":"string","value":"(* Title: thys/Rec_Ex.thy\n Author: Franz Regensburger (FABR) 03/2022\n *)\n\nsection ‹Examples for Recursive Functions based on Rec\\_def›\n\ntheory Rec_Ex\n imports Rec_Def\nbegin\n\n\ndefinition plus_2 :: \"recf\"\n where\n \"plus_2 = (Cn 1 s [s])\"\n\nlemma \"rec_exec plus_2 [0] = Suc (Suc 0)\"\n unfolding plus_2_def\n (*goal: ‹rec_exec (Cn 1 s [s]) [0] = Suc (Suc 0)›*)\n proof (-)\n (*goal: ‹rec_exec (Cn 1 s [s]) [0] = Suc (Suc 0)›*)\n have \"rec_exec (Cn 1 s [s]) [0] = rec_exec s (map (λ a. rec_exec a [0]) [s])\"\n by auto\n also (*calculation: ‹rec_exec (Cn 1 s [s]) [0] = rec_exec s (map (λa. rec_exec a [0]) [s])›*) have \"... = rec_exec s [Suc 0]\"\n by auto\n also (*calculation: ‹rec_exec (Cn (1::nat) s [s]) [0::nat] = rec_exec s [Suc (0::nat)]›*) have \"... = Suc (Suc 0)\"\n by auto\n finally (*calculation: ‹rec_exec (Cn 1 s [s]) [0] = Suc (Suc 0)›*) show \"rec_exec (Cn 1 s [s]) [0] = Suc (Suc 0)\" .\n qed\n\nlemma \"rec_exec plus_2 [2] = 4\"\n unfolding plus_2_def\n (*goal: ‹rec_exec (Cn 1 s [s]) [2] = 4›*)\n by auto\n\nlemma \"rec_exec plus_2 [0] = 2\"\n unfolding plus_2_def\n (*goal: ‹rec_exec (Cn 1 s [s]) [0] = 2›*)\n by auto\n\ntext ‹The arity parameter given to the constructors of recursive functions\nis not checked during execution by the interpreter.\n\nSee the next example where we\nrun @{term \"pls_2\"} with two arguments instead of only one.\n›\n\nlemma \"rec_exec plus_2 [2,3] = 4\"\n unfolding plus_2_def\n (*goal: ‹rec_exec (Cn 1 s [s]) [2, 3] = 4›*)\n by auto\n\nlemma \"rec_exec plus_2 [2,3] = 4\"\n unfolding plus_2_def\n (*goal: ‹rec_exec (Cn 1 s [s]) [2, 3] = 4›*)\n by auto\n\ntext ‹What is the purpose of the arity parameter?\n\n The argument 1 of the constructors, which is supposed to be the arity,\n is completely ignored by @{term \"rec_exec\"}.\n However, for proofing termination, we need a correct arity specification.\n›\n\nlemma \"terminate plus_2 [2]\"\n unfolding plus_2_def\n (*goal: ‹terminate (Cn 1 s [s]) [2]›*)\n proof (rule Rec_Def.terminate.termi_cn (*‹⟦terminate ?f (map (λg. rec_exec g ?xs) ?gs); ∀g∈set ?gs. terminate g ?xs; length ?xs = ?n⟧ ⟹ terminate (Cn ?n ?f ?gs) ?xs›*))\n (*goals:\n 1. ‹terminate s (map (λg. rec_exec g [2]) [s])›\n 2. ‹∀g∈set [s]. terminate g [2]›\n 3. ‹length [2] = 1›*)\n show \"terminate s (map (λg. rec_exec g [2]) [s])\"\n proof (-)\n (*goal: ‹terminate s (map (λg. rec_exec g [2]) [s])›*)\n have \"(map (λg. rec_exec g [2]) [s]) = [rec_exec s [2]]\"\n by auto\n then show \"?thesis\"\n (*goal: ‹terminate s (map (λg. rec_exec g [2]) [s])›*)\n using termi_s (*‹terminate s [?n]›*) by auto\n qed\n next\n (*goals:\n 1. ‹∀g∈set [s]. terminate g [2]›\n 2. ‹length [2] = 1›*)\n show \"∀g∈set [s]. terminate g [2]\"\n by (auto simp add: termi_s (*‹terminate s [?n]›*))\n next\n (*goal: ‹length [2] = 1›*)\n show \"length [2] = 1\"\n by auto\n qed\n\nlemma \"terminate plus_2 [2]\"\n unfolding plus_2_def\n (*goal: ‹terminate (Cn 1 s [s]) [2]›*)\n apply (rule Rec_Def.terminate.termi_cn (*‹⟦terminate ?f (map (λg. rec_exec g ?xs) ?gs); ∀g∈set ?gs. terminate g ?xs; length ?xs = ?n⟧ ⟹ terminate (Cn ?n ?f ?gs) ?xs›*))\n (*goals:\n 1. ‹terminate s (map (λg. rec_exec g [2]) [s])›\n 2. ‹∀g∈set [s]. terminate g [2]›\n 3. ‹length [2] = 1›\n discuss goal 1*)\n apply ((auto simp add: termi_s (*‹terminate s [?n]›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: termi_s (*‹terminate s [?n]›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: termi_s (*‹terminate s [?n]›*))[1])\n (*proven 3 subgoals*) .\n\ntext ‹If we try to proof termination of a run with superfluous arguments, we are stuck.\nWe need the correct arity for proving the predicate termination.›\n\nlemma \"terminate plus_2 [2,3]\"\n unfolding plus_2_def\nproof (rule Rec_Def.terminate.termi_cn)\n show \"terminate s (map (λg. rec_exec g [2, 3]) [s])\"\n by (auto simp add: termi_s)\nnext\n show \"∀g∈set [s]. terminate g [2, 3]\"\n proof\n fix g\n assume \"g ∈ set [s]\"\n then show \"terminate g [2, 3]\"\n proof -\n have \"terminate s [2, 3]\"\n (* this, we cannot prove based on the inductive definition of s *)\n oops\n\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Universal_Turing_Machine/Rec_Ex.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"3c144c4459806b0aac3a4a44eedb1b0a749b93f8143dc042c67cc9251bccc3e8\"\n}"}}},{"rowIdx":172,"cells":{"text":{"kind":"string","value":"(*******************************************************************************\n\n Project: Development of Security Protocols by Refinement\n\n Module: Key_establish/m3_kerberos4.thy (Isabelle/HOL 2016-1)\n ID: $Id: m3_kerberos4.thy 132890 2016-12-24 10:25:57Z csprenge $\n Authors Ivano Somaini, ETH Zurich <somainii@student.ethz.ch>\n Christoph Sprenger, ETH Zurich <sprenger@inf.ethz.ch>\n\n Key distribution protocols\n Third refinement: core Kerberos IV\n\n Copyright (c) 2009-2016 Ivano Somaini, Christoph Sprenger\n Licence: LGPL\n\n*******************************************************************************)\n\nsection ‹Core Kerberos 4 (L3)›\n\ntheory m3_kerberos4 imports m2_kerberos \"../Refinement/Message\"\nbegin\n\ntext ‹\nWe model the core Kerberos 4 protocol:\n\\[\n\\begin{array}{lll}\n \\mathrm{M1.} & A \\rightarrow S: & A, B \\\\ \n \\mathrm{M2.} & S \\rightarrow A: & \\{Kab, B, Ts, Na, \\{Kab, A, Ts\\}_{Kbs} \\}_{Kas} \\\\\n \\mathrm{M3.} & A \\rightarrow B: & \\{A, Ta\\}_{Kab}, \\{Kab, A, Ts\\}_{Kbs} \\\\\n \\mathrm{M4.} & B \\rightarrow A: & \\{Ta\\}_{Kab} \\\\\n\\end{array}\n\\]\n›\n\ntext ‹Proof tool configuration. Avoid annoying automatic unfolding of\n‹dom›.›\n\ndeclare domIff [simp, iff del]\n\n\n(******************************************************************************)\nsubsection ‹Setup›\n(******************************************************************************)\n\ntext ‹Now we can define the initial key knowledge.›\n\noverloading ltkeySetup' ≡ ltkeySetup begin\ndefinition ltkeySetup_def: \"ltkeySetup' ≡ {(sharK C, A) | C A. A = C ∨ A = Sv}\"\nend\n\nlemma corrKey_shrK_bad [simp]: \"corrKey = shrK`bad\"\nby (auto simp add: keySetup_def (*‹keySetup ≡ {(ltK K, A) |K A. (K, A) ∈ ltkeySetup}›*) ltkeySetup_def (*‹ltkeySetup ≡ {(sharK C, A) |C A. A = C ∨ A = Sv}›*) corrKey_def (*‹corrKey ≡ keySetup¯ `` bad›*))\n\n\n(******************************************************************************)\nsubsection ‹State›\n(******************************************************************************)\n\ntext ‹The secure channels are star-shaped to/from the server. Therefore,\nwe have only one agent in the relation.›\n\nrecord m3_state = \"m1_state\" +\n IK :: \"msg set\" ― ‹intruder knowledge›\n\n\ntext ‹Observable state:\n@{term \"runs\"}, @{term \"clk\"}, and @{term \"cache\"}.›\n\ntype_synonym\n m3_obs = \"m2_obs\"\n\ndefinition\n m3_obs :: \"m3_state ⇒ m3_obs\" where\n \"m3_obs s ≡ ⦇ runs = runs s, leak = leak s, clk = clk s, cache = cache s ⦈\"\n\ntype_synonym\n m3_pred = \"m3_state set\"\n\ntype_synonym\n m3_trans = \"(m3_state × m3_state) set\"\n\n\n(******************************************************************************)\nsubsection ‹Events›\n(******************************************************************************)\n\ntext ‹Protocol events.›\n\ndefinition ― ‹by @{term \"A\"}, refines @{term \"m2_step1\"}›\n m3_step1 :: \"[rid_t, agent, agent, nonce] ⇒ m3_trans\"\nwhere\n \"m3_step1 Ra A B Na ≡ {(s, s1).\n ― ‹guards:›\n Ra ∉ dom (runs s) ∧ ― ‹‹Ra› is fresh›\n Na = Ra$na ∧ ― ‹generated nonce›\n\n ― ‹actions:›\n s1 = s⦇\n runs := (runs s)(Ra ↦ (Init, [A, B], [])),\n IK := insert ⦃Agent A, Agent B, Nonce Na⦄ (IK s) ― ‹send ‹M1››\n ⦈\n }\"\n\ndefinition ― ‹by @{term \"B\"}, refines @{term \"m2_step2\"}›\n m3_step2 :: \"[rid_t, agent, agent] ⇒ m3_trans\"\nwhere\n \"m3_step2 ≡ m1_step2\"\n\ndefinition ― ‹by @{text \"Server\"}, refines @{term m2_step3}›\n m3_step3 :: \"[rid_t, agent, agent, key, nonce, time] ⇒ m3_trans\"\nwhere\n \"m3_step3 Rs A B Kab Na Ts ≡ {(s, s1).\n ― ‹guards:›\n Rs ∉ dom (runs s) ∧ ― ‹fresh server run›\n Kab = sesK (Rs$sk) ∧ ― ‹fresh session key›\n\n ⦃Agent A, Agent B, Nonce Na⦄ ∈ IK s ∧ ― ‹recv ‹M1››\n Ts = clk s ∧ ― ‹fresh timestamp›\n\n ― ‹actions:›\n ― ‹record session key and send ‹M2››\n s1 = s⦇\n runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum Ts])),\n IK := insert (Crypt (shrK A) ― ‹send ‹M2››\n ⦃Key Kab, Agent B, Number Ts, Nonce Na,\n Crypt (shrK B) ⦃Key Kab, Agent A, Number Ts⦄⦄)\n (IK s)\n ⦈\n }\"\n\ndefinition ― ‹by @{term \"A\"}, refines @{term m2_step4}›\n m3_step4 :: \"[rid_t, agent, agent, nonce, key, time, time, msg] ⇒ m3_trans\"\nwhere\n \"m3_step4 Ra A B Na Kab Ts Ta X ≡ {(s, s1).\n\n ― ‹guards:›\n runs s Ra = Some (Init, [A, B], []) ∧ ― ‹key not yet recv'd›\n Na = Ra$na ∧ ― ‹generated nonce›\n\n Crypt (shrK A) ― ‹recv ‹M2››\n ⦃Key Kab, Agent B, Number Ts, Nonce Na, X⦄ ∈ IK s ∧\n\n ― ‹read current time›\n Ta = clk s ∧\n\n ― ‹check freshness of session key›\n clk s < Ts + Ls ∧\n\n ― ‹actions:›\n ― ‹record session key and send ‹M3››\n s1 = s⦇\n runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum Ta])),\n IK := insert ⦃Crypt Kab ⦃Agent A, Number Ta⦄, X⦄ (IK s) ― ‹‹M3››\n ⦈\n }\"\n\ndefinition ― ‹by @{term \"B\"}, refines @{term m2_step5}›\n m3_step5 :: \"[rid_t, agent, agent, key, time, time] ⇒ m3_trans\"\nwhere\n \"m3_step5 Rb A B Kab Ts Ta ≡ {(s, s1).\n ― ‹guards:›\n runs s Rb = Some (Resp, [A, B], []) ∧ ― ‹key not yet recv'd›\n\n ⦃Crypt Kab ⦃Agent A, Number Ta⦄, ― ‹recv ‹M3››\n Crypt (shrK B) ⦃Key Kab, Agent A, Number Ts⦄⦄ ∈ IK s ∧\n\n ― ‹ensure freshness of session key›\n clk s < Ts + Ls ∧\n\n ― ‹check authenticator's validity and replay; 'replays' with fresh authenticator ok!›\n clk s < Ta + La ∧\n (B, Kab, Ta) ∉ cache s ∧\n\n ― ‹actions:›\n ― ‹record session key›\n s1 = s⦇\n runs := (runs s)(Rb ↦ (Resp, [A, B], [aKey Kab, aNum Ts, aNum Ta])),\n cache := insert (B, Kab, Ta) (cache s),\n IK := insert (Crypt Kab (Number Ta)) (IK s) ― ‹send ‹M4››\n ⦈\n }\"\n\ndefinition ― ‹by @{term \"A\"}, refines @{term m2_step6}›\n m3_step6 :: \"[rid_t, agent, agent, nonce, key, time, time] ⇒ m3_trans\"\nwhere\n \"m3_step6 Ra A B Na Kab Ts Ta ≡ {(s, s').\n ― ‹guards:›\n runs s Ra = Some (Init, [A, B], [aKey Kab, aNum Ts, aNum Ta]) ∧ ― ‹knows key›\n Na = Ra$na ∧ ― ‹generated nonce›\n clk s < Ts + Ls ∧ ― ‹check session key's recentness›\n\n Crypt Kab (Number Ta) ∈ IK s ∧ ― ‹recv ‹M4››\n\n ― ‹actions:›\n s' = s⦇\n runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum Ta, END]))\n ⦈\n }\"\n\ntext ‹Clock tick event›\n\ndefinition ― ‹refines @{term \"m2_tick\"}›\n m3_tick :: \"time ⇒ m3_trans\"\nwhere\n \"m3_tick ≡ m1_tick\"\n\n\ntext ‹Purge event: purge cache of expired timestamps›\n\ndefinition ― ‹refines @{term \"m2_purge\"}›\n m3_purge :: \"agent ⇒ m3_trans\"\nwhere\n \"m3_purge ≡ m1_purge\"\n\n\ntext ‹Session key compromise.›\n\ndefinition ― ‹refines @{term m2_leak}›\n m3_leak :: \"[rid_t, agent, agent, nonce, time] ⇒ m3_trans\"\nwhere\n \"m3_leak Rs A B Na Ts ≡ {(s, s1).\n ― ‹guards:›\n runs s Rs = Some (Serv, [A, B], [aNon Na, aNum Ts]) ∧\n (clk s ≥ Ts + Ls) ∧ ― ‹only compromise 'old' session keys!›\n\n ― ‹actions:›\n ― ‹record session key as leaked and add it to intruder knowledge›\n s1 = s⦇ leak := insert (sesK (Rs$sk), A, B, Na, Ts) (leak s),\n IK := insert (Key (sesK (Rs$sk))) (IK s) ⦈\n }\"\n\n\ntext ‹Intruder fake event. The following \"Dolev-Yao\" event generates all\nintruder-derivable messages.›\n\ndefinition ― ‹refines @{term \"m2_fake\"}›\n m3_DY_fake :: \"m3_trans\"\nwhere\n \"m3_DY_fake ≡ {(s, s1).\n\n ― ‹actions:›\n s1 = s⦇ IK := synth (analz (IK s)) ⦈ ― ‹take DY closure›\n }\"\n\n\n(******************************************************************************)\nsubsection ‹Transition system›\n(******************************************************************************)\n\ndefinition\n m3_init :: \"m3_pred\"\nwhere\n \"m3_init ≡ { ⦇\n runs = Map.empty,\n leak = shrK`bad × {undefined},\n clk = 0,\n cache = {},\n IK = Key`shrK`bad\n ⦈ }\"\n\ndefinition\n m3_trans :: \"m3_trans\" where\n \"m3_trans ≡ (⋃A B Ra Rb Rs Na Kab Ts Ta T X.\n m3_step1 Ra A B Na ∪\n m3_step2 Rb A B ∪\n m3_step3 Rs A B Kab Na Ts ∪\n m3_step4 Ra A B Na Kab Ts Ta X ∪\n m3_step5 Rb A B Kab Ts Ta ∪\n m3_step6 Ra A B Na Kab Ts Ta ∪\n m3_tick T ∪\n m3_purge A ∪\n m3_leak Rs A B Na Ts ∪\n m3_DY_fake ∪\n Id\n )\"\n\ndefinition\n m3 :: \"(m3_state, m3_obs) spec\" where\n \"m3 ≡ ⦇\n init = m3_init,\n trans = m3_trans,\n obs = m3_obs\n ⦈\"\n\nlemmas m3_loc_defs =\n m3_def m3_init_def m3_trans_def m3_obs_def\n m3_step1_def m3_step2_def m3_step3_def m3_step4_def m3_step5_def\n m3_step6_def m3_tick_def m3_purge_def m3_leak_def m3_DY_fake_def\n\nlemmas m3_defs = m3_loc_defs m2_defs\n\n\n(******************************************************************************)\nsubsection ‹Invariants›\n(******************************************************************************)\n\ntext ‹Specialized injection that we can apply more aggressively.›\n\nlemmas analz_Inj_IK = analz.Inj [where H=\"IK s\" for s]\nlemmas parts_Inj_IK = parts.Inj [where H=\"IK s\" for s]\n\ndeclare parts_Inj_IK [dest!]\n\ndeclare analz_into_parts [dest]\n\n\nsubsubsection ‹inv4: Secrecy of pre-distributed shared keys›\n(*inv**************************************************************************)\n\ndefinition\n m3_inv4_lkeysec :: \"m3_pred\"\nwhere\n \"m3_inv4_lkeysec ≡ {s. ∀C.\n (Key (shrK C) ∈ parts (IK s) ⟶ C ∈ bad) ∧\n (C ∈ bad ⟶ Key (shrK C) ∈ IK s)\n }\"\n\nlemmas m3_inv4_lkeysecI = m3_inv4_lkeysec_def [THEN setc_def_to_intro, rule_format]\nlemmas m3_inv4_lkeysecE [elim] = m3_inv4_lkeysec_def [THEN setc_def_to_elim, rule_format]\nlemmas m3_inv4_lkeysecD = m3_inv4_lkeysec_def [THEN setc_def_to_dest, rule_format]\n\n\ntext ‹Invariance proof.›\n\nlemma PO_m3_inv4_lkeysec_init [iff]:\n \"init m3 ⊆ m3_inv4_lkeysec\"\nby (auto simp add: m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: m3_inv4_lkeysecI (*‹(⋀C. (Key (shrK C) ∈ parts (IK ?x) ⟶ C ∈ bad) ∧ (C ∈ bad ⟶ Key (shrK C) ∈ IK ?x)) ⟹ ?x ∈ m3_inv4_lkeysec›*))\n\nlemma PO_m3_inv4_lkeysec_trans [iff]:\n \"{m3_inv4_lkeysec} trans m3 {> m3_inv4_lkeysec}\"\napply (auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: m3_inv4_lkeysecI (*‹(⋀C. (Key (shrK C) ∈ parts (IK ?x) ⟶ C ∈ bad) ∧ (C ∈ bad ⟶ Key (shrK C) ∈ IK ?x)) ⟹ ?x ∈ m3_inv4_lkeysec›*))\n(*goals:\n 1. ‹⋀C xa xb xc xd xh xi xk. ⟦xa ∈ m3_inv4_lkeysec; runs xa xd = Some (Init, [xb, xc], []); Crypt (shrK xb) ⦃Key xh, msg.Agent xc, Number xi, Nonce (xd $ na), xk⦄ ∈ parts (IK xa); clk xa < xi + Ls; Key (shrK C) ∈ parts (insert xk (IK xa))⟧ ⟹ C ∈ bad›\n 2. ‹⋀C xa. ⟦xa ∈ m3_inv4_lkeysec; Key (shrK C) ∈ parts (IK xa)⟧ ⟹ C ∈ bad›\n 3. ‹⋀C xa. ⟦xa ∈ m3_inv4_lkeysec; Key (shrK C) ∈ analz (IK xa)⟧ ⟹ C ∈ bad›\ndiscuss goal 1*)\n apply ((auto dest!: Body (*‹Crypt (?K::key) (?X::msg) ∈ parts (?H::msg set) ⟹ ?X ∈ parts ?H›*))[1])\n(*discuss goal 2*)\n apply ((auto dest!: Body (*‹Crypt ?K ?X ∈ parts ?H ⟹ ?X ∈ parts ?H›*))[1])\n(*discuss goal 3*)\n apply ((auto dest!: Body (*‹Crypt ?K ?X ∈ parts ?H ⟹ ?X ∈ parts ?H›*))[1])\n(*proven 3 subgoals*) .\n\nlemma PO_m3_inv4_lkeysec [iff]: \"reach m3 ⊆ m3_inv4_lkeysec\"\napply (rule inv_rule_incr (*‹⟦init ?T ⊆ ?I; {?I ∩ ?J} TS.trans ?T {> ?I}; reach ?T ⊆ ?J⟧ ⟹ reach ?T ⊆ ?I›*))\n(*goals:\n 1. ‹init m3 ⊆ m3_inv4_lkeysec›\n 2. ‹{m3_inv4_lkeysec ∩ ?J} TS.trans m3 {> m3_inv4_lkeysec}›\n 3. ‹reach m3 ⊆ ?J›\ndiscuss goal 1*)\n apply fast\n(*discuss goal 2*)\n apply fast\n(*discuss goal 3*)\n apply fast\n(*proven 3 subgoals*) .\n\n\ntext ‹Useful simplifier lemmas›\n\nlemma m3_inv4_lkeysec_for_parts [simp]:\n \"⟦ s ∈ m3_inv4_lkeysec ⟧ ⟹ Key (shrK C) ∈ parts (IK s) ⟷ C ∈ bad\"\nby auto\n\nlemma m3_inv4_lkeysec_for_analz [simp]:\n \"⟦ s ∈ m3_inv4_lkeysec ⟧ ⟹ Key (shrK C) ∈ analz (IK s) ⟷ C ∈ bad\"\nby auto\n\n\nsubsubsection ‹inv6: Ticket shape for honestly encrypted M2›\n(*inv**************************************************************************)\n\ndefinition\n m3_inv6_ticket :: \"m3_pred\"\nwhere\n \"m3_inv6_ticket ≡ {s. ∀A B T K N X.\n A ∉ bad ⟶\n Crypt (shrK A) ⦃Key K, Agent B, Number T, Nonce N, X⦄ ∈ parts (IK s) ⟶\n X = Crypt (shrK B) ⦃Key K, Agent A, Number T⦄ ∧ K ∈ range sesK\n }\"\n\nlemmas m3_inv6_ticketI = m3_inv6_ticket_def [THEN setc_def_to_intro, rule_format]\nlemmas m3_inv6_ticketE [elim] = m3_inv6_ticket_def [THEN setc_def_to_elim, rule_format]\nlemmas m3_inv6_ticketD = m3_inv6_ticket_def [THEN setc_def_to_dest, rule_format, rotated -1]\n\n\ntext ‹Invariance proof.›\n\nlemma PO_m3_inv6_ticket_init [iff]:\n \"init m3 ⊆ m3_inv6_ticket\"\nby (auto simp add: m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: m3_inv6_ticketI (*‹(⋀A B T K N X. ⟦A ∉ bad; Crypt (shrK A) ⦃Key K, msg.Agent B, Number T, Nonce N, X⦄ ∈ parts (IK ?x)⟧ ⟹ X = Crypt (shrK B) ⦃Key K, msg.Agent A, Number T⦄ ∧ K ∈ range sesK) ⟹ ?x ∈ m3_inv6_ticket›*))\n\nlemma PO_m3_inv6_ticket_trans [iff]:\n \"{m3_inv6_ticket ∩ m3_inv4_lkeysec} trans m3 {> m3_inv6_ticket}\"\napply (auto simp add: PO_hoare_defs m3_defs intro!: m3_inv6_ticketI)\napply (auto dest: m3_inv6_ticketD)\n― ‹2 subgoals›\napply (drule parts_cut, drule Body, auto dest: m3_inv6_ticketD)+\ndone\n\nlemma PO_m3_inv6_ticket [iff]: \"reach m3 ⊆ m3_inv6_ticket\"\napply (rule inv_rule_incr (*‹⟦init ?T ⊆ ?I; {?I ∩ ?J} TS.trans ?T {> ?I}; reach ?T ⊆ ?J⟧ ⟹ reach ?T ⊆ ?I›*))\n(*goals:\n 1. ‹init m3 ⊆ m3_inv6_ticket›\n 2. ‹{m3_inv6_ticket ∩ ?J} TS.trans m3 {> m3_inv6_ticket}›\n 3. ‹reach m3 ⊆ ?J›\ndiscuss goal 1*)\n apply ((auto del: subsetI (*‹(⋀x. x ∈ ?A ⟹ x ∈ ?B) ⟹ ?A ⊆ ?B›*))[1])\n(*discuss goal 2*)\n apply ((auto del: subsetI (*‹(⋀x. x ∈ ?A ⟹ x ∈ ?B) ⟹ ?A ⊆ ?B›*))[1])\n(*discuss goal 3*)\n apply ((auto del: subsetI (*‹(⋀x. x ∈ ?A ⟹ x ∈ ?B) ⟹ ?A ⊆ ?B›*))[1])\n(*proven 3 subgoals*) .\n\n\nsubsubsection ‹inv7: Session keys not used to encrypt other session keys›\n(*inv**************************************************************************)\n\ntext ‹Session keys are not used to encrypt other keys. Proof requires\ngeneralization to sets of session keys.\n\nNOTE: For Kerberos 4, this invariant cannot be inherited from the corresponding\nL2 invariant. The simulation relation is only an implication not an equivalence.\n›\n\ndefinition\n m3_inv7a_sesK_compr :: \"m3_pred\"\nwhere\n \"m3_inv7a_sesK_compr ≡ {s. ∀K KK.\n KK ⊆ range sesK ⟶\n (Key K ∈ analz (Key`KK ∪ (IK s))) = (K ∈ KK ∨ Key K ∈ analz (IK s))\n }\"\n\nlemmas m3_inv7a_sesK_comprI = m3_inv7a_sesK_compr_def [THEN setc_def_to_intro, rule_format]\nlemmas m3_inv7a_sesK_comprE = m3_inv7a_sesK_compr_def [THEN setc_def_to_elim, rule_format]\nlemmas m3_inv7a_sesK_comprD = m3_inv7a_sesK_compr_def [THEN setc_def_to_dest, rule_format]\n\ntext ‹Additional lemma›\nlemmas insert_commute_Key = insert_commute [where x=\"Key K\" for K]\n\nlemmas m3_inv7a_sesK_compr_simps =\n m3_inv7a_sesK_comprD\n m3_inv7a_sesK_comprD [where KK=\"insert Kab KK\" for Kab KK, simplified]\n m3_inv7a_sesK_comprD [where KK=\"{Kab}\" for Kab, simplified]\n insert_commute_Key\n\n\ntext ‹Invariance proof.›\n\nlemma PO_m3_inv7a_sesK_compr_step4:\n \"{m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec}\n m3_step4 Ra A B Na Kab Ts Ta X\n {> m3_inv7a_sesK_compr}\"\nproof (-)\n (*goal: ‹{m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step4 Ra A B Na Kab Ts Ta X {> m3_inv7a_sesK_compr}›*)\n {\n fix K and KK and s\n assume H: \"s ∈ m3_inv4_lkeysec\" \"s ∈ m3_inv7a_sesK_compr\" \"s ∈ m3_inv6_ticket\" \"runs s Ra = Some (Init, [A, B], [])\" \"Na = Ra$na\" \"KK ⊆ range sesK\" \"Crypt (shrK A) ⦃Key Kab, Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK s)\" (*‹(s::m3_state) ∈ m3_inv4_lkeysec› ‹(s::m3_state) ∈ m3_inv7a_sesK_compr› ‹(s::m3_state) ∈ m3_inv6_ticket› ‹runs (s::m3_state) (Ra::fid_t) = Some (Init, [A::agent, B::agent], [])› ‹(Na::fresh_t) = (Ra::fid_t) $ na› ‹(KK::key set) ⊆ range sesK› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (s::m3_state))›*)\n have \"(Key K ∈ analz (insert X (Key ` KK ∪ IK s))) =\n (K ∈ KK ∨ Key K ∈ analz (insert X (IK s)))\"\n proof (cases \"A ∈ bad\")\n (*goals:\n 1. ‹A ∈ bad ⟹ (Key K ∈ analz (insert X (Key ` KK ∪ IK s))) = (K ∈ KK ∨ Key K ∈ analz (insert X (IK s)))›\n 2. ‹A ∉ bad ⟹ (Key K ∈ analz (insert X (Key ` KK ∪ IK s))) = (K ∈ KK ∨ Key K ∈ analz (insert X (IK s)))›*)\n case True (*‹A ∈ bad›*)\n with H (*‹(s::m3_state) ∈ m3_inv4_lkeysec› ‹s ∈ m3_inv7a_sesK_compr› ‹s ∈ m3_inv6_ticket› ‹runs s Ra = Some (Init, [A, B], [])› ‹(Na::fresh_t) = (Ra::fid_t) $ na› ‹KK ⊆ range sesK› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (s::m3_state))›*) have \"X ∈ analz (IK s)\"\n by (auto dest!: Decrypt (*‹⟦Crypt ?K ?X ∈ analz ?H; Key (invKey ?K) ∈ analz ?H⟧ ⟹ ?X ∈ analz ?H›*))\n moreover with H (*‹(s::m3_state) ∈ m3_inv4_lkeysec› ‹s ∈ m3_inv7a_sesK_compr› ‹(s::m3_state) ∈ m3_inv6_ticket› ‹runs s Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹(KK::key set) ⊆ range sesK› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK s)›*) have \"X ∈ analz (Key ` KK ∪ IK s)\"\n by (auto intro: analz_monotonic (*‹⟦?c ∈ analz ?G; ?G ⊆ ?H⟧ ⟹ ?c ∈ analz ?H›*))\n ultimately show \"?thesis\"\n (*goal: ‹(Key K ∈ analz (insert X (Key ` KK ∪ IK s))) = (K ∈ KK ∨ Key K ∈ analz (insert X (IK s)))›*)\n using H (*‹(s::m3_state) ∈ m3_inv4_lkeysec› ‹s ∈ m3_inv7a_sesK_compr› ‹s ∈ m3_inv6_ticket› ‹runs s Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹KK ⊆ range sesK› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK s)›*) by (auto simp add: m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) analz_insert_eq (*‹?X ∈ analz ?H ⟹ analz (insert ?X ?H) = analz ?H›*))\n next\n (*goal: ‹A ∉ bad ⟹ (Key K ∈ analz (insert X (Key ` KK ∪ IK s))) = (K ∈ KK ∨ Key K ∈ analz (insert X (IK s)))›*)\n case False (*‹A ∉ bad›*)\n thus \"?thesis\"\n (*goal: ‹(Key K ∈ analz (insert X (Key ` KK ∪ IK s))) = (K ∈ KK ∨ Key K ∈ analz (insert X (IK s)))›*)\n using H (*‹s ∈ m3_inv4_lkeysec› ‹s ∈ m3_inv7a_sesK_compr› ‹s ∈ m3_inv6_ticket› ‹runs (s::m3_state) (Ra::fid_t) = Some (Init, [A::agent, B::agent], [])› ‹Na = Ra $ na› ‹(KK::key set) ⊆ range sesK› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK s)›*) by (fastforce simp add: m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) dest!: m3_inv6_ticketD [OF analz_into_parts] (*‹⟦Crypt (shrK ?A) ⦃Key ?K, msg.Agent ?B, Number ?T, Nonce ?N, ?X⦄ ∈ analz (IK ?x); ?x ∈ m3_inv6_ticket; ?A ∉ bad⟧ ⟹ ?X = Crypt (shrK ?B) ⦃Key ?K, msg.Agent ?A, Number ?T⦄ ∧ ?K ∈ range sesK›*))\n qed\n }\n thus \"?thesis\"\n (*goal: ‹{m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step4 Ra A B Na Kab Ts Ta X {> m3_inv7a_sesK_compr}›*)\n by (auto simp add: PO_hoare_defs (*‹{?pre::?'s set} ?R::(?'s × ?'s) set {> ?post::?'s set} ≡ ?R `` ?pre ⊆ ?post› ‹(?r::(?'a × ?'b) set) `` (?s::?'a set) = {y::?'b. ∃x::?'a∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: m3_inv7a_sesK_comprI (*‹(⋀(K::key) KK::key set. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK (?x::m3_state))) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*) dest!: analz_Inj_IK (*‹(?X::msg) ∈ IK (?s::?'a m3_state_scheme) ⟹ ?X ∈ analz (IK ?s)›*))\nqed\n\ntext ‹All together now.›\n\nlemmas PO_m3_inv7a_sesK_compr_trans_lemmas =\n PO_m3_inv7a_sesK_compr_step4\n\nlemma PO_m3_inv7a_sesK_compr_init [iff]:\n \"init m3 ⊆ m3_inv7a_sesK_compr\"\nby (auto simp add: m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))\n\nlemma PO_m3_inv7a_sesK_compr_trans [iff]:\n \"{m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec}\n trans m3\n {> m3_inv7a_sesK_compr}\"\napply (auto simp add: m3_def (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈›*) m3_trans_def (*‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼›*) intro!: PO_m3_inv7a_sesK_compr_trans_lemmas (*‹{m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> m3_inv7a_sesK_compr}›*))\n(*goals:\n 1. ‹⋀x xa xb xe. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step1 xb x xa xe {> m3_inv7a_sesK_compr}›\n 2. ‹⋀x xa xc. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step2 xc x xa {> m3_inv7a_sesK_compr}›\n 3. ‹⋀x xa xd xe xf xg. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step3 xd x xa xf xe xg {> m3_inv7a_sesK_compr}›\n 4. ‹⋀x xa xc xf xg xh. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step5 xc x xa xf xg xh {> m3_inv7a_sesK_compr}›\n 5. ‹⋀x xa xb xe xf xg xh. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step6 xb x xa xe xf xg xh {> m3_inv7a_sesK_compr}›\n 6. ‹⋀xh. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_tick xh {> m3_inv7a_sesK_compr}›\n 7. ‹⋀x. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_purge x {> m3_inv7a_sesK_compr}›\n 8. ‹⋀x xa xd xe xg. {m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_leak xd x xa xe xg {> m3_inv7a_sesK_compr}›\n 9. ‹{m3_inv7a_sesK_compr ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_DY_fake {> m3_inv7a_sesK_compr}›\ndiscuss goal 1*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 2*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 3*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 4*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 5*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre::?'s set} ?R::(?'s × ?'s) set {> ?post::?'s set} ≡ ?R `` ?pre ⊆ ?post› ‹(?r::(?'a × ?'b) set) `` (?s::?'a set) = {y::?'b. ∃x::?'a∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦(?x::m3_state) ∈ m3_inv7a_sesK_compr; (?KK::key set) ⊆ range sesK⟧ ⟹ (Key (?K::key) ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7a_sesK_compr; (?Kab::key) ∈ range sesK ∧ (?KK::key set) ⊆ range sesK⟧ ⟹ (Key (?K::key) ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7a_sesK_compr; (?Kab::key) ∈ range sesK⟧ ⟹ (Key (?K::key) ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key (?K::key)) (insert (?y::msg) (?A::msg set)) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀(K::key) KK::key set. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK (?x::m3_state))) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 6*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 7*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 8*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*discuss goal 9*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7a_sesK_comprI (*‹(⋀K KK. KK ⊆ range sesK ⟹ (Key K ∈ analz (Key ` KK ∪ IK ?x)) = (K ∈ KK ∨ Key K ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7a_sesK_compr›*))[1])\n(*proven 9 subgoals*) .\n\nlemma PO_m3_inv7a_sesK_compr [iff]: \"reach m3 ⊆ m3_inv7a_sesK_compr\"\napply (rule_tac J=\"m3_inv6_ticket ∩ m3_inv4_lkeysec\" in inv_rule_incr (*‹⟦init ?T ⊆ ?I; {?I ∩ ?J} TS.trans ?T {> ?I}; reach ?T ⊆ ?J⟧ ⟹ reach ?T ⊆ ?I›*))\n(*goals:\n 1. ‹init m3 ⊆ m3_inv7a_sesK_compr›\n 2. ‹{m3_inv7a_sesK_compr ∩ (m3_inv6_ticket ∩ m3_inv4_lkeysec)} TS.trans m3 {> m3_inv7a_sesK_compr}›\n 3. ‹reach m3 ⊆ m3_inv6_ticket ∩ m3_inv4_lkeysec›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*proven 3 subgoals*) .\n\n\nsubsubsection ‹inv7b: Session keys not used to encrypt nonces›\n(*inv**************************************************************************)\n\ntext ‹Session keys are not used to encrypt nonces. The proof requires a\ngeneralization to sets of session keys.›\n\ndefinition\n m3_inv7b_sesK_compr_non :: \"m3_pred\"\nwhere\n \"m3_inv7b_sesK_compr_non ≡ {s. ∀N KK.\n KK ⊆ range sesK ⟶ (Nonce N ∈ analz (Key`KK ∪ (IK s))) = (Nonce N ∈ analz (IK s))\n }\"\n\nlemmas m3_inv7b_sesK_compr_nonI = m3_inv7b_sesK_compr_non_def [THEN setc_def_to_intro, rule_format]\nlemmas m3_inv7b_sesK_compr_nonE = m3_inv7b_sesK_compr_non_def [THEN setc_def_to_elim, rule_format]\nlemmas m3_inv7b_sesK_compr_nonD = m3_inv7b_sesK_compr_non_def [THEN setc_def_to_dest, rule_format]\n\nlemmas m3_inv7b_sesK_compr_non_simps =\n m3_inv7b_sesK_compr_nonD\n m3_inv7b_sesK_compr_nonD [where KK=\"insert Kab KK\" for Kab KK, simplified]\n m3_inv7b_sesK_compr_nonD [where KK=\"{Kab}\" for Kab, simplified]\n insert_commute_Key\n\n\ntext ‹Invariance proof.›\n\nlemma PO_m3_inv7b_sesK_compr_non_step3:\n \"{m3_inv7b_sesK_compr_non} m3_step3 Rs A B Kab Na Ts {> m3_inv7b_sesK_compr_non}\"\nby (auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀N KK. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK ?x)) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*) dest!: analz_Inj_IK (*‹?X ∈ IK ?s ⟹ ?X ∈ analz (IK ?s)›*))\n\nlemma PO_m3_inv7b_sesK_compr_non_step4:\n \"{m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec}\n m3_step4 Ra A B Na Kab Ts Ta X\n {> m3_inv7b_sesK_compr_non}\"\nproof (-)\n (*goal: ‹{m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step4 Ra A B Na Kab Ts Ta X {> m3_inv7b_sesK_compr_non}›*)\n {\n fix N and KK and s\n assume H: \"s ∈ m3_inv4_lkeysec\" \"s ∈ m3_inv7b_sesK_compr_non\" \"s ∈ m3_inv6_ticket\" \"runs s Ra = Some (Init, [A, B], [])\" \"Na = Ra$na\" \"KK ⊆ range sesK\" \"Crypt (shrK A) ⦃Key Kab, Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK s)\" (*‹(s::m3_state) ∈ m3_inv4_lkeysec› ‹(s::m3_state) ∈ m3_inv7b_sesK_compr_non› ‹(s::m3_state) ∈ m3_inv6_ticket› ‹runs (s::m3_state) (Ra::fid_t) = Some (Init, [A::agent, B::agent], [])› ‹(Na::fresh_t) = (Ra::fid_t) $ na› ‹(KK::key set) ⊆ range sesK› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (s::m3_state))›*)\n have \"(Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) =\n (Nonce N ∈ analz (insert X (IK s)))\"\n proof (cases)\n (*goals:\n 1. ‹?P ⟹ (Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) = (Nonce N ∈ analz (insert X (IK s)))›\n 2. ‹¬ ?P ⟹ (Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) = (Nonce N ∈ analz (insert X (IK s)))›*)\n assume \"A ∈ bad\" (*‹(A::agent) ∈ bad›*)\n show \"?thesis\"\n (*goal: ‹(Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) = (Nonce N ∈ analz (insert X (IK s)))›*)\n proof (-)\n (*goal: ‹(Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) = (Nonce N ∈ analz (insert X (IK s)))›*)\n from ‹A ∈ bad› (*‹(A::agent) ∈ bad›*) have \"X ∈ analz (IK s)\"\n using H (*‹s ∈ m3_inv4_lkeysec› ‹s ∈ m3_inv7b_sesK_compr_non› ‹s ∈ m3_inv6_ticket› ‹runs s Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹KK ⊆ range sesK› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (s::m3_state))›*) by (auto dest!: Decrypt (*‹⟦Crypt ?K ?X ∈ analz ?H; Key (invKey ?K) ∈ analz ?H⟧ ⟹ ?X ∈ analz ?H›*))\n moreover hence \"X ∈ analz (Key ` KK ∪ IK s)\"\n by (auto intro: analz_monotonic (*‹⟦?c ∈ analz ?G; ?G ⊆ ?H⟧ ⟹ ?c ∈ analz ?H›*))\n ultimately show \"?thesis\"\n (*goal: ‹(Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) = (Nonce N ∈ analz (insert X (IK s)))›*)\n using H (*‹s ∈ m3_inv4_lkeysec› ‹s ∈ m3_inv7b_sesK_compr_non› ‹s ∈ m3_inv6_ticket› ‹runs (s::m3_state) (Ra::fid_t) = Some (Init, [A::agent, B::agent], [])› ‹Na = Ra $ na› ‹KK ⊆ range sesK› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (s::m3_state))›*) by (auto simp add: m3_inv7b_sesK_compr_non_simps (*‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) analz_insert_eq (*‹?X ∈ analz ?H ⟹ analz (insert ?X ?H) = analz ?H›*))\n qed\n next\n (*goal: ‹A ∉ bad ⟹ (Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) = (Nonce N ∈ analz (insert X (IK s)))›*)\n assume \"A ∉ bad\" (*‹(A::agent) ∉ bad›*)\n thus \"?thesis\"\n (*goal: ‹(Nonce N ∈ analz (insert X (Key ` KK ∪ IK s))) = (Nonce N ∈ analz (insert X (IK s)))›*)\n using H (*‹s ∈ m3_inv4_lkeysec› ‹s ∈ m3_inv7b_sesK_compr_non› ‹(s::m3_state) ∈ m3_inv6_ticket› ‹runs s Ra = Some (Init, [A, B], [])› ‹(Na::fresh_t) = (Ra::fid_t) $ na› ‹KK ⊆ range sesK› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK s)›*) by (fastforce simp add: m3_inv7b_sesK_compr_non_simps (*‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK ∧ (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key (?K::key)) (insert (?y::msg) (?A::msg set)) = insert ?y (insert (Key ?K) ?A)›*) dest!: m3_inv6_ticketD [OF analz_into_parts] (*‹⟦Crypt (shrK (?A::agent)) ⦃Key (?K::key), msg.Agent (?B::agent), Number (?T::nat), Nonce (?N::fresh_t), ?X::msg⦄ ∈ analz (IK (?x::m3_state)); ?x ∈ m3_inv6_ticket; ?A ∉ bad⟧ ⟹ ?X = Crypt (shrK ?B) ⦃Key ?K, msg.Agent ?A, Number ?T⦄ ∧ ?K ∈ range sesK›*))\n qed\n }\n thus \"?thesis\"\n (*goal: ‹{m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step4 Ra A B Na Kab Ts Ta X {> m3_inv7b_sesK_compr_non}›*)\n by (auto simp add: PO_hoare_defs (*‹{?pre::?'s set} ?R::(?'s × ?'s) set {> ?post::?'s set} ≡ ?R `` ?pre ⊆ ?post› ‹(?r::(?'a × ?'b) set) `` (?s::?'a set) = {y::?'b. ∃x::?'a∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀(N::fresh_t) KK::key set. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK (?x::m3_state))) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*) dest!: analz_Inj_IK (*‹(?X::msg) ∈ IK (?s::?'a m3_state_scheme) ⟹ ?X ∈ analz (IK ?s)›*))\nqed\n\n\ntext ‹All together now.›\n\nlemmas PO_m3_inv7b_sesK_compr_non_trans_lemmas =\n PO_m3_inv7b_sesK_compr_non_step3 PO_m3_inv7b_sesK_compr_non_step4\n\nlemma PO_m3_inv7b_sesK_compr_non_init [iff]:\n \"init m3 ⊆ m3_inv7b_sesK_compr_non\"\nby (auto simp add: m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀N KK. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK ?x)) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))\n\nlemma PO_m3_inv7b_sesK_compr_non_trans [iff]:\n \"{m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec}\n trans m3\n {> m3_inv7b_sesK_compr_non}\"\napply (auto simp add: m3_def (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈›*) m3_trans_def (*‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼›*) intro!: PO_m3_inv7b_sesK_compr_non_trans_lemmas (*‹{m3_inv7b_sesK_compr_non} m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) {> m3_inv7b_sesK_compr_non}› ‹{m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) {> m3_inv7b_sesK_compr_non}›*))\n(*goals:\n 1. ‹⋀x xa xb xe. {m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step1 xb x xa xe {> m3_inv7b_sesK_compr_non}›\n 2. ‹⋀x xa xc. {m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step2 xc x xa {> m3_inv7b_sesK_compr_non}›\n 3. ‹⋀x xa xc xf xg xh. {m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step5 xc x xa xf xg xh {> m3_inv7b_sesK_compr_non}›\n 4. ‹⋀x xa xb xe xf xg xh. {m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_step6 xb x xa xe xf xg xh {> m3_inv7b_sesK_compr_non}›\n 5. ‹⋀xh. {m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_tick xh {> m3_inv7b_sesK_compr_non}›\n 6. ‹⋀x. {m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_purge x {> m3_inv7b_sesK_compr_non}›\n 7. ‹⋀x xa xd xe xg. {m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_leak xd x xa xe xg {> m3_inv7b_sesK_compr_non}›\n 8. ‹{m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec} m3_DY_fake {> m3_inv7b_sesK_compr_non}›\ndiscuss goal 1*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀N KK. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK ?x)) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*discuss goal 2*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre::?'s set} ?R::(?'s × ?'s) set {> ?post::?'s set} ≡ ?R `` ?pre ⊆ ?post› ‹(?r::(?'a × ?'b) set) `` (?s::?'a set) = {y::?'b. ∃x::?'a∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK ∧ (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key (?K::key)) (insert (?y::msg) (?A::msg set)) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀(N::fresh_t) KK::key set. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK (?x::m3_state))) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*discuss goal 3*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀N KK. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK ?x)) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*discuss goal 4*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀N KK. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK ?x)) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*discuss goal 5*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre::?'s set} ?R::(?'s × ?'s) set {> ?post::?'s set} ≡ ?R `` ?pre ⊆ ?post› ‹(?r::(?'a × ?'b) set) `` (?s::?'a set) = {y::?'b. ∃x::?'a∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK ∧ (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key (?K::key)) (insert (?y::msg) (?A::msg set)) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀(N::fresh_t) KK::key set. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK (?x::m3_state))) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*discuss goal 6*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre::?'s::type set} ?R::(?'s::type × ?'s::type) set {> ?post::?'s::type set} ≡ ?R `` ?pre ⊆ ?post› ‹(?r::(?'a::type × ?'b::type) set) `` (?s::?'a::type set) = {y::?'b::type. ∃x::?'a::type∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK ∧ (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key (?K::key)) (insert (?y::msg) (?A::msg set)) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀(N::fresh_t) KK::key set. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK (?x::m3_state))) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*discuss goal 7*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre::?'s set} ?R::(?'s × ?'s) set {> ?post::?'s set} ≡ ?R `` ?pre ⊆ ?post› ‹(?r::(?'a × ?'b) set) `` (?s::?'a set) = {y::?'b. ∃x::?'a∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK ∧ (?KK::key set) ⊆ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7b_sesK_compr_non; (?Kab::key) ∈ range sesK⟧ ⟹ (Nonce (?N::fresh_t) ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key (?K::key)) (insert (?y::msg) (?A::msg set)) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀(N::fresh_t) KK::key set. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK (?x::m3_state))) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*discuss goal 8*)\n apply ((auto simp add: PO_hoare_defs (*‹{?pre} ?R {> ?post} ≡ ?R `` ?pre ⊆ ?post› ‹?r `` ?s = {y. ∃x∈?s. (x, y) ∈ ?r}›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) m3_inv7b_sesK_compr_non_simps (*‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) intro!: m3_inv7b_sesK_compr_nonI (*‹(⋀N KK. KK ⊆ range sesK ⟹ (Nonce N ∈ analz (Key ` KK ∪ IK ?x)) = (Nonce N ∈ analz (IK ?x))) ⟹ ?x ∈ m3_inv7b_sesK_compr_non›*))[1])\n(*proven 8 subgoals*) .\n\nlemma PO_m3_inv7b_sesK_compr_non [iff]: \"reach m3 ⊆ m3_inv7b_sesK_compr_non\"\napply (rule_tac J=\"m3_inv6_ticket ∩ m3_inv4_lkeysec\" in inv_rule_incr (*‹⟦init ?T ⊆ ?I; {?I ∩ ?J} TS.trans ?T {> ?I}; reach ?T ⊆ ?J⟧ ⟹ reach ?T ⊆ ?I›*))\n(*goals:\n 1. ‹init m3 ⊆ m3_inv7b_sesK_compr_non›\n 2. ‹{m3_inv7b_sesK_compr_non ∩ (m3_inv6_ticket ∩ m3_inv4_lkeysec)} TS.trans m3 {> m3_inv7b_sesK_compr_non}›\n 3. ‹reach m3 ⊆ m3_inv6_ticket ∩ m3_inv4_lkeysec›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*proven 3 subgoals*) .\n\n\n(******************************************************************************)\nsubsection ‹Refinement›\n(******************************************************************************)\n\nsubsubsection ‹Message abstraction and simulation relation›\n(******************************************************************************)\n\ntext ‹Abstraction function on sets of messages.›\n\ninductive_set\n abs_msg :: \"msg set ⇒ chmsg set\"\n for H :: \"msg set\"\nwhere\n am_M1:\n \"⦃Agent A, Agent B, Nonce N⦄ ∈ H\n ⟹ Insec A B (Msg [aNon N]) ∈ abs_msg H\"\n| am_M2a:\n \"Crypt (shrK C) ⦃Key K, Agent B, Number T, Nonce N, X⦄ ∈ H\n ⟹ Secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]) ∈ abs_msg H\"\n| am_M2b:\n \"Crypt (shrK C) ⦃Key K, Agent A, Number T⦄ ∈ H\n ⟹ Secure Sv C (Msg [aKey K, aAgt A, aNum T]) ∈ abs_msg H\"\n| am_M3:\n \"Crypt K ⦃Agent A, Number T⦄ ∈ H\n ⟹ dAuth K (Msg [aAgt A, aNum T]) ∈ abs_msg H\"\n| am_M4:\n \"Crypt K (Number T) ∈ H\n ⟹ dAuth K (Msg [aNum T]) ∈ abs_msg H\"\n\n\ntext ‹R23: The simulation relation. This is a data refinement of\nthe insecure and secure channels of refinement 2.›\n\ndefinition\n R23_msgs :: \"(m2_state × m3_state) set\" where\n \"R23_msgs ≡ {(s, t). abs_msg (parts (IK t)) ⊆ chan s }\"\n\ndefinition\n R23_keys :: \"(m2_state × m3_state) set\" where\n \"R23_keys ≡ {(s, t). ∀KK K. KK ⊆ range sesK ⟶\n Key K ∈ analz (Key`KK ∪ (IK t)) ⟶ aKey K ∈ extr (aKey`KK ∪ ik0) (chan s)\n }\"\n\ndefinition\n R23_non :: \"(m2_state × m3_state) set\" where\n \"R23_non ≡ {(s, t). ∀KK N. KK ⊆ range sesK ⟶\n Nonce N ∈ analz (Key`KK ∪ (IK t)) ⟶ aNon N ∈ extr (aKey`KK ∪ ik0) (chan s)\n }\"\n\ndefinition\n R23_pres :: \"(m2_state × m3_state) set\" where\n \"R23_pres ≡ {(s, t). runs s = runs t ∧ leak s = leak t ∧ clk s = clk t ∧ cache s = cache t}\"\n\ndefinition\n R23 :: \"(m2_state × m3_state) set\" where\n \"R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres\"\n\nlemmas R23_defs =\n R23_def R23_msgs_def R23_keys_def R23_non_def R23_pres_def\n\n\ntext ‹The mediator function is the identity here.›\n\ndefinition\n med32 :: \"m3_obs ⇒ m2_obs\" where\n \"med32 ≡ id\"\n\n\nlemmas R23_msgsI = R23_msgs_def [THEN rel_def_to_intro, simplified, rule_format]\nlemmas R23_msgsE [elim] = R23_msgs_def [THEN rel_def_to_elim, simplified, rule_format]\nlemmas R23_msgsE' [elim] =\n R23_msgs_def [THEN rel_def_to_dest, simplified, rule_format, THEN subsetD]\n\nlemmas R23_keysI = R23_keys_def [THEN rel_def_to_intro, simplified, rule_format]\nlemmas R23_keysE [elim] = R23_keys_def [THEN rel_def_to_elim, simplified, rule_format]\nlemmas R23_keysD = R23_keys_def [THEN rel_def_to_dest, simplified, rule_format, rotated 2]\n\nlemmas R23_nonI = R23_non_def [THEN rel_def_to_intro, simplified, rule_format]\nlemmas R23_nonE [elim] = R23_non_def [THEN rel_def_to_elim, simplified, rule_format]\nlemmas R23_nonD = R23_non_def [THEN rel_def_to_dest, simplified, rule_format, rotated 2]\n\nlemmas R23_presI = R23_pres_def [THEN rel_def_to_intro, simplified, rule_format]\nlemmas R23_presE [elim] = R23_pres_def [THEN rel_def_to_elim, simplified, rule_format]\n\nlemmas R23_intros = R23_msgsI R23_keysI R23_nonI R23_presI\n\n\ntext ‹Lemmas for various instantiations (keys and nonces).›\n\nlemmas R23_keys_dests =\n R23_keysD\n R23_keysD [where KK=\"{}\", simplified]\n R23_keysD [where KK=\"{K}\" for K, simplified]\n R23_keysD [where KK=\"insert K KK\" for K KK, simplified, OF _ _ conjI]\n\nlemmas R23_non_dests =\n R23_nonD\n R23_nonD [where KK=\"{}\", simplified]\n R23_nonD [where KK=\"{K}\" for K, simplified]\n R23_nonD [where KK=\"insert K KK\" for K KK, simplified, OF _ _ conjI]\n\nlemmas R23_dests = R23_keys_dests R23_non_dests\n\n\nsubsubsection ‹General lemmas›\n(******************************************************************************)\n\ntext ‹General facts about @{term \"abs_msg\"}›\n\ndeclare abs_msg.intros [intro!]\ndeclare abs_msg.cases [elim!]\n\nlemma abs_msg_empty: \"abs_msg {} = {}\"\nby auto\n\nlemma abs_msg_Un [simp]:\n \"abs_msg (G ∪ H) = abs_msg G ∪ abs_msg H\"\nby auto\n\nlemma abs_msg_mono [elim]:\n \"⟦ m ∈ abs_msg G; G ⊆ H ⟧ ⟹ m ∈ abs_msg H\"\nby auto\n\nlemma abs_msg_insert_mono [intro]:\n \"⟦ m ∈ abs_msg H ⟧ ⟹ m ∈ abs_msg (insert m' H)\"\nby auto\n\n\ntext ‹Facts about @{term \"abs_msg\"} concerning abstraction of fakeable\nmessages. This is crucial for proving the refinement of the intruder event.›\n\nlemma abs_msg_DY_subset_fakeable:\n \"⟦ (s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec ⟧\n ⟹ abs_msg (synth (analz (IK t))) ⊆ fake ik0 (dom (runs s)) (chan s)\"\napply auto\n(*goals:\n 1. ‹⋀(A::agent) (B::agent) N::fresh_t. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Nonce N ∈ analz (IK t)⟧ ⟹ Insec A B (Msg [aNon N]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 2. ‹⋀(C::agent) (K::key) (B::agent) (T::nat) (N::fresh_t) X::msg. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Crypt (shrK C) ⦃Key K, msg.Agent B, Number T, Nonce N, X⦄ ∈ analz (IK t)⟧ ⟹ StatCh secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 3. ‹⋀(C::agent) (K::key) (B::agent) (T::nat) (N::fresh_t) X::msg. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; C ∈ bad; Key K ∈ analz (IK t); X ∈ synth (analz (IK t)); Nonce N ∈ analz (IK t)⟧ ⟹ StatCh secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 4. ‹⋀(C::agent) (K::key) (A::agent) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Crypt (shrK C) ⦃Key K, msg.Agent A, Number T⦄ ∈ analz (IK t)⟧ ⟹ StatCh secure Sv C (Msg [aKey K, aAgt A, aNum T]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 5. ‹⋀(C::agent) (K::key) (A::agent) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; C ∈ bad; Key K ∈ analz (IK t)⟧ ⟹ StatCh secure Sv C (Msg [aKey K, aAgt A, aNum T]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 6. ‹⋀(K::key) (A::agent) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Crypt K ⦃msg.Agent A, Number T⦄ ∈ analz (IK t)⟧ ⟹ dAuth K (Msg [aAgt A, aNum T]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 7. ‹⋀(K::key) (A::agent) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Key K ∈ analz (IK t)⟧ ⟹ dAuth K (Msg [aAgt A, aNum T]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 8. ‹⋀(K::key) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Crypt K (Number T) ∈ analz (IK t)⟧ ⟹ dAuth K (Msg [aNum T]) ∈ fake ik0 (dom (runs s)) (chan s)›\n 9. ‹⋀(K::key) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Key K ∈ analz (IK t)⟧ ⟹ dAuth K (Msg [aNum T]) ∈ fake ik0 (dom (runs s)) (chan s)›\ndiscuss goal 1*)\n apply (intro fake_StatCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∨ ?A ∈ bad ∨ ?B ∈ bad⟧ ⟹ StatCh ?c ?A ?B (Msg ?M) ∈ fake ?T ?U ?H›*) fake_DynCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∧ ?K ∈ rkeys ?U ∨ aKey ?K ∈ extr ?T ?H⟧ ⟹ DynCh ?c ?K (Msg ?M) ∈ fake ?T ?U ?H›*))\n (*goals:\n 1. ‹⋀A B N. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Nonce N ∈ analz (IK t)⟧ ⟹ set [aNon N] ⊆ extr ik0 (chan s)›\n 2. ‹⋀A B N. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Nonce N ∈ analz (IK t)⟧ ⟹ auth ∉ {} ∨ A ∈ bad ∨ B ∈ bad›\n discuss goal 1*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply blast\n(*discuss goal 3*)\n apply (intro fake_StatCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∨ ?A ∈ bad ∨ ?B ∈ bad⟧ ⟹ StatCh ?c ?A ?B (Msg ?M) ∈ fake ?T ?U ?H›*) fake_DynCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∧ ?K ∈ rkeys ?U ∨ aKey ?K ∈ extr ?T ?H⟧ ⟹ DynCh ?c ?K (Msg ?M) ∈ fake ?T ?U ?H›*))\n (*goals:\n 1. ‹⋀C K B T N X. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; C ∈ bad; Key K ∈ analz (IK t); X ∈ synth (analz (IK t)); Nonce N ∈ analz (IK t)⟧ ⟹ set [aKey K, aAgt B, aNum T, aNon N] ⊆ extr ik0 (chan s)›\n 2. ‹⋀C K B T N X. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; C ∈ bad; Key K ∈ analz (IK t); X ∈ synth (analz (IK t)); Nonce N ∈ analz (IK t)⟧ ⟹ auth ∉ secure ∨ Sv ∈ bad ∨ C ∈ bad›\n discuss goal 1*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*proven 2 subgoals*)\n(*discuss goal 4*)\n apply blast\n(*discuss goal 5*)\n apply (intro fake_StatCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∨ ?A ∈ bad ∨ ?B ∈ bad⟧ ⟹ StatCh ?c ?A ?B (Msg ?M) ∈ fake ?T ?U ?H›*) fake_DynCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∧ ?K ∈ rkeys ?U ∨ aKey ?K ∈ extr ?T ?H⟧ ⟹ DynCh ?c ?K (Msg ?M) ∈ fake ?T ?U ?H›*))\n (*goals:\n 1. ‹⋀C K A T. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; C ∈ bad; Key K ∈ analz (IK t)⟧ ⟹ set [aKey K, aAgt A, aNum T] ⊆ extr ik0 (chan s)›\n 2. ‹⋀C K A T. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; C ∈ bad; Key K ∈ analz (IK t)⟧ ⟹ auth ∉ secure ∨ Sv ∈ bad ∨ C ∈ bad›\n discuss goal 1*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*proven 2 subgoals*)\n(*discuss goal 6*)\n apply blast\n(*discuss goal 7*)\n apply (intro fake_StatCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∨ ?A ∈ bad ∨ ?B ∈ bad⟧ ⟹ StatCh ?c ?A ?B (Msg ?M) ∈ fake ?T ?U ?H›*) fake_DynCh (*‹⟦set ?M ⊆ extr ?T ?H; auth ∉ ?c ∧ ?K ∈ rkeys ?U ∨ aKey ?K ∈ extr ?T ?H⟧ ⟹ DynCh ?c ?K (Msg ?M) ∈ fake ?T ?U ?H›*))\n (*goals:\n 1. ‹⋀(K::key) (A::agent) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Key K ∈ analz (IK t)⟧ ⟹ set [aAgt A, aNum T] ⊆ extr ik0 (chan s)›\n 2. ‹⋀(K::key) (A::agent) T::nat. ⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Key K ∈ analz (IK t)⟧ ⟹ auth ∉ {auth} ∧ K ∈ rkeys (dom (runs s)) ∨ aKey K ∈ extr ik0 (chan s)›\n discuss goal 1*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*proven 2 subgoals*)\n(*discuss goal 8*)\n apply blast\n(*discuss goal 9*)\n apply (intro fake_StatCh (*‹⟦set (?M::atom list) ⊆ extr (?T::atom set) (?H::chmsg set); auth ∉ (?c::secprop set) ∨ (?A::agent) ∈ bad ∨ (?B::agent) ∈ bad⟧ ⟹ StatCh ?c ?A ?B (Msg ?M) ∈ fake ?T (?U::fid_t set) ?H›*) fake_DynCh (*‹⟦set (?M::atom list) ⊆ extr (?T::atom set) (?H::chmsg set); auth ∉ (?c::secprop set) ∧ (?K::key) ∈ rkeys (?U::fid_t set) ∨ aKey ?K ∈ extr ?T ?H⟧ ⟹ DynCh ?c ?K (Msg ?M) ∈ fake ?T ?U ?H›*))\n (*goals:\n 1. ‹⋀K T. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Key K ∈ analz (IK t)⟧ ⟹ set [aNum T] ⊆ extr ik0 (chan s)›\n 2. ‹⋀K T. ⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; t ∈ m3_inv4_lkeysec; Key K ∈ analz (IK t)⟧ ⟹ auth ∉ {auth} ∧ K ∈ rkeys (dom (runs s)) ∨ aKey K ∈ extr ik0 (chan s)›\n discuss goal 1*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n (*proven 2 subgoals*)\n(*proven 9 subgoals*) .\n\n\nsubsubsection ‹Refinement proof›\n(******************************************************************************)\n\ntext ‹Pair decomposition. These were set to \\texttt{elim!}, which is too\nagressive here.›\n\ndeclare MPair_analz [rule del, elim]\ndeclare MPair_parts [rule del, elim]\n\n\ntext ‹Protocol events.›\n\nlemma PO_m3_step1_refines_m2_step1:\n \"{R23}\n (m2_step1 Ra A B Na), (m3_step1 Ra A B Na)\n {> R23}\"\napply (auto simp add: PO_rhoare_defs (*‹{?pre::(?'s::type × ?'t::type) set} ?Ra::(?'s::type × ?'s::type) set, ?Rc::(?'t::type × ?'t::type) set {> ?post::(?'s::type × ?'t::type) set} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹(?r::(?'a::type × ?'c::type) set) O (?s::(?'c::type × ?'b::type) set) = {(x::?'a::type, z::?'b::type). ∃y::?'c::type. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK (?t::m3_state))) ⊆ chan (?s::m2_state) ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀(KK::key set) K::key. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀(KK::key set) N::fresh_t. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_non› ‹runs (?s::m2_state) = runs (?t::m3_state) ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n(*goals:\n 1. ‹⋀a y Aa B N. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; Ra ∉ dom (runs y); Na = Ra $ na; Insec Aa B (Msg [aNon N]) ∉ chan a; ⦃msg.Agent Aa, msg.Agent B, Nonce N⦄ ∈ parts (IK y)⟧ ⟹ Aa = A›\n 2. ‹⋀a y A Ba N. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; Ra ∉ dom (runs y); Na = Ra $ na; Insec A Ba (Msg [aNon N]) ∉ chan a; ⦃msg.Agent A, msg.Agent Ba, Nonce N⦄ ∈ parts (IK y)⟧ ⟹ Ba = B›\n 3. ‹⋀a y A B N. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; Ra ∉ dom (runs y); Na = Ra $ na; Insec A B (Msg [aNon N]) ∉ chan a; ⦃msg.Agent A, msg.Agent B, Nonce N⦄ ∈ parts (IK y)⟧ ⟹ N = Ra $ na›\n 4. ‹⋀a y KK N. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; Ra ∉ dom (runs y); Na = Ra $ na; KK ⊆ range sesK; aNon N ∉ extr (aKey ` KK ∪ ik0) (chan a); Nonce N ∈ analz (Key ` KK ∪ IK y)⟧ ⟹ N = Ra $ na›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*discuss goal 4*)\n apply ((auto)[1])\n(*proven 4 subgoals*) .\n\nlemma PO_m3_step2_refines_m2_step2:\n \"{R23}\n (m2_step2 Rb A B), (m3_step2 Rb A B)\n {> R23}\"\nby (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n\nlemma PO_m3_step3_refines_m2_step3:\n \"{R23 ∩ (m2_inv3a_sesK_compr) × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)}\n (m2_step3 Rs A B Kab Na Ts), (m3_step3 Rs A B Kab Na Ts)\n {> R23}\"\nproof (-)\n (*goal: ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 Rs A B Kab Na Ts, m3_step3 Rs A B Kab Na Ts {> R23}›*)\n {\n fix s and t\n assume H: \"(s, t) ∈ R23_msgs\" \"(s, t) ∈ R23_keys\" \"(s, t) ∈ R23_non\" \"(s, t) ∈ R23_pres\" \"s ∈ m2_inv3a_sesK_compr\" \"t ∈ m3_inv7a_sesK_compr\" \"t ∈ m3_inv4_lkeysec\" \"Kab = sesK (Rs$sk)\" \"Rs ∉ dom (runs t)\" \"⦃Agent A, Agent B, Nonce Na⦄ ∈ parts (IK t)\" (*‹(s::m2_state, t::m3_state) ∈ R23_msgs› ‹(s::m2_state, t::m3_state) ∈ R23_keys› ‹(s::m2_state, t::m3_state) ∈ R23_non› ‹(s::m2_state, t::m3_state) ∈ R23_pres› ‹(s::m2_state) ∈ m2_inv3a_sesK_compr› ‹(t::m3_state) ∈ m3_inv7a_sesK_compr› ‹(t::m3_state) ∈ m3_inv4_lkeysec› ‹(Kab::key) = sesK ((Rs::fid_t) $ sk)› ‹(Rs::fid_t) ∉ dom (runs (t::m3_state))› ‹⦃msg.Agent (A::agent), msg.Agent (B::agent), Nonce (Na::fresh_t)⦄ ∈ parts (IK (t::m3_state))›*)\n let ?s' = \"s⦇ runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])),\n chan := insert (Secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na]))\n (insert (Secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s)) ⦈\"\n let ?t' = \"t⦇ runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])),\n IK := insert\n (Crypt (shrK A)\n ⦃ Key Kab, Agent B, Number (clk t), Nonce Na,\n Crypt (shrK B) ⦃ Key Kab, Agent A, Number (clk t) ⦄⦄)\n (IK t) ⦈\"\n have \"(?s', ?t') ∈ R23_msgs\"\n using H (*‹(s::m2_state, t::m3_state) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s::m2_state, t::m3_state) ∈ R23_non› ‹(s::m2_state, t::m3_state) ∈ R23_pres› ‹(s::m2_state) ∈ m2_inv3a_sesK_compr› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv4_lkeysec› ‹Kab = sesK (Rs $ sk)› ‹Rs ∉ dom (runs t)› ‹⦃msg.Agent A, msg.Agent B, Nonce Na⦄ ∈ parts (IK t)›*) apply -\n (*goal: ‹(s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_msgs›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; s ∈ m2_inv3a_sesK_compr; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv4_lkeysec; Kab = sesK (Rs $ sk); Rs ∉ dom (runs t); ⦃msg.Agent A, msg.Agent B, Nonce Na⦄ ∈ parts (IK t)⟧ ⟹ (s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_msgs›*)\n by auto\n moreover have \"(?s', ?t') ∈ R23_keys\"\n using H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹s ∈ m2_inv3a_sesK_compr› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv4_lkeysec› ‹Kab = sesK (Rs $ sk)› ‹Rs ∉ dom (runs t)› ‹⦃msg.Agent A, msg.Agent B, Nonce Na⦄ ∈ parts (IK t)›*) apply -\n (*goal: ‹(s⦇runs := (runs s)(Rs::fid_t ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv (A::agent) (Msg [aKey (Kab::key), aAgt (B::agent), aNum (clk (t::m3_state)), aNon (Na::fresh_t)])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan (s::m2_state)))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_keys›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; s ∈ m2_inv3a_sesK_compr; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv4_lkeysec; (Kab::key) = sesK ((Rs::fid_t) $ sk); Rs ∉ dom (runs t); ⦃msg.Agent (A::agent), msg.Agent (B::agent), Nonce (Na::fresh_t)⦄ ∈ parts (IK t)⟧ ⟹ (s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_keys›*)\n by (auto simp add: m2_inv3a_sesK_compr_simps (*‹(?x::m2_state) ∈ m2_inv3a_sesK_compr ⟹ (aKey (?K::key) ∈ extr (aKey ` (?KK::key set) ∪ ik0) (chan ?x)) = (?K ∈ ?KK ∨ aKey ?K ∈ extr ik0 (chan ?x))› ‹(?x::m2_state) ∈ m2_inv3a_sesK_compr ⟹ (aKey (?K::key) ∈ extr (insert (aKey (?Kab::key)) (aKey ` (?KK::key set) ∪ ik0)) (chan ?x)) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ aKey ?K ∈ extr ik0 (chan ?x))› ‹(?x::m2_state) ∈ m2_inv3a_sesK_compr ⟹ (aKey (?K::key) ∈ extr (insert (aKey (?Kab::key)) ik0) (chan ?x)) = (?K = ?Kab ∨ aKey ?K ∈ extr ik0 (chan ?x))› ‹insert (aKey (?K::key)) (insert (?y::atom) (?A::atom set)) = insert ?y (insert (aKey ?K) ?A)›*) m3_inv7a_sesK_compr_simps (*‹⟦(?x::m3_state) ∈ m3_inv7a_sesK_compr; (?KK::key set) ⊆ range sesK⟧ ⟹ (Key (?K::key) ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7a_sesK_compr; (?Kab::key) ∈ range sesK ∧ (?KK::key set) ⊆ range sesK⟧ ⟹ (Key (?K::key) ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦(?x::m3_state) ∈ m3_inv7a_sesK_compr; (?Kab::key) ∈ range sesK⟧ ⟹ (Key (?K::key) ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key (?K::key)) (insert (?y::msg) (?A::msg set)) = insert ?y (insert (Key ?K) ?A)›*) dest: R23_keys_dests (*‹⟦Key (?K::key) ∈ analz (Key ` (?KK::key set) ∪ IK (?t::m3_state)); (?s::m2_state, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key (?K::key) ∈ analz (IK (?t::m3_state)); (?s::m2_state, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key (?K::key) ∈ analz (insert (Key (?Ka::key)) (IK (?t::m3_state))); (?s::m2_state, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key (?K::key) ∈ analz (insert (Key (?Ka::key)) (Key ` (?KK::key set) ∪ IK (?t::m3_state))); (?s::m2_state, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))\n moreover have \"(?s', ?t') ∈ R23_non\"\n using H (*‹(s::m2_state, t::m3_state) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s::m2_state, t::m3_state) ∈ R23_non› ‹(s::m2_state, t::m3_state) ∈ R23_pres› ‹s ∈ m2_inv3a_sesK_compr› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv4_lkeysec› ‹Kab = sesK (Rs $ sk)› ‹(Rs::fid_t) ∉ dom (runs (t::m3_state))› ‹⦃msg.Agent A, msg.Agent B, Nonce Na⦄ ∈ parts (IK t)›*) apply -\n (*goal: ‹(s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_non›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; s ∈ m2_inv3a_sesK_compr; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv4_lkeysec; Kab = sesK (Rs $ sk); Rs ∉ dom (runs t); ⦃msg.Agent A, msg.Agent B, Nonce Na⦄ ∈ parts (IK t)⟧ ⟹ (s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_non›*)\n by (auto simp add: m2_inv3a_sesK_compr_simps (*‹?x ∈ m2_inv3a_sesK_compr ⟹ (aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?x)) = (?K ∈ ?KK ∨ aKey ?K ∈ extr ik0 (chan ?x))› ‹?x ∈ m2_inv3a_sesK_compr ⟹ (aKey ?K ∈ extr (insert (aKey ?Kab) (aKey ` ?KK ∪ ik0)) (chan ?x)) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ aKey ?K ∈ extr ik0 (chan ?x))› ‹?x ∈ m2_inv3a_sesK_compr ⟹ (aKey ?K ∈ extr (insert (aKey ?Kab) ik0) (chan ?x)) = (?K = ?Kab ∨ aKey ?K ∈ extr ik0 (chan ?x))› ‹insert (aKey ?K) (insert ?y ?A) = insert ?y (insert (aKey ?K) ?A)›*) m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) dest: R23_non_dests (*‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))\n moreover have \"(?s', ?t') ∈ R23_pres\"\n using H (*‹(s::m2_state, t::m3_state) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s::m2_state, t::m3_state) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹(s::m2_state) ∈ m2_inv3a_sesK_compr› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv4_lkeysec› ‹Kab = sesK (Rs $ sk)› ‹Rs ∉ dom (runs t)› ‹⦃msg.Agent A, msg.Agent B, Nonce Na⦄ ∈ parts (IK t)›*) apply -\n (*goal: ‹(s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_pres›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; s ∈ m2_inv3a_sesK_compr; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv4_lkeysec; Kab = sesK (Rs $ sk); Rs ∉ dom (runs t); ⦃msg.Agent A, msg.Agent B, Nonce Na⦄ ∈ parts (IK t)⟧ ⟹ (s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_pres›*)\n by auto\n moreover note calculation (*‹(s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_msgs› ‹(s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_keys› ‹(s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_non› ‹(s⦇runs := (runs s)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), chan := insert (StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum (clk t), aNon Na])) (insert (StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum (clk t)])) (chan s))⦈, t⦇runs := (runs t)(Rs ↦ (Serv, [A, B], [aNon Na, aNum (clk t)])), IK := insert (Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number (clk t), Nonce Na, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (clk t)⦄⦄) (IK t)⦈) ∈ R23_pres›*)\n }\n thus \"?thesis\"\n (*goal: ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 Rs A B Kab Na Ts, m3_step3 Rs A B Kab Na Ts {> R23}›*)\n by (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*))\nqed\n\nlemma PO_m3_step4_refines_m2_step4:\n \"{R23 ∩ (UNIV)\n × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)}\n (m2_step4 Ra A B Na Kab Ts Ta), (m3_step4 Ra A B Na Kab Ts Ta X)\n {> R23}\"\nproof (-)\n (*goal: ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 Ra A B Na Kab Ts Ta, m3_step4 Ra A B Na Kab Ts Ta X {> R23}›*)\n {\n fix s and t\n assume H: \"(s, t) ∈ R23_msgs\" \"(s, t) ∈ R23_keys\" \"(s, t) ∈ R23_non\" \"(s, t) ∈ R23_pres\" \"t ∈ m3_inv7a_sesK_compr\" \"t ∈ m3_inv7b_sesK_compr_non\" \"t ∈ m3_inv6_ticket\" \"t ∈ m3_inv4_lkeysec\" \"runs t Ra = Some (Init, [A, B], [])\" \"Na = Ra$na\" \"Crypt (shrK A) ⦃Key Kab, Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)\" (*‹(s::m2_state, t::m3_state) ∈ R23_msgs› ‹(s::m2_state, t::m3_state) ∈ R23_keys› ‹(s::m2_state, t::m3_state) ∈ R23_non› ‹(s::m2_state, t::m3_state) ∈ R23_pres› ‹(t::m3_state) ∈ m3_inv7a_sesK_compr› ‹(t::m3_state) ∈ m3_inv7b_sesK_compr_non› ‹(t::m3_state) ∈ m3_inv6_ticket› ‹(t::m3_state) ∈ m3_inv4_lkeysec› ‹runs (t::m3_state) (Ra::fid_t) = Some (Init, [A::agent, B::agent], [])› ‹(Na::fresh_t) = (Ra::fid_t) $ na› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (t::m3_state))›*)\n let ?s' = \"s⦇ runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])),\n chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s) ⦈\" and ?t' = \"t⦇ runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])),\n IK := insert ⦃ Crypt Kab ⦃Agent A, Number (clk t)⦄, X ⦄ (IK t) ⦈\"\n from H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s::m2_state, t::m3_state) ∈ R23_pres› ‹(t::m3_state) ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) have \"Secure Sv A (Msg [aKey Kab, aAgt B, aNum Ts, aNon Na]) ∈ chan s\"\n by auto\n moreover have \"X ∈ parts (IK t)\"\n using H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹(Na::fresh_t) = (Ra::fid_t) $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) by (auto dest!: Body (*‹Crypt (?K::key) (?X::msg) ∈ parts (?H::msg set) ⟹ ?X ∈ parts ?H›*) MPair_parts (*‹⟦⦃?X::msg, ?Y::msg⦄ ∈ parts (?H::msg set); ⟦?X ∈ parts ?H; ?Y ∈ parts ?H⟧ ⟹ ?P::bool⟧ ⟹ ?P›*))\n hence \"(?s', ?t') ∈ R23_msgs\"\n using H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s::m2_state, t::m3_state) ∈ R23_non› ‹(s::m2_state, t::m3_state) ∈ R23_pres› ‹(t::m3_state) ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (t::m3_state))›*) apply (auto intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goals:\n 1. ‹⋀(K::key) (Aa::agent) T::nat. ⟦(X::msg) ∈ parts (IK (t::m3_state)); (s::m2_state, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t (Ra::fid_t) = Some (Init, [A::agent, B::agent], []); (Na::fresh_t) = Ra $ na; Crypt (shrK A) ⦃Key (Kab::key), msg.Agent B, Number (Ts::nat), Nonce (Ra $ na), X⦄ ∈ analz (IK t); dAuth K (Msg [aAgt Aa, aNum T]) ∉ chan s; Crypt K ⦃msg.Agent Aa, Number T⦄ ∈ parts (IK t)⟧ ⟹ K = Kab›\n 2. ‹⋀(K::key) (Aa::agent) T::nat. ⟦(X::msg) ∈ parts (IK (t::m3_state)); (s::m2_state, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t (Ra::fid_t) = Some (Init, [A::agent, B::agent], []); (Na::fresh_t) = Ra $ na; Crypt (shrK A) ⦃Key (Kab::key), msg.Agent B, Number (Ts::nat), Nonce (Ra $ na), X⦄ ∈ analz (IK t); dAuth K (Msg [aAgt Aa, aNum T]) ∉ chan s; Crypt K ⦃msg.Agent Aa, Number T⦄ ∈ parts (IK t)⟧ ⟹ Aa = A›\n 3. ‹⋀(K::key) (Aa::agent) T::nat. ⟦(X::msg) ∈ parts (IK (t::m3_state)); (s::m2_state, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t (Ra::fid_t) = Some (Init, [A::agent, B::agent], []); (Na::fresh_t) = Ra $ na; Crypt (shrK A) ⦃Key (Kab::key), msg.Agent B, Number (Ts::nat), Nonce (Ra $ na), X⦄ ∈ analz (IK t); dAuth K (Msg [aAgt Aa, aNum T]) ∉ chan s; Crypt K ⦃msg.Agent Aa, Number T⦄ ∈ parts (IK t)⟧ ⟹ T = clk t›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n moreover have \"(?s', ?t') ∈ R23_keys\"\n proof (cases)\n (*goals:\n 1. ‹?P ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›\n 2. ‹¬ ?P ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n assume \"A ∈ bad\" (*‹(A::agent) ∈ bad›*)\n with H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s::m2_state, t::m3_state) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹(Na::fresh_t) = (Ra::fid_t) $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) have \"X ∈ analz (IK t)\"\n apply -\n (*goal: ‹X ∈ analz (IK t)›*)\n apply (drule Decrypt (*‹⟦Crypt ?K ?X ∈ analz ?H; Key (invKey ?K) ∈ analz ?H⟧ ⟹ ?X ∈ analz ?H›*))\n (*goals:\n 1. ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; A ∈ bad⟧ ⟹ Key (invKey (shrK A)) ∈ analz (IK t)›\n 2. ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; A ∈ bad; ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)⟧ ⟹ X ∈ analz (IK t)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n with H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹(t::m3_state) ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹(t::m3_state) ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) show \"?thesis\"\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n apply -\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK (?t::m3_state))) ⊆ chan (?s::m2_state) ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀(KK::key set) K::key. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀(KK::key set) N::fresh_t. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_non› ‹runs (?s::m2_state) = runs (?t::m3_state) ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t); X ∈ analz (IK t)⟧ ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n by (auto dest!: analz_cut (*‹⟦(?Y::msg) ∈ analz (insert (?X::msg) (?H::msg set)); ?X ∈ analz ?H⟧ ⟹ ?Y ∈ analz ?H›*) intro: analz_monotonic (*‹⟦(?c::msg) ∈ analz (?G::msg set); ?G ⊆ (?H::msg set)⟧ ⟹ ?c ∈ analz ?H›*))\n next\n (*goal: ‹A ∉ bad ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n assume \"A ∉ bad\" (*‹(A::agent) ∉ bad›*)\n show \"?thesis\"\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n proof (-)\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n note H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*)\n moreover with ‹A ∉ bad› (*‹(A::agent) ∉ bad›*) have \"X = Crypt (shrK B) ⦃Key Kab, Agent A, Number Ts ⦄ ∧ Kab ∈ range sesK\"\n by (auto dest!: m3_inv6_ticketD (*‹⟦Crypt (shrK ?A) ⦃Key ?K, msg.Agent ?B, Number ?T, Nonce ?N, ?X⦄ ∈ parts (IK ?x); ?x ∈ m3_inv6_ticket; ?A ∉ bad⟧ ⟹ ?X = Crypt (shrK ?B) ⦃Key ?K, msg.Agent ?A, Number ?T⦄ ∧ ?K ∈ range sesK›*))\n moreover {\n assume H1: \"Key (shrK B) ∈ analz (IK t)\" (*‹Key (shrK (B::agent)) ∈ analz (IK (t::m3_state))›*)\n have \"aKey Kab ∈ extr ik0 (chan s)\"\n proof (-)\n (*goal: ‹aKey Kab ∈ extr ik0 (chan s)›*)\n note calculation (*‹(s, t) ∈ R23_msgs› ‹(s::m2_state, t::m3_state) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹(t::m3_state) ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK (A::agent)) ⦃Key (Kab::key), msg.Agent (B::agent), Number (Ts::nat), Nonce (Na::fresh_t), X::msg⦄ ∈ analz (IK (t::m3_state))› ‹(X::msg) = Crypt (shrK (B::agent)) ⦃Key (Kab::key), msg.Agent (A::agent), Number (Ts::nat)⦄ ∧ Kab ∈ range sesK›*)\n moreover hence \"Secure Sv B (Msg [aKey Kab, aAgt A, aNum Ts]) ∈ chan s\"\n apply -\n (*goal: ‹StatCh secure Sv (B::agent) (Msg [aKey (Kab::key), aAgt (A::agent), aNum (Ts::nat)]) ∈ chan (s::m2_state)›*)\n apply (drule analz_into_parts (*‹?c ∈ analz ?H ⟹ ?c ∈ parts ?H›*))\n (*goal: ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t); X = Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number Ts⦄ ∧ Kab ∈ range sesK⟧ ⟹ StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum Ts]) ∈ chan s›*)\n apply (drule Body (*‹Crypt ?K ?X ∈ parts ?H ⟹ ?X ∈ parts ?H›*))\n (*goal: ‹⟦(s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; X = Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number Ts⦄ ∧ Kab ∈ range sesK; Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ parts (IK t)⟧ ⟹ StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum Ts]) ∈ chan s›*)\n apply (elim MPair_parts (*‹⟦⦃?X, ?Y⦄ ∈ parts ?H; ⟦?X ∈ parts ?H; ?Y ∈ parts ?H⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goal: ‹⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t (Ra::fid_t) = Some (Init, [A::agent, B::agent], []); (Na::fresh_t) = Ra $ na; (X::msg) = Crypt (shrK B) ⦃Key (Kab::key), msg.Agent A, Number (Ts::nat)⦄ ∧ Kab ∈ range sesK; ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ parts (IK t)⟧ ⟹ StatCh secure Sv B (Msg [aKey Kab, aAgt A, aNum Ts]) ∈ chan s›*)\n by auto\n ultimately show \"?thesis\"\n (*goal: ‹aKey Kab ∈ extr ik0 (chan s)›*)\n using H1 (*‹Key (shrK (B::agent)) ∈ analz (IK (t::m3_state))›*) by auto\n qed\n }\n ultimately show \"?thesis\"\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n apply -\n (*goal: ‹(s⦇runs := (runs s)(Ra::fid_t ↦ (Init, [A, B::agent], [aKey Kab, aNum (Ts::nat), aNum (clk t)])), chan := insert (dAuth (Kab::key) (Msg [aAgt (A::agent), aNum (clk (t::m3_state))])) (chan (s::m2_state))⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X::msg⦄ (IK t)⦈) ∈ R23_keys›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK (?t::m3_state))) ⊆ chan (?s::m2_state) ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀(KK::key set) K::key. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀(KK::key set) N::fresh_t. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_non› ‹runs (?s::m2_state) = runs (?t::m3_state) ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦(s::m2_state, t::m3_state) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t (Ra::fid_t) = Some (Init, [A::agent, B::agent], []); (Na::fresh_t) = Ra $ na; Crypt (shrK A) ⦃Key (Kab::key), msg.Agent B, Number (Ts::nat), Nonce Na, X::msg⦄ ∈ analz (IK t); X = Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number Ts⦄ ∧ Kab ∈ range sesK; Key (shrK B) ∈ analz (IK t) ⟹ aKey Kab ∈ extr ik0 (chan s)⟧ ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_keys›*)\n by (auto simp add: m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*))\n qed\n qed\n moreover have \"(?s', ?t') ∈ R23_non\"\n proof (cases)\n (*goals:\n 1. ‹?P::bool ⟹ (s⦇runs := (runs s)(Ra::fid_t ↦ (Init, [A, B::agent], [aKey Kab, aNum (Ts::nat), aNum (clk t)])), chan := insert (dAuth (Kab::key) (Msg [aAgt (A::agent), aNum (clk (t::m3_state))])) (chan (s::m2_state))⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X::msg⦄ (IK t)⦈) ∈ R23_non›\n 2. ‹¬ (?P::bool) ⟹ (s⦇runs := (runs s)(Ra::fid_t ↦ (Init, [A, B::agent], [aKey Kab, aNum (Ts::nat), aNum (clk t)])), chan := insert (dAuth (Kab::key) (Msg [aAgt (A::agent), aNum (clk (t::m3_state))])) (chan (s::m2_state))⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X::msg⦄ (IK t)⦈) ∈ R23_non›*)\n assume \"A ∈ bad\" (*‹(A::agent) ∈ bad›*)\n hence \"X ∈ analz (IK t)\"\n using H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹(t::m3_state) ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) apply -\n (*goal: ‹X ∈ analz (IK t)›*)\n apply (drule Decrypt (*‹⟦Crypt ?K ?X ∈ analz ?H; Key (invKey ?K) ∈ analz ?H⟧ ⟹ ?X ∈ analz ?H›*))\n (*goals:\n 1. ‹⟦A ∈ bad; (s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na⟧ ⟹ Key (invKey (shrK A)) ∈ analz (IK t)›\n 2. ‹⟦A ∈ bad; (s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)⟧ ⟹ X ∈ analz (IK t)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_non›*)\n using H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹(t::m3_state) ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) apply -\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_non›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK (?t::m3_state))) ⊆ chan (?s::m2_state) ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀(KK::key set) K::key. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀(KK::key set) N::fresh_t. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_non› ‹runs (?s::m2_state) = runs (?t::m3_state) ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦X ∈ analz (IK t); (s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)⟧ ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_non›*)\n by (auto dest!: analz_cut (*‹⟦?Y ∈ analz (insert ?X ?H); ?X ∈ analz ?H⟧ ⟹ ?Y ∈ analz ?H›*) intro: analz_monotonic (*‹⟦?c ∈ analz ?G; ?G ⊆ ?H⟧ ⟹ ?c ∈ analz ?H›*))\n next\n (*goal: ‹A ∉ bad ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_non›*)\n assume \"A ∉ bad\" (*‹(A::agent) ∉ bad›*)\n hence \"X = Crypt (shrK B) ⦃Key Kab, Agent A, Number Ts ⦄ ∧ Kab ∈ range sesK\"\n using H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) by (auto dest!: m3_inv6_ticketD (*‹⟦Crypt (shrK ?A) ⦃Key ?K, msg.Agent ?B, Number ?T, Nonce ?N, ?X⦄ ∈ parts (IK ?x); ?x ∈ m3_inv6_ticket; ?A ∉ bad⟧ ⟹ ?X = Crypt (shrK ?B) ⦃Key ?K, msg.Agent ?A, Number ?T⦄ ∧ ?K ∈ range sesK›*))\n thus \"?thesis\"\n (*goal: ‹(s⦇runs := (runs s)(Ra::fid_t ↦ (Init, [A, B::agent], [aKey Kab, aNum (Ts::nat), aNum (clk t)])), chan := insert (dAuth (Kab::key) (Msg [aAgt (A::agent), aNum (clk (t::m3_state))])) (chan (s::m2_state))⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X::msg⦄ (IK t)⦈) ∈ R23_non›*)\n using H (*‹(s, t) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹t ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) apply -\n (*goal: ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_non›*)\n apply (rule R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n (*goal: ‹⟦X = Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number Ts⦄ ∧ Kab ∈ range sesK; (s, t) ∈ R23_msgs; (s, t) ∈ R23_keys; (s, t) ∈ R23_non; (s, t) ∈ R23_pres; t ∈ m3_inv7a_sesK_compr; t ∈ m3_inv7b_sesK_compr_non; t ∈ m3_inv6_ticket; t ∈ m3_inv4_lkeysec; runs t Ra = Some (Init, [A, B], []); Na = Ra $ na; Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)⟧ ⟹ (s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_non›*)\n by (auto simp add: m3_inv7a_sesK_compr_simps (*‹⟦?x ∈ m3_inv7a_sesK_compr; ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (Key ` ?KK ∪ IK ?x)) = (?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (?K = ?Kab ∨ ?K ∈ ?KK ∨ Key ?K ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7a_sesK_compr; ?Kab ∈ range sesK⟧ ⟹ (Key ?K ∈ analz (insert (Key ?Kab) (IK ?x))) = (?K = ?Kab ∨ Key ?K ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*) m3_inv7b_sesK_compr_non_simps (*‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?x)) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK ∧ ?KK ⊆ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (Key ` ?KK ∪ IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹⟦?x ∈ m3_inv7b_sesK_compr_non; ?Kab ∈ range sesK⟧ ⟹ (Nonce ?N ∈ analz (insert (Key ?Kab) (IK ?x))) = (Nonce ?N ∈ analz (IK ?x))› ‹insert (Key ?K) (insert ?y ?A) = insert ?y (insert (Key ?K) ?A)›*))\n qed\n moreover have \"(?s', ?t') ∈ R23_pres\"\n using H (*‹(s::m2_state, t::m3_state) ∈ R23_msgs› ‹(s, t) ∈ R23_keys› ‹(s, t) ∈ R23_non› ‹(s, t) ∈ R23_pres› ‹t ∈ m3_inv7a_sesK_compr› ‹(t::m3_state) ∈ m3_inv7b_sesK_compr_non› ‹t ∈ m3_inv6_ticket› ‹t ∈ m3_inv4_lkeysec› ‹runs t Ra = Some (Init, [A, B], [])› ‹Na = Ra $ na› ‹Crypt (shrK A) ⦃Key Kab, msg.Agent B, Number Ts, Nonce Na, X⦄ ∈ analz (IK t)›*) by (auto intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n moreover note calculation (*‹StatCh secure Sv A (Msg [aKey Kab, aAgt B, aNum Ts, aNon Na]) ∈ chan s› ‹(s⦇runs := (runs s)(Ra::fid_t ↦ (Init, [A, B::agent], [aKey Kab, aNum (Ts::nat), aNum (clk t)])), chan := insert (dAuth (Kab::key) (Msg [aAgt (A::agent), aNum (clk (t::m3_state))])) (chan (s::m2_state))⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X::msg⦄ (IK t)⦈) ∈ R23_msgs› ‹(s⦇runs := (runs s)(Ra::fid_t ↦ (Init, [A, B::agent], [aKey Kab, aNum (Ts::nat), aNum (clk t)])), chan := insert (dAuth (Kab::key) (Msg [aAgt (A::agent), aNum (clk (t::m3_state))])) (chan (s::m2_state))⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X::msg⦄ (IK t)⦈) ∈ R23_keys› ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_non› ‹(s⦇runs := (runs s)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), chan := insert (dAuth Kab (Msg [aAgt A, aNum (clk t)])) (chan s)⦈, t⦇runs := (runs t)(Ra ↦ (Init, [A, B], [aKey Kab, aNum Ts, aNum (clk t)])), IK := insert ⦃Crypt Kab ⦃msg.Agent A, Number (clk t)⦄, X⦄ (IK t)⦈) ∈ R23_pres›*)\n }\n thus \"?thesis\"\n (*goal: ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 Ra A B Na Kab Ts Ta, m3_step4 Ra A B Na Kab Ts Ta X {> R23}›*)\n by (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) dest!: analz_Inj_IK (*‹?X ∈ IK ?s ⟹ ?X ∈ analz (IK ?s)›*))\nqed\n\nlemma PO_m3_step5_refines_m2_step5:\n \"{R23}\n (m2_step5 Rb A B Kab Ts Ta), (m3_step5 Rb A B Kab Ts Ta)\n {> R23}\"\napply (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n(*goals:\n 1. ‹⋀(a::m2_state) (y::m3_state) (K::key) T::nat. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; runs y (Rb::fid_t) = Some (Resp, [A::agent, B::agent], []); ⦃Crypt (Kab::key) ⦃msg.Agent A, Number (Ta::nat)⦄, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (Ts::nat)⦄⦄ ∈ parts (IK y); clk y < Ts + Ls; clk y < Ta + La; (B, Kab, Ta) ∉ cache y; dAuth K (Msg [aNum T]) ∉ chan a; Crypt K (Number T) ∈ parts (IK y)⟧ ⟹ K = Kab›\n 2. ‹⋀(a::m2_state) (y::m3_state) (K::key) T::nat. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; runs y (Rb::fid_t) = Some (Resp, [A::agent, B::agent], []); ⦃Crypt (Kab::key) ⦃msg.Agent A, Number (Ta::nat)⦄, Crypt (shrK B) ⦃Key Kab, msg.Agent A, Number (Ts::nat)⦄⦄ ∈ parts (IK y); clk y < Ts + Ls; clk y < Ta + La; (B, Kab, Ta) ∉ cache y; dAuth K (Msg [aNum T]) ∉ chan a; Crypt K (Number T) ∈ parts (IK y)⟧ ⟹ T = Ta›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\nlemma PO_m3_step6_refines_m2_step6:\n \"{R23}\n (m2_step6 Ra A B Na Kab Ts Ta), (m3_step6 Ra A B Na Kab Ts Ta)\n {> R23}\"\nby (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n\nlemma PO_m3_tick_refines_m2_tick:\n \"{R23}\n (m2_tick T), (m3_tick T)\n {>R23}\"\nby (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n\nlemma PO_m3_purge_refines_m2_purge:\n \"{R23}\n (m2_purge A), (m3_purge A)\n {>R23}\"\nby (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n\n\ntext ‹Intruder events.›\n\nlemma PO_m3_leak_refines_m2_leak:\n \"{R23}\n (m2_leak Rs A B Na Ts), (m3_leak Rs A B Na Ts)\n {>R23}\"\napply (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n(*goals:\n 1. ‹⋀a y KK K. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; runs y Rs = Some (Serv, [A, B], [aNon Na, aNum Ts]); Ts + Ls ≤ clk y; KK ⊆ range sesK; Key K ∈ analz (insert (Key (sesK (Rs $ sk))) (Key ` KK ∪ IK y))⟧ ⟹ aKey K ∈ extr (insert (aKey (sesK (Rs $ sk))) (aKey ` KK ∪ ik0)) (chan a)›\n 2. ‹⋀a y KK N. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; runs y Rs = Some (Serv, [A, B], [aNon Na, aNum Ts]); Ts + Ls ≤ clk y; KK ⊆ range sesK; Nonce N ∈ analz (insert (Key (sesK (Rs $ sk))) (Key ` KK ∪ IK y))⟧ ⟹ aNon N ∈ extr (insert (aKey (sesK (Rs $ sk))) (aKey ` KK ∪ ik0)) (chan a)›\ndiscuss goal 1*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n(*discuss goal 2*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n(*proven 2 subgoals*) .\n\nlemma PO_m3_DY_fake_refines_m2_fake:\n \"{R23 ∩ UNIV × (m3_inv4_lkeysec)}\n m2_fake, m3_DY_fake\n {> R23}\"\napply (auto simp add: PO_rhoare_defs (*‹{?pre} ?Ra, ?Rc {> ?post} ≡ ?pre O ?Rc ⊆ ?Ra O ?post› ‹?r O ?s = {(x, z). ∃y. (x, y) ∈ ?r ∧ (y, z) ∈ ?s}›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK ?t)) ⊆ chan ?s ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀KK K. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀KK N. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK ?t)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan ?s)) ⟹ (?s, ?t) ∈ R23_non› ‹runs ?s = runs ?t ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*) del: abs_msg.cases (*‹⟦?a ∈ abs_msg ?H; ⋀A B N. ⟦?a = Insec A B (Msg [aNon N]); ⦃msg.Agent A, msg.Agent B, Nonce N⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K B T N X. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]); Crypt (shrK C) ⦃Key K, msg.Agent B, Number T, Nonce N, X⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K A T. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt A, aNum T]); Crypt (shrK C) ⦃Key K, msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K A T. ⟦?a = dAuth K (Msg [aAgt A, aNum T]); Crypt K ⦃msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K T. ⟦?a = dAuth K (Msg [aNum T]); Crypt K (Number T) ∈ ?H⟧ ⟹ ?P⟧ ⟹ ?P›*))\n(*goals:\n 1. ‹⋀a y x. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; y ∈ m3_inv4_lkeysec; x ∈ abs_msg (parts (IK y))⟧ ⟹ x ∈ fake ik0 (dom (runs a)) (chan a)›\n 2. ‹⋀a y x. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; y ∈ m3_inv4_lkeysec; x ∈ abs_msg (synth (analz (IK y)))⟧ ⟹ x ∈ fake ik0 (dom (runs a)) (chan a)›\n 3. ‹⋀a y KK K. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; y ∈ m3_inv4_lkeysec; KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK y)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan a)›\n 4. ‹⋀a y KK K. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; y ∈ m3_inv4_lkeysec; KK ⊆ range sesK; Key K ∈ analz (IK y)⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan a)›\n 5. ‹⋀a y KK N. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; y ∈ m3_inv4_lkeysec; KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK y)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan a)›\n 6. ‹⋀a y KK N. ⟦(a, y) ∈ R23_msgs; (a, y) ∈ R23_keys; (a, y) ∈ R23_non; (a, y) ∈ R23_pres; y ∈ m3_inv4_lkeysec; KK ⊆ range sesK; Nonce N ∈ analz (IK y)⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan a)›\ndiscuss goal 1*)\n apply ((auto intro: abs_msg_DY_subset_fakeable [THEN subsetD] (*‹⟦(?s1, ?t1) ∈ R23_msgs; (?s1, ?t1) ∈ R23_keys; (?s1, ?t1) ∈ R23_non; ?t1 ∈ m3_inv4_lkeysec; ?c ∈ abs_msg (synth (analz (IK ?t1)))⟧ ⟹ ?c ∈ fake ik0 (dom (runs ?s1)) (chan ?s1)›*) del: abs_msg.cases (*‹⟦?a ∈ abs_msg ?H; ⋀A B N. ⟦?a = Insec A B (Msg [aNon N]); ⦃msg.Agent A, msg.Agent B, Nonce N⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K B T N X. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]); Crypt (shrK C) ⦃Key K, msg.Agent B, Number T, Nonce N, X⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K A T. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt A, aNum T]); Crypt (shrK C) ⦃Key K, msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K A T. ⟦?a = dAuth K (Msg [aAgt A, aNum T]); Crypt K ⦃msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K T. ⟦?a = dAuth K (Msg [aNum T]); Crypt K (Number T) ∈ ?H⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n(*discuss goal 2*)\n apply ((auto intro: abs_msg_DY_subset_fakeable [THEN subsetD] (*‹⟦(?s1, ?t1) ∈ R23_msgs; (?s1, ?t1) ∈ R23_keys; (?s1, ?t1) ∈ R23_non; ?t1 ∈ m3_inv4_lkeysec; ?c ∈ abs_msg (synth (analz (IK ?t1)))⟧ ⟹ ?c ∈ fake ik0 (dom (runs ?s1)) (chan ?s1)›*) del: abs_msg.cases (*‹⟦?a ∈ abs_msg ?H; ⋀A B N. ⟦?a = Insec A B (Msg [aNon N]); ⦃msg.Agent A, msg.Agent B, Nonce N⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K B T N X. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]); Crypt (shrK C) ⦃Key K, msg.Agent B, Number T, Nonce N, X⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K A T. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt A, aNum T]); Crypt (shrK C) ⦃Key K, msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K A T. ⟦?a = dAuth K (Msg [aAgt A, aNum T]); Crypt K ⦃msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K T. ⟦?a = dAuth K (Msg [aNum T]); Crypt K (Number T) ∈ ?H⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n(*discuss goal 3*)\n apply ((auto intro: abs_msg_DY_subset_fakeable [THEN subsetD] (*‹⟦(?s1::m2_state, ?t1::m3_state) ∈ R23_msgs; (?s1, ?t1) ∈ R23_keys; (?s1, ?t1) ∈ R23_non; ?t1 ∈ m3_inv4_lkeysec; (?c::chmsg) ∈ abs_msg (synth (analz (IK ?t1)))⟧ ⟹ ?c ∈ fake ik0 (dom (runs ?s1)) (chan ?s1)›*) del: abs_msg.cases (*‹⟦(?a::chmsg) ∈ abs_msg (?H::msg set); ⋀(A::agent) (B::agent) N::fresh_t. ⟦?a = Insec A B (Msg [aNon N]); ⦃msg.Agent A, msg.Agent B, Nonce N⦄ ∈ ?H⟧ ⟹ ?P::bool; ⋀(C::agent) (K::key) (B::agent) (T::nat) (N::fresh_t) X::msg. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]); Crypt (shrK C) ⦃Key K, msg.Agent B, Number T, Nonce N, X⦄ ∈ ?H⟧ ⟹ ?P; ⋀(C::agent) (K::key) (A::agent) T::nat. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt A, aNum T]); Crypt (shrK C) ⦃Key K, msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀(K::key) (A::agent) T::nat. ⟦?a = dAuth K (Msg [aAgt A, aNum T]); Crypt K ⦃msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀(K::key) T::nat. ⟦?a = dAuth K (Msg [aNum T]); Crypt K (Number T) ∈ ?H⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n(*discuss goal 4*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n(*discuss goal 5*)\n apply ((auto intro: abs_msg_DY_subset_fakeable [THEN subsetD] (*‹⟦(?s1, ?t1) ∈ R23_msgs; (?s1, ?t1) ∈ R23_keys; (?s1, ?t1) ∈ R23_non; ?t1 ∈ m3_inv4_lkeysec; ?c ∈ abs_msg (synth (analz (IK ?t1)))⟧ ⟹ ?c ∈ fake ik0 (dom (runs ?s1)) (chan ?s1)›*) del: abs_msg.cases (*‹⟦?a ∈ abs_msg ?H; ⋀A B N. ⟦?a = Insec A B (Msg [aNon N]); ⦃msg.Agent A, msg.Agent B, Nonce N⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K B T N X. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt B, aNum T, aNon N]); Crypt (shrK C) ⦃Key K, msg.Agent B, Number T, Nonce N, X⦄ ∈ ?H⟧ ⟹ ?P; ⋀C K A T. ⟦?a = StatCh secure Sv C (Msg [aKey K, aAgt A, aNum T]); Crypt (shrK C) ⦃Key K, msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K A T. ⟦?a = dAuth K (Msg [aAgt A, aNum T]); Crypt K ⦃msg.Agent A, Number T⦄ ∈ ?H⟧ ⟹ ?P; ⋀K T. ⟦?a = dAuth K (Msg [aNum T]); Crypt K (Number T) ∈ ?H⟧ ⟹ ?P⟧ ⟹ ?P›*))[1])\n(*discuss goal 6*)\n apply ((auto dest: R23_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)› ‹⟦Nonce ?N ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_non; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (IK ?t); (?s, ?t) ∈ R23_non⟧ ⟹ aNon ?N ∈ extr ik0 (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) ik0) (chan ?s)› ‹⟦Nonce ?N ∈ analz (insert (Key ?K) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_non; ?K ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aNon ?N ∈ extr (insert (aKey ?K) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))[1])\n(*proven 6 subgoals*) .\n\n\ntext ‹All together now...›\n\nlemmas PO_m3_trans_refines_m2_trans =\n PO_m3_step1_refines_m2_step1 PO_m3_step2_refines_m2_step2\n PO_m3_step3_refines_m2_step3 PO_m3_step4_refines_m2_step4\n PO_m3_step5_refines_m2_step5 PO_m3_step6_refines_m2_step6\n PO_m3_tick_refines_m2_tick PO_m3_purge_refines_m2_purge\n PO_m3_leak_refines_m2_leak PO_m3_DY_fake_refines_m2_fake\n\n\nlemma PO_m3_refines_init_m2 [iff]:\n \"init m3 ⊆ R23``(init m2)\"\nby (auto simp add: R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx::fid_t. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0::nat, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃(A::agent) (B::agent) (Ra::fid_t) (Rb::fid_t) (Rs::fid_t) (Na::fresh_t) (Kab::key) (Ts::nat) (Ta::nat) (T::nat) X::msg. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs (?s::m3_state) ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) ≡ {(s::m3_state, s1::m3_state). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat) ≡ {(s::m3_state, s1::m3_state). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) (?X::msg) ≡ {(s::m3_state, s1::m3_state). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s1::m3_state). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat) ≡ {(s::m3_state, s'::m3_state). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*) intro!: R23_intros (*‹abs_msg (parts (IK (?t::m3_state))) ⊆ chan (?s::m2_state) ⟹ (?s, ?t) ∈ R23_msgs› ‹(⋀(KK::key set) K::key. ⟦KK ⊆ range sesK; Key K ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aKey K ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_keys› ‹(⋀(KK::key set) N::fresh_t. ⟦KK ⊆ range sesK; Nonce N ∈ analz (Key ` KK ∪ IK (?t::m3_state))⟧ ⟹ aNon N ∈ extr (aKey ` KK ∪ ik0) (chan (?s::m2_state))) ⟹ (?s, ?t) ∈ R23_non› ‹runs (?s::m2_state) = runs (?t::m3_state) ∧ m1_state.leak ?s = m1_state.leak ?t ∧ clk ?s = clk ?t ∧ cache ?s = cache ?t ⟹ (?s, ?t) ∈ R23_pres›*))\n\nlemma PO_m3_refines_trans_m2 [iff]:\n \"{R23 ∩ (m2_inv3a_sesK_compr)\n × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)}\n (trans m2), (trans m3)\n {> R23}\"\napply (auto simp add: m3_def (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈›*) m3_trans_def (*‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼›*) m2_def (*‹m2 ≡ ⦇init = m2_init, trans = m2_trans, obs = m2_obs⦈›*) m2_trans_def (*‹m2_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T. (m2_step1 Ra A B Na ∪ m2_step2 Rb A B ∪ m2_step3 Rs A B Kab Na Ts ∪ m2_step4 Ra A B Na Kab Ts Ta ∪ m2_step5 Rb A B Kab Ts Ta ∪ m2_step6 Ra A B Na Kab Ts Ta ∪ m2_tick T ∪ m2_purge A ∪ m2_leak Rs A B Na Ts ∪ m2_fake)⁼›*))\n(*goals:\n 1. ‹⋀(x::agent) (xa::agent) (xb::fid_t) xe::fresh_t. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_step1 xb x xa xe {> R23}›\n 2. ‹⋀(x::agent) (xa::agent) xc::fid_t. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_step2 xc x xa {> R23}›\n 3. ‹⋀(x::agent) (xa::agent) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_step3 xd x xa xf xe xg {> R23}›\n 4. ‹⋀(x::agent) (xa::agent) (xb::fid_t) (xe::fresh_t) (xf::key) (xg::nat) (xh::nat) xi::msg. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_step4 xb x xa xe xf xg xh xi {> R23}›\n 5. ‹⋀(x::agent) (xa::agent) (xc::fid_t) (xf::key) (xg::nat) xh::nat. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_step5 xc x xa xf xg xh {> R23}›\n 6. ‹⋀(x::agent) (xa::agent) (xb::fid_t) (xe::fresh_t) (xf::key) (xg::nat) xh::nat. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_step6 xb x xa xe xf xg xh {> R23}›\n 7. ‹⋀xh::nat. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_tick xh {> R23}›\n 8. ‹⋀x::agent. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_purge x {> R23}›\n 9. ‹⋀(x::agent) (xa::agent) (xd::fid_t) (xe::fresh_t) xg::nat. {R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_leak xd x xa xe xg {> R23}›\n 10. ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} ((⋃(x::agent) (xa::agent) (xb::fid_t) (xc::fid_t) (xd::fid_t) (xe::fresh_t) (xf::key) xg::nat. (⋃xh::nat. m2_step1 xb x xa xe ∪ m2_step2 xc x xa ∪ m2_step3 xd x xa xf xe xg ∪ m2_step4 xb x xa xe xf xg xh ∪ m2_step5 xc x xa xf xg xh ∪ m2_step6 xb x xa xe xf xg xh) ∪ ⋃ (range m2_tick) ∪ m2_purge x ∪ m2_leak xd x xa xe xg) ∪ m2_fake)⁼, m3_DY_fake {> R23}›\ndiscuss goal 1*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 2*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 3*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 4*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 5*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 6*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 7*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 8*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 9*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 ?Ra ?A ?B ?Na, m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 ?Rb ?A ?B, m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 ?Rs ?A ?B ?Kab ?Na ?Ts, m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X {> R23}› ‹{R23} m2_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta, m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta, m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick ?T, m3_tick ?T {> R23}› ‹{R23} m2_purge ?A, m3_purge ?A {> R23}› ‹{R23} m2_leak ?Rs ?A ?B ?Na ?Ts, m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*discuss goal 10*)\n apply (blast intro!: PO_m3_trans_refines_m2_trans (*‹{R23} m2_step1 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t), m3_step1 ?Ra ?A ?B ?Na {> R23}› ‹{R23} m2_step2 (?Rb::fid_t) (?A::agent) (?B::agent), m3_step2 ?Rb ?A ?B {> R23}› ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv4_lkeysec)} m2_step3 (?Rs::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Na::fresh_t) (?Ts::nat), m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} m2_step4 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat), m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta (?X::msg) {> R23}› ‹{R23} m2_step5 (?Rb::fid_t) (?A::agent) (?B::agent) (?Kab::key) (?Ts::nat) (?Ta::nat), m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_step6 (?Ra::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Kab::key) (?Ts::nat) (?Ta::nat), m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta {> R23}› ‹{R23} m2_tick (?T::nat), m3_tick ?T {> R23}› ‹{R23} m2_purge (?A::agent), m3_purge ?A {> R23}› ‹{R23} m2_leak (?Rs::fid_t) (?A::agent) (?B::agent) (?Na::fresh_t) (?Ts::nat), m3_leak ?Rs ?A ?B ?Na ?Ts {> R23}› ‹{R23 ∩ UNIV × m3_inv4_lkeysec} m2_fake, m3_DY_fake {> R23}›*))\n(*proven 10 subgoals*) .\n\nlemma PO_m3_observation_consistent [iff]:\n \"obs_consistent R23 med32 m2 m3\"\nby (auto simp add: obs_consistent_def (*‹obs_consistent ?R ?pi ?Sa ?Sc ≡ ∀s t. (s, t) ∈ ?R ⟶ ?pi (obs ?Sc t) = obs ?Sa s›*) R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) med32_def (*‹med32 ≡ id›*) m3_defs (*‹m3 ≡ ⦇init = m3_init, trans = m3_trans, obs = m3_obs⦈› ‹m3_init ≡ {⦇runs = λx. None, m1_state.leak = shrK ` bad × {undefined}, clk = 0, cache = {}, IK = Key ` shrK ` bad⦈}› ‹m3_trans ≡ ⋃A B Ra Rb Rs Na Kab Ts Ta T X. (m3_step1 Ra A B Na ∪ m3_step2 Rb A B ∪ m3_step3 Rs A B Kab Na Ts ∪ m3_step4 Ra A B Na Kab Ts Ta X ∪ m3_step5 Rb A B Kab Ts Ta ∪ m3_step6 Ra A B Na Kab Ts Ta ∪ m3_tick T ∪ m3_purge A ∪ m3_leak Rs A B Na Ts ∪ m3_DY_fake)⁼› ‹m3_obs ?s ≡ ⦇runs = runs ?s, m1_state.leak = m1_state.leak ?s, clk = clk ?s, cache = cache ?s⦈› ‹m3_step1 ?Ra ?A ?B ?Na ≡ {(s, s1). ?Ra ∉ dom (runs s) ∧ ?Na = ?Ra $ na ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [])), IK := insert ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ (IK s)⦈}› ‹m3_step2 ≡ m1_step2› ‹m3_step3 ?Rs ?A ?B ?Kab ?Na ?Ts ≡ {(s, s1). ?Rs ∉ dom (runs s) ∧ ?Kab = sesK (?Rs $ sk) ∧ ⦃msg.Agent ?A, msg.Agent ?B, Nonce ?Na⦄ ∈ IK s ∧ ?Ts = clk s ∧ s1 = s⦇runs := (runs s)(?Rs ↦ (Serv, [?A, ?B], [aNon ?Na, aNum ?Ts])), IK := insert (Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄) (IK s)⦈}› ‹m3_step4 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ?X ≡ {(s, s1). runs s ?Ra = Some (Init, [?A, ?B], []) ∧ ?Na = ?Ra $ na ∧ Crypt (shrK ?A) ⦃Key ?Kab, msg.Agent ?B, Number ?Ts, Nonce ?Na, ?X⦄ ∈ IK s ∧ ?Ta = clk s ∧ clk s < ?Ts + Ls ∧ s1 = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), IK := insert ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, ?X⦄ (IK s)⦈}› ‹m3_step5 ?Rb ?A ?B ?Kab ?Ts ?Ta ≡ {(s, s1). runs s ?Rb = Some (Resp, [?A, ?B], []) ∧ ⦃Crypt ?Kab ⦃msg.Agent ?A, Number ?Ta⦄, Crypt (shrK ?B) ⦃Key ?Kab, msg.Agent ?A, Number ?Ts⦄⦄ ∈ IK s ∧ clk s < ?Ts + Ls ∧ clk s < ?Ta + La ∧ (?B, ?Kab, ?Ta) ∉ cache s ∧ s1 = s⦇runs := (runs s)(?Rb ↦ (Resp, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta])), cache := insert (?B, ?Kab, ?Ta) (cache s), IK := insert (Crypt ?Kab (Number ?Ta)) (IK s)⦈}› ‹m3_step6 ?Ra ?A ?B ?Na ?Kab ?Ts ?Ta ≡ {(s, s'). runs s ?Ra = Some (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta]) ∧ ?Na = ?Ra $ na ∧ clk s < ?Ts + Ls ∧ Crypt ?Kab (Number ?Ta) ∈ IK s ∧ s' = s⦇runs := (runs s)(?Ra ↦ (Init, [?A, ?B], [aKey ?Kab, aNum ?Ts, aNum ?Ta, END]))⦈}› ‹m3_tick ≡ m1_tick› ‹m3_purge ≡ m1_purge› and more 46 facts*))\n\n\ntext ‹Refinement result.›\n\nlemma m3_refines_m2 [iff]:\n \"refines\n (R23 ∩\n (m2_inv3a_sesK_compr) ×\n (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec))\n med32 m2 m3\"\napply (rule Refinement_using_invariants (*‹⟦{?R ∩ ?I × ?J} TS.trans ?Sa, TS.trans ?Sc {> ?R}; {?I ∩ Domain (?R ∩ UNIV × ?J)} TS.trans ?Sa {> ?I}; {?J ∩ Range (?R ∩ ?I × UNIV)} TS.trans ?Sc {> ?J}; init ?Sc ⊆ ?R `` init ?Sa; init ?Sa ⊆ ?I; init ?Sc ⊆ ?J; obs_consistent (?R ∩ ?I × ?J) ?pi ?Sa ?Sc⟧ ⟹ refines (?R ∩ ?I × ?J) ?pi ?Sa ?Sc›*))\n(*goals:\n 1. ‹{R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)} TS.trans m2, TS.trans m3 {> R23}›\n 2. ‹{m2_inv3a_sesK_compr ∩ Domain (R23 ∩ UNIV × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec))} TS.trans m2 {> m2_inv3a_sesK_compr}›\n 3. ‹{m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec ∩ Range (R23 ∩ m2_inv3a_sesK_compr × UNIV)} TS.trans m3 {> m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec}›\n 4. ‹init m3 ⊆ R23 `` init m2›\n 5. ‹init m2 ⊆ m2_inv3a_sesK_compr›\n 6. ‹init m3 ⊆ m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec›\n 7. ‹obs_consistent (R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)) med32 m2 m3›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*discuss goal 4*)\n apply ((auto)[1])\n(*discuss goal 5*)\n apply ((auto)[1])\n(*discuss goal 6*)\n apply ((auto)[1])\n(*discuss goal 7*)\n apply ((auto)[1])\n(*proven 7 subgoals*) .\n\nlemma m3_implements_m2 [iff]:\n \"implements med32 m2 m3\"\napply (rule refinement_soundness (*‹refines ?R ?pi ?Sa ?Sc ⟹ implements ?pi ?Sa ?Sc›*))\n(*goal: ‹implements med32 m2 m3›*)\nby auto\n\n\nsubsection ‹Inherited invariants›\n(******************************************************************************)\n\nsubsubsection ‹inv3 (derived): Key secrecy for initiator›\n(*invh*************************************************************************)\n\ndefinition\n m3_inv3_ikk_init :: \"m3_state set\"\nwhere\n \"m3_inv3_ikk_init ≡ {s. ∀A B Ra K Ts nl.\n runs s Ra = Some (Init, [A, B], aKey K # aNum Ts # nl) ⟶ A ∈ good ⟶ B ∈ good ⟶\n Key K ∈ analz (IK s) ⟶\n (K, A, B, Ra$na, Ts) ∈ leak s\n }\"\n\nlemmas m3_inv3_ikk_initI = m3_inv3_ikk_init_def [THEN setc_def_to_intro, rule_format]\nlemmas m3_inv3_ikk_initE [elim] = m3_inv3_ikk_init_def [THEN setc_def_to_elim, rule_format]\nlemmas m3_inv3_ikk_initD = m3_inv3_ikk_init_def [THEN setc_def_to_dest, rule_format, rotated 1]\n\nlemma PO_m3_inv3_ikk_init: \"reach m3 ⊆ m3_inv3_ikk_init\"\nproof (rule INV_from_Refinement_using_invariants [OF m3_refines_m2] (*‹⟦Range (R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec) ∩ ?I × ?J) ⊆ ?K; reach m2 ⊆ ?I; reach m3 ⊆ ?J⟧ ⟹ reach m3 ⊆ ?K›*))\n (*goals:\n 1. ‹Range (R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec) ∩ (?I::m2_state set) × (?J::m3_state set)) ⊆ m3_inv3_ikk_init›\n 2. ‹reach m2 ⊆ (?I::m2_state set)›\n 3. ‹reach m3 ⊆ (?J::m3_state set)›*)\n show \"Range (R23 ∩\n m2_inv3a_sesK_compr\n × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)\n ∩ m2_inv6_ikk_init × UNIV)\n ⊆ m3_inv3_ikk_init\"\n apply (auto simp add: R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) R23_pres_def (*‹R23_pres ≡ {(s, t). runs s = runs t ∧ m1_state.leak s = m1_state.leak t ∧ clk s = clk t ∧ cache s = cache t}›*) intro!: m3_inv3_ikk_initI (*‹(⋀A B Ra K Ts nl. ⟦runs ?x Ra = Some (Init, [A, B], aKey K # aNum Ts # nl); A ∈ good; B ∈ good; Key K ∈ analz (IK ?x)⟧ ⟹ (K, A, B, Ra $ na, Ts) ∈ m1_state.leak ?x) ⟹ ?x ∈ m3_inv3_ikk_init›*))\n (*goal: ‹Range (R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec) ∩ m2_inv6_ikk_init × UNIV) ⊆ m3_inv3_ikk_init›*)\n apply (elim m2_inv6_ikk_initE (*‹⟦?x ∈ m2_inv6_ikk_init; (⋀A B Ra K Ts nl. ⟦runs ?x Ra = Some (Init, [A, B], aKey K # aNum Ts # nl); A ∈ good; B ∈ good; aKey K ∈ extr ik0 (chan ?x)⟧ ⟹ (K, A, B, Ra $ na, Ts) ∈ m1_state.leak ?x) ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀x A B Ra K Ts nl a. ⟦runs x Ra = Some (Init, [A, B], aKey K # aNum Ts # nl); Key K ∈ analz (IK x); (a, x) ∈ R23_non; (a, x) ∈ R23_msgs; (a, x) ∈ R23_keys; A ∉ bad; B ∉ bad; a ∈ m2_inv6_ikk_init; a ∈ m2_inv3a_sesK_compr; x ∈ m3_inv4_lkeysec; x ∈ m3_inv6_ticket; x ∈ m3_inv7a_sesK_compr; x ∈ m3_inv7b_sesK_compr_non; runs a = runs x; m1_state.leak a = m1_state.leak x; clk a = clk x; cache a = cache x⟧ ⟹ (K, A, B, Ra $ na, Ts) ∈ m1_state.leak x›*)\n by (auto dest: R23_keys_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))\nqed (auto)\n(*solves the remaining goals:\n 1. ‹reach m2 ⊆ m2_inv6_ikk_init›\n 2. ‹reach m3 ⊆ UNIV›*)\n\n\nsubsubsection ‹inv4 (derived): Key secrecy for responder›\n(*invh*************************************************************************)\n\ndefinition\n m3_inv4_ikk_resp :: \"m3_state set\"\nwhere\n \"m3_inv4_ikk_resp ≡ {s. ∀A B Rb K Ts nl.\n runs s Rb = Some (Resp, [A, B], aKey K # aNum Ts # nl) ⟶ A ∈ good ⟶ B ∈ good ⟶\n Key K ∈ analz (IK s) ⟶\n (∃Na. (K, A, B, Na, Ts) ∈ leak s)\n }\"\n\nlemmas m3_inv4_ikk_respI = m3_inv4_ikk_resp_def [THEN setc_def_to_intro, rule_format]\nlemmas m3_inv4_ikk_respE [elim] = m3_inv4_ikk_resp_def [THEN setc_def_to_elim, rule_format]\nlemmas m3_inv4_ikk_respD = m3_inv4_ikk_resp_def [THEN setc_def_to_dest, rule_format, rotated 1]\n\nlemma PO_m3_inv4_ikk_resp: \"reach m3 ⊆ m3_inv4_ikk_resp\"\nproof (rule INV_from_Refinement_using_invariants [OF m3_refines_m2] (*‹⟦Range (R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec) ∩ (?I::m2_state set) × (?J::m3_state set)) ⊆ (?K::m3_state set); reach m2 ⊆ ?I; reach m3 ⊆ ?J⟧ ⟹ reach m3 ⊆ ?K›*))\n (*goals:\n 1. ‹Range (R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec) ∩ ?I × ?J) ⊆ m3_inv4_ikk_resp›\n 2. ‹reach m2 ⊆ ?I›\n 3. ‹reach m3 ⊆ ?J›*)\n show \"Range (R23 ∩ m2_inv3a_sesK_compr\n × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec)\n ∩ m2_inv7_ikk_resp × UNIV)\n ⊆ m3_inv4_ikk_resp\"\n apply (auto simp add: R23_def (*‹R23 ≡ R23_msgs ∩ R23_keys ∩ R23_non ∩ R23_pres›*) R23_pres_def (*‹R23_pres ≡ {(s, t). runs s = runs t ∧ m1_state.leak s = m1_state.leak t ∧ clk s = clk t ∧ cache s = cache t}›*) intro!: m3_inv4_ikk_respI (*‹(⋀A B Rb K Ts nl. ⟦runs ?x Rb = Some (Resp, [A, B], aKey K # aNum Ts # nl); A ∈ good; B ∈ good; Key K ∈ analz (IK ?x)⟧ ⟹ ∃Na. (K, A, B, Na, Ts) ∈ m1_state.leak ?x) ⟹ ?x ∈ m3_inv4_ikk_resp›*))\n (*goal: ‹Range (R23 ∩ m2_inv3a_sesK_compr × (m3_inv7a_sesK_compr ∩ m3_inv7b_sesK_compr_non ∩ m3_inv6_ticket ∩ m3_inv4_lkeysec) ∩ m2_inv7_ikk_resp × UNIV) ⊆ m3_inv4_ikk_resp›*)\n apply (elim m2_inv7_ikk_respE (*‹⟦?x ∈ m2_inv7_ikk_resp; (⋀A B Rb K Ts nl. ⟦runs ?x Rb = Some (Resp, [A, B], aKey K # aNum Ts # nl); A ∈ good; B ∈ good; aKey K ∈ extr ik0 (chan ?x)⟧ ⟹ ∃Na. (K, A, B, Na, Ts) ∈ m1_state.leak ?x) ⟹ PROP ?W⟧ ⟹ PROP ?W›*))\n (*goal: ‹⋀x A B Rb K Ts nl a. ⟦runs x Rb = Some (Resp, [A, B], aKey K # aNum Ts # nl); Key K ∈ analz (IK x); (a, x) ∈ R23_non; (a, x) ∈ R23_msgs; (a, x) ∈ R23_keys; A ∉ bad; B ∉ bad; a ∈ m2_inv7_ikk_resp; a ∈ m2_inv3a_sesK_compr; x ∈ m3_inv4_lkeysec; x ∈ m3_inv6_ticket; x ∈ m3_inv7a_sesK_compr; x ∈ m3_inv7b_sesK_compr_non; runs a = runs x; m1_state.leak a = m1_state.leak x; clk a = clk x; cache a = cache x⟧ ⟹ ∃Na. (K, A, B, Na, Ts) ∈ m1_state.leak x›*)\n by (auto dest: R23_keys_dests (*‹⟦Key ?K ∈ analz (Key ` ?KK ∪ IK ?t); (?s, ?t) ∈ R23_keys; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (aKey ` ?KK ∪ ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (IK ?t); (?s, ?t) ∈ R23_keys⟧ ⟹ aKey ?K ∈ extr ik0 (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) ik0) (chan ?s)› ‹⟦Key ?K ∈ analz (insert (Key ?Ka) (Key ` ?KK ∪ IK ?t)); (?s, ?t) ∈ R23_keys; ?Ka ∈ range sesK; ?KK ⊆ range sesK⟧ ⟹ aKey ?K ∈ extr (insert (aKey ?Ka) (aKey ` ?KK ∪ ik0)) (chan ?s)›*))\nqed (auto)\n(*solves the remaining goals:\n 1. ‹reach m2 ⊆ m2_inv7_ikk_resp›\n 2. ‹reach m3 ⊆ UNIV›*)\n\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Security_Protocol_Refinement/Key_establish/m3_kerberos4.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"cc5c60af1a6b222168e01518c1ebf504ff3bdc1c6dc0addbeb832e8a18c987bc\"\n}"}}},{"rowIdx":173,"cells":{"text":{"kind":"string","value":"(* \n Title: Code_Cayley_Hamilton.thy\n Author: Jose Divasón <jose.divasonm at unirioja.es>\n Author: Jesús Aransay <jesus-maria.aransay at unirioja.es>\n*)\n\nsection‹Code Cayley Hamilton›\n\ntheory Code_Cayley_Hamilton\n imports \n \"HOL-Computational_Algebra.Polynomial\"\n Cayley_Hamilton_Compatible\n Gauss_Jordan.Code_Matrix\nbegin\n\nsubsection‹Code equations for the definitions presented in the Cayley-Hamilton development›\n\ndefinition \"scalar_matrix_mult_row c A i = (χ j. c * (A $ i $ j))\"\n\nlemma scalar_matrix_mult_row_code [code abstract]:\n \"vec_nth (scalar_matrix_mult_row c A i) =(% j. c * (A $ i $ j))\"\n by (simp add: scalar_matrix_mult_row_def (*‹scalar_matrix_mult_row ?c ?A ?i = (χj. ?c * ?A $ ?i $ j)›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*))\n\nlemma scalar_matrix_mult_code [code abstract]: \"vec_nth (c *k A) = scalar_matrix_mult_row c A\"\n unfolding matrix_scalar_mult_def scalar_matrix_mult_row_def[abs_def]\n (*goal: ‹($) (χ(i::'a::finite) j::'c::finite. (c::'b::ab_semigroup_mult) * (A::(('b::ab_semigroup_mult, 'c::finite) vec, 'a::finite) vec) $ i $ j) = (λi::'a::finite. χj::'c::finite. c * A $ i $ j)›*)\n using vec_lambda_beta (*‹vec_lambda ?g $ ?i = ?g ?i›*) by auto\n\n\ndefinition \"minorM_row A i j k= vec_lambda (%l. if k = i ∧ l = j then 1 else\n if k = i ∨ l = j then 0 else A$k$l)\"\n\nlemma minorM_row_code [code abstract]:\n \"vec_nth (minorM_row A i j k) =(%l. if k = i ∧ l = j then 1 else\n if k = i ∨ l = j then 0 else A$k$l)\"\n by (simp add: minorM_row_def (*‹minorM_row ?A ?i ?j ?k = (χl. if ?k = ?i ∧ l = ?j then 1 else if ?k = ?i ∨ l = ?j then 0 else ?A $ ?k $ l)›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*))\n\nlemma minorM_code [code abstract]: \"vec_nth (minorM A i j) = minorM_row A i j\"\n unfolding minorM_def\n (*goal: ‹($) (χk l. if k = i ∧ l = j then 1 else if k = i ∨ l = j then 0 else A $ k $ l) = minorM_row A i j›*)\n apply transfer\n (*goal: ‹($) (χk l. if k = i ∧ l = j then 1 else if k = i ∨ l = j then 0 else A $ k $ l) = minorM_row A i j›*)\n by (auto simp: vec_eq_iff (*‹(?x = ?y) = (∀i. ?x $ i = ?y $ i)›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) minorM_row_def (*‹minorM_row ?A ?i ?j ?k = (χl. if ?k = ?i ∧ l = ?j then 1 else if ?k = ?i ∨ l = ?j then 0 else ?A $ ?k $ l)›*))\n\ndefinition \"cofactorM_row A i = vec_lambda (λj. cofactorM A $ i $ j)\"\n\nlemma cofactorM_row_code [code abstract]: \"vec_nth (cofactorM_row A i) = cofactor A i\"\n by (simp add: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) cofactorM_row_def (*‹cofactorM_row ?A ?i = (χj. cofactorM ?A $ ?i $ j)›*) cofactor_def (*‹cofactor ?A ?i ?j = det (minorM ?A ?i ?j)›*) cofactorM_def (*‹cofactorM ?A = (χi j. cofactor ?A i j)›*))\n\nlemma cofactorM_code [code abstract]: \"vec_nth (cofactorM A) = cofactorM_row A\"\n by (simp add: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) cofactorM_row_def (*‹cofactorM_row ?A ?i = (χj. cofactorM ?A $ ?i $ j)›*) vec_eq_iff (*‹(?x = ?y) = (∀i. ?x $ i = ?y $ i)›*))\n\nlemmas cofactor_def[code_unfold]\n\ndefinition mat2matofpoly_row\n where \"mat2matofpoly_row A i = vec_lambda (λj. [: A $ i $ j :])\"\n\nlemma mat2matofpoly_row_code [code abstract]:\n \"vec_nth (mat2matofpoly_row A i) = (%j. [: A $ i $ j :])\" \n unfolding mat2matofpoly_row_def\n (*goal: ‹($) (χj. [:A $ i $ j:]) = (λj. [:A $ i $ j:])›*)\n by auto\n\nlemma [code abstract]: \"vec_nth (mat2matofpoly k) = mat2matofpoly_row k\"\n unfolding mat2matofpoly_def\n (*goal: ‹($) (χi j. [:k $ i $ j:]) = mat2matofpoly_row k›*)\n unfolding mat2matofpoly_row_def[abs_def]\n (*goal: ‹($) (χ(i::'a) j::'c. [:(k::(('b, 'c) vec, 'a) vec) $ i $ j:]) = (λi::'a. χj::'c. [:k $ i $ j:])›*)\n by auto\n\nprimrec matpow :: \"'a::semiring_1^'n^'n ⇒ nat ⇒ 'a^'n^'n\" where\n matpow_0: \"matpow A 0 = mat 1\" |\n matpow_Suc: \"matpow A (Suc n) = A ** (matpow A n)\"\n\ndefinition evalmat :: \"'a::comm_ring_1 poly ⇒ 'a^'n^'n ⇒ 'a^'n^'n\" where\n \"evalmat P A = (∑ i ∈ { n::nat . n ≤ ( degree P ) } . (coeff P i) *k (matpow A i) )\"\n\nlemma evalmat_unfold:\n \"evalmat P A = (∑i = 0..degree P. coeff P i *k matpow A i)\"\n by (simp add: evalmat_def (*‹evalmat ?P ?A = (∑i∈{n. n ≤ degree ?P}. coeff ?P i *k matpow ?A i)›*) atMost_def [symmetric] (*‹{x. x ≤ ?u} ≡ {..?u}›*) atMost_atLeast0 (*‹{..?n} = {0..?n}›*))\n\nlemma evalmat_code[code]:\n \"evalmat P A = (∑i←[0..int (degree P)]. coeff P (nat i) *k matpow A (nat i))\" (is \"_ = ?rhs\")\nproof (-)\n (*goal: ‹evalmat P A = (∑i←[0..int (degree P)]. coeff P (nat i) *k matpow A (nat i))›*)\n let ?t = \"λn. coeff P n *k matpow A n\"\n have \"(∑i = 0..degree P. coeff P i *k matpow A i) = (∑i∈{0..int (degree P)}. coeff P (nat i) *k matpow A (nat i))\"\n apply (rule sum.reindex_cong [of nat] (*‹⟦inj_on nat (?B::int set); (?A::nat set) = nat ` ?B; ⋀x::int. x ∈ ?B ⟹ (?g::nat ⇒ ?'a::comm_monoid_add) (nat x) = (?h::int ⇒ ?'a::comm_monoid_add) x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹inj_on nat {0..int (degree P)}›\n 2. ‹{0..degree P} = nat ` {0..int (degree P)}›\n 3. ‹⋀x. x ∈ {0..int (degree P)} ⟹ coeff P (nat x) *k matpow A (nat x) = coeff P (nat x) *k matpow A (nat x)›\n discuss goal 1*)\n apply ((auto simp add: eq_nat_nat_iff (*‹⟦0 ≤ ?z; 0 ≤ ?z'⟧ ⟹ (nat ?z = nat ?z') = (?z = ?z')›*) image_iff (*‹(?z ∈ ?f ` ?A) = (∃x∈?A. ?z = ?f x)›*) intro: inj_onI (*‹(⋀x y. ⟦x ∈ ?A; y ∈ ?A; ?f x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: eq_nat_nat_iff (*‹⟦(0::int) ≤ (?z::int); (0::int) ≤ (?z'::int)⟧ ⟹ (nat ?z = nat ?z') = (?z = ?z')›*) image_iff (*‹((?z::?'a) ∈ (?f::?'b ⇒ ?'a) ` (?A::?'b set)) = (∃x::?'b∈?A. ?z = ?f x)›*) intro: inj_onI (*‹(⋀(x::?'a) y::?'a. ⟦x ∈ (?A::?'a set); y ∈ ?A; (?f::?'a ⇒ ?'b) x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*))[1])\n (*top goal: ‹{0..degree P} = nat ` {0..int (degree P)}› and 1 goal remains*)\n apply presburger\n (*discuss goal 3*)\n apply ((auto simp add: eq_nat_nat_iff (*‹⟦0 ≤ ?z; 0 ≤ ?z'⟧ ⟹ (nat ?z = nat ?z') = (?z = ?z')›*) image_iff (*‹(?z ∈ ?f ` ?A) = (∃x∈?A. ?z = ?f x)›*) intro: inj_onI (*‹(⋀x y. ⟦x ∈ ?A; y ∈ ?A; ?f x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*))[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹(∑i = 0..degree P. coeff P i *k matpow A i) = (∑i = 0..int (degree P). coeff P (nat i) *k matpow A (nat i))›*) have \"… = ?rhs\"\n by (simp add: sum_set_upto_conv_sum_list_int [symmetric] (*‹sum_list (map (?f::int ⇒ ?'a::comm_monoid_add) [?i::int..?j::int]) = sum ?f (set [?i..?j])›*))\n finally (*calculation: ‹(∑i = 0..degree P. coeff P i *k matpow A i) = (∑i←[0..int (degree P)]. coeff P (nat i) *k matpow A (nat i))›*) show \"?thesis\"\n (*goal: ‹evalmat P A = (∑i←[0..int (degree P)]. coeff P (nat i) *k matpow A (nat i))›*)\n by (simp add: evalmat_unfold (*‹evalmat ?P ?A = (∑i = 0..degree ?P. coeff ?P i *k matpow ?A i)›*))\nqed\n\ndefinition coeffM_zero :: \"'a poly^'n^'n ⇒ 'a::zero^'n^'n\" where\n \"coeffM_zero A = (χ i j. (coeff (A $ i $ j) 0))\"\n\ndefinition \"coeffM_zero_row A i = (χ j. (coeff (A $ i $ j) 0))\"\n\ndefinition coeffM :: \"'a poly^'n^'n ⇒ nat ⇒ 'a::zero^'n^'n\" where\n \"coeffM A n = (χ i j. coeff (A $ i $ j) n)\"\n\nlemma coeffM_zero_row_code [code abstract]:\n \"vec_nth (coeffM_zero_row A i) = (% j. (coeff (A $ i $ j) 0))\"\n by (simp add: coeffM_zero_row_def (*‹coeffM_zero_row ?A ?i = (χj. coeff (?A $ ?i $ j) 0)›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*))\n\nlemma coeffM_zero_code [code abstract]: \"vec_nth (coeffM_zero A) = coeffM_zero_row A\"\n unfolding coeffM_zero_def coeffM_zero_row_def[abs_def]\n (*goal: ‹($) (χi j. coeff (A $ i $ j) 0) = (λi. χj. coeff (A $ i $ j) 0)›*)\n using vec_lambda_beta (*‹vec_lambda ?g $ ?i = ?g ?i›*) by auto\n\ndefinition\n \"coeffM_row A n i = (χ j. coeff (A $ i $ j) n)\"\n\nlemma coeffM_row_code [code abstract]:\n \"vec_nth (coeffM_row A n i) = (% j. coeff (A $ i $ j) n)\"\n by (simp add: coeffM_row_def (*‹coeffM_row ?A ?n ?i = (χj. coeff (?A $ ?i $ j) ?n)›*) coeffM_def (*‹coeffM ?A ?n = (χi j. coeff (?A $ i $ j) ?n)›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*))\n\nlemma coeffM_code [code abstract]: \"vec_nth (coeffM A n) = coeffM_row A n\"\n unfolding coeffM_def coeffM_row_def[abs_def]\n (*goal: ‹($) (χ(i::'a::finite) j::'a::finite. coeff ((A::(('b::zero poly, 'a::finite) vec, 'a::finite) vec) $ i $ j) (n::nat)) = (λi::'a::finite. χj::'a::finite. coeff (A $ i $ j) n)›*)\n using vec_lambda_beta (*‹vec_lambda ?g $ ?i = ?g ?i›*) by auto\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Echelon_Form/Code_Cayley_Hamilton.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"8d855fb052e7861ee0eefb95b9aad3cb33fbe43c789aa25233aae000623377a0\"\n}"}}},{"rowIdx":174,"cells":{"text":{"kind":"string","value":"(*<*)\n(*\n * Copyright 2015, NICTA\n *\n * This software may be distributed and modified according to the terms of\n * the BSD 2-Clause license. Note that NO WARRANTY is provided.\n * See \"LICENSE_BSD2.txt\" for details.\n *\n * @TAG(NICTA_BSD)\n *)\n\ntheory Global_Invariants_Lemmas\nimports\n Global_Invariants\nbegin\n\ndeclare subst_all [simp del] [[simproc del: defined_all]]\n\n(*>*)\nsection‹Global invariants lemma bucket›\n\n(* FIXME this file should be about the invs. This split makes it easier to move things around, maybe. *)\n\ndeclare mut_m.mutator_phase_inv_aux.simps[simp]\ncase_of_simps mutator_phase_inv_aux_case: mut_m.mutator_phase_inv_aux.simps\ncase_of_simps sys_phase_inv_aux_case: sys_phase_inv_aux.simps\n\n\nsubsection‹TSO invariants›\n\nlemma tso_store_inv_eq_imp:\n \"eq_imp (λp s. mem_store_buffers (s sys) p)\n tso_store_inv\"\nby (simp add: eq_imp_def (*‹eq_imp (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) (?g::?'b::type ⇒ ?'e::type) ≡ ∀(s::?'b::type) s'::?'b::type. (∀x::?'a::type. ?f x s = ?f x s') ⟶ ?g s = ?g s'›*) tso_store_inv_def (*‹tso_store_inv = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (∀x::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀(x::?'mut::type) xa::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*))\n\nlemmas tso_store_inv_fun_upd[simp] = eq_imp_fun_upd[OF tso_store_inv_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma tso_store_invD[simp]:\n \"tso_store_inv s ⟹ ¬sys_mem_store_buffers gc s = mw_Mutate r f r' # ws\"\n \"tso_store_inv s ⟹ ¬sys_mem_store_buffers gc s = mw_Mutate_Payload r f pl # ws\"\n \"tso_store_inv s ⟹ ¬sys_mem_store_buffers (mutator m) s = mw_fA fl # ws\"\n \"tso_store_inv s ⟹ ¬sys_mem_store_buffers (mutator m) s = mw_fM fl # ws\"\n \"tso_store_inv s ⟹ ¬sys_mem_store_buffers (mutator m) s = mw_Phase ph # ws\"\n\n(*goals:\n 1. ‹tso_store_inv s ⟹ sys_mem_store_buffers gc s ≠ mw_Mutate r f r' # ws›\n 2. ‹tso_store_inv s ⟹ sys_mem_store_buffers gc s ≠ mw_Mutate_Payload r f pl # ws›\n 3. ‹tso_store_inv s ⟹ sys_mem_store_buffers (mutator m) s ≠ mw_fA fl # ws›\n 4. ‹tso_store_inv s ⟹ sys_mem_store_buffers (mutator m) s ≠ mw_fM fl # ws›\n 5. ‹tso_store_inv s ⟹ sys_mem_store_buffers (mutator m) s ≠ mw_Phase ph # ws›\ndiscuss goal 1*)\n apply ((auto simp: tso_store_inv_def (*‹tso_store_inv = (λs. (∀x. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀x xa. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*) dest!: spec[where x=m] (*‹∀x. ?P x ⟹ ?P m›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: tso_store_inv_def (*‹tso_store_inv = (λs. (∀x. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀x xa. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*) dest!: spec[where x=m] (*‹∀x. ?P x ⟹ ?P m›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: tso_store_inv_def (*‹tso_store_inv = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (∀x::(?'field, ?'payload, ?'ref) mem_store_action. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀(x::?'mut) xa::(?'field, ?'payload, ?'ref) mem_store_action. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*) dest!: spec[where x=m] (*‹∀x::'a. (?P::'a ⇒ bool) x ⟹ ?P (m::'a)›*))[1])\n(*discuss goal 4*)\n apply ((auto simp: tso_store_inv_def (*‹tso_store_inv = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (∀x::(?'field, ?'payload, ?'ref) mem_store_action. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀(x::?'mut) xa::(?'field, ?'payload, ?'ref) mem_store_action. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*) dest!: spec[where x=m] (*‹∀x::'a. (?P::'a ⇒ bool) x ⟹ ?P (m::'a)›*))[1])\n(*discuss goal 5*)\n apply ((auto simp: tso_store_inv_def (*‹tso_store_inv = (λs. (∀x. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀x xa. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*) dest!: spec[where x=m] (*‹∀x. ?P x ⟹ ?P m›*))[1])\n(*proven 5 subgoals*) .\n\nlemma mut_do_store_action[simp]:\n \"⟦ sys_mem_store_buffers (mutator m) s = w # ws; tso_store_inv s ⟧ ⟹ fA (do_store_action w (s sys)) = sys_fA s\"\n \"⟦ sys_mem_store_buffers (mutator m) s = w # ws; tso_store_inv s ⟧ ⟹ fM (do_store_action w (s sys)) = sys_fM s\"\n \"⟦ sys_mem_store_buffers (mutator m) s = w # ws; tso_store_inv s ⟧ ⟹ phase (do_store_action w (s sys)) = sys_phase s\"\n\n(*goals:\n 1. ‹⟦sys_mem_store_buffers (mutator m) s = w # ws; tso_store_inv s⟧ ⟹ fA (do_store_action w (s sys)) = sys_fA s›\n 2. ‹⟦sys_mem_store_buffers (mutator m) s = w # ws; tso_store_inv s⟧ ⟹ fM (do_store_action w (s sys)) = sys_fM s›\n 3. ‹⟦sys_mem_store_buffers (mutator m) s = w # ws; tso_store_inv s⟧ ⟹ phase (do_store_action w (s sys)) = sys_phase s›\ndiscuss goal 1*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma tso_store_inv_sys_load_Mut[simp]:\n assumes \"tso_store_inv s\"\n assumes \"(ract, v) ∈ { (mr_fM, mv_Mark (Some (sys_fM s))), (mr_fA, mv_Mark (Some (sys_fA s))), (mr_Phase, mv_Phase (sys_phase s)) }\"\n shows \"sys_load (mutator m) ract (s sys) = v\"\nusing assms (*‹tso_store_inv s› ‹(ract, v) ∈ {(mr_fM, mv_Mark (Some (sys_fM s))), (mr_fA, mv_Mark (Some (sys_fA s))), (mr_Phase, mv_Phase (sys_phase s))}›*) apply (clarsimp simp: sys_load_def (*‹sys_load ?p ?ract = do_load_action ?ract ∘ processors_view_of_memory ?p›*) fold_stores_def (*‹fold_stores ?ws = fold (λw. (∘) (do_store_action w)) ?ws id›*))\n(*goal: ‹sys_load (mutator m) ract (s sys) = v›*)\napply (rule fold_invariant[where P=\"λfr. do_load_action ract (fr (s sys)) = v\" and Q=\"mut_writes\"] (*‹⟦⋀x. x ∈ set ?xs ⟹ mut_writes x; do_load_action ract (?s (s sys)) = v; ⋀x sa. ⟦mut_writes x; do_load_action ract (sa (s sys)) = v⟧ ⟹ do_load_action ract (?f x sa (s sys)) = v⟧ ⟹ do_load_action ract (fold ?f ?xs ?s (s sys)) = v›*))\n(*goals:\n 1. ‹⋀x::('b, 'c, 'd) mem_store_action. ⟦tso_store_inv (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state); (ract::('b, 'd) mem_load_action) = mr_fM ∧ (v::('b, 'c, 'd) response) = mv_Mark (Some (sys_fM s)) ∨ ract = mr_fA ∧ v = mv_Mark (Some (sys_fA s)) ∨ ract = mr_Phase ∧ v = mv_Phase (sys_phase s); x ∈ set (sys_mem_store_buffers (mutator (m::'a)) s)⟧ ⟹ mut_writes x›\n 2. ‹⟦tso_store_inv (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state); (ract::('b, 'd) mem_load_action) = mr_fM ∧ (v::('b, 'c, 'd) response) = mv_Mark (Some (sys_fM s)) ∨ ract = mr_fA ∧ v = mv_Mark (Some (sys_fA s)) ∨ ract = mr_Phase ∧ v = mv_Phase (sys_phase s)⟧ ⟹ do_load_action ract (id (s sys)) = v›\n 3. ‹⋀(x::('b, 'c, 'd) mem_store_action) sa::('b, 'a, 'c, 'd) Model.local_state ⇒ ('b, 'a, 'c, 'd) Model.local_state. ⟦tso_store_inv (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state); (ract::('b, 'd) mem_load_action) = mr_fM ∧ (v::('b, 'c, 'd) response) = mv_Mark (Some (sys_fM s)) ∨ ract = mr_fA ∧ v = mv_Mark (Some (sys_fA s)) ∨ ract = mr_Phase ∧ v = mv_Phase (sys_phase s); mut_writes x; do_load_action ract (sa (s sys)) = v⟧ ⟹ do_load_action ract ((do_store_action x ∘ sa) (s sys)) = v›\ndiscuss goal 1*)\n apply (fastforce simp: tso_store_inv_def (*‹tso_store_inv = (λs. (∀x. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀x xa. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*))\n(*discuss goal 2*)\n apply ((auto simp: do_load_action_def (*‹do_load_action ?ract = (λs. case ?ract of mr_Ref r f ⇒ mv_Ref (Option.bind (heap s r) (λobj. obj_fields obj f)) | mr_Payload r f ⇒ mv_Payload (Option.bind (heap s r) (λobj. obj_payload obj f)) | mr_Mark r ⇒ mv_Mark (map_option obj_mark (heap s r)) | mr_Phase ⇒ mv_Phase (phase s) | mr_fM ⇒ mv_Mark (Some (fM s)) | mr_fA ⇒ mv_Mark (Some (fA s)))›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: do_load_action_def (*‹do_load_action ?ract = (λs. case ?ract of mr_Ref r f ⇒ mv_Ref (Option.bind (heap s r) (λobj. obj_fields obj f)) | mr_Payload r f ⇒ mv_Payload (Option.bind (heap s r) (λobj. obj_payload obj f)) | mr_Mark r ⇒ mv_Mark (map_option obj_mark (heap s r)) | mr_Phase ⇒ mv_Phase (phase s) | mr_fM ⇒ mv_Mark (Some (fM s)) | mr_fA ⇒ mv_Mark (Some (fA s)))›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma tso_store_inv_sys_load_GC[simp]:\n assumes \"tso_store_inv s\"\n shows \"sys_load gc (mr_Ref r f) (s sys) = mv_Ref (Option.bind (sys_heap s r) (λobj. obj_fields obj f))\" (is \"?lhs = mv_Ref ?rhs\")\nusing assms (*‹tso_store_inv (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state)›*) unfolding sys_load_def fold_stores_def\n(*goal: ‹(do_load_action (mr_Ref r f) ∘ (λs. fold (λw. (∘) (do_store_action w)) (mem_store_buffers s gc) id s)) (s sys) = mv_Ref (Option.bind (sys_heap s r) (λobj. obj_fields obj f))›*)\napply clarsimp\n(*goal: ‹(do_load_action (mr_Ref r f) ∘ (λs. fold (λw. (∘) (do_store_action w)) (mem_store_buffers s gc) id s)) (s sys) = mv_Ref (Option.bind (sys_heap s r) (λobj. obj_fields obj f))›*)\napply (rule fold_invariant[where P=\"λfr. Option.bind (heap (fr (s sys)) r) (λobj. obj_fields obj f) = ?rhs\" and Q=\"λw. ∀r f r'. w ≠ mw_Mutate r f r'\"] (*‹⟦⋀x::(?'h1::type, ?'i1::type, ?'j1::type) mem_store_action. x ∈ set (?xs::(?'h1::type, ?'i1::type, ?'j1::type) mem_store_action list) ⟹ ∀(r::?'j1::type) (f::?'h1::type) r'::?'j1::type option. x ≠ mw_Mutate r f r'; Option.bind (heap ((?s::('b, 'a, 'c, 'd) Model.local_state ⇒ ('b, ?'e1, ?'f1, 'd, ?'g1) local_state_scheme) ((s::'a::type process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) sys)) (r::'d::type)) (λobj::('b, ?'f1, 'd) object. obj_fields obj (f::'b::type)) = Option.bind (sys_heap s r) (λobj::('b, 'c, 'd) object. obj_fields obj f); ⋀(x::(?'h1::type, ?'i1::type, ?'j1::type) mem_store_action) sa::('b, 'a, 'c, 'd) Model.local_state ⇒ ('b, ?'e1, ?'f1, 'd, ?'g1) local_state_scheme. ⟦∀(r::?'j1::type) (f::?'h1::type) r'::?'j1::type option. x ≠ mw_Mutate r f r'; Option.bind (heap (sa (s sys)) r) (λobj::('b, ?'f1, 'd) object. obj_fields obj f) = Option.bind (sys_heap s r) (λobj::('b, 'c, 'd) object. obj_fields obj f)⟧ ⟹ Option.bind (heap ((?f::(?'h1::type, ?'i1::type, ?'j1::type) mem_store_action ⇒ (('b, 'a, 'c, 'd) Model.local_state ⇒ ('b, ?'e1, ?'f1, 'd, ?'g1) local_state_scheme) ⇒ ('b, 'a, 'c, 'd) Model.local_state ⇒ ('b, ?'e1, ?'f1, 'd, ?'g1) local_state_scheme) x sa (s sys)) r) (λobj::('b, ?'f1, 'd) object. obj_fields obj f) = Option.bind (sys_heap s r) (λobj::('b, 'c, 'd) object. obj_fields obj f)⟧ ⟹ Option.bind (heap (fold ?f ?xs ?s (s sys)) r) (λobj::('b, ?'f1, 'd) object. obj_fields obj f) = Option.bind (sys_heap s r) (λobj::('b, 'c, 'd) object. obj_fields obj f)›*))\n(*goals:\n 1. ‹⋀x. ⟦tso_store_inv s; x ∈ set (sys_mem_store_buffers gc s)⟧ ⟹ ∀r f r'. x ≠ mw_Mutate r f r'›\n 2. ‹tso_store_inv s ⟹ Option.bind (heap (id (s sys)) r) (λobj. obj_fields obj f) = Option.bind (sys_heap s r) (λobj. obj_fields obj f)›\n 3. ‹⋀x sa. ⟦tso_store_inv s; ∀r f r'. x ≠ mw_Mutate r f r'; Option.bind (heap (sa (s sys)) r) (λobj. obj_fields obj f) = Option.bind (sys_heap s r) (λobj. obj_fields obj f)⟧ ⟹ Option.bind (heap ((do_store_action x ∘ sa) (s sys)) r) (λobj. obj_fields obj f) = Option.bind (sys_heap s r) (λobj. obj_fields obj f)›\ndiscuss goal 1*)\n apply (fastforce simp: tso_store_inv_def (*‹tso_store_inv = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (∀x::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action. x ∈ set (sys_mem_store_buffers gc s) ⟶ gc_writes x) ∧ (∀(x::?'mut::type) xa::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action. xa ∈ set (sys_mem_store_buffers (mutator x) s) ⟶ mut_writes xa))›*))\n(*discuss goal 2*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma tso_no_pending_marksD[simp]:\n assumes \"tso_pending_mark p s = []\"\n shows \"sys_load p (mr_Mark r) (s sys) = mv_Mark (map_option obj_mark (sys_heap s r))\"\nusing assms (*‹filter is_mw_Mark (sys_mem_store_buffers (p::'d::type process_name) (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)) = []›*) unfolding sys_load_def fold_stores_def\n(*goal: ‹(do_load_action (mr_Mark r) ∘ (λs. fold (λw. (∘) (do_store_action w)) (mem_store_buffers s p) id s)) (s sys) = mv_Mark (map_option obj_mark (sys_heap s r))›*)\napply clarsimp\n(*goal: ‹(do_load_action (mr_Mark r) ∘ (λs. fold (λw. (∘) (do_store_action w)) (mem_store_buffers s p) id s)) (s sys) = mv_Mark (map_option obj_mark (sys_heap s r))›*)\napply (rule fold_invariant[where P=\"λfr. map_option obj_mark (heap (fr (s sys)) r) = map_option obj_mark (sys_heap s r)\" and Q=\"λw. ∀fl. w ≠ mw_Mark r fl\"] (*‹⟦⋀x. x ∈ set ?xs ⟹ ∀fl. x ≠ mw_Mark r fl; map_option obj_mark (heap (?s (s sys)) r) = map_option obj_mark (sys_heap s r); ⋀x sa. ⟦∀fl. x ≠ mw_Mark r fl; map_option obj_mark (heap (sa (s sys)) r) = map_option obj_mark (sys_heap s r)⟧ ⟹ map_option obj_mark (heap (?f x sa (s sys)) r) = map_option obj_mark (sys_heap s r)⟧ ⟹ map_option obj_mark (heap (fold ?f ?xs ?s (s sys)) r) = map_option obj_mark (sys_heap s r)›*))\n(*goals:\n 1. ‹⋀x. ⟦filter is_mw_Mark (sys_mem_store_buffers p s) = []; x ∈ set (sys_mem_store_buffers p s)⟧ ⟹ ∀fl. x ≠ mw_Mark r fl›\n 2. ‹filter is_mw_Mark (sys_mem_store_buffers p s) = [] ⟹ map_option obj_mark (heap (id (s sys)) r) = map_option obj_mark (sys_heap s r)›\n 3. ‹⋀x sa. ⟦filter is_mw_Mark (sys_mem_store_buffers p s) = []; ∀fl. x ≠ mw_Mark r fl; map_option obj_mark (heap (sa (s sys)) r) = map_option obj_mark (sys_heap s r)⟧ ⟹ map_option obj_mark (heap ((do_store_action x ∘ sa) (s sys)) r) = map_option obj_mark (sys_heap s r)›\ndiscuss goal 1*)\n apply ((auto simp: map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma no_pending_phase_sys_load[simp]:\n assumes \"tso_pending_phase p s = []\"\n shows \"sys_load p mr_Phase (s sys) = mv_Phase (sys_phase s)\"\nusing assms (*‹filter is_mw_Phase (sys_mem_store_buffers p s) = []›*) apply (clarsimp simp: sys_load_def (*‹sys_load (?p::?'mut::type process_name) (?ract::(?'field::type, ?'ref::type) mem_load_action) = do_load_action ?ract ∘ processors_view_of_memory ?p›*) fold_stores_def (*‹fold_stores (?ws::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action list) = fold (λw::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action. (∘) (do_store_action w)) ?ws id›*))\n(*goal: ‹sys_load p mr_Phase (s sys) = mv_Phase (sys_phase s)›*)\napply (rule fold_invariant[where P=\"λfr. phase (fr (s sys)) = sys_phase s\" and Q=\"λw. ∀ph. w ≠ mw_Phase ph\"] (*‹⟦⋀x. x ∈ set ?xs ⟹ ∀ph. x ≠ mw_Phase ph; phase (?s (s sys)) = sys_phase s; ⋀x sa. ⟦∀ph. x ≠ mw_Phase ph; phase (sa (s sys)) = sys_phase s⟧ ⟹ phase (?f x sa (s sys)) = sys_phase s⟧ ⟹ phase (fold ?f ?xs ?s (s sys)) = sys_phase s›*))\n(*goals:\n 1. ‹⋀x::('a::type, 'b::type, 'c::type) mem_store_action. ⟦filter is_mw_Phase (sys_mem_store_buffers (p::'d::type process_name) (s::'e::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)) = []; x ∈ set (sys_mem_store_buffers p s)⟧ ⟹ ∀ph::gc_phase. x ≠ mw_Phase ph›\n 2. ‹filter is_mw_Phase (sys_mem_store_buffers (p::'d::type process_name) (s::'e::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)) = [] ⟹ phase (id (s sys)) = sys_phase s›\n 3. ‹⋀(x::('a::type, 'b::type, 'c::type) mem_store_action) sa::('a, 'd, 'b, 'c) Model.local_state ⇒ ('a, 'd, 'b, 'c) Model.local_state. ⟦filter is_mw_Phase (sys_mem_store_buffers (p::'d::type process_name) (s::'e::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)) = []; ∀ph::gc_phase. x ≠ mw_Phase ph; phase (sa (s sys)) = sys_phase s⟧ ⟹ phase ((do_store_action x ∘ sa) (s sys)) = sys_phase s›\ndiscuss goal 1*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: do_store_action_def (*‹do_store_action (?wact::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action) = (λs::(?'field, ?'mut, ?'payload, ?'ref) Model.local_state. case ?wact of mw_Mark (r::?'ref::type) (gc_mark::bool) ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_::bool. gc_mark)) (heap s r))⦈ | mw_Mutate (r::?'ref::type) (f::?'field::type) (new_r::?'ref::type option) ⇒ s⦇heap := (heap s)(r := map_option (λobj::(?'field, ?'payload, ?'ref) object. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload (r::?'ref::type) (f::?'field::type) (pl::?'payload::type option) ⇒ s⦇heap := (heap s)(r := map_option (λobj::(?'field, ?'payload, ?'ref) object. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA (gc_mark::bool) ⇒ s⦇fA := gc_mark⦈ | mw_fM (gc_mark::bool) ⇒ s⦇fM := gc_mark⦈ | mw_Phase (gc_phase::gc_phase) ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter (?P::?'a::type ⇒ bool) (?xs::?'a::type list) = []) = (∀x::?'a::type∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹(?P::?'a::type ⇒ bool) (case ?mem_store_action::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action of mw_Mark (x::?'ref::type) (xa::bool) ⇒ (?f1.0::?'ref::type ⇒ bool ⇒ ?'a::type) x xa | mw_Mutate (x::?'ref::type) (xa::?'field::type) (xb::?'ref::type option) ⇒ (?f2.0::?'ref::type ⇒ ?'field::type ⇒ ?'ref::type option ⇒ ?'a::type) x xa xb | mw_Mutate_Payload (x::?'ref::type) (xa::?'field::type) (xb::?'payload::type option) ⇒ (?f3.0::?'ref::type ⇒ ?'field::type ⇒ ?'payload::type option ⇒ ?'a::type) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a::type) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a::type) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a::type) x) = ((∀(x11::?'ref::type) x12::bool. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀(x21::?'ref::type) (x22::?'field::type) x23::?'ref::type option. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀(x31::?'ref::type) (x32::?'field::type) x33::?'payload::type option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4::bool. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5::bool. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6::gc_phase. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹(?P::?'a::type ⇒ bool) (case ?mem_store_action::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action of mw_Mark (x::?'ref::type) (xa::bool) ⇒ (?f1.0::?'ref::type ⇒ bool ⇒ ?'a::type) x xa | mw_Mutate (x::?'ref::type) (xa::?'field::type) (xb::?'ref::type option) ⇒ (?f2.0::?'ref::type ⇒ ?'field::type ⇒ ?'ref::type option ⇒ ?'a::type) x xa xb | mw_Mutate_Payload (x::?'ref::type) (xa::?'field::type) (xb::?'payload::type option) ⇒ (?f3.0::?'ref::type ⇒ ?'field::type ⇒ ?'payload::type option ⇒ ?'a::type) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a::type) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a::type) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a::type) x) = (¬ ((∃(x11::?'ref::type) x12::bool. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃(x21::?'ref::type) (x22::?'field::type) x23::?'ref::type option. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃(x31::?'ref::type) (x32::?'field::type) x33::?'payload::type option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4::bool. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5::bool. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6::gc_phase. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma gc_no_pending_fM_write[simp]:\n assumes \"tso_pending_fM gc s = []\"\n shows \"sys_load gc mr_fM (s sys) = mv_Mark (Some (sys_fM s))\"\nusing assms (*‹filter is_mw_fM (sys_mem_store_buffers gc s) = []›*) apply (clarsimp simp: sys_load_def (*‹sys_load (?p::?'mut::type process_name) (?ract::(?'field::type, ?'ref::type) mem_load_action) = do_load_action ?ract ∘ processors_view_of_memory ?p›*) fold_stores_def (*‹fold_stores (?ws::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action list) = fold (λw::(?'field::type, ?'payload::type, ?'ref::type) mem_store_action. (∘) (do_store_action w)) ?ws id›*))\n(*goal: ‹sys_load gc mr_fM ((s::'e process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) sys) = mv_Mark (Some (sys_fM s))›*)\napply (rule fold_invariant[where P=\"λfr. fM (fr (s sys)) = sys_fM s\" and Q=\"λw. ∀fl. w ≠ mw_fM fl\"] (*‹⟦⋀x. x ∈ set ?xs ⟹ ∀fl. x ≠ mw_fM fl; fM (?s (s sys)) = sys_fM s; ⋀x sa. ⟦∀fl. x ≠ mw_fM fl; fM (sa (s sys)) = sys_fM s⟧ ⟹ fM (?f x sa (s sys)) = sys_fM s⟧ ⟹ fM (fold ?f ?xs ?s (s sys)) = sys_fM s›*))\n(*goals:\n 1. ‹⋀x. ⟦filter is_mw_fM (sys_mem_store_buffers gc s) = []; x ∈ set (sys_mem_store_buffers gc s)⟧ ⟹ ∀fl. x ≠ mw_fM fl›\n 2. ‹filter is_mw_fM (sys_mem_store_buffers gc s) = [] ⟹ fM (id (s sys)) = sys_fM s›\n 3. ‹⋀x sa. ⟦filter is_mw_fM (sys_mem_store_buffers gc s) = []; ∀fl. x ≠ mw_fM fl; fM (sa (s sys)) = sys_fM s⟧ ⟹ fM ((do_store_action x ∘ sa) (s sys)) = sys_fM s›\ndiscuss goal 1*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: do_store_action_def (*‹do_store_action (?wact::(?'field, ?'payload, ?'ref) mem_store_action) = (λs::(?'field, ?'mut, ?'payload, ?'ref) Model.local_state. case ?wact of mw_Mark (r::?'ref) (gc_mark::bool) ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_::bool. gc_mark)) (heap s r))⦈ | mw_Mutate (r::?'ref) (f::?'field) (new_r::?'ref option) ⇒ s⦇heap := (heap s)(r := map_option (λobj::(?'field, ?'payload, ?'ref) object. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload (r::?'ref) (f::?'field) (pl::?'payload option) ⇒ s⦇heap := (heap s)(r := map_option (λobj::(?'field, ?'payload, ?'ref) object. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA (gc_mark::bool) ⇒ s⦇fA := gc_mark⦈ | mw_fM (gc_mark::bool) ⇒ s⦇fM := gc_mark⦈ | mw_Phase (gc_phase::gc_phase) ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter (?P::?'a ⇒ bool) (?xs::?'a list) = []) = (∀x::?'a∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹(?P::?'a ⇒ bool) (case ?mem_store_action::(?'field, ?'payload, ?'ref) mem_store_action of mw_Mark (x::?'ref) (xa::bool) ⇒ (?f1.0::?'ref ⇒ bool ⇒ ?'a) x xa | mw_Mutate (x::?'ref) (xa::?'field) (xb::?'ref option) ⇒ (?f2.0::?'ref ⇒ ?'field ⇒ ?'ref option ⇒ ?'a) x xa xb | mw_Mutate_Payload (x::?'ref) (xa::?'field) (xb::?'payload option) ⇒ (?f3.0::?'ref ⇒ ?'field ⇒ ?'payload option ⇒ ?'a) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a) x) = ((∀(x11::?'ref) x12::bool. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀(x21::?'ref) (x22::?'field) x23::?'ref option. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀(x31::?'ref) (x32::?'field) x33::?'payload option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4::bool. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5::bool. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6::gc_phase. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹(?P::?'a ⇒ bool) (case ?mem_store_action::(?'field, ?'payload, ?'ref) mem_store_action of mw_Mark (x::?'ref) (xa::bool) ⇒ (?f1.0::?'ref ⇒ bool ⇒ ?'a) x xa | mw_Mutate (x::?'ref) (xa::?'field) (xb::?'ref option) ⇒ (?f2.0::?'ref ⇒ ?'field ⇒ ?'ref option ⇒ ?'a) x xa xb | mw_Mutate_Payload (x::?'ref) (xa::?'field) (xb::?'payload option) ⇒ (?f3.0::?'ref ⇒ ?'field ⇒ ?'payload option ⇒ ?'a) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a) x) = (¬ ((∃(x11::?'ref) x12::bool. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃(x21::?'ref) (x22::?'field) x23::?'ref option. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃(x31::?'ref) (x32::?'field) x33::?'payload option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4::bool. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5::bool. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6::gc_phase. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma tso_store_refs_simps[simp]:\n \"mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇roots := roots'⦈))\n = mut_m.tso_store_refs m s\"\n \"mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇ghost_honorary_root := {}⦈,\n sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := sys_mem_store_buffers (mutator m') s @ [mw_Mutate r f opt_r'])⦈))\n = mut_m.tso_store_refs m s ∪ (if m' = m then store_refs (mw_Mutate r f opt_r') else {})\"\n \"mut_m.tso_store_refs m (s(sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := sys_mem_store_buffers (mutator m') s @ [mw_Mutate_Payload r f pl])⦈))\n = mut_m.tso_store_refs m s ∪ (if m' = m then store_refs (mw_Mutate_Payload r f pl) else {})\"\n \"mut_m.tso_store_refs m (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈))\n = mut_m.tso_store_refs m s\"\n \"mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ obj)⦈))\n = mut_m.tso_store_refs m s\"\n \"mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇ghost_honorary_root := Option.set_option opt_r', ref := opt_r'⦈))\n = mut_m.tso_store_refs m s\"\n \"mut_m.tso_store_refs m (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := opt_r')⦈) (sys_heap s r)),\n mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))\n = (if p = mutator m then ⋃w ∈ set ws. store_refs w else mut_m.tso_store_refs m s)\"\n \"mut_m.tso_store_refs m (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)),\n mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))\n = (if p = mutator m then ⋃w ∈ set ws. store_refs w else mut_m.tso_store_refs m s)\"\n \"sys_mem_store_buffers p s = mw_Mark r fl # ws\n⟹ mut_m.tso_store_refs m (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))\n = mut_m.tso_store_refs m s\"\nunfolding mut_m.tso_store_refs_def\n(*goals:\n 1. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m'::'b) := s (mutator m')⦇roots := roots'::'a set⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 2. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m'::'b) := s (mutator m')⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := sys_mem_store_buffers (mutator m') s @ [mw_Mutate (r::'a) (f::'c) (opt_r'::'a option)])⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)) ∪ (if m' = m then store_refs (mw_Mutate r f opt_r') else {})›\n 3. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator (m'::'b) := sys_mem_store_buffers (mutator m') s @ [mw_Mutate_Payload (r::'a) (f::'c) (pl::'d option)])⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)) ∪ (if m' = m then store_refs (mw_Mutate_Payload r f pl) else {})›\n 4. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r'::'a := None)⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 5. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m'::'b) := s (mutator m')⦇roots := insert (r::'a) (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ obj::('c, 'd, 'a) object)⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 6. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m'::'b) := s (mutator m')⦇ghost_honorary_root := set_option opt_r', ref := opt_r'::'a option⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 7. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r::'a := map_option (λobj::('c, 'd, 'a) object. obj⦇obj_fields := (obj_fields obj)(f::'c := opt_r'::'a option)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p::'b process_name := ws::('c, 'd, 'a) mem_store_action list)⦈)))) = (if p = mutator m then ⋃ (store_refs ` set ws) else ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)))›\n 8. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r::'a := map_option (λobj::('c, 'd, 'a) object. obj⦇obj_payload := (obj_payload obj)(f::'c := pl::'d option)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p::'b process_name := ws::('c, 'd, 'a) mem_store_action list)⦈)))) = (if p = mutator m then ⋃ (store_refs ` set ws) else ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)))›\n 9. ‹sys_mem_store_buffers (p::'b process_name) (s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) = mw_Mark (r::'a) (fl::bool) # (ws::('c, 'd, 'a) mem_store_action list) ⟹ ⋃ (store_refs ` set (sys_mem_store_buffers (mutator (m::'b)) (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_::bool. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›*)\n(*goals:\n 1. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(mutator m' := s (mutator m')⦇roots := roots'⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 2. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(mutator m' := s (mutator m')⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := sys_mem_store_buffers (mutator m') s @ [mw_Mutate r f opt_r'])⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)) ∪ (if m' = m then store_refs (mw_Mutate r f opt_r') else {})›\n 3. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := sys_mem_store_buffers (mutator m') s @ [mw_Mutate_Payload r f pl])⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)) ∪ (if m' = m then store_refs (mw_Mutate_Payload r f pl) else {})›\n 4. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 5. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ obj)⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 6. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(mutator m' := s (mutator m')⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\n 7. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := opt_r')⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)))) = (if p = mutator m then ⋃ (store_refs ` set ws) else ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)))›\n 8. ‹⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)))) = (if p = mutator m then ⋃ (store_refs ` set ws) else ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)))›\n 9. ‹sys_mem_store_buffers p s = mw_Mark r fl # ws ⟹ ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)))) = ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s))›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 4*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 5*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 6*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 7*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 8*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 9*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*proven 9 subgoals*) .\n\nlemma fold_stores_points_to[rule_format, simplified conj_explode]:\n \"heap (fold_stores ws (s sys)) r = Some obj ∧ obj_fields obj f = Some r'\n ⟶ (r points_to r') s ∨ (∃w ∈ set ws. r' ∈ store_refs w)\" (is \"?P (fold_stores ws) obj\")\nunfolding fold_stores_def\n(*goal: ‹heap (fold (λw. (∘) (do_store_action w)) ws id (s sys)) r = Some obj ∧ obj_fields obj f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)›*)\napply (rule spec[OF fold_invariant[where P=\"λfr. ∀obj. ?P fr obj\" and Q=\"λw. w ∈ set ws\"]] (*‹⟦⋀x. x ∈ set ?xs1 ⟹ x ∈ set ws; ∀x. heap (?s1 (s sys)) r = Some x ∧ obj_fields x f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w); ⋀x sa. ⟦x ∈ set ws; ∀x. heap (sa (s sys)) r = Some x ∧ obj_fields x f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)⟧ ⟹ ∀xa. heap (?f1 x sa (s sys)) r = Some xa ∧ obj_fields xa f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)⟧ ⟹ heap (fold ?f1 ?xs1 ?s1 (s sys)) r = Some ?x ∧ obj_fields ?x f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)›*))\n(*goals:\n 1. ‹⋀x. x ∈ set ws ⟹ x ∈ set ws›\n 2. ‹∀x. heap (id (s sys)) r = Some x ∧ obj_fields x f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)›\n 3. ‹⋀x sa. ⟦x ∈ set ws; ∀x. heap (sa (s sys)) r = Some x ∧ obj_fields x f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)⟧ ⟹ ∀xa. heap ((do_store_action x ∘ sa) (s sys)) r = Some xa ∧ obj_fields xa f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)›\ndiscuss goal 1*)\n apply fastforce\n(*discuss goal 2*)\n apply (fastforce simp: ran_def (*‹ran ?m = {b. ∃a. ?m a = Some b}›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*discuss goal 3*)\n apply clarsimp\n (*goal: ‹⋀x sa. ⟦x ∈ set ws; ∀x. heap (sa (s sys)) r = Some x ∧ obj_fields x f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)⟧ ⟹ ∀xa. heap ((do_store_action x ∘ sa) (s sys)) r = Some xa ∧ obj_fields xa f = Some r' ⟶ (r points_to r') s ∨ (∃w∈set ws. r' ∈ store_refs w)›*)\n apply (drule (1) bspec (*‹⟦∀x∈?A. ?P x; ?x ∈ ?A⟧ ⟹ ?P ?x›*))\n (*goal: ‹⋀x sa xa. ⟦x ∈ set ws; heap (do_store_action x (sa (s sys))) r = Some xa; obj_fields xa f = Some r'; ∀w∈set ws. r' ∉ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}); ∀x. heap (sa (s sys)) r = Some x ⟶ obj_fields x f ≠ Some r'⟧ ⟹ (r points_to r') s›*)\n apply (clarsimp simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.split_asm (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n(*proven 3 subgoals*) .\n\nlemma points_to_Mutate:\n \"(x points_to y)\n (s(sys := (s sys)⦇ heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := opt_r')⦈) (sys_heap s r)),\n mem_store_buffers := (mem_store_buffers (s sys))(p := ws) ⦈))\n ⟷ (r ≠ x ∧ (x points_to y) s) ∨ (r = x ∧ valid_ref r s ∧ (opt_r' = Some y ∨ ( (x points_to y) s ∧ obj_at (λobj. ∃f'. obj_fields obj f' = Some y ∧ f ≠ f') r s)))\"\nunfolding ran_def\n(*goal: ‹obj_at (λobj::('a, 'b, 'c) object. (y::'c) ∈ {b::'c. ∃a::'a. obj_fields obj a = Some b}) (x::'c) ((s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r::'c := map_option (λobj::('a, 'b, 'c) object. obj⦇obj_fields := (obj_fields obj)(f::'a := opt_r'::'c option)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p::'d process_name := ws::('a, 'b, 'c) mem_store_action list)⦈)) = (r ≠ x ∧ obj_at (λobj::('a, 'b, 'c) object. y ∈ {b::'c. ∃a::'a. obj_fields obj a = Some b}) x s ∨ r = x ∧ obj_at (λs::('a, 'b, 'c) object. True) r s ∧ (opt_r' = Some y ∨ obj_at (λobj::('a, 'b, 'c) object. y ∈ {b::'c. ∃a::'a. obj_fields obj a = Some b}) x s ∧ obj_at (λobj::('a, 'b, 'c) object. ∃f'::'a. obj_fields obj f' = Some y ∧ f ≠ f') r s))›*)\nby (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\n\nsubsection‹ FIXME mutator handshake facts ›\n\nlemma ― ‹Sanity›\n \"hp' = hs_step hp ⟹ ∃in' ht. (in', ht, hp', hp) ∈ hp_step_rel\"\napply (cases hp)\n(*goals:\n 1. ‹⟦hp' = hs_step hp; hp = hp_Idle⟧ ⟹ ∃in' ht. (in', ht, hp', hp) ∈ hp_step_rel›\n 2. ‹⟦hp' = hs_step hp; hp = hp_IdleInit⟧ ⟹ ∃in' ht. (in', ht, hp', hp) ∈ hp_step_rel›\n 3. ‹⟦hp' = hs_step hp; hp = hp_InitMark⟧ ⟹ ∃in' ht. (in', ht, hp', hp) ∈ hp_step_rel›\n 4. ‹⟦hp' = hs_step hp; hp = hp_Mark⟧ ⟹ ∃in' ht. (in', ht, hp', hp) ∈ hp_step_rel›\n 5. ‹⟦hp' = hs_step hp; hp = hp_IdleMarkSweep⟧ ⟹ ∃in' ht. (in', ht, hp', hp) ∈ hp_step_rel›\ndiscuss goal 1*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*discuss goal 4*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*discuss goal 5*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp::hs_phase. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*proven 5 subgoals*) .\n\nlemma ― ‹Sanity›\n \"(False, ht, hp', hp) ∈ hp_step_rel ⟹ hp' = hp_step ht hp\"\napply (cases ht)\n(*goals:\n 1. ‹⟦(False, ht, hp', hp) ∈ hp_step_rel; ht = ht_NOOP⟧ ⟹ hp' = hp_step ht hp›\n 2. ‹⟦(False, ht, hp', hp) ∈ hp_step_rel; ht = ht_GetRoots⟧ ⟹ hp' = hp_step ht hp›\n 3. ‹⟦(False, ht, hp', hp) ∈ hp_step_rel; ht = ht_GetWork⟧ ⟹ hp' = hp_step ht hp›\ndiscuss goal 1*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))[1])\n(*proven 3 subgoals*) .\n\nlemma (in mut_m) handshake_phase_invD:\n assumes \"handshake_phase_inv s\"\n shows \"(sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, mut_ghost_hs_phase s) ∈ hp_step_rel\n ∧ (sys_hs_pending m s ⟶ ¬sys_ghost_hs_in_sync m s)\"\nusing assms (*‹handshake_phase_inv (s::'mut::type process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state)›*) unfolding handshake_phase_inv_def\n(*goal: ‹(sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, mut_ghost_hs_phase s) ∈ hp_step_rel ∧ (sys_hs_pending m s ⟶ ¬ sys_ghost_hs_in_sync m s)›*)\nby simp\n\nlemma handshake_in_syncD:\n \"⟦ All (ghost_hs_in_sync (s sys)); handshake_phase_inv s ⟧\n ⟹ ∀m'. mut_m.mut_ghost_hs_phase m' s = sys_ghost_hs_phase s\"\napply clarsimp\n(*goal: ‹⟦All (ghost_hs_in_sync (s sys)); handshake_phase_inv s⟧ ⟹ ∀m'. ghost_hs_phase (s (mutator m')) = sys_ghost_hs_phase s›*)\nby (auto simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp::hs_phase. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*) dest!: mut_m.handshake_phase_invD (*‹handshake_phase_inv (?s::?'mut process_name ⇒ (?'a, ?'mut, ?'b, ?'c) Model.local_state) ⟹ (sys_ghost_hs_in_sync (?m::?'mut) ?s, sys_hs_type ?s, sys_ghost_hs_phase ?s, ghost_hs_phase (?s (mutator ?m))) ∈ hp_step_rel ∧ (sys_hs_pending ?m ?s ⟶ ¬ sys_ghost_hs_in_sync ?m ?s)›*))\n\nlemmas fM_rel_invD = iffD1[OF fun_cong[OF fM_rel_inv_def[simplified atomize_eq]]]\n\ntext‹\n\nRelate @{const \"sys_ghost_hs_phase\"}, @{const \"gc_phase\"},\n@{const \"sys_phase\"} and writes to the phase in the GC's TSO buffer.\n\n›\n\nsimps_of_case handshake_phase_rel_simps[simp]: handshake_phase_rel_def (splits: hs_phase.split)\n\nlemma phase_rel_invD:\n assumes \"phase_rel_inv s\"\n shows \"(∀m. sys_ghost_hs_in_sync m s, sys_ghost_hs_phase s, gc_phase s, sys_phase s, tso_pending_phase gc s) ∈ phase_rel\"\nusing assms (*‹phase_rel_inv s›*) unfolding phase_rel_inv_def\n(*goal: ‹(∀m. sys_ghost_hs_in_sync m s, sys_ghost_hs_phase s, gc_phase s, sys_phase s, filter is_mw_Phase (sys_mem_store_buffers gc s)) ∈ phase_rel›*)\nby simp\n\nlemma mut_m_not_idle_no_fM_write:\n \"⟦ ghost_hs_phase (s (mutator m)) ≠ hp_Idle; fM_rel_inv s; handshake_phase_inv s; tso_store_inv s; p ≠ sys ⟧\n ⟹ ¬sys_mem_store_buffers p s = mw_fM fl # ws\"\napply (drule mut_m.handshake_phase_invD[where m=m] (*‹handshake_phase_inv ?s ⟹ (sys_ghost_hs_in_sync m ?s, sys_hs_type ?s, sys_ghost_hs_phase ?s, ghost_hs_phase (?s (mutator m))) ∈ hp_step_rel ∧ (sys_hs_pending m ?s ⟶ ¬ sys_ghost_hs_in_sync m ?s)›*))\n(*goal: ‹⟦ghost_hs_phase (s (mutator m)) ≠ hp_Idle; fM_rel_inv s; handshake_phase_inv s; tso_store_inv s; p ≠ sys⟧ ⟹ sys_mem_store_buffers p s ≠ mw_fM fl # ws›*)\napply (drule fM_rel_invD (*‹fM_rel_inv ?x ⟹ (∀x. sys_ghost_hs_in_sync x ?x, sys_ghost_hs_phase ?x, gc_fM ?x, sys_fM ?x, filter is_mw_fM (sys_mem_store_buffers gc ?x), sys_mem_lock ?x = Some gc) ∈ fM_rel›*))\n(*goal: ‹⟦ghost_hs_phase (s (mutator m)) ≠ hp_Idle; fM_rel_inv s; tso_store_inv s; p ≠ sys; (sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, ghost_hs_phase (s (mutator m))) ∈ hp_step_rel ∧ (sys_hs_pending m s ⟶ ¬ sys_ghost_hs_in_sync m s)⟧ ⟹ sys_mem_store_buffers p s ≠ mw_fM fl # ws›*)\napply (clarsimp simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*) fM_rel_def (*‹fM_rel = {(in_sync, hp, fM, fM, [], l) |fM hp in_sync l. hp = hp_Idle ⟶ ¬ in_sync} ∪ {(in_sync, hp_Idle, fM, fM', [], l) |fM fM' in_sync l. in_sync} ∪ {(in_sync, hp_Idle, ¬ fM, fM, [mw_fM (¬ fM)], False) |fM in_sync. in_sync}›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) p_not_sys (*‹(?p ≠ sys) = (?p = gc ∨ (∃m. ?p = mutator m))›*))\n(*goal: ‹⟦ghost_hs_phase (s (mutator m)) ≠ hp_Idle; tso_store_inv s; p ≠ sys; (sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, ghost_hs_phase (s (mutator m))) ∈ hp_step_rel ∧ (sys_hs_pending m s ⟶ ¬ sys_ghost_hs_in_sync m s); (∀x. sys_ghost_hs_in_sync x s, sys_ghost_hs_phase s, gc_fM s, sys_fM s, filter is_mw_fM (sys_mem_store_buffers gc s), sys_mem_lock s = Some gc) ∈ fM_rel⟧ ⟹ sys_mem_store_buffers p s ≠ mw_fM fl # ws›*)\nby (metis list.set_intros( (*‹?x21.0 ∈ set (?x21.0 # ?x22.0)›*) 1) tso_store_invD( (*‹tso_store_inv ?s ⟹ sys_mem_store_buffers (mutator ?m) ?s ≠ mw_fM ?fl # ?ws›*) 4))\n\nlemma (in mut_m) mut_ghost_handshake_phase_idle:\n \"⟦ mut_ghost_hs_phase s = hp_Idle; handshake_phase_inv s; phase_rel_inv s ⟧\n ⟹ sys_phase s = ph_Idle\"\napply (drule phase_rel_invD (*‹phase_rel_inv ?s ⟹ (∀m. sys_ghost_hs_in_sync m ?s, sys_ghost_hs_phase ?s, gc_phase ?s, sys_phase ?s, filter is_mw_Phase (sys_mem_store_buffers gc ?s)) ∈ phase_rel›*))\n(*goal: ‹⟦mut_ghost_hs_phase s = hp_Idle; handshake_phase_inv s; phase_rel_inv s⟧ ⟹ sys_phase s = ph_Idle›*)\napply (drule handshake_phase_invD (*‹handshake_phase_inv ?s ⟹ (sys_ghost_hs_in_sync m ?s, sys_hs_type ?s, sys_ghost_hs_phase ?s, mut_ghost_hs_phase ?s) ∈ hp_step_rel ∧ (sys_hs_pending m ?s ⟶ ¬ sys_ghost_hs_in_sync m ?s)›*))\n(*goal: ‹⟦mut_ghost_hs_phase s = hp_Idle; handshake_phase_inv s; (∀m. sys_ghost_hs_in_sync m s, sys_ghost_hs_phase s, gc_phase s, sys_phase s, filter is_mw_Phase (sys_mem_store_buffers gc s)) ∈ phase_rel⟧ ⟹ sys_phase s = ph_Idle›*)\nby (auto simp: phase_rel_def (*‹phase_rel = {(in_sync, hp, ph, ph, []) |in_sync hp ph. handshake_phase_rel hp in_sync ph} ∪ {True} × {(hp_IdleInit, ph_Init, ph_Idle, [mw_Phase ph_Init]), (hp_InitMark, ph_Mark, ph_Init, [mw_Phase ph_Mark]), (hp_IdleMarkSweep, ph_Sweep, ph_Mark, [mw_Phase ph_Sweep]), (hp_IdleMarkSweep, ph_Idle, ph_Mark, [mw_Phase ph_Sweep, mw_Phase ph_Idle]), (hp_IdleMarkSweep, ph_Idle, ph_Sweep, [mw_Phase ph_Idle])}›*) hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*))\n\nlemma mut_m_not_idle_no_fM_writeD:\n \"⟦ sys_mem_store_buffers p s = mw_fM fl # ws; ghost_hs_phase (s (mutator m)) ≠ hp_Idle; fM_rel_inv s; handshake_phase_inv s; tso_store_inv s; p ≠ sys ⟧\n ⟹ False\"\napply (drule mut_m.handshake_phase_invD[where m=m] (*‹handshake_phase_inv ?s ⟹ (sys_ghost_hs_in_sync m ?s, sys_hs_type ?s, sys_ghost_hs_phase ?s, ghost_hs_phase (?s (mutator m))) ∈ hp_step_rel ∧ (sys_hs_pending m ?s ⟶ ¬ sys_ghost_hs_in_sync m ?s)›*))\n(*goal: ‹⟦sys_mem_store_buffers p s = mw_fM fl # ws; ghost_hs_phase (s (mutator m)) ≠ hp_Idle; fM_rel_inv s; handshake_phase_inv s; tso_store_inv s; p ≠ sys⟧ ⟹ False›*)\napply (drule fM_rel_invD (*‹fM_rel_inv (?x::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state) ⟹ (∀x::?'mut. sys_ghost_hs_in_sync x ?x, sys_ghost_hs_phase ?x, gc_fM ?x, sys_fM ?x, filter is_mw_fM (sys_mem_store_buffers gc ?x), sys_mem_lock ?x = Some gc) ∈ fM_rel›*))\n(*goal: ‹⟦sys_mem_store_buffers p s = mw_fM fl # ws; ghost_hs_phase (s (mutator m)) ≠ hp_Idle; fM_rel_inv s; tso_store_inv s; p ≠ sys; (sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, ghost_hs_phase (s (mutator m))) ∈ hp_step_rel ∧ (sys_hs_pending m s ⟶ ¬ sys_ghost_hs_in_sync m s)⟧ ⟹ False›*)\napply (clarsimp simp: hp_step_rel_def (*‹hp_step_rel = {True} × ({(ht_NOOP, hp, hp) |hp. hp ∈ {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} ∪ {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) ∪ {False} × {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}›*) fM_rel_def (*‹fM_rel = {(in_sync, hp, fM, fM, [], l) |fM hp in_sync l. hp = hp_Idle ⟶ ¬ in_sync} ∪ {(in_sync, hp_Idle, fM, fM', [], l) |fM fM' in_sync l. in_sync} ∪ {(in_sync, hp_Idle, ¬ fM, fM, [mw_fM (¬ fM)], False) |fM in_sync. in_sync}›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) p_not_sys (*‹(?p ≠ sys) = (?p = gc ∨ (∃m. ?p = mutator m))›*))\n(*goal: ‹⟦sys_mem_store_buffers (p::'d process_name) (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) = mw_fM (fl::bool) # (ws::('a, 'b, 'c) mem_store_action list); ghost_hs_phase (s (mutator (m::'d))) ≠ hp_Idle; tso_store_inv s; p ≠ sys; (sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, ghost_hs_phase (s (mutator m))) ∈ hp_step_rel ∧ (sys_hs_pending m s ⟶ ¬ sys_ghost_hs_in_sync m s); (∀x::'d. sys_ghost_hs_in_sync x s, sys_ghost_hs_phase s, gc_fM s, sys_fM s, filter is_mw_fM (sys_mem_store_buffers gc s), sys_mem_lock s = Some gc) ∈ fM_rel⟧ ⟹ False›*)\nby (metis list.set_intros( (*‹?x21.0 ∈ set (?x21.0 # ?x22.0)›*) 1) tso_store_invD( (*‹tso_store_inv ?s ⟹ sys_mem_store_buffers (mutator ?m) ?s ≠ mw_fM ?fl # ?ws›*) 4))\n\n\nsubsection‹points to, reaches, reachable mut›\n\nlemma (in mut_m) reachable_eq_imp:\n \"eq_imp (λr'. mut_roots ❙⊗ mut_ghost_honorary_root ❙⊗ (λs. ⋃(ran ` obj_fields ` set_option (sys_heap s r')))\n ❙⊗ tso_pending_mutate (mutator m))\n (reachable r)\"\nunfolding eq_imp_def reachable_def tso_store_refs_def\n(*goal: ‹∀s s'. (∀x. (mut_roots s, mut_ghost_honorary_root s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s)) = (mut_roots s', mut_ghost_honorary_root s', ⋃ (ran ` obj_fields ` set_option (sys_heap s' x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'))) ⟶ (∃x. x ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)) ∧ (x reaches r) s) = (∃x. x ∈ mut_roots s' ∪ mut_ghost_honorary_root s' ∪ ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s')) ∧ (x reaches r) s')›*)\napply ((clarsimp)[1])\n(*goal: ‹∀s s'. (∀x. (mut_roots s, mut_ghost_honorary_root s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s)) = (mut_roots s', mut_ghost_honorary_root s', ⋃ (ran ` obj_fields ` set_option (sys_heap s' x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'))) ⟶ (∃x. x ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)) ∧ (x reaches r) s) = (∃x. x ∈ mut_roots s' ∪ mut_ghost_honorary_root s' ∪ ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s')) ∧ (x reaches r) s')›*)\napply ((rename_tac s s')[1])\n(*goal: ‹⋀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x::'a. (⋃a::('b, 'c, 'a) object∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a::('b, 'c, 'a) object∈set_option (sys_heap s' x). ran (obj_fields a)); filter (λs::('b, 'c, 'a) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator (m::'mut)) s) = filter (λs::('b, 'c, 'a) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (∃x::'a. (x ∈ mut_roots s' ∨ x ∈ mut_ghost_honorary_root s' ∨ (∃xa::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {}))) ∧ (x reaches (r::'a)) s) = (∃x::'a. (x ∈ mut_roots s' ∨ x ∈ mut_ghost_honorary_root s' ∨ (∃xa::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s'). x ∈ (case xa of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {}))) ∧ (x reaches r) s')›*)\napply ((subgoal_tac \"∀r'. (∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ store_refs w) ⟷ (∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ store_refs w)\")[1])\n(*goals:\n 1. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); ∀r'. (∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ store_refs w) = (∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ store_refs w)⟧ ⟹ (∃x. (x ∈ mut_roots s' ∨ x ∈ mut_ghost_honorary_root s' ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (x reaches r) s) = (∃x. (x ∈ mut_roots s' ∨ x ∈ mut_ghost_honorary_root s' ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s'). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (x reaches r) s')›\n 2. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ ∀r'. (∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ store_refs w) = (∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ store_refs w)›\ndiscuss goal 1*)\n apply ((subgoal_tac \"∀x. (x reaches r) s ⟷ (x reaches r) s'\")[1])\n (*goals:\n 1. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); ∀r'. (∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ store_refs w) = (∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ store_refs w); ∀x. (x reaches r) s = (x reaches r) s'⟧ ⟹ (∃x. (x ∈ mut_roots s' ∨ x ∈ mut_ghost_honorary_root s' ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (x reaches r) s) = (∃x. (x ∈ mut_roots s' ∨ x ∈ mut_ghost_honorary_root s' ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s'). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (x reaches r) s')›\n 2. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); ∀r'. (∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ store_refs w) = (∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ store_refs w)⟧ ⟹ ∀x. (x reaches r) s = (x reaches r) s'›\n discuss goal 1*)\n apply ((clarsimp)[1])\n (*discuss goal 2*)\n apply ((auto simp: reaches_fields (*‹⟦(?x reaches ?y) ?s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap ?s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap ?s r'))⟧ ⟹ (?x reaches ?y) ?s›*))[1])\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply ((drule arg_cong[where f=set] (*‹?x = ?y ⟹ set ?x = set ?y›*))[1])\n (*goal: ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ ∀r'. (∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ store_refs w) = (∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ store_refs w)›*)\n apply ((clarsimp simp: set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*))[1])\n (*goal: ‹⋀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x::'a. (⋃a::('b, 'c, 'a) object∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a::('b, 'c, 'a) object∈set_option (sys_heap s' x). ran (obj_fields a)); set (filter (λs::('b, 'c, 'a) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator (m::'mut)) s)) = set (filter (λs::('b, 'c, 'a) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'))⟧ ⟹ ∀r'::'a. (∃w::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). r' ∈ store_refs w) = (∃w::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ store_refs w)›*)\n apply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹⋀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) r'::'a. ⟦∀x::'a. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x::'a. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀(x::'a) xa::'a. (∃x::('b, 'c, 'a) object∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x::('b, 'c, 'a) object∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x::('b, 'c, 'a) mem_store_action. (x ∈ set (sys_mem_store_buffers (mutator (m::'mut)) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); ∃w::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). r' ∈ (case w of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ (case w of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {})›\n 2. ‹⋀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) r'::'a. ⟦∀x::'a. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x::'a. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀(x::'a) xa::'a. (∃x::('b, 'c, 'a) object∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x::('b, 'c, 'a) object∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x::('b, 'c, 'a) mem_store_action. (x ∈ set (sys_mem_store_buffers (mutator (m::'mut)) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); ∃w::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ (case w of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). r' ∈ (case w of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {})›\n discuss goal 1*)\n apply ((clarsimp)[1])\n (*top goal: ‹⋀s s' r'. ⟦∀x. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀x xa. (∃x∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x. (x ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); ∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})› and 1 goal remains*)\n apply ((rename_tac s s' r' w)[1])\n (*top goal: ‹⋀s s' r' w. ⟦∀x. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀x xa. (∃x∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x. (x ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); w ∈ set (sys_mem_store_buffers (mutator m) s); r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})› and 1 goal remains*)\n apply ((drule_tac x=w in spec (*‹∀x. ?P x ⟹ ?P ?x›*))[1])\n (*top goal: ‹⋀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (r'::'a) w::('b, 'c, 'a) mem_store_action. ⟦∀x::'a. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x::'a. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀(x::'a) xa::'a. (∃x::('b, 'c, 'a) object∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x::('b, 'c, 'a) object∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x::('b, 'c, 'a) mem_store_action. (x ∈ set (sys_mem_store_buffers (mutator (m::'mut)) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); w ∈ set (sys_mem_store_buffers (mutator m) s); r' ∈ (case w of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w::('b, 'c, 'a) mem_store_action∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ (case w of mw_Mutate (r::'a) (f::'b) (r'::'a option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a) (f::'b) (pl::'c option) ⇒ {r} | _ ⇒ {})› and 1 goal remains*)\n apply ((rule_tac x=w in bexI (*‹⟦(?P::?'a ⇒ bool) (?x::?'a); ?x ∈ (?A::?'a set)⟧ ⟹ ∃x::?'a∈?A. ?P x›*))[1])\n (*goals:\n 1. ‹⋀(s::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (s'::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (r'::'a::type) w::('b::type, 'c::type, 'a::type) mem_store_action. ⟦∀x::'a::type. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x::'a::type. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀(x::'a::type) xa::'a::type. (∃x::('b, 'c, 'a) object∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x::('b, 'c, 'a) object∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); w ∈ set (sys_mem_store_buffers (mutator (m::'mut::type)) s); r' ∈ (case w of mw_Mutate (r::'a::type) (f::'b::type) (r'::'a::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a::type) (f::'b::type) (pl::'c::type option) ⇒ {r} | _ ⇒ {}); (w ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w)) = (w ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w))⟧ ⟹ r' ∈ (case w of mw_Mutate (r::'a::type) (f::'b::type) (r'::'a::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a::type) (f::'b::type) (pl::'c::type option) ⇒ {r} | _ ⇒ {})›\n 2. ‹⋀(s::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (s'::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (r'::'a::type) w::('b::type, 'c::type, 'a::type) mem_store_action. ⟦∀x::'a::type. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x::'a::type. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀(x::'a::type) xa::'a::type. (∃x::('b, 'c, 'a) object∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x::('b, 'c, 'a) object∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); w ∈ set (sys_mem_store_buffers (mutator (m::'mut::type)) s); r' ∈ (case w of mw_Mutate (r::'a::type) (f::'b::type) (r'::'a::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'a::type) (f::'b::type) (pl::'c::type option) ⇒ {r} | _ ⇒ {}); (w ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w)) = (w ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w))⟧ ⟹ w ∈ set (sys_mem_store_buffers (mutator m) s')›\n discuss goal 1*)\n apply ((clarsimp)[1])\n (*discuss goal 2*)\n apply ((clarsimp split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply ((clarsimp)[1])\n (*goal: ‹⋀s s' r'. ⟦∀x. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀x xa. (∃x∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x. (x ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); ∃w∈set (sys_mem_store_buffers (mutator m) s'). r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})›*)\n apply ((rename_tac s s' r' w)[1])\n (*goal: ‹⋀s s' r' w. ⟦∀x. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀x xa. (∃x∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x. (x ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); w ∈ set (sys_mem_store_buffers (mutator m) s'); r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})›*)\n apply ((drule_tac x=w in spec (*‹∀x. ?P x ⟹ ?P ?x›*))[1])\n (*goal: ‹⋀s s' r' w. ⟦∀x. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀x xa. (∃x∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); ∀x. (x ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)) = (x ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate x ∨ is_mw_Mutate_Payload x)); w ∈ set (sys_mem_store_buffers (mutator m) s'); r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃w∈set (sys_mem_store_buffers (mutator m) s). r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})›*)\n apply ((rule_tac x=w in bexI (*‹⟦?P ?x; ?x ∈ ?A⟧ ⟹ ∃x∈?A. ?P x›*))[1])\n (*goals:\n 1. ‹⋀s s' r' w. ⟦∀x. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀x xa. (∃x∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); w ∈ set (sys_mem_store_buffers (mutator m) s'); r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}); (w ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w)) = (w ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w))⟧ ⟹ r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})›\n 2. ‹⋀s s' r' w. ⟦∀x. (x ∈ mut_roots s) = (x ∈ mut_roots s'); ∀x. (x ∈ mut_ghost_honorary_root s) = (x ∈ mut_ghost_honorary_root s'); ∀x xa. (∃x∈set_option (sys_heap s x). xa ∈ ran (obj_fields x)) = (∃x∈set_option (sys_heap s' x). xa ∈ ran (obj_fields x)); w ∈ set (sys_mem_store_buffers (mutator m) s'); r' ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}); (w ∈ set (sys_mem_store_buffers (mutator m) s) ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w)) = (w ∈ set (sys_mem_store_buffers (mutator m) s') ∧ (is_mw_Mutate w ∨ is_mw_Mutate_Payload w))⟧ ⟹ w ∈ set (sys_mem_store_buffers (mutator m) s)›\n discuss goal 1*)\n apply ((clarsimp)[1])\n (*discuss goal 2*)\n apply ((clarsimp split: mem_store_action.splits (*‹(?P::?'a ⇒ bool) (case ?mem_store_action::(?'field, ?'payload, ?'ref) mem_store_action of mw_Mark (x::?'ref) (xa::bool) ⇒ (?f1.0::?'ref ⇒ bool ⇒ ?'a) x xa | mw_Mutate (x::?'ref) (xa::?'field) (xb::?'ref option) ⇒ (?f2.0::?'ref ⇒ ?'field ⇒ ?'ref option ⇒ ?'a) x xa xb | mw_Mutate_Payload (x::?'ref) (xa::?'field) (xb::?'payload option) ⇒ (?f3.0::?'ref ⇒ ?'field ⇒ ?'payload option ⇒ ?'a) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a) x) = ((∀(x11::?'ref) x12::bool. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀(x21::?'ref) (x22::?'field) x23::?'ref option. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀(x31::?'ref) (x32::?'field) x33::?'payload option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4::bool. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5::bool. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6::gc_phase. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹(?P::?'a ⇒ bool) (case ?mem_store_action::(?'field, ?'payload, ?'ref) mem_store_action of mw_Mark (x::?'ref) (xa::bool) ⇒ (?f1.0::?'ref ⇒ bool ⇒ ?'a) x xa | mw_Mutate (x::?'ref) (xa::?'field) (xb::?'ref option) ⇒ (?f2.0::?'ref ⇒ ?'field ⇒ ?'ref option ⇒ ?'a) x xa xb | mw_Mutate_Payload (x::?'ref) (xa::?'field) (xb::?'payload option) ⇒ (?f3.0::?'ref ⇒ ?'field ⇒ ?'payload option ⇒ ?'a) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a) x) = (¬ ((∃(x11::?'ref) x12::bool. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃(x21::?'ref) (x22::?'field) x23::?'ref option. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃(x31::?'ref) (x32::?'field) x33::?'payload option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4::bool. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5::bool. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6::gc_phase. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))[1])\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n(*proven 2 subgoals*) .\n\nlemmas reachable_fun_upd[simp] = eq_imp_fun_upd[OF mut_m.reachable_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma reachableI[intro]:\n \"x ∈ mut_m.mut_roots m s ⟹ mut_m.reachable m x s\"\n \"x ∈ mut_m.tso_store_refs m s ⟹ mut_m.reachable m x s\"\n\n(*goals:\n 1. ‹x ∈ roots (s (mutator m)) ⟹ mut_m.reachable m x s›\n 2. ‹x ∈ mut_m.tso_store_refs m s ⟹ mut_m.reachable m x s›\ndiscuss goal 1*)\n apply ((auto simp: mut_m.reachable_def (*‹mut_m.reachable (?m::?'mut::type) (?y::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref::type. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) reaches_def (*‹(?x::?'ref::type) reaches (?y::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (λ(x::?'ref::type) y::?'ref::type. (x points_to y) s)⇧*⇧* ?x ?y)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) reaches_def (*‹?x reaches ?y = (λs. (λx y. (x points_to y) s)⇧*⇧* ?x ?y)›*))[1])\n(*proven 2 subgoals*) .\n\nlemma reachable_points_to[elim]:\n \"⟦ (x points_to y) s; mut_m.reachable m x s ⟧ ⟹ mut_m.reachable m y s\"\nby (auto simp: mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) reaches_def (*‹?x reaches ?y = (λs. (λx y. (x points_to y) s)⇧*⇧* ?x ?y)›*) elim: rtranclp.intros( (*‹⟦?r⇧*⇧* ?a ?b; ?r ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2))\n\nlemma (in mut_m) mut_reachableE[consumes 1, case_names mut_root tso_store_refs]:\n \"⟦ reachable y s;\n ⋀x. ⟦ (x reaches y) s; x ∈ mut_roots s ⟧ ⟹ Q;\n ⋀x. ⟦ (x reaches y) s; x ∈ mut_ghost_honorary_root s ⟧ ⟹ Q;\n ⋀x. ⟦ (x reaches y) s; x ∈ tso_store_refs s ⟧ ⟹ Q ⟧ ⟹ Q\"\nby (auto simp: reachable_def (*‹reachable (?y::?'ref) = (λs::'mut process_name ⇒ (?'field, 'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. x ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (x reaches ?y) s)›*))\n\nlemma reachable_induct[consumes 1, case_names root ghost_honorary_root tso_root reaches]:\n assumes r: \"mut_m.reachable m y s\"\n assumes root: \"⋀x. ⟦ x ∈ mut_m.mut_roots m s ⟧ ⟹ P x\"\n assumes ghost_honorary_root: \"⋀x. ⟦ x ∈ mut_m.mut_ghost_honorary_root m s ⟧ ⟹ P x\"\n assumes tso_root: \"⋀x. x ∈ mut_m.tso_store_refs m s ⟹ P x\"\n assumes reaches: \"⋀x y. ⟦ mut_m.reachable m x s; (x points_to y) s; P x ⟧ ⟹ P y\"\n shows \"P y\"\nusing r (*‹mut_m.reachable m y s›*) unfolding mut_m.reachable_def\n(*goal: ‹P y›*)\nproof (clarify)\n (*goal: ‹⋀x. ⟦x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s; (x reaches y) s⟧ ⟹ P y›*)\n fix x\n assume \"(x reaches y) s\" and \"x ∈ mut_m.mut_roots m s ∪ mut_m.mut_ghost_honorary_root m s ∪ mut_m.tso_store_refs m s\" (*‹((x::'b) reaches (y::'b)) (s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state)› ‹(x::'b) ∈ roots ((s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (mutator (m::'a))) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s›*)\n then show \"P y\"\n unfolding reaches_def\n (*goal: ‹P y›*)\n proof (induct)\n (*goals:\n 1. ‹x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s ⟹ P x›\n 2. ‹⋀y z. ⟦(λx y. (x points_to y) s)⇧*⇧* x y; (y points_to z) s; x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s ⟹ P y; x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s⟧ ⟹ P z›*)\n case base (*‹(x::'b) ∈ roots ((s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (mutator (m::'a))) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s›*)\n with root (*‹?x ∈ roots (s (mutator m)) ⟹ P ?x›*) ghost_honorary_root (*‹?x ∈ ghost_honorary_root (s (mutator m)) ⟹ P ?x›*) tso_root (*‹?x ∈ mut_m.tso_store_refs m s ⟹ P ?x›*) show \"?case\"\n (*goal: ‹P x›*)\n by blast\n next\n (*goal: ‹⋀(y::'b) z::'b. ⟦(λ(x::'b) y::'b. (x points_to y) (s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state))⇧*⇧* (x::'b) y; (y points_to z) s; x ∈ roots (s (mutator (m::'a))) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s ⟹ (P::'b ⇒ bool) y; x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s⟧ ⟹ P z›*)\n case (step y z) (*‹(λx y. (x points_to y) s)⇧*⇧* x y› ‹(y points_to z) s› ‹x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s ⟹ P y› ‹(x::'b) ∈ roots ((s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (mutator (m::'a))) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s›*)\n with reaches (*‹⟦mut_m.reachable m ?x s; (?x points_to ?y) s; P ?x⟧ ⟹ P ?y›*) show \"?case\"\n (*goal: ‹(P::'b::type ⇒ bool) (z::'b::type)›*)\n unfolding mut_m.reachable_def reaches_def\n (*goal: ‹P z›*)\n by meson\n qed\nqed\n\nlemma mutator_reachable_tso:\n \"sys_mem_store_buffers (mutator m) s = mw_Mutate r f opt_r' # ws\n ⟹ mut_m.reachable m r s ∧ (∀r'. opt_r' = Some r' ⟶ mut_m.reachable m r' s)\"\n \"sys_mem_store_buffers (mutator m) s = mw_Mutate_Payload r f pl # ws\n ⟹ mut_m.reachable m r s\"\n\n(*goals:\n 1. ‹sys_mem_store_buffers (mutator m) s = mw_Mutate r f opt_r' # ws ⟹ mut_m.reachable m r s ∧ (∀r'. opt_r' = Some r' ⟶ mut_m.reachable m r' s)›\n 2. ‹sys_mem_store_buffers (mutator m) s = mw_Mutate_Payload r f pl # ws ⟹ mut_m.reachable m r s›\ndiscuss goal 1*)\n apply ((auto simp: mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*))[1])\n(*proven 2 subgoals*) .\n\n\nsubsection‹ Colours ›\n\nlemma greyI[intro]:\n \"r ∈ ghost_honorary_grey (s p) ⟹ grey r s\"\n \"r ∈ W (s p) ⟹ grey r s\"\n \"r ∈ WL p s ⟹ grey r s\"\nunfolding grey_def WL_def\n(*goals:\n 1. ‹r ∈ ghost_honorary_grey (s p) ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 2. ‹r ∈ W (s p) ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 3. ‹r ∈ W (s p) ∪ ghost_honorary_grey (s p) ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›*)\n(*goals:\n 1. ‹r ∈ ghost_honorary_grey (s p) ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 2. ‹r ∈ W (s p) ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 3. ‹r ∈ W (s p) ∪ ghost_honorary_grey (s p) ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\ndiscuss goal 1*)\n apply ((case_tac [!] p)[1])\n (*goals:\n 1. ‹⋀x1. ⟦r ∈ ghost_honorary_grey (s p); p = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 2. ‹⟦r ∈ ghost_honorary_grey (s p); p = gc⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 3. ‹⟦r ∈ ghost_honorary_grey (s p); p = sys⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*)\n(*discuss goal 2*)\n apply ((case_tac [!] p)[1])\n (*goals:\n 1. ‹⋀x1. ⟦r ∈ W (s p); p = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 2. ‹⟦r ∈ W (s p); p = gc⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 3. ‹⟦r ∈ W (s p); p = sys⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*)\n(*discuss goal 3*)\n apply ((case_tac [!] p)[1])\n (*goals:\n 1. ‹⋀x1. ⟦r ∈ W (s p) ∪ ghost_honorary_grey (s p); p = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 2. ‹⟦r ∈ W (s p) ∪ ghost_honorary_grey (s p); p = gc⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n 3. ‹⟦r ∈ W (s p) ∪ ghost_honorary_grey (s p); p = sys⟧ ⟹ ∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*)\n(*proven 3 subgoals*) .\n\nlemma blackD[dest]:\n \"black r s ⟹ marked r s\"\n \"black r s ⟹ r ∉ WL p s\"\nunfolding black_def grey_def\n(*goals:\n 1. ‹marked r s ∧ (∄x. r ∈ WL x s) ⟹ marked r s›\n 2. ‹marked r s ∧ (∄x. r ∈ WL x s) ⟹ r ∉ WL p s›*)\n(*goals:\n 1. ‹marked r s ∧ (∄x. r ∈ WL x s) ⟹ marked r s›\n 2. ‹marked r s ∧ (∄x. r ∈ WL x s) ⟹ r ∉ WL p s›\ndiscuss goal 1*)\n apply simp\n(*discuss goal 2*)\n apply simp\n(*proven 2 subgoals*) .\n\nlemma whiteI[intro]: (* FIXME simp normal form of def *)\n \"obj_at (λobj. obj_mark obj = (¬ sys_fM s)) r s ⟹ white r s\"\nunfolding white_def\n(*goal: ‹obj_at (λobj. obj_mark obj = (¬ sys_fM s)) r s ⟹ obj_at (λobj. obj_mark obj ≠ sys_fM s) r s›*)\nby simp\n\nlemma marked_not_white[dest]:\n \"white r s ⟹ ¬marked r s\"\nunfolding white_def\n(*goal: ‹obj_at (λobj. obj_mark obj ≠ sys_fM s) r s ⟹ ¬ marked r s›*)\nby (simp split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma white_valid_ref[elim!]:\n \"white r s ⟹ valid_ref r s\"\nunfolding white_def\n(*goal: ‹obj_at (λobj. obj_mark obj ≠ sys_fM s) r s ⟹ obj_at (λs. True) r s›*)\nby (simp split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma not_white_marked[elim!]:\n \"⟦¬ white r s; valid_ref r s⟧ ⟹ marked r s\"\nunfolding white_def\n(*goal: ‹⟦¬ obj_at (λobj. obj_mark obj ≠ sys_fM s) r s; obj_at (λs. True) r s⟧ ⟹ marked r s›*)\nby (simp split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma black_eq_imp:\n \"eq_imp (λ_::unit. (λs. r ∈ (⋃p. WL p s)) ❙⊗ sys_fM ❙⊗ (λs. map_option obj_mark (sys_heap s r)))\n (black r)\"\nunfolding eq_imp_def black_def grey_def\n(*goal: ‹∀s s'. (∀x. (r ∈ (⋃p. WL p s), sys_fM s, map_option obj_mark (sys_heap s r)) = (r ∈ (⋃p. WL p s'), sys_fM s', map_option obj_mark (sys_heap s' r))) ⟶ (marked r s ∧ (∄x. r ∈ WL x s)) = (marked r s' ∧ (∄x. r ∈ WL x s'))›*)\nby (auto split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma grey_eq_imp:\n \"eq_imp (λ_::unit. (λs. r ∈ (⋃p. WL p s)))\n (grey r)\"\nunfolding eq_imp_def grey_def\n(*goal: ‹∀s s'. (∀x. (r ∈ (⋃p. WL p s)) = (r ∈ (⋃p. WL p s'))) ⟶ (∃x. r ∈ WL x s) = (∃x. r ∈ WL x s')›*)\nby auto\n\nlemma white_eq_imp:\n \"eq_imp (λ_::unit. sys_fM ❙⊗ (λs. map_option obj_mark (sys_heap s r)))\n (white r)\"\nunfolding eq_imp_def white_def\n(*goal: ‹∀(s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) s'::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state. (∀x::unit. (sys_fM s, map_option obj_mark (sys_heap s (r::'d))) = (sys_fM s', map_option obj_mark (sys_heap s' r))) ⟶ obj_at (λobj::('b, 'c, 'd) object. obj_mark obj ≠ sys_fM s) r s = obj_at (λobj::('b, 'c, 'd) object. obj_mark obj ≠ sys_fM s') r s'›*)\nby (auto split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemmas black_fun_upd[simp] = eq_imp_fun_upd[OF black_eq_imp, simplified eq_imp_simps, rule_format]\nlemmas grey_fun_upd[simp] = eq_imp_fun_upd[OF grey_eq_imp, simplified eq_imp_simps, rule_format]\nlemmas white_fun_upd[simp] = eq_imp_fun_upd[OF white_eq_imp, simplified eq_imp_simps, rule_format]\n\n\ntext‹coloured heaps›\n\nlemma black_heap_eq_imp:\n \"eq_imp (λr'. (λs. ⋃p. WL p s) ❙⊗ sys_fM ❙⊗ (λs. map_option obj_mark (sys_heap s r')))\n black_heap\"\napply ((clarsimp simp: eq_imp_def (*‹eq_imp (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) (?g::?'b::type ⇒ ?'e::type) ≡ ∀(s::?'b::type) s'::?'b::type. (∀x::?'a::type. ?f x s = ?f x s') ⟶ ?g s = ?g s'›*) black_heap_def (*‹black_heap = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref::type. obj_at (λs::(?'field, ?'payload, ?'ref) object. True) x s ⟶ black x s)›*) black_def (*‹black (?r::?'ref::type) ≡ λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. marked ?r s ∧ ¬ grey ?r s›*) grey_def (*‹grey (?r::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'mut::type process_name. ?r ∈ WL x s)›*) all_conj_distrib (*‹(∀x::?'a::type. (?P::?'a::type ⇒ bool) x ∧ (?Q::?'a::type ⇒ bool) x) = ((∀x::?'a::type. ?P x) ∧ (∀x::?'a::type. ?Q x))›*) fun_eq_iff (*‹((?f::?'a::type ⇒ ?'b::type) = (?g::?'a::type ⇒ ?'b::type)) = (∀x::?'a::type. ?f x = ?g x)›*) split: option.splits (*‹(?P::?'b::type ⇒ bool) (case ?option::?'a::type option of None ⇒ ?f1.0::?'b::type | Some (x::?'a::type) ⇒ (?f2.0::?'a::type ⇒ ?'b::type) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a::type. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b::type ⇒ bool) (case ?option::?'a::type option of None ⇒ ?f1.0::?'b::type | Some (x::?'a::type) ⇒ (?f2.0::?'a::type ⇒ ?'b::type) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a::type. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*goal: ‹eq_imp (λr' s. (⋃p. WL p s, sys_fM s, map_option obj_mark (sys_heap s r'))) black_heap›*)\napply ((rename_tac s s')[1])\n(*goal: ‹⋀s s'. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (∀x. obj_at (λs. True) x s ⟶ obj_at (λobj. obj_mark obj = sys_fM s') x s ∧ (∀xa. x ∉ WL xa s)) = (∀x. obj_at (λs. True) x s' ⟶ marked x s' ∧ (∀xa. x ∉ WL xa s'))›*)\napply ((subgoal_tac \"∀x. marked x s ⟷ marked x s'\")[1])\n(*goals:\n 1. ‹⋀(s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state. ⟦(⋃p::'b process_name. WL p s) = (⋃p::'b process_name. WL p s'); sys_fM s = sys_fM s'; ∀x::'a. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x::'a. marked x s = marked x s'⟧ ⟹ (∀x::'a. obj_at (λs::('c, 'd, 'a) object. True) x s ⟶ obj_at (λobj::('c, 'd, 'a) object. obj_mark obj = sys_fM s') x s ∧ (∀xa::'b process_name. x ∉ WL xa s)) = (∀x::'a. obj_at (λs::('c, 'd, 'a) object. True) x s' ⟶ marked x s' ∧ (∀xa::'b process_name. x ∉ WL xa s'))›\n 2. ‹⋀(s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state. ⟦(⋃p::'b process_name. WL p s) = (⋃p::'b process_name. WL p s'); sys_fM s = sys_fM s'; ∀x::'a. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀x::'a. marked x s = marked x s'›\ndiscuss goal 1*)\n apply ((subgoal_tac \"∀x. valid_ref x s ⟷ valid_ref x s'\")[1])\n (*goals:\n 1. ‹⋀s s'. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x. marked x s = marked x s'; ∀x. obj_at (λs. True) x s = obj_at (λs. True) x s'⟧ ⟹ (∀x. obj_at (λs. True) x s ⟶ obj_at (λobj. obj_mark obj = sys_fM s') x s ∧ (∀xa. x ∉ WL xa s)) = (∀x. obj_at (λs. True) x s' ⟶ marked x s' ∧ (∀xa. x ∉ WL xa s'))›\n 2. ‹⋀s s'. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x. marked x s = marked x s'⟧ ⟹ ∀x. obj_at (λs. True) x s = obj_at (λs. True) x s'›\n discuss goal 1*)\n apply ((subgoal_tac \"∀x. (∀p. x ∉ WL p s) ⟷ (∀p. x ∉ WL p s')\")[1])\n (*goals:\n 1. ‹⋀(s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state. ⟦(⋃p::'b process_name. WL p s) = (⋃p::'b process_name. WL p s'); sys_fM s = sys_fM s'; ∀x::'a. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x::'a. marked x s = marked x s'; ∀x::'a. obj_at (λs::('c, 'd, 'a) object. True) x s = obj_at (λs::('c, 'd, 'a) object. True) x s'; ∀x::'a. (∀p::'b process_name. x ∉ WL p s) = (∀p::'b process_name. x ∉ WL p s')⟧ ⟹ (∀x::'a. obj_at (λs::('c, 'd, 'a) object. True) x s ⟶ obj_at (λobj::('c, 'd, 'a) object. obj_mark obj = sys_fM s') x s ∧ (∀xa::'b process_name. x ∉ WL xa s)) = (∀x::'a. obj_at (λs::('c, 'd, 'a) object. True) x s' ⟶ marked x s' ∧ (∀xa::'b process_name. x ∉ WL xa s'))›\n 2. ‹⋀(s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state. ⟦(⋃p::'b process_name. WL p s) = (⋃p::'b process_name. WL p s'); sys_fM s = sys_fM s'; ∀x::'a. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x::'a. marked x s = marked x s'; ∀x::'a. obj_at (λs::('c, 'd, 'a) object. True) x s = obj_at (λs::('c, 'd, 'a) object. True) x s'⟧ ⟹ ∀x::'a. (∀p::'b process_name. x ∉ WL p s) = (∀p::'b process_name. x ∉ WL p s')›\n discuss goal 1*)\n apply ((clarsimp)[1])\n (*discuss goal 2*)\n apply ((auto simp: set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*))[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply ((clarsimp)[1])\n (*top goal: ‹⋀s s'. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x. marked x s = marked x s'⟧ ⟹ ∀x. obj_at (λs. True) x s = obj_at (λs. True) x s'› and 1 goal remains*)\n apply ((rename_tac x)[1])\n (*top goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'⟧ ⟹ obj_at (λs. True) x s = obj_at (λs. True) x s'› and 1 goal remains*)\n apply (rule eq_impD[OF obj_at_eq_imp] (*‹∀x. map_option ?P1 (sys_heap ?s ?r1) = map_option ?P1 (sys_heap ?s' ?r1) ⟹ obj_at ?P1 ?r1 ?s = obj_at ?P1 ?r1 ?s'›*))\n (*top goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'⟧ ⟹ obj_at (λs. True) x s = obj_at (λs. True) x s'› and 1 goal remains*)\n apply ((drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))[1])\n (*top goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'⟧ ⟹ ∀xa. map_option (λs. True) (sys_heap s x) = map_option (λs. True) (sys_heap s' x)› and 1 goal remains*)\n apply ((drule_tac f=\"map_option ⟨True⟩\" in arg_cong (*‹(?x::?'a::type) = (?y::?'a::type) ⟹ (?f::?'a::type ⇒ ?'b::type) ?x = ?f ?y›*))[1])\n (*top goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'; map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀xa. map_option (λs. True) (sys_heap s x) = map_option (λs. True) (sys_heap s' x)› and 1 goal remains*)\n apply ((fastforce)[1])\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply ((clarsimp)[1])\n (*goal: ‹⋀s s'. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀x. marked x s = marked x s'›*)\n apply (rule eq_impD[OF obj_at_eq_imp] (*‹∀x. map_option ?P1 (sys_heap ?s ?r1) = map_option ?P1 (sys_heap ?s' ?r1) ⟹ obj_at ?P1 ?r1 ?s = obj_at ?P1 ?r1 ?s'›*))\n (*goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'›*)\n apply ((clarsimp)[1])\n (*goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀xa. map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s x) = map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s' x)›*)\n apply ((rename_tac x)[1])\n (*goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s x) = map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s' x)›*)\n apply ((drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))[1])\n (*goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s x) = map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s' x)›*)\n apply ((drule_tac f=\"map_option (λfl. fl = sys_fM s)\" in arg_cong (*‹?x = ?y ⟹ ?f ?x = ?f ?y›*))[1])\n (*goal: ‹⋀s s' x. ⟦(⋃p. WL p s) = (⋃p. WL p s'); sys_fM s = sys_fM s'; map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s x) = map_option (λobj. obj_mark obj = sys_fM s') (sys_heap s' x)›*)\n apply simp\n(*proven 2 subgoals*) .\n\nlemma white_heap_eq_imp:\n \"eq_imp (λr'. sys_fM ❙⊗ (λs. map_option obj_mark (sys_heap s r')))\n white_heap\"\napply (clarsimp simp: all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*) eq_imp_def (*‹eq_imp ?f ?g ≡ ∀s s'. (∀x. ?f x s = ?f x s') ⟶ ?g s = ?g s'›*) white_def (*‹white ?r ?s ≡ obj_at (λobj. obj_mark obj ≠ sys_fM ?s) ?r ?s›*) white_heap_def (*‹white_heap = (λs. ∀x. obj_at (λs. True) x s ⟶ white x s)›*) obj_at_def (*‹obj_at ?P ?r ≡ λs. case sys_heap s ?r of None ⇒ False | Some obj ⇒ ?P obj›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n(*goal: ‹eq_imp (λ(r'::'a::type) s::'b::type process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state. (sys_fM s, map_option obj_mark (sys_heap s r'))) white_heap›*)\napply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n(*goals:\n 1. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x x2. sys_heap s x = Some x2 ⟶ obj_mark x2 = (¬ sys_fM s')⟧ ⟹ ∀x x2. sys_heap s' x = Some x2 ⟶ obj_mark x2 = (¬ sys_fM s')›\n 2. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x x2. sys_heap s' x = Some x2 ⟶ obj_mark x2 = (¬ sys_fM s')⟧ ⟹ ∀x x2. sys_heap s x = Some x2 ⟶ obj_mark x2 = (¬ sys_fM s')›\ndiscuss goal 1*)\n apply (metis (opaque_lifting, no_types) map_option_eq_Some (*‹(map_option (?f::?'b ⇒ ?'a) (?xo::?'b option) = Some (?y::?'a)) = (∃z::?'b. ?xo = Some z ∧ ?f z = ?y)›*))\n(*discuss goal 2*)\n apply (metis (opaque_lifting, no_types) map_option_eq_Some (*‹(map_option ?f ?xo = Some ?y) = (∃z. ?xo = Some z ∧ ?f z = ?y)›*))\n(*proven 2 subgoals*) .\n\nlemma no_black_refs_eq_imp:\n \"eq_imp (λr'. (λs. (⋃p. WL p s)) ❙⊗ sys_fM ❙⊗ (λs. map_option obj_mark (sys_heap s r')))\n no_black_refs\"\napply (clarsimp simp add: eq_imp_def (*‹eq_imp ?f ?g ≡ ∀s s'. (∀x. ?f x s = ?f x s') ⟶ ?g s = ?g s'›*) no_black_refs_def (*‹no_black_refs = (λs. ∀x. ¬ black x s)›*) black_def (*‹black ?r ≡ λs. marked ?r s ∧ ¬ grey ?r s›*) grey_def (*‹grey ?r = (λs. ∃x. ?r ∈ WL x s)›*) all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*) split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n(*goal: ‹eq_imp (λr' s. (⋃p. WL p s, sys_fM s, map_option obj_mark (sys_heap s r'))) no_black_refs›*)\napply (rename_tac s s')\n(*goal: ‹⋀s s'. ⟦∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s ⟶ (∃xa. x ∈ WL xa s')) = (∀x. marked x s' ⟶ (∃xa. x ∈ WL xa s'))›*)\napply (subgoal_tac \"∀x. marked x s ⟷ marked x s'\")\n(*goals:\n 1. ‹⋀s s'. ⟦∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); ∀x. marked x s = marked x s'⟧ ⟹ (∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s ⟶ (∃xa. x ∈ WL xa s')) = (∀x. marked x s' ⟶ (∃xa. x ∈ WL xa s'))›\n 2. ‹⋀s s'. ⟦∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀x. marked x s = marked x s'›\ndiscuss goal 1*)\n apply clarsimp\n(*discuss goal 2*)\n apply (clarsimp split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*goal: ‹⋀s s'. ⟦∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀x. marked x s = marked x s'›*)\n apply (rename_tac x)\n (*goal: ‹⋀s s' x. ⟦∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (drule_tac x=x in spec (*‹∀x::?'a. (?P::?'a ⇒ bool) x ⟹ ?P (?x::?'a)›*))\n (*goal: ‹⋀s s' x. ⟦∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))\n (*goal: ‹⋀s s' x. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s')⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (auto split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*proven 2 subgoals*) .\n\nlemmas black_heap_fun_upd[simp] = eq_imp_fun_upd[OF black_heap_eq_imp, simplified eq_imp_simps, rule_format]\nlemmas white_heap_fun_upd[simp] = eq_imp_fun_upd[OF white_heap_eq_imp, simplified eq_imp_simps, rule_format]\nlemmas no_black_refs_fun_upd[simp] = eq_imp_fun_upd[OF no_black_refs_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma white_heap_imp_no_black_refs[elim!]:\n \"white_heap s ⟹ no_black_refs s\"\napply (clarsimp simp: white_def (*‹white ?r ?s ≡ obj_at (λobj. obj_mark obj ≠ sys_fM ?s) ?r ?s›*) white_heap_def (*‹white_heap = (λs. ∀x. obj_at (λs. True) x s ⟶ white x s)›*) no_black_refs_def (*‹no_black_refs = (λs. ∀x. ¬ black x s)›*) black_def (*‹black ?r ≡ λs. marked ?r s ∧ ¬ grey ?r s›*))\n(*goal: ‹white_heap s ⟹ no_black_refs s›*)\napply (rename_tac x)\n(*goal: ‹⋀x::'d. ⟦∀x::'d. obj_at (λs::('b, 'c, 'd) object. True) x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ⟶ obj_at (λobj::('b, 'c, 'd) object. obj_mark obj = (¬ sys_fM s)) x s; marked x s⟧ ⟹ grey x s›*)\napply (drule_tac x=x in spec (*‹∀x::?'a. (?P::?'a ⇒ bool) x ⟹ ?P (?x::?'a)›*))\n(*goal: ‹⋀x. ⟦∀x. obj_at (λs. True) x s ⟶ obj_at (λobj. obj_mark obj = (¬ sys_fM s)) x s; marked x s⟧ ⟹ grey x s›*)\nby (clarsimp split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma black_heap_no_greys[elim]:\n \"⟦ no_grey_refs s; ∀r. marked r s ∨ ¬valid_ref r s ⟧ ⟹ black_heap s\"\nunfolding black_def black_heap_def no_grey_refs_def\n(*goal: ‹⟦∀x::'d. ¬ grey x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state); ∀r::'d. marked r s ∨ ¬ obj_at (λs::('b, 'c, 'd) object. True) r s⟧ ⟹ ∀x::'d. obj_at (λs::('b, 'c, 'd) object. True) x s ⟶ marked x s ∧ ¬ grey x s›*)\nby fastforce\n\nlemma heap_colours_colours:\n \"black_heap s ⟹ ¬white r s\"\n \"white_heap s ⟹ ¬black r s\"\n\n(*goals:\n 1. ‹black_heap s ⟹ ¬ white r s›\n 2. ‹white_heap s ⟹ ¬ black r s›\ndiscuss goal 1*)\n apply ((auto simp: black_heap_def (*‹black_heap = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref::type. obj_at (λs::(?'field, ?'payload, ?'ref) object. True) x s ⟶ black x s)›*) white_def (*‹white (?r::?'ref::type) (?s::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state) ≡ obj_at (λobj::(?'field, ?'payload, ?'ref) object. obj_mark obj ≠ sys_fM ?s) ?r ?s›*) white_heap_def (*‹white_heap = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref::type. obj_at (λs::(?'field, ?'payload, ?'ref) object. True) x s ⟶ white x s)›*) dest!: spec[where x=r] (*‹∀x::'d::type. (?P::'d::type ⇒ bool) x ⟹ ?P (r::'d::type)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c::type) (?s::?'d::type process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c::type) (?s::?'d::type process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: black_heap_def (*‹black_heap = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref. obj_at (λs::(?'field, ?'payload, ?'ref) object. True) x s ⟶ black x s)›*) white_def (*‹white (?r::?'ref) (?s::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state) ≡ obj_at (λobj::(?'field, ?'payload, ?'ref) object. obj_mark obj ≠ sys_fM ?s) ?r ?s›*) white_heap_def (*‹white_heap = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref. obj_at (λs::(?'field, ?'payload, ?'ref) object. True) x s ⟶ white x s)›*) dest!: spec[where x=r] (*‹∀x::'d. (?P::'d ⇒ bool) x ⟹ ?P (r::'d)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*proven 2 subgoals*) .\n\n\ntext‹The strong-tricolour invariant ›\n\nlemma strong_tricolour_invD:\n \"⟦ black x s; (x points_to y) s; valid_ref y s; strong_tricolour_inv s ⟧\n ⟹ marked y s\"\nunfolding strong_tricolour_inv_def\n(*goal: ‹⟦black x s; (x points_to y) s; obj_at (λs. True) y s; ∀x xa. black x s ⟶ ¬ ((x points_to xa) s ∧ white xa s)⟧ ⟹ marked y s›*)\nby fastforce\n\nlemma no_black_refsD:\n \"no_black_refs s ⟹ ¬black r s\"\nunfolding no_black_refs_def\n(*goal: ‹∀x. ¬ black x s ⟹ ¬ black r s›*)\nby simp\n\nlemma has_white_path_to_induct[consumes 1, case_names refl step, induct set: has_white_path_to]:\n assumes \"(x has_white_path_to y) s\"\n assumes \"⋀x. P x x\"\n assumes \"⋀x y z. ⟦(x has_white_path_to y) s; P x y; (y points_to z) s; white z s⟧ ⟹ P x z\"\n shows \"P x y\"\nusing assms (*‹(x has_white_path_to y) s› ‹(P::'a ⇒ 'a ⇒ bool) (?x::'a) ?x› ‹⟦(?x has_white_path_to ?y) s; P ?x ?y; (?y points_to ?z) s; white ?z s⟧ ⟹ P ?x ?z›*) unfolding has_white_path_to_def\n(*goal: ‹(P::'a::type ⇒ 'a::type ⇒ bool) (x::'a::type) (y::'a::type)›*)\napply (rule rtranclp.induct (*‹⟦?r⇧*⇧* ?x1.0 ?x2.0; ⋀a. ?P a a; ⋀a b c. ⟦?r⇧*⇧* a b; ?P a b; ?r b c⟧ ⟹ ?P a c⟧ ⟹ ?P ?x1.0 ?x2.0›*))\n(*goals:\n 1. ‹⋀a b c. ⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* a b; P a b; (b points_to c) s ∧ white c s⟧ ⟹ (λx y. (x points_to y) s ∧ white y s)⇧*⇧* a (?y1 a b c)›\n 2. ‹⋀a b c. ⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* a b; P a b; (b points_to c) s ∧ white c s⟧ ⟹ P a (?y1 a b c)›\n 3. ‹⋀a b c. ⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* a b; P a b; (b points_to c) s ∧ white c s⟧ ⟹ (?y1 a b c points_to c) s›\n 4. ‹⋀a b c. ⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* a b; P a b; (b points_to c) s ∧ white c s⟧ ⟹ white c s›\ndiscuss goal 1*)\n apply blast\n(*discuss goal 2*)\n apply blast\n(*discuss goal 3*)\n apply blast\n(*discuss goal 4*)\n apply blast\n(*proven 4 subgoals*) .\n\nlemma has_white_path_toD[dest]:\n \"(x has_white_path_to y) s ⟹ white y s ∨ x = y\"\nunfolding has_white_path_to_def\n(*goal: ‹(λ(x::'a) y::'a. (x points_to y) (s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) ∧ white y s)⇧*⇧* (x::'a) (y::'a) ⟹ white y s ∨ x = y›*)\nby (fastforce elim: rtranclp.cases (*‹⟦?r⇧*⇧* ?a1.0 ?a2.0; ⋀a. ⟦?a1.0 = a; ?a2.0 = a⟧ ⟹ ?P; ⋀a b c. ⟦?a1.0 = a; ?a2.0 = c; ?r⇧*⇧* a b; ?r b c⟧ ⟹ ?P⟧ ⟹ ?P›*))\n\nlemma has_white_path_to_refl[iff]:\n \"(x has_white_path_to x) s\"\nunfolding has_white_path_to_def\n(*goal: ‹(λx y. (x points_to y) s ∧ white y s)⇧*⇧* x x›*)\nby simp\n\nlemma has_white_path_to_step[intro]:\n \"⟦(x has_white_path_to y) s; (y points_to z) s; white z s⟧ ⟹ (x has_white_path_to z) s\"\n \"⟦(y has_white_path_to z) s; (x points_to y) s; white y s⟧ ⟹ (x has_white_path_to z) s\"\nunfolding has_white_path_to_def\n(*goals:\n 1. ‹⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* x y; (y points_to z) s; white z s⟧ ⟹ (λx y. (x points_to y) s ∧ white y s)⇧*⇧* x z›\n 2. ‹⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* y z; (x points_to y) s; white y s⟧ ⟹ (λx y. (x points_to y) s ∧ white y s)⇧*⇧* x z›*)\n(*goals:\n 1. ‹⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* x y; (y points_to z) s; white z s⟧ ⟹ (λx y. (x points_to y) s ∧ white y s)⇧*⇧* x z›\n 2. ‹⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* y z; (x points_to y) s; white y s⟧ ⟹ (λx y. (x points_to y) s ∧ white y s)⇧*⇧* x z›\ndiscuss goal 1*)\n apply (simp add: rtranclp.rtrancl_into_rtrancl (*‹⟦?r⇧*⇧* ?a ?b; ?r ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*))\n(*discuss goal 2*)\n apply (simp add: converse_rtranclp_into_rtranclp (*‹⟦?r ?a ?b; ?r⇧*⇧* ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*))\n(*proven 2 subgoals*) .\n\nlemma has_white_path_toE[elim!]:\n \"⟦ (x points_to y) s; white y s ⟧ ⟹ (x has_white_path_to y) s\"\nunfolding has_white_path_to_def\n(*goal: ‹⟦(x points_to y) s; white y s⟧ ⟹ (λx y. (x points_to y) s ∧ white y s)⇧*⇧* x y›*)\nby (auto elim: rtranclp.intros( (*‹⟦(?r::?'a ⇒ ?'a ⇒ bool)⇧*⇧* (?a::?'a) (?b::?'a); ?r ?b (?c::?'a)⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2))\n\nlemma has_white_path_to_reaches[elim]:\n \"(x has_white_path_to y) s ⟹ (x reaches y) s\"\nunfolding has_white_path_to_def reaches_def\n(*goal: ‹(λ(x::'a) y::'a. (x points_to y) (s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) ∧ white y s)⇧*⇧* (x::'a) (y::'a) ⟹ (λ(x::'a) y::'a. (x points_to y) s)⇧*⇧* x y›*)\napply (induct rule: rtranclp.induct (*‹⟦(?r::?'a::type ⇒ ?'a::type ⇒ bool)⇧*⇧* (?x1.0::?'a::type) (?x2.0::?'a::type); ⋀a::?'a::type. (?P::?'a::type ⇒ ?'a::type ⇒ bool) a a; ⋀(a::?'a::type) (b::?'a::type) c::?'a::type. ⟦?r⇧*⇧* a b; ?P a b; ?r b c⟧ ⟹ ?P a c⟧ ⟹ ?P ?x1.0 ?x2.0›*))\n(*goals:\n 1. ‹⋀a. (λx y. (x points_to y) s)⇧*⇧* a a›\n 2. ‹⋀a b c. ⟦(λx y. (x points_to y) s ∧ white y s)⇧*⇧* a b; (λx y. (x points_to y) s)⇧*⇧* a b; (b points_to c) s ∧ white c s⟧ ⟹ (λx y. (x points_to y) s)⇧*⇧* a c›\ndiscuss goal 1*)\n apply ((auto intro: rtranclp.intros( (*‹⟦?r⇧*⇧* ?a ?b; ?r ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2))[1])\n(*discuss goal 2*)\n apply ((auto intro: rtranclp.intros( (*‹⟦?r⇧*⇧* ?a ?b; ?r ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2))[1])\n(*proven 2 subgoals*) .\n\nlemma has_white_path_to_blacken[simp]:\n \"(x has_white_path_to w) (s(gc := s gc⦇ W := gc_W s - rs ⦈)) ⟷ (x has_white_path_to w) s\"\nunfolding has_white_path_to_def\n(*goal: ‹(λx y. (x points_to y) (s(gc := s gc⦇W := gc_W s - rs⦈)) ∧ white y (s(gc := s gc⦇W := gc_W s - rs⦈)))⇧*⇧* x w = (λx y. (x points_to y) s ∧ white y s)⇧*⇧* x w›*)\nby (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n\nlemma has_white_path_to_eq_imp': ― ‹Complicated condition takes care of ‹alloc›: collapses no object and object with no fields›\n assumes \"(x has_white_path_to y) s'\"\n assumes \"∀r'. ⋃(ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃(ran ` obj_fields ` set_option (sys_heap s r'))\"\n assumes \"∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r')\"\n assumes \"sys_fM s' = sys_fM s\"\n shows \"(x has_white_path_to y) s\"\nusing assms (*‹(x has_white_path_to y) s'› ‹∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r'))› ‹∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r')› ‹sys_fM s' = sys_fM s›*) proof (induct)\n (*goals:\n 1. ‹⋀x. ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to x) s›\n 2. ‹⋀x y z. ⟦(x has_white_path_to y) s'; ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s; (y points_to z) s'; white z s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to z) s›*)\n case (step x y z) (*‹(x has_white_path_to y) s'› ‹⟦∀r'::'a. ⋃ (ran ` obj_fields ` set_option (sys_heap (s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap (s::'g process_name ⇒ ('e, 'g, 'f, 'a) Model.local_state) r')); ∀r'::'a. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ ((x::'a) has_white_path_to (y::'a)) s› ‹(y points_to z) s'› ‹white z s'› ‹∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r'))› ‹∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r')› ‹sys_fM s' = sys_fM s›*)\n then have \"(y points_to z) s\"\n apply (cases \"sys_heap s y\")\n (*goals:\n 1. ‹⟦(x has_white_path_to y) s'; ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s; (y points_to z) s'; white z s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s; sys_heap s y = None⟧ ⟹ (y points_to z) s›\n 2. ‹⋀a. ⟦(x has_white_path_to y) s'; ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s; (y points_to z) s'; white z s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s; sys_heap s y = Some a⟧ ⟹ (y points_to z) s›\n discuss goal 1*)\n apply ((auto 10 10 simp: ran_def (*‹ran ?m = {b. ∃a. ?m a = Some b}›*) obj_at_def (*‹obj_at ?P ?r ≡ λs. case sys_heap s ?r of None ⇒ False | Some obj ⇒ ?P obj›*) split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) dest!: spec[where x=y] (*‹∀x. ?P x ⟹ ?P y›*))[1])\n (*discuss goal 2*)\n apply ((auto 10 10 simp: ran_def (*‹ran ?m = {b. ∃a. ?m a = Some b}›*) obj_at_def (*‹obj_at ?P ?r ≡ λs. case sys_heap s ?r of None ⇒ False | Some obj ⇒ ?P obj›*) split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) dest!: spec[where x=y] (*‹∀x. ?P x ⟹ ?P y›*))[1])\n (*proven 2 subgoals*) .\n with step (*‹(x has_white_path_to y) s'› ‹⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s› ‹((y::'a) points_to (z::'a)) (s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)› ‹white z s'› ‹∀r'::'a::type. ⋃ (ran ` obj_fields ` set_option (sys_heap (s'::'b::type process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap (s::'g::type process_name ⇒ ('e, 'g, 'f, 'a) Model.local_state) r'))› ‹∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r')› ‹sys_fM s' = sys_fM s›*) show \"?case\"\n (*goal: ‹(x has_white_path_to z) s›*)\n apply -\n (*goal: ‹(x has_white_path_to z) s›*)\n apply (rule has_white_path_to_step (*‹⟦(?x has_white_path_to ?y) ?s; (?y points_to ?z) ?s; white ?z ?s⟧ ⟹ (?x has_white_path_to ?z) ?s› ‹⟦(?y has_white_path_to ?z) ?s; (?x points_to ?y) ?s; white ?y ?s⟧ ⟹ (?x has_white_path_to ?z) ?s›*))\n (*goals:\n 1. ‹⟦(x has_white_path_to y) s'; ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s; (y points_to z) s'; white z s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s; (y points_to z) s⟧ ⟹ (x has_white_path_to ?y8) s›\n 2. ‹⟦(x has_white_path_to y) s'; ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s; (y points_to z) s'; white z s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s; (y points_to z) s⟧ ⟹ (?y8 points_to z) s›\n 3. ‹⟦(x has_white_path_to y) s'; ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s; (y points_to z) s'; white z s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s; (y points_to z) s⟧ ⟹ white z s›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply (clarsimp simp: white_def (*‹white ?r ?s ≡ obj_at (λobj. obj_mark obj ≠ sys_fM ?s) ?r ?s›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*goal: ‹⟦(x has_white_path_to y) s'; ⟦∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to y) s; (y points_to z) s'; white z s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r')); ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s; (y points_to z) s⟧ ⟹ white z s›*)\n apply (metis map_option_eq_Some (*‹(map_option ?f ?xo = Some ?y) = (∃z. ?xo = Some z ∧ ?f z = ?y)›*) option.sel (*‹the (Some ?x2.0) = ?x2.0›*))\n (*proven 3 subgoals*) .\nqed (simp)\n(*solved the remaining goal: ‹⋀x::'a. ⟦∀r'::'a. ⋃ (ran ` obj_fields ` set_option (sys_heap (s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap (s::'g process_name ⇒ ('e, 'g, 'f, 'a) Model.local_state) r')); ∀r'::'a. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r'); sys_fM s' = sys_fM s⟧ ⟹ (x has_white_path_to x) s›*)\n\nlemma has_white_path_to_eq_imp:\n \"eq_imp (λr'. sys_fM ❙⊗ (λs. ⋃(ran ` obj_fields ` set_option (sys_heap s r'))) ❙⊗ (λs. map_option obj_mark (sys_heap s r')))\n (x has_white_path_to y)\"\nunfolding eq_imp_def\n(*goal: ‹∀(s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) s'::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state. (∀x::'a. (sys_fM s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), map_option obj_mark (sys_heap s x)) = (sys_fM s', ⋃ (ran ` obj_fields ` set_option (sys_heap s' x)), map_option obj_mark (sys_heap s' x))) ⟶ ((x::'a) has_white_path_to (y::'a)) s = (x has_white_path_to y) s'›*)\napply ((clarsimp simp: all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*))[1])\n(*goal: ‹∀s s'. (∀x. (sys_fM s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), map_option obj_mark (sys_heap s x)) = (sys_fM s', ⋃ (ran ` obj_fields ` set_option (sys_heap s' x)), map_option obj_mark (sys_heap s' x))) ⟶ (x has_white_path_to y) s = (x has_white_path_to y) s'›*)\napply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n(*goals:\n 1. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); (x has_white_path_to y) s⟧ ⟹ (x has_white_path_to y) s'›\n 2. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); (x has_white_path_to y) s'⟧ ⟹ (x has_white_path_to y) s›\ndiscuss goal 1*)\n apply ((erule has_white_path_to_eq_imp' (*‹⟦(?x has_white_path_to ?y) ?s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap ?s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap ?s r')); ∀r'. map_option obj_mark (sys_heap ?s' r') = map_option obj_mark (sys_heap ?s r'); sys_fM ?s' = sys_fM ?s⟧ ⟹ (?x has_white_path_to ?y) ?s›*))[1])\n (*goals:\n 1. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))›\n 2. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀r'. map_option obj_mark (sys_heap s r') = map_option obj_mark (sys_heap s' r')›\n 3. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ sys_fM s = sys_fM s'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*)\n(*discuss goal 2*)\n apply ((erule has_white_path_to_eq_imp' (*‹⟦((?x::?'a::type) has_white_path_to (?y::?'a::type)) (?s'::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state); ∀r'::?'a::type. ⋃ (ran ` obj_fields ` set_option (sys_heap ?s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap (?s::?'g::type process_name ⇒ (?'e, ?'g, ?'f, ?'a) Model.local_state) r')); ∀r'::?'a::type. map_option obj_mark (sys_heap ?s' r') = map_option obj_mark (sys_heap ?s r'); sys_fM ?s' = sys_fM ?s⟧ ⟹ (?x has_white_path_to ?y) ?s›*))[1])\n (*goals:\n 1. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s' r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s r'))›\n 2. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ ∀r'. map_option obj_mark (sys_heap s' r') = map_option obj_mark (sys_heap s r')›\n 3. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ sys_fM s' = sys_fM s›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*)\n(*proven 2 subgoals*) .\n\nlemmas has_white_path_to_fun_upd[simp] = eq_imp_fun_upd[OF has_white_path_to_eq_imp, simplified eq_imp_simps, rule_format]\n\n\ntext‹grey protects white›\n\nlemma grey_protects_whiteD[dest]:\n \"(g grey_protects_white w) s ⟹ grey g s ∧ (g = w ∨ white w s)\"\nby (auto simp: grey_protects_white_def (*‹?g grey_protects_white ?w = (λs. grey ?g s ∧ (?g has_white_path_to ?w) s)›*))\n\nlemma grey_protects_whiteI[iff]:\n \"grey g s ⟹ (g grey_protects_white g) s\"\nby (simp add: grey_protects_white_def (*‹?g grey_protects_white ?w = (λs. grey ?g s ∧ (?g has_white_path_to ?w) s)›*))\n\nlemma grey_protects_whiteE[elim!]:\n \"⟦ (g points_to w) s; grey g s; white w s ⟧ ⟹ (g grey_protects_white w) s\"\n \"⟦ (g grey_protects_white y) s; (y points_to w) s; white w s ⟧ ⟹ (g grey_protects_white w) s\"\n\n(*goals:\n 1. ‹⟦(g points_to w) s; grey g s; white w s⟧ ⟹ (g grey_protects_white w) s›\n 2. ‹⟦(g grey_protects_white y) s; (y points_to w) s; white w s⟧ ⟹ (g grey_protects_white w) s›\ndiscuss goal 1*)\n apply ((auto simp: grey_protects_white_def (*‹(?g::?'ref::type) grey_protects_white (?w::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. grey ?g s ∧ (?g has_white_path_to ?w) s)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: grey_protects_white_def (*‹?g grey_protects_white ?w = (λs. grey ?g s ∧ (?g has_white_path_to ?w) s)›*))[1])\n(*proven 2 subgoals*) .\n\nlemma grey_protects_white_reaches[elim]:\n \"(g grey_protects_white w) s ⟹ (g reaches w) s\"\nby (auto simp: grey_protects_white_def (*‹?g grey_protects_white ?w = (λs. grey ?g s ∧ (?g has_white_path_to ?w) s)›*))\n\nlemma grey_protects_white_induct[consumes 1, case_names refl step, induct set: grey_protects_white]:\n assumes \"(g grey_protects_white w) s\"\n assumes \"⋀x. grey x s ⟹ P x x\"\n assumes \"⋀x y z. ⟦(x has_white_path_to y) s; P x y; (y points_to z) s; white z s⟧ ⟹ P x z\"\n shows \"P g w\"\nusing assms (*‹(g grey_protects_white w) s› ‹grey ?x s ⟹ P ?x ?x› ‹⟦((?x::'a) has_white_path_to (?y::'a)) (s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state); (P::'a ⇒ 'a ⇒ bool) ?x ?y; (?y points_to (?z::'a)) s; white ?z s⟧ ⟹ P ?x ?z›*) unfolding grey_protects_white_def\n(*goal: ‹P g w›*)\napply -\n(*goal: ‹P g w›*)\napply (elim conjE (*‹⟦?P ∧ ?Q; ⟦?P; ?Q⟧ ⟹ ?R⟧ ⟹ ?R›*))\n(*goal: ‹⟦grey g s ∧ (g has_white_path_to w) s; ⋀x. grey x s ⟹ P x x; ⋀x y z. ⟦(x has_white_path_to y) s; P x y; (y points_to z) s; white z s⟧ ⟹ P x z⟧ ⟹ P g w›*)\napply (rotate_tac -1)\n(*goal: ‹⟦⋀x::'a. grey x (s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state) ⟹ (P::'a ⇒ 'a ⇒ bool) x x; ⋀(x::'a) (y::'a) z::'a. ⟦(x has_white_path_to y) s; P x y; (y points_to z) s; white z s⟧ ⟹ P x z; grey (g::'a) s; (g has_white_path_to (w::'a)) s⟧ ⟹ P g w›*)\napply (induct rule: has_white_path_to_induct (*‹⟦(?x has_white_path_to ?y) ?s; ⋀x. ?P x x; ⋀x y z. ⟦(x has_white_path_to y) ?s; ?P x y; (y points_to z) ?s; white z ?s⟧ ⟹ ?P x z⟧ ⟹ ?P ?x ?y›*))\n(*goals:\n 1. ‹⋀x. ⟦⋀x. grey x s ⟹ P x x; ⋀x y z. ⟦(x has_white_path_to y) s; P x y; (y points_to z) s; white z s⟧ ⟹ P x z; grey x s⟧ ⟹ P x x›\n 2. ‹⋀x y z. ⟦(x has_white_path_to y) s; ⟦⋀x. grey x s ⟹ P x x; ⋀x y z. ⟦(x has_white_path_to y) s; P x y; (y points_to z) s; white z s⟧ ⟹ P x z; grey x s⟧ ⟹ P x y; (y points_to z) s; white z s; ⋀x. grey x s ⟹ P x x; ⋀x y z. ⟦(x has_white_path_to y) s; P x y; (y points_to z) s; white z s⟧ ⟹ P x z; grey x s⟧ ⟹ P x z›\ndiscuss goal 1*)\n apply blast\n(*discuss goal 2*)\n apply blast\n(*proven 2 subgoals*) .\n\n\nsubsection‹ @{term \"valid_W_inv\"} ›\n\nlemma valid_W_inv_sys_ghg_empty_iff[elim!]:\n \"valid_W_inv s ⟹ sys_ghost_honorary_grey s = {}\"\nunfolding valid_W_inv_def\n(*goal: ‹(∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]) ⟹ sys_ghost_honorary_grey s = {}›*)\nby simp\n\nlemma WLI[intro]:\n \"r ∈ W (s p) ⟹ r ∈ WL p s\"\n \"r ∈ ghost_honorary_grey (s p) ⟹ r ∈ WL p s\"\nunfolding WL_def\n(*goals:\n 1. ‹(r::'a) ∈ W ((s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) (p::'c process_name)) ⟹ r ∈ W (s p) ∪ ghost_honorary_grey (s p)›\n 2. ‹(r::'a) ∈ ghost_honorary_grey ((s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) (p::'c process_name)) ⟹ r ∈ W (s p) ∪ ghost_honorary_grey (s p)›*)\n(*goals:\n 1. ‹r ∈ W (s p) ⟹ r ∈ W (s p) ∪ ghost_honorary_grey (s p)›\n 2. ‹r ∈ ghost_honorary_grey (s p) ⟹ r ∈ W (s p) ∪ ghost_honorary_grey (s p)›\ndiscuss goal 1*)\n apply simp\n(*discuss goal 2*)\n apply simp\n(*proven 2 subgoals*) .\n\nlemma WL_eq_imp:\n \"eq_imp (λ(_::unit) s. (ghost_honorary_grey (s p), W (s p)))\n (WL p)\"\nunfolding eq_imp_def WL_def\n(*goal: ‹∀(s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) s'::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state. (∀x::unit. (ghost_honorary_grey (s (p::'a process_name)), W (s p)) = (ghost_honorary_grey (s' p), W (s' p))) ⟶ W (s p) ∪ ghost_honorary_grey (s p) = W (s' p) ∪ ghost_honorary_grey (s' p)›*)\nby simp\n\nlemmas WL_fun_upd[simp] = eq_imp_fun_upd[OF WL_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma valid_W_inv_eq_imp:\n \"eq_imp (λ(p, r). (λs. W (s p)) ❙⊗ (λs. ghost_honorary_grey (s p)) ❙⊗ sys_fM ❙⊗ (λs. map_option obj_mark (sys_heap s r)) ❙⊗ sys_mem_lock ❙⊗ tso_pending_mark p)\n valid_W_inv\"\napply (clarsimp simp: eq_imp_def (*‹eq_imp ?f ?g ≡ ∀s s'. (∀x. ?f x s = ?f x s') ⟶ ?g s = ?g s'›*) valid_W_inv_def (*‹valid_W_inv = (λs. (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]))›*) fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*) white_def (*‹white ?r ?s ≡ obj_at (λobj. obj_mark obj ≠ sys_fM ?s) ?r ?s›*))\n(*goal: ‹eq_imp (λ(p, r) s. (W (s p), ghost_honorary_grey (s p), sys_fM s, map_option obj_mark (sys_heap s r), sys_mem_lock s, filter is_mw_Mark (sys_mem_store_buffers p s))) valid_W_inv›*)\napply (rename_tac s s')\n(*goal: ‹⋀(s::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) s'::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state. ⟦∀a::'a::type process_name. W (s a) = W (s' a); ∀a::'a::type process_name. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b::'b::type. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a::'a::type process_name. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s')⟧ ⟹ ((∀(x::'a::type process_name) xa::'b::type. xa ∈ W (s' x) ⟶ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = sys_fM s') xa s) ∧ (∀(x::'a::type process_name) xa::'b::type. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = sys_fM s') xa s) ∧ (∀(x::'a::type process_name) xa::'a::type process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'a::type process_name) (xa::'a::type process_name) xb::'b::type. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀(x::'a::type process_name) (xa::'b::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = (¬ sys_fM s')) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb])) = ((∀(x::'a::type process_name) xa::'b::type. xa ∈ W (s' x) ⟶ marked xa s') ∧ (∀(x::'a::type process_name) xa::'b::type. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ marked xa s') ∧ (∀(x::'a::type process_name) xa::'a::type process_name. x ≠ xa ⟶ WL x s' ∩ WL xa s' = {}) ∧ (∀(x::'a::type process_name) (xa::'a::type process_name) xb::'b::type. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀(x::'a::type process_name) (xa::'b::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s') ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = (¬ sys_fM s')) xa s' ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb]))›*)\napply (subgoal_tac \"∀p. WL p s = WL p s'\")\n(*goal: ‹⋀s s'. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s')⟧ ⟹ ((∀x xa. xa ∈ W (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb])) = ((∀x xa. xa ∈ W (s' x) ⟶ marked xa s') ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ marked xa s') ∧ (∀x xa. x ≠ xa ⟶ WL x s' ∩ WL xa s' = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s') ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s' ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb]))›*)\napply (subgoal_tac \"∀x. marked x s ⟷ marked x s'\")\n(*top goal: ‹⋀(s::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) s'::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state. ⟦∀a::'a::type process_name. W (s a) = W (s' a); ∀a::'a::type process_name. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b::'b::type. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a::'a::type process_name. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p::'a::type process_name. WL p s = WL p s'⟧ ⟹ ((∀(x::'a::type process_name) xa::'b::type. xa ∈ W (s' x) ⟶ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = sys_fM s') xa s) ∧ (∀(x::'a::type process_name) xa::'b::type. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = sys_fM s') xa s) ∧ (∀(x::'a::type process_name) xa::'a::type process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'a::type process_name) (xa::'a::type process_name) xb::'b::type. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀(x::'a::type process_name) (xa::'b::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = (¬ sys_fM s')) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb])) = ((∀(x::'a::type process_name) xa::'b::type. xa ∈ W (s' x) ⟶ marked xa s') ∧ (∀(x::'a::type process_name) xa::'b::type. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ marked xa s') ∧ (∀(x::'a::type process_name) xa::'a::type process_name. x ≠ xa ⟶ WL x s' ∩ WL xa s' = {}) ∧ (∀(x::'a::type process_name) (xa::'a::type process_name) xb::'b::type. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀(x::'a::type process_name) (xa::'b::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s') ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = (¬ sys_fM s')) xa s' ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb]))› and 1 goal remains*)\napply (subgoal_tac \"∀x. obj_at (λobj. obj_mark obj = (¬sys_fM s')) x s ⟷ obj_at (λobj. obj_mark obj = (¬sys_fM s')) x s'\")\n(*top goal: ‹⋀s s'. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; ∀x. marked x s = marked x s'⟧ ⟹ ((∀x xa. xa ∈ W (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb])) = ((∀x xa. xa ∈ W (s' x) ⟶ marked xa s') ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ marked xa s') ∧ (∀x xa. x ≠ xa ⟶ WL x s' ∩ WL xa s' = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s') ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s' ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb]))› and 2 goals remain*)\napply (subgoal_tac \"∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟷ mw_Mark xa xb ∈ set (sys_mem_store_buffers x s')\")\n(*top goal: ‹⋀s s'. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; ∀x. marked x s = marked x s'; ∀x. obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s = obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s'⟧ ⟹ ((∀x xa. xa ∈ W (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb])) = ((∀x xa. xa ∈ W (s' x) ⟶ marked xa s') ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ marked xa s') ∧ (∀x xa. x ≠ xa ⟶ WL x s' ∩ WL xa s' = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s') ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s' ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb]))› and 3 goals remain*)\napply (simp; fail)\n(*top goal: ‹⋀s s'. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; ∀x. marked x s = marked x s'; ∀x. obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s = obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s'; ∀x xa xb. (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s)) = (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s'))⟧ ⟹ ((∀x xa. xa ∈ W (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ obj_at (λobj. obj_mark obj = sys_fM s') xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb])) = ((∀x xa. xa ∈ W (s' x) ⟶ marked xa s') ∧ (∀x xa. sys_mem_lock s' ≠ Some x ∧ xa ∈ ghost_honorary_grey (s' x) ⟶ marked xa s') ∧ (∀x xa. x ≠ xa ⟶ WL x s' ∩ WL xa s' = {}) ∧ (∀x xa xb. xb ∈ ghost_honorary_grey (s' x) ⟶ xb ∉ W (s' xa)) ∧ sys_ghost_honorary_grey s' = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s') ⟶ xb = sys_fM s' ∧ xa ∈ ghost_honorary_grey (s' x) ∧ sys_mem_lock s' = Some x ∧ obj_at (λobj. obj_mark obj = (¬ sys_fM s')) xa s' ∧ filter is_mw_Mark (sys_mem_store_buffers x s') = [mw_Mark xa xb]))› and 4 goals remain*)\napply clarsimp\n(*top goal: ‹⋀s s'. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; ∀x. marked x s = marked x s'; ∀x. obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s = obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s'⟧ ⟹ ∀x xa xb. (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s)) = (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s'))› and 3 goals remain*)\napply (rename_tac x xa xb)\n(*top goal: ‹⋀s s' x xa xb. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'; ∀x. obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s = obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s'⟧ ⟹ (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s)) = (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s'))› and 3 goals remain*)\napply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*), drule arg_cong[where f=set] (*‹?x = ?y ⟹ set ?x = set ?y›*), fastforce)\n(*top goal: ‹⋀s s' x xa xb. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'; ∀x. obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s = obj_at (λobj. obj_mark obj = (¬ sys_fM s')) x s'⟧ ⟹ (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s)) = (mw_Mark xa xb ∈ set (sys_mem_store_buffers x s'))› and 3 goals remain*)\napply (clarsimp split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*top goal: ‹⋀(s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) s'::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state. ⟦∀a::'a process_name. W (s a) = W (s' a); ∀a::'a process_name. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b::'b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a::'a process_name. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p::'a process_name. WL p s = WL p s'; ∀x::'b. marked x s = marked x s'⟧ ⟹ ∀x::'b. obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = (¬ sys_fM s')) x s = obj_at (λobj::('c, 'd, 'b) object. obj_mark obj = (¬ sys_fM s')) x s'› and 2 goals remain*)\napply (rename_tac x)\n(*top goal: ‹⋀s s' x. ⟦∀x. ((∃y. sys_heap s x = Some y) ∨ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ ((∃y. sys_heap s' x = Some y) ∨ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s'))); ∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = sys_fM s')) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = sys_fM s') ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = (¬ sys_fM s')) = (obj_mark obja = (¬ sys_fM s'))))› and 2 goals remain*)\napply (((drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))+)[1])\n(*top goal: ‹⋀(s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (s'::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) x::'b. ⟦∀x::'b. ((∃y::('c, 'd, 'b) object. sys_heap s x = Some y) ∨ (∀obj::('c, 'd, 'b) object. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj::('c, 'd, 'b) object. sys_heap s x = Some obj ⟶ ((∃y::('c, 'd, 'b) object. sys_heap s' x = Some y) ∨ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja::('c, 'd, 'b) object. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s'))); ∀a::'a process_name. W (s a) = W (s' a); ∀a::'a process_name. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b::'b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a::'a process_name. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p::'a process_name. WL p s = WL p s'⟧ ⟹ (sys_heap s x = None ⟶ (∀obj::('c, 'd, 'b) object. sys_heap s' x = Some obj ⟶ obj_mark obj = sys_fM s')) ∧ (∀obj::('c, 'd, 'b) object. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = sys_fM s') ∧ (∀obja::('c, 'd, 'b) object. sys_heap s' x = Some obja ⟶ (obj_mark obj = (¬ sys_fM s')) = (obj_mark obja = (¬ sys_fM s'))))› and 2 goals remain*)\napply (case_tac \"sys_heap s x\", simp_all)\n(*top goal: ‹⋀s s' x. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; ((∃y. sys_heap s x = Some y) ∨ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ ((∃y. sys_heap s' x = Some y) ∨ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s'))); map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = sys_fM s')) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = sys_fM s') ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = (¬ sys_fM s')) = (obj_mark obja = (¬ sys_fM s'))))› and 2 goals remain*)\napply ((case_tac \"sys_heap s' x\", auto)[1])\n(*top goal: ‹⋀(s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (s'::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (x::'b) a::('c, 'd, 'b) object. ⟦∀a::'a process_name. W (s a) = W (s' a); ∀a::'a process_name. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; sys_mem_lock s = sys_mem_lock s'; ∀a::'a process_name. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p::'a process_name. WL p s = WL p s'; ((∃y::('c, 'd, 'b) object. sys_heap s' x = Some y) ∨ obj_mark a = (¬ sys_fM s')) ∧ (∀obj::('c, 'd, 'b) object. sys_heap s' x = Some obj ⟶ (obj_mark a = sys_fM s') = (obj_mark obj = sys_fM s')); Some (obj_mark a) = map_option obj_mark (sys_heap s' x); sys_heap s x = Some a⟧ ⟹ (sys_heap s' x = None ⟶ obj_mark a = sys_fM s') ∧ (∀obj::('c, 'd, 'b) object. sys_heap s' x = Some obj ⟶ (obj_mark a = (¬ sys_fM s')) = (obj_mark obj = (¬ sys_fM s')))› and 2 goals remain*)\napply (clarsimp split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*top goal: ‹⋀s s'. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'⟧ ⟹ ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'› and 1 goal remains*)\napply (rename_tac x)\n(*top goal: ‹⋀s s' x. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))› and 1 goal remains*)\napply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))\n(*top goal: ‹⋀s s' x. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; ∀b. map_option obj_mark (sys_heap s b) = map_option obj_mark (sys_heap s' b); sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))› and 1 goal remains*)\napply (case_tac \"sys_heap s x\", simp_all)\n(*top goal: ‹⋀(s::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (s'::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) x::'b::type. ⟦∀a::'a::type process_name. W (s a) = W (s' a); ∀a::'a::type process_name. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; sys_mem_lock s = sys_mem_lock s'; ∀a::'a::type process_name. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p::'a::type process_name. WL p s = WL p s'; map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (sys_heap s x = None ⟶ (∀obj::('c, 'd, 'b) object. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj::('c, 'd, 'b) object. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja::('c, 'd, 'b) object. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))› and 1 goal remains*)\napply (case_tac \"sys_heap s' x\", simp_all)\n(*top goal: ‹⋀s s' x a. ⟦∀a. W (s a) = W (s' a); ∀a. ghost_honorary_grey (s a) = ghost_honorary_grey (s' a); sys_fM s = sys_fM s'; sys_mem_lock s = sys_mem_lock s'; ∀a. filter is_mw_Mark (sys_mem_store_buffers a s) = filter is_mw_Mark (sys_mem_store_buffers a s'); ∀p. WL p s = WL p s'; Some (obj_mark a) = map_option obj_mark (sys_heap s' x); sys_heap s x = Some a⟧ ⟹ (sys_heap s' x = None ⟶ obj_mark a = (¬ sys_fM s')) ∧ (∀obj. sys_heap s' x = Some obj ⟶ (obj_mark a = sys_fM s') = (obj_mark obj = sys_fM s'))› and 1 goal remains*)\nby (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*))\n\nlemmas valid_W_inv_fun_upd[simp] = eq_imp_fun_upd[OF valid_W_inv_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma valid_W_invE[elim!]:\n \"⟦ r ∈ W (s p); valid_W_inv s ⟧ ⟹ marked r s\"\n \"⟦ r ∈ ghost_honorary_grey (s p); sys_mem_lock s ≠ Some p; valid_W_inv s ⟧ ⟹ marked r s\"\n \"⟦ r ∈ W (s p); valid_W_inv s ⟧ ⟹ valid_ref r s\"\n \"⟦ r ∈ ghost_honorary_grey (s p); sys_mem_lock s ≠ Some p; valid_W_inv s ⟧ ⟹ valid_ref r s\"\n \"⟦ mw_Mark r fl ∈ set (sys_mem_store_buffers p s); valid_W_inv s ⟧ ⟹ r ∈ ghost_honorary_grey (s p)\"\nunfolding valid_W_inv_def\n(*goals:\n 1. ‹⟦r ∈ W (s p); (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ marked r s›\n 2. ‹⟦r ∈ ghost_honorary_grey (s p); sys_mem_lock s ≠ Some p; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ marked r s›\n 3. ‹⟦r ∈ W (s p); (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ obj_at (λs. True) r s›\n 4. ‹⟦r ∈ ghost_honorary_grey (s p); sys_mem_lock s ≠ Some p; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ obj_at (λs. True) r s›\n 5. ‹⟦mw_Mark r fl ∈ set (sys_mem_store_buffers p s); (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ r ∈ ghost_honorary_grey (s p)›*)\n(*goals:\n 1. ‹⟦r ∈ W (s p); (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ marked r s›\n 2. ‹⟦r ∈ ghost_honorary_grey (s p); sys_mem_lock s ≠ Some p; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ marked r s›\n 3. ‹⟦r ∈ W (s p); (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ obj_at (λs. True) r s›\n 4. ‹⟦r ∈ ghost_honorary_grey (s p); sys_mem_lock s ≠ Some p; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ obj_at (λs. True) r s›\n 5. ‹⟦mw_Mark r fl ∈ set (sys_mem_store_buffers p s); (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ r ∈ ghost_honorary_grey (s p)›\ndiscuss goal 1*)\n apply (simp add: split: (*‹(case (?a::?'b, ?b::?'c) of (c::?'b, d::?'c) ⇒ (?f::?'b ⇒ ?'c ⇒ ?'a) c d) = ?f ?a ?b›*) obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*discuss goal 2*)\n apply (simp add: split: (*‹(case (?a, ?b) of (c, d) ⇒ ?f c d) = ?f ?a ?b›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*discuss goal 3*)\n apply (simp add: split: (*‹(case (?a, ?b) of (c, d) ⇒ ?f c d) = ?f ?a ?b›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦r ∈ W (s p); (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ obj_at (λs. True) r s› and 2 goals remain*)\n apply blast\n(*discuss goal 4*)\n apply (simp add: split: (*‹(case (?a, ?b) of (c, d) ⇒ ?f c d) = ?f ?a ?b›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦r ∈ ghost_honorary_grey (s p); sys_mem_lock s ≠ Some p; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ obj_at (λs. True) r s› and 1 goal remains*)\n apply blast\n(*discuss goal 5*)\n apply (simp add: split: (*‹(case (?a, ?b) of (c, d) ⇒ ?f c d) = ?f ?a ?b›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*proven 5 subgoals*) .\n\nlemma valid_W_invD:\n \"⟦ sys_mem_store_buffers p s = mw_Mark r fl # ws; valid_W_inv s ⟧\n ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ tso_locked_by p s ∧ white r s ∧ filter is_mw_Mark ws = []\"\n \"⟦ mw_Mark r fl ∈ set (sys_mem_store_buffers p s); valid_W_inv s ⟧\n ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ tso_locked_by p s ∧ white r s ∧ filter is_mw_Mark (sys_mem_store_buffers p s) = [mw_Mark r fl]\"\nunfolding valid_W_inv_def white_def\n(*goals:\n 1. ‹⟦sys_mem_store_buffers (p::'d::type process_name) (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) = mw_Mark (r::'c::type) (fl::bool) # (ws::('a::type, 'b::type, 'c::type) mem_store_action list); (∀(x::'d::type process_name) xa::'c::type. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀(x::'d::type process_name) xa::'d::type process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'d::type process_name) (xa::'d::type process_name) xb::'c::type. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀(x::'d::type process_name) (xa::'c::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ sys_mem_lock s = Some p ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) r s ∧ filter is_mw_Mark ws = []›\n 2. ‹⟦mw_Mark (r::'c::type) (fl::bool) ∈ set (sys_mem_store_buffers (p::'d::type process_name) (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)); (∀(x::'d::type process_name) xa::'c::type. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀(x::'d::type process_name) xa::'d::type process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'d::type process_name) (xa::'d::type process_name) xb::'c::type. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀(x::'d::type process_name) (xa::'c::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ sys_mem_lock s = Some p ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) r s ∧ filter is_mw_Mark (sys_mem_store_buffers p s) = [mw_Mark r fl]›*)\n(*goals:\n 1. ‹⟦sys_mem_store_buffers (p::'d::type process_name) (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) = mw_Mark (r::'c::type) (fl::bool) # (ws::('a::type, 'b::type, 'c::type) mem_store_action list); (∀(x::'d::type process_name) xa::'c::type. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀(x::'d::type process_name) xa::'d::type process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'d::type process_name) (xa::'d::type process_name) xb::'c::type. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀(x::'d::type process_name) (xa::'c::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ sys_mem_lock s = Some p ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) r s ∧ filter is_mw_Mark ws = []›\n 2. ‹⟦mw_Mark (r::'c::type) (fl::bool) ∈ set (sys_mem_store_buffers (p::'d::type process_name) (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)); (∀(x::'d::type process_name) xa::'c::type. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀(x::'d::type process_name) xa::'d::type process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'d::type process_name) (xa::'d::type process_name) xb::'c::type. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀(x::'d::type process_name) (xa::'c::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ sys_mem_lock s = Some p ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) r s ∧ filter is_mw_Mark (sys_mem_store_buffers p s) = [mw_Mark r fl]›\ndiscuss goal 1*)\n apply (clarsimp dest!: spec[where x=p] (*‹∀x::'d process_name. (?P::'d process_name ⇒ bool) x ⟹ ?P (p::'d process_name)›*))\n (*top goal: ‹⟦sys_mem_store_buffers p s = mw_Mark r fl # ws; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ obj_at (λobj. obj_mark obj ≠ sys_fM s) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ sys_mem_lock s = Some p ∧ obj_at (λobj. obj_mark obj ≠ sys_fM s) r s ∧ filter is_mw_Mark ws = []› and 1 goal remains*)\n apply blast\n(*discuss goal 2*)\n apply (clarsimp dest!: spec[where x=p] (*‹∀x. ?P x ⟹ ?P p›*))\n (*goal: ‹⟦mw_Mark (r::'c) (fl::bool) ∈ set (sys_mem_store_buffers (p::'d process_name) (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)); (∀(x::'d process_name) xa::'c. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀(x::'d process_name) xa::'d process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'d process_name) (xa::'d process_name) xb::'c. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀(x::'d process_name) (xa::'c) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ fl = sys_fM s ∧ r ∈ ghost_honorary_grey (s p) ∧ sys_mem_lock s = Some p ∧ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) r s ∧ filter is_mw_Mark (sys_mem_store_buffers p s) = [mw_Mark r fl]›*)\n apply blast\n(*proven 2 subgoals*) .\n\nlemma valid_W_inv_colours:\n \"⟦white x s; valid_W_inv s⟧ ⟹ x ∉ W (s p)\"\nusing marked_not_white (*‹white ?r ?s ⟹ ¬ marked ?r ?s›*) valid_W_invE(1) (*‹⟦(?r::?'a::type) ∈ W ((?s::?'c::type process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state) (?p::?'c::type process_name)); valid_W_inv ?s⟧ ⟹ marked ?r ?s›*) by force\n\nlemma valid_W_inv_no_mark_stores_invD:\n \"⟦ sys_mem_lock s ≠ Some p; valid_W_inv s ⟧\n ⟹ tso_pending p is_mw_Mark s = []\"\nby (auto dest: valid_W_invD( (*‹⟦mw_Mark ?r ?fl ∈ set (sys_mem_store_buffers ?p ?s); valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark (sys_mem_store_buffers ?p ?s) = [mw_Mark ?r ?fl]›*) 2) intro!: filter_False (*‹∀x∈set ?xs. ¬ ?P x ⟹ filter ?P ?xs = []›*))\n\nlemma valid_W_inv_sys_load[simp]:\n \"⟦ sys_mem_lock s ≠ Some p; valid_W_inv s ⟧\n ⟹ sys_load p (mr_Mark r) (s sys) = mv_Mark (map_option obj_mark (sys_heap s r))\"\nunfolding sys_load_def fold_stores_def\n(*goal: ‹⟦sys_mem_lock s ≠ Some p; valid_W_inv s⟧ ⟹ (do_load_action (mr_Mark r) ∘ (λs. fold (λw. (∘) (do_store_action w)) (mem_store_buffers s p) id s)) (s sys) = mv_Mark (map_option obj_mark (sys_heap s r))›*)\napply clarsimp\n(*goal: ‹⟦sys_mem_lock (s::'a::type process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ≠ Some (p::'a::type process_name); valid_W_inv s⟧ ⟹ (do_load_action (mr_Mark (r::'d::type)) ∘ (λs::('b, 'a, 'c, 'd) Model.local_state. fold (λw::('b::type, 'c::type, 'd::type) mem_store_action. (∘) (do_store_action w)) (mem_store_buffers s p) id s)) (s sys) = mv_Mark (map_option obj_mark (sys_heap s r))›*)\napply (rule fold_invariant[where P=\"λfr. map_option obj_mark (heap (fr (s sys)) r) = map_option obj_mark (sys_heap s r)\" and Q=\"λw. ∀r fl. w ≠ mw_Mark r fl\"] (*‹⟦⋀x. x ∈ set ?xs ⟹ ∀r fl. x ≠ mw_Mark r fl; map_option obj_mark (heap (?s (s sys)) r) = map_option obj_mark (sys_heap s r); ⋀x sa. ⟦∀r fl. x ≠ mw_Mark r fl; map_option obj_mark (heap (sa (s sys)) r) = map_option obj_mark (sys_heap s r)⟧ ⟹ map_option obj_mark (heap (?f x sa (s sys)) r) = map_option obj_mark (sys_heap s r)⟧ ⟹ map_option obj_mark (heap (fold ?f ?xs ?s (s sys)) r) = map_option obj_mark (sys_heap s r)›*))\n(*goals:\n 1. ‹⋀x. ⟦sys_mem_lock s ≠ Some p; valid_W_inv s; x ∈ set (sys_mem_store_buffers p s)⟧ ⟹ ∀r fl. x ≠ mw_Mark r fl›\n 2. ‹⟦sys_mem_lock s ≠ Some p; valid_W_inv s⟧ ⟹ map_option obj_mark (heap (id (s sys)) r) = map_option obj_mark (sys_heap s r)›\n 3. ‹⋀x sa. ⟦sys_mem_lock s ≠ Some p; valid_W_inv s; ∀r fl. x ≠ mw_Mark r fl; map_option obj_mark (heap (sa (s sys)) r) = map_option obj_mark (sys_heap s r)⟧ ⟹ map_option obj_mark (heap ((do_store_action x ∘ sa) (s sys)) r) = map_option obj_mark (sys_heap s r)›\ndiscuss goal 1*)\n apply ((auto simp: map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) dest: valid_W_invD( (*‹⟦mw_Mark ?r ?fl ∈ set (sys_mem_store_buffers ?p ?s); valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark (sys_mem_store_buffers ?p ?s) = [mw_Mark ?r ?fl]›*) 2) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) dest: valid_W_invD( (*‹⟦mw_Mark ?r ?fl ∈ set (sys_mem_store_buffers ?p ?s); valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark (sys_mem_store_buffers ?p ?s) = [mw_Mark ?r ?fl]›*) 2) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: map_option_case (*‹map_option ?f ?y = (case ?y of None ⇒ None | Some x ⇒ Some (?f x))›*) do_store_action_def (*‹do_store_action ?wact = (λs. case ?wact of mw_Mark r gc_mark ⇒ s⦇heap := (heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (heap s r))⦈ | mw_Mutate r f new_r ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s⦇heap := (heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (heap s r))⦈ | mw_fA gc_mark ⇒ s⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s⦇phase := gc_phase⦈)›*) filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) dest: valid_W_invD( (*‹⟦mw_Mark ?r ?fl ∈ set (sys_mem_store_buffers ?p ?s); valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark (sys_mem_store_buffers ?p ?s) = [mw_Mark ?r ?fl]›*) 2) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*proven 3 subgoals*) .\n\n\nsubsection‹ ‹grey_reachable› ›\n\nlemma grey_reachable_eq_imp:\n \"eq_imp (λr'. (λs. ⋃p. WL p s) ❙⊗ (λs. Set.bind (Option.set_option (sys_heap s r')) (ran ∘ obj_fields)))\n (grey_reachable r)\"\nby (msorry)\n\nlemmas grey_reachable_fun_upd[simp] = eq_imp_fun_upd[OF grey_reachable_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma grey_reachableI[intro]:\n \"grey g s ⟹ grey_reachable g s\"\nunfolding grey_reachable_def reaches_def\n(*goal: ‹grey g s ⟹ ∃x. grey x s ∧ (λx y. (x points_to y) s)⇧*⇧* x g›*)\nby blast\n\nlemma grey_reachableE:\n \"⟦ (g points_to y) s; grey_reachable g s ⟧ ⟹ grey_reachable y s\"\nunfolding grey_reachable_def reaches_def\n(*goal: ‹⟦(g points_to y) s; ∃x. grey x s ∧ (λx y. (x points_to y) s)⇧*⇧* x g⟧ ⟹ ∃x. grey x s ∧ (λx y. (x points_to y) s)⇧*⇧* x y›*)\nby (auto elim: rtranclp.intros( (*‹⟦?r⇧*⇧* ?a ?b; ?r ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2))\n\n\nsubsection‹valid refs inv›\n\nlemma valid_refs_invI:\n \"⟦ ⋀m x y. ⟦ (x reaches y) s; mut_m.root m x s ∨ grey x s ⟧ ⟹ valid_ref y s\n ⟧ ⟹ valid_refs_inv s\"\nby (auto simp: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*))\n\nlemma valid_refs_inv_eq_imp:\n \"eq_imp (λ(m', r'). (λs. roots (s (mutator m'))) ❙⊗ (λs. ghost_honorary_root (s (mutator m'))) ❙⊗ (λs. map_option obj_fields (sys_heap s r')) ❙⊗ tso_pending_mutate (mutator m') ❙⊗ (λs. ⋃p. WL p s))\n valid_refs_inv\"\napply (clarsimp simp: eq_imp_def (*‹eq_imp ?f ?g ≡ ∀s s'. (∀x. ?f x s = ?f x s') ⟶ ?g s = ?g s'›*) valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*))\n(*goal: ‹eq_imp (λ(m', r') s. (roots (s (mutator m')), ghost_honorary_root (s (mutator m')), map_option obj_fields (sys_heap s r'), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m') s), ⋃p. WL p s)) valid_refs_inv›*)\napply (rename_tac s s')\n(*goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s')⟧ ⟹ ((∀x xa. mut_m.reachable x xa s ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa s ∧ (xa reaches x) s) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))›*)\napply (subgoal_tac \"∀r'. valid_ref r' s ⟷ valid_ref r' s'\")\n(*goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s')⟧ ⟹ ((∀x xa. mut_m.reachable x xa s ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa s ∧ (xa reaches x) s) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))›*)\napply (subgoal_tac \"∀r'. ⋃(ran ` obj_fields ` set_option (sys_heap s r')) = ⋃(ran ` obj_fields ` set_option (sys_heap s' r'))\")\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'⟧ ⟹ ((∀x xa. mut_m.reachable x xa s ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa s ∧ (xa reaches x) s) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))› and 1 goal remains*)\napply (subst eq_impD[OF mut_m.reachable_eq_imp] (*‹∀x::?'a. (roots ((?s::?'mut1 process_name ⇒ (?'b1, ?'mut1, ?'c1, ?'a) Model.local_state) (mutator (?m1::?'mut1))), ghost_honorary_root (?s (mutator ?m1)), ⋃ (ran ` obj_fields ` set_option (sys_heap ?s x)), filter (λs::(?'b1, ?'c1, ?'a) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator ?m1) ?s)) = (roots ((?s'::?'mut1 process_name ⇒ (?'b1, ?'mut1, ?'c1, ?'a) Model.local_state) (mutator ?m1)), ghost_honorary_root (?s' (mutator ?m1)), ⋃ (ran ` obj_fields ` set_option (sys_heap ?s' x)), filter (λs::(?'b1, ?'c1, ?'a) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator ?m1) ?s')) ⟹ mut_m.reachable ?m1 (?r1::?'a) ?s = mut_m.reachable ?m1 ?r1 ?s'›*))\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ((∀x xa. mut_m.reachable x xa s ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa s ∧ (xa reaches x) s) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))› and 2 goals remain*)\ndefer 1\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ((∀x xa. mut_m.reachable x xa (?s'15 s s') ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa s ∧ (xa reaches x) s) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))› and 3 goals remain*)\napply (subst eq_impD[OF grey_eq_imp] (*‹∀x. (?r1 ∈ (⋃p. WL p ?s)) = (?r1 ∈ (⋃p. WL p ?s')) ⟹ grey ?r1 ?s = grey ?r1 ?s'›*))\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ((∀x xa. mut_m.reachable x xa (?s'15 s s') ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa s ∧ (xa reaches x) s) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))› and 3 goals remain*)\ndefer 1\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ((∀x xa. mut_m.reachable x xa (?s'a18 s s' s s') ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa (?s'18 s s') ∧ (xa reaches x) s) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))› and 4 goals remain*)\napply (subst eq_impD[OF reaches_eq_imp] (*‹∀x. ⋃ (ran ` obj_fields ` set_option (sys_heap ?s x)) = ⋃ (ran ` obj_fields ` set_option (sys_heap ?s' x)) ⟹ (?x1 reaches ?y1) ?s = (?x1 reaches ?y1) ?s'›*))\n(*top goal: ‹⋀(s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) s'::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state. ⟦∀a::'a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a::'a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b::'b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a::'a. filter (λs::('c, 'd, 'b) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs::('c, 'd, 'b) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p::'a process_name. WL p s) = (⋃p::'a process_name. WL p s'); ∀r'::'b. obj_at (λs::('c, 'd, 'b) object. True) r' s = obj_at (λs::('c, 'd, 'b) object. True) r' s'; ∀r'::'b. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ((∀(x::'a) xa::'b. mut_m.reachable x xa ((?s'a18::('a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) ⇒ ('a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) ⇒ ('a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) ⇒ ('a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) ⇒ 'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) s s' s s') ⟶ obj_at (λs::('c, 'd, 'b) object. True) xa s) ∧ (∀x::'b. (∃xa::'b. grey xa ((?s'18::('a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) ⇒ ('a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) ⇒ 'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) s s') ∧ (xa reaches x) s) ⟶ obj_at (λs::('c, 'd, 'b) object. True) x s)) = ((∀(x::'a) xa::'b. mut_m.reachable x xa s' ⟶ obj_at (λs::('c, 'd, 'b) object. True) xa s') ∧ (∀x::'b. (∃xa::'b. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs::('c, 'd, 'b) object. True) x s'))› and 4 goals remain*)\ndefer 1\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ((∀x xa. mut_m.reachable x xa (?s'aa23 s s' s s' s s') ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa (?s'a23 s s' s s') ∧ (xa reaches x) (?s'23 s s')) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))› and 5 goals remain*)\napply force\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ((∀x xa. mut_m.reachable x xa (?s'aa23 s s' s s' s s') ⟶ obj_at (λs. True) xa s) ∧ (∀x. (∃xa. grey xa (?s'a23 s s' s s') ∧ (xa reaches x) (?s'23 s s')) ⟶ obj_at (λs. True) x s)) = ((∀x xa. mut_m.reachable x xa s' ⟶ obj_at (λs. True) xa s') ∧ (∀x. (∃xa. grey xa s' ∧ (xa reaches x) s') ⟶ obj_at (λs. True) x s'))› and 5 goals remain*)\napply (metis option.set_map (*‹set_option (map_option ?f ?v) = ?f ` set_option ?v›*))\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'⟧ ⟹ ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))› and 4 goals remain*)\napply (clarsimp split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*top goal: ‹⋀s s'. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s')⟧ ⟹ ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'› and 3 goals remain*)\napply (metis (no_types, opaque_lifting) None_eq_map_option_iff (*‹(None = map_option ?f ?x) = (?x = None)›*) option.exhaust (*‹⟦?y = None ⟹ ?P; ⋀x2. ?y = Some x2 ⟹ ?P⟧ ⟹ ?P›*))\n(*top goal: ‹⋀(s::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (s'::'a process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) r'::'b. ⟦∀a::'a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a::'a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b::'b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a::'a. filter (λs::('c, 'd, 'b) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs::('c, 'd, 'b) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p::'a process_name. WL p s) = (⋃p::'a process_name. WL p s')⟧ ⟹ (sys_heap s r' = None ⟶ sys_heap s' r' = None) ∧ ((∃obj::('c, 'd, 'b) object. sys_heap s r' = Some obj) ⟶ (∃y::('c, 'd, 'b) object. sys_heap s' r' = Some y))› and 3 goals remain*)\napply clarsimp\n(*top goal: ‹⋀s s' xa x. ⟦∀a. roots (s (mutator a)) = roots (s' (mutator a)); ∀a. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a. filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p. WL p s) = (⋃p. WL p s'); ∀r'. obj_at (λs. True) r' s = obj_at (λs. True) r' s'; ∀r'. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ∀xa. (roots (s (mutator x)), ghost_honorary_root (s (mutator x)), ⋃ (ran ` obj_fields ` set_option (sys_heap s xa)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator x) s)) = (roots (s' (mutator x)), ghost_honorary_root (s' (mutator x)), ⋃ (ran ` obj_fields ` set_option (sys_heap s' xa)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator x) s'))› and 2 goals remain*)\napply clarsimp\n(*top goal: ‹⋀(s::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (s'::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state) (xa::'b::type) x::'b::type. ⟦∀a::'a::type. roots (s (mutator a)) = roots (s' (mutator a)); ∀a::'a::type. ghost_honorary_root (s (mutator a)) = ghost_honorary_root (s' (mutator a)); ∀b::'b::type. map_option obj_fields (sys_heap s b) = map_option obj_fields (sys_heap s' b); ∀a::'a::type. filter (λs::('c::type, 'd::type, 'b::type) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s) = filter (λs::('c::type, 'd::type, 'b::type) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator a) s'); (⋃p::'a::type process_name. WL p s) = (⋃p::'a::type process_name. WL p s'); ∀r'::'b::type. obj_at (λs::('c, 'd, 'b) object. True) r' s = obj_at (λs::('c, 'd, 'b) object. True) r' s'; ∀r'::'b::type. ⋃ (ran ` obj_fields ` set_option (sys_heap s r')) = ⋃ (ran ` obj_fields ` set_option (sys_heap s' r'))⟧ ⟹ ∀x::unit. (xa ∈ (⋃p::'a::type process_name. WL p s)) = (xa ∈ (⋃p::'a::type process_name. WL p s'))› and 1 goal remains*)\nby clarsimp\n\nlemmas valid_refs_inv_fun_upd[simp] = eq_imp_fun_upd[OF valid_refs_inv_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma valid_refs_invD[elim]:\n \"⟦ x ∈ mut_m.mut_roots m s; (x reaches y) s; valid_refs_inv s ⟧ ⟹ valid_ref y s\"\n \"⟦ x ∈ mut_m.mut_roots m s; (x reaches y) s; valid_refs_inv s ⟧ ⟹ ∃obj. sys_heap s y = Some obj\"\n \"⟦ x ∈ mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s ⟧ ⟹ valid_ref y s\"\n \"⟦ x ∈ mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s ⟧ ⟹ ∃obj. sys_heap s y = Some obj\"\n \"⟦ w ∈ set (sys_mem_store_buffers (mutator m) s); x ∈ store_refs w; (x reaches y) s; valid_refs_inv s ⟧ ⟹ valid_ref y s\"\n \"⟦ w ∈ set (sys_mem_store_buffers (mutator m) s); x ∈ store_refs w; (x reaches y) s; valid_refs_inv s ⟧ ⟹ ∃obj. sys_heap s y = Some obj\"\n \"⟦ grey x s; (x reaches y) s; valid_refs_inv s ⟧ ⟹ valid_ref y s\"\n \"⟦ mut_m.reachable m x s; valid_refs_inv s ⟧ ⟹ valid_ref x s\"\n \"⟦ mut_m.reachable m x s; valid_refs_inv s ⟧ ⟹ ∃obj. sys_heap s x = Some obj\"\n \"⟦ x ∈ mut_m.mut_ghost_honorary_root m s; (x reaches y) s; valid_refs_inv s ⟧ ⟹ valid_ref y s\"\n \"⟦ x ∈ mut_m.mut_ghost_honorary_root m s; (x reaches y) s; valid_refs_inv s ⟧ ⟹ ∃obj. sys_heap s y = Some obj\"\n\n(*goals:\n 1. ‹⟦x ∈ roots (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s›\n 2. ‹⟦x ∈ roots (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj›\n 3. ‹⟦x ∈ mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s›\n 4. ‹⟦x ∈ mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj›\n 5. ‹⟦w ∈ set (sys_mem_store_buffers (mutator m) s); x ∈ store_refs w; (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s›\n 6. ‹⟦w ∈ set (sys_mem_store_buffers (mutator m) s); x ∈ store_refs w; (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj›\n 7. ‹⟦grey x s; (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s›\n 8. ‹⟦mut_m.reachable m x s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) x s›\n 9. ‹⟦mut_m.reachable m x s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s x = Some obj›\n 10. ‹⟦x ∈ ghost_honorary_root (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s›\n 11. ‹⟦x ∈ ghost_honorary_root (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj›\ndiscuss goal 1*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀(x::?'mut) xa::?'ref. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs::(?'field, ?'payload, ?'ref) object. True) xa s)›*) grey_reachable_def (*‹grey_reachable (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable (?m::?'mut) (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦x ∈ roots (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s› and 10 goals remain*)\n apply blast\n(*discuss goal 2*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦x ∈ roots (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj› and 9 goals remain*)\n apply blast\n(*discuss goal 3*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦x ∈ mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s› and 8 goals remain*)\n apply blast\n(*discuss goal 4*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦x ∈ mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj› and 7 goals remain*)\n apply blast\n(*discuss goal 5*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀(x::?'mut) xa::?'ref. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs::(?'field, ?'payload, ?'ref) object. True) xa s)›*) grey_reachable_def (*‹grey_reachable (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable (?m::?'mut) (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦(w::('b::type, 'd::type, 'a::type) mem_store_action) ∈ set (sys_mem_store_buffers (mutator (m::'c::type)) (s::'c::type process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state)); (x::'a::type) ∈ store_refs w; (x reaches (y::'a::type)) s; valid_refs_inv s⟧ ⟹ obj_at (λs::('b, 'd, 'a) object. True) y s› and 6 goals remain*)\n apply blast\n(*discuss goal 6*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦w ∈ set (sys_mem_store_buffers (mutator m) s); x ∈ store_refs w; (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj› and 5 goals remain*)\n apply blast\n(*discuss goal 7*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀(x::?'mut) xa::?'ref. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs::(?'field, ?'payload, ?'ref) object. True) xa s)›*) grey_reachable_def (*‹grey_reachable (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable (?m::?'mut) (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦grey x s; (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s› and 4 goals remain*)\n apply blast\n(*discuss goal 8*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*discuss goal 9*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀(x::?'mut) xa::?'ref. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs::(?'field, ?'payload, ?'ref) object. True) xa s)›*) grey_reachable_def (*‹grey_reachable (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable (?m::?'mut) (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦mut_m.reachable m x s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s x = Some obj› and 2 goals remain*)\n apply blast\n(*discuss goal 10*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦x ∈ ghost_honorary_root (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ obj_at (λs. True) y s› and 1 goal remains*)\n apply blast\n(*discuss goal 11*)\n apply (simp add: valid_refs_inv_def (*‹valid_refs_inv = (λs. ∀x xa. mut_m.reachable x xa s ∨ grey_reachable xa s ⟶ obj_at (λs. True) xa s)›*) grey_reachable_def (*‹grey_reachable ?y = (λs. ∃x. grey x s ∧ (x reaches ?y) s)›*) mut_m.reachable_def (*‹mut_m.reachable ?m ?y = (λs. ∃x. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*goal: ‹⟦x ∈ ghost_honorary_root (s (mutator m)); (x reaches y) s; valid_refs_inv s⟧ ⟹ ∃obj. sys_heap s y = Some obj›*)\n apply blast\n(*proven 11 subgoals*) .\n\n\ntext‹reachable snapshot inv›\n\ncontext mut_m\nbegin\n\nlemma reachable_snapshot_invI[intro]:\n \"(⋀y. reachable y s ⟹ in_snapshot y s) ⟹ reachable_snapshot_inv s\"\nby (simp add: reachable_snapshot_inv_def (*‹reachable_snapshot_inv = (λs. ∀x. reachable x s ⟶ in_snapshot x s)›*))\n\nlemma reachable_snapshot_inv_eq_imp:\n \"eq_imp (λr'. mut_roots ❙⊗ mut_ghost_honorary_root ❙⊗ (λs. r' ∈ (⋃p. WL p s)) ❙⊗ sys_fM\n ❙⊗ (λs. ⋃(ran ` obj_fields ` set_option (sys_heap s r'))) ❙⊗ (λs. map_option obj_mark (sys_heap s r'))\n ❙⊗ tso_pending_mutate (mutator m))\n reachable_snapshot_inv\"\nunfolding eq_imp_def mut_m.reachable_snapshot_inv_def in_snapshot_def grey_protects_white_def black_def grey_def\n(*goal: ‹∀s s'. (∀x. (mut_roots s, mut_ghost_honorary_root s, x ∈ (⋃p. WL p s), sys_fM s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), map_option obj_mark (sys_heap s x), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s)) = (mut_roots s', mut_ghost_honorary_root s', x ∈ (⋃p. WL p s'), sys_fM s', ⋃ (ran ` obj_fields ` set_option (sys_heap s' x)), map_option obj_mark (sys_heap s' x), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'))) ⟶ (∀x. reachable x s ⟶ marked x s ∧ (∄xa. x ∈ WL xa s) ∨ (∃xa. (∃x. xa ∈ WL x s) ∧ (xa has_white_path_to x) s)) = (∀x. reachable x s' ⟶ marked x s' ∧ (∄xa. x ∈ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s'))›*)\napply (clarsimp simp: all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*))\n(*goal: ‹∀s s'. (∀x. (mut_roots s, mut_ghost_honorary_root s, x ∈ (⋃p. WL p s), sys_fM s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), map_option obj_mark (sys_heap s x), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s)) = (mut_roots s', mut_ghost_honorary_root s', x ∈ (⋃p. WL p s'), sys_fM s', ⋃ (ran ` obj_fields ` set_option (sys_heap s' x)), map_option obj_mark (sys_heap s' x), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'))) ⟶ (∀x. reachable x s ⟶ marked x s ∧ (∄xa. x ∈ WL xa s) ∨ (∃xa. (∃x. xa ∈ WL x s) ∧ (xa has_white_path_to x) s)) = (∀x. reachable x s' ⟶ marked x s' ∧ (∄xa. x ∈ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s'))›*)\napply (rename_tac s s')\n(*goal: ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (∀x. reachable x s ⟶ obj_at (λobj. obj_mark obj = sys_fM s') x s ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s)) = (∀x. reachable x s' ⟶ marked x s' ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s'))›*)\napply (subst (1) eq_impD[OF has_white_path_to_eq_imp] (*‹∀x. (sys_fM ?s, ⋃ (ran ` obj_fields ` set_option (sys_heap ?s x)), map_option obj_mark (sys_heap ?s x)) = (sys_fM ?s', ⋃ (ran ` obj_fields ` set_option (sys_heap ?s' x)), map_option obj_mark (sys_heap ?s' x)) ⟹ (?x1 has_white_path_to ?y1) ?s = (?x1 has_white_path_to ?y1) ?s'›*))\n(*goals:\n 1. ‹⋀s s' xa x. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ ∀x. (sys_fM s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), map_option obj_mark (sys_heap s x)) = (sys_fM (?s'22 s s'), ⋃ (ran ` obj_fields ` set_option (sys_heap (?s'22 s s') x)), map_option obj_mark (sys_heap (?s'22 s s') x))›\n 2. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (∀x. reachable x s ⟶ obj_at (λobj. obj_mark obj = sys_fM s') x s ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) (?s'22 s s'))) = (∀x. reachable x s' ⟶ marked x s' ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s'))›\ndiscuss goal 1*)\n apply force\n(*discuss goal 2*)\n apply (subst eq_impD[OF reachable_eq_imp] (*‹∀x. (mut_roots ?s, mut_ghost_honorary_root ?s, ⋃ (ran ` obj_fields ` set_option (sys_heap ?s x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) ?s)) = (mut_roots ?s', mut_ghost_honorary_root ?s', ⋃ (ran ` obj_fields ` set_option (sys_heap ?s' x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) ?s')) ⟹ reachable ?r1 ?s = reachable ?r1 ?s'›*))\n (*goals:\n 1. ‹⋀s s' x. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ ∀x. (mut_roots s, mut_ghost_honorary_root s, ⋃ (ran ` obj_fields ` set_option (sys_heap s x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s)) = (mut_roots (?s'40 s s'), mut_ghost_honorary_root (?s'40 s s'), ⋃ (ran ` obj_fields ` set_option (sys_heap (?s'40 s s') x)), filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) (?s'40 s s')))›\n 2. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (∀x. reachable x (?s'40 s s') ⟶ obj_at (λobj. obj_mark obj = sys_fM s') x s ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s')) = (∀x. reachable x s' ⟶ marked x s' ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s'))›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply (subgoal_tac \"∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s ⟷ obj_at (λobj. obj_mark obj = sys_fM s') x s'\")\n (*goals:\n 1. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'⟧ ⟹ (∀x. reachable x s' ⟶ obj_at (λobj. obj_mark obj = sys_fM s') x s ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s')) = (∀x. reachable x s' ⟶ marked x s' ∧ (∀xa. x ∉ WL xa s') ∨ (∃xa. (∃x. xa ∈ WL x s') ∧ (xa has_white_path_to x) s'))›\n 2. ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply (clarsimp split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*goal: ‹⋀s s'. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'›*)\n apply (rename_tac x)\n (*goal: ‹⋀s s' x. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))\n (*goal: ‹⋀s s' x. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; ∀x. (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (drule_tac x=x in spec (*‹∀x::?'a. (?P::?'a ⇒ bool) x ⟹ ?P (?x::?'a)›*))\n (*goal: ‹⋀s s' x. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; sys_fM s = sys_fM s'; ∀x. (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s')⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))\n (*goal: ‹⋀s s' x. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a))⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (case_tac \"sys_heap s x\")\n (*goals:\n 1. ‹⋀s s' x. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; sys_fM s = sys_fM s'; filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); sys_heap s x = None⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›\n 2. ‹⋀s s' x a. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; sys_fM s = sys_fM s'; filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); sys_heap s x = Some a⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*goal: ‹⋀s s' x a. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; sys_fM s = sys_fM s'; filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); (⋃a∈set_option (sys_heap s x). ran (obj_fields a)) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); sys_heap s x = Some a⟧ ⟹ (sys_heap s x = None ⟶ (∀obj. sys_heap s' x = Some obj ⟶ obj_mark obj = (¬ sys_fM s'))) ∧ (∀obj. sys_heap s x = Some obj ⟶ (sys_heap s' x = None ⟶ obj_mark obj = (¬ sys_fM s')) ∧ (∀obja. sys_heap s' x = Some obja ⟶ (obj_mark obj = sys_fM s') = (obj_mark obja = sys_fM s')))›*)\n apply (case_tac \"sys_heap s' x\")\n (*goals:\n 1. ‹⋀s s' x a. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; sys_fM s = sys_fM s'; filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); ran (obj_fields a) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); Some (obj_mark a) = map_option obj_mark (sys_heap s' x); sys_heap s x = Some a; sys_heap s' x = None⟧ ⟹ (sys_heap s' x = None ⟶ obj_mark a = (¬ sys_fM s')) ∧ (∀obj. sys_heap s' x = Some obj ⟶ (obj_mark a = sys_fM s') = (obj_mark obj = sys_fM s'))›\n 2. ‹⋀s s' x a aa. ⟦mut_roots s = mut_roots s'; mut_ghost_honorary_root s = mut_ghost_honorary_root s'; sys_fM s = sys_fM s'; filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s'); (∃xa. x ∈ WL xa s) = (∃xa. x ∈ WL xa s'); ran (obj_fields a) = (⋃a∈set_option (sys_heap s' x). ran (obj_fields a)); Some (obj_mark a) = map_option obj_mark (sys_heap s' x); sys_heap s x = Some a; sys_heap s' x = Some aa⟧ ⟹ (sys_heap s' x = None ⟶ obj_mark a = (¬ sys_fM s')) ∧ (∀obj. sys_heap s' x = Some obj ⟶ (obj_mark a = sys_fM s') = (obj_mark obj = sys_fM s'))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n(*proven 2 subgoals*) .\n\nend\n\nlemmas reachable_snapshot_fun_upd[simp] = eq_imp_fun_upd[OF mut_m.reachable_snapshot_inv_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma in_snapshotI[intro]:\n \"black r s ⟹ in_snapshot r s\"\n \"grey r s ⟹ in_snapshot r s\"\n \"⟦ white w s; (g grey_protects_white w) s ⟧ ⟹ in_snapshot w s\"\n\n(*goals:\n 1. ‹black r s ⟹ in_snapshot r s›\n 2. ‹grey r s ⟹ in_snapshot r s›\n 3. ‹⟦white w s; (g grey_protects_white w) s⟧ ⟹ in_snapshot w s›\ndiscuss goal 1*)\n apply ((auto simp: in_snapshot_def (*‹in_snapshot ?r = (λs. black ?r s ∨ (∃x. (x grey_protects_white ?r) s))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: in_snapshot_def (*‹in_snapshot ?r = (λs. black ?r s ∨ (∃x. (x grey_protects_white ?r) s))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: in_snapshot_def (*‹in_snapshot (?r::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. black ?r s ∨ (∃x::?'ref::type. (x grey_protects_white ?r) s))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma ― ‹Sanity›\n \"in_snapshot r s ⟹ black r s ∨ grey r s ∨ white r s\"\nby (auto simp: in_snapshot_def (*‹in_snapshot ?r = (λs. black ?r s ∨ (∃x. (x grey_protects_white ?r) s))›*))\n\nlemma in_snapshot_valid_ref:\n \"⟦ in_snapshot r s; valid_refs_inv s ⟧ ⟹ valid_ref r s\"\nby (metis blackD( (*‹black ?r ?s ⟹ marked ?r ?s›*) 1) grey_protects_whiteD (*‹(?g grey_protects_white ?w) ?s ⟹ grey ?g ?s ∧ (?g = ?w ∨ white ?w ?s)›*) grey_protects_white_reaches (*‹(?g grey_protects_white ?w) ?s ⟹ (?g reaches ?w) ?s›*) in_snapshot_def (*‹in_snapshot ?r = (λs. black ?r s ∨ (∃x. (x grey_protects_white ?r) s))›*) obj_at_cong (*‹⟦⋀obj. sys_heap ?s ?r = Some obj ⟹ ?P obj = ?P' obj; ?r = ?r'; ?s = ?s'⟧ ⟹ obj_at ?P ?r ?s = obj_at ?P' ?r' ?s'›*) obj_at_def (*‹obj_at ?P ?r ≡ λs. case sys_heap s ?r of None ⇒ False | Some obj ⇒ ?P obj›*) option.case( (*‹(case Some ?x2.0 of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ?f2.0 ?x2.0›*) 2) valid_refs_invD( (*‹⟦grey ?x ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s›*) 7))\n\nlemma reachableI2[intro]:\n \"x ∈ mut_m.mut_ghost_honorary_root m s ⟹ mut_m.reachable m x s\"\nunfolding mut_m.reachable_def reaches_def\n(*goal: ‹(x::'a) ∈ ghost_honorary_root ((s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) (mutator (m::'c))) ⟹ ∃xa::'a. xa ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s ∧ (λ(x::'a) y::'a. (x points_to y) s)⇧*⇧* xa x›*)\nby blast\n\nlemma tso_pending_mw_mutate_cong:\n \"⟦ filter is_mw_Mutate (sys_mem_store_buffers p s) = filter is_mw_Mutate (sys_mem_store_buffers p s');\n ⋀r f r'. P r f r' ⟷ Q r f r' ⟧\n ⟹ (∀r f r'. mw_Mutate r f r' ∈ set (sys_mem_store_buffers p s) ⟶ P r f r')\n ⟷ (∀r f r'. mw_Mutate r f r' ∈ set (sys_mem_store_buffers p s') ⟶ Q r f r')\"\napply (intro iff_allI (*‹(⋀x. ?P x = ?Q x) ⟹ (∀x. ?P x) = (∀x. ?Q x)›*))\n(*goal: ‹⟦filter is_mw_Mutate (sys_mem_store_buffers p s) = filter is_mw_Mutate (sys_mem_store_buffers p s'); ⋀r f r'. P r f r' = Q r f r'⟧ ⟹ (∀r f r'. mw_Mutate r f r' ∈ set (sys_mem_store_buffers p s) ⟶ P r f r') = (∀r f r'. mw_Mutate r f r' ∈ set (sys_mem_store_buffers p s') ⟶ Q r f r')›*)\nby (auto dest!: arg_cong[where f=set] (*‹?x = ?y ⟹ set ?x = set ?y›*))\n\nlemma (in mut_m) marked_insertions_eq_imp:\n \"eq_imp (λr'. sys_fM ❙⊗ (λs. map_option obj_mark (sys_heap s r')) ❙⊗ tso_pending_mw_mutate (mutator m))\n marked_insertions\"\nunfolding eq_imp_def marked_insertions_def obj_at_def\n(*goal: ‹∀s s'. (∀x. (sys_fM s, map_option obj_mark (sys_heap s x), filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s)) = (sys_fM s', map_option obj_mark (sys_heap s' x), filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s'))) ⟶ (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f None ⇒ True | mw_Mutate r f (Some r') ⇒ case sys_heap s r' of None ⇒ False | Some obj ⇒ obj_mark obj = sys_fM s | _ ⇒ True)) = (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f None ⇒ True | mw_Mutate r f (Some r') ⇒ case sys_heap s' r' of None ⇒ False | Some obj ⇒ obj_mark obj = sys_fM s' | _ ⇒ True))›*)\napply (clarsimp split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))\n(*goal: ‹∀s s'. (∀x. (sys_fM s, map_option obj_mark (sys_heap s x), filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s)) = (sys_fM s', map_option obj_mark (sys_heap s' x), filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s'))) ⟶ (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f None ⇒ True | mw_Mutate r f (Some r') ⇒ case sys_heap s r' of None ⇒ False | Some obj ⇒ obj_mark obj = sys_fM s | _ ⇒ True)) = (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f None ⇒ True | mw_Mutate r f (Some r') ⇒ case sys_heap s' r' of None ⇒ False | Some obj ⇒ obj_mark obj = sys_fM s' | _ ⇒ True))›*)\napply (erule tso_pending_mw_mutate_cong (*‹⟦filter is_mw_Mutate (sys_mem_store_buffers (?p::?'d process_name) (?s::?'f process_name ⇒ (?'a, ?'d, ?'b, ?'c, ?'e) local_state_scheme)) = filter is_mw_Mutate (sys_mem_store_buffers ?p (?s'::?'h process_name ⇒ (?'a, ?'d, ?'b, ?'c, ?'g) local_state_scheme)); ⋀(r::?'c) (f::?'a) r'::?'c option. (?P::?'c ⇒ ?'a ⇒ ?'c option ⇒ bool) r f r' = (?Q::?'c ⇒ ?'a ⇒ ?'c option ⇒ bool) r f r'⟧ ⟹ (∀(r::?'c) (f::?'a) r'::?'c option. mw_Mutate r f r' ∈ set (sys_mem_store_buffers ?p ?s) ⟶ ?P r f r') = (∀(r::?'c) (f::?'a) r'::?'c option. mw_Mutate r f r' ∈ set (sys_mem_store_buffers ?p ?s') ⟶ ?Q r f r')›*))\n(*goal: ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s) = filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (∀x21 x22 x23. mw_Mutate x21 x22 x23 ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x23 of None ⇒ True | Some r' ⇒ case sys_heap s r' of None ⇒ False | Some obj ⇒ obj_mark obj = sys_fM s)) = (∀x21 x22 x23. mw_Mutate x21 x22 x23 ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x23 of None ⇒ True | Some r' ⇒ case sys_heap s' r' of None ⇒ False | Some obj ⇒ obj_mark obj = sys_fM s'))›*)\napply (clarsimp split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*goal: ‹⋀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (x21::'a) (x22::'b) x23::'a option. ⟦sys_fM s = sys_fM s'; ∀x::'a. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (case x23 of None ⇒ True | Some (r'::'a) ⇒ case sys_heap s r' of None ⇒ False | Some (obj::('b, 'c, 'a) object) ⇒ obj_mark obj = sys_fM s) = (case x23 of None ⇒ True | Some (r'::'a) ⇒ case sys_heap s' r' of None ⇒ False | Some (obj::('b, 'c, 'a) object) ⇒ obj_mark obj = sys_fM s')›*)\napply (rename_tac s s' opt x)\n(*goal: ‹⋀s s' x23 x2. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (sys_heap s x2 = None ⟶ (∀x2a. sys_heap s' x2 = Some x2a ⟶ x23 = Some x2 ⟶ obj_mark x2a = (¬ sys_fM s'))) ∧ (∀x2a. sys_heap s x2 = Some x2a ⟶ (sys_heap s' x2 = None ⟶ x23 = Some x2 ⟶ obj_mark x2a = (¬ sys_fM s')) ∧ (∀x2b. sys_heap s' x2 = Some x2b ⟶ x23 = Some x2 ⟶ (obj_mark x2a = sys_fM s') = (obj_mark x2b = sys_fM s')))›*)\napply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))\n(*goal: ‹⋀s s' opt x. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x)⟧ ⟹ (sys_heap s x = None ⟶ (∀x2. sys_heap s' x = Some x2 ⟶ opt = Some x ⟶ obj_mark x2 = (¬ sys_fM s'))) ∧ (∀x2. sys_heap s x = Some x2 ⟶ (sys_heap s' x = None ⟶ opt = Some x ⟶ obj_mark x2 = (¬ sys_fM s')) ∧ (∀x2a. sys_heap s' x = Some x2a ⟶ opt = Some x ⟶ (obj_mark x2 = sys_fM s') = (obj_mark x2a = sys_fM s')))›*)\nby auto\n\nlemmas marked_insertions_fun_upd[simp] = eq_imp_fun_upd[OF mut_m.marked_insertions_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma marked_insertionD[elim!]:\n \"⟦ sys_mem_store_buffers (mutator m) s = mw_Mutate r f (Some r') # ws; mut_m.marked_insertions m s ⟧\n ⟹ marked r' s\"\nby (auto simp: mut_m.marked_insertions_def (*‹mut_m.marked_insertions ?m = (λs. ∀x. x ∈ set (sys_mem_store_buffers (mutator ?m) s) ⟶ marked_insertion x s)›*))\n\nlemma marked_insertions_store_buffer_empty[intro]:\n \"tso_pending_mutate (mutator m) s = [] ⟹ mut_m.marked_insertions m s\"\nunfolding mut_m.marked_insertions_def\n(*goal: ‹filter (λs::('a, 'b, 'c) mem_store_action. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator (m::'d)) (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)) = [] ⟹ ∀x::('a, 'b, 'c) mem_store_action. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ marked_insertion x s›*)\nby (auto simp: filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))\n\n(* marked_deletions *)\n\nlemma (in mut_m) marked_deletions_eq_imp:\n \"eq_imp (λr'. sys_fM ❙⊗ (λs. map_option obj_fields (sys_heap s r')) ❙⊗ (λs. map_option obj_mark (sys_heap s r')) ❙⊗ tso_pending_mw_mutate (mutator m))\n marked_deletions\"\nunfolding eq_imp_def marked_deletions_def obj_at_field_on_heap_def ran_def\n(*goal: ‹∀s s'. (∀x. (sys_fM s, map_option obj_fields (sys_heap s x), map_option obj_mark (sys_heap s x), filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s)) = (sys_fM s', map_option obj_fields (sys_heap s' x), map_option obj_mark (sys_heap s' x), filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s'))) ⟶ (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f opt_r' ⇒ case map_option obj_fields (sys_heap s r) of None ⇒ False | Some fs ⇒ case fs f of None ⇒ True | Some r' ⇒ marked r' s | _ ⇒ True)) = (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f opt_r' ⇒ case map_option obj_fields (sys_heap s' r) of None ⇒ False | Some fs ⇒ case fs f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True))›*)\napply (clarsimp simp: all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*))\n(*goal: ‹∀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state. (∀x::'a. (sys_fM s, map_option obj_fields (sys_heap s x), map_option obj_mark (sys_heap s x), filter is_mw_Mutate (sys_mem_store_buffers (mutator (m::'mut)) s)) = (sys_fM s', map_option obj_fields (sys_heap s' x), map_option obj_mark (sys_heap s' x), filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s'))) ⟶ (∀x::('b, 'c, 'a) mem_store_action. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate (r::'a) (f::'b) (opt_r'::'a option) ⇒ case map_option obj_fields (sys_heap s r) of None ⇒ False | Some (fs::'b ⇒ 'a option) ⇒ case fs f of None ⇒ True | Some (r'::'a) ⇒ marked r' s | _ ⇒ True)) = (∀x::('b, 'c, 'a) mem_store_action. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate (r::'a) (f::'b) (opt_r'::'a option) ⇒ case map_option obj_fields (sys_heap s' r) of None ⇒ False | Some (fs::'b ⇒ 'a option) ⇒ case fs f of None ⇒ True | Some (r'::'a) ⇒ marked r' s' | _ ⇒ True))›*)\napply (drule arg_cong[where f=set] (*‹?x = ?y ⟹ set ?x = set ?y›*))\n(*goal: ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s) = filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s')⟧ ⟹ (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x ⇒ ((λfs. case fs f of None ⇒ True | Some r' ⇒ marked r' s) ∘ obj_fields) x | _ ⇒ True)) = (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x ⇒ ((λfs. case fs f of None ⇒ True | Some r' ⇒ marked r' s') ∘ obj_fields) x | _ ⇒ True))›*)\napply (subgoal_tac \"∀x. marked x s ⟷ marked x s'\")\n(*goals:\n 1. ‹⋀(s::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) s'::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state. ⟦sys_fM s = sys_fM s'; ∀x::'a::type. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x::'a::type. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); set (filter is_mw_Mutate (sys_mem_store_buffers (mutator (m::'mut::type)) s)) = set (filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s')); ∀x::'a::type. marked x s = marked x s'⟧ ⟹ (∀x::('b::type, 'c::type, 'a::type) mem_store_action. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate (r::'a::type) (f::'b::type) (opt_r'::'a::type option) ⇒ case sys_heap s' r of None ⇒ False | Some (x::('b, 'c, 'a) object) ⇒ ((λfs::'b::type ⇒ 'a::type option. case fs f of None ⇒ True | Some (r'::'a::type) ⇒ marked r' s) ∘ obj_fields) x | _ ⇒ True)) = (∀x::('b::type, 'c::type, 'a::type) mem_store_action. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate (r::'a::type) (f::'b::type) (opt_r'::'a::type option) ⇒ case sys_heap s' r of None ⇒ False | Some (x::('b, 'c, 'a) object) ⇒ ((λfs::'b::type ⇒ 'a::type option. case fs f of None ⇒ True | Some (r'::'a::type) ⇒ marked r' s') ∘ obj_fields) x | _ ⇒ True))›\n 2. ‹⋀(s::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) s'::'mut::type process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state. ⟦sys_fM s = sys_fM s'; ∀x::'a::type. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x::'a::type. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); set (filter is_mw_Mutate (sys_mem_store_buffers (mutator (m::'mut::type)) s)) = set (filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s'))⟧ ⟹ ∀x::'a::type. marked x s = marked x s'›\ndiscuss goal 1*)\n apply (clarsimp cong: option.case_cong (*‹⟦?option = ?option'; ?option' = None ⟹ ?f1.0 = ?g1.0; ⋀x2. ?option' = Some x2 ⟹ ?f2.0 x2 = ?g2.0 x2⟧ ⟹ (case ?option of None ⇒ ?f1.0 | Some x2 ⇒ ?f2.0 x2) = (case ?option' of None ⇒ ?g1.0 | Some x2 ⇒ ?g2.0 x2)›*))\n (*top goal: ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); set (filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s)) = set (filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s')); ∀x. marked x s = marked x s'⟧ ⟹ (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x ⇒ ((λfs. case fs f of None ⇒ True | Some r' ⇒ marked r' s) ∘ obj_fields) x | _ ⇒ True)) = (∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x ⇒ ((λfs. case fs f of None ⇒ True | Some r' ⇒ marked r' s') ∘ obj_fields) x | _ ⇒ True))› and 1 goal remains*)\n apply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); {x ∈ set (sys_mem_store_buffers (mutator m) s). is_mw_Mutate x} = {x ∈ set (sys_mem_store_buffers (mutator m) s'). is_mw_Mutate x}; ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'; ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)⟧ ⟹ ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)›\n 2. ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); {x ∈ set (sys_mem_store_buffers (mutator m) s). is_mw_Mutate x} = {x ∈ set (sys_mem_store_buffers (mutator m) s'). is_mw_Mutate x}; ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'; ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)⟧ ⟹ ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)›\n discuss goal 1*)\n apply (clarsimp simp: set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*) split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))\n (*top goal: ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); {x ∈ set (sys_mem_store_buffers (mutator m) s). is_mw_Mutate x} = {x ∈ set (sys_mem_store_buffers (mutator m) s'). is_mw_Mutate x}; ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'; ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)⟧ ⟹ ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)› and 2 goals remain*)\n apply blast\n (*discuss goal 2*)\n apply (clarsimp simp: set_eq_iff (*‹((?A::?'a set) = (?B::?'a set)) = (∀x::?'a. (x ∈ ?A) = (x ∈ ?B))›*) split: option.splits (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) mem_store_action.splits (*‹(?P::?'a ⇒ bool) (case ?mem_store_action::(?'field, ?'payload, ?'ref) mem_store_action of mw_Mark (x::?'ref) (xa::bool) ⇒ (?f1.0::?'ref ⇒ bool ⇒ ?'a) x xa | mw_Mutate (x::?'ref) (xa::?'field) (xb::?'ref option) ⇒ (?f2.0::?'ref ⇒ ?'field ⇒ ?'ref option ⇒ ?'a) x xa xb | mw_Mutate_Payload (x::?'ref) (xa::?'field) (xb::?'payload option) ⇒ (?f3.0::?'ref ⇒ ?'field ⇒ ?'payload option ⇒ ?'a) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a) x) = ((∀(x11::?'ref) x12::bool. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀(x21::?'ref) (x22::?'field) x23::?'ref option. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀(x31::?'ref) (x32::?'field) x33::?'payload option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4::bool. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5::bool. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6::gc_phase. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹(?P::?'a ⇒ bool) (case ?mem_store_action::(?'field, ?'payload, ?'ref) mem_store_action of mw_Mark (x::?'ref) (xa::bool) ⇒ (?f1.0::?'ref ⇒ bool ⇒ ?'a) x xa | mw_Mutate (x::?'ref) (xa::?'field) (xb::?'ref option) ⇒ (?f2.0::?'ref ⇒ ?'field ⇒ ?'ref option ⇒ ?'a) x xa xb | mw_Mutate_Payload (x::?'ref) (xa::?'field) (xb::?'payload option) ⇒ (?f3.0::?'ref ⇒ ?'field ⇒ ?'payload option ⇒ ?'a) x xa xb | mw_fA (x::bool) ⇒ (?f4.0::bool ⇒ ?'a) x | mw_fM (x::bool) ⇒ (?f5.0::bool ⇒ ?'a) x | mw_Phase (x::gc_phase) ⇒ (?f6.0::gc_phase ⇒ ?'a) x) = (¬ ((∃(x11::?'ref) x12::bool. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃(x21::?'ref) (x22::?'field) x23::?'ref option. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃(x31::?'ref) (x32::?'field) x33::?'payload option. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4::bool. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5::bool. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6::gc_phase. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))\n (*top goal: ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); {x ∈ set (sys_mem_store_buffers (mutator m) s). is_mw_Mutate x} = {x ∈ set (sys_mem_store_buffers (mutator m) s'). is_mw_Mutate x}; ∀x. obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'; ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s') ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)⟧ ⟹ ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ (case x of mw_Mutate r f opt_r' ⇒ case sys_heap s' r of None ⇒ False | Some x2 ⇒ case obj_fields x2 f of None ⇒ True | Some r' ⇒ marked r' s' | _ ⇒ True)› and 1 goal remains*)\n apply blast\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply clarsimp\n (*goal: ‹⋀s s'. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); set (filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s)) = set (filter is_mw_Mutate (sys_mem_store_buffers (mutator m) s'))⟧ ⟹ ∀x. marked x s = marked x s'›*)\n apply (rename_tac s s' x)\n (*goal: ‹⋀(s::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) (s'::'mut process_name ⇒ ('b, 'mut, 'c, 'a) Model.local_state) x::'a. ⟦sys_fM s = sys_fM s'; ∀x::'a. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x::'a. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); {x::('b, 'c, 'a) mem_store_action ∈ set (sys_mem_store_buffers (mutator (m::'mut)) s). is_mw_Mutate x} = {x::('b, 'c, 'a) mem_store_action ∈ set (sys_mem_store_buffers (mutator m) s'). is_mw_Mutate x}⟧ ⟹ obj_at (λobj::('b, 'c, 'a) object. obj_mark obj = sys_fM s') x s = marked x s'›*)\n apply (drule_tac x=x in spec (*‹∀x::?'a. (?P::?'a ⇒ bool) x ⟹ ?P (?x::?'a)›*))\n (*goal: ‹⋀s s' x. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x); ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); {x ∈ set (sys_mem_store_buffers (mutator m) s). is_mw_Mutate x} = {x ∈ set (sys_mem_store_buffers (mutator m) s'). is_mw_Mutate x}⟧ ⟹ obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'›*)\n apply (drule_tac x=x in spec (*‹∀x::?'a. (?P::?'a ⇒ bool) x ⟹ ?P (?x::?'a)›*))\n (*goal: ‹⋀s s' x. ⟦sys_fM s = sys_fM s'; ∀x. map_option obj_mark (sys_heap s x) = map_option obj_mark (sys_heap s' x); {x ∈ set (sys_mem_store_buffers (mutator m) s). is_mw_Mutate x} = {x ∈ set (sys_mem_store_buffers (mutator m) s'). is_mw_Mutate x}; map_option obj_fields (sys_heap s x) = map_option obj_fields (sys_heap s' x)⟧ ⟹ obj_at (λobj. obj_mark obj = sys_fM s') x s = marked x s'›*)\n apply (force split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*proven 2 subgoals*) .\n\nlemmas marked_deletions_fun_upd[simp] = eq_imp_fun_upd[OF mut_m.marked_deletions_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma marked_deletions_store_buffer_empty[intro]:\n \"tso_pending_mutate (mutator m) s = [] ⟹ mut_m.marked_deletions m s\"\nunfolding mut_m.marked_deletions_def\n(*goal: ‹filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s) = [] ⟹ ∀x. x ∈ set (sys_mem_store_buffers (mutator m) s) ⟶ marked_deletion x s›*)\nby (auto simp: filter_empty_conv (*‹(filter ?P ?xs = []) = (∀x∈set ?xs. ¬ ?P x)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))\n\n\nsubsection‹Location-specific simplification rules›\n\nlemma obj_at_ref_sweep_loop_free[simp]:\n \"obj_at P r (s(sys := (s sys)⦇heap := (sys_heap s)(r' := None)⦈)) ⟷ obj_at P r s ∧ r ≠ r'\"\nby (clarsimp simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma obj_at_alloc[simp]:\n \"sys_heap s r' = None\n ⟹ obj_at P r (s(m := mut_m_s', sys := (s sys)⦇ heap := (sys_heap s)(r' ↦ obj) ⦈))\n ⟷ (obj_at P r s ∨ (r = r' ∧ P obj))\"\nunfolding ran_def\n(*goal: ‹sys_heap s r' = None ⟹ obj_at P r (s(m := mut_m_s', sys := s sys⦇heap := (sys_heap s)(r' ↦ obj)⦈)) = (obj_at P r s ∨ r = r' ∧ P obj)›*)\nby (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma valid_ref_valid_null_ref_simps[simp]:\n \"valid_ref r (s(sys := do_store_action w (s sys)⦇mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)) ⟷ valid_ref r s\"\n \"valid_null_ref r' (s(sys := do_store_action w (s sys)⦇mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)) ⟷ valid_null_ref r' s\"\n \"valid_null_ref r' (s(mutator m := mut_s', sys := (s sys)⦇ heap := (heap (s sys))(r'' ↦ obj) ⦈)) ⟷ valid_null_ref r' s ∨ r' = Some r''\"\nunfolding do_store_action_def valid_null_ref_def\n(*goals:\n 1. ‹obj_at (λs::('a, 'b, 'c) object. True) (r::'c::type) ((s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)(sys := (case w::('a::type, 'b::type, 'c::type) mem_store_action of mw_Mark (r::'c::type) (gc_mark::bool) ⇒ s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_::bool. gc_mark)) (sys_heap s r))⦈ | mw_Mutate (r::'c::type) (f::'a::type) (new_r::'c::type option) ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj::('a, 'b, 'c) object. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (sys_heap s r))⦈ | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj::('a, 'b, 'c) object. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r))⦈ | mw_fA (gc_mark::bool) ⇒ s sys⦇fA := gc_mark⦈ | mw_fM (gc_mark::bool) ⇒ s sys⦇fM := gc_mark⦈ | mw_Phase (gc_phase::gc_phase) ⇒ s sys⦇phase := gc_phase⦈)⦇mem_store_buffers := (mem_store_buffers (s sys))(p::'d::type process_name := ws::('a::type, 'b::type, 'c::type) mem_store_action list)⦈)) = obj_at (λs::('a, 'b, 'c) object. True) r s›\n 2. ‹(case r'::'c::type option of None ⇒ λs::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state. True | Some (r'::'c::type) ⇒ obj_at (λs::('a, 'b, 'c) object. True) r') ((s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)(sys := (case w::('a::type, 'b::type, 'c::type) mem_store_action of mw_Mark (r::'c::type) (gc_mark::bool) ⇒ s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_::bool. gc_mark)) (sys_heap s r))⦈ | mw_Mutate (r::'c::type) (f::'a::type) (new_r::'c::type option) ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj::('a, 'b, 'c) object. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (sys_heap s r))⦈ | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj::('a, 'b, 'c) object. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r))⦈ | mw_fA (gc_mark::bool) ⇒ s sys⦇fA := gc_mark⦈ | mw_fM (gc_mark::bool) ⇒ s sys⦇fM := gc_mark⦈ | mw_Phase (gc_phase::gc_phase) ⇒ s sys⦇phase := gc_phase⦈)⦇mem_store_buffers := (mem_store_buffers (s sys))(p::'d::type process_name := ws::('a::type, 'b::type, 'c::type) mem_store_action list)⦈)) = (case r' of None ⇒ λs::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state. True | Some (r'::'c::type) ⇒ obj_at (λs::('a, 'b, 'c) object. True) r') s›\n 3. ‹(case r'::'c::type option of None ⇒ λs::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state. True | Some (r'::'c::type) ⇒ obj_at (λs::('a, 'b, 'c) object. True) r') ((s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)(mutator (m::'d::type) := mut_s'::('a, 'd, 'b, 'c) Model.local_state, sys := s sys⦇heap := (sys_heap s)(r''::'c::type ↦ obj::('a, 'b, 'c) object)⦈)) = ((case r' of None ⇒ λs::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state. True | Some (r'::'c::type) ⇒ obj_at (λs::('a, 'b, 'c) object. True) r') s ∨ r' = Some r'')›*)\n(*goals:\n 1. ‹obj_at (λs. True) r (s(sys := (case w of mw_Mark r gc_mark ⇒ s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (sys_heap s r))⦈ | mw_Mutate r f new_r ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (sys_heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r))⦈ | mw_fA gc_mark ⇒ s sys⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s sys⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s sys⦇phase := gc_phase⦈)⦇mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)) = obj_at (λs. True) r s›\n 2. ‹(case r' of None ⇒ λs. True | Some r' ⇒ obj_at (λs. True) r') (s(sys := (case w of mw_Mark r gc_mark ⇒ s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. gc_mark)) (sys_heap s r))⦈ | mw_Mutate r f new_r ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := new_r)⦈) (sys_heap s r))⦈ | mw_Mutate_Payload r f pl ⇒ s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r))⦈ | mw_fA gc_mark ⇒ s sys⦇fA := gc_mark⦈ | mw_fM gc_mark ⇒ s sys⦇fM := gc_mark⦈ | mw_Phase gc_phase ⇒ s sys⦇phase := gc_phase⦈)⦇mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)) = (case r' of None ⇒ λs. True | Some r' ⇒ obj_at (λs. True) r') s›\n 3. ‹(case r' of None ⇒ λs. True | Some r' ⇒ obj_at (λs. True) r') (s(mutator m := mut_s', sys := s sys⦇heap := (sys_heap s)(r'' ↦ obj)⦈)) = ((case r' of None ⇒ λs. True | Some r' ⇒ obj_at (λs. True) r') s ∨ r' = Some r'')›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*) obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n(*proven 3 subgoals*) .\n\ncontext mut_m\nbegin\n\nlemma reachable_load[simp]:\n assumes \"sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref r'\"\n assumes \"r ∈ mut_roots s\"\n shows \"mut_m.reachable m' y (s(mutator m := s (mutator m)⦇ roots := mut_roots s ∪ Option.set_option r' ⦈)) ⟷ mut_m.reachable m' y s\" (is \"?lhs = ?rhs\")\nproof (cases \"m' = m\")\n (*goals:\n 1. ‹m' = m ⟹ mut_m.reachable m' y (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) = mut_m.reachable m' y s›\n 2. ‹m' ≠ m ⟹ mut_m.reachable m' y (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) = mut_m.reachable m' y s›*)\n case True (*‹m' = m›*)\n show \"?thesis\"\n (*goal: ‹mut_m.reachable m' y (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) = mut_m.reachable m' y s›*)\n proof (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹mut_m.reachable m' y (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) ⟹ mut_m.reachable m' y s›\n 2. ‹mut_m.reachable m' y s ⟹ mut_m.reachable m' y (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈))›*)\n assume \"?lhs\" (*‹mut_m.reachable (m'::'mut) (y::'c) ((s::'mut process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state)(mutator (m::'mut) := s (mutator m)⦇roots := mut_roots s ∪ set_option (r'::'c option)⦈))›*)\n with assms (*‹sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref r'› ‹r ∈ mut_roots s›*) True (*‹(m'::'mut::type) = (m::'mut::type)›*) show \"?rhs\"\n unfolding sys_load_def\n (*goal: ‹mut_m.reachable m' y s›*)\n apply clarsimp\n (*goal: ‹mut_m.reachable m' y s›*)\n apply (clarsimp simp: reachable_def (*‹reachable ?y = (λs. ∃x. x ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (x reaches ?y) s)›*) reaches_def (*‹?x reaches ?y = (λs. (λx y. (x points_to y) s)⇧*⇧* ?x ?y)›*) tso_store_refs_def (*‹tso_store_refs = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator m) s)))›*) sys_load_def (*‹sys_load ?p ?ract = do_load_action ?ract ∘ processors_view_of_memory ?p›*) fold_stores_def (*‹fold_stores ?ws = fold (λw. (∘) (do_store_action w)) ?ws id›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n (*goal: ‹⟦(r::'c::type) ∈ mut_roots (s::'mut::type process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); (m'::'mut::type) = (m::'mut::type); reachable (y::'c::type) (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ (⋃x::('a, 'b, 'c) object∈set_option (heap (processors_view_of_memory (mutator m) (s sys)) r). set_option (obj_fields x (f::'a::type)))⦈)); (r'::'c::type option) = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj::('a, 'b, 'c) object. obj_fields obj f)⟧ ⟹ reachable y s›*)\n apply (elim disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹⋀x::'c::type. ⟦(r::'c::type) ∈ mut_roots (s::'mut::type process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); (m'::'mut::type) = (m::'mut::type); (r'::'c::type option) = Option.bind (heap (fold (λw::('a::type, 'b::type, 'c::type) mem_store_action. (∘) (do_store_action w)) (sys_mem_store_buffers (mutator m) s) id (s sys)) r) (λobj::('a, 'b, 'c) object. obj_fields obj (f::'a::type)); (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x (y::'c::type); x ∈ mut_roots s⟧ ⟹ ∃x::'c::type. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa::('a::type, 'b::type, 'c::type) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate (r::'c::type) (f::'a::type) (r'::'c::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ {r} | _ ⇒ {}))) ∧ (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x y›\n 2. ‹⋀x::'c::type. ⟦(r::'c::type) ∈ mut_roots (s::'mut::type process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); (m'::'mut::type) = (m::'mut::type); (r'::'c::type option) = Option.bind (heap (fold (λw::('a::type, 'b::type, 'c::type) mem_store_action. (∘) (do_store_action w)) (sys_mem_store_buffers (mutator m) s) id (s sys)) r) (λobj::('a, 'b, 'c) object. obj_fields obj (f::'a::type)); (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x (y::'c::type); ∃xa::('a, 'b, 'c) object∈set_option (heap (fold (λw::('a::type, 'b::type, 'c::type) mem_store_action. (∘) (do_store_action w)) (sys_mem_store_buffers (mutator m) s) id (s sys)) r). obj_fields xa f = Some x⟧ ⟹ ∃x::'c::type. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa::('a::type, 'b::type, 'c::type) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate (r::'c::type) (f::'a::type) (r'::'c::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ {r} | _ ⇒ {}))) ∧ (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x y›\n 3. ‹⋀x::'c::type. ⟦(r::'c::type) ∈ mut_roots (s::'mut::type process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); (m'::'mut::type) = (m::'mut::type); (r'::'c::type option) = Option.bind (heap (fold (λw::('a::type, 'b::type, 'c::type) mem_store_action. (∘) (do_store_action w)) (sys_mem_store_buffers (mutator m) s) id (s sys)) r) (λobj::('a, 'b, 'c) object. obj_fields obj (f::'a::type)); (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x (y::'c::type); x ∈ mut_ghost_honorary_root s⟧ ⟹ ∃x::'c::type. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa::('a::type, 'b::type, 'c::type) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate (r::'c::type) (f::'a::type) (r'::'c::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ {r} | _ ⇒ {}))) ∧ (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x y›\n 4. ‹⋀x::'c::type. ⟦(r::'c::type) ∈ mut_roots (s::'mut::type process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); (m'::'mut::type) = (m::'mut::type); (r'::'c::type option) = Option.bind (heap (fold (λw::('a::type, 'b::type, 'c::type) mem_store_action. (∘) (do_store_action w)) (sys_mem_store_buffers (mutator m) s) id (s sys)) r) (λobj::('a, 'b, 'c) object. obj_fields obj (f::'a::type)); (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x (y::'c::type); ∃xa::('a::type, 'b::type, 'c::type) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate (r::'c::type) (f::'a::type) (r'::'c::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃x::'c::type. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa::('a::type, 'b::type, 'c::type) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate (r::'c::type) (f::'a::type) (r'::'c::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ {r} | _ ⇒ {}))) ∧ (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x y›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply (fold fold_stores_def)\n (*top goal: ‹⋀x. ⟦r ∈ mut_roots s; m' = m; r' = Option.bind (heap (fold (λw. (∘) (do_store_action w)) (sys_mem_store_buffers (mutator m) s) id (s sys)) r) (λobj. obj_fields obj f); (λx y. (x points_to y) s)⇧*⇧* x y; ∃xa∈set_option (heap (fold (λw. (∘) (do_store_action w)) (sys_mem_store_buffers (mutator m) s) id (s sys)) r). obj_fields xa f = Some x⟧ ⟹ ∃x. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (λx y. (x points_to y) s)⇧*⇧* x y› and 2 goals remain*)\n apply clarsimp\n (*top goal: ‹⋀x::'c::type. ⟦(r::'c::type) ∈ mut_roots (s::'mut::type process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); (m'::'mut::type) = (m::'mut::type); (r'::'c::type option) = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj::('a, 'b, 'c) object. obj_fields obj (f::'a::type)); (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x (y::'c::type); ∃xa::('a, 'b, 'c) object∈set_option (heap (processors_view_of_memory (mutator m) (s sys)) r). obj_fields xa f = Some x⟧ ⟹ ∃x::'c::type. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa::('a::type, 'b::type, 'c::type) mem_store_action∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate (r::'c::type) (f::'a::type) (r'::'c::type option) ⇒ insert r (set_option r') | mw_Mutate_Payload (r::'c::type) (f::'a::type) (pl::'b::type option) ⇒ {r} | _ ⇒ {}))) ∧ (λ(x::'c::type) y::'c::type. (x points_to y) s)⇧*⇧* x y› and 2 goals remain*)\n apply (drule (1) fold_stores_points_to (*‹⟦heap (fold_stores ?ws (?s sys)) ?r = Some ?obj; obj_fields ?obj ?f = Some ?r'⟧ ⟹ (?r points_to ?r') ?s ∨ (∃w∈set ?ws. ?r' ∈ store_refs w)›*))\n (*top goal: ‹⋀x xa. ⟦r ∈ mut_roots s; m' = m; r' = Some x; (λx y. (x points_to y) s)⇧*⇧* x y; obj_fields xa f = Some x; heap (processors_view_of_memory (mutator m) (s sys)) r = Some xa⟧ ⟹ ∃x. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (λx y. (x points_to y) s)⇧*⇧* x y› and 2 goals remain*)\n apply (erule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹⋀x xa. ⟦r ∈ mut_roots s; m' = m; r' = Some x; (λx y. (x points_to y) s)⇧*⇧* x y; obj_fields xa f = Some x; (r points_to x) s⟧ ⟹ ∃x. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n 2. ‹⋀x xa. ⟦r ∈ mut_roots s; m' = m; r' = Some x; (λx y. (x points_to y) s)⇧*⇧* x y; obj_fields xa f = Some x; ∃w∈set (sys_mem_store_buffers (mutator m) s). x ∈ store_refs w⟧ ⟹ ∃x. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n discuss goal 1*)\n apply (fastforce elim!: converse_rtranclp_into_rtranclp[rotated] (*‹⟦?r⇧*⇧* ?b ?c; ?r ?a ?b⟧ ⟹ ?r⇧*⇧* ?a ?c›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) intro!: ranI (*‹?m ?a = Some ?b ⟹ ?b ∈ ran ?m›*))\n (*discuss goal 2*)\n apply (clarsimp split: mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))\n (*top goal: ‹⋀x xa. ⟦r ∈ mut_roots s; m' = m; r' = Some x; (λx y. (x points_to y) s)⇧*⇧* x y; obj_fields xa f = Some x; ∃w∈set (sys_mem_store_buffers (mutator m) s). x ∈ store_refs w⟧ ⟹ ∃x. (x ∈ mut_roots s ∨ x ∈ mut_ghost_honorary_root s ∨ (∃xa∈set (sys_mem_store_buffers (mutator m) s). x ∈ (case xa of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {}))) ∧ (λx y. (x points_to y) s)⇧*⇧* x y› and 2 goals remain*)\n apply meson\n (*proven 2 subgoals*)\n (*discuss goal 3*)\n apply blast\n (*discuss goal 4*)\n apply blast\n (*proven 4 subgoals*) .\n next\n (*goal: ‹mut_m.reachable m' y s ⟹ mut_m.reachable m' y (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈))›*)\n assume \"?rhs\" (*‹mut_m.reachable (m'::'mut) (y::'c) (s::'mut process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state)›*)\n with True (*‹m' = m›*) show \"?lhs\"\n unfolding mut_m.reachable_def\n (*goal: ‹∃x. x ∈ roots ((s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) (mutator m')) ∪ ghost_honorary_root ((s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) (mutator m')) ∪ mut_m.tso_store_refs m' (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) ∧ (x reaches y) (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈))›*)\n by (fastforce simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n qed\nqed (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n(*solved the remaining goal: ‹m' ≠ m ⟹ mut_m.reachable m' y (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) = mut_m.reachable m' y s›*)\n\nend\n\ntext‹WL›\n\nlemma WL_blacken[simp]:\n \"gc_ghost_honorary_grey s = {}\n ⟹ WL p (s(gc := s gc⦇ W := gc_W s - rs ⦈)) = WL p s - { r |r. p = gc ∧ r ∈ rs }\"\nunfolding WL_def\n(*goal: ‹gc_ghost_honorary_grey s = {} ⟹ W ((s(gc := s gc⦇W := gc_W s - rs⦈)) p) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := gc_W s - rs⦈)) p) = W (s p) ∪ ghost_honorary_grey (s p) - {r |r. p = gc ∧ r ∈ rs}›*)\nby (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n\nlemma WL_hs_done[simp]:\n \"ghost_honorary_grey (s (mutator m)) = {}\n ⟹ WL p (s(mutator m := s (mutator m)⦇ W := {}, ghost_hs_phase := hp' ⦈,\n sys := s sys⦇ hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)),\n ghost_hs_in_sync := in' ⦈))\n = (case p of gc ⇒ WL gc s | mutator m' ⇒ (if m' = m then {} else WL (mutator m') s) | sys ⇒ WL sys s ∪ WL (mutator m) s)\"\n \"ghost_honorary_grey (s (mutator m)) = {}\n ⟹ WL p (s(mutator m := s (mutator m)⦇ W := {} ⦈,\n sys := s sys⦇ hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)),\n ghost_hs_in_sync := in' ⦈))\n = (case p of gc ⇒ WL gc s | mutator m' ⇒ (if m' = m then {} else WL (mutator m') s) | sys ⇒ WL sys s ∪ WL (mutator m) s)\"\nunfolding WL_def\n(*goals:\n 1. ‹ghost_honorary_grey (s (mutator m)) = {} ⟹ W ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hp'⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hp'⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) = (case p of mutator m' ⇒ if m' = m then {} else W (s (mutator m')) ∪ ghost_honorary_grey (s (mutator m')) | gc ⇒ gc_W s ∪ gc_ghost_honorary_grey s | sys ⇒ sys_W s ∪ sys_ghost_honorary_grey s ∪ (W (s (mutator m)) ∪ ghost_honorary_grey (s (mutator m))))›\n 2. ‹ghost_honorary_grey (s (mutator m)) = {} ⟹ W ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) = (case p of mutator m' ⇒ if m' = m then {} else W (s (mutator m')) ∪ ghost_honorary_grey (s (mutator m')) | gc ⇒ gc_W s ∪ gc_ghost_honorary_grey s | sys ⇒ sys_W s ∪ sys_ghost_honorary_grey s ∪ (W (s (mutator m)) ∪ ghost_honorary_grey (s (mutator m))))›*)\n(*goals:\n 1. ‹ghost_honorary_grey (s (mutator m)) = {} ⟹ W ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hp'⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hp'⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) = (case p of mutator m' ⇒ if m' = m then {} else W (s (mutator m')) ∪ ghost_honorary_grey (s (mutator m')) | gc ⇒ gc_W s ∪ gc_ghost_honorary_grey s | sys ⇒ sys_W s ∪ sys_ghost_honorary_grey s ∪ (W (s (mutator m)) ∪ ghost_honorary_grey (s (mutator m))))›\n 2. ‹ghost_honorary_grey (s (mutator m)) = {} ⟹ W ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hsp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := in'⦈)) p) = (case p of mutator m' ⇒ if m' = m then {} else W (s (mutator m')) ∪ ghost_honorary_grey (s (mutator m')) | gc ⇒ gc_W s ∪ gc_ghost_honorary_grey s | sys ⇒ sys_W s ∪ sys_ghost_honorary_grey s ∪ (W (s (mutator m)) ∪ ghost_honorary_grey (s (mutator m))))›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: process_name.splits (*‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = ((∀x1. ?process_name = mutator x1 ⟶ ?P (?f1.0 x1)) ∧ (?process_name = gc ⟶ ?P ?f2.0) ∧ (?process_name = sys ⟶ ?P ?f3.0))› ‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = (¬ ((∃x1. ?process_name = mutator x1 ∧ ¬ ?P (?f1.0 x1)) ∨ ?process_name = gc ∧ ¬ ?P ?f2.0 ∨ ?process_name = sys ∧ ¬ ?P ?f3.0))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: process_name.splits (*‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = ((∀x1. ?process_name = mutator x1 ⟶ ?P (?f1.0 x1)) ∧ (?process_name = gc ⟶ ?P ?f2.0) ∧ (?process_name = sys ⟶ ?P ?f3.0))› ‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = (¬ ((∃x1. ?process_name = mutator x1 ∧ ¬ ?P (?f1.0 x1)) ∨ ?process_name = gc ∧ ¬ ?P ?f2.0 ∨ ?process_name = sys ∧ ¬ ?P ?f3.0))›*))[1])\n(*proven 2 subgoals*) .\n\nlemma colours_load_W[iff]:\n \"gc_W s = {} ⟹ black r (s(gc := (s gc)⦇W := W (s sys)⦈, sys := (s sys)⦇W := {}⦈)) ⟷ black r s\"\n \"gc_W s = {} ⟹ grey r (s(gc := (s gc)⦇W := W (s sys)⦈, sys := (s sys)⦇W := {}⦈)) ⟷ grey r s\"\nunfolding black_def grey_def WL_def\n(*goals:\n 1. ‹gc_W s = {} ⟹ (marked r (s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) ∧ (∄x. r ∈ W ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x))) = (marked r s ∧ (∄x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))›\n 2. ‹gc_W s = {} ⟹ (∃x. r ∈ W ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x)) = (∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x))›*)\n(*goals:\n 1. ‹gc_W s = {} ⟹ (marked r (s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) ∧ (∄x. r ∈ W ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x))) = (marked r s ∧ (∄x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))›\n 2. ‹gc_W s = {} ⟹ (∃x. r ∈ W ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x)) = (∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x))›\ndiscuss goal 1*)\n apply (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n (*top goal: ‹gc_W s = {} ⟹ (marked r (s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) ∧ (∄x. r ∈ W ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x))) = (marked r s ∧ (∄x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))› and 1 goal remains*)\n apply ((safe)[1])\n (*goals:\n 1. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ W (s x)⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ ghost_honorary_grey (s x)⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ sys_W s⟧ ⟹ False›\n 4. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ gc_ghost_honorary_grey s⟧ ⟹ False›\n 5. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); sys ≠ gc; r ∈ sys_ghost_honorary_grey s⟧ ⟹ False›\n 6. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ W (s x)⟧ ⟹ False›\n 7. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x)⟧ ⟹ False›\n discuss goal 1*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ W (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ W (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ W (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 2*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. (x = gc ⟶ r ∉ sys_W s ∧ r ∉ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∉ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x))); r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 3*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ sys_W s; x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ sys_W s; x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ sys_W s; x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 4*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ gc_ghost_honorary_grey s; x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ gc_ghost_honorary_grey s; x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); r ∈ gc_ghost_honorary_grey s; x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 5*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); sys ≠ gc; r ∈ sys_ghost_honorary_grey s; x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); sys ≠ gc; r ∈ sys_ghost_honorary_grey s; x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); sys ≠ gc; r ∈ sys_ghost_honorary_grey s; x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 6*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ W (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ W (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; marked r s; ∀x. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ W (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 7*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀(x::'c process_name) x1::'c. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; marked (r::'a) s; ∀x::'c process_name. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x::'c process_name. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; marked (r::'a) s; ∀x::'c process_name. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x::'c process_name. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; marked (r::'a) s; ∀x::'c process_name. r ∉ W (s x) ∧ r ∉ ghost_honorary_grey (s x); x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*proven 7 subgoals*)\n(*discuss goal 2*)\n apply (simp add: fun_upd_apply (*‹((?f::?'b::type ⇒ ?'a::type)(?x::?'b::type := ?y::?'a::type)) (?z::?'b::type) = (if ?z = ?x then ?y else ?f ?z)›*))\n (*goal: ‹gc_W (s::'c::type process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {} ⟹ (∃x::'c::type process_name. (r::'a::type) ∈ W ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) x)) = (∃x::'c::type process_name. r ∈ W (s x) ∪ ghost_honorary_grey (s x))›*)\n apply ((safe)[1])\n (*goals:\n 1. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ W (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; sys ≠ gc; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)⟧ ⟹ False›\n 4. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 5. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 6. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)⟧ ⟹ False›\n 7. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; x ≠ sys; r ∈ W (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 8. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; x ≠ sys; r ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 9. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)⟧ ⟹ False›\n 10. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 11. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 12. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)⟧ ⟹ False›\n 13. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ W (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 14. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 15. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)⟧ ⟹ False›\n 16. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 17. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 18. ‹⋀x. ⟦gc_W s = {}; r ∈ W (s x)⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n 19. ‹⋀x. ⟦gc_W s = {}; r ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n discuss goal 1*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ W (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ W (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ W (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 2*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 3*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; sys ≠ gc; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; sys ≠ gc; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; sys ≠ gc; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 4*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 5*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; x ≠ gc; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 6*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ sys_W s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 7*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀(x::'c process_name) x1::'c. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; (r::'a) ∈ sys_W s; x ≠ sys; r ∈ W (s x); x = mutator x1⟧ ⟹ ∃x::'c process_name. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x::'c process_name. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; (r::'a) ∈ sys_W s; x ≠ sys; r ∈ W (s x); x = gc⟧ ⟹ ∃x::'c process_name. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x::'c process_name. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; (r::'a) ∈ sys_W s; x ≠ sys; r ∈ W (s x); x = sys⟧ ⟹ ∃x::'c process_name. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 8*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ sys_W s; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 9*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 10*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 11*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ sys_W s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 12*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 13*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ W (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ W (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ W (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 14*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; x ≠ sys; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 15*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ False›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ False›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; ∄x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ False›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 16*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = mutator x1⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = gc⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ W (s x); x = sys⟧ ⟹ ∃x. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 17*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀(x::'c process_name) x1::'c. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; (r::'a) ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ ∃x::'c process_name. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x::'c process_name. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; (r::'a) ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ ∃x::'c process_name. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n 3. ‹⋀x::'c process_name. ⟦gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {}; (r::'a) ∈ gc_ghost_honorary_grey s; r ∈ sys_ghost_honorary_grey s; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ ∃x::'c process_name. r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 18*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ W (s x); x = mutator x1⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ W (s x); x = gc⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ W (s x); x = sys⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*discuss goal 19*)\n apply ((case_tac [!] x)[1])\n (*goals:\n 1. ‹⋀x x1. ⟦gc_W s = {}; r ∈ ghost_honorary_grey (s x); x = mutator x1⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n 2. ‹⋀x. ⟦gc_W s = {}; r ∈ ghost_honorary_grey (s x); x = gc⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n 3. ‹⋀x. ⟦gc_W s = {}; r ∈ ghost_honorary_grey (s x); x = sys⟧ ⟹ ∃x. (x = gc ⟶ r ∈ sys_W s ∨ r ∈ gc_ghost_honorary_grey s) ∧ (x ≠ gc ⟶ (x = sys ⟶ r ∈ sys_ghost_honorary_grey s) ∧ (x ≠ sys ⟶ r ∈ W (s x) ∨ r ∈ ghost_honorary_grey (s x)))›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*)\n (*proven 19 subgoals*)\n(*proven 2 subgoals*) .\n\nlemma WL_load_W[simp]:\n \"gc_W s = {}\n ⟹ (WL p (s(gc := (s gc)⦇W := sys_W s⦈, sys := (s sys)⦇W := {}⦈)))\n = (case p of gc ⇒ WL gc s ∪ sys_W s | mutator m ⇒ WL (mutator m) s | sys ⇒ sys_ghost_honorary_grey s)\"\nunfolding WL_def\n(*goal: ‹gc_W (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) = {} ⟹ W ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) (p::'c process_name)) ∪ ghost_honorary_grey ((s(gc := s gc⦇W := sys_W s⦈, sys := s sys⦇W := {}⦈)) p) = (case p of mutator (m::'c) ⇒ W (s (mutator m)) ∪ ghost_honorary_grey (s (mutator m)) | gc ⇒ gc_W s ∪ gc_ghost_honorary_grey s ∪ sys_W s | sys ⇒ sys_ghost_honorary_grey s)›*)\nby (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: process_name.splits (*‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = ((∀x1. ?process_name = mutator x1 ⟶ ?P (?f1.0 x1)) ∧ (?process_name = gc ⟶ ?P ?f2.0) ∧ (?process_name = sys ⟶ ?P ?f3.0))› ‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = (¬ ((∃x1. ?process_name = mutator x1 ∧ ¬ ?P (?f1.0 x1)) ∨ ?process_name = gc ∧ ¬ ?P ?f2.0 ∨ ?process_name = sys ∧ ¬ ?P ?f3.0))›*))\n\ntext‹no grey refs›\n\nlemma no_grey_refs_eq_imp:\n \"eq_imp (λ(_::unit). (λs. ⋃p. WL p s))\n no_grey_refs\"\nby (auto simp add: eq_imp_def (*‹eq_imp ?f ?g ≡ ∀s s'. (∀x. ?f x s = ?f x s') ⟶ ?g s = ?g s'›*) grey_def (*‹grey ?r = (λs. ∃x. ?r ∈ WL x s)›*) no_grey_refs_def (*‹no_grey_refs = (λs. ∀x. ¬ grey x s)›*) set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*))\n\nlemmas no_grey_refs_fun_upd[simp] = eq_imp_fun_upd[OF no_grey_refs_eq_imp, simplified eq_imp_simps, rule_format]\n\nlemma no_grey_refs_no_pending_marks:\n \"⟦ no_grey_refs s; valid_W_inv s ⟧ ⟹ tso_no_pending_marks s\"\nunfolding no_grey_refs_def\n(*goal: ‹⟦∀x::'d. ¬ grey x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state); valid_W_inv s⟧ ⟹ ∀x::'a process_name. filter is_mw_Mark (sys_mem_store_buffers x s) = []›*)\nby (auto intro!: filter_False (*‹∀x∈set ?xs. ¬ ?P x ⟹ filter ?P ?xs = []›*) dest: valid_W_invD( (*‹⟦mw_Mark ?r ?fl ∈ set (sys_mem_store_buffers ?p ?s); valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark (sys_mem_store_buffers ?p ?s) = [mw_Mark ?r ?fl]›*) 2))\n\nlemma no_grey_refs_not_grey_reachableD:\n \"no_grey_refs s ⟹ ¬grey_reachable x s\"\nby (clarsimp simp: no_grey_refs_def (*‹no_grey_refs = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref::type. ¬ grey x s)›*) grey_reachable_def (*‹grey_reachable (?y::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref::type. grey x s ∧ (x reaches ?y) s)›*))\n\nlemma no_grey_refsD:\n \"no_grey_refs s ⟹ r ∉ W (s p)\"\n \"no_grey_refs s ⟹ r ∉ WL p s\"\n \"no_grey_refs s ⟹ r ∉ ghost_honorary_grey (s p)\"\n\n(*goals:\n 1. ‹no_grey_refs s ⟹ r ∉ W (s p)›\n 2. ‹no_grey_refs s ⟹ r ∉ WL p s›\n 3. ‹no_grey_refs s ⟹ r ∉ ghost_honorary_grey (s p)›\ndiscuss goal 1*)\n apply ((auto simp: no_grey_refs_def (*‹no_grey_refs = (λs. ∀x. ¬ grey x s)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: no_grey_refs_def (*‹no_grey_refs = (λs. ∀x. ¬ grey x s)›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: no_grey_refs_def (*‹no_grey_refs = (λs. ∀x. ¬ grey x s)›*))[1])\n(*proven 3 subgoals*) .\n\nlemma no_grey_refs_marked[dest]:\n \"⟦ marked r s; no_grey_refs s ⟧ ⟹ black r s\"\nby (auto simp: no_grey_refs_def (*‹no_grey_refs = (λs. ∀x. ¬ grey x s)›*) black_def (*‹black ?r ≡ λs. marked ?r s ∧ ¬ grey ?r s›*))\n\nlemma no_grey_refs_bwD[dest]:\n \"⟦ heap (s sys) r = Some obj; no_grey_refs s ⟧ ⟹ black r s ∨ white r s\"\nby (clarsimp simp: black_def (*‹black ?r ≡ λs. marked ?r s ∧ ¬ grey ?r s›*) grey_def (*‹grey ?r = (λs. ∃x. ?r ∈ WL x s)›*) no_grey_refs_def (*‹no_grey_refs = (λs. ∀x. ¬ grey x s)›*) white_def (*‹white ?r ?s ≡ obj_at (λobj. obj_mark obj ≠ sys_fM ?s) ?r ?s›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\ncontext mut_m\nbegin\n\nlemma reachable_blackD:\n \"⟦ no_grey_refs s; reachable_snapshot_inv s; reachable r s ⟧ ⟹ black r s\"\nby (simp add: no_grey_refs_def (*‹no_grey_refs = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref::type. ¬ grey x s)›*) reachable_snapshot_inv_def (*‹reachable_snapshot_inv = (λs::'mut::type process_name ⇒ (?'field, 'mut, ?'payload, ?'ref) Model.local_state. ∀x::?'ref::type. reachable x s ⟶ in_snapshot x s)›*) in_snapshot_def (*‹in_snapshot (?r::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. black ?r s ∨ (∃x::?'ref::type. (x grey_protects_white ?r) s))›*) grey_protects_white_def (*‹(?g::?'ref::type) grey_protects_white (?w::?'ref::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. grey ?g s ∧ (?g has_white_path_to ?w) s)›*))\n\nlemma no_grey_refs_not_reachable:\n \"⟦ no_grey_refs s; reachable_snapshot_inv s; white r s ⟧ ⟹ ¬reachable r s\"\nby (fastforce simp: no_grey_refs_def (*‹no_grey_refs = (λs. ∀x. ¬ grey x s)›*) reachable_snapshot_inv_def (*‹reachable_snapshot_inv = (λs. ∀x. reachable x s ⟶ in_snapshot x s)›*) in_snapshot_def (*‹in_snapshot ?r = (λs. black ?r s ∨ (∃x. (x grey_protects_white ?r) s))›*) grey_protects_white_def (*‹?g grey_protects_white ?w = (λs. grey ?g s ∧ (?g has_white_path_to ?w) s)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma no_grey_refs_not_rootD:\n \"⟦ no_grey_refs s; reachable_snapshot_inv s; white r s ⟧\n ⟹ r ∉ mut_roots s ∧ r ∉ mut_ghost_honorary_root s ∧ r ∉ tso_store_refs s\"\napply (drule (2) no_grey_refs_not_reachable (*‹⟦no_grey_refs ?s; reachable_snapshot_inv ?s; white ?r ?s⟧ ⟹ ¬ reachable ?r ?s›*))\n(*goal: ‹⟦no_grey_refs (s::'mut process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); reachable_snapshot_inv s; white (r::'c) s⟧ ⟹ r ∉ mut_roots s ∧ r ∉ mut_ghost_honorary_root s ∧ r ∉ tso_store_refs s›*)\nby (force simp: reachable_def (*‹reachable (?y::?'ref) = (λs::'mut process_name ⇒ (?'field, 'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. x ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (x reaches ?y) s)›*) reaches_def (*‹(?x::?'ref) reaches (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (λ(x::?'ref) y::?'ref. (x points_to y) s)⇧*⇧* ?x ?y)›*))\n\nlemma reachable_snapshot_inv_white_root:\n \"⟦ white w s; w ∈ mut_roots s ∨ w ∈ mut_ghost_honorary_root s; reachable_snapshot_inv s ⟧ ⟹ ∃g. (g grey_protects_white w) s\"\nunfolding reachable_snapshot_inv_def in_snapshot_def reachable_def grey_protects_white_def reaches_def\n(*goal: ‹⟦white w s; w ∈ mut_roots s ∨ w ∈ mut_ghost_honorary_root s; ∀x. (∃xa. xa ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (λx y. (x points_to y) s)⇧*⇧* xa x) ⟶ black x s ∨ (∃xa. grey xa s ∧ (xa has_white_path_to x) s)⟧ ⟹ ∃g. grey g s ∧ (g has_white_path_to w) s›*)\nby auto\n\nend\n\n(* colours *)\n\nlemma black_dequeue_Mark[simp]:\n \"black b (s(sys := (s sys)⦇ heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws) ⦈))\n⟷ (black b s ∧ b ≠ r) ∨ (b = r ∧ fl = sys_fM s ∧ valid_ref r s ∧ ¬grey r s)\"\nunfolding black_def\n(*goal: ‹(marked b (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)) ∧ ¬ grey b (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))) = ((marked b s ∧ ¬ grey b s) ∧ b ≠ r ∨ b = r ∧ fl = sys_fM s ∧ obj_at (λs. True) r s ∧ ¬ grey r s)›*)\nby (auto simp: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n\nlemma colours_sweep_loop_free[iff]:\n \"black r (s(sys := s sys⦇heap := (heap (s sys))(r' := None)⦈)) ⟷ (black r s ∧ r ≠ r')\"\n \"grey r (s(sys := s sys⦇heap := (heap (s sys))(r' := None)⦈)) ⟷ (grey r s)\"\n \"white r (s(sys := s sys⦇heap := (heap (s sys))(r' := None)⦈)) ⟷ (white r s ∧ r ≠ r')\"\nunfolding black_def grey_def white_def\n(*goals:\n 1. ‹(marked r (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) ∧ (∄x. r ∈ WL x (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)))) = ((marked r s ∧ (∄x. r ∈ WL x s)) ∧ r ≠ r')›\n 2. ‹(∃x. r ∈ WL x (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈))) = (∃x. r ∈ WL x s)›\n 3. ‹obj_at (λobj. obj_mark obj ≠ sys_fM (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈))) r (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) = (obj_at (λobj. obj_mark obj ≠ sys_fM s) r s ∧ r ≠ r')›*)\n(*goals:\n 1. ‹(marked r (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) ∧ (∄x. r ∈ WL x (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)))) = ((marked r s ∧ (∄x. r ∈ WL x s)) ∧ r ≠ r')›\n 2. ‹(∃x. r ∈ WL x (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈))) = (∃x. r ∈ WL x s)›\n 3. ‹obj_at (λobj. obj_mark obj ≠ sys_fM (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈))) r (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) = (obj_at (λobj. obj_mark obj ≠ sys_fM s) r s ∧ r ≠ r')›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma colours_get_work_done[simp]:\n \"black r (s(mutator m := (s (mutator m))⦇W := {}⦈,\n sys := (s sys)⦇ hs_pending := hp', W := W (s sys) ∪ W (s (mutator m)),\n ghost_hs_in_sync := his' ⦈)) ⟷ black r s\"\n \"grey r (s(mutator m := (s (mutator m))⦇W := {}⦈,\n sys := (s sys)⦇ hs_pending := hp', W := W (s sys) ∪ W (s (mutator m)),\n ghost_hs_in_sync := his' ⦈)) ⟷ grey r s\"\n \"white r (s(mutator m := (s (mutator m))⦇W := {}⦈,\n sys := (s sys)⦇ hs_pending := hp', W := W (s sys) ∪ W (s (mutator m)),\n ghost_hs_in_sync := his' ⦈)) ⟷ white r s\"\nunfolding black_def grey_def WL_def\n(*goals:\n 1. ‹(marked r (s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) ∧ (∄x. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x))) = (marked r s ∧ (∄x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))›\n 2. ‹(∃x. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x)) = (∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x))›\n 3. ‹white r (s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) = white r s›*)\n(*goals:\n 1. ‹(marked (r::'a::type) ((s::'b::type process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m::'b::type) := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp'::'b::type ⇒ bool, W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'::'b::type ⇒ bool⦈)) ∧ (∄x::'b::type process_name. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x))) = (marked r s ∧ (∄x::'b::type process_name. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))›\n 2. ‹(∃x::'b::type process_name. (r::'a::type) ∈ W (((s::'b::type process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m::'b::type) := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp'::'b::type ⇒ bool, W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'::'b::type ⇒ bool⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x)) = (∃x::'b::type process_name. r ∈ W (s x) ∪ ghost_honorary_grey (s x))›\n 3. ‹white (r::'a::type) ((s::'b::type process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m::'b::type) := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp'::'b::type ⇒ bool, W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'::'b::type ⇒ bool⦈)) = white r s›\ndiscuss goal 1*)\n apply (simp add: fun_upd_apply (*‹((?f::?'b::type ⇒ ?'a::type)(?x::?'b::type := ?y::?'a::type)) (?z::?'b::type) = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c::type) (?s::?'d::type process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c::type) (?s::?'d::type process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹(marked (r::'a) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m::'b) := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp'::'b ⇒ bool, W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'::'b ⇒ bool⦈)) ∧ (∄x::'b process_name. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x))) = (marked r s ∧ (∄x::'b process_name. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))› and 2 goals remain*)\n apply blast\n(*discuss goal 2*)\n apply (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹(∃x. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x)) = (∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x))› and 1 goal remains*)\n apply (metis process_name.distinct( (*‹mutator ?x1.0 ≠ sys›*) 3))\n(*discuss goal 3*)\n apply (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*proven 3 subgoals*) .\n\nlemma colours_get_roots_done[simp]:\n \"black r (s(mutator m := (s (mutator m))⦇ W := {}, ghost_hs_phase := hs' ⦈,\n sys := (s sys)⦇ hs_pending := hp', W := W (s sys) ∪ W (s (mutator m)),\n ghost_hs_in_sync := his' ⦈)) ⟷ black r s\"\n \"grey r (s(mutator m := (s (mutator m))⦇ W := {}, ghost_hs_phase := hs' ⦈,\n sys := (s sys)⦇ hs_pending := hp', W := W (s sys) ∪ W (s (mutator m)),\n ghost_hs_in_sync := his' ⦈)) ⟷ grey r s\"\n \"white r (s(mutator m := (s (mutator m))⦇ W := {}, ghost_hs_phase := hs' ⦈,\n sys := (s sys)⦇ hs_pending := hp', W := W (s sys) ∪ W (s (mutator m)),\n ghost_hs_in_sync := his' ⦈)) ⟷ white r s\"\nunfolding black_def grey_def WL_def\n(*goals:\n 1. ‹(marked r (s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) ∧ (∄x. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x))) = (marked r s ∧ (∄x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))›\n 2. ‹(∃x. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x)) = (∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x))›\n 3. ‹white r (s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) = white r s›*)\n(*goals:\n 1. ‹(marked (r::'a) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m::'b) := s (mutator m)⦇W := {}, ghost_hs_phase := hs'::hs_phase⦈, sys := s sys⦇hs_pending := hp'::'b ⇒ bool, W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'::'b ⇒ bool⦈)) ∧ (∄x::'b process_name. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x))) = (marked r s ∧ (∄x::'b process_name. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))›\n 2. ‹(∃x::'b process_name. (r::'a) ∈ W (((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m::'b) := s (mutator m)⦇W := {}, ghost_hs_phase := hs'::hs_phase⦈, sys := s sys⦇hs_pending := hp'::'b ⇒ bool, W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'::'b ⇒ bool⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x)) = (∃x::'b process_name. r ∈ W (s x) ∪ ghost_honorary_grey (s x))›\n 3. ‹white (r::'a) ((s::'b process_name ⇒ ('c, 'b, 'd, 'a) Model.local_state)(mutator (m::'b) := s (mutator m)⦇W := {}, ghost_hs_phase := hs'::hs_phase⦈, sys := s sys⦇hs_pending := hp'::'b ⇒ bool, W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'::'b ⇒ bool⦈)) = white r s›\ndiscuss goal 1*)\n apply (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹(marked r (s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) ∧ (∄x. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x))) = (marked r s ∧ (∄x. r ∈ W (s x) ∪ ghost_honorary_grey (s x)))› and 2 goals remain*)\n apply blast\n(*discuss goal 2*)\n apply (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹(∃x. r ∈ W ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x) ∪ ghost_honorary_grey ((s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) x)) = (∃x. r ∈ W (s x) ∪ ghost_honorary_grey (s x))› and 1 goal remains*)\n apply (metis process_name.distinct( (*‹mutator ?x1.0 ≠ sys›*) 3))\n(*discuss goal 3*)\n apply (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n(*proven 3 subgoals*) .\n\nlemma colours_flip_fM[simp]:\n \"fl ≠ sys_fM s ⟹ black b (s(sys := (s sys)⦇fM := fl, mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)) ⟷ white b s ∧ ¬grey b s\"\nunfolding black_def white_def\n(*goal: ‹fl ≠ sys_fM s ⟹ (marked b (s(sys := s sys⦇fM := fl, mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈)) ∧ ¬ grey b (s(sys := s sys⦇fM := fl, mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))) = (obj_at (λobj. obj_mark obj ≠ sys_fM s) b s ∧ ¬ grey b s)›*)\nby (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n\nlemma colours_alloc[simp]:\n \"heap (s sys) r' = None\n ⟹ black r (s(mutator m := (s (mutator m))⦇ roots := roots' ⦈, sys := (s sys)⦇heap := (heap (s sys))(r' ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ black r s ∨ (r' = r ∧ fl = sys_fM s ∧ ¬grey r' s)\"\n \"grey r (s(mutator m := (s (mutator m))⦇ roots := roots' ⦈, sys := (s sys)⦇heap := (heap (s sys))(r' ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ grey r s\"\n \"heap (s sys) r' = None\n ⟹ white r (s(mutator m := (s (mutator m))⦇ roots := roots' ⦈, sys := (s sys)⦇heap := (heap (s sys))(r' ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ white r s ∨ (r' = r ∧ fl ≠ sys_fM s)\"\nunfolding black_def white_def\n(*goals:\n 1. ‹sys_heap (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) (r'::'c::type) = None ⟹ (marked (r::'c::type) (s(mutator (m::'d::type) := s (mutator m)⦇roots := roots'::'c::type set⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl::bool, obj_fields = λx::'a::type. None, obj_payload = λx::'a::type. None⦈)⦈)) ∧ ¬ grey r (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx::'a::type. None, obj_payload = λx::'a::type. None⦈)⦈))) = (marked r s ∧ ¬ grey r s ∨ r' = r ∧ fl = sys_fM s ∧ ¬ grey r' s)›\n 2. ‹grey (r::'c::type) ((s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)(mutator (m::'d::type) := s (mutator m)⦇roots := roots'::'c::type set⦈, sys := s sys⦇heap := (sys_heap s)(r'::'c::type ↦ ⦇obj_mark = fl::bool, obj_fields = λx::'a::type. None, obj_payload = λx::'a::type. None⦈)⦈)) = grey r s›\n 3. ‹sys_heap (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) (r'::'c::type) = None ⟹ obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM (s(mutator (m::'d::type) := s (mutator m)⦇roots := roots'::'c::type set⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl::bool, obj_fields = λx::'a::type. None, obj_payload = λx::'a::type. None⦈)⦈))) (r::'c::type) (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx::'a::type. None, obj_payload = λx::'a::type. None⦈)⦈)) = (obj_at (λobj::('a, 'b, 'c) object. obj_mark obj ≠ sys_fM s) r s ∨ r' = r ∧ fl ≠ sys_fM s)›*)\n(*goals:\n 1. ‹sys_heap s r' = None ⟹ (marked r (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ∧ ¬ grey r (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = (marked r s ∧ ¬ grey r s ∨ r' = r ∧ fl = sys_fM s ∧ ¬ grey r' s)›\n 2. ‹grey r (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) = grey r s›\n 3. ‹sys_heap s r' = None ⟹ obj_at (λobj. obj_mark obj ≠ sys_fM (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) r (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) = (obj_at (λobj. obj_mark obj ≠ sys_fM s) r s ∨ r' = r ∧ fl ≠ sys_fM s)›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n(*proven 3 subgoals*) .\n\nlemma heap_colours_alloc[simp]:\n \"⟦ heap (s sys) r' = None; valid_refs_inv s ⟧\n ⟹ black_heap (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ black_heap s ∧ fl = sys_fM s\"\n \"heap (s sys) r' = None\n ⟹ white_heap (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ white_heap s ∧ fl ≠ sys_fM s\"\nunfolding black_heap_def white_def white_heap_def\n(*goals:\n 1. ‹⟦sys_heap s r' = None; valid_refs_inv s⟧ ⟹ (∀x. obj_at (λs. True) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟶ black x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = ((∀x. obj_at (λs. True) x s ⟶ black x s) ∧ fl = sys_fM s)›\n 2. ‹sys_heap s r' = None ⟹ (∀x. obj_at (λs. True) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟶ obj_at (λobj. obj_mark obj ≠ sys_fM (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = ((∀x. obj_at (λs. True) x s ⟶ obj_at (λobj. obj_mark obj ≠ sys_fM s) x s) ∧ fl ≠ sys_fM s)›*)\n(*goals:\n 1. ‹⟦sys_heap s r' = None; valid_refs_inv s⟧ ⟹ (∀x. obj_at (λs. True) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟶ black x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = ((∀x. obj_at (λs. True) x s ⟶ black x s) ∧ fl = sys_fM s)›\n 2. ‹sys_heap s r' = None ⟹ (∀x. obj_at (λs. True) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟶ obj_at (λobj. obj_mark obj ≠ sys_fM (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = ((∀x. obj_at (λs. True) x s ⟶ obj_at (λobj. obj_mark obj ≠ sys_fM s) x s) ∧ fl ≠ sys_fM s)›\ndiscuss goal 1*)\n apply (simp add: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*top goal: ‹⟦sys_heap s r' = None; valid_refs_inv s⟧ ⟹ (∀x. obj_at (λs. True) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟶ black x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = ((∀x. obj_at (λs. True) x s ⟶ black x s) ∧ fl = sys_fM s)› and 1 goal remains*)\n apply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹⟦sys_heap s r' = None; valid_refs_inv s; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s) ∧ ((∃obj. sys_heap s x = Some obj) ⟶ (black x s ∨ r' = x ∧ fl = sys_fM s ∧ ¬ grey r' s) ∧ (x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s))⟧ ⟹ (∀x. (∃obj. sys_heap s x = Some obj) ⟶ black x s) ∧ fl = sys_fM s›\n 2. ‹⟦sys_heap s r' = None; valid_refs_inv s; (∀x. (∃obj. sys_heap s x = Some obj) ⟶ black x s) ∧ fl = sys_fM s⟧ ⟹ ∀x. (sys_heap s x = None ⟶ x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s) ∧ ((∃obj. sys_heap s x = Some obj) ⟶ (black x s ∨ r' = x ∧ fl = sys_fM s ∧ ¬ grey r' s) ∧ (x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s))›\n discuss goal 1*)\n apply (intro allI (*‹(⋀x::?'a::type. (?P::?'a::type ⇒ bool) x) ⟹ ∀x::?'a::type. ?P x›*) conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*))\n (*goals:\n 1. ‹⋀x. ⟦sys_heap s r' = None; valid_refs_inv s; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s) ∧ ((∃obj. sys_heap s x = Some obj) ⟶ (black x s ∨ r' = x ∧ fl = sys_fM s ∧ ¬ grey r' s) ∧ (x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s)); ∃obj. sys_heap s x = Some obj⟧ ⟹ black x s›\n 2. ‹⟦sys_heap s r' = None; valid_refs_inv s; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s) ∧ ((∃obj. sys_heap s x = Some obj) ⟶ (black x s ∨ r' = x ∧ fl = sys_fM s ∧ ¬ grey r' s) ∧ (x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s))⟧ ⟹ fl = sys_fM s›\n discuss goal 1*)\n apply (rename_tac x)\n (*top goal: ‹⋀x::'c. ⟦sys_heap (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) (r'::'c) = None; valid_refs_inv s; ∀x::'c. (sys_heap s x = None ⟶ x = r' ⟶ black r' s ∨ (fl::bool) = sys_fM s ∧ ¬ grey r' s) ∧ ((∃obj::('a, 'b, 'c) object. sys_heap s x = Some obj) ⟶ (black x s ∨ r' = x ∧ fl = sys_fM s ∧ ¬ grey r' s) ∧ (x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s)); ∃obj::('a, 'b, 'c) object. sys_heap s x = Some obj⟧ ⟹ black x s› and 3 goals remain*)\n apply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))\n (*top goal: ‹⋀x::'c::type. ⟦sys_heap (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) (r'::'c::type) = None; valid_refs_inv s; ∀x::'c::type. (sys_heap s x = None ⟶ x = r' ⟶ black r' s ∨ (fl::bool) = sys_fM s ∧ ¬ grey r' s) ∧ ((∃obj::('a, 'b, 'c) object. sys_heap s x = Some obj) ⟶ (black x s ∨ r' = x ∧ fl = sys_fM s ∧ ¬ grey r' s) ∧ (x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s)); ∃obj::('a, 'b, 'c) object. sys_heap s x = Some obj⟧ ⟹ black x s› and 3 goals remain*)\n apply clarsimp\n (*discuss goal 2*)\n apply (drule spec[where x=r'] (*‹∀x. ?P x ⟹ ?P r'›*))\n (*top goal: ‹⟦sys_heap s r' = None; valid_refs_inv s; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s) ∧ ((∃obj. sys_heap s x = Some obj) ⟶ (black x s ∨ r' = x ∧ fl = sys_fM s ∧ ¬ grey r' s) ∧ (x = r' ⟶ black r' s ∨ fl = sys_fM s ∧ ¬ grey r' s))⟧ ⟹ fl = sys_fM s› and 2 goals remain*)\n apply ((auto simp: reaches_def (*‹(?x::?'ref) reaches (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (λ(x::?'ref) y::?'ref. (x points_to y) s)⇧*⇧* ?x ?y)›*) dest!: valid_refs_invD (*‹⟦(?x::?'a) ∈ roots ((?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state) (mutator (?m::?'c))); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs::(?'b, ?'d, ?'a) object. True) ?y ?s› ‹⟦(?x::?'a) ∈ roots ((?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state) (mutator (?m::?'c))); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj::(?'b, ?'d, ?'a) object. sys_heap ?s ?y = Some obj› ‹⟦(?x::?'a) ∈ mut_m.tso_store_refs (?m::?'c) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs::(?'b, ?'d, ?'a) object. True) ?y ?s› ‹⟦(?x::?'a) ∈ mut_m.tso_store_refs (?m::?'c) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj::(?'b, ?'d, ?'a) object. sys_heap ?s ?y = Some obj› ‹⟦(?w::(?'b, ?'d, ?'a) mem_store_action) ∈ set (sys_mem_store_buffers (mutator (?m::?'c)) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state)); (?x::?'a) ∈ store_refs ?w; (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs::(?'b, ?'d, ?'a) object. True) ?y ?s› ‹⟦(?w::(?'b, ?'d, ?'a) mem_store_action) ∈ set (sys_mem_store_buffers (mutator (?m::?'c)) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state)); (?x::?'a) ∈ store_refs ?w; (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj::(?'b, ?'d, ?'a) object. sys_heap ?s ?y = Some obj› ‹⟦grey (?x::?'a) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs::(?'b, ?'d, ?'a) object. True) ?y ?s› ‹⟦mut_m.reachable (?m::?'c) (?x::?'a) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state); valid_refs_inv ?s⟧ ⟹ obj_at (λs::(?'b, ?'d, ?'a) object. True) ?x ?s› ‹⟦mut_m.reachable (?m::?'c) (?x::?'a) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state); valid_refs_inv ?s⟧ ⟹ ∃obj::(?'b, ?'d, ?'a) object. sys_heap ?s ?x = Some obj› ‹⟦(?x::?'a) ∈ ghost_honorary_root ((?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state) (mutator (?m::?'c))); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs::(?'b, ?'d, ?'a) object. True) ?y ?s› ‹⟦(?x::?'a) ∈ ghost_honorary_root ((?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state) (mutator (?m::?'c))); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj::(?'b, ?'d, ?'a) object. sys_heap ?s ?y = Some obj›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply ((auto simp: reaches_def (*‹?x reaches ?y = (λs. (λx y. (x points_to y) s)⇧*⇧* ?x ?y)›*) dest!: valid_refs_invD (*‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦grey ?x ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?x ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?x = Some obj› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n (*proven 2 subgoals*)\n(*discuss goal 2*)\n apply (simp add: fun_upd_apply (*‹((?f::?'b::type ⇒ ?'a::type)(?x::?'b::type := ?y::?'a::type)) (?z::?'b::type) = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c::type) (?s::?'d::type process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c::type) (?s::?'d::type process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n (*goal: ‹sys_heap s r' = None ⟹ (∀x. obj_at (λs. True) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟶ obj_at (λobj. obj_mark obj ≠ sys_fM (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) x (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = ((∀x. obj_at (λs. True) x s ⟶ obj_at (λobj. obj_mark obj ≠ sys_fM s) x s) ∧ fl ≠ sys_fM s)›*)\n apply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹⟦sys_heap s r' = None; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ fl = (¬ sys_fM s)) ∧ (∀obj. sys_heap s x = Some obj ⟶ (obj_mark obj = (¬ sys_fM s) ∨ x = r' ∧ fl = (¬ sys_fM s)) ∧ (x = r' ⟶ fl = (¬ sys_fM s)))⟧ ⟹ (∀x obj. sys_heap s x = Some obj ⟶ obj_mark obj = (¬ sys_fM s)) ∧ fl = (¬ sys_fM s)›\n 2. ‹⟦sys_heap s r' = None; (∀x obj. sys_heap s x = Some obj ⟶ obj_mark obj = (¬ sys_fM s)) ∧ fl = (¬ sys_fM s)⟧ ⟹ ∀x. (sys_heap s x = None ⟶ x = r' ⟶ fl = (¬ sys_fM s)) ∧ (∀obj. sys_heap s x = Some obj ⟶ (obj_mark obj = (¬ sys_fM s) ∨ x = r' ∧ fl = (¬ sys_fM s)) ∧ (x = r' ⟶ fl = (¬ sys_fM s)))›\n discuss goal 1*)\n apply (intro allI (*‹(⋀x::?'a. (?P::?'a ⇒ bool) x) ⟹ ∀x::?'a. ?P x›*) conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*))\n (*goals:\n 1. ‹⋀x obj. ⟦sys_heap s r' = None; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ fl = (¬ sys_fM s)) ∧ (∀obj. sys_heap s x = Some obj ⟶ (obj_mark obj = (¬ sys_fM s) ∨ x = r' ∧ fl = (¬ sys_fM s)) ∧ (x = r' ⟶ fl = (¬ sys_fM s))); sys_heap s x = Some obj⟧ ⟹ obj_mark obj = (¬ sys_fM s)›\n 2. ‹⟦sys_heap s r' = None; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ fl = (¬ sys_fM s)) ∧ (∀obj. sys_heap s x = Some obj ⟶ (obj_mark obj = (¬ sys_fM s) ∨ x = r' ∧ fl = (¬ sys_fM s)) ∧ (x = r' ⟶ fl = (¬ sys_fM s)))⟧ ⟹ fl = (¬ sys_fM s)›\n discuss goal 1*)\n apply (rename_tac x obj)\n (*top goal: ‹⋀x obj. ⟦sys_heap s r' = None; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ fl = (¬ sys_fM s)) ∧ (∀obj. sys_heap s x = Some obj ⟶ (obj_mark obj = (¬ sys_fM s) ∨ x = r' ∧ fl = (¬ sys_fM s)) ∧ (x = r' ⟶ fl = (¬ sys_fM s))); sys_heap s x = Some obj⟧ ⟹ obj_mark obj = (¬ sys_fM s)› and 2 goals remain*)\n apply (drule_tac x=x in spec (*‹∀x. ?P x ⟹ ?P ?x›*))\n (*top goal: ‹⋀x obj. ⟦sys_heap s r' = None; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ fl = (¬ sys_fM s)) ∧ (∀obj. sys_heap s x = Some obj ⟶ (obj_mark obj = (¬ sys_fM s) ∨ x = r' ∧ fl = (¬ sys_fM s)) ∧ (x = r' ⟶ fl = (¬ sys_fM s))); sys_heap s x = Some obj⟧ ⟹ obj_mark obj = (¬ sys_fM s)› and 2 goals remain*)\n apply clarsimp\n (*discuss goal 2*)\n apply (drule spec[where x=r'] (*‹∀x::'c. (?P::'c ⇒ bool) x ⟹ ?P (r'::'c)›*))\n (*top goal: ‹⟦sys_heap s r' = None; ∀x. (sys_heap s x = None ⟶ x = r' ⟶ fl = (¬ sys_fM s)) ∧ (∀obj. sys_heap s x = Some obj ⟶ (obj_mark obj = (¬ sys_fM s) ∨ x = r' ∧ fl = (¬ sys_fM s)) ∧ (x = r' ⟶ fl = (¬ sys_fM s)))⟧ ⟹ fl = (¬ sys_fM s)› and 1 goal remains*)\n apply ((auto dest!: valid_refs_invD (*‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦grey ?x ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?x ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?x = Some obj› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply ((auto dest!: valid_refs_invD (*‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦grey ?x ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?x ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?x = Some obj› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))[1])\n (*proven 2 subgoals*)\n(*proven 2 subgoals*) .\n\nlemma grey_protects_white_hs_done[simp]:\n \"(g grey_protects_white w) (s(mutator m := s (mutator m)⦇ W := {}, ghost_hs_phase := hs' ⦈,\n sys := s sys⦇ hs_pending := hp', W := sys_W s ∪ W (s (mutator m)),\n ghost_hs_in_sync := his' ⦈))\n ⟷ (g grey_protects_white w) s\"\nunfolding grey_protects_white_def\n(*goal: ‹(grey g (s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈)) ∧ (g has_white_path_to w) (s(mutator m := s (mutator m)⦇W := {}, ghost_hs_phase := hs'⦈, sys := s sys⦇hs_pending := hp', W := sys_W s ∪ W (s (mutator m)), ghost_hs_in_sync := his'⦈))) = (grey g s ∧ (g has_white_path_to w) s)›*)\nby (simp add: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n\nlemma grey_protects_white_alloc[simp]:\n \"⟦ fl = sys_fM s; sys_heap s r = None ⟧\n ⟹ (g grey_protects_white w) (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ (g grey_protects_white w) s\"\nunfolding grey_protects_white_def has_white_path_to_def\n(*goal: ‹⟦fl = sys_fM s; sys_heap s r = None⟧ ⟹ (grey g (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ∧ (λx y. (x points_to y) (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ∧ white y (s(mutator m := s (mutator m)⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)))⇧*⇧* g w) = (grey g s ∧ (λx y. (x points_to y) s ∧ white y s)⇧*⇧* g w)›*)\nby simp\n\nlemma (in mut_m) reachable_snapshot_inv_sweep_loop_free:\n fixes s :: \"('field, 'mut, 'payload, 'ref) lsts\"\n assumes nmr: \"white r s\"\n assumes ngs: \"no_grey_refs s\"\n assumes rsi: \"reachable_snapshot_inv s\"\n shows \"reachable_snapshot_inv (s(sys := (s sys)⦇heap := (heap (s sys))(r := None)⦈))\" (is \"reachable_snapshot_inv ?s'\")\nproof (standard)\n (*goal: ‹⋀y. reachable y (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ⟹ in_snapshot y (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈))›*)\n fix y :: 'ref\n assume rx: \"reachable y ?s'\" (*‹reachable (y::'ref) ((s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r::'ref := None)⦈))›*)\n then have \"black y s ∧ y ≠ r\"\n proof (induct rule: reachable_induct (*‹⟦mut_m.reachable ?m ?y ?s; ⋀x. x ∈ roots (?s (mutator ?m)) ⟹ ?P x; ⋀x. x ∈ ghost_honorary_root (?s (mutator ?m)) ⟹ ?P x; ⋀x. x ∈ mut_m.tso_store_refs ?m ?s ⟹ ?P x; ⋀x y. ⟦mut_m.reachable ?m x ?s; (x points_to y) ?s; ?P x⟧ ⟹ ?P y⟧ ⟹ ?P ?y›*))\n (*goals:\n 1. ‹⋀x. x ∈ mut_roots (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ⟹ black x s ∧ x ≠ r›\n 2. ‹⋀x. x ∈ mut_ghost_honorary_root (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ⟹ black x s ∧ x ≠ r›\n 3. ‹⋀x. x ∈ tso_store_refs (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ⟹ black x s ∧ x ≠ r›\n 4. ‹⋀x y. ⟦reachable x (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); (x points_to y) (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); black x s ∧ x ≠ r⟧ ⟹ black y s ∧ y ≠ r›*)\n case (root x) (*‹(x::'ref) ∈ mut_roots ((s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r::'ref := None)⦈))›*)\n with ngs (*‹no_grey_refs s›*) nmr (*‹white r s›*) rsi (*‹reachable_snapshot_inv (s::'mut::type process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state)›*) show \"?case\"\n (*goal: ‹black x s ∧ x ≠ r›*)\n by (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) dest: reachable_blackD (*‹⟦no_grey_refs ?s; reachable_snapshot_inv ?s; reachable ?r ?s⟧ ⟹ black ?r ?s›*))\n next\n (*goals:\n 1. ‹⋀x. x ∈ mut_ghost_honorary_root (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ⟹ black x s ∧ x ≠ r›\n 2. ‹⋀x. x ∈ tso_store_refs (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ⟹ black x s ∧ x ≠ r›\n 3. ‹⋀x y. ⟦reachable x (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); (x points_to y) (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); black x s ∧ x ≠ r⟧ ⟹ black y s ∧ y ≠ r›*)\n case (ghost_honorary_root x) (*‹x ∈ mut_ghost_honorary_root (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈))›*)\n with ngs (*‹no_grey_refs (s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state)›*) nmr (*‹white r s›*) rsi (*‹reachable_snapshot_inv s›*) show \"?case\"\n (*goal: ‹black x s ∧ x ≠ r›*)\n unfolding reachable_def reaches_def\n (*goal: ‹black x s ∧ x ≠ r›*)\n by (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) dest: reachable_blackD (*‹⟦no_grey_refs ?s; reachable_snapshot_inv ?s; reachable ?r ?s⟧ ⟹ black ?r ?s›*))\n next\n (*goals:\n 1. ‹⋀x. x ∈ tso_store_refs (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ⟹ black x s ∧ x ≠ r›\n 2. ‹⋀x y. ⟦reachable x (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); (x points_to y) (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); black x s ∧ x ≠ r⟧ ⟹ black y s ∧ y ≠ r›*)\n case (tso_root x) (*‹x ∈ tso_store_refs (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈))›*)\n with ngs (*‹no_grey_refs s›*) nmr (*‹white r s›*) rsi (*‹reachable_snapshot_inv (s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state)›*) show \"?case\"\n (*goal: ‹black x s ∧ x ≠ r›*)\n unfolding reachable_def reaches_def\n (*goal: ‹black x s ∧ x ≠ r›*)\n by (auto simp: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) dest: reachable_blackD (*‹⟦no_grey_refs (?s::'mut process_name ⇒ (?'a, 'mut, ?'b, ?'c) Model.local_state); reachable_snapshot_inv ?s; reachable (?r::?'c) ?s⟧ ⟹ black ?r ?s›*))\n next\n (*goal: ‹⋀x y. ⟦reachable x (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); (x points_to y) (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)); black x s ∧ x ≠ r⟧ ⟹ black y s ∧ y ≠ r›*)\n case (reaches x y) (*‹reachable x (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈))› ‹(x points_to y) (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈))› ‹black x s ∧ x ≠ r›*)\n with ngs (*‹no_grey_refs s›*) nmr (*‹white r s›*) rsi (*‹reachable_snapshot_inv s›*) show \"?case\"\n (*goal: ‹black y s ∧ y ≠ r›*)\n unfolding reachable_def reaches_def\n (*goal: ‹black y s ∧ y ≠ r›*)\n apply (clarsimp simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n (*goal: ‹black y s ∧ y ≠ r›*)\n apply (drule predicate2D[OF rtranclp_mono[where s=\"λx y. (x points_to y) s\", OF predicate2I], rotated] (*‹⟦?r1⇧*⇧* ?x ?y; ⋀x y. ?r1 x y ⟹ (x points_to y) s⟧ ⟹ (λx y. (x points_to y) s)⇧*⇧* ?x ?y›*))\n (*goals:\n 1. ‹⋀xa xaa y. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to y) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (xaa points_to y) s ∧ xaa ≠ r⟧ ⟹ (xaa points_to y) s›\n 2. ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to y) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ black y s ∧ y ≠ r›\n discuss goal 1*)\n apply (clarsimp split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) if_splits (*‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 2*)\n apply (rule conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to y) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ black y s›\n 2. ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to y) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ y ≠ r›\n discuss goal 1*)\n apply (rule reachable_blackD (*‹⟦no_grey_refs ?s; reachable_snapshot_inv ?s; reachable ?r ?s⟧ ⟹ black ?r ?s›*))\n (*goals:\n 1. ‹⋀xa::'ref. ⟦no_grey_refs (s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state); white (r::'ref) s; reachable_snapshot_inv s; ((x::'ref) points_to (y::'ref)) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λ(x::'ref) y::'ref. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ no_grey_refs s›\n 2. ‹⋀xa::'ref. ⟦no_grey_refs (s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state); white (r::'ref) s; reachable_snapshot_inv s; ((x::'ref) points_to (y::'ref)) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λ(x::'ref) y::'ref. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ reachable_snapshot_inv s›\n 3. ‹⋀xa::'ref. ⟦no_grey_refs (s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state); white (r::'ref) s; reachable_snapshot_inv s; ((x::'ref) points_to (y::'ref)) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λ(x::'ref) y::'ref. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ reachable y s›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply (simp add: reachable_def (*‹reachable ?y = (λs. ∃x. x ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (x reaches ?y) s)›*) reaches_def (*‹?x reaches ?y = (λs. (λx y. (x points_to y) s)⇧*⇧* ?x ?y)›*))\n (*top goal: ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to y) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ reachable y s› and 1 goal remains*)\n apply (blast intro: rtranclp.intros( (*‹⟦?r⇧*⇧* ?a ?b; ?r ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2))\n (*proven 3 subgoals*)\n (*discuss goal 2*)\n apply clarsimp\n (*goal: ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to y) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x⟧ ⟹ y ≠ r›*)\n apply (frule (1) reachable_blackD[where r=r] (*‹⟦no_grey_refs ?s; reachable_snapshot_inv ?s; reachable r ?s⟧ ⟹ black r ?s›*))\n (*goals:\n 1. ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to r) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x; y = r⟧ ⟹ reachable r s›\n 2. ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to r) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x; y = r; black r s⟧ ⟹ False›\n discuss goal 1*)\n apply (simp add: reachable_def (*‹reachable ?y = (λs. ∃x. x ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (x reaches ?y) s)›*) reaches_def (*‹?x reaches ?y = (λs. (λx y. (x points_to y) s)⇧*⇧* ?x ?y)›*))\n (*top goal: ‹⋀xa. ⟦no_grey_refs s; white r s; reachable_snapshot_inv s; (x points_to r) s; black x s; x ≠ r; xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s; (λx y. (x points_to y) s)⇧*⇧* xa x; y = r⟧ ⟹ reachable r s› and 1 goal remains*)\n apply (blast intro: rtranclp.intros( (*‹⟦?r⇧*⇧* ?a ?b; ?r ?b ?c⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2))\n (*discuss goal 2*)\n apply auto\n (*proven 2 subgoals*)\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n qed\n then show \"in_snapshot y ?s'\"\n unfolding in_snapshot_def\n (*goal: ‹black y (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)) ∨ (∃x. (x grey_protects_white y) (s(sys := s sys⦇heap := (sys_heap s)(r := None)⦈)))›*)\n by simp\nqed\n\nlemma reachable_alloc[simp]:\n assumes rn: \"sys_heap s r = None\"\n shows \"mut_m.reachable m r' (s(mutator m' := (s (mutator m'))⦇roots := insert r (roots (s (mutator m')))⦈, sys := (s sys)⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ mut_m.reachable m r' s ∨ (m' = m ∧ r' = r)\" (is \"?lhs ⟷ ?rhs\")\nproof (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟹ mut_m.reachable m r' s ∨ m' = m ∧ r' = r›\n 2. ‹mut_m.reachable m r' s ∨ m' = m ∧ r' = r ⟹ mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n assume \"?lhs\" (*‹mut_m.reachable (m::'d) (r'::'c) ((s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)(mutator (m'::'d) := s (mutator m')⦇roots := insert (r::'c) (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl::bool, obj_fields = λx::'a. None, obj_payload = λx::'a. None⦈)⦈))›*)\n from this (*‹mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*) assms (*‹sys_heap (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) (r::'c) = None›*) show \"?rhs\"\n proof (induct rule: reachable_induct (*‹⟦mut_m.reachable (?m::?'a) (?y::?'b) (?s::?'a process_name ⇒ (?'c, ?'a, ?'d, ?'b) Model.local_state); ⋀x::?'b. x ∈ roots (?s (mutator ?m)) ⟹ (?P::?'b ⇒ bool) x; ⋀x::?'b. x ∈ ghost_honorary_root (?s (mutator ?m)) ⟹ ?P x; ⋀x::?'b. x ∈ mut_m.tso_store_refs ?m ?s ⟹ ?P x; ⋀(x::?'b) y::?'b. ⟦mut_m.reachable ?m x ?s; (x points_to y) ?s; ?P x⟧ ⟹ ?P y⟧ ⟹ ?P ?y›*))\n (*goals:\n 1. ‹⋀x. ⟦x ∈ roots ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)); sys_heap s r = None⟧ ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r›\n 2. ‹⋀x. ⟦x ∈ ghost_honorary_root ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)); sys_heap s r = None⟧ ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r›\n 3. ‹⋀x. ⟦x ∈ mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)); sys_heap s r = None⟧ ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r›\n 4. ‹⋀x y. ⟦mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)); (x points_to y) (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)); sys_heap s r = None ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r; sys_heap s r = None⟧ ⟹ mut_m.reachable m y s ∨ m' = m ∧ y = r›*)\n case (reaches x y) (*‹mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))› ‹(x points_to y) (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))› ‹sys_heap s r = None ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r› ‹sys_heap s r = None›*)\n then show \"?case\"\n (*goal: ‹mut_m.reachable m y s ∨ m' = m ∧ y = r›*)\n apply clarsimp\n (*goal: ‹mut_m.reachable (m::'d::type) (y::'c::type) (s::'d::type process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) ∨ (m'::'d::type) = m ∧ y = (r::'c::type)›*)\n by (fastforce simp: mut_m.reachable_def (*‹mut_m.reachable (?m::?'mut) (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ∃x::?'ref. x ∈ roots (s (mutator ?m)) ∪ ghost_honorary_root (s (mutator ?m)) ∪ mut_m.tso_store_refs ?m s ∧ (x reaches ?y) s)›*) reaches_def (*‹(?x::?'ref) reaches (?y::?'ref) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. (λ(x::?'ref) y::?'ref. (x points_to y) s)⇧*⇧* ?x ?y)›*) elim: rtranclp.intros( (*‹⟦(?r::?'a ⇒ ?'a ⇒ bool)⇧*⇧* (?a::?'a) (?b::?'a); ?r ?b (?c::?'a)⟧ ⟹ ?r⇧*⇧* ?a ?c›*) 2) split: obj_at_splits (*‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹(?Q::bool ⇒ bool) (obj_at (?P::(?'a, ?'b, ?'c) object ⇒ bool) (?r::?'c) (?s::?'d process_name ⇒ (?'a, ?'d, ?'b, ?'c) Model.local_state)) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj::(?'a, ?'b, ?'c) object. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*))\n qed (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*solves the remaining goals:\n 1. ‹⋀x. ⟦x ∈ roots ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)); sys_heap s r = None⟧ ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r›\n 2. ‹⋀x. ⟦x ∈ ghost_honorary_root ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)); sys_heap s r = None⟧ ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r›\n 3. ‹⋀x. ⟦x ∈ mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)); sys_heap s r = None⟧ ⟹ mut_m.reachable m x s ∨ m' = m ∧ x = r›*)\nnext\n (*goal: ‹mut_m.reachable m r' s ∨ m' = m ∧ r' = r ⟹ mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n assume \"?rhs\" (*‹mut_m.reachable (m::'d) (r'::'c) (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) ∨ (m'::'d) = m ∧ r' = (r::'c)›*)\n then show \"?lhs\"\n proof (rule disjE (*‹⟦(?P::bool) ∨ (?Q::bool); ?P ⟹ ?R::bool; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹mut_m.reachable m r' s ⟹ mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›\n 2. ‹m' = m ∧ r' = r ⟹ mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n assume \"mut_m.reachable m r' s\" (*‹mut_m.reachable (m::'d) (r'::'c) (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)›*)\n then show \"?thesis\"\n (*goal: ‹mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n proof (induct rule: reachable_induct (*‹⟦mut_m.reachable ?m ?y ?s; ⋀x. x ∈ roots (?s (mutator ?m)) ⟹ ?P x; ⋀x. x ∈ ghost_honorary_root (?s (mutator ?m)) ⟹ ?P x; ⋀x. x ∈ mut_m.tso_store_refs ?m ?s ⟹ ?P x; ⋀x y. ⟦mut_m.reachable ?m x ?s; (x points_to y) ?s; ?P x⟧ ⟹ ?P y⟧ ⟹ ?P ?y›*))\n (*goals:\n 1. ‹⋀x. x ∈ roots (s (mutator m)) ⟹ mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›\n 2. ‹⋀x. x ∈ ghost_honorary_root (s (mutator m)) ⟹ mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›\n 3. ‹⋀x. x ∈ mut_m.tso_store_refs m s ⟹ mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›\n 4. ‹⋀x y. ⟦mut_m.reachable m x s; (x points_to y) s; mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))⟧ ⟹ mut_m.reachable m y (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n case (tso_root x) (*‹x ∈ mut_m.tso_store_refs m s›*)\n then show \"?case\"\n (*goal: ‹mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n unfolding mut_m.reachable_def\n (*goal: ‹∃xa. xa ∈ roots ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)) ∪ ghost_honorary_root ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)) ∪ mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ∧ (xa reaches x) (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n by fastforce\n next\n (*goals:\n 1. ‹⋀x. x ∈ roots (s (mutator m)) ⟹ mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›\n 2. ‹⋀x. x ∈ ghost_honorary_root (s (mutator m)) ⟹ mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›\n 3. ‹⋀x y. ⟦mut_m.reachable m x s; (x points_to y) s; mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))⟧ ⟹ mut_m.reachable m y (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n case (reaches x y) (*‹mut_m.reachable m x s› ‹(x points_to y) s› ‹mut_m.reachable m x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n with rn (*‹sys_heap s r = None›*) show \"?case\"\n (*goal: ‹mut_m.reachable m y (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n unfolding mut_m.reachable_def\n (*goal: ‹∃x. x ∈ roots ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)) ∪ ghost_honorary_root ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) (mutator m)) ∪ mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ∧ (x reaches y) (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n by fastforce\n qed (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n (*solves the remaining goals:\n 1. ‹⋀x::'c. x ∈ roots ((s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) (mutator (m::'d))) ⟹ mut_m.reachable m x (s(mutator (m'::'d) := s (mutator m')⦇roots := insert (r::'c) (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl::bool, obj_fields = λx::'a. None, obj_payload = λx::'a. None⦈)⦈))›\n 2. ‹⋀x::'c. x ∈ ghost_honorary_root ((s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) (mutator (m::'d))) ⟹ mut_m.reachable m x (s(mutator (m'::'d) := s (mutator m')⦇roots := insert (r::'c) (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl::bool, obj_fields = λx::'a. None, obj_payload = λx::'a. None⦈)⦈))›*)\n next\n (*goal: ‹m' = m ∧ r' = r ⟹ mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n assume \"m' = m ∧ r' = r\" (*‹(m'::'d) = (m::'d) ∧ (r'::'c) = (r::'c)›*)\n with rn (*‹sys_heap s r = None›*) show \"?thesis\"\n (*goal: ‹mut_m.reachable m r' (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))›*)\n unfolding mut_m.reachable_def\n (*goal: ‹∃x::'c. x ∈ roots (((s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state)(mutator (m'::'d) := s (mutator m')⦇roots := insert (r::'c) (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl::bool, obj_fields = λx::'a. None, obj_payload = λx::'a. None⦈)⦈)) (mutator (m::'d))) ∪ ghost_honorary_root ((s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx::'a. None, obj_payload = λx::'a. None⦈)⦈)) (mutator m)) ∪ mut_m.tso_store_refs m (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx::'a. None, obj_payload = λx::'a. None⦈)⦈)) ∧ (x reaches (r'::'c)) (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx::'a. None, obj_payload = λx::'a. None⦈)⦈))›*)\n by (fastforce simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n qed\nqed\n\ncontext mut_m\nbegin\n\nlemma reachable_snapshot_inv_alloc[simp, elim!]:\n fixes s :: \"('field, 'mut, 'payload, 'ref) lsts\"\n assumes rsi: \"reachable_snapshot_inv s\"\n assumes rn: \"sys_heap s r = None\"\n assumes fl: \"fl = sys_fM s\"\n assumes vri: \"valid_refs_inv s\"\n shows \"reachable_snapshot_inv (s(mutator m' := (s (mutator m'))⦇roots := insert r (roots (s (mutator m')))⦈, sys := (s sys)⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\"\n(is \"reachable_snapshot_inv ?s'\")\nusing assms (*‹reachable_snapshot_inv s› ‹sys_heap s r = None› ‹fl = sys_fM s› ‹valid_refs_inv (s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state)›*) unfolding reachable_snapshot_inv_def in_snapshot_def\n(*goal: ‹∀x. reachable x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ⟶ black x (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)) ∨ (∃xa. (xa grey_protects_white x) (s(mutator m' := s (mutator m')⦇roots := insert r (roots (s (mutator m')))⦈, sys := s sys⦇heap := (sys_heap s)(r ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈)))›*)\nby (auto simp del: reachable_fun_upd (*‹⟦roots ((?s(?fld := ?val)) (mutator ?m)) = roots (?s (mutator ?m)); ghost_honorary_root ((?s(?fld := ?val)) (mutator ?m)) = ghost_honorary_root (?s (mutator ?m)); ⋀x. ⋃ (ran ` obj_fields ` set_option (sys_heap (?s(?fld := ?val)) x)) = ⋃ (ran ` obj_fields ` set_option (sys_heap ?s x)); filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator ?m) (?s(?fld := ?val))) = filter (λs. is_mw_Mutate s ∨ is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator ?m) ?s)⟧ ⟹ mut_m.reachable ?m ?r (?s(?fld := ?val)) = mut_m.reachable ?m ?r ?s›*))\n\nlemma reachable_snapshot_inv_discard_roots[simp]:\n \"⟦ reachable_snapshot_inv s; roots' ⊆ roots (s (mutator m)) ⟧\n ⟹ reachable_snapshot_inv (s(mutator m := (s (mutator m))⦇roots := roots'⦈))\"\nunfolding reachable_snapshot_inv_def reachable_def in_snapshot_def grey_protects_white_def\n(*goal: ‹⟦∀x. (∃xa. xa ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (xa reaches x) s) ⟶ black x s ∨ (∃xa. grey xa s ∧ (xa has_white_path_to x) s); roots' ⊆ mut_roots s⟧ ⟹ ∀x. (∃xa. xa ∈ mut_roots (s(mutator m := s (mutator m)⦇roots := roots'⦈)) ∪ mut_ghost_honorary_root (s(mutator m := s (mutator m)⦇roots := roots'⦈)) ∪ tso_store_refs (s(mutator m := s (mutator m)⦇roots := roots'⦈)) ∧ (xa reaches x) (s(mutator m := s (mutator m)⦇roots := roots'⦈))) ⟶ black x (s(mutator m := s (mutator m)⦇roots := roots'⦈)) ∨ (∃xa. grey xa (s(mutator m := s (mutator m)⦇roots := roots'⦈)) ∧ (xa has_white_path_to x) (s(mutator m := s (mutator m)⦇roots := roots'⦈)))›*)\nby (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n\nlemma reachable_snapshot_inv_load[simp]:\n \"⟦ reachable_snapshot_inv s; sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref r'; r ∈ mut_roots s ⟧\n ⟹ reachable_snapshot_inv (s(mutator m := s (mutator m)⦇ roots := mut_roots s ∪ Option.set_option r' ⦈))\"\nunfolding reachable_snapshot_inv_def in_snapshot_def grey_protects_white_def\n(*goal: ‹⟦∀x. reachable x s ⟶ black x s ∨ (∃xa. grey xa s ∧ (xa has_white_path_to x) s); sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref r'; r ∈ mut_roots s⟧ ⟹ ∀x. reachable x (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) ⟶ black x (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) ∨ (∃xa. grey xa (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)) ∧ (xa has_white_path_to x) (s(mutator m := s (mutator m)⦇roots := mut_roots s ∪ set_option r'⦈)))›*)\nby (simp add: fun_upd_apply (*‹((?f::?'b::type ⇒ ?'a::type)(?x::?'b::type := ?y::?'a::type)) (?z::?'b::type) = (if ?z = ?x then ?y else ?f ?z)›*))\n\nlemma reachable_snapshot_inv_store_ins[simp]:\n \"⟦ reachable_snapshot_inv s; r ∈ mut_roots s; (∃r'. opt_r' = Some r') ⟶ the opt_r' ∈ mut_roots s ⟧\n ⟹ reachable_snapshot_inv (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈,\n sys := s sys⦇ mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r']) ⦈))\"\nunfolding reachable_snapshot_inv_def in_snapshot_def grey_protects_white_def reachable_def\n(*goal: ‹⟦∀x::'c. (∃xa::'c. xa ∈ mut_roots (s::'mut process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state) ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (xa reaches x) s) ⟶ black x s ∨ (∃xa::'c. grey xa s ∧ (xa has_white_path_to x) s); (r::'c) ∈ mut_roots s; (∃r'::'c. (opt_r'::'c option) = Some r') ⟶ the opt_r' ∈ mut_roots s⟧ ⟹ ∀x::'c. (∃xa::'c. xa ∈ mut_roots (s(mutator (m::'mut) := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r (f::'a) opt_r'])⦈)) ∪ mut_ghost_honorary_root (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∪ tso_store_refs (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∧ (xa reaches x) (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈))) ⟶ black x (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∨ (∃xa::'c. grey xa (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∧ (xa has_white_path_to x) (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)))›*)\napply clarsimp\n(*goal: ‹⟦∀x. (∃xa. xa ∈ mut_roots s ∪ mut_ghost_honorary_root s ∪ tso_store_refs s ∧ (xa reaches x) s) ⟶ black x s ∨ (∃xa. grey xa s ∧ (xa has_white_path_to x) s); r ∈ mut_roots s; (∃r'. opt_r' = Some r') ⟶ the opt_r' ∈ mut_roots s⟧ ⟹ ∀x. (∃xa. xa ∈ mut_roots (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∪ mut_ghost_honorary_root (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∪ tso_store_refs (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∧ (xa reaches x) (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈))) ⟶ black x (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∨ (∃xa. grey xa (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ∧ (xa has_white_path_to x) (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)))›*)\napply (drule_tac x=x in spec (*‹∀x::?'a. (?P::?'a ⇒ bool) x ⟹ ?P (?x::?'a)›*))\n(*goal: ‹⋀x xa. ⟦∀x. (∃xa. (xa ∈ mut_roots s ∨ xa ∈ mut_ghost_honorary_root s ∨ xa ∈ tso_store_refs s) ∧ (xa reaches x) s) ⟶ black x s ∨ (∃xa. grey xa s ∧ (xa has_white_path_to x) s); r ∈ mut_roots s; (∃r'. opt_r' = Some r') ⟶ the opt_r' ∈ mut_roots s; xa ∈ mut_roots s ∨ xa ∈ tso_store_refs (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)); (xa reaches x) s; ∀xa. grey xa (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)) ⟶ ¬ (xa has_white_path_to x) s⟧ ⟹ black x (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈))›*)\napply (auto simp: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*))\n(*goals:\n 1. ‹⋀x xa. ⟦r ∈ mut_roots s; (xa reaches x) s; ∀xa. grey xa s ⟶ ¬ (xa has_white_path_to x) s; xa ∈ tso_store_refs (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f None])⦈)); opt_r' = None; ∀xa. xa ∉ mut_roots s ∧ xa ∉ mut_ghost_honorary_root s ∧ xa ∉ tso_store_refs s ∨ ¬ (xa reaches x) s⟧ ⟹ black x s›\n 2. ‹⋀x xa. ⟦r ∈ mut_roots s; (xa reaches x) s; ∀xa. grey xa s ⟶ ¬ (xa has_white_path_to x) s; xa ∈ tso_store_refs (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f opt_r'])⦈)); the opt_r' ∈ mut_roots s; ∀xa. xa ∉ mut_roots s ∧ xa ∉ mut_ghost_honorary_root s ∧ xa ∉ tso_store_refs s ∨ ¬ (xa reaches x) s⟧ ⟹ black x s›\ndiscuss goal 1*)\n apply (subst (asm) tso_store_refs_simps (*‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b::type) := ?s (mutator ?m')⦇roots := ?roots'::?'a::type set⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b::type) := ?s (mutator ?m')⦇ghost_honorary_root := {}⦈, sys := ?s sys⦇mem_store_buffers := (mem_store_buffers (?s sys))(mutator ?m' := sys_mem_store_buffers (mutator ?m') ?s @ [mw_Mutate (?r::?'a::type) (?f::?'c::type) (?opt_r'::?'a::type option)])⦈)) = mut_m.tso_store_refs ?m ?s ∪ (if ?m' = ?m then store_refs (mw_Mutate ?r ?f ?opt_r') else {})› ‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇mem_store_buffers := (mem_store_buffers (?s sys))(mutator (?m'::?'b::type) := sys_mem_store_buffers (mutator ?m') ?s @ [mw_Mutate_Payload (?r::?'a::type) (?f::?'c::type) (?pl::?'d::type option)])⦈)) = mut_m.tso_store_refs ?m ?s ∪ (if ?m' = ?m then store_refs (mw_Mutate_Payload ?r ?f ?pl) else {})› ‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇heap := (sys_heap ?s)(?r'::?'a::type := None)⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b::type) := ?s (mutator ?m')⦇roots := insert (?r::?'a::type) (roots (?s (mutator ?m')))⦈, sys := ?s sys⦇heap := (sys_heap ?s)(?r ↦ ?obj::(?'c, ?'d, ?'a) object)⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b::type) := ?s (mutator ?m')⦇ghost_honorary_root := set_option ?opt_r', ref := ?opt_r'::?'a::type option⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇heap := (sys_heap ?s)(?r::?'a::type := map_option (λobj::(?'c, ?'d, ?'a) object. obj⦇obj_fields := (obj_fields obj)(?f::?'c::type := ?opt_r'::?'a::type option)⦈) (sys_heap ?s ?r)), mem_store_buffers := (mem_store_buffers (?s sys))(?p::?'b::type process_name := ?ws::(?'c::type, ?'d::type, ?'a::type) mem_store_action list)⦈)) = (if ?p = mutator ?m then ⋃ (store_refs ` set ?ws) else mut_m.tso_store_refs ?m ?s)› ‹mut_m.tso_store_refs (?m::?'b::type) ((?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇heap := (sys_heap ?s)(?r::?'a::type := map_option (λobj::(?'c, ?'d, ?'a) object. obj⦇obj_payload := (obj_payload obj)(?f::?'c::type := ?pl::?'d::type option)⦈) (sys_heap ?s ?r)), mem_store_buffers := (mem_store_buffers (?s sys))(?p::?'b::type process_name := ?ws::(?'c::type, ?'d::type, ?'a::type) mem_store_action list)⦈)) = (if ?p = mutator ?m then ⋃ (store_refs ` set ?ws) else mut_m.tso_store_refs ?m ?s)› ‹sys_mem_store_buffers (?p::?'b::type process_name) (?s::?'b::type process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state) = mw_Mark (?r::?'a::type) (?fl::bool) # (?ws::(?'c::type, ?'d::type, ?'a::type) mem_store_action list) ⟹ mut_m.tso_store_refs (?m::?'b::type) (?s(sys := ?s sys⦇heap := (sys_heap ?s)(?r := map_option (obj_mark_update (λ_::bool. ?fl)) (sys_heap ?s ?r)), mem_store_buffers := (mem_store_buffers (?s sys))(?p := ?ws)⦈)) = mut_m.tso_store_refs ?m ?s›*))\n (*top goal: ‹⋀x xa. ⟦r ∈ mut_roots s; (xa reaches x) s; ∀xa. grey xa s ⟶ ¬ (xa has_white_path_to x) s; xa ∈ tso_store_refs (s(mutator m := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r f None])⦈)); opt_r' = None; ∀xa. xa ∉ mut_roots s ∧ xa ∉ mut_ghost_honorary_root s ∧ xa ∉ tso_store_refs s ∨ ¬ (xa reaches x) s⟧ ⟹ black x s› and 1 goal remains*)\n apply force\n(*discuss goal 2*)\n apply (subst (asm) tso_store_refs_simps (*‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b) := ?s (mutator ?m')⦇roots := ?roots'::?'a set⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b) := ?s (mutator ?m')⦇ghost_honorary_root := {}⦈, sys := ?s sys⦇mem_store_buffers := (mem_store_buffers (?s sys))(mutator ?m' := sys_mem_store_buffers (mutator ?m') ?s @ [mw_Mutate (?r::?'a) (?f::?'c) (?opt_r'::?'a option)])⦈)) = mut_m.tso_store_refs ?m ?s ∪ (if ?m' = ?m then store_refs (mw_Mutate ?r ?f ?opt_r') else {})› ‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇mem_store_buffers := (mem_store_buffers (?s sys))(mutator (?m'::?'b) := sys_mem_store_buffers (mutator ?m') ?s @ [mw_Mutate_Payload (?r::?'a) (?f::?'c) (?pl::?'d option)])⦈)) = mut_m.tso_store_refs ?m ?s ∪ (if ?m' = ?m then store_refs (mw_Mutate_Payload ?r ?f ?pl) else {})› ‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇heap := (sys_heap ?s)(?r'::?'a := None)⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b) := ?s (mutator ?m')⦇roots := insert (?r::?'a) (roots (?s (mutator ?m')))⦈, sys := ?s sys⦇heap := (sys_heap ?s)(?r ↦ ?obj::(?'c, ?'d, ?'a) object)⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(mutator (?m'::?'b) := ?s (mutator ?m')⦇ghost_honorary_root := set_option ?opt_r', ref := ?opt_r'::?'a option⦈)) = mut_m.tso_store_refs ?m ?s› ‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇heap := (sys_heap ?s)(?r::?'a := map_option (λobj::(?'c, ?'d, ?'a) object. obj⦇obj_fields := (obj_fields obj)(?f::?'c := ?opt_r'::?'a option)⦈) (sys_heap ?s ?r)), mem_store_buffers := (mem_store_buffers (?s sys))(?p::?'b process_name := ?ws::(?'c, ?'d, ?'a) mem_store_action list)⦈)) = (if ?p = mutator ?m then ⋃ (store_refs ` set ?ws) else mut_m.tso_store_refs ?m ?s)› ‹mut_m.tso_store_refs (?m::?'b) ((?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state)(sys := ?s sys⦇heap := (sys_heap ?s)(?r::?'a := map_option (λobj::(?'c, ?'d, ?'a) object. obj⦇obj_payload := (obj_payload obj)(?f::?'c := ?pl::?'d option)⦈) (sys_heap ?s ?r)), mem_store_buffers := (mem_store_buffers (?s sys))(?p::?'b process_name := ?ws::(?'c, ?'d, ?'a) mem_store_action list)⦈)) = (if ?p = mutator ?m then ⋃ (store_refs ` set ?ws) else mut_m.tso_store_refs ?m ?s)› ‹sys_mem_store_buffers (?p::?'b process_name) (?s::?'b process_name ⇒ (?'c, ?'b, ?'d, ?'a) Model.local_state) = mw_Mark (?r::?'a) (?fl::bool) # (?ws::(?'c, ?'d, ?'a) mem_store_action list) ⟹ mut_m.tso_store_refs (?m::?'b) (?s(sys := ?s sys⦇heap := (sys_heap ?s)(?r := map_option (obj_mark_update (λ_::bool. ?fl)) (sys_heap ?s ?r)), mem_store_buffers := (mem_store_buffers (?s sys))(?p := ?ws)⦈)) = mut_m.tso_store_refs ?m ?s›*))\n (*goal: ‹⋀(x::'c) xa::'c. ⟦(r::'c) ∈ mut_roots (s::'mut process_name ⇒ ('a, 'mut, 'b, 'c) Model.local_state); (xa reaches x) s; ∀xa::'c. grey xa s ⟶ ¬ (xa has_white_path_to x) s; xa ∈ tso_store_refs (s(mutator (m::'mut) := s (mutator m)⦇ghost_honorary_root := {}⦈, sys := s sys⦇mem_store_buffers := (mem_store_buffers (s sys))(mutator m := sys_mem_store_buffers (mutator m) s @ [mw_Mutate r (f::'a) (opt_r'::'c option)])⦈)); the opt_r' ∈ mut_roots s; ∀xa::'c. xa ∉ mut_roots s ∧ xa ∉ mut_ghost_honorary_root s ∧ xa ∉ tso_store_refs s ∨ ¬ (xa reaches x) s⟧ ⟹ black x s›*)\n apply force\n(*proven 2 subgoals*) .\n\nend\n\nlemma WL_mo_co_mark[simp]:\n \"ghost_honorary_grey (s p) = {}\n ⟹ WL p' (s(p := s p⦇ ghost_honorary_grey := rs ⦈)) = WL p' s ∪ { r |r. p' = p ∧ r ∈ rs}\"\nunfolding WL_def\n(*goal: ‹ghost_honorary_grey ((s::'c::type process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) (p::'c::type process_name)) = {} ⟹ W ((s(p := s p⦇ghost_honorary_grey := rs::'a::type set⦈)) (p'::'c::type process_name)) ∪ ghost_honorary_grey ((s(p := s p⦇ghost_honorary_grey := rs⦈)) p') = W (s p') ∪ ghost_honorary_grey (s p') ∪ {r |r::'a::type. p' = p ∧ r ∈ rs}›*)\nby (simp add: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*))\n\nlemma ghost_honorary_grey_mo_co_mark[simp]:\n \"⟦ ghost_honorary_grey (s p) = {} ⟧ ⟹ black b (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ⟷ black b s ∧ b ≠ r\"\n \"⟦ ghost_honorary_grey (s p) = {} ⟧ ⟹ grey g (s(p := (s p)⦇ghost_honorary_grey := {r}⦈)) ⟷ grey g s ∨ g = r\"\n \"⟦ ghost_honorary_grey (s p) = {} ⟧ ⟹ white w (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ⟷ white w s\"\nunfolding black_def grey_def\n(*goals:\n 1. ‹ghost_honorary_grey (s p) = {} ⟹ (marked b (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ∧ (∄x. b ∈ WL x (s(p := s p⦇ghost_honorary_grey := {r}⦈)))) = ((marked b s ∧ (∄x. b ∈ WL x s)) ∧ b ≠ r)›\n 2. ‹ghost_honorary_grey (s p) = {} ⟹ (∃x. g ∈ WL x (s(p := s p⦇ghost_honorary_grey := {r}⦈))) = ((∃x. g ∈ WL x s) ∨ g = r)›\n 3. ‹ghost_honorary_grey (s p) = {} ⟹ white w (s(p := s p⦇ghost_honorary_grey := {r}⦈)) = white w s›*)\n(*goals:\n 1. ‹ghost_honorary_grey (s p) = {} ⟹ (marked b (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ∧ (∄x. b ∈ WL x (s(p := s p⦇ghost_honorary_grey := {r}⦈)))) = ((marked b s ∧ (∄x. b ∈ WL x s)) ∧ b ≠ r)›\n 2. ‹ghost_honorary_grey (s p) = {} ⟹ (∃x. g ∈ WL x (s(p := s p⦇ghost_honorary_grey := {r}⦈))) = ((∃x. g ∈ WL x s) ∨ g = r)›\n 3. ‹ghost_honorary_grey (s p) = {} ⟹ white w (s(p := s p⦇ghost_honorary_grey := {r}⦈)) = white w s›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹((?f::?'b::type ⇒ ?'a::type)(?x::?'b::type := ?y::?'a::type)) (?z::?'b::type) = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*discuss goal 3*)\n apply ((auto simp: fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*proven 3 subgoals*) .\n\nlemma ghost_honorary_grey_mo_co_W[simp]:\n \"ghost_honorary_grey (s p') = {r}\n ⟹ (WL p (s(p' := (s p')⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈))) = (WL p s)\"\n \"ghost_honorary_grey (s p') = {r}\n ⟹ grey g (s(p' := (s p')⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) ⟷ grey g s\"\nunfolding grey_def WL_def\n(*goals:\n 1. ‹ghost_honorary_grey (s p') = {r} ⟹ W ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) p) ∪ ghost_honorary_grey ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) p) = W (s p) ∪ ghost_honorary_grey (s p)›\n 2. ‹ghost_honorary_grey (s p') = {r} ⟹ (∃x. g ∈ W ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) x) ∪ ghost_honorary_grey ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) x)) = (∃x. g ∈ W (s x) ∪ ghost_honorary_grey (s x))›*)\n(*goals:\n 1. ‹ghost_honorary_grey (s p') = {r} ⟹ W ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) p) ∪ ghost_honorary_grey ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) p) = W (s p) ∪ ghost_honorary_grey (s p)›\n 2. ‹ghost_honorary_grey (s p') = {r} ⟹ (∃x. g ∈ W ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) x) ∪ ghost_honorary_grey ((s(p' := s p'⦇W := insert r (W (s p')), ghost_honorary_grey := {}⦈)) x)) = (∃x. g ∈ W (s x) ∪ ghost_honorary_grey (s x))›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: process_name.splits (*‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = ((∀x1. ?process_name = mutator x1 ⟶ ?P (?f1.0 x1)) ∧ (?process_name = gc ⟶ ?P ?f2.0) ∧ (?process_name = sys ⟶ ?P ?f3.0))› ‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = (¬ ((∃x1. ?process_name = mutator x1 ∧ ¬ ?P (?f1.0 x1)) ∨ ?process_name = gc ∧ ¬ ?P ?f2.0 ∨ ?process_name = sys ∧ ¬ ?P ?f3.0))›*) if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: process_name.splits (*‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = ((∀x1. ?process_name = mutator x1 ⟶ ?P (?f1.0 x1)) ∧ (?process_name = gc ⟶ ?P ?f2.0) ∧ (?process_name = sys ⟶ ?P ?f3.0))› ‹?P (case ?process_name of mutator x ⇒ ?f1.0 x | gc ⇒ ?f2.0 | sys ⇒ ?f3.0) = (¬ ((∃x1. ?process_name = mutator x1 ∧ ¬ ?P (?f1.0 x1)) ∨ ?process_name = gc ∧ ¬ ?P ?f2.0 ∨ ?process_name = sys ∧ ¬ ?P ?f3.0))›*) if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))[1])\n(*proven 2 subgoals*) .\n\nlemma reachable_sweep_loop_free:\n \"mut_m.reachable m r (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈))\n ⟹ mut_m.reachable m r s\"\nunfolding mut_m.reachable_def reaches_def\n(*goal: ‹∃x. x ∈ roots ((s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) (mutator m)) ∪ ghost_honorary_root ((s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) (mutator m)) ∪ mut_m.tso_store_refs m (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) ∧ (λx y. (x points_to y) (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)))⇧*⇧* x r ⟹ ∃x. x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s ∧ (λx y. (x points_to y) s)⇧*⇧* x r›*)\napply (clarsimp simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n(*goal: ‹∃x::'b::type. x ∈ roots (((s::'a::type process_name ⇒ ('c, 'a, 'd, 'b) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r'::'b::type := None)⦈)) (mutator (m::'a::type))) ∪ ghost_honorary_root ((s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) (mutator m)) ∪ mut_m.tso_store_refs m (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)) ∧ (λ(x::'b::type) y::'b::type. (x points_to y) (s(sys := s sys⦇heap := (sys_heap s)(r' := None)⦈)))⇧*⇧* x (r::'b::type) ⟹ ∃x::'b::type. x ∈ roots (s (mutator m)) ∪ ghost_honorary_root (s (mutator m)) ∪ mut_m.tso_store_refs m s ∧ (λ(x::'b::type) y::'b::type. (x points_to y) s)⇧*⇧* x r›*)\nby (metis (no_types, lifting) mono_rtranclp (*‹(⋀a b. ?x a b ⟶ ?y a b) ⟹ ?x⇧*⇧* ?a ?b ⟶ ?y⇧*⇧* ?a ?b›*))\n\nlemma reachable_deref_del[simp]:\n \"⟦ sys_load (mutator m) (mr_Ref r f) (s sys) = mv_Ref opt_r'; r ∈ mut_m.mut_roots m s; mut_m.mut_ghost_honorary_root m s = {} ⟧\n ⟹ mut_m.reachable m' y (s(mutator m := s (mutator m)⦇ ghost_honorary_root := Option.set_option opt_r', ref := opt_r' ⦈))\n ⟷ mut_m.reachable m' y s\"\nunfolding mut_m.reachable_def reaches_def sys_load_def\n(*goal: ‹⟦(do_load_action (mr_Ref (r::'c) (f::'a)) ∘ processors_view_of_memory (mutator (m::'d))) ((s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) sys) = mv_Ref (opt_r'::'c option); r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}⟧ ⟹ (∃x::'c. x ∈ roots ((s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)) (mutator (m'::'d))) ∪ ghost_honorary_root ((s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)) (mutator m')) ∪ mut_m.tso_store_refs m' (s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)) ∧ (λ(x::'c) y::'c. (x points_to y) (s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)))⇧*⇧* x (y::'c)) = (∃x::'c. x ∈ roots (s (mutator m')) ∪ ghost_honorary_root (s (mutator m')) ∪ mut_m.tso_store_refs m' s ∧ (λ(x::'c) y::'c. (x points_to y) s)⇧*⇧* x y)›*)\napply ((clarsimp simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n(*goal: ‹⟦(do_load_action (mr_Ref r f) ∘ processors_view_of_memory (mutator m)) (s sys) = mv_Ref opt_r'; r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}⟧ ⟹ (∃x. x ∈ roots ((s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)) (mutator m')) ∪ ghost_honorary_root ((s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)) (mutator m')) ∪ mut_m.tso_store_refs m' (s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)) ∧ (λx y. (x points_to y) (s(mutator m := s (mutator m)⦇ghost_honorary_root := set_option opt_r', ref := opt_r'⦈)))⇧*⇧* x y) = (∃x. x ∈ roots (s (mutator m')) ∪ ghost_honorary_root (s (mutator m')) ∪ mut_m.tso_store_refs m' s ∧ (λx y. (x points_to y) s)⇧*⇧* x y)›*)\napply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n(*goals:\n 1. ‹⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f); m' = m; ∃x. (x ∈ roots (s (mutator m)) ∨ Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f) = Some x ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n 2. ‹⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f); m' = m; ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f) = Some x ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\ndiscuss goal 1*)\n apply ((clarsimp)[1])\n (*top goal: ‹⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f); m' = m; ∃x. (x ∈ roots (s (mutator m)) ∨ Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f) = Some x ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y› and 1 goal remains*)\n apply ((elim disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))[1])\n (*goals:\n 1. ‹⋀x. ⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f); m' = m; (λx y. (x points_to y) s)⇧*⇧* x y; x ∈ roots (s (mutator m))⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n 2. ‹⋀x. ⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f); m' = m; (λx y. (x points_to y) s)⇧*⇧* x y; Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f) = Some x⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n 3. ‹⋀x. ⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f); m' = m; (λx y. (x points_to y) s)⇧*⇧* x y; x ∈ mut_m.tso_store_refs m s⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n discuss goal 1*)\n apply ((metis)[1])\n (*discuss goal 2*)\n apply ((erule option_bind_invE (*‹⟦Option.bind ?f ?g = None; ⋀a. ⟦?f = Some a; ?g a = None⟧ ⟹ ?Q; ?f = None ⟹ ?Q⟧ ⟹ ?Q› ‹⟦Option.bind ?f ?g = Some ?x; ⋀a. ⟦?f = Some a; ?g a = Some ?x⟧ ⟹ ?Q⟧ ⟹ ?Q›*))[1])\n (*top goal: ‹⋀x. ⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f); m' = m; (λx y. (x points_to y) s)⇧*⇧* x y; Option.bind (heap (processors_view_of_memory (mutator m) (s sys)) r) (λobj. obj_fields obj f) = Some x⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y› and 2 goals remain*)\n apply ((auto dest!: fold_stores_points_to (*‹⟦heap (fold_stores ?ws (?s sys)) ?r = Some ?obj; obj_fields ?obj ?f = Some ?r'⟧ ⟹ (?r points_to ?r') ?s ∨ (∃w∈set ?ws. ?r' ∈ store_refs w)›*))[1])\n (*goals:\n 1. ‹⋀x a. ⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Some x; m' = m; (λx y. (x points_to y) s)⇧*⇧* x y; obj_fields a f = Some x; (r points_to x) s⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n 2. ‹⋀x a w. ⟦r ∈ roots (s (mutator m)); ghost_honorary_root (s (mutator m)) = {}; opt_r' = Some x; m' = m; (λx y. (x points_to y) s)⇧*⇧* x y; obj_fields a f = Some x; w ∈ set (sys_mem_store_buffers (mutator m) s); x ∈ (case w of mw_Mutate r f r' ⇒ insert r (set_option r') | mw_Mutate_Payload r f pl ⇒ {r} | _ ⇒ {})⟧ ⟹ ∃x. (x ∈ roots (s (mutator m)) ∨ x ∈ mut_m.tso_store_refs m s) ∧ (λx y. (x points_to y) s)⇧*⇧* x y›\n discuss goal 1*)\n apply ((auto elim!: converse_rtranclp_into_rtranclp[rotated] (*‹⟦(?r::?'a::type ⇒ ?'a::type ⇒ bool)⇧*⇧* (?b::?'a::type) (?c::?'a::type); ?r (?a::?'a::type) ?b⟧ ⟹ ?r⇧*⇧* ?a ?c›*) simp: mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut::type) = (λs::?'mut::type process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*))[1])\n (*discuss goal 2*)\n apply ((auto elim!: converse_rtranclp_into_rtranclp[rotated] (*‹⟦?r⇧*⇧* ?b ?c; ?r ?a ?b⟧ ⟹ ?r⇧*⇧* ?a ?c›*) simp: mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*))[1])\n (*proven 2 subgoals*)\n (*discuss goal 3*)\n apply ((auto elim!: converse_rtranclp_into_rtranclp[rotated] (*‹⟦(?r::?'a ⇒ ?'a ⇒ bool)⇧*⇧* (?b::?'a) (?c::?'a); ?r (?a::?'a) ?b⟧ ⟹ ?r⇧*⇧* ?a ?c›*) simp: mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*))[1])\n (*proven 3 subgoals*)\n(*discuss goal 2*)\n apply ((auto elim!: converse_rtranclp_into_rtranclp[rotated] (*‹⟦?r⇧*⇧* ?b ?c; ?r ?a ?b⟧ ⟹ ?r⇧*⇧* ?a ?c›*) simp: mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs ?m = (λs. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*))[1])\n(*proven 2 subgoals*) .\n\nlemma no_black_refs_dequeue[simp]:\n \"⟦ sys_mem_store_buffers p s = mw_Mark r fl # ws; no_black_refs s; valid_W_inv s ⟧\n ⟹ no_black_refs (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))\"\n \"⟦ sys_mem_store_buffers p s = mw_Mutate r f r' # ws; no_black_refs s ⟧\n ⟹ no_black_refs (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := r')⦈) (sys_heap s r)),\n mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))\"\nunfolding no_black_refs_def\n(*goals:\n 1. ‹⟦sys_mem_store_buffers p s = mw_Mark r fl # ws; ∀x. ¬ black x s; valid_W_inv s⟧ ⟹ ∀x. ¬ black x (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))›\n 2. ‹⟦sys_mem_store_buffers p s = mw_Mutate r f r' # ws; ∀x. ¬ black x s⟧ ⟹ ∀x. ¬ black x (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := r')⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))›*)\n(*goals:\n 1. ‹⟦sys_mem_store_buffers (p::'d process_name) (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) = mw_Mark (r::'c) (fl::bool) # (ws::('a, 'b, 'c) mem_store_action list); ∀x::'c. ¬ black x s; valid_W_inv s⟧ ⟹ ∀x::'c. ¬ black x (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (obj_mark_update (λ_::bool. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))›\n 2. ‹⟦sys_mem_store_buffers (p::'d process_name) (s::'d process_name ⇒ ('a, 'd, 'b, 'c) Model.local_state) = mw_Mutate (r::'c) (f::'a) (r'::'c option) # (ws::('a, 'b, 'c) mem_store_action list); ∀x::'c. ¬ black x s⟧ ⟹ ∀x::'c. ¬ black x (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj::('a, 'b, 'c) object. obj⦇obj_fields := (obj_fields obj)(f := r')⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)⦈))›\ndiscuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) dest: valid_W_invD (*‹⟦sys_mem_store_buffers ?p ?s = mw_Mark ?r ?fl # ?ws; valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark ?ws = []› ‹⟦mw_Mark ?r ?fl ∈ set (sys_mem_store_buffers ?p ?s); valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark (sys_mem_store_buffers ?p ?s) = [mw_Mark ?r ?fl]›*))[1])\n(*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) dest: valid_W_invD (*‹⟦sys_mem_store_buffers ?p ?s = mw_Mark ?r ?fl # ?ws; valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark ?ws = []› ‹⟦mw_Mark ?r ?fl ∈ set (sys_mem_store_buffers ?p ?s); valid_W_inv ?s⟧ ⟹ ?fl = sys_fM ?s ∧ ?r ∈ ghost_honorary_grey (?s ?p) ∧ sys_mem_lock ?s = Some ?p ∧ white ?r ?s ∧ filter is_mw_Mark (sys_mem_store_buffers ?p ?s) = [mw_Mark ?r ?fl]›*))[1])\n(*proven 2 subgoals*) .\n\nlemma colours_blacken[simp]:\n \"valid_W_inv s ⟹ black b (s(gc := s gc⦇W := gc_W s - {r}⦈)) ⟷ black b s ∨ (r ∈ gc_W s ∧ b = r)\"\n \"⟦ r ∈ gc_W s; valid_W_inv s ⟧ ⟹ grey g (s(gc := s gc⦇W := gc_W s - {r}⦈)) ⟷ (grey g s ∧ g ≠ r)\"\n (* \"white w (s(gc := s gc⦇W := gc_W s - {r}⦈)) ⟷ white w s\" is redundant *)\nunfolding black_def grey_def valid_W_inv_def\n(*goals:\n 1. ‹(∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]) ⟹ (marked b (s(gc := s gc⦇W := gc_W s - {r}⦈)) ∧ (∄x. b ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈)))) = (marked b s ∧ (∄x. b ∈ WL x s) ∨ r ∈ gc_W s ∧ b = r)›\n 2. ‹⟦r ∈ gc_W s; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ (∃x. g ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))) = ((∃x. g ∈ WL x s) ∧ g ≠ r)›*)\n(*goals:\n 1. ‹(∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]) ⟹ (marked b (s(gc := s gc⦇W := gc_W s - {r}⦈)) ∧ (∄x. b ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈)))) = (marked b s ∧ (∄x. b ∈ WL x s) ∨ r ∈ gc_W s ∧ b = r)›\n 2. ‹⟦r ∈ gc_W s; (∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ (∃x. g ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))) = ((∃x. g ∈ WL x s) ∧ g ≠ r)›\ndiscuss goal 1*)\n apply (simp add: all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*top goal: ‹(∀x xa. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀x xa xb. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]) ⟹ (marked b (s(gc := s gc⦇W := gc_W s - {r}⦈)) ∧ (∄x. b ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈)))) = (marked b s ∧ (∄x. b ∈ WL x s) ∨ r ∈ gc_W s ∧ b = r)› and 1 goal remains*)\n apply ((safe)[1])\n (*goals:\n 1. ‹⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (r::'d) = None; r ∈ gc_W s; (b::'d) = r⟧ ⟹ False›\n 2. ‹⋀(obj::('b, 'c, 'd) object) x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (b::'d) = Some obj; ∀x::'a process_name. b ∉ WL x (s(gc := s gc⦇W := gc_W s - {r::'d}⦈)); obj_mark obj; sys_fM s; r ∉ gc_W s; b ∈ WL x s⟧ ⟹ False›\n 3. ‹⋀(obj::('b, 'c, 'd) object) x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (b::'d) = Some obj; ∀x::'a process_name. b ∉ WL x (s(gc := s gc⦇W := gc_W s - {r::'d}⦈)); obj_mark obj; sys_fM s; b ≠ r; b ∈ WL x s⟧ ⟹ False›\n 4. ‹⋀(obj::('b, 'c, 'd) object) x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (b::'d) = Some obj; ∀x::'a process_name. b ∉ WL x (s(gc := s gc⦇W := gc_W s - {r::'d}⦈)); ¬ obj_mark obj; ¬ sys_fM s; r ∉ gc_W s; b ∈ WL x s⟧ ⟹ False›\n 5. ‹⋀(obj::('b, 'c, 'd) object) x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (b::'d) = Some obj; ∀x::'a process_name. b ∉ WL x (s(gc := s gc⦇W := gc_W s - {r::'d}⦈)); ¬ obj_mark obj; ¬ sys_fM s; b ≠ r; b ∈ WL x s⟧ ⟹ False›\n 6. ‹⋀(obj::('b, 'c, 'd) object) x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (b::'d) = Some obj; ∀x::'a process_name. b ∉ WL x s; obj_mark obj; sys_fM s; b ∈ WL x (s(gc := s gc⦇W := gc_W s - {r::'d}⦈))⟧ ⟹ False›\n 7. ‹⋀(obj::('b, 'c, 'd) object) x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (b::'d) = Some obj; ∀x::'a process_name. b ∉ WL x s; ¬ obj_mark obj; ¬ sys_fM s; b ∈ WL x (s(gc := s gc⦇W := gc_W s - {r::'d}⦈))⟧ ⟹ False›\n 8. ‹⋀obj::('b, 'c, 'd) object. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (r::'d) = Some obj; r ∈ gc_W s; (b::'d) = r; obj_mark obj⟧ ⟹ sys_fM s›\n 9. ‹⋀obj::('b, 'c, 'd) object. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (r::'d) = Some obj; r ∈ gc_W s; (b::'d) = r; sys_fM s⟧ ⟹ obj_mark obj›\n 10. ‹⋀(obj::('b, 'c, 'd) object) x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (r::'d) = Some obj; r ∈ gc_W s; (b::'d) = r; r ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))⟧ ⟹ False›\n discuss goal 1*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*top goal: ‹⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (r::'d) = None; r ∈ gc_W s; (b::'d) = r⟧ ⟹ False› and 10 goals remain*)\n apply (metis option.distinct( (*‹None ≠ Some (?x2.0::?'a::type)›*) 1))\n (*discuss goal 2*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 3*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*top goal: ‹⋀(obj::('b, 'c, 'd) object) x::'a::type process_name. ⟦∀(x::'a::type process_name) xa::'a::type process_name. x ≠ xa ⟶ WL x (s::'a::type process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a::type process_name) (xa::'a::type process_name) xb::'d::type. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a::type process_name) (xa::'d::type) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀(x::'a::type process_name) xa::'d::type. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a::type process_name) xa::'d::type. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a::type process_name) xa::'d::type. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a::type process_name) xa::'d::type. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s (b::'d::type) = Some obj; ∀x::'a::type process_name. b ∉ WL x (s(gc := s gc⦇W := gc_W s - {r::'d::type}⦈)); obj_mark obj; sys_fM s; b ≠ r; b ∈ WL x s⟧ ⟹ False› and 8 goals remain*)\n apply blast\n (*discuss goal 4*)\n apply (simp add: WL_def (*‹WL (?p::?'mut process_name) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 5*)\n apply (simp add: WL_def (*‹WL (?p::?'mut process_name) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*top goal: ‹⋀obj x. ⟦∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s b = Some obj; ∀x. b ∉ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈)); ¬ obj_mark obj; ¬ sys_fM s; b ≠ r; b ∈ WL x s⟧ ⟹ False› and 6 goals remain*)\n apply blast\n (*discuss goal 6*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 7*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 8*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 9*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*discuss goal 10*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*top goal: ‹⋀obj x. ⟦∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); sys_heap s r = Some obj; r ∈ gc_W s; b = r; r ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))⟧ ⟹ False› and 1 goal remains*)\n apply blast\n (*proven 10 subgoals*)\n(*discuss goal 2*)\n apply (simp add: all_conj_distrib (*‹(∀x. ?P x ∧ ?Q x) = ((∀x. ?P x) ∧ (∀x. ?Q x))›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⟦(r::'d) ∈ gc_W (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state); (∀(x::'a process_name) xa::'d. xa ∈ W (s x) ∨ sys_mem_lock s ≠ Some x ∧ xa ∈ ghost_honorary_grey (s x) ⟶ marked xa s) ∧ (∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x s ∩ WL xa s = {}) ∧ (∀(x::'a process_name) (xa::'a process_name) xb::'d. ¬ (xb ∈ ghost_honorary_grey (s x) ∧ xb ∈ W (s xa))) ∧ sys_ghost_honorary_grey s = {} ∧ (∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb])⟧ ⟹ (∃x::'a process_name. (g::'d) ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))) = ((∃x::'a process_name. g ∈ WL x s) ∧ g ≠ r)›*)\n apply ((safe)[1])\n (*goals:\n 1. ‹⋀x. ⟦∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; r ∈ gc_W s; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); g ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))⟧ ⟹ ∃x. g ∈ WL x s›\n 2. ‹⋀x. ⟦∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; r ∈ gc_W s; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); r ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈)); g = r⟧ ⟹ False›\n 3. ‹⋀x. ⟦∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; r ∈ gc_W s; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); g ≠ r; g ∈ WL x s⟧ ⟹ ∃x. g ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))›\n discuss goal 1*)\n apply (simp add: WL_def (*‹WL (?p::?'mut process_name) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goals:\n 1. ‹⋀x. ⟦∀x xa. x ≠ xa ⟶ (W (s x) ∪ ghost_honorary_grey (s x)) ∩ (W (s xa) ∪ ghost_honorary_grey (s xa)) = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; r ∈ gc_W s; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); x = gc; g ∈ gc_W s ∧ g ≠ r ∨ g ∈ gc_ghost_honorary_grey s⟧ ⟹ ∃x. g ∈ W (s x) ∨ g ∈ ghost_honorary_grey (s x)›\n 2. ‹⋀x. ⟦∀x xa. x ≠ xa ⟶ (W (s x) ∪ ghost_honorary_grey (s x)) ∩ (W (s xa) ∪ ghost_honorary_grey (s xa)) = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; r ∈ gc_W s; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); x ≠ gc; g ∈ W (s x) ∨ g ∈ ghost_honorary_grey (s x)⟧ ⟹ ∃x. g ∈ W (s x) ∨ g ∈ ghost_honorary_grey (s x)›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (simp add: WL_def (*‹WL (?p::?'mut process_name) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹((?f::?'b ⇒ ?'a)(?x::?'b := ?y::?'a)) (?z::?'b) = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹(?P::?'a ⇒ bool) (if ?Q::bool then ?x::?'a else (?y::?'a)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*top goal: ‹⋀x::'a process_name. ⟦∀(x::'a process_name) xa::'a process_name. x ≠ xa ⟶ WL x (s::'a process_name ⇒ ('b, 'a, 'c, 'd) Model.local_state) ∩ WL xa s = {}; ∀(x::'a process_name) (xa::'a process_name) xb::'d. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀(x::'a process_name) (xa::'d) xb::bool. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; (r::'d) ∈ gc_W s; ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀(x::'a process_name) xa::'d. xa ∈ W (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀(x::'a process_name) xa::'d. (∃y::('b, 'c, 'd) object. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀(x::'a process_name) xa::'d. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj::('b, 'c, 'd) object. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); r ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈)); (g::'d) = r⟧ ⟹ False› and 1 goal remains*)\n apply blast\n (*discuss goal 3*)\n apply (simp add: WL_def (*‹WL ?p = (λs. W (s ?p) ∪ ghost_honorary_grey (s ?p))›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: if_splits (*‹?P (if ?Q then ?x else ?y) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀x. ⟦∀x xa. x ≠ xa ⟶ WL x s ∩ WL xa s = {}; ∀x xa xb. xb ∈ ghost_honorary_grey (s x) ⟶ xb ∉ W (s xa); sys_ghost_honorary_grey s = {}; ∀x xa xb. mw_Mark xa xb ∈ set (sys_mem_store_buffers x s) ⟶ xb = sys_fM s ∧ xa ∈ ghost_honorary_grey (s x) ∧ sys_mem_lock s = Some x ∧ white xa s ∧ filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]; r ∈ gc_W s; ∀x xa. (∃y. sys_heap s xa = Some y) ∨ xa ∉ W (s x); ∀x xa. xa ∈ W (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ obj_mark obj = sys_fM s); ∀x xa. (∃y. sys_heap s xa = Some y) ∨ sys_mem_lock s = Some x ∨ xa ∉ ghost_honorary_grey (s x); ∀x xa. xa ∈ ghost_honorary_grey (s x) ⟶ (∀obj. sys_heap s xa = Some obj ⟶ sys_mem_lock s = Some x ∨ obj_mark obj = sys_fM s); g ≠ r; g ∈ WL x s⟧ ⟹ ∃x. g ∈ WL x (s(gc := s gc⦇W := gc_W s - {r}⦈))›*)\n apply metis\n (*proven 3 subgoals*)\n(*proven 2 subgoals*) .\n(* FIXME\napply (auto simp: black_def grey_def WL_def split: obj_at_splits)\napply metis+\ndone\n*)\n\nlemma no_black_refs_alloc[simp]:\n \"⟦ heap (s sys) r' = None; no_black_refs s ⟧\n ⟹ no_black_refs (s(mutator m' := s (mutator m')⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = Map.empty, obj_payload = Map.empty⦈)⦈))\n ⟷ fl ≠ sys_fM s ∨ grey r' s\"\nunfolding no_black_refs_def\n(*goal: ‹⟦sys_heap s r' = None; ∀x. ¬ black x s⟧ ⟹ (∀x. ¬ black x (s(mutator m' := s (mutator m')⦇roots := roots'⦈, sys := s sys⦇heap := (sys_heap s)(r' ↦ ⦇obj_mark = fl, obj_fields = λx. None, obj_payload = λx. None⦈)⦈))) = (fl ≠ sys_fM s ∨ grey r' s)›*)\nby simp\n\nlemma no_black_refs_mo_co_mark[simp]:\n \"⟦ ghost_honorary_grey (s p) = {}; white r s ⟧\n ⟹ no_black_refs (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ⟷ no_black_refs s\"\nunfolding no_black_refs_def\n(*goal: ‹⟦ghost_honorary_grey (s p) = {}; white r s⟧ ⟹ (∀x. ¬ black x (s(p := s p⦇ghost_honorary_grey := {r}⦈))) = (∀x. ¬ black x s)›*)\nby auto\n\nlemma grey_protects_white_mark[simp]:\n assumes ghg: \"ghost_honorary_grey (s p) = {}\"\n shows \"(∃g. (g grey_protects_white w) (s(p := s p⦇ ghost_honorary_grey := {r} ⦈)))\n ⟷ (∃g'. (g' grey_protects_white w) s) ∨ (r has_white_path_to w) s\" (is \"?lhs ⟷ ?rhs\")\nproof (standard)\n (*goals:\n 1. ‹∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ⟹ (∃g'. (g' grey_protects_white w) s) ∨ (r has_white_path_to w) s›\n 2. ‹(∃g'. (g' grey_protects_white w) s) ∨ (r has_white_path_to w) s ⟹ ∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n assume \"?lhs\" (*‹∃g::'a. (g grey_protects_white (w::'a)) ((s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state)(p::'c process_name := s p⦇ghost_honorary_grey := {r::'a}⦈))›*)\n then obtain g where \"(g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))\"\n (*goal: ‹(⋀g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ⟹ thesis) ⟹ thesis›*)\n by blast\n from this (*‹(g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*) ghg (*‹ghost_honorary_grey (s p) = {}›*) show \"?rhs\"\n apply induct\n (*goals:\n 1. ‹⋀x. ⟦grey x (s(p := s p⦇ghost_honorary_grey := {r}⦈)); ghost_honorary_grey (s p) = {}⟧ ⟹ (∃g'. (g' grey_protects_white x) s) ∨ (r has_white_path_to x) s›\n 2. ‹⋀x y z. ⟦(x has_white_path_to y) (s(p := s p⦇ghost_honorary_grey := {r}⦈)); ghost_honorary_grey (s p) = {} ⟹ (∃g'. (g' grey_protects_white y) s) ∨ (r has_white_path_to y) s; (y points_to z) (s(p := s p⦇ghost_honorary_grey := {r}⦈)); white z (s(p := s p⦇ghost_honorary_grey := {r}⦈)); ghost_honorary_grey (s p) = {}⟧ ⟹ (∃g'. (g' grey_protects_white z) s) ∨ (r has_white_path_to z) s›\n discuss goal 1*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))[1])\n (*proven 2 subgoals*) .\nnext\n (*goal: ‹(∃g'. (g' grey_protects_white w) s) ∨ (r has_white_path_to w) s ⟹ ∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n assume \"?rhs\" (*‹(∃g'::'a. (g' grey_protects_white (w::'a)) (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state)) ∨ ((r::'a) has_white_path_to w) s›*)\n then show \"?lhs\"\n proof (safe)\n (*goals:\n 1. ‹⋀g'. (g' grey_protects_white w) s ⟹ ∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›\n 2. ‹(r has_white_path_to w) s ⟹ ∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n fix g\n assume \"(g grey_protects_white w) s\" (*‹((g::'a) grey_protects_white (w::'a)) (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state)›*)\n from this (*‹(g grey_protects_white w) s›*) ghg (*‹ghost_honorary_grey ((s::'c::type process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) (p::'c::type process_name)) = {}›*) show \"?thesis\"\n (*goal: ‹∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n apply induct\n (*goal: ‹∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n apply force\n (*top goal: ‹⋀x. ⟦grey x s; ghost_honorary_grey (s p) = {}⟧ ⟹ ∃g. (g grey_protects_white x) (s(p := s p⦇ghost_honorary_grey := {r}⦈))› and 1 goal remains*)\n unfolding grey_protects_white_def\n (*goal: ‹⋀x y z. ⟦(x has_white_path_to y) s; ghost_honorary_grey (s p) = {} ⟹ ∃g. grey g (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ∧ (g has_white_path_to y) (s(p := s p⦇ghost_honorary_grey := {r}⦈)); (y points_to z) s; white z s; ghost_honorary_grey (s p) = {}⟧ ⟹ ∃g. grey g (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ∧ (g has_white_path_to z) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n by (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n next\n (*goal: ‹(r has_white_path_to w) s ⟹ ∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n assume \"(r has_white_path_to w) s\" (*‹((r::'a) has_white_path_to (w::'a)) (s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state)›*)\n with ghg (*‹ghost_honorary_grey ((s::'c process_name ⇒ ('b, 'c, 'd, 'a) Model.local_state) (p::'c process_name)) = {}›*) show \"?thesis\"\n (*goal: ‹∃g. (g grey_protects_white w) (s(p := s p⦇ghost_honorary_grey := {r}⦈))›*)\n unfolding grey_protects_white_def has_white_path_to_def\n (*goal: ‹∃g. grey g (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ∧ (λx y. (x points_to y) (s(p := s p⦇ghost_honorary_grey := {r}⦈)) ∧ white y (s(p := s p⦇ghost_honorary_grey := {r}⦈)))⇧*⇧* g w›*)\n by (auto simp: fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*))\n qed\nqed\n\nlemma valid_refs_inv_dequeue_Mutate:\n fixes s :: \"('field, 'mut, 'payload, 'ref) lsts\"\n assumes vri: \"valid_refs_inv s\"\n assumes sb: \"sys_mem_store_buffers (mutator m') s = mw_Mutate r f opt_r' # ws\"\n shows \"valid_refs_inv (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_fields := (obj_fields obj)(f := opt_r')⦈) (sys_heap s r)),\n mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := ws)⦈))\" (is \"valid_refs_inv ?s'\")\nproof(rule valid_refs_invI)\n fix m\n let ?root = \"λm x. mut_m.root m x ❙∨ grey x\"\n fix x y assume xy: \"(x reaches y) ?s'\" and x: \"?root m x ?s'\"\n from xy have \"(∃m x. ?root m x s ∧ (x reaches y) s) ∧ valid_ref y ?s'\"\n unfolding reaches_def proof induct\n case base with x sb vri show ?case\n apply -\n apply (subst obj_at_fun_upd)\n apply (auto simp: mut_m.tso_store_refs_def reaches_def fun_upd_apply split: if_splits intro: valid_refs_invD(5)[where m=m])\n apply (metis list.set_intros(2) rtranclp.rtrancl_refl)\n done (* FIXME rules *)\n next\n case (step y z)\n with sb vri show ?case\n apply -\n apply (subst obj_at_fun_upd, clarsimp simp: fun_upd_apply)\n apply (subst (asm) obj_at_fun_upd, fastforce simp: fun_upd_apply)\n apply (clarsimp simp: points_to_Mutate fun_upd_apply)\n apply (fastforce elim: rtranclp.intros(2) simp: mut_m.tso_store_refs_def reaches_def fun_upd_apply intro: exI[where x=m'] valid_refs_invD(5)[where m=m'])\n done\n qed\n then show \"valid_ref y ?s'\" by blast\nqed\n\nlemma valid_refs_inv_dequeue_Mutate_Payload:\n notes if_split_asm[split del]\n fixes s :: \"('field, 'mut, 'payload, 'ref) lsts\"\n assumes vri: \"valid_refs_inv s\"\n assumes sb: \"sys_mem_store_buffers (mutator m') s = mw_Mutate_Payload r f pl # ws\"\n shows \"valid_refs_inv (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)),\n mem_store_buffers := (mem_store_buffers (s sys))(mutator m := ws)⦈))\" (is \"valid_refs_inv ?s'\")\napply (rule valid_refs_invI (*‹(⋀m x y. ⟦(x reaches y) ?s; x ∈ roots (?s (mutator m)) ∪ ghost_honorary_root (?s (mutator m)) ∪ mut_m.tso_store_refs m ?s ∨ grey x ?s⟧ ⟹ obj_at (λs. True) y ?s) ⟹ valid_refs_inv ?s›*))\n(*goal: ‹valid_refs_inv ((s::'mut process_name ⇒ ('field, 'mut, 'payload, 'ref) Model.local_state)(sys := s sys⦇heap := (sys_heap s)(r::'ref := map_option (λobj::('field, 'payload, 'ref) object. obj⦇obj_payload := (obj_payload obj)(f::'field := pl::'payload option)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator (m::'mut) := ws::('field, 'payload, 'ref) mem_store_action list)⦈))›*)\nusing assms (*‹valid_refs_inv s› ‹sys_mem_store_buffers (mutator m') s = mw_Mutate_Payload r f pl # ws›*) apply (clarsimp simp: valid_refs_invD (*‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ roots (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ mut_m.tso_store_refs ?m ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?w ∈ set (sys_mem_store_buffers (mutator ?m) ?s); ?x ∈ store_refs ?w; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj› ‹⟦grey ?x ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?x ?s› ‹⟦mut_m.reachable ?m ?x ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?x = Some obj› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s› ‹⟦?x ∈ ghost_honorary_root (?s (mutator ?m)); (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj. sys_heap ?s ?y = Some obj›*) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) split: obj_at_splits (*‹?Q (obj_at ?P ?r ?s) = ((sys_heap ?s ?r = None ⟶ ?Q False) ∧ (∀obj. sys_heap ?s ?r = Some obj ⟶ ?Q (?P obj)))› ‹?Q (obj_at ?P ?r ?s) = (¬ (sys_heap ?s ?r = None ∧ ¬ ?Q False ∨ (∃obj. sys_heap ?s ?r = Some obj ∧ ¬ ?Q (?P obj))))›*) mem_store_action.splits (*‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = ((∀x11 x12. ?mem_store_action = mw_Mark x11 x12 ⟶ ?P (?f1.0 x11 x12)) ∧ (∀x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ⟶ ?P (?f2.0 x21 x22 x23)) ∧ (∀x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ⟶ ?P (?f3.0 x31 x32 x33)) ∧ (∀x4. ?mem_store_action = mw_fA x4 ⟶ ?P (?f4.0 x4)) ∧ (∀x5. ?mem_store_action = mw_fM x5 ⟶ ?P (?f5.0 x5)) ∧ (∀x6. ?mem_store_action = mw_Phase x6 ⟶ ?P (?f6.0 x6)))› ‹?P (case ?mem_store_action of mw_Mark x xa ⇒ ?f1.0 x xa | mw_Mutate x xa xb ⇒ ?f2.0 x xa xb | mw_Mutate_Payload x xa xb ⇒ ?f3.0 x xa xb | mw_fA x ⇒ ?f4.0 x | mw_fM x ⇒ ?f5.0 x | mw_Phase x ⇒ ?f6.0 x) = (¬ ((∃x11 x12. ?mem_store_action = mw_Mark x11 x12 ∧ ¬ ?P (?f1.0 x11 x12)) ∨ (∃x21 x22 x23. ?mem_store_action = mw_Mutate x21 x22 x23 ∧ ¬ ?P (?f2.0 x21 x22 x23)) ∨ (∃x31 x32 x33. ?mem_store_action = mw_Mutate_Payload x31 x32 x33 ∧ ¬ ?P (?f3.0 x31 x32 x33)) ∨ (∃x4. ?mem_store_action = mw_fA x4 ∧ ¬ ?P (?f4.0 x4)) ∨ (∃x5. ?mem_store_action = mw_fM x5 ∧ ¬ ?P (?f5.0 x5)) ∨ (∃x6. ?mem_store_action = mw_Phase x6 ∧ ¬ ?P (?f6.0 x6))))›*))\n(*goal: ‹⋀ma x y. ⟦(x reaches y) (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m := ws)⦈)); x ∈ roots ((s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m := ws)⦈)) (mutator ma)) ∪ ghost_honorary_root ((s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m := ws)⦈)) (mutator ma)) ∪ mut_m.tso_store_refs ma (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m := ws)⦈)) ∨ grey x (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m := ws)⦈))⟧ ⟹ obj_at (λs. True) y (s(sys := s sys⦇heap := (sys_heap s)(r := map_option (λobj. obj⦇obj_payload := (obj_payload obj)(f := pl)⦈) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m := ws)⦈))›*)\napply auto\n(*goals:\n 1. ‹⋀ma x y. ⟦(x reaches y) s; valid_refs_inv s; sys_mem_store_buffers (mutator m') s = mw_Mutate_Payload r f pl # ws; x ∈ (if m = ma then ⋃ (store_refs ` set ws) else mut_m.tso_store_refs ma s)⟧ ⟹ ∃ya. sys_heap s y = Some ya›\n 2. ‹⋀x y. ⟦(x reaches y) s; valid_refs_inv s; sys_mem_store_buffers (mutator m') s = mw_Mutate_Payload r f pl # ws; grey x s⟧ ⟹ ∃ya. sys_heap s y = Some ya›\ndiscuss goal 1*)\n apply (metis (mono_tags, lifting) UN_insert (*‹⋃ ((?B::?'b ⇒ ?'a set) ` insert (?a::?'b) (?A::?'b set)) = ?B ?a ∪ ⋃ (?B ` ?A)›*) Un_iff (*‹((?c::?'a) ∈ (?A::?'a set) ∪ (?B::?'a set)) = (?c ∈ ?A ∨ ?c ∈ ?B)›*) list.simps( (*‹set ((?x21.0::?'a) # (?x22.0::?'a list)) = insert ?x21.0 (set ?x22.0)›*) 15) mut_m.tso_store_refs_def (*‹mut_m.tso_store_refs (?m::?'mut) = (λs::?'mut process_name ⇒ (?'field, ?'mut, ?'payload, ?'ref) Model.local_state. ⋃ (store_refs ` set (sys_mem_store_buffers (mutator ?m) s)))›*) valid_refs_invD( (*‹⟦(?x::?'a) ∈ mut_m.tso_store_refs (?m::?'c) (?s::?'c process_name ⇒ (?'b, ?'c, ?'d, ?'a) Model.local_state); (?x reaches (?y::?'a)) ?s; valid_refs_inv ?s⟧ ⟹ ∃obj::(?'b, ?'d, ?'a) object. sys_heap ?s ?y = Some obj›*) 4))\n(*discuss goal 2*)\n apply (metis case_optionE (*‹⟦case ?x of None ⇒ ?P | Some y ⇒ ?Q y; ⟦?x = None; ?P⟧ ⟹ ?thesis; ⋀y. ⟦?x = Some y; ?Q y⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) obj_at_def (*‹obj_at ?P ?r ≡ λs. case sys_heap s ?r of None ⇒ False | Some obj ⇒ ?P obj›*) valid_refs_invD( (*‹⟦grey ?x ?s; (?x reaches ?y) ?s; valid_refs_inv ?s⟧ ⟹ obj_at (λs. True) ?y ?s›*) 7))\n(*proven 2 subgoals*) .\n\n(*<*)\n\nend\n(*>*)\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/ConcurrentGC/Global_Invariants_Lemmas.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"4ea1e5e6692fa2030b4eb70521c06ddf447e61b364d6fa64a18cf5ec294ed573\"\n}"}}},{"rowIdx":175,"cells":{"text":{"kind":"string","value":"theory Preliminary_Library\n imports Main \"HOL-Library.Word\" \"Word_Lib.Word_Lib_Sumo\" \"HOL-Library.Countable\"\nbegin\n\nsection ‹Preliminary Theories›\ntext ‹In this subsection, we provide the type and value system used by the CHERI-C Memory Model.\n We also provide proofs for the conversion between large words (i.e. bits) and a list of bytes.\n For primitive bytes that are not $U8_\\tau$ or $S8_\\tau$, we need to be able to convert between\n their normal representation and list of bytes so that storing values work as intended. \n The high-level detail is given in the paper~\\cite{park_2022}.›\nsubsection ‹Types and Values›\ntext ‹We first formalise the capability type. We first define \\textit{memory capabilities} as a record,\n then we define \\textit{tagged capabilities} by extending the record. We state the class \n comp\\_countable for future work, but countable is sufficient for the block\\_id type.\n For the permissions, we present only those used by the memory model.›\nclass comp_countable = countable + zero + ord\n\nrecord ('a :: comp_countable) mem_capability =\n block_id :: \"'a\" (* block_id *)\n offset :: int (* offset/address *)\n base :: nat (* metadata *)\n len :: nat\n perm_load :: bool\n perm_cap_load :: bool\n perm_store :: bool\n perm_cap_store :: bool\n perm_cap_store_local :: bool\n perm_global :: bool\n\nrecord ('a :: comp_countable) capability = \"'a mem_capability\" + \n tag :: bool (* tag *)\n\ntext ‹cctype corresponds to $\\tau$ in the paper~\\cite{park_2022}, where $\\tau$ is the type system.›\ndatatype cctype = \n Uint8\n | Sint8\n | Uint16\n | Sint16\n | Uint32\n | Sint32\n | Uint64\n | Sint64\n | Cap\n\ntext ‹$'a\\ ccval$ corresponds to $\\mathcal{V}_\\mathcal{C}$ in the paper~\\cite{park_2022}. $'a$ in \n this instance must be countable. ›\ndatatype 'a ccval = \n Uint8_v \"8 word\"\n | Sint8_v \"8 sword\"\n | Uint16_v \"16 word\"\n | Sint16_v \"16 sword\"\n | Uint32_v \"32 word\"\n | Sint32_v \"32 sword\"\n | Uint64_v \"64 word\"\n | Sint64_v \"64 sword\"\n | Cap_v \"'a capability\"\n | Cap_v_frag \"'a capability\" \"nat\"\n | Undef\n\ntext ‹memval\\_type infers the type of a value. ›\nfun memval_type :: \"'a ccval ⇒ cctype\"\n where\n \"memval_type v = (case v of \n Uint8_v _ ⇒ Uint8\n | Sint8_v _ ⇒ Sint8\n | Uint16_v _ ⇒ Uint16\n | Sint16_v _ ⇒ Sint16\n | Uint32_v _ ⇒ Uint32\n | Sint32_v _ ⇒ Sint32\n | Uint64_v _ ⇒ Uint64\n | Sint64_v _ ⇒ Sint64\n | Cap_v _ ⇒ Cap\n | Cap_v_frag _ _ ⇒ Uint8)\"\n\n\nsubsection ‹Primitive Value Conversion and Cast Proof›\ntext ‹In this subsection, we provide proofs for the conversion between words and list of words.\n We also provide proofs that casting primitive values is correct. These will be used by the\n \\texttt{load} and \\texttt{store} operations in the memory model.›\nabbreviation encode_u8 :: \"nat ⇒ 8 word\"\n where\n \"encode_u8 x ≡ word_of_nat x\"\n\nabbreviation decode_u8 :: \"8 word ⇒ nat\"\n where\n \"decode_u8 b ≡ unat b\"\n\nabbreviation encode_u8_list :: \"8 word ⇒ 8 word list\"\n where\n \"encode_u8_list w ≡ [w]\"\n\nabbreviation decode_u8_list :: \"8 word list ⇒ 8 word\"\n where\n \"decode_u8_list ls ≡ hd ls\"\n\nlemma encode_decode_u8_list:\n \"ls = [b] ⟹ ls = encode_u8_list (decode_u8_list ls)\"\n by simp\n\nlemma decode_encode_u8_list:\n \"w = decode_u8_list (encode_u8_list w)\"\n by simp\n\nlemma encode_decode_u8:\n \"w = encode_u8 (decode_u8 w)\"\n by simp\n\nlemma decode_encode_u8:\n assumes \"i ≤ 2 ^ LENGTH(8) - 1\"\n shows \"i = decode_u8 (encode_u8 i)\"\n by (metis assms (*‹i ≤ 2 ^ LENGTH(8) - 1›*) le_unat_uoi (*‹?y ≤ unat ?z ⟹ unat (word_of_nat ?y) = ?y›*) unat_minus_one_word (*‹unat (- 1) = 2 ^ LENGTH(?'a) - 1›*))\n \nabbreviation u64_split :: \"64 word ⇒ 32 word list\"\n where\n \"u64_split x ≡ (word_rsplit :: 64 word ⇒ 32 word list) x\"\n\nabbreviation u32_split :: \"32 word ⇒ 16 word list\"\n where\n \"u32_split x ≡ (word_rsplit :: 32 word ⇒ 16 word list) x\"\n\nabbreviation u16_split :: \"16 word ⇒ 8 word list\"\n where\n \"u16_split x ≡ (word_rsplit :: 16 word ⇒ 8 word list) x\"\n\nabbreviation cat_u16 :: \"8 word list ⇒ 16 word\"\n where\n \"cat_u16 x ≡ (word_rcat :: 8 word list ⇒ 16 word) x\"\n\nabbreviation encode_u16 :: \"nat ⇒ 8 word list\"\n where \n \"encode_u16 x ≡ u16_split (word_of_nat x)\"\n\nabbreviation decode_u16 :: \"8 word list ⇒ nat\"\n where\n \"decode_u16 x ≡ unat (cat_u16 x)\"\n\nlemma flatten_u16_length:\n \"length (u16_split vs) = 2\"\n by (simp add: length_word_rsplit_even_size (*‹⟦?n = LENGTH(?'a); size ?w = ?m * ?n⟧ ⟹ length (word_rsplit ?w) = ?m›*) wsst_TYs( (*‹size ?w = LENGTH(?'a)›*) 3))\n\nlemma rsplit_rcat_eq:\n assumes \"LENGTH(('b::len)) mod LENGTH(('a::len)) = 0\"\n and \"length w = LENGTH('b) div LENGTH('a)\"\n shows \"(word_rsplit :: 'b word ⇒ 'a word list) ((word_rcat :: 'a word list ⇒ 'b word) w) = w\"\n by (simp add: assms (*‹LENGTH('b) mod LENGTH('a) = 0› ‹length w = LENGTH('b) div LENGTH('a)›*) mod_0_imp_dvd (*‹?a mod ?b = 0 ⟹ ?b dvd ?a›*) size_word.rep_eq (*‹size ?x = LENGTH(?'a)›*) word_rsplit_rcat_size (*‹size (word_rcat ?ws) = length ?ws * LENGTH(?'a) ⟹ word_rsplit (word_rcat ?ws) = ?ws›*))\n\nlemma rsplit_rcat_u16_eq:\n assumes \"w = [a1, a2]\"\n shows \"(word_rsplit :: 16 word ⇒ 8 word list) ((word_rcat :: 8 word list ⇒ 16 word) w) = w\"\nproof (-)\n (*goal: ‹u16_split (cat_u16 (w::8 word list)) = w›*)\n have l1: \"length w * 8 = 16\"\n using assms (*‹(w::8 word list) = (a1::8 word) # encode_u8_list (a2::8 word)›*) by clarsimp\n moreover have l2: \"size ((word_rcat :: 8 word list ⇒ 16 word) w) = 16\"\n using assms (*‹w = a1 # encode_u8_list a2›*) by (simp add: size_word.rep_eq (*‹size ?x = LENGTH(?'a)›*))\n from l1 (*‹length (w::8 word list) * (8::nat) = (16::nat)›*) l2 (*‹size (cat_u16 w) = 16›*) have \"length w * 8 = size ((word_rcat :: 8 word list ⇒ 16 word) w)\"\n by argo\n thus \"?thesis\"\n (*goal: ‹u16_split (cat_u16 w) = w›*)\n by (metis l1 (*‹length (w::8 word list) * (8::nat) = (16::nat)›*) l2 (*‹size (cat_u16 (w::8 word list)) = (16::nat)›*) len8 (*‹len_of (?x::8 itself) = (8::nat)›*) word_rsplit_rcat_size (*‹size (word_rcat (?ws::?'a word list)) = length ?ws * LENGTH(?'a) ⟹ word_rsplit (word_rcat ?ws) = ?ws›*))\nqed\n \n\nlemma encode_decode_u16:\n assumes \"w = [a, b]\"\n shows \"w = encode_u16 (decode_u16 w)\"\n by (simp add: assms (*‹w = a # encode_u8_list b›*) rsplit_rcat_eq (*‹⟦LENGTH(?'b) mod LENGTH(?'a) = 0; length ?w = LENGTH(?'b) div LENGTH(?'a)⟧ ⟹ word_rsplit (word_rcat ?w) = ?w›*))\n\nlemma cat_flatten_u16_eq:\n \"cat_u16 (u16_split w) = w\" \n by (simp add: word_rcat_rsplit (*‹word_rcat (word_rsplit ?w) = ?w›*))\n\nlemma decode_encode_u16:\n assumes \"i ≤ 2 ^ LENGTH(16) - 1\"\n shows \"i = decode_u16 (encode_u16 i)\" \n by (metis assms (*‹i ≤ 2 ^ LENGTH(16) - 1›*) cat_flatten_u16_eq (*‹cat_u16 (u16_split ?w) = ?w›*) le_unat_uoi (*‹?y ≤ unat ?z ⟹ unat (word_of_nat ?y) = ?y›*) unat_minus_one_word (*‹unat (- 1) = 2 ^ LENGTH(?'a) - 1›*))\n\nabbreviation flatten_u32 :: \"32 word ⇒ 8 word list\"\n where\n \"flatten_u32 x ≡ (word_rsplit :: 32 word ⇒ 8 word list) x\"\n\nabbreviation cat_u32 :: \"8 word list ⇒ 32 word\"\n where\n \"cat_u32 x ≡ (word_rcat :: 8 word list ⇒ 32 word) x\"\n\nabbreviation encode_u32 :: \"nat ⇒ 8 word list\"\n where\n \"encode_u32 x ≡ flatten_u32 (word_of_nat x)\"\n\nabbreviation decode_u32 :: \"8 word list ⇒ nat\"\n where\n \"decode_u32 i ≡ unat (cat_u32 i)\"\n\nlemma flatten_u32_length:\n \"length (flatten_u32 vs) = 4\"\n by (simp add: length_word_rsplit_even_size (*‹⟦?n = LENGTH(?'a); size ?w = ?m * ?n⟧ ⟹ length (word_rsplit ?w) = ?m›*) wsst_TYs( (*‹size ?w = LENGTH(?'a)›*) 3))\n\nlemma rsplit_rcat_u32_eq:\n assumes \"w = [a1, a2, b1, b2]\"\n shows \"(word_rsplit :: 32 word ⇒ 8 word list) ((word_rcat :: 8 word list ⇒ 32 word) w) = w\" \n using rsplit_rcat_eq (*‹⟦LENGTH(?'b) mod LENGTH(?'a) = 0; length ?w = LENGTH(?'b) div LENGTH(?'a)⟧ ⟹ word_rsplit (word_rcat ?w) = ?w›*) assms (*‹w = a1 # a2 # b1 # encode_u8_list b2›*) by force\n\nlemma encode_decode_u32:\n assumes \"w = [a1, a2, b1, b2]\"\n shows \"w = encode_u32 (decode_u32 w)\" \n using assms (*‹(w::8 word list) = (a1::8 word) # (a2::8 word) # (b1::8 word) # encode_u8_list (b2::8 word)›*) by (simp add: rsplit_rcat_u32_eq (*‹?w = ?a1.0 # ?a2.0 # ?b1.0 # encode_u8_list ?b2.0 ⟹ flatten_u32 (cat_u32 ?w) = ?w›*))\n\nlemma cat_flatten_u32_eq:\n \"cat_u32 (flatten_u32 w) = w\" \n by (simp add: word_rcat_rsplit (*‹word_rcat (word_rsplit ?w) = ?w›*))\n\nlemma decode_encode_u32:\n assumes \"i ≤ 2 ^ LENGTH(32) - 1\"\n shows \"i = decode_u32 (encode_u32 i)\" \n by (metis assms (*‹i ≤ 2 ^ LENGTH(32) - 1›*) le_unat_uoi (*‹?y ≤ unat ?z ⟹ unat (word_of_nat ?y) = ?y›*) unat_minus_one_word (*‹unat (- 1) = 2 ^ LENGTH(?'a) - 1›*) word_rcat_rsplit (*‹word_rcat (word_rsplit ?w) = ?w›*))\n\nabbreviation flatten_u64 :: \"64 word ⇒ 8 word list\"\n where\n \"flatten_u64 x ≡ (word_rsplit :: 64 word ⇒ 8 word list) x\"\n\nabbreviation cat_u64 :: \"8 word list ⇒ 64 word\"\n where\n \"cat_u64 x ≡ word_rcat x\"\n\nabbreviation encode_u64 :: \"nat ⇒ 8 word list\"\n where\n \"encode_u64 x ≡ flatten_u64 (word_of_nat x)\"\n\nabbreviation decode_u64 :: \"8 word list ⇒ nat\"\n where\n \"decode_u64 x ≡ unat (cat_u64 x)\"\n\nlemma flatten_u64_length:\n \"length (flatten_u64 vs) = 8\"\n by (simp add: length_word_rsplit_even_size (*‹⟦?n = LENGTH(?'a); size ?w = ?m * ?n⟧ ⟹ length (word_rsplit ?w) = ?m›*) wsst_TYs( (*‹size ?w = LENGTH(?'a)›*) 3))\n\nlemma encode_decode_u64:\n assumes \"w = [a1, a2, b1, b2, c1, c2, d1, d2]\"\n shows \"w = encode_u64 (decode_u64 w)\"\n using assms (*‹w = a1 # a2 # b1 # b2 # c1 # c2 # d1 # encode_u8_list d2›*) by (simp add: rsplit_rcat_eq (*‹⟦LENGTH(?'b) mod LENGTH(?'a) = 0; length ?w = LENGTH(?'b) div LENGTH(?'a)⟧ ⟹ word_rsplit (word_rcat ?w) = ?w›*))\n\nlemma cat_flatten_u64_eq:\n \"cat_u64 (flatten_u64 w) = w\" \n by (simp add: word_rcat_rsplit (*‹word_rcat (word_rsplit ?w) = ?w›*))\n\nlemma decode_encode_u64:\n assumes \"i ≤ 2 ^ LENGTH(64) - 1\"\n shows \"i = decode_u64 (encode_u64 i)\" \n by (metis assms (*‹i ≤ 2 ^ LENGTH(64) - 1›*) le_unat_uoi (*‹?y ≤ unat ?z ⟹ unat (word_of_nat ?y) = ?y›*) unat_minus_one_word (*‹unat (- 1) = 2 ^ LENGTH(?'a) - 1›*) word_rcat_rsplit (*‹word_rcat (word_rsplit ?w) = ?w›*))\n\nabbreviation encode_s8 :: \"int ⇒ 8 sword\"\n where\n \"encode_s8 x ≡ word_of_int x\"\n\nabbreviation decode_s8 :: \"8 sword ⇒ int\"\n where\n \"decode_s8 b ≡ sint b\"\n\nabbreviation encode_s8_list :: \"8 sword ⇒ 8 word list\"\n where\n \"encode_s8_list w ≡ [SCAST(8 signed → 8) w]\"\n\nabbreviation decode_s8_list :: \"8 word list ⇒ 8 sword\"\n where\n \"decode_s8_list ls ≡ UCAST(8 → 8 signed) (hd ls)\"\n\nlemma encode_decode_s8_list:\n \"ls = [b] ⟹ ls = encode_s8_list (decode_s8_list ls)\"\n by simp\n\nlemma decode_encode_s8_list:\n \"w = decode_s8_list (encode_s8_list w)\"\n by simp\n\nlemma encode_decode_s8:\n \"w = encode_s8 (decode_s8 w)\"\n by simp\n\nlemma decode_encode_s8:\n assumes \"- (2 ^ (LENGTH(8) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(8) - 1)\"\n shows \"i = decode_s8 (encode_s8 i)\" \n by (metis assms (*‹- (2 ^ (LENGTH(8) - 1)) ≤ i› ‹i < 2 ^ (LENGTH(8) - 1)›*) More_Word.sint_of_int_eq (*‹⟦- (2 ^ (LENGTH(?'a) - 1)) ≤ ?x; ?x < 2 ^ (LENGTH(?'a) - 1)⟧ ⟹ sint (word_of_int ?x) = ?x›*) len_signed (*‹len_of ?x = LENGTH(?'a)›*))\n \nabbreviation s64_split :: \"64 sword ⇒ 32 word list\"\n where\n \"s64_split x ≡ (word_rsplit :: 64 sword ⇒ 32 word list) x\"\n\nabbreviation s32_split :: \"32 sword ⇒ 16 word list\"\n where\n \"s32_split x ≡ (word_rsplit :: 32 sword ⇒ 16 word list) x\"\n\nabbreviation s16_split :: \"16 sword ⇒ 8 word list\"\n where\n \"s16_split x ≡ (word_rsplit :: 16 sword ⇒ 8 word list) x\"\n\nabbreviation cat_s16 :: \"8 word list ⇒ 16 sword\"\n where\n \"cat_s16 x ≡ (word_rcat :: 8 word list ⇒ 16 sword) x\"\n\nabbreviation encode_s16 :: \"int ⇒ 8 word list\"\n where \n \"encode_s16 x ≡ s16_split (word_of_int x)\"\n\nabbreviation decode_s16 :: \"8 word list ⇒ int\"\n where\n \"decode_s16 x ≡ sint (cat_s16 x)\"\n\nlemma flatten_s16_length:\n \"length (s16_split vs) = 2\"\n by (simp add: length_word_rsplit_even_size (*‹⟦?n = LENGTH(?'a); size ?w = ?m * ?n⟧ ⟹ length (word_rsplit ?w) = ?m›*) wsst_TYs( (*‹size ?w = LENGTH(?'a)›*) 3))\n\nlemma rsplit_rcat_s16_eq:\n assumes \"w = [a1, a2]\"\n shows \"(word_rsplit :: 16 sword ⇒ 8 word list) ((word_rcat :: 8 word list ⇒ 16 sword) w) = w\"\nproof (-)\n (*goal: ‹s16_split (cat_s16 w) = w›*)\n have l1: \"length w * 8 = 16\"\n using assms (*‹w = a1 # encode_u8_list a2›*) by clarsimp\n moreover have l2: \"size ((word_rcat :: 8 word list ⇒ 16 sword) w) = 16\"\n using assms (*‹w = a1 # encode_u8_list a2›*) by (simp add: size_word.rep_eq (*‹size (?x::?'a word) = LENGTH(?'a)›*))\n from l1 (*‹length w * 8 = 16›*) l2 (*‹size (cat_s16 (w::8 word list)) = (16::nat)›*) have \"length w * 8 = size ((word_rcat :: 8 word list ⇒ 16 sword) w)\"\n by argo\n thus \"?thesis\"\n (*goal: ‹s16_split (cat_s16 (w::8 word list)) = w›*)\n by (simp add: word_rsplit_rcat_size (*‹size (word_rcat ?ws) = length ?ws * LENGTH(?'a) ⟹ word_rsplit (word_rcat ?ws) = ?ws›*))\nqed\n \n\nlemma encode_decode_s16:\n assumes \"w = [a, b]\"\n shows \"w = encode_s16 (decode_s16 w)\"\n by (simp add: assms (*‹w = a # encode_u8_list b›*) rsplit_rcat_eq (*‹⟦LENGTH(?'b) mod LENGTH(?'a) = 0; length ?w = LENGTH(?'b) div LENGTH(?'a)⟧ ⟹ word_rsplit (word_rcat ?w) = ?w›*))\n\nlemma cat_flatten_s16_eq:\n \"cat_s16 (s16_split w) = w\" \n by (simp add: word_rcat_rsplit (*‹word_rcat (word_rsplit ?w) = ?w›*))\n\nlemma decode_encode_s16:\n assumes \"- (2 ^ (LENGTH(16) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(16) - 1)\"\n shows \"i = decode_s16 (encode_s16 i)\" \n by (metis assms (*‹- (2 ^ (LENGTH(16) - 1)) ≤ i› ‹i < 2 ^ (LENGTH(16) - 1)›*) cat_flatten_s16_eq (*‹cat_s16 (s16_split ?w) = ?w›*) len_signed (*‹len_of ?x = LENGTH(?'a)›*) sint_of_int_eq (*‹⟦- (2 ^ (LENGTH(?'a) - 1)) ≤ ?x; ?x < 2 ^ (LENGTH(?'a) - 1)⟧ ⟹ sint (word_of_int ?x) = ?x›*))\n\nabbreviation flatten_s32 :: \"32 sword ⇒ 8 word list\"\n where\n \"flatten_s32 x ≡ (word_rsplit :: 32 sword ⇒ 8 word list) x\"\n\nabbreviation cat_s32 :: \"8 word list ⇒ 32 sword\"\n where\n \"cat_s32 x ≡ (word_rcat :: 8 word list ⇒ 32 sword) x\"\n\nabbreviation encode_s32 :: \"int ⇒ 8 word list\"\n where\n \"encode_s32 x ≡ flatten_s32 (word_of_int x)\"\n\nabbreviation decode_s32 :: \"8 word list ⇒ int\"\n where\n \"decode_s32 i ≡ sint (cat_s32 i)\"\n\nlemma flatten_s32_length:\n \"length (flatten_s32 vs) = 4\"\n by (simp add: length_word_rsplit_even_size (*‹⟦?n = LENGTH(?'a); size ?w = ?m * ?n⟧ ⟹ length (word_rsplit ?w) = ?m›*) wsst_TYs( (*‹size ?w = LENGTH(?'a)›*) 3))\n\nlemma rsplit_rcat_s32_eq:\n assumes \"w = [a1, a2, b1, b2]\"\n shows \"(word_rsplit :: 32 sword ⇒ 8 word list) ((word_rcat :: 8 word list ⇒ 32 sword) w) = w\" \n using rsplit_rcat_eq (*‹⟦LENGTH(?'b) mod LENGTH(?'a) = 0; length ?w = LENGTH(?'b) div LENGTH(?'a)⟧ ⟹ word_rsplit (word_rcat ?w) = ?w›*) assms (*‹(w::8 word list) = (a1::8 word) # (a2::8 word) # (b1::8 word) # encode_u8_list (b2::8 word)›*) by force\n\nlemma encode_decode_s32:\n assumes \"w = [a1, a2, b1, b2]\"\n shows \"w = encode_s32 (decode_s32 w)\" \n using assms (*‹w = a1 # a2 # b1 # encode_u8_list b2›*) by (simp add: rsplit_rcat_s32_eq (*‹?w = ?a1.0 # ?a2.0 # ?b1.0 # encode_u8_list ?b2.0 ⟹ flatten_s32 (cat_s32 ?w) = ?w›*))\n\nlemma decode_encode_s32:\n assumes \"- (2 ^ (LENGTH(32) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(32) - 1)\"\n shows \"i = decode_s32 (encode_s32 i)\" \n by (metis assms (*‹- (2 ^ (LENGTH(32) - 1)) ≤ i› ‹i < 2 ^ (LENGTH(32) - 1)›*) len_signed (*‹len_of ?x = LENGTH(?'a)›*) sint_of_int_eq (*‹⟦- (2 ^ (LENGTH(?'a) - 1)) ≤ ?x; ?x < 2 ^ (LENGTH(?'a) - 1)⟧ ⟹ sint (word_of_int ?x) = ?x›*) word_rcat_rsplit (*‹word_rcat (word_rsplit ?w) = ?w›*))\n \nabbreviation flatten_s64 :: \"64 sword ⇒ 8 word list\"\n where\n \"flatten_s64 x ≡ (word_rsplit :: 64 sword ⇒ 8 word list) x\"\n\nlemma flatten_s64_length:\n \"length (flatten_s64 vs) = 8\"\n by (simp add: length_word_rsplit_even_size (*‹⟦?n = LENGTH(?'a); size ?w = ?m * ?n⟧ ⟹ length (word_rsplit ?w) = ?m›*) wsst_TYs( (*‹size ?w = LENGTH(?'a)›*) 3))\n\nabbreviation cat_s64 :: \"8 word list ⇒ 64 sword\"\n where\n \"cat_s64 x ≡ word_rcat x\"\n\nabbreviation encode_s64 :: \"int ⇒ 8 word list\"\n where\n \"encode_s64 x ≡ flatten_s64 (word_of_int x)\"\n\nabbreviation decode_s64 :: \"8 word list ⇒ int\"\n where\n \"decode_s64 x ≡ sint (cat_s64 x)\"\n\nlemma encode_decode_s64:\n assumes \"w = [a1, a2, b1, b2, c1, c2, d1, d2]\"\n shows \"w = encode_s64 (decode_s64 w)\"\n using assms (*‹w = a1 # a2 # b1 # b2 # c1 # c2 # d1 # encode_u8_list d2›*) by (simp add: rsplit_rcat_eq (*‹⟦LENGTH(?'b::len) mod LENGTH(?'a::len) = (0::nat); length (?w::?'a::len word list) = LENGTH(?'b::len) div LENGTH(?'a::len)⟧ ⟹ word_rsplit (word_rcat ?w) = ?w›*))\n\nlemma decode_encode_s64:\n assumes \"- (2 ^ (LENGTH(64) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(64) - 1)\"\n shows \"i = decode_s64 (encode_s64 i)\" \n by (metis assms (*‹- ((2::int) ^ (LENGTH(64) - (1::nat))) ≤ (i::int)› ‹(i::int) < (2::int) ^ (LENGTH(64) - (1::nat))›*) len_signed (*‹len_of (?x::?'a signed itself) = LENGTH(?'a)›*) sint_of_int_eq (*‹⟦- ((2::int) ^ (LENGTH(?'a) - (1::nat))) ≤ (?x::int); ?x < (2::int) ^ (LENGTH(?'a) - (1::nat))⟧ ⟹ sint (word_of_int ?x) = ?x›*) word_rcat_rsplit (*‹word_rcat (word_rsplit (?w::?'a word)) = ?w›*))\n\ndefinition word_of_integer :: \"integer ⇒ 'a::len word\"\n where\n \"word_of_integer x ≡ word_of_int (int_of_integer x)\"\n\ndefinition sword_of_integer :: \"integer ⇒ 'a::len sword\"\n where\n \"sword_of_integer x ≡ word_of_int (int_of_integer x)\"\n\ndefinition integer_of_word :: \"'a::len word ⇒ integer\"\n where\n \"integer_of_word x ≡ integer_of_int (uint x)\"\n\ndefinition integer_of_sword :: \"'a::len sword ⇒ integer\"\n where\n \"integer_of_sword x ≡ integer_of_int (sint x)\"\n\nlemma word_integer_eq:\n \"word_of_integer (integer_of_word w) = w\"\n unfolding word_of_integer_def integer_of_word_def\n (*goal: ‹word_of_int (int_of_integer (integer_of_int (uint w))) = w›*)\n by (metis int_of_integer_of_int (*‹int_of_integer (of_int ?k) = ?k›*) integer_of_int_eq_of_int (*‹integer_of_int = of_int›*) word_uint.Rep_inverse' (*‹uint ?a = ?r ⟹ word_of_int ?r = ?a›*))\n\nlemma sword_integer_eq:\n \"sword_of_integer (integer_of_sword w) = w\"\n unfolding sword_of_integer_def integer_of_sword_def\n (*goal: ‹word_of_int (int_of_integer (integer_of_int (sint w))) = w›*)\n by (metis int_of_integer_of_int (*‹int_of_integer (of_int ?k) = ?k›*) integer_of_int_eq_of_int (*‹integer_of_int = of_int›*) word_sint.Rep_inverse' (*‹sint ?a = ?r ⟹ word_of_int ?r = ?a›*))\n\nlemma integer_word_bounded_eq:\n assumes \"0 ≤ i\"\n assumes \"i ≤ 2 ^ LENGTH('a::len) - 1\"\n shows \"integer_of_word ((word_of_integer :: integer ⇒ 'a word) i) = i\"\n unfolding integer_of_word_def word_of_integer_def\n (*goal: ‹integer_of_int (uint (word_of_int (int_of_integer i))) = i›*)\n using assms (*‹0 ≤ i› ‹i ≤ 2 ^ LENGTH('a) - 1›*) by (metis integer_less_eq_iff (*‹(?k ≤ ?l) = (int_of_integer ?k ≤ int_of_integer ?l)›*) integer_of_int_eq_of_int (*‹integer_of_int = of_int›*) minus_integer.rep_eq (*‹int_of_integer (?x - ?xa) = int_of_integer ?x - int_of_integer ?xa›*) of_int_0_le_iff (*‹(0 ≤ of_int ?z) = (0 ≤ ?z)›*) of_int_eq_1_iff (*‹(of_int ?z = 1) = (?z = 1)›*) of_int_eq_numeral_power_cancel_iff (*‹(of_int ?y = numeral ?x ^ ?n) = (?y = numeral ?x ^ ?n)›*) of_int_integer_of (*‹of_int (int_of_integer ?k) = ?k›*) word_of_int_inverse (*‹⟦word_of_int ?r = ?a; 0 ≤ ?r; ?r < 2 ^ LENGTH(?'a)⟧ ⟹ uint ?a = ?r›*) zle_diff1_eq (*‹(?w ≤ ?z - 1) = (?w < ?z)›*))\n\nlemma integer_sword_bounded_eq:\n assumes \"- (2 ^ (LENGTH('a::len) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH('a) - 1)\"\n shows \"integer_of_sword ((sword_of_integer :: integer ⇒ 'a sword) i) = i\"\n unfolding integer_of_sword_def sword_of_integer_def\n (*goal: ‹integer_of_int (sint (word_of_int (int_of_integer i))) = i›*)\n using signed_take_bit_int_eq_self (*‹⟦- (2 ^ ?n) ≤ ?k; ?k < 2 ^ ?n⟧ ⟹ signed_take_bit ?n ?k = ?k›*) assms (*‹- (2 ^ (LENGTH('a) - 1)) ≤ i› ‹(i::integer) < (2::integer) ^ (LENGTH('a::len) - (1::nat))›*) by (smt (verit) diff_numeral_special( (*‹1 - - 1 = 2›*) 11) int_of_integer_numeral (*‹int_of_integer (numeral ?k) = numeral ?k›*) integer_less_eq_iff (*‹(?k ≤ ?l) = (int_of_integer ?k ≤ int_of_integer ?l)›*) integer_of_int_eq_of_int (*‹integer_of_int = of_int›*) len_signed (*‹len_of ?x = LENGTH(?'a)›*) minus_integer.rep_eq (*‹int_of_integer (?x - ?xa) = int_of_integer ?x - int_of_integer ?xa›*) numeral_power_eq_of_int_cancel_iff (*‹(numeral ?x ^ ?n = of_int ?y) = (numeral ?x ^ ?n = ?y)›*) of_int_integer_of (*‹of_int (int_of_integer ?k) = ?k›*) of_int_power_less_of_int_cancel_iff (*‹(of_int ?x < of_int ?b ^ ?w) = (?x < ?b ^ ?w)›*) one_integer.rep_eq (*‹int_of_integer 1 = 1›*) sint_of_int_eq (*‹⟦- (2 ^ (LENGTH(?'a) - 1)) ≤ ?x; ?x < 2 ^ (LENGTH(?'a) - 1)⟧ ⟹ sint (word_of_int ?x) = ?x›*) uminus_integer.rep_eq (*‹int_of_integer (- ?x) = - int_of_integer ?x›*))\n\ndefinition word8_of_integer :: \"integer ⇒ 8 word\"\n where\n \"word8_of_integer ≡ word_of_integer\"\n\ndefinition word16_of_integer :: \"integer ⇒ 16 word\"\n where\n \"word16_of_integer ≡ word_of_integer\"\n\ndefinition word32_of_integer :: \"integer ⇒ 32 word\"\n where\n \"word32_of_integer ≡ word_of_integer\"\n\ndefinition word64_of_integer :: \"integer ⇒ 64 word\"\n where\n \"word64_of_integer ≡ word_of_integer\"\n\ndefinition integer_of_word8 :: \"8 word ⇒ integer\"\n where\n \"integer_of_word8 ≡ integer_of_word\"\n \ndefinition integer_of_word16 :: \"16 word ⇒ integer\"\n where\n \"integer_of_word16 ≡ integer_of_word\"\n\ndefinition integer_of_word32 :: \"32 word ⇒ integer\"\n where\n \"integer_of_word32 ≡ integer_of_word\"\n\ndefinition integer_of_word64 :: \"64 word ⇒ integer\"\n where\n \"integer_of_word64 ≡ integer_of_word\"\n\nlemma word8_integer_eq:\n \"word8_of_integer (integer_of_word8 w) = w\" \n unfolding word8_of_integer_def integer_of_word8_def\n (*goal: ‹word_of_integer (integer_of_word w) = w›*)\n using word_integer_eq (*‹word_of_integer (integer_of_word ?w) = ?w›*) by blast\n\nlemma word16_integer_eq:\n \"word16_of_integer (integer_of_word16 w) = w\" \n unfolding word16_of_integer_def integer_of_word16_def\n (*goal: ‹word_of_integer (integer_of_word (w::16 word)) = w›*)\n using word_integer_eq (*‹word_of_integer (integer_of_word ?w) = ?w›*) by blast\n\nlemma word32_integer_eq:\n \"word32_of_integer (integer_of_word32 w) = w\" \n unfolding word32_of_integer_def integer_of_word32_def\n (*goal: ‹word_of_integer (integer_of_word w) = w›*)\n using word_integer_eq (*‹word_of_integer (integer_of_word ?w) = ?w›*) by blast\n\nlemma word64_integer_eq:\n \"word64_of_integer (integer_of_word64 w) = w\"\n unfolding word64_of_integer_def integer_of_word64_def\n (*goal: ‹word_of_integer (integer_of_word w) = w›*)\n using word_integer_eq (*‹word_of_integer (integer_of_word (?w::?'a word)) = ?w›*) by blast\n\nlemma integer_word8_bounded_eq:\n assumes \"0 ≤ i\"\n and \"i ≤ 2 ^ LENGTH(8) - 1\"\n shows \"integer_of_word8 (word8_of_integer i) = i\"\n unfolding word8_of_integer_def integer_of_word8_def\n (*goal: ‹integer_of_word (word_of_integer i) = i›*)\n using integer_word_bounded_eq (*‹⟦(0::integer) ≤ (?i::integer); ?i ≤ (2::integer) ^ LENGTH(?'a::len) - (1::integer)⟧ ⟹ integer_of_word (word_of_integer ?i) = ?i›*) assms (*‹(0::integer) ≤ (i::integer)› ‹i ≤ 2 ^ LENGTH(8) - 1›*) by blast\n\nlemma integer_word16_bounded_eq:\n assumes \"0 ≤ i\"\n and \"i ≤ 2 ^ LENGTH(16) - 1\"\n shows \"integer_of_word16 (word16_of_integer i) = i\"\n unfolding word16_of_integer_def integer_of_word16_def\n (*goal: ‹integer_of_word (word_of_integer i) = i›*)\n using integer_word_bounded_eq (*‹⟦0 ≤ ?i; ?i ≤ 2 ^ LENGTH(?'a) - 1⟧ ⟹ integer_of_word (word_of_integer ?i) = ?i›*) assms (*‹0 ≤ i› ‹i ≤ 2 ^ LENGTH(16) - 1›*) by blast\n\nlemma integer_word32_bounded_eq:\n assumes \"0 ≤ i\"\n and \"i ≤ 2 ^ LENGTH(32) - 1\"\n shows \"integer_of_word32 (word32_of_integer i) = i\"\n unfolding word32_of_integer_def integer_of_word32_def\n (*goal: ‹integer_of_word (word_of_integer i) = i›*)\n using integer_word_bounded_eq (*‹⟦0 ≤ ?i; ?i ≤ 2 ^ LENGTH(?'a) - 1⟧ ⟹ integer_of_word (word_of_integer ?i) = ?i›*) assms (*‹0 ≤ i› ‹(i::integer) ≤ (2::integer) ^ LENGTH(32) - (1::integer)›*) by blast\n\nlemma integer_word64_bounded_eq:\n assumes \"0 ≤ i\"\n and \"i ≤ 2 ^ LENGTH(64) - 1\"\n shows \"integer_of_word64 (word64_of_integer i) = i\"\n unfolding word64_of_integer_def integer_of_word64_def\n (*goal: ‹integer_of_word (word_of_integer i) = i›*)\n using integer_word_bounded_eq (*‹⟦0 ≤ ?i; ?i ≤ 2 ^ LENGTH(?'a) - 1⟧ ⟹ integer_of_word (word_of_integer ?i) = ?i›*) assms (*‹0 ≤ i› ‹i ≤ 2 ^ LENGTH(64) - 1›*) by blast\n\ndefinition sword8_of_integer :: \"integer ⇒ 8 sword\"\n where\n \"sword8_of_integer ≡ sword_of_integer\"\n\ndefinition sword16_of_integer :: \"integer ⇒ 16 sword\"\n where\n \"sword16_of_integer ≡ sword_of_integer\"\n\ndefinition sword32_of_integer :: \"integer ⇒ 32 sword\"\n where\n \"sword32_of_integer ≡ sword_of_integer\"\n\ndefinition sword64_of_integer :: \"integer ⇒ 64 sword\"\n where\n \"sword64_of_integer ≡ sword_of_integer\"\n\ndefinition integer_of_sword8 :: \"8 sword ⇒ integer\"\n where\n \"integer_of_sword8 ≡ integer_of_sword\"\n\ndefinition integer_of_sword16 :: \"16 sword ⇒ integer\"\n where\n \"integer_of_sword16 ≡ integer_of_sword\"\n\ndefinition integer_of_sword32 :: \"32 sword ⇒ integer\"\n where\n \"integer_of_sword32 ≡ integer_of_sword\"\n\ndefinition integer_of_sword64 :: \"64 sword ⇒ integer\"\n where\n \"integer_of_sword64 ≡ integer_of_sword\"\n\nlemma sword8_integer_eq:\n \"sword8_of_integer (integer_of_sword8 w) = w\" \n unfolding sword8_of_integer_def integer_of_sword8_def\n (*goal: ‹sword_of_integer (integer_of_sword w) = w›*)\n using sword_integer_eq (*‹sword_of_integer (integer_of_sword ?w) = ?w›*) by blast\n\nlemma sword16_integer_eq:\n \"sword16_of_integer (integer_of_sword16 w) = w\" \n unfolding sword16_of_integer_def integer_of_sword16_def\n (*goal: ‹sword_of_integer (integer_of_sword w) = w›*)\n using sword_integer_eq (*‹sword_of_integer (integer_of_sword ?w) = ?w›*) by blast\n\nlemma sword32_integer_eq:\n \"sword32_of_integer (integer_of_sword32 w) = w\" \n unfolding sword32_of_integer_def integer_of_sword32_def\n (*goal: ‹sword_of_integer (integer_of_sword w) = w›*)\n using sword_integer_eq (*‹sword_of_integer (integer_of_sword (?w::?'a signed word)) = ?w›*) by blast\n\nlemma sword64_integer_eq:\n \"sword64_of_integer (integer_of_sword64 w) = w\"\n unfolding sword64_of_integer_def integer_of_sword64_def\n (*goal: ‹sword_of_integer (integer_of_sword w) = w›*)\n using sword_integer_eq (*‹sword_of_integer (integer_of_sword ?w) = ?w›*) by blast\n\nlemma integer_sword8_bounded_eq:\n assumes \"- (2 ^ (LENGTH(8) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(8) - 1)\"\n shows \"integer_of_sword8 (sword8_of_integer i) = i\"\n unfolding sword8_of_integer_def integer_of_sword8_def\n (*goal: ‹integer_of_sword (sword_of_integer i) = i›*)\n using integer_sword_bounded_eq (*‹⟦- ((2::integer) ^ (LENGTH(?'a) - (1::nat))) ≤ (?i::integer); ?i < (2::integer) ^ (LENGTH(?'a) - (1::nat))⟧ ⟹ integer_of_sword (sword_of_integer ?i) = ?i›*) assms (*‹- (2 ^ (LENGTH(8) - 1)) ≤ i› ‹i < 2 ^ (LENGTH(8) - 1)›*) by metis \n\nlemma integer_sword16_bounded_eq:\n assumes \"- (2 ^ (LENGTH(16) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(16) - 1)\"\n shows \"integer_of_sword16 (sword16_of_integer i) = i\"\n unfolding sword16_of_integer_def integer_of_sword16_def\n (*goal: ‹integer_of_sword (sword_of_integer (i::integer)) = i›*)\n using integer_sword_bounded_eq (*‹⟦- ((2::integer) ^ (LENGTH(?'a::len) - (1::nat))) ≤ (?i::integer); ?i < (2::integer) ^ (LENGTH(?'a::len) - (1::nat))⟧ ⟹ integer_of_sword (sword_of_integer ?i) = ?i›*) assms (*‹- (2 ^ (LENGTH(16) - 1)) ≤ i› ‹i < 2 ^ (LENGTH(16) - 1)›*) by metis\n\nlemma integer_sword32_bounded_eq:\n assumes \"- (2 ^ (LENGTH(32) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(32) - 1)\"\n shows \"integer_of_sword32 (sword32_of_integer i) = i\"\n unfolding sword32_of_integer_def integer_of_sword32_def\n (*goal: ‹integer_of_sword (sword_of_integer i) = i›*)\n using integer_sword_bounded_eq (*‹⟦- (2 ^ (LENGTH(?'a) - 1)) ≤ ?i; ?i < 2 ^ (LENGTH(?'a) - 1)⟧ ⟹ integer_of_sword (sword_of_integer ?i) = ?i›*) assms (*‹- ((2::integer) ^ (LENGTH(32) - (1::nat))) ≤ (i::integer)› ‹(i::integer) < (2::integer) ^ (LENGTH(32) - (1::nat))›*) by metis\n\nlemma integer_sword64_bounded_eq:\n assumes \"- (2 ^ (LENGTH(64) - 1)) ≤ i\"\n and \"i < 2 ^ (LENGTH(64) - 1)\"\n shows \"integer_of_sword64 (sword64_of_integer i) = i\"\n unfolding sword64_of_integer_def integer_of_sword64_def\n (*goal: ‹integer_of_sword (sword_of_integer (i::integer)) = i›*)\n using integer_sword_bounded_eq (*‹⟦- (2 ^ (LENGTH(?'a) - 1)) ≤ ?i; ?i < 2 ^ (LENGTH(?'a) - 1)⟧ ⟹ integer_of_sword (sword_of_integer ?i) = ?i›*) assms (*‹- (2 ^ (LENGTH(64) - 1)) ≤ i› ‹(i::integer) < (2::integer) ^ (LENGTH(64) - (1::nat))›*) by metis\n\nlemmas flatten_types_length = flatten_u16_length flatten_s16_length flatten_u32_length flatten_s32_length flatten_u64_length flatten_s64_length\n\ntext ‹cast\\_val is an executable code that ensures easy casting of values. This value cast function\n is used within the Gillian \n framework~\\cite{old_gillian_2020, gillian_2021, maksimovic_2021, fragoso_2020}.›\ndefinition cast_val :: \"String.literal ⇒ integer ⇒ integer\"\n where\n \"cast_val s i ≡\n if s = STR ''uint8'' then integer_of_word8 (word8_of_integer i) \n else if s = STR ''int8'' then integer_of_sword8 (sword8_of_integer i)\n else if s = STR ''uint16'' then integer_of_word16 (word16_of_integer i)\n else if s = STR ''int16'' then integer_of_sword16 (sword16_of_integer i)\n else if s = STR ''uint32'' then integer_of_word32 (word32_of_integer i)\n else if s = STR ''int32'' then integer_of_sword32 (sword32_of_integer i)\n else if s = STR ''uint64'' then integer_of_word64 (word64_of_integer i)\n else if s = STR ''int64'' then integer_of_sword64 (sword64_of_integer i)\n else i\"\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/CHERI-C_Memory_Model/Preliminary_Library.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"55cf8e557dfaa50709c34261001ee6ed8aec56b5d004da0e2b1daeb1506adde1\"\n}"}}},{"rowIdx":176,"cells":{"text":{"kind":"string","value":"(*\n * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)\n * Copyright (c) 2022 Apple Inc. All rights reserved.\n *\n * SPDX-License-Identifier: BSD-2-Clause\n *)\n\ntheory globals_fn\nimports \"AutoCorres2.CTranslation\"\nbegin\n\ninstall_C_file \"globals_fn.c\"\n\nprint_locale globals_fn_global_addresses\n\ncontext globals_fn_simpl\nbegin\n thm f_body_def\n thm gupdate_body_def\n thm update_body_def\n thm test1_body_def\n thm test2_body_def\n thm test3_body_def\n\nend\n\nlemma (in test2_impl) returns40:\n \"Γ ⊢ ⦃ True ⦄ ´ret' :== PROC test2() ⦃ ´ret' = 40 ⦄\"\n apply vcg\n (*goal: ‹Γ⊢ ⦃True⦄ Call globals_fn.test2 ⦃´(ret'::32 signed word) = 0x28⦄›*)\n by auto\n\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/AutoCorres2/tests/c-parser/globals_fn.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"984dfbcabafc5efc6359f71af19030e24272ffcd7d2f0012761076b833aea042\"\n}"}}},{"rowIdx":177,"cells":{"text":{"kind":"string","value":"(* Title: Right group actions\n Author: Tim Makarios <tjm1983 at gmail.com>, 2012\n Maintainer: Tim Makarios <tjm1983 at gmail.com>\n*)\n\n(* After Isabelle 2012, this may be moved to ~~/src/HOL/Algebra *)\n\nsection \"Right group actions\"\n\ntheory Action\n imports \"HOL-Algebra.Group\"\nbegin\n\nlocale action = group +\n fixes act :: \"'b ⇒ 'a ⇒ 'b\" (infixl \"<o\" 69)\n assumes id_act [simp]: \"b <o 𝟭 = b\"\n and act_act':\n \"g ∈ carrier G ∧ h ∈ carrier G ⟶ (b <o g) <o h = b <o (g ⊗ h)\"\nbegin\n\nlemma act_act:\n assumes \"g ∈ carrier G\" and \"h ∈ carrier G\"\n shows \"(b <o g) <o h = b <o (g ⊗ h)\"\nproof (-)\n (*goal: ‹b <o g <o h = b <o g ⊗ h›*)\n from ‹g ∈ carrier G› (*‹g ∈ carrier G›*) ‹h ∈ carrier G› (*‹h ∈ carrier G›*) act_act' (*‹?g ∈ carrier G ∧ ?h ∈ carrier G ⟶ ?b <o ?g <o ?h = ?b <o ?g ⊗ ?h›*) show \"(b <o g) <o h = b <o (g ⊗ h)\"\n by simp\nqed\n\nlemma act_act_inv [simp]:\n assumes \"g ∈ carrier G\"\n shows \"b <o g <o inv g = b\"\nproof (-)\n (*goal: ‹b <o g <o inv g = b›*)\n from ‹g ∈ carrier G› (*‹g ∈ carrier G›*) have \"inv g ∈ carrier G\"\n by (rule inv_closed (*‹?x ∈ carrier G ⟹ inv ?x ∈ carrier G›*))\n with ‹g ∈ carrier G› (*‹g ∈ carrier G›*) have \"b <o g <o inv g = b <o g ⊗ inv g\"\n by (rule act_act (*‹⟦?g ∈ carrier G; ?h ∈ carrier G⟧ ⟹ ?b <o ?g <o ?h = ?b <o ?g ⊗ ?h›*))\n with ‹g ∈ carrier G› (*‹g ∈ carrier G›*) show \"b <o g <o inv g = b\"\n by simp\nqed\n\nlemma act_inv_act [simp]:\n assumes \"g ∈ carrier G\"\n shows \"b <o inv g <o g = b\"\n using ‹g ∈ carrier G› (*‹(g::'a) ∈ carrier G›*) act_act_inv[of \"inv g\"] (*‹inv g ∈ carrier G ⟹ ?b <o inv g <o inv (inv g) = ?b›*) by simp\n\nlemma act_inv_iff:\n assumes \"g ∈ carrier G\"\n shows \"b <o inv g = c ⟷ b = c <o g\"\nproof (standard)\n (*goals:\n 1. ‹(b::'b) <o inv (g::'a) = (c::'b) ⟹ b = c <o g›\n 2. ‹(b::'b) = (c::'b) <o (g::'a) ⟹ b <o inv g = c›*)\n assume \"b <o inv g = c\" (*‹(b::'b) <o inv (g::'a) = (c::'b)›*)\n hence \"b <o inv g <o g = c <o g\"\n by simp\n with ‹g ∈ carrier G› (*‹g ∈ carrier G›*) show \"b = c <o g\"\n by simp\nnext\n (*goal: ‹b = c <o g ⟹ b <o inv g = c›*)\n assume \"b = c <o g\" (*‹(b::'b) = (c::'b) <o (g::'a)›*)\n hence \"b <o inv g = c <o g <o inv g\"\n by simp\n with ‹g ∈ carrier G› (*‹g ∈ carrier G›*) show \"b <o inv g = c\"\n by simp\nqed\n\nend\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Tarskis_Geometry/Action.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"c75ff23224eb19c3e6b9c15003666208d1b709c7d4bfd550cb6bcb4b3a46c361\"\n}"}}},{"rowIdx":178,"cells":{"text":{"kind":"string","value":"(* Title: HOL/Statespace/DistinctTreeProver.thy\n Author: Norbert Schirmer, TU Muenchen\n*)\n\nsection ‹Distinctness of Names in a Binary Tree \\label{sec:DistinctTreeProver}›\n\ntheory DistinctTreeProver \nimports Main\nbegin\n\ntext ‹A state space manages a set of (abstract) names and assumes\nthat the names are distinct. The names are stored as parameters of a\nlocale and distinctness as an assumption. The most common request is\nto proof distinctness of two given names. We maintain the names in a\nbalanced binary tree and formulate a predicate that all nodes in the\ntree have distinct names. This setup leads to logarithmic certificates.\n›\n\nsubsection ‹The Binary Tree›\n\ndatatype 'a tree = Node \"'a tree\" 'a bool \"'a tree\" | Tip\n\n\ntext ‹The boolean flag in the node marks the content of the node as\ndeleted, without having to build a new tree. We prefer the boolean\nflag to an option type, so that the ML-layer can still use the node\ncontent to facilitate binary search in the tree. The ML code keeps the\nnodes sorted using the term order. We do not have to push ordering to\nthe HOL level.›\n\nsubsection ‹Distinctness of Nodes›\n\n\nprimrec set_of :: \"'a tree ⇒ 'a set\"\nwhere\n \"set_of Tip = {}\"\n| \"set_of (Node l x d r) = (if d then {} else {x}) ∪ set_of l ∪ set_of r\"\n\nprimrec all_distinct :: \"'a tree ⇒ bool\"\nwhere\n \"all_distinct Tip = True\"\n| \"all_distinct (Node l x d r) =\n ((d ∨ (x ∉ set_of l ∧ x ∉ set_of r)) ∧ \n set_of l ∩ set_of r = {} ∧\n all_distinct l ∧ all_distinct r)\"\n\ntext ‹Given a binary tree \\<^term>‹t› for which \n\\<^const>‹all_distinct› holds, given two different nodes contained in the tree,\nwe want to write a ML function that generates a logarithmic\ncertificate that the content of the nodes is distinct. We use the\nfollowing lemmas to achieve this.› \n\nlemma all_distinct_left: \"all_distinct (Node l x b r) ⟹ all_distinct l\"\n by simp\n\nlemma all_distinct_right: \"all_distinct (Node l x b r) ⟹ all_distinct r\"\n by simp\n\nlemma distinct_left: \"all_distinct (Node l x False r) ⟹ y ∈ set_of l ⟹ x ≠ y\"\n by auto\n\nlemma distinct_right: \"all_distinct (Node l x False r) ⟹ y ∈ set_of r ⟹ x ≠ y\"\n by auto\n\nlemma distinct_left_right:\n \"all_distinct (Node l z b r) ⟹ x ∈ set_of l ⟹ y ∈ set_of r ⟹ x ≠ y\"\n by auto\n\nlemma in_set_root: \"x ∈ set_of (Node l x False r)\"\n by simp\n\nlemma in_set_left: \"y ∈ set_of l ⟹ y ∈ set_of (Node l x False r)\"\n by simp\n\nlemma in_set_right: \"y ∈ set_of r ⟹ y ∈ set_of (Node l x False r)\"\n by simp\n\nlemma swap_neq: \"x ≠ y ⟹ y ≠ x\"\n by blast\n\nlemma neq_to_eq_False: \"x≠y ⟹ (x=y)≡False\"\n by simp\n\nsubsection ‹Containment of Trees›\n\ntext ‹When deriving a state space from other ones, we create a new\nname tree which contains all the names of the parent state spaces and\nassume the predicate \\<^const>‹all_distinct›. We then prove that the new\nlocale interprets all parent locales. Hence we have to show that the\nnew distinctness assumption on all names implies the distinctness\nassumptions of the parent locales. This proof is implemented in ML. We\ndo this efficiently by defining a kind of containment check of trees\nby ``subtraction''. We subtract the parent tree from the new tree. If\nthis succeeds we know that \\<^const>‹all_distinct› of the new tree\nimplies \\<^const>‹all_distinct› of the parent tree. The resulting\ncertificate is of the order \\<^term>‹n * log(m)› where \\<^term>‹n› is\nthe size of the (smaller) parent tree and \\<^term>‹m› the size of the\n(bigger) new tree.›\n\n\nprimrec delete :: \"'a ⇒ 'a tree ⇒ 'a tree option\"\nwhere\n \"delete x Tip = None\"\n| \"delete x (Node l y d r) = (case delete x l of\n Some l' ⇒\n (case delete x r of \n Some r' ⇒ Some (Node l' y (d ∨ (x=y)) r')\n | None ⇒ Some (Node l' y (d ∨ (x=y)) r))\n | None ⇒\n (case delete x r of \n Some r' ⇒ Some (Node l y (d ∨ (x=y)) r')\n | None ⇒ if x=y ∧ ¬d then Some (Node l y True r)\n else None))\"\n\n\nlemma delete_Some_set_of: \"delete x t = Some t' ⟹ set_of t' ⊆ set_of t\"\nproof (induct t arbitrary: t')\n (*goals:\n 1. ‹⋀(t1::'a tree) (x2::'a) (x3::bool) (t2::'a tree) t'::'a tree. ⟦⋀t'::'a tree. delete (x::'a) t1 = Some t' ⟹ set_of t' ⊆ set_of t1; ⋀t'::'a tree. delete x t2 = Some t' ⟹ set_of t' ⊆ set_of t2; delete x (Node t1 x2 x3 t2) = Some t'⟧ ⟹ set_of t' ⊆ set_of (Node t1 x2 x3 t2)›\n 2. ‹⋀t'::'a tree. delete (x::'a) Tip = Some t' ⟹ set_of t' ⊆ set_of Tip›*)\n case Tip (*‹delete x Tip = Some t'›*)\n thus \"?case\"\n (*goal: ‹set_of t' ⊆ set_of Tip›*)\n by simp\nnext\n (*goal: ‹⋀t1 x2 x3 t2 t'. ⟦⋀t'. delete x t1 = Some t' ⟹ set_of t' ⊆ set_of t1; ⋀t'. delete x t2 = Some t' ⟹ set_of t' ⊆ set_of t2; delete x (Node t1 x2 x3 t2) = Some t'⟧ ⟹ set_of t' ⊆ set_of (Node t1 x2 x3 t2)›*)\n case (Node l y d r) (*‹delete x l = Some ?t' ⟹ set_of ?t' ⊆ set_of l› ‹delete (x::'a::type) (r::'a::type tree) = Some (?t'::'a::type tree) ⟹ set_of ?t' ⊆ set_of r› ‹delete x (Node l y d r) = Some t'›*)\n have del: \"delete x (Node l y d r) = Some t'\"\n by fact\n show \"?case\"\n (*goal: ‹set_of t' ⊆ set_of (Node l y d r)›*)\n proof (cases \"delete x l\")\n (*goals:\n 1. ‹delete x l = None ⟹ set_of t' ⊆ set_of (Node l y d r)›\n 2. ‹⋀a. delete x l = Some a ⟹ set_of t' ⊆ set_of (Node l y d r)›*)\n case (Some l') (*‹delete x l = Some l'›*)\n note x_l_Some = this (*‹delete x l = Some l'›*)\n with Node.hyps (*‹delete x l = Some ?t' ⟹ set_of ?t' ⊆ set_of l› ‹delete (x::'a) (r::'a tree) = Some (?t'::'a tree) ⟹ set_of ?t' ⊆ set_of r›*) have l'_l: \"set_of l' ⊆ set_of l\"\n by simp\n show \"?thesis\"\n (*goal: ‹set_of (t'::'a tree) ⊆ set_of (Node (l::'a tree) (y::'a) (d::bool) (r::'a tree))›*)\n proof (cases \"delete x r\")\n (*goals:\n 1. ‹delete x r = None ⟹ set_of t' ⊆ set_of (Node l y d r)›\n 2. ‹⋀a. delete x r = Some a ⟹ set_of t' ⊆ set_of (Node l y d r)›*)\n case (Some r') (*‹delete x r = Some r'›*)\n with Node.hyps (*‹delete x l = Some ?t' ⟹ set_of ?t' ⊆ set_of l› ‹delete x r = Some ?t' ⟹ set_of ?t' ⊆ set_of r›*) have \"set_of r' ⊆ set_of r\"\n by simp\n with l'_l (*‹set_of l' ⊆ set_of l›*) Some (*‹delete x r = Some r'›*) x_l_Some (*‹delete x l = Some l'›*) del (*‹delete x (Node l y d r) = Some t'›*) show \"?thesis\"\n (*goal: ‹set_of t' ⊆ set_of (Node l y d r)›*)\n by (auto split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n next\n (*goal: ‹delete x r = None ⟹ set_of t' ⊆ set_of (Node l y d r)›*)\n case None (*‹delete (x::'a::type) (r::'a::type tree) = None›*)\n with l'_l (*‹set_of l' ⊆ set_of l›*) Some (*‹delete x l = Some l'›*) x_l_Some (*‹delete x l = Some l'›*) del (*‹delete x (Node l y d r) = Some t'›*) show \"?thesis\"\n (*goal: ‹set_of t' ⊆ set_of (Node l y d r)›*)\n by (fastforce split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n qed\n next\n (*goal: ‹delete (x::'a) (l::'a tree) = None ⟹ set_of (t'::'a tree) ⊆ set_of (Node l (y::'a) (d::bool) (r::'a tree))›*)\n case None (*‹delete x l = None›*)\n note x_l_None = this (*‹delete x l = None›*)\n show \"?thesis\"\n (*goal: ‹set_of t' ⊆ set_of (Node l y d r)›*)\n proof (cases \"delete x r\")\n (*goals:\n 1. ‹delete x r = None ⟹ set_of t' ⊆ set_of (Node l y d r)›\n 2. ‹⋀a. delete x r = Some a ⟹ set_of t' ⊆ set_of (Node l y d r)›*)\n case (Some r') (*‹delete x r = Some r'›*)\n with Node.hyps (*‹delete x l = Some ?t' ⟹ set_of ?t' ⊆ set_of l› ‹delete x r = Some ?t' ⟹ set_of ?t' ⊆ set_of r›*) have \"set_of r' ⊆ set_of r\"\n by simp\n with Some (*‹delete x r = Some r'›*) x_l_None (*‹delete x l = None›*) del (*‹delete x (Node l y d r) = Some t'›*) show \"?thesis\"\n (*goal: ‹set_of t' ⊆ set_of (Node l y d r)›*)\n by (fastforce split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n next\n (*goal: ‹delete x r = None ⟹ set_of t' ⊆ set_of (Node l y d r)›*)\n case None (*‹delete x r = None›*)\n with x_l_None (*‹delete (x::'a::type) (l::'a::type tree) = None›*) del (*‹delete x (Node l y d r) = Some t'›*) show \"?thesis\"\n (*goal: ‹set_of t' ⊆ set_of (Node l y d r)›*)\n by (fastforce split: if_split_asm (*‹(?P::?'a::type ⇒ bool) (if ?Q::bool then ?x::?'a::type else (?y::?'a::type)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n qed\n qed\nqed\n\nlemma delete_Some_all_distinct:\n \"delete x t = Some t' ⟹ all_distinct t ⟹ all_distinct t'\"\nproof (induct t arbitrary: t')\n (*goals:\n 1. ‹⋀t1 x2 x3 t2 t'. ⟦⋀t'. ⟦delete x t1 = Some t'; all_distinct t1⟧ ⟹ all_distinct t'; ⋀t'. ⟦delete x t2 = Some t'; all_distinct t2⟧ ⟹ all_distinct t'; delete x (Node t1 x2 x3 t2) = Some t'; all_distinct (Node t1 x2 x3 t2)⟧ ⟹ all_distinct t'›\n 2. ‹⋀t'. ⟦delete x Tip = Some t'; all_distinct Tip⟧ ⟹ all_distinct t'›*)\n case Tip (*‹delete x Tip = Some t'› ‹all_distinct Tip›*)\n thus \"?case\"\n (*goal: ‹all_distinct t'›*)\n by simp\nnext\n (*goal: ‹⋀t1 x2 x3 t2 t'. ⟦⋀t'. ⟦delete x t1 = Some t'; all_distinct t1⟧ ⟹ all_distinct t'; ⋀t'. ⟦delete x t2 = Some t'; all_distinct t2⟧ ⟹ all_distinct t'; delete x (Node t1 x2 x3 t2) = Some t'; all_distinct (Node t1 x2 x3 t2)⟧ ⟹ all_distinct t'›*)\n case (Node l y d r) (*‹⟦delete x l = Some ?t'; all_distinct l⟧ ⟹ all_distinct ?t'› ‹⟦delete (x::'a) (r::'a tree) = Some (?t'::'a tree); all_distinct r⟧ ⟹ all_distinct ?t'› ‹delete x (Node l y d r) = Some t'› ‹all_distinct (Node (l::'a tree) (y::'a) (d::bool) (r::'a tree))›*)\n have del: \"delete x (Node l y d r) = Some t'\"\n by fact\n have \"all_distinct (Node l y d r)\"\n by fact\n then obtain dist_l: \"all_distinct l\" and dist_r: \"all_distinct r\" and d: \"d ∨ (y ∉ set_of l ∧ y ∉ set_of r)\" and dist_l_r: \"set_of l ∩ set_of r = {}\"\n (*goal: ‹(⟦all_distinct l; all_distinct r; d ∨ y ∉ set_of l ∧ y ∉ set_of r; set_of l ∩ set_of r = {}⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n show \"?case\"\n (*goal: ‹all_distinct t'›*)\n proof (cases \"delete x l\")\n (*goals:\n 1. ‹delete x l = None ⟹ all_distinct t'›\n 2. ‹⋀a. delete x l = Some a ⟹ all_distinct t'›*)\n case (Some l') (*‹delete (x::'a) (l::'a tree) = Some (l'::'a tree)›*)\n note x_l_Some = this (*‹delete (x::'a) (l::'a tree) = Some (l'::'a tree)›*)\n from Node.hyps(1)[OF Some dist_l] (*‹all_distinct l'›*) have dist_l': \"all_distinct l'\"\n by simp\n from delete_Some_set_of[OF x_l_Some] (*‹set_of l' ⊆ set_of l›*) have l'_l: \"set_of l' ⊆ set_of l\" .\n show \"?thesis\"\n (*goal: ‹all_distinct t'›*)\n proof (cases \"delete x r\")\n (*goals:\n 1. ‹delete x r = None ⟹ all_distinct t'›\n 2. ‹⋀a. delete x r = Some a ⟹ all_distinct t'›*)\n case (Some r') (*‹delete (x::'a) (r::'a tree) = Some (r'::'a tree)›*)\n from Node.hyps(2)[OF Some dist_r] (*‹all_distinct r'›*) have dist_r': \"all_distinct r'\"\n by simp\n from delete_Some_set_of[OF Some] (*‹set_of r' ⊆ set_of r›*) have \"set_of r' ⊆ set_of r\" .\n with dist_l' (*‹all_distinct l'›*) dist_r' (*‹all_distinct r'›*) l'_l (*‹set_of l' ⊆ set_of l›*) Some (*‹delete x r = Some r'›*) x_l_Some (*‹delete (x::'a) (l::'a tree) = Some (l'::'a tree)›*) del (*‹delete x (Node l y d r) = Some t'›*) d (*‹d ∨ y ∉ set_of l ∧ y ∉ set_of r›*) dist_l_r (*‹set_of l ∩ set_of r = {}›*) show \"?thesis\"\n (*goal: ‹all_distinct t'›*)\n by fastforce\n next\n (*goal: ‹delete x r = None ⟹ all_distinct t'›*)\n case None (*‹delete x r = None›*)\n with l'_l (*‹set_of l' ⊆ set_of l›*) dist_l' (*‹all_distinct (l'::'a::type tree)›*) x_l_Some (*‹delete (x::'a) (l::'a tree) = Some (l'::'a tree)›*) del (*‹delete x (Node l y d r) = Some t'›*) d (*‹d ∨ y ∉ set_of l ∧ y ∉ set_of r›*) dist_l_r (*‹set_of l ∩ set_of r = {}›*) dist_r (*‹all_distinct r›*) show \"?thesis\"\n (*goal: ‹all_distinct (t'::'a tree)›*)\n by fastforce\n qed\n next\n (*goal: ‹delete x l = None ⟹ all_distinct t'›*)\n case None (*‹delete x l = None›*)\n note x_l_None = this (*‹delete x l = None›*)\n show \"?thesis\"\n (*goal: ‹all_distinct t'›*)\n proof (cases \"delete x r\")\n (*goals:\n 1. ‹delete x r = None ⟹ all_distinct t'›\n 2. ‹⋀a. delete x r = Some a ⟹ all_distinct t'›*)\n case (Some r') (*‹delete x r = Some r'›*)\n with Node.hyps(2)[OF Some dist_r] (*‹all_distinct r'›*) have dist_r': \"all_distinct r'\"\n by simp\n from delete_Some_set_of[OF Some] (*‹set_of r' ⊆ set_of r›*) have \"set_of r' ⊆ set_of r\" .\n with Some (*‹delete x r = Some r'›*) dist_r' (*‹all_distinct r'›*) x_l_None (*‹delete (x::'a) (l::'a tree) = None›*) del (*‹delete x (Node l y d r) = Some t'›*) dist_l (*‹all_distinct l›*) d (*‹d ∨ y ∉ set_of l ∧ y ∉ set_of r›*) dist_l_r (*‹set_of (l::'a tree) ∩ set_of (r::'a tree) = {}›*) show \"?thesis\"\n (*goal: ‹all_distinct t'›*)\n by fastforce\n next\n (*goal: ‹delete x r = None ⟹ all_distinct t'›*)\n case None (*‹delete x r = None›*)\n with x_l_None (*‹delete (x::'a) (l::'a tree) = None›*) del (*‹delete (x::'a::type) (Node (l::'a::type tree) (y::'a::type) (d::bool) (r::'a::type tree)) = Some (t'::'a::type tree)›*) dist_l (*‹all_distinct (l::'a tree)›*) dist_r (*‹all_distinct r›*) d (*‹d ∨ y ∉ set_of l ∧ y ∉ set_of r›*) dist_l_r (*‹set_of l ∩ set_of r = {}›*) show \"?thesis\"\n (*goal: ‹all_distinct t'›*)\n by (fastforce split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n qed\n qed\nqed\n\nlemma delete_None_set_of_conv: \"delete x t = None = (x ∉ set_of t)\"\nproof (induct t)\n (*goals:\n 1. ‹⋀t1 x2 x3 t2. ⟦(delete x t1 = None) = (x ∉ set_of t1); (delete x t2 = None) = (x ∉ set_of t2)⟧ ⟹ (delete x (Node t1 x2 x3 t2) = None) = (x ∉ set_of (Node t1 x2 x3 t2))›\n 2. ‹(delete x Tip = None) = (x ∉ set_of Tip)›*)\n case Tip (*no hyothesis introduced yet*)\n thus \"?case\"\n (*goal: ‹(delete x Tip = None) = (x ∉ set_of Tip)›*)\n by simp\nnext\n (*goal: ‹⋀(t1::'a tree) (x2::'a) (x3::bool) t2::'a tree. ⟦(delete (x::'a) t1 = None) = (x ∉ set_of t1); (delete x t2 = None) = (x ∉ set_of t2)⟧ ⟹ (delete x (Node t1 x2 x3 t2) = None) = (x ∉ set_of (Node t1 x2 x3 t2))›*)\n case (Node l y d r) (*‹(delete (x::'a) (l::'a tree) = None) = (x ∉ set_of l)› ‹(delete (x::'a) (r::'a tree) = None) = (x ∉ set_of r)›*)\n thus \"?case\"\n (*goal: ‹(delete x (Node l y d r) = None) = (x ∉ set_of (Node l y d r))›*)\n by (auto split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\nqed\n\nlemma delete_Some_x_set_of:\n \"delete x t = Some t' ⟹ x ∈ set_of t ∧ x ∉ set_of t'\"\nproof (induct t arbitrary: t')\n (*goals:\n 1. ‹⋀t1 x2 x3 t2 t'. ⟦⋀t'. delete x t1 = Some t' ⟹ x ∈ set_of t1 ∧ x ∉ set_of t'; ⋀t'. delete x t2 = Some t' ⟹ x ∈ set_of t2 ∧ x ∉ set_of t'; delete x (Node t1 x2 x3 t2) = Some t'⟧ ⟹ x ∈ set_of (Node t1 x2 x3 t2) ∧ x ∉ set_of t'›\n 2. ‹⋀t'. delete x Tip = Some t' ⟹ x ∈ set_of Tip ∧ x ∉ set_of t'›*)\n case Tip (*‹delete (x::'a) Tip = Some (t'::'a tree)›*)\n thus \"?case\"\n (*goal: ‹x ∈ set_of Tip ∧ x ∉ set_of t'›*)\n by simp\nnext\n (*goal: ‹⋀t1 x2 x3 t2 t'. ⟦⋀t'. delete x t1 = Some t' ⟹ x ∈ set_of t1 ∧ x ∉ set_of t'; ⋀t'. delete x t2 = Some t' ⟹ x ∈ set_of t2 ∧ x ∉ set_of t'; delete x (Node t1 x2 x3 t2) = Some t'⟧ ⟹ x ∈ set_of (Node t1 x2 x3 t2) ∧ x ∉ set_of t'›*)\n case (Node l y d r) (*‹delete x l = Some ?t' ⟹ x ∈ set_of l ∧ x ∉ set_of ?t'› ‹delete x r = Some ?t' ⟹ x ∈ set_of r ∧ x ∉ set_of ?t'› ‹delete (x::'a) (Node (l::'a tree) (y::'a) (d::bool) (r::'a tree)) = Some (t'::'a tree)›*)\n have del: \"delete x (Node l y d r) = Some t'\"\n by fact\n show \"?case\"\n (*goal: ‹(x::'a::type) ∈ set_of (Node (l::'a::type tree) (y::'a::type) (d::bool) (r::'a::type tree)) ∧ x ∉ set_of (t'::'a::type tree)›*)\n proof (cases \"delete x l\")\n (*goals:\n 1. ‹delete x l = None ⟹ x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›\n 2. ‹⋀a. delete x l = Some a ⟹ x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n case (Some l') (*‹delete x l = Some l'›*)\n note x_l_Some = this (*‹delete x l = Some l'›*)\n from Node.hyps(1)[OF Some] (*‹x ∈ set_of l ∧ x ∉ set_of l'›*) obtain x_l: \"x ∈ set_of l\" \"x ∉ set_of l'\"\n (*goal: ‹(⟦x ∈ set_of l; x ∉ set_of l'⟧ ⟹ thesis) ⟹ thesis›*)\n by simp\n show \"?thesis\"\n (*goal: ‹x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n proof (cases \"delete x r\")\n (*goals:\n 1. ‹delete x r = None ⟹ x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›\n 2. ‹⋀a. delete x r = Some a ⟹ x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n case (Some r') (*‹delete x r = Some r'›*)\n from Node.hyps(2)[OF Some] (*‹x ∈ set_of r ∧ x ∉ set_of r'›*) obtain x_r: \"x ∈ set_of r\" \"x ∉ set_of r'\"\n (*goal: ‹(⟦x ∈ set_of r; x ∉ set_of r'⟧ ⟹ thesis) ⟹ thesis›*)\n by simp\n from x_r (*‹x ∈ set_of r› ‹x ∉ set_of r'›*) x_l (*‹x ∈ set_of l› ‹x ∉ set_of l'›*) Some (*‹delete (x::'a) (r::'a tree) = Some (r'::'a tree)›*) x_l_Some (*‹delete x l = Some l'›*) del (*‹delete (x::'a) (Node (l::'a tree) (y::'a) (d::bool) (r::'a tree)) = Some (t'::'a tree)›*) show \"?thesis\"\n (*goal: ‹x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n by (clarsimp split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n next\n (*goal: ‹delete x r = None ⟹ x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n case None (*‹delete x r = None›*)\n then have \"x ∉ set_of r\"\n by (simp add: delete_None_set_of_conv (*‹(delete ?x ?t = None) = (?x ∉ set_of ?t)›*))\n with x_l (*‹x ∈ set_of l› ‹x ∉ set_of l'›*) None (*‹delete x r = None›*) x_l_Some (*‹delete x l = Some l'›*) del (*‹delete x (Node l y d r) = Some t'›*) show \"?thesis\"\n (*goal: ‹x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n by (clarsimp split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n qed\n next\n (*goal: ‹delete x l = None ⟹ x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n case None (*‹delete x l = None›*)\n note x_l_None = this (*‹delete (x::'a::type) (l::'a::type tree) = None›*)\n then have x_notin_l: \"x ∉ set_of l\"\n by (simp add: delete_None_set_of_conv (*‹(delete (?x::?'a) (?t::?'a tree) = None) = (?x ∉ set_of ?t)›*))\n show \"?thesis\"\n (*goal: ‹x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n proof (cases \"delete x r\")\n (*goals:\n 1. ‹delete (x::'a) (r::'a tree) = None ⟹ x ∈ set_of (Node (l::'a tree) (y::'a) (d::bool) r) ∧ x ∉ set_of (t'::'a tree)›\n 2. ‹⋀a::'a tree. delete (x::'a) (r::'a tree) = Some a ⟹ x ∈ set_of (Node (l::'a tree) (y::'a) (d::bool) r) ∧ x ∉ set_of (t'::'a tree)›*)\n case (Some r') (*‹delete x r = Some r'›*)\n from Node.hyps(2)[OF Some] (*‹x ∈ set_of r ∧ x ∉ set_of r'›*) obtain x_r: \"x ∈ set_of r\" \"x ∉ set_of r'\"\n (*goal: ‹(⟦(x::'a) ∈ set_of (r::'a tree); x ∉ set_of (r'::'a tree)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by simp\n from x_r (*‹x ∈ set_of r› ‹(x::'a) ∉ set_of (r'::'a tree)›*) x_notin_l (*‹x ∉ set_of l›*) Some (*‹delete (x::'a) (r::'a tree) = Some (r'::'a tree)›*) x_l_None (*‹delete x l = None›*) del (*‹delete x (Node l y d r) = Some t'›*) show \"?thesis\"\n (*goal: ‹x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n by (clarsimp split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n next\n (*goal: ‹delete x r = None ⟹ x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n case None (*‹delete x r = None›*)\n then have \"x ∉ set_of r\"\n by (simp add: delete_None_set_of_conv (*‹(delete ?x ?t = None) = (?x ∉ set_of ?t)›*))\n with None (*‹delete x r = None›*) x_l_None (*‹delete x l = None›*) x_notin_l (*‹x ∉ set_of l›*) del (*‹delete x (Node l y d r) = Some t'›*) show \"?thesis\"\n (*goal: ‹x ∈ set_of (Node l y d r) ∧ x ∉ set_of t'›*)\n by (clarsimp split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n qed\n qed\nqed\n\n\nprimrec subtract :: \"'a tree ⇒ 'a tree ⇒ 'a tree option\"\nwhere\n \"subtract Tip t = Some t\"\n| \"subtract (Node l x b r) t =\n (case delete x t of\n Some t' ⇒ (case subtract l t' of \n Some t'' ⇒ subtract r t''\n | None ⇒ None)\n | None ⇒ None)\"\n\nlemma subtract_Some_set_of_res: \n \"subtract t₁ t₂ = Some t ⟹ set_of t ⊆ set_of t₂\"\nproof (induct t₁ arbitrary: t₂ t)\n (*goals:\n 1. ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. subtract t₁1 t₂ = Some t ⟹ set_of t ⊆ set_of t₂; ⋀t₂ t. subtract t₁2 t₂ = Some t ⟹ set_of t ⊆ set_of t₂; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t⟧ ⟹ set_of t ⊆ set_of t₂›\n 2. ‹⋀t₂ t. subtract Tip t₂ = Some t ⟹ set_of t ⊆ set_of t₂›*)\n case Tip (*‹subtract Tip t₂ = Some t›*)\n thus \"?case\"\n (*goal: ‹set_of t ⊆ set_of t₂›*)\n by simp\nnext\n (*goal: ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. subtract t₁1 t₂ = Some t ⟹ set_of t ⊆ set_of t₂; ⋀t₂ t. subtract t₁2 t₂ = Some t ⟹ set_of t ⊆ set_of t₂; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t⟧ ⟹ set_of t ⊆ set_of t₂›*)\n case (Node l x b r) (*‹subtract l ?t₂ = Some ?t ⟹ set_of ?t ⊆ set_of ?t₂› ‹subtract r ?t₂ = Some ?t ⟹ set_of ?t ⊆ set_of ?t₂› ‹subtract (Node l x b r) t₂ = Some t›*)\n have sub: \"subtract (Node l x b r) t₂ = Some t\"\n by fact\n show \"?case\"\n (*goal: ‹set_of (t::'a tree) ⊆ set_of (t₂::'a tree)›*)\n proof (cases \"delete x t₂\")\n (*goals:\n 1. ‹delete (x::'a) (t₂::'a tree) = None ⟹ set_of (t::'a tree) ⊆ set_of t₂›\n 2. ‹⋀a::'a tree. delete (x::'a) (t₂::'a tree) = Some a ⟹ set_of (t::'a tree) ⊆ set_of t₂›*)\n case (Some t₂') (*‹delete x t₂ = Some t₂'›*)\n note del_x_Some = this (*‹delete (x::'a) (t₂::'a tree) = Some (t₂'::'a tree)›*)\n from delete_Some_set_of[OF Some] (*‹set_of t₂' ⊆ set_of t₂›*) have t2'_t2: \"set_of t₂' ⊆ set_of t₂\" .\n show \"?thesis\"\n (*goal: ‹set_of t ⊆ set_of t₂›*)\n proof (cases \"subtract l t₂'\")\n (*goals:\n 1. ‹subtract l t₂' = None ⟹ set_of t ⊆ set_of t₂›\n 2. ‹⋀a. subtract l t₂' = Some a ⟹ set_of t ⊆ set_of t₂›*)\n case (Some t₂'') (*‹subtract l t₂' = Some t₂''›*)\n note sub_l_Some = this (*‹subtract l t₂' = Some t₂''›*)\n from Node.hyps(1)[OF Some] (*‹set_of t₂'' ⊆ set_of t₂'›*) have t2''_t2': \"set_of t₂'' ⊆ set_of t₂'\" .\n show \"?thesis\"\n (*goal: ‹set_of t ⊆ set_of t₂›*)\n proof (cases \"subtract r t₂''\")\n (*goals:\n 1. ‹subtract r t₂'' = None ⟹ set_of t ⊆ set_of t₂›\n 2. ‹⋀a. subtract r t₂'' = Some a ⟹ set_of t ⊆ set_of t₂›*)\n case (Some t₂''') (*‹subtract r t₂'' = Some t₂'''›*)\n from Node.hyps(2)[OF Some] (*‹set_of t₂''' ⊆ set_of t₂''›*) have \"set_of t₂''' ⊆ set_of t₂''\" .\n with Some (*‹subtract (r::'a::type tree) (t₂''::'a::type tree) = Some (t₂'''::'a::type tree)›*) sub_l_Some (*‹subtract l t₂' = Some t₂''›*) del_x_Some (*‹delete x t₂ = Some t₂'›*) sub (*‹subtract (Node l x b r) t₂ = Some t›*) t2''_t2' (*‹set_of t₂'' ⊆ set_of t₂'›*) t2'_t2 (*‹set_of (t₂'::'a tree) ⊆ set_of (t₂::'a tree)›*) show \"?thesis\"\n (*goal: ‹set_of t ⊆ set_of t₂›*)\n by simp\n next\n (*goal: ‹subtract r t₂'' = None ⟹ set_of t ⊆ set_of t₂›*)\n case None (*‹subtract r t₂'' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub_l_Some (*‹subtract (l::'a::type tree) (t₂'::'a::type tree) = Some (t₂''::'a::type tree)›*) sub (*‹subtract (Node (l::'a::type tree) (x::'a::type) (b::bool) (r::'a::type tree)) (t₂::'a::type tree) = Some (t::'a::type tree)›*) show \"?thesis\"\n (*goal: ‹set_of t ⊆ set_of t₂›*)\n by simp\n qed\n next\n (*goal: ‹subtract l t₂' = None ⟹ set_of t ⊆ set_of t₂›*)\n case None (*‹subtract l t₂' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub (*‹subtract (Node l x b r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of t ⊆ set_of t₂›*)\n by simp\n qed\n next\n (*goal: ‹delete x t₂ = None ⟹ set_of t ⊆ set_of t₂›*)\n case None (*‹delete x t₂ = None›*)\n with sub (*‹subtract (Node l x b r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of (t::'a::type tree) ⊆ set_of (t₂::'a::type tree)›*)\n by simp\n qed\nqed\n\nlemma subtract_Some_set_of: \n \"subtract t₁ t₂ = Some t ⟹ set_of t₁ ⊆ set_of t₂\"\nproof (induct t₁ arbitrary: t₂ t)\n (*goals:\n 1. ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. subtract t₁1 t₂ = Some t ⟹ set_of t₁1 ⊆ set_of t₂; ⋀t₂ t. subtract t₁2 t₂ = Some t ⟹ set_of t₁2 ⊆ set_of t₂; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t⟧ ⟹ set_of (Node t₁1 x2 x3 t₁2) ⊆ set_of t₂›\n 2. ‹⋀t₂ t. subtract Tip t₂ = Some t ⟹ set_of Tip ⊆ set_of t₂›*)\n case Tip (*‹subtract Tip t₂ = Some t›*)\n thus \"?case\"\n (*goal: ‹set_of Tip ⊆ set_of t₂›*)\n by simp\nnext\n (*goal: ‹⋀(t₁1::'a tree) (x2::'a) (x3::bool) (t₁2::'a tree) (t₂::'a tree) t::'a tree. ⟦⋀(t₂::'a tree) t::'a tree. subtract t₁1 t₂ = Some t ⟹ set_of t₁1 ⊆ set_of t₂; ⋀(t₂::'a tree) t::'a tree. subtract t₁2 t₂ = Some t ⟹ set_of t₁2 ⊆ set_of t₂; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t⟧ ⟹ set_of (Node t₁1 x2 x3 t₁2) ⊆ set_of t₂›*)\n case (Node l x d r) (*‹subtract (l::'a tree) (?t₂::'a tree) = Some (?t::'a tree) ⟹ set_of l ⊆ set_of ?t₂› ‹subtract r ?t₂ = Some ?t ⟹ set_of r ⊆ set_of ?t₂› ‹subtract (Node (l::'a tree) (x::'a) (d::bool) (r::'a tree)) (t₂::'a tree) = Some (t::'a tree)›*)\n have sub: \"subtract (Node l x d r) t₂ = Some t\"\n by fact\n show \"?case\"\n (*goal: ‹set_of (Node l x d r) ⊆ set_of t₂›*)\n proof (cases \"delete x t₂\")\n (*goals:\n 1. ‹delete x t₂ = None ⟹ set_of (Node l x d r) ⊆ set_of t₂›\n 2. ‹⋀a. delete x t₂ = Some a ⟹ set_of (Node l x d r) ⊆ set_of t₂›*)\n case (Some t₂') (*‹delete (x::'a::type) (t₂::'a::type tree) = Some (t₂'::'a::type tree)›*)\n note del_x_Some = this (*‹delete x t₂ = Some t₂'›*)\n from delete_Some_set_of[OF Some] (*‹set_of t₂' ⊆ set_of t₂›*) have t2'_t2: \"set_of t₂' ⊆ set_of t₂\" .\n from delete_None_set_of_conv[of x t₂] (*‹(delete x t₂ = None) = (x ∉ set_of t₂)›*) Some (*‹delete x t₂ = Some t₂'›*) have x_t2: \"x ∈ set_of t₂\"\n by simp\n show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ⊆ set_of t₂›*)\n proof (cases \"subtract l t₂'\")\n (*goals:\n 1. ‹subtract l t₂' = None ⟹ set_of (Node l x d r) ⊆ set_of t₂›\n 2. ‹⋀a. subtract l t₂' = Some a ⟹ set_of (Node l x d r) ⊆ set_of t₂›*)\n case (Some t₂'') (*‹subtract l t₂' = Some t₂''›*)\n note sub_l_Some = this (*‹subtract l t₂' = Some t₂''›*)\n from Node.hyps(1)[OF Some] (*‹set_of l ⊆ set_of t₂'›*) have l_t2': \"set_of l ⊆ set_of t₂'\" .\n from subtract_Some_set_of_res[OF Some] (*‹set_of t₂'' ⊆ set_of t₂'›*) have t2''_t2': \"set_of t₂'' ⊆ set_of t₂'\" .\n show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ⊆ set_of t₂›*)\n proof (cases \"subtract r t₂''\")\n (*goals:\n 1. ‹subtract r t₂'' = None ⟹ set_of (Node l x d r) ⊆ set_of t₂›\n 2. ‹⋀a. subtract r t₂'' = Some a ⟹ set_of (Node l x d r) ⊆ set_of t₂›*)\n case (Some t₂''') (*‹subtract r t₂'' = Some t₂'''›*)\n from Node.hyps(2)[OF Some] (*‹set_of r ⊆ set_of t₂''›*) have \"r_t₂''\": \"set_of r ⊆ set_of t₂''\" .\n from Some (*‹subtract r t₂'' = Some t₂'''›*) sub_l_Some (*‹subtract l t₂' = Some t₂''›*) del_x_Some (*‹delete x t₂ = Some t₂'›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) \"r_t₂''\" (*‹set_of r ⊆ set_of t₂''›*) l_t2' (*‹set_of (l::'a::type tree) ⊆ set_of (t₂'::'a::type tree)›*) t2'_t2 (*‹set_of t₂' ⊆ set_of t₂›*) t2''_t2' (*‹set_of t₂'' ⊆ set_of t₂'›*) x_t2 (*‹x ∈ set_of t₂›*) show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ⊆ set_of t₂›*)\n by auto\n next\n (*goal: ‹subtract (r::'a tree) (t₂''::'a tree) = None ⟹ set_of (Node (l::'a tree) (x::'a) (d::bool) r) ⊆ set_of (t₂::'a tree)›*)\n case None (*‹subtract r t₂'' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub_l_Some (*‹subtract (l::'a tree) (t₂'::'a tree) = Some (t₂''::'a tree)›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of (Node (l::'a tree) (x::'a) (d::bool) (r::'a tree)) ⊆ set_of (t₂::'a tree)›*)\n by simp\n qed\n next\n (*goal: ‹subtract l t₂' = None ⟹ set_of (Node l x d r) ⊆ set_of t₂›*)\n case None (*‹subtract l t₂' = None›*)\n with del_x_Some (*‹delete (x::'a::type) (t₂::'a::type tree) = Some (t₂'::'a::type tree)›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of (Node (l::'a tree) (x::'a) (d::bool) (r::'a tree)) ⊆ set_of (t₂::'a tree)›*)\n by simp\n qed\n next\n (*goal: ‹delete x t₂ = None ⟹ set_of (Node l x d r) ⊆ set_of t₂›*)\n case None (*‹delete x t₂ = None›*)\n with sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ⊆ set_of t₂›*)\n by simp\n qed\nqed\n\nlemma subtract_Some_all_distinct_res: \n \"subtract t₁ t₂ = Some t ⟹ all_distinct t₂ ⟹ all_distinct t\"\nproof (induct t₁ arbitrary: t₂ t)\n (*goals:\n 1. ‹⋀(t₁1::'a::type tree) (x2::'a::type) (x3::bool) (t₁2::'a::type tree) (t₂::'a::type tree) t::'a::type tree. ⟦⋀(t₂::'a::type tree) t::'a::type tree. ⟦subtract t₁1 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t; ⋀(t₂::'a::type tree) t::'a::type tree. ⟦subtract t₁2 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t›\n 2. ‹⋀(t₂::'a::type tree) t::'a::type tree. ⟦subtract Tip t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t›*)\n case Tip (*‹subtract Tip t₂ = Some t› ‹all_distinct t₂›*)\n thus \"?case\"\n (*goal: ‹all_distinct t›*)\n by simp\nnext\n (*goal: ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. ⟦subtract t₁1 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t; ⋀t₂ t. ⟦subtract t₁2 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t›*)\n case (Node l x d r) (*‹⟦subtract l ?t₂ = Some ?t; all_distinct ?t₂⟧ ⟹ all_distinct ?t› ‹⟦subtract r ?t₂ = Some ?t; all_distinct ?t₂⟧ ⟹ all_distinct ?t› ‹subtract (Node (l::'a tree) (x::'a) (d::bool) (r::'a tree)) (t₂::'a tree) = Some (t::'a tree)› ‹all_distinct t₂›*)\n have sub: \"subtract (Node l x d r) t₂ = Some t\"\n by fact\n have dist_t2: \"all_distinct t₂\"\n by fact\n show \"?case\"\n (*goal: ‹all_distinct t›*)\n proof (cases \"delete x t₂\")\n (*goals:\n 1. ‹delete (x::'a::type) (t₂::'a::type tree) = None ⟹ all_distinct (t::'a::type tree)›\n 2. ‹⋀a::'a::type tree. delete (x::'a::type) (t₂::'a::type tree) = Some a ⟹ all_distinct (t::'a::type tree)›*)\n case (Some t₂') (*‹delete (x::'a) (t₂::'a tree) = Some (t₂'::'a tree)›*)\n note del_x_Some = this (*‹delete x t₂ = Some t₂'›*)\n from delete_Some_all_distinct[OF Some dist_t2] (*‹all_distinct (t₂'::'a::type tree)›*) have dist_t2': \"all_distinct t₂'\" .\n show \"?thesis\"\n (*goal: ‹all_distinct (t::'a tree)›*)\n proof (cases \"subtract l t₂'\")\n (*goals:\n 1. ‹subtract l t₂' = None ⟹ all_distinct t›\n 2. ‹⋀a. subtract l t₂' = Some a ⟹ all_distinct t›*)\n case (Some t₂'') (*‹subtract l t₂' = Some t₂''›*)\n note sub_l_Some = this (*‹subtract l t₂' = Some t₂''›*)\n from Node.hyps(1)[OF Some dist_t2'] (*‹all_distinct t₂''›*) have dist_t2'': \"all_distinct t₂''\" .\n show \"?thesis\"\n (*goal: ‹all_distinct (t::'a tree)›*)\n proof (cases \"subtract r t₂''\")\n (*goals:\n 1. ‹subtract r t₂'' = None ⟹ all_distinct t›\n 2. ‹⋀a. subtract r t₂'' = Some a ⟹ all_distinct t›*)\n case (Some t₂''') (*‹subtract r t₂'' = Some t₂'''›*)\n from Node.hyps(2)[OF Some dist_t2''] (*‹all_distinct t₂'''›*) have dist_t2''': \"all_distinct t₂'''\" .\n from Some (*‹subtract r t₂'' = Some t₂'''›*) sub_l_Some (*‹subtract l t₂' = Some t₂''›*) del_x_Some (*‹delete (x::'a) (t₂::'a tree) = Some (t₂'::'a tree)›*) sub (*‹subtract (Node (l::'a::type tree) (x::'a::type) (d::bool) (r::'a::type tree)) (t₂::'a::type tree) = Some (t::'a::type tree)›*) dist_t2''' (*‹all_distinct t₂'''›*) show \"?thesis\"\n (*goal: ‹all_distinct t›*)\n by simp\n next\n (*goal: ‹subtract r t₂'' = None ⟹ all_distinct t›*)\n case None (*‹subtract r t₂'' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub_l_Some (*‹subtract l t₂' = Some t₂''›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹all_distinct t›*)\n by simp\n qed\n next\n (*goal: ‹subtract l t₂' = None ⟹ all_distinct t›*)\n case None (*‹subtract l t₂' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹all_distinct t›*)\n by simp\n qed\n next\n (*goal: ‹delete (x::'a) (t₂::'a tree) = None ⟹ all_distinct (t::'a tree)›*)\n case None (*‹delete x t₂ = None›*)\n with sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹all_distinct t›*)\n by simp\n qed\nqed\n\n\nlemma subtract_Some_dist_res: \n \"subtract t₁ t₂ = Some t ⟹ set_of t₁ ∩ set_of t = {}\"\nproof (induct t₁ arbitrary: t₂ t)\n (*goals:\n 1. ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. subtract t₁1 t₂ = Some t ⟹ set_of t₁1 ∩ set_of t = {}; ⋀t₂ t. subtract t₁2 t₂ = Some t ⟹ set_of t₁2 ∩ set_of t = {}; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t⟧ ⟹ set_of (Node t₁1 x2 x3 t₁2) ∩ set_of t = {}›\n 2. ‹⋀t₂ t. subtract Tip t₂ = Some t ⟹ set_of Tip ∩ set_of t = {}›*)\n case Tip (*‹subtract Tip (t₂::'a tree) = Some (t::'a tree)›*)\n thus \"?case\"\n (*goal: ‹set_of Tip ∩ set_of t = {}›*)\n by simp\nnext\n (*goal: ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. subtract t₁1 t₂ = Some t ⟹ set_of t₁1 ∩ set_of t = {}; ⋀t₂ t. subtract t₁2 t₂ = Some t ⟹ set_of t₁2 ∩ set_of t = {}; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t⟧ ⟹ set_of (Node t₁1 x2 x3 t₁2) ∩ set_of t = {}›*)\n case (Node l x d r) (*‹subtract l ?t₂ = Some ?t ⟹ set_of l ∩ set_of ?t = {}› ‹subtract r ?t₂ = Some ?t ⟹ set_of r ∩ set_of ?t = {}› ‹subtract (Node l x d r) t₂ = Some t›*)\n have sub: \"subtract (Node l x d r) t₂ = Some t\"\n by fact\n show \"?case\"\n (*goal: ‹set_of (Node l x d r) ∩ set_of t = {}›*)\n proof (cases \"delete x t₂\")\n (*goals:\n 1. ‹delete x t₂ = None ⟹ set_of (Node l x d r) ∩ set_of t = {}›\n 2. ‹⋀a. delete x t₂ = Some a ⟹ set_of (Node l x d r) ∩ set_of t = {}›*)\n case (Some t₂') (*‹delete (x::'a) (t₂::'a tree) = Some (t₂'::'a tree)›*)\n note del_x_Some = this (*‹delete x t₂ = Some t₂'›*)\n from delete_Some_x_set_of[OF Some] (*‹x ∈ set_of t₂ ∧ x ∉ set_of t₂'›*) obtain x_t2: \"x ∈ set_of t₂\" and x_not_t2': \"x ∉ set_of t₂'\"\n (*goal: ‹(⟦x ∈ set_of t₂; x ∉ set_of t₂'⟧ ⟹ thesis) ⟹ thesis›*)\n by simp\n from delete_Some_set_of[OF Some] (*‹set_of t₂' ⊆ set_of t₂›*) have t2'_t2: \"set_of t₂' ⊆ set_of t₂\" .\n show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ∩ set_of t = {}›*)\n proof (cases \"subtract l t₂'\")\n (*goals:\n 1. ‹subtract (l::'a tree) (t₂'::'a tree) = None ⟹ set_of (Node l (x::'a) (d::bool) (r::'a tree)) ∩ set_of (t::'a tree) = {}›\n 2. ‹⋀a::'a tree. subtract (l::'a tree) (t₂'::'a tree) = Some a ⟹ set_of (Node l (x::'a) (d::bool) (r::'a tree)) ∩ set_of (t::'a tree) = {}›*)\n case (Some t₂'') (*‹subtract l t₂' = Some t₂''›*)\n note sub_l_Some = this (*‹subtract (l::'a tree) (t₂'::'a tree) = Some (t₂''::'a tree)›*)\n from Node.hyps(1)[OF Some] (*‹set_of l ∩ set_of t₂'' = {}›*) have dist_l_t2'': \"set_of l ∩ set_of t₂'' = {}\" .\n from subtract_Some_set_of_res[OF Some] (*‹set_of t₂'' ⊆ set_of t₂'›*) have t2''_t2': \"set_of t₂'' ⊆ set_of t₂'\" .\n show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ∩ set_of t = {}›*)\n proof (cases \"subtract r t₂''\")\n (*goals:\n 1. ‹subtract r t₂'' = None ⟹ set_of (Node l x d r) ∩ set_of t = {}›\n 2. ‹⋀a. subtract r t₂'' = Some a ⟹ set_of (Node l x d r) ∩ set_of t = {}›*)\n case (Some t₂''') (*‹subtract r t₂'' = Some t₂'''›*)\n from Node.hyps(2)[OF Some] (*‹set_of r ∩ set_of t₂''' = {}›*) have dist_r_t2''': \"set_of r ∩ set_of t₂''' = {}\" .\n from subtract_Some_set_of_res[OF Some] (*‹set_of (t₂'''::'a::type tree) ⊆ set_of (t₂''::'a::type tree)›*) have t2'''_t2'': \"set_of t₂''' ⊆ set_of t₂''\" .\n from Some (*‹subtract r t₂'' = Some t₂'''›*) sub_l_Some (*‹subtract l t₂' = Some t₂''›*) del_x_Some (*‹delete x t₂ = Some t₂'›*) sub (*‹subtract (Node (l::'a tree) (x::'a) (d::bool) (r::'a tree)) (t₂::'a tree) = Some (t::'a tree)›*) t2'''_t2'' (*‹set_of t₂''' ⊆ set_of t₂''›*) dist_l_t2'' (*‹set_of l ∩ set_of t₂'' = {}›*) dist_r_t2''' (*‹set_of r ∩ set_of t₂''' = {}›*) t2''_t2' (*‹set_of t₂'' ⊆ set_of t₂'›*) t2'_t2 (*‹set_of t₂' ⊆ set_of t₂›*) x_not_t2' (*‹x ∉ set_of t₂'›*) show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ∩ set_of t = {}›*)\n by auto\n next\n (*goal: ‹subtract (r::'a::type tree) (t₂''::'a::type tree) = None ⟹ set_of (Node (l::'a::type tree) (x::'a::type) (d::bool) r) ∩ set_of (t::'a::type tree) = {}›*)\n case None (*‹subtract r t₂'' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub_l_Some (*‹subtract (l::'a tree) (t₂'::'a tree) = Some (t₂''::'a tree)›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ∩ set_of t = {}›*)\n by simp\n qed\n next\n (*goal: ‹subtract l t₂' = None ⟹ set_of (Node l x d r) ∩ set_of t = {}›*)\n case None (*‹subtract l t₂' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ∩ set_of t = {}›*)\n by simp\n qed\n next\n (*goal: ‹delete x t₂ = None ⟹ set_of (Node l x d r) ∩ set_of t = {}›*)\n case None (*‹delete x t₂ = None›*)\n with sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹set_of (Node l x d r) ∩ set_of t = {}›*)\n by simp\n qed\nqed\n \nlemma subtract_Some_all_distinct:\n \"subtract t₁ t₂ = Some t ⟹ all_distinct t₂ ⟹ all_distinct t₁\"\nproof (induct t₁ arbitrary: t₂ t)\n (*goals:\n 1. ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. ⟦subtract t₁1 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t₁1; ⋀t₂ t. ⟦subtract t₁2 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t₁2; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct (Node t₁1 x2 x3 t₁2)›\n 2. ‹⋀t₂ t. ⟦subtract Tip t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct Tip›*)\n case Tip (*‹subtract Tip t₂ = Some t› ‹all_distinct t₂›*)\n thus \"?case\"\n (*goal: ‹all_distinct Tip›*)\n by simp\nnext\n (*goal: ‹⋀t₁1 x2 x3 t₁2 t₂ t. ⟦⋀t₂ t. ⟦subtract t₁1 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t₁1; ⋀t₂ t. ⟦subtract t₁2 t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct t₁2; subtract (Node t₁1 x2 x3 t₁2) t₂ = Some t; all_distinct t₂⟧ ⟹ all_distinct (Node t₁1 x2 x3 t₁2)›*)\n case (Node l x d r) (*‹⟦subtract l ?t₂ = Some ?t; all_distinct ?t₂⟧ ⟹ all_distinct l› ‹⟦subtract r ?t₂ = Some ?t; all_distinct ?t₂⟧ ⟹ all_distinct r› ‹subtract (Node l x d r) t₂ = Some t› ‹all_distinct t₂›*)\n have sub: \"subtract (Node l x d r) t₂ = Some t\"\n by fact\n have dist_t2: \"all_distinct t₂\"\n by fact\n show \"?case\"\n (*goal: ‹all_distinct (Node l x d r)›*)\n proof (cases \"delete x t₂\")\n (*goals:\n 1. ‹delete x t₂ = None ⟹ all_distinct (Node l x d r)›\n 2. ‹⋀a. delete x t₂ = Some a ⟹ all_distinct (Node l x d r)›*)\n case (Some t₂') (*‹delete x t₂ = Some t₂'›*)\n note del_x_Some = this (*‹delete x t₂ = Some t₂'›*)\n from delete_Some_all_distinct[OF Some dist_t2] (*‹all_distinct t₂'›*) have dist_t2': \"all_distinct t₂'\" .\n from delete_Some_set_of[OF Some] (*‹set_of (t₂'::'a tree) ⊆ set_of (t₂::'a tree)›*) have t2'_t2: \"set_of t₂' ⊆ set_of t₂\" .\n from delete_Some_x_set_of[OF Some] (*‹x ∈ set_of t₂ ∧ x ∉ set_of t₂'›*) obtain x_t2: \"x ∈ set_of t₂\" and x_not_t2': \"x ∉ set_of t₂'\"\n (*goal: ‹(⟦(x::'a::type) ∈ set_of (t₂::'a::type tree); x ∉ set_of (t₂'::'a::type tree)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by simp\n show \"?thesis\"\n (*goal: ‹all_distinct (Node (l::'a tree) (x::'a) (d::bool) (r::'a tree))›*)\n proof (cases \"subtract l t₂'\")\n (*goals:\n 1. ‹subtract (l::'a::type tree) (t₂'::'a::type tree) = None ⟹ all_distinct (Node l (x::'a::type) (d::bool) (r::'a::type tree))›\n 2. ‹⋀a::'a::type tree. subtract (l::'a::type tree) (t₂'::'a::type tree) = Some a ⟹ all_distinct (Node l (x::'a::type) (d::bool) (r::'a::type tree))›*)\n case (Some t₂'') (*‹subtract l t₂' = Some t₂''›*)\n note sub_l_Some = this (*‹subtract l t₂' = Some t₂''›*)\n from Node.hyps(1)[OF Some dist_t2'] (*‹all_distinct l›*) have dist_l: \"all_distinct l\" .\n from subtract_Some_all_distinct_res[OF Some dist_t2'] (*‹all_distinct t₂''›*) have dist_t2'': \"all_distinct t₂''\" .\n from subtract_Some_set_of[OF Some] (*‹set_of l ⊆ set_of t₂'›*) have l_t2': \"set_of l ⊆ set_of t₂'\" .\n from subtract_Some_set_of_res[OF Some] (*‹set_of t₂'' ⊆ set_of t₂'›*) have t2''_t2': \"set_of t₂'' ⊆ set_of t₂'\" .\n from subtract_Some_dist_res[OF Some] (*‹set_of l ∩ set_of t₂'' = {}›*) have dist_l_t2'': \"set_of l ∩ set_of t₂'' = {}\" .\n show \"?thesis\"\n (*goal: ‹all_distinct (Node l x d r)›*)\n proof (cases \"subtract r t₂''\")\n (*goals:\n 1. ‹subtract r t₂'' = None ⟹ all_distinct (Node l x d r)›\n 2. ‹⋀a. subtract r t₂'' = Some a ⟹ all_distinct (Node l x d r)›*)\n case (Some t₂''') (*‹subtract r t₂'' = Some t₂'''›*)\n from Node.hyps(2)[OF Some dist_t2''] (*‹all_distinct r›*) have dist_r: \"all_distinct r\" .\n from subtract_Some_set_of[OF Some] (*‹set_of r ⊆ set_of t₂''›*) have r_t2'': \"set_of r ⊆ set_of t₂''\" .\n from subtract_Some_dist_res[OF Some] (*‹set_of r ∩ set_of t₂''' = {}›*) have dist_r_t2''': \"set_of r ∩ set_of t₂''' = {}\" .\n from dist_l (*‹all_distinct l›*) dist_r (*‹all_distinct r›*) Some (*‹subtract (r::'a tree) (t₂''::'a tree) = Some (t₂'''::'a tree)›*) sub_l_Some (*‹subtract l t₂' = Some t₂''›*) del_x_Some (*‹delete (x::'a::type) (t₂::'a::type tree) = Some (t₂'::'a::type tree)›*) r_t2'' (*‹set_of (r::'a::type tree) ⊆ set_of (t₂''::'a::type tree)›*) l_t2' (*‹set_of l ⊆ set_of t₂'›*) x_t2 (*‹(x::'a::type) ∈ set_of (t₂::'a::type tree)›*) x_not_t2' (*‹x ∉ set_of t₂'›*) t2''_t2' (*‹set_of t₂'' ⊆ set_of t₂'›*) dist_l_t2'' (*‹set_of l ∩ set_of t₂'' = {}›*) dist_r_t2''' (*‹set_of r ∩ set_of t₂''' = {}›*) show \"?thesis\"\n (*goal: ‹all_distinct (Node l x d r)›*)\n by auto\n next\n (*goal: ‹subtract (r::'a tree) (t₂''::'a tree) = None ⟹ all_distinct (Node (l::'a tree) (x::'a) (d::bool) r)›*)\n case None (*‹subtract (r::'a::type tree) (t₂''::'a::type tree) = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub_l_Some (*‹subtract l t₂' = Some t₂''›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹all_distinct (Node l x d r)›*)\n by simp\n qed\n next\n (*goal: ‹subtract l t₂' = None ⟹ all_distinct (Node l x d r)›*)\n case None (*‹subtract l t₂' = None›*)\n with del_x_Some (*‹delete x t₂ = Some t₂'›*) sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹all_distinct (Node (l::'a tree) (x::'a) (d::bool) (r::'a tree))›*)\n by simp\n qed\n next\n (*goal: ‹delete x t₂ = None ⟹ all_distinct (Node l x d r)›*)\n case None (*‹delete (x::'a) (t₂::'a tree) = None›*)\n with sub (*‹subtract (Node l x d r) t₂ = Some t›*) show \"?thesis\"\n (*goal: ‹all_distinct (Node l x d r)›*)\n by simp\n qed\nqed\n\n\nlemma delete_left:\n assumes dist: \"all_distinct (Node l y d r)\" \n assumes del_l: \"delete x l = Some l'\"\n shows \"delete x (Node l y d r) = Some (Node l' y d r)\"\nproof (-)\n (*goal: ‹delete x (Node l y d r) = Some (Node l' y d r)›*)\n from delete_Some_x_set_of[OF del_l] (*‹x ∈ set_of l ∧ x ∉ set_of l'›*) obtain x: \"x ∈ set_of l\"\n (*goal: ‹(x ∈ set_of l ⟹ thesis) ⟹ thesis›*)\n by simp\n with dist (*‹all_distinct (Node (l::'a::type tree) (y::'a::type) (d::bool) (r::'a::type tree))›*) have \"delete x r = None\"\n apply (cases \"delete x r\")\n (*goals:\n 1. ‹⟦all_distinct (Node l y d r); x ∈ set_of l; delete x r = None⟧ ⟹ delete x r = None›\n 2. ‹⋀a. ⟦all_distinct (Node l y d r); x ∈ set_of l; delete x r = Some a⟧ ⟹ delete x r = None›\n discuss goal 1*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete ?x ?t = Some ?t' ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*discuss goal 2*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete ?x ?t = Some ?t' ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*proven 2 subgoals*) .\n with x (*‹x ∈ set_of l›*) show \"?thesis\"\n (*goal: ‹delete x (Node l y d r) = Some (Node l' y d r)›*)\n using del_l (*‹delete x l = Some l'›*) dist (*‹all_distinct (Node l y d r)›*) by (auto split: option.splits (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\nqed\n\nlemma delete_right:\n assumes dist: \"all_distinct (Node l y d r)\" \n assumes del_r: \"delete x r = Some r'\"\n shows \"delete x (Node l y d r) = Some (Node l y d r')\"\nproof (-)\n (*goal: ‹delete x (Node l y d r) = Some (Node l y d r')›*)\n from delete_Some_x_set_of[OF del_r] (*‹x ∈ set_of r ∧ x ∉ set_of r'›*) obtain x: \"x ∈ set_of r\"\n (*goal: ‹(x ∈ set_of r ⟹ thesis) ⟹ thesis›*)\n by simp\n with dist (*‹all_distinct (Node l y d r)›*) have \"delete x l = None\"\n apply (cases \"delete x l\")\n (*goals:\n 1. ‹⟦all_distinct (Node l y d r); x ∈ set_of r; delete x l = None⟧ ⟹ delete x l = None›\n 2. ‹⋀a. ⟦all_distinct (Node l y d r); x ∈ set_of r; delete x l = Some a⟧ ⟹ delete x l = None›\n discuss goal 1*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete (?x::?'a) (?t::?'a tree) = Some (?t'::?'a tree) ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*discuss goal 2*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete ?x ?t = Some ?t' ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*proven 2 subgoals*) .\n with x (*‹x ∈ set_of r›*) show \"?thesis\"\n (*goal: ‹delete x (Node l y d r) = Some (Node l y d r')›*)\n using del_r (*‹delete x r = Some r'›*) dist (*‹all_distinct (Node l y d r)›*) by (auto split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\nqed\n\nlemma delete_root: \n assumes dist: \"all_distinct (Node l x False r)\" \n shows \"delete x (Node l x False r) = Some (Node l x True r)\"\nproof (-)\n (*goal: ‹delete x (Node l x False r) = Some (Node l x True r)›*)\n from dist (*‹all_distinct (Node l x False r)›*) have \"delete x r = None\"\n apply (cases \"delete x r\")\n (*goals:\n 1. ‹⟦all_distinct (Node (l::'a tree) (x::'a) False (r::'a tree)); delete x r = None⟧ ⟹ delete x r = None›\n 2. ‹⋀a::'a tree. ⟦all_distinct (Node (l::'a tree) (x::'a) False (r::'a tree)); delete x r = Some a⟧ ⟹ delete x r = None›\n discuss goal 1*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete ?x ?t = Some ?t' ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*discuss goal 2*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete ?x ?t = Some ?t' ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*proven 2 subgoals*) .\n moreover from dist (*‹all_distinct (Node l x False r)›*) have \"delete x l = None\"\n apply (cases \"delete x l\")\n (*goals:\n 1. ‹⟦all_distinct (Node l x False r); delete x l = None⟧ ⟹ delete x l = None›\n 2. ‹⋀a. ⟦all_distinct (Node l x False r); delete x l = Some a⟧ ⟹ delete x l = None›\n discuss goal 1*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete ?x ?t = Some ?t' ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*discuss goal 2*)\n apply ((auto dest:delete_Some_x_set_of (*‹delete (?x::?'a::type) (?t::?'a::type tree) = Some (?t'::?'a::type tree) ⟹ ?x ∈ set_of ?t ∧ ?x ∉ set_of ?t'›*))[1])\n (*proven 2 subgoals*) .\n ultimately show \"?thesis\"\n (*goal: ‹delete x (Node l x False r) = Some (Node l x True r)›*)\n using dist (*‹all_distinct (Node l x False r)›*) by (auto split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\nqed \n\nlemma subtract_Node:\n assumes del: \"delete x t = Some t'\" \n assumes sub_l: \"subtract l t' = Some t''\"\n assumes sub_r: \"subtract r t'' = Some t'''\"\n shows \"subtract (Node l x False r) t = Some t'''\"\nusing del (*‹delete (x::'a::type) (t::'a::type tree) = Some (t'::'a::type tree)›*) sub_l (*‹subtract l t' = Some t''›*) sub_r (*‹subtract (r::'a::type tree) (t''::'a::type tree) = Some (t'''::'a::type tree)›*) by simp\n\nlemma subtract_Tip: \"subtract Tip t = Some t\"\n by simp\n \ntext ‹Now we have all the theorems in place that are needed for the\ncertificate generating ML functions.›\n\nML_file ‹distinct_tree_prover.ML›\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/Statespace/DistinctTreeProver.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"0c52313faa5cbb9f00a0a692731277a8deccc2b21838a37e2b53ecc4a7c98e57\"\n}"}}},{"rowIdx":179,"cells":{"text":{"kind":"string","value":"(*\n Authors: Jose Divasón\n Sebastiaan Joosten\n René Thiemann\n Akihisa Yamada\n License: BSD\n*)\n\nsection ‹List representation›\n\ntheory List_Representation\n imports Main\nbegin\n\nlemma rev_take_Suc: assumes j: \"j < length xs\"\n shows \"rev (take (Suc j) xs) = xs ! j # rev (take j xs)\" \nproof (-)\n (*goal: ‹rev (take (Suc j) xs) = xs ! j # rev (take j xs)›*)\n from j (*‹j < length xs›*) have xs: \"xs = take j xs @ xs ! j # drop (Suc j) xs\"\n by (rule id_take_nth_drop (*‹?i < length ?xs ⟹ ?xs = take ?i ?xs @ ?xs ! ?i # drop (Suc ?i) ?xs›*))\n show \"?thesis\"\n (*goal: ‹rev (take (Suc j) xs) = xs ! j # rev (take j xs)›*)\n unfolding arg_cong[OF xs, of \"λ xs. rev (take (Suc j) xs)\"]\n (*goal: ‹rev (take (Suc (j::nat)) (take j (xs::'a list) @ xs ! j # drop (Suc j) xs)) = xs ! j # rev (take j xs)›*)\n by (simp add: min_def (*‹min ?a ?b = (if ?a ≤ ?b then ?a else ?b)›*))\nqed\n \ntype_synonym 'a list_repr = \"'a list × 'a list\" \n\ndefinition list_repr :: \"nat ⇒ 'a list_repr ⇒ 'a list ⇒ bool\" where\n \"list_repr i ba xs = (i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs)\" \n\ndefinition of_list_repr :: \"'a list_repr ⇒ 'a list\" where\n \"of_list_repr ba = (rev (fst ba) @ snd ba)\" \n \nlemma of_list_repr: \"list_repr i ba xs ⟹ of_list_repr ba = xs\" \n unfolding of_list_repr_def list_repr_def\n (*goal: ‹(i::nat) ≤ length (xs::'a list) ∧ fst (ba::'a list × 'a list) = rev (take i xs) ∧ snd ba = drop i xs ⟹ rev (fst ba) @ snd ba = xs›*)\n by auto\n \ndefinition get_nth_i :: \"'a list_repr ⇒ 'a\" where\n \"get_nth_i ba = hd (snd ba)\" \n\ndefinition get_nth_im1 :: \"'a list_repr ⇒ 'a\" where\n \"get_nth_im1 ba = hd (fst ba)\" \n \nlemma get_nth_i: \"list_repr i ba xs ⟹ i < length xs ⟹ get_nth_i ba = xs ! i\" \n unfolding list_repr_def get_nth_i_def\n (*goal: ‹⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i < length xs⟧ ⟹ hd (snd ba) = xs ! i›*)\n by (auto simp: hd_drop_conv_nth (*‹?n < length ?xs ⟹ hd (drop ?n ?xs) = ?xs ! ?n›*)) \n\nlemma get_nth_im1: \"list_repr i ba xs ⟹ i ≠ 0 ⟹ get_nth_im1 ba = xs ! (i - 1)\" \n unfolding list_repr_def get_nth_im1_def\n (*goal: ‹⟦(i::nat) ≤ length (xs::'a::type list) ∧ fst (ba::'a::type list × 'a::type list) = rev (take i xs) ∧ snd ba = drop i xs; i ≠ (0::nat)⟧ ⟹ hd (fst ba) = xs ! (i - (1::nat))›*)\n apply (cases i)\n (*goals:\n 1. ‹⟦(i::nat) ≤ length (xs::'a::type list) ∧ fst (ba::'a::type list × 'a::type list) = rev (take i xs) ∧ snd ba = drop i xs; i ≠ (0::nat); i = (0::nat)⟧ ⟹ hd (fst ba) = xs ! (i - (1::nat))›\n 2. ‹⋀nat::nat. ⟦(i::nat) ≤ length (xs::'a::type list) ∧ fst (ba::'a::type list × 'a::type list) = rev (take i xs) ∧ snd ba = drop i xs; i ≠ (0::nat); i = Suc nat⟧ ⟹ hd (fst ba) = xs ! (i - (1::nat))›\n discuss goal 1*)\n apply ((auto simp: rev_take_Suc (*‹?j < length ?xs ⟹ rev (take (Suc ?j) ?xs) = ?xs ! ?j # rev (take ?j ?xs)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: rev_take_Suc (*‹?j < length ?xs ⟹ rev (take (Suc ?j) ?xs) = ?xs ! ?j # rev (take ?j ?xs)›*))[1])\n (*proven 2 subgoals*) . \n \ndefinition update_i :: \"'a list_repr ⇒ 'a ⇒ 'a list_repr\" where\n \"update_i ba x = (fst ba, x # tl (snd ba))\" \n \nlemma Cons_tl_drop_update: \"i < length xs ⟹ x # tl (drop i xs) = drop i (xs[i := x])\"\nproof (induct i arbitrary: xs)\n (*goals:\n 1. ‹⋀xs. 0 < length xs ⟹ x # tl (drop 0 xs) = drop 0 (xs[0 := x])›\n 2. ‹⋀i xs. ⟦⋀xs. i < length xs ⟹ x # tl (drop i xs) = drop i (xs[i := x]); Suc i < length xs⟧ ⟹ x # tl (drop (Suc i) xs) = drop (Suc i) (xs[Suc i := x])›*)\n case (0 xs) (*‹0 < length xs›*)\n thus \"?case\"\n (*goal: ‹(x::'a) # tl (drop (0::nat) (xs::'a list)) = drop (0::nat) (xs[0::nat := x])›*)\n apply (cases xs)\n (*goals:\n 1. ‹⟦0 < length xs; xs = []⟧ ⟹ x # tl (drop 0 xs) = drop 0 (xs[0 := x])›\n 2. ‹⋀a list. ⟦0 < length xs; xs = a # list⟧ ⟹ x # tl (drop 0 xs) = drop 0 (xs[0 := x])›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nnext\n (*goal: ‹⋀i xs. ⟦⋀xs. i < length xs ⟹ x # tl (drop i xs) = drop i (xs[i := x]); Suc i < length xs⟧ ⟹ x # tl (drop (Suc i) xs) = drop (Suc i) (xs[Suc i := x])›*)\n case (Suc i xs) (*‹(i::nat) < length (?xs::'a list) ⟹ (x::'a) # tl (drop i ?xs) = drop i (?xs[i := x])› ‹Suc i < length xs›*)\n thus \"?case\"\n (*goal: ‹x # tl (drop (Suc i) xs) = drop (Suc i) (xs[Suc i := x])›*)\n apply (cases xs)\n (*goals:\n 1. ‹⟦⋀xs. i < length xs ⟹ x # tl (drop i xs) = drop i (xs[i := x]); Suc i < length xs; xs = []⟧ ⟹ x # tl (drop (Suc i) xs) = drop (Suc i) (xs[Suc i := x])›\n 2. ‹⋀a list. ⟦⋀xs. i < length xs ⟹ x # tl (drop i xs) = drop i (xs[i := x]); Suc i < length xs; xs = a # list⟧ ⟹ x # tl (drop (Suc i) xs) = drop (Suc i) (xs[Suc i := x])›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nqed\n \nlemma update_i: \"list_repr i ba xs ⟹ i < length xs ⟹ list_repr i (update_i ba x) (xs [i := x])\" \n unfolding update_i_def list_repr_def\n (*goal: ‹⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i < length xs⟧ ⟹ i ≤ length (xs[i := x]) ∧ fst (fst ba, x # tl (snd ba)) = rev (take i (xs[i := x])) ∧ snd (fst ba, x # tl (snd ba)) = drop i (xs[i := x])›*)\n by (auto simp: Cons_tl_drop_update (*‹?i < length ?xs ⟹ ?x # tl (drop ?i ?xs) = drop ?i (?xs[?i := ?x])›*))\n \ndefinition update_im1 :: \"'a list_repr ⇒ 'a ⇒ 'a list_repr\" where\n \"update_im1 ba x = (x # tl (fst ba), snd ba)\" \n\nlemma update_im1: \"list_repr i ba xs ⟹ i ≠ 0 ⟹ list_repr i (update_im1 ba x) (xs [i - 1 := x])\" \n unfolding update_im1_def list_repr_def\n (*goal: ‹⟦(i::nat) ≤ length (xs::'a list) ∧ fst (ba::'a list × 'a list) = rev (take i xs) ∧ snd ba = drop i xs; i ≠ (0::nat)⟧ ⟹ i ≤ length (xs[i - (1::nat) := x::'a]) ∧ fst (x # tl (fst ba), snd ba) = rev (take i (xs[i - (1::nat) := x])) ∧ snd (x # tl (fst ba), snd ba) = drop i (xs[i - (1::nat) := x])›*)\n apply (cases i)\n (*goals:\n 1. ‹⟦(i::nat) ≤ length (xs::'a list) ∧ fst (ba::'a list × 'a list) = rev (take i xs) ∧ snd ba = drop i xs; i ≠ (0::nat); i = (0::nat)⟧ ⟹ i ≤ length (xs[i - (1::nat) := x::'a]) ∧ fst (x # tl (fst ba), snd ba) = rev (take i (xs[i - (1::nat) := x])) ∧ snd (x # tl (fst ba), snd ba) = drop i (xs[i - (1::nat) := x])›\n 2. ‹⋀nat::nat. ⟦(i::nat) ≤ length (xs::'a list) ∧ fst (ba::'a list × 'a list) = rev (take i xs) ∧ snd ba = drop i xs; i ≠ (0::nat); i = Suc nat⟧ ⟹ i ≤ length (xs[i - (1::nat) := x::'a]) ∧ fst (x # tl (fst ba), snd ba) = rev (take i (xs[i - (1::nat) := x])) ∧ snd (x # tl (fst ba), snd ba) = drop i (xs[i - (1::nat) := x])›\n discuss goal 1*)\n apply ((auto simp: rev_take_Suc (*‹(?j::nat) < length (?xs::?'a::type list) ⟹ rev (take (Suc ?j) ?xs) = ?xs ! ?j # rev (take ?j ?xs)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: rev_take_Suc (*‹?j < length ?xs ⟹ rev (take (Suc ?j) ?xs) = ?xs ! ?j # rev (take ?j ?xs)›*))[1])\n (*proven 2 subgoals*) .\n \nlemma tl_drop_Suc: \"tl (drop i xs) = drop (Suc i) xs\"\nproof (induct i arbitrary: xs)\n (*goals:\n 1. ‹⋀xs. tl (drop 0 xs) = drop (Suc 0) xs›\n 2. ‹⋀i xs. (⋀xs. tl (drop i xs) = drop (Suc i) xs) ⟹ tl (drop (Suc i) xs) = drop (Suc (Suc i)) xs›*)\n case (0 xs) (*no hyothesis introduced yet*)\n thus \"?case\"\n (*goal: ‹tl (drop 0 xs) = drop (Suc 0) xs›*)\n apply (cases xs)\n (*goals:\n 1. ‹xs = [] ⟹ tl (drop 0 xs) = drop (Suc 0) xs›\n 2. ‹⋀a list. xs = a # list ⟹ tl (drop 0 xs) = drop (Suc 0) xs›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nnext\n (*goal: ‹⋀i xs. (⋀xs. tl (drop i xs) = drop (Suc i) xs) ⟹ tl (drop (Suc i) xs) = drop (Suc (Suc i)) xs›*)\n case (Suc i xs) (*‹tl (drop i ?xs) = drop (Suc i) ?xs›*)\n thus \"?case\"\n (*goal: ‹tl (drop (Suc i) xs) = drop (Suc (Suc i)) xs›*)\n apply (cases xs)\n (*goals:\n 1. ‹⟦⋀xs::'a list. tl (drop (i::nat) xs) = drop (Suc i) xs; (xs::'a list) = []⟧ ⟹ tl (drop (Suc i) xs) = drop (Suc (Suc i)) xs›\n 2. ‹⋀(a::'a) list::'a list. ⟦⋀xs::'a list. tl (drop (i::nat) xs) = drop (Suc i) xs; (xs::'a list) = a # list⟧ ⟹ tl (drop (Suc i) xs) = drop (Suc (Suc i)) xs›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nqed\n \ndefinition inc_i :: \"'a list_repr ⇒ 'a list_repr\" where\n \"inc_i ba = (case ba of (b,a) ⇒ (hd a # b, tl a))\" \n \nlemma inc_i: \"list_repr i ba xs ⟹ i < length xs ⟹ list_repr (Suc i) (inc_i ba) xs\" \n unfolding list_repr_def inc_i_def\n (*goal: ‹⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i < length xs⟧ ⟹ Suc i ≤ length xs ∧ fst (case ba of (b, a) ⇒ (hd a # b, tl a)) = rev (take (Suc i) xs) ∧ snd (case ba of (b, a) ⇒ (hd a # b, tl a)) = drop (Suc i) xs›*)\n apply (cases ba)\n (*goal: ‹⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i < length xs⟧ ⟹ Suc i ≤ length xs ∧ fst (case ba of (b, a) ⇒ (hd a # b, tl a)) = rev (take (Suc i) xs) ∧ snd (case ba of (b, a) ⇒ (hd a # b, tl a)) = drop (Suc i) xs›*)\n by (auto simp: rev_take_Suc (*‹?j < length ?xs ⟹ rev (take (Suc ?j) ?xs) = ?xs ! ?j # rev (take ?j ?xs)›*) hd_drop_conv_nth (*‹?n < length ?xs ⟹ hd (drop ?n ?xs) = ?xs ! ?n›*) tl_drop_Suc (*‹tl (drop ?i ?xs) = drop (Suc ?i) ?xs›*)) \n\ndefinition dec_i :: \"'a list_repr ⇒ 'a list_repr\" where\n \"dec_i ba = (case ba of (b,a) ⇒ (tl b, hd b # a))\" \n \nlemma dec_i: \"list_repr i ba xs ⟹ i ≠ 0 ⟹ list_repr (i - 1) (dec_i ba) xs\" \n unfolding list_repr_def dec_i_def\n (*goal: ‹⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i ≠ 0⟧ ⟹ i - 1 ≤ length xs ∧ fst (case ba of (b, a) ⇒ (tl b, hd b # a)) = rev (take (i - 1) xs) ∧ snd (case ba of (b, a) ⇒ (tl b, hd b # a)) = drop (i - 1) xs›*)\n apply (cases ba)\n (*goal: ‹⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i ≠ 0⟧ ⟹ i - 1 ≤ length xs ∧ fst (case ba of (b, a) ⇒ (tl b, hd b # a)) = rev (take (i - 1) xs) ∧ snd (case ba of (b, a) ⇒ (tl b, hd b # a)) = drop (i - 1) xs›*)\n apply (cases i)\n (*goals:\n 1. ‹⋀a b. ⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i ≠ 0; ba = (a, b); i = 0⟧ ⟹ i - 1 ≤ length xs ∧ fst (case ba of (b, a) ⇒ (tl b, hd b # a)) = rev (take (i - 1) xs) ∧ snd (case ba of (b, a) ⇒ (tl b, hd b # a)) = drop (i - 1) xs›\n 2. ‹⋀a b nat. ⟦i ≤ length xs ∧ fst ba = rev (take i xs) ∧ snd ba = drop i xs; i ≠ 0; ba = (a, b); i = Suc nat⟧ ⟹ i - 1 ≤ length xs ∧ fst (case ba of (b, a) ⇒ (tl b, hd b # a)) = rev (take (i - 1) xs) ∧ snd (case ba of (b, a) ⇒ (tl b, hd b # a)) = drop (i - 1) xs›\n discuss goal 1*)\n apply ((auto simp: rev_take_Suc (*‹?j < length ?xs ⟹ rev (take (Suc ?j) ?xs) = ?xs ! ?j # rev (take ?j ?xs)›*) hd_drop_conv_nth (*‹?n < length ?xs ⟹ hd (drop ?n ?xs) = ?xs ! ?n›*) Cons_nth_drop_Suc (*‹?i < length ?xs ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: rev_take_Suc (*‹?j < length ?xs ⟹ rev (take (Suc ?j) ?xs) = ?xs ! ?j # rev (take ?j ?xs)›*) hd_drop_conv_nth (*‹?n < length ?xs ⟹ hd (drop ?n ?xs) = ?xs ! ?n›*) Cons_nth_drop_Suc (*‹?i < length ?xs ⟹ ?xs ! ?i # drop (Suc ?i) ?xs = drop ?i ?xs›*))[1])\n (*proven 2 subgoals*) . \n\nlemma dec_i_Suc: \"list_repr (Suc i) ba xs ⟹ list_repr i (dec_i ba) xs\" \n using dec_i[of \"Suc i\" ba xs] (*‹⟦list_repr (Suc i) ba xs; Suc i ≠ 0⟧ ⟹ list_repr (Suc i - 1) (dec_i ba) xs›*) by auto\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/LLL_Basis_Reduction/List_Representation.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"714b9040bff54d3349330f77da605f811175cf05637d4b92f720d698f3d0d918\"\n}"}}},{"rowIdx":180,"cells":{"text":{"kind":"string","value":"subsection ‹Digit function is Diophantine›\n\ntheory Digit_Function\n imports Exponential_Relation \"Digit_Expansions.Bits_Digits\"\nbegin\n\ndefinition digit (\"[ _ = Digit _ _ _]\" [999] 1000)\n where \"[D = Digit AA K BASE] ≡ (QUATERNARY (λd a k b. b > 1\n ∧ d = nth_digit a k b) D AA K BASE)\"\nlemma mod_dioph2[dioph]:\n fixes A B C\n defines \"D ≡ (MOD A B C)\"\n shows \"is_dioph_rel D\"\nproof (-)\n (*goal: ‹is_dioph_rel D›*)\n define A' and B' and C' where pushed_defs: \"A' ≡ push_param A 2\" \"B' ≡ push_param B 2\" \"C' ≡ push_param C 2\"\n define DS where \"DS ≡ [∃2] (Param 0 [*] B' [+] C' [=] Param 1 [*] B' [+] A')\"\n have \"eval DS a = eval D a\" for a\n proof (standard)\n (*goals:\n 1. ‹eval DS a ⟹ eval D a›\n 2. ‹eval D a ⟹ eval DS a›*)\n show \"eval DS a ⟹ eval D a\"\n unfolding DS_def defs D_def mod_def\n (*goal: ‹eval ([∃2] NARY (λl. l ! 0 = l ! 1) [polynomial.Param 0 [*] B' [+] C', polynomial.Param 1 [*] B' [+] A']) a ⟹ eval (NARY (λl. l ! 0 mod l ! 1 = l ! 2 mod l ! 1) [A, B, C]) a›*)\n apply auto\n (*goal: ‹eval ([∃2] NARY (λl. l ! 0 = l ! 1) [polynomial.Param 0 [*] B' [+] C', polynomial.Param 1 [*] B' [+] A']) a ⟹ eval (NARY (λl. l ! 0 mod l ! 1 = l ! 2 mod l ! 1) [A, B, C]) a›*)\n by (metis add.commute (*‹(?a::?'a) + (?b::?'a) = ?b + ?a›*) mod_mult_self1 (*‹((?a::?'a) + (?c::?'a) * (?b::?'a)) mod ?b = ?a mod ?b›*) push_push_simp (*‹peval (push_param (?P::polynomial) (length (?ns::nat list))) (push_list (?a::nat ⇒ nat) ?ns) = peval ?P ?a›*) pushed_defs( (*‹A'::polynomial ≡ push_param (A::polynomial) (2::nat)›*) 1) pushed_defs( (*‹B'::polynomial ≡ push_param (B::polynomial) (2::nat)›*) 2) pushed_defs( (*‹C'::polynomial ≡ push_param (C::polynomial) (2::nat)›*) 3))\n show \"eval D a ⟹ eval DS a\"\n unfolding DS_def defs D_def mod_def\n (*goal: ‹eval (NARY (λl. l ! 0 mod l ! 1 = l ! 2 mod l ! 1) [A, B, C]) a ⟹ eval ([∃2] NARY (λl. l ! 0 = l ! 1) [polynomial.Param 0 [*] B' [+] C', polynomial.Param 1 [*] B' [+] A']) a›*)\n apply (auto simp: mod_repr (*‹(?a mod ?b = ?c mod ?b) = (∃x y. ?c + x * ?b = ?a + y * ?b)›*))\n (*goal: ‹eval (NARY (λl. l ! 0 mod l ! 1 = l ! 2 mod l ! 1) [A, B, C]) a ⟹ eval ([∃2] NARY (λl. l ! 0 = l ! 1) [polynomial.Param 0 [*] B' [+] C', polynomial.Param 1 [*] B' [+] A']) a›*)\n subgoal for x and y\n apply (rule exI[of _ \"[x, y]\"] (*‹(?P::nat list ⇒ bool) [x::nat, y::nat] ⟹ ∃x::nat list. ?P x›*))\n (*goal: ‹peval C a + x * peval B a = peval A a + y * peval B a ⟹ ∃ks. length ks = 2 ∧ push_list a ks 0 * peval B' (push_list a ks) + peval C' (push_list a ks) = push_list a ks (Suc 0) * peval B' (push_list a ks) + peval A' (push_list a ks)›*)\n unfolding pushed_defs\n (*goal: ‹peval C a + x * peval B a = peval A a + y * peval B a ⟹ length [x, y] = 2 ∧ push_list a [x, y] 0 * peval (push_param B 2) (push_list a [x, y]) + peval (push_param C 2) (push_list a [x, y]) = push_list a [x, y] (Suc 0) * peval (push_param B 2) (push_list a [x, y]) + peval (push_param A 2) (push_list a [x, y])›*)\n by (simp add: push_push[where ?n = 2] (*‹length ?ns = 2 ⟹ peval (push_param ?P 2) (push_list ?a ?ns) = peval ?P ?a›*) push_list_eval (*‹?i < length ?ns ⟹ push_list ?a ?ns ?i = ?ns ! ?i›*)) .\n qed\n moreover have \"is_dioph_rel DS\"\n unfolding DS_def\n (*goal: ‹is_dioph_rel ([∃2] polynomial.Param 0 [*] B' [+] C' [=] polynomial.Param 1 [*] B' [+] A')›*)\n by (simp add: dioph (*‹⟦is_dioph_rel ?A; is_dioph_rel ?B⟧ ⟹ is_dioph_rel (?A [∨] ?B)› ‹⟦is_dioph_rel ?A; is_dioph_rel ?B⟧ ⟹ is_dioph_rel (?A [∧] ?B)› ‹is_dioph_rel TRUE› ‹is_dioph_rel (?Q [=] ?R)› ‹is_dioph_rel (?Q [<] ?R)› ‹is_dioph_rel ([0=] ?Q)› ‹is_dioph_rel (?Q [>] ?R)› ‹is_dioph_rel (?Q [≤] ?R)› ‹is_dioph_rel (?Q [≥] ?R)› ‹list_all (is_dioph_rel ∘ ?DF) ?L ⟹ is_dioph_rel [∀ in ?L] ?DF› ‹∀i<?n. is_dioph_rel (?DF i) ⟹ is_dioph_rel [∀<?n] ?DF› ‹is_dioph_rel ?D ⟹ is_dioph_rel ([∃?n] ?D)› and more 5 facts*))\n ultimately show \"?thesis\"\n (*goal: ‹is_dioph_rel D›*)\n by (auto simp: is_dioph_rel_def (*‹is_dioph_rel ?DR = (∃P₁ P₂. ∀a. eval ?DR a = (∃v. ppeval P₁ a v = ppeval P₂ a v))›*))\nqed\n\nlemma digit_dioph[dioph]:\n fixes D A B K :: polynomial\n defines \"DR ≡ [D = Digit A K B]\"\n shows \"is_dioph_rel DR\"\nproof (-)\n (*goal: ‹is_dioph_rel DR›*)\n define D' and A' and B' and K' where pushed_defs: \"D' == (push_param D 4)\" \"A' == (push_param A 4)\" \"B' == (push_param B 4)\" \"K' == (push_param K 4)\"\n define x and y where param_defs: \"x == (Param 0)\" \"y == (Param 1)\"\n define DS where \"DS ≡ [∃4] ( B' [>] Const 1 [∧]\n [(Param 2) = B' ^ (K' [+] Const 1)] [∧]\n [(Param 3) = B' ^ K'] [∧] \n A' [=] x [*] (Param 2) [+] D' [*] (Param 3) [+] y [∧] \n D' [<] B' [∧] \n y [<] (Param 3)) \"\n have \"eval DS a = eval DR a\" for a\n proof (standard)\n (*goals:\n 1. ‹eval DS a ⟹ eval DR a›\n 2. ‹eval DR a ⟹ eval DS a›*)\n show \"eval DS a ⟹ eval DR a\"\n unfolding DS_def defs DR_def digit_def\n (*goal: ‹eval ([∃4] NARY (λl. l ! 0 < l ! 1) [polynomial.Const 1, B'] [∧] NARY (λl. l ! 0 = l ! 1 ^ l ! 2) [polynomial.Param 2, B', K' [+] polynomial.Const 1] [∧] NARY (λl. l ! 0 = l ! 1 ^ l ! 2) [polynomial.Param 3, B', K'] [∧] NARY (λl. l ! 0 = l ! 1) [A', x [*] polynomial.Param 2 [+] D' [*] polynomial.Param 3 [+] y] [∧] NARY (λl. l ! 0 < l ! 1) [D', B'] [∧] NARY (λl. l ! 0 < l ! 1) [y, polynomial.Param 3]) a ⟹ eval (NARY (λl. 1 < l ! 3 ∧ l ! 0 = nth_digit (l ! 1) (l ! 2) (l ! 3)) [D, A, K, B]) a›*)\n apply auto\n (*goal: ‹eval ([∃4::nat] NARY (λl::nat list. l ! (0::nat) < l ! (1::nat)) [polynomial.Const (1::nat), B'::polynomial] [∧] NARY (λl::nat list. l ! (0::nat) = l ! (1::nat) ^ l ! (2::nat)) [polynomial.Param (2::nat), B', (K'::polynomial) [+] polynomial.Const (1::nat)] [∧] NARY (λl::nat list. l ! (0::nat) = l ! (1::nat) ^ l ! (2::nat)) [polynomial.Param (3::nat), B', K'] [∧] NARY (λl::nat list. l ! (0::nat) = l ! (1::nat)) [A'::polynomial, (x::polynomial) [*] polynomial.Param (2::nat) [+] (D'::polynomial) [*] polynomial.Param (3::nat) [+] (y::polynomial)] [∧] NARY (λl::nat list. l ! (0::nat) < l ! (1::nat)) [D', B'] [∧] NARY (λl::nat list. l ! (0::nat) < l ! (1::nat)) [y, polynomial.Param (3::nat)]) (a::nat ⇒ nat) ⟹ eval (NARY (λl::nat list. (1::nat) < l ! (3::nat) ∧ l ! (0::nat) = nth_digit (l ! (1::nat)) (l ! (2::nat)) (l ! (3::nat))) [D::polynomial, A::polynomial, K::polynomial, B::polynomial]) a›*)\n unfolding pushed_defs push_push\n (*goals:\n 1. ‹⋀ks. ⟦length ks = 4; Suc 0 < peval B a; push_list a ks 2 = peval B a * peval B a ^ peval K a; push_list a ks 3 = peval B a ^ peval K a; peval A a = peval x (push_list a ks) * (peval B a * peval B a ^ peval K a) + peval D a * peval B a ^ peval K a + peval y (push_list a ks); peval D a < peval B a; peval y (push_list a ks) < peval B a ^ peval K a⟧ ⟹ Suc 0 < peval B a›\n 2. ‹⋀ks. ⟦length ks = 4; Suc 0 < peval B a; push_list a ks 2 = peval B a * peval B a ^ peval K a; push_list a ks 3 = peval B a ^ peval K a; peval A a = peval x (push_list a ks) * (peval B a * peval B a ^ peval K a) + peval D a * peval B a ^ peval K a + peval y (push_list a ks); peval D a < peval B a; peval y (push_list a ks) < peval B a ^ peval K a⟧ ⟹ peval D a = nth_digit (peval A a) (peval K a) (peval B a)›*)\n using pushed_defs (*‹D' ≡ push_param D 4› ‹A' ≡ push_param A 4› ‹B'::polynomial ≡ push_param (B::polynomial) (4::nat)› ‹K' ≡ push_param K 4›*) push_push (*‹length ?ns = ?n ⟹ peval (push_param ?P ?n) (push_list ?a ?ns) = peval ?P ?a›*) digit_gen_equiv (*‹1 < ?b ⟹ (?d = nth_digit ?a ?k ?b) = (∃x y. ?a = x * ?b ^ (?k + 1) + ?d * ?b ^ ?k + y ∧ ?d < ?b ∧ y < ?b ^ ?k)›*) apply -\n (*goals:\n 1. ‹⋀ks. ⟦length ks = 4; Suc 0 < peval B a; push_list a ks 2 = peval B a * peval B a ^ peval K a; push_list a ks 3 = peval B a ^ peval K a; peval A a = peval x (push_list a ks) * (peval B a * peval B a ^ peval K a) + peval D a * peval B a ^ peval K a + peval y (push_list a ks); peval D a < peval B a; peval y (push_list a ks) < peval B a ^ peval K a; D' ≡ push_param D 4; A' ≡ push_param A 4; B' ≡ push_param B 4; K' ≡ push_param K 4; ⋀ns n P a. length ns = n ⟹ peval (push_param P n) (push_list a ns) = peval P a; ⋀b d a k. 1 < b ⟹ (d = nth_digit a k b) = (∃x y. a = x * b ^ (k + 1) + d * b ^ k + y ∧ d < b ∧ y < b ^ k)⟧ ⟹ Suc 0 < peval B a›\n 2. ‹⋀ks. ⟦length ks = 4; Suc 0 < peval B a; push_list a ks 2 = peval B a * peval B a ^ peval K a; push_list a ks 3 = peval B a ^ peval K a; peval A a = peval x (push_list a ks) * (peval B a * peval B a ^ peval K a) + peval D a * peval B a ^ peval K a + peval y (push_list a ks); peval D a < peval B a; peval y (push_list a ks) < peval B a ^ peval K a; D' ≡ push_param D 4; A' ≡ push_param A 4; B' ≡ push_param B 4; K' ≡ push_param K 4; ⋀ns n P a. length ns = n ⟹ peval (push_param P n) (push_list a ns) = peval P a; ⋀b d a k. 1 < b ⟹ (d = nth_digit a k b) = (∃x y. a = x * b ^ (k + 1) + d * b ^ k + y ∧ d < b ∧ y < b ^ k)⟧ ⟹ peval D a = nth_digit (peval A a) (peval K a) (peval B a)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n assume asm: \"eval DR a\" (*‹eval (DR::relation) (a::nat ⇒ nat)›*)\n then obtain x_val and y_val where cond: \"(peval A a) = x_val * (peval B a)^( (peval K a)+1) \n + (peval D a) * (peval B a)^ (peval K a) + y_val\n ∧ (peval D a) < (peval B a) \n ∧ y_val < (peval B a)^ (peval K a)\"\n (*goal: ‹(⋀x_val y_val. peval A a = x_val * peval B a ^ (peval K a + 1) + peval D a * peval B a ^ peval K a + y_val ∧ peval D a < peval B a ∧ y_val < peval B a ^ peval K a ⟹ thesis) ⟹ thesis›*)\n unfolding DS_def defs DR_def digit_def\n (*goal: ‹(⋀(x_val::nat) y_val::nat. peval (A::polynomial) (a::nat ⇒ nat) = x_val * peval (B::polynomial) a ^ (peval (K::polynomial) a + (1::nat)) + peval (D::polynomial) a * peval B a ^ peval K a + y_val ∧ peval D a < peval B a ∧ y_val < peval B a ^ peval K a ⟹ thesis::bool) ⟹ thesis›*)\n using digit_gen_equiv (*‹1 < ?b ⟹ (?d = nth_digit ?a ?k ?b) = (∃x y. ?a = x * ?b ^ (?k + 1) + ?d * ?b ^ ?k + y ∧ ?d < ?b ∧ y < ?b ^ ?k)›*) apply auto\n (*goal: ‹(⋀x_val y_val. peval A a = x_val * peval B a ^ (peval K a + 1) + peval D a * peval B a ^ peval K a + y_val ∧ peval D a < peval B a ∧ y_val < peval B a ^ peval K a ⟹ thesis) ⟹ thesis›*)\n by metis\n show \"eval DS a\"\n using asm (*‹eval DR a›*) unfolding DS_def defs DR_def digit_def\n (*goal: ‹eval ([∃4::nat] NARY (λl::nat list. l ! (0::nat) < l ! (1::nat)) [polynomial.Const (1::nat), B'::polynomial] [∧] NARY (λl::nat list. l ! (0::nat) = l ! (1::nat) ^ l ! (2::nat)) [polynomial.Param (2::nat), B', (K'::polynomial) [+] polynomial.Const (1::nat)] [∧] NARY (λl::nat list. l ! (0::nat) = l ! (1::nat) ^ l ! (2::nat)) [polynomial.Param (3::nat), B', K'] [∧] NARY (λl::nat list. l ! (0::nat) = l ! (1::nat)) [A'::polynomial, (x::polynomial) [*] polynomial.Param (2::nat) [+] (D'::polynomial) [*] polynomial.Param (3::nat) [+] (y::polynomial)] [∧] NARY (λl::nat list. l ! (0::nat) < l ! (1::nat)) [D', B'] [∧] NARY (λl::nat list. l ! (0::nat) < l ! (1::nat)) [y, polynomial.Param (3::nat)]) (a::nat ⇒ nat)›*)\n apply auto\n (*goal: ‹eval ([∃4] NARY (λl. l ! 0 < l ! 1) [polynomial.Const 1, B'] [∧] NARY (λl. l ! 0 = l ! 1 ^ l ! 2) [polynomial.Param 2, B', K' [+] polynomial.Const 1] [∧] NARY (λl. l ! 0 = l ! 1 ^ l ! 2) [polynomial.Param 3, B', K'] [∧] NARY (λl. l ! 0 = l ! 1) [A', x [*] polynomial.Param 2 [+] D' [*] polynomial.Param 3 [+] y] [∧] NARY (λl. l ! 0 < l ! 1) [D', B'] [∧] NARY (λl. l ! 0 < l ! 1) [y, polynomial.Param 3]) a›*)\n apply (rule exI[of _ \"[x_val, y_val, (peval B a) ^ ((peval K a) + 1),\n (peval B a) ^ (peval K a)]\" ] (*‹(?P::nat list ⇒ bool) [x_val::nat, y_val::nat, peval (B::polynomial) (a::nat ⇒ nat) ^ (peval (K::polynomial) a + (1::nat)), peval B a ^ peval K a] ⟹ ∃x::nat list. ?P x›*))\n (*goal: ‹⟦Suc 0 < peval B a; peval D a = nth_digit (peval A a) (peval K a) (peval B a)⟧ ⟹ ∃ks. length ks = 4 ∧ Suc 0 < peval B' (push_list a ks) ∧ push_list a ks 2 = peval B' (push_list a ks) * peval B' (push_list a ks) ^ peval K' (push_list a ks) ∧ push_list a ks 3 = peval B' (push_list a ks) ^ peval K' (push_list a ks) ∧ peval A' (push_list a ks) = peval x (push_list a ks) * push_list a ks 2 + peval D' (push_list a ks) * push_list a ks 3 + peval y (push_list a ks) ∧ peval D' (push_list a ks) < peval B' (push_list a ks) ∧ peval y (push_list a ks) < push_list a ks 3›*)\n unfolding pushed_defs\n (*goal: ‹⟦Suc 0 < peval B a; peval D a = nth_digit (peval A a) (peval K a) (peval B a)⟧ ⟹ length [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a] = 4 ∧ Suc 0 < peval (push_param B 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) ∧ push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a] 2 = peval (push_param B 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) * peval (push_param B 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) ^ peval (push_param K 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) ∧ push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a] 3 = peval (push_param B 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) ^ peval (push_param K 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) ∧ peval (push_param A 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) = peval x (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) * push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a] 2 + peval (push_param D 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) * push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a] 3 + peval y (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) ∧ peval (push_param D 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) < peval (push_param B 4) (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) ∧ peval y (push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a]) < push_list a [x_val, y_val, peval B a ^ (peval K a + 1), peval B a ^ peval K a] 3›*)\n using param_defs (*‹x::polynomial ≡ polynomial.Param (0::nat)› ‹y ≡ polynomial.Param 1›*) push_push (*‹length ?ns = ?n ⟹ peval (push_param ?P ?n) (push_list ?a ?ns) = peval ?P ?a›*) push_list_def (*‹push_list ?a ?ns ?i = (if ?i < length ?ns then ?ns ! ?i else ?a (?i - length ?ns))›*) cond (*‹peval A a = x_val * peval B a ^ (peval K a + 1) + peval D a * peval B a ^ peval K a + y_val ∧ peval D a < peval B a ∧ y_val < peval B a ^ peval K a›*) by auto\n qed\n moreover have \"is_dioph_rel DS\"\n unfolding DS_def\n (*goal: ‹is_dioph_rel ([∃4] B' [>] polynomial.Const 1 [∧] [polynomial.Param 2 = B' ^ K' [+] polynomial.Const 1] [∧] [polynomial.Param 3 = B' ^ K'] [∧] A' [=] x [*] polynomial.Param 2 [+] D' [*] polynomial.Param 3 [+] y [∧] D' [<] B' [∧] y [<] polynomial.Param 3)›*)\n by (simp add: dioph (*‹⟦is_dioph_rel ?A; is_dioph_rel ?B⟧ ⟹ is_dioph_rel (?A [∨] ?B)› ‹⟦is_dioph_rel ?A; is_dioph_rel ?B⟧ ⟹ is_dioph_rel (?A [∧] ?B)› ‹is_dioph_rel TRUE› ‹is_dioph_rel (?Q [=] ?R)› ‹is_dioph_rel (?Q [<] ?R)› ‹is_dioph_rel ([0=] ?Q)› ‹is_dioph_rel (?Q [>] ?R)› ‹is_dioph_rel (?Q [≤] ?R)› ‹is_dioph_rel (?Q [≥] ?R)› ‹list_all (is_dioph_rel ∘ ?DF) ?L ⟹ is_dioph_rel [∀ in ?L] ?DF› ‹∀i<?n. is_dioph_rel (?DF i) ⟹ is_dioph_rel [∀<?n] ?DF› ‹is_dioph_rel ?D ⟹ is_dioph_rel ([∃?n] ?D)› and more 5 facts*))\n ultimately show \"?thesis\"\n (*goal: ‹is_dioph_rel DR›*)\n by (auto simp: is_dioph_rel_def (*‹is_dioph_rel ?DR = (∃P₁ P₂. ∀a. eval ?DR a = (∃v. ppeval P₁ a v = ppeval P₂ a v))›*))\nqed\n\ndeclare digit_def[defs]\n\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/DPRM_Theorem/Diophantine/Digit_Function.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"0910ea9c8ad34eb4ad10e8185df932eff86748beaf6628e81f8c24c6f458c450\"\n}"}}},{"rowIdx":181,"cells":{"text":{"kind":"string","value":"section ‹Semantics of SM›\ntheory SM_Semantics\nimports SM_State SM_Cfg Gen_Scheduler\nbegin\n\n section ‹Evaluation of Expressions›\n subsection ‹Basic operations›\n text ‹Attention: Silently overflows, using 2's complement.\n This should match Java-semantics. In C, signed overflow is undefined!›\n\n context\n includes bit_operations_syntax\n begin\n\n primrec eval_bin_op :: \"bin_op ⇒ val ⇒ val ⇒ val\" where\n \"eval_bin_op bo_plus v1 v2 = v1+v2\"\n | \"eval_bin_op bo_minus v1 v2 = v1-v2\"\n | \"eval_bin_op bo_mul v1 v2 = v1*v2\"\n | \"eval_bin_op bo_div v1 v2 = v1 sdiv v2\"\n | \"eval_bin_op bo_mod v1 v2 = v1 smod v2\"\n | \"eval_bin_op bo_less v1 v2 = val_of_bool (v1 < v2)\"\n | \"eval_bin_op bo_less_eq v1 v2 = val_of_bool (v1 ≤ v2)\"\n | \"eval_bin_op bo_eq v1 v2 = val_of_bool (v1 = v2)\"\n | \"eval_bin_op bo_and v1 v2 = v1 AND v2\"\n | \"eval_bin_op bo_or v1 v2 = v1 OR v2\"\n | \"eval_bin_op bo_xor v1 v2 = v1 XOR v2\"\n\n primrec eval_un_op :: \"un_op ⇒ val ⇒ val\" where\n \"eval_un_op uo_minus v = -v\" ― ‹Attention: Silently overflows. @{term \"-min_int = min_int\"}!›\n | \"eval_un_op uo_not v = NOT v\"\n\n end\n\n subsection ‹Expressions›\n abbreviation exists_var :: \"valuation ⇒ ident ⇒ bool\" where\n \"exists_var s x ≡ x ∈ dom s\"\n\n abbreviation update_var :: \"ident ⇒ val ⇒ valuation ⇒ valuation\" where\n \"update_var x v s ≡ s(x↦v)\"\n\n fun eval_exp :: \"exp ⇒ focused_state ⇀ val\" where\n \"eval_exp (e_var x) (ls,gs) = do {\n let lv = local_state.variables ls; let gv = global_state.variables gs;\n case lv x of\n Some v ⇒ Some v\n | None ⇒ (case gv x of\n Some v ⇒ Some v\n | None ⇒ None)\n }\"\n | \"eval_exp (e_localvar x) (ls,gs) = local_state.variables ls x\"\n | \"eval_exp (e_globalvar x) (ls,gs) = global_state.variables gs x\"\n | \"eval_exp (e_const n) fs = do {\n assert_option (n≥min_signed ∧ n≤max_signed);\n Some (signed_of_int n)\n }\"\n | \"eval_exp (e_bin bop e1 e2) fs = do {\n v1←eval_exp e1 fs;\n v2←eval_exp e2 fs;\n Some (eval_bin_op bop v1 v2)\n }\"\n | \"eval_exp (e_un uop e) fs = do {\n v←eval_exp e fs;\n Some (eval_un_op uop v)\n }\"\n\n subsection ‹Local Actions›\n\n text ‹Enabledness and effects of actions›\n primrec la_en :: \"focused_state ⇒ action ⇀ bool\" where\n \"la_en fs (AAssign c _ _) = do { v ← eval_exp c fs; Some (bool_of_val v)}\"\n | \"la_en fs (AAssign_local c _ _) = do { v ← eval_exp c fs; Some (bool_of_val v)}\"\n | \"la_en fs (AAssign_global c _ _) = do { v ← eval_exp c fs; Some (bool_of_val v)}\"\n | \"la_en fs (ATest e) = do {\n v ← eval_exp e fs;\n Some (bool_of_val v)}\"\n | \"la_en _ (ASkip) = Some True\"\n\n text ‹Extension (for run, handshake): Let action have scheduler effect›\n fun la_ex :: \"focused_state ⇒ action ⇀ focused_state\" where\n \"la_ex (ls,gs) (AAssign c x e) = do {\n u ← eval_exp c (ls,gs);\n assert_option (bool_of_val u);\n v ← eval_exp e (ls,gs);\n if exists_var (local_state.variables ls) x then do {\n let ls = local_state.variables_update (update_var x v) ls;\n Some (ls,gs)\n } else do {\n assert_option (exists_var (global_state.variables gs) x);\n let gs = global_state.variables_update (update_var x v) gs;\n Some (ls,gs)\n }\n }\"\n | \"la_ex (ls,gs) (AAssign_local c x e) = do {\n u ← eval_exp c (ls,gs);\n assert_option (bool_of_val u);\n v ← eval_exp e (ls,gs);\n assert_option (exists_var (local_state.variables ls) x);\n let ls = local_state.variables_update (update_var x v) ls;\n Some (ls,gs)\n }\"\n | \"la_ex (ls,gs) (AAssign_global c x e) = do {\n u ← eval_exp c (ls,gs);\n assert_option (bool_of_val u);\n v ← eval_exp e (ls,gs);\n assert_option (exists_var (global_state.variables gs) x);\n let gs = global_state.variables_update (update_var x v) gs;\n Some (ls,gs)\n }\"\n | \"la_ex fs (ATest e) = do {\n v ← eval_exp e fs;\n assert_option (bool_of_val v); \n Some fs\n }\"\n | \"la_ex fs ASkip = Some fs\"\n\n\n subsection \"Scheduling\"\n type_synonym local_config = \"(cmd,local_state) local_config\"\n type_synonym global_config = \"(cmd,local_state,global_state) global_config\"\n \n interpretation li: Gen_Scheduler cfg la_en la_ex .\n\n subsection ‹Initial State›\n definition init_valuation :: \"var_decl list ⇒ valuation\" where\n \"init_valuation vd x ≡ if x ∈ set vd then Some 0 else None\"\n\n lemma init_valuation_eq_Some[simp]: \"init_valuation vd x = Some v ⟷ x∈set vd ∧ v=0\" \n unfolding init_valuation_def\n (*goal: ‹((if (x::String.literal) ∈ set (vd::String.literal list) then Some (0::32 Word.word) else None) = Some (v::32 Word.word)) = (x ∈ set vd ∧ v = (0::32 Word.word))›*)\n by auto\n\n lemma init_valuation_eq_None[simp]: \"init_valuation vd x = None ⟷ x∉set vd\" \n unfolding init_valuation_def\n (*goal: ‹((if x ∈ set vd then Some 0 else None) = None) = (x ∉ set vd)›*)\n by auto\n\n definition init_pc :: \"proc_decl ⇒ local_config\" where\n \"init_pc pd ≡ ⦇\n local_config.command = proc_decl.body pd,\n local_config.state = ⦇\n local_state.variables = init_valuation (proc_decl.local_vars pd)\n ⦈\n ⦈\"\n\n definition init_gc :: \"program ⇒ global_config\" where\n \"init_gc prog ≡ ⦇\n global_config.processes = mset (map init_pc (program.processes prog)),\n global_config.state = ⦇\n global_state.variables = init_valuation (program.global_vars prog)\n ⦈\n ⦈\"\n\n subsection \"Semantics Reference Point\"\n \n interpretation li: Gen_Scheduler_linit cfg la_en la_ex \"{init_gc prog}\" global_config.state\n for prog .\n \n abbreviation \"ref_is_run ≡ li.sa.is_run\"\n abbreviation \"ref_accept ≡ li.sa.accept\"\n\nend\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/CAVA_LTL_Modelchecker/SM/RefPoint/SM_Semantics.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"d3543cf221b283ad9e9482ff5caac4f5f53f819aa92a5f9db691fe0816e16993\"\n}"}}},{"rowIdx":182,"cells":{"text":{"kind":"string","value":"(* Title: Prim's Minimum Spanning Tree Algorithm\n Author: Walter Guttmann\n Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>\n*)\n\nsection ‹Prim's Minimum Spanning Tree Algorithm›\n\ntext ‹\nIn this theory we prove total correctness of Prim's minimum spanning tree algorithm.\nThe proof has the same overall structure as the total-correctness proof of Kruskal's algorithm \\<^cite>‹\"Guttmann2018c\"›.\nThe partial-correctness proof of Prim's algorithm is discussed in \\<^cite>‹\"Guttmann2016c\" and \"Guttmann2018b\"›.\n›\n\ntheory Prim\n\nimports \"HOL-Hoare.Hoare_Logic\" Aggregation_Algebras.Aggregation_Algebras\n\nbegin\n\ncontext m_kleene_algebra\nbegin\n\nabbreviation \"component g r ≡ r⇧T * (--g)⇧⋆\"\ndefinition \"spanning_tree t g r ≡ forest t ∧ t ≤ (component g r)⇧T * (component g r) ⊓ --g ∧ component g r ≤ r⇧T * t⇧⋆ ∧ regular t\"\ndefinition \"minimum_spanning_tree t g r ≡ spanning_tree t g r ∧ (∀u . spanning_tree u g r ⟶ sum (t ⊓ g) ≤ sum (u ⊓ g))\"\ndefinition \"prim_precondition g r ≡ g = g⇧T ∧ injective r ∧ vector r ∧ regular r\"\ndefinition \"prim_spanning_invariant t v g r ≡ prim_precondition g r ∧ v⇧T = r⇧T * t⇧⋆ ∧ spanning_tree t (v * v⇧T ⊓ g) r\"\ndefinition \"prim_invariant t v g r ≡ prim_spanning_invariant t v g r ∧ (∃w . minimum_spanning_tree w g r ∧ t ≤ w)\"\n\nlemma span_tree_split:\n assumes \"vector r\"\n shows \"t ≤ (component g r)⇧T * (component g r) ⊓ --g ⟷ (t ≤ (component g r)⇧T ∧ t ≤ component g r ∧ t ≤ --g)\"\nproof (-)\n (*goal: ‹(t ≤ component g r⇧T * component g r ⊓ - - g) = (t ≤ component g r⇧T ∧ t ≤ component g r ∧ t ≤ - - g)›*)\n have \"(component g r)⇧T * (component g r) = (component g r)⇧T ⊓ component g r\"\n by (metis assms (*‹vector (r::'a)›*) conv_involutive (*‹(?x::'a)⇧T⇧T = ?x›*) covector_mult_closed (*‹covector (?x::'a) ⟹ covector (?x * (?y::'a))›*) vector_conv_covector (*‹vector (?v::'a) = covector (?v⇧T)›*) vector_covector (*‹⟦vector (?v::'a); vector (?w::'a)⟧ ⟹ ?v ⊓ ?w⇧T = ?v * ?w⇧T›*))\n thus \"?thesis\"\n (*goal: ‹(t ≤ component g r⇧T * component g r ⊓ - - g) = (t ≤ component g r⇧T ∧ t ≤ component g r ∧ t ≤ - - g)›*)\n by simp\nqed\n\nlemma span_tree_component:\n assumes \"spanning_tree t g r\"\n shows \"component g r = component t r\"\n using assms (*‹spanning_tree t g r›*) by (simp add: order.antisym (*‹⟦(?a::'a::type) ≤ (?b::'a::type); ?b ≤ ?a⟧ ⟹ ?a = ?b›*) mult_right_isotone (*‹(?x::'a::type) ≤ (?y::'a::type) ⟹ (?z::'a::type) * ?x ≤ ?z * ?y›*) star_isotone (*‹(?x::'a::type) ≤ (?y::'a::type) ⟹ ?x⇧⋆ ≤ ?y⇧⋆›*) spanning_tree_def (*‹spanning_tree (?t::'a::type) (?g::'a::type) (?r::'a::type) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n\ntext ‹\nWe first show three verification conditions which are used in both correctness proofs.\n›\n\nlemma prim_vc_1:\n assumes \"prim_precondition g r\"\n shows \"prim_spanning_invariant bot r g r\"\nproof (unfold prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹prim_precondition g r›\n 2. ‹r⇧T = r⇧T * bot⇧⋆›\n 3. ‹spanning_tree bot (r * r⇧T ⊓ g) r›*)\n show \"prim_precondition g r\"\n using assms (*‹prim_precondition g r›*) by simp\nnext\n (*goals:\n 1. ‹r⇧T = r⇧T * bot⇧⋆›\n 2. ‹spanning_tree bot (r * r⇧T ⊓ g) r›*)\n show \"r⇧T = r⇧T * bot⇧⋆\"\n by (simp add: star_absorb (*‹?x * ?y = bot ⟹ ?x * ?y⇧⋆ = ?x›*))\nnext\n (*goal: ‹spanning_tree bot (r * r⇧T ⊓ g) r›*)\n let ?ss = \"r * r⇧T ⊓ g\"\n show \"spanning_tree bot ?ss r\"\n proof (unfold spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹injective bot›\n 2. ‹pd_kleene_allegory_class.acyclic bot›\n 3. ‹bot ≤ component (r * r⇧T ⊓ g) r⇧T * component (r * r⇧T ⊓ g) r ⊓ - - (r * r⇧T ⊓ g)›\n 4. ‹component (r * r⇧T ⊓ g) r ≤ r⇧T * bot⇧⋆›\n 5. ‹regular bot›*)\n show \"injective bot\"\n by simp\n next\n (*goals:\n 1. ‹pd_kleene_allegory_class.acyclic bot›\n 2. ‹bot ≤ component ((r::'a::type) * r⇧T ⊓ (g::'a::type)) r⇧T * component (r * r⇧T ⊓ g) r ⊓ - - (r * r⇧T ⊓ g)›\n 3. ‹component ((r::'a::type) * r⇧T ⊓ (g::'a::type)) r ≤ r⇧T * bot⇧⋆›\n 4. ‹regular bot›*)\n show \"acyclic bot\"\n by simp\n next\n (*goals:\n 1. ‹bot ≤ component ((r::'a::type) * r⇧T ⊓ (g::'a::type)) r⇧T * component (r * r⇧T ⊓ g) r ⊓ - - (r * r⇧T ⊓ g)›\n 2. ‹component ((r::'a::type) * r⇧T ⊓ (g::'a::type)) r ≤ r⇧T * bot⇧⋆›\n 3. ‹regular bot›*)\n show \"bot ≤ (component ?ss r)⇧T * (component ?ss r) ⊓ --?ss\"\n by simp\n next\n (*goals:\n 1. ‹component (r * r⇧T ⊓ g) r ≤ r⇧T * bot⇧⋆›\n 2. ‹regular bot›*)\n have \"component ?ss r ≤ component (r * r⇧T) r\"\n by (simp add: mult_right_isotone (*‹?x ≤ ?y ⟹ ?z * ?x ≤ ?z * ?y›*) star_isotone (*‹?x ≤ ?y ⟹ ?x⇧⋆ ≤ ?y⇧⋆›*))\n also (*calculation: ‹component ((r::'a) * r⇧T ⊓ (g::'a)) r ≤ component (r * r⇧T) r›*) have \"... ≤ r⇧T * 1⇧⋆\"\n using assms (*‹prim_precondition (g::'a) (r::'a)›*) by (metis order.eq_iff (*‹(?a = ?b) = (?a ≤ ?b ∧ ?b ≤ ?a)›*) p_antitone (*‹?x ≤ ?y ⟹ - ?y ≤ - ?x›*) regular_one_closed (*‹regular 1›*) star_sub_one (*‹coreflexive ?x ⟹ ?x⇧⋆ = 1›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*))\n also (*calculation: ‹component (r * r⇧T ⊓ g) r ≤ r⇧T * 1⇧⋆›*) have \"... = r⇧T * bot⇧⋆\"\n by (simp add: star.circ_zero (*‹bot⇧⋆ = (1::'a)›*) star_one (*‹(1::'a)⇧⋆ = (1::'a)›*))\n finally (*calculation: ‹component (r * r⇧T ⊓ g) r ≤ r⇧T * bot⇧⋆›*) show \"component ?ss r ≤ r⇧T * bot⇧⋆\" .\n next\n (*goal: ‹regular bot›*)\n show \"regular bot\"\n by simp\n qed\nqed\n\nlemma prim_vc_2:\n assumes \"prim_spanning_invariant t v g r\"\n and \"v * -v⇧T ⊓ g ≠ bot\"\n shows \"prim_spanning_invariant (t ⊔ minarc (v * -v⇧T ⊓ g)) (v ⊔ minarc (v * -v⇧T ⊓ g)⇧T * top) g r ∧ card { x . regular x ∧ x ≤ component g r ∧ x ≤ -(v ⊔ minarc (v * -v⇧T ⊓ g)⇧T * top)⇧T } < card { x . regular x ∧ x ≤ component g r ∧ x ≤ -v⇧T }\"\nproof (-)\n (*goal: ‹prim_spanning_invariant (t ⊔ minarc (v * - v⇧T ⊓ g)) (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) g r ∧ card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›*)\n let ?vcv = \"v * -v⇧T ⊓ g\"\n let ?e = \"minarc ?vcv\"\n let ?t = \"t ⊔ ?e\"\n let ?v = \"v ⊔ ?e⇧T * top\"\n let ?c = \"component g r\"\n let ?g = \"--g\"\n let ?n1 = \"card { x . regular x ∧ x ≤ ?c ∧ x ≤ -v⇧T }\"\n let ?n2 = \"card { x . regular x ∧ x ≤ ?c ∧ x ≤ -?v⇧T }\"\n have 1: \"regular v ∧ regular (v * v⇧T) ∧ regular (?v * ?v⇧T) ∧ regular (top * ?e)\"\n using assms(1) (*‹prim_spanning_invariant (t::'a::type) (v::'a::type) (g::'a::type) (r::'a::type)›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) regular_conv_closed (*‹regular ?x ⟹ regular (?x⇧T)›*) regular_closed_star (*‹regular ?x ⟹ regular (?x⇧⋆)›*) regular_mult_closed (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x * ?y)›*) conv_involutive (*‹?x⇧T⇧T = ?x›*) regular_closed_top (*‹regular top›*) regular_closed_sup (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x ⊔ ?y)›*) minarc_regular (*‹regular (minarc ?x)›*))\n hence 2: \"t ≤ v * v⇧T ⊓ ?g\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant (?t::'a) (?v::'a) (?g::'a) (?r::'a) ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree (?t::'a) (?g::'a) (?r::'a) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) inf_pp_commute (*‹- - (- - (?x::'a) ⊓ (?y::'a)) = - - ?x ⊓ - - ?y›*) inf.boundedE (*‹⟦(?a::'a) ≤ (?b::'a) ⊓ (?c::'a); ⟦?a ≤ ?b; ?a ≤ ?c⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*))\n hence 3: \"t ≤ v * v⇧T\"\n by simp\n have 4: \"t ≤ ?g\"\n using \"2\" (*‹t ≤ v * v⇧T ⊓ - - g›*) by simp\n have 5: \"?e ≤ v * -v⇧T ⊓ ?g\"\n using \"1\" (*‹regular v ∧ regular (v * v⇧T) ∧ regular ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T) ∧ regular (top * minarc (v * - v⇧T ⊓ g))›*) by (metis minarc_below (*‹minarc ?x ≤ - - ?x›*) pp_dist_inf (*‹- - (?x ⊓ ?y) = - - ?x ⊓ - - ?y›*) regular_mult_closed (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x * ?y)›*) regular_closed_p (*‹regular (- ?x)›*))\n hence 6: \"?e ≤ v * -v⇧T\"\n by simp\n have 7: \"vector v\"\n using assms(1) (*‹prim_spanning_invariant (t::'a::type) (v::'a::type) (g::'a::type) (r::'a::type)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) by (simp add: covector_mult_closed (*‹covector (?x::'a) ⟹ covector (?x * (?y::'a))›*) vector_conv_covector (*‹vector (?v::'a) = covector (?v⇧T)›*))\n hence 8: \"?e ≤ v\"\n using \"6\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) by (metis conv_complement (*‹(- ?x)⇧T = - ?x⇧T›*) inf.boundedE (*‹⟦?a ≤ ?b ⊓ ?c; ⟦?a ≤ ?b; ?a ≤ ?c⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) vector_complement_closed (*‹vector ?x ⟹ vector (- ?x)›*) vector_covector (*‹⟦vector ?v; vector ?w⟧ ⟹ ?v ⊓ ?w⇧T = ?v * ?w⇧T›*))\n have 9: \"?e * t = bot\"\n using \"3\" (*‹(t::'a) ≤ (v::'a) * v⇧T›*) \"6\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"7\" (*‹vector v›*) et(1) (*‹⟦vector ?v; ?e ≤ ?v * - ?v⇧T; ?t ≤ ?v * ?v⇧T⟧ ⟹ ?e * ?t = bot›*) by blast\n have 10: \"?e * t⇧T = bot\"\n using \"3\" (*‹t ≤ v * v⇧T›*) \"6\" (*‹minarc ((v::'a) * - v⇧T ⊓ (g::'a)) ≤ v * - v⇧T›*) \"7\" (*‹vector v›*) et(2) (*‹⟦vector ?v; ?e ≤ ?v * - ?v⇧T; ?t ≤ ?v * ?v⇧T⟧ ⟹ ?e * ?t⇧T = bot›*) by simp\n have 11: \"arc ?e\"\n using assms(2) (*‹v * - v⇧T ⊓ g ≠ bot›*) minarc_arc (*‹?x ≠ bot ⟶ arc (minarc ?x)›*) by simp\n have \"r⇧T ≤ r⇧T * t⇧⋆\"\n by (metis mult_right_isotone (*‹?x ≤ ?y ⟹ ?z * ?x ≤ ?z * ?y›*) order_refl (*‹?x ≤ ?x›*) semiring.mult_not_zero (*‹?a * ?b ≠ bot ⟹ ?a ≠ bot ∧ ?b ≠ bot›*) star.circ_separate_mult_1 (*‹?y * ?x ≤ ?x * ?y ⟹ (?x * ?y)⇧⋆ ≤ ?x⇧⋆ * ?y⇧⋆›*) star_absorb (*‹?x * ?y = bot ⟹ ?x * ?y⇧⋆ = ?x›*))\n hence 12: \"r⇧T ≤ v⇧T\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) by (simp add: prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*))\n have 13: \"vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) minimum_spanning_tree_def (*‹minimum_spanning_tree ?t ?g ?r ≡ spanning_tree ?t ?g ?r ∧ (∀u. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) reachable_restrict (*‹vector ?r ⟹ ?r⇧T * ?g⇧⋆ = ?r⇧T * ((?r⇧T * ?g⇧⋆)⇧T * (?r⇧T * ?g⇧⋆) ⊓ ?g)⇧⋆›*) by simp\n have \"g = g⇧T\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) by simp\n hence 14: \"?g⇧T = ?g\"\n using conv_complement (*‹(- ?x)⇧T = - ?x⇧T›*) by simp\n show \"prim_spanning_invariant ?t ?v g r ∧ ?n2 < ?n1\"\n proof (rule conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹prim_spanning_invariant (t ⊔ minarc (v * - v⇧T ⊓ g)) (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) g r›\n 2. ‹card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›*)\n show \"prim_spanning_invariant ?t ?v g r\"\n proof (unfold prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹prim_precondition g r›\n 2. ‹(v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T = r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›\n 3. ‹spanning_tree (t ⊔ minarc (v * - v⇧T ⊓ g)) ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r›*)\n show \"prim_precondition g r\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) prim_spanning_invariant_def (*‹prim_spanning_invariant (?t::'a::type) (?v::'a::type) (?g::'a::type) (?r::'a::type) ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) by simp\n next\n (*goals:\n 1. ‹(v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T = r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›\n 2. ‹spanning_tree (t ⊔ minarc (v * - v⇧T ⊓ g)) ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r›*)\n show \"?v⇧T = r⇧T * ?t⇧⋆\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) \"6\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"7\" (*‹vector (v::'a::type)›*) \"9\" (*‹minarc (v * - v⇧T ⊓ g) * t = bot›*) by (simp add: reachable_inv (*‹⟦vector ?v; ?e ≤ ?v * - ?v⇧T; ?e * ?t = bot; ?v⇧T = ?r⇧T * ?t⇧⋆⟧ ⟹ (?v ⊔ ?e⇧T * top)⇧T = ?r⇧T * (?t ⊔ ?e)⇧⋆›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n next\n (*goal: ‹spanning_tree (t ⊔ minarc (v * - v⇧T ⊓ g)) ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r›*)\n let ?G = \"?v * ?v⇧T ⊓ g\"\n show \"spanning_tree ?t ?G r\"\n proof (unfold spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹injective (t ⊔ minarc (v * - v⇧T ⊓ g))›\n 2. ‹pd_kleene_allegory_class.acyclic (t ⊔ minarc (v * - v⇧T ⊓ g))›\n 3. ‹t ⊔ minarc (v * - v⇧T ⊓ g) ≤ component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r⇧T * component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ⊓ - - ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g)›\n 4. ‹component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ≤ r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›\n 5. ‹regular (t ⊔ minarc (v * - v⇧T ⊓ g))›*)\n show \"injective ?t\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) \"10\" (*‹minarc (v * - v⇧T ⊓ g) * t⇧T = bot›*) \"11\" (*‹arc (minarc (v * - v⇧T ⊓ g))›*) by (simp add: injective_inv (*‹⟦injective ?t; ?e * ?t⇧T = bot; arc ?e⟧ ⟹ injective (?t ⊔ ?e)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n next\n (*goals:\n 1. ‹pd_kleene_allegory_class.acyclic (t ⊔ minarc (v * - v⇧T ⊓ g))›\n 2. ‹t ⊔ minarc (v * - v⇧T ⊓ g) ≤ component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r⇧T * component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ⊓ - - ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g)›\n 3. ‹component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ≤ r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›\n 4. ‹regular (t ⊔ minarc (v * - v⇧T ⊓ g))›*)\n show \"acyclic ?t\"\n using assms(1) (*‹prim_spanning_invariant (t::'a) (v::'a) (g::'a) (r::'a)›*) \"3\" (*‹(t::'a) ≤ (v::'a) * v⇧T›*) \"6\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"7\" (*‹vector v›*) acyclic_inv (*‹⟦pd_kleene_allegory_class.acyclic (?t::'a); vector (?v::'a); (?e::'a) ≤ ?v * - ?v⇧T; ?t ≤ ?v * ?v⇧T⟧ ⟹ pd_kleene_allegory_class.acyclic (?t ⊔ ?e)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) by simp\n next\n (*goals:\n 1. ‹(t::'a) ⊔ minarc ((v::'a) * - v⇧T ⊓ (g::'a)) ≤ component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) (r::'a)⇧T * component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ⊓ - - ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g)›\n 2. ‹component (((v::'a) ⊔ minarc (v * - v⇧T ⊓ (g::'a))⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) (r::'a) ≤ r⇧T * ((t::'a) ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›\n 3. ‹regular ((t::'a) ⊔ minarc ((v::'a) * - v⇧T ⊓ (g::'a)))›*)\n show \"?t ≤ (component ?G r)⇧T * (component ?G r) ⊓ --?G\"\n using \"1\" (*‹regular v ∧ regular (v * v⇧T) ∧ regular ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T) ∧ regular (top * minarc (v * - v⇧T ⊓ g))›*) \"2\" (*‹t ≤ v * v⇧T ⊓ - - g›*) \"5\" (*‹minarc ((v::'a::type) * - v⇧T ⊓ (g::'a::type)) ≤ v * - v⇧T ⊓ - - g›*) \"7\" (*‹vector v›*) \"13\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆›*) prim_subgraph_inv (*‹⟦?e ≤ ?v * - ?v⇧T ⊓ ?g; ?t ≤ ?v * ?v⇧T ⊓ ?g⟧ ⟹ ?t ⊔ ?e ≤ (?v ⊔ ?e⇧T * top) * (?v ⊔ ?e⇧T * top)⇧T ⊓ ?g›*) inf_pp_commute (*‹- - (- - (?x::'a) ⊓ (?y::'a)) = - - ?x ⊓ - - ?y›*) mst_subgraph_inv_2 (*‹⟦regular ((?v::'a::type) * ?v⇧T); (?t::'a::type) ≤ ?v * ?v⇧T ⊓ - - (?g::'a::type); ?v⇧T = (?r::'a::type)⇧T * ?t⇧⋆; (?e::'a::type) ≤ ?v * - ?v⇧T ⊓ - - ?g; vector ?v; regular ((?v ⊔ ?e⇧T * top) * (?v ⊔ ?e⇧T * top)⇧T)⟧ ⟹ ?t ⊔ ?e ≤ component ((?v ⊔ ?e⇧T * top) * (?v ⊔ ?e⇧T * top)⇧T ⊓ ?g) ?r⇧T * component ((?v ⊔ ?e⇧T * top) * (?v ⊔ ?e⇧T * top)⇧T ⊓ ?g) ?r›*) by auto\n next\n (*goals:\n 1. ‹component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ≤ r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›\n 2. ‹regular (t ⊔ minarc (v * - v⇧T ⊓ g))›*)\n show \"component (?v * ?v⇧T ⊓ g) r ≤ r⇧T * ?t⇧⋆\"\n proof (-)\n (*goal: ‹component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ≤ r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›*)\n have 15: \"r⇧T * (v * v⇧T ⊓ ?g)⇧⋆ ≤ r⇧T * t⇧⋆\"\n using assms(1) (*‹prim_spanning_invariant (t::'a) (v::'a) (g::'a) (r::'a)›*) \"1\" (*‹regular (v::'a) ∧ regular (v * v⇧T) ∧ regular ((v ⊔ minarc (v * - v⇧T ⊓ (g::'a))⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T) ∧ regular (top * minarc (v * - v⇧T ⊓ g))›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) inf_pp_commute (*‹- - (- - ?x ⊓ ?y) = - - ?x ⊓ - - ?y›*))\n have \"component (?v * ?v⇧T ⊓ g) r = r⇧T * (?v * ?v⇧T ⊓ ?g)⇧⋆\"\n using \"1\" (*‹regular v ∧ regular (v * v⇧T) ∧ regular ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T) ∧ regular (top * minarc (v * - v⇧T ⊓ g))›*) by simp\n also (*calculation: ‹component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r = r⇧T * ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ - - g)⇧⋆›*) have \"... ≤ r⇧T * ?t⇧⋆\"\n using \"2\" (*‹(t::'a) ≤ (v::'a) * v⇧T ⊓ - - (g::'a)›*) \"6\" (*‹minarc ((v::'a) * - v⇧T ⊓ (g::'a)) ≤ v * - v⇧T›*) \"7\" (*‹vector v›*) \"11\" (*‹arc (minarc (v * - v⇧T ⊓ g))›*) \"12\" (*‹(r::'a::type)⇧T ≤ (v::'a::type)⇧T›*) \"13\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆›*) \"14\" (*‹symmetric (- - g)›*) \"15\" (*‹r⇧T * (v * v⇧T ⊓ - - g)⇧⋆ ≤ r⇧T * t⇧⋆›*) by (metis span_inv (*‹⟦?e ≤ ?v * - ?v⇧T; vector ?v; arc ?e; ?t ≤ ?v * ?v⇧T ⊓ ?g; symmetric ?g; ?v⇧T = ?r⇧T * ?t⇧⋆; injective ?r; ?r⇧T ≤ ?v⇧T; ?r⇧T * (?v * ?v⇧T ⊓ ?g)⇧⋆ ≤ ?r⇧T * ?t⇧⋆⟧ ⟹ ?r⇧T * ((?v ⊔ ?e⇧T * top) * (?v ⊔ ?e⇧T * top)⇧T ⊓ ?g)⇧⋆ ≤ ?r⇧T * (?t ⊔ ?e)⇧⋆›*))\n finally (*calculation: ‹component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ≤ r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›*) show \"?thesis\"\n (*goal: ‹component ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T ⊓ g) r ≤ r⇧T * (t ⊔ minarc (v * - v⇧T ⊓ g))⇧⋆›*) .\n qed\n next\n (*goal: ‹regular ((t::'a) ⊔ minarc ((v::'a) * - v⇧T ⊓ (g::'a)))›*)\n show \"regular ?t\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) regular_closed_sup (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x ⊔ ?y)›*) minarc_regular (*‹regular (minarc ?x)›*))\n qed\n qed\n next\n (*goal: ‹card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›*)\n have 16: \"top * ?e ≤ ?c\"\n proof (-)\n (*goal: ‹top * minarc (v * - v⇧T ⊓ g) ≤ component g r›*)\n have \"top * ?e = top * ?e⇧T * ?e\"\n using \"11\" (*‹arc (minarc (v * - v⇧T ⊓ g))›*) by (metis arc_top_edge (*‹arc ?x ⟹ ?x⇧T * top * ?x = ?x⇧T * ?x›*) mult_assoc (*‹?a * ?b * ?c = ?a * (?b * ?c)›*))\n also (*calculation: ‹top * minarc (v * - v⇧T ⊓ g) = top * minarc (v * - v⇧T ⊓ g)⇧T * minarc (v * - v⇧T ⊓ g)›*) have \"... ≤ v⇧T * ?e\"\n using \"7\" (*‹vector v›*) \"8\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v›*) by (metis conv_dist_comp (*‹(?x * ?y)⇧T = ?y⇧T * ?x⇧T›*) conv_isotone (*‹?x ≤ ?y ⟹ ?x⇧T ≤ ?y⇧T›*) mult_left_isotone (*‹?x ≤ ?y ⟹ ?x * ?z ≤ ?y * ?z›*) symmetric_top_closed (*‹symmetric top›*))\n also (*calculation: ‹top * minarc (v * - v⇧T ⊓ g) ≤ v⇧T * minarc (v * - v⇧T ⊓ g)›*) have \"... ≤ v⇧T * ?g\"\n using \"5\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T ⊓ - - g›*) mult_right_isotone (*‹(?x::'a) ≤ (?y::'a) ⟹ (?z::'a) * ?x ≤ ?z * ?y›*) by auto\n also (*calculation: ‹top * minarc (v * - v⇧T ⊓ g) ≤ v⇧T * - - g›*) have \"... = r⇧T * t⇧⋆ * ?g\"\n using \"13\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆›*) by simp\n also (*calculation: ‹top * minarc (v * - v⇧T ⊓ g) ≤ r⇧T * t⇧⋆ * - - g›*) have \"... ≤ r⇧T * ?g⇧⋆ * ?g\"\n using \"4\" (*‹t ≤ - - g›*) by (simp add: mult_left_isotone (*‹?x ≤ ?y ⟹ ?x * ?z ≤ ?y * ?z›*) mult_right_isotone (*‹?x ≤ ?y ⟹ ?z * ?x ≤ ?z * ?y›*) star_isotone (*‹?x ≤ ?y ⟹ ?x⇧⋆ ≤ ?y⇧⋆›*))\n also (*calculation: ‹top * minarc (v * - v⇧T ⊓ g) ≤ component g r * - - g›*) have \"... ≤ ?c\"\n by (simp add: comp_associative (*‹?x * ?y * ?z = ?x * (?y * ?z)›*) mult_right_isotone (*‹?x ≤ ?y ⟹ ?z * ?x ≤ ?z * ?y›*) star.right_plus_below_circ (*‹?x⇧⋆ * ?x ≤ ?x⇧⋆›*))\n finally (*calculation: ‹top * minarc (v * - v⇧T ⊓ g) ≤ component g r›*) show \"?thesis\"\n (*goal: ‹top * minarc (v * - v⇧T ⊓ g) ≤ component g r›*)\n by simp\n qed\n have 17: \"top * ?e ≤ -v⇧T\"\n using \"6\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"7\" (*‹vector (v::'a)›*) by (simp add: schroeder_4_p (*‹(?x * ?y ≤ - ?z) = (?z * ?y⇧T ≤ - ?x)›*) vTeT (*‹⟦vector ?v; ?e ≤ ?v * - ?v⇧T⟧ ⟹ ?v⇧T * ?e⇧T = bot›*))\n have 18: \"¬ top * ?e ≤ -(top * ?e)\"\n by (metis assms( (*‹v * - v⇧T ⊓ g ≠ bot›*) 2) inf.orderE (*‹⟦?a ≤ ?b; ?a = ?a ⊓ ?b ⟹ ?thesis⟧ ⟹ ?thesis›*) minarc_bot_iff (*‹(minarc ?x = bot) = (?x = bot)›*) conv_complement_sub_inf (*‹?x⇧T * - (?x * ?y) ⊓ ?y = bot›*) inf_p (*‹?x ⊓ - ?x = bot›*) inf_top.left_neutral (*‹top ⊓ ?a = ?a›*) p_bot (*‹- bot = top›*) symmetric_top_closed (*‹symmetric top›*) vector_top_closed (*‹surjective top›*))\n have 19: \"-?v⇧T = -v⇧T ⊓ -(top * ?e)\"\n by (simp add: conv_dist_comp (*‹(?x * ?y)⇧T = ?y⇧T * ?x⇧T›*) conv_dist_sup (*‹(?x ⊔ ?y)⇧T = ?x⇧T ⊔ ?y⇧T›*))\n hence 20: \"¬ top * ?e ≤ -?v⇧T\"\n using \"18\" (*‹¬ top * minarc ((v::'a::type) * - v⇧T ⊓ (g::'a::type)) ≤ - (top * minarc (v * - v⇧T ⊓ g))›*) by simp\n show \"?n2 < ?n1\"\n apply (rule psubset_card_mono (*‹⟦finite ?B; ?A ⊂ ?B⟧ ⟹ card ?A < card ?B›*))\n (*goal: ‹card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›*)\n using finite_regular (*‹finite {x::'a. regular x}›*) apply simp\n (*top goal: ‹finite {x::'a. regular x ∧ x ≤ component (g::'a) (r::'a) ∧ x ≤ - (v::'a)⇧T}› and 1 goal remains*)\n using \"1\" (*‹regular v ∧ regular (v * v⇧T) ∧ regular ((v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) * (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T) ∧ regular (top * minarc (v * - v⇧T ⊓ g))›*) \"16\" (*‹top * minarc ((v::'a) * - v⇧T ⊓ (g::'a)) ≤ component g (r::'a)›*) \"17\" (*‹top * minarc (v * - v⇧T ⊓ g) ≤ - v⇧T›*) \"19\" (*‹- (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T = - v⇧T ⊓ - (top * minarc (v * - v⇧T ⊓ g))›*) \"20\" (*‹¬ top * minarc ((v::'a) * - v⇧T ⊓ (g::'a)) ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T›*) by auto\n qed\nqed\n\nlemma prim_vc_3:\n assumes \"prim_spanning_invariant t v g r\"\n and \"v * -v⇧T ⊓ g = bot\"\n shows \"spanning_tree t g r\"\nproof (-)\n (*goal: ‹spanning_tree t g r›*)\n let ?g = \"--g\"\n have 1: \"regular v ∧ regular (v * v⇧T)\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant (?t::'a) (?v::'a) (?g::'a) (?r::'a) ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree (?t::'a) (?g::'a) (?r::'a) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) prim_precondition_def (*‹prim_precondition (?g::'a) (?r::'a) ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) regular_conv_closed (*‹regular (?x::'a) ⟹ regular (?x⇧T)›*) regular_closed_star (*‹regular (?x::'a) ⟹ regular (?x⇧⋆)›*) regular_mult_closed (*‹⟦regular (?x::'a); regular (?y::'a)⟧ ⟹ regular (?x * ?y)›*) conv_involutive (*‹(?x::'a)⇧T⇧T = ?x›*))\n have 2: \"v * -v⇧T ⊓ ?g = bot\"\n using assms(2) (*‹v * - v⇧T ⊓ g = bot›*) pp_inf_bot_iff (*‹(?x ⊓ ?y = bot) = (- - ?x ⊓ ?y = bot)›*) pp_pp_inf_bot_iff (*‹(?x ⊓ ?y = bot) = (- - ?x ⊓ - - ?y = bot)›*) by simp\n have 3: \"v⇧T = r⇧T * t⇧⋆ ∧ vector v\"\n using assms(1) (*‹prim_spanning_invariant (t::'a) (v::'a) (g::'a) (r::'a)›*) by (simp add: covector_mult_closed (*‹covector ?x ⟹ covector (?x * ?y)›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) vector_conv_covector (*‹vector ?v = covector (?v⇧T)›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*))\n have 4: \"t ≤ v * v⇧T ⊓ ?g\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) \"1\" (*‹regular v ∧ regular (v * v⇧T)›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) inf_pp_commute (*‹- - (- - ?x ⊓ ?y) = - - ?x ⊓ - - ?y›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) inf.boundedE (*‹⟦?a ≤ ?b ⊓ ?c; ⟦?a ≤ ?b; ?a ≤ ?c⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n have \"r⇧T * (v * v⇧T ⊓ ?g)⇧⋆ ≤ r⇧T * t⇧⋆\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) \"1\" (*‹regular v ∧ regular (v * v⇧T)›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant (?t::'a::type) (?v::'a::type) (?g::'a::type) (?r::'a::type) ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) inf_pp_commute (*‹- - (- - (?x::'a::type) ⊓ (?y::'a::type)) = - - ?x ⊓ - - ?y›*) spanning_tree_def (*‹spanning_tree (?t::'a::type) (?g::'a::type) (?r::'a::type) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n hence 5: \"component g r = v⇧T\"\n using \"1\" (*‹regular v ∧ regular (v * v⇧T)›*) \"2\" (*‹v * - v⇧T ⊓ - - g = bot›*) \"3\" (*‹(v::'a)⇧T = (r::'a)⇧T * (t::'a)⇧⋆ ∧ vector v›*) \"4\" (*‹t ≤ v * v⇧T ⊓ - - g›*) by (metis span_post (*‹⟦regular ?v; vector ?v; ?v⇧T = ?r⇧T * ?t⇧⋆; ?v * - ?v⇧T ⊓ ?g = bot; ?t ≤ ?v * ?v⇧T ⊓ ?g; ?r⇧T * (?v * ?v⇧T ⊓ ?g)⇧⋆ ≤ ?r⇧T * ?t⇧⋆⟧ ⟹ ?v⇧T = ?r⇧T * ?g⇧⋆›*))\n have \"regular (v * v⇧T)\"\n using assms(1) (*‹prim_spanning_invariant t v g r›*) by (metis prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) regular_conv_closed (*‹regular ?x ⟹ regular (?x⇧T)›*) regular_closed_star (*‹regular ?x ⟹ regular (?x⇧⋆)›*) regular_mult_closed (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x * ?y)›*) conv_involutive (*‹?x⇧T⇧T = ?x›*))\n hence 6: \"t ≤ v * v⇧T ⊓ ?g\"\n by (metis assms( (*‹prim_spanning_invariant t v g r›*) 1) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) inf_pp_commute (*‹- - (- - ?x ⊓ ?y) = - - ?x ⊓ - - ?y›*) inf.boundedE (*‹⟦?a ≤ ?b ⊓ ?c; ⟦?a ≤ ?b; ?a ≤ ?c⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n show \"spanning_tree t g r\"\n apply (unfold spanning_tree_def (*‹spanning_tree (?t::'a::type) (?g::'a::type) (?r::'a::type) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*), intro conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*))\n (*goal: ‹spanning_tree t g r›*)\n using assms(1) (*‹prim_spanning_invariant t v g r›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree (?t::'a) (?g::'a) (?r::'a) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) apply simp\n (*top goal: ‹injective t› and 4 goals remain*)\n using assms(1) (*‹prim_spanning_invariant t v g r›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree (?t::'a) (?g::'a) (?r::'a) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) apply simp\n (*top goal: ‹pd_kleene_allegory_class.acyclic (t::'a::type)› and 3 goals remain*)\n using \"5\" (*‹component g r = v⇧T›*) \"6\" (*‹t ≤ v * v⇧T ⊓ - - g›*) apply simp\n (*top goal: ‹(t::'a) ≤ component (g::'a) (r::'a)⇧T * component g r ⊓ - - g› and 2 goals remain*)\n using assms(1) (*‹prim_spanning_invariant t v g r›*) \"5\" (*‹component g r = v⇧T›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) apply simp\n (*top goal: ‹component (g::'a::type) (r::'a::type) ≤ r⇧T * (t::'a::type)⇧⋆› and 1 goal remains*)\n using assms(1) (*‹prim_spanning_invariant t v g r›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) by simp\nqed\n\ntext ‹\nThe following result shows that Prim's algorithm terminates and constructs a spanning tree.\nWe cannot yet show that this is a minimum spanning tree.\n›\n\ntheorem prim_spanning:\n \"VARS t v e\n [ prim_precondition g r ]\n t := bot;\n v := r;\n WHILE v * -v⇧T ⊓ g ≠ bot\n INV { prim_spanning_invariant t v g r }\n VAR { card { x . regular x ∧ x ≤ component g r ⊓ -v⇧T } }\n DO e := minarc (v * -v⇧T ⊓ g);\n t := t ⊔ e;\n v := v ⊔ e⇧T * top\n OD\n [ spanning_tree t g r ]\"\n apply vcg_tc_simp\n (*goal: ‹[prim_precondition g r]\n t := bot; v := r; WHILE v * - v⇧T ⊓ g ≠ bot INV {prim_spanning_invariant t v g r} VAR {card {x. regular x ∧ x ≤ component g r ⊓ - v⇧T}} \n DO e := minarc (v * - v⇧T ⊓ g); t := t ⊔ e; v := v ⊔ e⇧T * top OD\n [spanning_tree t g r]›*)\n apply (simp add: prim_vc_1 (*‹prim_precondition ?g ?r ⟹ prim_spanning_invariant bot ?r ?g ?r›*))\n (*top goal: ‹prim_precondition (g::'a::type) (r::'a::type) ⟹ prim_spanning_invariant bot r g r› and 2 goals remain*)\n using prim_vc_2 (*‹⟦prim_spanning_invariant ?t ?v ?g ?r; ?v * - ?v⇧T ⊓ ?g ≠ bot⟧ ⟹ prim_spanning_invariant (?t ⊔ minarc (?v * - ?v⇧T ⊓ ?g)) (?v ⊔ minarc (?v * - ?v⇧T ⊓ ?g)⇧T * top) ?g ?r ∧ card {x. regular x ∧ x ≤ component ?g ?r ∧ x ≤ - (?v ⊔ minarc (?v * - ?v⇧T ⊓ ?g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component ?g ?r ∧ x ≤ - ?v⇧T}›*) apply blast\n (*top goal: ‹⋀t v. prim_spanning_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot ⟹ prim_spanning_invariant (t ⊔ minarc (v * - v⇧T ⊓ g)) (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) g r ∧ card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}› and 1 goal remains*)\n using prim_vc_3 (*‹⟦prim_spanning_invariant ?t ?v ?g ?r; ?v * - ?v⇧T ⊓ ?g = bot⟧ ⟹ spanning_tree ?t ?g ?r›*) by auto\n\ntext ‹\nBecause we have shown total correctness, we conclude that a spanning tree exists.\n›\n\nlemma prim_exists_spanning:\n \"prim_precondition g r ⟹ ∃t . spanning_tree t g r\"\n using tc_extract_function (*‹[?p]\n ?c\n [?q] ⟹ ∃f. ∀s. s ∈ ?p ⟶ f s ∈ ?q›*) prim_spanning (*‹[prim_precondition ?g ?r]\n t := bot; v := ?r; WHILE v * - v⇧T ⊓ ?g ≠ bot INV {prim_spanning_invariant t v ?g ?r} VAR {card {x. regular x ∧ x ≤ component ?g ?r ⊓ - v⇧T}} \n DO e := minarc (v * - v⇧T ⊓ ?g); t := t ⊔ e; v := v ⊔ e⇧T * top OD\n [spanning_tree t ?g ?r]›*) by blast\n\ntext ‹\nThis implies that a minimum spanning tree exists, which is used in the subsequent correctness proof.\n›\n\nlemma prim_exists_minimal_spanning:\n assumes \"prim_precondition g r\"\n shows \"∃t . minimum_spanning_tree t g r\"\nsorry\n\ntext ‹\nPrim's minimum spanning tree algorithm terminates and is correct.\nThis is the same algorithm that is used in the previous correctness proof, with the same precondition and variant, but with a different invariant and postcondition.\n›\n\ntheorem prim:\n \"VARS t v e\n [ prim_precondition g r ∧ (∃w . minimum_spanning_tree w g r) ]\n t := bot;\n v := r;\n WHILE v * -v⇧T ⊓ g ≠ bot\n INV { prim_invariant t v g r }\n VAR { card { x . regular x ∧ x ≤ component g r ⊓ -v⇧T } }\n DO e := minarc (v * -v⇧T ⊓ g);\n t := t ⊔ e;\n v := v ⊔ e⇧T * top\n OD\n [ minimum_spanning_tree t g r ]\"\nproof (vcg_tc_simp)\n (*goals:\n 1. ‹prim_precondition g r ∧ (∃w. minimum_spanning_tree w g r) ⟹ prim_invariant bot r g r›\n 2. ‹⋀t v. prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot ⟹ prim_invariant (t ⊔ minarc (v * - v⇧T ⊓ g)) (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) g r ∧ card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›\n 3. ‹⋀t v. prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot ⟹ minimum_spanning_tree t g r›*)\n assume \"prim_precondition g r ∧ (∃w . minimum_spanning_tree w g r)\" (*‹prim_precondition (g::'a) (r::'a) ∧ (∃w::'a. minimum_spanning_tree w g r)›*)\n thus \"prim_invariant bot r g r\"\n using prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_vc_1 (*‹prim_precondition ?g ?r ⟹ prim_spanning_invariant bot ?r ?g ?r›*) by simp\nnext\n (*goals:\n 1. ‹⋀t v. prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot ⟹ prim_invariant (t ⊔ minarc (v * - v⇧T ⊓ g)) (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) g r ∧ card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›\n 2. ‹⋀t v. prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot ⟹ minimum_spanning_tree t g r›*)\n fix t and v\n let ?vcv = \"v * -v⇧T ⊓ g\"\n let ?vv = \"v * v⇧T ⊓ g\"\n let ?e = \"minarc ?vcv\"\n let ?t = \"t ⊔ ?e\"\n let ?v = \"v ⊔ ?e⇧T * top\"\n let ?c = \"component g r\"\n let ?g = \"--g\"\n let ?n1 = \"card { x . regular x ∧ x ≤ ?c ∧ x ≤ -v⇧T }\"\n let ?n2 = \"card { x . regular x ∧ x ≤ ?c ∧ x ≤ -?v⇧T }\"\n assume 1: \"prim_invariant t v g r ∧ ?vcv ≠ bot\" (*‹prim_invariant (t::'a) (v::'a) (g::'a) (r::'a) ∧ v * - v⇧T ⊓ g ≠ bot›*)\n hence 2: \"regular v ∧ regular (v * v⇧T)\"\n by (metis (no_types, opaque_lifting) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) regular_conv_closed (*‹regular ?x ⟹ regular (?x⇧T)›*) regular_closed_star (*‹regular ?x ⟹ regular (?x⇧⋆)›*) regular_mult_closed (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x * ?y)›*) conv_involutive (*‹?x⇧T⇧T = ?x›*))\n have 3: \"t ≤ v * v⇧T ⊓ ?g\"\n using \"1\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot›*) \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) by (metis (no_types, opaque_lifting) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) inf_pp_commute (*‹- - (- - ?x ⊓ ?y) = - - ?x ⊓ - - ?y›*) inf.boundedE (*‹⟦?a ≤ ?b ⊓ ?c; ⟦?a ≤ ?b; ?a ≤ ?c⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n hence 4: \"t ≤ v * v⇧T\"\n by simp\n have 5: \"t ≤ ?g\"\n using \"3\" (*‹t ≤ v * v⇧T ⊓ - - g›*) by simp\n have 6: \"?e ≤ v * -v⇧T ⊓ ?g\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) by (metis minarc_below (*‹minarc ?x ≤ - - ?x›*) pp_dist_inf (*‹- - (?x ⊓ ?y) = - - ?x ⊓ - - ?y›*) regular_mult_closed (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x * ?y)›*) regular_closed_p (*‹regular (- ?x)›*))\n hence 7: \"?e ≤ v * -v⇧T\"\n by simp\n have 8: \"vector v\"\n using \"1\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition (?g::'a) (?r::'a) ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) by (simp add: covector_mult_closed (*‹covector ?x ⟹ covector (?x * ?y)›*) vector_conv_covector (*‹vector ?v = covector (?v⇧T)›*))\n have 9: \"arc ?e\"\n using \"1\" (*‹prim_invariant (t::'a) (v::'a) (g::'a) (r::'a) ∧ v * - v⇧T ⊓ g ≠ bot›*) minarc_arc (*‹(?x::'a) ≠ bot ⟶ arc (minarc ?x)›*) by simp\n from \"1\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot›*) obtain w where 10: \"minimum_spanning_tree w g r ∧ t ≤ w\"\n (*goal: ‹(⋀w. minimum_spanning_tree w g r ∧ t ≤ w ⟹ thesis) ⟹ thesis›*)\n by (metis prim_invariant_def (*‹prim_invariant (?t::'a) (?v::'a) (?g::'a) (?r::'a) ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w::'a. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*))\n hence 11: \"vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆ ∧ forest w ∧ t ≤ w ∧ w ≤ ?c⇧T * ?c ⊓ ?g ∧ r⇧T * (?c⇧T * ?c ⊓ ?g)⇧⋆ ≤ r⇧T * w⇧⋆\"\n using \"1\" (*‹prim_invariant (t::'a::type) (v::'a::type) (g::'a::type) (r::'a::type) ∧ v * - v⇧T ⊓ g ≠ bot›*) \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) minimum_spanning_tree_def (*‹minimum_spanning_tree ?t ?g ?r ≡ spanning_tree ?t ?g ?r ∧ (∀u. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) reachable_restrict (*‹vector ?r ⟹ ?r⇧T * ?g⇧⋆ = ?r⇧T * ((?r⇧T * ?g⇧⋆)⇧T * (?r⇧T * ?g⇧⋆) ⊓ ?g)⇧⋆›*) by simp\n hence 12: \"w * v ≤ v\"\n using predecessors_reachable (*‹⟦vector ?r; injective ?r; ?v⇧T = ?r⇧T * ?t⇧⋆; forest ?w; ?t ≤ ?w; ?w ≤ (?r⇧T * ?g⇧⋆)⇧T * (?r⇧T * ?g⇧⋆) ⊓ ?g; ?r⇧T * ?g⇧⋆ ≤ ?r⇧T * ?w⇧⋆⟧ ⟹ ?w * ?v ≤ ?v›*) reachable_restrict (*‹vector ?r ⟹ ?r⇧T * ?g⇧⋆ = ?r⇧T * ((?r⇧T * ?g⇧⋆)⇧T * (?r⇧T * ?g⇧⋆) ⊓ ?g)⇧⋆›*) by auto\n have 13: \"g = g⇧T\"\n using \"1\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) by simp\n hence 14: \"?g⇧T = ?g\"\n using conv_complement (*‹(- ?x)⇧T = - ?x⇧T›*) by simp\n show \"prim_invariant ?t ?v g r ∧ ?n2 < ?n1\"\n proof (unfold prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹prim_spanning_invariant (t ⊔ minarc (v * - v⇧T ⊓ g)) (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top) g r›\n 2. ‹∃w. minimum_spanning_tree w g r ∧ t ⊔ minarc (v * - v⇧T ⊓ g) ≤ w›\n 3. ‹card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›*)\n show \"prim_spanning_invariant ?t ?v g r\"\n using \"1\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_vc_2 (*‹⟦prim_spanning_invariant ?t ?v ?g ?r; ?v * - ?v⇧T ⊓ ?g ≠ bot⟧ ⟹ prim_spanning_invariant (?t ⊔ minarc (?v * - ?v⇧T ⊓ ?g)) (?v ⊔ minarc (?v * - ?v⇧T ⊓ ?g)⇧T * top) ?g ?r ∧ card {x. regular x ∧ x ≤ component ?g ?r ∧ x ≤ - (?v ⊔ minarc (?v * - ?v⇧T ⊓ ?g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component ?g ?r ∧ x ≤ - ?v⇧T}›*) by blast\n next\n (*goals:\n 1. ‹∃w. minimum_spanning_tree w g r ∧ t ⊔ minarc (v * - v⇧T ⊓ g) ≤ w›\n 2. ‹card {x. regular x ∧ x ≤ component g r ∧ x ≤ - (v ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›*)\n show \"∃w . minimum_spanning_tree w g r ∧ ?t ≤ w\"\n proof (standard)\n (*goal: ‹minimum_spanning_tree ?w g r ∧ t ⊔ minarc (v * - v⇧T ⊓ g) ≤ ?w›*)\n let ?f = \"w ⊓ v * -v⇧T ⊓ top * ?e * w⇧T⇧⋆\"\n let ?p = \"w ⊓ -v * -v⇧T ⊓ top * ?e * w⇧T⇧⋆\"\n let ?fp = \"w ⊓ -v⇧T ⊓ top * ?e * w⇧T⇧⋆\"\n let ?w = \"(w ⊓ -?fp) ⊔ ?p⇧T ⊔ ?e\"\n have 15: \"regular ?f ∧ regular ?fp ∧ regular ?w\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) \"10\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) by (metis regular_conv_closed (*‹regular (?x::'a) ⟹ regular (?x⇧T)›*) regular_closed_star (*‹regular (?x::'a) ⟹ regular (?x⇧⋆)›*) regular_mult_closed (*‹⟦regular (?x::'a); regular (?y::'a)⟧ ⟹ regular (?x * ?y)›*) regular_closed_top (*‹regular top›*) regular_closed_inf (*‹⟦regular (?x::'a); regular (?y::'a)⟧ ⟹ regular (?x ⊓ ?y)›*) regular_closed_sup (*‹⟦regular (?x::'a); regular (?y::'a)⟧ ⟹ regular (?x ⊔ ?y)›*) minarc_regular (*‹regular (minarc (?x::'a))›*) minimum_spanning_tree_def (*‹minimum_spanning_tree (?t::'a) (?g::'a) (?r::'a) ≡ spanning_tree ?t ?g ?r ∧ (∀u::'a. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree (?t::'a) (?g::'a) (?r::'a) ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n show \"minimum_spanning_tree ?w g r ∧ ?t ≤ ?w\"\n proof (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹minimum_spanning_tree (prim_W (w::'a) (v::'a) (minarc (v * - v⇧T ⊓ (g::'a)))) g (r::'a)›\n 2. ‹(t::'a) ⊔ minarc ((v::'a) * - v⇧T ⊓ (g::'a)) ≤ prim_W (w::'a) v (minarc (v * - v⇧T ⊓ g))›*)\n show \"minimum_spanning_tree ?w g r\"\n proof (unfold minimum_spanning_tree_def (*‹minimum_spanning_tree (?t::'a::type) (?g::'a::type) (?r::'a::type) ≡ spanning_tree ?t ?g ?r ∧ (∀u::'a::type. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*), intro conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹spanning_tree (prim_W w v (minarc (v * - v⇧T ⊓ g))) g r›\n 2. ‹∀u. spanning_tree u g r ⟶ sum_class.sum (prim_W w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) ≤ sum_class.sum (u ⊓ g)›*)\n show \"spanning_tree ?w g r\"\n proof (unfold spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹injective (prim_W w v (minarc (v * - v⇧T ⊓ g)))›\n 2. ‹pd_kleene_allegory_class.acyclic (prim_W w v (minarc (v * - v⇧T ⊓ g)))›\n 3. ‹prim_W w v (minarc (v * - v⇧T ⊓ g)) ≤ component g r⇧T * component g r ⊓ - - g›\n 4. ‹component g r ≤ r⇧T * prim_W w v (minarc (v * - v⇧T ⊓ g))⇧⋆›\n 5. ‹regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*)\n show \"injective ?w\"\n using \"7\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"8\" (*‹vector v›*) \"9\" (*‹arc (minarc (v * - v⇧T ⊓ g))›*) \"11\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆ ∧ forest w ∧ t ≤ w ∧ w ≤ component g r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) exchange_injective (*‹⟦arc ?e; ?e ≤ ?v * - ?v⇧T; forest ?w; vector ?v⟧ ⟹ injective (prim_W ?w ?v ?e)›*) by blast\n next\n (*goals:\n 1. ‹pd_kleene_allegory_class.acyclic (prim_W w v (minarc (v * - v⇧T ⊓ g)))›\n 2. ‹prim_W w v (minarc (v * - v⇧T ⊓ g)) ≤ component g r⇧T * component g r ⊓ - - g›\n 3. ‹component g r ≤ r⇧T * prim_W w v (minarc (v * - v⇧T ⊓ g))⇧⋆›\n 4. ‹regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*)\n show \"acyclic ?w\"\n using \"7\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"8\" (*‹vector v›*) \"11\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆ ∧ forest w ∧ t ≤ w ∧ w ≤ component g r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) \"12\" (*‹w * v ≤ v›*) exchange_acyclic (*‹⟦vector ?v; ?e ≤ ?v * - ?v⇧T; ?w * ?v ≤ ?v; pd_kleene_allegory_class.acyclic ?w⟧ ⟹ pd_kleene_allegory_class.acyclic (prim_W ?w ?v ?e)›*) by blast\n next\n (*goals:\n 1. ‹prim_W w v (minarc (v * - v⇧T ⊓ g)) ≤ component g r⇧T * component g r ⊓ - - g›\n 2. ‹component g r ≤ r⇧T * prim_W w v (minarc (v * - v⇧T ⊓ g))⇧⋆›\n 3. ‹regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*)\n show \"?w ≤ ?c⇧T * ?c ⊓ --g\"\n proof (-)\n (*goal: ‹prim_W w v (minarc (v * - v⇧T ⊓ g)) ≤ component g r⇧T * component g r ⊓ - - g›*)\n have 16: \"w ⊓ -?fp ≤ ?c⇧T * ?c ⊓ --g\"\n using \"10\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) by (simp add: le_infI1 (*‹?a ≤ ?x ⟹ ?a ⊓ ?b ≤ ?x›*) minimum_spanning_tree_def (*‹minimum_spanning_tree ?t ?g ?r ≡ spanning_tree ?t ?g ?r ∧ (∀u. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n have \"?p⇧T ≤ w⇧T\"\n by (simp add: conv_isotone (*‹(?x::'a::type) ≤ (?y::'a::type) ⟹ ?x⇧T ≤ ?y⇧T›*) inf.sup_monoid.add_assoc (*‹(?a::'a::type) ⊓ (?b::'a::type) ⊓ (?c::'a::type) = ?a ⊓ (?b ⊓ ?c)›*))\n also (*calculation: ‹prim_P (w::'a) (v::'a) (minarc (v * - v⇧T ⊓ (g::'a)))⇧T ≤ w⇧T›*) have \"... ≤ (?c⇧T * ?c ⊓ --g)⇧T\"\n using \"11\" (*‹vector (r::'a) ∧ injective r ∧ (v::'a)⇧T = r⇧T * (t::'a)⇧⋆ ∧ forest (w::'a) ∧ t ≤ w ∧ w ≤ component (g::'a) r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) conv_order (*‹(?x ≤ ?y) = (?x⇧T ≤ ?y⇧T)›*) by simp\n also (*calculation: ‹prim_P w v (minarc (v * - v⇧T ⊓ g))⇧T ≤ (component g r⇧T * component g r ⊓ - - g)⇧T›*) have \"... = ?c⇧T * ?c ⊓ --g\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) \"14\" (*‹symmetric (- - g)›*) conv_dist_comp (*‹(?x * ?y)⇧T = ?y⇧T * ?x⇧T›*) conv_dist_inf (*‹(?x ⊓ ?y)⇧T = ?x⇧T ⊓ ?y⇧T›*) by simp\n finally (*calculation: ‹prim_P w v (minarc (v * - v⇧T ⊓ g))⇧T ≤ component g r⇧T * component g r ⊓ - - g›*) have 17: \"?p⇧T ≤ ?c⇧T * ?c ⊓ --g\" .\n have \"?e ≤ ?c⇧T * ?c ⊓ ?g\"\n using \"5\" (*‹t ≤ - - g›*) \"6\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T ⊓ - - g›*) \"11\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆ ∧ forest w ∧ t ≤ w ∧ w ≤ component g r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) mst_subgraph_inv (*‹⟦?e ≤ ?v * - ?v⇧T ⊓ ?g; ?t ≤ ?g; ?v⇧T = ?r⇧T * ?t⇧⋆⟧ ⟹ ?e ≤ (?r⇧T * ?g⇧⋆)⇧T * (?r⇧T * ?g⇧⋆) ⊓ ?g›*) by auto\n thus \"?thesis\"\n (*goal: ‹prim_W (w::'a::type) (v::'a::type) (minarc (v * - v⇧T ⊓ (g::'a::type))) ≤ component g (r::'a::type)⇧T * component g r ⊓ - - g›*)\n using \"16\" (*‹w ⊓ - prim_EP w v (minarc (v * - v⇧T ⊓ g)) ≤ component g r⇧T * component g r ⊓ - - g›*) \"17\" (*‹prim_P w v (minarc (v * - v⇧T ⊓ g))⇧T ≤ component g r⇧T * component g r ⊓ - - g›*) by simp\n qed\n next\n (*goals:\n 1. ‹component (g::'a) (r::'a) ≤ r⇧T * prim_W (w::'a) (v::'a) (minarc (v * - v⇧T ⊓ g))⇧⋆›\n 2. ‹regular (prim_W (w::'a) (v::'a) (minarc (v * - v⇧T ⊓ (g::'a))))›*)\n show \"?c ≤ r⇧T * ?w⇧⋆\"\n proof (-)\n (*goal: ‹component g r ≤ r⇧T * prim_W w v (minarc (v * - v⇧T ⊓ g))⇧⋆›*)\n have \"?c ≤ r⇧T * w⇧⋆\"\n using \"10\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) minimum_spanning_tree_def (*‹minimum_spanning_tree (?t::'a) (?g::'a) (?r::'a) ≡ spanning_tree ?t ?g ?r ∧ (∀u::'a. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) by simp\n also (*calculation: ‹component g r ≤ r⇧T * w⇧⋆›*) have \"... ≤ r⇧T * ?w⇧⋆\"\n using \"4\" (*‹(t::'a) ≤ (v::'a) * v⇧T›*) \"7\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"8\" (*‹vector (v::'a)›*) \"10\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) \"11\" (*‹vector (r::'a) ∧ injective r ∧ (v::'a)⇧T = r⇧T * (t::'a)⇧⋆ ∧ forest (w::'a) ∧ t ≤ w ∧ w ≤ component (g::'a) r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) \"12\" (*‹w * v ≤ v›*) \"15\" (*‹regular (w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆) ∧ regular (prim_EP w v (minarc (v * - v⇧T ⊓ g))) ∧ regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*) by (metis mst_reachable_inv (*‹⟦regular (prim_EP ?w ?v ?e); vector ?r; ?e ≤ ?v * - ?v⇧T; vector ?v; ?v⇧T = ?r⇧T * ?t⇧⋆; ?t ≤ ?w; ?t ≤ ?v * ?v⇧T; ?w * ?v ≤ ?v⟧ ⟹ ?r⇧T * ?w⇧⋆ ≤ ?r⇧T * prim_W ?w ?v ?e⇧⋆›*))\n finally (*calculation: ‹component g r ≤ r⇧T * prim_W w v (minarc (v * - v⇧T ⊓ g))⇧⋆›*) show \"?thesis\"\n (*goal: ‹component g r ≤ r⇧T * prim_W w v (minarc (v * - v⇧T ⊓ g))⇧⋆›*) .\n qed\n next\n (*goal: ‹regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*)\n show \"regular ?w\"\n using \"15\" (*‹regular ((w::'a) ⊓ (v::'a) * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ (g::'a)) * w⇧T⇧⋆) ∧ regular (prim_EP w v (minarc (v * - v⇧T ⊓ g))) ∧ regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*) by simp\n qed\n next\n (*goal: ‹∀u. spanning_tree u g r ⟶ sum_class.sum (prim_W w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) ≤ sum_class.sum (u ⊓ g)›*)\n have 18: \"?f ⊔ ?p = ?fp\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) \"8\" (*‹vector v›*) epm_1 (*‹vector ?v ⟹ prim_E ?w ?v ?e ⊔ prim_P ?w ?v ?e = prim_EP ?w ?v ?e›*) by fastforce\n have \"arc (w ⊓ --v * -v⇧T ⊓ top * ?e * w⇧T⇧⋆)\"\n using \"5\" (*‹t ≤ - - g›*) \"6\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T ⊓ - - g›*) \"8\" (*‹vector (v::'a)›*) \"9\" (*‹arc (minarc (v * - v⇧T ⊓ g))›*) \"11\" (*‹vector (r::'a) ∧ injective r ∧ (v::'a)⇧T = r⇧T * (t::'a)⇧⋆ ∧ forest (w::'a) ∧ t ≤ w ∧ w ≤ component (g::'a) r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) \"12\" (*‹w * v ≤ v›*) reachable_restrict (*‹vector ?r ⟹ ?r⇧T * ?g⇧⋆ = ?r⇧T * ((?r⇧T * ?g⇧⋆)⇧T * (?r⇧T * ?g⇧⋆) ⊓ ?g)⇧⋆›*) arc_edge (*‹⟦?e ≤ ?v * - ?v⇧T ⊓ ?g; vector ?v; ?v⇧T = ?r⇧T * ?t⇧⋆; ?t ≤ ?g; ?r⇧T * ?g⇧⋆ ≤ ?r⇧T * ?w⇧⋆; ?w * ?v ≤ ?v; injective ?w; arc ?e⟧ ⟹ arc (prim_E ?w ?v ?e)›*) by auto\n hence 19: \"arc ?f\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) by simp\n hence \"?f = bot ⟶ top = bot\"\n by (metis mult_left_zero (*‹bot * ?x = bot›*) mult_right_zero (*‹?x * bot = bot›*))\n hence \"?f ≠ bot\"\n using \"1\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g ≠ bot›*) le_bot (*‹?a ≤ bot ⟹ ?a = bot›*) by auto\n hence \"?f ⊓ v * -v⇧T ⊓ ?g ≠ bot\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) \"11\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆ ∧ forest w ∧ t ≤ w ∧ w ≤ component g r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) by (simp add: inf.absorb1 (*‹?a ≤ ?b ⟹ ?a ⊓ ?b = ?a›*) le_infI1 (*‹?a ≤ ?x ⟹ ?a ⊓ ?b ≤ ?x›*))\n hence \"g ⊓ (?f ⊓ v * -v⇧T) ≠ bot\"\n using inf_commute (*‹?x ⊓ ?y = ?y ⊓ ?x›*) pp_inf_bot_iff (*‹(?x ⊓ ?y = bot) = (- - ?x ⊓ ?y = bot)›*) by simp\n hence 20: \"?f ⊓ ?vcv ≠ bot\"\n by (simp add: inf_assoc (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) inf_commute (*‹?x ⊓ ?y = ?y ⊓ ?x›*))\n hence 21: \"?f ⊓ g = ?f ⊓ ?vcv\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) by (simp add: inf_assoc (*‹(?x::'a) ⊓ (?y::'a) ⊓ (?z::'a) = ?x ⊓ (?y ⊓ ?z)›*) inf_commute (*‹(?x::'a) ⊓ (?y::'a) = ?y ⊓ ?x›*) inf_left_commute (*‹(?x::'a) ⊓ ((?y::'a) ⊓ (?z::'a)) = ?y ⊓ (?x ⊓ ?z)›*))\n have 22: \"?e ⊓ g = minarc ?vcv ⊓ ?vcv\"\n using \"7\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) by (simp add: inf.absorb2 (*‹?b ≤ ?a ⟹ ?a ⊓ ?b = ?b›*) inf.assoc (*‹?a ⊓ ?b ⊓ ?c = ?a ⊓ (?b ⊓ ?c)›*) inf.commute (*‹?a ⊓ ?b = ?b ⊓ ?a›*))\n hence 23: \"sum (?e ⊓ g) ≤ sum (?f ⊓ g)\"\n using \"15\" (*‹regular (w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆) ∧ regular (prim_EP w v (minarc (v * - v⇧T ⊓ g))) ∧ regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*) \"19\" (*‹arc (w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆)›*) \"20\" (*‹(w::'a) ⊓ (v::'a) * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ (g::'a)) * w⇧T⇧⋆ ⊓ (v * - v⇧T ⊓ g) ≠ bot›*) \"21\" (*‹(w::'a::type) ⊓ (v::'a::type) * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ (g::'a::type)) * w⇧T⇧⋆ ⊓ g = w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆ ⊓ (v * - v⇧T ⊓ g)›*) by (simp add: minarc_min (*‹arc ?y ∧ ?y ⊓ ?x ≠ bot ⟶ sum_class.sum (minarc ?x ⊓ ?x) ≤ sum_class.sum (?y ⊓ ?x)›*))\n have \"?e ≠ bot\"\n using \"20\" (*‹w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆ ⊓ (v * - v⇧T ⊓ g) ≠ bot›*) comp_inf.semiring.mult_not_zero (*‹?a ⊓ ?b ≠ bot ⟹ ?a ≠ bot ∧ ?b ≠ bot›*) semiring.mult_not_zero (*‹?a * ?b ≠ bot ⟹ ?a ≠ bot ∧ ?b ≠ bot›*) by blast\n hence 24: \"?e ⊓ g ≠ bot\"\n using \"22\" (*‹minarc (v * - v⇧T ⊓ g) ⊓ g = minarc (v * - v⇧T ⊓ g) ⊓ (v * - v⇧T ⊓ g)›*) minarc_meet_bot (*‹minarc ?x ⊓ ?x = bot ⟹ minarc ?x = bot›*) by auto\n have \"sum (?w ⊓ g) = sum (w ⊓ -?fp ⊓ g) + sum (?p⇧T ⊓ g) + sum (?e ⊓ g)\"\n using \"7\" (*‹minarc (v * - v⇧T ⊓ g) ≤ v * - v⇧T›*) \"8\" (*‹vector v›*) \"10\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) by (metis sum_disjoint_3 (*‹⟦?w ⊓ ?x = bot; ?w ⊓ ?y = bot; ?x ⊓ ?y = bot⟧ ⟹ sum_class.sum ((?w ⊔ ?x ⊔ ?y) ⊓ ?z) = sum_class.sum (?w ⊓ ?z) + sum_class.sum (?x ⊓ ?z) + sum_class.sum (?y ⊓ ?z)›*) epm_8 (*‹pd_kleene_allegory_class.acyclic ?w ⟹ ?w ⊓ - prim_EP ?w ?v ?e ⊓ prim_P ?w ?v ?e⇧T = bot›*) epm_9 (*‹⟦?e ≤ ?v * - ?v⇧T; vector ?v⟧ ⟹ ?w ⊓ - prim_EP ?w ?v ?e ⊓ ?e = bot›*) epm_10 (*‹⟦?e ≤ ?v * - ?v⇧T; vector ?v⟧ ⟹ prim_P ?w ?v ?e⇧T ⊓ ?e = bot›*) minimum_spanning_tree_def (*‹minimum_spanning_tree ?t ?g ?r ≡ spanning_tree ?t ?g ?r ∧ (∀u. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n also (*calculation: ‹sum_class.sum (prim_W w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) = sum_class.sum (w ⊓ - prim_EP w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) + sum_class.sum (prim_P w v (minarc (v * - v⇧T ⊓ g))⇧T ⊓ g) + sum_class.sum (minarc (v * - v⇧T ⊓ g) ⊓ g)›*) have \"... = sum (((w ⊓ -?fp) ⊔ ?p⇧T) ⊓ g) + sum (?e ⊓ g)\"\n using \"11\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆ ∧ forest w ∧ t ≤ w ∧ w ≤ component g r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) by (metis epm_8 (*‹pd_kleene_allegory_class.acyclic ?w ⟹ ?w ⊓ - prim_EP ?w ?v ?e ⊓ prim_P ?w ?v ?e⇧T = bot›*) sum_disjoint (*‹?x ⊓ ?y = bot ⟹ sum_class.sum ((?x ⊔ ?y) ⊓ ?z) = sum_class.sum (?x ⊓ ?z) + sum_class.sum (?y ⊓ ?z)›*))\n also (*calculation: ‹sum_class.sum (prim_W w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) = sum_class.sum ((w ⊓ - prim_EP w v (minarc (v * - v⇧T ⊓ g)) ⊔ prim_P w v (minarc (v * - v⇧T ⊓ g))⇧T) ⊓ g) + sum_class.sum (minarc (v * - v⇧T ⊓ g) ⊓ g)›*) have \"... ≤ sum (((w ⊓ -?fp) ⊔ ?p⇧T) ⊓ g) + sum (?f ⊓ g)\"\n using \"23\" (*‹sum_class.sum (minarc (v * - v⇧T ⊓ g) ⊓ g) ≤ sum_class.sum (w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆ ⊓ g)›*) \"24\" (*‹minarc ((v::'a) * - v⇧T ⊓ (g::'a)) ⊓ g ≠ bot›*) by (simp add: sum_plus_right_isotone (*‹?x ≠ bot ∧ sum_class.sum ?x ≤ sum_class.sum ?y ⟶ sum_class.sum ?z + sum_class.sum ?x ≤ sum_class.sum ?z + sum_class.sum ?y›*))\n also (*calculation: ‹sum_class.sum (prim_W w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) ≤ sum_class.sum ((w ⊓ - prim_EP w v (minarc (v * - v⇧T ⊓ g)) ⊔ prim_P w v (minarc (v * - v⇧T ⊓ g))⇧T) ⊓ g) + sum_class.sum (w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆ ⊓ g)›*) have \"... = sum (w ⊓ -?fp ⊓ g) + sum (?p⇧T ⊓ g) + sum (?f ⊓ g)\"\n using \"11\" (*‹vector r ∧ injective r ∧ v⇧T = r⇧T * t⇧⋆ ∧ forest w ∧ t ≤ w ∧ w ≤ component g r⇧T * component g r ⊓ - - g ∧ r⇧T * (component g r⇧T * component g r ⊓ - - g)⇧⋆ ≤ r⇧T * w⇧⋆›*) by (metis epm_8 (*‹pd_kleene_allegory_class.acyclic ?w ⟹ ?w ⊓ - prim_EP ?w ?v ?e ⊓ prim_P ?w ?v ?e⇧T = bot›*) sum_disjoint (*‹?x ⊓ ?y = bot ⟹ sum_class.sum ((?x ⊔ ?y) ⊓ ?z) = sum_class.sum (?x ⊓ ?z) + sum_class.sum (?y ⊓ ?z)›*))\n also (*calculation: ‹sum_class.sum (prim_W w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) ≤ sum_class.sum (w ⊓ - prim_EP w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) + sum_class.sum (prim_P w v (minarc (v * - v⇧T ⊓ g))⇧T ⊓ g) + sum_class.sum (w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆ ⊓ g)›*) have \"... = sum (w ⊓ -?fp ⊓ g) + sum (?p ⊓ g) + sum (?f ⊓ g)\"\n using \"13\" (*‹g = g⇧T›*) sum_symmetric (*‹?y = ?y⇧T ⟹ sum_class.sum (?x⇧T ⊓ ?y) = sum_class.sum (?x ⊓ ?y)›*) by auto\n also (*calculation: ‹sum_class.sum (prim_W (w::'a) (v::'a) (minarc (v * - v⇧T ⊓ (g::'a))) ⊓ g) ≤ sum_class.sum (w ⊓ - prim_EP w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) + sum_class.sum (prim_P w v (minarc (v * - v⇧T ⊓ g)) ⊓ g) + sum_class.sum (w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆ ⊓ g)›*) have \"... = sum (((w ⊓ -?fp) ⊔ ?p ⊔ ?f) ⊓ g)\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) \"8\" (*‹vector v›*) by (metis sum_disjoint_3 (*‹⟦?w ⊓ ?x = bot; ?w ⊓ ?y = bot; ?x ⊓ ?y = bot⟧ ⟹ sum_class.sum ((?w ⊔ ?x ⊔ ?y) ⊓ ?z) = sum_class.sum (?w ⊓ ?z) + sum_class.sum (?x ⊓ ?z) + sum_class.sum (?y ⊓ ?z)›*) epm_11 (*‹vector ?v ⟹ ?w ⊓ - prim_EP ?w ?v ?e ⊓ prim_P ?w ?v ?e = bot›*) epm_12 (*‹vector ?v ⟹ ?w ⊓ - prim_EP ?w ?v ?e ⊓ prim_E ?w ?v ?e = bot›*) epm_13 (*‹vector ?v ⟹ prim_P ?w ?v ?e ⊓ prim_E ?w ?v ?e = bot›*))\n also (*calculation: ‹sum_class.sum (prim_W (w::'a) (v::'a) (minarc (v * - v⇧T ⊓ (g::'a))) ⊓ g) ≤ sum_class.sum ((w ⊓ - prim_EP w v (minarc (v * - v⇧T ⊓ g)) ⊔ prim_P w v (minarc (v * - v⇧T ⊓ g)) ⊔ w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆) ⊓ g)›*) have \"... = sum (w ⊓ g)\"\n using \"2\" (*‹regular v ∧ regular (v * v⇧T)›*) \"8\" (*‹vector (v::'a)›*) \"15\" (*‹regular ((w::'a::type) ⊓ (v::'a::type) * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ (g::'a::type)) * w⇧T⇧⋆) ∧ regular (prim_EP w v (minarc (v * - v⇧T ⊓ g))) ∧ regular (prim_W w v (minarc (v * - v⇧T ⊓ g)))›*) \"18\" (*‹w ⊓ v * - v⇧T ⊓ top * minarc (v * - v⇧T ⊓ g) * w⇧T⇧⋆ ⊔ prim_P w v (minarc (v * - v⇧T ⊓ g)) = prim_EP w v (minarc (v * - v⇧T ⊓ g))›*) epm_2 (*‹⟦regular (prim_EP ?w ?v ?e); vector ?v⟧ ⟹ ?w ⊓ - prim_EP ?w ?v ?e ⊔ prim_P ?w ?v ?e ⊔ prim_E ?w ?v ?e = ?w›*) by force\n finally (*calculation: ‹sum_class.sum (prim_W (w::'a) (v::'a) (minarc (v * - v⇧T ⊓ (g::'a))) ⊓ g) ≤ sum_class.sum (w ⊓ g)›*) have \"sum (?w ⊓ g) ≤ sum (w ⊓ g)\" .\n thus \"∀u . spanning_tree u g r ⟶ sum (?w ⊓ g) ≤ sum (u ⊓ g)\"\n using \"10\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) order_lesseq_imp (*‹(∀z≥?x. ?y ≤ z) = (?y ≤ ?x)›*) minimum_spanning_tree_def (*‹minimum_spanning_tree ?t ?g ?r ≡ spanning_tree ?t ?g ?r ∧ (∀u. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) by auto\n qed\n next\n (*goal: ‹(t::'a::type) ⊔ minarc ((v::'a::type) * - v⇧T ⊓ (g::'a::type)) ≤ prim_W (w::'a::type) v (minarc (v * - v⇧T ⊓ g))›*)\n show \"?t ≤ ?w\"\n using \"4\" (*‹t ≤ v * v⇧T›*) \"8\" (*‹vector v›*) \"10\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) mst_extends_new_tree (*‹⟦?t ≤ ?w; ?t ≤ ?v * ?v⇧T; vector ?v⟧ ⟹ ?t ⊔ ?e ≤ prim_W ?w ?v ?e›*) by simp\n qed\n qed\n next\n (*goal: ‹card {x::'a. regular x ∧ x ≤ component (g::'a) (r::'a) ∧ x ≤ - ((v::'a) ⊔ minarc (v * - v⇧T ⊓ g)⇧T * top)⇧T} < card {x::'a. regular x ∧ x ≤ component g r ∧ x ≤ - v⇧T}›*)\n show \"?n2 < ?n1\"\n using \"1\" (*‹prim_invariant (t::'a) (v::'a) (g::'a) (r::'a) ∧ v * - v⇧T ⊓ g ≠ bot›*) prim_invariant_def (*‹prim_invariant (?t::'a) (?v::'a) (?g::'a) (?r::'a) ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w::'a. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_vc_2 (*‹⟦prim_spanning_invariant ?t ?v ?g ?r; ?v * - ?v⇧T ⊓ ?g ≠ bot⟧ ⟹ prim_spanning_invariant (?t ⊔ minarc (?v * - ?v⇧T ⊓ ?g)) (?v ⊔ minarc (?v * - ?v⇧T ⊓ ?g)⇧T * top) ?g ?r ∧ card {x. regular x ∧ x ≤ component ?g ?r ∧ x ≤ - (?v ⊔ minarc (?v * - ?v⇧T ⊓ ?g)⇧T * top)⇧T} < card {x. regular x ∧ x ≤ component ?g ?r ∧ x ≤ - ?v⇧T}›*) by auto\n qed\nnext\n (*goal: ‹⋀t v. prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot ⟹ minimum_spanning_tree t g r›*)\n fix t and v\n let ?g = \"--g\"\n assume 25: \"prim_invariant t v g r ∧ v * -v⇧T ⊓ g = bot\" (*‹prim_invariant (t::'a) (v::'a) (g::'a) (r::'a) ∧ v * - v⇧T ⊓ g = bot›*)\n hence 26: \"regular v\"\n by (metis prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) regular_conv_closed (*‹regular ?x ⟹ regular (?x⇧T)›*) regular_closed_star (*‹regular ?x ⟹ regular (?x⇧⋆)›*) regular_mult_closed (*‹⟦regular ?x; regular ?y⟧ ⟹ regular (?x * ?y)›*) conv_involutive (*‹?x⇧T⇧T = ?x›*))\n from \"25\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot›*) obtain w where 27: \"minimum_spanning_tree w g r ∧ t ≤ w\"\n (*goal: ‹(⋀w. minimum_spanning_tree w g r ∧ t ≤ w ⟹ thesis) ⟹ thesis›*)\n by (metis prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*))\n have \"spanning_tree t g r\"\n using \"25\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_vc_3 (*‹⟦prim_spanning_invariant (?t::'a) (?v::'a) (?g::'a) (?r::'a); ?v * - ?v⇧T ⊓ ?g = bot⟧ ⟹ spanning_tree ?t ?g ?r›*) by blast\n hence \"component g r = v⇧T\"\n by (metis 25 (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) span_tree_component (*‹spanning_tree ?t ?g ?r ⟹ component ?g ?r = component ?t ?r›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n hence 28: \"w ≤ v * v⇧T\"\n using \"26\" (*‹regular v›*) \"27\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) by (simp add: minimum_spanning_tree_def (*‹minimum_spanning_tree ?t ?g ?r ≡ spanning_tree ?t ?g ?r ∧ (∀u. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*) inf_pp_commute (*‹- - (- - ?x ⊓ ?y) = - - ?x ⊓ - - ?y›*))\n have \"vector r ∧ injective r ∧ forest w\"\n using \"25\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot›*) \"27\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) by (simp add: prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) prim_precondition_def (*‹prim_precondition ?g ?r ≡ ?g = ?g⇧T ∧ injective ?r ∧ vector ?r ∧ regular ?r›*) minimum_spanning_tree_def (*‹minimum_spanning_tree ?t ?g ?r ≡ spanning_tree ?t ?g ?r ∧ (∀u. spanning_tree u ?g ?r ⟶ sum_class.sum (?t ⊓ ?g) ≤ sum_class.sum (u ⊓ ?g))›*) spanning_tree_def (*‹spanning_tree ?t ?g ?r ≡ forest ?t ∧ ?t ≤ component ?g ?r⇧T * component ?g ?r ⊓ - - ?g ∧ component ?g ?r ≤ ?r⇧T * ?t⇧⋆ ∧ regular ?t›*))\n hence \"w = t\"\n using \"25\" (*‹prim_invariant t v g r ∧ v * - v⇧T ⊓ g = bot›*) \"27\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) \"28\" (*‹(w::'a) ≤ (v::'a) * v⇧T›*) prim_invariant_def (*‹prim_invariant ?t ?v ?g ?r ≡ prim_spanning_invariant ?t ?v ?g ?r ∧ (∃w. minimum_spanning_tree w ?g ?r ∧ ?t ≤ w)›*) prim_spanning_invariant_def (*‹prim_spanning_invariant ?t ?v ?g ?r ≡ prim_precondition ?g ?r ∧ ?v⇧T = ?r⇧T * ?t⇧⋆ ∧ spanning_tree ?t (?v * ?v⇧T ⊓ ?g) ?r›*) mst_post (*‹⟦vector ?r; injective ?r; ?v⇧T = ?r⇧T * ?t⇧⋆; forest ?w; ?t ≤ ?w; ?w ≤ ?v * ?v⇧T⟧ ⟹ ?w = ?t›*) by blast\n thus \"minimum_spanning_tree t g r\"\n using \"27\" (*‹minimum_spanning_tree w g r ∧ t ≤ w›*) by simp\nqed\n\nend\n\nend\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Relational_Minimum_Spanning_Trees/Prim.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"601c8d58cc563cdb74514b4b08523b5a4650ef7c863860bee2ee39e935af0156\"\n}"}}},{"rowIdx":183,"cells":{"text":{"kind":"string","value":"(*<*)\ntheory Event_Data\n imports\n \"HOL-Library.Char_ord\"\n Code_Double\n Deriving.Derive\nbegin\n(*>*)\n\nsection ‹Event parameters›\n\ndefinition div_to_zero :: \"integer ⇒ integer ⇒ integer\" where\n \"div_to_zero x y = (let z = fst (Code_Numeral.divmod_abs x y) in\n if (x < 0) ≠ (y < 0) then - z else z)\"\n\ndefinition mod_to_zero :: \"integer ⇒ integer ⇒ integer\" where\n \"mod_to_zero x y = (let z = snd (Code_Numeral.divmod_abs x y) in\n if x < 0 then - z else z)\"\n\nlemma \"b ≠ 0 ⟹ div_to_zero a b * b + mod_to_zero a b = a\"\n unfolding div_to_zero_def mod_to_zero_def Let_def\n (*goal: ‹(b::integer) ≠ (0::integer) ⟹ (if ((a::integer) < (0::integer)) ≠ (b < (0::integer)) then - fst (Code_Numeral.divmod_abs a b) else fst (Code_Numeral.divmod_abs a b)) * b + (if a < (0::integer) then - snd (Code_Numeral.divmod_abs a b) else snd (Code_Numeral.divmod_abs a b)) = a›*)\n by (auto simp: minus_mod_eq_mult_div[symmetric] (*‹(?b::?'a) * ((?a::?'a) div ?b) = ?a - ?a mod ?b›*) div_minus_right (*‹(?a::?'a) div - (?b::?'a) = - ?a div ?b›*) mod_minus_right (*‹(?a::?'a) mod - (?b::?'a) = - (- ?a mod ?b)›*) ac_simps (*‹(?a::?'a) + (?b::?'a) + (?c::?'a) = ?a + (?b + ?c)› ‹(?a::?'a) + (?b::?'a) = ?b + ?a› ‹(?b::?'a) + ((?a::?'a) + (?c::?'a)) = ?a + (?b + ?c)› ‹(?a::?'a) * (?b::?'a) * (?c::?'a) = ?a * (?b * ?c)› ‹(?a::?'a) * (?b::?'a) = ?b * ?a› ‹(?b::?'a) * ((?a::?'a) * (?c::?'a)) = ?a * (?b * ?c)› ‹(((?a::bool) ∧ (?b::bool)) ∧ (?c::bool)) = (?a ∧ ?b ∧ ?c)› ‹((?a::bool) ∧ (?b::bool)) = (?b ∧ ?a)› ‹((?b::bool) ∧ (?a::bool) ∧ (?c::bool)) = (?a ∧ ?b ∧ ?c)› ‹(((?a::bool) ∨ (?b::bool)) ∨ (?c::bool)) = (?a ∨ ?b ∨ ?c)› ‹((?a::bool) ∨ (?b::bool)) = (?b ∨ ?a)› ‹((?b::bool) ∨ (?a::bool) ∨ (?c::bool)) = (?a ∨ ?b ∨ ?c)› and more 42 facts*))\n\n\ndatatype event_data = EInt integer | EFloat double | EString String.literal\n\nderive (eq) ceq event_data\nderive ccompare event_data\n\ninstantiation event_data :: \"{ord, plus, minus, uminus, times, divide, modulo}\"\nbegin\n\nfun less_eq_event_data where\n \"EInt x ≤ EInt y ⟷ x ≤ y\"\n| \"EInt x ≤ EFloat y ⟷ double_of_integer x ≤ y\"\n| \"EInt _ ≤ EString _ ⟷ False\"\n| \"EFloat x ≤ EInt y ⟷ x ≤ double_of_integer y\"\n| \"EFloat x ≤ EFloat y ⟷ x ≤ y\"\n| \"EFloat _ ≤ EString _ ⟷ False\"\n| \"EString x ≤ EString y ⟷ lexordp_eq (String.explode x) (String.explode y)\"\n| \"EString _ ≤ _ ⟷ False\"\n\ndefinition less_event_data :: \"event_data ⇒ event_data ⇒ bool\" where\n \"less_event_data x y ⟷ x ≤ y ∧ ¬ y ≤ x\"\n\nfun plus_event_data where\n \"EInt x + EInt y = EInt (x + y)\"\n| \"EInt x + EFloat y = EFloat (double_of_integer x + y)\"\n| \"EFloat x + EInt y = EFloat (x + double_of_integer y)\"\n| \"EFloat x + EFloat y = EFloat (x + y)\"\n| \"(_::event_data) + _ = EFloat nan\"\n\nfun minus_event_data where\n \"EInt x - EInt y = EInt (x - y)\"\n| \"EInt x - EFloat y = EFloat (double_of_integer x - y)\"\n| \"EFloat x - EInt y = EFloat (x - double_of_integer y)\"\n| \"EFloat x - EFloat y = EFloat (x - y)\"\n| \"(_::event_data) - _ = EFloat nan\"\n\nfun uminus_event_data where\n \"- EInt x = EInt (- x)\"\n| \"- EFloat x = EFloat (- x)\"\n| \"- (_::event_data) = EFloat nan\"\n\nfun times_event_data where\n \"EInt x * EInt y = EInt (x * y)\"\n| \"EInt x * EFloat y = EFloat (double_of_integer x * y)\"\n| \"EFloat x * EInt y = EFloat (x * double_of_integer y)\"\n| \"EFloat x * EFloat y = EFloat (x * y)\"\n| \"(_::event_data) * _ = EFloat nan\"\n\nfun divide_event_data where\n \"EInt x div EInt y = EInt (div_to_zero x y)\"\n| \"EInt x div EFloat y = EFloat (double_of_integer x div y)\"\n| \"EFloat x div EInt y = EFloat (x div double_of_integer y)\"\n| \"EFloat x div EFloat y = EFloat (x div y)\"\n| \"(_::event_data) div _ = EFloat nan\"\n\nfun modulo_event_data where\n \"EInt x mod EInt y = EInt (mod_to_zero x y)\"\n| \"(_::event_data) mod _ = EFloat nan\"\n\ninstance ..\n\nend\n\nprimrec integer_of_event_data :: \"event_data ⇒ integer\" where\n \"integer_of_event_data (EInt x) = x\"\n| \"integer_of_event_data (EFloat x) = integer_of_double x\"\n| \"integer_of_event_data (EString _) = 0\"\n\nprimrec double_of_event_data :: \"event_data ⇒ double\" where\n \"double_of_event_data (EInt x) = double_of_integer x\"\n| \"double_of_event_data (EFloat x) = x\"\n| \"double_of_event_data (EString _) = nan\"\n\n(*<*)\nend\n(*>*)\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/MFODL_Monitor_Optimized/Event_Data.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"2823f1fdcd57d42f007fe66fe4378addb2f70e8e7c7788720fdd294d237442ae\"\n}"}}},{"rowIdx":184,"cells":{"text":{"kind":"string","value":"(* Author: Alexander Maletzky *)\n\nsection ‹Macaulay Matrices›\n\ntheory Macaulay_Matrix\n imports More_MPoly_Type_Class Jordan_Normal_Form.Gauss_Jordan_Elimination\nbegin\n\ntext ‹We build upon vectors and matrices represented by dimension and characteristic function, because\n later on we need to quantify the dimensions of certain matrices existentially. This is not possible\n (at least not easily possible) with a type-based approach, as in HOL-Multivariate Analysis.›\n\nsubsection ‹More about Vectors›\n\nlemma vec_of_list_alt: \"vec_of_list xs = vec (length xs) (nth xs)\"\n apply transfer\n (*goal: ‹vec_of_list xs = vec (length xs) ((!) xs)›*)\n by (rule refl (*‹?t = ?t›*))\n\nlemma vec_cong:\n assumes \"n = m\" and \"⋀i. i < m ⟹ f i = g i\"\n shows \"vec n f = vec m g\"\n using assms (*‹n = m› ‹?i < m ⟹ f ?i = g ?i›*) by auto\n\nlemma scalar_prod_comm:\n assumes \"dim_vec v = dim_vec w\"\n shows \"v ∙ w = w ∙ (v::'a::comm_semiring_0 vec)\"\n apply (simp add: scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $ i * ?w $ i›*) assms (*‹dim_vec v = dim_vec w›*))\n (*goal: ‹v ∙ w = w ∙ v›*)\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0..<dim_vec w} = {0..<dim_vec w}›\n 2. ‹⋀x. x ∈ {0..<dim_vec w} ⟹ v $ x * w $ x = w $ x * v $ x›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply (simp only: ac_simps (*‹?a + ?b + ?c = ?a + (?b + ?c)› ‹?a + ?b = ?b + ?a› ‹?b + (?a + ?c) = ?a + (?b + ?c)› ‹?a * ?b * ?c = ?a * (?b * ?c)› ‹?a * ?b = ?b * ?a› ‹?b * (?a * ?c) = ?a * (?b * ?c)› ‹((?a ∧ ?b) ∧ ?c) = (?a ∧ ?b ∧ ?c)› ‹(?a ∧ ?b) = (?b ∧ ?a)› ‹(?b ∧ ?a ∧ ?c) = (?a ∧ ?b ∧ ?c)› ‹((?a ∨ ?b) ∨ ?c) = (?a ∨ ?b ∨ ?c)› ‹(?a ∨ ?b) = (?b ∨ ?a)› ‹(?b ∨ ?a ∨ ?c) = (?a ∨ ?b ∨ ?c)› and more 46 facts*))\n (*proven 2 subgoals*) .\n\nlemma vec_scalar_mult_fun: \"vec n (λx. c * f x) = c ⋅⇩v vec n f\"\n apply (simp add: smult_vec_def (*‹?a ⋅⇩v ?v ≡ vec (dim_vec ?v) (λi. ?a * ?v $ i)›*))\n (*goal: ‹vec n (λx. c * f x) = c ⋅⇩v vec n f›*)\n apply (rule vec_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ vec ?n ?f = vec ?m ?g›*))\n (*goals:\n 1. ‹n = n›\n 2. ‹⋀i. i < n ⟹ c * f i = c * vec n f $ i›\n discuss goal 1*)\n apply (rule refl (*‹(?t::?'a) = ?t›*))\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\ndefinition mult_vec_mat :: \"'a vec ⇒ 'a :: semiring_0 mat ⇒ 'a vec\" (infixl \"⇩v*\" 70)\n where \"v ⇩v* A ≡ vec (dim_col A) (λj. v ∙ col A j)\"\n\ndefinition resize_vec :: \"nat ⇒ 'a vec ⇒ 'a vec\"\n where \"resize_vec n v = vec n (vec_index v)\"\n\nlemma dim_resize_vec[simp]: \"dim_vec (resize_vec n v) = n\"\n by (simp add: resize_vec_def (*‹resize_vec ?n ?v = vec ?n (($) ?v)›*))\n\nlemma resize_vec_carrier: \"resize_vec n v ∈ carrier_vec n\"\n by (simp add: carrier_dim_vec (*‹(?v ∈ carrier_vec ?n) = (dim_vec ?v = ?n)›*))\n\nlemma resize_vec_dim[simp]: \"resize_vec (dim_vec v) v = v\"\n by (simp add: resize_vec_def (*‹resize_vec (?n::nat) (?v::?'a::type vec) = vec ?n (($) ?v)›*) eq_vecI (*‹⟦⋀i::nat. i < dim_vec (?w::?'a::type vec) ⟹ (?v::?'a::type vec) $ i = ?w $ i; dim_vec ?v = dim_vec ?w⟧ ⟹ ?v = ?w›*))\n\nlemma resize_vec_index:\n assumes \"i < n\"\n shows \"resize_vec n v $ i = v $ i\"\n using assms (*‹i < n›*) by (simp add: resize_vec_def (*‹resize_vec ?n ?v = vec ?n (($) ?v)›*))\n\nlemma mult_mat_vec_resize:\n \"v ⇩v* A = (resize_vec (dim_row A) v) ⇩v* A\"\n apply (simp add: mult_vec_mat_def (*‹?v ⇩v* ?A ≡ vec (dim_col ?A) (λj. ?v ∙ col ?A j)›*) scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $ i * ?w $ i›*))\n (*goal: ‹v ⇩v* A = resize_vec (dim_row A) v ⇩v* A›*)\n apply (rule arg_cong2[of _ _ _ _ vec] (*‹⟦?a = ?b; ?c = ?d⟧ ⟹ vec ?a ?c = vec ?b ?d›*))\n (*goals:\n 1. ‹dim_col A = dim_col A›\n 2. ‹(λj. ∑i = 0..<dim_row A. v $ i * col A j $ i) = (λj. ∑i = 0..<dim_row A. resize_vec (dim_row A) v $ i * col A j $ i)›\n discuss goal 1*)\n apply rule\n (*discuss goal 2*)\n apply rule\n (*goal: ‹(λj::nat. ∑i::nat = 0::nat..<dim_row A. (v::'a::semiring_0 vec) $ i * col (A::'a::semiring_0 mat) j $ i) = (λj::nat. ∑i::nat = 0::nat..<dim_row A. resize_vec (dim_row A) v $ i * col A j $ i)›*)\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹⋀j. {0..<dim_row A} = {0..<dim_row A}›\n 2. ‹⋀j x. x ∈ {0..<dim_row A} ⟹ v $ x * col A j $ x = resize_vec (dim_row A) v $ x * col A j $ x›\n discuss goal 1*)\n apply rule\n (*discuss goal 2*)\n apply (simp add: resize_vec_index (*‹?i < ?n ⟹ resize_vec ?n ?v $ ?i = ?v $ ?i›*))\n (*proven 2 subgoals*)\n (*proven 2 subgoals*) .\n\nlemma assoc_mult_vec_mat:\n assumes \"v ∈ carrier_vec n1\" and \"A ∈ carrier_mat n1 n2\" and \"B ∈ carrier_mat n2 n3\"\n shows \"v ⇩v* (A * B) = (v ⇩v* A) ⇩v* B\"\n using assms (*‹v ∈ carrier_vec n1› ‹A ∈ carrier_mat n1 n2› ‹B ∈ carrier_mat n2 n3›*) apply (intro eq_vecI (*‹⟦⋀i. i < dim_vec ?w ⟹ ?v $ i = ?w $ i; dim_vec ?v = dim_vec ?w⟧ ⟹ ?v = ?w›*))\n (*goals:\n 1. ‹⋀i::nat. ⟦(v::'a vec) ∈ carrier_vec (n1::nat); (A::'a mat) ∈ carrier_mat n1 (n2::nat); (B::'a mat) ∈ carrier_mat n2 (n3::nat); i < dim_vec (v ⇩v* A ⇩v* B)⟧ ⟹ (v ⇩v* (A * B)) $ i = (v ⇩v* A ⇩v* B) $ i›\n 2. ‹⟦(v::'a vec) ∈ carrier_vec (n1::nat); (A::'a mat) ∈ carrier_mat n1 (n2::nat); (B::'a mat) ∈ carrier_mat n2 (n3::nat)⟧ ⟹ dim_vec (v ⇩v* (A * B)) = dim_vec (v ⇩v* A ⇩v* B)›\n discuss goal 1*)\n apply ((auto simp add: mult_vec_mat_def (*‹?v ⇩v* ?A ≡ vec (dim_col ?A) (λj. ?v ∙ col ?A j)›*) mult_mat_vec_def (*‹?A *⇩v ?v ≡ vec (dim_row ?A) (λi. row ?A i ∙ ?v)›*) assoc_scalar_prod (*‹⟦?v₁ ∈ carrier_vec ?nr; ?A ∈ carrier_mat ?nr ?nc; ?v₂ ∈ carrier_vec ?nc⟧ ⟹ vec ?nc (λj. ?v₁ ∙ col ?A j) ∙ ?v₂ = ?v₁ ∙ vec ?nr (λi. row ?A i ∙ ?v₂)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: mult_vec_mat_def (*‹?v ⇩v* ?A ≡ vec (dim_col ?A) (λj. ?v ∙ col ?A j)›*) mult_mat_vec_def (*‹?A *⇩v ?v ≡ vec (dim_row ?A) (λi. row ?A i ∙ ?v)›*) assoc_scalar_prod (*‹⟦?v₁ ∈ carrier_vec ?nr; ?A ∈ carrier_mat ?nr ?nc; ?v₂ ∈ carrier_vec ?nc⟧ ⟹ vec ?nc (λj. ?v₁ ∙ col ?A j) ∙ ?v₂ = ?v₁ ∙ vec ?nr (λi. row ?A i ∙ ?v₂)›*))[1])\n (*proven 2 subgoals*) .\n\nlemma mult_vec_mat_transpose:\n assumes \"dim_vec v = dim_row A\"\n shows \"v ⇩v* A = (transpose_mat A) *⇩v (v::'a::comm_semiring_0 vec)\"\nproof (simp add: mult_vec_mat_def (*‹?v ⇩v* ?A ≡ vec (dim_col ?A) (λj. ?v ∙ col ?A j)›*) mult_mat_vec_def (*‹?A *⇩v ?v ≡ vec (dim_row ?A) (λi. row ?A i ∙ ?v)›*), rule vec_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ vec ?n ?f = vec ?m ?g›*), rule refl (*‹?t = ?t›*), simp)\n (*goal: ‹⋀i. i < dim_col A ⟹ v ∙ col A i = col A i ∙ v›*)\n fix j\n show \"v ∙ col A j = col A j ∙ v\"\n apply (rule scalar_prod_comm (*‹dim_vec ?v = dim_vec ?w ⟹ ?v ∙ ?w = ?w ∙ ?v›*))\n (*goal: ‹(v::'a vec) ∙ col (A::'a mat) (j::nat) = col A j ∙ v›*)\n by (simp add: assms (*‹dim_vec v = dim_row A›*))\nqed\n\nsubsection ‹More about Matrices›\n\ndefinition nzrows :: \"'a::zero mat ⇒ 'a vec list\"\n where \"nzrows A = filter (λr. r ≠ 0⇩v (dim_col A)) (rows A)\"\n\ndefinition row_space :: \"'a mat ⇒ 'a::semiring_0 vec set\"\n where \"row_space A = (λv. mult_vec_mat v A) ` (carrier_vec (dim_row A))\"\n\ndefinition row_echelon :: \"'a mat ⇒ 'a::field mat\"\n where \"row_echelon A = fst (gauss_jordan A (1⇩m (dim_row A)))\"\n\nsubsubsection ‹@{const nzrows}›\n\nlemma length_nzrows: \"length (nzrows A) ≤ dim_row A\"\n by (simp add: nzrows_def (*‹nzrows (?A::?'a mat) = filter (λr::?'a vec. r ≠ 0⇩v (dim_col ?A)) (rows ?A)›*) length_rows[symmetric] (*‹dim_row (?A::?'a mat) = length (rows ?A)›*) del: length_rows (*‹length (rows (?A::?'a mat)) = dim_row ?A›*))\n\nlemma set_nzrows: \"set (nzrows A) = set (rows A) - {0⇩v (dim_col A)}\"\n by (auto simp add: nzrows_def (*‹nzrows ?A = filter (λr. r ≠ 0⇩v (dim_col ?A)) (rows ?A)›*))\n\nlemma nzrows_nth_not_zero:\n assumes \"i < length (nzrows A)\"\n shows \"nzrows A ! i ≠ 0⇩v (dim_col A)\"\n using assms (*‹(i::nat) < length (nzrows (A::'a::zero mat))›*) unfolding nzrows_def\n (*goal: ‹filter (λr. r ≠ 0⇩v (dim_col A)) (rows A) ! i ≠ 0⇩v (dim_col A)›*)\n using nth_mem (*‹?n < length ?xs ⟹ ?xs ! ?n ∈ set ?xs›*) by force\n\nsubsubsection ‹@{const row_space}›\n\nlemma row_spaceI:\n assumes \"x = v ⇩v* A\"\n shows \"x ∈ row_space A\"\n unfolding row_space_def assms\n (*goal: ‹v ⇩v* A ∈ (λv. v ⇩v* A) ` carrier_vec (dim_row A)›*)\n apply rule\n (*goals:\n 1. ‹(v::'a vec) ⇩v* (A::'a mat) = (?x::'a vec) ⇩v* A›\n 2. ‹(?x::'a vec) ∈ carrier_vec (dim_row (A::'a mat))›\n discuss goal 1*)\n apply (fact mult_mat_vec_resize (*‹?v ⇩v* ?A = resize_vec (dim_row ?A) ?v ⇩v* ?A›*))\n (*discuss goal 2*)\n apply (fact resize_vec_carrier (*‹resize_vec ?n ?v ∈ carrier_vec ?n›*))\n (*proven 2 subgoals*) .\n\nlemma row_spaceE:\n assumes \"x ∈ row_space A\"\n obtains v where \"v ∈ carrier_vec (dim_row A)\" and \"x = v ⇩v* A\"\n using assms (*‹x ∈ row_space A›*) unfolding row_space_def\n (*goal: ‹(⋀v. ⟦v ∈ carrier_vec (dim_row A); x = v ⇩v* A⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n\nlemma row_space_alt: \"row_space A = range (λv. mult_vec_mat v A)\"\nproof (standard)\n (*goals:\n 1. ‹row_space A ⊆ range (λv. v ⇩v* A)›\n 2. ‹range (λv. v ⇩v* A) ⊆ row_space A›*)\n show \"row_space A ⊆ range (λv. v ⇩v* A)\"\n unfolding row_space_def\n (*goal: ‹(λv. v ⇩v* A) ` carrier_vec (dim_row A) ⊆ range (λv. v ⇩v* A)›*)\n by auto\nnext\n (*goal: ‹range (λv. v ⇩v* A) ⊆ row_space A›*)\n show \"range (λv. v ⇩v* A) ⊆ row_space A\"\n proof (standard)\n (*goal: ‹⋀x. x ∈ range (λv. v ⇩v* A) ⟹ x ∈ row_space A›*)\n fix x\n assume \"x ∈ range (λv. v ⇩v* A)\" (*‹(x::'a vec) ∈ range (λv::'a vec. v ⇩v* (A::'a mat))›*)\n then obtain v where \"x = v ⇩v* A\"\n (*goal: ‹(⋀v. x = v ⇩v* A ⟹ thesis) ⟹ thesis›*)\n by standard\n thus \"x ∈ row_space A\"\n by (rule row_spaceI (*‹?x = ?v ⇩v* ?A ⟹ ?x ∈ row_space ?A›*))\n qed\nqed\n\nlemma row_space_mult:\n assumes \"A ∈ carrier_mat nr nc\" and \"B ∈ carrier_mat nr nr\"\n shows \"row_space (B * A) ⊆ row_space A\"\nproof (standard)\n (*goal: ‹⋀x. x ∈ row_space (B * A) ⟹ x ∈ row_space A›*)\n from assms(2) (*‹B ∈ carrier_mat nr nr›*) assms(1) (*‹A ∈ carrier_mat nr nc›*) have \"B * A ∈ carrier_mat nr nc\"\n by (rule mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*))\n hence \"nr = dim_row (B * A)\"\n by blast\n fix x\n assume \"x ∈ row_space (B * A)\" (*‹(x::'a vec) ∈ row_space ((B::'a mat) * (A::'a mat))›*)\n then obtain v where \"v ∈ carrier_vec nr\" and x: \"x = v ⇩v* (B * A)\"\n (*goal: ‹(⋀v. ⟦v ∈ carrier_vec nr; x = v ⇩v* (B * A)⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding ‹nr = dim_row (B * A)›\n (*goal: ‹(⋀v. ⟦v ∈ carrier_vec (dim_row (B * A)); x = v ⇩v* (B * A)⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule row_spaceE (*‹⟦(?x::?'a vec) ∈ row_space (?A::?'a mat); ⋀v::?'a vec. ⟦v ∈ carrier_vec (dim_row ?A); ?x = v ⇩v* ?A⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*))\n from this(1) (*‹v ∈ carrier_vec nr›*) assms(2) (*‹(B::'a::semiring_0 mat) ∈ carrier_mat (nr::nat) nr›*) assms(1) (*‹(A::'a mat) ∈ carrier_mat (nr::nat) (nc::nat)›*) have \"x = (v ⇩v* B) ⇩v* A\"\n unfolding x\n (*goal: ‹v ⇩v* (B * A) = v ⇩v* B ⇩v* A›*)\n by (rule assoc_mult_vec_mat (*‹⟦?v ∈ carrier_vec ?n1.0; ?A ∈ carrier_mat ?n1.0 ?n2.0; ?B ∈ carrier_mat ?n2.0 ?n3.0⟧ ⟹ ?v ⇩v* (?A * ?B) = ?v ⇩v* ?A ⇩v* ?B›*))\n thus \"x ∈ row_space A\"\n by (rule row_spaceI (*‹?x = ?v ⇩v* ?A ⟹ ?x ∈ row_space ?A›*))\nqed\n\nlemma row_space_mult_unit:\n assumes \"P ∈ Units (ring_mat TYPE('a::semiring_1) (dim_row A) b)\"\n shows \"row_space (P * A) = row_space A\"\nproof (-)\n (*goal: ‹row_space ((P::'a mat) * (A::'a mat)) = row_space A›*)\n have A: \"A ∈ carrier_mat (dim_row A) (dim_col A)\"\n by simp\n from assms (*‹P ∈ Units (ring_mat TYPE('a) (dim_row A) b)›*) have P: \"P ∈ carrier (ring_mat TYPE('a) (dim_row A) b)\" and \"*\": \"∃Q∈(carrier (ring_mat TYPE('a) (dim_row A) b)).\n Q ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ P = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙\"\n unfolding Units_def\n (*goals:\n 1. ‹P ∈ carrier (ring_mat TYPE('a) (dim_row A) b)›\n 2. ‹∃Q∈carrier (ring_mat TYPE('a) (dim_row A) b). Q ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ P = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙›*)\n apply -\n (*goals:\n 1. ‹P ∈ {y ∈ carrier (ring_mat TYPE('a) (dim_row A) b). ∃x∈carrier (ring_mat TYPE('a) (dim_row A) b). x ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ y = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙ ∧ y ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ x = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙} ⟹ P ∈ carrier (ring_mat TYPE('a) (dim_row A) b)›\n 2. ‹P ∈ {y ∈ carrier (ring_mat TYPE('a) (dim_row A) b). ∃x∈carrier (ring_mat TYPE('a) (dim_row A) b). x ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ y = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙ ∧ y ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ x = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙} ⟹ ∃Q∈carrier (ring_mat TYPE('a) (dim_row A) b). Q ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ P = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from P (*‹P ∈ carrier (ring_mat TYPE('a) (dim_row A) b)›*) have P_in: \"P ∈ carrier_mat (dim_row A) (dim_row A)\"\n by (simp add: ring_mat_def (*‹ring_mat ?ty ?n ?b ≡ ⦇carrier = carrier_mat ?n ?n, monoid.mult = (*), one = 1⇩m ?n, zero = 0⇩m ?n ?n, add = (+), … = ?b⦈›*))\n from \"*\" (*‹∃Q::'a mat∈carrier (ring_mat TYPE('a) (dim_row (A::'a mat)) (b::'b)). Q ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ (P::'a mat) = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙›*) obtain Q where \"Q ∈ carrier (ring_mat TYPE('a) (dim_row A) b)\" and \"Q ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ P = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙\"\n (*goal: ‹(⋀Q. ⟦Q ∈ carrier (ring_mat TYPE('a) (dim_row A) b); Q ⊗⇘ring_mat TYPE('a) (dim_row A) b⇙ P = 𝟭⇘ring_mat TYPE('a) (dim_row A) b⇙⟧ ⟹ thesis) ⟹ thesis›*)\n by standard\n hence Q_in: \"Q ∈ carrier_mat (dim_row A) (dim_row A)\" and QP: \"Q * P = 1⇩m (dim_row A)\"\n apply -\n (*goals:\n 1. ‹⟦(Q::'a::semiring_1 mat) ∈ carrier (ring_mat TYPE('a::semiring_1) (dim_row (A::'a::semiring_1 mat)) (b::'b::type)); Q ⊗⇘ring_mat TYPE('a::semiring_1) (dim_row A) b⇙ (P::'a::semiring_1 mat) = 𝟭⇘ring_mat TYPE('a::semiring_1) (dim_row A) b⇙⟧ ⟹ Q ∈ carrier_mat (dim_row A) (dim_row A)›\n 2. ‹⟦(Q::'a::semiring_1 mat) ∈ carrier (ring_mat TYPE('a::semiring_1) (dim_row (A::'a::semiring_1 mat)) (b::'b::type)); Q ⊗⇘ring_mat TYPE('a::semiring_1) (dim_row A) b⇙ (P::'a::semiring_1 mat) = 𝟭⇘ring_mat TYPE('a::semiring_1) (dim_row A) b⇙⟧ ⟹ Q * P = 1⇩m (dim_row A)›\n discuss goal 1*)\n apply (simp add: ring_mat_def (*‹ring_mat ?ty ?n ?b ≡ ⦇carrier = carrier_mat ?n ?n, monoid.mult = (*), one = 1⇩m ?n, zero = 0⇩m ?n ?n, add = (+), … = ?b⦈›*))\n (*discuss goal 2*)\n apply (simp add: ring_mat_def (*‹ring_mat ?ty ?n ?b ≡ ⦇carrier = carrier_mat ?n ?n, monoid.mult = (*), one = 1⇩m ?n, zero = 0⇩m ?n ?n, add = (+), … = ?b⦈›*))\n (*proven 2 subgoals*) .\n show \"?thesis\"\n (*goal: ‹row_space (P * A) = row_space A›*)\n proof (standard)\n (*goals:\n 1. ‹row_space (P * A) ⊆ row_space A›\n 2. ‹row_space A ⊆ row_space (P * A)›*)\n from A (*‹(A::'a mat) ∈ carrier_mat (dim_row A) (dim_col A)›*) P_in (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) show \"row_space (P * A) ⊆ row_space A\"\n by (rule row_space_mult (*‹⟦?A ∈ carrier_mat ?nr ?nc; ?B ∈ carrier_mat ?nr ?nr⟧ ⟹ row_space (?B * ?A) ⊆ row_space ?A›*))\n next\n (*goal: ‹row_space A ⊆ row_space (P * A)›*)\n from A (*‹A ∈ carrier_mat (dim_row A) (dim_col A)›*) P_in (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) Q_in (*‹Q ∈ carrier_mat (dim_row A) (dim_row A)›*) have \"Q * (P * A) = (Q * P) * A\"\n by (simp only: assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*))\n also (*calculation: ‹Q * (P * A) = Q * P * A›*) from A (*‹A ∈ carrier_mat (dim_row A) (dim_col A)›*) have \"... = A\"\n by (simp add: QP (*‹Q * P = 1⇩m (dim_row A)›*))\n finally (*calculation: ‹Q * (P * A) = A›*) have eq: \"row_space A = row_space (Q * (P * A))\"\n by simp\n show \"row_space A ⊆ row_space (P * A)\"\n unfolding eq\n (*goal: ‹row_space (Q * (P * A)) ⊆ row_space (P * A)›*)\n apply (rule row_space_mult (*‹⟦?A ∈ carrier_mat ?nr ?nc; ?B ∈ carrier_mat ?nr ?nr⟧ ⟹ row_space (?B * ?A) ⊆ row_space ?A›*))\n (*goals:\n 1. ‹P * A ∈ carrier_mat ?nr ?nc›\n 2. ‹Q ∈ carrier_mat ?nr ?nr›\n discuss goal 1*)\n apply (rule mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*))\n (*goals:\n 1. ‹P ∈ carrier_mat ?nr ?n3›\n 2. ‹A ∈ carrier_mat ?n3 ?nc›\n discuss goal 1*)\n apply fact\n (*discuss goal 2*)\n apply fact\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply fact\n (*proven 2 subgoals*) .\n qed\nqed\n\nsubsubsection ‹@{const row_echelon}›\n\nlemma row_eq_zero_iff_pivot_fun:\n assumes \"pivot_fun A f (dim_col A)\" and \"i < dim_row (A::'a::zero_neq_one mat)\"\n shows \"(row A i = 0⇩v (dim_col A)) ⟷ (f i = dim_col A)\"\nproof (-)\n (*goal: ‹(row A i = 0⇩v (dim_col A)) = (f i = dim_col A)›*)\n have \"*\": \"dim_row A = dim_row A\"\n by standard\n show \"?thesis\"\n (*goal: ‹(row A i = 0⇩v (dim_col A)) = (f i = dim_col A)›*)\n proof (standard)\n (*goals:\n 1. ‹row A i = 0⇩v (dim_col A) ⟹ f i = dim_col A›\n 2. ‹f i = dim_col A ⟹ row A i = 0⇩v (dim_col A)›*)\n assume a: \"row A i = 0⇩v (dim_col A)\" (*‹row (A::'a mat) (i::nat) = 0⇩v (dim_col A)›*)\n show \"f i = dim_col A\"\n proof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹(f::nat ⇒ nat) (i::nat) ≠ dim_col (A::'a mat) ⟹ False›*)\n assume \"f i ≠ dim_col A\" (*‹(f::nat ⇒ nat) (i::nat) ≠ dim_col (A::'a mat)›*)\n with pivot_funD(1)[OF * assms] (*‹f i ≤ dim_col A›*) have \"**\": \"f i < dim_col A\"\n by simp\n with \"*\" (*‹dim_row A = dim_row A›*) assms (*‹pivot_fun A f (dim_col A)› ‹(i::nat) < dim_row (A::'a::zero_neq_one mat)›*) have \"A $$ (i, f i) = 1\"\n by (rule pivot_funD (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?j < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = 0› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = 1› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*))\n with \"**\" (*‹(f::nat ⇒ nat) (i::nat) < dim_col (A::'a mat)›*) assms(2) (*‹i < dim_row A›*) have \"row A i $ (f i) = 1\"\n by simp\n hence \"(1::'a) = (0⇩v (dim_col A)) $ (f i)\"\n by (simp only: a (*‹row A i = 0⇩v (dim_col A)›*))\n also (*calculation: ‹1 = 0⇩v (dim_col A) $ f i›*) have \"... = (0::'a)\"\n using \"**\" (*‹f i < dim_col A›*) by simp\n finally (*calculation: ‹1 = 0›*) show False\n by simp\n qed\n next\n (*goal: ‹(f::nat ⇒ nat) (i::nat) = dim_col (A::'a mat) ⟹ row A i = 0⇩v (dim_col A)›*)\n assume a: \"f i = dim_col A\" (*‹(f::nat ⇒ nat) (i::nat) = dim_col (A::'a mat)›*)\n show \"row A i = 0⇩v (dim_col A)\"\n proof (rule, simp_all add: assms(2))\n (*goal: ‹⋀ia::nat. ia < dim_col (A::'a mat) ⟹ A $$ (i::nat, ia) = (0::'a)›*)\n fix j\n assume \"j < dim_col A\" (*‹(j::nat) < dim_col (A::'a mat)›*)\n hence \"j < f i\"\n by (simp only: a (*‹f i = dim_col A›*))\n with \"*\" (*‹dim_row A = dim_row A›*) assms (*‹pivot_fun A f (dim_col A)› ‹i < dim_row A›*) show \"A $$ (i, j) = 0\"\n by (rule pivot_funD (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?j < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = 0› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = 1› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*))\n qed\n qed\nqed\n\nlemma row_not_zero_iff_pivot_fun:\n assumes \"pivot_fun A f (dim_col A)\" and \"i < dim_row (A::'a::zero_neq_one mat)\"\n shows \"(row A i ≠ 0⇩v (dim_col A)) ⟷ (f i < dim_col A)\"\nproof (simp only: row_eq_zero_iff_pivot_fun[OF assms] (*‹(row A i = 0⇩v (dim_col A)) = (f i = dim_col A)›*))\n (*goal: ‹((f::nat ⇒ nat) (i::nat) ≠ dim_col (A::'a mat)) = (f i < dim_col A)›*)\n have \"f i ≤ dim_col A\"\n apply (rule pivot_funD[where ?f = f] (*‹⟦dim_row (?A::?'a::{one,zero} mat) = (?nr::nat); pivot_fun ?A (f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr⟧ ⟹ f ?i ≤ ?nc› ‹⟦dim_row (?A::?'a::{one,zero} mat) = (?nr::nat); pivot_fun ?A (f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; (?j::nat) < f ?i⟧ ⟹ ?A $$ (?i, ?j) = (0::?'a::{one,zero})› ‹⟦dim_row (?A::?'a::{one,zero} mat) = (?nr::nat); pivot_fun ?A (f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; Suc ?i < ?nr⟧ ⟹ f ?i < f (Suc ?i) ∨ f (Suc ?i) = ?nc› ‹⟦dim_row (?A::?'a::{one,zero} mat) = (?nr::nat); pivot_fun ?A (f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; f ?i < ?nc⟧ ⟹ ?A $$ (?i, f ?i) = (1::?'a::{one,zero})› ‹⟦dim_row (?A::?'a::{one,zero} mat) = (?nr::nat); pivot_fun ?A (f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; f ?i < ?nc; (?i'::nat) < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', f ?i) = (0::?'a::{one,zero})›*))\n (*goals:\n 1. ‹dim_row (?A::?'a::{one,zero} mat) = (?nr::nat)›\n 2. ‹pivot_fun (?A::?'a::{one,zero} mat) (f::nat ⇒ nat) (dim_col (A::'a::zero_neq_one mat))›\n 3. ‹(i::nat) < (?nr::nat)›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*proven 3 subgoals*) .\n thus \"(f i ≠ dim_col A) = (f i < dim_col A)\"\n by auto\nqed\n\nlemma pivot_fun_stabilizes:\n assumes \"pivot_fun A f nc\" and \"i1 ≤ i2\" and \"i2 < dim_row A\" and \"nc ≤ f i1\"\n shows \"f i2 = nc\"\nproof (-)\n (*goal: ‹(f::nat ⇒ nat) (i2::nat) = (nc::nat)›*)\n from assms(2) (*‹i1 ≤ i2›*) have \"i2 = i1 + (i2 - i1)\"\n by simp\n then obtain k where \"i2 = i1 + k\"\n (*goal: ‹(⋀k. i2 = i1 + k ⟹ thesis) ⟹ thesis›*)\n by standard\n from assms(3) (*‹i2 < dim_row A›*) assms(4) (*‹nc ≤ f i1›*) show \"?thesis\"\n (*goal: ‹f i2 = nc›*)\n unfolding ‹i2 = i1 + k›\n (*goal: ‹f (i1 + k) = nc›*)\n proof (induct k arbitrary: i1)\n (*goals:\n 1. ‹⋀i1. ⟦i1 + 0 < dim_row A; nc ≤ f i1⟧ ⟹ f (i1 + 0) = nc›\n 2. ‹⋀k i1. ⟦⋀i1. ⟦i1 + k < dim_row A; nc ≤ f i1⟧ ⟹ f (i1 + k) = nc; i1 + Suc k < dim_row A; nc ≤ f i1⟧ ⟹ f (i1 + Suc k) = nc›*)\n case 0 (*‹i1 + 0 < dim_row A› ‹nc ≤ f i1›*)\n from this(1) (*‹i1 + 0 < dim_row A›*) have \"i1 < dim_row A\"\n by simp\n from \"_\" (*‹PROP ?psi ⟹ PROP ?psi›*) assms(1) (*‹pivot_fun (A::'a mat) (f::nat ⇒ nat) (nc::nat)›*) this (*‹i1 < dim_row A›*) have \"f i1 ≤ nc\"\n apply (rule pivot_funD (*‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; (?j::nat) < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = (0::?'a)› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = (1::?'a)› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; ?f ?i < ?nc; (?i'::nat) < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = (0::?'a)›*))\n (*goal: ‹f i1 ≤ nc›*)\n by (intro refl (*‹?t = ?t›*))\n with ‹nc ≤ f i1› (*‹nc ≤ f i1›*) show \"?case\"\n (*goal: ‹f (i1 + 0) = nc›*)\n by simp\n next\n (*goal: ‹⋀k i1. ⟦⋀i1. ⟦i1 + k < dim_row A; nc ≤ f i1⟧ ⟹ f (i1 + k) = nc; i1 + Suc k < dim_row A; nc ≤ f i1⟧ ⟹ f (i1 + Suc k) = nc›*)\n case (Suc k) (*‹⟦?i1.0 + k < dim_row A; nc ≤ f ?i1.0⟧ ⟹ f (?i1.0 + k) = nc› ‹i1 + Suc k < dim_row A› ‹(nc::nat) ≤ (f::nat ⇒ nat) (i1::nat)›*)\n from Suc(2) (*‹i1 + Suc k < dim_row A›*) have \"Suc (i1 + k) < dim_row A\"\n by simp\n hence \"Suc i1 + k < dim_row A\"\n by simp\n hence \"Suc i1 < dim_row A\"\n by simp\n hence \"i1 < dim_row A\"\n by simp\n have \"nc ≤ f (Suc i1)\"\n proof (-)\n (*goal: ‹nc ≤ f (Suc i1)›*)\n have \"f i1 < f (Suc i1) ∨ f (Suc i1) = nc\"\n apply (rule pivot_funD (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?j < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = 0› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = 1› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*))\n (*goals:\n 1. ‹dim_row ?A = ?nr›\n 2. ‹pivot_fun ?A f nc›\n 3. ‹i1 < ?nr›\n 4. ‹Suc i1 < ?nr›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n with Suc(3) (*‹(nc::nat) ≤ (f::nat ⇒ nat) (i1::nat)›*) show \"?thesis\"\n (*goal: ‹nc ≤ f (Suc i1)›*)\n by auto\n qed\n with ‹Suc i1 + k < dim_row A› (*‹Suc i1 + k < dim_row A›*) have \"f (Suc i1 + k) = nc\"\n by (rule Suc( (*‹⟦?i1.0 + k < dim_row A; nc ≤ f ?i1.0⟧ ⟹ f (?i1.0 + k) = nc›*) 1))\n thus \"?case\"\n (*goal: ‹f (i1 + Suc k) = nc›*)\n by simp\n qed\nqed\n\nlemma pivot_fun_mono_strict:\n assumes \"pivot_fun A f nc\" and \"i1 < i2\" and \"i2 < dim_row A\" and \"f i1 < nc\"\n shows \"f i1 < f i2\"\nproof (-)\n (*goal: ‹f i1 < f i2›*)\n from assms(2) (*‹i1 < i2›*) have \"i2 - i1 ≠ 0\" and \"i2 = i1 + (i2 - i1)\"\n apply -\n (*goals:\n 1. ‹(i1::nat) < (i2::nat) ⟹ i2 - i1 ≠ (0::nat)›\n 2. ‹(i1::nat) < (i2::nat) ⟹ i2 = i1 + (i2 - i1)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n then obtain k where \"k ≠ 0\" and \"i2 = i1 + k\"\n (*goal: ‹(⋀k. ⟦k ≠ 0; i2 = i1 + k⟧ ⟹ thesis) ⟹ thesis›*)\n by standard\n from this(1) (*‹k ≠ 0›*) assms(3) (*‹i2 < dim_row A›*) assms(4) (*‹f i1 < nc›*) show \"?thesis\"\n (*goal: ‹f i1 < f i2›*)\n unfolding ‹i2 = i1 + k›\n (*goal: ‹(f::nat ⇒ nat) (i1::nat) < f (i1 + (k::nat))›*)\n proof (induct k arbitrary: i1)\n (*goals:\n 1. ‹⋀i1. ⟦0 ≠ 0; i1 + 0 < dim_row A; f i1 < nc⟧ ⟹ f i1 < f (i1 + 0)›\n 2. ‹⋀k i1. ⟦⋀i1. ⟦k ≠ 0; i1 + k < dim_row A; f i1 < nc⟧ ⟹ f i1 < f (i1 + k); Suc k ≠ 0; i1 + Suc k < dim_row A; f i1 < nc⟧ ⟹ f i1 < f (i1 + Suc k)›*)\n case 0 (*‹0 ≠ 0› ‹(i1::nat) + (0::nat) < dim_row (A::'a::{one,zero} mat)› ‹f i1 < nc›*)\n thus \"?case\"\n (*goal: ‹f i1 < f (i1 + 0)›*)\n by simp\n next\n (*goal: ‹⋀k i1. ⟦⋀i1. ⟦k ≠ 0; i1 + k < dim_row A; f i1 < nc⟧ ⟹ f i1 < f (i1 + k); Suc k ≠ 0; i1 + Suc k < dim_row A; f i1 < nc⟧ ⟹ f i1 < f (i1 + Suc k)›*)\n case (Suc k) (*‹⟦k ≠ 0; ?i1.0 + k < dim_row A; f ?i1.0 < nc⟧ ⟹ f ?i1.0 < f (?i1.0 + k)› ‹Suc k ≠ 0› ‹i1 + Suc k < dim_row A› ‹f i1 < nc›*)\n from Suc(3) (*‹i1 + Suc k < dim_row A›*) have \"Suc (i1 + k) < dim_row A\"\n by simp\n hence \"Suc i1 + k < dim_row A\"\n by simp\n hence \"Suc i1 < dim_row A\"\n by simp\n hence \"i1 < dim_row A\"\n by simp\n have \"*\": \"f i1 < f (Suc i1)\"\n proof (-)\n (*goal: ‹f i1 < f (Suc i1)›*)\n have \"f i1 < f (Suc i1) ∨ f (Suc i1) = nc\"\n apply (rule pivot_funD (*‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; (?j::nat) < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = (0::?'a)› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = (1::?'a)› ‹⟦dim_row (?A::?'a mat) = (?nr::nat); pivot_fun ?A (?f::nat ⇒ nat) (?nc::nat); (?i::nat) < ?nr; ?f ?i < ?nc; (?i'::nat) < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = (0::?'a)›*))\n (*goals:\n 1. ‹dim_row ?A = ?nr›\n 2. ‹pivot_fun ?A f nc›\n 3. ‹i1 < ?nr›\n 4. ‹Suc i1 < ?nr›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n with Suc(4) (*‹f i1 < nc›*) show \"?thesis\"\n (*goal: ‹f i1 < f (Suc i1)›*)\n by auto\n qed\n show \"?case\"\n (*goal: ‹(f::nat ⇒ nat) (i1::nat) < f (i1 + Suc (k::nat))›*)\n proof (simp, cases \"k = 0\")\n (*goals:\n 1. ‹k = 0 ⟹ f i1 < f (Suc (i1 + k))›\n 2. ‹k ≠ 0 ⟹ f i1 < f (Suc (i1 + k))›*)\n case True (*‹(k::nat) = (0::nat)›*)\n show \"f i1 < f (Suc (i1 + k))\"\n by (simp add: True (*‹k = 0›*) * (*‹f i1 < f (Suc i1)›*))\n next\n (*goal: ‹k ≠ 0 ⟹ f i1 < f (Suc (i1 + k))›*)\n case False (*‹(k::nat) ≠ (0::nat)›*)\n have \"f (Suc i1) ≤ f (Suc i1 + k)\"\n proof (cases \"f (Suc i1) < nc\")\n (*goals:\n 1. ‹f (Suc i1) < nc ⟹ f (Suc i1) ≤ f (Suc i1 + k)›\n 2. ‹¬ f (Suc i1) < nc ⟹ f (Suc i1) ≤ f (Suc i1 + k)›*)\n case True (*‹f (Suc i1) < nc›*)\n from False (*‹k ≠ 0›*) ‹Suc i1 + k < dim_row A› (*‹Suc i1 + k < dim_row A›*) True (*‹f (Suc i1) < nc›*) have \"f (Suc i1) < f (Suc i1 + k)\"\n by (rule Suc( (*‹⟦k ≠ 0; ?i1.0 + k < dim_row A; f ?i1.0 < nc⟧ ⟹ f ?i1.0 < f (?i1.0 + k)›*) 1))\n thus \"?thesis\"\n (*goal: ‹f (Suc i1) ≤ f (Suc i1 + k)›*)\n by simp\n next\n (*goal: ‹¬ f (Suc i1) < nc ⟹ f (Suc i1) ≤ f (Suc i1 + k)›*)\n case False (*‹¬ f (Suc i1) < nc›*)\n hence \"nc ≤ f (Suc i1)\"\n by simp\n from assms(1) (*‹pivot_fun A f nc›*) \"_\" (*‹PROP ?psi ⟹ PROP ?psi›*) ‹Suc i1 + k < dim_row A› (*‹Suc i1 + k < dim_row A›*) this (*‹nc ≤ f (Suc i1)›*) have \"f (Suc i1 + k) = nc\"\n apply (rule pivot_fun_stabilizes[where ?f=f] (*‹⟦pivot_fun ?A f ?nc; ?i1.0 ≤ ?i2.0; ?i2.0 < dim_row ?A; ?nc ≤ f ?i1.0⟧ ⟹ f ?i2.0 = ?nc›*))\n (*goal: ‹f (Suc i1 + k) = nc›*)\n by simp\n moreover have \"f (Suc i1) = nc\"\n apply (rule pivot_fun_stabilizes[where ?f=f] (*‹⟦pivot_fun ?A f ?nc; ?i1.0 ≤ ?i2.0; ?i2.0 < dim_row ?A; ?nc ≤ f ?i1.0⟧ ⟹ f ?i2.0 = ?nc›*))\n (*goals:\n 1. ‹pivot_fun ?A f nc›\n 2. ‹?i1.0 ≤ Suc i1›\n 3. ‹Suc i1 < dim_row ?A›\n 4. ‹nc ≤ f ?i1.0›\n discuss goal 1*)\n apply fact\n (*discuss goal 2*)\n apply (rule le_refl (*‹?n ≤ ?n›*))\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n ultimately show \"?thesis\"\n (*goal: ‹(f::nat ⇒ nat) (Suc (i1::nat)) ≤ f (Suc i1 + (k::nat))›*)\n by simp\n qed\n also (*calculation: ‹f (Suc i1) ≤ f (Suc i1 + k)›*) have \"... = f (i1 + Suc k)\"\n by simp\n finally (*calculation: ‹(f::nat ⇒ nat) (Suc (i1::nat)) ≤ f (i1 + Suc (k::nat))›*) have \"f (Suc i1) ≤ f (i1 + Suc k)\" .\n with \"*\" (*‹f i1 < f (Suc i1)›*) show \"f i1 < f (Suc (i1 + k))\"\n by simp\n qed\n qed\nqed\n\nlemma pivot_fun_mono:\n assumes \"pivot_fun A f nc\" and \"i1 ≤ i2\" and \"i2 < dim_row A\"\n shows \"f i1 ≤ f i2\"\nproof (-)\n (*goal: ‹f i1 ≤ f i2›*)\n from assms(2) (*‹i1 ≤ i2›*) have \"i1 < i2 ∨ i1 = i2\"\n by auto\n thus \"?thesis\"\n (*goal: ‹f i1 ≤ f i2›*)\n proof (standard)\n (*goals:\n 1. ‹i1 < i2 ⟹ f i1 ≤ f i2›\n 2. ‹i1 = i2 ⟹ f i1 ≤ f i2›*)\n assume \"i1 < i2\" (*‹(i1::nat) < (i2::nat)›*)\n show \"?thesis\"\n (*goal: ‹f i1 ≤ f i2›*)\n proof (cases \"f i1 < nc\")\n (*goals:\n 1. ‹(f::nat ⇒ nat) (i1::nat) < (nc::nat) ⟹ f i1 ≤ f (i2::nat)›\n 2. ‹¬ (f::nat ⇒ nat) (i1::nat) < (nc::nat) ⟹ f i1 ≤ f (i2::nat)›*)\n case True (*‹f i1 < nc›*)\n from assms(1) (*‹pivot_fun A f nc›*) ‹i1 < i2› (*‹i1 < i2›*) assms(3) (*‹i2 < dim_row A›*) this (*‹(f::nat ⇒ nat) (i1::nat) < (nc::nat)›*) have \"f i1 < f i2\"\n by (rule pivot_fun_mono_strict (*‹⟦pivot_fun ?A ?f ?nc; ?i1.0 < ?i2.0; ?i2.0 < dim_row ?A; ?f ?i1.0 < ?nc⟧ ⟹ ?f ?i1.0 < ?f ?i2.0›*))\n thus \"?thesis\"\n (*goal: ‹(f::nat ⇒ nat) (i1::nat) ≤ f (i2::nat)›*)\n by simp\n next\n (*goal: ‹¬ f i1 < nc ⟹ f i1 ≤ f i2›*)\n case False (*‹¬ (f::nat ⇒ nat) (i1::nat) < (nc::nat)›*)\n hence \"nc ≤ f i1\"\n by simp\n from assms(1) (*‹pivot_fun A f nc›*) \"_\" (*‹PROP ?psi ⟹ PROP ?psi›*) \"_\" (*‹PROP ?psi::prop ⟹ PROP ?psi›*) this (*‹nc ≤ f i1›*) have \"f i1 = nc\"\n proof (rule pivot_fun_stabilizes[where ?f=f] (*‹⟦pivot_fun ?A f ?nc; ?i1.0 ≤ ?i2.0; ?i2.0 < dim_row ?A; ?nc ≤ f ?i1.0⟧ ⟹ f ?i2.0 = ?nc›*), simp)\n (*goal: ‹i1 < dim_row A›*)\n from assms(2) (*‹i1 ≤ i2›*) assms(3) (*‹i2 < dim_row A›*) show \"i1 < dim_row A\"\n by (rule le_less_trans (*‹⟦?x ≤ ?y; ?y < ?z⟧ ⟹ ?x < ?z›*))\n qed\n moreover have \"f i2 = nc\"\n apply (rule pivot_fun_stabilizes[where ?f=f] (*‹⟦pivot_fun ?A f ?nc; ?i1.0 ≤ ?i2.0; ?i2.0 < dim_row ?A; ?nc ≤ f ?i1.0⟧ ⟹ f ?i2.0 = ?nc›*))\n (*goals:\n 1. ‹pivot_fun ?A f nc›\n 2. ‹?i1.0 ≤ i2›\n 3. ‹i2 < dim_row ?A›\n 4. ‹nc ≤ f ?i1.0›\n discuss goal 1*)\n apply fact\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n ultimately show \"?thesis\"\n (*goal: ‹f i1 ≤ f i2›*)\n by simp\n qed\n next\n (*goal: ‹i1 = i2 ⟹ f i1 ≤ f i2›*)\n assume \"i1 = i2\" (*‹(i1::nat) = (i2::nat)›*)\n thus \"?thesis\"\n (*goal: ‹f i1 ≤ f i2›*)\n by simp\n qed\nqed\n\nlemma row_echelon_carrier:\n assumes \"A ∈ carrier_mat nr nc\"\n shows \"row_echelon A ∈ carrier_mat nr nc\"\nproof (-)\n (*goal: ‹row_echelon A ∈ carrier_mat nr nc›*)\n from assms (*‹(A::'a mat) ∈ carrier_mat (nr::nat) (nc::nat)›*) have \"dim_row A = nr\"\n by simp\n let ?B = \"1⇩m (dim_row A)\"\n note assms (*‹A ∈ carrier_mat nr nc›*)\n moreover have \"?B ∈ carrier_mat nr nr\"\n by (simp add: ‹dim_row A = nr›)\n moreover from surj_pair (*‹∃x y. ?p = (x, y)›*) obtain A' and B' where \"*\": \"gauss_jordan A ?B = (A', B')\"\n (*goal: ‹(⋀(A'::'a mat) B'::'a mat. gauss_jordan (A::'a mat) (1⇩m (dim_row A)) = (A', B') ⟹ thesis::bool) ⟹ thesis›*)\n by metis\n ultimately have \"A' ∈ carrier_mat nr nc\"\n by (rule gauss_jordan_carrier (*‹⟦?A ∈ carrier_mat ?nr ?nc; ?B ∈ carrier_mat ?nr ?nc'; gauss_jordan ?A ?B = (?A', ?B')⟧ ⟹ ?A' ∈ carrier_mat ?nr ?nc› ‹⟦?A ∈ carrier_mat ?nr ?nc; ?B ∈ carrier_mat ?nr ?nc'; gauss_jordan ?A ?B = (?A', ?B')⟧ ⟹ ?B' ∈ carrier_mat ?nr ?nc'›*))\n thus \"?thesis\"\n (*goal: ‹row_echelon A ∈ carrier_mat nr nc›*)\n by (simp add: row_echelon_def (*‹row_echelon ?A = fst (gauss_jordan ?A (1⇩m (dim_row ?A)))›*) * (*‹gauss_jordan A (1⇩m (dim_row A)) = (A', B')›*))\nqed\n\nlemma dim_row_echelon[simp]:\n shows \"dim_row (row_echelon A) = dim_row A\" and \"dim_col (row_echelon A) = dim_col A\"\nproof (-)\n (*goals:\n 1. ‹dim_row (row_echelon A) = dim_row A›\n 2. ‹dim_col (row_echelon A) = dim_col A›*)\n have \"A ∈ carrier_mat (dim_row A) (dim_col A)\"\n by simp\n hence \"row_echelon A ∈ carrier_mat (dim_row A) (dim_col A)\"\n by (rule row_echelon_carrier (*‹?A ∈ carrier_mat ?nr ?nc ⟹ row_echelon ?A ∈ carrier_mat ?nr ?nc›*))\n thus \"dim_row (row_echelon A) = dim_row A\" and \"dim_col (row_echelon A) = dim_col A\"\n apply -\n (*goals:\n 1. ‹row_echelon A ∈ carrier_mat (dim_row A) (dim_col A) ⟹ dim_row (row_echelon A) = dim_row A›\n 2. ‹row_echelon A ∈ carrier_mat (dim_row A) (dim_col A) ⟹ dim_col (row_echelon A) = dim_col A›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\nqed\n\nlemma row_echelon_transform:\n obtains P where \"P ∈ Units (ring_mat TYPE('a::field) (dim_row A) b)\" and \"row_echelon A = P * A\"\nproof (-)\n (*goal: ‹(⋀P. ⟦P ∈ Units (ring_mat TYPE('a) (dim_row A) b); row_echelon A = P * A⟧ ⟹ thesis) ⟹ thesis›*)\n let ?B = \"1⇩m (dim_row A)\"\n have \"A ∈ carrier_mat (dim_row A) (dim_col A)\"\n by simp\n moreover have \"?B ∈ carrier_mat (dim_row A) (dim_row A)\"\n by simp\n moreover from surj_pair (*‹∃x y. ?p = (x, y)›*) obtain A' and B' where \"*\": \"gauss_jordan A ?B = (A', B')\"\n (*goal: ‹(⋀A' B'. gauss_jordan A (1⇩m (dim_row A)) = (A', B') ⟹ thesis) ⟹ thesis›*)\n by metis\n ultimately have \"∃P∈Units (ring_mat TYPE('a) (dim_row A) b). A' = P * A ∧ B' = P * ?B\"\n by (rule gauss_jordan_transform (*‹⟦?A ∈ carrier_mat ?nr ?nc; ?B ∈ carrier_mat ?nr ?nc'; gauss_jordan ?A ?B = (?A', ?B')⟧ ⟹ ∃P∈Units (ring_mat TYPE(?'a) ?nr ?b). ?A' = P * ?A ∧ ?B' = P * ?B›*))\n then obtain P where \"P ∈ Units (ring_mat TYPE('a) (dim_row A) b)\" and \"**\": \"A' = P * A ∧ B' = P * ?B\"\n (*goal: ‹(⋀P::'a mat. ⟦P ∈ Units (ring_mat TYPE('a) (dim_row (A::'a mat)) (b::'b)); (A'::'a mat) = P * A ∧ (B'::'a mat) = P * 1⇩m (dim_row A)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by standard\n from this(1) (*‹P ∈ Units (ring_mat TYPE('a) (dim_row A) b)›*) show \"?thesis\"\n (*goal: ‹thesis::bool›*)\n proof (standard)\n (*goal: ‹row_echelon A = P * A›*)\n from \"**\" (*‹(A'::'a mat) = (P::'a mat) * (A::'a mat) ∧ (B'::'a mat) = P * 1⇩m (dim_row A)›*) have \"A' = P * A\"\n by standard\n thus \"row_echelon A = P * A\"\n by (simp add: row_echelon_def (*‹row_echelon (?A::?'a mat) = fst (gauss_jordan ?A (1⇩m (dim_row ?A)))›*) * (*‹gauss_jordan (A::'a mat) (1⇩m (dim_row A)) = (A'::'a mat, B'::'a mat)›*))\n qed\nqed\n\nlemma row_space_row_echelon[simp]: \"row_space (row_echelon A) = row_space A\"\nproof (-)\n (*goal: ‹row_space (row_echelon A) = row_space A›*)\n obtain P where \"*\": \"P ∈ Units (ring_mat TYPE('a::field) (dim_row A) Nil)\" and \"**\": \"row_echelon A = P * A\"\n (*goal: ‹(⋀P::'a mat. ⟦P ∈ Units (ring_mat TYPE('a) (dim_row (A::'a mat)) []); row_echelon A = P * A⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by (rule row_echelon_transform (*‹(⋀P. ⟦P ∈ Units (ring_mat TYPE(?'a) (dim_row ?A) ?b); row_echelon ?A = P * ?A⟧ ⟹ ?thesis) ⟹ ?thesis›*))\n from \"*\" (*‹P ∈ Units (ring_mat TYPE('a) (dim_row A) [])›*) have \"row_space (P * A) = row_space A\"\n by (rule row_space_mult_unit (*‹?P ∈ Units (ring_mat TYPE(?'a) (dim_row ?A) ?b) ⟹ row_space (?P * ?A) = row_space ?A›*))\n thus \"?thesis\"\n (*goal: ‹row_space (row_echelon A) = row_space A›*)\n by (simp only: ** (*‹row_echelon A = P * A›*))\nqed\n\nlemma row_echelon_pivot_fun:\n obtains f where \"pivot_fun (row_echelon A) f (dim_col (row_echelon A))\"\nproof (-)\n (*goal: ‹(⋀f. pivot_fun (row_echelon A) f (dim_col (row_echelon A)) ⟹ thesis) ⟹ thesis›*)\n let ?B = \"1⇩m (dim_row A)\"\n have \"A ∈ carrier_mat (dim_row A) (dim_col A)\"\n by simp\n moreover from surj_pair (*‹∃(x::?'a::type) y::?'b::type. (?p::?'a::type × ?'b::type) = (x, y)›*) obtain A' and B' where \"*\": \"gauss_jordan A ?B = (A', B')\"\n (*goal: ‹(⋀(A'::'a mat) B'::'a mat. gauss_jordan (A::'a mat) (1⇩m (dim_row A)) = (A', B') ⟹ thesis::bool) ⟹ thesis›*)\n by metis\n ultimately have \"row_echelon_form A'\"\n by (rule gauss_jordan_row_echelon (*‹⟦?A ∈ carrier_mat ?nr ?nc; gauss_jordan ?A ?B = (?A', ?B')⟧ ⟹ row_echelon_form ?A'›*))\n then obtain f where \"pivot_fun A' f (dim_col A')\"\n (*goal: ‹(⋀f::nat ⇒ nat. pivot_fun (A'::'a mat) f (dim_col A') ⟹ thesis::bool) ⟹ thesis›*)\n unfolding row_echelon_form_def\n (*goal: ‹(⋀f::nat ⇒ nat. pivot_fun (A'::'a mat) f (dim_col A') ⟹ thesis::bool) ⟹ thesis›*)\n by standard\n hence \"pivot_fun (row_echelon A) f (dim_col (row_echelon A))\"\n by (simp add: row_echelon_def (*‹row_echelon ?A = fst (gauss_jordan ?A (1⇩m (dim_row ?A)))›*) * (*‹gauss_jordan A (1⇩m (dim_row A)) = (A', B')›*))\n thus \"?thesis\"\n (*goal: ‹thesis::bool›*)\n by standard\nqed\n\nlemma distinct_nzrows_row_echelon: \"distinct (nzrows (row_echelon A))\"\n unfolding nzrows_def\n (*goal: ‹distinct (filter (λr. r ≠ 0⇩v (dim_col (row_echelon A))) (rows (row_echelon A)))›*)\n apply (rule distinct_filterI (*‹(⋀i j. ⟦i < j; i < length ?xs; j < length ?xs; ?P (?xs ! i); ?P (?xs ! j)⟧ ⟹ ?xs ! i ≠ ?xs ! j) ⟹ distinct (filter ?P ?xs)›*))\n (*goal: ‹distinct (filter (λr. r ≠ 0⇩v (dim_col (row_echelon A))) (rows (row_echelon A)))›*)\n proof (simp del: dim_row_echelon (*‹dim_row (row_echelon ?A) = dim_row ?A› ‹dim_col (row_echelon ?A) = dim_col ?A›*))\n (*goal: ‹⋀i j. ⟦i < j; j < dim_row (row_echelon A); row (row_echelon A) i ≠ 0⇩v (dim_col (row_echelon A)); row (row_echelon A) j ≠ 0⇩v (dim_col (row_echelon A))⟧ ⟹ row (row_echelon A) i ≠ row (row_echelon A) j›*)\n let ?B = \"row_echelon A\"\n fix i :: nat and j :: nat\n assume \"i < j\" and \"j < dim_row ?B\" (*‹(i::nat) < (j::nat)› ‹(j::nat) < dim_row (row_echelon (A::'a mat))›*)\n hence \"i ≠ j\" and \"i < dim_row ?B\"\n apply -\n (*goals:\n 1. ‹⟦(i::nat) < (j::nat); j < dim_row (row_echelon (A::'a mat))⟧ ⟹ i ≠ j›\n 2. ‹⟦(i::nat) < (j::nat); j < dim_row (row_echelon (A::'a mat))⟧ ⟹ i < dim_row (row_echelon A)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n assume ri: \"row ?B i ≠ 0⇩v (dim_col ?B)\" and rj: \"row ?B j ≠ 0⇩v (dim_col ?B)\" (*‹row (row_echelon (A::'a mat)) (i::nat) ≠ 0⇩v (dim_col (row_echelon A))› ‹row (row_echelon (A::'a mat)) (j::nat) ≠ 0⇩v (dim_col (row_echelon A))›*)\n obtain f where pf: \"pivot_fun ?B f (dim_col ?B)\"\n (*goal: ‹(⋀f. pivot_fun (row_echelon A) f (dim_col (row_echelon A)) ⟹ thesis) ⟹ thesis›*)\n by (fact row_echelon_pivot_fun (*‹(⋀f. pivot_fun (row_echelon ?A) f (dim_col (row_echelon ?A)) ⟹ ?thesis) ⟹ ?thesis›*))\n from rj (*‹row (row_echelon A) j ≠ 0⇩v (dim_col (row_echelon A))›*) have \"f j < dim_col ?B\"\n by (simp only: row_not_zero_iff_pivot_fun[OF pf ‹j < dim_row ?B›] (*‹(row (row_echelon A) j ≠ 0⇩v (dim_col (row_echelon A))) = (f j < dim_col (row_echelon A))›*))\n from \"_\" (*‹PROP ?psi ⟹ PROP ?psi›*) pf (*‹pivot_fun (row_echelon A) f (dim_col (row_echelon A))›*) ‹j < dim_row ?B› (*‹(j::nat) < dim_row (row_echelon (A::'a mat))›*) this (*‹f j < dim_col (row_echelon A)›*) ‹i < dim_row ?B› (*‹i < dim_row (row_echelon A)›*) ‹i ≠ j› (*‹(i::nat) ≠ (j::nat)›*) have \"*\": \"?B $$ (i, f j) = 0\"\n apply (rule pivot_funD( (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*) 5))\n (*goal: ‹row_echelon (A::'a mat) $$ (i::nat, (f::nat ⇒ nat) (j::nat)) = (0::'a)›*)\n by (intro refl (*‹?t = ?t›*))\n show \"row ?B i ≠ row ?B j\"\n proof (standard)\n (*goal: ‹row (row_echelon A) i = row (row_echelon A) j ⟹ False›*)\n assume \"row ?B i = row ?B j\" (*‹row (row_echelon (A::'a mat)) (i::nat) = row (row_echelon A) (j::nat)›*)\n hence \"row ?B i $ (f j) = row ?B j $ (f j)\"\n by simp\n with ‹i < dim_row ?B› (*‹i < dim_row (row_echelon A)›*) ‹j < dim_row ?B› (*‹j < dim_row (row_echelon A)›*) ‹f j < dim_col ?B› (*‹f j < dim_col (row_echelon A)›*) have \"?B $$ (i, f j) = ?B $$ (j, f j)\"\n by simp\n also (*calculation: ‹row_echelon A $$ (i, f j) = row_echelon A $$ (j, f j)›*) from \"_\" (*‹PROP ?psi ⟹ PROP ?psi›*) pf (*‹pivot_fun (row_echelon A) f (dim_col (row_echelon A))›*) ‹j < dim_row ?B› (*‹(j::nat) < dim_row (row_echelon (A::'a mat))›*) ‹f j < dim_col ?B› (*‹f j < dim_col (row_echelon A)›*) have \"... = 1\"\n apply (rule pivot_funD (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?j < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = 0› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = 1› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*))\n (*goal: ‹row_echelon (A::'a mat) $$ (j::nat, (f::nat ⇒ nat) j) = (1::'a)›*)\n by (intro refl (*‹(?t::?'a) = ?t›*))\n finally (*calculation: ‹row_echelon A $$ (i, f j) = 1›*) show False\n by (simp add: * (*‹row_echelon A $$ (i, f j) = 0›*))\n qed\n qed\n\nsubsection ‹Converting Between Polynomials and Macaulay Matrices›\n\ndefinition poly_to_row :: \"'a list ⇒ ('a ⇒₀ 'b::zero) ⇒ 'b vec\" where\n \"poly_to_row ts p = vec_of_list (map (lookup p) ts)\"\n\ndefinition polys_to_mat :: \"'a list ⇒ ('a ⇒₀ 'b::zero) list ⇒ 'b mat\" where\n \"polys_to_mat ts ps = mat_of_rows (length ts) (map (poly_to_row ts) ps)\"\n\ndefinition list_to_fun :: \"'a list ⇒ ('b::zero) list ⇒ 'a ⇒ 'b\" where\n \"list_to_fun ts cs t = (case map_of (zip ts cs) t of Some c ⇒ c | None ⇒ 0)\"\n\ndefinition list_to_poly :: \"'a list ⇒ 'b list ⇒ ('a ⇒₀ 'b::zero)\" where\n \"list_to_poly ts cs = Abs_poly_mapping (list_to_fun ts cs)\"\n\ndefinition row_to_poly :: \"'a list ⇒ 'b vec ⇒ ('a ⇒₀ 'b::zero)\" where\n \"row_to_poly ts r = list_to_poly ts (list_of_vec r)\"\n\ndefinition mat_to_polys :: \"'a list ⇒ 'b mat ⇒ ('a ⇒₀ 'b::zero) list\" where\n \"mat_to_polys ts A = map (row_to_poly ts) (rows A)\"\n\nlemma dim_poly_to_row: \"dim_vec (poly_to_row ts p) = length ts\"\n by (simp add: poly_to_row_def (*‹poly_to_row (?ts::?'a list) (?p::?'a ⇒₀ ?'b) = vec_of_list (map (lookup ?p) ?ts)›*))\n\nlemma poly_to_row_index:\n assumes \"i < length ts\"\n shows \"poly_to_row ts p $ i = lookup p (ts ! i)\"\n by (simp add: poly_to_row_def (*‹poly_to_row ?ts ?p = vec_of_list (map (lookup ?p) ?ts)›*) vec_of_list_index (*‹vec_of_list ?xs $ ?j = ?xs ! ?j›*) assms (*‹i < length ts›*))\n\ncontext term_powerprod\nbegin\n\nlemma poly_to_row_scalar_mult:\n assumes \"keys p ⊆ set ts\"\n shows \"row_to_poly ts (c ⋅⇩v (poly_to_row ts p)) = c ⋅ p\"\nproof (-)\n (*goal: ‹row_to_poly ts (c ⋅⇩v poly_to_row ts p) = c ⋅ p›*)\n have eq: \"(vec (length ts) (λi. c * poly_to_row ts p $ i)) =\n (vec (length ts) (λi. c * lookup p (ts ! i)))\"\n apply (rule vec_cong (*‹⟦(?n::nat) = (?m::nat); ⋀i::nat. i < ?m ⟹ (?f::nat ⇒ ?'a) i = (?g::nat ⇒ ?'a) i⟧ ⟹ vec ?n ?f = vec ?m ?g›*))\n (*goals:\n 1. ‹length ts = length ts›\n 2. ‹⋀i. i < length ts ⟹ c * poly_to_row ts p $ i = c * lookup p (ts ! i)›\n discuss goal 1*)\n apply rule\n (*discuss goal 2*)\n apply (simp only: poly_to_row_index (*‹?i < length ?ts ⟹ poly_to_row ?ts ?p $ ?i = lookup ?p (?ts ! ?i)›*))\n (*proven 2 subgoals*) .\n have \"*\": \"list_to_fun ts (list_of_vec (c ⋅⇩v (poly_to_row ts p))) = (λt. c * lookup p t)\"\n apply rule\n (*goal: ‹list_to_fun ts (list_of_vec (c ⋅⇩v poly_to_row ts p)) = (λt. c * lookup p t)›*)\n apply (simp add: list_to_fun_def (*‹list_to_fun ?ts ?cs ?t = (case map_of (zip ?ts ?cs) ?t of None ⇒ 0 | Some c ⇒ c)›*) smult_vec_def (*‹?a ⋅⇩v ?v ≡ vec (dim_vec ?v) (λi. ?a * ?v $ i)›*) dim_poly_to_row (*‹dim_vec (poly_to_row ?ts ?p) = length ?ts›*) eq (*‹vec (length ts) (λi. c * poly_to_row ts p $ i) = vec (length ts) (λi. c * lookup p (ts ! i))›*))\n (*goal: ‹⋀t::'b. list_to_fun (ts::'b list) (list_of_vec ((c::'c) ⋅⇩v poly_to_row ts (p::'b ⇒₀ 'c))) t = c * lookup p t›*)\n apply (simp add: map_upt[of \"λx. c * lookup p x\"] (*‹map (λi. c * lookup p (?xs ! i)) [0..<length ?xs] = map (λx. c * lookup p x) ?xs›*) map_of_zip_map (*‹map_of (zip ?xs (map ?f ?xs)) = (λx. if x ∈ set ?xs then Some (?f x) else None)›*))\n (*goal: ‹⋀t. (case map_of (zip ts (map (λi. c * lookup p (ts ! i)) [0..<length ts])) t of None ⇒ 0 | Some c ⇒ c) = c * lookup p t›*)\n proof (rule)\n (*goal: ‹⋀t. t ∉ set ts ⟹ c * lookup p t = 0›*)\n fix t\n assume \"t ∉ set ts\" (*‹(t::'b) ∉ set (ts::'b list)›*)\n with assms(1) (*‹keys p ⊆ set ts›*) have \"t ∉ keys p\"\n by auto\n thus \"c * lookup p t = 0\"\n by (simp add: in_keys_iff (*‹(?x ∈ keys ?s) = (lookup ?s ?x ≠ 0)›*))\n qed\n have \"**\": \"lookup (Abs_poly_mapping (list_to_fun ts (list_of_vec (c ⋅⇩v (poly_to_row ts p))))) =\n (λt. c * lookup p t)\"\n proof (simp only: * (*‹list_to_fun ts (list_of_vec (c ⋅⇩v poly_to_row ts p)) = (λt. c * lookup p t)›*), rule Abs_poly_mapping_inverse (*‹?y ∈ {f. finite {x. f x ≠ 0}} ⟹ lookup (Abs_poly_mapping ?y) = ?y›*), simp, rule finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*), rule, simp)\n (*goals:\n 1. ‹⋀x. c * lookup p x ≠ 0 ⟹ x ∈ ?B3›\n 2. ‹finite ?B3›*)\n fix t\n assume \"c * lookup p t ≠ 0\" (*‹(c::'c) * lookup (p::'b ⇒₀ 'c) (t::'b) ≠ (0::'c)›*)\n hence \"lookup p t ≠ 0\"\n using mult_not_zero (*‹?a * ?b ≠ 0 ⟹ ?a ≠ 0 ∧ ?b ≠ 0›*) by blast\n thus \"t ∈ keys p\"\n by (simp add: in_keys_iff (*‹((?x::?'a) ∈ keys (?s::?'a ⇒₀ ?'b)) = (lookup ?s ?x ≠ (0::?'b))›*))\n qed (fact finite_keys (*‹finite (keys ?f)›*))\n (*solved the remaining goal: ‹finite (keys p)›*)\n show \"?thesis\"\n (*goal: ‹row_to_poly (ts::'b list) ((c::'c) ⋅⇩v poly_to_row ts (p::'b ⇒₀ 'c)) = c ⋅ p›*)\n unfolding row_to_poly_def\n (*goal: ‹list_to_poly ts (list_of_vec (c ⋅⇩v poly_to_row ts p)) = c ⋅ p›*)\n apply (rule poly_mapping_eqI (*‹(⋀k. lookup ?f k = lookup ?g k) ⟹ ?f = ?g›*))\n (*goal: ‹list_to_poly ts (list_of_vec (c ⋅⇩v poly_to_row ts p)) = c ⋅ p›*)\n by (simp only: list_to_poly_def (*‹list_to_poly (?ts::?'a list) (?cs::?'b list) = Abs_poly_mapping (list_to_fun ?ts ?cs)›*) ** (*‹lookup (Abs_poly_mapping (list_to_fun (ts::'b list) (list_of_vec ((c::'c) ⋅⇩v poly_to_row ts (p::'b ⇒₀ 'c))))) = (λt::'b. c * lookup p t)›*) lookup_map_scale (*‹lookup ((?c::?'b) ⋅ (?p::?'a ⇒₀ ?'b)) = (λx::?'a. ?c * lookup ?p x)›*))\nqed\n\nlemma poly_to_row_to_poly:\n assumes \"keys p ⊆ set ts\"\n shows \"row_to_poly ts (poly_to_row ts p) = (p::'t ⇒₀ 'b::semiring_1)\"\nproof (-)\n (*goal: ‹row_to_poly ts (poly_to_row ts p) = p›*)\n have \"1 ⋅⇩v (poly_to_row ts p) = poly_to_row ts p\"\n by simp\n thus \"?thesis\"\n (*goal: ‹row_to_poly ts (poly_to_row ts p) = p›*)\n using poly_to_row_scalar_mult[OF assms, of 1] (*‹row_to_poly ts (1 ⋅⇩v poly_to_row ts p) = 1 ⋅ p›*) by simp\nqed\n\nlemma lookup_list_to_poly: \"lookup (list_to_poly ts cs) = list_to_fun ts cs\"\n unfolding list_to_poly_def\n (*goal: ‹lookup (Abs_poly_mapping (list_to_fun ts cs)) = list_to_fun ts cs›*)\n proof (rule Abs_poly_mapping_inverse (*‹?y ∈ {f. finite {x. f x ≠ 0}} ⟹ lookup (Abs_poly_mapping ?y) = ?y›*), rule, rule finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*))\n (*goals:\n 1. ‹{x. list_to_fun ts cs x ≠ 0} ⊆ ?B4›\n 2. ‹finite ?B4›*)\n show \"{x. list_to_fun ts cs x ≠ 0} ⊆ set ts\"\n apply rule\n (*goal: ‹{x. list_to_fun ts cs x ≠ 0} ⊆ set ts›*)\n proof (simp)\n (*goal: ‹⋀x. list_to_fun ts cs x ≠ 0 ⟹ x ∈ set ts›*)\n fix t\n assume \"list_to_fun ts cs t ≠ 0\" (*‹list_to_fun (ts::'b list) (cs::'c list) (t::'b) ≠ (0::'c)›*)\n then obtain c where \"map_of (zip ts cs) t = Some c\"\n (*goal: ‹(⋀c. map_of (zip ts cs) t = Some c ⟹ thesis) ⟹ thesis›*)\n unfolding list_to_fun_def\n (*goal: ‹(⋀c. map_of (zip ts cs) t = Some c ⟹ thesis) ⟹ thesis›*)\n by fastforce\n thus \"t ∈ set ts\"\n by (meson in_set_zipE (*‹⟦(?x, ?y) ∈ set (zip ?xs ?ys); ⟦?x ∈ set ?xs; ?y ∈ set ?ys⟧ ⟹ ?R⟧ ⟹ ?R›*) map_of_SomeD (*‹map_of ?xs ?k = Some ?y ⟹ (?k, ?y) ∈ set ?xs›*))\n qed\n qed (simp)\n (*solved the remaining goal: ‹finite (set ts)›*)\n\nlemma list_to_fun_Nil [simp]: \"list_to_fun [] cs = 0\"\n apply (simp only: zero_fun_def (*‹(0::?'a ⇒ ?'b) = (λx::?'a. 0::?'b)›*))\n (*goal: ‹list_to_fun [] cs = 0›*)\n apply rule\n (*goal: ‹list_to_fun [] cs = (λx. 0)›*)\n by (simp add: list_to_fun_def (*‹list_to_fun ?ts ?cs ?t = (case map_of (zip ?ts ?cs) ?t of None ⇒ 0 | Some c ⇒ c)›*))\n\nlemma list_to_poly_Nil [simp]: \"list_to_poly [] cs = 0\"\n apply (rule poly_mapping_eqI (*‹(⋀k. lookup ?f k = lookup ?g k) ⟹ ?f = ?g›*))\n (*goal: ‹list_to_poly [] cs = 0›*)\n by (simp add: lookup_list_to_poly (*‹lookup (list_to_poly ?ts ?cs) = list_to_fun ?ts ?cs›*))\n\nlemma row_to_poly_Nil [simp]: \"row_to_poly [] r = 0\"\n apply (simp only: row_to_poly_def (*‹row_to_poly ?ts ?r = list_to_poly ?ts (list_of_vec ?r)›*))\n (*goal: ‹row_to_poly [] r = 0›*)\n by (fact list_to_poly_Nil (*‹list_to_poly [] (?cs::?'c::zero list) = (0::?'b::type ⇒₀ ?'c::zero)›*))\n\nlemma lookup_row_to_poly:\n assumes \"distinct ts\" and \"dim_vec r = length ts\" and \"i < length ts\"\n shows \"lookup (row_to_poly ts r) (ts ! i) = r $ i\"\nproof (simp only: row_to_poly_def (*‹row_to_poly ?ts ?r = list_to_poly ?ts (list_of_vec ?r)›*) lookup_list_to_poly (*‹lookup (list_to_poly ?ts ?cs) = list_to_fun ?ts ?cs›*))\n (*goal: ‹list_to_fun ts (list_of_vec r) (ts ! i) = r $ i›*)\n from assms(2) (*‹dim_vec r = length ts›*) assms(3) (*‹i < length ts›*) have \"i < dim_vec r\"\n by simp\n have \"map_of (zip ts (list_of_vec r)) (ts ! i) = Some ((list_of_vec r) ! i)\"\n apply (rule map_of_zip_nth (*‹⟦length ?xs = length ?ys; distinct ?xs; ?i < length ?ys⟧ ⟹ map_of (zip ?xs ?ys) (?xs ! ?i) = Some (?ys ! ?i)›*))\n (*goals:\n 1. ‹length ts = length (list_of_vec r)›\n 2. ‹distinct ts›\n 3. ‹i < length (list_of_vec r)›\n discuss goal 1*)\n apply (simp only: length_list_of_vec (*‹length (list_of_vec ?v) = dim_vec ?v›*) assms( (*‹dim_vec r = length ts›*) 2))\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply (simp only: length_list_of_vec (*‹length (list_of_vec ?v) = dim_vec ?v›*) assms( (*‹dim_vec r = length ts›*) 2))\n (*goal: ‹i < length (list_of_vec r)›*)\n apply fact\n (*proven 3 subgoals*) .\n also (*calculation: ‹map_of (zip ts (list_of_vec r)) (ts ! i) = Some (list_of_vec r ! i)›*) have \"... = Some (r $ i)\"\n by (simp only: list_of_vec_index (*‹list_of_vec ?v ! ?j = ?v $ ?j›*))\n finally (*calculation: ‹map_of (zip ts (list_of_vec r)) (ts ! i) = Some (r $ i)›*) show \"list_to_fun ts (list_of_vec r) (ts ! i) = r $ i\"\n by (simp add: list_to_fun_def (*‹list_to_fun ?ts ?cs ?t = (case map_of (zip ?ts ?cs) ?t of None ⇒ 0 | Some c ⇒ c)›*))\nqed\n\nlemma keys_row_to_poly: \"keys (row_to_poly ts r) ⊆ set ts\"\nproof (standard)\n (*goal: ‹⋀x. x ∈ keys (row_to_poly ts r) ⟹ x ∈ set ts›*)\n fix t\n assume \"t ∈ keys (row_to_poly ts r)\" (*‹(t::'b) ∈ keys (row_to_poly (ts::'b list) (r::'c vec))›*)\n hence \"lookup (row_to_poly ts r) t ≠ 0\"\n by (simp add: in_keys_iff (*‹(?x ∈ keys ?s) = (lookup ?s ?x ≠ 0)›*))\n thus \"t ∈ set ts\"\n proof (simp add: row_to_poly_def (*‹row_to_poly ?ts ?r = list_to_poly ?ts (list_of_vec ?r)›*) lookup_list_to_poly (*‹lookup (list_to_poly ?ts ?cs) = list_to_fun ?ts ?cs›*) list_to_fun_def (*‹list_to_fun ?ts ?cs ?t = (case map_of (zip ?ts ?cs) ?t of None ⇒ 0 | Some c ⇒ c)›*) del: lookup_not_eq_zero_eq_in_keys (*‹(lookup ?f ?k ≠ 0) = (?k ∈ keys ?f)›*) split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n (*goal: ‹⋀x2. ⟦map_of (zip ts (list_of_vec r)) t = Some x2; x2 ≠ 0⟧ ⟹ t ∈ set ts›*)\n fix c\n assume \"map_of (zip ts (list_of_vec r)) t = Some c\" (*‹map_of (zip (ts::'b list) (list_of_vec (r::'c vec))) (t::'b) = Some (c::'c)›*)\n thus \"t ∈ set ts\"\n by (meson in_set_zipE (*‹⟦(?x, ?y) ∈ set (zip ?xs ?ys); ⟦?x ∈ set ?xs; ?y ∈ set ?ys⟧ ⟹ ?R⟧ ⟹ ?R›*) map_of_SomeD (*‹map_of ?xs ?k = Some ?y ⟹ (?k, ?y) ∈ set ?xs›*))\n qed\nqed\n\nlemma lookup_row_to_poly_not_zeroE:\n assumes \"lookup (row_to_poly ts r) t ≠ 0\"\n obtains i where \"i < length ts\" and \"t = ts ! i\"\nproof (-)\n (*goal: ‹(⋀i. ⟦i < length ts; t = ts ! i⟧ ⟹ thesis) ⟹ thesis›*)\n from assms (*‹lookup (row_to_poly ts r) t ≠ 0›*) have \"t ∈ keys (row_to_poly ts r)\"\n by (simp add: in_keys_iff (*‹(?x ∈ keys ?s) = (lookup ?s ?x ≠ 0)›*))\n have \"t ∈ set ts\"\n apply rule\n (*goals:\n 1. ‹t ∈ ?A›\n 2. ‹?A ⊆ set ts›\n discuss goal 1*)\n apply fact\n (*discuss goal 2*)\n apply (fact keys_row_to_poly (*‹keys (row_to_poly ?ts ?r) ⊆ set ?ts›*))\n (*proven 2 subgoals*) .\n then obtain i where \"i < length ts\" and \"t = ts ! i\"\n (*goal: ‹(⋀i. ⟦i < length ts; t = ts ! i⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*))\n thus \"?thesis\"\n (*goal: ‹thesis›*)\n by standard\nqed\n\nlemma row_to_poly_zero [simp]: \"row_to_poly ts (0⇩v (length ts)) = (0::'t ⇒₀ 'b::zero)\"\nproof (-)\n (*goal: ‹row_to_poly ts (0⇩v (length ts)) = 0›*)\n have eq: \"map (λ_. 0::'b) [0..<length ts] = map (λ_. 0) ts\"\n by (simp add: map_replicate_const (*‹map (λx. ?k) ?lst = replicate (length ?lst) ?k›*))\n show \"?thesis\"\n (*goal: ‹row_to_poly ts (0⇩v (length ts)) = 0›*)\n apply (simp add: row_to_poly_def (*‹row_to_poly ?ts ?r = list_to_poly ?ts (list_of_vec ?r)›*) zero_vec_def (*‹0⇩v ?n ≡ vec ?n (λi. 0)›*))\n (*goal: ‹row_to_poly ts (0⇩v (length ts)) = 0›*)\n apply (rule poly_mapping_eqI (*‹(⋀k::?'a. lookup (?f::?'a ⇒₀ ?'b) k = lookup (?g::?'a ⇒₀ ?'b) k) ⟹ ?f = ?g›*))\n (*goal: ‹list_to_poly ts (map (λi. 0) [0..<length ts]) = 0›*)\n by (simp add: lookup_list_to_poly (*‹lookup (list_to_poly (?ts::?'b list) (?cs::?'c list)) = list_to_fun ?ts ?cs›*) list_to_fun_def (*‹list_to_fun (?ts::?'a list) (?cs::?'b list) (?t::?'a) = (case map_of (zip ?ts ?cs) ?t of None ⇒ 0::?'b | Some (c::?'b) ⇒ c)›*) eq (*‹map (λ_::nat. 0::'b) [0::nat..<length (ts::'t list)] = map (λ_::'t. 0::'b) ts›*) map_of_zip_map (*‹map_of (zip (?xs::?'a list) (map (?f::?'a ⇒ ?'b) ?xs)) = (λx::?'a. if x ∈ set ?xs then Some (?f x) else None)›*))\nqed\n\nlemma row_to_poly_zeroD:\n assumes \"distinct ts\" and \"dim_vec r = length ts\" and \"row_to_poly ts r = 0\"\n shows \"r = 0⇩v (length ts)\"\nproof (rule, simp_all add: assms(2))\n (*goal: ‹⋀i. i < length ts ⟹ r $ i = 0›*)\n fix i\n assume \"i < length ts\" (*‹(i::nat) < length (ts::'b list)›*)\n from assms(3) (*‹row_to_poly ts r = 0›*) have \"0 = lookup (row_to_poly ts r) (ts ! i)\"\n by simp\n also (*calculation: ‹0 = lookup (row_to_poly ts r) (ts ! i)›*) from assms(1) (*‹distinct ts›*) assms(2) (*‹dim_vec (r::'c vec) = length (ts::'b list)›*) ‹i < length ts› (*‹i < length ts›*) have \"... = r $ i\"\n by (rule lookup_row_to_poly (*‹⟦distinct ?ts; dim_vec ?r = length ?ts; ?i < length ?ts⟧ ⟹ lookup (row_to_poly ?ts ?r) (?ts ! ?i) = ?r $ ?i›*))\n finally (*calculation: ‹0 = r $ i›*) show \"r $ i = 0\"\n by simp\nqed\n\nlemma row_to_poly_inj:\n assumes \"distinct ts\" and \"dim_vec r1 = length ts\" and \"dim_vec r2 = length ts\"\n and \"row_to_poly ts r1 = row_to_poly ts r2\"\n shows \"r1 = r2\"\nproof (rule, simp_all add: assms(2) assms(3))\n (*goal: ‹⋀i. i < length ts ⟹ r1 $ i = r2 $ i›*)\n fix i\n assume \"i < length ts\" (*‹(i::nat) < length (ts::'b list)›*)\n have \"r1 $ i = lookup (row_to_poly ts r1) (ts ! i)\"\n by (simp only: lookup_row_to_poly[OF assms(1) assms(2) ‹i < length ts›] (*‹lookup (row_to_poly ts r1) (ts ! i) = r1 $ i›*))\n also (*calculation: ‹r1 $ i = lookup (row_to_poly ts r1) (ts ! i)›*) from assms(4) (*‹row_to_poly ts r1 = row_to_poly ts r2›*) have \"... = lookup (row_to_poly ts r2) (ts ! i)\"\n by simp\n also (*calculation: ‹r1 $ i = lookup (row_to_poly ts r2) (ts ! i)›*) from assms(1) (*‹distinct ts›*) assms(3) (*‹dim_vec r2 = length ts›*) ‹i < length ts› (*‹i < length ts›*) have \"... = r2 $ i\"\n by (rule lookup_row_to_poly (*‹⟦distinct ?ts; dim_vec ?r = length ?ts; ?i < length ?ts⟧ ⟹ lookup (row_to_poly ?ts ?r) (?ts ! ?i) = ?r $ ?i›*))\n finally (*calculation: ‹r1 $ i = r2 $ i›*) show \"r1 $ i = r2 $ i\" .\nqed\n\nlemma row_to_poly_vec_plus:\n assumes \"distinct ts\" and \"length ts = n\"\n shows \"row_to_poly ts (vec n (f1 + f2)) = row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)\"\nproof (rule poly_mapping_eqI (*‹(⋀k::?'a::type. lookup (?f::?'a::type ⇒₀ ?'b::zero) k = lookup (?g::?'a::type ⇒₀ ?'b::zero) k) ⟹ ?f = ?g›*))\n (*goal: ‹⋀k::'b::type. lookup (row_to_poly (ts::'b::type list) (vec (n::nat) ((f1::nat ⇒ 'c::monoid_add) + (f2::nat ⇒ 'c::monoid_add)))) k = lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) k›*)\n fix t\n show \"lookup (row_to_poly ts (vec n (f1 + f2))) t =\n lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) t\" (is \"lookup ?l t = lookup (?r1 + ?r2) t\")\n proof (cases \"t ∈ set ts\")\n (*goals:\n 1. ‹t ∈ set ts ⟹ lookup (row_to_poly ts (vec n (f1 + f2))) t = lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) t›\n 2. ‹t ∉ set ts ⟹ lookup (row_to_poly ts (vec n (f1 + f2))) t = lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) t›*)\n case True (*‹t ∈ set ts›*)\n then obtain j where j: \"j < length ts\" and t: \"t = ts ! j\"\n (*goal: ‹(⋀j. ⟦j < length ts; t = ts ! j⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*))\n have d1: \"dim_vec (vec n f1) = length ts\" and d2: \"dim_vec (vec n f2) = length ts\" and da: \"dim_vec (vec n (f1 + f2)) = length ts\"\n (*goals:\n 1. ‹dim_vec (vec n f1) = length ts›\n 2. ‹dim_vec (vec n f2) = length ts›\n 3. ‹dim_vec (vec n (f1 + f2)) = length ts›\n discuss goal 1*)\n apply (simp add: assms( (*‹length (ts::'b::type list) = (n::nat)›*) 2))\n (*discuss goal 2*)\n apply (simp add: assms( (*‹length ts = n›*) 2))\n (*discuss goal 3*)\n apply (simp add: assms( (*‹length ts = n›*) 2))\n (*proven 3 subgoals*) .\n from j (*‹(j::nat) < length (ts::'b list)›*) have j': \"j < n\"\n by (simp only: assms( (*‹length ts = n›*) 2))\n show \"?thesis\"\n (*goal: ‹lookup (row_to_poly ts (vec n (f1 + f2))) t = lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) t›*)\n apply (simp only: t (*‹t = ts ! j›*) lookup_add (*‹lookup (?f + ?g) ?k = lookup ?f ?k + lookup ?g ?k›*) lookup_row_to_poly[OF assms(1) d1 j] (*‹lookup (row_to_poly ts (vec n f1)) (ts ! j) = vec n f1 $ j›*) lookup_row_to_poly[OF assms(1) d2 j] (*‹lookup (row_to_poly ts (vec n f2)) (ts ! j) = vec n f2 $ j›*) lookup_row_to_poly[OF assms(1) da j] (*‹lookup (row_to_poly ts (vec n (f1 + f2))) (ts ! j) = vec n (f1 + f2) $ j›*) index_vec[OF j'] (*‹vec n ?f $ j = ?f j›*))\n (*goal: ‹lookup (row_to_poly ts (vec n (f1 + f2))) t = lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) t›*)\n by (simp only: plus_fun_def (*‹?f + ?g = (λx. ?f x + ?g x)›*))\n next\n (*goal: ‹t ∉ set ts ⟹ lookup (row_to_poly ts (vec n (f1 + f2))) t = lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) t›*)\n case False (*‹t ∉ set ts›*)\n with keys_row_to_poly[of ts \"vec n (f1 + f2)\"] (*‹keys (row_to_poly (ts::'b list) (vec (n::nat) ((f1::nat ⇒ 'c) + (f2::nat ⇒ 'c)))) ⊆ set ts›*) keys_row_to_poly[of ts \"vec n f1\"] (*‹keys (row_to_poly ts (vec n f1)) ⊆ set ts›*) keys_row_to_poly[of ts \"vec n f2\"] (*‹keys (row_to_poly ts (vec n f2)) ⊆ set ts›*) have \"t ∉ keys ?l\" and \"t ∉ keys ?r1\" and \"t ∉ keys ?r2\"\n apply -\n (*goals:\n 1. ‹⟦keys (row_to_poly ts (vec n (f1 + f2))) ⊆ set ts; keys (row_to_poly ts (vec n f1)) ⊆ set ts; keys (row_to_poly ts (vec n f2)) ⊆ set ts; t ∉ set ts⟧ ⟹ t ∉ keys (row_to_poly ts (vec n (f1 + f2)))›\n 2. ‹⟦keys (row_to_poly ts (vec n (f1 + f2))) ⊆ set ts; keys (row_to_poly ts (vec n f1)) ⊆ set ts; keys (row_to_poly ts (vec n f2)) ⊆ set ts; t ∉ set ts⟧ ⟹ t ∉ keys (row_to_poly ts (vec n f1))›\n 3. ‹⟦keys (row_to_poly ts (vec n (f1 + f2))) ⊆ set ts; keys (row_to_poly ts (vec n f1)) ⊆ set ts; keys (row_to_poly ts (vec n f2)) ⊆ set ts; t ∉ set ts⟧ ⟹ t ∉ keys (row_to_poly ts (vec n f2))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n from this(2) (*‹(t::'b) ∉ keys (row_to_poly (ts::'b list) (vec (n::nat) (f1::nat ⇒ 'c)))›*) this(3) (*‹t ∉ keys (row_to_poly ts (vec n f2))›*) have \"t ∉ keys (?r1 + ?r2)\"\n by (meson Poly_Mapping.keys_add (*‹keys (?f + ?g) ⊆ keys ?f ∪ keys ?g›*) UnE (*‹⟦?c ∈ ?A ∪ ?B; ?c ∈ ?A ⟹ ?P; ?c ∈ ?B ⟹ ?P⟧ ⟹ ?P›*) in_mono (*‹?A ⊆ ?B ⟹ ?x ∈ ?A ⟶ ?x ∈ ?B›*))\n with ‹t ∉ keys ?l› (*‹t ∉ keys (row_to_poly ts (vec n (f1 + f2)))›*) show \"?thesis\"\n (*goal: ‹lookup (row_to_poly ts (vec n (f1 + f2))) t = lookup (row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)) t›*)\n by (simp add: in_keys_iff (*‹((?x::?'a) ∈ keys (?s::?'a ⇒₀ ?'b)) = (lookup ?s ?x ≠ (0::?'b))›*))\n qed\nqed\n\nlemma row_to_poly_vec_sum:\n assumes \"distinct ts\" and \"length ts = n\"\n shows \"row_to_poly ts (vec n (λj. ∑i∈I. f i j)) = ((∑i∈I. row_to_poly ts (vec n (f i)))::'t ⇒₀ 'b::comm_monoid_add)\"\nproof (cases \"finite I\")\n (*goals:\n 1. ‹finite I ⟹ row_to_poly ts (vec n (λj. ∑i∈I. f i j)) = (∑i∈I. row_to_poly ts (vec n (f i)))›\n 2. ‹infinite I ⟹ row_to_poly ts (vec n (λj. ∑i∈I. f i j)) = (∑i∈I. row_to_poly ts (vec n (f i)))›*)\n case True (*‹finite (I::'c set)›*)\n thus \"?thesis\"\n (*goal: ‹row_to_poly ts (vec n (λj. ∑i∈I. f i j)) = (∑i∈I. row_to_poly ts (vec n (f i)))›*)\n proof (induct I)\n (*goals:\n 1. ‹row_to_poly ts (vec n (λj. ∑i∈{}. f i j)) = (∑i∈{}. row_to_poly ts (vec n (f i)))›\n 2. ‹⋀x F. ⟦finite F; x ∉ F; row_to_poly ts (vec n (λj. ∑i∈F. f i j)) = (∑i∈F. row_to_poly ts (vec n (f i)))⟧ ⟹ row_to_poly ts (vec n (λj. ∑i∈insert x F. f i j)) = (∑i∈insert x F. row_to_poly ts (vec n (f i)))›*)\n case empty (*no hyothesis introduced yet*)\n thus \"?case\"\n (*goal: ‹row_to_poly ts (vec n (λj. ∑i∈{}. f i j)) = (∑i∈{}. row_to_poly ts (vec n (f i)))›*)\n by (simp add: zero_vec_def[symmetric] (*‹vec ?n (λi. 0) ≡ 0⇩v ?n›*) assms(2)[symmetric] (*‹n = length ts›*))\n next\n (*goal: ‹⋀x F. ⟦finite F; x ∉ F; row_to_poly ts (vec n (λj. ∑i∈F. f i j)) = (∑i∈F. row_to_poly ts (vec n (f i)))⟧ ⟹ row_to_poly ts (vec n (λj. ∑i∈insert x F. f i j)) = (∑i∈insert x F. row_to_poly ts (vec n (f i)))›*)\n case (insert x I) (*‹finite I› ‹x ∉ I› ‹row_to_poly ts (vec n (λj. ∑i∈I. f i j)) = (∑i∈I. row_to_poly ts (vec n (f i)))›*)\n have \"row_to_poly ts (vec n (λj. ∑i∈insert x I. f i j)) = row_to_poly ts (vec n (λj. f x j + (∑i∈I. f i j)))\"\n by (simp add: insert( (*‹finite I›*) 1) insert( (*‹x ∉ I›*) 2))\n also (*calculation: ‹row_to_poly ts (vec n (λj. ∑i∈insert x I. f i j)) = row_to_poly ts (vec n (λj. f x j + (∑i∈I. f i j)))›*) have \"... = row_to_poly ts (vec n (f x + (λj. (∑i∈I. f i j))))\"\n by (simp only: plus_fun_def (*‹?f + ?g = (λx. ?f x + ?g x)›*))\n also (*calculation: ‹row_to_poly ts (vec n (λj. ∑i∈insert x I. f i j)) = row_to_poly ts (vec n (f x + (λj. ∑i∈I. f i j)))›*) from assms (*‹distinct ts› ‹length ts = n›*) have \"... = row_to_poly ts (vec n (f x)) + row_to_poly ts (vec n (λj. (∑i∈I. f i j)))\"\n by (rule row_to_poly_vec_plus (*‹⟦distinct ?ts; length ?ts = ?n⟧ ⟹ row_to_poly ?ts (vec ?n (?f1.0 + ?f2.0)) = row_to_poly ?ts (vec ?n ?f1.0) + row_to_poly ?ts (vec ?n ?f2.0)›*))\n also (*calculation: ‹row_to_poly ts (vec n (λj. ∑i∈insert x I. f i j)) = row_to_poly ts (vec n (f x)) + row_to_poly ts (vec n (λj. ∑i∈I. f i j))›*) have \"... = row_to_poly ts (vec n (f x)) + (∑i∈I. row_to_poly ts (vec n (f i)))\"\n by (simp only: insert( (*‹row_to_poly ts (vec n (λj. ∑i∈I. f i j)) = (∑i∈I. row_to_poly ts (vec n (f i)))›*) 3))\n also (*calculation: ‹row_to_poly ts (vec n (λj. ∑i∈insert x I. f i j)) = row_to_poly ts (vec n (f x)) + (∑i∈I. row_to_poly ts (vec n (f i)))›*) have \"... = (∑i∈insert x I. row_to_poly ts (vec n (f i)))\"\n by (simp add: insert( (*‹finite I›*) 1) insert( (*‹x ∉ I›*) 2))\n finally (*calculation: ‹row_to_poly ts (vec n (λj. ∑i∈insert x I. f i j)) = (∑i∈insert x I. row_to_poly ts (vec n (f i)))›*) show \"?case\"\n (*goal: ‹row_to_poly ts (vec n (λj. ∑i∈insert x I. f i j)) = (∑i∈insert x I. row_to_poly ts (vec n (f i)))›*) .\n qed\nnext\n (*goal: ‹infinite I ⟹ row_to_poly ts (vec n (λj. ∑i∈I. f i j)) = (∑i∈I. row_to_poly ts (vec n (f i)))›*)\n case False (*‹infinite (I::'c::type set)›*)\n thus \"?thesis\"\n (*goal: ‹row_to_poly (ts::'t list) (vec (n::nat) (λj::nat. ∑i::'c∈(I::'c set). (f::'c ⇒ nat ⇒ 'b) i j)) = (∑i::'c∈I. row_to_poly ts (vec n (f i)))›*)\n by (simp add: zero_vec_def[symmetric] (*‹vec ?n (λi. 0) ≡ 0⇩v ?n›*) assms(2)[symmetric] (*‹n = length ts›*))\nqed\n\nlemma row_to_poly_smult:\n assumes \"distinct ts\" and \"dim_vec r = length ts\"\n shows \"row_to_poly ts (c ⋅⇩v r) = c ⋅ (row_to_poly ts r)\"\napply (rule poly_mapping_eqI (*‹(⋀k. lookup ?f k = lookup ?g k) ⟹ ?f = ?g›*))\n(*goal: ‹row_to_poly ts (c ⋅⇩v r) = c ⋅ row_to_poly ts r›*)\nproof (simp only: lookup_map_scale (*‹lookup (?c ⋅ ?p) = (λx. ?c * lookup ?p x)›*))\n (*goal: ‹⋀k. lookup (row_to_poly ts (c ⋅⇩v r)) k = c * lookup (row_to_poly ts r) k›*)\n fix t\n show \"lookup (row_to_poly ts (c ⋅⇩v r)) t = c * lookup (row_to_poly ts r) t\" (is \"lookup ?l t = c * lookup ?r t\")\n proof (cases \"t ∈ set ts\")\n (*goals:\n 1. ‹t ∈ set ts ⟹ lookup (row_to_poly ts (c ⋅⇩v r)) t = c * lookup (row_to_poly ts r) t›\n 2. ‹t ∉ set ts ⟹ lookup (row_to_poly ts (c ⋅⇩v r)) t = c * lookup (row_to_poly ts r) t›*)\n case True (*‹t ∈ set ts›*)\n then obtain j where j: \"j < length ts\" and t: \"t = ts ! j\"\n (*goal: ‹(⋀j. ⟦j < length ts; t = ts ! j⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*))\n from assms(2) (*‹dim_vec (r::'c vec) = length (ts::'b list)›*) have dm: \"dim_vec (c ⋅⇩v r) = length ts\"\n by simp\n from j (*‹(j::nat) < length (ts::'b list)›*) have j': \"j < dim_vec r\"\n by (simp only: assms( (*‹dim_vec r = length ts›*) 2))\n show \"?thesis\"\n (*goal: ‹lookup (row_to_poly ts (c ⋅⇩v r)) t = c * lookup (row_to_poly ts r) t›*)\n by (simp add: t (*‹(t::'b::type) = (ts::'b::type list) ! (j::nat)›*) lookup_row_to_poly[OF assms j] (*‹lookup (row_to_poly (ts::'b::type list) (r::'c::mult_zero vec)) (ts ! (j::nat)) = r $ j›*) lookup_row_to_poly[OF assms(1) dm j] (*‹lookup (row_to_poly (ts::'b::type list) ((c::'c::mult_zero) ⋅⇩v (r::'c::mult_zero vec))) (ts ! (j::nat)) = (c ⋅⇩v r) $ j›*) index_smult_vec(1)[OF j'] (*‹((?a::'c::mult_zero) ⋅⇩v (r::'c::mult_zero vec)) $ (j::nat) = ?a * r $ j›*))\n next\n (*goal: ‹t ∉ set ts ⟹ lookup (row_to_poly ts (c ⋅⇩v r)) t = c * lookup (row_to_poly ts r) t›*)\n case False (*‹t ∉ set ts›*)\n with keys_row_to_poly[of ts \"c ⋅⇩v r\"] (*‹keys (row_to_poly (ts::'b::type list) ((c::'c::mult_zero) ⋅⇩v (r::'c::mult_zero vec))) ⊆ set ts›*) keys_row_to_poly[of ts r] (*‹keys (row_to_poly ts r) ⊆ set ts›*) have \"t ∉ keys ?l\" and \"t ∉ keys ?r\"\n apply -\n (*goals:\n 1. ‹⟦keys (row_to_poly ts (c ⋅⇩v r)) ⊆ set ts; keys (row_to_poly ts r) ⊆ set ts; t ∉ set ts⟧ ⟹ t ∉ keys (row_to_poly ts (c ⋅⇩v r))›\n 2. ‹⟦keys (row_to_poly ts (c ⋅⇩v r)) ⊆ set ts; keys (row_to_poly ts r) ⊆ set ts; t ∉ set ts⟧ ⟹ t ∉ keys (row_to_poly ts r)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹lookup (row_to_poly (ts::'b list) ((c::'c) ⋅⇩v (r::'c vec))) (t::'b) = c * lookup (row_to_poly ts r) t›*)\n by (simp add: in_keys_iff (*‹((?x::?'a::type) ∈ keys (?s::?'a::type ⇒₀ ?'b::zero)) = (lookup ?s ?x ≠ (0::?'b::zero))›*))\n qed\nqed\n\nlemma poly_to_row_Nil [simp]: \"poly_to_row [] p = vec 0 f\"\nproof (-)\n (*goal: ‹poly_to_row [] p = vec 0 f›*)\n have \"dim_vec (poly_to_row [] p) = 0\"\n by (simp add: dim_poly_to_row (*‹dim_vec (poly_to_row ?ts ?p) = length ?ts›*))\n thus \"?thesis\"\n (*goal: ‹poly_to_row [] (p::'c::type ⇒₀ 'b::zero) = vec (0::nat) (f::nat ⇒ 'b::zero)›*)\n by auto\nqed\n\nlemma polys_to_mat_Nil [simp]: \"polys_to_mat ts [] = mat 0 (length ts) f\"\n by (simp add: polys_to_mat_def (*‹polys_to_mat (?ts::?'a list) (?ps::(?'a ⇒₀ ?'b) list) = mat_of_rows (length ?ts) (map (poly_to_row ?ts) ?ps)›*) mat_eq_iff (*‹((?x::?'a mat) = (?y::?'a mat)) = (dim_row ?x = dim_row ?y ∧ dim_col ?x = dim_col ?y ∧ (∀(i::nat) j::nat. i < dim_row ?y ⟶ j < dim_col ?y ⟶ ?x $$ (i, j) = ?y $$ (i, j)))›*))\n\nlemma dim_row_polys_to_mat[simp]: \"dim_row (polys_to_mat ts ps) = length ps\"\n by (simp add: polys_to_mat_def (*‹polys_to_mat ?ts ?ps = mat_of_rows (length ?ts) (map (poly_to_row ?ts) ?ps)›*))\n\nlemma dim_col_polys_to_mat[simp]: \"dim_col (polys_to_mat ts ps) = length ts\"\n by (simp add: polys_to_mat_def (*‹polys_to_mat (?ts::?'a list) (?ps::(?'a ⇒₀ ?'b) list) = mat_of_rows (length ?ts) (map (poly_to_row ?ts) ?ps)›*))\n\nlemma polys_to_mat_index:\n assumes \"i < length ps\" and \"j < length ts\"\n shows \"(polys_to_mat ts ps) $$ (i, j) = lookup (ps ! i) (ts ! j)\"\n apply (simp add: polys_to_mat_def (*‹polys_to_mat ?ts ?ps = mat_of_rows (length ?ts) (map (poly_to_row ?ts) ?ps)›*) index_mat(1)[OF assms] (*‹mat (length ps) (length ts) ?f $$ (i, j) = ?f (i, j)›*) mat_of_rows_def (*‹mat_of_rows ?n ?rs = mat (length ?rs) ?n (λ(i, j). ?rs ! i $ j)›*) nth_map[OF assms(1)] (*‹map ?f ps ! i = ?f (ps ! i)›*))\n (*goal: ‹polys_to_mat ts ps $$ (i, j) = lookup (ps ! i) (ts ! j)›*)\n apply (rule poly_to_row_index (*‹?i < length ?ts ⟹ poly_to_row ?ts ?p $ ?i = lookup ?p (?ts ! ?i)›*))\n (*goal: ‹poly_to_row ts (ps ! i) $ j = lookup (ps ! i) (ts ! j)›*)\n by fact\n\nlemma row_polys_to_mat:\n assumes \"i < length ps\"\n shows \"row (polys_to_mat ts ps) i = poly_to_row ts (ps ! i)\"\nproof (-)\n (*goal: ‹row (polys_to_mat ts ps) i = poly_to_row ts (ps ! i)›*)\n have \"row (polys_to_mat ts ps) i = (map (poly_to_row ts) ps) ! i\"\n unfolding polys_to_mat_def\n (*goal: ‹row (mat_of_rows (length ts) (map (poly_to_row ts) ps)) i = map (poly_to_row ts) ps ! i›*)\n proof (rule mat_of_rows_row (*‹⟦?i < length ?vs; ?vs ! ?i ∈ carrier_vec ?n⟧ ⟹ row (mat_of_rows ?n ?vs) ?i = ?vs ! ?i›*))\n (*goals:\n 1. ‹i < length (map (poly_to_row ts) ps)›\n 2. ‹map (poly_to_row ts) ps ! i ∈ carrier_vec (length ts)›*)\n from assms (*‹(i::nat) < length (ps::('b::type ⇒₀ 'c::zero) list)›*) show \"i < length (map (poly_to_row ts) ps)\"\n by simp\n next\n (*goal: ‹map (poly_to_row ts) ps ! i ∈ carrier_vec (length ts)›*)\n show \"map (poly_to_row ts) ps ! i ∈ carrier_vec (length ts)\"\n unfolding nth_map[OF assms]\n (*goal: ‹poly_to_row ts (ps ! i) ∈ carrier_vec (length ts)›*)\n apply (rule carrier_vecI (*‹dim_vec ?v = ?n ⟹ ?v ∈ carrier_vec ?n›*))\n (*goal: ‹poly_to_row ts (ps ! i) ∈ carrier_vec (length ts)›*)\n by (fact dim_poly_to_row (*‹dim_vec (poly_to_row ?ts ?p) = length ?ts›*))\n qed\n also (*calculation: ‹row (polys_to_mat ts ps) i = map (poly_to_row ts) ps ! i›*) from assms (*‹i < length ps›*) have \"... = poly_to_row ts (ps ! i)\"\n by (rule nth_map (*‹?n < length ?xs ⟹ map ?f ?xs ! ?n = ?f (?xs ! ?n)›*))\n finally (*calculation: ‹row (polys_to_mat ts ps) i = poly_to_row ts (ps ! i)›*) show \"?thesis\"\n (*goal: ‹row (polys_to_mat (ts::'b::type list) (ps::('b::type ⇒₀ 'c::zero) list)) (i::nat) = poly_to_row ts (ps ! i)›*) .\nqed\n\nlemma col_polys_to_mat:\n assumes \"j < length ts\"\n shows \"col (polys_to_mat ts ps) j = vec_of_list (map (λp. lookup p (ts ! j)) ps)\"\n apply (simp add: vec_of_list_alt (*‹vec_of_list ?xs = vec (length ?xs) ((!) ?xs)›*) col_def (*‹col ?A ?j = vec (dim_row ?A) (λi. ?A $$ (i, ?j))›*))\n (*goal: ‹col (polys_to_mat (ts::'b list) (ps::('b ⇒₀ 'c) list)) (j::nat) = vec_of_list (map (λp::'b ⇒₀ 'c. lookup p (ts ! j)) ps)›*)\n apply (rule vec_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ vec ?n ?f = vec ?m ?g›*))\n (*goals:\n 1. ‹length ps = length ps›\n 2. ‹⋀i. i < length ps ⟹ polys_to_mat ts ps $$ (i, j) = map (λp. lookup p (ts ! j)) ps ! i›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply (simp add: polys_to_mat_index (*‹⟦?i < length ?ps; ?j < length ?ts⟧ ⟹ polys_to_mat ?ts ?ps $$ (?i, ?j) = lookup (?ps ! ?i) (?ts ! ?j)›*) assms (*‹j < length ts›*))\n (*proven 2 subgoals*) .\n\nlemma length_mat_to_polys[simp]: \"length (mat_to_polys ts A) = dim_row A\"\n by (simp add: mat_to_polys_def (*‹mat_to_polys ?ts ?A = map (row_to_poly ?ts) (rows ?A)›*) mat_to_list_def (*‹mat_to_list ?A = map (λi. map (λj. ?A $$ (i, j)) [0..<dim_col ?A]) [0..<dim_row ?A]›*))\n\nlemma mat_to_polys_nth:\n assumes \"i < dim_row A\"\n shows \"(mat_to_polys ts A) ! i = row_to_poly ts (row A i)\"\nproof (-)\n (*goal: ‹mat_to_polys (ts::'c::type list) (A::'b::zero mat) ! (i::nat) = row_to_poly ts (row A i)›*)\n from assms (*‹i < dim_row A›*) have \"i < length (rows A)\"\n by (simp only: length_rows (*‹length (rows ?A) = dim_row ?A›*))\n thus \"?thesis\"\n (*goal: ‹mat_to_polys (ts::'c list) (A::'b mat) ! (i::nat) = row_to_poly ts (row A i)›*)\n by (simp add: mat_to_polys_def (*‹mat_to_polys (?ts::?'a list) (?A::?'b mat) = map (row_to_poly ?ts) (rows ?A)›*))\nqed\n\nlemma Keys_mat_to_polys: \"Keys (set (mat_to_polys ts A)) ⊆ set ts\"\nproof (standard)\n (*goal: ‹⋀x::'b. x ∈ Keys (set (mat_to_polys (ts::'b list) (A::'c mat))) ⟹ x ∈ set ts›*)\n fix t\n assume \"t ∈ Keys (set (mat_to_polys ts A))\" (*‹(t::'b) ∈ Keys (set (mat_to_polys (ts::'b list) (A::'c mat)))›*)\n then obtain p where \"p ∈ set (mat_to_polys ts A)\" and t: \"t ∈ keys p\"\n (*goal: ‹(⋀p. ⟦p ∈ set (mat_to_polys ts A); t ∈ keys p⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule in_KeysE (*‹⟦?s ∈ Keys ?F; ⋀f. ⟦?s ∈ keys f; f ∈ ?F⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n from this(1) (*‹p ∈ set (mat_to_polys ts A)›*) obtain i where \"i < length (mat_to_polys ts A)\" and p: \"p = (mat_to_polys ts A) ! i\"\n (*goal: ‹(⋀i. ⟦i < length (mat_to_polys ts A); p = mat_to_polys ts A ! i⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*))\n from this(1) (*‹i < length (mat_to_polys ts A)›*) have \"i < dim_row A\"\n by simp\n with p (*‹(p::'b ⇒₀ 'c) = mat_to_polys (ts::'b list) (A::'c mat) ! (i::nat)›*) have \"p = row_to_poly ts (row A i)\"\n by (simp only: mat_to_polys_nth (*‹?i < dim_row ?A ⟹ mat_to_polys ?ts ?A ! ?i = row_to_poly ?ts (row ?A ?i)›*))\n with t (*‹(t::'b) ∈ keys (p::'b ⇒₀ 'c)›*) have \"t ∈ keys (row_to_poly ts (row A i))\"\n by simp\n also (*calculation: ‹t ∈ keys (row_to_poly ts (row A i))›*) have \"... ⊆ set ts\"\n by (fact keys_row_to_poly (*‹keys (row_to_poly ?ts ?r) ⊆ set ?ts›*))\n finally (*calculation: ‹t ∈ set ts›*) show \"t ∈ set ts\" .\nqed\n\nlemma polys_to_mat_to_polys:\n assumes \"Keys (set ps) ⊆ set ts\"\n shows \"mat_to_polys ts (polys_to_mat ts ps) = (ps::('t ⇒₀ 'b::semiring_1) list)\"\n unfolding mat_to_polys_def mat_to_list_def\n (*goal: ‹map (row_to_poly ts) (rows (polys_to_mat ts ps)) = ps›*)\n proof (rule nth_equalityI (*‹⟦length ?xs = length ?ys; ⋀i. i < length ?xs ⟹ ?xs ! i = ?ys ! i⟧ ⟹ ?xs = ?ys›*), simp_all)\n (*goal: ‹⋀i::nat. i < length (ps::('t ⇒₀ 'b) list) ⟹ row_to_poly (ts::'t list) (row (polys_to_mat ts ps) i) = ps ! i›*)\n fix i\n assume \"i < length ps\" (*‹(i::nat) < length (ps::('t ⇒₀ 'b) list)›*)\n have \"*\": \"keys (ps ! i) ⊆ set ts\"\n using ‹i < length ps› (*‹i < length ps›*) assms (*‹Keys (set ps) ⊆ set ts›*) keys_subset_Keys (*‹?f ∈ ?F ⟹ keys ?f ⊆ Keys ?F›*) nth_mem (*‹?n < length ?xs ⟹ ?xs ! ?n ∈ set ?xs›*) by blast\n show \"row_to_poly ts (row (polys_to_mat ts ps) i) = ps ! i\"\n by (simp only: row_polys_to_mat[OF ‹i < length ps›] (*‹row (polys_to_mat ?ts ps) i = poly_to_row ?ts (ps ! i)›*) poly_to_row_to_poly[OF *] (*‹row_to_poly ts (poly_to_row ts (ps ! i)) = ps ! i›*))\n qed\n\nlemma mat_to_polys_to_mat:\n assumes \"distinct ts\" and \"length ts = dim_col A\"\n shows \"(polys_to_mat ts (mat_to_polys ts A)) = A\"\nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row A; j < dim_col A⟧ ⟹ polys_to_mat ts (mat_to_polys ts A) $$ (i, j) = A $$ (i, j)›\n 2. ‹dim_row (polys_to_mat ts (mat_to_polys ts A)) = dim_row A›\n 3. ‹dim_col (polys_to_mat ts (mat_to_polys ts A)) = dim_col A›*)\n fix i and j\n assume i: \"i < dim_row A\" and j: \"j < dim_col A\" (*‹(i::nat) < dim_row (A::'c mat)› ‹(j::nat) < dim_col (A::'c mat)›*)\n hence i': \"i < length (mat_to_polys ts A)\" and j': \"j < length ts\"\n apply -\n (*goals:\n 1. ‹⟦i < dim_row A; j < dim_col A⟧ ⟹ i < length (mat_to_polys ts A)›\n 2. ‹⟦i < dim_row A; j < dim_col A⟧ ⟹ j < length ts›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply (simp only: assms( (*‹length ts = dim_col A›*) 2))\n (*proven 2 subgoals*) .\n have r: \"dim_vec (row A i) = length ts\"\n by (simp add: assms( (*‹length ts = dim_col A›*) 2))\n show \"polys_to_mat ts (mat_to_polys ts A) $$ (i, j) = A $$ (i, j)\"\n by (simp only: polys_to_mat_index[OF i' j'] (*‹polys_to_mat (ts::'b list) (mat_to_polys ts (A::'c mat)) $$ (i::nat, j::nat) = lookup (mat_to_polys ts A ! i) (ts ! j)›*) mat_to_polys_nth[OF ‹i < dim_row A›] (*‹mat_to_polys (?ts::?'c list) (A::'c mat) ! (i::nat) = row_to_poly ?ts (row A i)›*) lookup_row_to_poly[OF assms(1) r j'] (*‹lookup (row_to_poly (ts::'b list) (row (A::'c mat) (i::nat))) (ts ! (j::nat)) = row A i $ j›*) index_row(1)[OF i j] (*‹row (A::'c mat) (i::nat) $ (j::nat) = A $$ (i, j)›*))\nqed (simp_all add: assms)\n(*solves the remaining goals:\n 1. ‹dim_row (polys_to_mat ts (mat_to_polys ts A)) = dim_row A›\n 2. ‹dim_col (polys_to_mat ts (mat_to_polys ts A)) = dim_col A›*)\n\nsubsection ‹Properties of Macaulay Matrices›\n\nlemma row_to_poly_vec_times:\n assumes \"distinct ts\" and \"length ts = dim_col A\"\n shows \"row_to_poly ts (v ⇩v* A) = ((∑i=0..<dim_row A. (v $ i) ⋅ (row_to_poly ts (row A i)))::'t ⇒₀ 'b::comm_semiring_0)\"\nproof (simp add: mult_vec_mat_def (*‹?v ⇩v* ?A ≡ vec (dim_col ?A) (λj. ?v ∙ col ?A j)›*) scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $ i * ?w $ i›*) row_to_poly_vec_sum[OF assms] (*‹row_to_poly ts (vec (dim_col A) (λj. ∑i∈?I. ?f i j)) = (∑i∈?I. row_to_poly ts (vec (dim_col A) (?f i)))›*), rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*), rule)\n (*goal: ‹⋀x. x ∈ {0..<dim_row A} ⟹ row_to_poly ts (vec (dim_col A) (λj. v $ x * col A j $ x)) = v $ x ⋅ row_to_poly ts (row A x)›*)\n fix i\n assume \"i ∈ {0..<dim_row A}\" (*‹(i::nat) ∈ {0::nat..<dim_row (A::'b mat)}›*)\n hence \"i < dim_row A\"\n by simp\n have \"dim_vec (row A i) = length ts\"\n by (simp add: assms( (*‹length ts = dim_col A›*) 2))\n have \"*\": \"vec (dim_col A) (λj. col A j $ i) = vec (dim_col A) (λj. A $$ (i, j))\"\n apply (rule vec_cong (*‹⟦?n = ?m; ⋀i. i < ?m ⟹ ?f i = ?g i⟧ ⟹ vec ?n ?f = vec ?m ?g›*))\n (*goals:\n 1. ‹dim_col A = dim_col A›\n 2. ‹⋀ia. ia < dim_col A ⟹ col A ia $ i = A $$ (i, ia)›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply (simp add: ‹i < dim_row A›)\n (*proven 2 subgoals*) .\n have \"vec (dim_col A) (λj. v $ i * col A j $ i) = v $ i ⋅⇩v vec (dim_col A) (λj. col A j $ i)\"\n by (simp only: vec_scalar_mult_fun (*‹vec ?n (λx. ?c * ?f x) = ?c ⋅⇩v vec ?n ?f›*))\n also (*calculation: ‹vec (dim_col A) (λj. v $ i * col A j $ i) = v $ i ⋅⇩v vec (dim_col A) (λj. col A j $ i)›*) have \"... = v $ i ⋅⇩v (row A i)\"\n by (simp only: * (*‹vec (dim_col A) (λj. col A j $ i) = vec (dim_col A) (λj. A $$ (i, j))›*) row_def[symmetric] (*‹vec (dim_col ?A) (λj. ?A $$ (?i, j)) = row ?A ?i›*))\n finally (*calculation: ‹vec (dim_col A) (λj. v $ i * col A j $ i) = v $ i ⋅⇩v row A i›*) show \"row_to_poly ts (vec (dim_col A) (λj. v $ i * col A j $ i)) =\n (v $ i) ⋅ (row_to_poly ts (row A i))\"\n by (simp add: row_to_poly_smult[OF assms(1) ‹dim_vec (row A i) = length ts›] (*‹row_to_poly ts (?c ⋅⇩v row A i) = ?c ⋅ row_to_poly ts (row A i)›*))\nqed\n\nlemma vec_times_polys_to_mat:\n assumes \"Keys (set ps) ⊆ set ts\" and \"v ∈ carrier_vec (length ps)\"\n shows \"row_to_poly ts (v ⇩v* (polys_to_mat ts ps)) = (∑(c, p)←zip (list_of_vec v) ps. c ⋅ p)\"\n (is \"?l = ?r\")\nproof (-)\n (*goal: ‹row_to_poly ts (v ⇩v* polys_to_mat ts ps) = sum_list (map2 (⋅) (list_of_vec v) ps)›*)\n from assms (*‹Keys (set (ps::('b ⇒₀ 'c) list)) ⊆ set (ts::'b list)› ‹v ∈ carrier_vec (length ps)›*) have \"*\": \"dim_vec v = length ps\"\n by (simp only: carrier_dim_vec (*‹(?v ∈ carrier_vec ?n) = (dim_vec ?v = ?n)›*))\n have eq: \"map (λi. v ∙ col (polys_to_mat ts ps) i) [0..<length ts] =\n map (λs. v ∙ (vec_of_list (map (λp. lookup p s) ps))) ts\"\n proof (rule nth_equalityI (*‹⟦length (?xs::?'a list) = length (?ys::?'a list); ⋀i::nat. i < length ?xs ⟹ ?xs ! i = ?ys ! i⟧ ⟹ ?xs = ?ys›*), simp_all)\n (*goal: ‹⋀i. i < length ts ⟹ v ∙ col (polys_to_mat ts ps) i = v ∙ map_vec (λp. lookup p (ts ! i)) (vec_of_list ps)›*)\n fix i\n assume \"i < length ts\" (*‹(i::nat) < length (ts::'b list)›*)\n hence \"col (polys_to_mat ts ps) i = vec_of_list (map (λp. lookup p (ts ! i)) ps)\"\n by (rule col_polys_to_mat (*‹?j < length ?ts ⟹ col (polys_to_mat ?ts ?ps) ?j = vec_of_list (map (λp. lookup p (?ts ! ?j)) ?ps)›*))\n thus \"v ∙ col (polys_to_mat ts ps) i = v ∙ map_vec (λp. lookup p (ts ! i)) (vec_of_list ps)\"\n by simp\n qed\n show \"?thesis\"\n (*goal: ‹row_to_poly ts (v ⇩v* polys_to_mat ts ps) = sum_list (map2 (⋅) (list_of_vec v) ps)›*)\n proof (rule poly_mapping_eqI (*‹(⋀k. lookup ?f k = lookup ?g k) ⟹ ?f = ?g›*), simp add: mult_vec_mat_def (*‹?v ⇩v* ?A ≡ vec (dim_col ?A) (λj. ?v ∙ col ?A j)›*) row_to_poly_def (*‹row_to_poly ?ts ?r = list_to_poly ?ts (list_of_vec ?r)›*) lookup_list_to_poly (*‹lookup (list_to_poly ?ts ?cs) = list_to_fun ?ts ?cs›*) eq (*‹map (λi. v ∙ col (polys_to_mat ts ps) i) [0..<length ts] = map (λs. v ∙ vec_of_list (map (λp. lookup p s) ps)) ts›*) list_to_fun_def (*‹list_to_fun ?ts ?cs ?t = (case map_of (zip ?ts ?cs) ?t of None ⇒ 0 | Some c ⇒ c)›*) map_of_zip_map (*‹map_of (zip ?xs (map ?f ?xs)) = (λx. if x ∈ set ?xs then Some (?f x) else None)›*) lookup_sum_list (*‹lookup (sum_list ?ps) ?a = (∑p←?ps. lookup p ?a)›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*), intro conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*))\n (*goals:\n 1. ‹⋀k. k ∈ set ts ⟹ v ∙ map_vec (λp. lookup p k) (vec_of_list ps) = (∑x←zip (list_of_vec v) ps. lookup (case x of (x, xa) ⇒ x ⋅ xa) k)›\n 2. ‹⋀k. k ∉ set ts ⟹ (∑x←zip (list_of_vec v) ps. lookup (case x of (x, xa) ⇒ x ⋅ xa) k) = 0›*)\n fix t\n assume \"t ∈ set ts\" (*‹(t::'b) ∈ set (ts::'b list)›*)\n have \"v ∙ vec_of_list (map (λp. lookup p t) ps) =\n (∑(c, p)←zip (list_of_vec v) ps. lookup (c ⋅ p) t)\"\n proof (simp add: scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $ i * ?w $ i›*) vec_of_list_index (*‹vec_of_list ?xs $ ?j = ?xs ! ?j›*))\n (*goal: ‹(∑i::nat = 0::nat..<length ps. (v::'c vec) $ i * lookup ((ps::('b ⇒₀ 'c) list) ! i) (t::'b)) = sum_list (map2 (λ(c::'c) p::'b ⇒₀ 'c. c * lookup p t) (list_of_vec v) ps)›*)\n have \"(∑i = 0..<length ps. v $ i * lookup (ps ! i) t) =\n (∑i = 0..<length ps. (list_of_vec v) ! i * lookup (ps ! i) t)\"\n apply (rule sum.cong (*‹⟦(?A::?'b set) = (?B::?'b set); ⋀x::?'b. x ∈ ?B ⟹ (?g::?'b ⇒ ?'a) x = (?h::?'b ⇒ ?'a) x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0..<length ps} = {0..<length ps}›\n 2. ‹⋀x. x ∈ {0..<length ps} ⟹ v $ x * lookup (ps ! x) t = list_of_vec v ! x * lookup (ps ! x) t›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply (simp add: * (*‹dim_vec (v::'c vec) = length (ps::('b ⇒₀ 'c) list)›*))\n (*proven 2 subgoals*) .\n also (*calculation: ‹(∑i = 0..<length ps. v $ i * lookup (ps ! i) t) = (∑i = 0..<length ps. list_of_vec v ! i * lookup (ps ! i) t)›*) have \"... = (∑(c, p)←zip (list_of_vec v) ps. c * lookup p t)\"\n apply (simp only: sum_set_upt_eq_sum_list (*‹sum ?f {?m..<?n} = sum_list (map ?f [?m..<?n])›*))\n (*goal: ‹(∑i = 0..<length ps. list_of_vec v ! i * lookup (ps ! i) t) = sum_list (map2 (λc p. c * lookup p t) (list_of_vec v) ps)›*)\n apply (rule sum_list_upt_zip (*‹length ?xs = length ?ys ⟹ (∑i←[0..<length ?ys]. ?f (?xs ! i) (?ys ! i)) = sum_list (map2 ?f ?xs ?ys)›*))\n (*goal: ‹(∑i←[0..<length ps]. list_of_vec v ! i * lookup (ps ! i) t) = sum_list (map2 (λc p. c * lookup p t) (list_of_vec v) ps)›*)\n by (simp only: length_list_of_vec (*‹length (list_of_vec ?v) = dim_vec ?v›*) * (*‹dim_vec v = length ps›*))\n finally (*calculation: ‹(∑i::nat = 0::nat..<length ps. (v::'c vec) $ i * lookup ((ps::('b ⇒₀ 'c) list) ! i) (t::'b)) = sum_list (map2 (λ(c::'c) p::'b ⇒₀ 'c. c * lookup p t) (list_of_vec v) ps)›*) show \"(∑i = 0..<length ps. v $ i * lookup (ps ! i) t) =\n (∑(c, p)←zip (list_of_vec v) ps. c * lookup p t)\" .\n qed\n thus \"v ∙ map_vec (λp. lookup p t) (vec_of_list ps) =\n (∑x←zip (list_of_vec v) ps. lookup (case x of (c, x) ⇒ c ⋅ x) t)\"\n by (metis (mono_tags, lifting) case_prod_conv (*‹(case (?a::?'b::type, ?b::?'c::type) of (c::?'b::type, d::?'c::type) ⇒ (?f::?'b::type ⇒ ?'c::type ⇒ ?'a::type) c d) = ?f ?a ?b›*) cond_case_prod_eta (*‹(⋀(x::?'a::type) y::?'b::type. (?f::?'a::type ⇒ ?'b::type ⇒ ?'c::type) x y = (?g::?'a::type × ?'b::type ⇒ ?'c::type) (x, y)) ⟹ (λ(x::?'a::type, y::?'b::type). ?f x y) = ?g›*) vec_of_list_map (*‹vec_of_list (map (?f::?'b::type ⇒ ?'a::type) (?xs::?'b::type list)) = map_vec ?f (vec_of_list ?xs)›*))\n next\n (*goal: ‹⋀k::'b. k ∉ set (ts::'b list) ⟹ (∑x::'c × ('b ⇒₀ 'c)←zip (list_of_vec (v::'c vec)) (ps::('b ⇒₀ 'c) list). lookup (case x of (x::'c, xa::'b ⇒₀ 'c) ⇒ x ⋅ xa) k) = (0::'c)›*)\n fix t\n assume \"t ∉ set ts\" (*‹(t::'b) ∉ set (ts::'b list)›*)\n with assms(1) (*‹Keys (set ps) ⊆ set ts›*) have \"t ∉ Keys (set ps)\"\n by auto\n have \"(∑(c, p)←zip (list_of_vec v) ps. lookup (c ⋅ p) t) = 0\"\n apply (rule sum_list_zeroI (*‹set ?xs ⊆ {0} ⟹ sum_list ?xs = 0›*))\n (*goal: ‹sum_list (map2 (λc p. lookup (c ⋅ p) t) (list_of_vec v) ps) = 0›*)\n apply rule\n (*goal: ‹set (map2 (λc p. lookup (c ⋅ p) t) (list_of_vec v) ps) ⊆ {0}›*)\n proof (simp)\n (*goal: ‹⋀x::'c. x ∈ (λx::'c × ('b ⇒₀ 'c). case x of (c::'c, p::'b ⇒₀ 'c) ⇒ c * lookup p (t::'b)) ` set (zip (list_of_vec (v::'c vec)) (ps::('b ⇒₀ 'c) list)) ⟹ x = (0::'c)›*)\n fix x\n assume \"x ∈ (λ(c, p). c * lookup p t) ` set (zip (list_of_vec v) ps)\" (*‹(x::'c) ∈ (λ(c::'c, p::'b ⇒₀ 'c). c * lookup p (t::'b)) ` set (zip (list_of_vec (v::'c vec)) (ps::('b ⇒₀ 'c) list))›*)\n then obtain c and p where cp: \"(c, p) ∈ set (zip (list_of_vec v) ps)\" and x: \"x = c * lookup p t\"\n (*goal: ‹(⋀c p. ⟦(c, p) ∈ set (zip (list_of_vec v) ps); x = c * lookup p t⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n from cp (*‹(c, p) ∈ set (zip (list_of_vec v) ps)›*) have \"p ∈ set ps\"\n by (rule set_zip_rightD (*‹(?x, ?y) ∈ set (zip ?xs ?ys) ⟹ ?y ∈ set ?ys›*))\n with ‹t ∉ Keys (set ps)› (*‹(t::'b) ∉ Keys (set (ps::('b ⇒₀ 'c) list))›*) have \"t ∉ keys p\"\n by (auto intro: in_KeysI (*‹⟦(?s::?'a) ∈ keys (?f::?'a ⇒₀ ?'b); ?f ∈ (?F::(?'a ⇒₀ ?'b) set)⟧ ⟹ ?s ∈ Keys ?F›*))\n thus \"x = 0\"\n by (simp add: x (*‹x = c * lookup p t›*) in_keys_iff (*‹(?x ∈ keys ?s) = (lookup ?s ?x ≠ 0)›*))\n qed\n thus \"(∑x←zip (list_of_vec v) ps. lookup (case x of (c, x) ⇒ c ⋅ x) t) = 0\"\n by (metis (mono_tags, lifting) case_prod_conv (*‹(case (?a::?'b, ?b::?'c) of (c::?'b, d::?'c) ⇒ (?f::?'b ⇒ ?'c ⇒ ?'a) c d) = ?f ?a ?b›*) cond_case_prod_eta (*‹(⋀(x::?'a) y::?'b. (?f::?'a ⇒ ?'b ⇒ ?'c) x y = (?g::?'a × ?'b ⇒ ?'c) (x, y)) ⟹ (λ(x::?'a, y::?'b). ?f x y) = ?g›*))\n qed\nqed\n\nlemma row_space_subset_phull:\n assumes \"Keys (set ps) ⊆ set ts\"\n shows \"row_to_poly ts ` row_space (polys_to_mat ts ps) ⊆ phull (set ps)\"\n (is \"?r ⊆ ?h\")\nproof (standard)\n (*goal: ‹⋀x. x ∈ row_to_poly ts ` row_space (polys_to_mat ts ps) ⟹ x ∈ phull (set ps)›*)\n fix q\n assume \"q ∈ ?r\" (*‹(q::'b ⇒₀ 'c) ∈ row_to_poly (ts::'b list) ` row_space (polys_to_mat ts (ps::('b ⇒₀ 'c) list))›*)\n then obtain x where x1: \"x ∈ row_space (polys_to_mat ts ps)\" and q1: \"q = row_to_poly ts x\"\n (*goal: ‹(⋀x. ⟦x ∈ row_space (polys_to_mat ts ps); q = row_to_poly ts x⟧ ⟹ thesis) ⟹ thesis›*)\n by standard\n from x1 (*‹x ∈ row_space (polys_to_mat ts ps)›*) obtain v where v: \"v ∈ carrier_vec (dim_row (polys_to_mat ts ps))\" and x: \"x = v ⇩v* polys_to_mat ts ps\"\n (*goal: ‹(⋀v. ⟦v ∈ carrier_vec (dim_row (polys_to_mat ts ps)); x = v ⇩v* polys_to_mat ts ps⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule row_spaceE (*‹⟦(?x::?'a vec) ∈ row_space (?A::?'a mat); ⋀v::?'a vec. ⟦v ∈ carrier_vec (dim_row ?A); ?x = v ⇩v* ?A⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*))\n from v (*‹v ∈ carrier_vec (dim_row (polys_to_mat ts ps))›*) have \"v ∈ carrier_vec (length ps)\"\n by (simp only: dim_row_polys_to_mat (*‹dim_row (polys_to_mat ?ts ?ps) = length ?ps›*))\n thm vec_times_polys_to_mat\n with x (*‹(x::'c vec) = (v::'c vec) ⇩v* polys_to_mat (ts::'b list) (ps::('b ⇒₀ 'c) list)›*) q1 (*‹(q::'b ⇒₀ 'c) = row_to_poly (ts::'b list) (x::'c vec)›*) have q: \"q = (∑(c, p)←zip (list_of_vec v) ps. c ⋅ p)\"\n by (simp add: vec_times_polys_to_mat[OF assms] (*‹?v ∈ carrier_vec (length ps) ⟹ row_to_poly ts (?v ⇩v* polys_to_mat ts ps) = sum_list (map2 (⋅) (list_of_vec ?v) ps)›*))\n show \"q ∈ ?h\"\n unfolding q\n (*goal: ‹sum_list (map2 (⋅) (list_of_vec (v::'c vec)) (ps::('b ⇒₀ 'c) list)) ∈ phull (set ps)›*)\n by (rule phull.span_listI (*‹sum_list (map2 (⋅) (?qs::?'a list) (?bs::(?'b ⇒₀ ?'a) list)) ∈ phull (set ?bs)›*))\nqed\n\nlemma phull_subset_row_space:\n assumes \"Keys (set ps) ⊆ set ts\"\n shows \"phull (set ps) ⊆ row_to_poly ts ` row_space (polys_to_mat ts ps)\"\n (is \"?h ⊆ ?r\")\nproof (standard)\n (*goal: ‹⋀x. x ∈ phull (set ps) ⟹ x ∈ row_to_poly ts ` row_space (polys_to_mat ts ps)›*)\n fix q\n assume \"q ∈ ?h\" (*‹(q::'b ⇒₀ 'c) ∈ phull (set (ps::('b ⇒₀ 'c) list))›*)\n then obtain cs where l: \"length cs = length ps\" and q: \"q = (∑(c, p)←zip cs ps. c ⋅ p)\"\n (*goal: ‹(⋀cs. ⟦length cs = length ps; q = sum_list (map2 (⋅) cs ps)⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule phull.span_listE (*‹⟦?p ∈ phull (set ?bs); ⋀qs. ⟦length qs = length ?bs; ?p = sum_list (map2 (⋅) qs ?bs)⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n let ?v = \"vec_of_list cs\"\n from l (*‹length cs = length ps›*) have \"*\": \"?v ∈ carrier_vec (length ps)\"\n by (simp only: carrier_dim_vec (*‹(?v ∈ carrier_vec ?n) = (dim_vec ?v = ?n)›*) dim_vec_of_list (*‹dim_vec (vec_of_list ?as) = length ?as›*))\n let ?q = \"?v ⇩v* polys_to_mat ts ps\"\n show \"q ∈ ?r\"\n proof (standard)\n (*goals:\n 1. ‹q = row_to_poly ts ?x›\n 2. ‹?x ∈ row_space (polys_to_mat ts ps)›*)\n show \"q = row_to_poly ts ?q\"\n by (simp add: vec_times_polys_to_mat[OF assms *] (*‹row_to_poly ts (vec_of_list cs ⇩v* polys_to_mat ts ps) = sum_list (map2 (⋅) (list_of_vec (vec_of_list cs)) ps)›*) q (*‹q = sum_list (map2 (⋅) cs ps)›*) list_vec (*‹list_of_vec (vec_of_list ?xs) = ?xs›*))\n next\n (*goal: ‹vec_of_list (cs::'c list) ⇩v* polys_to_mat (ts::'b list) (ps::('b ⇒₀ 'c) list) ∈ row_space (polys_to_mat ts ps)›*)\n show \"?q ∈ row_space (polys_to_mat ts ps)\"\n apply (rule row_spaceI (*‹(?x::?'a::semiring_0 vec) = (?v::?'a::semiring_0 vec) ⇩v* (?A::?'a::semiring_0 mat) ⟹ ?x ∈ row_space ?A›*))\n (*goal: ‹vec_of_list cs ⇩v* polys_to_mat ts ps ∈ row_space (polys_to_mat ts ps)›*)\n by rule\n qed\nqed\n\nlemma row_space_eq_phull:\n assumes \"Keys (set ps) ⊆ set ts\"\n shows \"row_to_poly ts ` row_space (polys_to_mat ts ps) = phull (set ps)\"\n apply rule\n (*goals:\n 1. ‹row_to_poly ts ` row_space (polys_to_mat ts ps) ⊆ phull (set ps)›\n 2. ‹phull (set ps) ⊆ row_to_poly ts ` row_space (polys_to_mat ts ps)›\n discuss goal 1*)\n apply (rule row_space_subset_phull (*‹Keys (set ?ps) ⊆ set ?ts ⟹ row_to_poly ?ts ` row_space (polys_to_mat ?ts ?ps) ⊆ phull (set ?ps)›*))\n (*top goal: ‹row_to_poly ts ` row_space (polys_to_mat ts ps) ⊆ phull (set ps)› and 1 goal remains*)\n apply fact\n (*discuss goal 2*)\n apply (rule phull_subset_row_space (*‹Keys (set ?ps) ⊆ set ?ts ⟹ phull (set ?ps) ⊆ row_to_poly ?ts ` row_space (polys_to_mat ?ts ?ps)›*))\n (*goal: ‹phull (set ps) ⊆ row_to_poly ts ` row_space (polys_to_mat ts ps)›*)\n apply fact\n (*proven 2 subgoals*) .\n\nlemma row_space_row_echelon_eq_phull:\n assumes \"Keys (set ps) ⊆ set ts\"\n shows \"row_to_poly ts ` row_space (row_echelon (polys_to_mat ts ps)) = phull (set ps)\"\n by (simp add: row_space_eq_phull[OF assms] (*‹row_to_poly (ts::'b::type list) ` row_space (polys_to_mat ts (ps::('b::type ⇒₀ 'c::field) list)) = phull (set ps)›*))\n\nlemma phull_row_echelon:\n assumes \"Keys (set ps) ⊆ set ts\" and \"distinct ts\"\n shows \"phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = phull (set ps)\"\nproof (-)\n (*goal: ‹phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = phull (set ps)›*)\n have len_ts: \"length ts = dim_col (row_echelon (polys_to_mat ts ps))\"\n by simp\n have \"*\": \"Keys (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) ⊆ set ts\"\n by (fact Keys_mat_to_polys (*‹Keys (set (mat_to_polys ?ts ?A)) ⊆ set ?ts›*))\n show \"?thesis\"\n (*goal: ‹phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = phull (set ps)›*)\n apply (simp only: row_space_eq_phull[OF *, symmetric] (*‹phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = row_to_poly ts ` row_space (polys_to_mat ts (mat_to_polys ts (row_echelon (polys_to_mat ts ps))))›*) mat_to_polys_to_mat[OF assms(2) len_ts] (*‹polys_to_mat ts (mat_to_polys ts (row_echelon (polys_to_mat ts ps))) = row_echelon (polys_to_mat ts ps)›*))\n (*goal: ‹phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = phull (set ps)›*)\n apply (rule row_space_row_echelon_eq_phull (*‹Keys (set ?ps) ⊆ set ?ts ⟹ row_to_poly ?ts ` row_space (row_echelon (polys_to_mat ?ts ?ps)) = phull (set ?ps)›*))\n (*goal: ‹row_to_poly (ts::'b::type list) ` row_space (row_echelon (polys_to_mat ts (ps::('b::type ⇒₀ 'c::field) list))) = phull (set ps)›*)\n by fact\nqed\n\nlemma pmdl_row_echelon:\n assumes \"Keys (set ps) ⊆ set ts\" and \"distinct ts\"\n shows \"pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = pmdl (set ps)\"\n (is \"?l = ?r\")\nproof (standard)\n (*goals:\n 1. ‹pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) ⊆ pmdl (set ps)›\n 2. ‹pmdl (set ps) ⊆ pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps))))›*)\n show \"?l ⊆ ?r\"\n apply (rule pmdl.span_subset_spanI (*‹?A ⊆ pmdl ?B ⟹ pmdl ?A ⊆ pmdl ?B›*))\n (*goal: ‹pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) ⊆ pmdl (set ps)›*)\n apply (rule subset_trans (*‹⟦?A ⊆ ?B; ?B ⊆ ?C⟧ ⟹ ?A ⊆ ?C›*))\n (*goals:\n 1. ‹set (mat_to_polys ts (row_echelon (polys_to_mat ts ps))) ⊆ ?B2›\n 2. ‹?B2 ⊆ pmdl (set ps)›\n discuss goal 1*)\n apply (rule phull.span_superset (*‹?S ⊆ phull ?S›*))\n (*discuss goal 2*)\n apply (simp only: phull_row_echelon[OF assms] (*‹phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = phull (set ps)›*) phull_subset_module (*‹phull ?B ⊆ pmdl ?B›*))\n (*proven 2 subgoals*) .\nnext\n (*goal: ‹pmdl (set ps) ⊆ pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps))))›*)\n show \"?r ⊆ ?l\"\n apply (rule pmdl.span_subset_spanI (*‹?A ⊆ pmdl ?B ⟹ pmdl ?A ⊆ pmdl ?B›*))\n (*goal: ‹pmdl (set ps) ⊆ pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps))))›*)\n apply (rule subset_trans (*‹⟦?A ⊆ ?B; ?B ⊆ ?C⟧ ⟹ ?A ⊆ ?C›*))\n (*goals:\n 1. ‹set ps ⊆ ?B2›\n 2. ‹?B2 ⊆ pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps))))›\n discuss goal 1*)\n apply (rule phull.span_superset (*‹?S ⊆ phull ?S›*))\n (*discuss goal 2*)\n apply (simp only: phull_row_echelon[OF assms, symmetric] (*‹phull (set ps) = phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps))))›*) phull_subset_module (*‹phull ?B ⊆ pmdl ?B›*))\n (*proven 2 subgoals*) .\nqed\n\nend (* term_powerprod *)\n\ncontext ordered_term\nbegin\n\nlemma lt_row_to_poly_pivot_fun:\n assumes \"card S = dim_col (A::'b::semiring_1 mat)\" and \"pivot_fun A f (dim_col A)\"\n and \"i < dim_row A\" and \"f i < dim_col A\"\n shows \"lt ((mat_to_polys (pps_to_list S) A) ! i) = (pps_to_list S) ! (f i)\"\nproof (-)\n (*goal: ‹lt (mat_to_polys (pps_to_list S) A ! i) = pps_to_list S ! f i›*)\n let ?ts = \"pps_to_list S\"\n have len_ts: \"length ?ts = dim_col A\"\n by (simp add: length_pps_to_list (*‹length (pps_to_list ?S) = card ?S›*) assms( (*‹card S = dim_col A›*) 1))\n show \"?thesis\"\n (*goal: ‹lt (mat_to_polys (pps_to_list S) A ! i) = pps_to_list S ! f i›*)\n proof (simp add: mat_to_polys_nth[OF assms(3)] (*‹mat_to_polys ?ts A ! i = row_to_poly ?ts (row A i)›*), rule lt_eqI (*‹⟦lookup (?p::'t::type ⇒₀ ?'b::zero) (?v::'t::type) ≠ (0::?'b::zero); ⋀u::'t::type. lookup ?p u ≠ (0::?'b::zero) ⟹ u ≼⇩t ?v⟧ ⟹ lt ?p = ?v›*))\n (*goals:\n 1. ‹lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) ≠ 0›\n 2. ‹⋀u. lookup (row_to_poly (pps_to_list S) (row A i)) u ≠ 0 ⟹ u ≼⇩t pps_to_list S ! f i›*)\n have \"lookup (row_to_poly ?ts (row A i)) (?ts ! f i) = (row A i) $ (f i)\"\n apply (rule lookup_row_to_poly (*‹⟦distinct ?ts; dim_vec ?r = length ?ts; ?i < length ?ts⟧ ⟹ lookup (row_to_poly ?ts ?r) (?ts ! ?i) = ?r $ ?i›*))\n (*goals:\n 1. ‹distinct (pps_to_list S)›\n 2. ‹dim_vec (row A i) = length (pps_to_list S)›\n 3. ‹f i < length (pps_to_list S)›\n discuss goal 1*)\n apply (fact distinct_pps_to_list (*‹distinct (pps_to_list ?S)›*))\n (*discuss goal 2*)\n apply (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*) assms( (*‹f i < dim_col A›*) 4))\n (*discuss goal 3*)\n apply (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*) assms( (*‹f i < dim_col A›*) 4))\n (*proven 3 subgoals*) .\n also (*calculation: ‹lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) = row A i $ f i›*) have \"... = A $$ (i, f i)\"\n using assms(3) (*‹i < dim_row A›*) assms(4) (*‹f i < dim_col A›*) by simp\n also (*calculation: ‹lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) = A $$ (i, f i)›*) have \"... = 1\"\n apply (rule pivot_funD (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?j < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = 0› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = 1› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*))\n (*goals:\n 1. ‹dim_row A = ?nr›\n 2. ‹pivot_fun A f ?nc›\n 3. ‹i < ?nr›\n 4. ‹f i < ?nc›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n finally (*calculation: ‹lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) = 1›*) show \"lookup (row_to_poly ?ts (row A i)) (?ts ! f i) ≠ 0\"\n by simp\n next\n (*goal: ‹⋀u. lookup (row_to_poly (pps_to_list S) (row A i)) u ≠ 0 ⟹ u ≼⇩t pps_to_list S ! f i›*)\n fix u\n assume a: \"lookup (row_to_poly ?ts (row A i)) u ≠ 0\" (*‹lookup (row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) (i::nat))) (u::'t) ≠ (0::'b)›*)\n then obtain j where j: \"j < length ?ts\" and u: \"u = ?ts ! j\"\n (*goal: ‹(⋀j::nat. ⟦j < length (pps_to_list (S::'t set)); (u::'t) = pps_to_list S ! j⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by (rule lookup_row_to_poly_not_zeroE (*‹⟦lookup (row_to_poly ?ts ?r) ?t ≠ 0; ⋀i. ⟦i < length ?ts; ?t = ?ts ! i⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n from j (*‹j < length (pps_to_list S)›*) have \"j < card S\" and \"j < dim_col A\"\n apply -\n (*goals:\n 1. ‹j < length (pps_to_list S) ⟹ j < card S›\n 2. ‹j < length (pps_to_list S) ⟹ j < dim_col A›\n discuss goal 1*)\n apply (simp only: length_pps_to_list (*‹length (pps_to_list ?S) = card ?S›*))\n (*discuss goal 2*)\n apply (simp only: len_ts (*‹length (pps_to_list (S::'t set)) = dim_col (A::'b mat)›*))\n (*proven 2 subgoals*) .\n from a (*‹lookup (row_to_poly (pps_to_list S) (row A i)) u ≠ 0›*) have \"0 ≠ lookup (row_to_poly ?ts (row A i)) (?ts ! j)\"\n by (simp add: u (*‹u = pps_to_list S ! j›*))\n also (*calculation: ‹0 ≠ lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! j)›*) have \"lookup (row_to_poly ?ts (row A i)) (?ts ! j) = (row A i) $ j\"\n apply (rule lookup_row_to_poly (*‹⟦distinct ?ts; dim_vec ?r = length ?ts; ?i < length ?ts⟧ ⟹ lookup (row_to_poly ?ts ?r) (?ts ! ?i) = ?r $ ?i›*))\n (*goals:\n 1. ‹distinct (pps_to_list S)›\n 2. ‹dim_vec (row A i) = length (pps_to_list S)›\n 3. ‹j < length (pps_to_list S)›\n discuss goal 1*)\n apply (fact distinct_pps_to_list (*‹distinct (pps_to_list (?S::'t set))›*))\n (*discuss goal 2*)\n apply (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*))\n (*discuss goal 3*)\n apply fact\n (*proven 3 subgoals*) .\n finally (*calculation: ‹0 ≠ row A i $ j›*) have \"A $$ (i, j) ≠ 0\"\n using assms(3) (*‹(i::nat) < dim_row (A::'b::semiring_1 mat)›*) ‹j < dim_col A› (*‹j < dim_col A›*) by simp\n from \"_\" (*‹PROP ?psi ⟹ PROP ?psi›*) ‹j < card S› (*‹j < card S›*) show \"u ≼⇩t ?ts ! f i\"\n unfolding u\n (*goal: ‹pps_to_list S ! j ≼⇩t pps_to_list S ! f i›*)\n proof (rule pps_to_list_nth_leI (*‹⟦?j ≤ ?i; ?i < card ?S⟧ ⟹ pps_to_list ?S ! ?i ≼⇩t pps_to_list ?S ! ?j›*))\n (*goal: ‹f i ≤ j›*)\n show \"f i ≤ j\"\n proof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹¬ f i ≤ j ⟹ False›*)\n assume \"¬ f i ≤ j\" (*‹¬ (f::nat ⇒ nat) (i::nat) ≤ (j::nat)›*)\n hence \"j < f i\"\n by simp\n have \"A $$ (i, j) = 0\"\n apply (rule pivot_funD (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?j < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = 0› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = 1› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*))\n (*goals:\n 1. ‹dim_row A = ?nr›\n 2. ‹pivot_fun A ?f ?nc›\n 3. ‹i < ?nr›\n 4. ‹j < ?f i›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n with ‹A $$ (i, j) ≠ 0› (*‹A $$ (i, j) ≠ 0›*) show False\n by standard\n qed\n qed\n qed\nqed\n\nlemma lc_row_to_poly_pivot_fun:\n assumes \"card S = dim_col (A::'b::semiring_1 mat)\" and \"pivot_fun A f (dim_col A)\"\n and \"i < dim_row A\" and \"f i < dim_col A\"\n shows \"lc ((mat_to_polys (pps_to_list S) A) ! i) = 1\"\nproof (-)\n (*goal: ‹lc (mat_to_polys (pps_to_list S) A ! i) = 1›*)\n let ?ts = \"pps_to_list S\"\n have len_ts: \"length ?ts = dim_col A\"\n by (simp only: length_pps_to_list (*‹length (pps_to_list ?S) = card ?S›*) assms( (*‹card S = dim_col A›*) 1))\n have \"lookup (row_to_poly ?ts (row A i)) (?ts ! f i) = (row A i) $ (f i)\"\n apply (rule lookup_row_to_poly (*‹⟦distinct (?ts::?'b list); dim_vec (?r::?'c vec) = length ?ts; (?i::nat) < length ?ts⟧ ⟹ lookup (row_to_poly ?ts ?r) (?ts ! ?i) = ?r $ ?i›*))\n (*goals:\n 1. ‹distinct (pps_to_list S)›\n 2. ‹dim_vec (row A i) = length (pps_to_list S)›\n 3. ‹f i < length (pps_to_list S)›\n discuss goal 1*)\n apply (fact distinct_pps_to_list (*‹distinct (pps_to_list ?S)›*))\n (*discuss goal 2*)\n apply (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*) assms( (*‹f i < dim_col A›*) 4))\n (*discuss goal 3*)\n apply (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*) assms( (*‹f i < dim_col A›*) 4))\n (*proven 3 subgoals*) .\n also (*calculation: ‹lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) = row A i $ f i›*) have \"... = A $$ (i, f i)\"\n using assms(3) (*‹i < dim_row A›*) assms(4) (*‹f i < dim_col A›*) by simp\n finally (*calculation: ‹lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) = A $$ (i, f i)›*) have eq: \"lookup (row_to_poly ?ts (row A i)) (?ts ! f i) = A $$ (i, f i)\" .\n show \"?thesis\"\n (*goal: ‹lc (mat_to_polys (pps_to_list S) A ! i) = 1›*)\n apply (simp only: lc_def (*‹lc (?p::'t ⇒₀ ?'b) = lookup ?p (lt ?p)›*) lt_row_to_poly_pivot_fun[OF assms] (*‹lt (mat_to_polys (pps_to_list (S::'t set)) (A::'b mat) ! (i::nat)) = pps_to_list S ! (f::nat ⇒ nat) i›*))\n (*goal: ‹lc (mat_to_polys (pps_to_list (S::'t set)) (A::'b mat) ! (i::nat)) = (1::'b)›*)\n apply (simp only: mat_to_polys_nth[OF assms(3)] (*‹mat_to_polys ?ts A ! i = row_to_poly ?ts (row A i)›*) eq (*‹lookup (row_to_poly (pps_to_list S) (row A i)) (pps_to_list S ! f i) = A $$ (i, f i)›*))\n (*goal: ‹lookup (mat_to_polys (pps_to_list S) A ! i) (pps_to_list S ! f i) = 1›*)\n apply (rule pivot_funD (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr⟧ ⟹ ?f ?i ≤ ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?j < ?f ?i⟧ ⟹ ?A $$ (?i, ?j) = 0› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; Suc ?i < ?nr⟧ ⟹ ?f ?i < ?f (Suc ?i) ∨ ?f (Suc ?i) = ?nc› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc⟧ ⟹ ?A $$ (?i, ?f ?i) = 1› ‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*))\n (*goals:\n 1. ‹dim_row A = ?nr2›\n 2. ‹pivot_fun A f ?nc2›\n 3. ‹i < ?nr2›\n 4. ‹f i < ?nc2›\n discuss goal 1*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\nqed\n\nlemma lt_row_to_poly_pivot_fun_less:\n assumes \"card S = dim_col (A::'b::semiring_1 mat)\" and \"pivot_fun A f (dim_col A)\"\n and \"i1 < i2\" and \"i2 < dim_row A\" and \"f i1 < dim_col A\" and \"f i2 < dim_col A\"\n shows \"(pps_to_list S) ! (f i2) ≺⇩t (pps_to_list S) ! (f i1)\"\nproof (-)\n (*goal: ‹pps_to_list S ! f i2 ≺⇩t pps_to_list S ! f i1›*)\n let ?ts = \"pps_to_list S\"\n have len_ts: \"length ?ts = dim_col A\"\n by (simp add: length_pps_to_list (*‹length (pps_to_list ?S) = card ?S›*) assms( (*‹card S = dim_col A›*) 1))\n from assms(3) (*‹(i1::nat) < (i2::nat)›*) assms(4) (*‹i2 < dim_row A›*) have \"i1 < dim_row A\"\n by simp\n show \"?thesis\"\n (*goal: ‹pps_to_list S ! f i2 ≺⇩t pps_to_list S ! f i1›*)\n apply (rule pps_to_list_nth_lessI (*‹⟦(?j::nat) < (?i::nat); ?i < card (?S::'t set)⟧ ⟹ pps_to_list ?S ! ?i ≺⇩t pps_to_list ?S ! ?j›*))\n (*goals:\n 1. ‹(f::nat ⇒ nat) (i1::nat) < f (i2::nat)›\n 2. ‹(f::nat ⇒ nat) (i2::nat) < card (S::'t set)›\n discuss goal 1*)\n apply (rule pivot_fun_mono_strict[where ?f=f] (*‹⟦pivot_fun (?A::?'a mat) (f::nat ⇒ nat) (?nc::nat); (?i1.0::nat) < (?i2.0::nat); ?i2.0 < dim_row ?A; f ?i1.0 < ?nc⟧ ⟹ f ?i1.0 < f ?i2.0›*))\n (*goals:\n 1. ‹pivot_fun ?A3 f ?nc3›\n 2. ‹i1 < i2›\n 3. ‹i2 < dim_row ?A3›\n 4. ‹f i1 < ?nc3›\n discuss goal 1*)\n apply fact\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*)\n (*discuss goal 2*)\n apply (simp only: assms( (*‹card S = dim_col A›*) 1) assms( (*‹f i2 < dim_col A›*) 6))\n (*proven 2 subgoals*) .\nqed\n\nlemma lt_row_to_poly_pivot_fun_eqD:\n assumes \"card S = dim_col (A::'b::semiring_1 mat)\" and \"pivot_fun A f (dim_col A)\"\n and \"i1 < dim_row A\" and \"i2 < dim_row A\" and \"f i1 < dim_col A\" and \"f i2 < dim_col A\"\n and \"(pps_to_list S) ! (f i1) = (pps_to_list S) ! (f i2)\"\n shows \"i1 = i2\"\nproof (rule linorder_cases (*‹⟦?x < ?y ⟹ ?P; ?x = ?y ⟹ ?P; ?y < ?x ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹?x < ?y ⟹ i1 = i2›\n 2. ‹?x = ?y ⟹ i1 = i2›\n 3. ‹?y < ?x ⟹ i1 = i2›*)\n assume \"i1 < i2\" (*‹(i1::nat) < (i2::nat)›*)\n from assms(1) (*‹card S = dim_col A›*) assms(2) (*‹pivot_fun (A::'b mat) (f::nat ⇒ nat) (dim_col A)›*) this (*‹(i1::nat) < (i2::nat)›*) assms(4) (*‹i2 < dim_row A›*) assms(5) (*‹f i1 < dim_col A›*) assms(6) (*‹f i2 < dim_col A›*) have \"(pps_to_list S) ! (f i2) ≺⇩t (pps_to_list S) ! (f i1)\"\n by (rule lt_row_to_poly_pivot_fun_less (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i1.0 < ?i2.0; ?i2.0 < dim_row ?A; ?f ?i1.0 < dim_col ?A; ?f ?i2.0 < dim_col ?A⟧ ⟹ pps_to_list ?S ! ?f ?i2.0 ≺⇩t pps_to_list ?S ! ?f ?i1.0›*))\n with assms(7) (*‹pps_to_list (S::'t set) ! (f::nat ⇒ nat) (i1::nat) = pps_to_list S ! f (i2::nat)›*) show \"?thesis\"\n (*goal: ‹i1 = i2›*)\n by auto\nnext\n (*goals:\n 1. ‹(i1::nat) = (i2::nat) ⟹ i1 = i2›\n 2. ‹(i2::nat) < (i1::nat) ⟹ i1 = i2›*)\n assume \"i2 < i1\" (*‹(i2::nat) < (i1::nat)›*)\n from assms(1) (*‹card S = dim_col A›*) assms(2) (*‹pivot_fun A f (dim_col A)›*) this (*‹i2 < i1›*) assms(3) (*‹i1 < dim_row A›*) assms(6) (*‹(f::nat ⇒ nat) (i2::nat) < dim_col (A::'b::semiring_1 mat)›*) assms(5) (*‹f i1 < dim_col A›*) have \"(pps_to_list S) ! (f i1) ≺⇩t (pps_to_list S) ! (f i2)\"\n by (rule lt_row_to_poly_pivot_fun_less (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i1.0 < ?i2.0; ?i2.0 < dim_row ?A; ?f ?i1.0 < dim_col ?A; ?f ?i2.0 < dim_col ?A⟧ ⟹ pps_to_list ?S ! ?f ?i2.0 ≺⇩t pps_to_list ?S ! ?f ?i1.0›*))\n with assms(7) (*‹pps_to_list S ! f i1 = pps_to_list S ! f i2›*) show \"?thesis\"\n (*goal: ‹i1 = i2›*)\n by auto\nqed\n\nlemma lt_row_to_poly_pivot_in_keysD:\n assumes \"card S = dim_col (A::'b::semiring_1 mat)\" and \"pivot_fun A f (dim_col A)\"\n and \"i1 < dim_row A\" and \"i2 < dim_row A\" and \"f i1 < dim_col A\"\n and \"(pps_to_list S) ! (f i1) ∈ keys ((mat_to_polys (pps_to_list S) A) ! i2)\"\n shows \"i1 = i2\"\nproof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹(i1::nat) ≠ (i2::nat) ⟹ False›*)\n assume \"i1 ≠ i2\" (*‹(i1::nat) ≠ (i2::nat)›*)\n hence \"i2 ≠ i1\"\n by simp\n let ?ts = \"pps_to_list S\"\n have len_ts: \"length ?ts = dim_col A\"\n by (simp only: length_pps_to_list (*‹length (pps_to_list ?S) = card ?S›*) assms( (*‹card S = dim_col A›*) 1))\n from assms(6) (*‹pps_to_list S ! f i1 ∈ keys (mat_to_polys (pps_to_list S) A ! i2)›*) have \"0 ≠ lookup (row_to_poly ?ts (row A i2)) (?ts ! (f i1))\"\n by (auto simp: mat_to_polys_nth[OF assms(4)] (*‹mat_to_polys ?ts A ! i2 = row_to_poly ?ts (row A i2)›*))\n also (*calculation: ‹0 ≠ lookup (row_to_poly (pps_to_list S) (row A i2)) (pps_to_list S ! f i1)›*) have \"lookup (row_to_poly ?ts (row A i2)) (?ts ! (f i1)) = (row A i2) $ (f i1)\"\n apply (rule lookup_row_to_poly (*‹⟦distinct ?ts; dim_vec ?r = length ?ts; ?i < length ?ts⟧ ⟹ lookup (row_to_poly ?ts ?r) (?ts ! ?i) = ?r $ ?i›*))\n (*goals:\n 1. ‹distinct (pps_to_list (S::'t::type set))›\n 2. ‹dim_vec (row (A::'b::semiring_1 mat) (i2::nat)) = length (pps_to_list (S::'t::type set))›\n 3. ‹(f::nat ⇒ nat) (i1::nat) < length (pps_to_list (S::'t::type set))›\n discuss goal 1*)\n apply (fact distinct_pps_to_list (*‹distinct (pps_to_list ?S)›*))\n (*discuss goal 2*)\n apply (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*) assms( (*‹f i1 < dim_col A›*) 5))\n (*discuss goal 3*)\n apply (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*) assms( (*‹f i1 < dim_col A›*) 5))\n (*proven 3 subgoals*) .\n finally (*calculation: ‹0 ≠ row A i2 $ f i1›*) have \"A $$ (i2, f i1) ≠ 0\"\n using assms(4) (*‹(i2::nat) < dim_row (A::'b mat)›*) assms(5) (*‹f i1 < dim_col A›*) by simp\n moreover have \"A $$ (i2, f i1) = 0\"\n by (rule pivot_funD( (*‹⟦dim_row ?A = ?nr; pivot_fun ?A ?f ?nc; ?i < ?nr; ?f ?i < ?nc; ?i' < ?nr; ?i' ≠ ?i⟧ ⟹ ?A $$ (?i', ?f ?i) = 0›*) 5), rule refl (*‹?t = ?t›*), (fact)+)\n ultimately show False\n by standard\nqed\n\nlemma lt_row_space_pivot_fun:\n assumes \"card S = dim_col (A::'b::{comm_semiring_0,semiring_1_no_zero_divisors} mat)\"\n and \"pivot_fun A f (dim_col A)\" and \"p ∈ row_to_poly (pps_to_list S) ` row_space A\" and \"p ≠ 0\"\n shows \"lt p ∈ lt_set (set (mat_to_polys (pps_to_list S) A))\"\nproof (-)\n (*goal: ‹lt p ∈ lt_set (set (mat_to_polys (pps_to_list S) A))›*)\n let ?ts = \"pps_to_list S\"\n let ?I = \"{0..<dim_row A}\"\n have len_ts: \"length ?ts = dim_col A\"\n by (simp add: length_pps_to_list (*‹length (pps_to_list (?S::'t::type set)) = card ?S›*) assms( (*‹card (S::'t::type set) = dim_col (A::'b::{comm_semiring_0,semiring_1_no_zero_divisors} mat)›*) 1))\n from assms(3) (*‹p ∈ row_to_poly (pps_to_list S) ` row_space A›*) obtain x where \"x ∈ row_space A\" and p: \"p = row_to_poly ?ts x\"\n (*goal: ‹(⋀x. ⟦x ∈ row_space A; p = row_to_poly (pps_to_list S) x⟧ ⟹ thesis) ⟹ thesis›*)\n by standard\n from this(1) (*‹x ∈ row_space A›*) obtain v where \"v ∈ carrier_vec (dim_row A)\" and x: \"x = v ⇩v* A\"\n (*goal: ‹(⋀v. ⟦v ∈ carrier_vec (dim_row A); x = v ⇩v* A⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule row_spaceE (*‹⟦?x ∈ row_space ?A; ⋀v. ⟦v ∈ carrier_vec (dim_row ?A); ?x = v ⇩v* ?A⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n have p': \"p = (∑i∈?I. (v $ i) ⋅ (row_to_poly ?ts (row A i)))\"\n unfolding p x\n (*goal: ‹row_to_poly (pps_to_list S) (v ⇩v* A) = (∑i = 0..<dim_row A. v $ i ⋅ row_to_poly (pps_to_list S) (row A i))›*)\n apply (rule row_to_poly_vec_times (*‹⟦distinct ?ts; length ?ts = dim_col ?A⟧ ⟹ row_to_poly ?ts (?v ⇩v* ?A) = (∑i = 0..<dim_row ?A. ?v $ i ⋅ row_to_poly ?ts (row ?A i))›*))\n (*goals:\n 1. ‹distinct (pps_to_list S)›\n 2. ‹length (pps_to_list S) = dim_col A›\n discuss goal 1*)\n apply (fact distinct_pps_to_list (*‹distinct (pps_to_list ?S)›*))\n (*discuss goal 2*)\n apply (fact len_ts (*‹length (pps_to_list S) = dim_col A›*))\n (*proven 2 subgoals*) .\n have \"lt (∑i = 0..<dim_row A. (v $ i) ⋅ (row_to_poly ?ts (row A i)))\n ∈ lt_set ((λi. (v $ i) ⋅ (row_to_poly ?ts (row A i))) ` {0..<dim_row A})\"\n proof (rule lt_sum_distinct_in_lt_set (*‹⟦finite ?I; sum ?p ?I ≠ 0; ⋀i1 i2. ⟦i1 ∈ ?I; i2 ∈ ?I; ?p i1 ≠ 0; ?p i2 ≠ 0; lt (?p i1) = lt (?p i2)⟧ ⟹ i1 = i2⟧ ⟹ lt (sum ?p ?I) ∈ lt_set (?p ` ?I)›*), rule, simp add: p'[symmetric] (*‹(∑i = 0..<dim_row A. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) = p›*) ‹p ≠ 0›)\n (*goal: ‹⋀i1 i2. ⟦i1 ∈ {0..<dim_row A}; i2 ∈ {0..<dim_row A}; v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1) ≠ 0; v $ i2 ⋅ row_to_poly (pps_to_list S) (row A i2) ≠ 0; lt (v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1)) = lt (v $ i2 ⋅ row_to_poly (pps_to_list S) (row A i2))⟧ ⟹ i1 = i2›*)\n fix i1 and i2\n let ?p1 = \"(v $ i1) ⋅ (row_to_poly ?ts (row A i1))\"\n let ?p2 = \"(v $ i2) ⋅ (row_to_poly ?ts (row A i2))\"\n assume \"i1 ∈ ?I\" and \"i2 ∈ ?I\" (*‹(i1::nat) ∈ {0::nat..<dim_row (A::'b mat)}› ‹(i2::nat) ∈ {0::nat..<dim_row (A::'b mat)}›*)\n hence \"i1 < dim_row A\" and \"i2 < dim_row A\"\n apply -\n (*goals:\n 1. ‹⟦i1 ∈ {0..<dim_row A}; i2 ∈ {0..<dim_row A}⟧ ⟹ i1 < dim_row A›\n 2. ‹⟦i1 ∈ {0..<dim_row A}; i2 ∈ {0..<dim_row A}⟧ ⟹ i2 < dim_row A›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n assume \"?p1 ≠ 0\" (*‹(v::'b vec) $ (i1::nat) ⋅ row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) i1) ≠ (0::'t ⇒₀ 'b)›*)\n hence \"v $ i1 ≠ 0\" and \"row_to_poly ?ts (row A i1) ≠ 0\"\n apply -\n (*goals:\n 1. ‹v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1) ≠ 0 ⟹ v $ i1 ≠ 0›\n 2. ‹v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1) ≠ 0 ⟹ row_to_poly (pps_to_list S) (row A i1) ≠ 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence \"row A i1 ≠ 0⇩v (length ?ts)\"\n by auto\n hence \"f i1 < dim_col A\"\n by (simp add: len_ts (*‹length (pps_to_list S) = dim_col A›*) row_not_zero_iff_pivot_fun[OF assms(2) ‹i1 < dim_row A›] (*‹(row A i1 ≠ 0⇩v (dim_col A)) = (f i1 < dim_col A)›*))\n have \"lt ?p1 = lt (row_to_poly ?ts (row A i1))\"\n apply (rule lt_map_scale (*‹?c ≠ 0 ⟹ lt (?c ⋅ ?p) = lt ?p›*))\n (*goal: ‹lt (v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1)) = lt (row_to_poly (pps_to_list S) (row A i1))›*)\n by fact\n also (*calculation: ‹lt (v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1)) = lt (row_to_poly (pps_to_list S) (row A i1))›*) have \"... = lt ((mat_to_polys ?ts A) ! i1)\"\n by (simp only: mat_to_polys_nth[OF ‹i1 < dim_row A›] (*‹mat_to_polys (?ts::?'c list) (A::'b mat) ! (i1::nat) = row_to_poly ?ts (row A i1)›*))\n also (*calculation: ‹lt (v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1)) = lt (mat_to_polys (pps_to_list S) A ! i1)›*) have \"... = ?ts ! (f i1)\"\n apply (rule lt_row_to_poly_pivot_fun (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i < dim_row ?A; ?f ?i < dim_col ?A⟧ ⟹ lt (mat_to_polys (pps_to_list ?S) ?A ! ?i) = pps_to_list ?S ! ?f ?i›*))\n (*goals:\n 1. ‹card S = dim_col A›\n 2. ‹pivot_fun A f (dim_col A)›\n 3. ‹i1 < dim_row A›\n 4. ‹f i1 < dim_col A›\n discuss goal 1*)\n apply fact\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n finally (*calculation: ‹lt (v $ i1 ⋅ row_to_poly (pps_to_list S) (row A i1)) = pps_to_list S ! f i1›*) have lt1: \"lt ?p1 = ?ts ! (f i1)\" .\n assume \"?p2 ≠ 0\" (*‹(v::'b vec) $ (i2::nat) ⋅ row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) i2) ≠ (0::'t ⇒₀ 'b)›*)\n hence \"v $ i2 ≠ 0\" and \"row_to_poly ?ts (row A i2) ≠ 0\"\n apply -\n (*goals:\n 1. ‹v $ i2 ⋅ row_to_poly (pps_to_list S) (row A i2) ≠ 0 ⟹ v $ i2 ≠ 0›\n 2. ‹v $ i2 ⋅ row_to_poly (pps_to_list S) (row A i2) ≠ 0 ⟹ row_to_poly (pps_to_list S) (row A i2) ≠ 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence \"row A i2 ≠ 0⇩v (length ?ts)\"\n by auto\n hence \"f i2 < dim_col A\"\n by (simp add: len_ts (*‹length (pps_to_list (S::'t::type set)) = dim_col (A::'b::{comm_semiring_0,semiring_1_no_zero_divisors} mat)›*) row_not_zero_iff_pivot_fun[OF assms(2) ‹i2 < dim_row A›] (*‹(row (A::'b::{comm_semiring_0,semiring_1_no_zero_divisors} mat) (i2::nat) ≠ 0⇩v (dim_col A)) = ((f::nat ⇒ nat) i2 < dim_col A)›*))\n have \"lt ?p2 = lt (row_to_poly ?ts (row A i2))\"\n apply (rule lt_map_scale (*‹?c ≠ 0 ⟹ lt (?c ⋅ ?p) = lt ?p›*))\n (*goal: ‹lt (v $ i2 ⋅ row_to_poly (pps_to_list S) (row A i2)) = lt (row_to_poly (pps_to_list S) (row A i2))›*)\n by fact\n also (*calculation: ‹lt (v $ i2 ⋅ row_to_poly (pps_to_list S) (row A i2)) = lt (row_to_poly (pps_to_list S) (row A i2))›*) have \"... = lt ((mat_to_polys ?ts A) ! i2)\"\n by (simp only: mat_to_polys_nth[OF ‹i2 < dim_row A›] (*‹mat_to_polys (?ts::?'c list) (A::'b mat) ! (i2::nat) = row_to_poly ?ts (row A i2)›*))\n also (*calculation: ‹lt ((v::'b vec) $ (i2::nat) ⋅ row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) i2)) = lt (mat_to_polys (pps_to_list S) A ! i2)›*) have \"... = ?ts ! (f i2)\"\n apply (rule lt_row_to_poly_pivot_fun (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i < dim_row ?A; ?f ?i < dim_col ?A⟧ ⟹ lt (mat_to_polys (pps_to_list ?S) ?A ! ?i) = pps_to_list ?S ! ?f ?i›*))\n (*goals:\n 1. ‹card S = dim_col A›\n 2. ‹pivot_fun A f (dim_col A)›\n 3. ‹i2 < dim_row A›\n 4. ‹f i2 < dim_col A›\n discuss goal 1*)\n apply fact\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n finally (*calculation: ‹lt (v $ i2 ⋅ row_to_poly (pps_to_list S) (row A i2)) = pps_to_list S ! f i2›*) have lt2: \"lt ?p2 = ?ts ! (f i2)\" .\n assume \"lt ?p1 = lt ?p2\" (*‹lt ((v::'b vec) $ (i1::nat) ⋅ row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) i1)) = lt (v $ (i2::nat) ⋅ row_to_poly (pps_to_list S) (row A i2))›*)\n with assms(1) (*‹card S = dim_col A›*) assms(2) (*‹pivot_fun A f (dim_col A)›*) ‹i1 < dim_row A› (*‹i1 < dim_row A›*) ‹i2 < dim_row A› (*‹i2 < dim_row A›*) ‹f i1 < dim_col A› (*‹f i1 < dim_col A›*) ‹f i2 < dim_col A› (*‹f i2 < dim_col A›*) show \"i1 = i2\"\n unfolding lt1 lt2\n (*goal: ‹i1 = i2›*)\n by (rule lt_row_to_poly_pivot_fun_eqD (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i1.0 < dim_row ?A; ?i2.0 < dim_row ?A; ?f ?i1.0 < dim_col ?A; ?f ?i2.0 < dim_col ?A; pps_to_list ?S ! ?f ?i1.0 = pps_to_list ?S ! ?f ?i2.0⟧ ⟹ ?i1.0 = ?i2.0›*))\n qed\n also (*calculation: ‹lt (∑i::nat = 0::nat..<dim_row A. (v::'b vec) $ i ⋅ row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) i)) ∈ lt_set ((λi::nat. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ` {0::nat..<dim_row A})›*) have \"... ⊆ lt_set ((λi. row_to_poly ?ts (row A i)) ` {0..<dim_row A})\"\n proof (standard)\n (*goal: ‹⋀x. x ∈ lt_set ((λi. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ` {0..<dim_row A}) ⟹ x ∈ lt_set ((λi. row_to_poly (pps_to_list S) (row A i)) ` {0..<dim_row A})›*)\n fix s\n assume \"s ∈ lt_set ((λi. (v $ i) ⋅ (row_to_poly ?ts (row A i))) ` {0..<dim_row A})\" (*‹(s::'t) ∈ lt_set ((λi::nat. (v::'b vec) $ i ⋅ row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) i)) ` {0::nat..<dim_row A})›*)\n then obtain f where \"f ∈ (λi. (v $ i) ⋅ (row_to_poly ?ts (row A i))) ` {0..<dim_row A}\" and \"f ≠ 0\" and \"lt f = s\"\n (*goal: ‹(⋀f. ⟦f ∈ (λi. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ` {0..<dim_row A}; f ≠ 0; lt f = s⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule lt_setE (*‹⟦?t ∈ lt_set ?F; ⋀f. ⟦f ∈ ?F; f ≠ 0; lt f = ?t⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n from this(1) (*‹f ∈ (λi. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ` {0..<dim_row A}›*) obtain i where \"i ∈ {0..<dim_row A}\" and f: \"f = (v $ i) ⋅ (row_to_poly ?ts (row A i))\"\n (*goal: ‹(⋀i::nat. ⟦i ∈ {0::nat..<dim_row (A::'b mat)}; (f::'t ⇒₀ 'b) = (v::'b vec) $ i ⋅ row_to_poly (pps_to_list (S::'t set)) (row A i)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by standard\n from this(2) (*‹f = v $ i ⋅ row_to_poly (pps_to_list S) (row A i)›*) ‹f ≠ 0› (*‹f ≠ 0›*) have \"v $ i ≠ 0\" and \"**\": \"row_to_poly ?ts (row A i) ≠ 0\"\n apply -\n (*goals:\n 1. ‹⟦f = v $ i ⋅ row_to_poly (pps_to_list S) (row A i); f ≠ 0⟧ ⟹ v $ i ≠ 0›\n 2. ‹⟦f = v $ i ⋅ row_to_poly (pps_to_list S) (row A i); f ≠ 0⟧ ⟹ row_to_poly (pps_to_list S) (row A i) ≠ 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from ‹lt f = s› (*‹lt f = s›*) have \"s = lt ((v $ i) ⋅ (row_to_poly ?ts (row A i)))\"\n by (simp only: f (*‹f = v $ i ⋅ row_to_poly (pps_to_list S) (row A i)›*))\n also (*calculation: ‹s = lt (v $ i ⋅ row_to_poly (pps_to_list S) (row A i))›*) from ‹v $ i ≠ 0› (*‹v $ i ≠ 0›*) have \"... = lt (row_to_poly ?ts (row A i))\"\n by (rule lt_map_scale (*‹?c ≠ 0 ⟹ lt (?c ⋅ ?p) = lt ?p›*))\n finally (*calculation: ‹s = lt (row_to_poly (pps_to_list S) (row A i))›*) have s: \"s = lt (row_to_poly ?ts (row A i))\" .\n show \"s ∈ lt_set ((λi. row_to_poly ?ts (row A i)) ` {0..<dim_row A})\"\n unfolding s\n (*goal: ‹lt (row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) (i::nat))) ∈ lt_set ((λi::nat. row_to_poly (pps_to_list S) (row A i)) ` {0::nat..<dim_row A})›*)\n apply (rule lt_setI (*‹⟦?f ∈ ?F; ?f ≠ 0⟧ ⟹ lt ?f ∈ lt_set ?F›*))\n (*goals:\n 1. ‹row_to_poly (pps_to_list S) (row A i) ∈ (λi. row_to_poly (pps_to_list S) (row A i)) ` {0..<dim_row A}›\n 2. ‹row_to_poly (pps_to_list S) (row A i) ≠ 0›\n discuss goal 1*)\n apply rule\n (*goals:\n 1. ‹row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) (i::nat)) = row_to_poly (pps_to_list S) (row A (?x3::nat))›\n 2. ‹(?x3::nat) ∈ {0::nat..<dim_row (A::'b mat)}›\n discuss goal 1*)\n apply (rule refl (*‹(?t::?'a) = ?t›*))\n (*discuss goal 2*)\n apply fact\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply fact\n (*proven 2 subgoals*) .\n qed\n also (*calculation: ‹lt (∑i = 0..<dim_row A. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ∈ lt_set ((λi. row_to_poly (pps_to_list S) (row A i)) ` {0..<dim_row A})›*) have \"... = lt_set ((λr. row_to_poly ?ts r) ` (row A ` {0..<dim_row A}))\"\n by (simp only: image_comp (*‹?f ` ?g ` ?r = (?f ∘ ?g) ` ?r›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n also (*calculation: ‹lt (∑i::nat = 0::nat..<dim_row A. (v::'b vec) $ i ⋅ row_to_poly (pps_to_list (S::'t set)) (row (A::'b mat) i)) ∈ lt_set (row_to_poly (pps_to_list S) ` row A ` {0::nat..<dim_row A})›*) have \"... = lt_set (set (map (λr. row_to_poly ?ts r) (map (row A) [0..<dim_row A])))\"\n by (metis image_set (*‹?f ` set ?xs = set (map ?f ?xs)›*) set_upt (*‹set [?i..<?j] = {?i..<?j}›*))\n also (*calculation: ‹lt (∑i = 0..<dim_row A. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ∈ lt_set (set (map (row_to_poly (pps_to_list S)) (map (row A) [0..<dim_row A])))›*) have \"... = lt_set (set (mat_to_polys ?ts A))\"\n by (simp only: mat_to_polys_def (*‹mat_to_polys (?ts::?'a list) (?A::?'b mat) = map (row_to_poly ?ts) (rows ?A)›*) rows_def (*‹rows (?A::?'a mat) = map (row ?A) [0::nat..<dim_row ?A]›*))\n finally (*calculation: ‹lt (∑i = 0..<dim_row A. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ∈ lt_set (set (mat_to_polys (pps_to_list S) A))›*) show \"?thesis\"\n (*goal: ‹lt p ∈ lt_set (set (mat_to_polys (pps_to_list S) A))›*)\n unfolding p'\n (*goal: ‹lt (∑i = 0..<dim_row A. v $ i ⋅ row_to_poly (pps_to_list S) (row A i)) ∈ lt_set (set (mat_to_polys (pps_to_list S) A))›*) .\nqed\n\nsubsection ‹Functions ‹Macaulay_mat› and ‹Macaulay_list››\n\ndefinition Macaulay_mat :: \"('t ⇒₀ 'b) list ⇒ 'b::field mat\"\n where \"Macaulay_mat ps = polys_to_mat (Keys_to_list ps) ps\"\n\ndefinition Macaulay_list :: \"('t ⇒₀ 'b) list ⇒ ('t ⇒₀ 'b::field) list\"\n where \"Macaulay_list ps =\n filter (λp. p ≠ 0) (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)))\"\n\nlemma dim_Macaulay_mat[simp]:\n \"dim_row (Macaulay_mat ps) = length ps\"\n \"dim_col (Macaulay_mat ps) = card (Keys (set ps))\"\n \n (*goals:\n 1. ‹dim_row (Macaulay_mat (ps::('t ⇒₀ 'b) list)) = length ps›\n 2. ‹dim_col (Macaulay_mat (ps::('t ⇒₀ 'b) list)) = card (Keys (set ps))›\n discuss goal 1*)\n apply (simp add: Macaulay_mat_def (*‹Macaulay_mat ?ps = polys_to_mat (Keys_to_list ?ps) ?ps›*) length_Keys_to_list (*‹length (Keys_to_list ?ps) = card (Keys (set ?ps))›*))\n (*discuss goal 2*)\n apply (simp add: Macaulay_mat_def (*‹Macaulay_mat ?ps = polys_to_mat (Keys_to_list ?ps) ?ps›*) length_Keys_to_list (*‹length (Keys_to_list ?ps) = card (Keys (set ?ps))›*))\n (*proven 2 subgoals*) .\n\nlemma Macaulay_list_Nil [simp]: \"Macaulay_list [] = ([]::('t ⇒₀ 'b::field) list)\" (is \"?l = _\")\nproof (-)\n (*goal: ‹Macaulay_list [] = []›*)\n have \"length ?l ≤ length (mat_to_polys (Keys_to_list ([]::('t ⇒₀ 'b) list))\n (row_echelon (Macaulay_mat ([]::('t ⇒₀ 'b) list))))\"\n unfolding Macaulay_list_def\n (*goal: ‹length (filter (λp. p ≠ 0) (mat_to_polys (Keys_to_list []) (row_echelon (Macaulay_mat [])))) ≤ length (mat_to_polys (Keys_to_list []) (row_echelon (Macaulay_mat [])))›*)\n by (fact length_filter_le (*‹length (filter (?P::?'a ⇒ bool) (?xs::?'a list)) ≤ length ?xs›*))\n also (*calculation: ‹length (Macaulay_list []) ≤ length (mat_to_polys (Keys_to_list []) (row_echelon (Macaulay_mat [])))›*) have \"... = 0\"\n by simp\n finally (*calculation: ‹length (Macaulay_list []) ≤ (0::nat)›*) show \"?thesis\"\n (*goal: ‹Macaulay_list [] = []›*)\n by simp\nqed\n\nlemma set_Macaulay_list:\n \"set (Macaulay_list ps) =\n set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))) - {0}\"\n by (auto simp add: Macaulay_list_def (*‹Macaulay_list ?ps = filter (λp. p ≠ 0) (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps)))›*))\n\nlemma Keys_Macaulay_list: \"Keys (set (Macaulay_list ps)) ⊆ Keys (set ps)\"\nproof (-)\n (*goal: ‹Keys (set (Macaulay_list ps)) ⊆ Keys (set ps)›*)\n have \"Keys (set (Macaulay_list ps)) ⊆ set (Keys_to_list ps)\"\n apply (simp only: set_Macaulay_list (*‹set (Macaulay_list ?ps) = set (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps))) - {0}›*) Keys_minus_zero (*‹Keys (?A - {0}) = Keys ?A›*))\n (*goal: ‹Keys (set (Macaulay_list ps)) ⊆ set (Keys_to_list ps)›*)\n by (fact Keys_mat_to_polys (*‹Keys (set (mat_to_polys ?ts ?A)) ⊆ set ?ts›*))\n also (*calculation: ‹Keys (set (Macaulay_list ps)) ⊆ set (Keys_to_list ps)›*) have \"... = Keys (set ps)\"\n by (fact set_Keys_to_list (*‹set (Keys_to_list ?ps) = Keys (set ?ps)›*))\n finally (*calculation: ‹Keys (set (Macaulay_list ps)) ⊆ Keys (set ps)›*) show \"?thesis\"\n (*goal: ‹Keys (set (Macaulay_list ps)) ⊆ Keys (set ps)›*) .\nqed\n\nlemma in_Macaulay_listE:\n assumes \"p ∈ set (Macaulay_list ps)\"\n and \"pivot_fun (row_echelon (Macaulay_mat ps)) f (dim_col (row_echelon (Macaulay_mat ps)))\"\n obtains i where \"i < dim_row (row_echelon (Macaulay_mat ps))\"\n and \"p = (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))) ! i\"\n and \"f i < dim_col (row_echelon (Macaulay_mat ps))\"\nproof (-)\n (*goal: ‹(⋀i. ⟦i < dim_row (row_echelon (Macaulay_mat ps)); p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i; f i < dim_col (row_echelon (Macaulay_mat ps))⟧ ⟹ thesis) ⟹ thesis›*)\n let ?ts = \"Keys_to_list ps\"\n let ?A = \"Macaulay_mat ps\"\n let ?E = \"row_echelon ?A\"\n from assms(1) (*‹p ∈ set (Macaulay_list ps)›*) have \"p ∈ set (mat_to_polys ?ts ?E) - {0}\"\n by (simp add: set_Macaulay_list (*‹set (Macaulay_list ?ps) = set (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps))) - {0}›*))\n hence \"p ∈ set (mat_to_polys ?ts ?E)\" and \"p ≠ 0\"\n apply -\n (*goals:\n 1. ‹p ∈ set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))) - {0} ⟹ p ∈ set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)))›\n 2. ‹p ∈ set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))) - {0} ⟹ p ≠ 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from this(1) (*‹(p::'t::type ⇒₀ 'b::field) ∈ set (mat_to_polys (Keys_to_list (ps::('t::type ⇒₀ 'b::field) list)) (row_echelon (Macaulay_mat ps)))›*) obtain i where \"i < length (mat_to_polys ?ts ?E)\" and p: \"p = (mat_to_polys ?ts ?E) ! i\"\n (*goal: ‹(⋀i. ⟦i < length (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))); p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis in_set_conv_nth (*‹((?x::?'a::type) ∈ set (?xs::?'a::type list)) = (∃i<length ?xs. ?xs ! i = ?x)›*))\n from this(1) (*‹i < length (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)))›*) have \"i < dim_row ?E\" and \"i < dim_row ?A\"\n apply -\n (*goals:\n 1. ‹i < length (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))) ⟹ i < dim_row (row_echelon (Macaulay_mat ps))›\n 2. ‹i < length (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))) ⟹ i < dim_row (Macaulay_mat ps)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n from this(1) (*‹i < dim_row (row_echelon (Macaulay_mat ps))›*) p (*‹p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i›*) show \"?thesis\"\n (*goal: ‹thesis›*)\n proof (standard)\n (*goal: ‹(f::nat ⇒ nat) (i::nat) < dim_col (row_echelon (Macaulay_mat (ps::('t ⇒₀ 'b) list)))›*)\n from ‹p ≠ 0› (*‹p ≠ 0›*) have \"0 ≠ (mat_to_polys ?ts ?E) ! i\"\n by (simp only: p (*‹p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i›*))\n also (*calculation: ‹0 ≠ mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i›*) have \"(mat_to_polys ?ts ?E) ! i = row_to_poly ?ts (row ?E i)\"\n by (simp only: Macaulay_list_def (*‹Macaulay_list ?ps = filter (λp. p ≠ 0) (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps)))›*) mat_to_polys_nth[OF ‹i < dim_row ?E›] (*‹mat_to_polys ?ts (row_echelon (Macaulay_mat ps)) ! i = row_to_poly ?ts (row (row_echelon (Macaulay_mat ps)) i)›*))\n finally (*calculation: ‹0 ≠ row_to_poly (Keys_to_list ps) (row (row_echelon (Macaulay_mat ps)) i)›*) have \"*\": \"row_to_poly ?ts (row ?E i) ≠ 0\"\n by simp\n have \"row ?E i ≠ 0⇩v (length ?ts)\"\n proof (standard)\n (*goal: ‹row (row_echelon (Macaulay_mat (ps::('t ⇒₀ 'b) list))) (i::nat) = 0⇩v (length (Keys_to_list ps)) ⟹ False›*)\n assume \"row ?E i = 0⇩v (length ?ts)\" (*‹row (row_echelon (Macaulay_mat (ps::('t ⇒₀ 'b) list))) (i::nat) = 0⇩v (length (Keys_to_list ps))›*)\n with \"*\" (*‹row_to_poly (Keys_to_list ps) (row (row_echelon (Macaulay_mat ps)) i) ≠ 0›*) show False\n by simp\n qed\n hence \"row ?E i ≠ 0⇩v (dim_col ?E)\"\n by (simp add: length_Keys_to_list (*‹length (Keys_to_list ?ps) = card (Keys (set ?ps))›*))\n thus \"f i < dim_col ?E\"\n by (simp only: row_not_zero_iff_pivot_fun[OF assms(2) ‹i < dim_row ?E›] (*‹(row (row_echelon (Macaulay_mat ps)) i ≠ 0⇩v (dim_col (row_echelon (Macaulay_mat ps)))) = (f i < dim_col (row_echelon (Macaulay_mat ps)))›*))\n qed\nqed\n\nlemma phull_Macaulay_list: \"phull (set (Macaulay_list ps)) = phull (set ps)\"\nproof (-)\n (*goal: ‹phull (set (Macaulay_list ps)) = phull (set ps)›*)\n have \"*\": \"Keys (set ps) ⊆ set (Keys_to_list ps)\"\n by (simp add: set_Keys_to_list (*‹set (Keys_to_list (?ps::('t ⇒₀ ?'b) list)) = Keys (set ?ps)›*))\n have \"phull (set (Macaulay_list ps)) =\n phull (set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))))\"\n by (simp only: set_Macaulay_list (*‹set (Macaulay_list ?ps) = set (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps))) - {0}›*) phull.span_Diff_zero (*‹phull (?B - {0}) = phull ?B›*))\n also (*calculation: ‹phull (set (Macaulay_list ps)) = phull (set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))))›*) have \"... = phull (set ps)\"\n by (simp only: Macaulay_mat_def (*‹Macaulay_mat ?ps = polys_to_mat (Keys_to_list ?ps) ?ps›*) phull_row_echelon[OF * distinct_Keys_to_list] (*‹phull (set (mat_to_polys (Keys_to_list ps) (row_echelon (polys_to_mat (Keys_to_list ps) ps)))) = phull (set ps)›*))\n finally (*calculation: ‹phull (set (Macaulay_list ps)) = phull (set ps)›*) show \"?thesis\"\n (*goal: ‹phull (set (Macaulay_list ps)) = phull (set ps)›*) .\nqed\n\nlemma pmdl_Macaulay_list: \"pmdl (set (Macaulay_list ps)) = pmdl (set ps)\"\nproof (-)\n (*goal: ‹pmdl (set (Macaulay_list ps)) = pmdl (set ps)›*)\n have \"*\": \"Keys (set ps) ⊆ set (Keys_to_list ps)\"\n by (simp add: set_Keys_to_list (*‹set (Keys_to_list ?ps) = Keys (set ?ps)›*))\n have \"pmdl (set (Macaulay_list ps)) =\n pmdl (set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))))\"\n by (simp only: set_Macaulay_list (*‹set (Macaulay_list ?ps) = set (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps))) - {0}›*) pmdl.span_Diff_zero (*‹pmdl (?B - {0}) = pmdl ?B›*))\n also (*calculation: ‹pmdl (set (Macaulay_list ps)) = pmdl (set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))))›*) have \"... = pmdl (set ps)\"\n by (simp only: Macaulay_mat_def (*‹Macaulay_mat ?ps = polys_to_mat (Keys_to_list ?ps) ?ps›*) pmdl_row_echelon[OF * distinct_Keys_to_list] (*‹pmdl (set (mat_to_polys (Keys_to_list ps) (row_echelon (polys_to_mat (Keys_to_list ps) ps)))) = pmdl (set ps)›*))\n finally (*calculation: ‹pmdl (set (Macaulay_list ps)) = pmdl (set ps)›*) show \"?thesis\"\n (*goal: ‹pmdl (set (Macaulay_list ps)) = pmdl (set ps)›*) .\nqed\n\nlemma Macaulay_list_is_monic_set: \"is_monic_set (set (Macaulay_list ps))\"\nproof (rule is_monic_setI (*‹(⋀b::'t ⇒₀ ?'b. ⟦b ∈ (?B::('t ⇒₀ ?'b) set); b ≠ (0::'t ⇒₀ ?'b)⟧ ⟹ lc b = (1::?'b)) ⟹ is_monic_set ?B›*))\n (*goal: ‹⋀b. ⟦b ∈ set (Macaulay_list ps); b ≠ 0⟧ ⟹ lc b = 1›*)\n let ?ts = \"Keys_to_list ps\"\n let ?E = \"row_echelon (Macaulay_mat ps)\"\n fix p\n assume \"p ∈ set (Macaulay_list ps)\" (*‹(p::'t ⇒₀ 'b) ∈ set (Macaulay_list (ps::('t ⇒₀ 'b) list))›*)\n obtain h where \"pivot_fun ?E h (dim_col ?E)\"\n (*goal: ‹(⋀h. pivot_fun (row_echelon (Macaulay_mat ps)) h (dim_col (row_echelon (Macaulay_mat ps))) ⟹ thesis) ⟹ thesis›*)\n by (rule row_echelon_pivot_fun (*‹(⋀f. pivot_fun (row_echelon ?A) f (dim_col (row_echelon ?A)) ⟹ ?thesis) ⟹ ?thesis›*))\n with ‹p ∈ set (Macaulay_list ps)› (*‹p ∈ set (Macaulay_list ps)›*) obtain i where \"i < dim_row ?E\" and p: \"p = (mat_to_polys ?ts ?E) ! i\" and \"h i < dim_col ?E\"\n (*goal: ‹(⋀i. ⟦i < dim_row (row_echelon (Macaulay_mat ps)); p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i; h i < dim_col (row_echelon (Macaulay_mat ps))⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule in_Macaulay_listE (*‹⟦?p ∈ set (Macaulay_list ?ps); pivot_fun (row_echelon (Macaulay_mat ?ps)) ?f (dim_col (row_echelon (Macaulay_mat ?ps))); ⋀i. ⟦i < dim_row (row_echelon (Macaulay_mat ?ps)); ?p = mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps)) ! i; ?f i < dim_col (row_echelon (Macaulay_mat ?ps))⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n show \"lc p = 1\"\n unfolding p Keys_to_list_eq_pps_to_list\n (*goal: ‹lc (mat_to_polys (pps_to_list (Keys (set ps))) (row_echelon (Macaulay_mat ps)) ! i) = 1›*)\n apply (rule lc_row_to_poly_pivot_fun (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i < dim_row ?A; ?f ?i < dim_col ?A⟧ ⟹ lc (mat_to_polys (pps_to_list ?S) ?A ! ?i) = 1›*))\n (*goals:\n 1. ‹card (Keys (set ps)) = dim_col (row_echelon (Macaulay_mat ps))›\n 2. ‹pivot_fun (row_echelon (Macaulay_mat ps)) ?f (dim_col (row_echelon (Macaulay_mat ps)))›\n 3. ‹i < dim_row (row_echelon (Macaulay_mat ps))›\n 4. ‹?f i < dim_col (row_echelon (Macaulay_mat ps))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\nqed\n\nlemma Macaulay_list_not_zero: \"0 ∉ set (Macaulay_list ps)\"\n by (simp add: Macaulay_list_def (*‹Macaulay_list (?ps::('t ⇒₀ ?'b) list) = filter (λp::'t ⇒₀ ?'b. p ≠ (0::'t ⇒₀ ?'b)) (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps)))›*))\n\nlemma Macaulay_list_distinct_lt:\n assumes \"x ∈ set (Macaulay_list ps)\" and \"y ∈ set (Macaulay_list ps)\"\n and \"x ≠ y\"\n shows \"lt x ≠ lt y\"\nproof (standard)\n (*goal: ‹lt x = lt y ⟹ False›*)\n let ?S = \"Keys (set ps)\"\n let ?ts = \"Keys_to_list ps\"\n let ?E = \"row_echelon (Macaulay_mat ps)\"\n assume \"lt x = lt y\" (*‹lt (x::'t ⇒₀ 'b) = lt (y::'t ⇒₀ 'b)›*)\n obtain h where pf: \"pivot_fun ?E h (dim_col ?E)\"\n (*goal: ‹(⋀h. pivot_fun (row_echelon (Macaulay_mat ps)) h (dim_col (row_echelon (Macaulay_mat ps))) ⟹ thesis) ⟹ thesis›*)\n by (rule row_echelon_pivot_fun (*‹(⋀f. pivot_fun (row_echelon ?A) f (dim_col (row_echelon ?A)) ⟹ ?thesis) ⟹ ?thesis›*))\n with assms(1) (*‹(x::'t::type ⇒₀ 'b::field) ∈ set (Macaulay_list (ps::('t::type ⇒₀ 'b::field) list))›*) obtain i1 where \"i1 < dim_row ?E\" and x: \"x = (mat_to_polys ?ts ?E) ! i1\" and \"h i1 < dim_col ?E\"\n (*goal: ‹(⋀i1. ⟦i1 < dim_row (row_echelon (Macaulay_mat ps)); x = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i1; h i1 < dim_col (row_echelon (Macaulay_mat ps))⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule in_Macaulay_listE (*‹⟦(?p::'t::type ⇒₀ ?'b::field) ∈ set (Macaulay_list (?ps::('t::type ⇒₀ ?'b::field) list)); pivot_fun (row_echelon (Macaulay_mat ?ps)) (?f::nat ⇒ nat) (dim_col (row_echelon (Macaulay_mat ?ps))); ⋀i::nat. ⟦i < dim_row (row_echelon (Macaulay_mat ?ps)); ?p = mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps)) ! i; ?f i < dim_col (row_echelon (Macaulay_mat ?ps))⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*))\n from assms(2) (*‹y ∈ set (Macaulay_list ps)›*) pf (*‹pivot_fun (row_echelon (Macaulay_mat ps)) h (dim_col (row_echelon (Macaulay_mat ps)))›*) obtain i2 where \"i2 < dim_row ?E\" and y: \"y = (mat_to_polys ?ts ?E) ! i2\" and \"h i2 < dim_col ?E\"\n (*goal: ‹(⋀i2. ⟦i2 < dim_row (row_echelon (Macaulay_mat ps)); y = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i2; h i2 < dim_col (row_echelon (Macaulay_mat ps))⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule in_Macaulay_listE (*‹⟦(?p::'t::type ⇒₀ ?'b::field) ∈ set (Macaulay_list (?ps::('t::type ⇒₀ ?'b::field) list)); pivot_fun (row_echelon (Macaulay_mat ?ps)) (?f::nat ⇒ nat) (dim_col (row_echelon (Macaulay_mat ?ps))); ⋀i::nat. ⟦i < dim_row (row_echelon (Macaulay_mat ?ps)); ?p = mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps)) ! i; ?f i < dim_col (row_echelon (Macaulay_mat ?ps))⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*))\n have \"lt x = ?ts ! (h i1)\"\n apply (simp only: x (*‹x = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i1›*) Keys_to_list_eq_pps_to_list (*‹Keys_to_list ?ps = pps_to_list (Keys (set ?ps))›*))\n (*goal: ‹lt x = Keys_to_list ps ! h i1›*)\n apply (rule lt_row_to_poly_pivot_fun (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i < dim_row ?A; ?f ?i < dim_col ?A⟧ ⟹ lt (mat_to_polys (pps_to_list ?S) ?A ! ?i) = pps_to_list ?S ! ?f ?i›*))\n (*goals:\n 1. ‹card (Keys (set (ps::('t ⇒₀ 'b) list))) = dim_col (row_echelon (Macaulay_mat ps))›\n 2. ‹pivot_fun (row_echelon (Macaulay_mat (ps::('t ⇒₀ 'b) list))) (h::nat ⇒ nat) (dim_col (row_echelon (Macaulay_mat ps)))›\n 3. ‹(i1::nat) < dim_row (row_echelon (Macaulay_mat (ps::('t ⇒₀ 'b) list)))›\n 4. ‹(h::nat ⇒ nat) (i1::nat) < dim_col (row_echelon (Macaulay_mat (ps::('t ⇒₀ 'b) list)))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n moreover have \"lt y = ?ts ! (h i2)\"\n apply (simp only: y (*‹(y::'t ⇒₀ 'b) = mat_to_polys (Keys_to_list (ps::('t ⇒₀ 'b) list)) (row_echelon (Macaulay_mat ps)) ! (i2::nat)›*) Keys_to_list_eq_pps_to_list (*‹Keys_to_list (?ps::('t ⇒₀ ?'b) list) = pps_to_list (Keys (set ?ps))›*))\n (*goal: ‹lt y = Keys_to_list ps ! h i2›*)\n apply (rule lt_row_to_poly_pivot_fun (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i < dim_row ?A; ?f ?i < dim_col ?A⟧ ⟹ lt (mat_to_polys (pps_to_list ?S) ?A ! ?i) = pps_to_list ?S ! ?f ?i›*))\n (*goals:\n 1. ‹card (Keys (set ps)) = dim_col (row_echelon (Macaulay_mat ps))›\n 2. ‹pivot_fun (row_echelon (Macaulay_mat ps)) h (dim_col (row_echelon (Macaulay_mat ps)))›\n 3. ‹i2 < dim_row (row_echelon (Macaulay_mat ps))›\n 4. ‹h i2 < dim_col (row_echelon (Macaulay_mat ps))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n ultimately have \"?ts ! (h i1) = ?ts ! (h i2)\"\n by (simp only: ‹lt x = lt y›)\n hence \"pps_to_list (Keys (set ps)) ! h i1 = pps_to_list (Keys (set ps)) ! h i2\"\n by (simp only: Keys_to_list_eq_pps_to_list (*‹Keys_to_list ?ps = pps_to_list (Keys (set ?ps))›*))\n have \"i1 = i2\"\n proof (rule lt_row_to_poly_pivot_fun_eqD (*‹⟦card ?S = dim_col ?A; pivot_fun ?A ?f (dim_col ?A); ?i1.0 < dim_row ?A; ?i2.0 < dim_row ?A; ?f ?i1.0 < dim_col ?A; ?f ?i2.0 < dim_col ?A; pps_to_list ?S ! ?f ?i1.0 = pps_to_list ?S ! ?f ?i2.0⟧ ⟹ ?i1.0 = ?i2.0›*))\n (*goals:\n 1. ‹card ?S = dim_col ?A›\n 2. ‹pivot_fun ?A ?f (dim_col ?A)›\n 3. ‹i1 < dim_row ?A›\n 4. ‹i2 < dim_row ?A›\n 5. ‹?f i1 < dim_col ?A›\n 6. ‹?f i2 < dim_col ?A›\n 7. ‹pps_to_list ?S ! ?f i1 = pps_to_list ?S ! ?f i2›*)\n show \"card ?S = dim_col ?E\"\n by simp\n qed ((fact)+)\n (*solves the remaining goals:\n 1. ‹pivot_fun (row_echelon (Macaulay_mat ps)) ?f (dim_col (row_echelon (Macaulay_mat ps)))›\n 2. ‹i1 < dim_row (row_echelon (Macaulay_mat ps))›\n 3. ‹i2 < dim_row (row_echelon (Macaulay_mat ps))›\n 4. ‹?f i1 < dim_col (row_echelon (Macaulay_mat ps))›\n 5. ‹?f i2 < dim_col (row_echelon (Macaulay_mat ps))›\n 6. ‹pps_to_list (Keys (set ps)) ! ?f i1 = pps_to_list (Keys (set ps)) ! ?f i2›*)\n hence \"x = y\"\n by (simp only: x (*‹x = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i1›*) y (*‹y = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i2›*))\n with ‹x ≠ y› (*‹x ≠ y›*) show False\n by standard\nqed\n\nlemma Macaulay_list_lt:\n assumes \"p ∈ phull (set ps)\" and \"p ≠ 0\"\n obtains g where \"g ∈ set (Macaulay_list ps)\" and \"g ≠ 0\" and \"lt p = lt g\"\nproof (-)\n (*goal: ‹(⋀g. ⟦g ∈ set (Macaulay_list ps); g ≠ 0; lt p = lt g⟧ ⟹ thesis) ⟹ thesis›*)\n let ?S = \"Keys (set ps)\"\n let ?ts = \"Keys_to_list ps\"\n let ?E = \"row_echelon (Macaulay_mat ps)\"\n let ?gs = \"mat_to_polys ?ts ?E\"\n have \"finite ?S\"\n apply (rule finite_Keys (*‹finite ?A ⟹ finite (Keys ?A)›*))\n (*goal: ‹finite (Keys (set ps))›*)\n by rule\n have \"?S ⊆ set ?ts\"\n by (simp only: set_Keys_to_list (*‹set (Keys_to_list ?ps) = Keys (set ?ps)›*))\n from assms(1) (*‹p ∈ phull (set ps)›*) ‹?S ⊆ set ?ts› (*‹Keys (set ps) ⊆ set (Keys_to_list ps)›*) have \"p ∈ row_to_poly ?ts ` row_space ?E\"\n by (simp only: Macaulay_mat_def (*‹Macaulay_mat ?ps = polys_to_mat (Keys_to_list ?ps) ?ps›*) row_space_row_echelon_eq_phull[symmetric] (*‹Keys (set ?ps) ⊆ set ?ts ⟹ phull (set ?ps) = row_to_poly ?ts ` row_space (row_echelon (polys_to_mat ?ts ?ps))›*))\n hence \"p ∈ row_to_poly (pps_to_list ?S) ` row_space ?E\"\n by (simp only: Keys_to_list_eq_pps_to_list (*‹Keys_to_list ?ps = pps_to_list (Keys (set ?ps))›*))\n obtain f where \"pivot_fun ?E f (dim_col ?E)\"\n (*goal: ‹(⋀f. pivot_fun (row_echelon (Macaulay_mat ps)) f (dim_col (row_echelon (Macaulay_mat ps))) ⟹ thesis) ⟹ thesis›*)\n by (rule row_echelon_pivot_fun (*‹(⋀f. pivot_fun (row_echelon ?A) f (dim_col (row_echelon ?A)) ⟹ ?thesis) ⟹ ?thesis›*))\n have \"lt p ∈ lt_set (set ?gs)\"\n unfolding Keys_to_list_eq_pps_to_list\n (*goal: ‹lt p ∈ lt_set (set (mat_to_polys (pps_to_list (Keys (set ps))) (row_echelon (Macaulay_mat ps))))›*)\n apply (rule lt_row_space_pivot_fun (*‹⟦card (?S::'t set) = dim_col (?A::?'b mat); pivot_fun ?A (?f::nat ⇒ nat) (dim_col ?A); (?p::'t ⇒₀ ?'b) ∈ row_to_poly (pps_to_list ?S) ` row_space ?A; ?p ≠ (0::'t ⇒₀ ?'b)⟧ ⟹ lt ?p ∈ lt_set (set (mat_to_polys (pps_to_list ?S) ?A))›*))\n (*goals:\n 1. ‹card (Keys (set (ps::('t::type ⇒₀ 'b::field) list))) = dim_col (row_echelon (Macaulay_mat ps))›\n 2. ‹pivot_fun (row_echelon (Macaulay_mat (ps::('t::type ⇒₀ 'b::field) list))) (?f::nat ⇒ nat) (dim_col (row_echelon (Macaulay_mat ps)))›\n 3. ‹(p::'t::type ⇒₀ 'b::field) ∈ row_to_poly (pps_to_list (Keys (set (ps::('t::type ⇒₀ 'b::field) list)))) ` row_space (row_echelon (Macaulay_mat ps))›\n 4. ‹(p::'t::type ⇒₀ 'b::field) ≠ (0::'t::type ⇒₀ 'b::field)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply fact\n (*discuss goal 3*)\n apply fact\n (*discuss goal 4*)\n apply fact\n (*proven 4 subgoals*) .\n then obtain g where \"g ∈ set ?gs\" and \"g ≠ 0\" and \"lt g = lt p\"\n (*goal: ‹(⋀g. ⟦g ∈ set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps))); g ≠ 0; lt g = lt p⟧ ⟹ thesis) ⟹ thesis›*)\n by (rule lt_setE (*‹⟦?t ∈ lt_set ?F; ⋀f. ⟦f ∈ ?F; f ≠ 0; lt f = ?t⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n show \"?thesis\"\n (*goal: ‹thesis›*)\n proof (standard)\n (*goals:\n 1. ‹(?g30::'t::type ⇒₀ 'b::field) ∈ set (Macaulay_list (ps::('t::type ⇒₀ 'b::field) list))›\n 2. ‹(?g30::'t::type ⇒₀ 'b::field) ≠ (0::'t::type ⇒₀ 'b::field)›\n 3. ‹lt (p::'t::type ⇒₀ 'b::field) = lt (?g30::'t::type ⇒₀ 'b::field)›*)\n from ‹g ∈ set ?gs› (*‹g ∈ set (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)))›*) ‹g ≠ 0› (*‹g ≠ 0›*) show \"g ∈ set (Macaulay_list ps)\"\n by (simp add: set_Macaulay_list (*‹set (Macaulay_list ?ps) = set (mat_to_polys (Keys_to_list ?ps) (row_echelon (Macaulay_mat ?ps))) - {0}›*))\n next\n (*goals:\n 1. ‹g ≠ 0›\n 2. ‹lt p = lt g›*)\n from ‹lt g = lt p› (*‹lt g = lt p›*) show \"lt p = lt g\"\n by simp\n qed (fact)\n (*solved the remaining goal: ‹g ≠ 0›*)\nqed\n\nend (* ordered_term *)\n\nend (* theory *)\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Groebner_Bases/Macaulay_Matrix.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"ae88c6316a514b4fadf51fcfe680a1a1468b2bfee64fb17f9583d27f9ffdcc02\"\n}"}}},{"rowIdx":185,"cells":{"text":{"kind":"string","value":"(*\n Author: Lawrence C Paulson, Cambridge University Computer Laboratory\n Tobias Nipkow, TUM\n*)\n\ntheory Tilings imports Main begin\n\nsection‹Inductive Tiling›\n\n\ninductive_set\n tiling :: \"'a set set ⇒ 'a set set\"\n for A :: \"'a set set\" where\nempty [simp, intro]: \"{} ∈ tiling A\" |\nUn [simp, intro]: \"⟦ a ∈ A; t ∈ tiling A; a ∩ t = {} ⟧\n ⟹ a ∪ t ∈ tiling A\"\n\n\nlemma tiling_UnI [intro]:\n \"⟦ t ∈ tiling A; u ∈ tiling A; t ∩ u = {} ⟧ ⟹ t ∪ u ∈ tiling A\"\napply (induct set: tiling)\n(*goals:\n 1. ‹⟦u ∈ tiling A; {} ∩ u = {}⟧ ⟹ {} ∪ u ∈ tiling A›\n 2. ‹⋀a t. ⟦a ∈ A; t ∈ tiling A; ⟦u ∈ tiling A; t ∩ u = {}⟧ ⟹ t ∪ u ∈ tiling A; a ∩ t = {}; u ∈ tiling A; (a ∪ t) ∩ u = {}⟧ ⟹ a ∪ t ∪ u ∈ tiling A›\ndiscuss goal 1*)\n apply ((auto simp add: Un_assoc (*‹?A ∪ ?B ∪ ?C = ?A ∪ (?B ∪ ?C)›*))[1])\n(*discuss goal 2*)\n apply ((auto simp add: Un_assoc (*‹?A ∪ ?B ∪ ?C = ?A ∪ (?B ∪ ?C)›*))[1])\n(*proven 2 subgoals*) .\n\nlemma tiling_Diff1E:\nassumes \"t-a ∈ tiling A\" and \"a ∈ A\" and \"a ⊆ t\"\nshows \"t ∈ tiling A\"\nproof (-)\n (*goal: ‹(t::'a set) ∈ tiling (A::'a set set)›*)\n from assms(2-3) (*‹a ∈ A› ‹a ⊆ t›*) have \"∃r. t = r Un a & r Int a = {}\"\n by (metis Diff_disjoint (*‹?A ∩ (?B - ?A) = {}›*) Int_commute (*‹?A ∩ ?B = ?B ∩ ?A›*) Un_Diff_cancel (*‹?A ∪ (?B - ?A) = ?A ∪ ?B›*) Un_absorb1 (*‹?A ⊆ ?B ⟹ ?A ∪ ?B = ?B›*) Un_commute (*‹?A ∪ ?B = ?B ∪ ?A›*))\n thus \"?thesis\"\n (*goal: ‹t ∈ tiling A›*)\n using assms(1,2) (*‹t - a ∈ tiling A› ‹a ∈ A›*) apply (auto simp:Un_Diff (*‹?A ∪ ?B - ?C = ?A - ?C ∪ (?B - ?C)›*))\n (*goal: ‹t ∈ tiling A›*)\n by (metis Compl_Diff_eq (*‹- ((?A::?'a set) - (?B::?'a set)) = - ?A ∪ ?B›*) Diff_Compl (*‹(?A::?'a set) - - (?B::?'a set) = ?A ∩ ?B›*) Diff_empty (*‹(?A::?'a set) - {} = ?A›*) Int_commute (*‹(?A::?'a set) ∩ (?B::?'a set) = ?B ∩ ?A›*) Un_Diff_cancel (*‹(?A::?'a set) ∪ ((?B::?'a set) - ?A) = ?A ∪ ?B›*) Un_commute (*‹(?A::?'a set) ∪ (?B::?'a set) = ?B ∪ ?A›*) double_complement (*‹- (- (?A::?'a set)) = ?A›*) tiling.Un (*‹⟦(?a::?'a set) ∈ (?A::?'a set set); (?t::?'a set) ∈ tiling ?A; ?a ∩ ?t = {}⟧ ⟹ ?a ∪ ?t ∈ tiling ?A›*))\nqed\n\nlemma tiling_finite:\n assumes \"⋀a. a ∈ A ⟹ finite a\"\n shows \"t ∈ tiling A ⟹ finite t\"\napply (induct set: tiling)\n(*goal: ‹(t::'a::type set) ∈ tiling (A::'a::type set set) ⟹ finite t›*)\nusing assms (*‹?a ∈ A ⟹ finite ?a›*) apply -\n(*goals:\n 1. ‹(⋀a. a ∈ A ⟹ finite a) ⟹ finite {}›\n 2. ‹⋀a t. ⟦a ∈ A; t ∈ tiling A; finite t; a ∩ t = {}; ⋀a. a ∈ A ⟹ finite a⟧ ⟹ finite (a ∪ t)›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\n\nsection‹The Mutilated Chess Board Cannot be Tiled by Dominoes›\n\ntext ‹The originator of this problem is Max Black, according to J A\nRobinson. It was popularized as the \\emph{Mutilated Checkerboard Problem} by\nJ McCarthy.›\n\ninductive_set domino :: \"(nat × nat) set set\" where\nhoriz [simp]: \"{(i, j), (i, Suc j)} ∈ domino\" |\nvertl [simp]: \"{(i, j), (Suc i, j)} ∈ domino\"\n\nlemma domino_finite: \"d ∈ domino ⟹ finite d\"\napply (erule domino.cases (*‹⟦(?a::(nat × nat) set) ∈ domino; ⋀(i::nat) j::nat. ?a = {(i, j), (i, Suc j)} ⟹ ?P::bool; ⋀(i::nat) j::nat. ?a = {(i, j), (Suc i, j)} ⟹ ?P⟧ ⟹ ?P›*))\n(*goals:\n 1. ‹⋀i j. d = {(i, j), (i, Suc j)} ⟹ finite d›\n 2. ‹⋀i j. d = {(i, j), (Suc i, j)} ⟹ finite d›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\ndeclare tiling_finite[OF domino_finite, simp]\n\ntext ‹\\medskip Sets of squares of the given colour›\n\ndefinition\n coloured :: \"nat ⇒ (nat × nat) set\" where\n \"coloured b = {(i, j). (i + j) mod 2 = b}\"\n\nabbreviation\n whites :: \"(nat × nat) set\" where\n \"whites ≡ coloured 0\"\n\nabbreviation\n blacks :: \"(nat × nat) set\" where\n \"blacks ≡ coloured (Suc 0)\"\n\n\ntext ‹\\medskip Chess boards›\n\nlemma Sigma_Suc1 [simp]:\n \"{0..< Suc n} × B = ({n} × B) ∪ ({0..<n} × B)\"\nby auto\n\nlemma Sigma_Suc2 [simp]:\n \"A × {0..< Suc n} = (A × {n}) ∪ (A × {0..<n})\"\nby auto\n\nlemma dominoes_tile_row [intro!]: \"{i} × {0..< 2*n} ∈ tiling domino\"\napply (induct n)\n(*goals:\n 1. ‹{i} × {0..<2 * 0} ∈ tiling domino›\n 2. ‹⋀n. {i} × {0..<2 * n} ∈ tiling domino ⟹ {i} × {0..<2 * Suc n} ∈ tiling domino›\ndiscuss goal 1*)\n apply (simp del:Un_insert_left (*‹insert ?a ?B ∪ ?C = insert ?a (?B ∪ ?C)›*) add: Un_assoc [symmetric] (*‹?A ∪ (?B ∪ ?C) = ?A ∪ ?B ∪ ?C›*))\n(*discuss goal 2*)\n apply (simp del:Un_insert_left (*‹insert ?a ?B ∪ ?C = insert ?a (?B ∪ ?C)›*) add: Un_assoc [symmetric] (*‹?A ∪ (?B ∪ ?C) = ?A ∪ ?B ∪ ?C›*))\n(*proven 2 subgoals*) .\n\nlemma dominoes_tile_matrix: \"{0..<m} × {0..< 2*n} ∈ tiling domino\"\napply (induct m)\n(*goals:\n 1. ‹{0..<0} × {0..<2 * n} ∈ tiling domino›\n 2. ‹⋀m. {0..<m} × {0..<2 * n} ∈ tiling domino ⟹ {0..<Suc m} × {0..<2 * n} ∈ tiling domino›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\n\ntext ‹\\medskip @{term coloured} and Dominoes›\n\nlemma coloured_insert [simp]:\n \"coloured b ∩ (insert (i, j) t) =\n (if (i + j) mod 2 = b then insert (i, j) (coloured b ∩ t)\n else coloured b ∩ t)\"\nby (auto simp add: coloured_def (*‹coloured ?b = {(i, j). (i + j) mod 2 = ?b}›*))\n\nlemma domino_singletons:\n \"d ∈ domino ⟹\n (∃i j. whites ∩ d = {(i, j)}) ∧\n (∃m n. blacks ∩ d = {(m, n)})\"\napply (erule domino.cases (*‹⟦(?a::(nat × nat) set) ∈ domino; ⋀(i::nat) j::nat. ?a = {(i, j), (i, Suc j)} ⟹ ?P::bool; ⋀(i::nat) j::nat. ?a = {(i, j), (Suc i, j)} ⟹ ?P⟧ ⟹ ?P›*))\n(*goals:\n 1. ‹⋀i j. d = {(i, j), (i, Suc j)} ⟹ (∃i j. whites ∩ d = {(i, j)}) ∧ (∃m n. blacks ∩ d = {(m, n)})›\n 2. ‹⋀i j. d = {(i, j), (Suc i, j)} ⟹ (∃i j. whites ∩ d = {(i, j)}) ∧ (∃m n. blacks ∩ d = {(m, n)})›\ndiscuss goal 1*)\n apply ((auto simp add: mod_Suc (*‹Suc ?m mod ?n = (if Suc (?m mod ?n) = ?n then 0 else Suc (?m mod ?n))›*))[1])\n(*discuss goal 2*)\n apply ((auto simp add: mod_Suc (*‹Suc (?m::nat) mod (?n::nat) = (if Suc (?m mod ?n) = ?n then 0::nat else Suc (?m mod ?n))›*))[1])\n(*proven 2 subgoals*) .\n\n\ntext ‹\\medskip Tilings of dominoes›\n\ndeclare\n Int_Un_distrib [simp]\n Diff_Int_distrib [simp]\n\nlemma tiling_domino_0_1:\n \"t ∈ tiling domino ==> card(whites ∩ t) = card(blacks ∩ t)\"\napply (induct set: tiling)\n(*goal: ‹(t::(nat × nat) set) ∈ tiling domino ⟹ card (whites ∩ t) = card (blacks ∩ t)›*)\napply (drule_tac [2] domino_singletons (*‹?d ∈ domino ⟹ (∃i j. whites ∩ ?d = {(i, j)}) ∧ (∃m n. blacks ∩ ?d = {(m, n)})›*))\n(*top goal: ‹card (whites ∩ {}) = card (blacks ∩ {})› and 1 goal remains*)\napply auto\n(*top goal: ‹card (whites ∩ {}) = card (blacks ∩ {})› and 1 goal remains*)\napply (subgoal_tac \"∀p C. C ∩ a = {p} --> p ∉ t\")\n(*goal: ‹⋀a t i m j n. ⟦t ∈ tiling domino; card (whites ∩ t) = card (blacks ∩ t); a ∩ t = {}; whites ∩ a = {(i, j)}; blacks ∩ a = {(m, n)}⟧ ⟹ card (insert (i, j) (whites ∩ t)) = card (insert (m, n) (blacks ∩ t))›*)\napply (simp (no_asm_simp))\n(*top goal: ‹⋀a t i m j n. ⟦t ∈ tiling domino; card (whites ∩ t) = card (blacks ∩ t); a ∩ t = {}; whites ∩ a = {(i, j)}; blacks ∩ a = {(m, n)}; ∀p C. C ∩ a = {p} ⟶ p ∉ t⟧ ⟹ card (insert (i, j) (whites ∩ t)) = card (insert (m, n) (blacks ∩ t))› and 1 goal remains*)\nby blast\n\n\ntext ‹\\medskip Final argument is surprisingly complex›\n\ntheorem gen_mutil_not_tiling:\n \"t ∈ tiling domino ==>\n (i + j) mod 2 = 0 ==> (m + n) mod 2 = 0 ==>\n {(i, j), (m, n)} ⊆ t\n ==> (t - {(i,j)} - {(m,n)}) ∉ tiling domino\"\napply (rule notI (*‹(?P ⟹ False) ⟹ ¬ ?P›*))\n(*goal: ‹⟦t ∈ tiling domino; (i + j) mod 2 = 0; (m + n) mod 2 = 0; {(i, j), (m, n)} ⊆ t⟧ ⟹ t - {(i, j)} - {(m, n)} ∉ tiling domino›*)\napply (subgoal_tac \"card (whites ∩ (t - {(i,j)} - {(m,n)})) <\n card (blacks ∩ (t - {(i,j)} - {(m,n)}))\")\n(*goals:\n 1. ‹⟦t ∈ tiling domino; (i + j) mod 2 = 0; (m + n) mod 2 = 0; {(i, j), (m, n)} ⊆ t; t - {(i, j)} - {(m, n)} ∈ tiling domino; card (whites ∩ (t - {(i, j)} - {(m, n)})) < card (blacks ∩ (t - {(i, j)} - {(m, n)}))⟧ ⟹ False›\n 2. ‹⟦t ∈ tiling domino; (i + j) mod 2 = 0; (m + n) mod 2 = 0; {(i, j), (m, n)} ⊆ t; t - {(i, j)} - {(m, n)} ∈ tiling domino⟧ ⟹ card (whites ∩ (t - {(i, j)} - {(m, n)})) < card (blacks ∩ (t - {(i, j)} - {(m, n)}))›\ndiscuss goal 1*)\n apply (force simp only: tiling_domino_0_1 (*‹?t ∈ tiling domino ⟹ card (whites ∩ ?t) = card (blacks ∩ ?t)›*))\n(*discuss goal 2*)\n apply (simp add: tiling_domino_0_1 [symmetric] (*‹?t ∈ tiling domino ⟹ card (blacks ∩ ?t) = card (whites ∩ ?t)›*))\n (*goal: ‹⟦t ∈ tiling domino; (i + j) mod 2 = 0; (m + n) mod 2 = 0; {(i, j), (m, n)} ⊆ t; t - {(i, j)} - {(m, n)} ∈ tiling domino⟧ ⟹ card (whites ∩ (t - {(i, j)} - {(m, n)})) < card (blacks ∩ (t - {(i, j)} - {(m, n)}))›*)\n apply (simp add: coloured_def (*‹coloured ?b = {(i, j). (i + j) mod 2 = ?b}›*) card_Diff2_less (*‹⟦finite ?A; ?x ∈ ?A; ?y ∈ ?A⟧ ⟹ card (?A - {?x} - {?y}) < card ?A›*))\n(*proven 2 subgoals*) .\n\ntext ‹Apply the general theorem to the well-known case›\n\ntheorem mutil_not_tiling:\n \"t = {0..< 2 * Suc m} × {0..< 2 * Suc n}\n ==> t - {(0,0)} - {(Suc(2 * m), Suc(2 * n))} ∉ tiling domino\"\napply (rule gen_mutil_not_tiling (*‹⟦?t ∈ tiling domino; (?i + ?j) mod 2 = 0; (?m + ?n) mod 2 = 0; {(?i, ?j), (?m, ?n)} ⊆ ?t⟧ ⟹ ?t - {(?i, ?j)} - {(?m, ?n)} ∉ tiling domino›*))\n(*goals:\n 1. ‹t = {0..<2 * Suc m} × {0..<2 * Suc n} ⟹ t ∈ tiling domino›\n 2. ‹t = {0..<2 * Suc m} × {0..<2 * Suc n} ⟹ (0 + 0) mod 2 = 0›\n 3. ‹t = {0..<2 * Suc m} × {0..<2 * Suc n} ⟹ (Suc (2 * m) + Suc (2 * n)) mod 2 = 0›\n 4. ‹t = {0..<2 * Suc m} × {0..<2 * Suc n} ⟹ {(0, 0), (Suc (2 * m), Suc (2 * n))} ⊆ t›\ndiscuss goal 1*)\n apply (blast intro!: dominoes_tile_matrix (*‹{0..<?m} × {0..<2 * ?n} ∈ tiling domino›*))\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*discuss goal 4*)\n apply ((auto)[1])\n(*proven 4 subgoals*) .\n\n\nsection‹The Mutilated Chess Board Can be Tiled by Ls›\n\ntext‹Remove a arbitrary square from a chess board of size $2^n \\times 2^n$.\nThe result can be tiled by L-shaped tiles:\n\\begin{picture}(8,8)\n\\put(0,0){\\framebox(4,4){}}\n\\put(4,0){\\framebox(4,4){}}\n\\put(0,4){\\framebox(4,4){}}\n\\end{picture}.\nThe four possible L-shaped tiles are obtained by dropping\none of the four squares from $\\{(x,y),(x+1,y),(x,y+1),(x+1,y+1)\\}$:›\n\ndefinition \"L2 (x::nat) (y::nat) = {(x,y), (x+1,y), (x, y+1)}\"\ndefinition \"L3 (x::nat) (y::nat) = {(x,y), (x+1,y), (x+1, y+1)}\"\ndefinition \"L0 (x::nat) (y::nat) = {(x+1,y), (x,y+1), (x+1, y+1)}\"\ndefinition \"L1 (x::nat) (y::nat) = {(x,y), (x,y+1), (x+1, y+1)}\"\n\ntext‹All tiles:›\n\ndefinition Ls :: \"(nat * nat) set set\" where\n\"Ls ≡ { L0 x y | x y. True} ∪ { L1 x y | x y. True} ∪\n { L2 x y | x y. True} ∪ { L3 x y | x y. True}\"\n\nlemma LinLs: \"L0 i j : Ls & L1 i j : Ls & L2 i j : Ls & L3 i j : Ls\"\nby (fastforce simp:Ls_def (*‹Ls ≡ {L0 x y |x y. True} ∪ {L1 x y |x y. True} ∪ {L2 x y |x y. True} ∪ {L3 x y |x y. True}›*))\n\n\ntext‹Square $2^n \\times 2^n$ grid, shifted by $i$ and $j$:›\n\ndefinition \"square2 (n::nat) (i::nat) (j::nat) = {i..< 2^n+i} × {j..< 2^n+j}\"\n\nlemma in_square2[simp]:\n \"(a,b) : square2 n i j ⟷ i≤a ∧ a<2^n+i ∧ j≤b ∧ b<2^n+j\"\nby (simp add:square2_def (*‹square2 ?n ?i ?j = {?i..<2 ^ ?n + ?i} × {?j..<2 ^ ?n + ?j}›*))\n\nlemma square2_Suc: \"square2 (Suc n) i j =\n square2 n i j ∪ square2 n (2^n + i) j ∪ square2 n i (2^n + j) ∪\n square2 n (2^n + i) (2^n + j)\"\nby (auto simp:square2_def (*‹square2 ?n ?i ?j = {?i..<2 ^ ?n + ?i} × {?j..<2 ^ ?n + ?j}›*))\n\nlemma square2_disj: \"square2 n i j ∩ square2 n x y = {} ⟷\n (2^n+i ≤ x ∨ 2^n+x ≤ i) ∨ (2^n+j ≤ y ∨ 2^n+y ≤ j)\" (is \"?A = ?B\")\nproof (-)\n (*goal: ‹(square2 n i j ∩ square2 n x y = {}) = ((2 ^ n + i ≤ x ∨ 2 ^ n + x ≤ i) ∨ 2 ^ n + j ≤ y ∨ 2 ^ n + y ≤ j)›*)\n {\n assume \"?B\" (*‹((2::nat) ^ (n::nat) + (i::nat) ≤ (x::nat) ∨ (2::nat) ^ n + x ≤ i) ∨ (2::nat) ^ n + (j::nat) ≤ (y::nat) ∨ (2::nat) ^ n + y ≤ j›*)\n hence \"?A\"\n by (auto simp:square2_def (*‹square2 ?n ?i ?j = {?i..<2 ^ ?n + ?i} × {?j..<2 ^ ?n + ?j}›*))\n }\n moreover {\n assume \"¬ ?B\" (*‹¬ (((2::nat) ^ (n::nat) + (i::nat) ≤ (x::nat) ∨ (2::nat) ^ n + x ≤ i) ∨ (2::nat) ^ n + (j::nat) ≤ (y::nat) ∨ (2::nat) ^ n + y ≤ j)›*)\n hence \"(max i x, max j y) : square2 n i j ∩ square2 n x y\"\n by simp\n hence \"¬ ?A\"\n by blast\n }\n ultimately show \"?thesis\"\n (*goal: ‹(square2 n i j ∩ square2 n x y = {}) = ((2 ^ n + i ≤ x ∨ 2 ^ n + x ≤ i) ∨ 2 ^ n + j ≤ y ∨ 2 ^ n + y ≤ j)›*)\n by blast\nqed\n\ntext‹Some specific lemmas:›\n\nlemma pos_pow2: \"(0::nat) < 2^(n::nat)\"\nby simp\n\ndeclare nat_zero_less_power_iff[simp del] zero_less_power[simp del]\n\nlemma Diff_insert_if: shows\n \"B ≠ {} ⟹ a:A ⟹ A - insert a B = (A-B - {a})\" and\n \"B ≠ {} ⟹ a ~: A ⟹ A - insert a B = A-B\"\n\n(*goals:\n 1. ‹⟦B ≠ {}; a ∈ A⟧ ⟹ A - insert a B = A - B - {a}›\n 2. ‹⟦B ≠ {}; a ∉ A⟧ ⟹ A - insert a B = A - B›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*proven 2 subgoals*) .\n\nlemma DisjI1: \"A Int B = {} ⟹ (A-X) Int B = {}\"\nby blast\nlemma DisjI2: \"A Int B = {} ⟹ A Int (B-X) = {}\"\nby blast\n\ntext‹The main theorem:›\n\ntheorem Ls_can_tile: \"i ≤ a ⟹ a < 2^n + i ⟹ j ≤ b ⟹ b < 2^n + j\n ⟹ square2 n i j - {(a,b)} : tiling Ls\"\nproof (induct n arbitrary: a b i j)\n (*goals:\n 1. ‹⋀a b i j. ⟦i ≤ a; a < 2 ^ 0 + i; j ≤ b; b < 2 ^ 0 + j⟧ ⟹ square2 0 i j - {(a, b)} ∈ tiling Ls›\n 2. ‹⋀n a b i j. ⟦⋀a b i j. ⟦i ≤ a; a < 2 ^ n + i; j ≤ b; b < 2 ^ n + j⟧ ⟹ square2 n i j - {(a, b)} ∈ tiling Ls; i ≤ a; a < 2 ^ Suc n + i; j ≤ b; b < 2 ^ Suc n + j⟧ ⟹ square2 (Suc n) i j - {(a, b)} ∈ tiling Ls›*)\n case 0 (*‹i ≤ a› ‹a < 2 ^ 0 + i› ‹j ≤ b› ‹b < 2 ^ 0 + j›*)\n thus \"?case\"\n (*goal: ‹square2 0 i j - {(a, b)} ∈ tiling Ls›*)\n by (simp add:square2_def (*‹square2 ?n ?i ?j = {?i..<2 ^ ?n + ?i} × {?j..<2 ^ ?n + ?j}›*))\nnext\n (*goal: ‹⋀n a b i j. ⟦⋀a b i j. ⟦i ≤ a; a < 2 ^ n + i; j ≤ b; b < 2 ^ n + j⟧ ⟹ square2 n i j - {(a, b)} ∈ tiling Ls; i ≤ a; a < 2 ^ Suc n + i; j ≤ b; b < 2 ^ Suc n + j⟧ ⟹ square2 (Suc n) i j - {(a, b)} ∈ tiling Ls›*)\n case (Suc n) (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls› ‹i ≤ a› ‹a < 2 ^ Suc n + i› ‹j ≤ b› ‹b < 2 ^ Suc n + j›*)\n note IH = Suc(1) (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*) and a = Suc(2-3) (*‹(i::nat) ≤ (a::nat)› ‹a < 2 ^ Suc n + i›*) and b = Suc(4-5) (*‹(j::nat) ≤ (b::nat)› ‹(b::nat) < (2::nat) ^ Suc (n::nat) + (j::nat)›*)\n hence \"a<2^n+i ∧ b<2^n+j ∨\n 2^n+i≤a ∧ a<2^(n+1)+i ∧ b<2^n+j ∨\n a<2^n+i ∧ 2^n+j≤b ∧ b<2^(n+1)+j ∨\n 2^n+i≤a ∧ a<2^(n+1)+i ∧ 2^n+j≤b ∧ b<2^(n+1)+j\" (is \"?A|?B|?C|?D\")\n apply simp\n (*goal: ‹(a::nat) < (2::nat) ^ (n::nat) + (i::nat) ∧ (b::nat) < (2::nat) ^ n + (j::nat) ∨ (2::nat) ^ n + i ≤ a ∧ a < (2::nat) ^ (n + (1::nat)) + i ∧ b < (2::nat) ^ n + j ∨ a < (2::nat) ^ n + i ∧ (2::nat) ^ n + j ≤ b ∧ b < (2::nat) ^ (n + (1::nat)) + j ∨ (2::nat) ^ n + i ≤ a ∧ a < (2::nat) ^ (n + (1::nat)) + i ∧ (2::nat) ^ n + j ≤ b ∧ b < (2::nat) ^ (n + (1::nat)) + j›*)\n by arith\n moreover {\n assume \"?A\" (*‹(a::nat) < (2::nat) ^ (n::nat) + (i::nat) ∧ (b::nat) < (2::nat) ^ n + (j::nat)›*)\n hence \"square2 n i j - {(a,b)} : tiling Ls\"\n using IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*) a (*‹i ≤ a› ‹(a::nat) < (2::nat) ^ Suc (n::nat) + (i::nat)›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) by auto\n moreover have \"square2 n (2^n+i) j - {(2^n+i,2^n+j - 1)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹2 ^ n + i ≤ 2 ^ n + i›\n 2. ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)›\n 3. ‹j ≤ 2 ^ n + j - 1›\n 4. ‹2 ^ n + j - 1 < 2 ^ n + j›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(2::nat) ^ (n::nat) + (i::nat) ≤ (2::nat) ^ n + i› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*top goal: ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*top goal: ‹j ≤ 2 ^ n + j - 1› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j - 1 < 2 ^ n + j›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n i (2^n+j) - {(2^n+i - 1, 2^n+j)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹i ≤ 2 ^ n + i - 1›\n 2. ‹2 ^ n + i - 1 < 2 ^ n + i›\n 3. ‹2 ^ n + j ≤ 2 ^ n + j›\n 4. ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹i ≤ 2 ^ n + i - 1› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i - 1 < 2 ^ n + i› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + j ≤ 2 ^ n + j› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n (2^n+i) (2^n+j) - {(2^n+i, 2^n+j)} : tiling Ls\"\n apply (rule IH (*‹⟦(?i::nat) ≤ (?a::nat); ?a < (2::nat) ^ (n::nat) + ?i; (?j::nat) ≤ (?b::nat); ?b < (2::nat) ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹2 ^ n + i ≤ 2 ^ n + i›\n 2. ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)›\n 3. ‹2 ^ n + j ≤ 2 ^ n + j›\n 4. ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i ≤ 2 ^ n + i› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(2::nat) ^ (n::nat) + (i::nat) < (2::nat) ^ n + ((2::nat) ^ n + i)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(2::nat) ^ (n::nat) + (j::nat) ≤ (2::nat) ^ n + j› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n ultimately have \"square2 (n+1) i j - {(a,b)} - L0 (2^n+i - 1) (2^n+j - 1) ∈ tiling Ls\"\n using a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) ‹?A› (*‹a < 2 ^ n + i ∧ b < 2 ^ n + j›*) apply -\n (*goal: ‹square2 (n + 1) i j - {(a, b)} - L0 (2 ^ n + i - 1) (2 ^ n + j - 1) ∈ tiling Ls›*)\n apply (clarsimp simp: square2_Suc (*‹square2 (Suc (?n::nat)) (?i::nat) (?j::nat) = square2 ?n ?i ?j ∪ square2 ?n ((2::nat) ^ ?n + ?i) ?j ∪ square2 ?n ?i ((2::nat) ^ ?n + ?j) ∪ square2 ?n ((2::nat) ^ ?n + ?i) ((2::nat) ^ ?n + ?j)›*) L0_def (*‹L0 (?x::nat) (?y::nat) = {(?x + (1::nat), ?y), (?x, ?y + (1::nat)), (?x + (1::nat), ?y + (1::nat))}›*) Un_Diff (*‹(?A::?'a set) ∪ (?B::?'a set) - (?C::?'a set) = ?A - ?C ∪ (?B - ?C)›*) Diff_insert_if (*‹⟦(?B::?'a set) ≠ {}; (?a::?'a) ∈ (?A::?'a set)⟧ ⟹ ?A - insert ?a ?B = ?A - ?B - {?a}› ‹⟦(?B::?'a set) ≠ {}; (?a::?'a) ∉ (?A::?'a set)⟧ ⟹ ?A - insert ?a ?B = ?A - ?B›*))\n (*goal: ‹⟦square2 n i j - {(a, b)} ∈ tiling Ls; square2 n (2 ^ n + i) j - {(2 ^ n + i, 2 ^ n + j - 1)} ∈ tiling Ls; square2 n i (2 ^ n + j) - {(2 ^ n + i - 1, 2 ^ n + j)} ∈ tiling Ls; square2 n (2 ^ n + i) (2 ^ n + j) - {(2 ^ n + i, 2 ^ n + j)} ∈ tiling Ls; i ≤ a; a < 2 ^ Suc n + i; j ≤ b; b < 2 ^ Suc n + j; a < 2 ^ n + i ∧ b < 2 ^ n + j⟧ ⟹ square2 (n + 1) i j - {(a, b)} - L0 (2 ^ n + i - 1) (2 ^ n + j - 1) ∈ tiling Ls›*)\n by (fastforce intro!: tiling_UnI (*‹⟦?t ∈ tiling ?A; ?u ∈ tiling ?A; ?t ∩ ?u = {}⟧ ⟹ ?t ∪ ?u ∈ tiling ?A›*) DisjI1 (*‹?A ∩ ?B = {} ⟹ (?A - ?X) ∩ ?B = {}›*) DisjI2 (*‹?A ∩ ?B = {} ⟹ ?A ∩ (?B - ?X) = {}›*) square2_disj[THEN iffD2] (*‹(2 ^ ?n1 + ?i1 ≤ ?x1 ∨ 2 ^ ?n1 + ?x1 ≤ ?i1) ∨ 2 ^ ?n1 + ?j1 ≤ ?y1 ∨ 2 ^ ?n1 + ?y1 ≤ ?j1 ⟹ square2 ?n1 ?i1 ?j1 ∩ square2 ?n1 ?x1 ?y1 = {}›*) simp:Int_Un_distrib2 (*‹(?B ∪ ?C) ∩ ?A = ?B ∩ ?A ∪ ?C ∩ ?A›*))\n }\n moreover {\n assume \"?B\" (*‹(2::nat) ^ (n::nat) + (i::nat) ≤ (a::nat) ∧ a < (2::nat) ^ (n + (1::nat)) + i ∧ (b::nat) < (2::nat) ^ n + (j::nat)›*)\n hence \"square2 n (2^n+i) j - {(a,b)} : tiling Ls\"\n using IH (*‹⟦(?i::nat) ≤ (?a::nat); ?a < (2::nat) ^ (n::nat) + ?i; (?j::nat) ≤ (?b::nat); ?b < (2::nat) ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*) a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) by auto\n moreover have \"square2 n i j - {(2^n+i - 1,2^n+j - 1)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹i ≤ 2 ^ n + i - 1›\n 2. ‹2 ^ n + i - 1 < 2 ^ n + i›\n 3. ‹j ≤ 2 ^ n + j - 1›\n 4. ‹2 ^ n + j - 1 < 2 ^ n + j›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹i ≤ 2 ^ n + i - 1› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i - 1 < 2 ^ n + i› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹j ≤ 2 ^ n + j - 1› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j - 1 < 2 ^ n + j›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n i (2^n+j) - {(2^n+i - 1, 2^n+j)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹i ≤ 2 ^ n + i - 1›\n 2. ‹2 ^ n + i - 1 < 2 ^ n + i›\n 3. ‹2 ^ n + j ≤ 2 ^ n + j›\n 4. ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹i ≤ 2 ^ n + i - 1› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*top goal: ‹2 ^ n + i - 1 < 2 ^ n + i› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*top goal: ‹2 ^ n + j ≤ 2 ^ n + j› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*goal: ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n (2^n+i) (2^n+j) - {(2^n+i, 2^n+j)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹2 ^ n + i ≤ 2 ^ n + i›\n 2. ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)›\n 3. ‹2 ^ n + j ≤ 2 ^ n + j›\n 4. ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i ≤ 2 ^ n + i› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + j ≤ 2 ^ n + j› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*goal: ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n ultimately have \"square2 (n+1) i j - {(a,b)} - L1 (2^n+i - 1) (2^n+j - 1) ∈ tiling Ls\"\n using a (*‹i ≤ a› ‹(a::nat) < (2::nat) ^ Suc (n::nat) + (i::nat)›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) ‹?B› (*‹2 ^ n + i ≤ a ∧ a < 2 ^ (n + 1) + i ∧ b < 2 ^ n + j›*) apply (simp add: square2_Suc (*‹square2 (Suc ?n) ?i ?j = square2 ?n ?i ?j ∪ square2 ?n (2 ^ ?n + ?i) ?j ∪ square2 ?n ?i (2 ^ ?n + ?j) ∪ square2 ?n (2 ^ ?n + ?i) (2 ^ ?n + ?j)›*) L1_def (*‹L1 ?x ?y = {(?x, ?y), (?x, ?y + 1), (?x + 1, ?y + 1)}›*) Un_Diff (*‹?A ∪ ?B - ?C = ?A - ?C ∪ (?B - ?C)›*) Diff_insert_if (*‹⟦?B ≠ {}; ?a ∈ ?A⟧ ⟹ ?A - insert ?a ?B = ?A - ?B - {?a}› ‹⟦?B ≠ {}; ?a ∉ ?A⟧ ⟹ ?A - insert ?a ?B = ?A - ?B›*) le_diff_conv2 (*‹?k ≤ ?j ⟹ (?i ≤ ?j - ?k) = (?i + ?k ≤ ?j)›*))\n (*goal: ‹square2 (n + 1) i j - {(a, b)} - L1 (2 ^ n + i - 1) (2 ^ n + j - 1) ∈ tiling Ls›*)\n by (fastforce intro!: tiling_UnI (*‹⟦?t ∈ tiling ?A; ?u ∈ tiling ?A; ?t ∩ ?u = {}⟧ ⟹ ?t ∪ ?u ∈ tiling ?A›*) DisjI1 (*‹?A ∩ ?B = {} ⟹ (?A - ?X) ∩ ?B = {}›*) DisjI2 (*‹?A ∩ ?B = {} ⟹ ?A ∩ (?B - ?X) = {}›*) square2_disj[THEN iffD2] (*‹(2 ^ ?n1 + ?i1 ≤ ?x1 ∨ 2 ^ ?n1 + ?x1 ≤ ?i1) ∨ 2 ^ ?n1 + ?j1 ≤ ?y1 ∨ 2 ^ ?n1 + ?y1 ≤ ?j1 ⟹ square2 ?n1 ?i1 ?j1 ∩ square2 ?n1 ?x1 ?y1 = {}›*) simp:Int_Un_distrib2 (*‹(?B ∪ ?C) ∩ ?A = ?B ∩ ?A ∪ ?C ∩ ?A›*))\n }\n moreover {\n assume \"?C\" (*‹(a::nat) < (2::nat) ^ (n::nat) + (i::nat) ∧ (2::nat) ^ n + (j::nat) ≤ (b::nat) ∧ b < (2::nat) ^ (n + (1::nat)) + j›*)\n hence \"square2 n i (2^n+j) - {(a,b)} : tiling Ls\"\n using IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*) a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) by auto\n moreover have \"square2 n i j - {(2^n+i - 1,2^n+j - 1)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹i ≤ 2 ^ n + i - 1›\n 2. ‹2 ^ n + i - 1 < 2 ^ n + i›\n 3. ‹j ≤ 2 ^ n + j - 1›\n 4. ‹2 ^ n + j - 1 < 2 ^ n + j›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹i ≤ 2 ^ n + i - 1› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i - 1 < 2 ^ n + i› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(j::nat) ≤ (2::nat) ^ (n::nat) + j - (1::nat)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j - 1 < 2 ^ n + j›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n (2^n+i) j - {(2^n+i, 2^n+j - 1)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹2 ^ n + i ≤ 2 ^ n + i›\n 2. ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)›\n 3. ‹j ≤ 2 ^ n + j - 1›\n 4. ‹2 ^ n + j - 1 < 2 ^ n + j›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i ≤ 2 ^ n + i› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(2::nat) ^ (n::nat) + (i::nat) < (2::nat) ^ n + ((2::nat) ^ n + i)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(j::nat) ≤ (2::nat) ^ (n::nat) + j - (1::nat)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j - 1 < 2 ^ n + j›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n (2^n+i) (2^n+j) - {(2^n+i, 2^n+j)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹2 ^ n + i ≤ 2 ^ n + i›\n 2. ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)›\n 3. ‹2 ^ n + j ≤ 2 ^ n + j›\n 4. ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i ≤ 2 ^ n + i› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*top goal: ‹2 ^ n + j ≤ 2 ^ n + j› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n ultimately have \"square2 (n+1) i j - {(a,b)} - L3 (2^n+i - 1) (2^n+j - 1) ∈ tiling Ls\"\n using a (*‹i ≤ a› ‹(a::nat) < (2::nat) ^ Suc (n::nat) + (i::nat)›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) ‹?C› (*‹a < 2 ^ n + i ∧ 2 ^ n + j ≤ b ∧ b < 2 ^ (n + 1) + j›*) apply (simp add: square2_Suc (*‹square2 (Suc (?n::nat)) (?i::nat) (?j::nat) = square2 ?n ?i ?j ∪ square2 ?n ((2::nat) ^ ?n + ?i) ?j ∪ square2 ?n ?i ((2::nat) ^ ?n + ?j) ∪ square2 ?n ((2::nat) ^ ?n + ?i) ((2::nat) ^ ?n + ?j)›*) L3_def (*‹L3 (?x::nat) (?y::nat) = {(?x, ?y), (?x + (1::nat), ?y), (?x + (1::nat), ?y + (1::nat))}›*) Un_Diff (*‹(?A::?'a set) ∪ (?B::?'a set) - (?C::?'a set) = ?A - ?C ∪ (?B - ?C)›*) Diff_insert_if (*‹⟦(?B::?'a set) ≠ {}; (?a::?'a) ∈ (?A::?'a set)⟧ ⟹ ?A - insert ?a ?B = ?A - ?B - {?a}› ‹⟦(?B::?'a set) ≠ {}; (?a::?'a) ∉ (?A::?'a set)⟧ ⟹ ?A - insert ?a ?B = ?A - ?B›*) le_diff_conv2 (*‹(?k::nat) ≤ (?j::nat) ⟹ ((?i::nat) ≤ ?j - ?k) = (?i + ?k ≤ ?j)›*))\n (*goal: ‹square2 (n + 1) i j - {(a, b)} - L3 (2 ^ n + i - 1) (2 ^ n + j - 1) ∈ tiling Ls›*)\n by (fastforce intro!: tiling_UnI (*‹⟦?t ∈ tiling ?A; ?u ∈ tiling ?A; ?t ∩ ?u = {}⟧ ⟹ ?t ∪ ?u ∈ tiling ?A›*) DisjI1 (*‹?A ∩ ?B = {} ⟹ (?A - ?X) ∩ ?B = {}›*) DisjI2 (*‹?A ∩ ?B = {} ⟹ ?A ∩ (?B - ?X) = {}›*) square2_disj[THEN iffD2] (*‹(2 ^ ?n1 + ?i1 ≤ ?x1 ∨ 2 ^ ?n1 + ?x1 ≤ ?i1) ∨ 2 ^ ?n1 + ?j1 ≤ ?y1 ∨ 2 ^ ?n1 + ?y1 ≤ ?j1 ⟹ square2 ?n1 ?i1 ?j1 ∩ square2 ?n1 ?x1 ?y1 = {}›*) simp:Int_Un_distrib2 (*‹(?B ∪ ?C) ∩ ?A = ?B ∩ ?A ∪ ?C ∩ ?A›*))\n }\n moreover {\n assume \"?D\" (*‹(2::nat) ^ (n::nat) + (i::nat) ≤ (a::nat) ∧ a < (2::nat) ^ (n + (1::nat)) + i ∧ (2::nat) ^ n + (j::nat) ≤ (b::nat) ∧ b < (2::nat) ^ (n + (1::nat)) + j›*)\n hence \"square2 n (2^n+i) (2^n+j) -{(a,b)} : tiling Ls\"\n using IH (*‹⟦(?i::nat) ≤ (?a::nat); ?a < (2::nat) ^ (n::nat) + ?i; (?j::nat) ≤ (?b::nat); ?b < (2::nat) ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*) a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) by auto\n moreover have \"square2 n i j - {(2^n+i - 1,2^n+j - 1)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹i ≤ 2 ^ n + i - 1›\n 2. ‹2 ^ n + i - 1 < 2 ^ n + i›\n 3. ‹j ≤ 2 ^ n + j - 1›\n 4. ‹2 ^ n + j - 1 < 2 ^ n + j›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(i::nat) ≤ (2::nat) ^ (n::nat) + i - (1::nat)› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i - 1 < 2 ^ n + i› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹j ≤ 2 ^ n + j - 1› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j - 1 < 2 ^ n + j›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n (2^n+i) j - {(2^n+i, 2^n+j - 1)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹2 ^ n + i ≤ 2 ^ n + i›\n 2. ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)›\n 3. ‹j ≤ 2 ^ n + j - 1›\n 4. ‹2 ^ n + j - 1 < 2 ^ n + j›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i ≤ 2 ^ n + i› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i < 2 ^ n + (2 ^ n + i)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹(0::nat) < (2::nat) ^ (n::nat)›*))[1])\n (*top goal: ‹(j::nat) ≤ (2::nat) ^ (n::nat) + j - (1::nat)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j - 1 < 2 ^ n + j›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n moreover have \"square2 n i (2^n+j) - {(2^n+i - 1, 2^n+j)} : tiling Ls\"\n apply (rule IH (*‹⟦?i ≤ ?a; ?a < 2 ^ n + ?i; ?j ≤ ?b; ?b < 2 ^ n + ?j⟧ ⟹ square2 n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n (*goals:\n 1. ‹i ≤ 2 ^ n + i - 1›\n 2. ‹2 ^ n + i - 1 < 2 ^ n + i›\n 3. ‹2 ^ n + j ≤ 2 ^ n + j›\n 4. ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›\n discuss goal 1*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹(i::nat) ≤ (2::nat) ^ (n::nat) + i - (1::nat)› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + i - 1 < 2 ^ n + i› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*top goal: ‹2 ^ n + j ≤ 2 ^ n + j› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert pos_pow2[of n] (*‹0 < 2 ^ n›*))[1])\n (*goal: ‹2 ^ n + j < 2 ^ n + (2 ^ n + j)›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n ultimately have \"square2 (n+1) i j - {(a,b)} - L2 (2^n+i - 1) (2^n+j - 1) ∈ tiling Ls\"\n using a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) ‹?D› (*‹2 ^ n + i ≤ a ∧ a < 2 ^ (n + 1) + i ∧ 2 ^ n + j ≤ b ∧ b < 2 ^ (n + 1) + j›*) by (msorry)\n }\n moreover have \"?A ⟹ L0 (2^n + i - 1) (2^n + j - 1) ⊆ square2 (n+1) i j - {(a, b)}\"\n using a (*‹(i::nat) ≤ (a::nat)› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) apply (simp add:L0_def (*‹L0 ?x ?y = {(?x + 1, ?y), (?x, ?y + 1), (?x + 1, ?y + 1)}›*))\n (*goal: ‹a < 2 ^ n + i ∧ b < 2 ^ n + j ⟹ L0 (2 ^ n + i - 1) (2 ^ n + j - 1) ⊆ square2 (n + 1) i j - {(a, b)}›*)\n by arith\n moreover have \"?B ⟹ L1 (2^n + i - 1) (2^n + j - 1) ⊆ square2 (n+1) i j - {(a, b)}\"\n using a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹(b::nat) < (2::nat) ^ Suc (n::nat) + (j::nat)›*) apply (simp add:L1_def (*‹L1 ?x ?y = {(?x, ?y), (?x, ?y + 1), (?x + 1, ?y + 1)}›*))\n (*goal: ‹(2::nat) ^ (n::nat) + (i::nat) ≤ (a::nat) ∧ a < (2::nat) ^ (n + (1::nat)) + i ∧ (b::nat) < (2::nat) ^ n + (j::nat) ⟹ L1 ((2::nat) ^ n + i - (1::nat)) ((2::nat) ^ n + j - (1::nat)) ⊆ square2 (n + (1::nat)) i j - {(a, b)}›*)\n by arith\n moreover have \"?C ⟹ L3 (2^n + i - 1) (2^n + j - 1) ⊆ square2 (n+1) i j - {(a, b)}\"\n using a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) apply (simp add:L3_def (*‹L3 ?x ?y = {(?x, ?y), (?x + 1, ?y), (?x + 1, ?y + 1)}›*))\n (*goal: ‹a < 2 ^ n + i ∧ 2 ^ n + j ≤ b ∧ b < 2 ^ (n + 1) + j ⟹ L3 (2 ^ n + i - 1) (2 ^ n + j - 1) ⊆ square2 (n + 1) i j - {(a, b)}›*)\n by arith\n moreover have \"?D ⟹ L2 (2^n + i - 1) (2^n + j - 1) ⊆ square2 (n+1) i j - {(a, b)}\"\n using a (*‹i ≤ a› ‹a < 2 ^ Suc n + i›*) b (*‹j ≤ b› ‹b < 2 ^ Suc n + j›*) apply (simp add:L2_def (*‹L2 ?x ?y = {(?x, ?y), (?x + 1, ?y), (?x, ?y + 1)}›*))\n (*goal: ‹2 ^ n + i ≤ a ∧ a < 2 ^ (n + 1) + i ∧ 2 ^ n + j ≤ b ∧ b < 2 ^ (n + 1) + j ⟹ L2 (2 ^ n + i - 1) (2 ^ n + j - 1) ⊆ square2 (n + 1) i j - {(a, b)}›*)\n by arith\n ultimately show \"?case\"\n (*goal: ‹square2 (Suc n) i j - {(a, b)} ∈ tiling Ls›*)\n apply simp\n (*goal: ‹square2 (Suc n) i j - {(a, b)} ∈ tiling Ls›*)\n by (metis LinLs (*‹L0 ?i ?j ∈ Ls ∧ L1 ?i ?j ∈ Ls ∧ L2 ?i ?j ∈ Ls ∧ L3 ?i ?j ∈ Ls›*) tiling_Diff1E (*‹⟦?t - ?a ∈ tiling ?A; ?a ∈ ?A; ?a ⊆ ?t⟧ ⟹ ?t ∈ tiling ?A›*))\nqed\n\ncorollary Ls_can_tile00:\n \"a < 2^n ⟹ b < 2^n ⟹ square2 n 0 0 - {(a, b)} ∈ tiling Ls\"\napply (rule Ls_can_tile (*‹⟦?i ≤ ?a; ?a < 2 ^ ?n + ?i; ?j ≤ ?b; ?b < 2 ^ ?n + ?j⟧ ⟹ square2 ?n ?i ?j - {(?a, ?b)} ∈ tiling Ls›*))\n(*goals:\n 1. ‹⟦(a::nat) < (2::nat) ^ (n::nat); (b::nat) < (2::nat) ^ n⟧ ⟹ (0::nat) ≤ a›\n 2. ‹⟦(a::nat) < (2::nat) ^ (n::nat); (b::nat) < (2::nat) ^ n⟧ ⟹ a < (2::nat) ^ n + (0::nat)›\n 3. ‹⟦(a::nat) < (2::nat) ^ (n::nat); (b::nat) < (2::nat) ^ n⟧ ⟹ (0::nat) ≤ b›\n 4. ‹⟦(a::nat) < (2::nat) ^ (n::nat); (b::nat) < (2::nat) ^ n⟧ ⟹ b < (2::nat) ^ n + (0::nat)›\ndiscuss goal 1*)\n apply ((auto)[1])\n(*discuss goal 2*)\n apply ((auto)[1])\n(*discuss goal 3*)\n apply ((auto)[1])\n(*discuss goal 4*)\n apply ((auto)[1])\n(*proven 4 subgoals*) .\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/FunWithTilings/Tilings.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"f3243f87e3dd4d519b177dad828ad35fbe31e080d674d75e963f1e2189c11e3b\"\n}"}}},{"rowIdx":186,"cells":{"text":{"kind":"string","value":"theory Valid_List\n imports Main \"../util/List_Util\"\nbegin\n\nsection ‹Valid List›\n\ndefinition \n valid_list :: \"('a :: {linorder, order_bot}) list ⇒ bool\"\nwhere\n \"valid_list s = (length s > 0 ∧ (∀i < length s - 1. s ! i ≠ bot) ∧ last s = bot)\"\n\nlemma valid_list_ex_def:\n fixes s ::\"('a :: {linorder, order_bot}) list\"\n shows \"(valid_list s) = \n (∃xs. s = xs @ [bot] ∧ \n (∀i < length xs. xs ! i ≠ bot))\"\nunfolding valid_list_def\n(*goal: ‹(0 < length s ∧ (∀i<length s - 1. s ! i ≠ bot) ∧ last s = bot) = (∃xs. s = xs @ [bot] ∧ (∀i<length xs. xs ! i ≠ bot))›*)\nproof (safe)\n (*goals:\n 1. ‹⟦s ≠ []; ∀i<length s - 1. s ! i ≠ bot; last s = bot⟧ ⟹ ∃xs. s = xs @ [bot] ∧ (∀i<length xs. xs ! i ≠ bot)›\n 2. ‹⋀xs i. ⟦∀i<length xs. xs ! i ≠ bot; s = xs @ [bot]; i < length (xs @ [bot]) - 1; (xs @ [bot]) ! i = bot⟧ ⟹ False›\n 3. ‹⋀xs. ⟦∀i<length xs. xs ! i ≠ bot; s = xs @ [bot]⟧ ⟹ last (xs @ [bot]) = bot›*)\n assume \"s ≠ []\" \"∀i<length s - 1. s ! i ≠ bot\" \"last s = bot\" (*‹(s::'a list) ≠ []› ‹∀i<length (s::'a list) - (1::nat). s ! i ≠ bot› ‹last (s::'a list) = bot›*)\n then show \"∃xs. s = xs @ [bot] ∧ \n (∀i<length xs. xs ! i ≠ bot)\"\n by (metis append_butlast_last_id (*‹?xs ≠ [] ⟹ butlast ?xs @ [last ?xs] = ?xs›*) length_butlast (*‹length (butlast ?xs) = length ?xs - 1›*) nth_butlast (*‹?n < length (butlast ?xs) ⟹ butlast ?xs ! ?n = ?xs ! ?n›*))\nqed ((simp add: nth_append (*‹(?xs @ ?ys) ! ?n = (if ?n < length ?xs then ?xs ! ?n else ?ys ! (?n - length ?xs))›*))+)\n(*solves the remaining goals:\n 1. ‹⋀xs i. ⟦∀i<length xs. xs ! i ≠ bot; s = xs @ [bot]; i < length (xs @ [bot]) - 1; (xs @ [bot]) ! i = bot⟧ ⟹ False›\n 2. ‹⋀xs. ⟦∀i<length xs. xs ! i ≠ bot; s = xs @ [bot]⟧ ⟹ last (xs @ [bot]) = bot›*)\n\nlemma valid_list_iff_butlast_app_last:\n fixes s :: \"('a :: {linorder, order_bot}) list\"\n shows \"valid_list s ⟷ \n s ≠ [] ∧ \n (∀x ∈ set (butlast s). x ≠ bot) ∧ \n last s = bot\"\n by (metis append_butlast_last_id (*‹?xs ≠ [] ⟹ butlast ?xs @ [last ?xs] = ?xs›*) butlast_snoc (*‹butlast (?xs @ [?x]) = ?xs›*) in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*) valid_list_def (*‹valid_list ?s = (0 < length ?s ∧ (∀i<length ?s - 1. ?s ! i ≠ bot) ∧ last ?s = bot)›*) valid_list_ex_def (*‹valid_list ?s = (∃xs. ?s = xs @ [bot] ∧ (∀i<length xs. xs ! i ≠ bot))›*) length_greater_0_conv (*‹(0 < length ?xs) = (?xs ≠ [])›*))\n\nlemma valid_list_consI:\n fixes s :: \"('a :: {linorder, order_bot}) list\"\n fixes a :: 'a\n assumes \"valid_list s\"\n and \"a ≠ bot\" \n shows \"valid_list (a # s)\"\n using assms (*‹valid_list s› ‹a ≠ bot›*) by (simp add: valid_list_iff_butlast_app_last (*‹valid_list ?s = (?s ≠ [] ∧ (∀x∈set (butlast ?s). x ≠ bot) ∧ last ?s = bot)›*))\n\nlemma valid_list_consD:\n fixes s :: \"('a :: {linorder, order_bot}) list\"\n fixes a :: 'a\n assumes \"valid_list (a # s)\"\n assumes \"s ≠ []\"\n shows \"valid_list s\"\n using assms (*‹valid_list (a # s)› ‹s ≠ []›*) by (simp add: valid_list_iff_butlast_app_last (*‹valid_list ?s = (?s ≠ [] ∧ (∀x∈set (butlast ?s). x ≠ bot) ∧ last ?s = bot)›*))\n\nlemma Min_valid_list:\n fixes s :: \"('a :: {linorder, order_bot}) list\"\n assumes \"valid_list s\"\n shows \"Min (set s) = bot\"\n by (metis assms (*‹valid_list (s::'a::{linorder,order_bot} list)›*) List.finite_set (*‹finite (set (?xs::?'a::type list))›*) Min.in_idem (*‹⟦finite (?A::?'a::linorder set); (?x::?'a::linorder) ∈ ?A⟧ ⟹ min ?x (Min ?A) = Min ?A›*) last_in_set (*‹(?as::?'a::type list) ≠ [] ⟹ last ?as ∈ set ?as›*) min_bot (*‹min bot (?x::?'a::order_bot) = bot›*) valid_list_iff_butlast_app_last (*‹valid_list (?s::?'a::{linorder,order_bot} list) = (?s ≠ [] ∧ (∀x::?'a::{linorder,order_bot}∈set (butlast ?s). x ≠ bot) ∧ last ?s = bot)›*))\n\nlemma valid_list_length:\n fixes s :: \"('a :: {linorder, order_bot}) list\"\n assumes \"valid_list s\"\n shows \"length s > 0\"\n using assms (*‹valid_list s›*) by (clarsimp simp: valid_list_ex_def (*‹valid_list ?s = (∃xs. ?s = xs @ [bot] ∧ (∀i<length xs. xs ! i ≠ bot))›*))\n\nlemma valid_list_length_ex:\n fixes s :: \"('a :: {linorder, order_bot}) list\"\n assumes \"valid_list s\"\n shows \"∃n. length s = Suc n\"\n using assms (*‹valid_list s›*) by (clarsimp simp: valid_list_ex_def (*‹valid_list (?s::?'a::{linorder,order_bot} list) = (∃xs::?'a::{linorder,order_bot} list. ?s = xs @ [bot] ∧ (∀i<length xs. xs ! i ≠ bot))›*))\n\nlemma valid_list_not_nil:\n fixes s :: \"('a :: {linorder, order_bot}) list\"\n assumes \"valid_list s\"\n shows \"s ≠ []\"\n using assms (*‹valid_list s›*) by (simp add: valid_list_def (*‹valid_list (?s::?'a::{linorder,order_bot} list) = ((0::nat) < length ?s ∧ (∀i<length ?s - (1::nat). ?s ! i ≠ bot) ∧ last ?s = bot)›*))\n\nlemma valid_list_Suc_mapping:\n fixes f :: \"'a ⇒ nat\"\n fixes s :: \"'a list\"\n shows \"valid_list ((map (λx. Suc (f x)) s) @ [bot])\"\nproof (induct s)\n (*goals:\n 1. ‹valid_list (map (λx. Suc (f x)) [] @ [bot])›\n 2. ‹⋀a s. valid_list (map (λx. Suc (f x)) s @ [bot]) ⟹ valid_list (map (λx. Suc (f x)) (a # s) @ [bot])›*)\n case (Cons a s) (*‹valid_list (map (λx. Suc (f x)) s @ [bot])›*)\n note IH = this (*‹valid_list (map (λx. Suc (f x)) s @ [bot])›*)\n have \"map (λx. Suc (f x)) (a # s) = (Suc (f a)) # map (λx. Suc (f x)) s\"\n by simp\n moreover have \"Suc (f a) ≠ bot\"\n by (simp add: bot_nat_def (*‹bot = 0›*))\n hence \"valid_list ((Suc (f a)) # (map (λx. Suc (f x)) s) @ [bot])\"\n by (simp add: IH (*‹valid_list (map (λx. Suc (f x)) s @ [bot])›*) valid_list_consI (*‹⟦valid_list ?s; ?a ≠ bot⟧ ⟹ valid_list (?a # ?s)›*))\n ultimately show \"?case\"\n (*goal: ‹valid_list (map (λx. Suc (f x)) (a # s) @ [bot])›*)\n by simp\nqed (simp add: valid_list_ex_def (*‹valid_list ?s = (∃xs. ?s = xs @ [bot] ∧ (∀i<length xs. xs ! i ≠ bot))›*))\n(*solved the remaining goal: ‹valid_list (map (λx. Suc (f x)) [] @ [bot])›*)\n\nlemma valid_list_app:\n assumes \"valid_list (xs @ y # ys)\"\n shows \"valid_list (y # ys)\"\n using assms (*‹valid_list (xs @ y # ys)›*) apply (induct xs)\n (*goals:\n 1. ‹valid_list ([] @ y # ys) ⟹ valid_list (y # ys)›\n 2. ‹⋀a xs. ⟦valid_list (xs @ y # ys) ⟹ valid_list (y # ys); valid_list ((a # xs) @ y # ys)⟧ ⟹ valid_list (y # ys)›\n discuss goal 1*)\n apply (simp add: valid_list_consD (*‹⟦valid_list ((?a::?'a::{linorder,order_bot}) # (?s::?'a::{linorder,order_bot} list)); ?s ≠ []⟧ ⟹ valid_list ?s›*))\n (*discuss goal 2*)\n apply (simp add: valid_list_consD (*‹⟦valid_list ((?a::?'a::{linorder,order_bot}) # (?s::?'a::{linorder,order_bot} list)); ?s ≠ []⟧ ⟹ valid_list ?s›*))\n (*proven 2 subgoals*) .\n\nlemma not_valid_list_app:\n assumes \"valid_list (xs @ y # ys)\"\n shows \"¬valid_list xs\"\n using assms (*‹valid_list (xs @ y # ys)›*) proof (induct xs)\n (*goals:\n 1. ‹valid_list ([] @ (y::'a) # (ys::'a list)) ⟹ ¬ valid_list []›\n 2. ‹⋀(a::'a) xs::'a list. ⟦valid_list (xs @ (y::'a) # (ys::'a list)) ⟹ ¬ valid_list xs; valid_list ((a # xs) @ y # ys)⟧ ⟹ ¬ valid_list (a # xs)›*)\n case Nil (*‹valid_list ([] @ y # ys)›*)\n then show \"?case\"\n (*goal: ‹¬ valid_list []›*)\n using valid_list_not_nil (*‹valid_list (?s::?'a list) ⟹ ?s ≠ []›*) by auto\n next\n (*goal: ‹⋀a xs. ⟦valid_list (xs @ y # ys) ⟹ ¬ valid_list xs; valid_list ((a # xs) @ y # ys)⟧ ⟹ ¬ valid_list (a # xs)›*)\n case (Cons a xs) (*‹valid_list (xs @ y # ys) ⟹ ¬ valid_list xs› ‹valid_list ((a # xs) @ y # ys)›*)\n then show \"?case\"\n (*goal: ‹¬ valid_list ((a::'a) # (xs::'a list))›*)\n by (metis Groups.add_ac( (*‹?a + ?b = ?b + ?a›*) 2) Nil_is_append_conv (*‹([] = ?xs @ ?ys) = (?xs = [] ∧ ?ys = [])›*) One_nat_def (*‹1 = Suc 0›*) add_diff_cancel_left' (*‹?a + ?b - ?a = ?b›*) append_Cons (*‹(?x # ?xs) @ ?ys = ?x # ?xs @ ?ys›*) last_ConsL (*‹?xs = [] ⟹ last (?x # ?xs) = ?x›*) list.discI (*‹?list = ?x21.0 # ?x22.0 ⟹ ?list ≠ []›*) list.size( (*‹length (?x21.0 # ?x22.0) = length ?x22.0 + Suc 0›*) 4) nth_Cons_0 (*‹(?x # ?xs) ! 0 = ?x›*) valid_list_consD (*‹⟦valid_list (?a # ?s); ?s ≠ []⟧ ⟹ valid_list ?s›*) valid_list_def (*‹valid_list ?s = (0 < length ?s ∧ (∀i<length ?s - 1. ?s ! i ≠ bot) ∧ last ?s = bot)›*))\n qed\n\nlemma valid_list_neqE:\n assumes \"valid_list xs\" \"valid_list ys\" \"xs ≠ ys\"\n shows \"∃x y as bs cs. xs = as @ x # bs ∧ ys = as @ y # cs ∧ x ≠ y\"\nproof (-)\n (*goal: ‹∃x y as bs cs. xs = as @ x # bs ∧ ys = as @ y # cs ∧ x ≠ y›*)\n note cases = list_neq_fc[OF assms ( 3 )] (*‹(∃z zs. xs = ys @ z # zs) ∨ (∃z zs. ys = xs @ z # zs) ∨ (∃x y as bs cs. xs = as @ x # bs ∧ ys = as @ y # cs ∧ x ≠ y)›*)\n moreover have \"¬(∃z zs. xs = ys @ z # zs)\"\n using assms(1) (*‹valid_list xs›*) assms(2) (*‹valid_list ys›*) not_valid_list_app (*‹valid_list ((?xs::?'a list) @ (?y::?'a) # (?ys::?'a list)) ⟹ ¬ valid_list ?xs›*) by blast\n moreover have \"¬(∃z zs. ys = xs @ z # zs)\"\n using assms(1) (*‹valid_list xs›*) assms(2) (*‹valid_list ys›*) not_valid_list_app (*‹valid_list ((?xs::?'a::{linorder,order_bot} list) @ (?y::?'a::{linorder,order_bot}) # (?ys::?'a::{linorder,order_bot} list)) ⟹ ¬ valid_list ?xs›*) by blast\n ultimately show \"?thesis\"\n (*goal: ‹∃x y as bs cs. xs = as @ x # bs ∧ ys = as @ y # cs ∧ x ≠ y›*)\n by blast\nqed\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/SuffixArray/order/Valid_List.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"0f3e229f371c2d3a44523088be1b2536311acc1e1dd001efd3dd6ee0509a7ca5\"\n}"}}},{"rowIdx":187,"cells":{"text":{"kind":"string","value":"section ‹Preliminary results›\n\ntheory Concentration_Inequalities_Preliminary\n imports Lp.Lp\nbegin\n\ntext ‹Version of Cauchy-Schwartz for the Lebesgue integral:›\nlemma cauchy_schwartz:\n fixes f g :: \"_ ⇒ real\"\n assumes \"f ∈ borel_measurable M\" \"g ∈ borel_measurable M\"\n assumes \"integrable M (λx. (f x) ^2)\" \"integrable M (λx. (g x) ^2)\"\n shows \"integrable M (λx. f x * g x)\" (is \"?A\")\n \"(∫x. f x * g x ∂M) ≤ (∫x. (f x)^2 ∂M) powr (1/2) * (∫x. (g x)^ 2 ∂M) powr (1/2)\"\n (is \"?L ≤ ?R\")\nproof (-)\n (*goals:\n 1. ‹integrable M (λx. f x * g x)›\n 2. ‹LINT x|M. f x * g x ≤ (LINT x|M. (f x)²) powr (1 / 2) * (LINT x|M. (g x)²) powr (1 / 2)›*)\n show 0: \"?A\"\n using assms (*‹f ∈ borel_measurable M› ‹g ∈ borel_measurable M› ‹integrable M (λx. (f x)²)› ‹integrable M (λx. (g x)²)›*) apply (intro Holder_inequality(1)[where p=\"2\" and q=\"2\"] (*‹⟦(0::real) < (2::real); (0::real) < (2::real); (1::real) / (2::real) + (1::real) / (2::real) = (1::real); (?f::?'a ⇒ real) ∈ borel_measurable (?M::?'a measure); (?g::?'a ⇒ real) ∈ borel_measurable ?M; integrable ?M (λx::?'a. ¦?f x¦ powr (2::real)); integrable ?M (λx::?'a. ¦?g x¦ powr (2::real))⟧ ⟹ integrable ?M (λx::?'a. ?f x * ?g x)›*))\n (*goals:\n 1. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ 0 < 2›\n 2. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ 0 < 2›\n 3. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ 1 / 2 + 1 / 2 = 1›\n 4. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ f ∈ borel_measurable M›\n 5. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ g ∈ borel_measurable M›\n 6. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ integrable M (λx. ¦f x¦ powr 2)›\n 7. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ integrable M (λx. ¦g x¦ powr 2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*proven 7 subgoals*) .\n have \"?L ≤ (∫x. ¦f x * g x¦ ∂M)\"\n using \"0\" (*‹integrable M (λx. f x * g x)›*) apply (intro integral_mono (*‹⟦integrable ?M ?f; integrable ?M ?g; ⋀x. x ∈ space ?M ⟹ ?f x ≤ ?g x⟧ ⟹ integral⇧L ?M ?f ≤ integral⇧L ?M ?g›*))\n (*goals:\n 1. ‹integrable M (λx. f x * g x) ⟹ integrable M (λx. f x * g x)›\n 2. ‹integrable M (λx. f x * g x) ⟹ integrable M (λx. ¦f x * g x¦)›\n 3. ‹⋀x. ⟦integrable M (λx. f x * g x); x ∈ space M⟧ ⟹ f x * g x ≤ ¦f x * g x¦›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹LINT x::'c|(M::'c measure). (f::'c ⇒ real) x * (g::'c ⇒ real) x ≤ LINT x::'c|M. ¦f x * g x¦›*) have \"... ≤ (∫x. ¦f x¦ powr 2 ∂M) powr (1/2) * (∫x. ¦g x¦ powr 2 ∂M) powr (1/2)\"\n using assms (*‹f ∈ borel_measurable M› ‹g ∈ borel_measurable M› ‹integrable (M::'c measure) (λx::'c. ((f::'c ⇒ real) x)²)› ‹integrable (M::'c measure) (λx::'c. ((g::'c ⇒ real) x)²)›*) apply (intro Holder_inequality( (*‹⟦0 < ?p; 0 < ?q; 1 / ?p + 1 / ?q = 1; ?f ∈ borel_measurable ?M; ?g ∈ borel_measurable ?M; integrable ?M (λx. ¦?f x¦ powr ?p); integrable ?M (λx. ¦?g x¦ powr ?q)⟧ ⟹ LINT x|?M. ¦?f x * ?g x¦ ≤ (LINT x|?M. ¦?f x¦ powr ?p) powr (1 / ?p) * (LINT x|?M. ¦?g x¦ powr ?q) powr (1 / ?q)›*) 2))\n (*goals:\n 1. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ 0 < 2›\n 2. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ 0 < 2›\n 3. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ 1 / 2 + 1 / 2 = 1›\n 4. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ f ∈ borel_measurable M›\n 5. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ g ∈ borel_measurable M›\n 6. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ integrable M (λx. ¦f x¦ powr 2)›\n 7. ‹⟦f ∈ borel_measurable M; g ∈ borel_measurable M; integrable M (λx. (f x)²); integrable M (λx. (g x)²)⟧ ⟹ integrable M (λx. ¦g x¦ powr 2)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*proven 7 subgoals*) .\n also (*calculation: ‹LINT x|M. f x * g x ≤ (LINT x|M. ¦f x¦ powr 2) powr (1 / 2) * (LINT x|M. ¦g x¦ powr 2) powr (1 / 2)›*) have \"... = ?R\"\n by simp\n finally (*calculation: ‹LINT x|M. f x * g x ≤ (LINT x|M. (f x)²) powr (1 / 2) * (LINT x|M. (g x)²) powr (1 / 2)›*) show \"?L ≤ ?R\"\n by simp\nqed\n\ntext ‹Generalization of @{thm [source] prob_space.indep_vars_iff_distr_eq_PiM'}:›\n\nlemma (in prob_space) indep_vars_iff_distr_eq_PiM'':\n fixes I :: \"'i set\" and X :: \"'i ⇒ 'a ⇒ 'b\"\n assumes rv: \"⋀i. i ∈ I ⟹ random_variable (M' i) (X i)\"\n shows \"indep_vars M' X I ⟷\n distr M (Π⇩M i∈I. M' i) (λx. λi∈I. X i x) = (Π⇩M i∈I. distr M (M' i) (X i))\"\nproof (cases \"I = {}\")\n (*goals:\n 1. ‹I = {} ⟹ indep_vars M' X I = (distr M (Pi⇩M I M') (λx. λi∈I. X i x) = Pi⇩M I (λi. distr M (M' i) (X i)))›\n 2. ‹I ≠ {} ⟹ indep_vars M' X I = (distr M (Pi⇩M I M') (λx. λi∈I. X i x) = Pi⇩M I (λi. distr M (M' i) (X i)))›*)\n case True (*‹I = {}›*)\n have 0: \" indicator A (λ_. undefined) = emeasure (count_space {λ_. undefined}) A\" (is \"?L = ?R\") if \"A ⊆ {λ_. undefined}\" for A :: \"('i ⇒ 'b) set\"\n proof (-)\n (*goal: ‹indicator A (λ_. undefined) = emeasure (count_space {λ_. undefined}) A›*)\n have 1: \"A ≠ {} ⟹ A = {λ_. undefined}\"\n using that (*‹(A::('i::type ⇒ 'b::type) set) ⊆ {λ_::'i::type. undefined}›*) by auto\n have \"?R = of_nat (card A)\"\n using finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) that (*‹(A::('i::type ⇒ 'b::type) set) ⊆ {λ_::'i::type. undefined}›*) apply (intro emeasure_count_space_finite (*‹⟦?X ⊆ ?A; finite ?X⟧ ⟹ emeasure (count_space ?A) ?X = of_nat (card ?X)›*) that (*‹A ⊆ {λ_. undefined}›*))\n (*goal: ‹emeasure (count_space {λ_. undefined}) A = of_nat (card A)›*)\n by auto\n also (*calculation: ‹emeasure (count_space {λ_. undefined}) A = of_nat (card A)›*) have \"... = ?L\"\n using \"1\" (*‹A ≠ {} ⟹ A = {λ_. undefined}›*) apply (cases \"A = {}\")\n (*goals:\n 1. ‹⟦A ≠ {} ⟹ A = {λ_. undefined}; A = {}⟧ ⟹ of_nat (card A) = indicator A (λ_. undefined)›\n 2. ‹⟦A ≠ {} ⟹ A = {λ_. undefined}; A ≠ {}⟧ ⟹ of_nat (card A) = indicator A (λ_. undefined)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n finally (*calculation: ‹emeasure (count_space {λ_. undefined}) A = indicator A (λ_. undefined)›*) show \"?thesis\"\n (*goal: ‹indicator A (λ_. undefined) = emeasure (count_space {λ_. undefined}) A›*)\n by simp\n qed\n have \"distr M (Π⇩M i∈I. M' i) (λx. λi∈I. X i x) =\n distr M (count_space {λ_. undefined}) (λ_. (λ_. undefined))\"\n unfolding True PiM_empty\n (*goal: ‹distr M (count_space {λ_. undefined}) (λx. λi∈{}. X i x) = distr M (count_space {λ_. undefined}) (λ_ _. undefined)›*)\n apply (intro distr_cong (*‹⟦?M = ?K; sets ?N = sets ?L; ⋀x. x ∈ space ?M ⟹ ?f x = ?g x⟧ ⟹ distr ?M ?N ?f = distr ?K ?L ?g›*))\n (*goals:\n 1. ‹M = M›\n 2. ‹sets (count_space {λ_. undefined}) = sets (count_space {λ_. undefined})›\n 3. ‹⋀x. x ∈ space M ⟹ (λi∈{}. X i x) = (λ_. undefined)›\n discuss goal 1*)\n apply ((auto simp:restrict_def (*‹restrict ?f ?A = (λx. if x ∈ ?A then ?f x else undefined)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp:restrict_def (*‹restrict ?f ?A = (λx. if x ∈ ?A then ?f x else undefined)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp:restrict_def (*‹restrict ?f ?A = (λx. if x ∈ ?A then ?f x else undefined)›*))[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹distr M (Pi⇩M I M') (λx. λi∈I. X i x) = distr M (count_space {λ_. undefined}) (λ_ _. undefined)›*) have \"... = return (count_space {λ_. undefined}) (λ_. undefined)\"\n apply (intro distr_const (*‹(?c::?'b::type) ∈ space (?N::?'b::type measure) ⟹ distr (M::'a::type measure) ?N (λx::'a::type. ?c) = return ?N ?c›*))\n (*goal: ‹distr M (count_space {λ_. undefined}) (λ_ _. undefined) = return (count_space {λ_. undefined}) (λ_. undefined)›*)\n by auto\n also (*calculation: ‹distr M (Pi⇩M I M') (λx. λi∈I. X i x) = return (count_space {λ_. undefined}) (λ_. undefined)›*) have \"... = count_space ({λ_. undefined} :: ('i ⇒ 'b) set) \"\n apply (intro measure_eqI (*‹⟦sets (?M::?'a::type measure) = sets (?N::?'a::type measure); ⋀A::?'a::type set. A ∈ sets ?M ⟹ emeasure ?M A = emeasure ?N A⟧ ⟹ ?M = ?N›*))\n (*goals:\n 1. ‹sets (return (count_space {λ_::'i::type. undefined}) (λ_::'i::type. undefined)) = sets (count_space {λ_::'i::type. undefined})›\n 2. ‹⋀A::('i::type ⇒ 'b::type) set. A ∈ sets (return (count_space {λ_::'i::type. undefined}) (λ_::'i::type. undefined)) ⟹ emeasure (return (count_space {λ_::'i::type. undefined}) (λ_::'i::type. undefined)) A = emeasure (count_space {λ_::'i::type. undefined}) A›\n discuss goal 1*)\n apply ((auto simp:0 (*‹?A12 ⊆ {λ_. undefined} ⟹ indicator ?A12 (λ_. undefined) = emeasure (count_space {λ_. undefined}) ?A12›*))[1])\n (*discuss goal 2*)\n apply ((auto simp:0 (*‹?A12 ⊆ {λ_. undefined} ⟹ indicator ?A12 (λ_. undefined) = emeasure (count_space {λ_. undefined}) ?A12›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹distr M (Pi⇩M I M') (λx. λi∈I. X i x) = count_space {λ_. undefined}›*) have \"... = (Π⇩M i∈I. distr M (M' i) (X i))\"\n unfolding True PiM_empty\n (*goal: ‹count_space {λ_. undefined} = count_space {λ_. undefined}›*)\n by simp\n finally (*calculation: ‹distr M (Pi⇩M I M') (λx. λi∈I. X i x) = Pi⇩M I (λi. distr M (M' i) (X i))›*) have \"distr M (Π⇩M i∈I. M' i) (λx. λi∈I. X i x)=(Π⇩M i∈I. distr M (M' i) (X i)) ⟷ True\"\n by simp\n also (*calculation: ‹(distr M (Pi⇩M I M') (λx. λi∈I. X i x) = Pi⇩M I (λi. distr M (M' i) (X i))) = True›*) have \"... ⟷ indep_vars M' X I\"\n unfolding indep_vars_def\n (*goal: ‹True = ((∀i::'i∈I::'i set. random_variable ((M'::'i ⇒ 'b measure) i) ((X::'i ⇒ 'a ⇒ 'b) i)) ∧ indep_sets (λi::'i. sigma_sets (space (M::'a measure)) {X i -` A ∩ space M |A::'b set. A ∈ sets (M' i)}) I)›*)\n apply (auto simp add: space_PiM (*‹space (Pi⇩M ?I ?M) = (Π⇩E i∈?I. space (?M i))›*) indep_sets_def (*‹indep_sets ?F ?I = ((∀i∈?I. ?F i ⊆ events) ∧ (∀J⊆?I. J ≠ {} ⟶ finite J ⟶ (∀A∈Pi J ?F. prob (⋂ (A ` J)) = (∏j∈J. prob (A j)))))›*))\n (*goals:\n 1. ‹⋀i. i ∈ I ⟹ random_variable (M' i) (X i)›\n 2. ‹⋀i x. ⟦i ∈ I; x ∈ sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}⟧ ⟹ x ∈ events›\n 3. ‹⋀J A x. ⟦J ⊆ I; finite J; A ∈ (Π i∈J. sigma_sets (space M) {X i -` A ∩ space M |A. A ∈ sets (M' i)}); prob (⋂ (A ` J)) ≠ (∏j∈J. prob (A j)); x ∈ J⟧ ⟹ False›\n discuss goal 1*)\n apply ((auto simp add:True (*‹(I::'i::type set) = {}›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add:True (*‹I = {}›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add:True (*‹I = {}›*))[1])\n (*proven 3 subgoals*) .\n finally (*calculation: ‹(distr (M::'a measure) (Pi⇩M (I::'i set) (M'::'i ⇒ 'b measure)) (λx::'a. λi::'i∈I. (X::'i ⇒ 'a ⇒ 'b) i x) = Pi⇩M I (λi::'i. distr M (M' i) (X i))) = indep_vars M' X I›*) show \"?thesis\"\n (*goal: ‹indep_vars M' X I = (distr M (Pi⇩M I M') (λx. λi∈I. X i x) = Pi⇩M I (λi. distr M (M' i) (X i)))›*)\n by simp\nnext\n (*goal: ‹I ≠ {} ⟹ indep_vars M' X I = (distr M (Pi⇩M I M') (λx. λi∈I. X i x) = Pi⇩M I (λi. distr M (M' i) (X i)))›*)\n case False (*‹I ≠ {}›*)\n thus \"?thesis\"\n (*goal: ‹indep_vars M' X I = (distr M (Pi⇩M I M') (λx. λi∈I. X i x) = Pi⇩M I (λi. distr M (M' i) (X i)))›*)\n apply (intro indep_vars_iff_distr_eq_PiM' (*‹⟦?I ≠ {}; ⋀i. i ∈ ?I ⟹ random_variable (?M' i) (?X i)⟧ ⟹ indep_vars ?M' ?X ?I = (distr M (Pi⇩M ?I ?M') (λx. λi∈?I. ?X i x) = Pi⇩M ?I (λi. distr M (?M' i) (?X i)))›*) assms (*‹?i12 ∈ I ⟹ random_variable (M' ?i12) (X ?i12)›*))\n (*goals:\n 1. ‹(I::'i set) ≠ {} ⟹ I ≠ {}›\n 2. ‹⋀i::'i. ⟦(I::'i set) ≠ {}; i ∈ I⟧ ⟹ i ∈ I›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nqed\n\nlemma proj_indep:\n assumes \"⋀i. i ∈ I ⟹ prob_space (M i)\"\n shows \"prob_space.indep_vars (PiM I M) M (λi ω. ω i) I\"\nproof (-)\n (*goal: ‹prob_space.indep_vars (Pi⇩M I M) M (λi ω. ω i) I›*)\n interpret prob_space \"(PiM I M)\"\n by (intro prob_space_PiM (*‹(⋀i. i ∈ ?I ⟹ prob_space (?M i)) ⟹ prob_space (Pi⇩M ?I ?M)›*) assms (*‹?i ∈ I ⟹ prob_space (M ?i)›*))\n have \"distr (Pi⇩M I M) (Pi⇩M I M) (λx. restrict x I) = PiM I M\"\n apply (intro distr_PiM_reindex (*‹⟦⋀i::?'a. i ∈ (?K::?'a set) ⟹ prob_space ((?M::?'a ⇒ ?'b measure) i); inj_on (?f::?'c ⇒ ?'a) (?I::?'c set); ?f ∈ ?I → ?K⟧ ⟹ distr (Pi⇩M ?K ?M) (Pi⇩M ?I (λi::?'c. ?M (?f i))) (λω::?'a ⇒ ?'b. λn::?'c∈?I. ω (?f n)) = Pi⇩M ?I (λi::?'c. ?M (?f i))›*) assms (*‹(?i::'a) ∈ (I::'a set) ⟹ prob_space ((M::'a ⇒ 'b measure) ?i)›*))\n (*goals:\n 1. ‹⋀i. i ∈ I ⟹ i ∈ I›\n 2. ‹inj_on (λi. i) I›\n 3. ‹(λi. i) ∈ I → I›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹distr (Pi⇩M I M) (Pi⇩M I M) (λx. restrict x I) = Pi⇩M I M›*) have \"... = Pi⇩M I (λi. distr (Pi⇩M I M) (M i) (λω. ω i))\"\n apply (intro PiM_cong (*‹⟦?I = ?J; ⋀x. x ∈ ?I ⟹ ?M x = ?N x⟧ ⟹ Pi⇩M ?I ?M = Pi⇩M ?J ?N›*) refl (*‹?t = ?t›*) distr_PiM_component[symmetric] (*‹⟦⋀i. i ∈ ?I ⟹ prob_space (?M i); ?i ∈ ?I⟧ ⟹ ?M ?i = distr (Pi⇩M ?I ?M) (?M ?i) (λω. ω ?i)›*) assms (*‹?i ∈ I ⟹ prob_space (M ?i)›*))\n (*goals:\n 1. ‹⋀(x::'a::type) i::'a::type. ⟦x ∈ (I::'a::type set); i ∈ I⟧ ⟹ i ∈ I›\n 2. ‹⋀x::'a::type. x ∈ (I::'a::type set) ⟹ x ∈ I›\n discuss goal 1*)\n apply ((assumption)[1])\n (*discuss goal 2*) .\n (*proven 2 subgoals*)\n finally (*calculation: ‹distr (Pi⇩M I M) (Pi⇩M I M) (λx. restrict x I) = Pi⇩M I (λi. distr (Pi⇩M I M) (M i) (λω. ω i))›*) have \"distr (Pi⇩M I M) (Pi⇩M I M) (λx. restrict x I) = Pi⇩M I (λi. distr (Pi⇩M I M) (M i) (λω. ω i))\"\n by simp\n thus \"indep_vars M (λi ω. ω i) I\"\n apply (intro iffD2[OF indep_vars_iff_distr_eq_PiM''] (*‹⟦⋀i::?'i1. i ∈ (?I1::?'i1 set) ⟹ random_variable ((?M'1::?'i1 ⇒ ?'b1 measure) i) ((?X1::?'i1 ⇒ ('a ⇒ 'b) ⇒ ?'b1) i); distr (Pi⇩M (I::'a set) (M::'a ⇒ 'b measure)) (Pi⇩M ?I1 ?M'1) (λx::'a ⇒ 'b. λi::?'i1∈?I1. ?X1 i x) = Pi⇩M ?I1 (λi::?'i1. distr (Pi⇩M I M) (?M'1 i) (?X1 i))⟧ ⟹ indep_vars ?M'1 ?X1 ?I1›*))\n (*goals:\n 1. ‹⋀i. ⟦distr (Pi⇩M I M) (Pi⇩M I M) (λx. restrict x I) = Pi⇩M I (λi. distr (Pi⇩M I M) (M i) (λω. ω i)); i ∈ I⟧ ⟹ random_variable (M i) (λω. ω i)›\n 2. ‹distr (Pi⇩M I M) (Pi⇩M I M) (λx. restrict x I) = Pi⇩M I (λi. distr (Pi⇩M I M) (M i) (λω. ω i)) ⟹ distr (Pi⇩M I M) (Pi⇩M I M) (λx. restrict x I) = Pi⇩M I (λi. distr (Pi⇩M I M) (M i) (λω. ω i))›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\nqed\n\nlemma forall_Pi_to_PiE:\n assumes \"⋀x. P x = P (restrict x I)\"\n shows \"(∀x ∈ Pi I A. P x) = (∀x ∈ PiE I A. P x)\"\n using assms (*‹P ?x = P (restrict ?x I)›*) apply (simp add:PiE_def (*‹Pi⇩E ?S ?T = Pi ?S ?T ∩ extensional ?S›*) Pi_def (*‹Pi ?A ?B = {f. ∀x. x ∈ ?A ⟶ f x ∈ ?B x}›*) set_eq_iff (*‹(?A = ?B) = (∀x. (x ∈ ?A) = (x ∈ ?B))›*))\n (*goal: ‹(∀x::'a ⇒ 'b∈Pi (I::'a set) (A::'a ⇒ 'b set). (P::('a ⇒ 'b) ⇒ bool) x) = (∀x::'a ⇒ 'b∈Pi⇩E I A. P x)›*)\n by force\n\nlemma PiE_reindex:\n assumes \"inj_on f I\"\n shows \"PiE I (A ∘ f) = (λa. restrict (a ∘ f) I) ` PiE (f ` I) A\" (is \"?lhs = ?g ` ?rhs\")\nproof (-)\n (*goal: ‹Pi⇩E I (A ∘ f) = (λa. restrict (a ∘ f) I) ` Pi⇩E (f ` I) A›*)\n have \"?lhs ⊆ ?g` ?rhs\"\n proof (rule subsetI (*‹(⋀x. x ∈ ?A ⟹ x ∈ ?B) ⟹ ?A ⊆ ?B›*))\n (*goal: ‹⋀x. x ∈ Pi⇩E I (A ∘ f) ⟹ x ∈ (λa. restrict (a ∘ f) I) ` Pi⇩E (f ` I) A›*)\n fix x\n assume a: \"x ∈ Pi⇩E I (A ∘ f)\" (*‹(x::'a ⇒ 'c) ∈ Pi⇩E (I::'a set) ((A::'b ⇒ 'c set) ∘ (f::'a ⇒ 'b))›*)\n define y where y_def: \"y = (λk. if k ∈ f ` I then x (the_inv_into I f k) else undefined)\"\n have b: \"y ∈ PiE (f ` I) A\"\n using a (*‹x ∈ Pi⇩E I (A ∘ f)›*) assms (*‹inj_on (f::'a::type ⇒ 'b::type) (I::'a::type set)›*) the_inv_into_f_eq[OF assms] (*‹⟦f ?x = ?y; ?x ∈ I⟧ ⟹ the_inv_into I f ?y = ?x›*) by (simp add: y_def (*‹(y::'b ⇒ 'c) = (λk::'b. if k ∈ (f::'a ⇒ 'b) ` (I::'a set) then (x::'a ⇒ 'c) (the_inv_into I f k) else undefined)›*) PiE_iff (*‹((?f::?'a ⇒ ?'b) ∈ Pi⇩E (?I::?'a set) (?X::?'a ⇒ ?'b set)) = ((∀i::?'a∈?I. ?f i ∈ ?X i) ∧ ?f ∈ extensional ?I)›*) extensional_def (*‹extensional (?A::?'a set) = {f::?'a ⇒ ?'b. ∀x::?'a. x ∉ ?A ⟶ f x = undefined}›*))\n have c: \"x = (λa. restrict (a ∘ f) I) y\"\n using a (*‹x ∈ Pi⇩E I (A ∘ f)›*) assms (*‹inj_on (f::'a ⇒ 'b) (I::'a set)›*) the_inv_into_f_eq (*‹⟦inj_on ?f ?A; ?f ?x = ?y; ?x ∈ ?A⟧ ⟹ the_inv_into ?A ?f ?y = ?x›*) extensional_arb (*‹⟦(?f::?'a::type ⇒ ?'b::type) ∈ extensional (?A::?'a::type set); (?x::?'a::type) ∉ ?A⟧ ⟹ ?f ?x = undefined›*) apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹x = restrict (y ∘ f) I›*)\n apply (simp add:y_def (*‹y = (λk. if k ∈ f ` I then x (the_inv_into I f k) else undefined)›*) PiE_iff (*‹(?f ∈ Pi⇩E ?I ?X) = ((∀i∈?I. ?f i ∈ ?X i) ∧ ?f ∈ extensional ?I)›*))\n (*goal: ‹⋀xa. ⟦x ∈ Pi⇩E I (A ∘ f); inj_on f I; ⋀f A x y. ⟦inj_on f A; f x = y; x ∈ A⟧ ⟹ the_inv_into A f y = x; ⋀f A x. ⟦f ∈ extensional A; x ∉ A⟧ ⟹ f x = undefined⟧ ⟹ x xa = restrict (y ∘ f) I xa›*)\n by fastforce\n show \"x ∈ ?g ` ?rhs\"\n using b (*‹y ∈ Pi⇩E (f ` I) A›*) c (*‹x = restrict (y ∘ f) I›*) by blast\n qed\n moreover have \"?g ` ?rhs ⊆ ?lhs\"\n apply (rule image_subsetI (*‹(⋀x. x ∈ ?A ⟹ ?f x ∈ ?B) ⟹ ?f ` ?A ⊆ ?B›*))\n (*goal: ‹(λa. restrict (a ∘ f) I) ` Pi⇩E (f ` I) A ⊆ Pi⇩E I (A ∘ f)›*)\n by (simp add:Pi_def (*‹Pi ?A ?B = {f. ∀x. x ∈ ?A ⟶ f x ∈ ?B x}›*) PiE_def (*‹Pi⇩E ?S ?T = Pi ?S ?T ∩ extensional ?S›*))\n ultimately show \"?thesis\"\n (*goal: ‹Pi⇩E I (A ∘ f) = (λa. restrict (a ∘ f) I) ` Pi⇩E (f ` I) A›*)\n by blast\nqed\n\ncontext prob_space\nbegin\n\nlemma indep_sets_reindex:\n assumes \"inj_on f I\"\n shows \"indep_sets A (f ` I) = indep_sets (λi. A (f i)) I\"\nproof (-)\n (*goal: ‹indep_sets (A::'c ⇒ 'a set set) ((f::'b ⇒ 'c) ` (I::'b set)) = indep_sets (λi::'b. A (f i)) I›*)\n have a: \"⋀J g. J ⊆ I ⟹ (∏j ∈ f ` J. g j) = (∏j ∈ J. g (f j))\"\n by (metis assms (*‹inj_on f I›*) prod.reindex_cong (*‹⟦inj_on ?l ?B; ?A = ?l ` ?B; ⋀x. x ∈ ?B ⟹ ?g (?l x) = ?h x⟧ ⟹ prod ?g ?A = prod ?h ?B›*) subset_inj_on (*‹⟦inj_on ?f ?B; ?A ⊆ ?B⟧ ⟹ inj_on ?f ?A›*))\n have b: \"J ⊆ I ⟹ (Π⇩E i ∈ J. A (f i)) = (λa. restrict (a ∘ f) J) ` PiE (f ` J) A\" for J\n using assms (*‹inj_on f I›*) inj_on_subset (*‹⟦inj_on ?f ?A; ?B ⊆ ?A⟧ ⟹ inj_on ?f ?B›*) apply (subst PiE_reindex[symmetric] (*‹inj_on (?f::?'a::type ⇒ ?'b::type) (?I::?'a::type set) ⟹ (λa::?'b::type ⇒ ?'c::type. restrict (a ∘ ?f) ?I) ` Pi⇩E (?f ` ?I) (?A::?'b::type ⇒ ?'c::type set) = Pi⇩E ?I (?A ∘ ?f)›*))\n (*goals:\n 1. ‹⟦(J::'b set) ⊆ (I::'b set); inj_on (f::'b ⇒ 'c) I; ⋀(f::?'a1 ⇒ ?'b1) (A::?'a1 set) B::?'a1 set. ⟦inj_on f A; B ⊆ A⟧ ⟹ inj_on f B⟧ ⟹ inj_on f J›\n 2. ‹⟦(J::'b set) ⊆ (I::'b set); inj_on (f::'b ⇒ 'c) I; ⋀(f::?'a1 ⇒ ?'b1) (A::?'a1 set) B::?'a1 set. ⟦inj_on f A; B ⊆ A⟧ ⟹ inj_on f B⟧ ⟹ (Π⇩E i::'b∈J. (A::'c ⇒ 'a set set) (f i)) = Pi⇩E J (A ∘ f)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have c: \"⋀J. J ⊆ I ⟹ finite (f ` J) = finite J\"\n by (meson assms (*‹inj_on f I›*) finite_image_iff (*‹inj_on ?f ?A ⟹ finite (?f ` ?A) = finite ?A›*) inj_on_subset (*‹⟦inj_on ?f ?A; ?B ⊆ ?A⟧ ⟹ inj_on ?f ?B›*))\n show \"?thesis\"\n (*goal: ‹indep_sets A (f ` I) = indep_sets (λi. A (f i)) I›*)\n apply (simp add:indep_sets_def (*‹indep_sets ?F ?I = ((∀i∈?I. ?F i ⊆ events) ∧ (∀J⊆?I. J ≠ {} ⟶ finite J ⟶ (∀A∈Pi J ?F. prob (⋂ (A ` J)) = (∏j∈J. prob (A j)))))›*) all_subset_image (*‹(∀B⊆?f ` ?A. ?P B) = (∀B⊆?A. ?P (?f ` B))›*) a (*‹?J12 ⊆ I ⟹ prod ?g12 (f ` ?J12) = (∏j∈?J12. ?g12 (f j))›*) c (*‹?J13 ⊆ I ⟹ finite (f ` ?J13) = finite ?J13›*))\n (*goal: ‹indep_sets A (f ` I) = indep_sets (λi. A (f i)) I›*)\n by (simp add:forall_Pi_to_PiE (*‹(⋀x. ?P x = ?P (restrict x ?I)) ⟹ (∀x∈Pi ?I ?A. ?P x) = (∀x∈Pi⇩E ?I ?A. ?P x)›*) b (*‹?J13 ⊆ I ⟹ (Π⇩E i∈?J13. A (f i)) = (λa. restrict (a ∘ f) ?J13) ` Pi⇩E (f ` ?J13) A›*))\nqed\n\nlemma indep_vars_reindex:\n assumes \"inj_on f I\"\n assumes \"indep_vars M' X' (f ` I)\"\n shows \"indep_vars (M' ∘ f) (λk ω. X' (f k) ω) I\"\n using assms (*‹inj_on f I› ‹indep_vars M' X' (f ` I)›*) by (simp add:indep_vars_def2 (*‹indep_vars ?M' ?X ?I = ((∀i∈?I. random_variable (?M' i) (?X i)) ∧ indep_sets (λi. {?X i -` A ∩ space M |A. A ∈ sets (?M' i)}) ?I)›*) indep_sets_reindex (*‹inj_on ?f ?I ⟹ indep_sets ?A (?f ` ?I) = indep_sets (λi. ?A (?f i)) ?I›*))\n\nlemma indep_vars_cong_AE:\n assumes \"AE x in M. (∀i ∈ I. X' i x = Y' i x)\"\n assumes \"indep_vars M' X' I\"\n assumes \"⋀i. i ∈ I ⟹ random_variable (M' i) (Y' i)\"\n shows \"indep_vars M' Y' I\"\nproof (-)\n (*goal: ‹indep_vars M' Y' I›*)\n have a: \"AE x in M. (λi∈I. Y' i x) = (λi∈I. X' i x)\"\n apply (rule AE_mp[OF assms(1)] (*‹AE x in M. (∀i∈I. X' i x = Y' i x) ⟶ ?Q x ⟹ almost_everywhere M ?Q›*))\n (*goal: ‹AE x in M. (λi∈I. Y' i x) = (λi∈I. X' i x)›*)\n apply (rule AE_I2 (*‹(⋀x. x ∈ space ?M ⟹ ?P x) ⟹ almost_everywhere ?M ?P›*))\n (*goal: ‹AE x in M. (∀i∈I. X' i x = Y' i x) ⟶ (λi∈I. Y' i x) = (λi∈I. X' i x)›*)\n by (simp cong:restrict_cong (*‹⟦(?I::?'a set) = (?J::?'a set); ⋀i::?'a. i ∈ ?J =simp=> (?f::?'a ⇒ ?'b) i = (?g::?'a ⇒ ?'b) i⟧ ⟹ restrict ?f ?I = restrict ?g ?J›*))\n have b: \"⋀i. i ∈ I ⟹ random_variable (M' i) (X' i)\"\n using assms(2) (*‹indep_vars M' X' I›*) by (simp add:indep_vars_def2 (*‹indep_vars ?M' ?X ?I = ((∀i∈?I. random_variable (?M' i) (?X i)) ∧ indep_sets (λi. {?X i -` A ∩ space M |A. A ∈ sets (?M' i)}) ?I)›*))\n have c: \"⋀x. x ∈ I ⟹ AE xa in M. X' x xa = Y' x xa\"\n apply (rule AE_mp[OF assms(1)] (*‹AE x in M. (∀i∈I. X' i x = Y' i x) ⟶ ?Q x ⟹ almost_everywhere M ?Q›*))\n (*goal: ‹⋀x::'b. x ∈ (I::'b set) ⟹ AE xa::'a in M::'a measure. (X'::'b ⇒ 'a ⇒ 'c) x xa = (Y'::'b ⇒ 'a ⇒ 'c) x xa›*)\n apply (rule AE_I2 (*‹(⋀x::?'a. x ∈ space (?M::?'a measure) ⟹ (?P::?'a ⇒ bool) x) ⟹ almost_everywhere ?M ?P›*))\n (*goal: ‹⋀x. x ∈ I ⟹ AE xa in M. (∀i∈I. X' i xa = Y' i xa) ⟶ X' x xa = Y' x xa›*)\n by simp\n have \"distr M (Pi⇩M I M') (λx. λi∈I. Y' i x) = distr M (Pi⇩M I M') (λx. λi∈I. X' i x)\"\n apply (intro distr_cong_AE (*‹⟦?M = ?K; sets ?N = sets ?L; AE x in ?M. ?f x = ?g x; ?f ∈ ?M →⇩M ?N; ?g ∈ ?K →⇩M ?L⟧ ⟹ distr ?M ?N ?f = distr ?K ?L ?g›*) measurable_restrict (*‹(⋀i. i ∈ ?I ⟹ ?X i ∈ ?N →⇩M ?M i) ⟹ (λx. λi∈?I. ?X i x) ∈ ?N →⇩M Pi⇩M ?I ?M›*) a (*‹AE x in M. (λi∈I. Y' i x) = (λi∈I. X' i x)›*) b (*‹?i12 ∈ I ⟹ random_variable (M' ?i12) (X' ?i12)›*) assms( (*‹?i12 ∈ I ⟹ random_variable (M' ?i12) (Y' ?i12)›*) 3))\n (*goals:\n 1. ‹(M::'a::type measure) = M›\n 2. ‹sets (Pi⇩M (I::'b::type set) (M'::'b::type ⇒ 'c::type measure)) = sets (Pi⇩M I M')›\n 3. ‹⋀i::'b::type. i ∈ (I::'b::type set) ⟹ i ∈ I›\n 4. ‹⋀i::'b::type. i ∈ (I::'b::type set) ⟹ i ∈ I›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n also (*calculation: ‹distr M (Pi⇩M I M') (λx. λi∈I. Y' i x) = distr M (Pi⇩M I M') (λx. λi∈I. X' i x)›*) have \"... = Pi⇩M I (λi. distr M (M' i) (X' i))\"\n using assms (*‹AE x in M. ∀i∈I. X' i x = Y' i x› ‹indep_vars M' X' I› ‹?i12 ∈ I ⟹ random_variable (M' ?i12) (Y' ?i12)›*) b (*‹(?i12::'b) ∈ (I::'b set) ⟹ random_variable ((M'::'b ⇒ 'c measure) ?i12) ((X'::'b ⇒ 'a ⇒ 'c) ?i12)›*) apply (subst indep_vars_iff_distr_eq_PiM''[symmetric] (*‹(⋀i. i ∈ ?I ⟹ random_variable (?M' i) (?X i)) ⟹ (distr M (Pi⇩M ?I ?M') (λx. λi∈?I. ?X i x) = Pi⇩M ?I (λi. distr M (?M' i) (?X i))) = indep_vars ?M' ?X ?I›*))\n (*goals:\n 1. ‹⋀i. ⟦AE x in M. ∀i∈I. X' i x = Y' i x; indep_vars M' X' I; ⋀i. i ∈ I ⟹ random_variable (M' i) (Y' i); ⋀i. i ∈ I ⟹ random_variable (M' i) (X' i); i ∈ I⟧ ⟹ random_variable (M' i) (X' i)›\n 2. ‹⟦AE x in M. ∀i∈I. X' i x = Y' i x; indep_vars M' X' I; ⋀i. i ∈ I ⟹ random_variable (M' i) (Y' i); ⋀i. i ∈ I ⟹ random_variable (M' i) (X' i)⟧ ⟹ indep_vars M' X' I›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹distr M (Pi⇩M I M') (λx. λi∈I. Y' i x) = Pi⇩M I (λi. distr M (M' i) (X' i))›*) have \"... = Pi⇩M I (λi. distr M (M' i) (Y' i))\"\n apply (intro PiM_cong (*‹⟦?I = ?J; ⋀x. x ∈ ?I ⟹ ?M x = ?N x⟧ ⟹ Pi⇩M ?I ?M = Pi⇩M ?J ?N›*) distr_cong_AE (*‹⟦?M = ?K; sets ?N = sets ?L; AE x in ?M. ?f x = ?g x; ?f ∈ ?M →⇩M ?N; ?g ∈ ?K →⇩M ?L⟧ ⟹ distr ?M ?N ?f = distr ?K ?L ?g›*) c (*‹?x12 ∈ I ⟹ AE xa in M. X' ?x12 xa = Y' ?x12 xa›*) assms( (*‹?i12 ∈ I ⟹ random_variable (M' ?i12) (Y' ?i12)›*) 3) b (*‹?i12 ∈ I ⟹ random_variable (M' ?i12) (X' ?i12)›*))\n (*goals:\n 1. ‹(I::'b set) = I›\n 2. ‹⋀x::'b. x ∈ (I::'b set) ⟹ (M::'a measure) = M›\n 3. ‹⋀x::'b. x ∈ (I::'b set) ⟹ sets ((M'::'b ⇒ 'c measure) x) = sets (M' x)›\n 4. ‹⋀x::'b. x ∈ (I::'b set) ⟹ x ∈ I›\n 5. ‹⋀x::'b. x ∈ (I::'b set) ⟹ x ∈ I›\n 6. ‹⋀x::'b. x ∈ (I::'b set) ⟹ x ∈ I›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*proven 6 subgoals*) .\n finally (*calculation: ‹distr (M::'a measure) (Pi⇩M (I::'b set) (M'::'b ⇒ 'c measure)) (λx::'a. λi::'b∈I. (Y'::'b ⇒ 'a ⇒ 'c) i x) = Pi⇩M I (λi::'b. distr M (M' i) (Y' i))›*) have \"distr M (Pi⇩M I M') (λx. λi∈I. Y' i x) = Pi⇩M I (λi. distr M (M' i) (Y' i))\"\n by simp\n thus \"?thesis\"\n (*goal: ‹indep_vars (M'::'b::type ⇒ 'c::type measure) (Y'::'b::type ⇒ 'a::type ⇒ 'c::type) (I::'b::type set)›*)\n using assms(3) (*‹?i12 ∈ I ⟹ random_variable (M' ?i12) (Y' ?i12)›*) apply (subst indep_vars_iff_distr_eq_PiM'' (*‹(⋀i. i ∈ ?I ⟹ random_variable (?M' i) (?X i)) ⟹ indep_vars ?M' ?X ?I = (distr M (Pi⇩M ?I ?M') (λx. λi∈?I. ?X i x) = Pi⇩M ?I (λi. distr M (?M' i) (?X i)))›*))\n (*goals:\n 1. ‹⋀i. ⟦distr M (Pi⇩M I M') (λx. λi∈I. Y' i x) = Pi⇩M I (λi. distr M (M' i) (Y' i)); ⋀i. i ∈ I ⟹ random_variable (M' i) (Y' i); i ∈ I⟧ ⟹ random_variable (M' i) (Y' i)›\n 2. ‹⟦distr M (Pi⇩M I M') (λx. λi∈I. Y' i x) = Pi⇩M I (λi. distr M (M' i) (Y' i)); ⋀i. i ∈ I ⟹ random_variable (M' i) (Y' i)⟧ ⟹ distr M (Pi⇩M I M') (λx. λi∈I. Y' i x) = Pi⇩M I (λi. distr M (M' i) (Y' i))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nqed\n\nend\n\ntext ‹Integrability of bounded functions on finite measure spaces:›\n\nlemma bounded_const: \"bounded ((λx. (c::real)) ` T)\"\n apply (intro boundedI[where B=\"norm c\"] (*‹(⋀x. x ∈ ?S ⟹ norm x ≤ norm c) ⟹ bounded ?S›*))\n (*goal: ‹bounded ((λx. c) ` T)›*)\n by auto\n\nlemma bounded_exp:\n fixes f :: \"'a ⇒ real\"\n assumes \"bounded ((λx. f x) ` T)\"\n shows \"bounded ((λx. exp (f x)) ` T)\"\nproof (-)\n (*goal: ‹bounded ((λx. exp (f x)) ` T)›*)\n obtain m where \"norm (f x) ≤ m\" if \"x ∈ T\" for x\n (*goal: ‹(⋀m::real. (⋀x::'a. x ∈ (T::'a set) ⟹ norm ((f::'a ⇒ real) x) ≤ m) ⟹ thesis::bool) ⟹ thesis›*)\n using assms (*‹bounded ((f::'a::type ⇒ real) ` (T::'a::type set))›*) unfolding bounded_iff\n (*goal: ‹(⋀m. (⋀x. x ∈ T ⟹ norm (f x) ≤ m) ⟹ thesis) ⟹ thesis›*)\n by auto\n thus \"?thesis\"\n (*goal: ‹bounded ((λx. exp (f x)) ` T)›*)\n apply (intro boundedI[where B=\"exp m\"] (*‹(⋀x. x ∈ ?S ⟹ norm x ≤ exp m) ⟹ bounded ?S›*))\n (*goal: ‹bounded ((λx. exp (f x)) ` T)›*)\n by fastforce\nqed\n\nlemma bounded_mult_comp:\n fixes f :: \"'a ⇒ real\"\n assumes \"bounded (f ` T)\" \"bounded (g ` T)\"\n shows \"bounded ((λx. (f x) * (g x)) ` T)\"\nproof (-)\n (*goal: ‹bounded ((λx. f x * g x) ` T)›*)\n obtain m1 where \"norm (f x) ≤ m1\" \"m1 ≥0\" if \"x ∈ T\" for x\n (*goal: ‹(⋀m1. ⟦⋀x. x ∈ T ⟹ norm (f x) ≤ m1; ⋀x. x ∈ T ⟹ 0 ≤ m1⟧ ⟹ thesis) ⟹ thesis›*)\n using assms (*‹bounded (f ` T)› ‹bounded (g ` T)›*) unfolding bounded_iff\n (*goal: ‹(⋀m1. ⟦⋀x. x ∈ T ⟹ norm (f x) ≤ m1; ⋀x. x ∈ T ⟹ 0 ≤ m1⟧ ⟹ thesis) ⟹ thesis›*)\n by fastforce\n moreover obtain m2 where \"norm (g x) ≤ m2\" \"m2 ≥0\" if \"x ∈ T\" for x\n (*goal: ‹(⋀m2. ⟦⋀x. x ∈ T ⟹ norm (g x) ≤ m2; ⋀x. x ∈ T ⟹ 0 ≤ m2⟧ ⟹ thesis) ⟹ thesis›*)\n using assms (*‹bounded (f ` T)› ‹bounded ((g::'a ⇒ real) ` (T::'a set))›*) unfolding bounded_iff\n (*goal: ‹(⋀m2. ⟦⋀x. x ∈ T ⟹ norm (g x) ≤ m2; ⋀x. x ∈ T ⟹ 0 ≤ m2⟧ ⟹ thesis) ⟹ thesis›*)\n by fastforce\n ultimately show \"?thesis\"\n (*goal: ‹bounded ((λx. f x * g x) ` T)›*)\n apply (intro boundedI[where B=\"m1 * m2\"] (*‹(⋀x. x ∈ ?S ⟹ norm x ≤ m1 * m2) ⟹ bounded ?S›*))\n (*goal: ‹bounded ((λx::'a. (f::'a ⇒ real) x * (g::'a ⇒ real) x) ` (T::'a set))›*)\n by (auto intro!: mult_mono (*‹⟦?a ≤ ?b; ?c ≤ ?d; 0 ≤ ?b; 0 ≤ ?c⟧ ⟹ ?a * ?c ≤ ?b * ?d›*) simp:abs_mult (*‹¦?a * ?b¦ = ¦?a¦ * ¦?b¦›*))\nqed\n\nlemma bounded_sum:\n fixes f :: \"'i ⇒ 'a ⇒ real\"\n assumes \"finite I\"\n assumes \"⋀i. i ∈ I ⟹ bounded (f i ` T)\"\n shows \"bounded ((λx. (∑i ∈ I. f i x)) ` T)\"\n using assms (*‹finite I› ‹?i ∈ I ⟹ bounded (f ?i ` T)›*) apply (induction I)\n (*goals:\n 1. ‹(⋀i. i ∈ {} ⟹ bounded (f i ` T)) ⟹ bounded ((λx. ∑i∈{}. f i x) ` T)›\n 2. ‹⋀x F. ⟦finite F; x ∉ F; (⋀i. i ∈ F ⟹ bounded (f i ` T)) ⟹ bounded ((λx. ∑i∈F. f i x) ` T); ⋀i. i ∈ insert x F ⟹ bounded (f i ` T)⟧ ⟹ bounded ((λxa. ∑i∈insert x F. f i xa) ` T)›\n discuss goal 1*)\n apply ((auto intro:bounded_plus_comp (*‹⟦bounded (?f ` ?S); bounded (?g ` ?S)⟧ ⟹ bounded ((λx. ?f x + ?g x) ` ?S)›*) bounded_const (*‹bounded ((λx. ?c) ` ?T)›*))[1])\n (*discuss goal 2*)\n apply ((auto intro:bounded_plus_comp (*‹⟦bounded ((?f::?'a::type ⇒ ?'b::real_normed_vector) ` (?S::?'a::type set)); bounded ((?g::?'a::type ⇒ ?'b::real_normed_vector) ` ?S)⟧ ⟹ bounded ((λx::?'a::type. ?f x + ?g x) ` ?S)›*) bounded_const (*‹bounded ((λx::?'a::type. ?c::real) ` (?T::?'a::type set))›*))[1])\n (*proven 2 subgoals*) .\n\nlemma bounded_pow:\n fixes f :: \"'a ⇒ real\"\n assumes \"bounded ((λx. f x) ` T)\"\n shows \"bounded ((λx. (f x)^n) ` T)\"\nproof (-)\n (*goal: ‹bounded ((λx. f x ^ n) ` T)›*)\n obtain m where \"norm (f x) ≤ m\" if \"x ∈ T\" for x\n (*goal: ‹(⋀m. (⋀x. x ∈ T ⟹ norm (f x) ≤ m) ⟹ thesis) ⟹ thesis›*)\n using assms (*‹bounded (f ` T)›*) unfolding bounded_iff\n (*goal: ‹(⋀m::real. (⋀x::'a. x ∈ (T::'a set) ⟹ norm ((f::'a ⇒ real) x) ≤ m) ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n hence \"norm ((f x)^n) ≤ m^n\" if \"x ∈ T\" for x\n using that (*‹x ∈ T›*) unfolding norm_power\n (*goal: ‹norm (f x) ^ n ≤ m ^ n›*)\n apply (intro power_mono (*‹⟦(?a::?'a) ≤ (?b::?'a); (0::?'a) ≤ ?a⟧ ⟹ ?a ^ (?n::nat) ≤ ?b ^ ?n›*))\n (*goals:\n 1. ‹⟦⋀x::'a. x ∈ (T::'a set) ⟹ norm ((f::'a ⇒ real) x) ≤ (m::real); (x::'a) ∈ T⟧ ⟹ norm (f x) ≤ m›\n 2. ‹⟦⋀x::'a. x ∈ (T::'a set) ⟹ norm ((f::'a ⇒ real) x) ≤ (m::real); (x::'a) ∈ T⟧ ⟹ (0::real) ≤ norm (f x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹bounded ((λx. f x ^ n) ` T)›*)\n apply (intro boundedI[where B=\"m^n\"] (*‹(⋀x. x ∈ ?S ⟹ norm x ≤ m ^ n) ⟹ bounded ?S›*))\n (*goal: ‹bounded ((λx. f x ^ n) ` T)›*)\n by auto\nqed\n\nlemma bounded_sum_list:\n fixes f :: \"'i ⇒ 'a ⇒ real\"\n assumes \"⋀y. y ∈ set ys ⟹ bounded (f y ` T)\"\n shows \"bounded ((λx. (∑y ← ys. f y x)) ` T)\"\n using assms (*‹?y ∈ set ys ⟹ bounded (f ?y ` T)›*) apply (induction ys)\n (*goals:\n 1. ‹(⋀y. y ∈ set [] ⟹ bounded (f y ` T)) ⟹ bounded ((λx. ∑y←[]. f y x) ` T)›\n 2. ‹⋀a ys. ⟦(⋀y. y ∈ set ys ⟹ bounded (f y ` T)) ⟹ bounded ((λx. ∑y←ys. f y x) ` T); ⋀y. y ∈ set (a # ys) ⟹ bounded (f y ` T)⟧ ⟹ bounded ((λx. ∑y←a # ys. f y x) ` T)›\n discuss goal 1*)\n apply ((auto intro:bounded_plus_comp (*‹⟦bounded (?f ` ?S); bounded (?g ` ?S)⟧ ⟹ bounded ((λx. ?f x + ?g x) ` ?S)›*) bounded_const (*‹bounded ((λx. ?c) ` ?T)›*))[1])\n (*discuss goal 2*)\n apply ((auto intro:bounded_plus_comp (*‹⟦bounded (?f ` ?S); bounded (?g ` ?S)⟧ ⟹ bounded ((λx. ?f x + ?g x) ` ?S)›*) bounded_const (*‹bounded ((λx. ?c) ` ?T)›*))[1])\n (*proven 2 subgoals*) .\n\nlemma (in finite_measure) bounded_int:\n fixes f :: \"'i ⇒ 'a ⇒ real\"\n assumes \"bounded ((λ x. f (fst x) (snd x)) ` (T × space M))\"\n shows \"bounded ((λx. (∫ω. (f x ω) ∂M)) ` T)\"\nproof (-)\n (*goal: ‹bounded ((λx. integral⇧L M (f x)) ` T)›*)\n obtain m where \"⋀x y. x ∈ T ⟹ y ∈ space M ⟹ norm (f x y) ≤ m\"\n (*goal: ‹(⋀m. (⋀x y. ⟦x ∈ T; y ∈ space M⟧ ⟹ norm (f x y) ≤ m) ⟹ thesis) ⟹ thesis›*)\n using assms (*‹bounded ((λx::'i × 'a. (f::'i ⇒ 'a ⇒ real) (fst x) (snd x)) ` ((T::'i set) × space (M::'a measure)))›*) unfolding bounded_iff\n (*goal: ‹(⋀m. (⋀x y. ⟦x ∈ T; y ∈ space M⟧ ⟹ norm (f x y) ≤ m) ⟹ thesis) ⟹ thesis›*)\n by auto\n hence m: \"⋀x y. x ∈ T ⟹ y ∈ space M ⟹ norm (f x y) ≤ max m 0\"\n by fastforce\n have \"norm (∫ω. (f x ω) ∂M) ≤ max m 0 * measure M (space M)\" (is \"?L ≤ ?R\") if \"x ∈ T\" for x\n proof (-)\n (*goal: ‹norm (integral⇧L M (f x)) ≤ max m 0 * Sigma_Algebra.measure M (space M)›*)\n have \"?L ≤ (∫ω. norm (f x ω) ∂M)\"\n by simp\n also (*calculation: ‹norm (integral⇧L M (f x)) ≤ LINT ω|M. norm (f x ω)›*) have \"... ≤ (∫ω. max m 0 ∂M)\"\n using that (*‹(x::'i) ∈ (T::'i set)›*) m (*‹⟦?x ∈ T; ?y ∈ space M⟧ ⟹ norm (f ?x ?y) ≤ max m 0›*) apply (intro integral_mono' (*‹⟦integrable ?M ?f; ⋀x. x ∈ space ?M ⟹ ?g x ≤ ?f x; ⋀x. x ∈ space ?M ⟹ 0 ≤ ?f x⟧ ⟹ integral⇧L ?M ?g ≤ integral⇧L ?M ?f›*))\n (*goals:\n 1. ‹⟦x ∈ T; ⋀x y. ⟦x ∈ T; y ∈ space M⟧ ⟹ norm (f x y) ≤ max m 0⟧ ⟹ integrable M (λx. max m 0)›\n 2. ‹⋀ω. ⟦x ∈ T; ⋀x y. ⟦x ∈ T; y ∈ space M⟧ ⟹ norm (f x y) ≤ max m 0; ω ∈ space M⟧ ⟹ norm (f x ω) ≤ max m 0›\n 3. ‹⋀ω. ⟦x ∈ T; ⋀x y. ⟦x ∈ T; y ∈ space M⟧ ⟹ norm (f x y) ≤ max m 0; ω ∈ space M⟧ ⟹ 0 ≤ max m 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹norm (integral⇧L M (f x)) ≤ LINT ω|M. max m 0›*) have \"... = ?R\"\n by simp\n finally (*calculation: ‹norm (integral⇧L M (f x)) ≤ max m 0 * Sigma_Algebra.measure M (space M)›*) show \"?thesis\"\n (*goal: ‹norm (integral⇧L M (f x)) ≤ max m 0 * Sigma_Algebra.measure M (space M)›*)\n by simp\n qed\n thus \"?thesis\"\n (*goal: ‹bounded ((λx. integral⇧L M (f x)) ` T)›*)\n apply (intro boundedI[where B=\"max m 0 * measure M (space M)\"] (*‹(⋀x. x ∈ ?S ⟹ norm x ≤ max m 0 * Sigma_Algebra.measure M (space M)) ⟹ bounded ?S›*))\n (*goal: ‹bounded ((λx. integral⇧L M (f x)) ` T)›*)\n by auto\nqed\n\nlemmas bounded_intros =\n bounded_minus_comp bounded_plus_comp bounded_mult_comp bounded_sum finite_measure.bounded_int\n bounded_const bounded_exp bounded_pow bounded_sum_list\n\nlemma (in prob_space) integrable_bounded:\n fixes f :: \"_ ⇒ ('b :: {banach,second_countable_topology})\"\n assumes \"bounded (f ` space M)\"\n assumes \"f ∈ M →⇩M borel\"\n shows \"integrable M f\"\nproof (-)\n (*goal: ‹integrable M f›*)\n obtain m where \"norm (f x) ≤ m\" if \"x ∈ space M\" for x\n (*goal: ‹(⋀m. (⋀x. x ∈ space M ⟹ norm (f x) ≤ m) ⟹ thesis) ⟹ thesis›*)\n using assms(1) (*‹bounded (f ` space M)›*) unfolding bounded_iff\n (*goal: ‹(⋀m. (⋀x. x ∈ space M ⟹ norm (f x) ≤ m) ⟹ thesis) ⟹ thesis›*)\n by auto\n thus \"?thesis\"\n (*goal: ‹integrable M f›*)\n by (intro integrable_const_bound[where B=\"m\"] (*‹⟦AE x in M. norm (?f x) ≤ m; random_variable borel ?f⟧ ⟹ integrable M ?f›*) AE_I2 (*‹(⋀x. x ∈ space ?M ⟹ ?P x) ⟹ almost_everywhere ?M ?P›*) assms( (*‹random_variable borel f›*) 2))\nqed\n\nlemma integrable_bounded_pmf:\n fixes f :: \"_ ⇒ ('b :: {banach,second_countable_topology})\"\n assumes \"bounded (f ` set_pmf M)\"\n shows \"integrable (measure_pmf M) f\"\nproof (-)\n (*goal: ‹integrable (measure_pmf M) f›*)\n obtain m where \"norm (f x) ≤ m\" if \"x ∈ set_pmf M\" for x\n (*goal: ‹(⋀m. (⋀x. x ∈ set_pmf M ⟹ norm (f x) ≤ m) ⟹ thesis) ⟹ thesis›*)\n using assms(1) (*‹bounded (f ` set_pmf M)›*) unfolding bounded_iff\n (*goal: ‹(⋀m. (⋀x. x ∈ set_pmf M ⟹ norm (f x) ≤ m) ⟹ thesis) ⟹ thesis›*)\n by auto\n thus \"?thesis\"\n (*goal: ‹integrable (measure_pmf M) f›*)\n apply (intro measure_pmf.integrable_const_bound[where B=\"m\"] (*‹⟦AE x in measure_pmf ?M. norm (?f x) ≤ m; measure_pmf.random_variable ?M borel ?f⟧ ⟹ integrable (measure_pmf ?M) ?f›*) AE_pmfI (*‹(⋀y. y ∈ set_pmf ?M ⟹ ?P y) ⟹ almost_everywhere (measure_pmf ?M) ?P›*))\n (*goals:\n 1. ‹⋀y. ⟦⋀x. x ∈ set_pmf M ⟹ norm (f x) ≤ m; y ∈ set_pmf M⟧ ⟹ norm (f y) ≤ m›\n 2. ‹(⋀x. x ∈ set_pmf M ⟹ norm (f x) ≤ m) ⟹ measure_pmf.random_variable M borel f›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nqed\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Concentration_Inequalities/Concentration_Inequalities_Preliminary.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"a05f62e524f21cfcc7ddeeb8b6b99905b128eca5029059a145f7725768952400\"\n}"}}},{"rowIdx":188,"cells":{"text":{"kind":"string","value":"(* Author: Dmitriy Traytel *)\n\nsection ‹Normalization of WS1S Formulas›\n\n(*<*)\ntheory WS1S_Normalization\nimports WS1S\nbegin\n(*>*)\n\nfun nNot where\n \"nNot (FNot φ) = φ\"\n| \"nNot (FAnd φ1 φ2) = FOr (nNot φ1) (nNot φ2)\"\n| \"nNot (FOr φ1 φ2) = FAnd (nNot φ1) (nNot φ2)\"\n| \"nNot φ = FNot φ\"\n\nprimrec norm where\n \"norm (FQ a m) = FQ a m\"\n| \"norm (FLess m n) = FLess m n\"\n| \"norm (FIn m M) = FIn m M\"\n| \"norm (FOr φ ψ) = FOr (norm φ) (norm ψ)\"\n| \"norm (FAnd φ ψ) = FAnd (norm φ) (norm ψ)\"\n| \"norm (FNot φ) = nNot (norm φ)\"\n| \"norm (FExists φ) = FExists (norm φ)\"\n| \"norm (FEXISTS φ) = FEXISTS (norm φ)\"\n\ncontext formula\nbegin\n\nlemma satisfies_nNot[simp]: \"(w, I) ⊨ nNot φ ⟷ (w, I) ⊨ FNot φ\"\n apply (induct φ rule: nNot.induct (*‹⟦⋀φ. ?P (FNot φ); ⋀φ1 φ2. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FAnd φ1 φ2); ⋀φ1 φ2. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FOr φ1 φ2); ⋀v va. ?P (FQ v va); ⋀v va. ?P (FLess v va); ⋀v va. ?P (FIn v va); ⋀v. ?P (FExists v); ⋀v. ?P (FEXISTS v)⟧ ⟹ ?P ?a0.0›*))\n (*goals:\n 1. ‹⋀φ. ((w, I) ⊨ nNot (FNot φ)) = ((w, I) ⊨ FNot (FNot φ))›\n 2. ‹⋀φ1 φ2. ⟦((w, I) ⊨ nNot φ1) = ((w, I) ⊨ FNot φ1); ((w, I) ⊨ nNot φ2) = ((w, I) ⊨ FNot φ2)⟧ ⟹ ((w, I) ⊨ nNot (FAnd φ1 φ2)) = ((w, I) ⊨ FNot (FAnd φ1 φ2))›\n 3. ‹⋀φ1 φ2. ⟦((w, I) ⊨ nNot φ1) = ((w, I) ⊨ FNot φ1); ((w, I) ⊨ nNot φ2) = ((w, I) ⊨ FNot φ2)⟧ ⟹ ((w, I) ⊨ nNot (FOr φ1 φ2)) = ((w, I) ⊨ FNot (FOr φ1 φ2))›\n 4. ‹⋀v va. ((w, I) ⊨ nNot (FQ v va)) = ((w, I) ⊨ FNot (FQ v va))›\n 5. ‹⋀v va. ((w, I) ⊨ nNot (FLess v va)) = ((w, I) ⊨ FNot (FLess v va))›\n 6. ‹⋀v va. ((w, I) ⊨ nNot (FIn v va)) = ((w, I) ⊨ FNot (FIn v va))›\n 7. ‹⋀v. ((w, I) ⊨ nNot (FExists v)) = ((w, I) ⊨ FNot (FExists v))›\n 8. ‹⋀v. ((w, I) ⊨ nNot (FEXISTS v)) = ((w, I) ⊨ FNot (FEXISTS v))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma FOV_nNot[simp]: \"FOV (nNot φ) = FOV (FNot φ)\"\n apply (induct φ rule: nNot.induct (*‹⟦⋀φ::?'a::type formula. (?P::?'a::type formula ⇒ bool) (FNot φ); ⋀(φ1::?'a::type formula) φ2::?'a::type formula. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FAnd φ1 φ2); ⋀(φ1::?'a::type formula) φ2::?'a::type formula. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FOr φ1 φ2); ⋀(v::?'a::type) va::nat. ?P (FQ v va); ⋀(v::nat) va::nat. ?P (FLess v va); ⋀(v::nat) va::nat. ?P (FIn v va); ⋀v::?'a::type formula. ?P (FExists v); ⋀v::?'a::type formula. ?P (FEXISTS v)⟧ ⟹ ?P (?a0.0::?'a::type formula)›*))\n (*goals:\n 1. ‹⋀φ. FOV (nNot (FNot φ)) = FOV (FNot (FNot φ))›\n 2. ‹⋀φ1 φ2. ⟦FOV (nNot φ1) = FOV (FNot φ1); FOV (nNot φ2) = FOV (FNot φ2)⟧ ⟹ FOV (nNot (FAnd φ1 φ2)) = FOV (FNot (FAnd φ1 φ2))›\n 3. ‹⋀φ1 φ2. ⟦FOV (nNot φ1) = FOV (FNot φ1); FOV (nNot φ2) = FOV (FNot φ2)⟧ ⟹ FOV (nNot (FOr φ1 φ2)) = FOV (FNot (FOr φ1 φ2))›\n 4. ‹⋀v va. FOV (nNot (FQ v va)) = FOV (FNot (FQ v va))›\n 5. ‹⋀v va. FOV (nNot (FLess v va)) = FOV (FNot (FLess v va))›\n 6. ‹⋀v va. FOV (nNot (FIn v va)) = FOV (FNot (FIn v va))›\n 7. ‹⋀v. FOV (nNot (FExists v)) = FOV (FNot (FExists v))›\n 8. ‹⋀v. FOV (nNot (FEXISTS v)) = FOV (FNot (FEXISTS v))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma SOV_nNot[simp]: \"SOV (nNot φ) = SOV (FNot φ)\"\n apply (induct φ rule: nNot.induct (*‹⟦⋀φ::?'a formula. (?P::?'a formula ⇒ bool) (FNot φ); ⋀(φ1::?'a formula) φ2::?'a formula. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FAnd φ1 φ2); ⋀(φ1::?'a formula) φ2::?'a formula. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FOr φ1 φ2); ⋀(v::?'a) va::nat. ?P (FQ v va); ⋀(v::nat) va::nat. ?P (FLess v va); ⋀(v::nat) va::nat. ?P (FIn v va); ⋀v::?'a formula. ?P (FExists v); ⋀v::?'a formula. ?P (FEXISTS v)⟧ ⟹ ?P (?a0.0::?'a formula)›*))\n (*goals:\n 1. ‹⋀φ. SOV (nNot (FNot φ)) = SOV (FNot (FNot φ))›\n 2. ‹⋀φ1 φ2. ⟦SOV (nNot φ1) = SOV (FNot φ1); SOV (nNot φ2) = SOV (FNot φ2)⟧ ⟹ SOV (nNot (FAnd φ1 φ2)) = SOV (FNot (FAnd φ1 φ2))›\n 3. ‹⋀φ1 φ2. ⟦SOV (nNot φ1) = SOV (FNot φ1); SOV (nNot φ2) = SOV (FNot φ2)⟧ ⟹ SOV (nNot (FOr φ1 φ2)) = SOV (FNot (FOr φ1 φ2))›\n 4. ‹⋀v va. SOV (nNot (FQ v va)) = SOV (FNot (FQ v va))›\n 5. ‹⋀v va. SOV (nNot (FLess v va)) = SOV (FNot (FLess v va))›\n 6. ‹⋀v va. SOV (nNot (FIn v va)) = SOV (FNot (FIn v va))›\n 7. ‹⋀v. SOV (nNot (FExists v)) = SOV (FNot (FExists v))›\n 8. ‹⋀v. SOV (nNot (FEXISTS v)) = SOV (FNot (FEXISTS v))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma pre_wf_formula_nNot[simp]: \"pre_wf_formula n (nNot φ) = pre_wf_formula n (FNot φ)\"\n apply (induct φ rule: nNot.induct (*‹⟦⋀φ. ?P (FNot φ); ⋀φ1 φ2. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FAnd φ1 φ2); ⋀φ1 φ2. ⟦?P φ1; ?P φ2⟧ ⟹ ?P (FOr φ1 φ2); ⋀v va. ?P (FQ v va); ⋀v va. ?P (FLess v va); ⋀v va. ?P (FIn v va); ⋀v. ?P (FExists v); ⋀v. ?P (FEXISTS v)⟧ ⟹ ?P ?a0.0›*))\n (*goals:\n 1. ‹⋀φ::'a::linorder formula. pre_wf_formula (n::nat) (nNot (FNot φ)) = pre_wf_formula n (FNot (FNot φ))›\n 2. ‹⋀(φ1::'a::linorder formula) φ2::'a::linorder formula. ⟦pre_wf_formula (n::nat) (nNot φ1) = pre_wf_formula n (FNot φ1); pre_wf_formula n (nNot φ2) = pre_wf_formula n (FNot φ2)⟧ ⟹ pre_wf_formula n (nNot (FAnd φ1 φ2)) = pre_wf_formula n (FNot (FAnd φ1 φ2))›\n 3. ‹⋀(φ1::'a::linorder formula) φ2::'a::linorder formula. ⟦pre_wf_formula (n::nat) (nNot φ1) = pre_wf_formula n (FNot φ1); pre_wf_formula n (nNot φ2) = pre_wf_formula n (FNot φ2)⟧ ⟹ pre_wf_formula n (nNot (FOr φ1 φ2)) = pre_wf_formula n (FNot (FOr φ1 φ2))›\n 4. ‹⋀(v::'a::linorder) va::nat. pre_wf_formula (n::nat) (nNot (FQ v va)) = pre_wf_formula n (FNot (FQ v va))›\n 5. ‹⋀(v::nat) va::nat. pre_wf_formula (n::nat) (nNot (FLess v va)) = pre_wf_formula n (FNot (FLess v va))›\n 6. ‹⋀(v::nat) va::nat. pre_wf_formula (n::nat) (nNot (FIn v va)) = pre_wf_formula n (FNot (FIn v va))›\n 7. ‹⋀v::'a::linorder formula. pre_wf_formula (n::nat) (nNot (FExists v)) = pre_wf_formula n (FNot (FExists v))›\n 8. ‹⋀v::'a::linorder formula. pre_wf_formula (n::nat) (nNot (FEXISTS v)) = pre_wf_formula n (FNot (FEXISTS v))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma FOV_norm[simp]: \"FOV (norm φ) = FOV φ\"\n apply (induct φ)\n (*goals:\n 1. ‹⋀x1 x2. FOV (norm (FQ x1 x2)) = FOV (FQ x1 x2)›\n 2. ‹⋀x1 x2. FOV (norm (FLess x1 x2)) = FOV (FLess x1 x2)›\n 3. ‹⋀x1 x2. FOV (norm (FIn x1 x2)) = FOV (FIn x1 x2)›\n 4. ‹⋀φ. FOV (norm φ) = FOV φ ⟹ FOV (norm (FNot φ)) = FOV (FNot φ)›\n 5. ‹⋀φ1 φ2. ⟦FOV (norm φ1) = FOV φ1; FOV (norm φ2) = FOV φ2⟧ ⟹ FOV (norm (FOr φ1 φ2)) = FOV (FOr φ1 φ2)›\n 6. ‹⋀φ1 φ2. ⟦FOV (norm φ1) = FOV φ1; FOV (norm φ2) = FOV φ2⟧ ⟹ FOV (norm (FAnd φ1 φ2)) = FOV (FAnd φ1 φ2)›\n 7. ‹⋀φ. FOV (norm φ) = FOV φ ⟹ FOV (norm (FExists φ)) = FOV (FExists φ)›\n 8. ‹⋀φ. FOV (norm φ) = FOV φ ⟹ FOV (norm (FEXISTS φ)) = FOV (FEXISTS φ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma SOV_norm[simp]: \"SOV (norm φ) = SOV φ\"\n apply (induct φ)\n (*goals:\n 1. ‹⋀x1 x2. SOV (norm (FQ x1 x2)) = SOV (FQ x1 x2)›\n 2. ‹⋀x1 x2. SOV (norm (FLess x1 x2)) = SOV (FLess x1 x2)›\n 3. ‹⋀x1 x2. SOV (norm (FIn x1 x2)) = SOV (FIn x1 x2)›\n 4. ‹⋀φ. SOV (norm φ) = SOV φ ⟹ SOV (norm (FNot φ)) = SOV (FNot φ)›\n 5. ‹⋀φ1 φ2. ⟦SOV (norm φ1) = SOV φ1; SOV (norm φ2) = SOV φ2⟧ ⟹ SOV (norm (FOr φ1 φ2)) = SOV (FOr φ1 φ2)›\n 6. ‹⋀φ1 φ2. ⟦SOV (norm φ1) = SOV φ1; SOV (norm φ2) = SOV φ2⟧ ⟹ SOV (norm (FAnd φ1 φ2)) = SOV (FAnd φ1 φ2)›\n 7. ‹⋀φ. SOV (norm φ) = SOV φ ⟹ SOV (norm (FExists φ)) = SOV (FExists φ)›\n 8. ‹⋀φ. SOV (norm φ) = SOV φ ⟹ SOV (norm (FEXISTS φ)) = SOV (FEXISTS φ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma pre_wf_formula_norm[simp]: \"pre_wf_formula n (norm φ) = pre_wf_formula n φ\"\n apply (induct φ arbitrary: n)\n (*goals:\n 1. ‹⋀(x1::'a::linorder) (x2::nat) n::nat. pre_wf_formula n (norm (FQ x1 x2)) = pre_wf_formula n (FQ x1 x2)›\n 2. ‹⋀(x1::nat) (x2::nat) n::nat. pre_wf_formula n (norm (FLess x1 x2)) = pre_wf_formula n (FLess x1 x2)›\n 3. ‹⋀(x1::nat) (x2::nat) n::nat. pre_wf_formula n (norm (FIn x1 x2)) = pre_wf_formula n (FIn x1 x2)›\n 4. ‹⋀(φ::'a::linorder formula) n::nat. (⋀n::nat. pre_wf_formula n (norm φ) = pre_wf_formula n φ) ⟹ pre_wf_formula n (norm (FNot φ)) = pre_wf_formula n (FNot φ)›\n 5. ‹⋀(φ1::'a::linorder formula) (φ2::'a::linorder formula) n::nat. ⟦⋀n::nat. pre_wf_formula n (norm φ1) = pre_wf_formula n φ1; ⋀n::nat. pre_wf_formula n (norm φ2) = pre_wf_formula n φ2⟧ ⟹ pre_wf_formula n (norm (FOr φ1 φ2)) = pre_wf_formula n (FOr φ1 φ2)›\n 6. ‹⋀(φ1::'a::linorder formula) (φ2::'a::linorder formula) n::nat. ⟦⋀n::nat. pre_wf_formula n (norm φ1) = pre_wf_formula n φ1; ⋀n::nat. pre_wf_formula n (norm φ2) = pre_wf_formula n φ2⟧ ⟹ pre_wf_formula n (norm (FAnd φ1 φ2)) = pre_wf_formula n (FAnd φ1 φ2)›\n 7. ‹⋀(φ::'a::linorder formula) n::nat. (⋀n::nat. pre_wf_formula n (norm φ) = pre_wf_formula n φ) ⟹ pre_wf_formula n (norm (FExists φ)) = pre_wf_formula n (FExists φ)›\n 8. ‹⋀(φ::'a::linorder formula) n::nat. (⋀n::nat. pre_wf_formula n (norm φ) = pre_wf_formula n φ) ⟹ pre_wf_formula n (norm (FEXISTS φ)) = pre_wf_formula n (FEXISTS φ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma satisfies_norm[simp]: \"wI ⊨ norm φ ⟷ wI ⊨ φ\"\n apply (induct φ arbitrary: wI)\n (*goals:\n 1. ‹⋀x1 x2 wI. (wI ⊨ norm (FQ x1 x2)) = (wI ⊨ FQ x1 x2)›\n 2. ‹⋀x1 x2 wI. (wI ⊨ norm (FLess x1 x2)) = (wI ⊨ FLess x1 x2)›\n 3. ‹⋀x1 x2 wI. (wI ⊨ norm (FIn x1 x2)) = (wI ⊨ FIn x1 x2)›\n 4. ‹⋀φ wI. (⋀wI. (wI ⊨ norm φ) = (wI ⊨ φ)) ⟹ (wI ⊨ norm (FNot φ)) = (wI ⊨ FNot φ)›\n 5. ‹⋀φ1 φ2 wI. ⟦⋀wI. (wI ⊨ norm φ1) = (wI ⊨ φ1); ⋀wI. (wI ⊨ norm φ2) = (wI ⊨ φ2)⟧ ⟹ (wI ⊨ norm (FOr φ1 φ2)) = (wI ⊨ FOr φ1 φ2)›\n 6. ‹⋀φ1 φ2 wI. ⟦⋀wI. (wI ⊨ norm φ1) = (wI ⊨ φ1); ⋀wI. (wI ⊨ norm φ2) = (wI ⊨ φ2)⟧ ⟹ (wI ⊨ norm (FAnd φ1 φ2)) = (wI ⊨ FAnd φ1 φ2)›\n 7. ‹⋀φ wI. (⋀wI. (wI ⊨ norm φ) = (wI ⊨ φ)) ⟹ (wI ⊨ norm (FExists φ)) = (wI ⊨ FExists φ)›\n 8. ‹⋀φ wI. (⋀wI. (wI ⊨ norm φ) = (wI ⊨ φ)) ⟹ (wI ⊨ norm (FEXISTS φ)) = (wI ⊨ FEXISTS φ)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*discuss goal 7*)\n apply ((auto)[1])\n (*discuss goal 8*)\n apply ((auto)[1])\n (*proven 8 subgoals*) .\n\nlemma lang⇩W⇩S₁⇩S_norm[simp]: \"lang⇩W⇩S₁⇩S n (norm φ) = lang⇩W⇩S₁⇩S n φ\"\n unfolding \"lang⇩W⇩S₁⇩S_def\"\n (*goal: ‹⋃ {enc (w, I) |w I. length I = n ∧ wf_interp_for_formula (w, I) (norm φ) ∧ (w, I) ⊨ norm φ} = ⋃ {enc (w, I) |w I. length I = n ∧ wf_interp_for_formula (w, I) φ ∧ (w, I) ⊨ φ}›*)\n by auto\n\nend\n\n(*<*)\nend\n(*>*)\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/MSO_Regex_Equivalence/WS1S_Normalization.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"deee4b7c960f5c82bfad81a1a4ccde6f8bbe13eb607dbc47c1ccc1b4ca31a515\"\n}"}}},{"rowIdx":189,"cells":{"text":{"kind":"string","value":"(* Title: Subset Boolean Algebras\n Authors: Walter Guttmann, Bernhard Möller\n Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>\n*)\n\ntheory Subset_Boolean_Algebras\n\nimports Stone_Algebras.P_Algebras\n\nbegin\n\nsection ‹Boolean Algebras›\n\ntext ‹\nWe show that Isabelle/HOL's ‹boolean_algebra› class is equivalent to Huntington's axioms \\<^cite>‹\"Huntington1933\"›.\nSee \\<^cite>‹\"WamplerDoty2016\"› for related results.\n›\n\nsubsection ‹Huntington's Axioms›\n\ntext ‹Definition 1›\n\nclass huntington = sup + uminus +\n assumes associative: \"x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n assumes commutative: \"x ⊔ y = y ⊔ x\"\n assumes huntington: \"x = -(-x ⊔ y) ⊔ -(-x ⊔ -y)\"\nbegin\n\nlemma top_unique:\n \"x ⊔ -x = y ⊔ -y\"\nproof (-)\n (*goal: ‹x ⊔ - x = y ⊔ - y›*)\n have \"x ⊔ -x = y ⊔ -(--y ⊔ -x) ⊔ -(--y ⊔ --x)\"\n by (smt associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*))\n thus \"?thesis\"\n (*goal: ‹x ⊔ - x = y ⊔ - y›*)\n by (metis associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*))\nqed\n\nend\n\nsubsection ‹Equivalence to ‹boolean_algebra› Class›\n\ntext ‹Definition 2›\n\nclass extended = sup + inf + minus + uminus + bot + top + ord +\n assumes top_def: \"top = (THE x . ∀y . x = y ⊔ -y)\" (* define without imposing uniqueness *)\n assumes bot_def: \"bot = -(THE x . ∀y . x = y ⊔ -y)\"\n assumes inf_def: \"x ⊓ y = -(-x ⊔ -y)\"\n assumes minus_def: \"x - y = -(-x ⊔ y)\"\n assumes less_eq_def: \"x ≤ y ⟷ x ⊔ y = y\"\n assumes less_def: \"x < y ⟷ x ⊔ y = y ∧ ¬ (y ⊔ x = x)\"\n\nclass huntington_extended = huntington + extended\nbegin\n\nlemma top_char:\n \"top = x ⊔ -x\"\n using top_def (*‹top = (THE x::'a::type. ∀y::'a::type. x = y ⊔ - y)›*) top_unique (*‹(?x::'a) ⊔ - ?x = (?y::'a) ⊔ - ?y›*) by auto\n\nlemma bot_char:\n \"bot = -top\"\n by (simp add: bot_def (*‹bot = - (THE x::'a. ∀y::'a. x = y ⊔ - y)›*) top_def (*‹top = (THE x::'a. ∀y::'a. x = y ⊔ - y)›*))\n\nsubclass boolean_algebra\nproof\n show 1: \"⋀x y. (x < y) = (x ≤ y ∧ ¬ y ≤ x)\"\n by (simp add: less_def less_eq_def)\n show 2: \"⋀x. x ≤ x\"\n proof -\n fix x\n have \"x ⊔ top = top ⊔ --x\"\n by (metis (full_types) associative top_char)\n thus \"x ≤ x\"\n by (metis (no_types) associative huntington less_eq_def top_char)\n qed\n show 3: \"⋀x y z. x ≤ y ⟹ y ≤ z ⟹ x ≤ z\"\n by (metis associative less_eq_def)\n show 4: \"⋀x y. x ≤ y ⟹ y ≤ x ⟹ x = y\"\n by (simp add: commutative less_eq_def)\n show 5: \"⋀x y. x ⊓ y ≤ x\"\n using 2 by (metis associative huntington inf_def less_eq_def)\n show 6: \"⋀x y. x ⊓ y ≤ y\"\n using 5 commutative inf_def by fastforce\n show 8: \"⋀x y. x ≤ x ⊔ y\"\n using 2 associative less_eq_def by auto\n show 9: \"⋀y x. y ≤ x ⊔ y\"\n using 8 commutative by fastforce\n show 10: \"⋀y x z. y ≤ x ⟹ z ≤ x ⟹ y ⊔ z ≤ x\"\n by (metis associative less_eq_def)\n show 11: \"⋀x. bot ≤ x\"\n using 8 by (metis bot_char huntington top_char)\n show 12: \"⋀x. x ≤ top\"\n using 6 11 by (metis huntington bot_def inf_def less_eq_def top_def)\n show 13: \"⋀x y z. x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)\"\n proof -\n have 2: \"⋀x y z . x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n by (simp add: associative)\n have 3: \"⋀x y z . (x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)\"\n using 2 by metis\n have 4: \"⋀x y . x ⊔ y = y ⊔ x\"\n by (simp add: commutative)\n have 5: \"⋀x y . x = - (- x ⊔ y) ⊔ - (- x ⊔ - y)\"\n by (simp add: huntington)\n have 6: \"⋀x y . - (- x ⊔ y) ⊔ - (- x ⊔ - y) = x\"\n using 5 by metis\n have 7: \"⋀x y . x ⊓ y = - (- x ⊔ - y)\"\n by (simp add: inf_def)\n have 10: \"⋀x y z . x ⊔ (y ⊔ z) = y ⊔ (x ⊔ z)\"\n using 3 4 by metis\n have 11: \"⋀x y z . - (- x ⊔ y) ⊔ (- (- x ⊔ - y) ⊔ z) = x ⊔ z\"\n using 3 6 by metis\n have 12: \"⋀x y . - (x ⊔ - y) ⊔ - (- y ⊔ - x) = y\"\n using 4 6 by metis\n have 13: \"⋀x y . - (- x ⊔ y) ⊔ - (- y ⊔ - x) = x\"\n using 4 6 by metis\n have 14: \"⋀x y . - x ⊔ - (- (- x ⊔ y) ⊔ - - (- x ⊔ - y)) = - x ⊔ y\"\n using 6 by metis\n have 18: \"⋀x y z . - (x ⊔ - y) ⊔ (- (- y ⊔ - x) ⊔ z) = y ⊔ z\"\n using 3 12 by metis\n have 20: \"⋀x y . - (- x ⊔ - y) ⊔ - (y ⊔ - x) = x\"\n using 4 12 by metis\n have 21: \"⋀x y . - (x ⊔ - y) ⊔ - (- x ⊔ - y) = y\"\n using 4 12 by metis\n have 22: \"⋀x y . - x ⊔ - (- (y ⊔ - x) ⊔ - - (- x ⊔ - y)) = y ⊔ - x\"\n using 6 12 by metis\n have 23: \"⋀x y . - x ⊔ - (- x ⊔ (- y ⊔ - (y ⊔ - x))) = y ⊔ - x\"\n using 3 4 6 12 by metis\n have 24: \"⋀x y . - x ⊔ - (- (- x ⊔ - y) ⊔ - - (- x ⊔ y)) = - x ⊔ - y\"\n using 6 12 by metis\n have 28: \"⋀x y . - (- x ⊔ - y) ⊔ - (- y ⊔ x) = y\"\n using 4 13 by metis\n have 30: \"⋀x y . - x ⊔ - (- y ⊔ (- x ⊔ - (- x ⊔ y))) = - x ⊔ y\"\n using 3 4 6 13 by metis\n have 32: \"⋀x y z . - (- x ⊔ y) ⊔ (z ⊔ - (- y ⊔ - x)) = z ⊔ x\"\n using 10 13 by metis\n have 37: \"⋀x y z . - (- x ⊔ - y) ⊔ (- (y ⊔ - x) ⊔ z) = x ⊔ z\"\n using 3 20 by metis\n have 39: \"⋀x y z . - (- x ⊔ - y) ⊔ (z ⊔ - (y ⊔ - x)) = z ⊔ x\"\n using 10 20 by metis\n have 40: \"⋀x y z . - (x ⊔ - y) ⊔ (- (- x ⊔ - y) ⊔ z) = y ⊔ z\"\n using 3 21 by metis\n have 43: \"⋀x y . - x ⊔ - (- y ⊔ (- x ⊔ - (y ⊔ - x))) = y ⊔ - x\"\n using 3 4 6 21 by metis\n have 47: \"⋀x y z . - (x ⊔ y) ⊔ - (- (- x ⊔ z) ⊔ - (- (- x ⊔ - z) ⊔ y)) = - x ⊔ z\"\n using 6 11 by metis\n have 55: \"⋀x y . x ⊔ - (- y ⊔ - - x) = y ⊔ - (- x ⊔ y)\"\n using 4 11 12 by metis\n have 58: \"⋀x y . x ⊔ - (- - y ⊔ - x) = x ⊔ - (- x ⊔ y)\"\n using 4 11 13 by metis\n have 63: \"⋀x y . x ⊔ - (- - x ⊔ - y) = y ⊔ - (- x ⊔ y)\"\n using 4 11 21 by metis\n have 71: \"⋀x y . x ⊔ - (- y ⊔ x) = y ⊔ - (- x ⊔ y)\"\n using 4 11 28 by metis\n have 75: \"⋀x y . x ⊔ - (- y ⊔ x) = y ⊔ - (y ⊔ - x)\"\n using 4 71 by metis\n have 78: \"⋀x y . - x ⊔ (y ⊔ - (- x ⊔ (y ⊔ - - (- x ⊔ - y)))) = - x ⊔ - (- x ⊔ - y)\"\n using 3 4 6 71 by metis\n have 86: \"⋀x y . - (- x ⊔ - (- y ⊔ x)) ⊔ - (y ⊔ - (- x ⊔ y)) = - y ⊔ x\"\n using 4 20 71 by metis\n have 172: \"⋀x y . - x ⊔ - (- x ⊔ - y) = y ⊔ - (- - x ⊔ y)\"\n using 14 75 by metis\n have 201: \"⋀x y . x ⊔ - (- y ⊔ - - x) = y ⊔ - (y ⊔ - x)\"\n using 4 55 by metis\n have 236: \"⋀x y . x ⊔ - (- - y ⊔ - x) = x ⊔ - (y ⊔ - x)\"\n using 4 58 by metis\n have 266: \"⋀x y . - x ⊔ - (- (- x ⊔ - (y ⊔ - - x)) ⊔ - - (- x ⊔ - - (- - x ⊔ y))) = - x ⊔ - (- - x ⊔ y)\"\n using 14 58 236 by metis\n have 678: \"⋀x y z . - (- x ⊔ - (- y ⊔ x)) ⊔ (- (y ⊔ - (- x ⊔ y)) ⊔ z) = - y ⊔ (x ⊔ z)\"\n using 3 4 37 71 by smt\n have 745: \"⋀x y z . - (- x ⊔ - (- y ⊔ x)) ⊔ (z ⊔ - (y ⊔ - (- x ⊔ y))) = z ⊔ (- y ⊔ x)\"\n using 4 39 71 by metis\n have 800: \"⋀x y . - - x ⊔ (- y ⊔ (- (y ⊔ - - x) ⊔ - (- x ⊔ (- - x ⊔ (- y ⊔ - (y ⊔ - - x)))))) = x ⊔ - (y ⊔ - - x)\"\n using 3 23 63 by metis\n have 944: \"⋀x y . x ⊔ - (x ⊔ - - (- (- x ⊔ - y) ⊔ - - (- x ⊔ y))) = - (- x ⊔ - y) ⊔ - (- (- x ⊔ - y) ⊔ - - (- x ⊔ y))\"\n using 4 24 71 by metis\n have 948: \"⋀x y . - x ⊔ - (- (y ⊔ - (y ⊔ - - x)) ⊔ - - (- x ⊔ (- y ⊔ - x))) = - x ⊔ - (- y ⊔ - x)\"\n using 24 75 by metis\n have 950: \"⋀x y . - x ⊔ - (- (y ⊔ - (- - x ⊔ y)) ⊔ - - (- x ⊔ (- x ⊔ - y))) = - x ⊔ - (- x ⊔ - y)\"\n using 24 75 by metis\n have 961: \"⋀x y . - x ⊔ - (- (y ⊔ - (- - x ⊔ y)) ⊔ - - (- x ⊔ (- - - x ⊔ - y))) = y ⊔ - (- - x ⊔ y)\"\n using 24 63 by metis\n have 966: \"⋀x y . - x ⊔ - (- (y ⊔ - (y ⊔ - - x)) ⊔ - - (- x ⊔ (- y ⊔ - - - x))) = y ⊔ - (y ⊔ - - x)\"\n using 24 201 by metis\n have 969: \"⋀x y . - x ⊔ - (- (- x ⊔ - (y ⊔ - - x)) ⊔ - - (- x ⊔ (- - y ⊔ - - x))) = - x ⊔ - (y ⊔ - - x)\"\n using 24 236 by metis\n have 1096: \"⋀x y z . - x ⊔ (- (- x ⊔ - y) ⊔ z) = y ⊔ (- (- - x ⊔ y) ⊔ z)\"\n using 3 172 by metis\n have 1098: \"⋀x y z . - x ⊔ (y ⊔ - (- x ⊔ - z)) = y ⊔ (z ⊔ - (- - x ⊔ z))\"\n using 10 172 by metis\n have 1105: \"⋀x y . x ⊔ - x = y ⊔ - y\"\n using 4 10 12 32 172 by metis\n have 1109: \"⋀x y z . x ⊔ (- x ⊔ y) = z ⊔ (- z ⊔ y)\"\n using 3 1105 by metis\n have 1110: \"⋀x y z . x ⊔ - x = y ⊔ (z ⊔ - (y ⊔ z))\"\n using 3 1105 by metis\n have 1114: \"⋀x y . - (- x ⊔ - - x) = - (y ⊔ - y)\"\n using 7 1105 by metis\n have 1115: \"⋀x y z . x ⊔ (y ⊔ - y) = z ⊔ (x ⊔ - z)\"\n using 10 1105 by metis\n have 1117: \"⋀x y . - (x ⊔ - - x) ⊔ - (y ⊔ - y) = - x\"\n using 4 13 1105 by metis\n have 1121: \"⋀x y . - (x ⊔ - x) ⊔ - (y ⊔ - - y) = - y\"\n using 4 28 1105 by metis\n have 1122: \"⋀x . - - x = x\"\n using 4 28 1105 1117 by metis\n have 1134: \"⋀x y z . - (x ⊔ - y) ⊔ (z ⊔ - z) = y ⊔ (- y ⊔ - x)\"\n using 18 1105 1122 by metis\n have 1140: \"⋀x . - x ⊔ - (x ⊔ (x ⊔ - x)) = - x ⊔ - x\"\n using 4 22 1105 1122 1134 by metis\n have 1143: \"⋀x y . x ⊔ (- x ⊔ y) = y ⊔ (x ⊔ - y)\"\n using 37 1105 1122 1134 by metis\n have 1155: \"⋀x y . - (x ⊔ - x) ⊔ - (y ⊔ y) = - y\"\n using 1121 1122 by metis\n have 1156: \"⋀x y . - (x ⊔ x) ⊔ - (y ⊔ - y) = - x\"\n using 1117 1122 by metis\n have 1157: \"⋀x y . - (x ⊔ - x) = - (y ⊔ - y)\"\n using 4 1114 1122 by metis\n have 1167: \"⋀x y z . - x ⊔ (y ⊔ - (- x ⊔ - z)) = y ⊔ (z ⊔ - (x ⊔ z))\"\n using 1098 1122 by metis\n have 1169: \"⋀x y z . - x ⊔ (- (- x ⊔ - y) ⊔ z) = y ⊔ (- (x ⊔ y) ⊔ z)\"\n using 1096 1122 by metis\n have 1227: \"⋀x y . - x ⊔ - (- x ⊔ (y ⊔ (x ⊔ - (- x ⊔ - (y ⊔ x))))) = - x ⊔ - (y ⊔ x)\"\n using 3 4 969 1122 by smt\n have 1230: \"⋀x y . - x ⊔ - (- x ⊔ (- y ⊔ (- x ⊔ - (y ⊔ - (y ⊔ x))))) = y ⊔ - (y ⊔ x)\"\n using 3 4 966 1122 by smt\n have 1234: \"⋀x y . - x ⊔ - (- x ⊔ (- x ⊔ (- y ⊔ - (y ⊔ - (x ⊔ y))))) = y ⊔ - (x ⊔ y)\"\n using 3 4 961 1122 by metis\n have 1239: \"⋀x y . - x ⊔ - (- x ⊔ - y) = y ⊔ - (x ⊔ y)\"\n using 3 4 950 1122 1234 by metis\n have 1240: \"⋀x y . - x ⊔ - (- y ⊔ - x) = y ⊔ - (y ⊔ x)\"\n using 3 4 948 1122 1230 by metis\n have 1244: \"⋀x y . x ⊔ - (x ⊔ (y ⊔ (y ⊔ - (x ⊔ y)))) = - (- x ⊔ - y) ⊔ - (y ⊔ (y ⊔ - (x ⊔ y)))\"\n using 3 4 944 1122 1167 by metis\n have 1275: \"⋀x y . x ⊔ (- y ⊔ (- (y ⊔ x) ⊔ - (x ⊔ (- x ⊔ (- y ⊔ - (y ⊔ x)))))) = x ⊔ - (y ⊔ x)\"\n using 10 800 1122 by metis\n have 1346: \"⋀x y . - x ⊔ - (x ⊔ (y ⊔ (y ⊔ (x ⊔ - (x ⊔ (y ⊔ x)))))) = - x ⊔ - (x ⊔ y)\"\n using 3 4 10 266 1122 1167 by smt\n have 1377: \"⋀x y . - x ⊔ (y ⊔ - (- x ⊔ (y ⊔ (- x ⊔ - y)))) = y ⊔ - (x ⊔ y)\"\n using 78 1122 1239 by metis\n have 1394: \"⋀x y . - (- x ⊔ - y) ⊔ - (y ⊔ (y ⊔ (- x ⊔ - (x ⊔ y)))) = x\"\n using 3 4 10 20 30 1122 1239 by smt\n have 1427: \"⋀x y . - (- x ⊔ - y) ⊔ - (y ⊔ - (x ⊔ (x ⊔ - (x ⊔ y)))) = x ⊔ (x ⊔ - (x ⊔ y))\"\n using 3 4 30 40 1240 by smt\n have 1436: \"⋀x . - x ⊔ - (x ⊔ (x ⊔ (- x ⊔ - x))) = - x ⊔ (- x ⊔ - (x ⊔ - x))\"\n using 3 4 30 1140 1239 by smt\n have 1437: \"⋀x y . - (x ⊔ y) ⊔ - (x ⊔ - y) = - x\"\n using 6 1122 by metis\n have 1438: \"⋀x y . - (x ⊔ y) ⊔ - (y ⊔ - x) = - y\"\n using 12 1122 by metis\n have 1439: \"⋀x y . - (x ⊔ y) ⊔ - (- y ⊔ x) = - x\"\n using 13 1122 by metis\n have 1440: \"⋀x y . - (x ⊔ - y) ⊔ - (y ⊔ x) = - x\"\n using 20 1122 by metis\n have 1441: \"⋀x y . - (x ⊔ y) ⊔ - (- x ⊔ y) = - y\"\n using 21 1122 by metis\n have 1568: \"⋀x y . x ⊔ (- y ⊔ - x) = y ⊔ (- y ⊔ x)\"\n using 10 1122 1143 by metis\n have 1598: \"⋀x . - x ⊔ - (x ⊔ (x ⊔ (x ⊔ - x))) = - x ⊔ (- x ⊔ - (x ⊔ - x))\"\n using 4 1436 1568 by metis\n have 1599: \"⋀x y . - x ⊔ (y ⊔ - (x ⊔ (- x ⊔ (- x ⊔ y)))) = y ⊔ - (x ⊔ y)\"\n using 10 1377 1568 by smt\n have 1617: \"⋀x . x ⊔ (- x ⊔ (- x ⊔ - (x ⊔ - x))) = x ⊔ - x\"\n using 3 4 10 71 1122 1155 1568 1598 by metis\n have 1632: \"⋀x y z . - (x ⊔ - x) ⊔ - (- y ⊔ (- (z ⊔ - z) ⊔ - (y ⊔ - (x ⊔ - x)))) = y ⊔ - (x ⊔ - x)\"\n using 43 1157 by metis\n have 1633: \"⋀x y z . - (x ⊔ - x) ⊔ - (- y ⊔ (- (x ⊔ - x) ⊔ - (y ⊔ - (z ⊔ - z)))) = y ⊔ - (x ⊔ - x)\"\n using 43 1157 by metis\n have 1636: \"⋀x y . x ⊔ - (y ⊔ (- y ⊔ - (x ⊔ x))) = x ⊔ x\"\n using 43 1109 1122 by metis\n have 1645: \"⋀x y . x ⊔ - x = y ⊔ (y ⊔ - y)\"\n using 3 1110 1156 by metis\n have 1648: \"⋀x y z . - (x ⊔ (y ⊔ (- y ⊔ - x))) ⊔ - (z ⊔ - z) = - (y ⊔ - y)\"\n using 3 1115 1156 by metis\n have 1657: \"⋀x y z . x ⊔ - x = y ⊔ (z ⊔ - z)\"\n using 1105 1645 by metis\n have 1664: \"⋀x y z . x ⊔ - x = y ⊔ (z ⊔ - y)\"\n using 1115 1645 by metis\n have 1672: \"⋀x y z . x ⊔ - x = y ⊔ (- y ⊔ z)\"\n using 3 4 1657 by metis\n have 1697: \"⋀x y z . - x ⊔ (y ⊔ x) = z ⊔ - z\"\n using 1122 1664 by metis\n have 1733: \"⋀x y z . - (x ⊔ y) ⊔ - (- (z ⊔ - z) ⊔ - (- (- x ⊔ - x) ⊔ y)) = x ⊔ - x\"\n using 4 47 1105 1122 by metis\n have 1791: \"⋀x y z . x ⊔ - (y ⊔ (- y ⊔ z)) = x ⊔ - (x ⊔ - x)\"\n using 4 71 1122 1672 by metis\n have 1818: \"⋀x y z . x ⊔ - (- y ⊔ (z ⊔ y)) = x ⊔ - (x ⊔ - x)\"\n using 4 71 1122 1697 by metis\n have 1861: \"⋀x y z . - (x ⊔ - x) ⊔ - (y ⊔ - (z ⊔ - z)) = - y\"\n using 1437 1657 by metis\n have 1867: \"⋀x y z . - (x ⊔ - x) ⊔ - (- y ⊔ - (z ⊔ y)) = y\"\n using 1122 1437 1697 by metis\n have 1868: \"⋀x y . x ⊔ - (y ⊔ - y) = x\"\n using 1122 1155 1633 1861 by metis\n have 1869: \"⋀x y z . - (x ⊔ - x) ⊔ - (- y ⊔ (- (z ⊔ - z) ⊔ - y)) = y\"\n using 1632 1868 by metis\n have 1870: \"⋀x y . - (x ⊔ - x) ⊔ - y = - y\"\n using 1861 1868 by metis\n have 1872: \"⋀x y z . x ⊔ - (- y ⊔ (z ⊔ y)) = x\"\n using 1818 1868 by metis\n have 1875: \"⋀x y z . x ⊔ - (y ⊔ (- y ⊔ z)) = x\"\n using 1791 1868 by metis\n have 1883: \"⋀x y . - (x ⊔ (y ⊔ (- y ⊔ - x))) = - (y ⊔ - y)\"\n using 1648 1868 by metis\n have 1885: \"⋀x . x ⊔ (x ⊔ - x) = x ⊔ - x\"\n using 4 1568 1617 1868 by metis\n have 1886: \"⋀x . - x ⊔ - x = - x\"\n using 1598 1868 1885 by metis\n have 1890: \"⋀x . - (x ⊔ x) = - x\"\n using 1156 1868 by metis\n have 1892: \"⋀x y . - (x ⊔ - x) ⊔ y = y\"\n using 1122 1869 1870 1886 by metis\n have 1893: \"⋀x y . - (- x ⊔ - (y ⊔ x)) = x\"\n using 1867 1892 by metis\n have 1902: \"⋀x y . x ⊔ (y ⊔ - (x ⊔ y)) = x ⊔ - x\"\n using 3 4 1122 1733 1886 1892 by metis\n have 1908: \"⋀x . x ⊔ x = x\"\n using 1636 1875 1890 by metis\n have 1910: \"⋀x y . x ⊔ - (y ⊔ x) = - y ⊔ x\"\n using 1599 1875 by metis\n have 1921: \"⋀x y . x ⊔ (- y ⊔ - (y ⊔ x)) = - y ⊔ x\"\n using 1275 1875 1910 by metis\n have 1951: \"⋀x y . - x ⊔ - (y ⊔ x) = - x\"\n using 1227 1872 1893 1908 by metis\n have 1954: \"⋀x y z . x ⊔ (y ⊔ - (x ⊔ z)) = y ⊔ (- z ⊔ x)\"\n using 745 1122 1910 1951 by metis\n have 1956: \"⋀x y z . x ⊔ (- (x ⊔ y) ⊔ z) = - y ⊔ (x ⊔ z)\"\n using 678 1122 1910 1951 by metis\n have 1959: \"⋀x y . x ⊔ - (x ⊔ y) = - y ⊔ x\"\n using 86 1122 1910 1951 by metis\n have 1972: \"⋀x y . x ⊔ (- x ⊔ y) = x ⊔ - x\"\n using 1902 1910 by metis\n have 2000: \"⋀x y . - (- x ⊔ - y) ⊔ - (y ⊔ (- x ⊔ y)) = x ⊔ - (y ⊔ (- x ⊔ y))\"\n using 4 1244 1910 1959 by metis\n have 2054: \"⋀x y . x ⊔ - (y ⊔ (- x ⊔ y)) = x\"\n using 1394 1921 2000 by metis\n have 2057: \"⋀x y . - (x ⊔ (y ⊔ - y)) = - (y ⊔ - y)\"\n using 1883 1972 by metis\n have 2061: \"⋀x y . x ⊔ (- y ⊔ x) = x ⊔ - y\"\n using 4 1122 1427 1910 1959 2054 by metis\n have 2090: \"⋀x y z . x ⊔ (- (y ⊔ x) ⊔ z) = x ⊔ (- y ⊔ z)\"\n using 1122 1169 1956 by metis\n have 2100: \"⋀x y . - x ⊔ - (x ⊔ y) = - x\"\n using 4 1346 1868 1885 1910 1959 1972 2057 by metis\n have 2144: \"⋀x y . x ⊔ - (y ⊔ - x) = x\"\n using 1122 1440 2000 2061 by metis\n have 2199: \"⋀x y . x ⊔ (x ⊔ y) = x ⊔ y\"\n using 3 1908 by metis\n have 2208: \"⋀x y z . x ⊔ (- (y ⊔ - x) ⊔ z) = x ⊔ z\"\n using 3 2144 by metis\n have 2349: \"⋀x y z . - (x ⊔ y) ⊔ - (x ⊔ (y ⊔ z)) = - (x ⊔ y)\"\n using 3 2100 by metis\n have 2432: \"⋀x y z . - (x ⊔ (y ⊔ z)) ⊔ - (y ⊔ (z ⊔ - x)) = - (y ⊔ z)\"\n using 3 1438 by metis\n have 2530: \"⋀x y z . - (- (x ⊔ y) ⊔ z) = - (y ⊔ (- x ⊔ z)) ⊔ - (- y ⊔ z)\"\n using 4 1122 1439 2090 2208 by smt\n have 3364: \"⋀x y z . - (- x ⊔ y) ⊔ (z ⊔ - (x ⊔ y)) = z ⊔ - y\"\n using 3 4 1122 1441 1910 1954 2199 by metis\n have 5763: \"⋀x y z . - (x ⊔ y) ⊔ - (- x ⊔ (y ⊔ z)) = - (x ⊔ y) ⊔ - (y ⊔ z)\"\n using 4 2349 3364 by metis\n have 6113: \"⋀x y z . - (x ⊔ (y ⊔ z)) ⊔ - (z ⊔ - x) = - (y ⊔ z) ⊔ - (z ⊔ - x)\"\n using 4 2432 3364 5763 by metis\n show \"⋀x y z. x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)\"\n proof -\n fix x y z\n have \"- (y ⊓ z ⊔ x) = - (- (- y ⊔ z) ⊔ - (- y ⊔ - z) ⊔ x) ⊔ - (x ⊔ - - z)\"\n using 1437 2530 6113 by (smt commutative inf_def)\n thus \"x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)\"\n using 12 1122 by (metis commutative inf_def)\n qed\n qed\n show 14: \"⋀x. x ⊓ - x = bot\"\n proof -\n fix x\n have \"(bot ⊔ x) ⊓ (bot ⊔ -x) = bot\"\n using huntington bot_def inf_def by auto\n thus \"x ⊓ -x = bot\"\n using 11 less_eq_def by force\n qed\n show 15: \"⋀x. x ⊔ - x = top\"\n using 5 14 by (metis (no_types, lifting) huntington bot_def less_eq_def top_def)\n show 16: \"⋀x y. x - y = x ⊓ - y\"\n using 15 by (metis commutative huntington inf_def minus_def)\n show 7: \"⋀x y z. x ≤ y ⟹ x ≤ z ⟹ x ≤ y ⊓ z\"\n by (simp add: 13 less_eq_def)\nqed\n\nend\n\ncontext boolean_algebra\nbegin\n\nsublocale ba_he: huntington_extended\nproof (standard)\n (*goals:\n 1. ‹top = (THE x. ∀y. x = y ⊔ - y)›\n 2. ‹bot = - (THE x. ∀y. x = y ⊔ - y)›\n 3. ‹⋀x y. x ⊓ y = - (- x ⊔ - y)›\n 4. ‹⋀x y. x - y = - (- x ⊔ y)›\n 5. ‹⋀x y. (x ≤ y) = (x ⊔ y = y)›\n 6. ‹⋀x y. (x < y) = (x ⊔ y = y ∧ y ⊔ x ≠ x)›\n 7. ‹⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z›\n 8. ‹⋀x y. x ⊔ y = y ⊔ x›\n 9. ‹⋀x y. x = - (- x ⊔ y) ⊔ - (- x ⊔ - y)›*)\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: sup_assoc (*‹(?x::'a) ⊔ (?y::'a) ⊔ (?z::'a) = ?x ⊔ (?y ⊔ ?z)›*))\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: sup_commute (*‹?x ⊔ ?y = ?y ⊔ ?x›*))\n show \"⋀x y. x = - (- x ⊔ y) ⊔ - (- x ⊔ - y)\"\n by simp\n show \"top = (THE x. ∀y. x = y ⊔ - y)\"\n by auto\n show \"bot = - (THE x. ∀y. x = y ⊔ - y)\"\n by auto\n show \"⋀x y. x ⊓ y = - (- x ⊔ - y)\"\n by simp\n show \"⋀x y. x - y = - (- x ⊔ y)\"\n by (simp add: diff_eq (*‹?x - ?y = ?x ⊓ - ?y›*))\n show \"⋀x y. (x ≤ y) = (x ⊔ y = y)\"\n by (simp add: le_iff_sup (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n show \"⋀x y. (x < y) = (x ⊔ y = y ∧ y ⊔ x ≠ x)\"\n using sup.strict_order_iff (*‹(?b < ?a) = (?a = ?a ⊔ ?b ∧ ?a ≠ ?b)›*) sup_commute (*‹(?x::'a::type) ⊔ (?y::'a::type) = ?y ⊔ ?x›*) by auto\nqed\n\nend\n\nsubsection ‹Stone Algebras›\n\ntext ‹\nWe relate Stone algebras to Boolean algebras.\n›\n\nclass stone_algebra_extended = stone_algebra + minus +\n assumes stone_minus_def[simp]: \"x - y = x ⊓ -y\"\n\nclass regular_stone_algebra = stone_algebra_extended +\n assumes double_complement[simp]: \"--x = x\"\nbegin\n\nsubclass boolean_algebra\nproof\n show \"⋀x. x ⊓ - x = bot\"\n by simp\n show \"⋀x. x ⊔ - x = top\"\n using regular_dense_top by fastforce\n show \"⋀x y. x - y = x ⊓ - y\"\n by simp\nqed\n\nend\n\ncontext boolean_algebra\nbegin\n\nsublocale ba_rsa: regular_stone_algebra\nproof (standard)\n (*goals:\n 1. ‹⋀(x::'a::type) y::'a::type. x - y = x ⊓ - y›\n 2. ‹⋀x::'a::type. - - x = x›*)\n show \"⋀x y. x - y = x ⊓ - y\"\n by (simp add: diff_eq (*‹?x - ?y = ?x ⊓ - ?y›*))\n show \"⋀x. - - x = x\"\n by simp\nqed\n\nend\n\nsection ‹Alternative Axiomatisations of Boolean Algebras›\n\ntext ‹\nWe consider four axiomatisations of Boolean algebras based only on join and complement.\nThe first three are from the literature and the fourth, a version using equational axioms, is new.\nThe motivation for Byrne's and the new axiomatisation is that the axioms are easier to understand than Huntington's third axiom.\nWe also include Meredith's axiomatisation.\n›\n\nsubsection ‹Lee Byrne's Formulation A›\n\ntext ‹\nThe following axiomatisation is from \\<^cite>‹‹Formulation A› in \"Byrne1946\"›; see also \\<^cite>‹\"Frink1941\"›.\n›\n\ntext ‹Theorem 3›\n\nclass boolean_algebra_1 = sup + uminus +\n assumes ba1_associative: \"x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n assumes ba1_commutative: \"x ⊔ y = y ⊔ x\"\n assumes ba1_complement: \"x ⊔ -y = z ⊔ -z ⟷ x ⊔ y = x\"\nbegin\n\nsubclass huntington\nproof\n show 1: \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: ba1_associative)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: ba1_commutative)\n show \"⋀x y. x = - (- x ⊔ y) ⊔ - (- x ⊔ - y)\"\n proof -\n have 2: \"∀x y. y ⊔ (y ⊔ x) = y ⊔ x\"\n using 1 by (metis ba1_complement)\n hence \"∀x. --x = x\"\n by (smt ba1_associative ba1_commutative ba1_complement)\n hence \"∀x y. y ⊔ -(y ⊔ -x) = y ⊔ x\"\n by (smt ba1_associative ba1_commutative ba1_complement)\n thus \"⋀x y. x = -(-x ⊔ y) ⊔ -(-x ⊔ - y)\"\n using 2 by (smt ba1_commutative ba1_complement)\n qed\nqed\n\nend\n\ncontext huntington\nbegin\n\nsublocale h_ba1: boolean_algebra_1\nproof (standard)\n (*goals:\n 1. ‹⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z›\n 2. ‹⋀x y. x ⊔ y = y ⊔ x›\n 3. ‹⋀x y z. (x ⊔ - y = z ⊔ - z) = (x ⊔ y = x)›*)\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*))\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*))\n show \"⋀x y z. (x ⊔ - y = z ⊔ - z) = (x ⊔ y = x)\"\n proof (standard)\n (*goals:\n 1. ‹⋀x y z. x ⊔ - y = z ⊔ - z ⟹ x ⊔ y = x›\n 2. ‹⋀x y z. x ⊔ y = x ⟹ x ⊔ - y = z ⊔ - z›*)\n fix x and y and z\n have 1: \"⋀x y z. -(-x ⊔ y) ⊔ (-(-x ⊔ -y) ⊔ z) = x ⊔ z\"\n using associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*) by force\n have 2: \"⋀x y. -(x ⊔ -y) ⊔ -(-y ⊔ -x) = y\"\n by (metis commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*))\n show \"x ⊔ - y = z ⊔ - z ⟹ x ⊔ y = x\"\n by (metis 1 (*‹- (- ?x ⊔ ?y) ⊔ (- (- ?x ⊔ - ?y) ⊔ ?z) = ?x ⊔ ?z›*) 2 (*‹- (?x ⊔ - ?y) ⊔ - (- ?y ⊔ - ?x) = ?y›*) associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) top_unique (*‹?x ⊔ - ?x = ?y ⊔ - ?y›*))\n show \"x ⊔ y = x ⟹ x ⊔ - y = z ⊔ - z\"\n by (metis associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*) commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) top_unique (*‹?x ⊔ - ?x = ?y ⊔ - ?y›*))\n qed\nqed\n\nend\n\nsubsection ‹Lee Byrne's Formulation B›\n\ntext ‹\nThe following axiomatisation is from \\<^cite>‹‹Formulation B› in \"Byrne1946\"›.\n›\n\ntext ‹Theorem 4›\n\nclass boolean_algebra_2 = sup + uminus +\n assumes ba2_associative_commutative: \"(x ⊔ y) ⊔ z = (y ⊔ z) ⊔ x\"\n assumes ba2_complement: \"x ⊔ -y = z ⊔ -z ⟷ x ⊔ y = x\"\nbegin\n\nsubclass boolean_algebra_1\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (smt ba2_associative_commutative ba2_complement)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (metis ba2_associative_commutative ba2_complement)\n show \"⋀x y z. (x ⊔ - y = z ⊔ - z) = (x ⊔ y = x)\"\n by (simp add: ba2_complement)\nqed\n\nend\n\ncontext boolean_algebra_1\nbegin\n\nsublocale ba1_ba2: boolean_algebra_2\nproof (standard)\n (*goals:\n 1. ‹⋀x y z. x ⊔ y ⊔ z = y ⊔ z ⊔ x›\n 2. ‹⋀x y z. (x ⊔ - y = z ⊔ - z) = (x ⊔ y = x)›*)\n show \"⋀x y z. x ⊔ y ⊔ z = y ⊔ z ⊔ x\"\n using ba1_associative (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?y ⊔ ?z›*) commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by force\n show \"⋀x y z. (x ⊔ - y = z ⊔ - z) = (x ⊔ y = x)\"\n by (simp add: ba1_complement (*‹(?x ⊔ - ?y = ?z ⊔ - ?z) = (?x ⊔ ?y = ?x)›*))\nqed\n\nend\n\nsubsection ‹Meredith's Equational Axioms›\n\ntext ‹\nThe following axiomatisation is from \\<^cite>‹‹page 221 (1) \\{A,N\\}› in \"MeredithPrior1968\"›.\n›\n\nclass boolean_algebra_mp = sup + uminus +\n assumes ba_mp_1: \"-(-x ⊔ y) ⊔ x = x\"\n assumes ba_mp_2: \"-(-x ⊔ y) ⊔ (z ⊔ y) = y ⊔ (z ⊔ x)\"\nbegin\n\nsubclass huntington\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (metis ba_mp_1 ba_mp_2)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (metis ba_mp_1 ba_mp_2)\n show \"⋀x y. x = - (- x ⊔ y) ⊔ - (- x ⊔ - y)\"\n by (metis ba_mp_1 ba_mp_2)\nqed\n\nend\n\ncontext huntington\nbegin\n\nsublocale mp_h: boolean_algebra_mp\nproof (standard)\n (*goals:\n 1. ‹⋀x y. - (- x ⊔ y) ⊔ x = x›\n 2. ‹⋀x y z. - (- x ⊔ y) ⊔ (z ⊔ y) = y ⊔ (z ⊔ x)›*)\n show 1: \"⋀x y. - (- x ⊔ y) ⊔ x = x\"\n by (metis h_ba1.ba1_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) h_ba1.ba1_complement (*‹(?x ⊔ - ?y = ?z ⊔ - ?z) = (?x ⊔ ?y = ?x)›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*))\n show \"⋀x y z. - (- x ⊔ y) ⊔ (z ⊔ y) = y ⊔ (z ⊔ x)\"\n proof (-)\n (*goal: ‹⋀(x::'a) (y::'a) z::'a. - (- x ⊔ y) ⊔ (z ⊔ y) = y ⊔ (z ⊔ x)›*)\n fix x and y and z\n have \"y = -(-x ⊔ -y) ⊔ y\"\n using \"1\" (*‹- (- ?x ⊔ ?y) ⊔ ?x = ?x›*) h_ba1.ba1_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by auto\n thus \"-(-x ⊔ y) ⊔ (z ⊔ y) = y ⊔ (z ⊔ x)\"\n by (metis h_ba1.ba1_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) h_ba1.ba1_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*))\n qed\nqed\n\nend\n\nsubsection ‹An Equational Axiomatisation based on Semilattices›\n\ntext ‹\nThe following version is an equational axiomatisation based on semilattices.\nWe add the double complement rule and that ‹top› is unique.\nThe final axiom ‹ba3_export› encodes the logical statement $P \\vee Q = P \\vee (\\neg P \\wedge Q)$.\nIts dual appears in \\<^cite>‹\"BalbesHorn1970\"›.\n›\n\ntext ‹Theorem 5›\n\nclass boolean_algebra_3 = sup + uminus +\n assumes ba3_associative: \"x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n assumes ba3_commutative: \"x ⊔ y = y ⊔ x\"\n assumes ba3_idempotent[simp]: \"x ⊔ x = x\"\n assumes ba3_double_complement[simp]: \"--x = x\"\n assumes ba3_top_unique: \"x ⊔ -x = y ⊔ -y\"\n assumes ba3_export: \"x ⊔ -(x ⊔ y) = x ⊔ -y\"\nbegin\n\nsubclass huntington\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: ba3_associative)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: ba3_commutative)\n show \"⋀x y. x = - (- x ⊔ y) ⊔ - (- x ⊔ - y)\"\n by (metis ba3_commutative ba3_double_complement ba3_export ba3_idempotent ba3_top_unique)\nqed\n\nend\n\ncontext huntington\nbegin\n\nsublocale h_ba3: boolean_algebra_3\nproof (standard)\n (*goals:\n 1. ‹⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z›\n 2. ‹⋀x y. x ⊔ y = y ⊔ x›\n 3. ‹⋀x. x ⊔ x = x›\n 4. ‹⋀x. - - x = x›\n 5. ‹⋀x y. x ⊔ - x = y ⊔ - y›\n 6. ‹⋀x y. x ⊔ - (x ⊔ y) = x ⊔ - y›*)\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: h_ba1.ba1_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*))\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: h_ba1.ba1_commutative (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*))\n show 3: \"⋀x. x ⊔ x = x\"\n using h_ba1.ba1_complement (*‹(?x ⊔ - ?y = ?z ⊔ - ?z) = (?x ⊔ ?y = ?x)›*) by blast\n show 4: \"⋀x. - - x = x\"\n by (metis h_ba1.ba1_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) huntington (*‹?x = - (- ?x ⊔ ?y) ⊔ - (- ?x ⊔ - ?y)›*) top_unique (*‹?x ⊔ - ?x = ?y ⊔ - ?y›*))\n show \"⋀x y. x ⊔ - x = y ⊔ - y\"\n by (simp add: top_unique (*‹(?x::'a) ⊔ - ?x = (?y::'a) ⊔ - ?y›*))\n show \"⋀x y. x ⊔ - (x ⊔ y) = x ⊔ - y\"\n using \"3\" (*‹(?x::'a::type) ⊔ ?x = ?x›*) \"4\" (*‹- - ?x = ?x›*) by (smt h_ba1.ba1_ba2.ba2_associative_commutative (*‹(?x::'a) ⊔ (?y::'a) ⊔ (?z::'a) = ?y ⊔ ?z ⊔ ?x›*) h_ba1.ba1_complement (*‹((?x::'a) ⊔ - (?y::'a) = (?z::'a) ⊔ - ?z) = (?x ⊔ ?y = ?x)›*))\nqed\n\nend\n\nsection ‹Subset Boolean Algebras›\n\ntext ‹\nWe apply Huntington's axioms to the range of a unary operation, which serves as complement on the range.\nThis gives a Boolean algebra structure on the range without imposing any further constraints on the set.\nThe obtained structure is used as a reference in the subsequent development and to inherit the results proved here.\nThis is taken from \\<^cite>‹\"Guttmann2012c\" and \"GuttmannStruthWeber2011b\"› and follows the development of Boolean algebras in \\<^cite>‹\"Maddux1996\"›.\n›\n\ntext ‹Definition 6›\n\nclass subset_boolean_algebra = sup + uminus +\n assumes sub_associative: \"-x ⊔ (-y ⊔ -z) = (-x ⊔ -y) ⊔ -z\"\n assumes sub_commutative: \"-x ⊔ -y = -y ⊔ -x\"\n assumes sub_complement: \"-x = -(--x ⊔ -y) ⊔ -(--x ⊔ --y)\"\n assumes sub_sup_closed: \"-x ⊔ -y = --(-x ⊔ -y)\"\nbegin\n\ntext ‹uniqueness of ‹top›, resulting in the lemma ‹top_def› to replace the assumption ‹sub_top_def››\n\nlemma top_unique:\n \"-x ⊔ --x = -y ⊔ --y\"\n by (metis sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_complement (*‹- ?x = - (- - ?x ⊔ - ?y) ⊔ - (- - ?x ⊔ - - ?y)›*))\n\ntext ‹consequences for join and complement›\n\nlemma double_negation[simp]:\n \"---x = -x\"\n by (metis sub_complement (*‹- (?x::'a::type) = - (- - ?x ⊔ - (?y::'a::type)) ⊔ - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- (?x::'a::type) ⊔ - (?y::'a::type) = - - (- ?x ⊔ - ?y)›*))\n\nlemma complement_1:\n \"--x = -(-x ⊔ -y) ⊔ -(-x ⊔ --y)\"\n by (metis double_negation (*‹- - - ?x = - ?x›*) sub_complement (*‹- ?x = - (- - ?x ⊔ - ?y) ⊔ - (- - ?x ⊔ - - ?y)›*))\n\nlemma sup_right_zero_var:\n \"-x ⊔ (-y ⊔ --y) = -z ⊔ --z\"\n by (smt complement_1 (*‹- - ?x = - (- ?x ⊔ - ?y) ⊔ - (- ?x ⊔ - - ?y)›*) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) top_unique (*‹- ?x ⊔ - - ?x = - ?y ⊔ - - ?y›*))\n\nlemma sup_right_unit_idempotent:\n \"-x ⊔ -x = -x ⊔ -(-y ⊔ --y)\"\n by (metis complement_1 (*‹- - ?x = - (- ?x ⊔ - ?y) ⊔ - (- ?x ⊔ - - ?y)›*) double_negation (*‹- - - ?x = - ?x›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sup_right_zero_var (*‹- ?x ⊔ (- ?y ⊔ - - ?y) = - ?z ⊔ - - ?z›*))\n\nlemma sup_idempotent[simp]:\n \"-x ⊔ -x = -x\"\n by (smt complement_1 (*‹- - ?x = - (- ?x ⊔ - ?y) ⊔ - (- ?x ⊔ - - ?y)›*) double_negation (*‹- - - ?x = - ?x›*) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sup_right_unit_idempotent (*‹- ?x ⊔ - ?x = - ?x ⊔ - (- ?y ⊔ - - ?y)›*))\n\nlemma complement_2:\n \"-x = -(-(-x ⊔ -y) ⊔ -(-x ⊔ --y))\"\n using complement_1 (*‹- - ?x = - (- ?x ⊔ - ?y) ⊔ - (- ?x ⊔ - - ?y)›*) by auto\n\nlemma sup_eq_cases:\n \"-x ⊔ -y = -x ⊔ -z ⟹ --x ⊔ -y = --x ⊔ -z ⟹ -y = -z\"\n by (metis complement_2 (*‹- (?x::'a::type) = - (- (- ?x ⊔ - (?y::'a::type)) ⊔ - (- ?x ⊔ - - ?y))›*) sub_commutative (*‹- (?x::'a::type) ⊔ - (?y::'a::type) = - ?y ⊔ - ?x›*))\n\nlemma sup_eq_cases_2:\n \"-y ⊔ -x = -z ⊔ -x ⟹ -y ⊔ --x = -z ⊔ --x ⟹ -y = -z\"\n using sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sup_eq_cases (*‹⟦- ?x ⊔ - ?y = - ?x ⊔ - ?z; - - ?x ⊔ - ?y = - - ?x ⊔ - ?z⟧ ⟹ - ?y = - ?z›*) by auto\n\nend\n\ntext ‹Definition 7›\n\nclass subset_extended = sup + inf + minus + uminus + bot + top + ord +\n assumes sub_top_def: \"top = (THE x . ∀y . x = -y ⊔ --y)\" (* define without imposing uniqueness *)\n assumes sub_bot_def: \"bot = -(THE x . ∀y . x = -y ⊔ --y)\"\n assumes sub_inf_def: \"-x ⊓ -y = -(--x ⊔ --y)\"\n assumes sub_minus_def: \"-x - -y = -(--x ⊔ -y)\"\n assumes sub_less_eq_def: \"-x ≤ -y ⟷ -x ⊔ -y = -y\"\n assumes sub_less_def: \"-x < -y ⟷ -x ⊔ -y = -y ∧ ¬ (-y ⊔ -x = -x)\"\n\nclass subset_boolean_algebra_extended = subset_boolean_algebra + subset_extended\nbegin\n\nlemma top_def:\n \"top = -x ⊔ --x\"\n using sub_top_def (*‹top = (THE x. ∀y. x = - y ⊔ - - y)›*) top_unique (*‹- ?x ⊔ - - ?x = - ?y ⊔ - - ?y›*) by blast\n\ntext ‹consequences for meet›\n\nlemma inf_closed:\n \"-x ⊓ -y = --(-x ⊓ -y)\"\n by (simp add: sub_inf_def (*‹- (?x::'a) ⊓ - (?y::'a) = - (- - ?x ⊔ - - ?y)›*))\n\nlemma inf_associative:\n \"-x ⊓ (-y ⊓ -z) = (-x ⊓ -y) ⊓ -z\"\n using sub_associative (*‹- (?x::'a) ⊔ (- (?y::'a) ⊔ - (?z::'a)) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) by auto\n\nlemma inf_commutative:\n \"-x ⊓ -y = -y ⊓ -x\"\n by (simp add: sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*))\n\nlemma inf_idempotent[simp]:\n \"-x ⊓ -x = -x\"\n by (simp add: sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*))\n\nlemma inf_absorb[simp]:\n \"(-x ⊔ -y) ⊓ -x = -x\"\n by (metis complement_1 (*‹- - ?x = - (- ?x ⊔ - ?y) ⊔ - (- ?x ⊔ - - ?y)›*) sup_idempotent (*‹- ?x ⊔ - ?x = - ?x›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*))\n\nlemma sup_absorb[simp]:\n \"-x ⊔ (-x ⊓ -y) = -x\"\n by (metis sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_complement (*‹- ?x = - (- - ?x ⊔ - ?y) ⊔ - (- - ?x ⊔ - - ?y)›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sup_idempotent (*‹- ?x ⊔ - ?x = - ?x›*))\n\nlemma inf_demorgan:\n \"-(-x ⊓ -y) = --x ⊔ --y\"\n using sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- (?x::'a) ⊔ - (?y::'a) = - - (- ?x ⊔ - ?y)›*) by auto\n\nlemma sub_sup_demorgan:\n \"-(-x ⊔ -y) = --x ⊓ --y\"\n by (simp add: sub_inf_def (*‹- (?x::'a) ⊓ - (?y::'a) = - (- - ?x ⊔ - - ?y)›*))\n\nlemma sup_cases:\n \"-x = (-x ⊓ -y) ⊔ (-x ⊓ --y)\"\n by (metis inf_closed (*‹- (?x::'a) ⊓ - (?y::'a) = - - (- ?x ⊓ - ?y)›*) inf_demorgan (*‹- (- (?x::'a) ⊓ - (?y::'a)) = - - ?x ⊔ - - ?y›*) sub_complement (*‹- (?x::'a) = - (- - ?x ⊔ - (?y::'a)) ⊔ - (- - ?x ⊔ - - ?y)›*))\n\nlemma inf_cases:\n \"-x = (-x ⊔ -y) ⊓ (-x ⊔ --y)\"\n by (metis complement_2 (*‹- ?x = - (- (- ?x ⊔ - ?y) ⊔ - (- ?x ⊔ - - ?y))›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_demorgan (*‹- (- ?x ⊔ - ?y) = - - ?x ⊓ - - ?y›*))\n\nlemma inf_complement_intro:\n \"(-x ⊔ -y) ⊓ --x = -y ⊓ --x\"\nproof (-)\n (*goal: ‹(- (x::'a) ⊔ - (y::'a)) ⊓ - - x = - y ⊓ - - x›*)\n have \"(-x ⊔ -y) ⊓ --x = (-x ⊔ -y) ⊓ (--x ⊔ -y) ⊓ --x\"\n by (metis inf_absorb (*‹(- ?x ⊔ - ?y) ⊓ - ?x = - ?x›*) inf_associative (*‹- ?x ⊓ (- ?y ⊓ - ?z) = - ?x ⊓ - ?y ⊓ - ?z›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*))\n also (*calculation: ‹(- x ⊔ - y) ⊓ - - x = (- x ⊔ - y) ⊓ (- - x ⊔ - y) ⊓ - - x›*) have \"... = -y ⊓ --x\"\n by (metis inf_cases (*‹- ?x = (- ?x ⊔ - ?y) ⊓ (- ?x ⊔ - - ?y)›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*))\n finally (*calculation: ‹(- x ⊔ - y) ⊓ - - x = - y ⊓ - - x›*) show \"?thesis\"\n (*goal: ‹(- x ⊔ - y) ⊓ - - x = - y ⊓ - - x›*) .\nqed\n\nlemma sup_complement_intro:\n \"-x ⊔ -y = -x ⊔ (--x ⊓ -y)\"\n by (metis inf_absorb (*‹(- ?x ⊔ - ?y) ⊓ - ?x = - ?x›*) inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) inf_complement_intro (*‹(- ?x ⊔ - ?y) ⊓ - - ?x = - ?y ⊓ - - ?x›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sup_cases (*‹- ?x = - ?x ⊓ - ?y ⊔ - ?x ⊓ - - ?y›*))\n\nlemma inf_left_dist_sup:\n \"-x ⊓ (-y ⊔ -z) = (-x ⊓ -y) ⊔ (-x ⊓ -z)\"\nproof (-)\n (*goal: ‹- (x::'a) ⊓ (- (y::'a) ⊔ - (z::'a)) = - x ⊓ - y ⊔ - x ⊓ - z›*)\n have \"-x ⊓ (-y ⊔ -z) = (-x ⊓ (-y ⊔ -z) ⊓ -y) ⊔ (-x ⊓ (-y ⊔ -z) ⊓ --y)\"\n by (metis sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sup_cases (*‹- ?x = - ?x ⊓ - ?y ⊔ - ?x ⊓ - - ?y›*))\n also (*calculation: ‹- x ⊓ (- y ⊔ - z) = - x ⊓ (- y ⊔ - z) ⊓ - y ⊔ - x ⊓ (- y ⊔ - z) ⊓ - - y›*) have \"... = (-x ⊓ -y) ⊔ (-x ⊓ -z ⊓ --y)\"\n by (metis inf_absorb (*‹(- ?x ⊔ - ?y) ⊓ - ?x = - ?x›*) inf_associative (*‹- ?x ⊓ (- ?y ⊓ - ?z) = - ?x ⊓ - ?y ⊓ - ?z›*) inf_complement_intro (*‹(- ?x ⊔ - ?y) ⊓ - - ?x = - ?y ⊓ - - ?x›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*))\n also (*calculation: ‹- x ⊓ (- y ⊔ - z) = - x ⊓ - y ⊔ - x ⊓ - z ⊓ - - y›*) have \"... = (-x ⊓ -y) ⊔ ((-x ⊓ -y ⊓ -z) ⊔ (-x ⊓ -z ⊓ --y))\"\n using sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sup_absorb (*‹- ?x ⊔ - ?x ⊓ - ?y = - ?x›*) by auto\n also (*calculation: ‹- x ⊓ (- y ⊔ - z) = - x ⊓ - y ⊔ (- x ⊓ - y ⊓ - z ⊔ - x ⊓ - z ⊓ - - y)›*) have \"... = (-x ⊓ -y) ⊔ ((-x ⊓ -z ⊓ -y) ⊔ (-x ⊓ -z ⊓ --y))\"\n by (metis inf_associative (*‹- ?x ⊓ (- ?y ⊓ - ?z) = - ?x ⊓ - ?y ⊓ - ?z›*) inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*))\n also (*calculation: ‹- x ⊓ (- y ⊔ - z) = - x ⊓ - y ⊔ (- x ⊓ - z ⊓ - y ⊔ - x ⊓ - z ⊓ - - y)›*) have \"... = (-x ⊓ -y) ⊔ (-x ⊓ -z)\"\n by (metis sub_inf_def (*‹- (?x::'a) ⊓ - (?y::'a) = - (- - ?x ⊔ - - ?y)›*) sup_cases (*‹- (?x::'a) = - ?x ⊓ - (?y::'a) ⊔ - ?x ⊓ - - ?y›*))\n finally (*calculation: ‹- x ⊓ (- y ⊔ - z) = - x ⊓ - y ⊔ - x ⊓ - z›*) show \"?thesis\"\n (*goal: ‹- x ⊓ (- y ⊔ - z) = - x ⊓ - y ⊔ - x ⊓ - z›*) .\nqed\n\nlemma sup_left_dist_inf:\n \"-x ⊔ (-y ⊓ -z) = (-x ⊔ -y) ⊓ (-x ⊔ -z)\"\nproof (-)\n (*goal: ‹- x ⊔ - y ⊓ - z = (- x ⊔ - y) ⊓ (- x ⊔ - z)›*)\n have \"-x ⊔ (-y ⊓ -z) = -(--x ⊓ (--y ⊔ --z))\"\n by (metis sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_demorgan (*‹- (- ?x ⊔ - ?y) = - - ?x ⊓ - - ?y›*))\n also (*calculation: ‹- x ⊔ - y ⊓ - z = - (- - x ⊓ (- - y ⊔ - - z))›*) have \"... = (-x ⊔ -y) ⊓ (-x ⊔ -z)\"\n by (metis inf_left_dist_sup (*‹- ?x ⊓ (- ?y ⊔ - ?z) = - ?x ⊓ - ?y ⊔ - ?x ⊓ - ?z›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_demorgan (*‹- (- ?x ⊔ - ?y) = - - ?x ⊓ - - ?y›*))\n finally (*calculation: ‹- x ⊔ - y ⊓ - z = (- x ⊔ - y) ⊓ (- x ⊔ - z)›*) show \"?thesis\"\n (*goal: ‹- x ⊔ - y ⊓ - z = (- x ⊔ - y) ⊓ (- x ⊔ - z)›*) .\nqed\n\nlemma sup_right_dist_inf:\n \"(-y ⊓ -z) ⊔ -x = (-y ⊔ -x) ⊓ (-z ⊔ -x)\"\n using sub_commutative (*‹- (?x::'a::type) ⊔ - (?y::'a::type) = - ?y ⊔ - ?x›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sup_left_dist_inf (*‹- ?x ⊔ - ?y ⊓ - ?z = (- ?x ⊔ - ?y) ⊓ (- ?x ⊔ - ?z)›*) by auto\n\nlemma inf_right_dist_sup:\n \"(-y ⊔ -z) ⊓ -x = (-y ⊓ -x) ⊔ (-z ⊓ -x)\"\n by (metis inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) inf_left_dist_sup (*‹- ?x ⊓ (- ?y ⊔ - ?z) = - ?x ⊓ - ?y ⊔ - ?x ⊓ - ?z›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*))\n\nlemma case_duality:\n \"(--x ⊓ -y) ⊔ (-x ⊓ -z) = (-x ⊔ -y) ⊓ (--x ⊔ -z)\"\nproof (-)\n (*goal: ‹- - x ⊓ - y ⊔ - x ⊓ - z = (- x ⊔ - y) ⊓ (- - x ⊔ - z)›*)\n have 1: \"-(--x ⊓ --y) ⊓ ----x = --x ⊓ -y\"\n using inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) inf_complement_intro (*‹(- ?x ⊔ - ?y) ⊓ - - ?x = - ?y ⊓ - - ?x›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_demorgan (*‹- (- (?x::'a::type) ⊔ - (?y::'a::type)) = - - ?x ⊓ - - ?y›*) by auto\n have 2: \"-(----x ⊔ -(--x ⊔ -z)) = -----x ⊓ ---z\"\n by (metis (no_types) double_negation (*‹- - - ?x = - ?x›*) sup_complement_intro (*‹- ?x ⊔ - ?y = - ?x ⊔ - - ?x ⊓ - ?y›*) sub_sup_demorgan (*‹- (- ?x ⊔ - ?y) = - - ?x ⊓ - - ?y›*))\n have 3: \"-(--x ⊓ --y) ⊓ -x = -x\"\n using inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) inf_left_dist_sup (*‹- ?x ⊓ (- ?y ⊔ - ?z) = - ?x ⊓ - ?y ⊔ - ?x ⊓ - ?z›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_demorgan (*‹- (- ?x ⊔ - ?y) = - - ?x ⊓ - - ?y›*) by auto\n hence \"-(--x ⊓ --y) = -x ⊔ -y\"\n using sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_demorgan (*‹- (- ?x ⊔ - ?y) = - - ?x ⊓ - - ?y›*) by auto\n thus \"?thesis\"\n (*goal: ‹- - x ⊓ - y ⊔ - x ⊓ - z = (- x ⊔ - y) ⊓ (- - x ⊔ - z)›*)\n by (metis double_negation (*‹- - - (?x::'a) = - ?x›*) 1 (*‹- (- - (x::'a) ⊓ - - (y::'a)) ⊓ - - - - x = - - x ⊓ - y›*) 2 (*‹- (- - - - (x::'a) ⊔ - (- - x ⊔ - (z::'a))) = - - - - - x ⊓ - - - z›*) 3 (*‹- (- - (x::'a) ⊓ - - (y::'a)) ⊓ - x = - x›*) inf_associative (*‹- (?x::'a) ⊓ (- (?y::'a) ⊓ - (?z::'a)) = - ?x ⊓ - ?y ⊓ - ?z›*) inf_left_dist_sup (*‹- (?x::'a) ⊓ (- (?y::'a) ⊔ - (?z::'a)) = - ?x ⊓ - ?y ⊔ - ?x ⊓ - ?z›*) sup_complement_intro (*‹- (?x::'a) ⊔ - (?y::'a) = - ?x ⊔ - - ?x ⊓ - ?y›*))\nqed\n\nlemma case_duality_2:\n \"(-x ⊓ -y) ⊔ (--x ⊓ -z) = (-x ⊔ -z) ⊓ (--x ⊔ -y)\"\n using case_duality (*‹- - ?x ⊓ - ?y ⊔ - ?x ⊓ - ?z = (- ?x ⊔ - ?y) ⊓ (- - ?x ⊔ - ?z)›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) by auto\n\nlemma complement_cases:\n \"((-v ⊓ -w) ⊔ (--v ⊓ -x)) ⊓ -((-v ⊓ -y) ⊔ (--v ⊓ -z)) = (-v ⊓ -w ⊓ --y) ⊔ (--v ⊓ -x ⊓ --z)\"\nproof (-)\n (*goal: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = - v ⊓ - w ⊓ - - y ⊔ - - v ⊓ - x ⊓ - - z›*)\n have 1: \"(--v ⊔ -w) = --(--v ⊔ -w) ∧ (-v ⊔ -x) = --(-v ⊔ -x) ∧ (--v ⊔ --y) = --(--v ⊔ --y) ∧ (-v ⊔ --z) = --(-v ⊔ --z)\"\n using sub_inf_def (*‹- (?x::'a) ⊓ - (?y::'a) = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) by auto\n have 2: \"(-v ⊔ (-x ⊓ --z)) = --(-v ⊔ (-x ⊓ --z))\"\n using sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- (?x::'a) ⊔ - (?y::'a) = - - (- ?x ⊔ - ?y)›*) by auto\n have \"((-v ⊓ -w) ⊔ (--v ⊓ -x)) ⊓ -((-v ⊓ -y) ⊔ (--v ⊓ -z)) = ((-v ⊓ -w) ⊔ (--v ⊓ -x)) ⊓ (-(-v ⊓ -y) ⊓ -(--v ⊓ -z))\"\n using sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) by auto\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = (- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ (- (- v ⊓ - y) ⊓ - (- - v ⊓ - z))›*) have \"... = ((-v ⊓ -w) ⊔ (--v ⊓ -x)) ⊓ ((--v ⊔ --y) ⊓ (-v ⊔ --z))\"\n using inf_demorgan (*‹- (- ?x ⊓ - ?y) = - - ?x ⊔ - - ?y›*) by auto\n also (*calculation: ‹(- (v::'a) ⊓ - (w::'a) ⊔ - - v ⊓ - (x::'a)) ⊓ - (- v ⊓ - (y::'a) ⊔ - - v ⊓ - (z::'a)) = (- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ ((- - v ⊔ - - y) ⊓ (- v ⊔ - - z))›*) have \"... = (--v ⊔ -w) ⊓ (-v ⊔ -x) ⊓ ((--v ⊔ --y) ⊓ (-v ⊔ --z))\"\n by (metis case_duality (*‹- - ?x ⊓ - ?y ⊔ - ?x ⊓ - ?z = (- ?x ⊔ - ?y) ⊓ (- - ?x ⊔ - ?z)›*) double_negation (*‹- - - ?x = - ?x›*))\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = (- - v ⊔ - w) ⊓ (- v ⊔ - x) ⊓ ((- - v ⊔ - - y) ⊓ (- v ⊔ - - z))›*) have \"... = (--v ⊔ -w) ⊓ ((-v ⊔ -x) ⊓ ((--v ⊔ --y) ⊓ (-v ⊔ --z)))\"\n by (metis 1 (*‹- - v ⊔ - w = - - (- - v ⊔ - w) ∧ - v ⊔ - x = - - (- v ⊔ - x) ∧ - - v ⊔ - - y = - - (- - v ⊔ - - y) ∧ - v ⊔ - - z = - - (- v ⊔ - - z)›*) inf_associative (*‹- ?x ⊓ (- ?y ⊓ - ?z) = - ?x ⊓ - ?y ⊓ - ?z›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*))\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = (- - v ⊔ - w) ⊓ ((- v ⊔ - x) ⊓ ((- - v ⊔ - - y) ⊓ (- v ⊔ - - z)))›*) have \"... = (--v ⊔ -w) ⊓ ((-v ⊔ -x) ⊓ (--v ⊔ --y) ⊓ (-v ⊔ --z))\"\n by (metis 1 (*‹- - v ⊔ - w = - - (- - v ⊔ - w) ∧ - v ⊔ - x = - - (- v ⊔ - x) ∧ - - v ⊔ - - y = - - (- - v ⊔ - - y) ∧ - v ⊔ - - z = - - (- v ⊔ - - z)›*) inf_associative (*‹- ?x ⊓ (- ?y ⊓ - ?z) = - ?x ⊓ - ?y ⊓ - ?z›*))\n also (*calculation: ‹(- (v::'a) ⊓ - (w::'a) ⊔ - - v ⊓ - (x::'a)) ⊓ - (- v ⊓ - (y::'a) ⊔ - - v ⊓ - (z::'a)) = (- - v ⊔ - w) ⊓ ((- v ⊔ - x) ⊓ (- - v ⊔ - - y) ⊓ (- v ⊔ - - z))›*) have \"... = (--v ⊔ -w) ⊓ ((--v ⊔ --y) ⊓ (-v ⊔ -x) ⊓ (-v ⊔ --z))\"\n by (metis 1 (*‹- - (v::'a) ⊔ - (w::'a) = - - (- - v ⊔ - w) ∧ - v ⊔ - (x::'a) = - - (- v ⊔ - x) ∧ - - v ⊔ - - (y::'a) = - - (- - v ⊔ - - y) ∧ - v ⊔ - - (z::'a) = - - (- v ⊔ - - z)›*) inf_commutative (*‹- (?x::'a) ⊓ - (?y::'a) = - ?y ⊓ - ?x›*))\n also (*calculation: ‹(- (v::'a) ⊓ - (w::'a) ⊔ - - v ⊓ - (x::'a)) ⊓ - (- v ⊓ - (y::'a) ⊔ - - v ⊓ - (z::'a)) = (- - v ⊔ - w) ⊓ ((- - v ⊔ - - y) ⊓ (- v ⊔ - x) ⊓ (- v ⊔ - - z))›*) have \"... = (--v ⊔ -w) ⊓ ((--v ⊔ --y) ⊓ ((-v ⊔ -x) ⊓ (-v ⊔ --z)))\"\n by (metis 1 (*‹- - (v::'a) ⊔ - (w::'a) = - - (- - v ⊔ - w) ∧ - v ⊔ - (x::'a) = - - (- v ⊔ - x) ∧ - - v ⊔ - - (y::'a) = - - (- - v ⊔ - - y) ∧ - v ⊔ - - (z::'a) = - - (- v ⊔ - - z)›*) inf_associative (*‹- (?x::'a) ⊓ (- (?y::'a) ⊓ - (?z::'a)) = - ?x ⊓ - ?y ⊓ - ?z›*))\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = (- - v ⊔ - w) ⊓ ((- - v ⊔ - - y) ⊓ ((- v ⊔ - x) ⊓ (- v ⊔ - - z)))›*) have \"... = (--v ⊔ -w) ⊓ ((--v ⊔ --y) ⊓ (-v ⊔ (-x ⊓ --z)))\"\n by (simp add: sup_left_dist_inf (*‹- ?x ⊔ - ?y ⊓ - ?z = (- ?x ⊔ - ?y) ⊓ (- ?x ⊔ - ?z)›*))\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = (- - v ⊔ - w) ⊓ ((- - v ⊔ - - y) ⊓ (- v ⊔ - x ⊓ - - z))›*) have \"... = (--v ⊔ -w) ⊓ (--v ⊔ --y) ⊓ (-v ⊔ (-x ⊓ --z))\"\n using \"1\" (*‹- - v ⊔ - w = - - (- - v ⊔ - w) ∧ - v ⊔ - x = - - (- v ⊔ - x) ∧ - - v ⊔ - - y = - - (- - v ⊔ - - y) ∧ - v ⊔ - - z = - - (- v ⊔ - - z)›*) \"2\" (*‹- v ⊔ - x ⊓ - - z = - - (- v ⊔ - x ⊓ - - z)›*) by (metis inf_associative (*‹- (?x::'a::type) ⊓ (- (?y::'a::type) ⊓ - (?z::'a::type)) = - ?x ⊓ - ?y ⊓ - ?z›*))\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = (- - v ⊔ - w) ⊓ (- - v ⊔ - - y) ⊓ (- v ⊔ - x ⊓ - - z)›*) have \"... = (--v ⊔ (-w ⊓ --y)) ⊓ (-v ⊔ (-x ⊓ --z))\"\n by (simp add: sup_left_dist_inf (*‹- ?x ⊔ - ?y ⊓ - ?z = (- ?x ⊔ - ?y) ⊓ (- ?x ⊔ - ?z)›*))\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = (- - v ⊔ - w ⊓ - - y) ⊓ (- v ⊔ - x ⊓ - - z)›*) have \"... = (-v ⊓ (-w ⊓ --y)) ⊔ (--v ⊓ (-x ⊓ --z))\"\n by (metis case_duality (*‹- - (?x::'a) ⊓ - (?y::'a) ⊔ - ?x ⊓ - (?z::'a) = (- ?x ⊔ - ?y) ⊓ (- - ?x ⊔ - ?z)›*) complement_1 (*‹- - (?x::'a) = - (- ?x ⊔ - (?y::'a)) ⊔ - (- ?x ⊔ - - ?y)›*) complement_2 (*‹- (?x::'a) = - (- (- ?x ⊔ - (?y::'a)) ⊔ - (- ?x ⊔ - - ?y))›*) sub_inf_def (*‹- (?x::'a) ⊓ - (?y::'a) = - (- - ?x ⊔ - - ?y)›*))\n also (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = - v ⊓ (- w ⊓ - - y) ⊔ - - v ⊓ (- x ⊓ - - z)›*) have \"... = (-v ⊓ -w ⊓ --y) ⊔ (--v ⊓ -x ⊓ --z)\"\n by (simp add: inf_associative (*‹- ?x ⊓ (- ?y ⊓ - ?z) = - ?x ⊓ - ?y ⊓ - ?z›*))\n finally (*calculation: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = - v ⊓ - w ⊓ - - y ⊔ - - v ⊓ - x ⊓ - - z›*) show \"?thesis\"\n (*goal: ‹(- v ⊓ - w ⊔ - - v ⊓ - x) ⊓ - (- v ⊓ - y ⊔ - - v ⊓ - z) = - v ⊓ - w ⊓ - - y ⊔ - - v ⊓ - x ⊓ - - z›*) .\nqed\n\nlemma inf_cases_2: \"--x = -(-x ⊓ -y) ⊓ -(-x ⊓ --y)\"\n using sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sup_cases (*‹- (?x::'a) = - ?x ⊓ - (?y::'a) ⊔ - ?x ⊓ - - ?y›*) by auto\n\ntext ‹consequences for ‹top› and ‹bot››\n\nlemma sup_complement[simp]:\n \"-x ⊔ --x = top\"\n using top_def (*‹top = - ?x ⊔ - - ?x›*) by auto\n\nlemma inf_complement[simp]:\n \"-x ⊓ --x = bot\"\n by (metis sub_bot_def (*‹bot = - (THE x. ∀y. x = - y ⊔ - - y)›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_top_def (*‹top = (THE x. ∀y. x = - y ⊔ - - y)›*) top_def (*‹top = - ?x ⊔ - - ?x›*))\n\nlemma complement_bot[simp]:\n \"-bot = top\"\n using inf_complement (*‹- (?x::'a) ⊓ - - ?x = bot›*) inf_demorgan (*‹- (- ?x ⊓ - ?y) = - - ?x ⊔ - - ?y›*) sup_complement (*‹- ?x ⊔ - - ?x = top›*) by fastforce\n\nlemma complement_top[simp]:\n \"-top = bot\"\n using sub_bot_def (*‹bot = - (THE x. ∀y. x = - y ⊔ - - y)›*) sub_top_def (*‹top = (THE x. ∀y. x = - y ⊔ - - y)›*) by blast\n\nlemma sup_right_zero[simp]:\n \"-x ⊔ top = top\"\n using sup_right_zero_var (*‹- ?x ⊔ (- ?y ⊔ - - ?y) = - ?z ⊔ - - ?z›*) by auto\n\nlemma sup_left_zero[simp]:\n \"top ⊔ -x = top\"\n by (metis complement_bot (*‹- bot = top›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sup_right_zero (*‹- ?x ⊔ top = top›*))\n\nlemma inf_right_unit[simp]:\n \"-x ⊓ bot = bot\"\n by (metis complement_bot (*‹- bot = top›*) complement_top (*‹- top = bot›*) double_negation (*‹- - - ?x = - ?x›*) sub_sup_demorgan (*‹- (- ?x ⊔ - ?y) = - - ?x ⊓ - - ?y›*) sup_right_zero (*‹- ?x ⊔ top = top›*))\n\nlemma inf_left_unit[simp]:\n \"bot ⊓ -x = bot\"\n by (metis complement_top (*‹- top = bot›*) inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) inf_right_unit (*‹- ?x ⊓ bot = bot›*))\n\nlemma sup_right_unit[simp]:\n \"-x ⊔ bot = -x\"\n using sup_right_unit_idempotent (*‹- ?x ⊔ - ?x = - ?x ⊔ - (- ?y ⊔ - - ?y)›*) by auto\n\nlemma sup_left_unit[simp]:\n \"bot ⊔ -x = -x\"\n by (metis complement_top (*‹- top = bot›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sup_right_unit (*‹- ?x ⊔ bot = - ?x›*))\n\nlemma inf_right_zero[simp]:\n \"-x ⊓ top = -x\"\n by (metis inf_left_dist_sup (*‹- ?x ⊓ (- ?y ⊔ - ?z) = - ?x ⊓ - ?y ⊔ - ?x ⊓ - ?z›*) sup_cases (*‹- ?x = - ?x ⊓ - ?y ⊔ - ?x ⊓ - - ?y›*) top_def (*‹top = - ?x ⊔ - - ?x›*))\n\nlemma sub_inf_left_zero[simp]:\n \"top ⊓ -x = -x\"\n using inf_absorb (*‹(- ?x ⊔ - ?y) ⊓ - ?x = - ?x›*) top_def (*‹top = - ?x ⊔ - - ?x›*) by fastforce\n\nlemma bot_double_complement[simp]:\n \"--bot = bot\"\n by simp\n\nlemma top_double_complement[simp]:\n \"--top = top\"\n by simp\n\ntext ‹consequences for the order›\n\nlemma reflexive:\n \"-x ≤ -x\"\n by (simp add: sub_less_eq_def (*‹(- (?x::'a) ≤ - (?y::'a)) = (- ?x ⊔ - ?y = - ?y)›*))\n\nlemma transitive:\n \"-x ≤ -y ⟹ -y ≤ -z ⟹ -x ≤ -z\"\n by (metis sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_less_eq_def (*‹(- ?x ≤ - ?y) = (- ?x ⊔ - ?y = - ?y)›*))\n\nlemma antisymmetric:\n \"-x ≤ -y ⟹ -y ≤ -x ⟹ -x = -y\"\n by (simp add: sub_commutative (*‹- (?x::'a) ⊔ - (?y::'a) = - ?y ⊔ - ?x›*) sub_less_eq_def (*‹(- (?x::'a) ≤ - (?y::'a)) = (- ?x ⊔ - ?y = - ?y)›*))\n\nlemma sub_bot_least:\n \"bot ≤ -x\"\n using sup_left_unit (*‹bot ⊔ - (?x::'a) = - ?x›*) complement_top (*‹- top = bot›*) sub_less_eq_def (*‹(- ?x ≤ - ?y) = (- ?x ⊔ - ?y = - ?y)›*) by blast\n\nlemma top_greatest:\n \"-x ≤ top\"\n using complement_bot (*‹- bot = top›*) sub_less_eq_def (*‹(- ?x ≤ - ?y) = (- ?x ⊔ - ?y = - ?y)›*) sup_right_zero (*‹- ?x ⊔ top = top›*) by blast\n\nlemma upper_bound_left:\n \"-x ≤ -x ⊔ -y\"\n by (metis sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_less_eq_def (*‹(- ?x ≤ - ?y) = (- ?x ⊔ - ?y = - ?y)›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sup_idempotent (*‹- ?x ⊔ - ?x = - ?x›*))\n\nlemma upper_bound_right:\n \"-y ≤ -x ⊔ -y\"\n using sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) upper_bound_left (*‹- ?x ≤ - ?x ⊔ - ?y›*) by fastforce\n\nlemma sub_sup_left_isotone:\n assumes \"-x ≤ -y\"\n shows \"-x ⊔ -z ≤ -y ⊔ -z\"\nproof (-)\n (*goal: ‹- x ⊔ - z ≤ - y ⊔ - z›*)\n have \"-x ⊔ -y = -y\"\n by (meson assms (*‹- x ≤ - y›*) sub_less_eq_def (*‹(- ?x ≤ - ?y) = (- ?x ⊔ - ?y = - ?y)›*))\n thus \"?thesis\"\n (*goal: ‹- (x::'a) ⊔ - (z::'a) ≤ - (y::'a) ⊔ - z›*)\n by (metis (full_types) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) upper_bound_left (*‹- ?x ≤ - ?x ⊔ - ?y›*))\nqed\n\nlemma sub_sup_right_isotone:\n \"-x ≤ -y ⟹ -z ⊔ -x ≤ -z ⊔ -y\"\n by (simp add: sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_sup_left_isotone (*‹- ?x ≤ - ?y ⟹ - ?x ⊔ - ?z ≤ - ?y ⊔ - ?z›*))\n\nlemma sup_isotone:\n assumes \"-p ≤ -q\"\n and \"-r ≤ -s\"\n shows \"-p ⊔ -r ≤ -q ⊔ -s\"\nproof (-)\n (*goal: ‹- (p::'a::type) ⊔ - (r::'a::type) ≤ - (q::'a::type) ⊔ - (s::'a::type)›*)\n have \"⋀x y. ¬ -x ≤ -y ⊔ -r ∨ -x ≤ -y ⊔ -s\"\n by (metis (full_types) assms( (*‹- r ≤ - s›*) 2) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_right_isotone (*‹- ?x ≤ - ?y ⟹ - ?z ⊔ - ?x ≤ - ?z ⊔ - ?y›*) transitive (*‹⟦- ?x ≤ - ?y; - ?y ≤ - ?z⟧ ⟹ - ?x ≤ - ?z›*))\n thus \"?thesis\"\n (*goal: ‹- p ⊔ - r ≤ - q ⊔ - s›*)\n by (metis (no_types) assms( (*‹- p ≤ - q›*) 1) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sub_sup_left_isotone (*‹- ?x ≤ - ?y ⟹ - ?x ⊔ - ?z ≤ - ?y ⊔ - ?z›*))\nqed\n\nlemma sub_complement_antitone:\n \"-x ≤ -y ⟹ --y ≤ --x\"\n by (metis inf_absorb (*‹(- ?x ⊔ - ?y) ⊓ - ?x = - ?x›*) inf_demorgan (*‹- (- ?x ⊓ - ?y) = - - ?x ⊔ - - ?y›*) sub_less_eq_def (*‹(- ?x ≤ - ?y) = (- ?x ⊔ - ?y = - ?y)›*))\n\nlemma less_eq_inf:\n \"-x ≤ -y ⟷ -x ⊓ -y = -x\"\n by (metis inf_absorb (*‹(- ?x ⊔ - ?y) ⊓ - ?x = - ?x›*) inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) sub_less_eq_def (*‹(- ?x ≤ - ?y) = (- ?x ⊔ - ?y = - ?y)›*) upper_bound_right (*‹- ?y ≤ - ?x ⊔ - ?y›*) sup_absorb (*‹- ?x ⊔ - ?x ⊓ - ?y = - ?x›*))\n\nlemma inf_complement_left_antitone:\n \"-x ≤ -y ⟹ -(-y ⊓ -z) ≤ -(-x ⊓ -z)\"\n by (simp add: sub_complement_antitone (*‹- ?x ≤ - ?y ⟹ - - ?y ≤ - - ?x›*) inf_demorgan (*‹- (- ?x ⊓ - ?y) = - - ?x ⊔ - - ?y›*) sub_sup_left_isotone (*‹- ?x ≤ - ?y ⟹ - ?x ⊔ - ?z ≤ - ?y ⊔ - ?z›*))\n\nlemma sub_inf_left_isotone:\n \"-x ≤ -y ⟹ -x ⊓ -z ≤ -y ⊓ -z\"\n using sub_complement_antitone (*‹- ?x ≤ - ?y ⟹ - - ?y ≤ - - ?x›*) inf_closed (*‹- ?x ⊓ - ?y = - - (- ?x ⊓ - ?y)›*) inf_complement_left_antitone (*‹- ?x ≤ - ?y ⟹ - (- ?y ⊓ - ?z) ≤ - (- ?x ⊓ - ?z)›*) by fastforce\n\nlemma sub_inf_right_isotone:\n \"-x ≤ -y ⟹ -z ⊓ -x ≤ -z ⊓ -y\"\n by (simp add: inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) sub_inf_left_isotone (*‹- ?x ≤ - ?y ⟹ - ?x ⊓ - ?z ≤ - ?y ⊓ - ?z›*))\n\nlemma inf_isotone:\n assumes \"-p ≤ -q\"\n and \"-r ≤ -s\"\n shows \"-p ⊓ -r ≤ -q ⊓ -s\"\nproof (-)\n (*goal: ‹- p ⊓ - r ≤ - q ⊓ - s›*)\n have \"∀w x y z. (-w ≤ -x ⊓ -y ∨ ¬ -w ≤ -x ⊓ -z) ∨ ¬ -z ≤ -y\"\n by (metis (no_types) inf_closed (*‹- ?x ⊓ - ?y = - - (- ?x ⊓ - ?y)›*) sub_inf_right_isotone (*‹- ?x ≤ - ?y ⟹ - ?z ⊓ - ?x ≤ - ?z ⊓ - ?y›*) transitive (*‹⟦- ?x ≤ - ?y; - ?y ≤ - ?z⟧ ⟹ - ?x ≤ - ?z›*))\n thus \"?thesis\"\n (*goal: ‹- (p::'a) ⊓ - (r::'a) ≤ - (q::'a) ⊓ - (s::'a)›*)\n by (metis (no_types) assms (*‹- p ≤ - q› ‹- r ≤ - s›*) inf_closed (*‹- ?x ⊓ - ?y = - - (- ?x ⊓ - ?y)›*) sub_inf_left_isotone (*‹- ?x ≤ - ?y ⟹ - ?x ⊓ - ?z ≤ - ?y ⊓ - ?z›*))\nqed\n\nlemma least_upper_bound:\n \"-x ≤ -z ∧ -y ≤ -z ⟷ -x ⊔ -y ≤ -z\"\n by (metis sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) transitive (*‹⟦- ?x ≤ - ?y; - ?y ≤ - ?z⟧ ⟹ - ?x ≤ - ?z›*) upper_bound_right (*‹- ?y ≤ - ?x ⊔ - ?y›*) sup_idempotent (*‹- ?x ⊔ - ?x = - ?x›*) sup_isotone (*‹⟦- ?p ≤ - ?q; - ?r ≤ - ?s⟧ ⟹ - ?p ⊔ - ?r ≤ - ?q ⊔ - ?s›*) upper_bound_left (*‹- ?x ≤ - ?x ⊔ - ?y›*))\n\nlemma lower_bound_left:\n \"-x ⊓ -y ≤ -x\"\n by (metis sub_inf_def (*‹- (?x::'a) ⊓ - (?y::'a) = - (- - ?x ⊔ - - ?y)›*) upper_bound_right (*‹- (?y::'a) ≤ - (?x::'a) ⊔ - ?y›*) sup_absorb (*‹- (?x::'a) ⊔ - ?x ⊓ - (?y::'a) = - ?x›*))\n\nlemma lower_bound_right:\n \"-x ⊓ -y ≤ -y\"\n using inf_commutative (*‹- (?x::'a) ⊓ - (?y::'a) = - ?y ⊓ - ?x›*) lower_bound_left (*‹- ?x ⊓ - ?y ≤ - ?x›*) by fastforce\n\nlemma greatest_lower_bound:\n \"-x ≤ -y ∧ -x ≤ -z ⟷ -x ≤ -y ⊓ -z\"\n by (metis inf_closed (*‹- ?x ⊓ - ?y = - - (- ?x ⊓ - ?y)›*) sub_inf_left_isotone (*‹- ?x ≤ - ?y ⟹ - ?x ⊓ - ?z ≤ - ?y ⊓ - ?z›*) less_eq_inf (*‹(- ?x ≤ - ?y) = (- ?x ⊓ - ?y = - ?x)›*) transitive (*‹⟦- ?x ≤ - ?y; - ?y ≤ - ?z⟧ ⟹ - ?x ≤ - ?z›*) lower_bound_left (*‹- ?x ⊓ - ?y ≤ - ?x›*) lower_bound_right (*‹- ?x ⊓ - ?y ≤ - ?y›*))\n\nlemma less_eq_sup_top:\n \"-x ≤ -y ⟷ --x ⊔ -y = top\"\n by (metis complement_1 (*‹- - ?x = - (- ?x ⊔ - ?y) ⊔ - (- ?x ⊔ - - ?y)›*) inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) inf_complement_intro (*‹(- ?x ⊔ - ?y) ⊓ - - ?x = - ?y ⊓ - - ?x›*) sub_inf_left_zero (*‹top ⊓ - ?x = - ?x›*) less_eq_inf (*‹(- ?x ≤ - ?y) = (- ?x ⊓ - ?y = - ?x)›*) sub_complement (*‹- ?x = - (- - ?x ⊔ - ?y) ⊔ - (- - ?x ⊔ - - ?y)›*) sup_complement_intro (*‹- ?x ⊔ - ?y = - ?x ⊔ - - ?x ⊓ - ?y›*) top_def (*‹top = - ?x ⊔ - - ?x›*))\n\nlemma less_eq_inf_bot:\n \"-x ≤ -y ⟷ -x ⊓ --y = bot\"\n by (metis complement_bot (*‹- bot = top›*) complement_top (*‹- top = bot›*) double_negation (*‹- - - ?x = - ?x›*) inf_demorgan (*‹- (- ?x ⊓ - ?y) = - - ?x ⊔ - - ?y›*) less_eq_sup_top (*‹(- ?x ≤ - ?y) = (- - ?x ⊔ - ?y = top)›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*))\n\nlemma shunting:\n \"-x ⊓ -y ≤ -z ⟷ -y ≤ --x ⊔ -z\"\nproof (cases \"--x ⊔ (-z ⊔ --y) = top\")\n (*goals:\n 1. ‹- - (x::'a::type) ⊔ (- (z::'a::type) ⊔ - - (y::'a::type)) = top ⟹ (- x ⊓ - y ≤ - z) = (- y ≤ - - x ⊔ - z)›\n 2. ‹- - (x::'a::type) ⊔ (- (z::'a::type) ⊔ - - (y::'a::type)) ≠ top ⟹ (- x ⊓ - y ≤ - z) = (- y ≤ - - x ⊔ - z)›*)\n case True (*‹- - x ⊔ (- z ⊔ - - y) = top›*)\n have \"∀v w. -v ≤ -w ∨ -w ⊔ --v ≠ top\"\n using less_eq_sup_top (*‹(- ?x ≤ - ?y) = (- - ?x ⊔ - ?y = top)›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) by blast\n thus \"?thesis\"\n (*goal: ‹(- x ⊓ - y ≤ - z) = (- y ≤ - - x ⊔ - z)›*)\n by (metis True (*‹- - (x::'a) ⊔ (- (z::'a) ⊔ - - (y::'a)) = top›*) sub_associative (*‹- (?x::'a) ⊔ (- (?y::'a) ⊔ - (?z::'a)) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_commutative (*‹- (?x::'a) ⊔ - (?y::'a) = - ?y ⊔ - ?x›*) sub_inf_def (*‹- (?x::'a) ⊓ - (?y::'a) = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- (?x::'a) ⊔ - (?y::'a) = - - (- ?x ⊔ - ?y)›*))\nnext\n (*goal: ‹- - x ⊔ (- z ⊔ - - y) ≠ top ⟹ (- x ⊓ - y ≤ - z) = (- y ≤ - - x ⊔ - z)›*)\n case False (*‹- - x ⊔ (- z ⊔ - - y) ≠ top›*)\n hence \"--x ⊔ (-z ⊔ --y) ≠ top ∧ ¬ -y ≤ -z ⊔ --x\"\n by (metis (no_types) less_eq_sup_top (*‹(- ?x ≤ - ?y) = (- - ?x ⊔ - ?y = top)›*) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*))\n thus \"?thesis\"\n (*goal: ‹(- x ⊓ - y ≤ - z) = (- y ≤ - - x ⊔ - z)›*)\n using less_eq_sup_top (*‹(- ?x ≤ - ?y) = (- - ?x ⊔ - ?y = top)›*) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_inf_def (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) sub_sup_closed (*‹- (?x::'a::type) ⊔ - (?y::'a::type) = - - (- ?x ⊔ - ?y)›*) by auto\nqed\n\nlemma shunting_right:\n \"-x ⊓ -y ≤ -z ⟷ -x ≤ -z ⊔ --y\"\n by (metis inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) shunting (*‹(- ?x ⊓ - ?y ≤ - ?z) = (- ?y ≤ - - ?x ⊔ - ?z)›*))\n\nlemma sup_less_eq_cases:\n assumes \"-z ≤ -x ⊔ -y\"\n and \"-z ≤ --x ⊔ -y\"\n shows \"-z ≤ -y\"\nproof (-)\n (*goal: ‹- z ≤ - y›*)\n have \"-z ≤ (-x ⊔ -y) ⊓ (--x ⊔ -y)\"\n by (metis assms (*‹- z ≤ - x ⊔ - y› ‹- z ≤ - - x ⊔ - y›*) greatest_lower_bound (*‹(- ?x ≤ - ?y ∧ - ?x ≤ - ?z) = (- ?x ≤ - ?y ⊓ - ?z)›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*))\n also (*calculation: ‹- z ≤ (- x ⊔ - y) ⊓ (- - x ⊔ - y)›*) have \"... = -y\"\n by (metis inf_cases (*‹- ?x = (- ?x ⊔ - ?y) ⊓ (- ?x ⊔ - - ?y)›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*))\n finally (*calculation: ‹- z ≤ - y›*) show \"?thesis\"\n (*goal: ‹- z ≤ - y›*) .\nqed\n\nlemma sup_less_eq_cases_2:\n \"-x ⊔ -y ≤ -x ⊔ -z ⟹ --x ⊔ -y ≤ --x ⊔ -z ⟹ -y ≤ -z\"\n by (metis least_upper_bound (*‹(- ?x ≤ - ?z ∧ - ?y ≤ - ?z) = (- ?x ⊔ - ?y ≤ - ?z)›*) sup_less_eq_cases (*‹⟦- ?z ≤ - ?x ⊔ - ?y; - ?z ≤ - - ?x ⊔ - ?y⟧ ⟹ - ?z ≤ - ?y›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*))\n\nlemma sup_less_eq_cases_3:\n \"-y ⊔ -x ≤ -z ⊔ -x ⟹ -y ⊔ --x ≤ -z ⊔ --x ⟹ -y ≤ -z\"\n by (simp add: sup_less_eq_cases_2 (*‹⟦- ?x ⊔ - ?y ≤ - ?x ⊔ - ?z; - - ?x ⊔ - ?y ≤ - - ?x ⊔ - ?z⟧ ⟹ - ?y ≤ - ?z›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*))\n\nlemma inf_less_eq_cases:\n \"-x ⊓ -y ≤ -z ⟹ --x ⊓ -y ≤ -z ⟹ -y ≤ -z\"\n by (simp add: shunting (*‹(- ?x ⊓ - ?y ≤ - ?z) = (- ?y ≤ - - ?x ⊔ - ?z)›*) sup_less_eq_cases (*‹⟦- ?z ≤ - ?x ⊔ - ?y; - ?z ≤ - - ?x ⊔ - ?y⟧ ⟹ - ?z ≤ - ?y›*))\n\nlemma inf_less_eq_cases_2:\n \"-x ⊓ -y ≤ -x ⊓ -z ⟹ --x ⊓ -y ≤ --x ⊓ -z ⟹ -y ≤ -z\"\n by (metis greatest_lower_bound (*‹(- ?x ≤ - ?y ∧ - ?x ≤ - ?z) = (- ?x ≤ - ?y ⊓ - ?z)›*) inf_closed (*‹- ?x ⊓ - ?y = - - (- ?x ⊓ - ?y)›*) inf_less_eq_cases (*‹⟦- ?x ⊓ - ?y ≤ - ?z; - - ?x ⊓ - ?y ≤ - ?z⟧ ⟹ - ?y ≤ - ?z›*))\n\nlemma inf_less_eq_cases_3:\n \"-y ⊓ -x ≤ -z ⊓ -x ⟹ -y ⊓ --x ≤ -z ⊓ --x ⟹ -y ≤ -z\"\n by (simp add: inf_commutative (*‹- (?x::'a) ⊓ - (?y::'a) = - ?y ⊓ - ?x›*) inf_less_eq_cases_2 (*‹⟦- (?x::'a) ⊓ - (?y::'a) ≤ - ?x ⊓ - (?z::'a); - - ?x ⊓ - ?y ≤ - - ?x ⊓ - ?z⟧ ⟹ - ?y ≤ - ?z›*))\n\nlemma inf_eq_cases:\n \"-x ⊓ -y = -x ⊓ -z ⟹ --x ⊓ -y = --x ⊓ -z ⟹ -y = -z\"\n by (metis inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) sup_cases (*‹- ?x = - ?x ⊓ - ?y ⊔ - ?x ⊓ - - ?y›*))\n\nlemma inf_eq_cases_2:\n \"-y ⊓ -x = -z ⊓ -x ⟹ -y ⊓ --x = -z ⊓ --x ⟹ -y = -z\"\n using inf_commutative (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) inf_eq_cases (*‹⟦- ?x ⊓ - ?y = - ?x ⊓ - ?z; - - ?x ⊓ - ?y = - - ?x ⊓ - ?z⟧ ⟹ - ?y = - ?z›*) by auto\n\nlemma wnf_lemma_1:\n \"((-x ⊔ -y) ⊓ (--x ⊔ -z)) ⊔ -x = -x ⊔ -y\"\nproof (-)\n (*goal: ‹(- x ⊔ - y) ⊓ (- - x ⊔ - z) ⊔ - x = - x ⊔ - y›*)\n have \"∀u v w. (-u ⊓ (-v ⊔ --w)) ⊔ -w = -u ⊔ -w\"\n by (metis inf_right_zero (*‹- ?x ⊓ top = - ?x›*) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sup_complement (*‹- ?x ⊔ - - ?x = top›*) sup_idempotent (*‹- ?x ⊔ - ?x = - ?x›*) sup_right_dist_inf (*‹- ?y ⊓ - ?z ⊔ - ?x = (- ?y ⊔ - ?x) ⊓ (- ?z ⊔ - ?x)›*))\n thus \"?thesis\"\n (*goal: ‹(- (x::'a) ⊔ - (y::'a)) ⊓ (- - x ⊔ - (z::'a)) ⊔ - x = - x ⊔ - y›*)\n by (metis (no_types) sub_associative (*‹- ?x ⊔ (- ?y ⊔ - ?z) = - ?x ⊔ - ?y ⊔ - ?z›*) sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) sub_sup_closed (*‹- ?x ⊔ - ?y = - - (- ?x ⊔ - ?y)›*) sup_idempotent (*‹- ?x ⊔ - ?x = - ?x›*))\nqed\n\nlemma wnf_lemma_2:\n \"((-x ⊔ -y) ⊓ (-z ⊔ --y)) ⊔ -y = -x ⊔ -y\"\n using sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) wnf_lemma_1 (*‹(- ?x ⊔ - ?y) ⊓ (- - ?x ⊔ - ?z) ⊔ - ?x = - ?x ⊔ - ?y›*) by fastforce\n\nlemma wnf_lemma_3:\n \"((-x ⊔ -z) ⊓ (--x ⊔ -y)) ⊔ --x = --x ⊔ -y\"\n by (metis case_duality (*‹- - (?x::'a) ⊓ - (?y::'a) ⊔ - ?x ⊓ - (?z::'a) = (- ?x ⊔ - ?y) ⊓ (- - ?x ⊔ - ?z)›*) case_duality_2 (*‹- (?x::'a) ⊓ - (?y::'a) ⊔ - - ?x ⊓ - (?z::'a) = (- ?x ⊔ - ?z) ⊓ (- - ?x ⊔ - ?y)›*) double_negation (*‹- - - (?x::'a) = - ?x›*) sub_commutative (*‹- (?x::'a) ⊔ - (?y::'a) = - ?y ⊔ - ?x›*) wnf_lemma_2 (*‹(- (?x::'a) ⊔ - (?y::'a)) ⊓ (- (?z::'a) ⊔ - - ?y) ⊔ - ?y = - ?x ⊔ - ?y›*))\n\nlemma wnf_lemma_4:\n \"((-z ⊔ -y) ⊓ (-x ⊔ --y)) ⊔ --y = -x ⊔ --y\"\n using sub_commutative (*‹- ?x ⊔ - ?y = - ?y ⊔ - ?x›*) wnf_lemma_3 (*‹(- ?x ⊔ - ?z) ⊓ (- - ?x ⊔ - ?y) ⊔ - - ?x = - - ?x ⊔ - ?y›*) by auto\n\nend\n\nclass subset_boolean_algebra' = sup + uminus +\n assumes sub_associative': \"-x ⊔ (-y ⊔ -z) = (-x ⊔ -y) ⊔ -z\"\n assumes sub_commutative': \"-x ⊔ -y = -y ⊔ -x\"\n assumes sub_complement': \"-x = -(--x ⊔ -y) ⊔ -(--x ⊔ --y)\"\n assumes sub_sup_closed': \"∃z . -x ⊔ -y = -z\"\nbegin\n\nsubclass subset_boolean_algebra\nproof\n show \"⋀x y z. - x ⊔ (- y ⊔ - z) = - x ⊔ - y ⊔ - z\"\n by (simp add: sub_associative')\n show \"⋀x y. - x ⊔ - y = - y ⊔ - x\"\n by (simp add: sub_commutative')\n show \"⋀x y. - x = - (- - x ⊔ - y) ⊔ - (- - x ⊔ - - y)\"\n by (simp add: sub_complement')\n show \"⋀x y. - x ⊔ - y = - - (- x ⊔ - y)\"\n proof -\n fix x y\n have \"∀x y. -y ⊔ (-(--y ⊔ -x) ⊔ -(---x ⊔ -y)) = -y ⊔ --x\"\n by (metis (no_types) sub_associative' sub_commutative' sub_complement')\n hence \"∀x. ---x = -x\"\n by (metis (no_types) sub_commutative' sub_complement')\n thus \"-x ⊔ -y = --(-x ⊔ -y)\"\n by (metis sub_sup_closed')\n qed\nqed\n\nend\n\ntext ‹\nWe introduce a type for the range of complement and show that it is an instance of ‹boolean_algebra›.\n›\n\ntypedef (overloaded) 'a boolean_subset = \"{ x::'a::uminus . ∃y . x = -y }\"\n by auto\n\nlemma simp_boolean_subset[simp]:\n \"∃y . Rep_boolean_subset x = -y\"\n using Rep_boolean_subset (*‹Rep_boolean_subset ?x ∈ {x. ∃y. x = - y}›*) by simp\n\nsetup_lifting type_definition_boolean_subset\n\ntext ‹Theorem 8.1›\n\ninstantiation boolean_subset :: (subset_boolean_algebra) huntington\nbegin\n\nlift_definition sup_boolean_subset :: \"'a boolean_subset ⇒ 'a boolean_subset ⇒ 'a boolean_subset\" is sup\n using sub_sup_closed by auto\n\nlift_definition uminus_boolean_subset :: \"'a boolean_subset ⇒ 'a boolean_subset\" is uminus\n by auto\n\ninstance\nproof\n show \"⋀x y z::'a boolean_subset. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n apply transfer\n using sub_associative by blast\n show \"⋀x y::'a boolean_subset. x ⊔ y = y ⊔ x\"\n apply transfer\n using sub_commutative by blast\n show \"⋀x y::'a boolean_subset. x = - (- x ⊔ y) ⊔ - (- x ⊔ - y)\"\n apply transfer\n using sub_complement by blast\nqed\n\nend\n\ntext ‹Theorem 8.2›\n\ninstantiation boolean_subset :: (subset_boolean_algebra_extended) huntington_extended\nbegin\n\nlift_definition inf_boolean_subset :: \"'a boolean_subset ⇒ 'a boolean_subset ⇒ 'a boolean_subset\" is inf\n using inf_closed by auto\n\nlift_definition minus_boolean_subset :: \"'a boolean_subset ⇒ 'a boolean_subset ⇒ 'a boolean_subset\" is minus\n using sub_minus_def by auto\n\nlift_definition bot_boolean_subset :: \"'a boolean_subset\" is bot\n by (metis complement_top)\n\nlift_definition top_boolean_subset :: \"'a boolean_subset\" is top\n by (metis complement_bot)\n\nlift_definition less_eq_boolean_subset :: \"'a boolean_subset ⇒ 'a boolean_subset ⇒ bool\" is less_eq .\n\nlift_definition less_boolean_subset :: \"'a boolean_subset ⇒ 'a boolean_subset ⇒ bool\" is less .\n\ninstance\nproof\n show 1: \"top = (THE x. ∀y::'a boolean_subset. x = y ⊔ - y)\"\n proof (rule the_equality[symmetric])\n show \"∀y::'a boolean_subset. top = y ⊔ - y\"\n apply transfer\n by auto\n show \"⋀x::'a boolean_subset. ∀y. x = y ⊔ - y ⟹ x = top\"\n apply transfer\n by force\n qed\n have \"(bot::'a boolean_subset) = - top\"\n apply transfer\n by simp\n thus \"bot = - (THE x. ∀y::'a boolean_subset. x = y ⊔ - y)\"\n using 1 by simp\n show \"⋀x y::'a boolean_subset. x ⊓ y = - (- x ⊔ - y)\"\n apply transfer\n using sub_inf_def by blast\n show \"⋀x y::'a boolean_subset. x - y = - (- x ⊔ y)\"\n apply transfer\n using sub_minus_def by blast\n show \"⋀x y::'a boolean_subset. (x ≤ y) = (x ⊔ y = y)\"\n apply transfer\n using sub_less_eq_def by blast\n show \"⋀x y::'a boolean_subset. (x < y) = (x ⊔ y = y ∧ y ⊔ x ≠ x)\"\n apply transfer\n using sub_less_def by blast\nqed\n\nend\n\nsection ‹Subset Boolean algebras with Additional Structure›\n\ntext ‹\nWe now discuss axioms that make the range of a unary operation a Boolean algebra, but add further properties that are common to the intended models.\nIn the intended models, the unary operation can be a complement, a pseudocomplement or the antidomain operation.\nFor simplicity, we mostly call the unary operation `complement'.\n\nWe first look at structures based only on join and complement, and then add axioms for the remaining operations of Boolean algebras.\nIn the intended models, the operation that is meet on the range of the complement can be a meet in the whole algebra or composition.\n›\n\nsubsection ‹Axioms Derived from the New Axiomatisation›\n\ntext ‹\nThe axioms of the first algebra are based on ‹boolean_algebra_3›.\n›\n\ntext ‹Definition 9›\n\nclass subset_boolean_algebra_1 = sup + uminus +\n assumes sba1_associative: \"x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n assumes sba1_commutative: \"x ⊔ y = y ⊔ x\"\n assumes sba1_idempotent[simp]: \"x ⊔ x = x\"\n assumes sba1_double_complement[simp]: \"---x = -x\"\n assumes sba1_bot_unique: \"-(x ⊔ -x) = -(y ⊔ -y)\"\n assumes sba1_export: \"-x ⊔ -(-x ⊔ y) = -x ⊔ -y\"\nbegin\n\ntext ‹Theorem 11.1›\n\nsubclass subset_boolean_algebra\nproof\n show \"⋀x y z. - x ⊔ (- y ⊔ - z) = - x ⊔ - y ⊔ - z\"\n by (simp add: sba1_associative)\n show \"⋀x y. - x ⊔ - y = - y ⊔ - x\"\n by (simp add: sba1_commutative)\n show \"⋀x y. - x = - (- - x ⊔ - y) ⊔ - (- - x ⊔ - - y)\"\n by (smt sba1_bot_unique sba1_commutative sba1_double_complement sba1_export sba1_idempotent)\n thus \"⋀x y. - x ⊔ - y = - - (- x ⊔ - y)\"\n by (metis sba1_double_complement sba1_export)\nqed\n\ndefinition \"sba1_bot ≡ THE x . ∀z . x = -(z ⊔ -z)\"\n\nlemma sba1_bot:\n \"sba1_bot = -(z ⊔ -z)\"\n using sba1_bot_def (*‹sba1_bot ≡ THE x. ∀z. x = - (z ⊔ - z)›*) sba1_bot_unique (*‹- ((?x::'a) ⊔ - ?x) = - ((?y::'a) ⊔ - ?y)›*) by auto\n\nend\n\ntext ‹Boolean algebra operations based on join and complement›\n\ntext ‹Definition 10›\n\nclass subset_extended_1 = sup + inf + minus + uminus + bot + top + ord +\n assumes ba_bot: \"bot = (THE x . ∀z . x = -(z ⊔ -z))\"\n assumes ba_top: \"top = -(THE x . ∀z . x = -(z ⊔ -z))\"\n assumes ba_inf: \"-x ⊓ -y = -(--x ⊔ --y)\"\n assumes ba_minus: \"-x - -y = -(--x ⊔ -y)\"\n assumes ba_less_eq: \"x ≤ y ⟷ x ⊔ y = y\"\n assumes ba_less: \"x < y ⟷ x ⊔ y = y ∧ ¬ (y ⊔ x = x)\"\n\nclass subset_extended_2 = subset_extended_1 +\n assumes ba_bot_unique: \"-(x ⊔ -x) = -(y ⊔ -y)\"\nbegin\n\nlemma ba_bot_def:\n \"bot = -(z ⊔ -z)\"\n using ba_bot (*‹bot = (THE x. ∀z. x = - (z ⊔ - z))›*) ba_bot_unique (*‹- (?x ⊔ - ?x) = - (?y ⊔ - ?y)›*) by auto\n\nlemma ba_top_def:\n \"top = --(z ⊔ -z)\"\n using ba_bot_def (*‹bot = - (?z ⊔ - ?z)›*) ba_top (*‹top = - (THE x. ∀z. x = - (z ⊔ - z))›*) by simp\n\nend\n\ntext ‹Subset forms Boolean Algebra, extended by Boolean algebra operations›\n\nclass subset_boolean_algebra_1_extended = subset_boolean_algebra_1 + subset_extended_1\nbegin\n\nsubclass subset_extended_2\nproof\n show \"⋀x y. - (x ⊔ - x) = - (y ⊔ - y)\"\n by (simp add: sba1_bot_unique)\nqed\n\nsubclass semilattice_sup\nproof\n show \"⋀x y. (x < y) = (x ≤ y ∧ ¬ y ≤ x)\"\n by (simp add: ba_less ba_less_eq)\n show \"⋀x. x ≤ x\"\n by (simp add: ba_less_eq)\n show \"⋀x y z. x ≤ y ⟹ y ≤ z ⟹ x ≤ z\"\n by (metis sba1_associative ba_less_eq)\n show \"⋀x y. x ≤ y ⟹ y ≤ x ⟹ x = y\"\n by (simp add: sba1_commutative ba_less_eq)\n show \"⋀x y. x ≤ x ⊔ y\"\n by (simp add: sba1_associative ba_less_eq)\n thus \"⋀y x. y ≤ x ⊔ y\"\n by (simp add: sba1_commutative)\n show \"⋀y x z. y ≤ x ⟹ z ≤ x ⟹ y ⊔ z ≤ x\"\n by (metis sba1_associative ba_less_eq)\nqed\n\ntext ‹Theorem 11.2›\n\nsubclass subset_boolean_algebra_extended\nproof\n show \"top = (THE x. ∀y. x = - y ⊔ - - y)\"\n by (smt ba_bot ba_bot_def ba_top sub_sup_closed the_equality)\n thus \"bot = - (THE x. ∀y. x = - y ⊔ - - y)\"\n using ba_bot_def ba_top_def by force\n show \"⋀x y. - x ⊓ - y = - (- - x ⊔ - - y)\"\n by (simp add: ba_inf)\n show \"⋀x y. - x - - y = - (- - x ⊔ - y)\"\n by (simp add: ba_minus)\n show \"⋀x y. (- x ≤ - y) = (- x ⊔ - y = - y)\"\n using le_iff_sup by auto\n show \"⋀x y. (- x < - y) = (- x ⊔ - y = - y ∧ - y ⊔ - x ≠ - x)\"\n by (simp add: ba_less)\nqed\n\nend\n\nsubsection ‹Stronger Assumptions based on Join and Complement›\n\ntext ‹\nWe add further axioms covering properties common to the antidomain and (pseudo)complement instances.\n›\n\ntext ‹Definition 12›\n\nclass subset_boolean_algebra_2 = sup + uminus +\n assumes sba2_associative: \"x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n assumes sba2_commutative: \"x ⊔ y = y ⊔ x\"\n assumes sba2_idempotent[simp]: \"x ⊔ x = x\"\n assumes sba2_bot_unit: \"x ⊔ -(y ⊔ -y) = x\"\n assumes sba2_sub_sup_demorgan: \"-(x ⊔ y) = -(--x ⊔ --y)\"\n assumes sba2_export: \"-x ⊔ -(-x ⊔ y) = -x ⊔ -y\"\nbegin\n\ntext ‹Theorem 13.1›\n\nsubclass subset_boolean_algebra_1\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: sba2_associative)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: sba2_commutative)\n show \"⋀x. x ⊔ x = x\"\n by simp\n show \"⋀x. - - - x = - x\"\n by (metis sba2_idempotent sba2_sub_sup_demorgan)\n show \"⋀x y. - (x ⊔ - x) = - (y ⊔ - y)\"\n by (metis sba2_bot_unit sba2_commutative)\n show \"⋀x y. - x ⊔ - (- x ⊔ y) = - x ⊔ - y\"\n by (simp add: sba2_export)\nqed\n\ntext ‹Theorem 13.2›\n\nlemma double_complement_dist_sup:\n \"--(x ⊔ y) = --x ⊔ --y\"\n by (metis sba2_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) sba2_export (*‹- ?x ⊔ - (- ?x ⊔ ?y) = - ?x ⊔ - ?y›*) sba2_idempotent (*‹?x ⊔ ?x = ?x›*) sba2_sub_sup_demorgan (*‹- (?x ⊔ ?y) = - (- - ?x ⊔ - - ?y)›*))\n\nlemma maddux_3_3[simp]:\n \"-(x ⊔ y) ⊔ -(x ⊔ -y) = -x\"\n by (metis double_complement_dist_sup (*‹- - (?x ⊔ ?y) = - - ?x ⊔ - - ?y›*) sba1_double_complement (*‹- - - ?x = - ?x›*) sba2_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) sub_complement (*‹- ?x = - (- - ?x ⊔ - ?y) ⊔ - (- - ?x ⊔ - - ?y)›*))\n\nlemma huntington_3_pp[simp]:\n \"-(-x ⊔ -y) ⊔ -(-x ⊔ y) = --x\"\n using sba2_commutative (*‹(?x::'a::type) ⊔ (?y::'a::type) = ?y ⊔ ?x›*) maddux_3_3 (*‹- (?x ⊔ ?y) ⊔ - (?x ⊔ - ?y) = - ?x›*) by fastforce\n\nend\n\nclass subset_boolean_algebra_2_extended = subset_boolean_algebra_2 + subset_extended_1\nbegin\n\nsubclass subset_boolean_algebra_1_extended ..\n\nsubclass bounded_semilattice_sup_bot\nproof\n show \"⋀x. bot ≤ x\"\n using sba2_bot_unit ba_bot_def sup_right_divisibility by auto\nqed\n\ntext ‹Theorem 13.3›\n\nlemma complement_antitone:\n \"x ≤ y ⟹ -y ≤ -x\"\n by (metis le_iff_sup (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) maddux_3_3 (*‹- (?x ⊔ ?y) ⊔ - (?x ⊔ - ?y) = - ?x›*) sba2_export (*‹- ?x ⊔ - (- ?x ⊔ ?y) = - ?x ⊔ - ?y›*) sup_monoid.add_commute (*‹?a ⊔ ?b = ?b ⊔ ?a›*))\n\nlemma double_complement_isotone:\n \"x ≤ y ⟹ --x ≤ --y\"\n by (simp add: complement_antitone (*‹?x ≤ ?y ⟹ - ?y ≤ - ?x›*))\n\nlemma sup_demorgan:\n \"-(x ⊔ y) = -x ⊓ -y\"\n using sba2_sub_sup_demorgan (*‹- (?x ⊔ ?y) = - (- - ?x ⊔ - - ?y)›*) ba_inf (*‹- ?x ⊓ - ?y = - (- - ?x ⊔ - - ?y)›*) by auto\n\nend\n\nsubsection ‹Axioms for Meet›\n\ntext ‹\nWe add further axioms of ‹inf› covering properties common to the antidomain and pseudocomplement instances.\nWe omit the left distributivity rule and the right zero rule as they do not hold in some models.\nIn particular, the operation ‹inf› does not have to be commutative.\n›\n\ntext ‹Definition 14›\n\nclass subset_boolean_algebra_3_extended = subset_boolean_algebra_2_extended +\n assumes sba3_inf_associative: \"x ⊓ (y ⊓ z) = (x ⊓ y) ⊓ z\"\n assumes sba3_inf_right_dist_sup: \"(x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)\"\n assumes sba3_inf_complement_bot: \"-x ⊓ x = bot\"\n assumes sba3_inf_left_unit[simp]: \"top ⊓ x = x\"\n assumes sba3_complement_inf_double_complement: \"-(x ⊓ --y) = -(x ⊓ y)\"\nbegin\n\ntext ‹Theorem 15›\n\nlemma inf_left_zero:\n \"bot ⊓ x = bot\"\n by (metis inf_right_unit (*‹- ?x ⊓ bot = bot›*) sba3_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) sba3_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*))\n\nlemma inf_double_complement_export:\n \"--(--x ⊓ y) = --x ⊓ --y\"\n by (metis inf_closed (*‹- ?x ⊓ - ?y = - - (- ?x ⊓ - ?y)›*) sba3_complement_inf_double_complement (*‹- (?x ⊓ - - ?y) = - (?x ⊓ ?y)›*))\n\nlemma inf_left_isotone:\n \"x ≤ y ⟹ x ⊓ z ≤ y ⊓ z\"\n using sba3_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) sup_right_divisibility (*‹(?x ≤ ?y) = (∃z. z ⊔ ?x = ?y)›*) by auto\n\nlemma inf_complement_export:\n \"--(-x ⊓ y) = -x ⊓ --y\"\n by (metis inf_double_complement_export (*‹- - (- - ?x ⊓ ?y) = - - ?x ⊓ - - ?y›*) sba1_double_complement (*‹- - - ?x = - ?x›*))\n\nlemma double_complement_above:\n \"--x ⊓ x = x\"\n by (metis sup_monoid.add_0_right (*‹?a ⊔ bot = ?a›*) complement_bot (*‹- bot = top›*) inf_demorgan (*‹- (- ?x ⊓ - ?y) = - - ?x ⊔ - - ?y›*) sba1_double_complement (*‹- - - ?x = - ?x›*) sba3_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) sba3_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) sba3_inf_left_unit (*‹top ⊓ ?x = ?x›*))\n\nlemma \"x ≤ y ⟹ z ⊓ x ≤ z ⊓ y\" nitpick [expect=genuine] oops\nlemma \"x ⊓ top = x\" nitpick [expect=genuine] oops\nlemma \"x ⊓ y = y ⊓ x\" nitpick [expect=genuine] oops\n\nend\n\nsubsection ‹Stronger Assumptions for Meet›\n\ntext ‹\nThe following axioms also hold in both models, but follow from the axioms of ‹subset_boolean_algebra_5_operations›.\n›\n\ntext ‹Definition 16›\n\nclass subset_boolean_algebra_4_extended = subset_boolean_algebra_3_extended +\n assumes sba4_inf_right_unit[simp]: \"x ⊓ top = x\"\n assumes inf_right_isotone: \"x ≤ y ⟹ z ⊓ x ≤ z ⊓ y\"\nbegin\n\nlemma \"x ⊔ top = top\" nitpick [expect=genuine] oops\nlemma \"x ⊓ bot = bot\" nitpick [expect=genuine] oops\nlemma \"x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)\" nitpick [expect=genuine] oops\nlemma \"(x ⊓ y = bot) = (x ≤ - y)\" nitpick [expect=genuine] oops\n\nend\n\nsection ‹Boolean Algebras in Stone Algebras›\n\ntext ‹\nWe specialise ‹inf› to meet and complement to pseudocomplement.\nThis puts Stone algebras into the picture; for these it is well known that regular elements form a Boolean subalgebra \\<^cite>‹\"Graetzer1971\"›.\n›\n\ntext ‹Definition 17›\n\nclass subset_boolean_algebra_5_extended = subset_boolean_algebra_3_extended +\n assumes sba5_inf_commutative: \"x ⊓ y = y ⊓ x\"\n assumes sba5_inf_absorb: \"x ⊓ (x ⊔ y) = x\"\nbegin\n\nsubclass distrib_lattice_bot\nproof\n show \"⋀x y. x ⊓ y ≤ x\"\n by (metis sba5_inf_commutative sba3_inf_right_dist_sup sba5_inf_absorb sup_right_divisibility)\n show \"⋀x y. x ⊓ y ≤ y\"\n by (metis inf_left_isotone sba5_inf_absorb sba5_inf_commutative sup_ge2)\n show \"⋀x y z. x ≤ y ⟹ x ≤ z ⟹ x ≤ y ⊓ z\"\n by (metis inf_left_isotone sba5_inf_absorb sup.orderE sup_monoid.add_commute)\n show \"⋀x y z. x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z) \"\n by (metis sba3_inf_right_dist_sup sba5_inf_absorb sba5_inf_commutative sup_assoc)\nqed\n\nlemma inf_demorgan_2:\n \"-(x ⊓ y) = -x ⊔ -y\"\n using sba3_complement_inf_double_complement (*‹- ((?x::'a::type) ⊓ - - (?y::'a::type)) = - (?x ⊓ ?y)›*) sba5_inf_commutative (*‹?x ⊓ ?y = ?y ⊓ ?x›*) sub_sup_closed (*‹- (?x::'a) ⊔ - (?y::'a) = - - (- ?x ⊔ - ?y)›*) sub_sup_demorgan (*‹- (- (?x::'a) ⊔ - (?y::'a)) = - - ?x ⊓ - - ?y›*) by auto\n\nlemma inf_export:\n \"x ⊓ -(x ⊓ y) = x ⊓ -y\"\n using inf_demorgan_2 (*‹- (?x ⊓ ?y) = - ?x ⊔ - ?y›*) sba3_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) sba3_inf_right_dist_sup (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) sba5_inf_commutative (*‹?x ⊓ ?y = ?y ⊓ ?x›*) by auto\n\nlemma complement_inf[simp]:\n \"x ⊓ -x = bot\"\n using sba3_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) sba5_inf_commutative (*‹?x ⊓ ?y = ?y ⊓ ?x›*) by auto\n\ntext ‹Theorem 18.2›\n\nsubclass stone_algebra\nproof\n show \"⋀x. x ≤ top\"\n by (simp add: inf.absorb_iff2)\n show \"⋀x y. (x ⊓ y = bot) = (x ≤ - y)\"\n by (metis (full_types) complement_bot complement_inf inf.cobounded1 inf.order_iff inf_export sba3_complement_inf_double_complement sba3_inf_left_unit)\n show \"⋀x. - x ⊔ - - x = top\"\n by simp\nqed\n\ntext ‹Theorem 18.1›\n\nsubclass subset_boolean_algebra_4_extended\nproof\n show \"⋀x. x ⊓ top = x\"\n by simp\n show \"⋀x y z. x ≤ y ⟹ z ⊓ x ≤ z ⊓ y\"\n using inf.sup_right_isotone by blast\nqed\n\nend\n\ncontext stone_algebra_extended\nbegin\n\ntext ‹Theorem 18.3›\n\nsubclass subset_boolean_algebra_5_extended\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n using sup_assoc by auto\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: sup_commute)\n show \"⋀x. x ⊔ x = x\"\n by simp\n show \"⋀x y. x ⊔ - (y ⊔ - y) = x\"\n by simp\n show \"⋀x y. - (x ⊔ y) = - (- - x ⊔ - - y)\"\n by auto\n show \"⋀x y. - x ⊔ - (- x ⊔ y) = - x ⊔ - y\"\n by (metis maddux_3_21_pp p_dist_sup regular_closed_p)\n show \"bot = (THE x. ∀z. x = - (z ⊔ - z))\"\n by simp\n thus \"top = - (THE x. ∀z. x = - (z ⊔ - z))\"\n using p_bot by blast\n show \"⋀x y. - x ⊓ - y = - (- - x ⊔ - - y)\"\n by simp\n show \"⋀x y. - x - - y = - (- - x ⊔ - y)\"\n by auto\n show \"⋀x y. (x ≤ y) = (x ⊔ y = y)\"\n by (simp add: le_iff_sup)\n thus \"⋀x y. (x < y) = (x ⊔ y = y ∧ y ⊔ x ≠ x)\"\n by (simp add: less_le_not_le)\n show \"⋀x y z. x ⊓ (y ⊓ z) = x ⊓ y ⊓ z\"\n by (simp add: inf.sup_monoid.add_assoc)\n show \"⋀x y z. (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z\"\n by (simp add: inf_sup_distrib2)\n show \"⋀x. - x ⊓ x = bot\"\n by simp\n show \"⋀x. top ⊓ x = x\"\n by simp\n show \"⋀x y. - (x ⊓ - - y) = - (x ⊓ y)\"\n by simp\n show \"⋀x y. x ⊓ y = y ⊓ x\"\n by (simp add: inf_commute)\n show \"⋀x y. x ⊓ (x ⊔ y) = x\"\n by simp\nqed\n\nend\n\nsection ‹Domain Semirings›\n\ntext ‹\nThe following development of tests in IL-semirings, prepredomain semirings, predomain semirings and domain semirings is mostly based on \\<^cite>‹\"MoellerDesharnais2019\"›; see also \\<^cite>‹\"DesharnaisMoeller2014\"›.\nSee \\<^cite>‹\"DesharnaisMoellerStruth2006b\"› for domain axioms in idempotent semirings.\nSee \\<^cite>‹\"DesharnaisJipsenStruth2009\" and \"JacksonStokes2004\"› for domain axioms in semigroups and monoids.\nSome variants have been implemented in \\<^cite>‹\"GomesGuttmannHoefnerStruthWeber2016\"›.\n›\n\nsubsection ‹Idempotent Left Semirings›\n\ntext ‹Definition 19›\n\nclass il_semiring = sup + inf + bot + top + ord +\n assumes il_associative: \"x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n assumes il_commutative: \"x ⊔ y = y ⊔ x\"\n assumes il_idempotent[simp]: \"x ⊔ x = x\"\n assumes il_bot_unit: \"x ⊔ bot = x\"\n assumes il_inf_associative: \"x ⊓ (y ⊓ z) = (x ⊓ y) ⊓ z\"\n assumes il_inf_right_dist_sup: \"(x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)\"\n assumes il_inf_left_unit[simp]: \"top ⊓ x = x\"\n assumes il_inf_right_unit[simp]: \"x ⊓ top = x\"\n assumes il_sub_inf_left_zero[simp]: \"bot ⊓ x = bot\"\n assumes il_sub_inf_right_isotone: \"x ≤ y ⟹ z ⊓ x ≤ z ⊓ y\"\n assumes il_less_eq: \"x ≤ y ⟷ x ⊔ y = y\"\n assumes il_less_def: \"x < y ⟷ x ≤ y ∧ ¬(y ≤ x)\"\nbegin\n\nlemma il_unit_bot: \"bot ⊔ x = x\"\n using il_bot_unit (*‹?x ⊔ bot = ?x›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by fastforce\n\nsubclass order\nproof\n show \"⋀x y. (x < y) = (x ≤ y ∧ ¬ y ≤ x)\"\n by (simp add: il_less_def)\n show \"⋀x. x ≤ x\"\n by (simp add: il_less_eq)\n show \"⋀x y z. x ≤ y ⟹ y ≤ z ⟹ x ≤ z\"\n by (metis il_associative il_less_eq)\n show \"⋀x y. x ≤ y ⟹ y ≤ x ⟹ x = y\"\n by (simp add: il_commutative il_less_eq)\nqed\n\nlemma il_sub_inf_right_isotone_var:\n \"(x ⊓ y) ⊔ (x ⊓ z) ≤ x ⊓ (y ⊔ z)\"\n by (smt il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_idempotent (*‹?x ⊔ ?x = ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*))\n\nlemma il_sub_inf_left_isotone:\n \"x ≤ y ⟹ x ⊓ z ≤ y ⊓ z\"\n by (metis il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n\nlemma il_sub_inf_left_isotone_var:\n \"(y ⊓ x) ⊔ (z ⊓ x) ≤ (y ⊔ z) ⊓ x\"\n by (simp add: il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n\nlemma sup_left_isotone:\n \"x ≤ y ⟹ x ⊔ z ≤ y ⊔ z\"\n by (smt il_associative (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?y ⊔ ?z›*) il_commutative (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) il_idempotent (*‹(?x::'a) ⊔ ?x = ?x›*) il_less_eq (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*))\n\nlemma sup_right_isotone:\n \"x ≤ y ⟹ z ⊔ x ≤ z ⊔ y\"\n by (simp add: il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) sup_left_isotone (*‹?x ≤ ?y ⟹ ?x ⊔ ?z ≤ ?y ⊔ ?z›*))\n\nlemma bot_least:\n \"bot ≤ x\"\n by (simp add: il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_unit_bot (*‹bot ⊔ ?x = ?x›*))\n\nlemma less_eq_bot:\n \"x ≤ bot ⟷ x = bot\"\n by (simp add: il_bot_unit (*‹?x ⊔ bot = ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n\nabbreviation are_complementary :: \"'a ⇒ 'a ⇒ bool\"\n where \"are_complementary x y ≡ x ⊔ y = top ∧ x ⊓ y = bot ∧ y ⊓ x = bot\"\n\nabbreviation test :: \"'a ⇒ bool\"\n where \"test x ≡ ∃y . are_complementary x y\"\n\ndefinition tests :: \"'a set\"\n where \"tests = { x . test x }\"\n\nlemma bot_test:\n \"test bot\"\n by (simp add: il_unit_bot (*‹bot ⊔ ?x = ?x›*))\n\nlemma top_test:\n \"test top\"\n by (simp add: il_bot_unit (*‹?x ⊔ bot = ?x›*))\n\nlemma test_sub_identity:\n \"test x ⟹ x ≤ top\"\n using il_associative (*‹(?x::'a::type) ⊔ ((?y::'a::type) ⊔ (?z::'a::type)) = ?x ⊔ ?y ⊔ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by auto\n\nlemma neg_unique:\n \"are_complementary x y ⟹ are_complementary x z ⟹ y = z\"\n by (metis order.antisym (*‹⟦(?a::'a) ≤ (?b::'a); ?b ≤ ?a⟧ ⟹ ?a = ?b›*) il_inf_left_unit (*‹top ⊓ (?x::'a) = ?x›*) il_inf_right_dist_sup (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_inf_right_unit (*‹(?x::'a) ⊓ top = ?x›*) il_sub_inf_right_isotone_var (*‹(?x::'a) ⊓ (?y::'a) ⊔ ?x ⊓ (?z::'a) ≤ ?x ⊓ (?y ⊔ ?z)›*))\n\ndefinition neg :: \"'a ⇒ 'a\" (\"!\")\n where \"!x ≡ THE y . are_complementary x y\"\n\nlemma neg_char:\n assumes \"test x\"\n shows \"are_complementary x (!x)\"\nproof (unfold neg_def (*‹! ?x ≡ THE y. are_complementary ?x y›*))\n (*goal: ‹are_complementary x (THE y. are_complementary x y)›*)\n from assms (*‹test x›*) obtain y where 1: \"are_complementary x y\"\n (*goal: ‹(⋀y. are_complementary x y ⟹ thesis) ⟹ thesis›*)\n by auto\n show \"are_complementary x (THE y. are_complementary x y)\"\n proof (rule theI (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ?P (THE x. ?P x)›*))\n (*goals:\n 1. ‹are_complementary x ?a›\n 2. ‹⋀xa. are_complementary x xa ⟹ xa = ?a›*)\n show \"are_complementary x y\"\n using \"1\" (*‹are_complementary x y›*) by simp\n show \"⋀z. are_complementary x z ⟹ z = y\"\n using \"1\" (*‹are_complementary x y›*) neg_unique (*‹⟦are_complementary ?x ?y; are_complementary ?x ?z⟧ ⟹ ?y = ?z›*) by blast\n qed\nqed\n\nlemma are_complementary_symmetric:\n \"are_complementary x y ⟷ are_complementary y x\"\n using il_commutative (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) by auto\n\nlemma neg_test:\n \"test x ⟹ test (!x)\"\n using are_complementary_symmetric (*‹are_complementary ?x ?y = are_complementary ?y ?x›*) neg_char (*‹test (?x::'a::type) ⟹ are_complementary ?x (! ?x)›*) by blast\n\nlemma are_complementary_test:\n \"test x ⟹ are_complementary x y ⟹ test y\"\n using il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by auto\n\nlemma neg_involutive:\n \"test x ⟹ !(!x) = x\"\n using are_complementary_symmetric (*‹are_complementary ?x ?y = are_complementary ?y ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) neg_unique (*‹⟦are_complementary ?x ?y; are_complementary ?x ?z⟧ ⟹ ?y = ?z›*) by blast\n\nlemma test_inf_left_below:\n \"test x ⟹ x ⊓ y ≤ y\"\n by (metis il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_idempotent (*‹?x ⊔ ?x = ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n\nlemma test_inf_right_below:\n \"test x ⟹ y ⊓ x ≤ y\"\n by (metis il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*))\n\nlemma neg_bot:\n \"!bot = top\"\n using il_unit_bot (*‹bot ⊔ ?x = ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) by fastforce\n\nlemma neg_top:\n \"!top = bot\"\n using bot_test (*‹test bot›*) neg_bot (*‹! bot = top›*) neg_involutive (*‹test (?x::'a) ⟹ ! (! ?x) = ?x›*) by fastforce\n\nlemma test_inf_idempotent:\n \"test x ⟹ x ⊓ x = x\"\n by (metis il_bot_unit (*‹?x ⊔ bot = ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n\nlemma test_inf_semicommutative:\n assumes \"test x\"\n and \"test y\"\n shows \"x ⊓ y ≤ y ⊓ x\"\nproof (-)\n (*goal: ‹x ⊓ y ≤ y ⊓ x›*)\n have \"x ⊓ y = (y ⊓ x ⊓ y) ⊔ (!y ⊓ x ⊓ y)\"\n by (metis assms( (*‹test y›*) 2) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n also (*calculation: ‹x ⊓ y = y ⊓ x ⊓ y ⊔ ! y ⊓ x ⊓ y›*) have \"... ≤ (y ⊓ x ⊓ y) ⊔ (!y ⊓ y)\"\n proof (-)\n (*goal: ‹y ⊓ x ⊓ y ⊔ ! y ⊓ x ⊓ y ≤ y ⊓ x ⊓ y ⊔ ! y ⊓ y›*)\n obtain z where \"are_complementary y z\"\n (*goal: ‹(⋀z::'a. are_complementary (y::'a) z ⟹ thesis::bool) ⟹ thesis›*)\n using assms(2) (*‹test y›*) by blast\n hence \"y ⊓ (x ⊓ y) ⊔ !y ⊓ (x ⊓ y) ≤ y ⊓ (x ⊓ y)\"\n by (metis assms( (*‹test (x::'a)›*) 1) calculation (*‹(x::'a) ⊓ (y::'a) = y ⊓ x ⊓ y ⊔ ! y ⊓ x ⊓ y›*) il_sub_inf_left_isotone (*‹(?x::'a) ≤ (?y::'a) ⟹ ?x ⊓ (?z::'a) ≤ ?y ⊓ ?z›*) il_bot_unit (*‹(?x::'a) ⊔ bot = ?x›*) il_idempotent (*‹(?x::'a) ⊔ ?x = ?x›*) il_inf_associative (*‹(?x::'a) ⊓ ((?y::'a) ⊓ (?z::'a)) = ?x ⊓ ?y ⊓ ?z›*) il_less_eq (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*) neg_char (*‹test (?x::'a) ⟹ are_complementary ?x (! ?x)›*) test_inf_right_below (*‹test (?x::'a) ⟹ (?y::'a) ⊓ ?x ≤ ?y›*))\n thus \"?thesis\"\n (*goal: ‹y ⊓ x ⊓ y ⊔ ! y ⊓ x ⊓ y ≤ y ⊓ x ⊓ y ⊔ ! y ⊓ y›*)\n by (simp add: il_associative (*‹(?x::'a::type) ⊔ ((?y::'a::type) ⊔ (?z::'a::type)) = ?x ⊔ ?y ⊔ ?z›*) il_inf_associative (*‹(?x::'a::type) ⊓ ((?y::'a::type) ⊓ (?z::'a::type)) = ?x ⊓ ?y ⊓ ?z›*) il_less_eq (*‹((?x::'a::type) ≤ (?y::'a::type)) = (?x ⊔ ?y = ?y)›*))\n qed\n also (*calculation: ‹x ⊓ y ≤ y ⊓ x ⊓ y ⊔ ! y ⊓ y›*) have \"... ≤ (y ⊓ x) ⊔ (!y ⊓ y)\"\n by (metis assms( (*‹test (y::'a::type)›*) 2) il_bot_unit (*‹(?x::'a::type) ⊔ bot = ?x›*) il_inf_right_unit (*‹(?x::'a::type) ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹(?x::'a::type) ≤ (?y::'a::type) ⟹ (?z::'a::type) ⊓ ?x ≤ ?z ⊓ ?y›*) neg_char (*‹test (?x::'a::type) ⟹ are_complementary ?x (! ?x)›*) test_sub_identity (*‹test (?x::'a::type) ⟹ ?x ≤ top›*))\n also (*calculation: ‹x ⊓ y ≤ y ⊓ x ⊔ ! y ⊓ y›*) have \"... = y ⊓ x\"\n by (simp add: assms( (*‹test y›*) 2) il_bot_unit (*‹?x ⊔ bot = ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n finally (*calculation: ‹x ⊓ y ≤ y ⊓ x›*) show \"?thesis\"\n (*goal: ‹x ⊓ y ≤ y ⊓ x›*) .\nqed\n\nlemma test_inf_commutative:\n \"test x ⟹ test y ⟹ x ⊓ y = y ⊓ x\"\n by (simp add: order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*) test_inf_semicommutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y ≤ ?y ⊓ ?x›*))\n\nlemma test_inf_bot:\n \"test x ⟹ x ⊓ bot = bot\"\n using il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) test_inf_idempotent (*‹test ?x ⟹ ?x ⊓ ?x = ?x›*) by fastforce\n\nlemma test_absorb_1:\n \"test x ⟹ test y ⟹ x ⊔ (x ⊓ y) = x\"\n using il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) test_inf_right_below (*‹test ?x ⟹ ?y ⊓ ?x ≤ ?y›*) by auto\n\nlemma test_absorb_2:\n \"test x ⟹ test y ⟹ x ⊔ (y ⊓ x) = x\"\n by (metis test_absorb_1 (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊔ ?x ⊓ ?y = ?x›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\n\nlemma test_absorb_3:\n \"test x ⟹ test y ⟹ x ⊓ (x ⊔ y) = x\"\n apply (rule order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹⟦test x; test y⟧ ⟹ x ⊓ (x ⊔ y) ≤ x›\n 2. ‹⟦test x; test y⟧ ⟹ x ≤ x ⊓ (x ⊔ y)›\n discuss goal 1*)\n apply (metis il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*))\n (*discuss goal 2*)\n apply (metis il_sub_inf_right_isotone_var (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*) test_absorb_1 (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊔ ?x ⊓ ?y = ?x›*) test_inf_idempotent (*‹test ?x ⟹ ?x ⊓ ?x = ?x›*))\n (*proven 2 subgoals*) .\n\nlemma test_absorb_4:\n \"test x ⟹ test y ⟹ (x ⊔ y) ⊓ x = x\"\n by (smt il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) test_inf_idempotent (*‹test ?x ⟹ ?x ⊓ ?x = ?x›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) test_inf_left_below (*‹test ?x ⟹ ?x ⊓ ?y ≤ ?y›*))\n\nlemma test_import_1:\n assumes \"test x\"\n and \"test y\"\n shows \"x ⊔ (!x ⊓ y) = x ⊔ y\"\nproof (-)\n (*goal: ‹x ⊔ ! x ⊓ y = x ⊔ y›*)\n have \"x ⊔ (!x ⊓ y) = x ⊔ ((y ⊔ !y) ⊓ x) ⊔ (!x ⊓ y)\"\n by (simp add: assms( (*‹test (y::'a::type)›*) 2) neg_char (*‹test (?x::'a::type) ⟹ are_complementary ?x (! ?x)›*))\n also (*calculation: ‹x ⊔ ! x ⊓ y = x ⊔ (y ⊔ ! y) ⊓ x ⊔ ! x ⊓ y›*) have \"... = x ⊔ (!y ⊓ x) ⊔ (x ⊓ y) ⊔ (!x ⊓ y)\"\n by (smt assms (*‹test x› ‹test y›*) il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\n also (*calculation: ‹x ⊔ ! x ⊓ y = x ⊔ ! y ⊓ x ⊔ x ⊓ y ⊔ ! x ⊓ y›*) have \"... = x ⊔ ((x ⊔ !x) ⊓ y)\"\n by (smt calculation (*‹x ⊔ ! x ⊓ y = x ⊔ ! y ⊓ x ⊔ x ⊓ y ⊔ ! x ⊓ y›*) il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_idempotent (*‹?x ⊔ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n also (*calculation: ‹x ⊔ ! x ⊓ y = x ⊔ (x ⊔ ! x) ⊓ y›*) have \"... = x ⊔ y\"\n by (simp add: assms( (*‹test x›*) 1) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n finally (*calculation: ‹x ⊔ ! x ⊓ y = x ⊔ y›*) show \"?thesis\"\n (*goal: ‹x ⊔ ! x ⊓ y = x ⊔ y›*) .\nqed\n\nlemma test_import_2:\n assumes \"test x\"\n and \"test y\"\n shows \"x ⊔ (y ⊓ !x) = x ⊔ y\"\nproof (-)\n (*goal: ‹x ⊔ y ⊓ ! x = x ⊔ y›*)\n obtain z where 1: \"are_complementary y z\"\n (*goal: ‹(⋀z::'a. are_complementary (y::'a) z ⟹ thesis::bool) ⟹ thesis›*)\n using assms(2) (*‹test y›*) by blast\n obtain w where 2: \"are_complementary x w\"\n (*goal: ‹(⋀w. are_complementary x w ⟹ thesis) ⟹ thesis›*)\n using assms(1) (*‹test (x::'a)›*) by auto\n hence \"x ⊓ !x = bot\"\n using neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) by blast\n hence \"!x ⊓ y = y ⊓ !x\"\n using \"1\" (*‹are_complementary y z›*) \"2\" (*‹are_complementary x w›*) by (metis il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\n thus \"?thesis\"\n (*goal: ‹x ⊔ y ⊓ ! x = x ⊔ y›*)\n using \"1\" (*‹are_complementary y z›*) \"2\" (*‹are_complementary x w›*) by (metis test_import_1 (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊔ ! ?x ⊓ ?y = ?x ⊔ ?y›*))\nqed\n\nlemma test_import_3:\n assumes \"test x\"\n shows \"(!x ⊔ y) ⊓ x = y ⊓ x\"\n by (simp add: assms( (*‹test x›*) 1) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_unit_bot (*‹bot ⊔ ?x = ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n\nlemma test_import_4:\n assumes \"test x\"\n and \"test y\"\n shows \"(!x ⊔ y) ⊓ x = x ⊓ y\"\n by (metis assms (*‹test x› ‹test y›*) test_import_3 (*‹test ?x ⟹ (! ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\n\nlemma test_inf:\n \"test x ⟹ test y ⟹ test z ⟹ z ≤ x ⊓ y ⟷ z ≤ x ∧ z ≤ y\"\n apply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goal: ‹⟦test (x::'a); test (y::'a); test (z::'a)⟧ ⟹ (z ≤ x ⊓ y) = (z ≤ x ∧ z ≤ y)›*)\n using dual_order.trans (*‹⟦(?b::'a) ≤ (?a::'a); (?c::'a) ≤ ?b⟧ ⟹ ?c ≤ ?a›*) test_inf_left_below (*‹test ?x ⟹ ?x ⊓ ?y ≤ ?y›*) test_inf_right_below (*‹test ?x ⟹ ?y ⊓ ?x ≤ ?y›*)\n (*goals:\n 1. ‹⟦test x; test y; test z; z ≤ x ⊓ y⟧ ⟹ z ≤ x ∧ z ≤ y›\n 2. ‹⟦test x; test y; test z; z ≤ x ∧ z ≤ y⟧ ⟹ z ≤ x ⊓ y›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply (smt il_less_eq (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone (*‹(?x::'a) ≤ (?y::'a) ⟹ (?z::'a) ⊓ ?x ≤ ?z ⊓ ?y›*) test_absorb_4 (*‹⟦test (?x::'a); test (?y::'a)⟧ ⟹ (?x ⊔ ?y) ⊓ ?x = ?x›*))\n (*proven 2 subgoals*) .\n\nlemma test_shunting:\n assumes \"test x\"\n and \"test y\"\n shows \"x ⊓ y ≤ z ⟷ x ≤ !y ⊔ z\"\nproof (standard)\n (*goals:\n 1. ‹x ⊓ y ≤ z ⟹ x ≤ ! y ⊔ z›\n 2. ‹x ≤ ! y ⊔ z ⟹ x ⊓ y ≤ z›*)\n assume 1: \"x ⊓ y ≤ z\" (*‹(x::'a) ⊓ (y::'a) ≤ (z::'a)›*)\n have \"x = (!y ⊓ x) ⊔ (y ⊓ x)\"\n by (metis assms( (*‹test y›*) 2) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n also (*calculation: ‹x = ! y ⊓ x ⊔ y ⊓ x›*) have \"... ≤ !y ⊔ (y ⊓ x)\"\n by (simp add: assms( (*‹test x›*) 1) sup_left_isotone (*‹?x ≤ ?y ⟹ ?x ⊔ ?z ≤ ?y ⊔ ?z›*) test_inf_right_below (*‹test ?x ⟹ ?y ⊓ ?x ≤ ?y›*))\n also (*calculation: ‹x ≤ ! y ⊔ y ⊓ x›*) have \"... ≤ !y ⊔ z\"\n using \"1\" (*‹x ⊓ y ≤ z›*) by (simp add: assms (*‹test x› ‹test y›*) sup_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊔ ?x ≤ ?z ⊔ ?y›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\n finally (*calculation: ‹x ≤ ! y ⊔ z›*) show \"x ≤ !y ⊔ z\" .\nnext\n (*goal: ‹x ≤ ! y ⊔ z ⟹ x ⊓ y ≤ z›*)\n assume \"x ≤ !y ⊔ z\" (*‹(x::'a) ≤ ! (y::'a) ⊔ (z::'a)›*)\n hence \"x ⊓ y ≤ (!y ⊔ z) ⊓ y\"\n using il_sub_inf_left_isotone (*‹?x ≤ ?y ⟹ ?x ⊓ ?z ≤ ?y ⊓ ?z›*) by blast\n also (*calculation: ‹x ⊓ y ≤ (! y ⊔ z) ⊓ y›*) have \"... = z ⊓ y\"\n by (simp add: assms( (*‹test (y::'a::type)›*) 2) test_import_3 (*‹test (?x::'a::type) ⟹ (! ?x ⊔ (?y::'a::type)) ⊓ ?x = ?y ⊓ ?x›*))\n also (*calculation: ‹(x::'a) ⊓ (y::'a) ≤ (z::'a) ⊓ y›*) have \"... ≤ z\"\n by (simp add: assms( (*‹test y›*) 2) test_inf_right_below (*‹test ?x ⟹ ?y ⊓ ?x ≤ ?y›*))\n finally (*calculation: ‹x ⊓ y ≤ z›*) show \"x ⊓ y ≤ z\" .\nqed\n\nlemma test_shunting_bot:\n assumes \"test x\"\n and \"test y\"\n shows \"x ≤ y ⟷ x ⊓ !y ≤ bot\"\n by (simp add: assms (*‹test x› ‹test y›*) il_bot_unit (*‹?x ⊔ bot = ?x›*) neg_involutive (*‹test ?x ⟹ ! (! ?x) = ?x›*) neg_test (*‹test ?x ⟹ test (! ?x)›*) test_shunting (*‹⟦test ?x; test ?y⟧ ⟹ (?x ⊓ ?y ≤ ?z) = (?x ≤ ! ?y ⊔ ?z)›*))\n\nlemma test_shunting_bot_eq:\n assumes \"test x\"\n and \"test y\"\n shows \"x ≤ y ⟷ x ⊓ !y = bot\"\n by (simp add: assms (*‹test x› ‹test y›*) test_shunting_bot (*‹⟦test ?x; test ?y⟧ ⟹ (?x ≤ ?y) = (?x ⊓ ! ?y ≤ bot)›*) less_eq_bot (*‹(?x ≤ bot) = (?x = bot)›*))\n\nlemma neg_antitone:\n assumes \"test x\"\n and \"test y\"\n and \"x ≤ y\"\n shows \"!y ≤ !x\"\nproof (-)\n (*goal: ‹! y ≤ ! x›*)\n have 1: \"x ⊓ !y = bot\"\n using assms (*‹test x› ‹test y› ‹x ≤ y›*) test_shunting_bot_eq (*‹⟦test ?x; test ?y⟧ ⟹ (?x ≤ ?y) = (?x ⊓ ! ?y = bot)›*) by blast\n have 2: \"x ⊔ !x = top\"\n by (simp add: assms( (*‹test x›*) 1) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n have \"are_complementary y (!y)\"\n by (simp add: assms( (*‹test y›*) 2) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n thus \"?thesis\"\n (*goal: ‹! y ≤ ! x›*)\n using \"1\" (*‹(x::'a) ⊓ ! (y::'a) = bot›*) \"2\" (*‹(x::'a::type) ⊔ ! x = top›*) by (metis il_unit_bot (*‹bot ⊔ ?x = ?x›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*))\nqed\n\nlemma test_sup_neg_1:\n assumes \"test x\"\n and \"test y\"\n shows \"(x ⊔ y) ⊔ (!x ⊓ !y) = top\"\nproof (-)\n (*goal: ‹(x::'a) ⊔ (y::'a) ⊔ ! x ⊓ ! y = top›*)\n have \"x ⊔ !x = top\"\n by (simp add: assms( (*‹test (x::'a)›*) 1) neg_char (*‹test (?x::'a) ⟹ are_complementary ?x (! ?x)›*))\n hence \"x ⊔ (y ⊔ !x) = top\"\n by (metis assms( (*‹test (y::'a)›*) 2) il_associative (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?y ⊔ ?z›*) il_commutative (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) il_idempotent (*‹(?x::'a) ⊔ ?x = ?x›*))\n hence \"x ⊔ (y ⊔ !x ⊓ !y) = top\"\n by (simp add: assms (*‹test x› ‹test y›*) neg_test (*‹test ?x ⟹ test (! ?x)›*) test_import_2 (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊔ ?y ⊓ ! ?x = ?x ⊔ ?y›*))\n thus \"?thesis\"\n (*goal: ‹(x::'a) ⊔ (y::'a) ⊔ ! x ⊓ ! y = top›*)\n by (simp add: il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*))\nqed\n\nlemma test_sup_neg_2:\n assumes \"test x\"\n and \"test y\"\n shows \"(x ⊔ y) ⊓ (!x ⊓ !y) = bot\"\nproof (-)\n (*goal: ‹(x ⊔ y) ⊓ (! x ⊓ ! y) = bot›*)\n have 1: \"are_complementary y (!y)\"\n by (simp add: assms( (*‹test y›*) 2) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n obtain z where 2: \"are_complementary x z\"\n (*goal: ‹(⋀z. are_complementary x z ⟹ thesis) ⟹ thesis›*)\n using assms(1) (*‹test x›*) by auto\n hence \"!x = z\"\n using neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) neg_unique (*‹⟦are_complementary ?x ?y; are_complementary ?x ?z⟧ ⟹ ?y = ?z›*) by blast\n thus \"?thesis\"\n (*goal: ‹((x::'a) ⊔ (y::'a)) ⊓ (! x ⊓ ! y) = bot›*)\n using \"1\" (*‹are_complementary y (! y)›*) \"2\" (*‹are_complementary x z›*) by (metis are_complementary_symmetric (*‹are_complementary ?x ?y = are_complementary ?y ?x›*) il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) neg_involutive (*‹test ?x ⟹ ! (! ?x) = ?x›*) test_import_3 (*‹test ?x ⟹ (! ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) test_inf_bot (*‹test ?x ⟹ ?x ⊓ bot = bot›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\nqed\n\nlemma de_morgan_1:\n assumes \"test x\"\n and \"test y\"\n and \"test (x ⊓ y)\"\n shows \"!(x ⊓ y) = !x ⊔ !y\"\nproof (rule order.antisym (*‹⟦(?a::'a) ≤ (?b::'a); ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹! (x ⊓ y) ≤ ! x ⊔ ! y›\n 2. ‹! x ⊔ ! y ≤ ! (x ⊓ y)›*)\n have 1: \"test (!(x ⊓ y))\"\n by (simp add: assms (*‹test (x::'a::type)› ‹test (y::'a::type)› ‹test ((x::'a::type) ⊓ (y::'a::type))›*) neg_test (*‹test (?x::'a::type) ⟹ test (! ?x)›*))\n have \"x ≤ (x ⊓ y) ⊔ !y\"\n by (metis (full_types) assms (*‹test x› ‹test y› ‹test (x ⊓ y)›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) test_shunting (*‹⟦test ?x; test ?y⟧ ⟹ (?x ⊓ ?y ≤ ?z) = (?x ≤ ! ?y ⊔ ?z)›*) test_shunting_bot_eq (*‹⟦test ?x; test ?y⟧ ⟹ (?x ≤ ?y) = (?x ⊓ ! ?y = bot)›*))\n hence \"x ⊓ !(x ⊓ y) ≤ !y\"\n using \"1\" (*‹test (! (x ⊓ y))›*) by (simp add: assms( (*‹test x› ‹test (x ⊓ y)›*) 1,3) neg_involutive (*‹test ?x ⟹ ! (! ?x) = ?x›*) test_shunting (*‹⟦test ?x; test ?y⟧ ⟹ (?x ⊓ ?y ≤ ?z) = (?x ≤ ! ?y ⊔ ?z)›*))\n hence \"!(x ⊓ y) ⊓ x ≤ !y\"\n using \"1\" (*‹test (! ((x::'a) ⊓ (y::'a)))›*) by (metis assms( (*‹test x›*) 1) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\n thus \"!(x ⊓ y) ≤ !x ⊔ !y\"\n using \"1\" (*‹test (! (x ⊓ y))›*) assms(1) (*‹test x›*) test_shunting (*‹⟦test ?x; test ?y⟧ ⟹ (?x ⊓ ?y ≤ ?z) = (?x ≤ ! ?y ⊔ ?z)›*) by blast\n have 2: \"!x ≤ !(x ⊓ y)\"\n by (simp add: assms (*‹test (x::'a)› ‹test (y::'a)› ‹test ((x::'a) ⊓ (y::'a))›*) neg_antitone (*‹⟦test (?x::'a); test (?y::'a); ?x ≤ ?y⟧ ⟹ ! ?y ≤ ! ?x›*) test_inf_right_below (*‹test (?x::'a) ⟹ (?y::'a) ⊓ ?x ≤ ?y›*))\n have \"!y ≤ !(x ⊓ y)\"\n by (simp add: assms (*‹test x› ‹test y› ‹test (x ⊓ y)›*) neg_antitone (*‹⟦test ?x; test ?y; ?x ≤ ?y⟧ ⟹ ! ?y ≤ ! ?x›*) test_inf_left_below (*‹test ?x ⟹ ?x ⊓ ?y ≤ ?y›*))\n thus \"!x ⊔ !y ≤ !(x ⊓ y)\"\n using \"2\" (*‹! (x::'a) ≤ ! (x ⊓ (y::'a))›*) by (metis il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\nqed\n\nlemma de_morgan_2:\n assumes \"test x\"\n and \"test y\"\n and \"test (x ⊔ y)\"\n shows \"!(x ⊔ y) = !x ⊓ !y\"\nproof (rule order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹! ((x::'a) ⊔ (y::'a)) ≤ ! x ⊓ ! y›\n 2. ‹! (x::'a) ⊓ ! (y::'a) ≤ ! (x ⊔ y)›*)\n have 1: \"!(x ⊔ y) ≤ !x\"\n by (metis assms (*‹test (x::'a)› ‹test (y::'a)› ‹test ((x::'a) ⊔ (y::'a))›*) il_inf_left_unit (*‹top ⊓ (?x::'a) = ?x›*) il_sub_inf_left_isotone (*‹(?x::'a) ≤ (?y::'a) ⟹ ?x ⊓ (?z::'a) ≤ ?y ⊓ ?z›*) neg_antitone (*‹⟦test (?x::'a); test (?y::'a); ?x ≤ ?y⟧ ⟹ ! ?y ≤ ! ?x›*) test_absorb_3 (*‹⟦test (?x::'a); test (?y::'a)⟧ ⟹ ?x ⊓ (?x ⊔ ?y) = ?x›*) test_sub_identity (*‹test (?x::'a) ⟹ ?x ≤ top›*))\n have \"!(x ⊔ y) ≤ !y\"\n by (metis assms (*‹test x› ‹test y› ‹test (x ⊔ y)›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_sub_inf_left_isotone (*‹?x ≤ ?y ⟹ ?x ⊓ ?z ≤ ?y ⊓ ?z›*) neg_antitone (*‹⟦test ?x; test ?y; ?x ≤ ?y⟧ ⟹ ! ?y ≤ ! ?x›*) test_absorb_3 (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ (?x ⊔ ?y) = ?x›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*))\n thus \"!(x ⊔ y) ≤ !x ⊓ !y\"\n using \"1\" (*‹! (x ⊔ y) ≤ ! x›*) by (simp add: assms (*‹test (x::'a)› ‹test (y::'a)› ‹test ((x::'a) ⊔ (y::'a))›*) neg_test (*‹test (?x::'a) ⟹ test (! ?x)›*) test_inf (*‹⟦test (?x::'a); test (?y::'a); test (?z::'a)⟧ ⟹ (?z ≤ ?x ⊓ ?y) = (?z ≤ ?x ∧ ?z ≤ ?y)›*))\n have \"top ≤ x ⊔ y ⊔ !(x ⊔ y)\"\n by (simp add: assms( (*‹test ((x::'a) ⊔ (y::'a))›*) 3) neg_char (*‹test (?x::'a) ⟹ are_complementary ?x (! ?x)›*))\n hence \"top ⊓ !x ≤ y ⊔ !(x ⊔ y)\"\n by (smt assms( (*‹test x›*) 1) assms( (*‹test (x ⊔ y)›*) 3) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) il_unit_bot (*‹bot ⊔ ?x = ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*))\n thus \"!x ⊓ !y ≤ !(x ⊔ y)\"\n by (simp add: assms( (*‹test (x::'a)›*) 1) assms( (*‹test (y::'a)›*) 2) neg_involutive (*‹test (?x::'a) ⟹ ! (! ?x) = ?x›*) neg_test (*‹test (?x::'a) ⟹ test (! ?x)›*) test_shunting (*‹⟦test (?x::'a); test (?y::'a)⟧ ⟹ (?x ⊓ ?y ≤ (?z::'a)) = (?x ≤ ! ?y ⊔ ?z)›*))\nqed\n\nlemma test_inf_closed_sup_complement:\n assumes \"test x\"\n and \"test y\"\n and \"∀u v . test u ∧ test v ⟶ test (u ⊓ v)\"\n shows \"!x ⊓ !y ⊓ (x ⊔ y) = bot\"\nproof (-)\n (*goal: ‹! (x::'a) ⊓ ! (y::'a) ⊓ (x ⊔ y) = bot›*)\n have 1: \"!(!x ⊓ !y) = x ⊔ y\"\n by (simp add: assms (*‹test x› ‹test y› ‹∀u v. test u ∧ test v ⟶ test (u ⊓ v)›*) de_morgan_1 (*‹⟦test ?x; test ?y; test (?x ⊓ ?y)⟧ ⟹ ! (?x ⊓ ?y) = ! ?x ⊔ ! ?y›*) neg_involutive (*‹test ?x ⟹ ! (! ?x) = ?x›*) neg_test (*‹test ?x ⟹ test (! ?x)›*))\n have \"test (!(!x ⊓ !y))\"\n by (metis assms (*‹test x› ‹test y› ‹∀u v. test u ∧ test v ⟶ test (u ⊓ v)›*) neg_test (*‹test ?x ⟹ test (! ?x)›*))\n thus \"?thesis\"\n (*goal: ‹! x ⊓ ! y ⊓ (x ⊔ y) = bot›*)\n using \"1\" (*‹! (! x ⊓ ! y) = x ⊔ y›*) by (metis assms( (*‹test (x::'a)› ‹test (y::'a)›*) 1,2) de_morgan_2 (*‹⟦test (?x::'a); test (?y::'a); test (?x ⊔ ?y)⟧ ⟹ ! (?x ⊔ ?y) = ! ?x ⊓ ! ?y›*) neg_char (*‹test (?x::'a) ⟹ are_complementary ?x (! ?x)›*))\nqed\n\nlemma test_sup_complement_sup_closed:\n assumes \"test x\"\n and \"test y\"\n and \"∀u v . test u ∧ test v ⟶ !u ⊓ !v ⊓ (u ⊔ v) = bot\"\n shows \"test (x ⊔ y)\"\n by (meson assms (*‹test x› ‹test y› ‹∀u v. test u ∧ test v ⟶ ! u ⊓ ! v ⊓ (u ⊔ v) = bot›*) test_sup_neg_1 (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊔ ?y ⊔ ! ?x ⊓ ! ?y = top›*) test_sup_neg_2 (*‹⟦test ?x; test ?y⟧ ⟹ (?x ⊔ ?y) ⊓ (! ?x ⊓ ! ?y) = bot›*))\n\nlemma test_inf_closed_sup_closed:\n assumes \"test x\"\n and \"test y\"\n and \"∀u v . test u ∧ test v ⟶ test (u ⊓ v)\"\n shows \"test (x ⊔ y)\"\n using assms (*‹test x› ‹test y› ‹∀u v. test u ∧ test v ⟶ test (u ⊓ v)›*) test_inf_closed_sup_complement (*‹⟦test ?x; test ?y; ∀u v. test u ∧ test v ⟶ test (u ⊓ v)⟧ ⟹ ! ?x ⊓ ! ?y ⊓ (?x ⊔ ?y) = bot›*) test_sup_complement_sup_closed (*‹⟦test (?x::'a); test (?y::'a); ∀(u::'a) v::'a. test u ∧ test v ⟶ ! u ⊓ ! v ⊓ (u ⊔ v) = bot⟧ ⟹ test (?x ⊔ ?y)›*) by simp\n\nend\n\nsubsection ‹Prepredomain Semirings›\n\nclass dom =\n fixes d :: \"'a ⇒ 'a\"\n\nclass ppd_semiring = il_semiring + dom +\n assumes d_closed: \"test (d x)\"\n assumes d1: \"x ≤ d x ⊓ x\"\nbegin\n\nlemma d_sub_identity:\n \"d x ≤ top\"\n using d_closed (*‹test (d (?x::'a))›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*) by blast\n\nlemma d1_eq:\n \"x = d x ⊓ x\"\nproof (-)\n (*goal: ‹x = d x ⊓ x›*)\n have \"x = (d x ⊔ top) ⊓ x\"\n using d_sub_identity (*‹d ?x ≤ top›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by auto\n thus \"?thesis\"\n (*goal: ‹x = d x ⊓ x›*)\n using d1 (*‹(?x::'a) ≤ d ?x ⊓ ?x›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_right_dist_sup (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by force\nqed\n\nlemma d_increasing_sub_identity:\n \"x ≤ top ⟹ x ≤ d x\"\n by (metis d1_eq (*‹?x = d ?x ⊓ ?x›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*))\n\nlemma d_top:\n \"d top = top\"\n by (simp add: d_increasing_sub_identity (*‹?x ≤ top ⟹ ?x ≤ d ?x›*) d_sub_identity (*‹d ?x ≤ top›*) dual_order.antisym (*‹⟦?b ≤ ?a; ?a ≤ ?b⟧ ⟹ ?a = ?b›*))\n\nlemma d_bot_only:\n \"d x = bot ⟹ x = bot\"\n by (metis d1_eq (*‹?x = d ?x ⊓ ?x›*) il_sub_inf_left_zero (*‹bot ⊓ ?x = bot›*))\n\nlemma d_strict: \"d bot ≤ bot\" nitpick [expect=genuine] oops\nlemma d_isotone_var: \"d x ≤ d (x ⊔ y)\" nitpick [expect=genuine] oops\nlemma d_fully_strict: \"d x = bot ⟷ x = bot\" nitpick [expect=genuine] oops\nlemma test_d_fixpoint: \"test x ⟹ d x = x\" nitpick [expect=genuine] oops\n\nend\n\nsubsection ‹Predomain Semirings›\n\nclass pd_semiring = ppd_semiring +\n assumes d2: \"test p ⟹ d (p ⊓ x) ≤ p\"\nbegin\n\nlemma d_strict:\n \"d bot ≤ bot\"\n using bot_test (*‹test bot›*) d2 (*‹test (?p::'a::type) ⟹ d (?p ⊓ (?x::'a::type)) ≤ ?p›*) by fastforce\n\nlemma d_strict_eq:\n \"d bot = bot\"\n using d_strict (*‹d bot ≤ bot›*) il_bot_unit (*‹?x ⊔ bot = ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by auto\n\nlemma test_d_fixpoint:\n \"test x ⟹ d x = x\"\n by (metis order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*) d1_eq (*‹?x = d ?x ⊓ ?x›*) d2 (*‹test ?p ⟹ d (?p ⊓ ?x) ≤ ?p›*) test_inf_idempotent (*‹test ?x ⟹ ?x ⊓ ?x = ?x›*) test_inf_right_below (*‹test ?x ⟹ ?y ⊓ ?x ≤ ?y›*))\n\nlemma d_surjective:\n \"test x ⟹ ∃y . d y = x\"\n using test_d_fixpoint (*‹test ?x ⟹ d ?x = ?x›*) by blast\n\nlemma test_d_fixpoint_iff:\n \"test x ⟷ d x = x\"\n by (metis d_closed (*‹test (d (?x::'a::type))›*) test_d_fixpoint (*‹test (?x::'a::type) ⟹ d ?x = ?x›*))\n\nlemma d_surjective_iff:\n \"test x ⟷ (∃y . d y = x)\"\n using d_surjective (*‹test ?x ⟹ ∃y. d y = ?x›*) d_closed (*‹test (d (?x::'a))›*) by blast\n\nlemma tests_d_range:\n \"tests = range d\"\n using tests_def (*‹tests = {x. test x}›*) image_def (*‹?f ` ?A = {y. ∃x∈?A. y = ?f x}›*) d_surjective_iff (*‹test (?x::'a::type) = (∃y::'a::type. d y = ?x)›*) by auto\n\nlemma llp:\n assumes \"test y\"\n shows \"d x ≤ y ⟷ x ≤ y ⊓ x\"\n by (metis assms (*‹test y›*) d1_eq (*‹?x = d ?x ⊓ ?x›*) d2 (*‹test ?p ⟹ d (?p ⊓ ?x) ≤ ?p›*) order.eq_iff (*‹(?a = ?b) = (?a ≤ ?b ∧ ?b ≤ ?a)›*) il_sub_inf_left_isotone (*‹?x ≤ ?y ⟹ ?x ⊓ ?z ≤ ?y ⊓ ?z›*) test_inf_left_below (*‹test ?x ⟹ ?x ⊓ ?y ≤ ?y›*))\n\nlemma gla:\n assumes \"test y\"\n shows \"y ≤ !(d x) ⟷ y ⊓ x ≤ bot\"\nproof (-)\n (*goal: ‹(y ≤ ! (d x)) = (y ⊓ x ≤ bot)›*)\n obtain ad where 1: \"∀x. are_complementary (d x) (ad x)\"\n (*goal: ‹(⋀ad. ∀x. are_complementary (d x) (ad x) ⟹ thesis) ⟹ thesis›*)\n using d_closed (*‹test (d ?x)›*) by moura\n hence 2: \"∀x y. d (d y ⊓ x) ≤ d y\"\n using d2 (*‹test (?p::'a::type) ⟹ d (?p ⊓ (?x::'a::type)) ≤ ?p›*) by blast\n have 3: \"∀x. ad x ⊓ x = bot\"\n using \"1\" (*‹∀x. are_complementary (d x) (ad x)›*) by (metis d1_eq (*‹(?x::'a) = d ?x ⊓ ?x›*) il_inf_associative (*‹(?x::'a) ⊓ ((?y::'a) ⊓ (?z::'a)) = ?x ⊓ ?y ⊓ ?z›*) il_sub_inf_left_zero (*‹bot ⊓ (?x::'a) = bot›*))\n have 4: \"∀x y. d y ⊓ x ⊔ ad y ⊓ x = top ⊓ x\"\n using \"1\" (*‹∀x. are_complementary (d x) (ad x)›*) by (metis il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n have 5: \"∀x y z. z ⊓ y ≤ x ⊓ y ∨ (z ⊔ x) ⊓ y ≠ x ⊓ y\"\n by (simp add: il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n have 6: \"∀x. !(d x) = ad x\"\n using \"1\" (*‹∀x::'a. are_complementary (d x) ((ad::'a ⇒ 'a) x)›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) neg_unique (*‹⟦are_complementary ?x ?y; are_complementary ?x ?z⟧ ⟹ ?y = ?z›*) by blast\n have 7: \"∀x. top ⊓ x = x\"\n by auto\n hence \"∀x. y ⊓ x ⊔ !y ⊓ x = x\"\n by (metis assms (*‹test y›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n thus \"?thesis\"\n (*goal: ‹(y ≤ ! (d x)) = (y ⊓ x ≤ bot)›*)\n using \"1\" (*‹∀x. are_complementary (d x) (ad x)›*) \"2\" (*‹∀x y. d (d y ⊓ x) ≤ d y›*) \"3\" (*‹∀x. ad x ⊓ x = bot›*) \"4\" (*‹∀x y. d y ⊓ x ⊔ ad y ⊓ x = top ⊓ x›*) \"5\" (*‹∀x y z. z ⊓ y ≤ x ⊓ y ∨ (z ⊔ x) ⊓ y ≠ x ⊓ y›*) \"6\" (*‹∀x. ! (d x) = ad x›*) \"7\" (*‹∀x. top ⊓ x = x›*) by (metis assms (*‹test y›*) d1_eq (*‹?x = d ?x ⊓ ?x›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) test_d_fixpoint (*‹test ?x ⟹ d ?x = ?x›*))\nqed\n\nlemma gla_var:\n \"test y ⟹ y ⊓ d x ≤ bot ⟷ y ⊓ x ≤ bot\"\n using gla (*‹test ?y ⟹ (?y ≤ ! (d ?x)) = (?y ⊓ ?x ≤ bot)›*) d_closed (*‹test (d (?x::'a))›*) il_bot_unit (*‹?x ⊔ bot = ?x›*) test_shunting (*‹⟦test (?x::'a); test (?y::'a)⟧ ⟹ (?x ⊓ ?y ≤ (?z::'a)) = (?x ≤ ! ?y ⊔ ?z)›*) by auto\n\nlemma llp_var:\n assumes \"test y\"\n shows \"y ≤ !(d x) ⟷ x ≤ !y ⊓ x\"\n apply (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹y ≤ ! (d x) ⟹ x ≤ ! y ⊓ x›\n 2. ‹x ≤ ! y ⊓ x ⟹ y ≤ ! (d x)›\n discuss goal 1*)\n apply (metis (no_types, opaque_lifting) assms (*‹test (y::'a)›*) gla (*‹test (?y::'a) ⟹ (?y ≤ ! (d (?x::'a))) = (?y ⊓ ?x ≤ bot)›*) Least_equality (*‹⟦(?P::'a ⇒ bool) (?x::'a); ⋀y::'a. ?P y ⟹ ?x ≤ y⟧ ⟹ Least ?P = ?x›*) il_inf_left_unit (*‹top ⊓ (?x::'a) = ?x›*) il_inf_right_dist_sup (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*) il_unit_bot (*‹bot ⊔ (?x::'a) = ?x›*) order.refl (*‹(?a::'a) ≤ ?a›*) neg_char (*‹test (?x::'a) ⟹ are_complementary ?x (! ?x)›*))\n (*discuss goal 2*)\n apply (metis assms (*‹test y›*) gla (*‹test ?y ⟹ (?y ≤ ! (d ?x)) = (?y ⊓ ?x ≤ bot)›*) gla_var (*‹test ?y ⟹ (?y ⊓ d ?x ≤ bot) = (?y ⊓ ?x ≤ bot)›*) llp (*‹test ?y ⟹ (d ?x ≤ ?y) = (?x ≤ ?y ⊓ ?x)›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n (*proven 2 subgoals*) .\n\nlemma d_idempotent:\n \"d (d x) = d x\"\n using d_closed (*‹test (d ?x)›*) test_d_fixpoint_iff (*‹test ?x = (d ?x = ?x)›*) by auto\n\nlemma d_neg:\n \"test x ⟹ d (!x) = !x\"\n using il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) test_d_fixpoint_iff (*‹test ?x = (d ?x = ?x)›*) by fastforce\n\nlemma d_fully_strict:\n \"d x = bot ⟷ x = bot\"\n using d_strict_eq (*‹d bot = bot›*) d_bot_only (*‹d (?x::'a) = bot ⟹ ?x = bot›*) by blast\n\nlemma d_ad_comp:\n \"!(d x) ⊓ x = bot\"\nproof (-)\n (*goal: ‹! (d x) ⊓ x = bot›*)\n have \"∀x. !(d x) ⊓ d x = bot\"\n by (simp add: d_closed (*‹test (d ?x)›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n thus \"?thesis\"\n (*goal: ‹! (d (x::'a)) ⊓ x = bot›*)\n by (metis d1_eq (*‹?x = d ?x ⊓ ?x›*) il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) il_sub_inf_left_zero (*‹bot ⊓ ?x = bot›*))\nqed\n\nlemma d_isotone:\n assumes \"x ≤ y\"\n shows \"d x ≤ d y\"\nproof (-)\n (*goal: ‹d x ≤ d y›*)\n obtain ad where 1: \"∀x. are_complementary (d x) (ad x)\"\n (*goal: ‹(⋀ad. ∀x. are_complementary (d x) (ad x) ⟹ thesis) ⟹ thesis›*)\n using d_closed (*‹test (d ?x)›*) by moura\n hence \"ad y ⊓ x ≤ bot\"\n by (metis assms (*‹x ≤ y›*) d1_eq (*‹?x = d ?x ⊓ ?x›*) il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) il_sub_inf_left_zero (*‹bot ⊓ ?x = bot›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*))\n thus \"?thesis\"\n (*goal: ‹d x ≤ d y›*)\n using \"1\" (*‹∀x. are_complementary (d x) (ad x)›*) by (metis d2 (*‹test ?p ⟹ d (?p ⊓ ?x) ≤ ?p›*) il_bot_unit (*‹?x ⊔ bot = ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\nqed\n\nlemma d_isotone_var:\n \"d x ≤ d (x ⊔ y)\"\n using d_isotone (*‹?x ≤ ?y ⟹ d ?x ≤ d ?y›*) il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by auto\n\nlemma d3_conv:\n \"d (x ⊓ y) ≤ d (x ⊓ d y)\"\n by (metis (mono_tags, opaque_lifting) d1_eq (*‹?x = d ?x ⊓ ?x›*) d2 (*‹test ?p ⟹ d (?p ⊓ ?x) ≤ ?p›*) d_closed (*‹test (d ?x)›*) il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*))\n\nlemma d_test_inf_idempotent:\n \"d x ⊓ d x = d x\"\n by (metis d_idempotent (*‹d (d ?x) = d ?x›*) d1_eq (*‹?x = d ?x ⊓ ?x›*))\n\nlemma d_test_inf_closed:\n assumes \"test x\"\n and \"test y\"\n shows \"d (x ⊓ y) = x ⊓ y\"\nproof (rule order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹d (x ⊓ y) ≤ x ⊓ y›\n 2. ‹x ⊓ y ≤ d (x ⊓ y)›*)\n have \"d (x ⊓ y) = d (x ⊓ y) ⊓ d (x ⊓ y)\"\n by (simp add: d_test_inf_idempotent (*‹d ?x ⊓ d ?x = d ?x›*))\n also (*calculation: ‹d (x ⊓ y) = d (x ⊓ y) ⊓ d (x ⊓ y)›*) have \"... ≤ x ⊓ d (x ⊓ y)\"\n by (simp add: assms( (*‹test x›*) 1) d2 (*‹test ?p ⟹ d (?p ⊓ ?x) ≤ ?p›*) il_sub_inf_left_isotone (*‹?x ≤ ?y ⟹ ?x ⊓ ?z ≤ ?y ⊓ ?z›*))\n also (*calculation: ‹d (x ⊓ y) ≤ x ⊓ d (x ⊓ y)›*) have \"... ≤ x ⊓ y\"\n by (metis assms (*‹test x› ‹test y›*) d_isotone (*‹?x ≤ ?y ⟹ d ?x ≤ d ?y›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) test_inf_left_below (*‹test ?x ⟹ ?x ⊓ ?y ≤ ?y›*) test_d_fixpoint (*‹test ?x ⟹ d ?x = ?x›*))\n finally (*calculation: ‹d (x ⊓ y) ≤ x ⊓ y›*) show \"d (x ⊓ y) ≤ x ⊓ y\" .\n show \"x ⊓ y ≤ d (x ⊓ y)\"\n using assms (*‹test x› ‹test (y::'a::type)›*) d_increasing_sub_identity (*‹?x ≤ top ⟹ ?x ≤ d ?x›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) test_inf_left_below (*‹test ?x ⟹ ?x ⊓ ?y ≤ ?y›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*) by blast\nqed\n\nlemma test_inf_closed:\n \"test x ⟹ test y ⟹ test (x ⊓ y)\"\n using d_test_inf_closed (*‹⟦test ?x; test ?y⟧ ⟹ d (?x ⊓ ?y) = ?x ⊓ ?y›*) test_d_fixpoint_iff (*‹test ?x = (d ?x = ?x)›*) by simp\n\nlemma test_sup_closed:\n \"test x ⟹ test y ⟹ test (x ⊔ y)\"\n using test_inf_closed (*‹⟦test (?x::'a); test (?y::'a)⟧ ⟹ test (?x ⊓ ?y)›*) test_inf_closed_sup_closed (*‹⟦test ?x; test ?y; ∀u v. test u ∧ test v ⟶ test (u ⊓ v)⟧ ⟹ test (?x ⊔ ?y)›*) by simp\n\nlemma d_export:\n assumes \"test x\"\n shows \"d (x ⊓ y) = x ⊓ d y\"\nproof (rule order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹d (x ⊓ y) ≤ x ⊓ d y›\n 2. ‹x ⊓ d y ≤ d (x ⊓ y)›*)\n have 1: \"d (x ⊓ y) ≤ x\"\n by (simp add: assms (*‹test x›*) d2 (*‹test ?p ⟹ d (?p ⊓ ?x) ≤ ?p›*))\n have \"d (x ⊓ y) ≤ d y\"\n by (metis assms (*‹test (x::'a::type)›*) d_isotone_var (*‹d (?x::'a::type) ≤ d (?x ⊔ (?y::'a::type))›*) il_inf_left_unit (*‹top ⊓ (?x::'a::type) = ?x›*) il_inf_right_dist_sup (*‹((?x::'a::type) ⊔ (?y::'a::type)) ⊓ (?z::'a::type) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n thus \"d (x ⊓ y) ≤ x ⊓ d y\"\n using \"1\" (*‹d (x ⊓ y) ≤ x›*) by (metis assms (*‹test x›*) d_idempotent (*‹d (d ?x) = d ?x›*) llp (*‹test ?y ⟹ (d ?x ≤ ?y) = (?x ≤ ?y ⊓ ?x)›*) dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*))\n have \"y = (!x ⊓ y) ⊔ (x ⊓ y)\"\n by (metis assms (*‹test x›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n also (*calculation: ‹y = ! x ⊓ y ⊔ x ⊓ y›*) have \"... = (!x ⊓ y) ⊔ (d (x ⊓ y) ⊓ x ⊓ y)\"\n by (metis d1_eq (*‹?x = d ?x ⊓ ?x›*) il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*))\n also (*calculation: ‹(y::'a) = ! (x::'a) ⊓ y ⊔ d (x ⊓ y) ⊓ x ⊓ y›*) have \"... = (!x ⊓ y) ⊔ (d (x ⊓ y) ⊓ y)\"\n using \"1\" (*‹d (x ⊓ y) ≤ x›*) by (smt calculation (*‹y = ! x ⊓ y ⊔ d (x ⊓ y) ⊓ x ⊓ y›*) d1_eq (*‹?x = d ?x ⊓ ?x›*) il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone_var (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*))\n also (*calculation: ‹y = ! x ⊓ y ⊔ d (x ⊓ y) ⊓ y›*) have \"... = (!x ⊔ d (x ⊓ y)) ⊓ y\"\n by (simp add: il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n finally (*calculation: ‹y = (! x ⊔ d (x ⊓ y)) ⊓ y›*) have \"y ≤ (!x ⊔ d (x ⊓ y)) ⊓ y\"\n by simp\n hence \"d y ≤ !x ⊔ d (x ⊓ y)\"\n using assms (*‹test x›*) llp (*‹test ?y ⟹ (d ?x ≤ ?y) = (?x ≤ ?y ⊓ ?x)›*) test_sup_closed (*‹⟦test ?x; test ?y⟧ ⟹ test (?x ⊔ ?y)›*) neg_test (*‹test (?x::'a) ⟹ test (! ?x)›*) d_closed (*‹test (d ?x)›*) by simp\n hence \"d y ⊓ x ≤ d (x ⊓ y)\"\n by (simp add: assms (*‹test x›*) d_closed (*‹test (d ?x)›*) test_shunting (*‹⟦test ?x; test ?y⟧ ⟹ (?x ⊓ ?y ≤ ?z) = (?x ≤ ! ?y ⊔ ?z)›*))\n thus \"x ⊓ d y ≤ d (x ⊓ y)\"\n by (metis assms (*‹test x›*) d_closed (*‹test (d ?x)›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*))\nqed\n\nlemma test_inf_left_dist_sup:\n assumes \"test x\"\n and \"test y\"\n and \"test z\"\n shows \"x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)\"\nproof (-)\n (*goal: ‹x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z›*)\n have \"x ⊓ (y ⊔ z) = (y ⊔ z) ⊓ x\"\n using assms (*‹test x› ‹test y› ‹test z›*) test_sup_closed (*‹⟦test ?x; test ?y⟧ ⟹ test (?x ⊔ ?y)›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*) by smt\n also (*calculation: ‹x ⊓ (y ⊔ z) = (y ⊔ z) ⊓ x›*) have \"... = (y ⊓ x) ⊔ (z ⊓ x)\"\n using il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) by simp\n also (*calculation: ‹x ⊓ (y ⊔ z) = y ⊓ x ⊔ z ⊓ x›*) have \"... = (x ⊓ y) ⊔ (x ⊓ z)\"\n using assms (*‹test x› ‹test (y::'a::type)› ‹test z›*) test_sup_closed (*‹⟦test ?x; test ?y⟧ ⟹ test (?x ⊔ ?y)›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*) by smt\n finally (*calculation: ‹x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z›*) show \"?thesis\"\n (*goal: ‹x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z›*) .\nqed\n\nlemma \"!x ⊔ !y = !(!(!x ⊔ !y))\" nitpick [expect=genuine] oops\nlemma \"d x = !(!x)\" nitpick [expect=genuine] oops\n\nsublocale subset_boolean_algebra where uminus = \"λ x . !(d x)\"\nproof (standard)\n (*goals:\n 1. ‹⋀(x::'a) (y::'a) z::'a. ! (d x) ⊔ (! (d y) ⊔ ! (d z)) = ! (d x) ⊔ ! (d y) ⊔ ! (d z)›\n 2. ‹⋀(x::'a) y::'a. ! (d x) ⊔ ! (d y) = ! (d y) ⊔ ! (d x)›\n 3. ‹⋀(x::'a) y::'a. ! (d x) = ! (d (! (d (! (d x))) ⊔ ! (d y))) ⊔ ! (d (! (d (! (d x))) ⊔ ! (d (! (d y)))))›\n 4. ‹⋀(x::'a) y::'a. ! (d x) ⊔ ! (d y) = ! (d (! (d (! (d x) ⊔ ! (d y)))))›*)\n show \"⋀x y z. !(d x) ⊔ (!(d y) ⊔ !(d z)) = !(d x) ⊔ !(d y) ⊔ !(d z)\"\n using il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) by blast\n show \"⋀x y. !(d x) ⊔ !(d y) = !(d y) ⊔ !(d x)\"\n by (simp add: il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*))\n show \"⋀x y. !(d x) ⊔ !(d y) = !(d (!(d (!(d x) ⊔ !(d y)))))\"\n proof (-)\n (*goal: ‹⋀x y. ! (d x) ⊔ ! (d y) = ! (d (! (d (! (d x) ⊔ ! (d y)))))›*)\n fix x and y\n have \"test (!(d x)) ∧ test (!(d y))\"\n by (simp add: d_closed (*‹test (d ?x)›*) neg_test (*‹test ?x ⟹ test (! ?x)›*))\n hence \"test (!(d x) ⊔ !(d y))\"\n by (simp add: test_sup_closed (*‹⟦test ?x; test ?y⟧ ⟹ test (?x ⊔ ?y)›*))\n thus \"!(d x) ⊔ !(d y) = !(d (!(d (!(d x) ⊔ !(d y)))))\"\n by (simp add: d_neg (*‹test ?x ⟹ d (! ?x) = ! ?x›*) neg_involutive (*‹test ?x ⟹ ! (! ?x) = ?x›*) test_d_fixpoint (*‹test ?x ⟹ d ?x = ?x›*))\n qed\n show \"⋀x y. !(d x) = !(d (!(d (!(d x))) ⊔ !(d y))) ⊔ !(d (!(d (!(d x))) ⊔ !(d (!(d y)))))\"\n proof (-)\n (*goal: ‹⋀(x::'a) y::'a. ! (d x) = ! (d (! (d (! (d x))) ⊔ ! (d y))) ⊔ ! (d (! (d (! (d x))) ⊔ ! (d (! (d y)))))›*)\n fix x and y\n have \"!(d (!(d (!(d x))) ⊔ !(d y))) ⊔ !(d (!(d (!(d x))) ⊔ !(d (!(d y))))) = !(d x ⊔ !(d y)) ⊔ !(d x ⊔ d y)\"\n using d_closed (*‹test (d ?x)›*) neg_test (*‹test ?x ⟹ test (! ?x)›*) test_sup_closed (*‹⟦test ?x; test ?y⟧ ⟹ test (?x ⊔ ?y)›*) neg_involutive (*‹test ?x ⟹ ! (! ?x) = ?x›*) test_d_fixpoint (*‹test ?x ⟹ d ?x = ?x›*) by auto\n also (*calculation: ‹! (d (! (d (! (d x))) ⊔ ! (d y))) ⊔ ! (d (! (d (! (d x))) ⊔ ! (d (! (d y))))) = ! (d x ⊔ ! (d y)) ⊔ ! (d x ⊔ d y)›*) have \"... = (!(d x) ⊓ d y) ⊔ (!(d x) ⊓ !(d y))\"\n using d_closed (*‹test (d ?x)›*) neg_test (*‹test ?x ⟹ test (! ?x)›*) test_sup_closed (*‹⟦test ?x; test ?y⟧ ⟹ test (?x ⊔ ?y)›*) neg_involutive (*‹test ?x ⟹ ! (! ?x) = ?x›*) de_morgan_2 (*‹⟦test ?x; test ?y; test (?x ⊔ ?y)⟧ ⟹ ! (?x ⊔ ?y) = ! ?x ⊓ ! ?y›*) by auto\n also (*calculation: ‹! (d (! (d (! (d x))) ⊔ ! (d y))) ⊔ ! (d (! (d (! (d x))) ⊔ ! (d (! (d y))))) = ! (d x) ⊓ d y ⊔ ! (d x) ⊓ ! (d y)›*) have \"... = !(d x) ⊓ (d y ⊔ !(d y))\"\n using d_closed (*‹test (d ?x)›*) neg_test (*‹test ?x ⟹ test (! ?x)›*) test_inf_left_dist_sup (*‹⟦test ?x; test ?y; test ?z⟧ ⟹ ?x ⊓ (?y ⊔ ?z) = ?x ⊓ ?y ⊔ ?x ⊓ ?z›*) by auto\n also (*calculation: ‹! (d (! (d (! (d x))) ⊔ ! (d y))) ⊔ ! (d (! (d (! (d x))) ⊔ ! (d (! (d y))))) = ! (d x) ⊓ (d y ⊔ ! (d y))›*) have \"... = !(d x) ⊓ top\"\n by (simp add: neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*) d_closed (*‹test (d ?x)›*))\n finally (*calculation: ‹! (d (! (d (! (d x))) ⊔ ! (d y))) ⊔ ! (d (! (d (! (d x))) ⊔ ! (d (! (d y))))) = ! (d x) ⊓ top›*) show \"!(d x) = !(d (!(d (!(d x))) ⊔ !(d y))) ⊔ !(d (!(d (!(d x))) ⊔ !(d (!(d y)))))\"\n by simp\n qed\nqed\n\nlemma d_dist_sup:\n \"d (x ⊔ y) = d x ⊔ d y\"\nproof (rule order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹d (x ⊔ y) ≤ d x ⊔ d y›\n 2. ‹d x ⊔ d y ≤ d (x ⊔ y)›*)\n have \"x ≤ d x ⊓ x\"\n by (simp add: d1 (*‹(?x::'a::type) ≤ d ?x ⊓ ?x›*))\n also (*calculation: ‹x ≤ d x ⊓ x›*) have \"... ≤ (d x ⊔ d y) ⊓ (x ⊔ y)\"\n using il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) by auto\n finally (*calculation: ‹x ≤ (d x ⊔ d y) ⊓ (x ⊔ y)›*) have 1: \"x ≤ (d x ⊔ d y) ⊓ (x ⊔ y)\" .\n have \"y ≤ d y ⊓ y\"\n by (simp add: d1 (*‹?x ≤ d ?x ⊓ ?x›*))\n also (*calculation: ‹y ≤ d y ⊓ y›*) have \"... ≤ (d y ⊔ d x) ⊓ (y ⊔ x)\"\n using il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) il_idempotent (*‹?x ⊔ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) by simp\n finally (*calculation: ‹y ≤ (d y ⊔ d x) ⊓ (y ⊔ x)›*) have \"y ≤ (d x ⊔ d y) ⊓ (x ⊔ y)\"\n using il_commutative (*‹(?x::'a::type) ⊔ (?y::'a::type) = ?y ⊔ ?x›*) by auto\n hence \"x ⊔ y ≤ (d x ⊔ d y) ⊓ (x ⊔ y)\"\n using \"1\" (*‹(x::'a) ≤ (d x ⊔ d (y::'a)) ⊓ (x ⊔ y)›*) by (metis il_associative (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?y ⊔ ?z›*) il_less_eq (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*))\n thus \"d (x ⊔ y) ≤ d x ⊔ d y\"\n using llp (*‹test (?y::'a::type) ⟹ (d (?x::'a::type) ≤ ?y) = (?x ≤ ?y ⊓ ?x)›*) test_sup_closed (*‹⟦test ?x; test ?y⟧ ⟹ test (?x ⊔ ?y)›*) neg_test (*‹test ?x ⟹ test (! ?x)›*) d_closed (*‹test (d (?x::'a::type))›*) by simp\n show \"d x ⊔ d y ≤ d (x ⊔ y)\"\n using d_isotone_var (*‹d ?x ≤ d (?x ⊔ ?y)›*) il_associative (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?y ⊔ ?z›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_less_eq (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*) by fastforce\nqed\n\nend\n\nclass pd_semiring_extended = pd_semiring + uminus +\n assumes uminus_def: \"-x = !(d x)\"\nbegin\n\nsubclass subset_boolean_algebra\n by (metis subset_boolean_algebra_axioms uminus_def ext)\n\nend\n\nsubsection ‹Domain Semirings›\n\nclass d_semiring = pd_semiring +\n assumes d3: \"d (x ⊓ d y) ≤ d (x ⊓ y)\"\nbegin\n\nlemma d3_eq: \"d (x ⊓ d y) = d (x ⊓ y)\"\n by (simp add: order.antisym (*‹⟦(?a::'a::type) ≤ (?b::'a::type); ?b ≤ ?a⟧ ⟹ ?a = ?b›*) d3 (*‹d ((?x::'a::type) ⊓ d (?y::'a::type)) ≤ d (?x ⊓ ?y)›*) d3_conv (*‹d ((?x::'a::type) ⊓ (?y::'a::type)) ≤ d (?x ⊓ d ?y)›*))\n\nend\n\ntext ‹\nAxioms (d1), (d2) and (d3) are independent in IL-semirings.\n›\n\ncontext il_semiring\nbegin\n\ncontext\n fixes d :: \"'a ⇒ 'a\"\n assumes d_closed: \"test (d x)\"\nbegin\n\ncontext\n assumes d1: \"x ≤ d x ⊓ x\"\n assumes d2: \"test p ⟹ d (p ⊓ x) ≤ p\"\nbegin\n\nlemma d3: \"d (x ⊓ d y) ≤ d (x ⊓ y)\" nitpick [expect=genuine] oops\n\nend\n\ncontext\n assumes d1: \"x ≤ d x ⊓ x\"\n assumes d3: \"d (x ⊓ d y) ≤ d (x ⊓ y)\"\nbegin\n\nlemma d2: \"test p ⟹ d (p ⊓ x) ≤ p\" nitpick [expect=genuine] oops\n\nend\n\ncontext\n assumes d2: \"test p ⟹ d (p ⊓ x) ≤ p\"\n assumes d3: \"d (x ⊓ d y) ≤ d (x ⊓ y)\"\nbegin\n\nlemma d1: \"x ≤ d x ⊓ x\" nitpick [expect=genuine] oops\n\nend\n\nend\n\nend\n\nclass d_semiring_var = ppd_semiring +\n assumes d3_var: \"d (x ⊓ d y) ≤ d (x ⊓ y)\"\n assumes d_strict_eq_var: \"d bot = bot\"\nbegin\n\nlemma d2_var:\n assumes \"test p\"\n shows \"d (p ⊓ x) ≤ p\"\nproof (-)\n (*goal: ‹d (p ⊓ x) ≤ p›*)\n have \"!p ⊓ p ⊓ x = bot\"\n by (simp add: assms (*‹test p›*) neg_char (*‹test ?x ⟹ are_complementary ?x (! ?x)›*))\n hence \"d (!p ⊓ p ⊓ x) = bot\"\n by (simp add: d_strict_eq_var (*‹d bot = bot›*))\n hence \"d (!p ⊓ d (p ⊓ x)) = bot\"\n by (metis d3_var (*‹d (?x ⊓ d ?y) ≤ d (?x ⊓ ?y)›*) il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) less_eq_bot (*‹(?x ≤ bot) = (?x = bot)›*))\n hence \"!p ⊓ d (p ⊓ x) = bot\"\n using d_bot_only (*‹d ?x = bot ⟹ ?x = bot›*) by blast\n thus \"?thesis\"\n (*goal: ‹d (p ⊓ x) ≤ p›*)\n by (metis (no_types, opaque_lifting) assms (*‹test (p::'a)›*) d_sub_identity (*‹d (?x::'a) ≤ top›*) il_bot_unit (*‹(?x::'a) ⊔ bot = ?x›*) il_inf_left_unit (*‹top ⊓ (?x::'a) = ?x›*) il_inf_right_dist_sup (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_inf_right_unit (*‹(?x::'a) ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹(?x::'a) ≤ (?y::'a) ⟹ (?z::'a) ⊓ ?x ≤ ?z ⊓ ?y›*) neg_char (*‹test (?x::'a) ⟹ are_complementary ?x (! ?x)›*))\nqed\n\nsubclass d_semiring\nproof\n show \"⋀p x. test p ⟹ d (p ⊓ x) ≤ p\"\n by (simp add: d2_var)\n show \"⋀x y. d (x ⊓ d y) ≤ d (x ⊓ y)\"\n by (simp add: d3_var)\nqed\n\nend\n\nsection ‹Antidomain Semirings›\n\ntext ‹\nWe now develop prepreantidomain semirings, preantidomain semirings and antidomain semirings.\nSee \\<^cite>‹\"DesharnaisStruth2008b\" and \"DesharnaisStruth2008a\" and \"DesharnaisStruth2011\"› for related work on internal axioms for antidomain.\n›\n\nsubsection ‹Prepreantidomain Semirings›\n\ntext ‹Definition 20›\n\nclass ppa_semiring = il_semiring + uminus +\n assumes a_inf_complement_bot: \"-x ⊓ x = bot\"\n assumes a_stone[simp]: \"-x ⊔ --x = top\"\nbegin\n\ntext ‹Theorem 21›\n\nlemma l1:\n \"-top = bot\"\n by (metis a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*))\n\nlemma l2:\n \"-bot = top\"\n by (metis l1 (*‹- top = bot›*) a_stone (*‹- ?x ⊔ - - ?x = top›*) il_unit_bot (*‹bot ⊔ ?x = ?x›*))\n\nlemma l3:\n \"-x ≤ -y ⟹ -x ⊓ y = bot\"\n by (metis a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) il_bot_unit (*‹?x ⊔ bot = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n\nlemma l5:\n \"--x ≤ --y ⟹ -y ≤ -x\"\n by (metis (mono_tags, opaque_lifting) l3 (*‹- ?x ≤ - ?y ⟹ - ?x ⊓ ?y = bot›*) a_stone (*‹- ?x ⊔ - - ?x = top›*) bot_least (*‹bot ≤ ?x›*) il_bot_unit (*‹?x ⊔ bot = ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) sup_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊔ ?x ≤ ?z ⊔ ?y›*))\n\nlemma l4:\n \"---x = -x\"\n by (metis l5 (*‹- - ?x ≤ - - ?y ⟹ - ?y ≤ - ?x›*) a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) a_stone (*‹- ?x ⊔ - - ?x = top›*) order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*) bot_least (*‹bot ≤ ?x›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) il_unit_bot (*‹bot ⊔ ?x = ?x›*))\n\nlemma l6:\n \"-x ⊓ --x = bot\"\n by (metis l3 (*‹- ?x ≤ - ?y ⟹ - ?x ⊓ ?y = bot›*) l5 (*‹- - ?x ≤ - - ?y ⟹ - ?y ≤ - ?x›*) a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) a_stone (*‹- ?x ⊔ - - ?x = top›*) il_inf_left_unit (*‹top ⊓ ?x = ?x›*) il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_inf_right_unit (*‹?x ⊓ top = ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) il_unit_bot (*‹bot ⊔ ?x = ?x›*))\n\nlemma l7:\n \"-x ⊓ -y = -y ⊓ -x\"\n using l6 (*‹- ?x ⊓ - - ?x = bot›*) a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) a_stone (*‹- ?x ⊔ - - ?x = top›*) test_inf_commutative (*‹⟦test ?x; test ?y⟧ ⟹ ?x ⊓ ?y = ?y ⊓ ?x›*) by blast\n\nlemma l8:\n \"x ≤ --x ⊓ x\"\n by (metis a_inf_complement_bot (*‹- (?x::'a) ⊓ ?x = bot›*) a_stone (*‹- (?x::'a) ⊔ - - ?x = top›*) il_idempotent (*‹(?x::'a) ⊔ ?x = ?x›*) il_inf_left_unit (*‹top ⊓ (?x::'a) = ?x›*) il_inf_right_dist_sup (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_less_eq (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*) il_unit_bot (*‹bot ⊔ (?x::'a) = ?x›*))\n\nsublocale ppa_ppd: ppd_semiring where d = \"λx . --x\"\nproof (standard)\n (*goals:\n 1. ‹⋀x. test (- - x)›\n 2. ‹⋀x. x ≤ - - x ⊓ x›*)\n show \"⋀x. test (- - x)\"\n using l4 (*‹- - - ?x = - ?x›*) l6 (*‹- ?x ⊓ - - ?x = bot›*) by force\n show \"⋀x. x ≤ - - x ⊓ x\"\n by (simp add: l8 (*‹?x ≤ - - ?x ⊓ ?x›*))\nqed\n\n(*\nThe following statements have counterexamples, but they take a while to find.\n\nlemma \"- x = - (- - x ⊔ - y) ⊔ - (- - x ⊔ - - y)\" nitpick [card=8, expect=genuine] oops\nlemma \"- x ⊔ - y = - - (- x ⊔ - y)\" nitpick [card=8, expect=genuine] oops\n*)\n\nend\n\nsubsection ‹Preantidomain Semirings›\n\ntext ‹Definition 22›\n\nclass pa_semiring = ppa_semiring +\n assumes pad2: \"--x ≤ -(-x ⊓ y)\"\nbegin\n\ntext ‹Theorem 23›\n\nlemma l10:\n \"-x ⊓ y = bot ⟹ -x ≤ -y\"\n by (metis a_stone (*‹- (?x::'a) ⊔ - - ?x = top›*) il_inf_left_unit (*‹top ⊓ (?x::'a) = ?x›*) il_inf_right_dist_sup (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) il_unit_bot (*‹bot ⊔ (?x::'a) = ?x›*) l4 (*‹- - - (?x::'a) = - ?x›*) pad2 (*‹- - (?x::'a) ≤ - (- ?x ⊓ (?y::'a))›*))\n\nlemma l10_iff:\n \"-x ⊓ y = bot ⟷ -x ≤ -y\"\n using l10 (*‹- ?x ⊓ ?y = bot ⟹ - ?x ≤ - ?y›*) l3 (*‹- ?x ≤ - ?y ⟹ - ?x ⊓ ?y = bot›*) by blast\n\nlemma l13:\n \"--(--x ⊓ y) ≤ --x\"\n by (metis l4 (*‹- - - ?x = - ?x›*) l5 (*‹- - ?x ≤ - - ?y ⟹ - ?y ≤ - ?x›*) pad2 (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*))\n\nlemma l14:\n \"-(x ⊓ --y) ≤ -(x ⊓ y)\"\n by (metis il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) l4 (*‹- - - ?x = - ?x›*) pad2 (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*) ppa_ppd.d1_eq (*‹?x = - - ?x ⊓ ?x›*))\n\nlemma l9:\n \"x ≤ y ⟹ -y ≤ -x\"\n by (metis l10 (*‹- ?x ⊓ ?y = bot ⟹ - ?x ≤ - ?y›*) a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*) il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*) il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) il_sub_inf_right_isotone (*‹?x ≤ ?y ⟹ ?z ⊓ ?x ≤ ?z ⊓ ?y›*) il_unit_bot (*‹bot ⊔ ?x = ?x›*))\n\nlemma l11:\n \"- x ⊔ - y = - (- - x ⊓ - - y)\"\nproof (-)\n (*goal: ‹- x ⊔ - y = - (- - x ⊓ - - y)›*)\n have 1: \"⋀x y . x ≤ y ⟷ x ⊔ y = y\"\n by (simp add: il_less_eq (*‹((?x::'a::type) ≤ (?y::'a::type)) = (?x ⊔ ?y = ?y)›*))\n have 4: \"⋀x y . ¬(x ≤ y) ∨ x ⊔ y = y\"\n using \"1\" (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by metis\n have 5: \"⋀x y z . (x ⊓ y) ⊔ (x ⊓ z) ≤ x ⊓ (y ⊔ z)\"\n by (simp add: il_sub_inf_right_isotone_var (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*))\n have 6: \"⋀x y . - - x ≤ - (- x ⊓ y)\"\n by (simp add: pad2 (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*))\n have 7: \"⋀x y z . x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n by (simp add: il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*))\n have 8: \"⋀x y z . (x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)\"\n using \"7\" (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) by metis\n have 9: \"⋀x y . x ⊔ y = y ⊔ x\"\n by (simp add: il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*))\n have 10: \"⋀x . x ⊔ bot = x\"\n by (simp add: il_bot_unit (*‹(?x::'a) ⊔ bot = ?x›*))\n have 11: \"⋀x . x ⊔ x = x\"\n by simp\n have 12: \"⋀x y z . x ⊓ (y ⊓ z) = (x ⊓ y) ⊓ z\"\n by (simp add: il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*))\n have 13: \"⋀x y z . (x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)\"\n using \"12\" (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) by metis\n have 14: \"⋀x . top ⊓ x = x\"\n by simp\n have 15: \"⋀x . x ⊓ top = x\"\n by simp\n have 16: \"⋀x y z . (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)\"\n by (simp add: il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n have 17: \"⋀x y z . (x ⊓ y) ⊔ (z ⊓ y) = (x ⊔ z) ⊓ y\"\n using \"16\" (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) by metis\n have 18: \"⋀x . bot ⊓ x = bot\"\n by simp\n have 19: \"⋀x . - x ⊔ - - x = top\"\n by simp\n have 20: \"⋀x . - x ⊓ x = bot\"\n by (simp add: a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*))\n have 23: \"⋀x y z . ((x ⊓ y) ⊔ (x ⊓ z)) ⊔ (x ⊓ (y ⊔ z)) = x ⊓ (y ⊔ z)\"\n using \"4\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"5\" (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 24: \"⋀x y z . (x ⊓ (y ⊔ z)) ⊔ ((x ⊓ y) ⊔ (x ⊓ z)) = x ⊓ (y ⊔ z)\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"23\" (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 25: \"⋀x y . - - x ⊔ - (- x ⊓ y) = - (- x ⊓ y)\"\n using \"4\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"6\" (*‹- - (?x::'a) ≤ - (- ?x ⊓ (?y::'a))›*) by metis\n have 26: \"⋀x y z . x ⊔ (y ⊔ z) = y ⊔ (x ⊔ z)\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by metis\n have 27: \"⋀x y z . (x ⊓ y) ⊔ ((x ⊓ z) ⊔ (x ⊓ (y ⊔ z))) = x ⊓ (y ⊔ z)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"24\" (*‹?x ⊓ (?y ⊔ ?z) ⊔ (?x ⊓ ?y ⊔ ?x ⊓ ?z) = ?x ⊓ (?y ⊔ ?z)›*) \"26\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) by metis\n have 30: \"⋀x . bot ⊔ x = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"10\" (*‹?x ⊔ bot = ?x›*) by metis\n have 31: \"⋀x y . x ⊔ (x ⊔ y) = x ⊔ y\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"11\" (*‹(?x::'a::type) ⊔ ?x = ?x›*) by metis\n have 34: \"⋀u x y z . ((x ⊔ y) ⊓ z) ⊔ u = (x ⊓ z) ⊔ ((y ⊓ z) ⊔ u)\"\n using \"8\" (*‹(?x::'a) ⊔ (?y::'a) ⊔ (?z::'a) = ?x ⊔ (?y ⊔ ?z)›*) \"17\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?z::'a) ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 35: \"⋀u x y z . (x ⊓ (y ⊓ z)) ⊔ (u ⊓ z) = ((x ⊓ y) ⊔ u) ⊓ z\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 36: \"⋀u x y z . (x ⊓ y) ⊔ (z ⊓ (u ⊓ y)) = (x ⊔ (z ⊓ u)) ⊓ y\"\n using \"13\" (*‹(?x::'a) ⊓ (?y::'a) ⊓ (?z::'a) = ?x ⊓ (?y ⊓ ?z)›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 39: \"⋀x y . - x ⊔ (- - x ⊔ y) = top ⊔ y\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) by metis\n have 41: \"⋀x y . - x ⊓ (x ⊓ y) = bot\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"18\" (*‹bot ⊓ (?x::'a) = bot›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) by metis\n have 42: \"- top = bot\"\n using \"15\" (*‹(?x::'a::type) ⊓ top = ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) by metis\n have 43: \"⋀x y . (- x ⊔ y) ⊓ x = y ⊓ x\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"30\" (*‹bot ⊔ (?x::'a::type) = ?x›*) by metis\n have 44: \"⋀x y . (x ⊔ - y) ⊓ y = x ⊓ y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"30\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 46: \"⋀x . - bot ⊔ - - x = - bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"20\" (*‹- (?x::'a) ⊓ ?x = bot›*) \"25\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) by metis\n have 50: \"- bot = top\"\n using \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"30\" (*‹bot ⊔ (?x::'a::type) = ?x›*) \"42\" (*‹- top = bot›*) by metis\n have 51: \"⋀x . top ⊔ - - x = top\"\n using \"46\" (*‹- bot ⊔ - - ?x = - bot›*) \"50\" (*‹- bot = top›*) by metis\n have 63: \"⋀x y . x ⊔ ((x ⊓ - y) ⊔ (x ⊓ - - y)) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"15\" (*‹?x ⊓ top = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"26\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"27\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 66: \"⋀x y . (- (x ⊔ y) ⊓ x) ⊔ (- (x ⊔ y) ⊓ y) = bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"27\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"30\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 67: \"⋀x y z . (x ⊓ - - y) ⊔ (x ⊓ - (- y ⊓ z)) = x ⊓ - (- y ⊓ z)\"\n using \"11\" (*‹?x ⊔ ?x = ?x›*) \"25\" (*‹- - (?x::'a) ⊔ - (- ?x ⊓ (?y::'a)) = - (- ?x ⊓ ?y)›*) \"27\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 70: \"⋀x y . x ⊔ (x ⊓ - - y) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"15\" (*‹(?x::'a::type) ⊓ top = ?x›*) \"27\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"31\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"51\" (*‹top ⊔ - - (?x::'a::type) = top›*) by metis\n have 82: \"⋀x . top ⊔ - x = top\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"31\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) by metis\n have 89: \"⋀x y . x ⊔ (- y ⊓ x) = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"82\" (*‹top ⊔ - (?x::'a) = top›*) by metis\n have 102: \"⋀x y z . x ⊔ (y ⊔ (x ⊓ - - z)) = y ⊔ x\"\n using \"26\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"70\" (*‹(?x::'a::type) ⊔ ?x ⊓ - - (?y::'a::type) = ?x›*) by metis\n have 104: \"⋀x y . x ⊔ (x ⊓ - y) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"63\" (*‹?x ⊔ (?x ⊓ - ?y ⊔ ?x ⊓ - - ?y) = ?x›*) \"102\" (*‹?x ⊔ (?y ⊔ ?x ⊓ - - ?z) = ?y ⊔ ?x›*) by metis\n have 112: \"⋀x y z . (- x ⊓ y) ⊔ ((- - x ⊓ y) ⊔ z) = y ⊔ z\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"34\" (*‹(?x ⊔ ?y) ⊓ ?z ⊔ ?u = ?x ⊓ ?z ⊔ (?y ⊓ ?z ⊔ ?u)›*) by metis\n have 117: \"⋀x y z . x ⊔ ((x ⊓ - y) ⊔ z) = x ⊔ z\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"104\" (*‹(?x::'a) ⊔ ?x ⊓ - (?y::'a) = ?x›*) by metis\n have 120: \"⋀x y z . x ⊔ (y ⊔ (x ⊓ - z)) = y ⊔ x\"\n using \"26\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"104\" (*‹(?x::'a::type) ⊔ ?x ⊓ - (?y::'a::type) = ?x›*) by metis\n have 124: \"⋀x . - - x ⊓ x = x\"\n using \"14\" (*‹top ⊓ (?x::'a) = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"43\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) by metis\n have 128: \"⋀x y . - - x ⊓ (x ⊓ y) = x ⊓ y\"\n using \"13\" (*‹(?x::'a) ⊓ (?y::'a) ⊓ (?z::'a) = ?x ⊓ (?y ⊓ ?z)›*) \"124\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 131: \"⋀x . - x ⊔ - - - x = - x\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"25\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"124\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 133: \"⋀x . - - - x = - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"104\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"124\" (*‹- - ?x ⊓ ?x = ?x›*) \"131\" (*‹- ?x ⊔ - - - ?x = - ?x›*) by metis\n have 135: \"⋀x y . - x ⊔ - (- - x ⊓ y) = - (- - x ⊓ y)\"\n using \"25\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"133\" (*‹- - - ?x = - ?x›*) by metis\n have 137: \"⋀x y . (- x ⊔ y) ⊓ - - x = y ⊓ - - x\"\n using \"43\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"133\" (*‹- - - ?x = - ?x›*) by metis\n have 145: \"⋀x y z . ((- (x ⊓ y) ⊓ x) ⊔ z) ⊓ y = z ⊓ y\"\n using \"20\" (*‹- ?x ⊓ ?x = bot›*) \"30\" (*‹bot ⊔ ?x = ?x›*) \"35\" (*‹(?x::'a) ⊓ ((?y::'a) ⊓ (?z::'a)) ⊔ (?u::'a) ⊓ ?z = (?x ⊓ ?y ⊔ ?u) ⊓ ?z›*) by metis\n have 183: \"⋀x y z . (x ⊔ (- - (y ⊓ z) ⊓ y)) ⊓ z = (x ⊔ y) ⊓ z\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"36\" (*‹?x ⊓ ?y ⊔ ?z ⊓ (?u ⊓ ?y) = (?x ⊔ ?z ⊓ ?u) ⊓ ?y›*) \"124\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 289: \"⋀x y . - x ⊔ - (- x ⊓ y) = top\"\n using \"25\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"39\" (*‹- ?x ⊔ (- - ?x ⊔ ?y) = top ⊔ ?y›*) \"82\" (*‹top ⊔ - ?x = top›*) by metis\n have 316: \"⋀x y . - (- x ⊓ y) ⊓ x = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"43\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"289\" (*‹- (?x::'a) ⊔ - (- ?x ⊓ (?y::'a)) = top›*) by metis\n have 317: \"⋀x y z . - (- x ⊓ y) ⊓ (x ⊓ z) = x ⊓ z\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"316\" (*‹- (- (?x::'a) ⊓ (?y::'a)) ⊓ ?x = ?x›*) by metis\n have 320: \"⋀x y . - x ⊔ - - (- x ⊓ y) = - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"25\" (*‹- - (?x::'a::type) ⊔ - (- ?x ⊓ (?y::'a::type)) = - (- ?x ⊓ ?y)›*) \"316\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) by metis\n have 321: \"⋀x y . - - (- x ⊓ y) ⊓ x = bot\"\n using \"41\" (*‹- ?x ⊓ (?x ⊓ ?y) = bot›*) \"316\" (*‹- (- (?x::'a) ⊓ (?y::'a)) ⊓ ?x = ?x›*) by metis\n have 374: \"⋀x y . - x ⊔ - (x ⊓ y) = - (x ⊓ y)\"\n using \"25\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"128\" (*‹- - (?x::'a::type) ⊓ (?x ⊓ (?y::'a::type)) = ?x ⊓ ?y›*) \"133\" (*‹- - - (?x::'a) = - ?x›*) by metis\n have 388: \"⋀x y . - (x ⊓ y) ⊓ - x = - x\"\n using \"128\" (*‹- - ?x ⊓ (?x ⊓ ?y) = ?x ⊓ ?y›*) \"316\" (*‹- (- (?x::'a) ⊓ (?y::'a)) ⊓ ?x = ?x›*) by metis\n have 389: \"⋀x y . - - (x ⊓ y) ⊓ - x = bot\"\n using \"128\" (*‹- - ?x ⊓ (?x ⊓ ?y) = ?x ⊓ ?y›*) \"321\" (*‹- - (- ?x ⊓ ?y) ⊓ ?x = bot›*) by metis\n have 405: \"⋀x y z . - (x ⊓ y) ⊓ (- x ⊓ z) = - x ⊓ z\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"388\" (*‹- ((?x::'a) ⊓ (?y::'a)) ⊓ - ?x = - ?x›*) by metis\n have 406: \"⋀x y z . - (x ⊓ (y ⊓ z)) ⊓ - (x ⊓ y) = - (x ⊓ y)\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"388\" (*‹- ((?x::'a) ⊓ (?y::'a)) ⊓ - ?x = - ?x›*) by metis\n have 420: \"⋀x y . - x ⊓ - - (- x ⊓ y) = - - (- x ⊓ y)\"\n using \"316\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) \"388\" (*‹- (?x ⊓ ?y) ⊓ - ?x = - ?x›*) by metis\n have 422: \"⋀x y z . - - (x ⊓ y) ⊓ (- x ⊓ z) = bot\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"18\" (*‹bot ⊓ (?x::'a) = bot›*) \"389\" (*‹- - (?x ⊓ ?y) ⊓ - ?x = bot›*) by metis\n have 758: \"⋀x y z . x ⊔ (x ⊓ (- y ⊓ - z)) = x\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"104\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"117\" (*‹(?x::'a) ⊔ (?x ⊓ - (?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?z›*) by metis\n have 1092: \"⋀x y . - (x ⊔ y) ⊓ x = bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"30\" (*‹bot ⊔ ?x = ?x›*) \"31\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"66\" (*‹- (?x ⊔ ?y) ⊓ ?x ⊔ - (?x ⊔ ?y) ⊓ ?y = bot›*) by metis\n have 1130: \"⋀x y z . (- (x ⊔ y) ⊔ z) ⊓ x = z ⊓ x\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"30\" (*‹bot ⊔ (?x::'a) = ?x›*) \"1092\" (*‹- (?x ⊔ ?y) ⊓ ?x = bot›*) by metis\n have 1156: \"⋀x y . - - x ⊓ - (- x ⊓ y) = - - x\"\n using \"67\" (*‹?x ⊓ - - ?y ⊔ ?x ⊓ - (- ?y ⊓ ?z) = ?x ⊓ - (- ?y ⊓ ?z)›*) \"104\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"124\" (*‹- - ?x ⊓ ?x = ?x›*) \"133\" (*‹- - - ?x = - ?x›*) by metis\n have 2098: \"⋀x y . - - (x ⊔ y) ⊓ x = x\"\n using \"14\" (*‹top ⊓ (?x::'a) = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"1130\" (*‹(- (?x ⊔ ?y) ⊔ ?z) ⊓ ?x = ?z ⊓ ?x›*) by metis\n have 2125: \"⋀x y . - - (x ⊔ y) ⊓ y = y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"2098\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2138: \"⋀x y . - x ⊔ - - (x ⊔ y) = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"289\" (*‹- ?x ⊔ - (- ?x ⊓ ?y) = top›*) \"2098\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2139: \"⋀x y . - x ⊓ - (x ⊔ y) = - (x ⊔ y)\"\n using \"316\" (*‹- (- (?x::'a::type) ⊓ (?y::'a::type)) ⊓ ?x = ?x›*) \"2098\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2192: \"⋀x y . - - x ⊓ (- y ⊓ x) = - y ⊓ x\"\n using \"89\" (*‹?x ⊔ - ?y ⊓ ?x = ?x›*) \"2125\" (*‹- - (?x ⊔ ?y) ⊓ ?y = ?y›*) by metis\n have 2202: \"⋀x y . - x ⊔ - - (y ⊔ x) = top\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"289\" (*‹- (?x::'a) ⊔ - (- ?x ⊓ (?y::'a)) = top›*) \"2125\" (*‹- - ((?x::'a::type) ⊔ (?y::'a::type)) ⊓ ?y = ?y›*) by metis\n have 2344: \"⋀x y . - (- x ⊓ y) ⊔ - - y = top\"\n using \"89\" (*‹(?x::'a) ⊔ - (?y::'a) ⊓ ?x = ?x›*) \"2202\" (*‹- ?x ⊔ - - (?y ⊔ ?x) = top›*) by metis\n have 2547: \"⋀x y z . - x ⊔ ((- - x ⊓ - y) ⊔ z) = - x ⊔ (- y ⊔ z)\"\n using \"112\" (*‹- ?x ⊓ ?y ⊔ (- - ?x ⊓ ?y ⊔ ?z) = ?y ⊔ ?z›*) \"117\" (*‹?x ⊔ (?x ⊓ - ?y ⊔ ?z) = ?x ⊔ ?z›*) by metis\n have 3023: \"⋀x y . - x ⊔ - (- y ⊓ - x) = top\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"133\" (*‹- - - ?x = - ?x›*) \"2344\" (*‹- (- ?x ⊓ ?y) ⊔ - - ?y = top›*) by metis\n have 3134: \"⋀x y . - (- x ⊓ - y) ⊓ y = y\"\n using \"14\" (*‹top ⊓ (?x::'a::type) = ?x›*) \"43\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"3023\" (*‹- ?x ⊔ - (- ?y ⊓ - ?x) = top›*) by metis\n have 3135: \"⋀x y . - x ⊓ (- y ⊓ - x) = - y ⊓ - x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"44\" (*‹(?x ⊔ - ?y) ⊓ ?y = ?x ⊓ ?y›*) \"3023\" (*‹- ?x ⊔ - (- ?y ⊓ - ?x) = top›*) by metis\n have 3962: \"⋀x y . - - (x ⊔ y) ⊓ - - x = - - x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"137\" (*‹(- (?x::'a) ⊔ (?y::'a)) ⊓ - - ?x = ?y ⊓ - - ?x›*) \"2138\" (*‹- (?x::'a) ⊔ - - (?x ⊔ (?y::'a)) = top›*) by metis\n have 5496: \"⋀x y z . - - (x ⊓ y) ⊓ - (x ⊔ z) = bot\"\n using \"422\" (*‹- - (?x ⊓ ?y) ⊓ (- ?x ⊓ ?z) = bot›*) \"2139\" (*‹- ?x ⊓ - (?x ⊔ ?y) = - (?x ⊔ ?y)›*) by metis\n have 9414: \"⋀x y . - - (- x ⊓ y) ⊓ y = - x ⊓ y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"104\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"183\" (*‹(?x ⊔ - - (?y ⊓ ?z) ⊓ ?y) ⊓ ?z = (?x ⊔ ?y) ⊓ ?z›*) \"320\" (*‹- ?x ⊔ - - (- ?x ⊓ ?y) = - ?x›*) by metis\n have 9520: \"⋀x y z . - - (- x ⊓ y) ⊓ - - (x ⊓ z) = bot\"\n using \"374\" (*‹- ?x ⊔ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"5496\" (*‹- - (?x ⊓ ?y) ⊓ - (?x ⊔ ?z) = bot›*) by metis\n have 11070: \"⋀x y z . - (- - x ⊓ y) ⊔ (- x ⊓ - z) = - (- - x ⊓ y)\"\n using \"317\" (*‹- (- ?x ⊓ ?y) ⊓ (?x ⊓ ?z) = ?x ⊓ ?z›*) \"758\" (*‹?x ⊔ ?x ⊓ (- ?y ⊓ - ?z) = ?x›*) by metis\n have 12371: \"⋀x y . - x ⊓ - (- - x ⊓ y) = - x\"\n using \"133\" (*‹- - - ?x = - ?x›*) \"1156\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) by metis\n have 12377: \"⋀x y . - x ⊓ - (x ⊓ y) = - x\"\n using \"128\" (*‹- - ?x ⊓ (?x ⊓ ?y) = ?x ⊓ ?y›*) \"133\" (*‹- - - ?x = - ?x›*) \"1156\" (*‹- - (?x::'a) ⊓ - (- ?x ⊓ (?y::'a)) = - - ?x›*) by metis\n have 12384: \"⋀x y . - (x ⊔ y) ⊓ - y = - (x ⊔ y)\"\n using \"133\" (*‹- - - ?x = - ?x›*) \"1156\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) \"2125\" (*‹- - (?x ⊔ ?y) ⊓ ?y = ?y›*) by metis\n have 12394: \"⋀x y . - - (- x ⊓ - y) = - x ⊓ - y\"\n using \"1156\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) \"3134\" (*‹- (- ?x ⊓ - ?y) ⊓ ?y = ?y›*) \"9414\" (*‹- - (- ?x ⊓ ?y) ⊓ ?y = - ?x ⊓ ?y›*) by metis\n have 12640: \"⋀x y . - x ⊓ - (- y ⊓ x) = - x\"\n using \"89\" (*‹?x ⊔ - ?y ⊓ ?x = ?x›*) \"12384\" (*‹- (?x ⊔ ?y) ⊓ - ?y = - (?x ⊔ ?y)›*) by metis\n have 24648: \"⋀x y . (- x ⊓ - y) ⊔ - (- x ⊓ - y) = top\"\n using \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"12394\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) by metis\n have 28270: \"⋀x y z . - - (x ⊓ y) ⊔ - (- x ⊓ z) = - (- x ⊓ z)\"\n using \"374\" (*‹- (?x::'a) ⊔ - (?x ⊓ (?y::'a)) = - (?x ⊓ ?y)›*) \"405\" (*‹- (?x ⊓ ?y) ⊓ (- ?x ⊓ ?z) = - ?x ⊓ ?z›*) by metis\n have 28339: \"⋀x y . - (- - (x ⊓ y) ⊓ x) = - (x ⊓ y)\"\n using \"124\" (*‹- - ?x ⊓ ?x = ?x›*) \"406\" (*‹- (?x ⊓ (?y ⊓ ?z)) ⊓ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"12371\" (*‹- (?x::'a) ⊓ - (- - ?x ⊓ (?y::'a)) = - ?x›*) by metis\n have 28423: \"⋀x y . - (- x ⊓ - y) = - (- y ⊓ - x)\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"3135\" (*‹- ?x ⊓ (- ?y ⊓ - ?x) = - ?y ⊓ - ?x›*) \"12394\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"28339\" (*‹- (- - (?x ⊓ ?y) ⊓ ?x) = - (?x ⊓ ?y)›*) by metis\n have 28487: \"⋀x y . - x ⊓ - y = - y ⊓ - x\"\n using \"2098\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) \"3962\" (*‹- - ((?x::'a) ⊔ (?y::'a)) ⊓ - - ?x = - - ?x›*) \"12394\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"28423\" (*‹- (- ?x ⊓ - ?y) = - (- ?y ⊓ - ?x)›*) by metis\n have 52423: \"⋀x y . - (- x ⊓ - (- x ⊓ y)) ⊓ y = y\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"145\" (*‹(- (?x ⊓ ?y) ⊓ ?x ⊔ ?z) ⊓ ?y = ?z ⊓ ?y›*) \"24648\" (*‹- (?x::'a) ⊓ - (?y::'a) ⊔ - (- ?x ⊓ - ?y) = top›*) \"28487\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) by metis\n have 52522: \"⋀x y . - x ⊓ - (- x ⊓ y) = - x ⊓ - y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"12377\" (*‹- (?x::'a) ⊓ - (?x ⊓ (?y::'a)) = - ?x›*) \"12394\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"12640\" (*‹- ?x ⊓ - (- ?y ⊓ ?x) = - ?x›*) \"28487\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"52423\" (*‹- (- ?x ⊓ - (- ?x ⊓ ?y)) ⊓ ?y = ?y›*) by metis\n have 61103: \"⋀x y z . - (- - x ⊓ y) ⊔ z = - x ⊔ (- y ⊔ z)\"\n using \"112\" (*‹- ?x ⊓ ?y ⊔ (- - ?x ⊓ ?y ⊔ ?z) = ?y ⊔ ?z›*) \"2547\" (*‹- ?x ⊔ (- - ?x ⊓ - ?y ⊔ ?z) = - ?x ⊔ (- ?y ⊔ ?z)›*) \"12371\" (*‹- ?x ⊓ - (- - ?x ⊓ ?y) = - ?x›*) \"52522\" (*‹- (?x::'a::type) ⊓ - (- ?x ⊓ (?y::'a::type)) = - ?x ⊓ - ?y›*) by metis\n have 61158: \"⋀x y . - - (- x ⊓ y) = - x ⊓ - - y\"\n using \"420\" (*‹- ?x ⊓ - - (- ?x ⊓ ?y) = - - (- ?x ⊓ ?y)›*) \"52522\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) by metis\n have 61231: \"⋀x y z . - x ⊓ (- - y ⊓ - (x ⊓ z)) = - x ⊓ - - y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"15\" (*‹?x ⊓ top = ?x›*) \"50\" (*‹- bot = top›*) \"133\" (*‹- - - ?x = - ?x›*) \"9520\" (*‹- - (- ?x ⊓ ?y) ⊓ - - (?x ⊓ ?z) = bot›*) \"52522\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) \"61158\" (*‹- - (- ?x ⊓ ?y) = - ?x ⊓ - - ?y›*) by metis\n have 61313: \"⋀x y . - x ⊔ - y = - (- - y ⊓ x)\"\n using \"120\" (*‹?x ⊔ (?y ⊔ ?x ⊓ - ?z) = ?y ⊔ ?x›*) \"11070\" (*‹- (- - ?x ⊓ ?y) ⊔ - ?x ⊓ - ?z = - (- - ?x ⊓ ?y)›*) \"61103\" (*‹- (- - ?x ⊓ ?y) ⊔ ?z = - ?x ⊔ (- ?y ⊔ ?z)›*) by metis\n have 61393: \"⋀x y . - (- x ⊓ - - y) = - (- x ⊓ y)\"\n using \"13\" (*‹(?x::'a) ⊓ (?y::'a) ⊓ (?z::'a) = ?x ⊓ (?y ⊓ ?z)›*) \"28270\" (*‹- - (?x ⊓ ?y) ⊔ - (- ?x ⊓ ?z) = - (- ?x ⊓ ?z)›*) \"61158\" (*‹- - (- ?x ⊓ ?y) = - ?x ⊓ - - ?y›*) \"61231\" (*‹- (?x::'a) ⊓ (- - (?y::'a) ⊓ - (?x ⊓ (?z::'a))) = - ?x ⊓ - - ?y›*) \"61313\" (*‹- ?x ⊔ - ?y = - (- - ?y ⊓ ?x)›*) by metis\n have 61422: \"⋀x y . - (- - x ⊓ y) = - (- - y ⊓ x)\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"135\" (*‹- ?x ⊔ - (- - ?x ⊓ ?y) = - (- - ?x ⊓ ?y)›*) \"2192\" (*‹- - ?x ⊓ (- ?y ⊓ ?x) = - ?y ⊓ ?x›*) \"61158\" (*‹- - (- (?x::'a) ⊓ (?y::'a)) = - ?x ⊓ - - ?y›*) \"61313\" (*‹- ?x ⊔ - ?y = - (- - ?y ⊓ ?x)›*) by metis\n show \"?thesis\"\n (*goal: ‹- x ⊔ - y = - (- - x ⊓ - - y)›*)\n using \"61313\" (*‹- ?x ⊔ - ?y = - (- - ?y ⊓ ?x)›*) \"61393\" (*‹- (- ?x ⊓ - - ?y) = - (- ?x ⊓ ?y)›*) \"61422\" (*‹- (- - ?x ⊓ ?y) = - (- - ?y ⊓ ?x)›*) by metis\nqed\n\nlemma l12:\n \"- x ⊓ - y = - (x ⊔ y)\"\nproof (-)\n (*goal: ‹- x ⊓ - y = - (x ⊔ y)›*)\n have 1: \"⋀x y . x ≤ y ⟷ x ⊔ y = y\"\n by (simp add: il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n have 4: \"⋀x y . ¬(x ≤ y) ∨ x ⊔ y = y\"\n using \"1\" (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by metis\n have 5: \"⋀x y z . (x ⊓ y) ⊔ (x ⊓ z) ≤ x ⊓ (y ⊔ z)\"\n by (simp add: il_sub_inf_right_isotone_var (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*))\n have 6: \"⋀x y . - - x ≤ - (- x ⊓ y)\"\n by (simp add: pad2 (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*))\n have 7: \"⋀x y z . x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n by (simp add: il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*))\n have 8: \"⋀x y z . (x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)\"\n using \"7\" (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) by metis\n have 9: \"⋀x y . x ⊔ y = y ⊔ x\"\n by (simp add: il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*))\n have 10: \"⋀x . x ⊔ bot = x\"\n by (simp add: il_bot_unit (*‹?x ⊔ bot = ?x›*))\n have 11: \"⋀x . x ⊔ x = x\"\n by simp\n have 12: \"⋀x y z . x ⊓ (y ⊓ z) = (x ⊓ y) ⊓ z\"\n by (simp add: il_inf_associative (*‹(?x::'a) ⊓ ((?y::'a) ⊓ (?z::'a)) = ?x ⊓ ?y ⊓ ?z›*))\n have 13: \"⋀x y z . (x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)\"\n using \"12\" (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) by metis\n have 14: \"⋀x . top ⊓ x = x\"\n by simp\n have 15: \"⋀x . x ⊓ top = x\"\n by simp\n have 16: \"⋀x y z . (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)\"\n by (simp add: il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n have 17: \"⋀x y z . (x ⊓ y) ⊔ (z ⊓ y) = (x ⊔ z) ⊓ y\"\n using \"16\" (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) by metis\n have 18: \"⋀x . bot ⊓ x = bot\"\n by simp\n have 19: \"⋀x . - x ⊔ - - x = top\"\n by simp\n have 20: \"⋀x . - x ⊓ x = bot\"\n by (simp add: a_inf_complement_bot (*‹- (?x::'a) ⊓ ?x = bot›*))\n have 22: \"⋀x y z . ((x ⊓ y) ⊔ (x ⊓ z)) ⊔ (x ⊓ (y ⊔ z)) = x ⊓ (y ⊔ z)\"\n using \"4\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"5\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ ?x ⊓ (?z::'a) ≤ ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 23: \"⋀x y z . (x ⊓ (y ⊔ z)) ⊔ ((x ⊓ y) ⊔ (x ⊓ z)) = x ⊓ (y ⊔ z)\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"22\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ ?x ⊓ (?z::'a) ⊔ ?x ⊓ (?y ⊔ ?z) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 24: \"⋀x y . - - x ⊔ - (- x ⊓ y) = - (- x ⊓ y)\"\n using \"4\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"6\" (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*) by metis\n have 25: \"⋀x y z . x ⊔ (y ⊔ z) = y ⊔ (x ⊔ z)\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by metis\n have 26: \"⋀x y z . (x ⊓ y) ⊔ ((x ⊓ z) ⊔ (x ⊓ (y ⊔ z))) = x ⊓ (y ⊔ z)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"23\" (*‹?x ⊓ (?y ⊔ ?z) ⊔ (?x ⊓ ?y ⊔ ?x ⊓ ?z) = ?x ⊓ (?y ⊔ ?z)›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) by metis\n have 29: \"⋀x . bot ⊔ x = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"10\" (*‹?x ⊔ bot = ?x›*) by metis\n have 30: \"⋀x y . x ⊔ (x ⊔ y) = x ⊔ y\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"11\" (*‹?x ⊔ ?x = ?x›*) by metis\n have 32: \"⋀x y . x ⊔ (y ⊔ x) = y ⊔ x\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"11\" (*‹?x ⊔ ?x = ?x›*) by metis\n have 33: \"⋀u x y z . ((x ⊔ y) ⊓ z) ⊔ u = (x ⊓ z) ⊔ ((y ⊓ z) ⊔ u)\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 34: \"⋀u x y z . (x ⊓ (y ⊓ z)) ⊔ (u ⊓ z) = ((x ⊓ y) ⊔ u) ⊓ z\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 35: \"⋀u x y z . (x ⊓ y) ⊔ (z ⊓ (u ⊓ y)) = (x ⊔ (z ⊓ u)) ⊓ y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"17\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?z::'a) ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 36: \"⋀x y . (top ⊔ x) ⊓ y = y ⊔ (x ⊓ y)\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 37: \"⋀x y . (x ⊔ top) ⊓ y = y ⊔ (x ⊓ y)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"14\" (*‹top ⊓ ?x = ?x›*) \"17\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?z::'a) ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 38: \"⋀x y . - x ⊔ (- - x ⊔ y) = top ⊔ y\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) by metis\n have 40: \"⋀x y . - x ⊓ (x ⊓ y) = bot\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"18\" (*‹bot ⊓ ?x = bot›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) by metis\n have 41: \"- top = bot\"\n using \"15\" (*‹?x ⊓ top = ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) by metis\n have 42: \"⋀x y . (- x ⊔ y) ⊓ x = y ⊓ x\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"29\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 43: \"⋀x y . (x ⊔ - y) ⊓ y = x ⊓ y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"17\" (*‹(?x::'a::type) ⊓ (?y::'a::type) ⊔ (?z::'a::type) ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"20\" (*‹- (?x::'a) ⊓ ?x = bot›*) \"29\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 45: \"⋀x . - bot ⊔ - - x = - bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) by metis\n have 46: \"⋀u x y z . (x ⊓ y) ⊔ (z ⊔ (u ⊓ y)) = z ⊔ ((x ⊔ u) ⊓ y)\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) by metis\n have 47: \"⋀x y . - x ⊔ (y ⊔ - - x) = y ⊔ top\"\n using \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"25\" (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?y ⊔ (?x ⊔ ?z)›*) by metis\n have 49: \"- bot = top\"\n using \"19\" (*‹- (?x::'a::type) ⊔ - - ?x = top›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"41\" (*‹- top = bot›*) by metis\n have 50: \"⋀x . top ⊔ - - x = top\"\n using \"45\" (*‹- bot ⊔ - - ?x = - bot›*) \"49\" (*‹- bot = top›*) by metis\n have 54: \"⋀u x y z . (x ⊓ y) ⊔ ((x ⊓ z) ⊔ ((x ⊓ (y ⊔ z)) ⊔ u)) = (x ⊓ (y ⊔ z)) ⊔ u\"\n using \"8\" (*‹(?x::'a) ⊔ (?y::'a) ⊔ (?z::'a) = ?x ⊔ (?y ⊔ ?z)›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 58: \"⋀u x y z . (x ⊓ (y ⊓ z)) ⊔ ((x ⊓ (y ⊓ u)) ⊔ (x ⊓ (y ⊓ (z ⊔ u)))) = x ⊓ (y ⊓ (z ⊔ u))\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 60: \"⋀x y . x ⊔ ((x ⊓ y) ⊔ (x ⊓ (y ⊔ top))) = x ⊓ (y ⊔ top)\"\n using \"15\" (*‹?x ⊓ top = ?x›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"26\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?x ⊓ (?z::'a) ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 62: \"⋀x y . x ⊔ ((x ⊓ - y) ⊔ (x ⊓ - - y)) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"15\" (*‹?x ⊓ top = ?x›*) \"19\" (*‹- (?x::'a) ⊔ - - ?x = top›*) \"25\" (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?y ⊔ (?x ⊔ ?z)›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 65: \"⋀x y . (- (x ⊔ y) ⊓ x) ⊔ (- (x ⊔ y) ⊓ y) = bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"29\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 66: \"⋀x y z . (x ⊓ - - y) ⊔ (x ⊓ - (- y ⊓ z)) = x ⊓ - (- y ⊓ z)\"\n using \"11\" (*‹?x ⊔ ?x = ?x›*) \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 69: \"⋀x y . x ⊔ (x ⊓ - - y) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"15\" (*‹?x ⊓ top = ?x›*) \"26\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?x ⊓ (?z::'a) ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"30\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"50\" (*‹top ⊔ - - ?x = top›*) by metis\n have 81: \"⋀x . top ⊔ - x = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"30\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) by metis\n have 82: \"⋀x y z . (x ⊓ y) ⊔ (x ⊓ (y ⊔ z)) = x ⊓ (y ⊔ z)\"\n using \"11\" (*‹(?x::'a) ⊔ ?x = ?x›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"30\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) by metis\n have 83: \"⋀x y . x ⊔ (x ⊓ (y ⊔ top)) = x ⊓ (y ⊔ top)\"\n using \"60\" (*‹?x ⊔ (?x ⊓ ?y ⊔ ?x ⊓ (?y ⊔ top)) = ?x ⊓ (?y ⊔ top)›*) \"82\" (*‹?x ⊓ ?y ⊔ ?x ⊓ (?y ⊔ ?z) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 88: \"⋀x y . x ⊔ (- y ⊓ x) = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"81\" (*‹top ⊔ - ?x = top›*) by metis\n have 89: \"⋀x y . top ⊔ (x ⊔ - y) = x ⊔ top\"\n using \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"81\" (*‹top ⊔ - ?x = top›*) by metis\n have 91: \"⋀x y z . x ⊔ (y ⊔ (z ⊔ x)) = y ⊔ (z ⊔ x)\"\n using \"8\" (*‹(?x::'a::type) ⊔ (?y::'a::type) ⊔ (?z::'a::type) = ?x ⊔ (?y ⊔ ?z)›*) \"32\" (*‹?x ⊔ (?y ⊔ ?x) = ?y ⊔ ?x›*) by metis\n have 94: \"⋀x y z . x ⊔ (y ⊔ (- z ⊓ x)) = y ⊔ x\"\n using \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"88\" (*‹?x ⊔ - ?y ⊓ ?x = ?x›*) by metis\n have 101: \"⋀x y z . x ⊔ (y ⊔ (x ⊓ - - z)) = y ⊔ x\"\n using \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"69\" (*‹(?x::'a) ⊔ ?x ⊓ - - (?y::'a) = ?x›*) by metis\n have 102: \"⋀x . x ⊔ (x ⊓ bot) = x\"\n using \"41\" (*‹- top = bot›*) \"49\" (*‹- bot = top›*) \"69\" (*‹?x ⊔ ?x ⊓ - - ?y = ?x›*) by metis\n have 103: \"⋀x y . x ⊔ (x ⊓ - y) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"62\" (*‹?x ⊔ (?x ⊓ - ?y ⊔ ?x ⊓ - - ?y) = ?x›*) \"101\" (*‹?x ⊔ (?y ⊔ ?x ⊓ - - ?z) = ?y ⊔ ?x›*) by metis\n have 109: \"⋀x y . x ⊔ (y ⊔ (x ⊓ bot)) = y ⊔ x\"\n using \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"102\" (*‹?x ⊔ ?x ⊓ bot = ?x›*) by metis\n have 111: \"⋀x y z . (- x ⊓ y) ⊔ ((- - x ⊓ y) ⊔ z) = y ⊔ z\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"33\" (*‹(?x ⊔ ?y) ⊓ ?z ⊔ ?u = ?x ⊓ ?z ⊔ (?y ⊓ ?z ⊔ ?u)›*) by metis\n have 116: \"⋀x y z . x ⊔ ((x ⊓ - y) ⊔ z) = x ⊔ z\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"103\" (*‹(?x::'a) ⊔ ?x ⊓ - (?y::'a) = ?x›*) by metis\n have 119: \"⋀x y z . x ⊔ (y ⊔ (x ⊓ - z)) = y ⊔ x\"\n using \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) by metis\n have 123: \"⋀x . - - x ⊓ x = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) by metis\n have 127: \"⋀x y . - - x ⊓ (x ⊓ y) = x ⊓ y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 130: \"⋀x . - x ⊔ - - - x = - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"24\" (*‹- - (?x::'a::type) ⊔ - (- ?x ⊓ (?y::'a::type)) = - (- ?x ⊓ ?y)›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 132: \"⋀x . - - - x = - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"130\" (*‹- ?x ⊔ - - - ?x = - ?x›*) by metis\n have 134: \"⋀x y . - x ⊔ - (- - x ⊓ y) = - (- - x ⊓ y)\"\n using \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"132\" (*‹- - - (?x::'a::type) = - ?x›*) by metis\n have 136: \"⋀x y . (- x ⊔ y) ⊓ - - x = y ⊓ - - x\"\n using \"42\" (*‹(- (?x::'a) ⊔ (?y::'a)) ⊓ ?x = ?y ⊓ ?x›*) \"132\" (*‹- - - (?x::'a) = - ?x›*) by metis\n have 138: \"⋀x . - x ⊓ - x = - x\"\n using \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"132\" (*‹- - - (?x::'a::type) = - ?x›*) by metis\n have 144: \"⋀x y z . ((- (x ⊓ y) ⊓ x) ⊔ z) ⊓ y = z ⊓ y\"\n using \"20\" (*‹- ?x ⊓ ?x = bot›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"34\" (*‹?x ⊓ (?y ⊓ ?z) ⊔ ?u ⊓ ?z = (?x ⊓ ?y ⊔ ?u) ⊓ ?z›*) by metis\n have 157: \"⋀x y . (- x ⊔ y) ⊓ - x = (top ⊔ y) ⊓ - x\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"36\" (*‹(top ⊔ ?x) ⊓ ?y = ?y ⊔ ?x ⊓ ?y›*) \"138\" (*‹- ?x ⊓ - ?x = - ?x›*) by metis\n have 182: \"⋀x y z . (x ⊔ (- - (y ⊓ z) ⊓ y)) ⊓ z = (x ⊔ y) ⊓ z\"\n using \"17\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?z::'a) ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"35\" (*‹?x ⊓ ?y ⊔ ?z ⊓ (?u ⊓ ?y) = (?x ⊔ ?z ⊓ ?u) ⊓ ?y›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 288: \"⋀x y . - x ⊔ - (- x ⊓ y) = top\"\n using \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"38\" (*‹- ?x ⊔ (- - ?x ⊔ ?y) = top ⊔ ?y›*) \"81\" (*‹top ⊔ - ?x = top›*) by metis\n have 315: \"⋀x y . - (- x ⊓ y) ⊓ x = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"288\" (*‹- ?x ⊔ - (- ?x ⊓ ?y) = top›*) by metis\n have 316: \"⋀x y z . - (- x ⊓ y) ⊓ (x ⊓ z) = x ⊓ z\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) by metis\n have 319: \"⋀x y . - x ⊔ - - (- x ⊓ y) = - x\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) by metis\n have 320: \"⋀x y . - - (- x ⊓ y) ⊓ x = bot\"\n using \"40\" (*‹- ?x ⊓ (?x ⊓ ?y) = bot›*) \"315\" (*‹- (- (?x::'a) ⊓ (?y::'a)) ⊓ ?x = ?x›*) by metis\n have 373: \"⋀x y . - x ⊔ - (x ⊓ y) = - (x ⊓ y)\"\n using \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"127\" (*‹- - ?x ⊓ (?x ⊓ ?y) = ?x ⊓ ?y›*) \"132\" (*‹- - - ?x = - ?x›*) by metis\n have 387: \"⋀x y . - (x ⊓ y) ⊓ - x = - x\"\n using \"127\" (*‹- - (?x::'a) ⊓ (?x ⊓ (?y::'a)) = ?x ⊓ ?y›*) \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) by metis\n have 388: \"⋀x y . - - (x ⊓ y) ⊓ - x = bot\"\n using \"127\" (*‹- - ?x ⊓ (?x ⊓ ?y) = ?x ⊓ ?y›*) \"320\" (*‹- - (- ?x ⊓ ?y) ⊓ ?x = bot›*) by metis\n have 404: \"⋀x y z . - (x ⊓ y) ⊓ (- x ⊓ z) = - x ⊓ z\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"387\" (*‹- ((?x::'a) ⊓ (?y::'a)) ⊓ - ?x = - ?x›*) by metis\n have 405: \"⋀x y z . - (x ⊓ (y ⊓ z)) ⊓ - (x ⊓ y) = - (x ⊓ y)\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"387\" (*‹- (?x ⊓ ?y) ⊓ - ?x = - ?x›*) by metis\n have 419: \"⋀x y . - x ⊓ - - (- x ⊓ y) = - - (- x ⊓ y)\"\n using \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) \"387\" (*‹- (?x ⊓ ?y) ⊓ - ?x = - ?x›*) by metis\n have 420: \"⋀x y . - - x ⊓ - - (x ⊓ y) = - - (x ⊓ y)\"\n using \"387\" (*‹- (?x ⊓ ?y) ⊓ - ?x = - ?x›*) by metis\n have 421: \"⋀x y z . - - (x ⊓ y) ⊓ (- x ⊓ z) = bot\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"18\" (*‹bot ⊓ (?x::'a) = bot›*) \"388\" (*‹- - (?x ⊓ ?y) ⊓ - ?x = bot›*) by metis\n have 536: \"⋀x y . (x ⊔ - - y) ⊓ y = (x ⊔ top) ⊓ y\"\n using \"42\" (*‹(- (?x::'a) ⊔ (?y::'a)) ⊓ ?x = ?y ⊓ ?x›*) \"47\" (*‹- ?x ⊔ (?y ⊔ - - ?x) = ?y ⊔ top›*) by metis\n have 662: \"⋀u x y z . (x ⊓ y) ⊔ ((x ⊓ (z ⊔ y)) ⊔ u) = (x ⊓ (z ⊔ y)) ⊔ u\"\n using \"9\" (*‹(?x::'a::type) ⊔ (?y::'a::type) = ?y ⊔ ?x›*) \"32\" (*‹(?x::'a) ⊔ ((?y::'a) ⊔ ?x) = ?y ⊔ ?x›*) \"54\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?x ⊓ (?z::'a) ⊔ (?x ⊓ (?y ⊔ ?z) ⊔ (?u::'a))) = ?x ⊓ (?y ⊔ ?z) ⊔ ?u›*) by metis\n have 705: \"⋀u x y z . (x ⊓ (y ⊔ z)) ⊔ ((x ⊓ (y ⊔ (z ⊓ bot))) ⊔ u) = (x ⊓ (y ⊔ z)) ⊔ u\"\n using \"25\" (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?y ⊔ (?x ⊔ ?z)›*) \"54\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ (?x ⊓ (?y ⊔ ?z) ⊔ ?u)) = ?x ⊓ (?y ⊔ ?z) ⊔ ?u›*) \"109\" (*‹?x ⊔ (?y ⊔ ?x ⊓ bot) = ?y ⊔ ?x›*) \"662\" (*‹?x ⊓ ?y ⊔ (?x ⊓ (?z ⊔ ?y) ⊔ ?u) = ?x ⊓ (?z ⊔ ?y) ⊔ ?u›*) by metis\n have 755: \"⋀x y z . (x ⊓ - y) ⊔ (z ⊔ x) = z ⊔ x\"\n using \"32\" (*‹?x ⊔ (?y ⊔ ?x) = ?y ⊔ ?x›*) \"91\" (*‹?x ⊔ (?y ⊔ (?z ⊔ ?x)) = ?y ⊔ (?z ⊔ ?x)›*) \"116\" (*‹(?x::'a::type) ⊔ (?x ⊓ - (?y::'a::type) ⊔ (?z::'a::type)) = ?x ⊔ ?z›*) by metis\n have 757: \"⋀x y z . x ⊔ (x ⊓ (- y ⊓ - z)) = x\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"116\" (*‹?x ⊔ (?x ⊓ - ?y ⊔ ?z) = ?x ⊔ ?z›*) by metis\n have 930: \"⋀x y z . (- (x ⊓ (y ⊔ z)) ⊓ (x ⊓ y)) ⊔ (- (x ⊓ (y ⊔ z)) ⊓ (x ⊓ z)) = bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"20\" (*‹- (?x::'a) ⊓ ?x = bot›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"58\" (*‹?x ⊓ (?y ⊓ ?z) ⊔ (?x ⊓ (?y ⊓ ?u) ⊔ ?x ⊓ (?y ⊓ (?z ⊔ ?u))) = ?x ⊓ (?y ⊓ (?z ⊔ ?u))›*) by metis\n have 1091: \"⋀x y . - (x ⊔ y) ⊓ x = bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"30\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"65\" (*‹- (?x ⊔ ?y) ⊓ ?x ⊔ - (?x ⊔ ?y) ⊓ ?y = bot›*) by metis\n have 1092: \"⋀x y . - (x ⊔ y) ⊓ y = bot\"\n using \"29\" (*‹bot ⊔ (?x::'a) = ?x›*) \"30\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"65\" (*‹- ((?x::'a) ⊔ (?y::'a)) ⊓ ?x ⊔ - (?x ⊔ ?y) ⊓ ?y = bot›*) \"1091\" (*‹- (?x ⊔ ?y) ⊓ ?x = bot›*) by metis\n have 1113: \"⋀u x y z . - (x ⊔ ((y ⊔ z) ⊓ u)) ⊓ (x ⊔ (z ⊓ u)) = bot\"\n using \"29\" (*‹bot ⊔ ?x = ?x›*) \"46\" (*‹?x ⊓ ?y ⊔ (?z ⊔ ?u ⊓ ?y) = ?z ⊔ (?x ⊔ ?u) ⊓ ?y›*) \"65\" (*‹- (?x ⊔ ?y) ⊓ ?x ⊔ - (?x ⊔ ?y) ⊓ ?y = bot›*) \"1091\" (*‹- (?x ⊔ ?y) ⊓ ?x = bot›*) by metis\n have 1117: \"⋀x y z . - (x ⊔ y) ⊓ (x ⊔ (- z ⊓ y)) = bot\"\n using \"29\" (*‹bot ⊔ ?x = ?x›*) \"65\" (*‹- (?x ⊔ ?y) ⊓ ?x ⊔ - (?x ⊔ ?y) ⊓ ?y = bot›*) \"94\" (*‹?x ⊔ (?y ⊔ - ?z ⊓ ?x) = ?y ⊔ ?x›*) \"1092\" (*‹- (?x ⊔ ?y) ⊓ ?y = bot›*) by metis\n have 1128: \"⋀x y z . - (x ⊔ (y ⊔ z)) ⊓ (x ⊔ y) = bot\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"1091\" (*‹- (?x ⊔ ?y) ⊓ ?x = bot›*) by metis\n have 1129: \"⋀x y z . (- (x ⊔ y) ⊔ z) ⊓ x = z ⊓ x\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"1091\" (*‹- (?x ⊔ ?y) ⊓ ?x = bot›*) by metis\n have 1155: \"⋀x y . - - x ⊓ - (- x ⊓ y) = - - x\"\n using \"66\" (*‹?x ⊓ - - ?y ⊔ ?x ⊓ - (- ?y ⊓ ?z) = ?x ⊓ - (- ?y ⊓ ?z)›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"132\" (*‹- - - ?x = - ?x›*) by metis\n have 1578: \"⋀x y z . - (x ⊓ (y ⊔ z)) ⊓ (x ⊓ y) = bot\"\n using \"82\" (*‹?x ⊓ ?y ⊔ ?x ⊓ (?y ⊔ ?z) = ?x ⊓ (?y ⊔ ?z)›*) \"1091\" (*‹- (?x ⊔ ?y) ⊓ ?x = bot›*) by metis\n have 1594: \"⋀x y z . - (x ⊓ (y ⊔ z)) ⊓ (x ⊓ z) = bot\"\n using \"29\" (*‹bot ⊔ ?x = ?x›*) \"930\" (*‹- (?x ⊓ (?y ⊔ ?z)) ⊓ (?x ⊓ ?y) ⊔ - (?x ⊓ (?y ⊔ ?z)) ⊓ (?x ⊓ ?z) = bot›*) \"1578\" (*‹- (?x ⊓ (?y ⊔ ?z)) ⊓ (?x ⊓ ?y) = bot›*) by metis\n have 2094: \"⋀x y z . - (x ⊔ (y ⊓ (z ⊔ top))) ⊓ (x ⊔ y) = bot\"\n using \"83\" (*‹?x ⊔ ?x ⊓ (?y ⊔ top) = ?x ⊓ (?y ⊔ top)›*) \"1128\" (*‹- (?x ⊔ (?y ⊔ ?z)) ⊓ (?x ⊔ ?y) = bot›*) by metis\n have 2097: \"⋀x y . - - (x ⊔ y) ⊓ x = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"1129\" (*‹(- (?x ⊔ ?y) ⊔ ?z) ⊓ ?x = ?z ⊓ ?x›*) by metis\n have 2124: \"⋀x y . - - (x ⊔ y) ⊓ y = y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"2097\" (*‹- - ((?x::'a) ⊔ (?y::'a)) ⊓ ?x = ?x›*) by metis\n have 2135: \"⋀x y . - - ((top ⊔ x) ⊓ y) ⊓ y = y\"\n using \"36\" (*‹(top ⊔ ?x) ⊓ ?y = ?y ⊔ ?x ⊓ ?y›*) \"2097\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2136: \"⋀x y . - - ((x ⊔ top) ⊓ y) ⊓ y = y\"\n using \"37\" (*‹(?x ⊔ top) ⊓ ?y = ?y ⊔ ?x ⊓ ?y›*) \"2097\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2137: \"⋀x y . - x ⊔ - - (x ⊔ y) = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"288\" (*‹- ?x ⊔ - (- ?x ⊓ ?y) = top›*) \"2097\" (*‹- - ((?x::'a) ⊔ (?y::'a)) ⊓ ?x = ?x›*) by metis\n have 2138: \"⋀x y . - x ⊓ - (x ⊔ y) = - (x ⊔ y)\"\n using \"315\" (*‹- (- (?x::'a) ⊓ (?y::'a)) ⊓ ?x = ?x›*) \"2097\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2151: \"⋀x y . - x ⊔ - (x ⊔ y) = - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"132\" (*‹- - - ?x = - ?x›*) \"373\" (*‹- ?x ⊔ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"2097\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2191: \"⋀x y . - - x ⊓ (- y ⊓ x) = - y ⊓ x\"\n using \"88\" (*‹?x ⊔ - ?y ⊓ ?x = ?x›*) \"2124\" (*‹- - (?x ⊔ ?y) ⊓ ?y = ?y›*) by metis\n have 2201: \"⋀x y . - x ⊔ - - (y ⊔ x) = top\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"288\" (*‹- ?x ⊔ - (- ?x ⊓ ?y) = top›*) \"2124\" (*‹- - (?x ⊔ ?y) ⊓ ?y = ?y›*) by metis\n have 2202: \"⋀x y . - x ⊓ - (y ⊔ x) = - (y ⊔ x)\"\n using \"315\" (*‹- (- (?x::'a::type) ⊓ (?y::'a::type)) ⊓ ?x = ?x›*) \"2124\" (*‹- - (?x ⊔ ?y) ⊓ ?y = ?y›*) by metis\n have 2320: \"⋀x y . - (x ⊓ (y ⊔ top)) = - x\"\n using \"83\" (*‹?x ⊔ ?x ⊓ (?y ⊔ top) = ?x ⊓ (?y ⊔ top)›*) \"373\" (*‹- ?x ⊔ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"2151\" (*‹- ?x ⊔ - (?x ⊔ ?y) = - ?x›*) by metis\n have 2343: \"⋀x y . - (- x ⊓ y) ⊔ - - y = top\"\n using \"88\" (*‹(?x::'a) ⊔ - (?y::'a) ⊓ ?x = ?x›*) \"2201\" (*‹- ?x ⊔ - - (?y ⊔ ?x) = top›*) by metis\n have 2546: \"⋀x y z . - x ⊔ ((- - x ⊓ - y) ⊔ z) = - x ⊔ (- y ⊔ z)\"\n using \"111\" (*‹- ?x ⊓ ?y ⊔ (- - ?x ⊓ ?y ⊔ ?z) = ?y ⊔ ?z›*) \"116\" (*‹?x ⊔ (?x ⊓ - ?y ⊔ ?z) = ?x ⊔ ?z›*) by metis\n have 2706: \"⋀x y z . - x ⊔ (y ⊔ - - ((top ⊔ z) ⊓ - x)) = y ⊔ - - ((top ⊔ z) ⊓ - x)\"\n using \"755\" (*‹?x ⊓ - ?y ⊔ (?z ⊔ ?x) = ?z ⊔ ?x›*) \"2135\" (*‹- - ((top ⊔ ?x) ⊓ ?y) ⊓ ?y = ?y›*) by metis\n have 2810: \"⋀x y . - x ⊓ - ((y ⊔ top) ⊓ x) = - ((y ⊔ top) ⊓ x)\"\n using \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) \"2136\" (*‹- - ((?x ⊔ top) ⊓ ?y) ⊓ ?y = ?y›*) by metis\n have 3022: \"⋀x y . - x ⊔ - (- y ⊓ - x) = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"132\" (*‹- - - ?x = - ?x›*) \"2343\" (*‹- (- ?x ⊓ ?y) ⊔ - - ?y = top›*) by metis\n have 3133: \"⋀x y . - (- x ⊓ - y) ⊓ y = y\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"3022\" (*‹- ?x ⊔ - (- ?y ⊓ - ?x) = top›*) by metis\n have 3134: \"⋀x y . - x ⊓ (- y ⊓ - x) = - y ⊓ - x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"43\" (*‹(?x ⊔ - ?y) ⊓ ?y = ?x ⊓ ?y›*) \"3022\" (*‹- (?x::'a) ⊔ - (- (?y::'a) ⊓ - ?x) = top›*) by metis\n have 3961: \"⋀x y . - - (x ⊔ y) ⊓ - - x = - - x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"136\" (*‹(- ?x ⊔ ?y) ⊓ - - ?x = ?y ⊓ - - ?x›*) \"2137\" (*‹- ?x ⊔ - - (?x ⊔ ?y) = top›*) by metis\n have 4644: \"⋀x y z . - (x ⊓ - y) ⊓ (x ⊓ - (y ⊔ z)) = bot\"\n using \"1594\" (*‹- (?x ⊓ (?y ⊔ ?z)) ⊓ (?x ⊓ ?z) = bot›*) \"2151\" (*‹- ?x ⊔ - (?x ⊔ ?y) = - ?x›*) by metis\n have 5495: \"⋀x y z . - - (x ⊓ y) ⊓ - (x ⊔ z) = bot\"\n using \"421\" (*‹- - (?x ⊓ ?y) ⊓ (- ?x ⊓ ?z) = bot›*) \"2138\" (*‹- ?x ⊓ - (?x ⊔ ?y) = - (?x ⊔ ?y)›*) by metis\n have 9413: \"⋀x y . - - (- x ⊓ y) ⊓ y = - x ⊓ y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"182\" (*‹((?x::'a) ⊔ - - ((?y::'a) ⊓ (?z::'a)) ⊓ ?y) ⊓ ?z = (?x ⊔ ?y) ⊓ ?z›*) \"319\" (*‹- ?x ⊔ - - (- ?x ⊓ ?y) = - ?x›*) by metis\n have 9519: \"⋀x y z . - - (- x ⊓ y) ⊓ - - (x ⊓ z) = bot\"\n using \"373\" (*‹- ?x ⊔ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"5495\" (*‹- - ((?x::'a) ⊓ (?y::'a)) ⊓ - (?x ⊔ (?z::'a)) = bot›*) by metis\n have 11069: \"⋀x y z . - (- - x ⊓ y) ⊔ (- x ⊓ - z) = - (- - x ⊓ y)\"\n using \"316\" (*‹- (- ?x ⊓ ?y) ⊓ (?x ⊓ ?z) = ?x ⊓ ?z›*) \"757\" (*‹?x ⊔ ?x ⊓ (- ?y ⊓ - ?z) = ?x›*) by metis\n have 12370: \"⋀x y . - x ⊓ - (- - x ⊓ y) = - x\"\n using \"132\" (*‹- - - (?x::'a) = - ?x›*) \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) by metis\n have 12376: \"⋀x y . - x ⊓ - (x ⊓ y) = - x\"\n using \"127\" (*‹- - ?x ⊓ (?x ⊓ ?y) = ?x ⊓ ?y›*) \"132\" (*‹- - - ?x = - ?x›*) \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) by metis\n have 12383: \"⋀x y . - (x ⊔ y) ⊓ - y = - (x ⊔ y)\"\n using \"132\" (*‹- - - ?x = - ?x›*) \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) \"2124\" (*‹- - (?x ⊔ ?y) ⊓ ?y = ?y›*) by metis\n have 12393: \"⋀x y . - - (- x ⊓ - y) = - x ⊓ - y\"\n using \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) \"3133\" (*‹- (- ?x ⊓ - ?y) ⊓ ?y = ?y›*) \"9413\" (*‹- - (- ?x ⊓ ?y) ⊓ ?y = - ?x ⊓ ?y›*) by metis\n have 12407: \"⋀x y . - - x ⊓ - - (x ⊔ y) = - - x\"\n using \"1155\" (*‹- - (?x::'a) ⊓ - (- ?x ⊓ (?y::'a)) = - - ?x›*) \"2138\" (*‹- ?x ⊓ - (?x ⊔ ?y) = - (?x ⊔ ?y)›*) by metis\n have 12639: \"⋀x y . - x ⊓ - (- y ⊓ x) = - x\"\n using \"88\" (*‹?x ⊔ - ?y ⊓ ?x = ?x›*) \"12383\" (*‹- ((?x::'a::type) ⊔ (?y::'a::type)) ⊓ - ?y = - (?x ⊔ ?y)›*) by metis\n have 24647: \"⋀x y . (- x ⊓ - y) ⊔ - (- x ⊓ - y) = top\"\n using \"19\" (*‹- (?x::'a) ⊔ - - ?x = top›*) \"12393\" (*‹- - (- (?x::'a) ⊓ - (?y::'a)) = - ?x ⊓ - ?y›*) by metis\n have 28269: \"⋀x y z . - - (x ⊓ y) ⊔ - (- x ⊓ z) = - (- x ⊓ z)\"\n using \"373\" (*‹- ?x ⊔ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"404\" (*‹- ((?x::'a) ⊓ (?y::'a)) ⊓ (- ?x ⊓ (?z::'a)) = - ?x ⊓ ?z›*) by metis\n have 28338: \"⋀x y . - (- - (x ⊓ y) ⊓ x) = - (x ⊓ y)\"\n using \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"405\" (*‹- (?x ⊓ (?y ⊓ ?z)) ⊓ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"12370\" (*‹- ?x ⊓ - (- - ?x ⊓ ?y) = - ?x›*) by metis\n have 28422: \"⋀x y . - (- x ⊓ - y) = - (- y ⊓ - x)\"\n using \"13\" (*‹(?x::'a::type) ⊓ (?y::'a::type) ⊓ (?z::'a::type) = ?x ⊓ (?y ⊓ ?z)›*) \"3134\" (*‹- ?x ⊓ (- ?y ⊓ - ?x) = - ?y ⊓ - ?x›*) \"12393\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"28338\" (*‹- (- - ((?x::'a::type) ⊓ (?y::'a::type)) ⊓ ?x) = - (?x ⊓ ?y)›*) by metis\n have 28485: \"⋀x y . - x ⊓ - y = - y ⊓ - x\"\n using \"2097\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) \"3961\" (*‹- - (?x ⊔ ?y) ⊓ - - ?x = - - ?x›*) \"12393\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"28422\" (*‹- (- ?x ⊓ - ?y) = - (- ?y ⊓ - ?x)›*) by metis\n have 30411: \"⋀x y . - x ⊓ (x ⊔ (x ⊓ y)) = bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"82\" (*‹(?x::'a::type) ⊓ (?y::'a::type) ⊔ ?x ⊓ (?y ⊔ (?z::'a::type)) = ?x ⊓ (?y ⊔ ?z)›*) \"2094\" (*‹- (?x ⊔ ?y ⊓ (?z ⊔ top)) ⊓ (?x ⊔ ?y) = bot›*) \"2320\" (*‹- (?x ⊓ (?y ⊔ top)) = - ?x›*) by metis\n have 30469: \"⋀x . - x ⊓ (x ⊔ - - x) = bot\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"132\" (*‹- - - ?x = - ?x›*) \"30411\" (*‹- ?x ⊓ (?x ⊔ ?x ⊓ ?y) = bot›*) by metis\n have 37513: \"⋀x y . - (- x ⊓ - y) ⊓ - (y ⊔ x) = bot\"\n using \"2202\" (*‹- ?x ⊓ - (?y ⊔ ?x) = - (?y ⊔ ?x)›*) \"4644\" (*‹- (?x ⊓ - ?y) ⊓ (?x ⊓ - (?y ⊔ ?z)) = bot›*) by metis\n have 52421: \"⋀x y . - (- x ⊓ - (- x ⊓ y)) ⊓ y = y\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"144\" (*‹(- (?x ⊓ ?y) ⊓ ?x ⊔ ?z) ⊓ ?y = ?z ⊓ ?y›*) \"24647\" (*‹- ?x ⊓ - ?y ⊔ - (- ?x ⊓ - ?y) = top›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) by metis\n have 52520: \"⋀x y . - x ⊓ - (- x ⊓ y) = - x ⊓ - y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"12376\" (*‹- (?x::'a) ⊓ - (?x ⊓ (?y::'a)) = - ?x›*) \"12393\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"12639\" (*‹- ?x ⊓ - (- ?y ⊓ ?x) = - ?x›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"52421\" (*‹- (- ?x ⊓ - (- ?x ⊓ ?y)) ⊓ ?y = ?y›*) by metis\n have 52533: \"⋀x y z . - - (x ⊔ (y ⊓ (z ⊔ top))) ⊓ (x ⊔ y) = x ⊔ y\"\n using \"15\" (*‹(?x::'a) ⊓ top = ?x›*) \"49\" (*‹- bot = top›*) \"2094\" (*‹- (?x ⊔ ?y ⊓ (?z ⊔ top)) ⊓ (?x ⊔ ?y) = bot›*) \"52421\" (*‹- (- ?x ⊓ - (- ?x ⊓ ?y)) ⊓ ?y = ?y›*) by metis\n have 61101: \"⋀x y z . - (- - x ⊓ y) ⊔ z = - x ⊔ (- y ⊔ z)\"\n using \"111\" (*‹- ?x ⊓ ?y ⊔ (- - ?x ⊓ ?y ⊔ ?z) = ?y ⊔ ?z›*) \"2546\" (*‹- (?x::'a) ⊔ (- - ?x ⊓ - (?y::'a) ⊔ (?z::'a)) = - ?x ⊔ (- ?y ⊔ ?z)›*) \"12370\" (*‹- ?x ⊓ - (- - ?x ⊓ ?y) = - ?x›*) \"52520\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) by metis\n have 61156: \"⋀x y . - - (- x ⊓ y) = - x ⊓ - - y\"\n using \"419\" (*‹- ?x ⊓ - - (- ?x ⊓ ?y) = - - (- ?x ⊓ ?y)›*) \"52520\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) by metis\n have 61162: \"⋀x y . - (x ⊔ (x ⊓ y)) = - x\"\n using \"15\" (*‹(?x::'a::type) ⊓ top = ?x›*) \"49\" (*‹- bot = top›*) \"2138\" (*‹- ?x ⊓ - (?x ⊔ ?y) = - (?x ⊔ ?y)›*) \"30411\" (*‹- ?x ⊓ (?x ⊔ ?x ⊓ ?y) = bot›*) \"52520\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) by metis\n have 61163: \"⋀x . - (x ⊔ - - x) = - x\"\n using \"15\" (*‹?x ⊓ top = ?x›*) \"49\" (*‹- bot = top›*) \"2138\" (*‹- ?x ⊓ - (?x ⊔ ?y) = - (?x ⊔ ?y)›*) \"30469\" (*‹- ?x ⊓ (?x ⊔ - - ?x) = bot›*) \"52520\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) by metis\n have 61229: \"⋀x y z . - x ⊓ (- - y ⊓ - (x ⊓ z)) = - x ⊓ - - y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"15\" (*‹?x ⊓ top = ?x›*) \"49\" (*‹- bot = top›*) \"132\" (*‹- - - (?x::'a::type) = - ?x›*) \"9519\" (*‹- - (- ?x ⊓ ?y) ⊓ - - (?x ⊓ ?z) = bot›*) \"52520\" (*‹- (?x::'a) ⊓ - (- ?x ⊓ (?y::'a)) = - ?x ⊓ - ?y›*) \"61156\" (*‹- - (- ?x ⊓ ?y) = - ?x ⊓ - - ?y›*) by metis\n have 61311: \"⋀x y . - x ⊔ - y = - (- - y ⊓ x)\"\n using \"119\" (*‹?x ⊔ (?y ⊔ ?x ⊓ - ?z) = ?y ⊔ ?x›*) \"11069\" (*‹- (- - ?x ⊓ ?y) ⊔ - ?x ⊓ - ?z = - (- - ?x ⊓ ?y)›*) \"61101\" (*‹- (- - ?x ⊓ ?y) ⊔ ?z = - ?x ⊔ (- ?y ⊔ ?z)›*) by metis\n have 61391: \"⋀x y . - (- x ⊓ - - y) = - (- x ⊓ y)\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"28269\" (*‹- - (?x ⊓ ?y) ⊔ - (- ?x ⊓ ?z) = - (- ?x ⊓ ?z)›*) \"61156\" (*‹- - (- ?x ⊓ ?y) = - ?x ⊓ - - ?y›*) \"61229\" (*‹- (?x::'a::type) ⊓ (- - (?y::'a::type) ⊓ - (?x ⊓ (?z::'a::type))) = - ?x ⊓ - - ?y›*) \"61311\" (*‹- (?x::'a) ⊔ - (?y::'a) = - (- - ?y ⊓ ?x)›*) by metis\n have 61420: \"⋀x y . - (- - x ⊓ y) = - (- - y ⊓ x)\"\n using \"13\" (*‹(?x::'a) ⊓ (?y::'a) ⊓ (?z::'a) = ?x ⊓ (?y ⊓ ?z)›*) \"134\" (*‹- ?x ⊔ - (- - ?x ⊓ ?y) = - (- - ?x ⊓ ?y)›*) \"2191\" (*‹- - ?x ⊓ (- ?y ⊓ ?x) = - ?y ⊓ ?x›*) \"61156\" (*‹- - (- ?x ⊓ ?y) = - ?x ⊓ - - ?y›*) \"61311\" (*‹- ?x ⊔ - ?y = - (- - ?y ⊓ ?x)›*) by metis\n have 61454: \"⋀x y . - (x ⊔ - (- y ⊓ - x)) = - y ⊓ - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"132\" (*‹- - - ?x = - ?x›*) \"3133\" (*‹- (- (?x::'a::type) ⊓ - (?y::'a::type)) ⊓ ?y = ?y›*) \"61156\" (*‹- - (- ?x ⊓ ?y) = - ?x ⊓ - - ?y›*) \"61162\" (*‹- (?x ⊔ ?x ⊓ ?y) = - ?x›*) by metis\n have 61648: \"⋀x y . - x ⊓ (x ⊔ (- y ⊓ - - x)) = bot\"\n using \"1117\" (*‹- (?x ⊔ ?y) ⊓ (?x ⊔ - ?z ⊓ ?y) = bot›*) \"61163\" (*‹- (?x ⊔ - - ?x) = - ?x›*) by metis\n have 62434: \"⋀x y . - (- - x ⊓ y) ⊓ x = - y ⊓ x\"\n using \"43\" (*‹(?x ⊔ - ?y) ⊓ ?y = ?x ⊓ ?y›*) \"61311\" (*‹- ?x ⊔ - ?y = - (- - ?y ⊓ ?x)›*) by metis\n have 63947: \"⋀x y . - (- x ⊓ y) ⊓ - (- y ⊔ x) = bot\"\n using \"37513\" (*‹- (- ?x ⊓ - ?y) ⊓ - (?y ⊔ ?x) = bot›*) \"61391\" (*‹- (- ?x ⊓ - - ?y) = - (- ?x ⊓ ?y)›*) by metis\n have 64227: \"⋀x y . - (x ⊔ (- y ⊓ - - x)) = - x\"\n using \"15\" (*‹(?x::'a) ⊓ top = ?x›*) \"49\" (*‹- bot = top›*) \"2138\" (*‹- ?x ⊓ - (?x ⊔ ?y) = - (?x ⊔ ?y)›*) \"52520\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) \"61648\" (*‹- ?x ⊓ (?x ⊔ - ?y ⊓ - - ?x) = bot›*) by metis\n have 64239: \"⋀x y . - (x ⊔ (- - x ⊔ y)) = - (x ⊔ y)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"12407\" (*‹- - ?x ⊓ - - (?x ⊔ ?y) = - - ?x›*) \"64227\" (*‹- (?x ⊔ - ?y ⊓ - - ?x) = - ?x›*) by metis\n have 64241: \"⋀x y . - (x ⊔ (- - x ⊓ - y)) = - x\"\n using \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"64227\" (*‹- (?x ⊔ - ?y ⊓ - - ?x) = - ?x›*) by metis\n have 64260: \"⋀x y . - (x ⊔ - - (x ⊓ y)) = - x\"\n using \"420\" (*‹- - ?x ⊓ - - (?x ⊓ ?y) = - - (?x ⊓ ?y)›*) \"64241\" (*‹- (?x ⊔ - - ?x ⊓ - ?y) = - ?x›*) by metis\n have 64271: \"⋀x y . - (- x ⊔ (y ⊔ - - (y ⊓ x))) = - (- x ⊔ y)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"64260\" (*‹- (?x ⊔ - - (?x ⊓ ?y)) = - ?x›*) by metis\n have 64281: \"⋀x y . - (- x ⊔ y) = - (y ⊔ - - ((top ⊔ y) ⊓ - x))\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"25\" (*‹(?x::'a::type) ⊔ ((?y::'a::type) ⊔ (?z::'a::type)) = ?y ⊔ (?x ⊔ ?z)›*) \"157\" (*‹(- ?x ⊔ ?y) ⊓ - ?x = (top ⊔ ?y) ⊓ - ?x›*) \"2706\" (*‹- ?x ⊔ (?y ⊔ - - ((top ⊔ ?z) ⊓ - ?x)) = ?y ⊔ - - ((top ⊔ ?z) ⊓ - ?x)›*) \"64260\" (*‹- (?x ⊔ - - (?x ⊓ ?y)) = - ?x›*) by metis\n have 64282: \"⋀x y . - (x ⊔ - - ((x ⊔ top) ⊓ y)) = - (x ⊔ - - y)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"132\" (*‹- - - ?x = - ?x›*) \"536\" (*‹(?x ⊔ - - ?y) ⊓ ?y = (?x ⊔ top) ⊓ ?y›*) \"2810\" (*‹- (?x::'a::type) ⊓ - (((?y::'a::type) ⊔ top) ⊓ ?x) = - ((?y ⊔ top) ⊓ ?x)›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"61311\" (*‹- ?x ⊔ - ?y = - (- - ?y ⊓ ?x)›*) \"64260\" (*‹- (?x ⊔ - - (?x ⊓ ?y)) = - ?x›*) by metis\n have 65110: \"⋀x y . - ((- x ⊓ y) ⊔ (- y ⊔ x)) = bot\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"14\" (*‹top ⊓ ?x = ?x›*) \"49\" (*‹- bot = top›*) \"37513\" (*‹- (- (?x::'a) ⊓ - (?y::'a)) ⊓ - (?y ⊔ ?x) = bot›*) \"63947\" (*‹- (- ?x ⊓ ?y) ⊓ - (- ?y ⊔ ?x) = bot›*) by metis\n have 65231: \"⋀x y . - (x ⊔ ((- x ⊓ y) ⊔ - y)) = bot\"\n using \"9\" (*‹(?x::'a::type) ⊔ (?y::'a::type) = ?y ⊔ ?x›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"65110\" (*‹- (- ?x ⊓ ?y ⊔ (- ?y ⊔ ?x)) = bot›*) by metis\n have 65585: \"⋀x y . - (x ⊔ - y) = - - y ⊓ - x\"\n using \"61311\" (*‹- ?x ⊔ - ?y = - (- - ?y ⊓ ?x)›*) \"61454\" (*‹- (?x ⊔ - (- ?y ⊓ - ?x)) = - ?y ⊓ - ?x›*) \"64239\" (*‹- (?x ⊔ (- - ?x ⊔ ?y)) = - (?x ⊔ ?y)›*) by metis\n have 65615: \"⋀x y . - x ⊓ - ((x ⊔ top) ⊓ y) = - y ⊓ - x\"\n using \"132\" (*‹- - - ?x = - ?x›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"64282\" (*‹- (?x ⊔ - - ((?x ⊔ top) ⊓ ?y)) = - (?x ⊔ - - ?y)›*) \"65585\" (*‹- (?x ⊔ - ?y) = - - ?y ⊓ - ?x›*) by metis\n have 65616: \"⋀x y . - (- x ⊔ y) = - y ⊓ - ((top ⊔ y) ⊓ - x)\"\n using \"132\" (*‹- - - (?x::'a) = - ?x›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"64281\" (*‹- (- (?x::'a::type) ⊔ (?y::'a::type)) = - (?y ⊔ - - ((top ⊔ ?y) ⊓ - ?x))›*) \"65585\" (*‹- (?x ⊔ - ?y) = - - ?y ⊓ - ?x›*) by metis\n have 65791: \"⋀x y . - x ⊓ - ((top ⊔ x) ⊓ - y) = - - y ⊓ - x\"\n using \"89\" (*‹top ⊔ (?x ⊔ - ?y) = ?x ⊔ top›*) \"132\" (*‹- - - ?x = - ?x›*) \"12376\" (*‹- ?x ⊓ - (?x ⊓ ?y) = - ?x›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"64271\" (*‹- (- ?x ⊔ (?y ⊔ - - (?y ⊓ ?x))) = - (- ?x ⊔ ?y)›*) \"65585\" (*‹- (?x ⊔ - ?y) = - - ?y ⊓ - ?x›*) \"65615\" (*‹- ?x ⊓ - ((?x ⊔ top) ⊓ ?y) = - ?y ⊓ - ?x›*) \"65616\" (*‹- (- ?x ⊔ ?y) = - ?y ⊓ - ((top ⊔ ?y) ⊓ - ?x)›*) by metis\n have 65933: \"⋀x y . - (- x ⊔ y) = - - x ⊓ - y\"\n using \"65616\" (*‹- (- ?x ⊔ ?y) = - ?y ⊓ - ((top ⊔ ?y) ⊓ - ?x)›*) \"65791\" (*‹- (?x::'a) ⊓ - ((top ⊔ ?x) ⊓ - (?y::'a)) = - - ?y ⊓ - ?x›*) by metis\n have 66082: \"⋀x y z . - (x ⊔ (y ⊔ - z)) = - - z ⊓ - (x ⊔ y)\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"65585\" (*‹- (?x ⊔ - ?y) = - - ?y ⊓ - ?x›*) by metis\n have 66204: \"⋀x y . - - x ⊓ - (y ⊔ (- y ⊓ x)) = bot\"\n using \"65231\" (*‹- (?x ⊔ (- ?x ⊓ ?y ⊔ - ?y)) = bot›*) \"66082\" (*‹- (?x ⊔ (?y ⊔ - ?z)) = - - ?z ⊓ - (?x ⊔ ?y)›*) by metis\n have 66281: \"⋀x y z . - (x ⊔ (- y ⊔ z)) = - - y ⊓ - (x ⊔ z)\"\n using \"25\" (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?y ⊔ (?x ⊔ ?z)›*) \"65933\" (*‹- (- ?x ⊔ ?y) = - - ?x ⊓ - ?y›*) by metis\n have 67527: \"⋀x y . - - (x ⊔ (- x ⊓ y)) ⊓ y = y\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"49\" (*‹- bot = top›*) \"62434\" (*‹- (- - (?x::'a) ⊓ (?y::'a)) ⊓ ?x = - ?y ⊓ ?x›*) \"66204\" (*‹- - (?x::'a) ⊓ - ((?y::'a) ⊔ - ?y ⊓ ?x) = bot›*) by metis\n have 67762: \"⋀x y . - (- - x ⊓ (y ⊔ (- y ⊓ x))) = - x\"\n using \"61420\" (*‹- (- - ?x ⊓ ?y) = - (- - ?y ⊓ ?x)›*) \"67527\" (*‹- - (?x ⊔ - ?x ⊓ ?y) ⊓ ?y = ?y›*) by metis\n have 68018: \"⋀x y z . - (x ⊔ y) ⊓ (x ⊔ (y ⊓ (z ⊔ top))) = bot\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"83\" (*‹?x ⊔ ?x ⊓ (?y ⊔ top) = ?x ⊓ (?y ⊔ top)›*) \"1113\" (*‹- ((?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) ⊓ (?u::'a)) ⊓ (?x ⊔ ?z ⊓ ?u) = bot›*) \"2320\" (*‹- (?x ⊓ (?y ⊔ top)) = - ?x›*) by metis\n have 71989: \"⋀x y z . - (x ⊔ (y ⊓ (z ⊔ top))) = - (x ⊔ y)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"52533\" (*‹- - ((?x::'a) ⊔ (?y::'a) ⊓ ((?z::'a) ⊔ top)) ⊓ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"67762\" (*‹- (- - ?x ⊓ (?y ⊔ - ?y ⊓ ?x)) = - ?x›*) \"68018\" (*‹- (?x ⊔ ?y) ⊓ (?x ⊔ ?y ⊓ (?z ⊔ top)) = bot›*) by metis\n have 71997: \"⋀x y z . - ((x ⊓ (y ⊔ top)) ⊔ z) = - (x ⊔ z)\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"2320\" (*‹- ((?x::'a::type) ⊓ ((?y::'a::type) ⊔ top)) = - ?x›*) \"71989\" (*‹- (?x ⊔ ?y ⊓ (?z ⊔ top)) = - (?x ⊔ ?y)›*) by metis\n have 72090: \"⋀x y z . - (x ⊔ ((x ⊓ y) ⊔ z)) = - (x ⊔ z)\"\n using \"10\" (*‹?x ⊔ bot = ?x›*) \"14\" (*‹top ⊓ ?x = ?x›*) \"705\" (*‹?x ⊓ (?y ⊔ ?z) ⊔ (?x ⊓ (?y ⊔ ?z ⊓ bot) ⊔ ?u) = ?x ⊓ (?y ⊔ ?z) ⊔ ?u›*) \"71997\" (*‹- ((?x::'a) ⊓ ((?y::'a) ⊔ top) ⊔ (?z::'a)) = - (?x ⊔ ?z)›*) by metis\n have 72139: \"⋀x y . - (x ⊔ y) = - x ⊓ - y\"\n using \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"132\" (*‹- - - (?x::'a) = - ?x›*) \"2138\" (*‹- ?x ⊓ - (?x ⊔ ?y) = - (?x ⊔ ?y)›*) \"65933\" (*‹- (- (?x::'a::type) ⊔ (?y::'a::type)) = - - ?x ⊓ - ?y›*) \"66281\" (*‹- (?x ⊔ (- ?y ⊔ ?z)) = - - ?y ⊓ - (?x ⊔ ?z)›*) \"72090\" (*‹- ((?x::'a) ⊔ (?x ⊓ (?y::'a) ⊔ (?z::'a))) = - (?x ⊔ ?z)›*) by metis\n show \"?thesis\"\n (*goal: ‹- x ⊓ - y = - (x ⊔ y)›*)\n using \"72139\" (*‹- (?x ⊔ ?y) = - ?x ⊓ - ?y›*) by metis\nqed\n\nlemma l15:\n \"--(x ⊔ y) = --x ⊔ --y\"\n by (simp add: l11 (*‹- ?x ⊔ - ?y = - (- - ?x ⊓ - - ?y)›*) l12 (*‹- ?x ⊓ - ?y = - (?x ⊔ ?y)›*) l4 (*‹- - - ?x = - ?x›*))\n\nlemma l13_var:\n \"- - (- x ⊓ y) = - x ⊓ - - y\"\nproof (-)\n (*goal: ‹- - (- x ⊓ y) = - x ⊓ - - y›*)\n have 1: \"⋀x y . x ≤ y ⟷ x ⊔ y = y\"\n by (simp add: il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n have 4: \"⋀x y . ¬(x ≤ y) ∨ x ⊔ y = y\"\n using \"1\" (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by metis\n have 5: \"⋀x y z . (x ⊓ y) ⊔ (x ⊓ z) ≤ x ⊓ (y ⊔ z)\"\n by (simp add: il_sub_inf_right_isotone_var (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*))\n have 6: \"⋀x y . - - x ≤ - (- x ⊓ y)\"\n by (simp add: pad2 (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*))\n have 7: \"⋀x y z . x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n by (simp add: il_associative (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?y ⊔ ?z›*))\n have 8: \"⋀x y z . (x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)\"\n using \"7\" (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*) by metis\n have 9: \"⋀x y . x ⊔ y = y ⊔ x\"\n by (simp add: il_commutative (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*))\n have 10: \"⋀x . x ⊔ bot = x\"\n by (simp add: il_bot_unit (*‹?x ⊔ bot = ?x›*))\n have 11: \"⋀x . x ⊔ x = x\"\n by simp\n have 12: \"⋀x y z . x ⊓ (y ⊓ z) = (x ⊓ y) ⊓ z\"\n by (simp add: il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*))\n have 13: \"⋀x y z . (x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)\"\n using \"12\" (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) by metis\n have 14: \"⋀x . top ⊓ x = x\"\n by simp\n have 15: \"⋀x . x ⊓ top = x\"\n by simp\n have 16: \"⋀x y z . (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)\"\n by (simp add: il_inf_right_dist_sup (*‹((?x::'a::type) ⊔ (?y::'a::type)) ⊓ (?z::'a::type) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n have 17: \"⋀x y z . (x ⊓ y) ⊔ (z ⊓ y) = (x ⊔ z) ⊓ y\"\n using \"16\" (*‹((?x::'a) ⊔ (?y::'a)) ⊓ (?z::'a) = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) by metis\n have 19: \"⋀x . - x ⊔ - - x = top\"\n by simp\n have 20: \"⋀x . - x ⊓ x = bot\"\n by (simp add: a_inf_complement_bot (*‹- ?x ⊓ ?x = bot›*))\n have 22: \"⋀x y z . ((x ⊓ y) ⊔ (x ⊓ z)) ⊔ (x ⊓ (y ⊔ z)) = x ⊓ (y ⊔ z)\"\n using \"4\" (*‹¬ (?x::'a) ≤ (?y::'a) ∨ ?x ⊔ ?y = ?y›*) \"5\" (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 23: \"⋀x y z . (x ⊓ (y ⊔ z)) ⊔ ((x ⊓ y) ⊔ (x ⊓ z)) = x ⊓ (y ⊔ z)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"22\" (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 24: \"⋀x y . - - x ⊔ - (- x ⊓ y) = - (- x ⊓ y)\"\n using \"4\" (*‹¬ (?x::'a) ≤ (?y::'a) ∨ ?x ⊔ ?y = ?y›*) \"6\" (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*) by metis\n have 25: \"⋀x y z . x ⊔ (y ⊔ z) = y ⊔ (x ⊔ z)\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by metis\n have 26: \"⋀x y z . (x ⊓ y) ⊔ ((x ⊓ z) ⊔ (x ⊓ (y ⊔ z))) = x ⊓ (y ⊔ z)\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"23\" (*‹(?x::'a) ⊓ ((?y::'a) ⊔ (?z::'a)) ⊔ (?x ⊓ ?y ⊔ ?x ⊓ ?z) = ?x ⊓ (?y ⊔ ?z)›*) \"25\" (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?y ⊔ (?x ⊔ ?z)›*) by metis\n have 29: \"⋀x . bot ⊔ x = x\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"10\" (*‹?x ⊔ bot = ?x›*) by metis\n have 30: \"⋀x y . x ⊔ (x ⊔ y) = x ⊔ y\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"11\" (*‹?x ⊔ ?x = ?x›*) by metis\n have 34: \"⋀u x y z . (x ⊓ (y ⊓ z)) ⊔ (u ⊓ z) = ((x ⊓ y) ⊔ u) ⊓ z\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 35: \"⋀u x y z . (x ⊓ y) ⊔ (z ⊓ (u ⊓ y)) = (x ⊔ (z ⊓ u)) ⊓ y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) by metis\n have 38: \"⋀x y . - x ⊔ (- - x ⊔ y) = top ⊔ y\"\n using \"8\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) by metis\n have 41: \"- top = bot\"\n using \"15\" (*‹?x ⊓ top = ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) by metis\n have 42: \"⋀x y . (- x ⊔ y) ⊓ x = y ⊓ x\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"29\" (*‹bot ⊔ (?x::'a) = ?x›*) by metis\n have 43: \"⋀x y . (x ⊔ - y) ⊓ y = x ⊓ y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"29\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 45: \"⋀x . - bot ⊔ - - x = - bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"24\" (*‹- - (?x::'a) ⊔ - (- ?x ⊓ (?y::'a)) = - (- ?x ⊓ ?y)›*) by metis\n have 49: \"- bot = top\"\n using \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"41\" (*‹- top = bot›*) by metis\n have 50: \"⋀x . top ⊔ - - x = top\"\n using \"45\" (*‹- bot ⊔ - - ?x = - bot›*) \"49\" (*‹- bot = top›*) by metis\n have 62: \"⋀x y . x ⊔ ((x ⊓ - y) ⊔ (x ⊓ - - y)) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"15\" (*‹?x ⊓ top = ?x›*) \"19\" (*‹- (?x::'a) ⊔ - - ?x = top›*) \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 65: \"⋀x y . (- (x ⊔ y) ⊓ x) ⊔ (- (x ⊔ y) ⊓ y) = bot\"\n using \"9\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"20\" (*‹- ?x ⊓ ?x = bot›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"29\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 66: \"⋀x y z . (x ⊓ - - y) ⊔ (x ⊓ - (- y ⊓ z)) = x ⊓ - (- y ⊓ z)\"\n using \"11\" (*‹?x ⊔ ?x = ?x›*) \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"26\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) by metis\n have 69: \"⋀x y . x ⊔ (x ⊓ - - y) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"15\" (*‹?x ⊓ top = ?x›*) \"26\" (*‹(?x::'a) ⊓ (?y::'a) ⊔ (?x ⊓ (?z::'a) ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"30\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"50\" (*‹top ⊔ - - (?x::'a::type) = top›*) by metis\n have 81: \"⋀x . top ⊔ - x = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"30\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) by metis\n have 88: \"⋀x y . x ⊔ (- y ⊓ x) = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"81\" (*‹top ⊔ - ?x = top›*) by metis\n have 101: \"⋀x y z . x ⊔ (y ⊔ (x ⊓ - - z)) = y ⊔ x\"\n using \"25\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"69\" (*‹?x ⊔ ?x ⊓ - - ?y = ?x›*) by metis\n have 103: \"⋀x y . x ⊔ (x ⊓ - y) = x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"62\" (*‹?x ⊔ (?x ⊓ - ?y ⊔ ?x ⊓ - - ?y) = ?x›*) \"101\" (*‹?x ⊔ (?y ⊔ ?x ⊓ - - ?z) = ?y ⊔ ?x›*) by metis\n have 123: \"⋀x . - - x ⊓ x = x\"\n using \"14\" (*‹top ⊓ (?x::'a::type) = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) by metis\n have 127: \"⋀x y . - - x ⊓ (x ⊓ y) = x ⊓ y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 130: \"⋀x . - x ⊔ - - - x = - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"24\" (*‹- - (?x::'a) ⊔ - (- ?x ⊓ (?y::'a)) = - (- ?x ⊓ ?y)›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) by metis\n have 132: \"⋀x . - - - x = - x\"\n using \"9\" (*‹(?x::'a::type) ⊔ (?y::'a::type) = ?y ⊔ ?x›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"123\" (*‹- - (?x::'a::type) ⊓ ?x = ?x›*) \"130\" (*‹- (?x::'a) ⊔ - - - ?x = - ?x›*) by metis\n have 136: \"⋀x y . (- x ⊔ y) ⊓ - - x = y ⊓ - - x\"\n using \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"132\" (*‹- - - ?x = - ?x›*) by metis\n have 144: \"⋀x y z . ((- (x ⊓ y) ⊓ x) ⊔ z) ⊓ y = z ⊓ y\"\n using \"20\" (*‹- ?x ⊓ ?x = bot›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"34\" (*‹?x ⊓ (?y ⊓ ?z) ⊔ ?u ⊓ ?z = (?x ⊓ ?y ⊔ ?u) ⊓ ?z›*) by metis\n have 182: \"⋀x y z . (x ⊔ (- - (y ⊓ z) ⊓ y)) ⊓ z = (x ⊔ y) ⊓ z\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"35\" (*‹?x ⊓ ?y ⊔ ?z ⊓ (?u ⊓ ?y) = (?x ⊔ ?z ⊓ ?u) ⊓ ?y›*) \"123\" (*‹- - (?x::'a) ⊓ ?x = ?x›*) by metis\n have 288: \"⋀x y . - x ⊔ - (- x ⊓ y) = top\"\n using \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"38\" (*‹- ?x ⊔ (- - ?x ⊔ ?y) = top ⊔ ?y›*) \"81\" (*‹top ⊔ - ?x = top›*) by metis\n have 315: \"⋀x y . - (- x ⊓ y) ⊓ x = x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"288\" (*‹- (?x::'a) ⊔ - (- ?x ⊓ (?y::'a)) = top›*) by metis\n have 319: \"⋀x y . - x ⊔ - - (- x ⊓ y) = - x\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"24\" (*‹- - ?x ⊔ - (- ?x ⊓ ?y) = - (- ?x ⊓ ?y)›*) \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) by metis\n have 387: \"⋀x y . - (x ⊓ y) ⊓ - x = - x\"\n using \"127\" (*‹- - (?x::'a) ⊓ (?x ⊓ (?y::'a)) = ?x ⊓ ?y›*) \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) by metis\n have 405: \"⋀x y z . - (x ⊓ (y ⊓ z)) ⊓ - (x ⊓ y) = - (x ⊓ y)\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"387\" (*‹- (?x ⊓ ?y) ⊓ - ?x = - ?x›*) by metis\n have 419: \"⋀x y . - x ⊓ - - (- x ⊓ y) = - - (- x ⊓ y)\"\n using \"315\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) \"387\" (*‹- (?x ⊓ ?y) ⊓ - ?x = - ?x›*) by metis\n have 1091: \"⋀x y . - (x ⊔ y) ⊓ x = bot\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"29\" (*‹bot ⊔ ?x = ?x›*) \"30\" (*‹(?x::'a) ⊔ (?x ⊔ (?y::'a)) = ?x ⊔ ?y›*) \"65\" (*‹- (?x ⊔ ?y) ⊓ ?x ⊔ - (?x ⊔ ?y) ⊓ ?y = bot›*) by metis\n have 1129: \"⋀x y z . (- (x ⊔ y) ⊔ z) ⊓ x = z ⊓ x\"\n using \"17\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"29\" (*‹bot ⊔ (?x::'a) = ?x›*) \"1091\" (*‹- (?x ⊔ ?y) ⊓ ?x = bot›*) by metis\n have 1155: \"⋀x y . - - x ⊓ - (- x ⊓ y) = - - x\"\n using \"66\" (*‹(?x::'a) ⊓ - - (?y::'a) ⊔ ?x ⊓ - (- ?y ⊓ (?z::'a)) = ?x ⊓ - (- ?y ⊓ ?z)›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"132\" (*‹- - - (?x::'a) = - ?x›*) by metis\n have 2097: \"⋀x y . - - (x ⊔ y) ⊓ x = x\"\n using \"14\" (*‹top ⊓ (?x::'a) = ?x›*) \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"1129\" (*‹(- ((?x::'a::type) ⊔ (?y::'a::type)) ⊔ (?z::'a::type)) ⊓ ?x = ?z ⊓ ?x›*) by metis\n have 2124: \"⋀x y . - - (x ⊔ y) ⊓ y = y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"2097\" (*‹- - ((?x::'a::type) ⊔ (?y::'a::type)) ⊓ ?x = ?x›*) by metis\n have 2137: \"⋀x y . - x ⊔ - - (x ⊔ y) = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"288\" (*‹- (?x::'a) ⊔ - (- ?x ⊓ (?y::'a)) = top›*) \"2097\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) by metis\n have 2201: \"⋀x y . - x ⊔ - - (y ⊔ x) = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"288\" (*‹- ?x ⊔ - (- ?x ⊓ ?y) = top›*) \"2124\" (*‹- - ((?x::'a) ⊔ (?y::'a)) ⊓ ?y = ?y›*) by metis\n have 2343: \"⋀x y . - (- x ⊓ y) ⊔ - - y = top\"\n using \"88\" (*‹?x ⊔ - ?y ⊓ ?x = ?x›*) \"2201\" (*‹- (?x::'a) ⊔ - - ((?y::'a) ⊔ ?x) = top›*) by metis\n have 3022: \"⋀x y . - x ⊔ - (- y ⊓ - x) = top\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"132\" (*‹- - - ?x = - ?x›*) \"2343\" (*‹- (- (?x::'a::type) ⊓ (?y::'a::type)) ⊔ - - ?y = top›*) by metis\n have 3133: \"⋀x y . - (- x ⊓ - y) ⊓ y = y\"\n using \"14\" (*‹top ⊓ (?x::'a) = ?x›*) \"42\" (*‹(- ?x ⊔ ?y) ⊓ ?x = ?y ⊓ ?x›*) \"3022\" (*‹- ?x ⊔ - (- ?y ⊓ - ?x) = top›*) by metis\n have 3134: \"⋀x y . - x ⊓ (- y ⊓ - x) = - y ⊓ - x\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"43\" (*‹(?x ⊔ - ?y) ⊓ ?y = ?x ⊓ ?y›*) \"3022\" (*‹- ?x ⊔ - (- ?y ⊓ - ?x) = top›*) by metis\n have 3961: \"⋀x y . - - (x ⊔ y) ⊓ - - x = - - x\"\n using \"14\" (*‹top ⊓ (?x::'a) = ?x›*) \"136\" (*‹(- (?x::'a) ⊔ (?y::'a)) ⊓ - - ?x = ?y ⊓ - - ?x›*) \"2137\" (*‹- ?x ⊔ - - (?x ⊔ ?y) = top›*) by metis\n have 9413: \"⋀x y . - - (- x ⊓ y) ⊓ y = - x ⊓ y\"\n using \"9\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"103\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"182\" (*‹((?x::'a) ⊔ - - ((?y::'a) ⊓ (?z::'a)) ⊓ ?y) ⊓ ?z = (?x ⊔ ?y) ⊓ ?z›*) \"319\" (*‹- ?x ⊔ - - (- ?x ⊓ ?y) = - ?x›*) by metis\n have 12370: \"⋀x y . - x ⊓ - (- - x ⊓ y) = - x\"\n using \"132\" (*‹- - - (?x::'a::type) = - ?x›*) \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) by metis\n have 12376: \"⋀x y . - x ⊓ - (x ⊓ y) = - x\"\n using \"127\" (*‹- - ?x ⊓ (?x ⊓ ?y) = ?x ⊓ ?y›*) \"132\" (*‹- - - ?x = - ?x›*) \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) by metis\n have 12383: \"⋀x y . - (x ⊔ y) ⊓ - y = - (x ⊔ y)\"\n using \"132\" (*‹- - - ?x = - ?x›*) \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) \"2124\" (*‹- - (?x ⊔ ?y) ⊓ ?y = ?y›*) by metis\n have 12393: \"⋀x y . - - (- x ⊓ - y) = - x ⊓ - y\"\n using \"1155\" (*‹- - ?x ⊓ - (- ?x ⊓ ?y) = - - ?x›*) \"3133\" (*‹- (- ?x ⊓ - ?y) ⊓ ?y = ?y›*) \"9413\" (*‹- - (- (?x::'a::type) ⊓ (?y::'a::type)) ⊓ ?y = - ?x ⊓ ?y›*) by metis\n have 12639: \"⋀x y . - x ⊓ - (- y ⊓ x) = - x\"\n using \"88\" (*‹?x ⊔ - ?y ⊓ ?x = ?x›*) \"12383\" (*‹- (?x ⊔ ?y) ⊓ - ?y = - (?x ⊔ ?y)›*) by metis\n have 24647: \"⋀x y . (- x ⊓ - y) ⊔ - (- x ⊓ - y) = top\"\n using \"19\" (*‹- ?x ⊔ - - ?x = top›*) \"12393\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) by metis\n have 28338: \"⋀x y . - (- - (x ⊓ y) ⊓ x) = - (x ⊓ y)\"\n using \"123\" (*‹- - ?x ⊓ ?x = ?x›*) \"405\" (*‹- (?x ⊓ (?y ⊓ ?z)) ⊓ - (?x ⊓ ?y) = - (?x ⊓ ?y)›*) \"12370\" (*‹- (?x::'a::type) ⊓ - (- - ?x ⊓ (?y::'a::type)) = - ?x›*) by metis\n have 28422: \"⋀x y . - (- x ⊓ - y) = - (- y ⊓ - x)\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"3134\" (*‹- ?x ⊓ (- ?y ⊓ - ?x) = - ?y ⊓ - ?x›*) \"12393\" (*‹- - (- (?x::'a) ⊓ - (?y::'a)) = - ?x ⊓ - ?y›*) \"28338\" (*‹- (- - (?x ⊓ ?y) ⊓ ?x) = - (?x ⊓ ?y)›*) by metis\n have 28485: \"⋀x y . - x ⊓ - y = - y ⊓ - x\"\n using \"2097\" (*‹- - (?x ⊔ ?y) ⊓ ?x = ?x›*) \"3961\" (*‹- - (?x ⊔ ?y) ⊓ - - ?x = - - ?x›*) \"12393\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"28422\" (*‹- (- (?x::'a) ⊓ - (?y::'a)) = - (- ?y ⊓ - ?x)›*) by metis\n have 52421: \"⋀x y . - (- x ⊓ - (- x ⊓ y)) ⊓ y = y\"\n using \"14\" (*‹top ⊓ ?x = ?x›*) \"144\" (*‹(- ((?x::'a) ⊓ (?y::'a)) ⊓ ?x ⊔ (?z::'a)) ⊓ ?y = ?z ⊓ ?y›*) \"24647\" (*‹- (?x::'a) ⊓ - (?y::'a) ⊔ - (- ?x ⊓ - ?y) = top›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) by metis\n have 52520: \"⋀x y . - x ⊓ - (- x ⊓ y) = - x ⊓ - y\"\n using \"13\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"12376\" (*‹- ?x ⊓ - (?x ⊓ ?y) = - ?x›*) \"12393\" (*‹- - (- ?x ⊓ - ?y) = - ?x ⊓ - ?y›*) \"12639\" (*‹- ?x ⊓ - (- ?y ⊓ ?x) = - ?x›*) \"28485\" (*‹- ?x ⊓ - ?y = - ?y ⊓ - ?x›*) \"52421\" (*‹- (- ?x ⊓ - (- ?x ⊓ ?y)) ⊓ ?y = ?y›*) by metis\n have 61156: \"⋀x y . - - (- x ⊓ y) = - x ⊓ - - y\"\n using \"419\" (*‹- ?x ⊓ - - (- ?x ⊓ ?y) = - - (- ?x ⊓ ?y)›*) \"52520\" (*‹- ?x ⊓ - (- ?x ⊓ ?y) = - ?x ⊓ - ?y›*) by metis\n show \"?thesis\"\n (*goal: ‹- - (- x ⊓ y) = - x ⊓ - - y›*)\n using \"61156\" (*‹- - (- ?x ⊓ ?y) = - ?x ⊓ - - ?y›*) by metis\nqed\n\ntext ‹Theorem 25.1›\n\nsubclass subset_boolean_algebra_2\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: il_associative)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: il_commutative)\n show \"⋀x. x ⊔ x = x\"\n by simp\n show \"⋀x y. x ⊔ - (y ⊔ - y) = x\"\n using il_bot_unit l12 l6 by auto\n show \"⋀x y. - (x ⊔ y) = - (- - x ⊔ - - y)\"\n by (metis l15 l4)\n show \"⋀x y. - x ⊔ - (- x ⊔ y) = - x ⊔ - y\"\n by (smt l11 l15 il_inf_right_dist_sup il_unit_bot l6 l7)\nqed\n\nlemma aa_test:\n \"p = --p ⟹ test p\"\n by (metis ppa_ppd.d_closed (*‹test (- - (?x::'a))›*))\n\nlemma test_aa_increasing:\n \"test p ⟹ p ≤ --p\"\n by (simp add: ppa_ppd.d_increasing_sub_identity (*‹?x ≤ top ⟹ ?x ≤ - - ?x›*) test_sub_identity (*‹test ?x ⟹ ?x ≤ top›*))\n\nlemma \"test p ⟹ - - (p ⊓ x) ≤ p\" nitpick [expect=genuine] oops\nlemma \"test p ⟹ --p ≤ p\" nitpick [expect=genuine] oops\n\nend\n\nclass pa_algebra = pa_semiring + minus +\n assumes pa_minus_def: \"-x - -y = -(--x ⊔ -y)\"\nbegin\n\nsubclass subset_boolean_algebra_2_extended\nproof\n show \"bot = (THE x. ∀z. x = - (z ⊔ - z))\"\n using l12 l6 by auto\n thus \"top = - (THE x. ∀z. x = - (z ⊔ - z))\"\n using l2 by blast\n show \"⋀x y. - x ⊓ - y = - (- - x ⊔ - - y)\"\n by (metis l12 l4)\n show \"⋀x y. - x - - y = - (- - x ⊔ - y)\"\n by (simp add: pa_minus_def)\n show \"⋀x y. (x ≤ y) = (x ⊔ y = y)\"\n by (simp add: il_less_eq)\n show \"⋀x y. (x < y) = (x ⊔ y = y ∧ y ⊔ x ≠ x)\"\n by (simp add: il_less_eq less_le_not_le)\nqed\n\nlemma \"⋀x y. - (x ⊓ - - y) = - (x ⊓ y)\" nitpick [expect=genuine] oops\n\nend\n\nsubsection ‹Antidomain Semirings›\n\ntext ‹Definition 24›\n\nclass a_semiring = ppa_semiring +\n assumes ad3: \"-(x ⊓ y) ≤ -(x ⊓ --y)\"\nbegin\n\nlemma l16:\n \"- - x ≤ - (- x ⊓ y)\"\nproof (-)\n (*goal: ‹- - x ≤ - (- x ⊓ y)›*)\n have 1: \"⋀x y . x ≤ y ⟷ x ⊔ y = y\"\n by (simp add: il_less_eq (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*))\n have 3: \"⋀x y z . x ⊔ (y ⊔ z) = (x ⊔ y) ⊔ z\"\n by (simp add: il_associative (*‹?x ⊔ (?y ⊔ ?z) = ?x ⊔ ?y ⊔ ?z›*))\n have 4: \"⋀x y z . (x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)\"\n using \"3\" (*‹(?x::'a) ⊔ ((?y::'a) ⊔ (?z::'a)) = ?x ⊔ ?y ⊔ ?z›*) by metis\n have 5: \"⋀x y . x ⊔ y = y ⊔ x\"\n by (simp add: il_commutative (*‹?x ⊔ ?y = ?y ⊔ ?x›*))\n have 6: \"⋀x . x ⊔ bot = x\"\n by (simp add: il_bot_unit (*‹?x ⊔ bot = ?x›*))\n have 7: \"⋀x . x ⊔ x = x\"\n by simp\n have 8: \"⋀x y . ¬(x ≤ y) ∨ x ⊔ y = y\"\n using \"1\" (*‹(?x ≤ ?y) = (?x ⊔ ?y = ?y)›*) by metis\n have 9: \"⋀x y . x ≤ y ∨ x ⊔ y ≠ y\"\n using \"1\" (*‹((?x::'a) ≤ (?y::'a)) = (?x ⊔ ?y = ?y)›*) by metis\n have 10: \"⋀x y z . x ⊓ (y ⊓ z) = (x ⊓ y) ⊓ z\"\n by (simp add: il_inf_associative (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*))\n have 11: \"⋀x y z . (x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)\"\n using \"10\" (*‹?x ⊓ (?y ⊓ ?z) = ?x ⊓ ?y ⊓ ?z›*) by metis\n have 12: \"⋀x . top ⊓ x = x\"\n by simp\n have 13: \"⋀x . x ⊓ top = x\"\n by simp\n have 14: \"⋀x y z . (x ⊓ y) ⊔ (x ⊓ z) ≤ x ⊓ (y ⊔ z)\"\n by (simp add: il_sub_inf_right_isotone_var (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*))\n have 15: \"⋀x y z . (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)\"\n by (simp add: il_inf_right_dist_sup (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*))\n have 16: \"⋀x y z . (x ⊓ y) ⊔ (z ⊓ y) = (x ⊔ z) ⊓ y\"\n using \"15\" (*‹(?x ⊔ ?y) ⊓ ?z = ?x ⊓ ?z ⊔ ?y ⊓ ?z›*) by metis\n have 17: \"⋀x . bot ⊓ x = bot\"\n by simp\n have 18: \"⋀x . - x ⊔ - - x = top\"\n by simp\n have 19: \"⋀x . - x ⊓ x = bot\"\n by (simp add: a_inf_complement_bot (*‹- (?x::'a) ⊓ ?x = bot›*))\n have 20: \"⋀x y . - (x ⊓ y) ≤ - (x ⊓ - - y)\"\n by (simp add: ad3 (*‹- (?x ⊓ ?y) ≤ - (?x ⊓ - - ?y)›*))\n have 22: \"⋀x y z . x ⊔ (y ⊔ z) = y ⊔ (x ⊔ z)\"\n using \"4\" (*‹(?x::'a::type) ⊔ (?y::'a::type) ⊔ (?z::'a::type) = ?x ⊔ (?y ⊔ ?z)›*) \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) by metis\n have 25: \"⋀x . bot ⊔ x = x\"\n using \"5\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"6\" (*‹?x ⊔ bot = ?x›*) by metis\n have 26: \"⋀x y . x ⊔ (x ⊔ y) = x ⊔ y\"\n using \"4\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"7\" (*‹?x ⊔ ?x = ?x›*) by metis\n have 33: \"⋀x y z . (x ⊓ y) ⊔ ((x ⊓ z) ⊔ (x ⊓ (y ⊔ z))) = x ⊓ (y ⊔ z)\"\n using \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"8\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"14\" (*‹?x ⊓ ?y ⊔ ?x ⊓ ?z ≤ ?x ⊓ (?y ⊔ ?z)›*) \"22\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) by metis\n have 47: \"⋀x y . - x ⊔ (- - x ⊔ y) = top ⊔ y\"\n using \"4\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"18\" (*‹- (?x::'a) ⊔ - - ?x = top›*) by metis\n have 48: \"⋀x y . - - x ⊔ (y ⊔ - x) = y ⊔ top\"\n using \"4\" (*‹?x ⊔ ?y ⊔ ?z = ?x ⊔ (?y ⊔ ?z)›*) \"5\" (*‹(?x::'a) ⊔ (?y::'a) = ?y ⊔ ?x›*) \"18\" (*‹- ?x ⊔ - - ?x = top›*) by metis\n have 51: \"⋀x y . - x ⊓ (x ⊓ y) = bot\"\n using \"11\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"17\" (*‹bot ⊓ ?x = bot›*) \"19\" (*‹- ?x ⊓ ?x = bot›*) by metis\n have 52: \"- top = bot\"\n using \"13\" (*‹?x ⊓ top = ?x›*) \"19\" (*‹- ?x ⊓ ?x = bot›*) by metis\n have 56: \"⋀x y . (- x ⊔ y) ⊓ x = y ⊓ x\"\n using \"16\" (*‹(?x::'a::type) ⊓ (?y::'a::type) ⊔ (?z::'a::type) ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"19\" (*‹- ?x ⊓ ?x = bot›*) \"25\" (*‹bot ⊔ ?x = ?x›*) by metis\n have 57: \"⋀x y . (x ⊔ - y) ⊓ y = x ⊓ y\"\n using \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"16\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"19\" (*‹- (?x::'a) ⊓ ?x = bot›*) \"25\" (*‹bot ⊔ (?x::'a) = ?x›*) by metis\n have 58: \"⋀x y . - (x ⊓ y) ⊔ - (x ⊓ - - y) = - (x ⊓ - - y)\"\n using \"8\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"20\" (*‹- (?x ⊓ ?y) ≤ - (?x ⊓ - - ?y)›*) by metis\n have 60: \"⋀x . - x ≤ - - - x\"\n using \"12\" (*‹top ⊓ ?x = ?x›*) \"20\" (*‹- ((?x::'a) ⊓ (?y::'a)) ≤ - (?x ⊓ - - ?y)›*) by metis\n have 69: \"- bot = top\"\n using \"18\" (*‹- ?x ⊔ - - ?x = top›*) \"25\" (*‹bot ⊔ ?x = ?x›*) \"52\" (*‹- top = bot›*) by metis\n have 74: \"⋀x y . x ≤ x ⊔ y\"\n using \"9\" (*‹?x ≤ ?y ∨ ?x ⊔ ?y ≠ ?y›*) \"26\" (*‹(?x::'a) ⊔ (?x ⊔ (?y::'a)) = ?x ⊔ ?y›*) by metis\n have 78: \"⋀x . top ⊔ - x = top\"\n using \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"18\" (*‹- (?x::'a) ⊔ - - ?x = top›*) \"26\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) by metis\n have 80: \"⋀x y . x ≤ y ⊔ x\"\n using \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"74\" (*‹?x ≤ ?x ⊔ ?y›*) by metis\n have 86: \"⋀x y z . x ⊔ y ≤ x ⊔ (z ⊔ y)\"\n using \"22\" (*‹?x ⊔ (?y ⊔ ?z) = ?y ⊔ (?x ⊔ ?z)›*) \"80\" (*‹?x ≤ ?y ⊔ ?x›*) by metis\n have 95: \"⋀x . - x ⊔ - - - x = - - - x\"\n using \"8\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"60\" (*‹- ?x ≤ - - - ?x›*) by metis\n have 143: \"⋀x y . x ⊔ (x ⊓ - y) = x\"\n using \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"13\" (*‹?x ⊓ top = ?x›*) \"26\" (*‹?x ⊔ (?x ⊔ ?y) = ?x ⊔ ?y›*) \"33\" (*‹?x ⊓ ?y ⊔ (?x ⊓ ?z ⊔ ?x ⊓ (?y ⊔ ?z)) = ?x ⊓ (?y ⊔ ?z)›*) \"78\" (*‹top ⊔ - ?x = top›*) by metis\n have 370: \"⋀x y z . x ⊔ (y ⊓ - z) ≤ x ⊔ y\"\n using \"86\" (*‹?x ⊔ ?y ≤ ?x ⊔ (?z ⊔ ?y)›*) \"143\" (*‹(?x::'a) ⊔ ?x ⊓ - (?y::'a) = ?x›*) by metis\n have 907: \"⋀x . - x ⊓ - x = - x\"\n using \"12\" (*‹top ⊓ ?x = ?x›*) \"18\" (*‹- (?x::'a) ⊔ - - ?x = top›*) \"57\" (*‹(?x ⊔ - ?y) ⊓ ?y = ?x ⊓ ?y›*) by metis\n have 928: \"⋀x y . - x ⊓ (- x ⊓ y) = - x ⊓ y\"\n using \"11\" (*‹?x ⊓ ?y ⊓ ?z = ?x ⊓ (?y ⊓ ?z)›*) \"907\" (*‹- ?x ⊓ - ?x = - ?x›*) by metis\n have 966: \"⋀x y . - (- x ⊓ - - (x ⊓ y)) = top\"\n using \"51\" (*‹- (?x::'a) ⊓ (?x ⊓ (?y::'a)) = bot›*) \"58\" (*‹- ((?x::'a) ⊓ (?y::'a)) ⊔ - (?x ⊓ - - ?y) = - (?x ⊓ - - ?y)›*) \"69\" (*‹- bot = top›*) \"78\" (*‹top ⊔ - ?x = top›*) by metis\n have 1535: \"⋀x . - x ⊔ - - - - x = top\"\n using \"47\" (*‹- ?x ⊔ (- - ?x ⊔ ?y) = top ⊔ ?y›*) \"78\" (*‹top ⊔ - ?x = top›*) \"95\" (*‹- ?x ⊔ - - - ?x = - - - ?x›*) by metis\n have 1630: \"⋀x y z . (x ⊔ y) ⊓ - z ≤ (x ⊓ - z) ⊔ y\"\n using \"16\" (*‹?x ⊓ ?y ⊔ ?z ⊓ ?y = (?x ⊔ ?z) ⊓ ?y›*) \"370\" (*‹?x ⊔ ?y ⊓ - ?z ≤ ?x ⊔ ?y›*) by metis\n have 2422: \"⋀x . - x ⊓ - - - x = - - - x\"\n using \"12\" (*‹top ⊓ ?x = ?x›*) \"57\" (*‹(?x ⊔ - ?y) ⊓ ?y = ?x ⊓ ?y›*) \"1535\" (*‹- (?x::'a) ⊔ - - - - ?x = top›*) by metis\n have 6567: \"⋀x y . - x ⊓ - - (x ⊓ y) = bot\"\n using \"12\" (*‹top ⊓ ?x = ?x›*) \"19\" (*‹- (?x::'a) ⊓ ?x = bot›*) \"966\" (*‹- (- (?x::'a) ⊓ - - (?x ⊓ (?y::'a))) = top›*) by metis\n have 18123: \"⋀x . - - - x = - x\"\n using \"95\" (*‹- (?x::'a) ⊔ - - - ?x = - - - ?x›*) \"143\" (*‹?x ⊔ ?x ⊓ - ?y = ?x›*) \"2422\" (*‹- ?x ⊓ - - - ?x = - - - ?x›*) by metis\n have 26264: \"⋀x y . - x ≤ (- y ⊓ - x) ⊔ - - y\"\n using \"12\" (*‹top ⊓ (?x::'a::type) = ?x›*) \"18\" (*‹- ?x ⊔ - - ?x = top›*) \"1630\" (*‹(?x ⊔ ?y) ⊓ - ?z ≤ ?x ⊓ - ?z ⊔ ?y›*) by metis\n have 26279: \"⋀x y . - - (x ⊓ y) ≤ - - x\"\n using \"25\" (*‹bot ⊔ ?x = ?x›*) \"6567\" (*‹- ?x ⊓ - - (?x ⊓ ?y) = bot›*) \"26264\" (*‹- ?x ≤ - ?y ⊓ - ?x ⊔ - - ?y›*) by metis\n have 26307: \"⋀x y . - - (- x ⊓ y) ≤ - x\"\n using \"928\" (*‹- ?x ⊓ (- ?x ⊓ ?y) = - ?x ⊓ ?y›*) \"18123\" (*‹- - - (?x::'a) = - ?x›*) \"26279\" (*‹- - (?x ⊓ ?y) ≤ - - ?x›*) by metis\n have 26339: \"⋀x y . - x ⊔ - - (- x ⊓ y) = - x\"\n using \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"8\" (*‹¬ ?x ≤ ?y ∨ ?x ⊔ ?y = ?y›*) \"26307\" (*‹- - (- ?x ⊓ ?y) ≤ - ?x›*) by metis\n have 26564: \"⋀x y . - x ⊔ - (- x ⊓ y) = top\"\n using \"5\" (*‹?x ⊔ ?y = ?y ⊔ ?x›*) \"48\" (*‹- - (?x::'a) ⊔ ((?y::'a) ⊔ - ?x) = ?y ⊔ top›*) \"78\" (*‹top ⊔ - ?x = top›*) \"18123\" (*‹- - - ?x = - ?x›*) \"26339\" (*‹- ?x ⊔ - - (- ?x ⊓ ?y) = - ?x›*) by metis\n have 26682: \"⋀x y . - (- x ⊓ y) ⊓ x = x\"\n using \"12\" (*‹top ⊓ ?x = ?x›*) \"56\" (*‹(- (?x::'a) ⊔ (?y::'a)) ⊓ ?x = ?y ⊓ ?x›*) \"26564\" (*‹- ?x ⊔ - (- ?x ⊓ ?y) = top›*) by metis\n have 26864: \"⋀x y . - - x ≤ - (- x ⊓ y)\"\n using \"18123\" (*‹- - - ?x = - ?x›*) \"26279\" (*‹- - ((?x::'a) ⊓ (?y::'a)) ≤ - - ?x›*) \"26682\" (*‹- (- ?x ⊓ ?y) ⊓ ?x = ?x›*) by metis\n show \"?thesis\"\n (*goal: ‹- - x ≤ - (- x ⊓ y)›*)\n using \"26864\" (*‹- - ?x ≤ - (- ?x ⊓ ?y)›*) by metis\nqed\n\ntext ‹Theorem 25.2›\n\nsubclass pa_semiring\nproof\n show \"⋀x y. - - x ≤ - (- x ⊓ y)\"\n by (rule l16)\nqed\n\nlemma l17:\n \"-(x ⊓ y) = -(x ⊓ --y)\"\n by (simp add: ad3 (*‹- (?x ⊓ ?y) ≤ - (?x ⊓ - - ?y)›*) order.antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*) l14 (*‹- (?x ⊓ - - ?y) ≤ - (?x ⊓ ?y)›*))\n\nlemma a_complement_inf_double_complement:\n \"-(x ⊓ --y) = -(x ⊓ y)\"\n using l17 (*‹- (?x ⊓ ?y) = - (?x ⊓ - - ?y)›*) by auto\n\nsublocale a_d: d_semiring_var where d = \"λx . --x\"\nproof (standard)\n (*goals:\n 1. ‹⋀(x::'a) y::'a. - - (x ⊓ - - y) ≤ - - (x ⊓ y)›\n 2. ‹- - bot = bot›*)\n show \"⋀x y. - - (x ⊓ - - y) ≤ - - (x ⊓ y)\"\n using l17 (*‹- (?x ⊓ ?y) = - (?x ⊓ - - ?y)›*) by auto\n show \"- - bot = bot\"\n by (simp add: l1 (*‹- top = bot›*) l2 (*‹- bot = top›*))\nqed\n\nlemma \"test p ⟹ - - (p ⊓ x) ≤ p\"\n by (fact a_d.d2 (*‹test ?p ⟹ - - (?p ⊓ ?x) ≤ ?p›*))\n\nend\n\nclass a_algebra = a_semiring + minus +\n assumes a_minus_def: \"-x - -y = -(--x ⊔ -y)\"\nbegin\n\nsubclass pa_algebra\nproof\n show \"⋀x y. - x - - y = - (- - x ⊔ - y)\"\n by (simp add: a_minus_def)\nqed\n\ntext ‹Theorem 25.4›\n\nsubclass subset_boolean_algebra_4_extended\nproof\n show \"⋀x y z. x ⊓ (y ⊓ z) = x ⊓ y ⊓ z\"\n by (simp add: il_inf_associative)\n show \"⋀x y z. (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z\"\n by (simp add: il_inf_right_dist_sup)\n show \"⋀x. - x ⊓ x = bot\"\n by (simp add: a_inf_complement_bot)\n show \"⋀x. top ⊓ x = x\"\n by simp\n show \"⋀x y. - (x ⊓ - - y) = - (x ⊓ y)\"\n using l17 by auto\n show \"⋀x. x ⊓ top = x\"\n by simp\n show \"⋀x y z. x ≤ y ⟹ z ⊓ x ≤ z ⊓ y\"\n by (simp add: il_sub_inf_right_isotone)\nqed\n\nend\n\ncontext subset_boolean_algebra_4_extended\nbegin\n\nsubclass il_semiring\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: sup_assoc)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: sup_commute)\n show \"⋀x. x ⊔ x = x\"\n by simp\n show \"⋀x. x ⊔ bot = x\"\n by simp\n show \"⋀x y z. x ⊓ (y ⊓ z) = x ⊓ y ⊓ z\"\n by (simp add: sba3_inf_associative)\n show \"⋀x y z. (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z\"\n by (simp add: sba3_inf_right_dist_sup)\n show \"⋀x. top ⊓ x = x\"\n by simp\n show \"⋀x. x ⊓ top = x\"\n by simp\n show \"⋀x. bot ⊓ x = bot\"\n by (simp add: inf_left_zero)\n show \"⋀x y z. x ≤ y ⟹ z ⊓ x ≤ z ⊓ y\"\n by (simp add: inf_right_isotone)\n show \"⋀x y. (x ≤ y) = (x ⊔ y = y)\"\n by (simp add: le_iff_sup)\n show \"⋀x y. (x < y) = (x ≤ y ∧ ¬ y ≤ x)\"\n by (simp add: less_le_not_le)\nqed\n\nsubclass a_semiring\nproof\n show \"⋀x. - x ⊓ x = bot\"\n by (simp add: sba3_inf_complement_bot)\n show \"⋀x. - x ⊔ - - x = top\"\n by simp\n show \"⋀x y. - (x ⊓ y) ≤ - (x ⊓ - - y)\"\n by (simp add: sba3_complement_inf_double_complement)\nqed\n\nsublocale sba4_a: a_algebra\nproof (standard)\n (*goal: ‹⋀(x::'a) y::'a. - x - - y = - (- - x ⊔ - y)›*)\n show \"⋀x y. - x - - y = - (- - x ⊔ - y)\"\n by (simp add: sub_minus_def (*‹- (?x::'a::type) - - (?y::'a::type) = - (- - ?x ⊔ - ?y)›*))\nqed\n\nend\n\ncontext stone_algebra\nbegin\n\ntext ‹Theorem 25.3›\n\nsubclass il_semiring\nproof\n show \"⋀x y z. x ⊔ (y ⊔ z) = x ⊔ y ⊔ z\"\n by (simp add: sup_assoc)\n show \"⋀x y. x ⊔ y = y ⊔ x\"\n by (simp add: sup_commute)\n show \"⋀x. x ⊔ x = x\"\n by simp\n show \"⋀x. x ⊔ bot = x\"\n by simp\n show \"⋀x y z. x ⊓ (y ⊓ z) = x ⊓ y ⊓ z\"\n by (simp add: inf.sup_monoid.add_assoc)\n show \"⋀x y z. (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z\"\n by (simp add: inf_sup_distrib2)\n show \"⋀x. top ⊓ x = x\"\n by simp\n show \"⋀x. x ⊓ top = x\"\n by simp\n show \"⋀x. bot ⊓ x = bot\"\n by simp\n show \"⋀x y z. x ≤ y ⟹ z ⊓ x ≤ z ⊓ y\"\n using inf.sup_right_isotone by blast\n show \"⋀x y. (x ≤ y) = (x ⊔ y = y)\"\n by (simp add: le_iff_sup)\n show \"⋀x y. (x < y) = (x ≤ y ∧ ¬ y ≤ x)\"\n by (simp add: less_le_not_le)\nqed\n\nsubclass a_semiring\nproof\n show \"⋀x. - x ⊓ x = bot\"\n by simp\n show \"⋀x. - x ⊔ - - x = top\"\n by simp\n show \"⋀x y. - (x ⊓ y) ≤ - (x ⊓ - - y)\"\n by simp\nqed\n\nend\n\nend\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Subset_Boolean_Algebras/Subset_Boolean_Algebras.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"04676f97d3db76c970437d6377d3988001df2eb5dd769fc2b89e355a1bad961a\"\n}"}}},{"rowIdx":190,"cells":{"text":{"kind":"string","value":"theory Aleph_Relative\n imports\n CardinalArith_Relative\nbegin\n\ndefinition\n HAleph :: \"[i,i] ⇒ i\" where\n \"HAleph(i,r) ≡ if(¬(Ord(i)),i,if(i=0, nat, if(¬Limit(i) ∧ i≠0,\n csucc(r`( ⋃ i )),\n ⋃j∈i. r`j)))\"\n\nreldb_add functional \"Limit\" \"Limit\"\nrelationalize \"Limit\" \"is_Limit\" external\nsynthesize \"is_Limit\" from_definition\narity_theorem for \"is_Limit_fm\"\n\nrelativize functional \"HAleph\" \"HAleph_rel\"\nrelationalize \"HAleph_rel\" \"is_HAleph\"\n\nsynthesize \"is_HAleph\" from_definition assuming \"nonempty\"\narity_theorem intermediate for \"is_HAleph_fm\"\n\nlemma arity_is_HAleph_fm_aux:\n assumes\n \"i ∈ nat\" \"r ∈ nat\"\n ― ‹NOTE: assumptions are ❙‹not› used, but if omitted, next lemma fails!›\n shows\n \"arity(Replace_fm(8 +⇩ω i, ⋅10 +⇩ω r`0 is 1⋅, 3)) = 9 +⇩ω i ∪ pred(pred(11 +⇩ω r))\"\n using arity_Replace_fm[of \"⋅ (10+⇩ωr)`0 is 1⋅\" \"8+⇩ωi\" 3 \"(11+⇩ωr) ∪ 1 ∪ 2\"] (*‹⟦⋅10 +⇩ω (r::i)`0 is 1⋅ ∈ formula; 8 +⇩ω (i::i) ∈ ω; 3 ∈ ω; 11 +⇩ω r ∪ 1 ∪ 2 ∈ ω; arity(⋅10 +⇩ω r`0 is 1⋅) = 11 +⇩ω r ∪ 1 ∪ 2⟧ ⟹ arity(Replace_fm(8 +⇩ω i, ⋅10 +⇩ω r`0 is 1⋅, 3)) = 4 ∪ succ(8 +⇩ω i) ∪ pred(pred(11 +⇩ω r ∪ 1 ∪ 2))›*) ord_simp_union (*‹⟦Ord(?i::i); Ord(?j::i)⟧ ⟹ ?i ∪ ?j = max(?i, ?j)› ‹⟦Ord(?i); Ord(?j)⟧ ⟹ Ord(max(?i, ?j))› ‹max(?x, ?y) ≡ if ?x ≤ ?y then ?y else ?x›*) by (auto simp:arity (*‹⟦?φ ∈ formula; ?t ∈ ω; ?f ∈ ω; ?r ∈ ω⟧ ⟹ arity(is_If_fm(?φ, ?t, ?f, ?r)) = arity(?φ) ∪ succ(?t) ∪ succ(?r) ∪ succ(?f)› ‹arity(⋅?p ∧ ?q⋅) = arity(?p) ∪ arity(?q)› ‹arity(⋅?p ∨ ?q⋅) = arity(?p) ∪ arity(?q)› ‹arity(⋅?p → ?q⋅) = arity(?p) ∪ arity(?q)› ‹arity(⋅?p ↔ ?q⋅) = arity(?p) ∪ arity(?q)› ‹arity((⋅∃?p⋅)) = pred(arity(?p))› ‹⟦?t1.0 ∈ ω; ?t2.0 ∈ ω; ?up ∈ ω⟧ ⟹ arity(⋅{?t1.0,?t2.0} is ?up ⋅) = ⋃{succ(?t1.0), succ(?t2.0), succ(?up)}› ‹⟦?x ∈ ω; ?y ∈ ω; ?z ∈ ω⟧ ⟹ arity(⋅⟨?x,?y⟩ is ?z ⋅) = succ(?x) ∪ succ(?y) ∪ succ(?z)› ‹⟦?r ∈ ω; ?s ∈ ω; ?t ∈ ω⟧ ⟹ arity(⋅?r ∘ ?s is ?t ⋅) = succ(?t) ∪ succ(?s) ∪ succ(?r)› ‹⟦?r ∈ ω; ?z ∈ ω⟧ ⟹ arity(⋅dom(?r) is ?z ⋅) = succ(?z) ∪ succ(?r)› ‹⟦?r ∈ ω; ?z ∈ ω⟧ ⟹ arity(⋅ran(?r) is ?z ⋅) = succ(?z) ∪ succ(?r)› ‹⟦?x ∈ ω; ?y ∈ ω; ?z ∈ ω⟧ ⟹ arity(⋅?x ∪ ?y is ?z ⋅) = succ(?z) ∪ succ(?x) ∪ succ(?y)› and more 80 facts*))\n\nlemma arity_is_HAleph_fm[arity]:\n assumes\n \"i ∈ nat\" \"r ∈ nat\" \"l ∈ nat\"\n shows\n \"arity(is_HAleph_fm(i, r, l)) = succ(i) ∪ succ(l) ∪ succ(r)\"\n using assms (*‹(i::i) ∈ ω› ‹r ∈ ω› ‹(l::i) ∈ ω›*) pred_Un (*‹⟦(?x::i) ∈ ω; (?y::i) ∈ ω⟧ ⟹ pred(succ(?x) ∪ ?y) = ?x ∪ pred(?y)› ‹⟦?x ∈ ω; ?y ∈ ω⟧ ⟹ pred(?x ∪ succ(?y)) = pred(?x) ∪ ?y›*) arity_is_HAleph_fm_aux (*‹⟦?i ∈ ω; ?r ∈ ω⟧ ⟹ arity(Replace_fm(8 +⇩ω ?i, ⋅10 +⇩ω ?r`0 is 1⋅, 3)) = 9 +⇩ω ?i ∪ pred(pred(11 +⇩ω ?r))›*) arity_is_HAleph_fm' (*‹⟦?i ∈ ω; ?r ∈ ω; ?l ∈ ω⟧ ⟹ arity(is_HAleph_fm(?i, ?r, ?l)) = succ(?i) ∪ succ(?l) ∪ (succ(?i) ∪ (succ(?i) ∪ (pred(pred(pred(pred(pred(pred(pred(pred(arity(Replace_fm(succ(succ(succ(succ(succ(succ(succ(succ(?i)))))))), ⋅succ(succ(succ(succ(succ(succ(succ(succ(succ(succ(?r))))))))))`0 is 1⋅, 3)))))))))) ∪ (succ(?r) ∪ succ(?i)))))›*) by auto\n\ndefinition\n Aleph' :: \"i => i\" where\n \"Aleph'(a) == transrec(a,λi r. HAleph(i,r))\"\n\nrelativize functional \"Aleph'\" \"Aleph_rel\"\nrelationalize \"Aleph_rel\" \"is_Aleph\"\n\ntxt‹The extra assumptions \\<^term>‹a < length(env)› and \\<^term>‹c < length(env)›\n in this schematic goal (and the following results on synthesis that\n depend on it) are imposed by @{thm [source] is_transrec_iff_sats}.›\nschematic_goal sats_is_Aleph_fm_auto:\n \"a ∈ nat ⟹ c ∈ nat ⟹ env ∈ list(A) ⟹\n a < length(env) ⟹ c < length(env) ⟹ 0 ∈ A ⟹\n is_Aleph(##A, nth(a, env), nth(c, env)) ⟷ A, env ⊨ ?fm(a, c)\"\n unfolding is_Aleph_def\n (*goal: ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ is_transrec(##A, is_HAleph(##A), nth(a, env), nth(c, env)) ⟷ A, env ⊨ ?fm(a, c)›*)\n proof (rule is_transrec_iff_sats (*‹⟦⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a0 ∈ ?A; a1 ∈ ?A; a2 ∈ ?A; a3 ∈ ?A; a4 ∈ ?A; a5 ∈ ?A; a6 ∈ ?A; a7 ∈ ?A⟧ ⟹ ?MH(a2, a1, a0) ⟷ ?A, Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, ?env)))))))) ⊨ ?p; nth(?i, ?env) = ?x; nth(?k, ?env) = ?z; ?i < length(?env); ?k < length(?env); ?env ∈ list(?A)⟧ ⟹ is_transrec(##?A, ?MH, ?x, ?z) ⟷ ?A, ?env ⊨ is_transrec_fm(?p, ?i, ?k)›*), rule_tac [1] is_HAleph_iff_sats (*‹⟦nth(?i, ?env) = ?ia; nth(?r, ?env) = ?ra; nth(?l, ?env) = ?la; ?i ∈ ω; ?r ∈ ω; ?l ∈ ω; ?env ∈ list(?A); 0 ∈ ?A⟧ ⟹ is_HAleph(##?A, ?ia, ?ra, ?la) ⟷ ?A, ?env ⊨ is_HAleph_fm(?i, ?r, ?l)›*))\n (*goals:\n 1. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ nth(?i17(a, c), Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))) = a2›\n 2. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ nth(?r16(a, c), Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))) = a1›\n 3. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ nth(?l15(a, c), Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))) = a0›\n 4. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ ?i17(a, c) ∈ ω›\n 5. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ ?r16(a, c) ∈ ω›\n 6. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ ?l15(a, c) ∈ ω›\n 7. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env)))))))) ∈ list(A)›\n 8. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ 0 ∈ A›\n 9. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ nth(?i6(a, c), env) = nth(a, env)›\n 10. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ nth(?k5(a, c), env) = nth(c, env)›\n 11. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ ?i6(a, c) < length(env)›\n 12. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ ?k5(a, c) < length(env)›\n 13. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ env ∈ list(A)›*)\n fix a0 and a1 and a2 and a3 and a4 and a5 and a6 and a7\n let ?env' = \"Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))\"\n show \"nth(2, ?env') = a2\" \"nth(1, ?env') = a1\" \"nth(0, ?env') = a0\" \"nth(c, env) = nth(c, env)\"\n (*goals:\n 1. ‹nth(2, Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))) = a2›\n 2. ‹nth(1, Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))) = a1›\n 3. ‹nth(0, Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env))))))))) = a0›\n 4. ‹nth(c, env) = nth(c, env)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*proven 4 subgoals*) .\n qed (simp_all)\n (*solves the remaining goals:\n 1. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ 2 ∈ ω›\n 2. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ 1 ∈ ω›\n 3. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ 0 ∈ ω›\n 4. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ Cons(a0, Cons(a1, Cons(a2, Cons(a3, Cons(a4, Cons(a5, Cons(a6, Cons(a7, env)))))))) ∈ list(A)›\n 5. ‹⋀a0 a1 a2 a3 a4 a5 a6 a7. ⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A; a0 ∈ A; a1 ∈ A; a2 ∈ A; a3 ∈ A; a4 ∈ A; a5 ∈ A; a6 ∈ A; a7 ∈ A⟧ ⟹ 0 ∈ A›\n 6. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ nth(?i6(a, c), env) = nth(a, env)›\n 7. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ ?i6(a, c) < length(env)›\n 8. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ c < length(env)›\n 9. ‹⟦a ∈ ω; c ∈ ω; env ∈ list(A); a < length(env); c < length(env); 0 ∈ A⟧ ⟹ env ∈ list(A)›*)\n\nsynthesize_notc \"is_Aleph\" from_schematic\n\nnotation is_Aleph_fm (‹⋅ℵ'(_') is _⋅›)\n\nlemma is_Aleph_fm_type [TC]: \"a ∈ nat ⟹ c ∈ nat ⟹ is_Aleph_fm(a, c) ∈ formula\"\n unfolding is_Aleph_fm_def\n (*goal: ‹⟦a ∈ ω; c ∈ ω⟧ ⟹ is_transrec_fm(is_HAleph_fm(2, 1, 0), a, c) ∈ formula›*)\n by simp\n\nlemma sats_is_Aleph_fm:\n assumes \"f∈nat\" \"r∈nat\" \"env ∈ list(A)\" \"0∈A\" \"f < length(env)\" \"r< length(env)\"\n shows \"is_Aleph(##A, nth(f, env), nth(r, env)) ⟷ A, env ⊨ is_Aleph_fm(f,r)\"\n using assms (*‹f ∈ ω› ‹r ∈ ω› ‹env ∈ list(A)› ‹0 ∈ A› ‹f < length(env)› ‹r < length(env)›*) sats_is_Aleph_fm_auto (*‹⟦?a ∈ ω; ?c ∈ ω; ?env ∈ list(?A); ?a < length(?env); ?c < length(?env); 0 ∈ ?A⟧ ⟹ is_Aleph(##?A, nth(?a, ?env), nth(?c, ?env)) ⟷ ?A, ?env ⊨ is_transrec_fm(is_HAleph_fm(2, 1, 0), ?a, ?c)›*) unfolding is_Aleph_def is_Aleph_fm_def\n (*goal: ‹is_transrec(##A, is_HAleph(##A), nth(f, env), nth(r, env)) ⟷ A, env ⊨ is_transrec_fm(is_HAleph_fm(2, 1, 0), f, r)›*)\n by simp\n\nlemma is_Aleph_iff_sats [iff_sats]:\n assumes\n \"nth(f, env) = fa\" \"nth(r, env) = ra\" \"f < length(env)\" \"r< length(env)\"\n \"f ∈ nat\" \"r ∈ nat\" \"env ∈ list(A)\" \"0∈A\"\n shows \"is_Aleph(##A,fa,ra) ⟷ A, env ⊨ is_Aleph_fm(f,r)\"\n using assms (*‹nth(f, env) = fa› ‹nth(r, env) = ra› ‹f < length(env)› ‹(r::i) < length(env::i)› ‹(f::i) ∈ ω› ‹r ∈ ω› ‹env ∈ list(A)› ‹0 ∈ (A::i)›*) sats_is_Aleph_fm[of f r env A] (*‹⟦f ∈ ω; r ∈ ω; env ∈ list(A); 0 ∈ A; f < length(env); r < length(env)⟧ ⟹ is_Aleph(##A, nth(f, env), nth(r, env)) ⟷ A, env ⊨ ⋅ℵ(f) is r⋅›*) by simp\n\narity_theorem for \"is_Aleph_fm\"\n\nlemma (in M_cardinal_arith_jump) is_Limit_iff:\n assumes \"M(a)\"\n shows \"is_Limit(M,a) ⟷ Limit(a)\"\n unfolding is_Limit_def Limit_def\n (*goal: ‹(∃aa[M]. (ordinal(M, a) ∧ lt_rel(M, aa, a) ∧ (∀y[M]. ∃b[M]. (lt_rel(M, y, a) ⟶ lt_rel(M, b, a)) ∧ successor(M, y, b))) ∧ empty(M, aa)) ⟷ Ord(a) ∧ 0 < a ∧ (∀y. y < a ⟶ succ(y) < a)›*)\n using lt_abs (*‹⟦M(?a); M(?b)⟧ ⟹ lt_rel(M, ?a, ?b) ⟷ ?a < ?b›*) transM[OF ltD ‹M(a)›] (*‹?y < a ⟹ M(?y)›*) assms (*‹M(a)›*) by auto\n\nlemma HAleph_eq_Aleph_recursive:\n \"Ord(i) ⟹ HAleph(i,r) = (if i = 0 then nat\n else if ∃j. i = succ(j) then csucc(r ` (THE j. i = succ(j))) else ⋃j<i. r ` j)\"\nproof (-)\n (*goal: ‹Ord(i) ⟹ HAleph(i, r) = (if i = 0 then ω else if ∃j. i = succ(j) then r ` (THE j. i = succ(j))⁺ else ⋃j<i. r ` j)›*)\n assume \"Ord(i)\" (*‹Ord(i::i)›*)\n moreover from this (*‹Ord(i)›*) have \"i = succ(j) ⟹ (⋃succ(j)) = j\" for j\n using Ord_Union_succ_eq (*‹Ord(?i::i) ⟹ ⋃succ(?i) = ?i›*) by simp\n moreover from ‹Ord(i)› (*‹Ord(i)›*) have \"(∃j. i = succ(j)) ⟷ ¬Limit(i) ∧ i ≠ 0\"\n using Ord_cases_disj (*‹Ord(?i) ⟹ ?i = 0 ∨ (∃j. Ord(j) ∧ ?i = succ(j)) ∨ Limit(?i)›*) by auto\n ultimately show \"?thesis\"\n (*goal: ‹HAleph(i, r) = (if i = 0 then ω else if ∃j. i = succ(j) then r ` (THE j. i = succ(j))⁺ else ⋃j<i. r ` j)›*)\n unfolding HAleph_def OUnion_def\n (*goal: ‹(if ¬ Ord(i) then i else if i = 0 then ω else if ¬ Limit(i) ∧ i ≠ 0 then r ` (⋃i)⁺ else ⋃j∈i. r ` j) = (if i = 0 then ω else if ∃j. i = succ(j) then r ` (THE j. i = succ(j))⁺ else {z ∈ ⋃x∈i. r ` x . Ord(i)})›*)\n by auto\nqed\n\nlemma Aleph'_eq_Aleph: \"Ord(a) ⟹ Aleph'(a) = Aleph(a)\"\n unfolding Aleph'_def Aleph_def transrec2_def\n (*goal: ‹Ord(a) ⟹ transrec(a, λi r. HAleph(i, r)) = transrec(a, λi r. if i = 0 then ω else if ∃j. i = succ(j) then r ` (THE j. i = succ(j))⁺ else ⋃j<i. r ` j)›*)\n using HAleph_eq_Aleph_recursive (*‹Ord(?i::i) ⟹ HAleph(?i, ?r::i) = (if ?i = 0 then ω else if ∃j::i. ?i = succ(j) then ?r ` (THE j::i. ?i = succ(j))⁺ else ⋃j::i<?i. ?r ` j)›*) apply (intro transrec_equal_on_Ord (*‹⟦⋀x f. Ord(x) ⟹ ?foo(x, f) = ?bar(x, f); Ord(?α)⟧ ⟹ transrec(?α, ?foo) = transrec(?α, ?bar)›*))\n (*goals:\n 1. ‹⋀x f. ⟦Ord(a); ⋀i r. Ord(i) ⟹ HAleph(i, r) = (if i = 0 then ω else if ∃j. i = succ(j) then r ` (THE j. i = succ(j))⁺ else ⋃j<i. r ` j); Ord(x)⟧ ⟹ HAleph(x, f) = (if x = 0 then ω else if ∃j. x = succ(j) then f ` (THE j. x = succ(j))⁺ else ⋃j<x. f ` j)›\n 2. ‹⟦Ord(a); ⋀i r. Ord(i) ⟹ HAleph(i, r) = (if i = 0 then ω else if ∃j. i = succ(j) then r ` (THE j. i = succ(j))⁺ else ⋃j<i. r ` j)⟧ ⟹ Ord(a)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nreldb_rem functional \"Aleph'\"\nreldb_rem relational \"is_Aleph\"\nreldb_add functional \"Aleph\" \"Aleph_rel\"\nreldb_add relational \"Aleph\" \"is_Aleph\"\n\nabbreviation\n Aleph_r :: \"[i,i⇒o] ⇒ i\" (‹ℵ⇘_⇙⇗_⇖›) where\n \"Aleph_r(a,M) ≡ Aleph_rel(M,a)\"\n\nabbreviation\n Aleph_r_set :: \"[i,i] ⇒ i\" (‹ℵ⇘_⇙⇗_⇖›) where\n \"Aleph_r_set(a,M) ≡ Aleph_rel(##M,a)\"\n\nlemma Aleph_rel_def': \"Aleph_rel(M,a) ≡ transrec(a, λi r. HAleph_rel(M, i, r))\"\n unfolding Aleph_rel_def\n (*goal: ‹transrec(a, HAleph_rel(M)) ≡ transrec(a, λi r. HAleph_rel(M, i, r))›*) .\n\nlemma succ_mem_Limit: \"Limit(j) ⟹ i ∈ j ⟹ succ(i) ∈ j\"\n using Limit_has_succ[THEN ltD] (*‹⟦Limit(?j); ?j1 < ?j⟧ ⟹ succ(?j1) ∈ ?j›*) ltI (*‹⟦?i ∈ ?j; Ord(?j)⟧ ⟹ ?i < ?j›*) Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) by auto\n\nlocale M_pre_aleph = M_eclose + M_cardinal_arith_jump +\n assumes\n haleph_transrec_replacement: \"M(a) ⟹ transrec_replacement(M,is_HAleph(M),a)\"\n\nbegin\n\nlemma aux_ex_Replace_funapply:\n assumes \"M(a)\" \"M(f)\"\n shows \"∃x[M]. is_Replace(M, a, λj y. f ` j = y, x)\"\nproof (-)\n (*goal: ‹∃x[M]. is_Replace(M, a, λj y. f ` j = y, x)›*)\n have \"{f`j . j∈a} = {y . j∈a , f ` j=y}\" \"{y . j∈a , f ` j=y} = {y . j∈a , y =f ` j}\"\n (*goals:\n 1. ‹{f ` j . j ∈ a} = {y . j ∈ a, f ` j = y}›\n 2. ‹{y . j ∈ a, f ` j = y} = {y . j ∈ a, y = f ` j}›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n moreover note assms (*‹(M::i ⇒ o)(a::i)› ‹M(f)›*)\n moreover from calculation (*‹{f ` j . j ∈ a} = {y . j ∈ a, f ` j = y}› ‹{y . j ∈ a, f ` j = y} = {y . j ∈ a, y = f ` j}› ‹M(a)› ‹(M::i ⇒ o)(f::i)›*) have \"x ∈ a ⟹ y = f `x ⟹ M(y)\" for x and y\n using transM[OF _ ‹M(a)›] (*‹?y ∈ a ⟹ M(?y)›*) by auto\n moreover from assms (*‹M(a)› ‹M(f)›*) have \"M({f`j . j∈a})\"\n using transM[OF _ ‹M(a)›] (*‹?y ∈ a ⟹ M(?y)›*) RepFun_closed[OF apply_replacement] (*‹⟦M(?S2); M(?A); ∀x∈?A. M(?S2 ` x)⟧ ⟹ M({?S2 ` x . x ∈ ?A})›*) by simp\n ultimately have 2: \"is_Replace(M, a, λj y. y = f ` j, {f`j . j∈a})\"\n using Replace_abs[of _ _ \"λj y. y = f ` j\", OF ‹M(a)›, THEN iffD2] (*‹⟦M(?z1); univalent(M, a, λj y. y = f ` j); ⋀x y. ⟦x ∈ a; y = f ` x⟧ ⟹ M(y); ?z1 = {y . j ∈ a, y = f ` j}⟧ ⟹ is_Replace(M, a, λj y. y = f ` j, ?z1)›*) by auto\n with ‹M({f`j . j∈a})› (*‹M({f ` j . j ∈ a})›*) show \"?thesis\"\n (*goal: ‹∃x[M]. is_Replace(M, a, λj y. f ` j = y, x)›*)\n using is_Replace_cong[of _ _ M \"λj y. y = f ` j\" \"λj y. f ` j = y\", THEN iffD1, OF _ _ _ 2] (*‹⟦a = ?A'1; ⋀x y. ⟦M(x); M(y)⟧ ⟹ y = f ` x ⟷ f ` x = y; {f ` j . j ∈ a} = ?z'1⟧ ⟹ is_Replace(M, ?A'1, λx y. f ` x = y, ?z'1)›*) by auto\nqed\n\nlemma is_HAleph_zero:\n assumes \"M(f)\"\n shows \"is_HAleph(M,0,f,res) ⟷ res = nat\"\n unfolding is_HAleph_def\n (*goal: ‹(∃b[M]. ∃c[M]. ∃d[M]. ∃e[M]. ∃fa[M]. ∃g[M]. ∃h[M]. ∃k[M]. is_If(M, ¬ ordinal(M, 0), 0, k, res) ∧ is_If(M, empty(M, 0), b, h, k) ∧ is_If(M, ¬ is_Limit(M, 0) ∧ ¬ empty(M, 0), e, g, h) ∧ big_union(M, fa, g) ∧ is_Replace(M, 0, λj a. M(a) ∧ M(j) ∧ is_apply(M, f, j, a), fa) ∧ is_csucc(M, d, e) ∧ is_apply(M, f, c, d) ∧ big_union(M, 0, c) ∧ omega(M, b)) ⟷ res = ω›*)\n using Ord_0 (*‹Ord(0)›*) If_abs (*‹is_If(M, ?b, ?t, ?f, ?r) ⟷ ?r = (if ?b then ?t else ?f)›*) is_Limit_iff (*‹M(?a) ⟹ is_Limit(M, ?a) ⟷ Limit(?a)›*) is_csucc_iff (*‹⟦M(?K); M(?res)⟧ ⟹ is_csucc(M, ?K, ?res) ⟷ ?res = (?K⁺)⇗M⇖›*) assms (*‹(M::i ⇒ o)(f::i)›*) aux_ex_Replace_funapply (*‹⟦M(?a); M(?f)⟧ ⟹ ∃x[M]. is_Replace(M, ?a, λj y. ?f ` j = y, x)›*) by auto\n\nlemma is_HAleph_succ:\n assumes \"M(f)\" \"M(x)\" \"Ord(x)\" \"M(res)\"\n shows \"is_HAleph(M,succ(x),f,res) ⟷ res = csucc_rel(M,f`x)\"\n unfolding is_HAleph_def\n (*goal: ‹(∃b[M]. ∃c[M]. ∃d[M]. ∃e[M]. ∃fa[M]. ∃g[M]. ∃h[M]. ∃k[M]. is_If(M, ¬ ordinal(M, succ(x)), succ(x), k, res) ∧ is_If(M, empty(M, succ(x)), b, h, k) ∧ is_If(M, ¬ is_Limit(M, succ(x)) ∧ ¬ empty(M, succ(x)), e, g, h) ∧ big_union(M, fa, g) ∧ is_Replace(M, succ(x), λj a. M(a) ∧ M(j) ∧ is_apply(M, f, j, a), fa) ∧ is_csucc(M, d, e) ∧ is_apply(M, f, c, d) ∧ big_union(M, succ(x), c) ∧ omega(M, b)) ⟷ res = (f ` x⁺)⇗M⇖›*)\n using assms (*‹M(f)› ‹(M::i ⇒ o)(x::i)› ‹Ord(x)› ‹(M::i ⇒ o)(res::i)›*) is_Limit_iff (*‹M(?a) ⟹ is_Limit(M, ?a) ⟷ Limit(?a)›*) is_csucc_iff (*‹⟦M(?K); M(?res)⟧ ⟹ is_csucc(M, ?K, ?res) ⟷ ?res = (?K⁺)⇗M⇖›*) aux_ex_Replace_funapply (*‹⟦M(?a); M(?f)⟧ ⟹ ∃x[M]. is_Replace(M, ?a, λj y. ?f ` j = y, x)›*) If_abs (*‹is_If(M, ?b, ?t, ?f, ?r) ⟷ ?r = (if ?b then ?t else ?f)›*) Ord_Union_succ_eq (*‹Ord(?i::i) ⟹ ⋃succ(?i) = ?i›*) by simp\n\nlemma is_HAleph_limit:\n assumes \"M(f)\" \"M(x)\" \"Limit(x)\" \"M(res)\"\n shows \"is_HAleph(M,x,f,res) ⟷ res = (⋃{y . i∈x ,M(i) ∧ M(y) ∧ y = f`i})\"\nproof (-)\n (*goal: ‹is_HAleph(M, x, f, res) ⟷ res = ⋃{y . i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}›*)\n from assms (*‹M(f)› ‹M(x)› ‹Limit(x)› ‹(M::i ⇒ o)(res::i)›*) have \"univalent(M, x, λj y. y = f ` j )\" \"(⋀xa y. xa ∈ x ⟹ f ` xa = y ⟹ M(y))\" \"{y . x ∈ x, f ` x = y} = {y . i∈x ,M(i) ∧ M(y) ∧ y = f`i}\"\n using univalent_triv[of M x \"λj .f ` j\"] (*‹univalent(M::i ⇒ o, x::i, λ(x::i) y::i. y = (f::i) ` x)›*) transM[OF _ ‹M(x)›] (*‹?y ∈ x ⟹ M(?y)›*) apply -\n (*goals:\n 1. ‹⟦(M::i ⇒ o)(f::i); M(x::i); Limit(x); M(res::i); univalent(M, x, λ(x::i) y::i. y = f ` x); ⋀y::i. y ∈ x ⟹ M(y)⟧ ⟹ univalent(M, x, λ(j::i) y::i. y = f ` j)›\n 2. ‹⋀(xa::i) y::i. ⟦xa ∈ (x::i); (f::i) ` xa = y; (M::i ⇒ o)(f); M(x); Limit(x); M(res::i); univalent(M, x, λ(x::i) y::i. y = f ` x); ⋀y::i. y ∈ x ⟹ M(y)⟧ ⟹ M(y)›\n 3. ‹⟦(M::i ⇒ o)(f::i); M(x::i); Limit(x); M(res::i); univalent(M, x, λ(x::i) y::i. y = f ` x); ⋀y::i. y ∈ x ⟹ M(y)⟧ ⟹ {y::i . x::i ∈ x, f ` x = y} = {y::i . i::i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n moreover from this (*‹univalent(M, x, λj y. y = f ` j)› ‹⟦(?xa::i) ∈ (x::i); (f::i) ` ?xa = (?y::i)⟧ ⟹ (M::i ⇒ o)(?y)› ‹{y . x ∈ x, f ` x = y} = {y . i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}›*) have \"univalent(M, x, λj y. f ` j = y )\"\n apply (rule_tac univalent_cong[of x x M \" λj y. y = f ` j\" \" λj y. f ` j=y\",THEN iffD1] (*‹⟦x = x; ⋀xa y. ⟦xa ∈ x; M(xa); M(y)⟧ ⟹ y = f ` xa ⟷ f ` xa = y; univalent(M, x, λx y. y = f ` x)⟧ ⟹ univalent(M, x, λx y. f ` x = y)›*))\n (*goals:\n 1. ‹⟦univalent(M, x, λj y. y = f ` j); ⋀xa y. ⟦xa ∈ x; f ` xa = y⟧ ⟹ M(y); {y . x ∈ x, f ` x = y} = {y . i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}⟧ ⟹ x = x›\n 2. ‹⋀j y. ⟦univalent(M, x, λj y. y = f ` j); ⋀xa y. ⟦xa ∈ x; f ` xa = y⟧ ⟹ M(y); {y . x ∈ x, f ` x = y} = {y . i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}; j ∈ x; M(j); M(y)⟧ ⟹ y = f ` j ⟷ f ` j = y›\n 3. ‹⟦univalent(M, x, λj y. y = f ` j); ⋀xa y. ⟦xa ∈ x; f ` xa = y⟧ ⟹ M(y); {y . x ∈ x, f ` x = y} = {y . i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}⟧ ⟹ univalent(M, x, λj y. y = f ` j)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n moreover from this (*‹univalent(M, x, λj y. f ` j = y)›*) have \"univalent(M, x, λj y. M(j) ∧ M(y) ∧ f ` j = y )\"\n by auto\n ultimately show \"?thesis\"\n (*goal: ‹is_HAleph(M, x, f, res) ⟷ res = ⋃{y . i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}›*)\n unfolding is_HAleph_def\n (*goal: ‹(∃b[M]. ∃c[M]. ∃d[M]. ∃e[M]. ∃fa[M]. ∃g[M]. ∃h[M]. ∃k[M]. is_If(M, ¬ ordinal(M, x), x, k, res) ∧ is_If(M, empty(M, x), b, h, k) ∧ is_If(M, ¬ is_Limit(M, x) ∧ ¬ empty(M, x), e, g, h) ∧ big_union(M, fa, g) ∧ is_Replace(M, x, λj a. M(a) ∧ M(j) ∧ is_apply(M, f, j, a), fa) ∧ is_csucc(M, d, e) ∧ is_apply(M, f, c, d) ∧ big_union(M, x, c) ∧ omega(M, b)) ⟷ res = ⋃{y . i ∈ x, M(i) ∧ M(y) ∧ y = f ` i}›*)\n using assms (*‹M(f)› ‹M(x)› ‹Limit(x)› ‹M(res)›*) is_Limit_iff (*‹(M::i ⇒ o)(?a::i) ⟹ is_Limit(M, ?a) ⟷ Limit(?a)›*) Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) zero_not_Limit (*‹¬ Limit(0)›*) If_abs (*‹is_If(M, ?b, ?t, ?f, ?r) ⟷ ?r = (if ?b then ?t else ?f)›*) is_csucc_iff (*‹⟦M(?K); M(?res)⟧ ⟹ is_csucc(M, ?K, ?res) ⟷ ?res = (?K⁺)⇗M⇖›*) Replace_abs (*‹⟦M(?A); M(?z); univalent(M, ?A, ?P); ⋀x y. ⟦x ∈ ?A; ?P(x, y)⟧ ⟹ M(y)⟧ ⟹ is_Replace(M, ?A, ?P, ?z) ⟷ ?z = Replace(?A, ?P)›*) apply_replacement (*‹M(?S) ⟹ strong_replacement(M, λx y. y = ?S ` x)›*) by auto\nqed\n\nlemma is_HAleph_iff:\n assumes \"M(a)\" \"M(f)\" \"M(res)\"\n shows \"is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)\"\nproof (cases \"Ord(a)\")\n (*goals:\n 1. ‹Ord(a) ⟹ is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›\n 2. ‹¬ Ord(a) ⟹ is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n case True (*‹Ord(a)›*)\n note Ord_cases[OF ‹Ord(a)›] (*‹⟦a = 0 ⟹ ?thesis; ⋀j. ⟦Ord(j); a = succ(j)⟧ ⟹ ?thesis; Limit(a) ⟹ ?thesis⟧ ⟹ ?thesis›*)\n then show \"?thesis\"\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n proof (cases)\n (*goals:\n 1. ‹a = 0 ⟹ is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›\n 2. ‹⋀j. ⟦Ord(j); a = succ(j)⟧ ⟹ is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›\n 3. ‹Limit(a) ⟹ is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n case 1 (*‹a = 0›*)\n with True (*‹Ord(a)›*) assms (*‹M(a)› ‹M(f)› ‹M(res)›*) show \"?thesis\"\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n using is_HAleph_zero (*‹M(?f) ⟹ is_HAleph(M, 0, ?f, ?res) ⟷ ?res = ω›*) unfolding HAleph_rel_def\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = (if ¬ Ord(a) then a else if a = 0 then ω else if ¬ Limit(a) ∧ a ≠ 0 then (f ` (⋃a)⁺)⇗M⇖ else ⋃{a . j ∈ a, M(a) ∧ M(j) ∧ a = f ` j})›*)\n by simp\n next\n (*goals:\n 1. ‹⋀j::i. ⟦Ord(j); (a::i) = succ(j)⟧ ⟹ is_HAleph(M::i ⇒ o, a, f::i, res::i) ⟷ res = HAleph_rel(M, a, f)›\n 2. ‹Limit(a::i) ⟹ is_HAleph(M::i ⇒ o, a, f::i, res::i) ⟷ res = HAleph_rel(M, a, f)›*)\n case (2 j) (*‹Ord(j)› ‹a = succ(j)›*)\n with True (*‹Ord(a)›*) assms (*‹M(a)› ‹M(f)› ‹M(res)›*) show \"?thesis\"\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n using is_HAleph_succ (*‹⟦M(?f); M(?x); Ord(?x); M(?res)⟧ ⟹ is_HAleph(M, succ(?x), ?f, ?res) ⟷ ?res = (?f ` ?x⁺)⇗M⇖›*) Ord_Union_succ_eq (*‹Ord(?i) ⟹ ⋃succ(?i) = ?i›*) unfolding HAleph_rel_def\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = (if ¬ Ord(a) then a else if a = 0 then ω else if ¬ Limit(a) ∧ a ≠ 0 then (f ` (⋃a)⁺)⇗M⇖ else ⋃{a . j ∈ a, M(a) ∧ M(j) ∧ a = f ` j})›*)\n by simp\n next\n (*goal: ‹Limit(a) ⟹ is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n case 3 (*‹Limit(a::i)›*)\n with assms (*‹M(a)› ‹M(f)› ‹M(res)›*) show \"?thesis\"\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n using is_HAleph_limit (*‹⟦M(?f); M(?x); Limit(?x); M(?res)⟧ ⟹ is_HAleph(M, ?x, ?f, ?res) ⟷ ?res = ⋃{y . i ∈ ?x, M(i) ∧ M(y) ∧ y = ?f ` i}›*) zero_not_Limit (*‹¬ Limit(0)›*) Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) unfolding HAleph_rel_def\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = (if ¬ Ord(a) then a else if a = 0 then ω else if ¬ Limit(a) ∧ a ≠ 0 then (f ` (⋃a)⁺)⇗M⇖ else ⋃{a . j ∈ a, M(a) ∧ M(j) ∧ a = f ` j})›*)\n by auto\n qed\nnext\n (*goal: ‹¬ Ord(a) ⟹ is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n case False (*‹¬ Ord(a)›*)\n then have \"¬Limit(a)\" \"a≠0\" \"⋀ x . Ord(x) ⟹ a≠succ(x)\"\n using Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) apply -\n (*goals:\n 1. ‹⟦¬ Ord(a::i); ⋀i::i. Limit(i) ⟹ Ord(i)⟧ ⟹ ¬ Limit(a)›\n 2. ‹⟦¬ Ord(a::i); ⋀i::i. Limit(i) ⟹ Ord(i)⟧ ⟹ a ≠ 0›\n 3. ‹⋀x::i. ⟦Ord(x); ¬ Ord(a::i); ⋀i::i. Limit(i) ⟹ Ord(i)⟧ ⟹ a ≠ succ(x)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n with False (*‹¬ Ord(a)›*) show \"?thesis\"\n (*goal: ‹is_HAleph(M, a, f, res) ⟷ res = HAleph_rel(M, a, f)›*)\n unfolding is_HAleph_def HAleph_rel_def\n (*goal: ‹(∃b[M]. ∃c[M]. ∃d[M]. ∃e[M]. ∃fa[M]. ∃g[M]. ∃h[M]. ∃k[M]. is_If(M, ¬ ordinal(M, a), a, k, res) ∧ is_If(M, empty(M, a), b, h, k) ∧ is_If(M, ¬ is_Limit(M, a) ∧ ¬ empty(M, a), e, g, h) ∧ big_union(M, fa, g) ∧ is_Replace(M, a, λj a. M(a) ∧ M(j) ∧ is_apply(M, f, j, a), fa) ∧ is_csucc(M, d, e) ∧ is_apply(M, f, c, d) ∧ big_union(M, a, c) ∧ omega(M, b)) ⟷ res = (if ¬ Ord(a) then a else if a = 0 then ω else if ¬ Limit(a) ∧ a ≠ 0 then (f ` (⋃a)⁺)⇗M⇖ else ⋃{a . j ∈ a, M(a) ∧ M(j) ∧ a = f ` j})›*)\n using assms (*‹M(a)› ‹M(f)› ‹M(res)›*) is_Limit_iff (*‹M(?a) ⟹ is_Limit(M, ?a) ⟷ Limit(?a)›*) If_abs (*‹is_If(M, ?b, ?t, ?f, ?r) ⟷ ?r = (if ?b then ?t else ?f)›*) is_csucc_iff (*‹⟦M(?K); M(?res)⟧ ⟹ is_csucc(M, ?K, ?res) ⟷ ?res = (?K⁺)⇗M⇖›*) aux_ex_Replace_funapply (*‹⟦M(?a); M(?f)⟧ ⟹ ∃x[M]. is_Replace(M, ?a, λj y. ?f ` j = y, x)›*) by auto\nqed\n\nlemma HAleph_rel_closed [intro,simp]:\n assumes \"function(f)\" \"M(a)\" \"M(f)\"\n shows \"M(HAleph_rel(M,a,f))\"\n unfolding HAleph_rel_def\n (*goal: ‹M(if ¬ Ord(a) then a else if a = 0 then ω else if ¬ Limit(a) ∧ a ≠ 0 then (f ` (⋃a)⁺)⇗M⇖ else ⋃{a . j ∈ a, M(a) ∧ M(j) ∧ a = f ` j})›*)\n using assms (*‹function(f)› ‹M(a)› ‹(M::i ⇒ o)(f::i)›*) apply_replacement (*‹M(?S) ⟹ strong_replacement(M, λx y. y = ?S ` x)›*) by simp\n\nlemma Aleph_rel_closed[intro, simp]:\n assumes \"Ord(a)\" \"M(a)\"\n shows \"M(Aleph_rel(M,a))\"\nproof (-)\n (*goal: ‹M(ℵ⇘a⇙⇗M⇖)›*)\n have \"relation2(M, is_HAleph(M), HAleph_rel(M))\"\n unfolding relation2_def\n (*goal: ‹∀x[M]. ∀y[M]. ∀z[M]. is_HAleph(M, x, y, z) ⟷ z = HAleph_rel(M, x, y)›*)\n using is_HAleph_iff (*‹⟦M(?a); M(?f); M(?res)⟧ ⟹ is_HAleph(M, ?a, ?f, ?res) ⟷ ?res = HAleph_rel(M, ?a, ?f)›*) assms (*‹Ord(a::i)› ‹(M::i ⇒ o)(a::i)›*) by simp\n moreover have \"∀x[M]. ∀g[M]. function(g) ⟶ M(HAleph_rel(M, x, g))\"\n using HAleph_rel_closed (*‹⟦function(?f); M(?a); M(?f)⟧ ⟹ M(HAleph_rel(M, ?a, ?f))›*) by simp\n moreover note assms (*‹Ord(a)› ‹(M::i ⇒ o)(a::i)›*)\n ultimately show \"?thesis\"\n (*goal: ‹M(ℵ⇘a⇙⇗M⇖)›*)\n unfolding Aleph_rel_def\n (*goal: ‹M(transrec(a, HAleph_rel(M)))›*)\n using transrec_closed[of \"is_HAleph(M)\" a \"HAleph_rel(M)\"] (*‹⟦transrec_replacement(M, is_HAleph(M), a); relation2(M, is_HAleph(M), HAleph_rel(M)); Ord(a); M(a); ∀x[M]. ∀g[M]. function(g) ⟶ M(HAleph_rel(M, x, g))⟧ ⟹ M(transrec(a, HAleph_rel(M)))›*) haleph_transrec_replacement (*‹M(?a) ⟹ transrec_replacement(M, is_HAleph(M), ?a)›*) by simp\nqed\n\nlemma Aleph_rel_zero: \"ℵ⇘0⇙⇗M⇖ = nat\"\n using def_transrec[OF Aleph_rel_def', of _ 0] (*‹ℵ⇘0⇙⇗?M2⇖ = HAleph_rel(?M2, 0, λx∈0. ℵ⇘x⇙⇗?M2⇖)›*) unfolding HAleph_rel_def\n (*goal: ‹ℵ⇘0⇙⇗M⇖ = ω›*)\n by simp\n\nlemma Aleph_rel_succ: \"Ord(α) ⟹ M(α) ⟹ ℵ⇘succ(α)⇙⇗M⇖ = (ℵ⇘α⇙⇗M⇖⁺)⇗M⇖\"\n using Ord_Union_succ_eq (*‹Ord(?i) ⟹ ⋃succ(?i) = ?i›*) apply (subst def_transrec [OF Aleph_rel_def'] (*‹ℵ⇘?a⇙⇗?M2⇖ = HAleph_rel(?M2, ?a, λx∈?a. ℵ⇘x⇙⇗?M2⇖)›*))\n (*goal: ‹⟦Ord(α::i); (M::i ⇒ o)(α)⟧ ⟹ ℵ⇘succ(α)⇙⇗M⇖ = (ℵ⇘α⇙⇗M⇖⁺)⇗M⇖›*)\n by (simp add:HAleph_rel_def (*‹HAleph_rel ≡ λ(N::i ⇒ o) (i::i) r::i. if ¬ Ord(i) then i else if i = 0 then ω else if ¬ Limit(i) ∧ i ≠ 0 then (r ` (⋃i)⁺)⇗N⇖ else ⋃{a::i . j::i ∈ i, N(a) ∧ N(j) ∧ a = r ` j}›*))\n\nlemma Aleph_rel_limit:\n assumes \"Limit(α)\" \"M(α)\"\n shows \"ℵ⇘α⇙⇗M⇖ = ⋃{ℵ⇘j⇙⇗M⇖ . j ∈ α}\"\nproof (-)\n (*goal: ‹ℵ⇘α⇙⇗M⇖ = (⋃j∈α. ℵ⇘j⇙⇗M⇖)›*)\n note trans = transM[OF _ ‹M(α)›] (*‹(?y::i) ∈ (α::i) ⟹ (M::i ⇒ o)(?y)›*)\n from ‹M(α)› (*‹M(α)›*) have \"ℵ⇘α⇙⇗M⇖ = HAleph_rel(M, α, λx∈α. ℵ⇘x⇙⇗M⇖)\"\n using def_transrec[OF Aleph_rel_def', of M α] (*‹ℵ⇘α::i⇙⇗M::i ⇒ o⇖ = HAleph_rel(M, α, λx::i∈α. ℵ⇘x⇙⇗M⇖)›*) by simp\n also (*calculation: ‹ℵ⇘α⇙⇗M⇖ = HAleph_rel(M, α, λx∈α. ℵ⇘x⇙⇗M⇖)›*) have \"... = ⋃{a . j ∈ α, M(a) ∧ a = ℵ⇘j⇙⇗M⇖}\"\n unfolding HAleph_rel_def\n (*goal: ‹(if ¬ Ord(α) then α else if α = 0 then ω else if ¬ Limit(α) ∧ α ≠ 0 then (Lambda(α, Aleph_rel(M)) ` (⋃α)⁺)⇗M⇖ else ⋃{a . j ∈ α, M(a) ∧ M(j) ∧ a = Lambda(α, Aleph_rel(M)) ` j}) = ⋃{a . j ∈ α, M(a) ∧ a = ℵ⇘j⇙⇗M⇖}›*)\n using assms (*‹Limit(α)› ‹M(α)›*) zero_not_Limit (*‹¬ Limit(0)›*) Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) trans (*‹(?y::i) ∈ (α::i) ⟹ (M::i ⇒ o)(?y)›*) by auto\n also (*calculation: ‹ℵ⇘α⇙⇗M⇖ = ⋃{a . j ∈ α, M(a) ∧ a = ℵ⇘j⇙⇗M⇖}›*) have \"... = ⋃{ℵ⇘j⇙⇗M⇖ . j ∈ α}\"\n using Aleph_rel_closed[OF _ trans] (*‹⟦Ord(?a); ?a ∈ α⟧ ⟹ M(ℵ⇘?a⇙⇗M⇖)›*) Ord_in_Ord (*‹⟦Ord(?i); ?j ∈ ?i⟧ ⟹ Ord(?j)›*) Limit_is_Ord[OF ‹Limit(α)›] (*‹Ord(α)›*) by auto\n finally (*calculation: ‹ℵ⇘α⇙⇗M⇖ = (⋃j∈α. ℵ⇘j⇙⇗M⇖)›*) show \"?thesis\"\n (*goal: ‹ℵ⇘α⇙⇗M⇖ = (⋃j∈α. ℵ⇘j⇙⇗M⇖)›*) .\nqed\n\nlemma is_Aleph_iff:\n assumes \"Ord(a)\" \"M(a)\" \"M(res)\"\n shows \"is_Aleph(M, a, res) ⟷ res = ℵ⇘a⇙⇗M⇖\"\nproof (-)\n (*goal: ‹is_Aleph(M, a, res) ⟷ res = ℵ⇘a⇙⇗M⇖›*)\n have \"relation2(M, is_HAleph(M), HAleph_rel(M))\"\n unfolding relation2_def\n (*goal: ‹∀x[M]. ∀y[M]. ∀z[M]. is_HAleph(M, x, y, z) ⟷ z = HAleph_rel(M, x, y)›*)\n using is_HAleph_iff (*‹⟦M(?a); M(?f); M(?res)⟧ ⟹ is_HAleph(M, ?a, ?f, ?res) ⟷ ?res = HAleph_rel(M, ?a, ?f)›*) assms (*‹Ord(a)› ‹M(a)› ‹M(res)›*) by simp\n moreover have \"∀x[M]. ∀g[M]. function(g) ⟶ M(HAleph_rel(M, x, g))\"\n using HAleph_rel_closed (*‹⟦function(?f::i); (M::i ⇒ o)(?a::i); M(?f)⟧ ⟹ M(HAleph_rel(M, ?a, ?f))›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹is_Aleph(M, a, res) ⟷ res = ℵ⇘a⇙⇗M⇖›*)\n using assms (*‹Ord(a)› ‹M(a)› ‹M(res)›*) transrec_abs (*‹⟦transrec_replacement(M::i ⇒ o, ?MH::i ⇒ i ⇒ i ⇒ o, ?i::i); relation2(M, ?MH, ?H::i ⇒ i ⇒ i); Ord(?i); M(?i); M(?z::i); ∀x::i[M]. ∀g::i[M]. function(g) ⟶ M(?H(x, g))⟧ ⟹ is_transrec(M, ?MH, ?i, ?z) ⟷ ?z = transrec(?i, ?H)›*) haleph_transrec_replacement (*‹(M::i ⇒ o)(?a::i) ⟹ transrec_replacement(M, is_HAleph(M), ?a)›*) unfolding is_Aleph_def Aleph_rel_def\n (*goal: ‹is_transrec(M, is_HAleph(M), a, res) ⟷ res = transrec(a, HAleph_rel(M))›*)\n by simp\nqed\n\nend ― ‹\\<^locale>‹M_pre_aleph››\n\nlocale M_aleph = M_pre_aleph +\n assumes\n aleph_rel_separation: \"Ord(x) ⟹ M(x) ⟹ separation(M, λy. ∃z∈x. y = ℵ⇘z⇙⇗M⇖)\"\nbegin\n\nlemma Aleph_rel_cont: \"Limit(l) ⟹ M(l) ⟹ ℵ⇘l⇙⇗M⇖ = (⋃i<l. ℵ⇘i⇙⇗M⇖)\"\n using Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) Aleph_rel_limit (*‹⟦Limit(?α); M(?α)⟧ ⟹ ℵ⇘?α⇙⇗M⇖ = (⋃j∈?α. ℵ⇘j⇙⇗M⇖)›*) by (simp add:OUnion_def (*‹OUnion(?i, ?B) ≡ {z ∈ ⋃x∈?i. ?B(x) . Ord(?i)}›*))\n\nlemma Ord_Aleph_rel:\n assumes \"Ord(a)\"\n shows \"M(a) ⟹ Ord(ℵ⇘a⇙⇗M⇖)\"\n using ‹Ord(a)› (*‹Ord(a)›*) proof (induct a rule:trans_induct3 (*‹⟦Ord(?i); ?P(0); ⋀x. ⟦Ord(x); ?P(x)⟧ ⟹ ?P(succ(x)); ⋀x. ⟦Limit(x); ⋀y. y ∈ x ⟹ ?P(y)⟧ ⟹ ?P(x)⟧ ⟹ ?P(?i)›*))\n (*goals:\n 1. ‹M(0) ⟹ Ord(ℵ⇘0⇙⇗M⇖)›\n 2. ‹⋀x. ⟦Ord(x); M(x) ⟹ Ord(ℵ⇘x⇙⇗M⇖); M(succ(x))⟧ ⟹ Ord(ℵ⇘succ(x)⇙⇗M⇖)›\n 3. ‹⋀x. ⟦Limit(x); ⋀y. ⟦y ∈ x; M(y)⟧ ⟹ Ord(ℵ⇘y⇙⇗M⇖); M(x)⟧ ⟹ Ord(ℵ⇘x⇙⇗M⇖)›*)\n case 0 (*‹M(0)›*)\n show \"?case\"\n (*goal: ‹Ord(ℵ⇘0⇙⇗λa. M(a)⇖)›*)\n using Aleph_rel_zero (*‹ℵ⇘0⇙⇗M⇖ = ω›*) by simp\n next\n (*goals:\n 1. ‹⋀x::i. ⟦Ord(x); (M::i ⇒ o)(x) ⟹ Ord(ℵ⇘x⇙⇗M⇖); M(succ(x))⟧ ⟹ Ord(ℵ⇘succ(x)⇙⇗M⇖)›\n 2. ‹⋀x::i. ⟦Limit(x); ⋀y::i. ⟦y ∈ x; (M::i ⇒ o)(y)⟧ ⟹ Ord(ℵ⇘y⇙⇗M⇖); M(x)⟧ ⟹ Ord(ℵ⇘x⇙⇗M⇖)›*)\n case (succ x) (*‹Ord(x)› ‹M(x) ⟹ Ord(ℵ⇘x⇙⇗λa. M(a)⇖)› ‹M(succ(x))›*)\n with ‹Ord(x)› (*‹Ord(x)›*) have \"M(x)\" \"Ord(ℵ⇘x⇙⇗M⇖)\"\n apply -\n (*goals:\n 1. ‹⟦Ord(x::i); Ord(x); (M::i ⇒ o)(x) ⟹ Ord(ℵ⇘x⇙⇗λa::i. M(a)⇖); M(succ(x))⟧ ⟹ M(x)›\n 2. ‹⟦Ord(x::i); Ord(x); (M::i ⇒ o)(x) ⟹ Ord(ℵ⇘x⇙⇗λa::i. M(a)⇖); M(succ(x))⟧ ⟹ Ord(ℵ⇘x⇙⇗M⇖)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n with ‹Ord(x)› (*‹Ord(x)›*) have \"Ord(csucc_rel(M,ℵ⇘x⇙⇗M⇖))\"\n using Card_rel_is_Ord (*‹⟦Card⇗M⇖(?i); M(?i)⟧ ⟹ Ord(?i)›*) Card_rel_csucc_rel (*‹⟦Ord(?K); M(?K)⟧ ⟹ Card⇗M⇖((?K⁺)⇗M⇖)›*) by simp\n with ‹Ord(x)› (*‹Ord(x)›*) ‹M(x)› (*‹M(x)›*) show \"?case\"\n (*goal: ‹Ord(ℵ⇘succ(x)⇙⇗λa. M(a)⇖)›*)\n using Aleph_rel_succ (*‹⟦Ord(?α); M(?α)⟧ ⟹ ℵ⇘succ(?α)⇙⇗M⇖ = (ℵ⇘?α⇙⇗M⇖⁺)⇗M⇖›*) by simp\n next\n (*goal: ‹⋀x. ⟦Limit(x); ⋀y. ⟦y ∈ x; M(y)⟧ ⟹ Ord(ℵ⇘y⇙⇗M⇖); M(x)⟧ ⟹ Ord(ℵ⇘x⇙⇗M⇖)›*)\n case (limit x) (*‹Limit(x)› ‹⟦?y ∈ x; M(?y)⟧ ⟹ Ord(ℵ⇘?y⇙⇗λa. M(a)⇖)› ‹M(x)›*)\n note trans = transM[OF _ ‹M(x)›] (*‹?y ∈ x ⟹ M(?y)›*)\n from limit (*‹Limit(x::i)› ‹⟦?y ∈ x; M(?y)⟧ ⟹ Ord(ℵ⇘?y⇙⇗λa. M(a)⇖)› ‹M(x)›*) have \"ℵ⇘x⇙⇗M⇖ = (⋃i∈x. ℵ⇘i⇙⇗M⇖)\"\n using Aleph_rel_cont (*‹⟦Limit(?l); M(?l)⟧ ⟹ ℵ⇘?l⇙⇗M⇖ = (⋃i<?l. ℵ⇘i⇙⇗M⇖)›*) OUnion_def (*‹OUnion(?i, ?B) ≡ {z ∈ ⋃x∈?i. ?B(x) . Ord(?i)}›*) Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) by auto\n with limit (*‹Limit(x)› ‹⟦?y ∈ x; M(?y)⟧ ⟹ Ord(ℵ⇘?y⇙⇗λa. M(a)⇖)› ‹M(x)›*) show \"?case\"\n (*goal: ‹Ord(ℵ⇘x⇙⇗λa. M(a)⇖)›*)\n using Ord_UN (*‹(⋀x. x ∈ ?A ⟹ Ord(?B(x))) ⟹ Ord(⋃x∈?A. ?B(x))›*) trans (*‹?y ∈ x ⟹ M(?y)›*) by auto\n qed\n\nlemma Aleph_rel_increasing:\n assumes \"a < b\" and types: \"M(a)\" \"M(b)\"\n shows \"ℵ⇘a⇙⇗M⇖ < ℵ⇘b⇙⇗M⇖\"\nproof (-)\n (*goal: ‹ℵ⇘a⇙⇗M⇖ < ℵ⇘b⇙⇗M⇖›*)\n {\n fix x\n from assms (*‹a < b› ‹M(a)› ‹(M::i ⇒ o)(b::i)›*) have \"Ord(b)\"\n by (blast intro: lt_Ord2 (*‹?j < ?i ⟹ Ord(?i)›*))\n moreover assume \"M(x)\" (*‹(M::i ⇒ o)(x::i)›*)\n moreover note ‹M(b)› (*‹M(b)›*)\n ultimately have \"x < b ⟹ ℵ⇘x⇙⇗M⇖ < ℵ⇘b⇙⇗M⇖\"\n proof (induct b arbitrary: x rule: trans_induct3)\n (*goals:\n 1. ‹⋀x. ⟦x < 0; M(x); M(0)⟧ ⟹ ℵ⇘x⇙⇗M⇖ < ℵ⇘0⇙⇗M⇖›\n 2. ‹⋀x xa. ⟦Ord(x); ⋀xa. ⟦xa < x; M(xa); M(x)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘x⇙⇗M⇖; xa ≤ x; M(xa); M(succ(x))⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘succ(x)⇙⇗M⇖›\n 3. ‹⋀x xa. ⟦Limit(x); ⋀y xa. ⟦y ∈ x; xa < y; M(xa); M(y)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘y⇙⇗M⇖; xa < x; M(xa); M(x)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘x⇙⇗M⇖›*)\n case 0 (*‹x < 0› ‹(M::i ⇒ o)(x::i)› ‹(M::i ⇒ o)(0)›*)\n thus \"?case\"\n (*goal: ‹ℵ⇘x⇙⇗λa. M(a)⇖ < ℵ⇘0⇙⇗λa. M(a)⇖›*)\n by simp\n next\n (*goals:\n 1. ‹⋀x xa. ⟦Ord(x); ⋀xa. ⟦xa < x; M(xa); M(x)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘x⇙⇗M⇖; xa ≤ x; M(xa); M(succ(x))⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘succ(x)⇙⇗M⇖›\n 2. ‹⋀x xa. ⟦Limit(x); ⋀y xa. ⟦y ∈ x; xa < y; M(xa); M(y)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘y⇙⇗M⇖; xa < x; M(xa); M(x)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘x⇙⇗M⇖›*)\n case (succ b) (*‹Ord(b)› ‹⟦?x < b; M(?x); M(b)⟧ ⟹ ℵ⇘?x⇙⇗λa. M(a)⇖ < ℵ⇘b⇙⇗λa. M(a)⇖› ‹(x::i) ≤ (b::i)› ‹(M::i ⇒ o)(x::i)› ‹M(succ(b))›*)\n then show \"?case\"\n (*goal: ‹ℵ⇘x⇙⇗λa. M(a)⇖ < ℵ⇘succ(b)⇙⇗λa. M(a)⇖›*)\n using Card_rel_csucc_rel (*‹⟦Ord(?K); M(?K)⟧ ⟹ Card⇗M⇖((?K⁺)⇗M⇖)›*) Ord_Aleph_rel (*‹⟦Ord(?a); M(?a)⟧ ⟹ Ord(ℵ⇘?a⇙⇗M⇖)›*) Ord_Union_succ_eq (*‹Ord(?i) ⟹ ⋃succ(?i) = ?i›*) lt_csucc_rel (*‹⟦Ord(?K::i); (M::i ⇒ o)(?K)⟧ ⟹ ?K < (?K⁺)⇗M⇖›*) lt_trans[of _ \"ℵ⇘b⇙⇗M⇖\" \"csucc⇗M⇖(ℵ⇘b⇙⇗M⇖)\"] (*‹⟦?i < ℵ⇘b⇙⇗M⇖; ℵ⇘b⇙⇗M⇖ < (ℵ⇘b⇙⇗M⇖⁺)⇗M⇖⟧ ⟹ ?i < (ℵ⇘b⇙⇗M⇖⁺)⇗M⇖›*) apply (subst (2) def_transrec[OF Aleph_rel_def'] (*‹ℵ⇘?a::i⇙⇗?M2::i ⇒ o⇖ = HAleph_rel(?M2, ?a, λx::i∈?a. ℵ⇘x⇙⇗?M2⇖)›*))\n (*goal: ‹ℵ⇘x::i⇙⇗λa::i. (M::i ⇒ o)(a)⇖ < ℵ⇘succ(b::i)⇙⇗λa::i. M(a)⇖›*)\n by (auto simp add: le_iff (*‹?i ≤ ?j ⟷ ?i < ?j ∨ ?i = ?j ∧ Ord(?j)›*) HAleph_rel_def (*‹HAleph_rel ≡ λN i r. if ¬ Ord(i) then i else if i = 0 then ω else if ¬ Limit(i) ∧ i ≠ 0 then (r ` (⋃i)⁺)⇗N⇖ else ⋃{a . j ∈ i, N(a) ∧ N(j) ∧ a = r ` j}›*))\n next\n (*goal: ‹⋀x xa. ⟦Limit(x); ⋀y xa. ⟦y ∈ x; xa < y; M(xa); M(y)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘y⇙⇗M⇖; xa < x; M(xa); M(x)⟧ ⟹ ℵ⇘xa⇙⇗M⇖ < ℵ⇘x⇙⇗M⇖›*)\n case (limit l) (*‹Limit(l::i)› ‹⟦?y ∈ l; ?x < ?y; M(?x); M(?y)⟧ ⟹ ℵ⇘?x⇙⇗λa. M(a)⇖ < ℵ⇘?y⇙⇗λa. M(a)⇖› ‹x < l› ‹M(x)› ‹M(l)›*)\n then have sc: \"succ(x) < l\"\n by (blast intro: Limit_has_succ (*‹⟦Limit(?i); ?j < ?i⟧ ⟹ succ(?j) < ?i›*))\n then have \"ℵ⇘x⇙⇗M⇖ < (⋃j<l. ℵ⇘j⇙⇗M⇖)\"\n using limit (*‹Limit(l)› ‹⟦?y ∈ l; ?x < ?y; M(?x); M(?y)⟧ ⟹ ℵ⇘?x⇙⇗λa. M(a)⇖ < ℵ⇘?y⇙⇗λa. M(a)⇖› ‹(x::i) < (l::i)› ‹M(x)› ‹M(l)›*) Ord_Aleph_rel (*‹⟦Ord(?a); M(?a)⟧ ⟹ Ord(ℵ⇘?a⇙⇗M⇖)›*) Ord_OUN (*‹(⋀x::i. x < (?A::i) ⟹ Ord((?B::i ⇒ i)(x))) ⟹ Ord(⋃x::i<?A. ?B(x))›*) proof (rule_tac OUN_upper_lt (*‹⟦?a < ?A; ?i < ?b(?a); Ord(⋃x<?A. ?b(x))⟧ ⟹ ?i < (⋃x<?A. ?b(x))›*), blast intro: Card_rel_is_Ord (*‹⟦Card⇗M⇖(?i); M(?i)⟧ ⟹ Ord(?i)›*) ltD (*‹?i < ?j ⟹ ?i ∈ ?j›*) lt_Ord (*‹?j < ?i ⟹ Ord(?j)›*))\n (*goals:\n 1. ‹⟦succ(x) < l; Limit(l); ⋀y x. ⟦y ∈ l; x < y; M(x); M(y)⟧ ⟹ ℵ⇘x⇙⇗λa. M(a)⇖ < ℵ⇘y⇙⇗λa. M(a)⇖; x < l; M(x); M(l); ⋀a. ⟦Ord(a); M(a)⟧ ⟹ Ord(ℵ⇘a⇙⇗M⇖); ⋀A B. (⋀x. x < A ⟹ Ord(B(x))) ⟹ Ord(⋃x<A. B(x))⟧ ⟹ ℵ⇘x⇙⇗M⇖ < ℵ⇘succ(x)⇙⇗M⇖›\n 2. ‹⟦succ(x) < l; Limit(l); ⋀y x. ⟦y ∈ l; x < y; M(x); M(y)⟧ ⟹ ℵ⇘x⇙⇗λa. M(a)⇖ < ℵ⇘y⇙⇗λa. M(a)⇖; x < l; M(x); M(l); ⋀a. ⟦Ord(a); M(a)⟧ ⟹ Ord(ℵ⇘a⇙⇗M⇖); ⋀A B. (⋀x. x < A ⟹ Ord(B(x))) ⟹ Ord(⋃x<A. B(x))⟧ ⟹ Ord(⋃j<l. ℵ⇘j⇙⇗M⇖)›*)\n from ‹x<l› (*‹x < l›*) ‹Limit(l)› (*‹Limit(l::i)›*) have \"Ord(x)\"\n using Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) Ord_in_Ord (*‹⟦Ord(?i); ?j ∈ ?i⟧ ⟹ Ord(?j)›*) by (auto dest!:ltD (*‹?i < ?j ⟹ ?i ∈ ?j›*))\n with ‹M(x)› (*‹M(x)›*) show \"ℵ⇘x⇙⇗M⇖ < ℵ⇘succ(x)⇙⇗M⇖\"\n using Card_rel_csucc_rel (*‹⟦Ord(?K::i); (M::i ⇒ o)(?K)⟧ ⟹ Card⇗M⇖((?K⁺)⇗M⇖)›*) Ord_Aleph_rel (*‹⟦Ord(?a); M(?a)⟧ ⟹ Ord(ℵ⇘?a⇙⇗M⇖)›*) lt_csucc_rel (*‹⟦Ord(?K); M(?K)⟧ ⟹ ?K < (?K⁺)⇗M⇖›*) ltD[THEN [ 2 ] Ord_in_Ord] (*‹⟦Ord(?i); ?j < ?i⟧ ⟹ Ord(?j)›*) succ_in_MI[OF ‹M(x)›] (*‹M(succ(x))›*) Aleph_rel_succ[of x] (*‹⟦Ord(x); M(x)⟧ ⟹ ℵ⇘succ(x)⇙⇗M⇖ = (ℵ⇘x⇙⇗M⇖⁺)⇗M⇖›*) by simp\n next\n (*goal: ‹⟦succ(x) < l; Limit(l); ⋀y x. ⟦y ∈ l; x < y; M(x); M(y)⟧ ⟹ ℵ⇘x⇙⇗λa. M(a)⇖ < ℵ⇘y⇙⇗λa. M(a)⇖; x < l; M(x); M(l); ⋀a. ⟦Ord(a); M(a)⟧ ⟹ Ord(ℵ⇘a⇙⇗M⇖); ⋀A B. (⋀x. x < A ⟹ Ord(B(x))) ⟹ Ord(⋃x<A. B(x))⟧ ⟹ Ord(⋃j<l. ℵ⇘j⇙⇗M⇖)›*)\n from ‹M(l)› (*‹M(l)›*) ‹Limit(l)› (*‹Limit(l)›*) show \"Ord(⋃j<l. ℵ⇘j⇙⇗M⇖)\"\n using Ord_Aleph_rel (*‹⟦Ord(?a); M(?a)⟧ ⟹ Ord(ℵ⇘?a⇙⇗M⇖)›*) lt_Ord (*‹?j < ?i ⟹ Ord(?j)›*) Limit_is_Ord (*‹Limit(?i::i) ⟹ Ord(?i)›*) Ord_in_Ord (*‹⟦Ord(?i::i); (?j::i) ∈ ?i⟧ ⟹ Ord(?j)›*) apply (rule_tac Ord_OUN (*‹(⋀x. x < ?A ⟹ Ord(?B(x))) ⟹ Ord(⋃x<?A. ?B(x))›*))\n (*goal: ‹Ord(⋃j<l. ℵ⇘j⇙⇗M⇖)›*)\n by (auto dest:transM (*‹⟦?y ∈ ?x; M(?x)⟧ ⟹ M(?y)›*) ltD (*‹?i < ?j ⟹ ?i ∈ ?j›*) intro!:Ord_Aleph_rel (*‹⟦Ord(?a); M(?a)⟧ ⟹ Ord(ℵ⇘?a⇙⇗M⇖)›*))\n qed\n then show \"?case\"\n (*goal: ‹ℵ⇘x⇙⇗λa. M(a)⇖ < ℵ⇘l⇙⇗λa. M(a)⇖›*)\n using limit (*‹Limit(l::i)› ‹⟦?y ∈ l; ?x < ?y; M(?x); M(?y)⟧ ⟹ ℵ⇘?x⇙⇗λa. M(a)⇖ < ℵ⇘?y⇙⇗λa. M(a)⇖› ‹x < l› ‹M(x)› ‹M(l)›*) Aleph_rel_cont (*‹⟦Limit(?l::i); (M::i ⇒ o)(?l)⟧ ⟹ ℵ⇘?l⇙⇗M⇖ = (⋃i::i<?l. ℵ⇘i⇙⇗M⇖)›*) by simp\n qed\n }\n with types (*‹(M::i ⇒ o)(a::i)› ‹M(b)›*) assms (*‹a < b› ‹M(a)› ‹M(b)›*) show \"?thesis\"\n (*goal: ‹ℵ⇘a::i⇙⇗M::i ⇒ o⇖ < ℵ⇘b::i⇙⇗M⇖›*)\n by simp\nqed\n\nlemma Card_rel_Aleph_rel [simp, intro]:\n assumes \"Ord(a)\" and types: \"M(a)\" shows \"Card⇗M⇖(ℵ⇘a⇙⇗M⇖)\"\n using assms (*‹Ord(a::i)› ‹M(a)›*) proof (induct rule:trans_induct3 (*‹⟦Ord(?i); ?P(0); ⋀x. ⟦Ord(x); ?P(x)⟧ ⟹ ?P(succ(x)); ⋀x. ⟦Limit(x); ⋀y. y ∈ x ⟹ ?P(y)⟧ ⟹ ?P(x)⟧ ⟹ ?P(?i)›*))\n (*goals:\n 1. ‹M(0) ⟹ Card⇗M⇖(ℵ⇘0⇙⇗M⇖)›\n 2. ‹⋀x. ⟦Ord(x); M(x) ⟹ Card⇗M⇖(ℵ⇘x⇙⇗M⇖); M(succ(x))⟧ ⟹ Card⇗M⇖(ℵ⇘succ(x)⇙⇗M⇖)›\n 3. ‹⋀x. ⟦Limit(x); ⋀y. ⟦y ∈ x; M(y)⟧ ⟹ Card⇗M⇖(ℵ⇘y⇙⇗M⇖); M(x)⟧ ⟹ Card⇗M⇖(ℵ⇘x⇙⇗M⇖)›*)\n case 0 (*‹M(0)›*)\n then show \"?case\"\n (*goal: ‹Card⇗λa::i. (M::i ⇒ o)(a)⇖(ℵ⇘0⇙⇗λa::i. M(a)⇖)›*)\n using Aleph_rel_zero (*‹ℵ⇘0⇙⇗M::i ⇒ o⇖ = ω›*) Card_rel_nat (*‹Card⇗M⇖(ω)›*) by simp\n next\n (*goals:\n 1. ‹⋀x. ⟦Ord(x); M(x) ⟹ Card⇗M⇖(ℵ⇘x⇙⇗M⇖); M(succ(x))⟧ ⟹ Card⇗M⇖(ℵ⇘succ(x)⇙⇗M⇖)›\n 2. ‹⋀x. ⟦Limit(x); ⋀y. ⟦y ∈ x; M(y)⟧ ⟹ Card⇗M⇖(ℵ⇘y⇙⇗M⇖); M(x)⟧ ⟹ Card⇗M⇖(ℵ⇘x⇙⇗M⇖)›*)\n case (succ x) (*‹Ord(x)› ‹M(x) ⟹ Card⇗λa. M(a)⇖(ℵ⇘x⇙⇗λa. M(a)⇖)› ‹M(succ(x))›*)\n then show \"?case\"\n (*goal: ‹Card⇗λa. M(a)⇖(ℵ⇘succ(x)⇙⇗λa. M(a)⇖)›*)\n using Card_rel_csucc_rel (*‹⟦Ord(?K); M(?K)⟧ ⟹ Card⇗M⇖((?K⁺)⇗M⇖)›*) Ord_Aleph_rel (*‹⟦Ord(?a); M(?a)⟧ ⟹ Ord(ℵ⇘?a⇙⇗M⇖)›*) Aleph_rel_succ (*‹⟦Ord(?α::i); (M::i ⇒ o)(?α)⟧ ⟹ ℵ⇘succ(?α)⇙⇗M⇖ = (ℵ⇘?α⇙⇗M⇖⁺)⇗M⇖›*) by simp\n next\n (*goal: ‹⋀x. ⟦Limit(x); ⋀y. ⟦y ∈ x; M(y)⟧ ⟹ Card⇗M⇖(ℵ⇘y⇙⇗M⇖); M(x)⟧ ⟹ Card⇗M⇖(ℵ⇘x⇙⇗M⇖)›*)\n case (limit x) (*‹Limit(x)› ‹⟦?y ∈ x; M(?y)⟧ ⟹ Card⇗λa. M(a)⇖(ℵ⇘?y⇙⇗λa. M(a)⇖)› ‹M(x)›*)\n moreover from this (*‹Limit(x)› ‹⟦?y ∈ x; M(?y)⟧ ⟹ Card⇗λa. M(a)⇖(ℵ⇘?y⇙⇗λa. M(a)⇖)› ‹M(x)›*) have \"Ord(x)\"\n using Limit_is_Ord (*‹Limit(?i::i) ⟹ Ord(?i)›*) by simp\n from this (*‹Ord(x)›*) have \"{y . z ∈ x, M(y) ∧ M(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y . z ∈ x, M(y) ∧ M(z) ∧ Ord(z) ∧ y = ℵ⇘z⇙⇗M⇖}\"\n using Ord_in_Ord (*‹⟦Ord(?i); ?j ∈ ?i⟧ ⟹ Ord(?j)›*) by simp\n moreover from ‹Ord(x)› (*‹Ord(x)›*) have \"{y . z ∈ x, M(y) ∧ M(z) ∧ Ord(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y . z ∈ x, M(z) ∧ y = ℵ⇘z⇙⇗M⇖}\"\n using Ord_in_Ord (*‹⟦Ord(?i::i); (?j::i) ∈ ?i⟧ ⟹ Ord(?j)›*) by blast\n moreover from ‹Ord(x)› (*‹Ord(x)›*) ‹M(x)› (*‹M(x)›*) have \"{y . z ∈ x, M(z) ∧ y = ℵ⇘z⇙⇗M⇖} ⊆ ℵ⇘x⇙⇗M⇖\"\n using Aleph_rel_increasing (*‹⟦?a < ?b; M(?a); M(?b)⟧ ⟹ ℵ⇘?a⇙⇗M⇖ < ℵ⇘?b⇙⇗M⇖›*) by (auto dest:ltD (*‹?i < ?j ⟹ ?i ∈ ?j›*) transM (*‹⟦?y ∈ ?x; M(?x)⟧ ⟹ M(?y)›*) intro:ltI (*‹⟦?i ∈ ?j; Ord(?j)⟧ ⟹ ?i < ?j›*))\n with calculation (*‹Limit(x)› ‹⟦?y ∈ x; M(?y)⟧ ⟹ Card⇗λa. M(a)⇖(ℵ⇘?y⇙⇗λa. M(a)⇖)› ‹(M::i ⇒ o)(x::i)› ‹{y . z ∈ x, M(y) ∧ M(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y . z ∈ x, M(y) ∧ M(z) ∧ Ord(z) ∧ y = ℵ⇘z⇙⇗M⇖}› ‹{y . z ∈ x, M(y) ∧ M(z) ∧ Ord(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y . z ∈ x, M(z) ∧ y = ℵ⇘z⇙⇗M⇖}›*) have \"{y . z ∈ x, M(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y ∈ ℵ⇘x⇙⇗M⇖ . (∃z ∈ x. y = ℵ⇘z⇙⇗M⇖)}\"\n by (blast dest:transM (*‹⟦?y ∈ ?x; M(?x)⟧ ⟹ M(?y)›*))\n with calculation (*‹Limit(x::i)› ‹⟦?y ∈ x; M(?y)⟧ ⟹ Card⇗λa. M(a)⇖(ℵ⇘?y⇙⇗λa. M(a)⇖)› ‹M(x)› ‹{y . z ∈ x, M(y) ∧ M(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y . z ∈ x, M(y) ∧ M(z) ∧ Ord(z) ∧ y = ℵ⇘z⇙⇗M⇖}› ‹{y::i . z::i ∈ x::i, (M::i ⇒ o)(y) ∧ M(z) ∧ Ord(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y::i . z::i ∈ x, M(z) ∧ y = ℵ⇘z⇙⇗M⇖}›*) have \"{y . z ∈ x, M(y) ∧ M(z) ∧ y = ℵ⇘z⇙⇗M⇖} = {y ∈ ℵ⇘x⇙⇗M⇖ . (∃z ∈ x. y = ℵ⇘z⇙⇗M⇖)}\"\n by simp\n moreover from ‹Ord(x)› (*‹Ord(x)›*) ‹M(x)› (*‹M(x)›*) have \"M({y ∈ ℵ⇘x⇙⇗M⇖ . (∃z ∈ x. y = ℵ⇘z⇙⇗M⇖)})\"\n using aleph_rel_separation (*‹⟦Ord(?x); M(?x)⟧ ⟹ separation(M, λy. ∃z∈?x. y = ℵ⇘z⇙⇗M⇖)›*) by simp\n ultimately show \"?case\"\n (*goal: ‹Card⇗λa. M(a)⇖(ℵ⇘x⇙⇗λa. M(a)⇖)›*)\n using Ord_Aleph_rel (*‹⟦Ord(?a); M(?a)⟧ ⟹ Ord(ℵ⇘?a⇙⇗M⇖)›*) Card_nat (*‹Card(ω)›*) Limit_is_Ord (*‹Limit(?i) ⟹ Ord(?i)›*) Card_relI (*‹⟦Ord(?i); ⋀j. ⟦j < ?i; M(j)⟧ ⟹ ¬ j ≈⇗M⇖ ?i; M(?i)⟧ ⟹ Card⇗M⇖(?i)›*) apply (subst def_transrec [OF Aleph_rel_def'] (*‹ℵ⇘?a⇙⇗?M2⇖ = HAleph_rel(?M2, ?a, λx∈?a. ℵ⇘x⇙⇗?M2⇖)›*))\n (*goal: ‹Card⇗λa. M(a)⇖(ℵ⇘x⇙⇗λa. M(a)⇖)›*)\n by (auto simp add:HAleph_rel_def (*‹HAleph_rel ≡ λ(N::i ⇒ o) (i::i) r::i. if ¬ Ord(i) then i else if i = 0 then ω else if ¬ Limit(i) ∧ i ≠ 0 then (r ` (⋃i)⁺)⇗N⇖ else ⋃{a::i . j::i ∈ i, N(a) ∧ N(j) ∧ a = r ` j}›*))\n qed\n\nlemmas nat_subset_Aleph_rel_1 =\n Ord_lt_subset[OF Ord_Aleph_rel[of 1] Aleph_rel_increasing[of 0 1,simplified],simplified]\n\nend ― ‹\\<^locale>‹M_aleph››\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Transitive_Models/Aleph_Relative.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"f927061b2269ec16878b11ffc3d6ed603a16de06f6b0c68d09f96c8521f7c926\"\n}"}}},{"rowIdx":191,"cells":{"text":{"kind":"string","value":"(*\n * Copyright (c) 2022 Apple Inc. All rights reserved.\n *\n * SPDX-License-Identifier: BSD-2-Clause\n *)\n\nsection ‹Pointers into Structures in Split Heap \\label{sec:open_struct}›\n\ntheory open_struct\n imports \"AutoCorres\"\nbegin\n\nsubsection ‹Overview›\n\ntext ‹\nHeap lifting in AutoCorres transforms the monolithic byte-level heap \\<^typ>‹heap_mem› with explicit\nterm level heap type description \\<^typ>‹heap_typ_desc› and typing constraints, like \\<^const>‹c_guard›,\n \\<^const>‹h_t_valid› and \\<^term>‹d ⊨⇩t p› to a more abstract split heap model with implicit HOL-typing.\nThe major advantage of the split heap model is that, updates to one particular heap do\nnot affect all other distinct heaps, removing the obligation to explicitly take care about aliasing.\nAliasing between different heaps is ruled out by the model already.\n\nThe question that arises is on which granularity do we split the byte-level heap?\nThe original AutoCorres splitted at the granularity of types. Each distinct c-type is mapped into\na separate heap. In that model you can still have intra-type aliasing (i.e. pointers of the\nsame type can alias), but no inter-type aliasing (i.e. pointers of different type are distinct, and\n cannot alias). As an example a pointer to a list structure refers to a different (split) heap as\na pointer to a tree structure. So when a function modifies lists, no pointer to a tree is affected.\n\nOf course not every C-program can be represented in this split-heap model. In those cases\nthe abstraction fails.\nIn particular think of a low-level memory allocator, that allocates some raw-bytes and\ndelivers a void-pointer, that is then retyped to point to the actual structure. To be able\nto combine those byte-level functions with high-level typed functions (in the split heap) one\ncan switch between both layers of abstraction by \\<^const>‹exec_concrete› and \\<^const>‹exec_abstract›.\nThe good thing is that this gives us a very expressive tool to switch between byte-level and\ntyped view. We can do the specification and proofs of different parts of the program on\nthe adequate level and still combine the results. But switching between the layers can be\ntedious and thus should be limited to the low-level functions that really inherently need\nbyte-level reasoning.\n\nUnfortunately the type granularity of the split heaps also rules out programs dealing with\npointers into a structure. For example a structure containing a ‹int› field ‹count› cannot pass a\n‹int› pointer to that field ‹count› to a another function, that might update it.\nThe complete structure has a separate\n(split) heap that is distinct from the heap containing ‹int›s. This would mean that we\nhave to resort to the byte-level heap to reason about those functions.\n\nWe extended the split heap model to support this common use-case. The core idea is that\nin addition to split heaps on the granularity of types you can also specify split heaps on the\ngranularity of structure fields. Conceptually you 'open' the structure into its \ncomponents (aka. fields) and each field gets a separate heap. Additionally you can specify which \nfields should be ∗‹addressable› and thus shared. In the example you can specify that ‹count› is addressable,\nso it is represented within the same heap as all other ‹int›. This means that an ‹int› pointer can\nnow point to a plain ‹int› or to a ‹count› field within a structure. You explicitly allow\nthis limited inter-type aliasing on ‹int› pointers. All other fields that are not\nexplicitly marked as ∗‹addressable› still have their own heap and thus cannot alias with another\npointer of the same field type. Note that mainly for performance reasons we actually avoid to \nhave a separate heap for each single field but try to minimise the number of heaps by clustering the \nfields that are treated the same. Some more details on this construction are below. \n›\n\nsubsection ‹Example Program and some Intuition›\n\ntext ‹We illustrate the approach with the following example specifying some structures,\nwith addressable and non-addressable fields. Note that for an addressable field that is\n of an array type, each element of the array is considered to be addressable.\n›\n\ninstall_C_file \"open_struct.c\"\n\ntext ‹›\n\ndeclare [[show_types=false, show_sorts=false,\n verbose = 0, verbose_timing = 0, ML_print_depth = 1000, goals_limit = 20]]\n\ninit-autocorres [addressable_fields =\n inner.fld1\n inner.fld2\n outer.inner\n outer_array.inner_array\n array.elements\n other.fy\n two_dimensional.matrix\n data_struct1.d1.y1\n data_struct1.d1.y2\n data_struct1.d1.y3\n data_struct1.d1.y4\n data_struct1.d2\n data_struct2.d\n data_array.array,\n single_threaded (* turn on to see timing info *)] \"open_struct.c\"\n\ntext ‹›\n\nautocorres open_struct.c\n\ntext ‹\nWhich split heaps are generated? Each split heap is a component of the record \\<^typ>‹lifted_globals›.\n›\n\nprint_record lifted_globals\n\n text ‹\n\nStructure ‹closed› does not have any addressable field. So there is a separate heap for that\ncomponent \\<^const>‹closed_C›.\n\nStructure ‹inner› has three ‹unsigned› fields \\<^const>‹fld1_C›, \\<^const>‹fld2_C› and \\<^const>‹fld3_C›,\nThe first two fields are addressable whereas the third one is not.\nThis means that the first two fields are put into the same common heap for unsigned integers\n\\<^const>‹heap_w32› whereas the third field is stored in a dedicated heap for all remaining fields\n \\<^const>‹dedicated_heap_inner_C›.\n\nWe use the following nomenclature to distinguish the various kinds of split heaps and types:\n▪ closed type or structure: There is no addressable field within the type / structure,\n e.g. \\<^typ>‹closed_C›.\n▪ open type or structure: There is at least one addressable field within the type / structure,\n e.g. \\<^typ>‹inner_C›.\n▪ atomic heap: Heap for a closed type, or a common heap that might be nested in an open structure.\n e.g. \\<^const>‹heap_closed_C›.\n▪ dedicated heap: Internal heap for all non-addressable fields of an open type,\n e.g. \\<^const>‹dedicated_heap_inner_C›. These dedicated heaps are used for the construction of a derived heap.\n▪ derived heap: Heap for an open type / structure. The name 'derived' indicates that it is not a fundamental heap\n but a composition of several common heaps and a dedicated heap,\n e.g. \\<^const>‹inner_C›\n▪ fundamental heap: those heaps that are directly present in the new global state \\<^typ>‹lifted_globals›.\n These are atomic heaps, and dedicated heaps.\n▪ virtual heap: heaps that represent the user level view. They are directly visible in\n the program after heap-lifting. This excludes dedicated heaps, which non-the-less may be used in the\n model as representation of that type.\n\nDedicated heaps play a special role, as they are subsumed within derived heaps. In a sense they are\nnot part of the 'API' for split heaps: After heap lifting the resulting program only directly\nmentions atomic, common or derived heaps. Nevertheless dedicated heaps are important for the\nconstruction of derived heaps and might show up after unfolding the definitions for derived heaps.\n\nAlso dedicated heaps may be used by the user as the actual heap for a specific type. Only the dedicated\nheaps and the atomic heaps commute, the derived heaps do not necessary commute with each other as\nthey might have common components that alias each other.\n›\n\nsubsection ‹Background›\n\ntext ‹\nThe starting point for the split heap abstraction is still the\nunified memory model (UMM) by Harvey Tuch\n(@{url ‹https://trustworthy.systems/publications/papers/Tuch%3Aphd.pdf›}). The UMM provides\nthe general concept of \\<^class>‹c_type› to convert between a byte-list representations of\nC values to abstract HOL-types. It also provides an explicit notion of typing and what a\nwell-typed byte level heap is (with respect to a heap type description). With these notions\nin place one can describe the set of valid pointers for a C program. The model is quite elaborate\nand respects the potential nesting of structures. A well-typed pointer to a structure in the heap\ngives rise to a bunch of other valid sub-pointers, pointing to sub-fields, which again could point to\na nested structure.\nIn the UMM, array types are modelled as a special case of a structure, where each index corresponds to\nan artificial field, where the field name is a unary encoding of the index. Hence, for most of\nthe explanation in this file it is sufficient to elaborate on structures. \nThe main limitation of the UMM is that it does not handle unions.\n\nFor the original split heap version of AutoCorres\n@{url ‹https://trustworthy.systems/publications/nicta_full_text/8758.pdf›}, David Greenaway\nput a simplified layer on top of the UMM focusing only on the root-pointers of a structure,\nignoring any nested pointers to a field. This provides a split heap abstraction on\nthe granularity of types.\n\nThe new split heap model takes an intermediate view. For closed structures it coincides with\nthe original split heap model. Only root-pointers to a closed structure are valid.\nIn addition for open structures also the nested pointers to addressable fields are welcome and valid.\nThis flexibility allows the user to carefully open up the structures only as far as needed.\nNote the trade-off between expressibility and \"proof-burden\" of open structures. Aliasing is\nby construction ruled out for distinct split heaps, but has to be dealt with explicitly on\nthe user level for common heaps.\n\nTo allow this flexibility some existing notions where slightly refined and some notions where\nintroduced.\n›\n\nsubsubsection ‹Footprints and valid pointers›\n\ntext ‹\nThe central typing judgement of UMM is \\<^const>‹h_t_valid›, with infix syntax \\<^term>‹d,g ⊨⇩t p›, meaning\npointer \\<^term>‹p› is valid with respect to heap type description \\<^term>‹d› and a guard \\<^term>‹g›.\nThe default guard is \\<^const>‹c_guard›, ensuring that the pointer is aligned, is not NULL and that\nthe referenced value fits into the address space, meaning that the addresses of the\nindividual bytes do not overflow the address space. This instantiation with \\<^const>‹c_guard›\nis abbreviated with \\<^term>‹d ⊨⇩t p›.\n›\nthm c_guard_def\nthm TypHeap.h_t_valid_def\nthm h_t_valid_valid_footprint\n\ntext ‹\nThe judgement \\<^term>‹d,g ⊨⇩t p› also ensures that the footprint is valid:\n\\<^term>‹valid_footprint d (ptr_val p) (typ_uinfo_t TYPE('a::c_type))›.\nNote that \\<^term>‹p› is a typed pointer, carrying the (phantom) type of the value it points\nto \\<^term>‹p::'a ptr›. This type annotation is no longer present in \\<^const>‹valid_footprint› which\nonly talks about the address of the pointer \\<^const>‹ptr_val› and relates it to a type tag\nvia \\<^const>‹typ_uinfo_t›.\n›\nthm valid_footprint_def\ntext ‹\nWithout going into details of the definition of \\<^const>‹valid_footprint› we get guarantees for\nall nested pointers of the structure which also have \\<^const>‹valid_footprint›. Moreover, what\nwe do not know is whether the pointer \\<^term>‹p› is itself a nested pointer of some enclosing\nstructure.\n\nTo get the additional knowledge that term \\<^term>‹p› is a root pointer, meaning it is not\nenclosed in a bigger structure, we have the notion \\<^const>‹valid_root_footprint›.\nThis notion was refined from the original version of AutoCorres \\<^const>‹valid_simple_footprint›\nto get the important property that every \\<^const>‹valid_root_footprint› is\nindeed also a \\<^const>‹valid_footprint›.\n›\nthm valid_root_footprint_valid_footprint\ntext ‹Also for all types that fit into the address space we can go from \\<^const>‹valid_root_footprint›\n to \\<^const>‹valid_simple_footprint›. The restriction on the address space size follows from the\n implicit property of the legacy\\<^const>‹valid_simple_footprint›: The first byte and all the rest have a\n different tag, so there must not be a complete wrap around in the address space.\n›\nthm valid_simple_footprint_size_td\nthm valid_root_footprint_valid_simple_footprint\n\ntext ‹\nThe corresponding judgment to \\<^term>‹d ⊨⇩t p› for root pointers is\n\\<^term>‹root_ptr_valid d p›. It implies \\<^term>‹c_guard p› as well as\n\\<^term>‹valid_root_footprint d (ptr_val p) (typ_uinfo_t TYPE('a::c_type))›.\n›\nthm root_ptr_valid_def\n\nsubsubsection ‹From byte heap to typed heaps›\ntext ‹\nIn UMM the core wrapper functions to provide a typed view on the raw byte level heap are:\n▪\\<^term>‹h_val::heap_mem ⇒ 'a::c_type ptr ⇒ 'a› to lookup typed values by typed pointers and\n▪\\<^term>‹heap_update::'a::c_type ptr ⇒ 'a ⇒ heap_mem ⇒ heap_mem› to update the heap\n on the location designated by a typed pointer with a typed value.\n\nProvided a byte level heap and a heap type description there is a function to lift the\nheap into a typed heap:\n▪\\<^term>‹lift_t::'a::c_type ptr_guard ⇒ heap_mem × heap_typ_desc ⇒ ('a ptr ⇀ 'a)›\n takes a pointer guard and a pair of byte level heap and heap type description and provides a\n partial function from typed pointers to typed values. The most prominent instance\n is abbreviation \\<^term>‹clift = lift_t c_guard›.\n\nThe resulting function is partial as it only lifts those pointers that actually point\nto valid addresses according to \\<^term>‹h_t_valid›. This fact and the connection to \\<^const>‹h_val›\nis reflected in @{thm lift_t_if}.\n›\nthm lift_t_if\n\ntext ‹To lift root pointers there is\n▪ \\<^term>‹simple_lift::heap_mem × heap_typ_desc ⇒ ('a ptr ⇀ 'a::c_type)›.\nIt yields only defined values for root pointers, according to \\<^const>‹root_ptr_valid›.\n›\nthm simple_lift_def\n\ntext ‹\nFor the derived heaps we introduce the family of functions \\<^const>‹plift›, which\nare defined in @{locale wf_ptr_valid} and instantiated for each collection of open types.\nLike \\<^term>‹clift› the definition is based on \\<^const>‹lift_t›, but instead of the plain \\<^const>‹c_guard› a custom\nvalidity-guard for the type is supplied \\<^const>‹ptr_valid›. The relevant definitions are collected as named theorems:\n›\nthm plift_defs\n\ntext ‹\nNote that we used to introduced a separate distinct instance of \\<^const>‹plift› and \\<^const>‹ptr_valid›\nfor each type. Now we have encapsulated this construction in @{locale open_types}.\nFrom a specification of which fields of a type should be addressable we first derive an\nvariant without phantom typing of all the concepts especially \\<^const>‹open_types.ptr_valid_u› and then\ndefine the phantom typed version based on that. This separation works around the limitation of the\nIsabelle type system that are also reflected in locales. It does not allow explicit quantification on type variables.\nSo we can only have fixed type \"variables\" in the locale assumptions. However, the lemmas derived within\na locale can still be polymorphic.\nSo we express the needed assumptions in the locale @{locale open_types} in the \"untyped\" world and\nderive the \"typed\" polymorphic versions as lemmas within that locale.\n›\nthm open_struct.ptr_valid_u.simps\nthm open_types.ptr_valid_u.simps\nthm open_struct.ptr_valid_def\nthm open_types.ptr_valid_def\n\n\n\ntext ‹\nLet us look into the definitions for the open structure \\<^typ>‹inner_C› which itself is part\nof the open structure \\<^typ>‹outer_C›.\n›\n\nlemma \"plift h = lift_t (ptr_valid (hrs_htd h)) h\"\n by (simp add: open_struct.plift_def (*‹plift ?h = lift_t (PTR_VALID(?'a) (hrs_htd ?h)) ?h›*))\n\n\nthm inner_C.ptr_valid.unfold\nlemma\n fixes p::\"inner_C ptr\"\n shows \"ptr_valid d p =\n (root_ptr_valid d p ∨\n (∃q::outer_C ptr. ptr_valid d q ∧ ptr_val p = &(q→[''inner_C''])) ∨\n (∃i<5. ∃q::(inner_C[5]) ptr. ptr_valid d q ∧ ptr_val p = &(q→[replicate i CHR ''1''])))\"\n by (rule inner_C.ptr_valid.unfold (*‹PTR_VALID(inner_C) ?d ?p = (root_ptr_valid ?d ?p ∨ (∃q. PTR_VALID(outer_C) ?d q ∧ ptr_val ?p = &(q→[''inner_C''])) ∨ (∃n<5. ∃q. PTR_VALID(inner_C[5]) ?d q ∧ ptr_val ?p = &(q→[replicate n CHR ''1''])))›*))\n\nlemma\n fixes p::\"outer_C ptr\"\n shows \"ptr_valid d p = root_ptr_valid d p\"\n by (simp add: ptr_valid (*‹PTR_VALID(32 word) ?d ?p = (root_ptr_valid ?d ?p ∨ (∃q. PTR_VALID(inner_C) ?d q ∧ ptr_val ?p = &(q→[''fld1_C''])) ∨ (∃q. PTR_VALID(inner_C) ?d q ∧ ptr_val ?p = &(q→[''fld2_C''])) ∨ (∃q. PTR_VALID(data_struct1_C) ?d q ∧ ptr_val ?p = &(q→[''d1_C'', ''y1_C''])) ∨ (∃q. PTR_VALID(data_struct1_C) ?d q ∧ ptr_val ?p = &(q→[''d1_C'', ''y2_C''])) ∨ (∃q. PTR_VALID(data_struct1_C) ?d q ∧ ptr_val ?p = &(q→[''d1_C'', ''y3_C''])) ∨ (∃q. PTR_VALID(data_struct1_C) ?d q ∧ ptr_val ?p = &(q→[''d1_C'', ''y4_C''])))› ‹PTR_VALID(8 word) ?d ?p = root_ptr_valid ?d ?p› ‹PTR_VALID(32 word ptr) ?d ?p = root_ptr_valid ?d ?p› ‹PTR_VALID(unit ptr) ?d ?p = root_ptr_valid ?d ?p› ‹PTR_VALID(data_C) ?d ?p = (root_ptr_valid ?d ?p ∨ (∃q. PTR_VALID(data_struct1_C) ?d q ∧ ptr_val ?p = &(q→[''d2_C''])) ∨ (∃n<10. ∃q. PTR_VALID(data_C[10]) ?d q ∧ ptr_val ?p = &(q→[replicate n CHR ''1''])))› ‹PTR_VALID(closed_C) ?d ?p = root_ptr_valid ?d ?p› ‹PTR_VALID(unpacked_C) ?d ?p = (root_ptr_valid ?d ?p ∨ (∃q. PTR_VALID(other_C) ?d q ∧ ptr_val ?p = &(q→[''fy_C''])) ∨ (∃n<3. ∃q. PTR_VALID(unpacked_C[3]) ?d q ∧ ptr_val ?p = &(q→[replicate n CHR ''1''])) ∨ (∃n<2. ∃q. PTR_VALID(unpacked_C[2]) ?d q ∧ ptr_val ?p = &(q→[replicate n CHR ''1''])))› ‹PTR_VALID(unpacked_C[3][2]) ?d ?p = (root_ptr_valid ?d ?p ∨ (∃q. PTR_VALID(two_dimensional_C) ?d q ∧ ptr_val ?p = &(q→[''matrix_C''])))› ‹PTR_VALID(two_dimensional_C) ?d ?p = root_ptr_valid ?d ?p› ‹PTR_VALID(data_struct1_C) ?d ?p = (root_ptr_valid ?d ?p ∨ (∃q. PTR_VALID(data_struct2_C) ?d q ∧ ptr_val ?p = &(q→[''d_C''])))› ‹PTR_VALID(data_struct2_C) ?d ?p = root_ptr_valid ?d ?p› ‹PTR_VALID(inner_C) ?d ?p = (root_ptr_valid ?d ?p ∨ (∃q. PTR_VALID(outer_C) ?d q ∧ ptr_val ?p = &(q→[''inner_C''])) ∨ (∃n<5. ∃q. PTR_VALID(inner_C[5]) ?d q ∧ ptr_val ?p = &(q→[replicate n CHR ''1''])))› and more 8 facts*))\n\n\ntext ‹A pointer to an \\<^typ>‹inner_C› is valid according to \\<^term>‹ptr_valid d p›, if it is either\n▪ a root pointer to an \\<^typ>‹inner_C›, or\n▪ there is a valid pointer to an enclosing pointer \\<^term>‹q::outer_C ptr› to\\<^typ>‹outer_C›, where\n \\<^term>‹p::inner_C ptr› is obtained by dereferencing field \\<^term>‹''inner_C''›, (Note that valid\n \\<^typ>‹outer_C› are already root pointers). or\n▪ there is a valid pointer to an array \\<^term>‹q::(inner_C[5]) ptr›, where \\<^term>‹p::inner_C ptr› is obtained\n by indexing into the array. Indexing into arrays is modelled by unary encoded field names\n \\<^term>‹replicate i CHR ''1''›. (Note that valid array pointers themselve might have \n various possible extensions to a root pointer of an enclosing structure)\n\nThese pointer-valid predicates are derived from the specification of addressable fields in the\ncorresponding @{command autocorres} command. They enumerate the possibilities to arrive\nat a valid pointer starting from a valid root-pointer. The specification is converted to\nthe parameter \\<^const>‹𝒯› of \\<^locale>‹open_types›. In our example program this is:\n›\n\n\nthm 𝒯_def\n\ntext ‹The \\<^const>‹𝒯› is an association list mapping a type tag \\<^const>‹typ_uinfo_t›\nof a \\<^class>‹mem_type› to the list of addressable fields within that structure. From this\nspecification we derive a lot of useful notions within the locale. Notably:\n›\nthm ptr_valid ― ‹These go up the chain to the root pointer.›\n\nsubsubsection ‹From typed heaps to split heaps›\n\ntext ‹\nNote that the functions in the previous subsubsection still are defined on the level of the\nmonolithic UMM memory model. They are central building blocks to finally arrive at the split\nheap model.\nEssentially there is a split heap component in the lifted globals corresponding\nto \\<^const>‹plift› for all the fundamental types.\n\nHowever, instead of partial functions as provided by \\<^const>‹wf_ptr_valid.plift›\nthe split heap separates definedness from the actual value, by providing a pair of a total function\nfrom pointer to value and a validity predicate which indicates definedness. This design choice\nwas introduced in the original AutoCorres.\n\nAs a prerequisite to lift the Functions into the split heap model, the new global state record\n@{typ lifted_globals} is defined. For each fundamental heap this record contains a field for\n the split heap e.g.\n\\<^term>‹heap_w32::lifted_globals ⇒ 32 word ptr ⇒ 32 word›.\n\nThese are the lookup functions provided by the record package, there are of course also\nthe corresponding update functions, e.g.:\n \\<^term>‹heap_w32_update:: ((32 word ptr ⇒ 32 word) ⇒ (32 word ptr ⇒ 32 word)) ⇒\n lifted_globals ⇒ lifted_globals›\nNote that the update function behaves like a \"map\", which maps an update function on the selected\nheap to an update function on \\<^typ>‹lifted_globals›.\n\nThe typing information (heap type description) is preserved from the monolithic heap in component\n\\<^term>‹heap_typing :: lifted_globals ⇒ heap_typ_desc›.\n\n›\nprint_record lifted_globals\n\ntext ‹For the derived heaps we provide a similar abstraction level by defining a derived heap lookup and\na derived heap update function. These functions are a composition of\nthe underlying functions for the fields of the structure.\n›\nthm derived_heap_defs\n\ntext ‹For example for \\<^typ>‹outer_C› we have.\n\n▪ Lookup: \\<^term>‹heap_outer_C:: lifted_globals ⇒ outer_C ptr ⇒ outer_C›\n▪ Pointwise update: \\<^term>‹heap_outer_C_map:: outer_C ptr ⇒ (outer_C ⇒ outer_C) ⇒ lifted_globals ⇒ lifted_globals›\n▪ Validity is generically based on the heap typing for all heaps:\n \\<^term>‹(λs. ptr_valid (heap_typing s)):: lifted_globals ⇒ 'a::mem_type ptr ⇒ bool›\n\nNote the difference in the signature of the derived update \\<^const>‹heap_outer_C_map› and\nthe fundamental \\<^const>‹heap_w32_update›. Besides the difference in naming between 'map' and\n'update', the derived update is defined pointwise instead of heap-wide. The reason is\nthat the pointwise update is what we actually need for autocorres and it is natural to\nbreak down a pointwise update defined via a function on\n\\<^term>‹f:: outer_C ⇒ outer_C› to the update functions on the fields of \\<^typ>‹outer_C›.\n›\nthm heap_outer_C_def\nthm heap_outer_C_map_def\n\n\ntext ‹\nThe connection between \\<^typ>‹globals› (containing the byte level heap) and \\<^typ>‹lifted_globals›\nis defined via the function \\<^term>‹lift_global_heap::globals => lifted_globals›.\nThe fundamental heaps are lifted via the corresponding \\<^const>‹wf_ptr_valid.plift›\nfunctions. Validity of pointers is maintained by literally preserving the heap typing from the\nmonolithic heap.\n›\nthm \"lift_global_heap_def\"\nthm \"lift_t_def\"\nthm plift_defs\n\ntext ‹›\nsubsection ‹User Level ›\n\ntext ‹\nDue to the abstraction layer of derived heap types, the structure of the programs after\nheap lifting is analogous for closed and opened structures.\n›\ncontext ts_definition_set_c1\nbegin\nthm ts_def\n\nlemma \"set_c1' p c ≡ do {guard (λs. IS_VALID(closed_C) s p);\n modify (heap_closed_C_update (λh. h(p := c1_C_update (λ_. c) (h p))))\n }\"\n by (rule ts_def (*‹set_c1' (?p::closed_C ptr) (?c::32 word) ≡ do {\n v::unit ← guard (λs::lifted_globals. IS_VALID(closed_C) s ?p);\n modify (heap_closed_C_update (λh::closed_C ptr ⇒ closed_C. h(?p := c1_C_update (λ_::32 word. ?c) (h ?p))))\n }›*))\n\nend\n\ncontext ts_definition_outer_fld1_upd\nbegin\nthm ts_def\nend\n\ncontext ts_definition_set_element\nbegin\nthm ts_def\nend\n\ncontext ts_definition_set_matrix_element\nbegin\nthm ts_def\nend\n\ntext ‹Of course, when it comes to specifications and proving properties about the programs,\nthe user has to take care about aliasing of addressable fields.\nAutoCorres generates a bunch of theorems and sets up the simpsets with some obvious rules for\nderived heaps, which hopefully helps in doing so. However, as a last resort one might still have\nto unfold the definitions and work with the parts.\nHere is some examples of some theorems or collections of theorems.\n›\nthm lifted_globals_ext_simps\nthm ptr_valid\n\nthm heap_inner_C.write_comp\nthm heap_inner_C.write_id\nthm heap_inner_C.write_other_commute\nthm heap_inner_C.write_same\nthm update_commute\nthm read_write_same\nthm read_write_other\nthm write_cong\nthm update_compose\nthm valid_implies_c_guard\nthm read_commutes\nthm write_commutes\n\nthm fg_cons_simps\nthm typ_info_simps\nthm td_names_simps\nthm typ_name_simps\nthm upd_lift_simps (* TODO: REMOVE EMPTY *)\nthm upd_other_simps (* TODO: REMOVE EMPTY *)\nthm size_align_simps\nthm fl_Some_simps\nthm fl_ti_simps\nthm typ_tag_defs\nthm size_simps\nthm typ_name_itself\n\nsubsubsection ‹\\<^term>‹addressable_fields›,\\<^term>‹merge_addressable_fields›,\\<^term>‹read_dedicated_heap›,\\<^term>‹write_dedicated_heap››\n\ntext ‹\nWe distinguish three aspects when thinking and reasoning about split heaps: the\nmonolithic heap stored in \\<^typ>‹globals› the split heap components in \\<^typ>‹lifted_globals› and\nthe function \\<^const>‹lift_global_heap› which transforms between both types. Autocorres does a \nrefinement proof that a program operating on \\<^typ>‹lifted_globals› refines a program \non \\<^typ>‹globals› where the heaps are related by function \\<^const>‹lift_global_heap›.\n\nFor open structures, the fields which are addressable have two representations in \\<^type>‹lifted_globals›:\n▸ the relevant value in the common heap, i.e. \\<^const>‹heap_w32› for the ‹inner.fld1› pointers\n▸ an unused value in the field of the dedicate heap, i.e. the ‹fld1_C› in the \\<^const>‹dedicated_heap_inner_C›.\nHaving the dedicated heap allows us to keep all non-addressable fields of a structure in a single split\nheap component of \\<^typ>‹lifted_globals›. The values of the addressable fields are irrelevant for\nthe virtual heap. When reading from a virtual heap we start with the value in the dedicated heap and\noverwrite the addressable fields by considering the common heaps.\n›\n\nthm \"heap_inner_C_def\"\n\ntext ‹When writing to a virtual heap we have a series of updates on \\<^typ>‹lifted_globals›. We start by \nupdating the non-addressable fields in the dedicated heap and then updating the common heaps with the\nnew values for the addressable fields:›\n\nthm \"heap_inner_C_map_def\"\n\ntext ‹To express \"update the non-addressable fields of the dedicated heap\" in an uniform way\nwe make use of the concept of ∗‹scenes› from the world of ∗‹lenses›. In contrast to lenses, \nscenes are formalised by properties of a single function \\<^typ>‹'a ⇒ 'a ⇒ 'a› which is referred to as\nË∗‹merge› or 'overrider' operation. The intuition of \\<^term>‹merge x y› is that we take certain\nportions of a compound value \\<^term>‹x› and merge (or override) them into another compound value\n \\<^term>‹y›. Scenes are formalised in locale \\<^locale>‹is_scene›. The notion ∗‹merge› emphasises the\nthe symmetric nature of the merge operation. Like a lense, a scene 'focuses' on a portion of a \ncompound value \\<^typ>‹'a›. The merge operation takes that portion from the first argument and the \ncomplement of that portion from the second argument. Think for example of a pair and the 'portion' being\nthe first component. Then @{term \"merge x y = (fst x, snd y)\"}. The notion ∗‹overrider› puts more \nemphasis on the merge as an update function, in particular this view becomes obvious if we think\nof a partial application @{term \"merge x\"}: this is an update function which overrides the 'portion'\nwithin an value @{term y} to that portion of @{term x}. You can also think of fixing the 'portion' to\nthose values in @{term x}. Analogously @{term \"λx. merge x y\"} fixes the complement of the 'portion' to\nthose values in @{term y}.\nScenes are closely related to lenses in the sense that it is straightforward to derive a \nscene from a lense @{thm is_scene_of_lense}.\n›\n\nthm is_scene_of_lense\n\ntext ‹For HOL the decisive advantage of scenes compared to lenses is that they only have one type parameter,\nwhich fixes the type of the compound value. But still you can formally 'talk' about different 'portions' of\nthat value in an uniform way. For example you can have a list or a set of scenes to describe their\ncomposition. This property can be used to uniformly describe \"all addressable fields\" of a value or \n\"all non-addressable fields\" of a value. \n\nThe relevant merging function for us is \\<^term>‹merge_addressable_fields› which is an \nabbreviation \\<^term>‹open_types.merge_addressable_fields›.\n›\n\nterm ‹open_types.merge_addressable_fields›\n\ntext ‹\nThe term\\<^term>‹merge_addressable_fields old new› has to effects:\n▪ The addressable fields are taken from \\<^term>‹old›\n▪ The non-addressable fields are taken from \\<^term>‹new›.\n›\n\ntext ‹To express the same semantics that we have just described for \\<^typ>‹lifted_globals› on the \nUMM heap in \\<^typ>‹globals› we make use of the functions \\<^const>‹read_dedicated_heap› and\n \\<^const>‹write_dedicated_heap›. These functions internally again make use of \\<^term>‹merge_addressable_fields›.\nWe zero out all parts of the compound value we are not actually interested in. This gives us\ncanonical values which we can easily relate to the \\<^typ>‹lifted_globals›. For example\nfor \\<^term>‹read_dedicated_heap› we zero out all addressable fields and also all values which\nare not even \\<^term>‹ptr_valid› (according to \\<^const>‹plift›).\n›\n\nlemma (in open_types) \n \"read_dedicated_heap h p = \n merge_addressable_fields ZERO('a::mem_type) (the_default ZERO('a) (plift h p))\"\n by (simp add: read_dedicated_heap_def (*‹local.read_dedicated_heap (?h::(32 word ⇒ 8 word) × (32 word ⇒ bool × (nat ⇒ ((8 word list ⇒ 8 word list, unit) typ_desc × bool) option))) (?p::?'a::mem_type ptr) = local.merge_addressable_fields ZERO(?'a::mem_type) (the_default ZERO(?'a::mem_type) (local.ptr_valid.plift ?h ?p))›*))\n\nthm open_types.read_dedicated_heap_def\nthm open_types.write_dedicated_heap_def\n\ntext ‹The lifting function \\<^const>‹lift_global_heap› directly uses \\<^const>‹read_dedicated_heap› to \nconstruct a dedicated heap from the UMM heap:›\nthm lift_global_heap_def\n\n\ntext ‹\nTo have a canonical representation we overwrite the fields in the dedicated heap with zeros, using the\ngeneric constants \\<^const>‹addressable_fields› and \\<^const>‹merge_addressable_fields› defined in \n\\<^locale>‹open_types›.\n\n\\<^term>‹read_dedicated_heap›,\\<^term>‹write_dedicated_heap› combine these with \\<^term>‹h_val›, to get the\ncanonical heap access.\n\n›\n\n\n\n\nsubsubsection ‹Why all the variants? \\<^term>‹h_val› / \\<^term>‹read_dedicated_heap› ›\n\n\nthm lifted_globals_ext_simps\nthm lift_global_heap_def\nthm open_types.read_dedicated_heap_def\ntext ‹\nWhen taking a close look in the definition of \\<^const>‹lift_global_heap› one sees that the\nabstraction for a split heap component from the monolithic heap is:\n\n\\<^term>‹read_dedicated_heap (t_hrs_' g)›\n\nSo this might seem a bit surprising at first sight. What is all the indirection useful for:\n▸ the \\<^const>‹read_dedicated_heap› adds an option layer to the plain \\<^const>‹h_val› and \\<^const>‹plift›\n▸ with \\<^term>‹t_hrs_'› we get the heap representation (including bytes and types)\n\nWhen we know that a pointer is valid there is a tight connection between \\<^const>‹read_dedicated_heap› and \\<^const>‹h_val›\n›\nthm open_struct.ptr_valid.ptr_valid_plift_Some_hval\nthm open_struct.read_dedicated_heap_def\n\ntext ‹\nThis connection directly carries over to lifting.\n›\nthm read_commutes\n\ntext ‹\nThe difference between \\<^const>‹read_dedicated_heap› and \\<^const>‹h_val› becomes apparent when looking at partial heaps and invalid\npointers. While \\<^const>‹read_dedicated_heap› always yields \\<^const>‹c_type_class.zero› the plain \\<^const>‹h_val› will still construct\nsome value out of the bytes in the heap. So in a sense the indirection to \\<^const>‹read_dedicated_heap› makes\nheaps equal if the agree on the valid pointers only. So we encode \"heap equality only on valid pointers\" in\nan ordinary equality on the complete lifted heap. This choice ensure compositionality for\nthe polymorphic \\<^const>‹c_type_class.zero›, in the sense that the \\<^const>‹c_type_class.zero› of a structure means that all subcomponents\nare \\<^const>‹c_type_class.zero›.\n\nMoreover \\<^const>‹read_dedicated_heap› only reads the fields which are not addressable, and overwrites the\naddressable fields with \\<^const>‹c_type_class.zero›. This gives us a canonical view on the partial heap.\n›\n\nthm zero_simps\nthm open_struct.ptr_valid.plift_None\n\n(* FIXME *)\nsubsubsection ‹I miss the typing in the split heap!›\n\ntext ‹\nA peculiar property of the original split heap model of autocorres was that you loose parts of\nthe type information that were directly available in the byte-level heap. There was no\nheap-type description in \\<^typ>‹lifted_globals› but only the more abstract ‹is_valid_<type>› fields for\neach type. For closed types this isn't so much of an issue as\nessentially all relevant type information is captured in the fact that all pointers to that\ntype have a distinct split heap, which is separate from all other split heaps.\nBut for shared types this is no longer true. Thus we maintain the typing information also\nin the split heap in the compponent \\<^const>‹heap_typing›. The relation to the typing information of\nthe original byte-level heap is encapsulated in \\<^const>‹lift_global_heap›. For low-level\noperations that are embedded via \\<^const>‹exec_concrete› you can directly connect the typing of the\nmonolithic and the split heap. In particular you can derive\n\\<^term>‹c_guard p› from that information. Or you can derive that if you have two valid pointers\nof the same type, where you only know that the address is different, you can still conclude that\nthe complete pointer span of the pointers do not overlap.\n\nBut keep in mind that typing is only a \"discipline\". As in the split heap you can manipulate the\nvalues on the heap or split heap independent of the typing information. When well-typedness or\n\\<^const>‹ptr_valid› is available you might be able to derive properties like distinctness of pointer\nspans. The distinctness of pointers spans is the actual reason that certain heap updates commute,\ne.g. @{thm heap_inner_C.write_other_commute}\n›\nthm heap_inner_C.write_other_commute\n\n\n\nsubsection ‹Simulation Proof›\n\ntext ‹\nThe simulation of a concrete program \\<^term>‹C::('a, 'b, globals) exn_monad› operating\non the byte-level heap by an\nabstract program \\<^term>‹A::('a, 'b, lifted_globals) exn_monad› operating on the split heaps\nis captured in predicate\\<^term>‹L2Tcorres lift_global_heap A C›. The proof for an instance\nof \\<^term>‹C› follows the syntactic structure of \\<^term>‹C› by applying introduction rules and\nsynthesizing the abstract program \\<^term>‹A›.\n\nIntuitively the core properties on which the simulation proof builds are:\n▪ Abstract programs only operate on abstract heap values not on byte-lists. Byte-level concepts\n like padding fields are irrelevant for the abstract program. The abstract heaps and\n programs don't distinguish between two byte-level heaps if they agree on all values\n of non-padding fields of properly allocated and typed portions of the heap.\n▪ Lookup and update via a pointer into a structure can be simulated by an lookup and update\n via the root pointer of the structure.\n▪ Each non-addressable field of a structure is mapped into exactly one split heap. For a 'closed' structure\n without any addressable fields this is the heap for the type.\n▪ For an 'open' structure an addressable field is mapped into a shared heap or multiple shared\n heaps in case the field is again an open structure.\n▪ The pointer spans of two valid root pointers of different types do not overlap.\n▪ The pointer spans of two valid root pointers to the same type might overlap only if the pointer value is\n the same.\n\nThe actual proof of a simulation is divided into three main steps.\n▪ First some general theorems relating byte-level and split heaps are derived. Most prominently\n \\<^const>‹typ_heap_simulation› and related predicates.\n▪ These theorems are used to instantiate the syntax driven introduction rules collected in\n named theorems @{thm heap_abs}.\n▪ The derived introduction rules are recursively applied to the program.\n\nThe first step is the one that was significantly extended to support addressable\nfields in open structures. The second and third step remained almost unchanged.\n›\nthm heap_abs\ntext ‹\nWhen thoroughly inspecting the rules @{thm heap_abs} and keeping in mind that they are\nused by recursively applying them as introduction rules to a concrete program \\<^term>‹C› and a\nschematic variable for the abstract program \\<^term>‹A› one can see the following strategy:\n▪ First some normalisation of expressions involving pointers is performed on the concrete\n program \\<^term>‹C›. In particular an update to a dereferenced pointer \\<^term>‹&(p→f)› by a value \\<^term>‹v›\n is transformed to an update on \\<^term>‹p›, where first the value of \\<^term>‹p› is fetched from memory and\n then field \\<^term>‹f› of this value is updated by \\<^term>‹v› and the resulting compound value is put\n back to memory.\n▪ Only after the normalisation on \\<^term>‹C›, the actual translation to an abstract \\<^term>‹A› on the\n split heap is performed.\n\nThe normalisation step is guided by predicates \\<^const>‹struct_rewrite_guard›,\n \\<^const>‹struct_rewrite_expr› and \\<^const>‹struct_rewrite_modifies›.\nAfter that normalisation the atomic building block for the simulation is provided by an\ninstance of \\<^const>‹typ_heap_simulation›, which tells how a lookup and update of a pointer of a\ngiven type in the byte heap is simulated in the split heap.\n\nThe proofs of the necessary instances of \\<^const>‹typ_heap_simulation› are provided once and forall in\nthe initialisation phase of @{command autocorres} or @{command \"init-autocorres\"}. They are provided\nfor atomic and shared heaps as well as derived heaps.\n›\n\nsubsubsection ‹Proving \\<^const>‹typ_heap_simulation››\n\ntext ‹As stated before this step was extended quite excessively and quite some infrastructure\nof both theorems as well as ML code were built to support it.\n\n›\n\nparagraph ‹Typed vs. Untyped Dialects/ Records vs. Byte-Lists›\n\ntext ‹\nFirst some general remarks on the general concepts and setup. The UMM was designed in the\nspirit to take advantage of HOL type and class inference to support reasoning about C-types and\npointers. As a consequence some of the definitions and theorems are rather subtle and somehow\nlive an the edge of what can be typed and expressed in the HOL type system. Sometimes the\nessence or the limitations of a theorem only become 'visible' when also looking at the\ntypes of expressions, especially the pointer types. Unfortunately writing about these\nconcepts is rather ambiguous with respect to the notion type. 'Type' might refer to\nthe original C-structure in the C program, this type is related to an abstract HOL-type, which\nis a record. This record is an instance of various type-classes: \\<^class>‹c_type›, \\<^class>‹mem_type›,\n \\<^class>‹xmem_type›. For types of \\<^class>‹c_type› we define overloaded \\<^const>‹typ_info_t›, which\nassociates a HOL-term describing the type (referred to as type-description or type-information). Moreover,\npointers represented in HOL carry the HOL-type they point to as phantom type.\n\nAn illustration for the subtle typing is the two terms\n \\<^term>‹&(p→f) ≠ &(q→g)› vs. \\<^term>‹p ≠ q ⟹ &(p→f) ≠ &(q→g)›.\nWhereas in first term the pointers \\<^term>‹p::'a ptr› and \\<^term>‹q::'b ptr› have a different type, in the second\nterm they both have the same type as the inequation in the precondition also makes the types equal.\nThis might not always be the intended meaning. When the type of the pointer is\nirrelevant one might resort to plain addresses instead of pointers \\<^term>‹ptr_val p = ptr_val q ⟹ &(p→f) = &(q→g)›.\nThis general theme occurs in various places. There often is a typed variant of a concept\nand also an untyped variant, based on addresses or byte lists, like \\<^term>‹p› vs. \\<^term>‹ptr_val p›.\nBoth are closely related. The typed variant somehow reflects the 'API' of the concept and\nis more abstract, whereas during a proof one actually resort to the untyped variant\nto gain flexibility. These two views on a concepts leads to a duplication of lemmas.\nMoreover, a lemma might not immediately be available in the form one expects, but can\nbe 'easily' derived from some related lemmas.\n\n\nThe central UMM HOL-datatype to reflect C-types into HOL terms is \\<^typ>‹('a, 'b) typ_desc› and is defined\nmutually recursive with \\<^typ>‹('a, 'b) typ_struct›. It is basically a tree describing the\nnested structure of an aggregate type. Field names, alignment and size information is encoded,\nand lookup and update of sub-fields can be derived from this information.\n\n\nThe most prominent instances of this type are the typed variant\\<^typ>‹'a xtyp_info› vs. the\n untyped variant \\<^typ>‹typ_uinfo›. The former can be transformed to the latter by\n \\<^const>‹export_uinfo›. As with pointer types the type variable in \\<^typ>‹'a xtyp_info› denotes\nthe abstract HOL-C-type that is described by the type information. So to derive a field-lookup or\nfield-update on HOL-types from the type information one needs to resort to the typed variant.\nExporting the type information via \\<^const>‹export_uinfo› maintains the shape of the tree, but\nremoves all the HOL-C-type dependent components. The resulting \\<^typ>‹typ_uinfo› has two main\nuse cases:\n▪ It can be seen as a type-tag identifying a C-type. It can be used to relate two C-types on\n the HOL-term level, e.g. ask if they are equal or if one is contained in the other.\n▪ It can be used to normalise a byte-list representing a value of that type. Normalisation means\n that all padding bytes in the byte-list are set to zero.\n\nFor each C-type the C-Parser also generates the type information which can be\naccessed via the overloaded function: \\<^term>‹typ_info_t TYPE('a::c_type)›.\n\nThe main use cases for \\<^term>‹typ_info_t TYPE('a::c_type)› are\n▪ Provide lenses for fields (access / update) on the abstract value (record),\n via \\<^const>‹access_ti›, \\<^const>‹update_ti› and \\<^const>‹field_lookup›\nThe main use cases for \\<^term>‹typ_uinfo_t TYPE('a::c_type)› are\n▪ comparison of type descriptors: equality and order \\<^term>‹s ≤⇩τ t›\n▪ normalisation of byte-lists \\<^const>‹norm_tu›, in particular setting all padding bytes to zero.\n›\nthm sub_typ_def\nthm typ_tag_le_def\ntext‹\nSo \\<^const>‹typ_info_t› is more related to the abstract view on a value (as a HOL record), whereas\n\\<^const>‹typ_uinfo_t› is more related to the byte-list encoding of the value.\nThis duality somehow also reflects the dual nature of types in C. On the one hand C is a\nstatically typed language and on the other hand it allows to break the abstraction and switch to\na low-level byte oriented view.\n›\n\ncontext\n fixes p::\"'a::c_type ptr\"\n fixes f::\"qualified_field_name\"\n fixes t::\"'a xtyp_info\"\n fixes n:: nat\nbegin\ntext ‹\nA central function on both typed and untyped typ-information is the\nfunction \\<^const>‹field_lookup› which retrieves the type-information for a field of a type.\nA common application of this function is to state some property on a dereferenced pointer:\n\\<^term>‹PTR('b) &(p→f)›. Note that \\<^term>‹p› is an pointer to an \\<^typ>‹'a›: \\<^term>‹p::'a ptr›.\nA typical precondition is to retrieve the type-information for field \\<^term>‹f› from the\ntype information of \\<^typ>‹'a›:\n \\<^term>‹field_lookup (typ_info_t TYPE('a)) f 0 = Some (t::'a xtyp_info, n)›\n\nNote that the retrieved type information \\<^term>‹t::'a xtyp_info› is still tagged with\n\\<^typ>‹'a›. The number \\<^term>‹n› is the offset of the selected field.\nTypically we want to relate \\<^term>‹t› to the type of the selected field \\<^typ>‹'b› (which\nonly happens to be the same \\<^typ>‹'a› in case the field is the empty list).\nThe relation can be established via \\<^term>‹export_uinfo›, namely\n \\<^term>‹export_uinfo t = export_uinfo (typ_info_t TYPE('b::c_type))›.\nNote that directly equating \\<^term>‹t› to \\<^term>‹typ_info_t TYPE('b::c_type)› is not even\na well-typed expression in HOL, as \\<^typ>‹'a› is not equal to \\<^typ>‹'b›.\n\nThe right hand side abbreviates to constant \\<^term>‹typ_uinfo_t TYPE('b::c_type)›.\n›\nend\n\ntext ‹A concrete example might give some further insight to the relation of \\<^const>‹typ_info_t› and\n\\<^const>‹typ_uinfo_t› and how the type-information is constructed. Let us consider the field\n \\<^term>‹''inner_C''› of \\<^typ>‹outer_C›, which selects a field of \\<^typ>‹inner_C›.\nWe can retrieve the information for the field:›\n\nlemma \"field_lookup (typ_info_t TYPE(outer_C)) [''inner_C''] 0 =\n Some (adjust_ti (typ_info_t TYPE(inner_C)) outer_C.inner_C (inner_C_update ∘ (λx _. x)), 0)\"\n by simp\n\ntext ‹\nNote that the retrieved type information is constructed from the nested type information\nfor \\<^typ>‹inner_C›, by adjusting it. Adjusting means that we say how we can lookup and update\nthe sub-field of the record \\<^typ>‹outer_C›. This adjustment only affects the typed-view of\nthe type information. Exporting the type information collapses to the adjusted inner type:\n›\n\nlemma \"export_uinfo (adjust_ti (typ_info_t TYPE(inner_C)) outer_C.inner_C (inner_C_update ∘ (λx _. x)))\n = export_uinfo (typ_info_t TYPE(inner_C))\"\n by simp\n\ntext ‹\nIn the realm of 'lenses\", \\<^const>‹adjust_ti› can be viewed as a form of composition of lenses.\nA lense for an inner type is transformed to a lense on the outer type.\nThe lense assotiated to type information is captured in \\<^const>‹access_ti›, for the 'lense-lookup'\naka. 'lense-get' part, and \\<^const>‹update_ti› for the 'lense-update' aka 'lense-put'.\n\nFunction \\<^term>‹field_lookup› is the essential building block to relate field names e.g. as\npart of dereferencing a pointer to their abstracts operations on the associated HOL-type.\nOne can convert between the typed and untyped version:\n›\nthm \"field_lookup_export_uinfo_Some_rev\"\nthm \"field_lookup_export_uinfo_Some\"\n\ntext ‹\nHere are some closely related, mostly derived concepts around \\<^term>‹field_lookup›:\n▪ \\<^const>‹field_lookup›, \\<^const>‹field_ti›, \\<^const>‹field_offset›, \\<^const>‹field_of›\n▪ \\<^const>‹td_set›, \\<^const>‹sub_typ› (\\<^term>‹ s ≤⇩τ t›\n▪ Family of field name functions: \\<^const>‹all_field_names›, \\<^const>‹field_names›, \\<^const>‹field_names_u›,\n \\<^const>‹field_names_no_padding›, \\<^const>‹all_field_names_no_padding›\n\nOften lemmas might not be available for all variants, but via some simple indirections, via\ndefinitions or conversion lemmas. In an Isar-proof, @{command sledgehammer} can often help to\nfind the connections.\n›\nthm td_set_field_lookupD\nthm td_set_field_lookup\nthm all_field_names_union_field_names_export_uinfo_conv\nthm set_field_names_all_field_names_conv\nthm field_names_u_field_names_export_uinfo_conv(1)\nthm set_field_names_no_padding_all_field_names_no_padding_conv\nthm all_field_names_no_padding_typ_uinfo_t_conv\n\nparagraph ‹Doing the Proof›\n\ntext ‹\nThe fundamental goal is to derive interpretations @{locale typ_heap_simulation} for every relevant\ntype. It states that the virtual heap for a type as obtained from @{typ lifted_globals} simulates the\noriginal UMM heap in @{typ globals}. Both states are connected by the abstraction \nfunction @{const lift_global_heap} which is the abstract function @{term st} in @{locale typ_heap_simulation}.\nTo facilitate the construction of the proofs we introduce intermediate helper locale \n@{locale typ_heap_simulation_open_types} and \nmake use of the infrastructure of lenses and scenes that we mentioned before.\n\nTo optimize the construction (in particular to minimise the number of heaps we have to put into into\n@{typ lifted_globals}) we distinguish three main cases:\n▪ A structure is closed and has no addressable fields: @{thm open_types.typ_heap_simulationI_no_addressable}\n In this case there is only one relevant heap in @{typ lifted_globals}. \n No overlay of a dedicated heap with some common heaps is necessary.\n▪ A structure is completely open, meaning that all fields are addressable: @{thm open_types.typ_heap_simulationI_all_addressable}.\n In this case we do not need a dedicated heap. The structure is described by a overlay of \n common heaps.\n▪ A structure is partially open, some fields are addressable and some not: @{thm open_types.typ_heap_simulationI_part_addressable}.\n In that case we need an overlay of a dedicated heap and some common heaps.\n\nOn an intuitive abstract level the lemmas and proof argue about composing field-level lookup and updates\nwithin some heap(s) to lookup and updates of the complete compound structure. We want to argue that lookup and\nupdate in the UMM heap is simulated by the overlayed updates of a dedicated heap and some common\nheaps in the split heap. To argue about different fields of a structure we build on the idea of scenes\nand also extend it to reads. Note that the general problem is that a structure is represented as\na HOL record and each field has a distinct type. Because of the limitations of the HOL type system\nwe cannot directly combine e.g. functions like readers for different fields like \\<^typ>‹'a ⇒ 'b1› \nand \\<^typ>‹'a ⇒ 'b2› in a HOL list. Here the scene idea to express everything via a merge or update\non \\<^typ>‹'a› is a helpful 'trick'. E.g. 'reading' the value of a field can in a sense be as well \nexpressed as a function \\<^typ>‹'a ⇒ 'a ⇒ 'a› that reads the field of the first argument and puts it into any\nstructure you give it as second argument:\n›\n\nthm lift_global_heap_def\nthm open_types.typ_heap_simulationI_all_addressable\nthm open_types.typ_heap_simulationI_part_addressable\nthm open_types.typ_heap_simulationI_no_addressable\nthm pointer_lense_def\nthm partial_pointer_lense_def\nthm typ_heap_simulation_of_field_def\ntext ‹\nOther important building blocks are:\n▪ @{locale pointer_lense} which describes a virtual heap.\n▪ @{locale partial_pointer_lense} which describes the effect of field lookup / updates on a virtual heap by\n combining a scene identifying the field and the pointer lense for the virtual heap.\n▪ @{thm typ_heap_simulation_of_field_def} which is used to break down @{locale typ_heap_simulation} to the \nlevel of fields in the structure.\n›\n\ntext ‹\nThe automation is implemented in @{ML ‹HeapLiftBase.gen_new_heap›}.\n\nThe following paragraphs describe some abstract arguments and lemma collections. In a previous\nversion the proof @{locale typ_heap_simulation} was more directly based on those arguments \nimplemented in ML. Meanwhile we were able to replace most parts of the ML code by lemmas \ndescribed before based on scenes and pointer lenses. Nevertheless the arguments are still valid\nand the lemma collections might be useful in other places.\n›\n\nparagraph ‹Fundamental heaps›\ntext ‹\nWe start with the fundamental heaps.\nThe simulation property for heap-updates is captured in @{locale write_simulation}. It\nis a commutation property. Provided we have a valid pointer in the split heap (obtained from\nlifting from the byte-level heap), we can\neither first perform the heap update in the byte-level heap and then lift the\nstate via \\<^const>‹lift_global_heap›, or first lift the byte-level heap into the split heap and\napply the corresponding update there. Both ways yield the same final state.\nSo we have to prove equality of two states of \\<^typ>‹lifted_globals›. The first question is\nwhich parts of the state did change? Intuitively only the affected split heap did change,\nall other heaps stayed the same. But all split heaps are derived from the same byte level heap.\nSo the proof decomposes into two main arguments, one is for\nthe affected heap where we have to show that both updates, the one on the byte-level heap and and the one on the\naffected split heap, are the same. The other argument is that all other split heaps remain unchanged.\nActually the first case is the more straight forward one and could be handled with theorems like\n @{thm plift_heap_update}.\n›\nthm plift_heap_update\n\ntext ‹\n The second case, to prove what\nis not changed is more involved. As a split heap might not only contain root pointers but\nalso valid pointers that are nested in other types we have to distinguish several cases.\nWe developed a general theory in \\<^locale>‹open_types› and the essential theorems for the commutation\nproof are collected in @{thm plift_simps}.\nWhen we update the heap at pointer \\<^term>‹p::'a ptr› and lookup the value of pointer \\<^term>‹q::'b ptr›\nwe can distinguish various cases by analysing the type relation (e.g. if one type is nested in the\nother). Note that by introducing guards into the code and the design of \\<^term>‹lift_global_heap›\nwe only have to care about the case were both pointers are valid according to \\<^const>‹ptr_valid›.\nTo be more specific. That the pointer that is updated is \\<^term>‹ptr_valid› is ensured by the\ncorresponding guard on the update. That the pointer we read from is actually \\<^term>‹ptr_valid› is more\nsubtle. Here we rely on the construction of \\<^const>‹lift_global_heap› were a split heap is\nconstructed by \\<^term>‹ λp. the_default c_type_class.zero (plift (t_hrs_' g) p)›. In case the pointer is\ninvalid we always obtain \\<^term>‹c_type_class.zero› and hence only the valid pointers may be affected by an heap update.\nThis reduction to valid pointers and \\<^term>‹h_val› is encapsulated in theorems @{thm plift_eqI}.\n›\n\nthm lift_global_heap_def\nthm the_plift_hval_eqI\nthm plift_eqI\nthm plift_simps\n\nparagraph ‹Treatment of Array Types›\n\ntext ‹\nAt the core there is no special type-information for array types. It is treated analogously\nto structures. Each index gets an individual field name which is the unary encoding of the index.\nThe good thing is that one does not need any new fundamental lemmas to deal with array types.\nBut it is not a good idea to unfold the type-information for arrays and to work on that\nexpanded version. On the one hand things like simplification might become slow, on the other\nhand we can make use of the regular structure of array types and once and forall derive\ngeneral lemmas.\n\nThe central theorems to work with field-lookup in arrays are @{thm field_lookup_array}\nand @{thm array_ptr_index_field_lvalue_conv}. The latter one allows to convert between\nan array-index-arithmetic based view of array pointers \\<^term>‹array_ptr_index p False i›\nand the field-name view \\<^term>‹Ptr &(p→[replicate i CHR ''1''])›.\n›\nthm field_lookup_array\nthm array_ptr_index_field_lvalue_conv\ntext ‹\nThese theorems are used to normalise array accesses towards symbolic field name\naccesses of the format term>‹Ptr &(p→[replicate i CHR ''1''])›. Note that we don't unfold\nthe definition of \\<^const>‹replicate› here, and index \\<^term>‹i› stays abstract, constrained\nby precondition that limits its range.\n›\nthm unpacked_C.ptr_valid.unfold\nthm unpacked_C'array_2.ptr_valid.unfold\n\ntext ‹\nThe instances of @{locale typ_heap_simulation} can be established from @{locale typ_heap_simulation}\nof the element types. We introduce the locales @{locale array_typ_heap_simulation} and\n@{locale two_dimensional_array_typ_heap_simulation} to automatically provide the sublocale relations when the\nelement type falls into \\<^class>‹array_outer_max_size› or \\<^class>‹array_inner_max_size›\nrespectively.\n›\n\nparagraph ‹Derived Heaps›\n\ntext ‹\nThe commutation proofs for derived heaps follow another path. For derived heaps we can assume\nthat we have all the instances of @{locale typ_heap_simulation} of the component types already\navailable. \nUnaddressable fields are never shared. Pointers only appear within a dedicated enclosing parent structure.\nValidity of the pointer coincides with validity of the parent pointer. Pointers that are mapped by the\nfield heap coincide with the parent pointers, there is no need to calculate the offset of the field.\nSo deriving the commutation proof from the available theorems boils down to their composition.\nThe central lemma connects an update of a derived heap to an fold over the updates of the\ntoplevel fields: @{thm heap_update_fold_toplevel_fields_pointless}.\nMoreover, we establish that @{const heap_update_padding} is equivalent to @{const heap_update} under\nthe state lifting function. Padding bytes become irrelevant in the split heap.\nFor each toplevel field we have the\ncommutation proof connecting the monolithic byte-level\nheap update to the split-heap update collected in @{thm write_commutes}.\n›\nthm heap_update_fold_toplevel_fields_pointless\nthm plift_heap_update_padding_heap_update_pointless_conv\nthm lift_heap_update_padding_heap_update_conv\nthm write_commutes\n\n\nsubsubsection \"No restriction to \\<^class>‹packed_type›!\"\n\ntext ‹\nIn the original version of AutoCorres there was a restriction of heap lifting to\ntypes that are also in \\<^class>‹packed_type›. Those are types that internally have no padding fields\nat all. The reason was that certain lemmas especially regarding heap update become more\ninvolved in the presence of padding fields. A \\<^const>‹heap_update› preserves the value of all\npadding bytes, whereas the abstract value obtained from a corresponding \\<^const>‹h_val› is independent\nof the value of the padding bytes. So in principle these notions fit together well but it\nseems that some formal language was missing to reason about padding bytes. Meanwhile we added\na theory to reason about those padding bytes in @{theory ‹AutoCorres2.Padding_Equivalence›}. This\nmade it possible to liberate heap lifting from the restriction to \\<^class>‹packed_type›.\nFor example @{thm heap_update_Array} for \\<^class>‹packed_type› holds in general\n@{thm heap_update_array}.\n›\nthm heap_update_array\nthm heap_update_Array\n\nsubsubsection ‹Padding Equivalence and \\<^class>‹xmem_type››\n\ntext ‹\nPadding bytes and padding fields are introduced via the construction of new C-Types via\nthe combinators \\<^const>‹ti_pad_combine› and \\<^const>‹ti_typ_pad_combine› to satisfy alignment\nproperties.\nHowever, the concept of\na padding byte is not a first-class citizen of a \\<^typ>‹('a, 'b) typ_desc›, but just happens\nto be a field with some special properties. In practice these fields are generated by\n\\<^const>‹ti_pad_combine›, so we know that the field name starts with \\<^term>‹''!pad''›, and the\nassociated 'lense' for lookup and update have the 'passthrough' properties of a padding field.\n\nWe made these properties explicit by supplying a theory to identify padding-bytes in a list\nof bytes associated with an C-Type, and having notions to compare such byte-lists, e.g. telling\nif two byte lists are equal up-to the padding bytes, or if they agree on the padding bytes.\nThis information is also backed into the properties of \\<^class>‹xmem_type›, which is a\nsubclass of \\<^class>‹mem_type›. All primitive word and pointer types as well as all types\nthat are constructed by the UMM module of the C-Parser are proved to belong to that class. The\nconstruction of array types propagates the class as expected.\n\nAgain the notions come in pairs of a 'typed' and a 'untyped' version. The 'typed' version\nassociated with \\<^const>‹typ_info_t› is a lense based formalisation, the 'untyped' version\nassociated with \\<^const>‹typ_uinfo_t› is based on byte lists.\n\nThe lense based version is introduced by @{locale padding_lense}. Access and update\nfollow the approach of \\<^typ>‹'a field_desc›. Lookup / access has type \\<^typ>‹'a ⇒ byte list ⇒ byte list›.\nIt takes an abstract value and a supply list of padding bytes and retrieves the bytes encoding the\nfield. The intuition of the padding byte list is the following. It is supposed to bridge the\ngap between the abstract value and the byte representation. The abstract value is independent\nof the padding bytes. Hence in general there is no one-to-one correspondence between a\nbyte list encoding of an abstract value and the abstract value. When converting from a byte list\nto an abstract value this is fine, the padding bytes are just ignored. But when we go the other\nway we have to invent the padding bytes. One solution could be to just fill up with zeros. Another\nsolution could be to yield a non-deterministic result for the padding bytes. The UMM model\nchose another way. The possible valuation for padding bytes is supplied as an additional\nargument. So whenever a padding byte is needed we just take it from the position in that list.\nThe update has type \\<^typ>‹byte list ⇒ 'a ⇒ 'a›. It takes the value of a field, encoded as a\nbyte list, and transforms it to an update on the abstract value.\n›\n\ncontext padding_lense\nbegin\ntext ‹\nThe concept of a @{locale padding_lense} follows the signature of \\<^typ>‹'a field_desc› and describes the\nnotions of padding bytes as semantic properties of the access function \\<^term>‹acc› and the\nupdate function \\<^term>‹upd›.\nBased on them it introduces the notions of\n▪ \\<^const>‹is_padding_byte›, is a byte a padding with respect to the lense?\n▪ \\<^const>‹is_value_byte›, is a byte a value byte with respect to the lense,\n i.e. does the abstract value associated with the byte list depend on that byte.\n▪ \\<^const>‹eq_padding›, access cannot distinguish the byte lists.\n▪ \\<^const>‹eq_upto_padding›, update cannot distinguish the byte list.\n›\nthm is_padding_byte_def\nthm is_value_byte_def\nthm eq_padding_def\nthm eq_upto_padding_def\ntext ‹\nAs the definitions are semantically defined the effect on \\<^term>‹acc› and \\<^term>‹upd› are rather\nimmediate. For example, if two byte lists are equal upto padding, then an update with \\<^term>‹upd›\nyields the same result. If the padding bytes within two supply byte lists are the same, then\n\\<^term>‹acc v› yields the same result for both byte lists.\n›\nend\n\ntext ‹\nIn the untyped \\<^term>‹typ_uinfo_t› we define the corresponding notions, as properties of the\nbyte list instead of the access and update functions.\n▪\\<^const>‹is_padding_byte›, a byte is a padding byte in a byte list, if normalisation\n of the byte list is independent of its value. Normalisation \\<^const>‹norm_tu› is defined on\n \\<^term>‹typ_uinfo_t› and sets all padding bytes to zero.\n▪\\<^const>‹is_value_byte›, normalisation depends on the value of the byte.\n▪\\<^const>‹eq_padding›, all padding bytes are the same.\n▪\\<^const>‹eq_upto_padding›, all value bytes are the same.\n›\nthm is_padding_byte_def\nthm is_value_byte_def\nthm eq_padding_def\nthm eq_upto_padding_def\n\ntext ‹For instances of \\<^class>‹xmem_type› we can go back and forth between both characterisations.›\n\nthm is_padding_byte_lense_conv\nthm field_lookup_is_padding_byte_lense_conv\nthm is_value_byte_lense_conv\nthm field_lookup_is_value_byte_lense_conv\nthm eq_padding_lense_conv\nthm field_lookup_eq_padding_lense_conv\nthm eq_upto_padding_lense_conv\nthm field_lookup_eq_upto_padding_lense_conv\n\ntext ‹With those theorems in place it is easy to show that padding fields only consist of\npadding bytes and thus do not account to the abstract value.›\nthm field_lookup_access_ti_to_bytes_field_conv\nthm access_ti_update_ti_eq_upto_padding\nthm field_lookup_qualified_padding_field_name(1)\nthm is_padding_tag_def padding_tag_def padding_desc_def\n\nsubsubsection ‹UMM-Simprocs Cache ›\n\ntext ‹\nTo solve sideconditions on UMM-Types we have implemented some simprocs within\n@{ML_structure UMM_Proofs}. Currently their scope is limited to the usecases\nwe have in the proofs described above. It should be straightforward to extend them\nto more usecases. Their purpose is not to support abstract reasoning on types but to\nprovide properties of concrete instances of C-types, for example deciding whether\n\\<^term>‹TYPE(32 word) ≤⇩τ TYPE(outer_C)› holds.\n\nThe benefit of the simprocs is that we do not have to guess and prove lots of\nlemmas already during the definition of a new UMM type, but can postpone it until we actually\nneed them. Once they are proven they are added to the cache, so they are only proven once.\nIn our use-case the lemmas we need depend on the addressable fields the user specifies on\nan @{command autocorres} invocation.\n›\n\nML_val ‹\nval _ = \\<^simproc>‹field_lookup›\nval _ = \\<^simproc>‹type_calculations›\nval _ = \\<^simproc>‹typuinfo_calculations›\n›\n\nML_val ‹\nCached_Theory_Simproc.trace_cache @{context}\n›\n\ntext ‹\nThe simprocs make use of a common cache. The cache itself is implemented as a simpset.\nCache lookup means we try to rewrite with the cache:\n▪ cache miss: term is unchanged.\n▪ cache hit: term is changed.\n\nIn case of a cache hit we are finished and return the resulting equation. In case of a\ncache miss we invoke the simplifier with a taylored simpset to derive a new equation. This\nequation is then added to the cache as well as returned from the simproc.\n\nSome fine points of this setup are related to context management. Conceptually we only prove\ntheory level theorems about an UMM type. So even if we prove them after the definition of the\nUMM type they should all be applicable as if we have immediately proven them at the point where\nthe UMM type was generated.\nHowever, the simproc is invoked in some context later on where the theory has already advanced.\nIn order to produce a result after a cache miss we have to somehow 'travel back in time' to the\noriginal theory context after the definition of\nthe UMM type, such that the simplifier properly works on it and the result is reusable in other\ncontexts. We maintain that original theory state and recertify terms before simplifying them.\n\nAnother point is, that the terms we attempt to simplify and cache might depend on each other.\nIn order not to miss to cache intermediate results one has to carefully craft the simpsets.\n\nIn general the simplifier tries simprocs only after unconditional and conditional rules.\nSo when a rewrite rule has already transformed the redex the simproc will never see that redex.\nThis has two main implications:\n▪ In order to have the simproc-based cache applied, there must not be a rule in the simpset that\n removes the redex before the cache actually has a chance to see it. This is why we maintain\n several simpsets to control that behaviour.\n▪ When a cache miss was encountered and we apply the simplifier recursively to rewrite the redex\n we have to take care that the simproc is not invoked again on the same redex. When there\n is an unconditional rule in the simpset that rewrites that redex we are on the safe side. But\n beware of ∗‹conditional› rules. When the solver fails to solve the conditions the rule can fail and\n then the simproc is invoked again on the same redex. To prevent the setup from looping in these\n cases we maintain the redexes the simproc has already seen.\n\nHere are some principles in the design of the simprocs and their simpsets:\n▪ The type descriptions obtained by \\<^const>‹typ_info_t› are not fully expanded.\n Instead we use a simplifier setup that works on the combinators, like\n \\<^const>‹empty_typ_info›, \\<^const>‹final_pad›, \\<^const>‹ti_typ_pad_combine›. This maintains\n the more compact representation of a type-descriptor as a combinator expression, which\n is anyways the way it is originally defined.\n▪ Fieldnames for array indexes stay symbolic: \\<^term>‹replicate i ''1''›. We employ derived\n rules for array indexes and do not have to expand the type-descriptor of arrays. For these\n rules to apply we typically need the information that the index is in the bounds of the\n array type. In those cases the simproc generates a conditional rewrite rule.\n▪ To test equality on \\<^term>‹export_uinfo t = export_uinfo s› we normalise towards\n \\<^term>‹export_uinfo (typ_info_t TYPE('a::c_type)) = export_uinfo (typ_info_t TYPE('b::c_type))›,\n where both \\<^typ>‹'a› and \\<^typ>‹'b› are concrete instances. Then when both expressions happen\n to be the same we have proved equality. Note that this approach is incomplete and\n in particular not sufficient to disprove the equality and prove the inequality. However,\n as these equality often appears as sidecondition in a conditional rewrite rule, not being\n able to prove the equality is somehow \"equivalent\" to disproving it: If we cannot prove\n the sidecondition the rule cannot be applied. The most prominent pattern for this\n is a \\<^const>‹field_lookup› yielding the type-description of the selected field which is\n then compared to some type-description that is derived from a pointer type.\n To also disprove equality we make use of rule @{thm export_uinfo_eq_sub_typ_conv} and\n use the simproc on \\<^const>‹sub_typ›.\n▪ To decide whether e.g. \\<^term>‹TYPE(32 word) ≤⇩τ TYPE(outer_C)› holds or not, we first\n try to disprove it by using type name information @{thm not_sub_typ_via_td_name}.\n both \\<^const>‹typ_name› and \\<^const>‹td_names› are supplied by the UMM package:\n @{thm typ_name_simps}, @{thm td_names_simps}. By construction every distinct type\n gets an individual name.\n If that does not work we try proving it instead, by using a tansitivity prover on\n the single step sub-type relations provided by @{thm sub_typ_simps} and the\n rule for arrays @{thm element_typ_subtyp_array_typ}.\n›\nthm not_sub_typ_via_td_name\nthm typ_name_simps\nthm td_names_simps\nthm sub_typ_simps\nthm element_typ_subtyp_array_typ\n\n\ntext ‹Here some examples›\nlemma \"export_uinfo (typ_info_t TYPE(array_C)) = export_uinfo (typ_info_t TYPE(other_C))\"\n apply simp\n oops\n\nschematic_goal \"field_lookup (typ_info_t TYPE(outer_C)) [''inner_C'', ''fld1_C''] 0 = ?X\"\n by simp\n\nlemma \"TYPE(8 word) ≤⇩τ TYPE(outer_C) = False\"\n by simp\n\nschematic_goal \"i < 2 ⟹ j < 3 ⟹\n field_ti TYPE(two_dimensional_C) [''matrix_C'', replicate i CHR ''1'', replicate j CHR ''1''] = ?X\"\n by simp\n\nlemma \"TYPE(8 word) ≤⇩τ TYPE(8 word[42])\"\n by simp\n\nML_val ‹\nCached_Theory_Simproc.trace_cache @{context}\n›\n\n\nsubsection \"Examples for normalisation of array indexes\"\n\nlemma\n fixes p::\"array_C ptr\"\n assumes bnd[simp]: \"i < 2\"\n shows \"(PTR(unpacked_C) &(p→[''elements_C'', replicate i CHR ''1''])) = PTR(unpacked_C) &(p→[''elements_C'']) +⇩p int i\"\n supply [[simp_trace=false, verbose=5]]\n apply (array_index_to_ptr_arith_simp)\n done\n\n\nlemma\n fixes p::\"array_C ptr\"\n assumes bnd: \"(i::32 word) < 2\"\n shows \"(PTR(unpacked_C) &(p→[''elements_C'', replicate (unat i) CHR ''1''])) = PTR(unpacked_C) &(p→[''elements_C'']) +⇩p uint i\"\n supply [[simp_trace=false, verbose=5]]\n apply (array_index_to_ptr_arith_simp simp: bnd)\n done\n\n\nlemma\n fixes p::\"array_C ptr\"\n shows \"do {_ <- guard (λ_. unat (i::32 word) < 2); \n return ((PTR(unpacked_C) &(p→[''elements_C'', replicate (unat i) CHR ''1''])))} = \n do {\n _ ← guard (λs. unat i < 2);\n return (PTR(unpacked_C) &(p→[''elements_C'']) +⇩p uint i)\n }\"\n supply [[simp_trace=false, verbose=6]]\n apply array_index_to_ptr_arith_simp\n done\n\nlemma\n fixes p::\"array_C ptr\"\n shows \"do {_ <- guard (λ_. (i::32 word) < 2); \n return ((PTR(unpacked_C) &(p→[''elements_C'', replicate (unat i) CHR ''1''])))} = \n do {\n _ ← guard (λs. i < 2);\n return (PTR(unpacked_C) &(p→[''elements_C'']) +⇩p uint i)\n }\"\n supply [[simp_trace=false, verbose=6]]\n apply array_index_to_ptr_arith_simp\n done\n\nlemma\n fixes p::\"array_C ptr\"\n assumes bnd[simp]: \"i < 2\"\n shows \"(PTR(unpacked_C) &(p→[''elements_C'', replicate i CHR ''1'', ''chr_C''])) = \n PTR(unpacked_C)\n &(PTR(unpacked_C) &(p→[''elements_C'']) +⇩p int i→[''chr_C''])\"\n supply [[simp_trace=false, verbose=5]]\n apply (array_index_to_ptr_arith_simp)\n done\n\nlemma \"(i::32 word) < 4 ⟹ unat i < 4\"\n supply [[array_bound_mksimps, verbose=5]]\n apply (simp)\n done\n\nlemma \"(i::32 signed word) <s 4 ∧ 0 ≤s i ⟹ nat (sint i) < 4\"\n supply [[array_bound_mksimps, verbose=5]]\n apply (simp)\n done\n\n\nsubsection \"Essence of Heap Lifting\"\n\ntext ‹Some birds eye view on what heap lifting is about. Consider the following C program.\n\n▩‹\ntypedef struct foo {\n int myint;\n bool mybool;\n} foo;\n\nint * p;\nfoo * q;\n\n(*p) = 42;\ni = q->myint;\nb = q->mybool;\n›\n\nIs value ‹i› and ‹b› affected by update to ‹*p›\n\nFrom the C-Parser we only get \\<^term>‹c_guard p› and \\<^term>‹c_guard q›:\n▩‹\n{c_guard p ∧ c_guard q}\n(*p) = 42;\ni = q->myint;\nb = q->mybool;\n›\n\nThis only tells us something about alignment of pointers and that the pointer span does not overflow,\nbut nothing about disjointness of the pointer spans.\n\nWhat about stronger guarantees, if we assume well-typedness of the pointers:\n▩‹\n{d ⊨⇩t p ∧ d ⊨⇩t p q}\n(*p) = 42;\ni = q->myint;\nb = q->mybool;\n›\n\nNow we know that ‹p› might alias with ‹q->myint› as they have the same type, but that pointer span\n ‹p› is actually disjoint form pointer span ‹q->mybool›.\nHence for the read of ‹b› we can skip the heap update via ‹p›.\n\nNow consider that the structure ‹foo› is a closed structure in the split heap model. This\nmeans that we have even stronger assumptions (which are guaranteed by guards in the lifted code):\n\n▩‹\n{root_ptr_valid d p ∧ root_ptr_valid p q}\n(*p) = 42;\ni = q->myint;\nb = q->mybool;\n›\n\nFrom this we can infer that the pointer spans ‹p› and the entire pointer span ‹q› don't overlap.\nhence reading ‹i› as well as ‹b› can skip the heap update via ‹p›.\n\nIn general the heap lifting phase introduces the potentially weaker guards for open types:\n▩‹\n{ptr_valid d p ∧ ptr_valid p q}\n(*p) = 42;\ni = q->myint;\nb = q->mybool;\n›\n\nSo whether ‹i› might be affected by ‹p› depends on whether field ‹myint› is addressable or not.\n\nThe essential semantic part of heap lifting is to introduce these guards at every pointer access.\nIn a second step these guards then allow a change in a representation from the monolitic heap\nto the split heap, with a separate heap for each atomic type. But in a sense this second\nstep could be viewed as optional or cosmetic in the semantic sense.\nThere is all the information available to perform the \"skipping\" steps\nas sketched above in the monolithic heap. This is what is performed in the simulation\nproof. So the question is if we could omit introduction of \"lifted globals\" completely and\nprovide the necessary automation to the user to simplify accesses on the monolithic heap that\nmimic that behaviour in the split heap. In a sense instead of doing the simulation proof upfront\nit is done adhoc at every heap access / update. This avoids to provide quadratically many\ncommutation lemmas (in the number of fields of open types) that are introduces for the lifted globals.\n›\n\ntext ‹\nFrom the perspective of the simulation proof alone such a \"virtual\" split heap, which does not\nintroduce a new lifted-globals type, seems to be equivalent to the one which introduces a new\ntype with concrete distinct record fields for each fundamental heap. The simulation proof\nonly works when every pointer access is guarded and under that assumption the two representations\nbehave the same.\n\nHowever, besides the simulation there are other aspects of the split heap that might pay off\nfor verification or further abstractions. The two models behave differently for invalid pointers:\nIn the split-heap (with different record fields), updates to one heap still do not\naffect other heaps but we cannot properly model such updates in the monolithic heap at all.\n›\n\ntext ‹\nAnother aspect was introduced later on for stack allocation: non-deterministic padding bytes.\nThe @{locale typ_heap_simulation} was generalised to state that lifting to the split heap is\ninvariant for arbitrary padding bytes, i.e. @{locale write_simulation}. The lifted heap only\ncares about the abstract values were the values of the padding bytes become irrelevant.\n›\nthm h_val_heap_update_padding\nthm plift_heap_update_padding_heap_update_pointless_conv\nthm lift_heap_update_padding_heap_update_conv\n\ncontext open_struct_all_corres\nbegin\nthm ts_def\nthm ac_corres\nend\nend\n\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/AutoCorres2/doc/open_struct.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"efdf615dc432b2089e73483df9535042e0ae4612b42e1bb4c942b00d1e8c9832\"\n}"}}},{"rowIdx":192,"cells":{"text":{"kind":"string","value":"(* Title: variants/c_gtobcast/Fresher.thy\n License: BSD 2-Clause. See LICENSE.\n Author: Timothy Bourke, Inria \n Author: Peter Höfner, NICTA\n*)\n\nsection \"Quality relations between routes\"\n\ntheory C_Fresher\nimports C_Aodv_Data\nbegin\n\nsubsection \"Net sequence numbers\"\n\nsubsubsection \"On individual routes\"\n\ndefinition\n nsqn⇩r :: \"r ⇒ sqn\"\nwhere\n \"nsqn⇩r r ≡ if π₄(r) = val ∨ π₂(r) = 0 then π₂(r) else (π₂(r) - 1)\"\n\nlemma nsqnr_def':\n \"nsqn⇩r r = (if π₄(r) = inv then π₂(r) - 1 else π₂(r))\"\n unfolding \"nsqn⇩r_def\"\n (*goal: ‹(if π₄ r = val ∨ π₂ r = 0 then π₂ r else π₂ r - 1) = (if π₄ r = Aodv_Basic.inv then π₂ r - 1 else π₂ r)›*)\n by simp\n\nlemma nsqn⇩r_zero [simp]:\n \"⋀dsn dsk flag hops nhip. nsqn⇩r (0, dsk, flag, hops, nhip) = 0\"\n unfolding \"nsqn⇩r_def\"\n (*goal: ‹⋀dsn dsk flag hops nhip. (if π₄ (0, dsk, flag, hops, nhip) = val ∨ π₂ (0, dsk, flag, hops, nhip) = 0 then π₂ (0, dsk, flag, hops, nhip) else π₂ (0, dsk, flag, hops, nhip) - 1) = 0›*)\n by clarsimp\n\nlemma nsqn⇩r_val [simp]:\n \"⋀dsn dsk hops nhip. nsqn⇩r (dsn, dsk, val, hops, nhip) = dsn\"\n unfolding \"nsqn⇩r_def\"\n (*goal: ‹⋀dsn dsk hops nhip. (if π₄ (dsn, dsk, val, hops, nhip) = val ∨ π₂ (dsn, dsk, val, hops, nhip) = 0 then π₂ (dsn, dsk, val, hops, nhip) else π₂ (dsn, dsk, val, hops, nhip) - 1) = dsn›*)\n by clarsimp\n\nlemma nsqn⇩r_inv [simp]:\n \"⋀dsn dsk hops nhip. nsqn⇩r (dsn, dsk, inv, hops, nhip) = dsn - 1\"\n unfolding \"nsqn⇩r_def\"\n (*goal: ‹⋀dsn dsk hops nhip. (if π₄ (dsn, dsk, Aodv_Basic.inv, hops, nhip) = val ∨ π₂ (dsn, dsk, Aodv_Basic.inv, hops, nhip) = 0 then π₂ (dsn, dsk, Aodv_Basic.inv, hops, nhip) else π₂ (dsn, dsk, Aodv_Basic.inv, hops, nhip) - 1) = dsn - 1›*)\n by clarsimp\n\nlemma nsqn⇩r_lte_dsn [simp]:\n \"⋀dsn dsk flag hops nhip. nsqn⇩r (dsn, dsk, flag, hops, nhip) ≤ dsn\"\n unfolding \"nsqn⇩r_def\"\n (*goal: ‹⋀dsn dsk flag hops nhip. (if π₄ (dsn, dsk, flag, hops, nhip) = val ∨ π₂ (dsn, dsk, flag, hops, nhip) = 0 then π₂ (dsn, dsk, flag, hops, nhip) else π₂ (dsn, dsk, flag, hops, nhip) - 1) ≤ dsn›*)\n by clarsimp\n\nsubsubsection \"On routes in routing tables\"\n\ndefinition\n nsqn :: \"rt ⇒ ip ⇒ sqn\"\nwhere\n \"nsqn ≡ λrt dip. case σ⇘route⇙(rt, dip) of None ⇒ 0 | Some r ⇒ nsqn⇩r(r)\"\n\nlemma nsqn_sqn_def:\n \"⋀rt dip. nsqn rt dip = (if flag rt dip = Some val ∨ sqn rt dip = 0\n then sqn rt dip else sqn rt dip - 1)\"\n unfolding nsqn_def sqn_def\n (*goal: ‹⋀rt dip. (case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x) = (if flag rt dip = Some val ∨ (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r) = 0 then case rt dip of None ⇒ 0 | Some r ⇒ π₂ r else (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r) - 1)›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma not_in_kD_nsqn [simp]:\n assumes \"dip ∉ kD(rt)\"\n shows \"nsqn rt dip = 0\"\n using assms (*‹(dip::nat) ∉ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)›*) unfolding nsqn_def\n (*goal: ‹(case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x) = 0›*)\n by simp\n\nlemma kD_nsqn:\n assumes \"dip ∈ kD(rt)\"\n shows \"nsqn rt dip = nsqn⇩r(the (σ⇘route⇙(rt, dip)))\"\n using assms[THEN kD_Some] (*‹∃dsn dsk flag hops nhip. rt dip = Some (dsn, dsk, flag, hops, nhip)›*) unfolding nsqn_def\n (*goal: ‹(case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x) = nsqn⇩r (the (rt dip))›*)\n by clarsimp\n\nlemma nsqnr_r_flag_pred [simp, intro]:\n fixes dsn dsk flag hops nhip\n assumes \"P (nsqn⇩r (dsn, dsk, val, hops, nhip))\"\n and \"P (nsqn⇩r (dsn, dsk, inv, hops, nhip))\"\n shows \"P (nsqn⇩r (dsn, dsk, flag, hops, nhip))\"\n sorry\n\nlemma sqn_nsqn:\n \"⋀rt dip. sqn rt dip - 1 ≤ nsqn rt dip\"\n unfolding sqn_def nsqn_def\n (*goal: ‹⋀rt dip. (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r) - 1 ≤ (case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x)›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma nsqn_sqn: \"nsqn rt dip ≤ sqn rt dip\"\n unfolding sqn_def nsqn_def\n (*goal: ‹(case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x) ≤ (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r)›*)\n apply (cases \"rt dip\")\n (*goals:\n 1. ‹rt dip = None ⟹ (case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x) ≤ (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r)›\n 2. ‹⋀a. rt dip = Some a ⟹ (case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x) ≤ (case rt dip of None ⇒ 0 | Some r ⇒ π₂ r)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma val_nsqn_sqn [elim, simp]:\n assumes \"ip∈kD(rt)\"\n and \"the (flag rt ip) = val\"\n shows \"nsqn rt ip = sqn rt ip\"\n using assms (*‹ip ∈ kD rt› ‹the (flag rt ip) = val›*) unfolding nsqn_sqn_def\n (*goal: ‹(if flag rt ip = Some val ∨ sqn rt ip = 0 then sqn rt ip else sqn rt ip - 1) = sqn rt ip›*)\n by auto\n\nlemma vD_nsqn_sqn [elim, simp]:\n assumes \"ip∈vD(rt)\"\n shows \"nsqn rt ip = sqn rt ip\"\n proof (-)\n (*goal: ‹nsqn rt ip = sqn rt ip›*)\n from ‹ip∈vD(rt)› (*‹ip ∈ vD rt›*) have \"ip∈kD(rt)\" and \"the (flag rt ip) = val\"\n apply -\n (*goals:\n 1. ‹ip ∈ vD rt ⟹ ip ∈ kD rt›\n 2. ‹ip ∈ vD rt ⟹ the (flag rt ip) = val›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹nsqn (rt::nat ⇒ (nat × k × f × nat × nat) option) (ip::nat) = sqn rt ip›*)\n by standard\n qed\n\nlemma inv_nsqn_sqn [elim, simp]:\n assumes \"ip∈kD(rt)\"\n and \"the (flag rt ip) = inv\"\n shows \"nsqn rt ip = sqn rt ip - 1\"\n using assms (*‹ip ∈ kD rt› ‹the (flag rt ip) = Aodv_Basic.inv›*) unfolding nsqn_sqn_def\n (*goal: ‹(if flag (rt::nat ⇒ (nat × k × f × nat × nat) option) (ip::nat) = Some val ∨ sqn rt ip = (0::nat) then sqn rt ip else sqn rt ip - (1::nat)) = sqn rt ip - (1::nat)›*)\n by auto\n\nlemma iD_nsqn_sqn [elim, simp]:\n assumes \"ip∈iD(rt)\"\n shows \"nsqn rt ip = sqn rt ip - 1\"\n proof (-)\n (*goal: ‹nsqn rt ip = sqn rt ip - 1›*)\n from ‹ip∈iD(rt)› (*‹(ip::nat) ∈ iD (rt::nat ⇒ (nat × k × f × nat × nat) option)›*) have \"ip∈kD(rt)\" and \"the (flag rt ip) = inv\"\n apply -\n (*goals:\n 1. ‹ip ∈ iD rt ⟹ ip ∈ kD rt›\n 2. ‹ip ∈ iD rt ⟹ the (flag rt ip) = Aodv_Basic.inv›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹nsqn (rt::nat ⇒ (nat × k × f × nat × nat) option) (ip::nat) = sqn rt ip - (1::nat)›*)\n by standard\n qed\n\nlemma nsqn_update_changed_kno_val [simp]: \"⋀rt ip dsn dsk hops nhip.\n rt ≠ update rt ip (dsn, kno, val, hops, nhip)\n ⟹ nsqn (update rt ip (dsn, kno, val, hops, nhip)) ip = dsn\"\n unfolding \"nsqn⇩r_def\" update_def\n (*goal: ‹⋀(rt::nat ⇒ (nat × k × f × nat × nat) option) (ip::nat) (dsn::nat) (dsk::'a) (hops::nat) nhip::nat. rt ≠ (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip)) | Some (s::nat × k × f × nat × nat) ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip) ∧ (π₅ (dsn, kno, val, hops, nhip) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₃ (dsn, kno, val, hops, nhip) = unk then rt(ip ↦ (π₂ s, snd (dsn, kno, val, hops, nhip))) else rt(ip ↦ s)) ⟹ nsqn (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip)) | Some (s::nat × k × f × nat × nat) ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip) ∧ (π₅ (dsn, kno, val, hops, nhip) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₃ (dsn, kno, val, hops, nhip) = unk then rt(ip ↦ (π₂ s, snd (dsn, kno, val, hops, nhip))) else rt(ip ↦ s)) ip = dsn›*)\n apply (clarsimp simp: kD_nsqn (*‹?dip ∈ kD ?rt ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*) split: option.split_asm (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*) option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*) if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n (*goal: ‹⋀rt ip dsn dsk hops nhip. rt ≠ (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip)) | Some s ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip) ∧ (π₅ (dsn, kno, val, hops, nhip) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₃ (dsn, kno, val, hops, nhip) = unk then rt(ip ↦ (π₂ s, snd (dsn, kno, val, hops, nhip))) else rt(ip ↦ s)) ⟹ nsqn (case rt ip of None ⇒ rt(ip ↦ (dsn, kno, val, hops, nhip)) | Some s ⇒ if π₂ s < π₂ (dsn, kno, val, hops, nhip) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₂ s = π₂ (dsn, kno, val, hops, nhip) ∧ (π₅ (dsn, kno, val, hops, nhip) < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ (dsn, kno, val, hops, nhip)) else if π₃ (dsn, kno, val, hops, nhip) = unk then rt(ip ↦ (π₂ s, snd (dsn, kno, val, hops, nhip))) else rt(ip ↦ s)) ip = dsn›*)\n by (metis fun_upd_triv (*‹?f(?x := ?f ?x) = ?f›*))\n\nlemma nsqn_update_other [simp]:\n fixes dsn dsk flag hops dip nhip rt ip\n assumes \"dip ≠ ip\"\n shows \"nsqn (update rt ip (dsn, dsk, flag, hops, nhip)) dip = nsqn rt dip\"\n using assms (*‹dip ≠ ip›*) unfolding nsqn_def\n (*goal: ‹(case update rt ip (dsn, dsk, flag, hops, nhip) dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x) = (case rt dip of None ⇒ 0 | Some x ⇒ nsqn⇩r x)›*)\n by (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n\nlemma nsqn_invalidate_eq:\n assumes \"dip ∈ kD(rt)\"\n and \"dests dip = Some rsn\"\n shows \"nsqn (invalidate rt dests) dip = rsn - 1\"\n using assms (*‹dip ∈ kD rt› ‹dests dip = Some rsn›*) proof (-)\n (*goal: ‹⟦dip ∈ kD rt; dests dip = Some rsn⟧ ⟹ nsqn (invalidate rt dests) dip = rsn - 1›*)\n from assms (*‹(dip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)› ‹dests dip = Some rsn›*) obtain dsk and hops and nhip and pre where \"invalidate rt dests dip = Some (rsn, dsk, inv, hops, nhip)\"\n (*goal: ‹(⋀dsk hops nhip. invalidate rt dests dip = Some (rsn, dsk, Aodv_Basic.inv, hops, nhip) ⟹ thesis) ⟹ thesis›*)\n unfolding invalidate_def\n (*goal: ‹(⋀dsk hops nhip. (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb)) = Some (rsn, dsk, Aodv_Basic.inv, hops, nhip) ⟹ thesis) ⟹ thesis›*)\n by auto\n moreover from ‹dip ∈ kD(rt)› (*‹dip ∈ kD rt›*) have \"dip ∈ kD(invalidate rt dests)\"\n by simp\n ultimately show \"?thesis\"\n (*goal: ‹nsqn (invalidate rt dests) dip = rsn - 1›*)\n using ‹dests dip = Some rsn› (*‹(dests::nat ⇒ nat option) (dip::nat) = Some (rsn::nat)›*) by simp\n qed\n\nlemma nsqn_invalidate_other [simp]:\n assumes \"dip∈kD(rt)\"\n and \"dip∉dom dests\"\n shows \"nsqn (invalidate rt dests) dip = nsqn rt dip\"\n using assms (*‹dip ∈ kD rt› ‹dip ∉ dom dests›*) by (clarsimp simp add: kD_nsqn (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*))\n\nsubsection \"Comparing routes \"\n\ndefinition\n fresher :: \"r ⇒ r ⇒ bool\" (\"(_/ ⊑ _)\" [51, 51] 50)\nwhere\n \"fresher r r' ≡ ((nsqn⇩r r < nsqn⇩r r') ∨ (nsqn⇩r r = nsqn⇩r r' ∧ π₅(r) ≥ π₅(r')))\"\n\nlemma fresherI1 [intro]:\n assumes \"nsqn⇩r r < nsqn⇩r r'\"\n shows \"r ⊑ r'\"\n unfolding fresher_def\n (*goal: ‹nsqn⇩r r < nsqn⇩r r' ∨ nsqn⇩r r = nsqn⇩r r' ∧ π₅ r' ≤ π₅ r›*)\n using assms (*‹nsqn⇩r (r::nat × k × f × nat × nat) < nsqn⇩r (r'::nat × k × f × nat × nat)›*) by simp\n\nlemma fresherI2 [intro]:\n assumes \"nsqn⇩r r = nsqn⇩r r'\"\n and \"π₅(r) ≥ π₅(r')\"\n shows \"r ⊑ r'\"\n unfolding fresher_def\n (*goal: ‹nsqn⇩r r < nsqn⇩r r' ∨ nsqn⇩r r = nsqn⇩r r' ∧ π₅ r' ≤ π₅ r›*)\n using assms (*‹nsqn⇩r r = nsqn⇩r r'› ‹π₅ r' ≤ π₅ r›*) by simp\n\nlemma fresherI [intro]:\n assumes \"(nsqn⇩r r < nsqn⇩r r') ∨ (nsqn⇩r r = nsqn⇩r r' ∧ π₅(r) ≥ π₅(r'))\"\n shows \"r ⊑ r'\"\n unfolding fresher_def\n (*goal: ‹nsqn⇩r r < nsqn⇩r r' ∨ nsqn⇩r r = nsqn⇩r r' ∧ π₅ r' ≤ π₅ r›*)\n using assms (*‹nsqn⇩r r < nsqn⇩r r' ∨ nsqn⇩r r = nsqn⇩r r' ∧ π₅ r' ≤ π₅ r›*) .\n\nlemma fresherE [elim]:\n assumes \"r ⊑ r'\"\n and \"nsqn⇩r r < nsqn⇩r r' ⟹ P r r'\"\n and \"nsqn⇩r r = nsqn⇩r r' ∧ π₅(r) ≥ π₅(r') ⟹ P r r'\"\n shows \"P r r'\"\n using assms (*‹r ⊑ r'› ‹nsqn⇩r (r::nat × k × f × nat × nat) < nsqn⇩r (r'::nat × k × f × nat × nat) ⟹ (P::nat × k × f × nat × nat ⇒ nat × k × f × nat × nat ⇒ bool) r r'› ‹nsqn⇩r r = nsqn⇩r r' ∧ π₅ r' ≤ π₅ r ⟹ P r r'›*) unfolding fresher_def\n (*goal: ‹P r r'›*)\n by auto\n\nlemma fresher_refl [simp]: \"r ⊑ r\"\n unfolding fresher_def\n (*goal: ‹nsqn⇩r r < nsqn⇩r r ∨ nsqn⇩r r = nsqn⇩r r ∧ π₅ r ≤ π₅ r›*)\n by simp\n\nlemma fresher_trans [elim, trans]:\n \"⟦ x ⊑ y; y ⊑ z ⟧ ⟹ x ⊑ z\"\n unfolding fresher_def\n (*goal: ‹⟦nsqn⇩r x < nsqn⇩r y ∨ nsqn⇩r x = nsqn⇩r y ∧ π₅ y ≤ π₅ x; nsqn⇩r y < nsqn⇩r z ∨ nsqn⇩r y = nsqn⇩r z ∧ π₅ z ≤ π₅ y⟧ ⟹ nsqn⇩r x < nsqn⇩r z ∨ nsqn⇩r x = nsqn⇩r z ∧ π₅ z ≤ π₅ x›*)\n by auto\n\nlemma not_fresher_trans [elim, trans]:\n \"⟦ ¬(x ⊑ y); ¬(z ⊑ x) ⟧ ⟹ ¬(z ⊑ y)\"\n unfolding fresher_def\n (*goal: ‹⟦¬ (nsqn⇩r x < nsqn⇩r y ∨ nsqn⇩r x = nsqn⇩r y ∧ π₅ y ≤ π₅ x); ¬ (nsqn⇩r z < nsqn⇩r x ∨ nsqn⇩r z = nsqn⇩r x ∧ π₅ x ≤ π₅ z)⟧ ⟹ ¬ (nsqn⇩r z < nsqn⇩r y ∨ nsqn⇩r z = nsqn⇩r y ∧ π₅ y ≤ π₅ z)›*)\n by auto\n\nlemma fresher_dsn_flag_hops_const [simp]:\n fixes dsn dsk dsk' flag hops nhip nhip'\n shows \"(dsn, dsk, flag, hops, nhip) ⊑ (dsn, dsk', flag, hops, nhip')\"\n unfolding fresher_def\n (*goal: ‹nsqn⇩r (dsn, dsk, flag, hops, nhip) < nsqn⇩r (dsn, dsk', flag, hops, nhip') ∨ nsqn⇩r (dsn, dsk, flag, hops, nhip) = nsqn⇩r (dsn, dsk', flag, hops, nhip') ∧ π₅ (dsn, dsk', flag, hops, nhip') ≤ π₅ (dsn, dsk, flag, hops, nhip)›*)\n apply (cases flag)\n (*goals:\n 1. ‹(flag::f) = val ⟹ nsqn⇩r (dsn::nat, dsk::k, flag, hops::nat, nhip::nat) < nsqn⇩r (dsn, dsk'::k, flag, hops, nhip'::nat) ∨ nsqn⇩r (dsn, dsk, flag, hops, nhip) = nsqn⇩r (dsn, dsk', flag, hops, nhip') ∧ π₅ (dsn, dsk', flag, hops, nhip') ≤ π₅ (dsn, dsk, flag, hops, nhip)›\n 2. ‹(flag::f) = Aodv_Basic.inv ⟹ nsqn⇩r (dsn::nat, dsk::k, flag, hops::nat, nhip::nat) < nsqn⇩r (dsn, dsk'::k, flag, hops, nhip'::nat) ∨ nsqn⇩r (dsn, dsk, flag, hops, nhip) = nsqn⇩r (dsn, dsk', flag, hops, nhip') ∧ π₅ (dsn, dsk', flag, hops, nhip') ≤ π₅ (dsn, dsk, flag, hops, nhip)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n\n\nsubsection \"Comparing routing tables \"\n\ndefinition\n rt_fresher :: \"ip ⇒ rt ⇒ rt ⇒ bool\"\nwhere\n \"rt_fresher ≡ λdip rt rt'. (the (σ⇘route⇙(rt, dip))) ⊑ (the (σ⇘route⇙(rt', dip)))\"\n\nabbreviation\n rt_fresher_syn :: \"rt ⇒ ip ⇒ rt ⇒ bool\" (\"(_/ ⊑⇘_⇙ _)\" [51, 999, 51] 50)\nwhere\n \"rt1 ⊑⇘i⇙ rt2 ≡ rt_fresher i rt1 rt2\"\n\nlemma rt_fresher_def':\n \"(rt₁ ⊑⇘i⇙ rt₂) = (nsqn⇩r (the (rt₁ i)) < nsqn⇩r (the (rt₂ i)) ∨\n nsqn⇩r (the (rt₁ i)) = nsqn⇩r (the (rt₂ i)) ∧ π₅ (the (rt₂ i)) ≤ π₅ (the (rt₁ i)))\"\n unfolding rt_fresher_def fresher_def\n (*goal: ‹(nsqn⇩r (the (rt₁ i)) < nsqn⇩r (the (rt₂ i)) ∨ nsqn⇩r (the (rt₁ i)) = nsqn⇩r (the (rt₂ i)) ∧ π₅ (the (rt₂ i)) ≤ π₅ (the (rt₁ i))) = (nsqn⇩r (the (rt₁ i)) < nsqn⇩r (the (rt₂ i)) ∨ nsqn⇩r (the (rt₁ i)) = nsqn⇩r (the (rt₂ i)) ∧ π₅ (the (rt₂ i)) ≤ π₅ (the (rt₁ i)))›*)\n by (rule refl (*‹?t = ?t›*))\n\nlemma single_rt_fresher [intro]:\n assumes \"the (rt1 ip) ⊑ the (rt2 ip)\"\n shows \"rt1 ⊑⇘ip⇙ rt2\"\n using assms (*‹the ((rt1::nat ⇒ (nat × k × f × nat × nat) option) (ip::nat)) ⊑ the ((rt2::nat ⇒ (nat × k × f × nat × nat) option) ip)›*) unfolding rt_fresher_def\n (*goal: ‹the (rt1 ip) ⊑ the (rt2 ip)›*) .\n\nlemma rt_fresher_single [intro]:\n assumes \"rt1 ⊑⇘ip⇙ rt2\"\n shows \"the (rt1 ip) ⊑ the (rt2 ip)\"\n using assms (*‹rt1 ⊑⇘ip⇙ rt2›*) unfolding rt_fresher_def\n (*goal: ‹the (rt1 ip) ⊑ the (rt2 ip)›*) .\n\nlemma rt_fresher_def2:\n assumes \"dip ∈ kD(rt1)\"\n and \"dip ∈ kD(rt2)\"\n shows \"(rt1 ⊑⇘dip⇙ rt2) = (nsqn rt1 dip < nsqn rt2 dip\n ∨ (nsqn rt1 dip = nsqn rt2 dip\n ∧ the (dhops rt1 dip) ≥ the (dhops rt2 dip)))\"\n using assms (*‹dip ∈ kD rt1› ‹dip ∈ kD rt2›*) unfolding rt_fresher_def fresher_def\n (*goal: ‹(nsqn⇩r (the (rt1 dip)) < nsqn⇩r (the (rt2 dip)) ∨ nsqn⇩r (the (rt1 dip)) = nsqn⇩r (the (rt2 dip)) ∧ π₅ (the (rt2 dip)) ≤ π₅ (the (rt1 dip))) = (nsqn rt1 dip < nsqn rt2 dip ∨ nsqn rt1 dip = nsqn rt2 dip ∧ the (dhops rt2 dip) ≤ the (dhops rt1 dip))›*)\n by (simp add: kD_nsqn (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*) proj5_eq_dhops (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ π₅ (the (?rt ?dip)) = the (dhops ?rt ?dip)›*))\n\nlemma rt_fresherI1 [intro]:\n assumes \"dip ∈ kD(rt1)\"\n and \"dip ∈ kD(rt2)\"\n and \"nsqn rt1 dip < nsqn rt2 dip\"\n shows \"rt1 ⊑⇘dip⇙ rt2\"\n unfolding rt_fresher_def2[OF assms ( 1 - 2 )]\n (*goal: ‹nsqn rt1 dip < nsqn rt2 dip ∨ nsqn rt1 dip = nsqn rt2 dip ∧ the (dhops rt2 dip) ≤ the (dhops rt1 dip)›*)\n using assms(3) (*‹nsqn rt1 dip < nsqn rt2 dip›*) by simp\n\nlemma rt_fresherI2 [intro]:\n assumes \"dip ∈ kD(rt1)\"\n and \"dip ∈ kD(rt2)\"\n and \"nsqn rt1 dip = nsqn rt2 dip\"\n and \"the (dhops rt1 dip) ≥ the (dhops rt2 dip)\"\n shows \"rt1 ⊑⇘dip⇙ rt2\"\n unfolding rt_fresher_def2[OF assms ( 1 - 2 )]\n (*goal: ‹nsqn rt1 dip < nsqn rt2 dip ∨ nsqn rt1 dip = nsqn rt2 dip ∧ the (dhops rt2 dip) ≤ the (dhops rt1 dip)›*)\n using assms(3-4) (*‹nsqn (rt1::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat) = nsqn (rt2::nat ⇒ (nat × k × f × nat × nat) option) dip› ‹the (dhops (rt2::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat)) ≤ the (dhops (rt1::nat ⇒ (nat × k × f × nat × nat) option) dip)›*) by simp\n\nlemma rt_fresherE [elim]:\n assumes \"rt1 ⊑⇘dip⇙ rt2\"\n and \"dip ∈ kD(rt1)\"\n and \"dip ∈ kD(rt2)\"\n and \"⟦ nsqn rt1 dip < nsqn rt2 dip ⟧ ⟹ P rt1 rt2 dip\"\n and \"⟦ nsqn rt1 dip = nsqn rt2 dip;\n the (dhops rt1 dip) ≥ the (dhops rt2 dip) ⟧ ⟹ P rt1 rt2 dip\"\n shows \"P rt1 rt2 dip\"\n using assms(1) (*‹rt1 ⊑⇘dip⇙ rt2›*) unfolding rt_fresher_def2[OF assms ( 2 - 3 )]\n (*goal: ‹(P::(nat ⇒ (nat × k × f × nat × nat) option) ⇒ (nat ⇒ (nat × k × f × nat × nat) option) ⇒ nat ⇒ bool) (rt1::nat ⇒ (nat × k × f × nat × nat) option) (rt2::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat)›*)\n using assms(4-5) (*‹nsqn rt1 dip < nsqn rt2 dip ⟹ P rt1 rt2 dip› ‹⟦nsqn (rt1::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat) = nsqn (rt2::nat ⇒ (nat × k × f × nat × nat) option) dip; the (dhops rt2 dip) ≤ the (dhops rt1 dip)⟧ ⟹ (P::(nat ⇒ (nat × k × f × nat × nat) option) ⇒ (nat ⇒ (nat × k × f × nat × nat) option) ⇒ nat ⇒ bool) rt1 rt2 dip›*) by auto\n\nlemma rt_fresher_refl [simp]: \"rt ⊑⇘dip⇙ rt\"\n unfolding rt_fresher_def\n (*goal: ‹the (rt dip) ⊑ the (rt dip)›*)\n by simp\n\nlemma rt_fresher_trans [elim, trans]:\n assumes \"rt1 ⊑⇘dip⇙ rt2\"\n and \"rt2 ⊑⇘dip⇙ rt3\"\n shows \"rt1 ⊑⇘dip⇙ rt3\"\n using assms (*‹rt1 ⊑⇘dip⇙ rt2› ‹rt2 ⊑⇘dip⇙ rt3›*) unfolding rt_fresher_def\n (*goal: ‹the (rt1 dip) ⊑ the (rt3 dip)›*)\n by auto\n\nlemma rt_fresher_if_Some [intro!]:\n assumes \"the (rt dip) ⊑ r\"\n shows \"rt ⊑⇘dip⇙ (λip. if ip = dip then Some r else rt ip)\"\n using assms (*‹the ((rt::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat)) ⊑ (r::nat × k × f × nat × nat)›*) unfolding rt_fresher_def\n (*goal: ‹the (rt dip) ⊑ the (if dip = dip then Some r else rt dip)›*)\n by simp\n\ndefinition rt_fresh_as :: \"ip ⇒ rt ⇒ rt ⇒ bool\"\nwhere\n \"rt_fresh_as ≡ λdip rt1 rt2. (rt1 ⊑⇘dip⇙ rt2) ∧ (rt2 ⊑⇘dip⇙ rt1)\"\n\nabbreviation\n rt_fresh_as_syn :: \"rt ⇒ ip ⇒ rt ⇒ bool\" (\"(_/ ≈⇘_⇙ _)\" [51, 999, 51] 50)\nwhere\n \"rt1 ≈⇘i⇙ rt2 ≡ rt_fresh_as i rt1 rt2\"\n\nlemma rt_fresh_as_refl [simp]: \"⋀rt dip. rt ≈⇘dip⇙ rt\"\n unfolding rt_fresh_as_def\n (*goal: ‹⋀(rt::nat ⇒ (nat × k × f × nat × nat) option) dip::nat. rt ⊑⇘dip⇙ rt ∧ rt ⊑⇘dip⇙ rt›*)\n by simp\n\nlemma rt_fresh_as_trans [simp, intro, trans]:\n \"⋀rt1 rt2 rt3 dip. ⟦ rt1 ≈⇘dip⇙ rt2; rt2 ≈⇘dip⇙ rt3 ⟧ ⟹ rt1 ≈⇘dip⇙ rt3\"\n unfolding rt_fresh_as_def rt_fresher_def\n (*goal: ‹⋀rt1 rt2 rt3 dip. ⟦the (rt1 dip) ⊑ the (rt2 dip) ∧ the (rt2 dip) ⊑ the (rt1 dip); the (rt2 dip) ⊑ the (rt3 dip) ∧ the (rt3 dip) ⊑ the (rt2 dip)⟧ ⟹ the (rt1 dip) ⊑ the (rt3 dip) ∧ the (rt3 dip) ⊑ the (rt1 dip)›*)\n by (metis (mono_tags) fresher_trans (*‹⟦?x ⊑ ?y; ?y ⊑ ?z⟧ ⟹ ?x ⊑ ?z›*))\n\nlemma rt_fresh_asI [intro!]:\n assumes \"rt1 ⊑⇘dip⇙ rt2\"\n and \"rt2 ⊑⇘dip⇙ rt1\"\n shows \"rt1 ≈⇘dip⇙ rt2\"\n using assms (*‹rt1 ⊑⇘dip⇙ rt2› ‹rt2 ⊑⇘dip⇙ rt1›*) unfolding rt_fresh_as_def\n (*goal: ‹rt1 ⊑⇘dip⇙ rt2 ∧ rt2 ⊑⇘dip⇙ rt1›*)\n by simp\n\nlemma rt_fresh_as_fresherI [intro]:\n assumes \"dip∈kD(rt1)\"\n and \"dip∈kD(rt2)\"\n and \"the (rt1 dip) ⊑ the (rt2 dip)\"\n and \"the (rt2 dip) ⊑ the (rt1 dip)\"\n shows \"rt1 ≈⇘dip⇙ rt2\"\n using assms (*‹(dip::nat) ∈ kD (rt1::nat ⇒ (nat × k × f × nat × nat) option)› ‹dip ∈ kD rt2› ‹the (rt1 dip) ⊑ the (rt2 dip)› ‹the (rt2 dip) ⊑ the (rt1 dip)›*) unfolding rt_fresh_as_def\n (*goal: ‹rt1 ⊑⇘dip⇙ rt2 ∧ rt2 ⊑⇘dip⇙ rt1›*)\n by (clarsimp dest!: single_rt_fresher (*‹the (?rt1.0 ?ip) ⊑ the (?rt2.0 ?ip) ⟹ ?rt1.0 ⊑⇘?ip⇙ ?rt2.0›*))\n\nlemma nsqn_rt_fresh_asI:\n assumes \"dip ∈ kD(rt)\"\n and \"dip ∈ kD(rt')\"\n and \"nsqn rt dip = nsqn rt' dip\"\n and \"π₅(the (rt dip)) = π₅(the (rt' dip))\"\n shows \"rt ≈⇘dip⇙ rt'\"\n proof (standard)\n (*goals:\n 1. ‹rt ⊑⇘dip⇙ rt'›\n 2. ‹rt' ⊑⇘dip⇙ rt›*)\n from assms(1-2,4) (*‹dip ∈ kD rt› ‹dip ∈ kD rt'› ‹π₅ (the (rt dip)) = π₅ (the (rt' dip))›*) have dhops': \"the (dhops rt' dip) ≤ the (dhops rt dip)\"\n by (simp add: proj5_eq_dhops (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ π₅ (the (?rt ?dip)) = the (dhops ?rt ?dip)›*))\n with assms(1-3) (*‹dip ∈ kD rt› ‹dip ∈ kD rt'› ‹nsqn rt dip = nsqn rt' dip›*) show \"rt ⊑⇘dip⇙ rt'\"\n by (rule rt_fresherI2 (*‹⟦?dip ∈ kD ?rt1.0; ?dip ∈ kD ?rt2.0; nsqn ?rt1.0 ?dip = nsqn ?rt2.0 ?dip; the (dhops ?rt2.0 ?dip) ≤ the (dhops ?rt1.0 ?dip)⟧ ⟹ ?rt1.0 ⊑⇘?dip⇙ ?rt2.0›*))\n next\n (*goal: ‹rt' ⊑⇘dip⇙ rt›*)\n from assms(1-2,4) (*‹dip ∈ kD rt› ‹(dip::nat) ∈ kD (rt'::nat ⇒ (nat × k × f × nat × nat) option)› ‹π₅ (the ((rt::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat))) = π₅ (the ((rt'::nat ⇒ (nat × k × f × nat × nat) option) dip))›*) have dhops: \"the (dhops rt dip) ≤ the (dhops rt' dip)\"\n by (simp add: proj5_eq_dhops (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ π₅ (the (?rt ?dip)) = the (dhops ?rt ?dip)›*))\n with assms(2,1) (*‹dip ∈ kD rt'› ‹(dip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)›*) assms(3)[symmetric] (*‹nsqn rt' dip = nsqn rt dip›*) show \"rt' ⊑⇘dip⇙ rt\"\n by (rule rt_fresherI2 (*‹⟦?dip ∈ kD ?rt1.0; ?dip ∈ kD ?rt2.0; nsqn ?rt1.0 ?dip = nsqn ?rt2.0 ?dip; the (dhops ?rt2.0 ?dip) ≤ the (dhops ?rt1.0 ?dip)⟧ ⟹ ?rt1.0 ⊑⇘?dip⇙ ?rt2.0›*))\n qed\n\nlemma rt_fresh_asE [elim]:\n assumes \"rt1 ≈⇘dip⇙ rt2\"\n and \"⟦ rt1 ⊑⇘dip⇙ rt2; rt2 ⊑⇘dip⇙ rt1 ⟧ ⟹ P rt1 rt2 dip\"\n shows \"P rt1 rt2 dip\"\n using assms (*‹rt1 ≈⇘dip⇙ rt2› ‹⟦rt1 ⊑⇘dip⇙ rt2; rt2 ⊑⇘dip⇙ rt1⟧ ⟹ P rt1 rt2 dip›*) unfolding rt_fresh_as_def\n (*goal: ‹(P::(nat ⇒ (nat × k × f × nat × nat) option) ⇒ (nat ⇒ (nat × k × f × nat × nat) option) ⇒ nat ⇒ bool) (rt1::nat ⇒ (nat × k × f × nat × nat) option) (rt2::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat)›*)\n by simp\n\nlemma rt_fresh_asD1 [dest]:\n assumes \"rt1 ≈⇘dip⇙ rt2\"\n shows \"rt1 ⊑⇘dip⇙ rt2\"\n using assms (*‹rt1 ≈⇘dip⇙ rt2›*) unfolding rt_fresh_as_def\n (*goal: ‹rt1 ⊑⇘dip⇙ rt2›*)\n by simp\n\nlemma rt_fresh_asD2 [dest]:\n assumes \"rt1 ≈⇘dip⇙ rt2\"\n shows \"rt2 ⊑⇘dip⇙ rt1\"\n using assms (*‹rt1 ≈⇘dip⇙ rt2›*) unfolding rt_fresh_as_def\n (*goal: ‹(rt2::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(dip::nat)⇙ (rt1::nat ⇒ (nat × k × f × nat × nat) option)›*)\n by simp\n\nlemma rt_fresh_as_sym:\n assumes \"rt1 ≈⇘dip⇙ rt2\"\n shows \"rt2 ≈⇘dip⇙ rt1\"\n using assms (*‹rt1 ≈⇘dip⇙ rt2›*) unfolding rt_fresh_as_def\n (*goal: ‹rt2 ⊑⇘dip⇙ rt1 ∧ rt1 ⊑⇘dip⇙ rt2›*)\n by simp\n\nlemma not_rt_fresh_asI1 [intro]:\n assumes \"¬ (rt1 ⊑⇘dip⇙ rt2)\"\n shows \"¬ (rt1 ≈⇘dip⇙ rt2)\"\n proof (standard)\n (*goal: ‹rt1 ≈⇘dip⇙ rt2 ⟹ False›*)\n assume \"rt1 ≈⇘dip⇙ rt2\" (*‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)›*)\n hence \"rt1 ⊑⇘dip⇙ rt2\"\n apply -\n (*goal: ‹rt1 ⊑⇘dip⇙ rt2›*)\n apply standard\n (*goals:\n 1. ‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option) ⟹ dip ∈ kD rt1›\n 2. ‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option) ⟹ dip ∈ kD rt2›\n 3. ‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option) ⟹ nsqn rt1 dip = nsqn rt2 dip›\n 4. ‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option) ⟹ the (dhops rt2 dip) ≤ the (dhops rt1 dip)›\n discuss goal 1*)\n apply ((msorry)[1])\n (*discuss goal 2*)\n apply ((msorry)[1])\n (*discuss goal 3*)\n apply ((msorry)[1])\n (*discuss goal 4*)\n apply ((msorry)[1])\n (*proven 4 subgoals*) .\n with ‹¬ (rt1 ⊑⇘dip⇙ rt2)› (*‹¬ rt1 ⊑⇘dip⇙ rt2›*) show False\n by standard\n qed\n\nlemma not_rt_fresh_asI2 [intro]:\n assumes \"¬ (rt2 ⊑⇘dip⇙ rt1)\"\n shows \"¬ (rt1 ≈⇘dip⇙ rt2)\"\n proof (standard)\n (*goal: ‹rt1 ≈⇘dip⇙ rt2 ⟹ False›*)\n assume \"rt1 ≈⇘dip⇙ rt2\" (*‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)›*)\n hence \"rt2 ⊑⇘dip⇙ rt1\"\n by standard\n with ‹¬ (rt2 ⊑⇘dip⇙ rt1)› (*‹¬ rt2 ⊑⇘dip⇙ rt1›*) show False\n by standard\n qed\n\nlemma not_single_rt_fresher [elim]:\n assumes \"¬(the (rt1 ip) ⊑ the (rt2 ip))\"\n shows \"¬(rt1 ⊑⇘ip⇙ rt2)\"\n proof (standard)\n (*goal: ‹rt1 ⊑⇘ip⇙ rt2 ⟹ False›*)\n assume \"rt1 ⊑⇘ip⇙ rt2\" (*‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(ip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)›*)\n hence \"the (rt1 ip) ⊑ the (rt2 ip)\"\n apply -\n (*goal: ‹the (rt1 ip) ⊑ the (rt2 ip)›*)\n by standard\n with ‹¬(the (rt1 ip) ⊑ the (rt2 ip))› (*‹¬ the (rt1 ip) ⊑ the (rt2 ip)›*) show False\n by standard\n qed\n\nlemmas not_single_rt_fresh_asI1 [intro] = not_rt_fresh_asI1 [OF not_single_rt_fresher]\nlemmas not_single_rt_fresh_asI2 [intro] = not_rt_fresh_asI2 [OF not_single_rt_fresher]\n\nlemma not_rt_fresher_single [elim]:\n assumes \"¬(rt1 ⊑⇘ip⇙ rt2)\"\n shows \"¬(the (rt1 ip) ⊑ the (rt2 ip))\"\n proof (standard)\n (*goal: ‹the (rt1 ip) ⊑ the (rt2 ip) ⟹ False›*)\n assume \"the (rt1 ip) ⊑ the (rt2 ip)\" (*‹the ((rt1::nat ⇒ (nat × k × f × nat × nat) option) (ip::nat)) ⊑ the ((rt2::nat ⇒ (nat × k × f × nat × nat) option) ip)›*)\n hence \"rt1 ⊑⇘ip⇙ rt2\"\n apply -\n (*goal: ‹rt1 ⊑⇘ip⇙ rt2›*)\n apply standard\n (*goals:\n 1. ‹the (rt1 ip) ⊑ the (rt2 ip) ⟹ ip ∈ kD rt1›\n 2. ‹the (rt1 ip) ⊑ the (rt2 ip) ⟹ ip ∈ kD rt2›\n 3. ‹the (rt1 ip) ⊑ the (rt2 ip) ⟹ nsqn rt1 ip = nsqn rt2 ip›\n 4. ‹the (rt1 ip) ⊑ the (rt2 ip) ⟹ the (dhops rt2 ip) ≤ the (dhops rt1 ip)›\n discuss goal 1*)\n apply ((msorry)[1])\n (*discuss goal 2*)\n apply ((msorry)[1])\n (*discuss goal 3*)\n apply ((msorry)[1])\n (*discuss goal 4*)\n apply ((msorry)[1])\n (*proven 4 subgoals*) .\n with ‹¬(rt1 ⊑⇘ip⇙ rt2)› (*‹¬ rt1 ⊑⇘ip⇙ rt2›*) show False\n by standard\n qed\n\nlemma rt_fresh_as_nsqnr:\n assumes \"dip ∈ kD(rt1)\"\n and \"dip ∈ kD(rt2)\"\n and \"rt1 ≈⇘dip⇙ rt2\"\n shows \"nsqn⇩r (the (rt2 dip)) = nsqn⇩r (the (rt1 dip))\"\n using assms(3) (*‹rt1 ≈⇘dip⇙ rt2›*) unfolding rt_fresh_as_def\n (*goal: ‹nsqn⇩r (the (rt2 dip)) = nsqn⇩r (the (rt1 dip))›*)\n by (auto simp: rt_fresher_def2 [OF ‹dip ∈ kD(rt1)› ‹dip ∈ kD(rt2)›] (*‹(rt1 ⊑⇘dip⇙ rt2) = (nsqn rt1 dip < nsqn rt2 dip ∨ nsqn rt1 dip = nsqn rt2 dip ∧ the (dhops rt2 dip) ≤ the (dhops rt1 dip))›*) rt_fresher_def2 [OF ‹dip ∈ kD(rt2)› ‹dip ∈ kD(rt1)›] (*‹(rt2 ⊑⇘dip⇙ rt1) = (nsqn rt2 dip < nsqn rt1 dip ∨ nsqn rt2 dip = nsqn rt1 dip ∧ the (dhops rt1 dip) ≤ the (dhops rt2 dip))›*) kD_nsqn [OF ‹dip ∈ kD(rt1)›] (*‹nsqn rt1 dip = nsqn⇩r (the (rt1 dip))›*) kD_nsqn [OF ‹dip ∈ kD(rt2)›] (*‹nsqn rt2 dip = nsqn⇩r (the (rt2 dip))›*))\n\nlemma rt_fresher_mapupd [intro!]:\n assumes \"dip∈kD(rt)\"\n and \"the (rt dip) ⊑ r\"\n shows \"rt ⊑⇘dip⇙ rt(dip ↦ r)\"\n using assms (*‹(dip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)› ‹the (rt dip) ⊑ r›*) unfolding rt_fresher_def\n (*goal: ‹the (rt dip) ⊑ the ((rt(dip ↦ r)) dip)›*)\n by simp\n\nlemma rt_fresher_map_update_other [intro!]:\n assumes \"dip∈kD(rt)\"\n and \"dip ≠ ip\"\n shows \"rt ⊑⇘dip⇙ rt(ip ↦ r)\"\n using assms (*‹(dip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)› ‹dip ≠ ip›*) unfolding rt_fresher_def\n (*goal: ‹the (rt dip) ⊑ the ((rt(ip ↦ r)) dip)›*)\n by simp\n\nlemma rt_fresher_update_other [simp]:\n assumes inkD: \"dip∈kD(rt)\"\n and \"dip ≠ ip\"\n shows \"rt ⊑⇘dip⇙ update rt ip r\"\n using assms (*‹dip ∈ kD rt› ‹dip ≠ ip›*) unfolding update_def\n (*goal: ‹rt ⊑⇘dip⇙ (case rt ip of None ⇒ rt(ip ↦ r) | Some s ⇒ if π₂ s < π₂ r then rt(ip ↦ r) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ r) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd r)) else rt(ip ↦ s))›*)\n apply (clarsimp split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n (*goal: ‹(rt::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(dip::nat)⇙ (case rt (ip::nat) of None ⇒ rt(ip ↦ r::nat × k × f × nat × nat) | Some (s::nat × k × f × nat × nat) ⇒ if π₂ s < π₂ r then rt(ip ↦ r) else if π₂ s = π₂ r ∧ (π₅ r < π₅ s ∨ π₄ s = Aodv_Basic.inv) then rt(ip ↦ r) else if π₃ r = unk then rt(ip ↦ (π₂ s, snd r)) else rt(ip ↦ s))›*)\n by fastforce\n\ntheorem rt_fresher_update [simp]:\n assumes \"dip∈kD(rt)\"\n and \"the (dhops rt dip) ≥ 1\"\n and \"update_arg_wf r\"\n shows \"rt ⊑⇘dip⇙ update rt ip r\"\n proof (cases \"dip = ip\")\n (*goals:\n 1. ‹dip = ip ⟹ rt ⊑⇘dip⇙ update rt ip r›\n 2. ‹dip ≠ ip ⟹ rt ⊑⇘dip⇙ update rt ip r›*)\n assume \"dip ≠ ip\" (*‹(dip::nat) ≠ (ip::nat)›*)\n with ‹dip∈kD(rt)› (*‹(dip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)›*) show \"?thesis\"\n (*goal: ‹rt ⊑⇘dip⇙ update rt ip r›*)\n by (rule rt_fresher_update_other (*‹⟦?dip ∈ kD ?rt; ?dip ≠ ?ip⟧ ⟹ ?rt ⊑⇘?dip⇙ update ?rt ?ip ?r›*))\n next\n (*goal: ‹dip = ip ⟹ rt ⊑⇘dip⇙ update rt ip r›*)\n assume \"dip = ip\" (*‹(dip::nat) = (ip::nat)›*)\n from ‹dip∈kD(rt)› (*‹dip ∈ kD rt›*) obtain dsn⇩n and dsk⇩n and f⇩n and hops⇩n and nhip⇩n where rtn[simp]: \"the (rt dip) = (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)\"\n (*goal: ‹(⋀dsn⇩n dsk⇩n f⇩n hops⇩n nhip⇩n. the (rt dip) = (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⟹ thesis) ⟹ thesis›*)\n by (metis prod_cases5 (*‹(⋀a b c d e. ?y = (a, b, c, d, e) ⟹ ?thesis) ⟹ ?thesis›*))\n with ‹the (dhops rt dip) ≥ 1› (*‹1 ≤ the (dhops rt dip)›*) ‹dip∈kD(rt)› (*‹dip ∈ kD rt›*) have \"hops⇩n ≥ 1\"\n by (metis proj5_eq_dhops (*‹?dip ∈ kD ?rt ⟹ π₅ (the (?rt ?dip)) = the (dhops ?rt ?dip)›*) projs( (*‹π₅ (?dsn, ?dsk, ?flag, ?hops, ?nhip) = ?hops›*) 4))\n from ‹dip∈kD(rt)› (*‹dip ∈ kD rt›*) rtn (*‹the ((rt::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat)) = (dsn⇩n::nat, dsk⇩n::k, f⇩n::f, hops⇩n::nat, nhip⇩n::nat)›*) have [simp]: \"sqn rt dip = dsn⇩n\" and [simp]: \"the (dhops rt dip) = hops⇩n\" and [simp]: \"the (flag rt dip) = f⇩n\"\n apply -\n (*goals:\n 1. ‹⟦dip ∈ kD rt; the (rt dip) = (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)⟧ ⟹ sqn rt dip = dsn⇩n›\n 2. ‹⟦dip ∈ kD rt; the (rt dip) = (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)⟧ ⟹ the (dhops rt dip) = hops⇩n›\n 3. ‹⟦dip ∈ kD rt; the (rt dip) = (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)⟧ ⟹ the (flag rt dip) = f⇩n›\n discuss goal 1*)\n apply (simp add: sqn_def (*‹sqn (?rt::nat ⇒ (nat × k × f × nat × nat) option) (?dip::nat) ≡ case ?rt ?dip of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat) ⇒ π₂ r›*) proj5_eq_dhops [symmetric] (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ the (dhops ?rt ?dip) = π₅ (the (?rt ?dip))›*) proj4_eq_flag [symmetric] (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ the (flag ?rt ?dip) = π₄ (the (?rt ?dip))›*))\n (*discuss goal 2*)\n apply (simp add: sqn_def (*‹sqn ?rt ?dip ≡ case ?rt ?dip of None ⇒ 0 | Some r ⇒ π₂ r›*) proj5_eq_dhops [symmetric] (*‹?dip ∈ kD ?rt ⟹ the (dhops ?rt ?dip) = π₅ (the (?rt ?dip))›*) proj4_eq_flag [symmetric] (*‹?dip ∈ kD ?rt ⟹ the (flag ?rt ?dip) = π₄ (the (?rt ?dip))›*))\n (*discuss goal 3*)\n apply (simp add: sqn_def (*‹sqn ?rt ?dip ≡ case ?rt ?dip of None ⇒ 0 | Some r ⇒ π₂ r›*) proj5_eq_dhops [symmetric] (*‹?dip ∈ kD ?rt ⟹ the (dhops ?rt ?dip) = π₅ (the (?rt ?dip))›*) proj4_eq_flag [symmetric] (*‹?dip ∈ kD ?rt ⟹ the (flag ?rt ?dip) = π₄ (the (?rt ?dip))›*))\n (*proven 3 subgoals*) .\n from ‹update_arg_wf r› (*‹update_arg_wf r›*) have \"(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)\n ⊑ the ((update rt dip r) dip)\"\n proof (rule wf_r_cases (*‹⟦update_arg_wf ?r; ⋀nhip. ?P (0, unk, val, Suc 0, nhip); ⋀dsn hops nhip. 0 < dsn ⟹ ?P (dsn, kno, val, hops, nhip)⟧ ⟹ ?P ?r›*))\n (*goals:\n 1. ‹⋀nhip. (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the (update rt dip (0, unk, val, Suc 0, nhip) dip)›\n 2. ‹⋀dsn hops nhip. 0 < dsn ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the (update rt dip (dsn, kno, val, hops, nhip) dip)›*)\n fix nhip and pre\n from ‹hops⇩n ≥ 1› (*‹1 ≤ hops⇩n›*) have \"⋀pre'. (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)\n ⊑ (dsn⇩n, unk, val, Suc 0, nhip)\"\n unfolding fresher_def sqn_def\n (*goal: ‹⋀pre'::'a. nsqn⇩r (dsn⇩n::nat, dsk⇩n::k, f⇩n::f, hops⇩n::nat, nhip⇩n::nat) < nsqn⇩r (dsn⇩n, unk, val, Suc (0::nat), nhip::nat) ∨ nsqn⇩r (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = nsqn⇩r (dsn⇩n, unk, val, Suc (0::nat), nhip) ∧ π₅ (dsn⇩n, unk, val, Suc (0::nat), nhip) ≤ π₅ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)›*)\n apply (cases f⇩n)\n (*goals:\n 1. ‹⋀pre'. ⟦1 ≤ hops⇩n; f⇩n = val⟧ ⟹ nsqn⇩r (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) < nsqn⇩r (dsn⇩n, unk, val, Suc 0, nhip) ∨ nsqn⇩r (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = nsqn⇩r (dsn⇩n, unk, val, Suc 0, nhip) ∧ π₅ (dsn⇩n, unk, val, Suc 0, nhip) ≤ π₅ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)›\n 2. ‹⋀pre'. ⟦1 ≤ hops⇩n; f⇩n = Aodv_Basic.inv⟧ ⟹ nsqn⇩r (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) < nsqn⇩r (dsn⇩n, unk, val, Suc 0, nhip) ∨ nsqn⇩r (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = nsqn⇩r (dsn⇩n, unk, val, Suc 0, nhip) ∧ π₅ (dsn⇩n, unk, val, Suc 0, nhip) ≤ π₅ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thus \"(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)\n ⊑ the (update rt dip (0, unk, val, Suc 0, nhip) dip)\"\n using ‹dip∈kD(rt)› (*‹dip ∈ kD rt›*) apply -\n (*goal: ‹(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the (update rt dip (0, unk, val, Suc 0, nhip) dip)›*)\n apply (rule update_cases_kD (*‹⟦(π₂ (?r::nat × k × f × nat × nat) = (0::nat)) = (π₃ ?r = unk); (?ip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option); sqn ?rt ?ip < π₂ ?r ⟹ (?P::(nat ⇒ (nat × k × f × nat × nat) option) ⇒ bool) (?rt(?ip ↦ ?r)); ⟦sqn ?rt ?ip = π₂ ?r; π₅ ?r < the (dhops ?rt ?ip)⟧ ⟹ ?P (?rt(?ip ↦ ?r)); ⟦sqn ?rt ?ip = π₂ ?r; the (flag ?rt ?ip) = Aodv_Basic.inv⟧ ⟹ ?P (?rt(?ip ↦ ?r)); π₃ ?r = unk ⟹ ?P (?rt(?ip ↦ (π₂ (the (?rt ?ip)), π₃ ?r, π₄ ?r, π₅ ?r, π₆ ?r))); ⟦π₂ ?r ≤ sqn ?rt ?ip; π₃ ?r = kno; sqn ?rt ?ip = π₂ ?r ⟹ the (dhops ?rt ?ip) ≤ π₅ ?r ∧ the (flag ?rt ?ip) = val⟧ ⟹ ?P (?rt(?ip ↦ the (?rt ?ip)))⟧ ⟹ ?P (update ?rt ?ip ?r)›*))\n (*goals:\n 1. ‹⟦(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn⇩n, unk, val, Suc 0, nhip); dip ∈ kD rt⟧ ⟹ (π₂ (0, unk, val, Suc 0, nhip) = 0) = (π₃ (0, unk, val, Suc 0, nhip) = unk)›\n 2. ‹⟦(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn⇩n, unk, val, Suc 0, nhip); dip ∈ kD rt⟧ ⟹ dip ∈ kD rt›\n 3. ‹⟦(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn⇩n, unk, val, Suc 0, nhip); dip ∈ kD rt; sqn rt dip < π₂ (0, unk, val, Suc 0, nhip)⟧ ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the ((rt(dip ↦ (0, unk, val, Suc 0, nhip))) dip)›\n 4. ‹⟦(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn⇩n, unk, val, Suc 0, nhip); dip ∈ kD rt; sqn rt dip = π₂ (0, unk, val, Suc 0, nhip); π₅ (0, unk, val, Suc 0, nhip) < the (dhops rt dip)⟧ ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the ((rt(dip ↦ (0, unk, val, Suc 0, nhip))) dip)›\n 5. ‹⟦(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn⇩n, unk, val, Suc 0, nhip); dip ∈ kD rt; sqn rt dip = π₂ (0, unk, val, Suc 0, nhip); the (flag rt dip) = Aodv_Basic.inv⟧ ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the ((rt(dip ↦ (0, unk, val, Suc 0, nhip))) dip)›\n 6. ‹⟦(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn⇩n, unk, val, Suc 0, nhip); dip ∈ kD rt; π₃ (0, unk, val, Suc 0, nhip) = unk⟧ ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the ((rt(dip ↦ (π₂ (the (rt dip)), π₃ (0, unk, val, Suc 0, nhip), π₄ (0, unk, val, Suc 0, nhip), π₅ (0, unk, val, Suc 0, nhip), π₆ (0, unk, val, Suc 0, nhip)))) dip)›\n 7. ‹⟦(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn⇩n, unk, val, Suc 0, nhip); dip ∈ kD rt; π₂ (0, unk, val, Suc 0, nhip) ≤ sqn rt dip; π₃ (0, unk, val, Suc 0, nhip) = kno; sqn rt dip = π₂ (0, unk, val, Suc 0, nhip) ⟹ the (dhops rt dip) ≤ π₅ (0, unk, val, Suc 0, nhip) ∧ the (flag rt dip) = val⟧ ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the ((rt(dip ↦ the (rt dip))) dip)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*discuss goal 6*)\n apply simp\n (*discuss goal 7*)\n apply simp\n (*proven 7 subgoals*) .\n next\n (*goal: ‹⋀dsn hops nhip. 0 < dsn ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the (update rt dip (dsn, kno, val, hops, nhip) dip)›*)\n fix dsn :: sqn and hops and nhip and pre\n assume \"0 < dsn\" (*‹(0::nat) < (dsn::nat)›*)\n show \"(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)\n ⊑ the (update rt dip (dsn, kno, val, hops, nhip) dip)\"\n proof (rule update_cases_kD [OF _ ‹dip∈kD(rt)›] (*‹⟦(π₂ ?r = 0) = (π₃ ?r = unk); sqn rt dip < π₂ ?r ⟹ ?P (rt(dip ↦ ?r)); ⟦sqn rt dip = π₂ ?r; π₅ ?r < the (dhops rt dip)⟧ ⟹ ?P (rt(dip ↦ ?r)); ⟦sqn rt dip = π₂ ?r; the (flag rt dip) = Aodv_Basic.inv⟧ ⟹ ?P (rt(dip ↦ ?r)); π₃ ?r = unk ⟹ ?P (rt(dip ↦ (π₂ (the (rt dip)), π₃ ?r, π₄ ?r, π₅ ?r, π₆ ?r))); ⟦π₂ ?r ≤ sqn rt dip; π₃ ?r = kno; sqn rt dip = π₂ ?r ⟹ the (dhops rt dip) ≤ π₅ ?r ∧ the (flag rt dip) = val⟧ ⟹ ?P (rt(dip ↦ the (rt dip)))⟧ ⟹ ?P (update rt dip ?r)›*), simp_all add: ‹0 < dsn›)\n (*goals:\n 1. ‹dsn⇩n < dsn ⟹ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn, kno, val, hops, nhip)›\n 2. ‹⟦dsn⇩n = dsn; hops < hops⇩n⟧ ⟹ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn, kno, val, hops, nhip)›\n 3. ‹⟦dsn⇩n = dsn; f⇩n = Aodv_Basic.inv⟧ ⟹ (dsn, dsk⇩n, Aodv_Basic.inv, hops⇩n, nhip⇩n) ⊑ (dsn, kno, val, hops, nhip)›*)\n assume \"dsn⇩n < dsn\" (*‹(dsn⇩n::nat) < (dsn::nat)›*)\n thus \"(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)\n ⊑ (dsn, kno, val, hops, nhip)\"\n unfolding fresher_def\n (*goal: ‹nsqn⇩r (dsn⇩n::nat, dsk⇩n::k, f⇩n::f, hops⇩n::nat, nhip⇩n::nat) < nsqn⇩r (dsn::nat, kno, val, hops::nat, nhip::nat) ∨ nsqn⇩r (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = nsqn⇩r (dsn, kno, val, hops, nhip) ∧ π₅ (dsn, kno, val, hops, nhip) ≤ π₅ (dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)›*)\n by auto\n next\n (*goals:\n 1. ‹⟦dsn⇩n = dsn; hops < hops⇩n⟧ ⟹ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ (dsn, kno, val, hops, nhip)›\n 2. ‹⟦dsn⇩n = dsn; f⇩n = Aodv_Basic.inv⟧ ⟹ (dsn, dsk⇩n, Aodv_Basic.inv, hops⇩n, nhip⇩n) ⊑ (dsn, kno, val, hops, nhip)›*)\n assume \"dsn⇩n = dsn\" and \"hops < hops⇩n\" (*‹(dsn⇩n::nat) = (dsn::nat)› ‹(hops::nat) < (hops⇩n::nat)›*)\n thus \"(dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)\n ⊑ (dsn, kno, val, hops, nhip)\"\n unfolding fresher_def \"nsqn⇩r_def\"\n (*goal: ‹(if π₄ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = val ∨ π₂ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = 0 then π₂ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) else π₂ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) - 1) < (if π₄ (dsn, kno, val, hops, nhip) = val ∨ π₂ (dsn, kno, val, hops, nhip) = 0 then π₂ (dsn, kno, val, hops, nhip) else π₂ (dsn, kno, val, hops, nhip) - 1) ∨ (if π₄ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = val ∨ π₂ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) = 0 then π₂ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) else π₂ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) - 1) = (if π₄ (dsn, kno, val, hops, nhip) = val ∨ π₂ (dsn, kno, val, hops, nhip) = 0 then π₂ (dsn, kno, val, hops, nhip) else π₂ (dsn, kno, val, hops, nhip) - 1) ∧ π₅ (dsn, kno, val, hops, nhip) ≤ π₅ (dsn, dsk⇩n, f⇩n, hops⇩n, nhip⇩n)›*)\n by simp\n next\n (*goal: ‹⟦dsn⇩n = dsn; f⇩n = Aodv_Basic.inv⟧ ⟹ (dsn, dsk⇩n, Aodv_Basic.inv, hops⇩n, nhip⇩n) ⊑ (dsn, kno, val, hops, nhip)›*)\n assume \"dsn⇩n = dsn\" (*‹(dsn⇩n::nat) = (dsn::nat)›*)\n with ‹0 < dsn› (*‹0 < dsn›*) show \"(dsn, dsk⇩n, inv, hops⇩n, nhip⇩n)\n ⊑ (dsn, kno, val, hops, nhip)\"\n unfolding fresher_def\n (*goal: ‹nsqn⇩r (dsn::nat, dsk⇩n::k, Aodv_Basic.inv, hops⇩n::nat, nhip⇩n::nat) < nsqn⇩r (dsn, kno, val, hops::nat, nhip::nat) ∨ nsqn⇩r (dsn, dsk⇩n, Aodv_Basic.inv, hops⇩n, nhip⇩n) = nsqn⇩r (dsn, kno, val, hops, nhip) ∧ π₅ (dsn, kno, val, hops, nhip) ≤ π₅ (dsn, dsk⇩n, Aodv_Basic.inv, hops⇩n, nhip⇩n)›*)\n by simp\n qed\n qed\n hence \"rt ⊑⇘dip⇙ update rt dip r\"\n apply -\n (*goal: ‹rt ⊑⇘dip⇙ update rt dip r›*)\n apply (rule single_rt_fresher (*‹the (?rt1.0 ?ip) ⊑ the (?rt2.0 ?ip) ⟹ ?rt1.0 ⊑⇘?ip⇙ ?rt2.0›*))\n (*goal: ‹(dsn⇩n, dsk⇩n, f⇩n, hops⇩n, nhip⇩n) ⊑ the (update rt dip r dip) ⟹ rt ⊑⇘dip⇙ update rt dip r›*)\n by simp\n with ‹dip = ip› (*‹dip = ip›*) show \"?thesis\"\n (*goal: ‹rt ⊑⇘dip⇙ update rt ip r›*)\n by simp\n qed\n\ntheorem rt_fresher_invalidate [simp]:\n assumes \"dip∈kD(rt)\"\n and indests: \"∀rip∈dom(dests). rip∈vD(rt) ∧ sqn rt rip < the (dests rip)\"\n shows \"rt ⊑⇘dip⇙ invalidate rt dests\"\n proof (cases \"dip∈dom(dests)\")\n (*goals:\n 1. ‹(dip::nat) ∈ dom (dests::nat ⇒ nat option) ⟹ (rt::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘dip⇙ invalidate rt dests›\n 2. ‹(dip::nat) ∉ dom (dests::nat ⇒ nat option) ⟹ (rt::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘dip⇙ invalidate rt dests›*)\n assume \"dip∉dom(dests)\" (*‹(dip::nat) ∉ dom (dests::nat ⇒ nat option)›*)\n thus \"?thesis\"\n (*goal: ‹rt ⊑⇘dip⇙ invalidate rt dests›*)\n using ‹dip∈kD(rt)› (*‹dip ∈ kD rt›*) apply -\n (*goal: ‹rt ⊑⇘dip⇙ invalidate rt dests›*)\n apply (rule single_rt_fresher (*‹the (?rt1.0 ?ip) ⊑ the (?rt2.0 ?ip) ⟹ ?rt1.0 ⊑⇘?ip⇙ ?rt2.0›*))\n (*goal: ‹⟦(dip::nat) ∉ dom (dests::nat ⇒ nat option); dip ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)⟧ ⟹ rt ⊑⇘dip⇙ invalidate rt dests›*)\n by simp\n next\n (*goal: ‹dip ∈ dom dests ⟹ rt ⊑⇘dip⇙ invalidate rt dests›*)\n assume \"dip∈dom(dests)\" (*‹(dip::nat) ∈ dom (dests::nat ⇒ nat option)›*)\n moreover with indests (*‹∀rip∈dom dests. rip ∈ vD rt ∧ sqn rt rip < the (dests rip)›*) have \"dip∈vD(rt)\" and \"sqn rt dip < the (dests dip)\"\n apply -\n (*goals:\n 1. ‹⟦∀rip::nat∈dom (dests::nat ⇒ nat option). rip ∈ vD (rt::nat ⇒ (nat × k × f × nat × nat) option) ∧ sqn rt rip < the (dests rip); (dip::nat) ∈ dom dests⟧ ⟹ dip ∈ vD rt›\n 2. ‹⟦∀rip::nat∈dom (dests::nat ⇒ nat option). rip ∈ vD (rt::nat ⇒ (nat × k × f × nat × nat) option) ∧ sqn rt rip < the (dests rip); (dip::nat) ∈ dom dests⟧ ⟹ sqn rt dip < the (dests dip)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n ultimately show \"?thesis\"\n (*goal: ‹(rt::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(dip::nat)⇙ invalidate rt (dests::nat ⇒ nat option)›*)\n unfolding invalidate_def sqn_def\n (*goal: ‹rt ⊑⇘dip⇙ (λip. case (rt ip, dests ip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))›*)\n apply -\n (*goal: ‹rt ⊑⇘dip⇙ (λip. case (rt ip, dests ip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))›*)\n apply (rule single_rt_fresher (*‹the ((?rt1.0::nat ⇒ (nat × k × f × nat × nat) option) (?ip::nat)) ⊑ the ((?rt2.0::nat ⇒ (nat × k × f × nat × nat) option) ?ip) ⟹ ?rt1.0 ⊑⇘?ip⇙ ?rt2.0›*))\n (*goal: ‹⟦(dip::nat) ∈ dom (dests::nat ⇒ nat option); dip ∈ vD (rt::nat ⇒ (nat × k × f × nat × nat) option); (case rt dip of None ⇒ 0::nat | Some (r::nat × k × f × nat × nat) ⇒ π₂ r) < the (dests dip)⟧ ⟹ rt ⊑⇘dip⇙ (λip::nat. case (rt ip, dests ip) of (None, x::nat option) ⇒ None | (Some (a::nat, ab::k, ad::f, af::nat, bb::nat), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a::nat, ab::k, ad::f, af::nat, bb::nat), Some (rsn::nat)) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))›*)\n by (auto simp: fresher_def (*‹(?r::nat × k × f × nat × nat) ⊑ (?r'::nat × k × f × nat × nat) ≡ nsqn⇩r ?r < nsqn⇩r ?r' ∨ nsqn⇩r ?r = nsqn⇩r ?r' ∧ π₅ ?r' ≤ π₅ ?r›*))\n qed\n\nlemma nsqn⇩r_invalidate [simp]:\n assumes \"dip∈kD(rt)\"\n and \"dip∈dom(dests)\"\n shows \"nsqn⇩r (the (invalidate rt dests dip)) = the (dests dip) - 1\"\n using assms (*‹dip ∈ kD rt› ‹dip ∈ dom dests›*) unfolding invalidate_def\n (*goal: ‹nsqn⇩r (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) = the (dests dip) - 1›*)\n by auto\n\nlemma rt_fresh_as_inc_invalidate [simp]:\n assumes \"dip∈kD(rt)\"\n and \"∀rip∈dom(dests). rip∈vD(rt) ∧ the (dests rip) = inc (sqn rt rip)\"\n shows \"rt ≈⇘dip⇙ invalidate rt dests\"\n proof (cases \"dip∈dom(dests)\")\n (*goals:\n 1. ‹dip ∈ dom dests ⟹ rt ≈⇘dip⇙ invalidate rt dests›\n 2. ‹dip ∉ dom dests ⟹ rt ≈⇘dip⇙ invalidate rt dests›*)\n assume \"dip∉dom(dests)\" (*‹(dip::nat) ∉ dom (dests::nat ⇒ nat option)›*)\n with ‹dip∈kD(rt)› (*‹(dip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)›*) have \"dip∈kD(invalidate rt dests)\"\n by simp\n with ‹dip∈kD(rt)› (*‹dip ∈ kD rt›*) show \"?thesis\"\n (*goal: ‹rt ≈⇘dip⇙ invalidate rt dests›*)\n apply rule\n (*goals:\n 1. ‹the (rt dip) ⊑ the (invalidate rt dests dip)›\n 2. ‹the (invalidate rt dests dip) ⊑ the (rt dip)›\n discuss goal 1*)\n apply (simp add: ‹dip∉dom(dests)›)\n (*discuss goal 2*)\n apply (simp add: ‹dip∉dom(dests)›)\n (*proven 2 subgoals*) .\n next\n (*goal: ‹dip ∈ dom dests ⟹ rt ≈⇘dip⇙ invalidate rt dests›*)\n assume \"dip∈dom(dests)\" (*‹(dip::nat) ∈ dom (dests::nat ⇒ nat option)›*)\n with assms(2) (*‹∀rip∈dom dests. rip ∈ vD rt ∧ the (dests rip) = inc (sqn rt rip)›*) have \"dip∈vD(rt)\" and \"the (dests dip) = inc (sqn rt dip)\"\n apply -\n (*goals:\n 1. ‹⟦∀rip∈dom dests. rip ∈ vD rt ∧ the (dests rip) = inc (sqn rt rip); dip ∈ dom dests⟧ ⟹ dip ∈ vD rt›\n 2. ‹⟦∀rip∈dom dests. rip ∈ vD rt ∧ the (dests rip) = inc (sqn rt rip); dip ∈ dom dests⟧ ⟹ the (dests dip) = inc (sqn rt dip)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from ‹dip∈vD(rt)› (*‹dip ∈ vD rt›*) have \"dip∈kD(rt)\"\n by simp\n moreover then have \"dip∈kD(invalidate rt dests)\"\n by simp\n ultimately show \"?thesis\"\n (*goal: ‹(rt::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ invalidate rt (dests::nat ⇒ nat option)›*)\n proof (rule nsqn_rt_fresh_asI (*‹⟦?dip ∈ kD ?rt; ?dip ∈ kD ?rt'; nsqn ?rt ?dip = nsqn ?rt' ?dip; π₅ (the (?rt ?dip)) = π₅ (the (?rt' ?dip))⟧ ⟹ ?rt ≈⇘?dip⇙ ?rt'›*))\n (*goals:\n 1. ‹nsqn rt dip = nsqn (invalidate rt dests) dip›\n 2. ‹π₅ (the (rt dip)) = π₅ (the (invalidate rt dests dip))›*)\n from ‹dip∈vD(rt)› (*‹(dip::nat) ∈ vD (rt::nat ⇒ (nat × k × f × nat × nat) option)›*) have \"nsqn rt dip = sqn rt dip\"\n by simp\n also (*calculation: ‹nsqn rt dip = sqn rt dip›*) have \"sqn rt dip = nsqn⇩r (the (invalidate rt dests dip))\"\n proof (-)\n (*goal: ‹sqn rt dip = nsqn⇩r (the (invalidate rt dests dip))›*)\n from ‹dip∈kD(rt)› (*‹dip ∈ kD rt›*) have \"nsqn⇩r (the (invalidate rt dests dip)) = the (dests dip) - 1\"\n using ‹dip∈dom(dests)› (*‹(dip::nat) ∈ dom (dests::nat ⇒ nat option)›*) by (rule nsqn⇩r_invalidate (*‹⟦?dip ∈ kD ?rt; ?dip ∈ dom ?dests⟧ ⟹ nsqn⇩r (the (invalidate ?rt ?dests ?dip)) = the (?dests ?dip) - 1›*))\n with ‹the (dests dip) = inc (sqn rt dip)› (*‹the (dests dip) = inc (sqn rt dip)›*) show \"sqn rt dip = nsqn⇩r (the (invalidate rt dests dip))\"\n by simp\n qed\n also (*calculation: ‹nsqn rt dip = nsqn⇩r (the (invalidate rt dests dip))›*) from ‹dip∈kD(invalidate rt dests)› (*‹dip ∈ kD (invalidate rt dests)›*) have \"nsqn⇩r (the (invalidate rt dests dip)) = nsqn (invalidate rt dests) dip\"\n by (simp add: kD_nsqn (*‹?dip ∈ kD ?rt ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*))\n finally (*calculation: ‹nsqn (rt::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat) = nsqn (invalidate rt (dests::nat ⇒ nat option)) dip›*) show \"nsqn rt dip = nsqn (invalidate rt dests) dip\" .\n qed (simp)\n (*solved the remaining goal: ‹π₅ (the (rt dip)) = π₅ (the (invalidate rt dests dip))›*)\n qed\n\nlemmas rt_fresher_inc_invalidate [simp] = rt_fresh_as_inc_invalidate [THEN rt_fresh_asD1]\n\n\nsubsection \"Strictly comparing routing tables \"\n\ndefinition rt_strictly_fresher :: \"ip ⇒ rt ⇒ rt ⇒ bool\"\nwhere\n \"rt_strictly_fresher ≡ λdip rt1 rt2. (rt1 ⊑⇘dip⇙ rt2) ∧ ¬(rt1 ≈⇘dip⇙ rt2)\"\n\nabbreviation\n rt_strictly_fresher_syn :: \"rt ⇒ ip ⇒ rt ⇒ bool\" (\"(_/ ⊏⇘_⇙ _)\" [51, 999, 51] 50)\nwhere\n \"rt1 ⊏⇘i⇙ rt2 ≡ rt_strictly_fresher i rt1 rt2\"\n\nlemma rt_strictly_fresher_def'':\n \"rt1 ⊏⇘i⇙ rt2 = ((rt1 ⊑⇘i⇙ rt2) ∧ ¬(rt2 ⊑⇘i⇙ rt1))\"\n unfolding rt_strictly_fresher_def rt_fresh_as_def\n (*goal: ‹(rt1 ⊑⇘i⇙ rt2 ∧ ¬ (rt1 ⊑⇘i⇙ rt2 ∧ rt2 ⊑⇘i⇙ rt1)) = (rt1 ⊑⇘i⇙ rt2 ∧ ¬ rt2 ⊑⇘i⇙ rt1)›*)\n by auto\n\nlemma rt_strictly_fresherI' [intro]:\n assumes \"rt1 ⊑⇘i⇙ rt2\"\n and \"¬(rt2 ⊑⇘i⇙ rt1)\"\n shows \"rt1 ⊏⇘i⇙ rt2\"\n using assms (*‹rt1 ⊑⇘i⇙ rt2› ‹¬ rt2 ⊑⇘i⇙ rt1›*) unfolding rt_strictly_fresher_def''\n (*goal: ‹rt1 ⊑⇘i⇙ rt2 ∧ ¬ rt2 ⊑⇘i⇙ rt1›*)\n by simp\n\nlemma rt_strictly_fresherE' [elim]:\n assumes \"rt1 ⊏⇘i⇙ rt2\"\n and \"⟦ rt1 ⊑⇘i⇙ rt2; ¬(rt2 ⊑⇘i⇙ rt1) ⟧ ⟹ P rt1 rt2 i\"\n shows \"P rt1 rt2 i\"\n using assms (*‹rt1 ⊏⇘i⇙ rt2› ‹⟦rt1 ⊑⇘i⇙ rt2; ¬ rt2 ⊑⇘i⇙ rt1⟧ ⟹ P rt1 rt2 i›*) unfolding rt_strictly_fresher_def''\n (*goal: ‹P rt1 rt2 i›*)\n by simp\n\nlemma rt_strictly_fresherI [intro]:\n assumes \"rt1 ⊑⇘i⇙ rt2\"\n and \"¬(rt1 ≈⇘i⇙ rt2)\"\n shows \"rt1 ⊏⇘i⇙ rt2\"\n unfolding rt_strictly_fresher_def\n (*goal: ‹rt1 ⊑⇘i⇙ rt2 ∧ ¬ rt1 ≈⇘i⇙ rt2›*)\n using assms (*‹rt1 ⊑⇘i⇙ rt2› ‹¬ rt1 ≈⇘i⇙ rt2›*) by standard\n\nlemmas rt_strictly_fresher_singleI [elim] = rt_strictly_fresherI [OF single_rt_fresher]\n\nlemma rt_strictly_fresherE [elim]:\n assumes \"rt1 ⊏⇘i⇙ rt2\"\n and \"⟦ rt1 ⊑⇘i⇙ rt2; ¬(rt1 ≈⇘i⇙ rt2) ⟧ ⟹ P rt1 rt2 i\"\n shows \"P rt1 rt2 i\"\n using assms(1) (*‹rt1 ⊏⇘i⇙ rt2›*) unfolding rt_strictly_fresher_def\n (*goal: ‹P rt1 rt2 i›*)\n apply rule\n (*goal: ‹P rt1 rt2 i›*)\n by (erule(1) assms( (*‹⟦rt1 ⊑⇘i⇙ rt2; ¬ rt1 ≈⇘i⇙ rt2⟧ ⟹ P rt1 rt2 i›*) 2))\n\nlemma rt_strictly_fresher_def':\n \"rt1 ⊏⇘i⇙ rt2 =\n (nsqn⇩r (the (rt1 i)) < nsqn⇩r (the (rt2 i))\n ∨ (nsqn⇩r (the (rt1 i)) = nsqn⇩r (the (rt2 i)) ∧ π₅(the (rt1 i)) > π₅(the (rt2 i))))\"\n unfolding rt_strictly_fresher_def'' rt_fresher_def fresher_def\n (*goal: ‹((nsqn⇩r (the (rt1 i)) < nsqn⇩r (the (rt2 i)) ∨ nsqn⇩r (the (rt1 i)) = nsqn⇩r (the (rt2 i)) ∧ π₅ (the (rt2 i)) ≤ π₅ (the (rt1 i))) ∧ ¬ (nsqn⇩r (the (rt2 i)) < nsqn⇩r (the (rt1 i)) ∨ nsqn⇩r (the (rt2 i)) = nsqn⇩r (the (rt1 i)) ∧ π₅ (the (rt1 i)) ≤ π₅ (the (rt2 i)))) = (nsqn⇩r (the (rt1 i)) < nsqn⇩r (the (rt2 i)) ∨ nsqn⇩r (the (rt1 i)) = nsqn⇩r (the (rt2 i)) ∧ π₅ (the (rt2 i)) < π₅ (the (rt1 i)))›*)\n by auto\n\nlemma rt_strictly_fresher_fresherD [dest]:\n assumes \"rt1 ⊏⇘dip⇙ rt2\"\n shows \"the (rt1 dip) ⊑ the (rt2 dip)\"\n using assms (*‹rt1 ⊏⇘dip⇙ rt2›*) unfolding rt_strictly_fresher_def rt_fresher_def\n (*goal: ‹the (rt1 dip) ⊑ the (rt2 dip)›*)\n by auto\n\nlemma rt_strictly_fresher_not_fresh_asD [dest]:\n assumes \"rt1 ⊏⇘dip⇙ rt2\"\n shows \"¬ rt1 ≈⇘dip⇙ rt2\"\n using assms (*‹rt1 ⊏⇘dip⇙ rt2›*) unfolding rt_strictly_fresher_def\n (*goal: ‹¬ rt1 ≈⇘dip⇙ rt2›*)\n by auto\n\nlemma rt_strictly_fresher_trans [elim, trans]:\n assumes \"rt1 ⊏⇘dip⇙ rt2\"\n and \"rt2 ⊏⇘dip⇙ rt3\"\n shows \"rt1 ⊏⇘dip⇙ rt3\"\n using assms (*‹rt1 ⊏⇘dip⇙ rt2› ‹rt2 ⊏⇘dip⇙ rt3›*) proof (-)\n (*goal: ‹⟦rt1 ⊏⇘dip⇙ rt2; rt2 ⊏⇘dip⇙ rt3⟧ ⟹ rt1 ⊏⇘dip⇙ rt3›*)\n from ‹rt1 ⊏⇘dip⇙ rt2› (*‹rt1 ⊏⇘dip⇙ rt2›*) obtain \"the (rt1 dip) ⊑ the (rt2 dip)\"\n (*goal: ‹(the (rt1 dip) ⊑ the (rt2 dip) ⟹ thesis) ⟹ thesis›*)\n by auto\n also (*calculation: ‹the (rt1 dip) ⊑ the (rt2 dip)›*) from ‹rt2 ⊏⇘dip⇙ rt3› (*‹rt2 ⊏⇘dip⇙ rt3›*) obtain \"the (rt2 dip) ⊑ the (rt3 dip)\"\n (*goal: ‹(the (rt2 dip) ⊑ the (rt3 dip) ⟹ thesis) ⟹ thesis›*)\n by auto\n finally (*calculation: ‹the (rt1 dip) ⊑ the (rt3 dip)›*) have \"the (rt1 dip) ⊑ the (rt3 dip)\" .\n moreover have \"¬ (rt1 ≈⇘dip⇙ rt3)\"\n proof (-)\n (*goal: ‹¬ rt1 ≈⇘dip⇙ rt3›*)\n from ‹rt1 ⊏⇘dip⇙ rt2› (*‹rt1 ⊏⇘dip⇙ rt2›*) obtain \"¬(the (rt2 dip) ⊑ the (rt1 dip))\"\n (*goal: ‹(¬ the (rt2 dip) ⊑ the (rt1 dip) ⟹ thesis) ⟹ thesis›*)\n by auto\n also (*calculation: ‹¬ the (rt2 dip) ⊑ the (rt1 dip)›*) from ‹rt2 ⊏⇘dip⇙ rt3› (*‹rt2 ⊏⇘dip⇙ rt3›*) obtain \"¬(the (rt3 dip) ⊑ the (rt2 dip))\"\n (*goal: ‹(¬ the (rt3 dip) ⊑ the (rt2 dip) ⟹ thesis) ⟹ thesis›*)\n by auto\n finally (*calculation: ‹¬ the (rt3 dip) ⊑ the (rt1 dip)›*) have \"¬(the (rt3 dip) ⊑ the (rt1 dip))\" .\n thus \"?thesis\"\n (*goal: ‹¬ rt1 ≈⇘dip⇙ rt3›*)\n apply -\n (*goal: ‹¬ rt1 ≈⇘dip⇙ rt3›*)\n apply standard\n (*goal: ‹¬ the ((rt3::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat)) ⊑ the ((rt1::nat ⇒ (nat × k × f × nat × nat) option) dip) ⟹ ¬ rt1 ≈⇘dip⇙ rt3›*)\n by (msorry)\n qed\n ultimately show \"rt1 ⊏⇘dip⇙ rt3\"\n by standard\n qed\n\nlemma rt_strictly_fresher_irefl [simp]: \"¬ (rt ⊏⇘dip⇙ rt)\"\n unfolding rt_strictly_fresher_def\n (*goal: ‹¬ (rt ⊑⇘dip⇙ rt ∧ ¬ rt ≈⇘dip⇙ rt)›*)\n by clarsimp\n\nlemma rt_fresher_trans_rt_strictly_fresher [elim, trans]:\n assumes \"rt1 ⊏⇘dip⇙ rt2\"\n and \"rt2 ⊑⇘dip⇙ rt3\"\n shows \"rt1 ⊏⇘dip⇙ rt3\"\n proof (-)\n (*goal: ‹rt1 ⊏⇘dip⇙ rt3›*)\n from ‹rt1 ⊏⇘dip⇙ rt2› (*‹rt1 ⊏⇘dip⇙ rt2›*) have \"rt1 ⊑⇘dip⇙ rt2\" and \"¬(rt2 ⊑⇘dip⇙ rt1)\"\n unfolding rt_strictly_fresher_def''\n (*goals:\n 1. ‹rt1 ⊑⇘dip⇙ rt2›\n 2. ‹¬ rt2 ⊑⇘dip⇙ rt1›*)\n apply -\n (*goals:\n 1. ‹rt1 ⊑⇘dip⇙ rt2 ∧ ¬ rt2 ⊑⇘dip⇙ rt1 ⟹ rt1 ⊑⇘dip⇙ rt2›\n 2. ‹rt1 ⊑⇘dip⇙ rt2 ∧ ¬ rt2 ⊑⇘dip⇙ rt1 ⟹ ¬ rt2 ⊑⇘dip⇙ rt1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from this(1) (*‹rt1 ⊑⇘dip⇙ rt2›*) ‹rt2 ⊑⇘dip⇙ rt3› (*‹rt2 ⊑⇘dip⇙ rt3›*) have \"rt1 ⊑⇘dip⇙ rt3\"\n by standard\n moreover from ‹¬(rt2 ⊑⇘dip⇙ rt1)› (*‹¬ (rt2::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(dip::nat)⇙ (rt1::nat ⇒ (nat × k × f × nat × nat) option)›*) have \"¬(rt3 ⊑⇘dip⇙ rt1)\"\n proof (rule contrapos_nn (*‹⟦¬ ?Q; ?P ⟹ ?Q⟧ ⟹ ¬ ?P›*))\n (*goal: ‹rt3 ⊑⇘dip⇙ rt1 ⟹ rt2 ⊑⇘dip⇙ rt1›*)\n assume \"rt3 ⊑⇘dip⇙ rt1\" (*‹(rt3::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(dip::nat)⇙ (rt1::nat ⇒ (nat × k × f × nat × nat) option)›*)\n with ‹rt2 ⊑⇘dip⇙ rt3› (*‹rt2 ⊑⇘dip⇙ rt3›*) show \"rt2 ⊑⇘dip⇙ rt1\"\n by standard\n qed\n ultimately show \"rt1 ⊏⇘dip⇙ rt3\"\n unfolding rt_strictly_fresher_def''\n (*goal: ‹rt1 ⊑⇘dip⇙ rt3 ∧ ¬ rt3 ⊑⇘dip⇙ rt1›*)\n by auto\n qed\n\nlemma rt_fresher_trans_rt_strictly_fresher' [elim, trans]:\n assumes \"rt1 ⊑⇘dip⇙ rt2\"\n and \"rt2 ⊏⇘dip⇙ rt3\"\n shows \"rt1 ⊏⇘dip⇙ rt3\"\n proof (-)\n (*goal: ‹rt1 ⊏⇘dip⇙ rt3›*)\n from ‹rt2 ⊏⇘dip⇙ rt3› (*‹rt2 ⊏⇘dip⇙ rt3›*) have \"rt2 ⊑⇘dip⇙ rt3\" and \"¬(rt3 ⊑⇘dip⇙ rt2)\"\n unfolding rt_strictly_fresher_def''\n (*goals:\n 1. ‹rt2 ⊑⇘dip⇙ rt3›\n 2. ‹¬ rt3 ⊑⇘dip⇙ rt2›*)\n apply -\n (*goals:\n 1. ‹rt2 ⊑⇘dip⇙ rt3 ∧ ¬ rt3 ⊑⇘dip⇙ rt2 ⟹ rt2 ⊑⇘dip⇙ rt3›\n 2. ‹rt2 ⊑⇘dip⇙ rt3 ∧ ¬ rt3 ⊑⇘dip⇙ rt2 ⟹ ¬ rt3 ⊑⇘dip⇙ rt2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from ‹rt1 ⊑⇘dip⇙ rt2› (*‹rt1 ⊑⇘dip⇙ rt2›*) this(1) (*‹rt2 ⊑⇘dip⇙ rt3›*) have \"rt1 ⊑⇘dip⇙ rt3\"\n by standard\n moreover from ‹¬(rt3 ⊑⇘dip⇙ rt2)› (*‹¬ (rt3::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)›*) have \"¬(rt3 ⊑⇘dip⇙ rt1)\"\n proof (rule contrapos_nn (*‹⟦¬ (?Q::bool); ?P::bool ⟹ ?Q⟧ ⟹ ¬ ?P›*))\n (*goal: ‹rt3 ⊑⇘dip⇙ rt1 ⟹ rt3 ⊑⇘dip⇙ rt2›*)\n assume \"rt3 ⊑⇘dip⇙ rt1\" (*‹(rt3::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(dip::nat)⇙ (rt1::nat ⇒ (nat × k × f × nat × nat) option)›*)\n thus \"rt3 ⊑⇘dip⇙ rt2\"\n using ‹rt1 ⊑⇘dip⇙ rt2› (*‹rt1 ⊑⇘dip⇙ rt2›*) by standard\n qed\n ultimately show \"rt1 ⊏⇘dip⇙ rt3\"\n unfolding rt_strictly_fresher_def''\n (*goal: ‹rt1 ⊑⇘dip⇙ rt3 ∧ ¬ rt3 ⊑⇘dip⇙ rt1›*)\n by auto\n qed\n\nlemma rt_fresher_imp_nsqn_le:\n assumes \"rt1 ⊑⇘ip⇙ rt2\"\n and \"ip ∈ kD rt1\"\n and \"ip ∈ kD rt2\"\n shows \"nsqn rt1 ip ≤ nsqn rt2 ip\"\n using assms(1) (*‹rt1 ⊑⇘ip⇙ rt2›*) by (auto simp add: rt_fresher_def2 [OF assms(2-3)] (*‹((rt1::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(ip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)) = (nsqn rt1 ip < nsqn rt2 ip ∨ nsqn rt1 ip = nsqn rt2 ip ∧ the (dhops rt2 ip) ≤ the (dhops rt1 ip))›*))\n\nlemma rt_strictly_fresher_ltI [intro]:\n assumes \"dip ∈ kD(rt1)\"\n and \"dip ∈ kD(rt2)\"\n and \"nsqn rt1 dip < nsqn rt2 dip\"\n shows \"rt1 ⊏⇘dip⇙ rt2\"\n proof (standard)\n (*goals:\n 1. ‹rt1 ⊑⇘dip⇙ rt2›\n 2. ‹¬ rt1 ≈⇘dip⇙ rt2›*)\n from assms (*‹dip ∈ kD rt1› ‹dip ∈ kD rt2› ‹nsqn rt1 dip < nsqn rt2 dip›*) show \"rt1 ⊑⇘dip⇙ rt2\"\n by standard\n next\n (*goal: ‹¬ rt1 ≈⇘dip⇙ rt2›*)\n show \"¬(rt1 ≈⇘dip⇙ rt2)\"\n proof (standard)\n (*goal: ‹rt1 ≈⇘dip⇙ rt2 ⟹ False›*)\n assume \"rt1 ≈⇘dip⇙ rt2\" (*‹(rt1::nat ⇒ (nat × k × f × nat × nat) option) ≈⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)›*)\n hence \"rt2 ⊑⇘dip⇙ rt1\"\n by standard\n hence \"nsqn rt2 dip ≤ nsqn rt1 dip\"\n using ‹dip ∈ kD(rt2)› (*‹dip ∈ kD rt2›*) ‹dip ∈ kD(rt1)› (*‹dip ∈ kD rt1›*) by (rule rt_fresher_imp_nsqn_le (*‹⟦(?rt1.0::nat ⇒ (nat × k × f × nat × nat) option) ⊑⇘(?ip::nat)⇙ (?rt2.0::nat ⇒ (nat × k × f × nat × nat) option); ?ip ∈ kD ?rt1.0; ?ip ∈ kD ?rt2.0⟧ ⟹ nsqn ?rt1.0 ?ip ≤ nsqn ?rt2.0 ?ip›*))\n with ‹nsqn rt1 dip < nsqn rt2 dip› (*‹nsqn rt1 dip < nsqn rt2 dip›*) show False\n by simp\n qed\n qed\n\nlemma rt_strictly_fresher_eqI [intro]:\n assumes \"i∈kD(rt1)\"\n and \"i∈kD(rt2)\"\n and \"nsqn rt1 i = nsqn rt2 i\"\n and \"π₅(the (rt2 i)) < π₅(the (rt1 i))\"\n shows \"rt1 ⊏⇘i⇙ rt2\"\n using assms (*‹i ∈ kD rt1› ‹i ∈ kD rt2› ‹nsqn (rt1::nat ⇒ (nat × k × f × nat × nat) option) (i::nat) = nsqn (rt2::nat ⇒ (nat × k × f × nat × nat) option) i› ‹π₅ (the (rt2 i)) < π₅ (the (rt1 i))›*) unfolding rt_strictly_fresher_def'\n (*goal: ‹nsqn⇩r (the ((rt1::nat ⇒ (nat × k × f × nat × nat) option) (i::nat))) < nsqn⇩r (the ((rt2::nat ⇒ (nat × k × f × nat × nat) option) i)) ∨ nsqn⇩r (the (rt1 i)) = nsqn⇩r (the (rt2 i)) ∧ π₅ (the (rt2 i)) < π₅ (the (rt1 i))›*)\n by (auto simp add: kD_nsqn (*‹?dip ∈ kD ?rt ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*))\n\nlemma invalidate_rtsf_left [simp]:\n \"⋀dests dip rt rt'. dests dip = None ⟹ (invalidate rt dests ⊏⇘dip⇙ rt') = (rt ⊏⇘dip⇙ rt')\"\n unfolding invalidate_def rt_strictly_fresher_def'\n (*goal: ‹⋀dests dip rt rt'. dests dip = None ⟹ (nsqn⇩r (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb)))) = (nsqn⇩r (the (rt dip)) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (rt dip)) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (rt dip)))›*)\n apply (rule iffI (*‹⟦?P::bool ⟹ ?Q::bool; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹⋀dests dip rt rt'. ⟦dests dip = None; nsqn⇩r (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb)))⟧ ⟹ nsqn⇩r (the (rt dip)) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (rt dip)) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (rt dip))›\n 2. ‹⋀dests dip rt rt'. ⟦dests dip = None; nsqn⇩r (the (rt dip)) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (rt dip)) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (rt dip))⟧ ⟹ nsqn⇩r (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (case (rt dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb)))›\n discuss goal 1*)\n apply ((auto split: option.split_asm (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: option.split_asm (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n (*proven 2 subgoals*) .\n\nlemma vD_invalidate_rt_strictly_fresher [simp]:\n assumes \"dip ∈ vD(invalidate rt1 dests)\"\n shows \"(invalidate rt1 dests ⊏⇘dip⇙ rt2) = (rt1 ⊏⇘dip⇙ rt2)\"\n proof (cases \"dip ∈ dom(dests)\")\n (*goals:\n 1. ‹(dip::nat) ∈ dom (dests::nat ⇒ nat option) ⟹ (invalidate (rt1::nat ⇒ (nat × k × f × nat × nat) option) dests ⊏⇘dip⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)) = (rt1 ⊏⇘dip⇙ rt2)›\n 2. ‹(dip::nat) ∉ dom (dests::nat ⇒ nat option) ⟹ (invalidate (rt1::nat ⇒ (nat × k × f × nat × nat) option) dests ⊏⇘dip⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)) = (rt1 ⊏⇘dip⇙ rt2)›*)\n assume \"dip ∈ dom(dests)\" (*‹(dip::nat) ∈ dom (dests::nat ⇒ nat option)›*)\n hence \"dip ∉ vD(invalidate rt1 dests)\"\n unfolding invalidate_def vD_def\n (*goal: ‹dip ∉ {dip. map_option π₄ (case (rt1 dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb)) = Some val}›*)\n apply clarsimp\n (*goal: ‹dip ∉ {dip. map_option π₄ (case (rt1 dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb)) = Some val}›*)\n by (metis assms (*‹(dip::nat) ∈ vD (invalidate (rt1::nat ⇒ (nat × k × f × nat × nat) option) (dests::nat ⇒ nat option))›*) option.simps( (*‹Some (?x2.0::?'a) ≠ None›*) 3) vD_invalidate_vD_not_dests (*‹(?dip::nat) ∈ vD (invalidate (?rt::nat ⇒ (nat × k × f × nat × nat) option) (?dests::nat ⇒ nat option)) ⟹ ?dip ∈ vD ?rt ∧ ?dests ?dip = None›*))\n with ‹dip ∈ vD(invalidate rt1 dests)› (*‹(dip::nat) ∈ vD (invalidate (rt1::nat ⇒ (nat × k × f × nat × nat) option) (dests::nat ⇒ nat option))›*) show \"?thesis\"\n (*goal: ‹(invalidate (rt1::nat ⇒ (nat × k × f × nat × nat) option) (dests::nat ⇒ nat option) ⊏⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)) = (rt1 ⊏⇘dip⇙ rt2)›*)\n by simp\n next\n (*goal: ‹dip ∉ dom dests ⟹ (invalidate rt1 dests ⊏⇘dip⇙ rt2) = (rt1 ⊏⇘dip⇙ rt2)›*)\n assume \"dip ∉ dom(dests)\" (*‹(dip::nat) ∉ dom (dests::nat ⇒ nat option)›*)\n hence \"dests dip = None\"\n by auto\n moreover with ‹dip ∈ vD(invalidate rt1 dests)› (*‹dip ∈ vD (invalidate rt1 dests)›*) have \"dip ∈ vD(rt1)\"\n unfolding invalidate_def vD_def\n (*goal: ‹dip ∈ {dip. flag rt1 dip = Some val}›*)\n apply clarsimp\n (*goal: ‹dip ∈ {dip. flag rt1 dip = Some val}›*)\n by (metis (opaque_lifting, no_types) assms (*‹dip ∈ vD (invalidate rt1 dests)›*) vD_Some (*‹?dip ∈ vD ?rt ⟹ ∃dsn dsk hops nhip. ?rt ?dip = Some (dsn, dsk, val, hops, nhip)›*) vD_invalidate_vD_not_dests (*‹?dip ∈ vD (invalidate ?rt ?dests) ⟹ ?dip ∈ vD ?rt ∧ ?dests ?dip = None›*))\n ultimately show \"?thesis\"\n (*goal: ‹(invalidate (rt1::nat ⇒ (nat × k × f × nat × nat) option) (dests::nat ⇒ nat option) ⊏⇘(dip::nat)⇙ (rt2::nat ⇒ (nat × k × f × nat × nat) option)) = (rt1 ⊏⇘dip⇙ rt2)›*)\n unfolding invalidate_def rt_strictly_fresher_def'\n (*goal: ‹(nsqn⇩r (the (case (rt1 dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) < nsqn⇩r (the (rt2 dip)) ∨ nsqn⇩r (the (case (rt1 dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb))) = nsqn⇩r (the (rt2 dip)) ∧ π₅ (the (rt2 dip)) < π₅ (the (case (rt1 dip, dests dip) of (None, x) ⇒ None | (Some (a, ab, ad, af, bb), None) ⇒ Some (a, ab, ad, af, bb) | (Some (a, ab, ad, af, bb), Some rsn) ⇒ Some (rsn, ab, Aodv_Basic.inv, af, bb)))) = (nsqn⇩r (the (rt1 dip)) < nsqn⇩r (the (rt2 dip)) ∨ nsqn⇩r (the (rt1 dip)) = nsqn⇩r (the (rt2 dip)) ∧ π₅ (the (rt2 dip)) < π₅ (the (rt1 dip)))›*)\n by auto\n qed\n\nlemma rt_strictly_fresher_update_other [elim!]:\n \"⋀dip ip rt r rt'. ⟦ dip ≠ ip; rt ⊏⇘dip⇙ rt' ⟧ ⟹ update rt ip r ⊏⇘dip⇙ rt'\"\n unfolding rt_strictly_fresher_def'\n (*goal: ‹⋀dip ip rt r rt'. ⟦dip ≠ ip; nsqn⇩r (the (rt dip)) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (rt dip)) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (rt dip))⟧ ⟹ nsqn⇩r (the (update rt ip r dip)) < nsqn⇩r (the (rt' dip)) ∨ nsqn⇩r (the (update rt ip r dip)) = nsqn⇩r (the (rt' dip)) ∧ π₅ (the (rt' dip)) < π₅ (the (update rt ip r dip))›*)\n by clarsimp\n\nlemma lt_sqn_imp_update_strictly_fresher:\n assumes \"dip ∈ vD (rt2 nhip)\"\n and *: \"osn < sqn (rt2 nhip) dip\"\n and **: \"rt ≠ update rt dip (osn, kno, val, hops, nhip)\"\n shows \"update rt dip (osn, kno, val, hops, nhip) ⊏⇘dip⇙ rt2 nhip\"\n unfolding rt_strictly_fresher_def'\n (*goal: ‹nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip)) < nsqn⇩r (the (rt2 nhip dip)) ∨ nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip)) = nsqn⇩r (the (rt2 nhip dip)) ∧ π₅ (the (rt2 nhip dip)) < π₅ (the (update rt dip (osn, kno, val, hops, nhip) dip))›*)\n proof (rule disjI1 (*‹?P ⟹ ?P ∨ ?Q›*))\n (*goal: ‹nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip)) < nsqn⇩r (the (rt2 nhip dip))›*)\n from \"**\" (*‹rt ≠ update rt dip (osn, kno, val, hops, nhip)›*) have \"nsqn (update rt dip (osn, kno, val, hops, nhip)) dip = osn\"\n by (rule nsqn_update_changed_kno_val (*‹?rt ≠ update ?rt ?ip (?dsn, kno, val, ?hops, ?nhip) ⟹ nsqn (update ?rt ?ip (?dsn, kno, val, ?hops, ?nhip)) ?ip = ?dsn›*))\n with ‹dip∈vD(rt2 nhip)› (*‹(dip::nat) ∈ vD ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) (nhip::nat))›*) have \"nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip)) = osn\"\n by (simp add: kD_nsqn (*‹?dip ∈ kD ?rt ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*))\n also (*calculation: ‹nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip)) = osn›*) have \"osn < sqn (rt2 nhip) dip\"\n by (rule * (*‹osn < sqn (rt2 nhip) dip›*))\n also (*calculation: ‹nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip)) < sqn (rt2 nhip) dip›*) have \"sqn (rt2 nhip) dip = nsqn⇩r (the (rt2 nhip dip))\"\n unfolding \"nsqn⇩r_def\"\n (*goal: ‹sqn (rt2 nhip) dip = (if π₄ (the (rt2 nhip dip)) = val ∨ π₂ (the (rt2 nhip dip)) = 0 then π₂ (the (rt2 nhip dip)) else π₂ (the (rt2 nhip dip)) - 1)›*)\n using ‹dip ∈ vD (rt2 nhip)› (*‹dip ∈ vD (rt2 nhip)›*) apply -\n (*goal: ‹sqn (rt2 nhip) dip = (if π₄ (the (rt2 nhip dip)) = val ∨ π₂ (the (rt2 nhip dip)) = 0 then π₂ (the (rt2 nhip dip)) else π₂ (the (rt2 nhip dip)) - 1)›*)\n by (metis vD_flag_val (*‹?dip ∈ vD ?rt ⟹ the (flag ?rt ?dip) = val›*) proj2_eq_sqn (*‹?dip ∈ kD ?rt ⟹ π₂ (the (?rt ?dip)) = sqn ?rt ?dip›*) proj4_eq_flag (*‹?dip ∈ kD ?rt ⟹ π₄ (the (?rt ?dip)) = the (flag ?rt ?dip)›*) vD_iD_gives_kD( (*‹?ip ∈ vD ?rt ⟹ ?ip ∈ kD ?rt›*) 1))\n finally (*calculation: ‹nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip)) < nsqn⇩r (the (rt2 nhip dip))›*) show \"nsqn⇩r (the (update rt dip (osn, kno, val, hops, nhip) dip))\n < nsqn⇩r (the (rt2 nhip dip))\" .\n qed\n\nlemma dhops_le_hops_imp_update_strictly_fresher:\n assumes \"dip ∈ vD(rt2 nhip)\"\n and sqn: \"sqn (rt2 nhip) dip = osn\"\n and hop: \"the (dhops (rt2 nhip) dip) ≤ hops\"\n and **: \"rt ≠ update rt dip (osn, kno, val, Suc hops, nhip)\"\n shows \"update rt dip (osn, kno, val, Suc hops, nhip) ⊏⇘dip⇙ rt2 nhip\"\n unfolding rt_strictly_fresher_def'\n (*goal: ‹nsqn⇩r (the (update rt dip (osn, kno, val, Suc hops, nhip) dip)) < nsqn⇩r (the (rt2 nhip dip)) ∨ nsqn⇩r (the (update rt dip (osn, kno, val, Suc hops, nhip) dip)) = nsqn⇩r (the (rt2 nhip dip)) ∧ π₅ (the (rt2 nhip dip)) < π₅ (the (update rt dip (osn, kno, val, Suc hops, nhip) dip))›*)\n proof (rule disjI2 (*‹?Q::bool ⟹ (?P::bool) ∨ ?Q›*), rule conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goals:\n 1. ‹nsqn⇩r (the (update (rt::nat ⇒ (nat × k × f × nat × nat) option) (dip::nat) (osn::nat, kno, val, Suc (hops::nat), nhip::nat) dip)) = nsqn⇩r (the ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) nhip dip))›\n 2. ‹π₅ (the ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) (nhip::nat) (dip::nat))) < π₅ (the (update (rt::nat ⇒ (nat × k × f × nat × nat) option) dip (osn::nat, kno, val, Suc (hops::nat), nhip) dip))›*)\n from \"**\" (*‹(rt::nat ⇒ (nat × k × f × nat × nat) option) ≠ update rt (dip::nat) (osn::nat, kno, val, Suc (hops::nat), nhip::nat)›*) have \"nsqn (update rt dip (osn, kno, val, Suc hops, nhip)) dip = osn\"\n by (rule nsqn_update_changed_kno_val (*‹?rt ≠ update ?rt ?ip (?dsn, kno, val, ?hops, ?nhip) ⟹ nsqn (update ?rt ?ip (?dsn, kno, val, ?hops, ?nhip)) ?ip = ?dsn›*))\n with ‹dip∈vD(rt2 nhip)› (*‹dip ∈ vD (rt2 nhip)›*) have \"nsqn⇩r (the (update rt dip (osn, kno, val, Suc hops, nhip) dip)) = osn\"\n by (simp add: kD_nsqn (*‹?dip ∈ kD ?rt ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*))\n also (*calculation: ‹nsqn⇩r (the (update rt dip (osn, kno, val, Suc hops, nhip) dip)) = osn›*) have \"osn = sqn (rt2 nhip) dip\"\n by (rule sqn [symmetric] (*‹osn = sqn (rt2 nhip) dip›*))\n also (*calculation: ‹nsqn⇩r (the (update rt dip (osn, kno, val, Suc hops, nhip) dip)) = sqn (rt2 nhip) dip›*) have \"sqn (rt2 nhip) dip = nsqn⇩r (the (rt2 nhip dip))\"\n unfolding \"nsqn⇩r_def\"\n (*goal: ‹sqn ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) (nhip::nat)) (dip::nat) = (if π₄ (the (rt2 nhip dip)) = val ∨ π₂ (the (rt2 nhip dip)) = (0::nat) then π₂ (the (rt2 nhip dip)) else π₂ (the (rt2 nhip dip)) - (1::nat))›*)\n using ‹dip ∈ vD(rt2 nhip)› (*‹dip ∈ vD (rt2 nhip)›*) apply -\n (*goal: ‹sqn ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) (nhip::nat)) (dip::nat) = (if π₄ (the (rt2 nhip dip)) = val ∨ π₂ (the (rt2 nhip dip)) = (0::nat) then π₂ (the (rt2 nhip dip)) else π₂ (the (rt2 nhip dip)) - (1::nat))›*)\n by (metis vD_flag_val (*‹(?dip::nat) ∈ vD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ the (flag ?rt ?dip) = val›*) proj2_eq_sqn (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ π₂ (the (?rt ?dip)) = sqn ?rt ?dip›*) proj4_eq_flag (*‹(?dip::nat) ∈ kD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ π₄ (the (?rt ?dip)) = the (flag ?rt ?dip)›*) vD_iD_gives_kD( (*‹(?ip::nat) ∈ vD (?rt::nat ⇒ (nat × k × f × nat × nat) option) ⟹ ?ip ∈ kD ?rt›*) 1))\n finally (*calculation: ‹nsqn⇩r (the (update rt dip (osn, kno, val, Suc hops, nhip) dip)) = nsqn⇩r (the (rt2 nhip dip))›*) show \"nsqn⇩r (the (update rt dip (osn, kno, val, Suc hops, nhip) dip))\n = nsqn⇩r (the (rt2 nhip dip))\" .\n next\n (*goal: ‹π₅ (the ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) (nhip::nat) (dip::nat))) < π₅ (the (update (rt::nat ⇒ (nat × k × f × nat × nat) option) dip (osn::nat, kno, val, Suc (hops::nat), nhip) dip))›*)\n have \"the (dhops (rt2 nhip) dip) ≤ hops\"\n by (rule hop (*‹the (dhops (rt2 nhip) dip) ≤ hops›*))\n also (*calculation: ‹the (dhops ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) (nhip::nat)) (dip::nat)) ≤ (hops::nat)›*) have \"hops < hops + 1\"\n by simp\n also (*calculation: ‹the (dhops (rt2 nhip) dip) < hops + 1›*) have \"hops + 1 = the (dhops (update rt dip (osn, kno, val, Suc hops, nhip)) dip)\"\n using \"**\" (*‹rt ≠ update rt dip (osn, kno, val, Suc hops, nhip)›*) by simp\n finally (*calculation: ‹the (dhops (rt2 nhip) dip) < the (dhops (update rt dip (osn, kno, val, Suc hops, nhip)) dip)›*) have \"the (dhops (rt2 nhip) dip)\n < the (dhops (update rt dip (osn, kno, val, Suc hops, nhip)) dip)\" .\n thus \"π₅ (the (rt2 nhip dip)) < π₅ (the (update rt dip (osn, kno, val, Suc hops, nhip) dip))\"\n using ‹dip ∈ vD(rt2 nhip)› (*‹(dip::nat) ∈ vD ((rt2::nat ⇒ nat ⇒ (nat × k × f × nat × nat) option) (nhip::nat))›*) by (simp add: proj5_eq_dhops (*‹?dip ∈ kD ?rt ⟹ π₅ (the (?rt ?dip)) = the (dhops ?rt ?dip)›*))\n qed\n\nlemma nsqn_invalidate:\n assumes \"dip ∈ kD(rt)\"\n and \"∀ip∈dom(dests). ip ∈ vD(rt) ∧ the (dests ip) = inc (sqn rt ip)\"\n shows \"nsqn (invalidate rt dests) dip = nsqn rt dip\"\n proof (-)\n (*goal: ‹nsqn (invalidate rt dests) dip = nsqn rt dip›*)\n from ‹dip ∈ kD(rt)› (*‹(dip::nat) ∈ kD (rt::nat ⇒ (nat × k × f × nat × nat) option)›*) have \"dip ∈ kD(invalidate rt dests)\"\n by simp\n from assms (*‹dip ∈ kD rt› ‹∀ip∈dom dests. ip ∈ vD rt ∧ the (dests ip) = inc (sqn rt ip)›*) have \"rt ≈⇘dip⇙ invalidate rt dests\"\n by (rule rt_fresh_as_inc_invalidate (*‹⟦?dip ∈ kD ?rt; ∀rip∈dom ?dests. rip ∈ vD ?rt ∧ the (?dests rip) = inc (sqn ?rt rip)⟧ ⟹ ?rt ≈⇘?dip⇙ invalidate ?rt ?dests›*))\n with ‹dip ∈ kD(rt)› (*‹dip ∈ kD rt›*) ‹dip ∈ kD(invalidate rt dests)› (*‹dip ∈ kD (invalidate rt dests)›*) show \"?thesis\"\n (*goal: ‹nsqn (invalidate rt dests) dip = nsqn rt dip›*)\n apply (simp add: kD_nsqn (*‹?dip ∈ kD ?rt ⟹ nsqn ?rt ?dip = nsqn⇩r (the (?rt ?dip))›*) del: invalidate_kD_inv (*‹kD (invalidate ?rt ?dests) = kD ?rt›*))\n (*goal: ‹nsqn (invalidate rt dests) dip = nsqn rt dip›*)\n by (erule(2) rt_fresh_as_nsqnr (*‹⟦?dip ∈ kD ?rt1.0; ?dip ∈ kD ?rt2.0; ?rt1.0 ≈⇘?dip⇙ ?rt2.0⟧ ⟹ nsqn⇩r (the (?rt2.0 ?dip)) = nsqn⇩r (the (?rt1.0 ?dip))›*))\n qed\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/AODV/variants/c_gtobcast/C_Fresher.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"f085a0e01f9ff9156e6505b35d720a9ed42661a632b5e138aca34964ed3bfb6f\"\n}"}}},{"rowIdx":193,"cells":{"text":{"kind":"string","value":"theory PathLemmas\nimports TheoremD14\nbegin\n\ncontext LocalLexing begin\n\nlemma characterize_𝒫:\n \"(∀ i < length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u) ⟹ admissible p ⟹\n ∃ u. p ∈ 𝒫 (charslength p) u\"\nproof (induct p rule: rev_induct (*‹⟦?P []; ⋀x xs. ?P xs ⟹ ?P (xs @ [x])⟧ ⟹ ?P ?xs›*))\n (*goals:\n 1. ‹⟦∀i<length []. ∃u. [] ! i ∈ 𝒵 (charslength (take i [])) u; admissible []⟧ ⟹ ∃u. [] ∈ 𝒫 (charslength []) u›\n 2. ‹⋀x xs. ⟦⟦∀i<length xs. ∃u. xs ! i ∈ 𝒵 (charslength (take i xs)) u; admissible xs⟧ ⟹ ∃u. xs ∈ 𝒫 (charslength xs) u; ∀i<length (xs @ [x]). ∃u. (xs @ [x]) ! i ∈ 𝒵 (charslength (take i (xs @ [x]))) u; admissible (xs @ [x])⟧ ⟹ ∃u. xs @ [x] ∈ 𝒫 (charslength (xs @ [x])) u›*)\n case Nil (*‹∀i<length []. ∃u. [] ! i ∈ 𝒵 (charslength (take i [])) u› ‹admissible []›*)\n show \"?case\"\n (*goal: ‹∃u. [] ∈ 𝒫 (charslength []) u›*)\n by simp\nnext\n (*goal: ‹⋀x xs. ⟦⟦∀i<length xs. ∃u. xs ! i ∈ 𝒵 (charslength (take i xs)) u; admissible xs⟧ ⟹ ∃u. xs ∈ 𝒫 (charslength xs) u; ∀i<length (xs @ [x]). ∃u. (xs @ [x]) ! i ∈ 𝒵 (charslength (take i (xs @ [x]))) u; admissible (xs @ [x])⟧ ⟹ ∃u. xs @ [x] ∈ 𝒫 (charslength (xs @ [x])) u›*)\n case (snoc a p) (*‹⟦∀i<length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u; admissible p⟧ ⟹ ∃u. p ∈ 𝒫 (charslength p) u› ‹∀i<length (p @ [a]). ∃u. (p @ [a]) ! i ∈ 𝒵 (charslength (take i (p @ [a]))) u› ‹admissible (p @ [a])›*)\n from snoc.prems (*‹∀i<length (p @ [a]). ∃u. (p @ [a]) ! i ∈ 𝒵 (charslength (take i (p @ [a]))) u› ‹admissible ((p::(symbol × character list) list) @ [a::symbol × character list])›*) have admissible_p: \"admissible p\"\n by (metis append_Nil2 (*‹?xs @ [] = ?xs›*) is_prefix_admissible (*‹⟦is_prefix ?a ?b; admissible ?b⟧ ⟹ admissible ?a›*) is_prefix_cancel (*‹is_prefix (?a @ ?b) (?a @ ?c) = is_prefix ?b ?c›*) is_prefix_empty (*‹is_prefix [] ?a›*))\n {\n fix i :: nat\n assume ilen: \"i < length p\" (*‹(i::nat) < length (p::(symbol × character list) list)›*)\n then have \"i < length (p@[a])\"\n by (simp add: Suc_leI (*‹?m < ?n ⟹ Suc ?m ≤ ?n›*) Suc_le_lessD (*‹Suc ?m ≤ ?n ⟹ ?m < ?n›*) trans_le_add1 (*‹?i ≤ ?j ⟹ ?i ≤ ?j + ?m›*))\n with snoc (*‹⟦∀i<length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u; admissible p⟧ ⟹ ∃u. p ∈ 𝒫 (charslength p) u› ‹∀i<length (p @ [a]). ∃u. (p @ [a]) ! i ∈ 𝒵 (charslength (take i (p @ [a]))) u› ‹admissible (p @ [a])›*) have \"∃u. (p @ [a]) ! i ∈ 𝒵 (charslength (take i (p @ [a]))) u\"\n by blast\n then obtain u where u: \"(p @ [a]) ! i ∈ 𝒵 (charslength (take i (p @ [a]))) u\"\n (*goal: ‹(⋀u. (p @ [a]) ! i ∈ 𝒵 (charslength (take i (p @ [a]))) u ⟹ thesis) ⟹ thesis›*)\n by blast\n from ilen (*‹(i::nat) < length (p::(symbol × character list) list)›*) have p_at: \"(p @ [a]) ! i = p ! i\"\n by (simp add: nth_append (*‹(?xs @ ?ys) ! ?n = (if ?n < length ?xs then ?xs ! ?n else ?ys ! (?n - length ?xs))›*))\n from ilen (*‹i < length p›*) have p_take: \"take i (p @ [a]) = take i p\"\n by (simp add: less_or_eq_imp_le (*‹?m < ?n ∨ ?m = ?n ⟹ ?m ≤ ?n›*))\n from u (*‹((p::(symbol × character list) list) @ [a::symbol × character list]) ! (i::nat) ∈ 𝒵 (charslength (take i (p @ [a]))) (u::nat)›*) p_at (*‹(p @ [a]) ! i = p ! i›*) p_take (*‹take i (p @ [a]) = take i p›*) have p_i: \"p ! i ∈ 𝒵 (charslength (take i p)) u\"\n by simp\n then have \"∃ u. p ! i ∈ 𝒵 (charslength (take i p)) u\"\n by blast\n }\n then have \"∀ i < length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u\"\n by auto\n with admissible_p (*‹admissible p›*) snoc.hyps (*‹⟦∀i<length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u; admissible p⟧ ⟹ ∃u. p ∈ 𝒫 (charslength p) u›*) obtain u where u: \"p ∈ 𝒫 (charslength p) u\"\n (*goal: ‹(⋀u. p ∈ 𝒫 (charslength p) u ⟹ thesis) ⟹ thesis›*)\n by blast\n have \"∃u. (p @ [a]) ! (length p) ∈ 𝒵 (charslength (take (length p) (p @ [a]))) u\"\n using snoc (*‹⟦∀i<length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u; admissible p⟧ ⟹ ∃u. p ∈ 𝒫 (charslength p) u› ‹∀i<length (p @ [a]). ∃u. (p @ [a]) ! i ∈ 𝒵 (charslength (take i (p @ [a]))) u› ‹admissible ((p::(symbol × character list) list) @ [a::symbol × character list])›*) by (metis (no_types, opaque_lifting) add_Suc_right (*‹(?m::nat) + Suc (?n::nat) = Suc (?m + ?n)›*) append_Nil2 (*‹(?xs::?'a list) @ [] = ?xs›*) length_Cons (*‹length ((?x::?'a) # (?xs::?'a list)) = Suc (length ?xs)›*) length_append (*‹length ((?xs::?'a list) @ (?ys::?'a list)) = length ?xs + length ?ys›*) less_Suc_eq_le (*‹((?m::nat) < Suc (?n::nat)) = (?m ≤ ?n)›*) less_or_eq_imp_le (*‹(?m::nat) < (?n::nat) ∨ ?m = ?n ⟹ ?m ≤ ?n›*))\n then obtain v where \"(p @ [a]) ! (length p) ∈ 𝒵 (charslength (take (length p) (p @ [a]))) v\"\n (*goal: ‹(⋀v. (p @ [a]) ! length p ∈ 𝒵 (charslength (take (length p) (p @ [a]))) v ⟹ thesis) ⟹ thesis›*)\n by blast\n then have v: \"a ∈ 𝒵 (charslength p) v\"\n by simp\n {\n assume v_leq_u: \"v ≤ u\" (*‹(v::nat) ≤ (u::nat)›*)\n then have \"a ∈ 𝒵 (charslength p) (Suc u)\"\n using v (*‹a ∈ 𝒵 (charslength p) v›*) by (meson LocalLexing.subset_fSuc (*‹⟦LocalLexing ?𝔑 ?𝔗 ?ℜ ?𝔖 ?Lex ?Sel; ⋀u. ?f u ⊆ ?f (Suc u); ?u ≤ ?v⟧ ⟹ ?f ?u ⊆ ?f ?v›*) LocalLexing_axioms (*‹LocalLexing 𝔑 𝔗 ℜ 𝔖 Lex Sel›*) 𝒵_subset_Suc (*‹𝒵 ?k ?n ⊆ 𝒵 ?k (Suc ?n)›*) subsetCE (*‹⟦?A ⊆ ?B; ?c ∉ ?A ⟹ ?P; ?c ∈ ?B ⟹ ?P⟧ ⟹ ?P›*))\n from path_append_token[OF u this snoc.prems ( 2 )] (*‹charslength p = charslength p ⟹ p @ [a] ∈ 𝒫 (charslength p) (Suc u)›*) have \"p @ [a] ∈ 𝒫 (charslength p) (Suc u)\"\n by blast\n then have \"?case\"\n using \"in_𝒫_charslength\" (*‹?p ∈ 𝒫 ?k ?u ⟹ ∃v. ?p ∈ 𝒫 (charslength ?p) v›*) by blast\n }\n note case_v_leq_u = this (*‹v ≤ u ⟹ ∃u. p @ [a] ∈ 𝒫 (charslength (p @ [a])) u›*)\n {\n assume u_less_v: \"u < v\" (*‹(u::nat) < (v::nat)›*)\n then obtain w where w: \"v = Suc w\"\n (*goal: ‹(⋀w. v = Suc w ⟹ thesis) ⟹ thesis›*)\n using less_imp_Suc_add (*‹?m < ?n ⟹ ∃k. ?n = Suc (?m + k)›*) by blast\n with u_less_v (*‹(u::nat) < (v::nat)›*) have \"u ≤ w\"\n by arith\n with u (*‹(p::(symbol × character list) list) ∈ 𝒫 (charslength p) (u::nat)›*) have \"p ∈ 𝒫 (charslength p) w\"\n by (meson subsetCE (*‹⟦?A ⊆ ?B; ?c ∉ ?A ⟹ ?P; ?c ∈ ?B ⟹ ?P⟧ ⟹ ?P›*) subset_𝒫k (*‹?u ≤ ?v ⟹ 𝒫 ?k ?u ⊆ 𝒫 ?k ?v›*))\n from v (*‹a ∈ 𝒵 (charslength p) v›*) w (*‹v = Suc w›*) path_append_token[OF this _ snoc.prems ( 2 )] (*‹⟦(a::symbol × character list) ∈ 𝒵 (charslength (p::(symbol × character list) list)) (Suc (w::nat)); charslength p = charslength p⟧ ⟹ p @ [a] ∈ 𝒫 (charslength p) (Suc w)›*) have \"p @ [a] ∈ 𝒫 (charslength p) (Suc w)\"\n by blast\n then have \"?case\"\n using \"in_𝒫_charslength\" (*‹?p ∈ 𝒫 ?k ?u ⟹ ∃v. ?p ∈ 𝒫 (charslength ?p) v›*) by blast\n }\n note case_u_less_v = this (*‹u < v ⟹ ∃u. p @ [a] ∈ 𝒫 (charslength (p @ [a])) u›*)\n show \"?case\"\n (*goal: ‹∃u. p @ [a] ∈ 𝒫 (charslength (p @ [a])) u›*)\n using case_v_leq_u (*‹(v::nat) ≤ (u::nat) ⟹ ∃u::nat. (p::(symbol × character list) list) @ [a::symbol × character list] ∈ 𝒫 (charslength (p @ [a])) u›*) case_u_less_v (*‹u < v ⟹ ∃u. p @ [a] ∈ 𝒫 (charslength (p @ [a])) u›*) not_le (*‹(¬ (?x::?'a) ≤ (?y::?'a)) = (?y < ?x)›*) by blast\nqed\n\nlemma drop_empty_tokens: \n assumes p: \"p ∈ 𝔓\"\n assumes r: \"r ≤ length p\"\n assumes empty: \"charslength (take r p) = 0\"\n assumes admissible: \"admissible (drop r p)\"\n shows \"drop r p ∈ 𝔓\"\nproof (-)\n (*goal: ‹drop r p ∈ 𝔓›*)\n have \"p_𝒵\": \"∀i<length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u\"\n using p (*‹p ∈ 𝔓›*) using \"tokens_nth_in_𝒵\" (*‹(?p::(symbol × character list) list) ∈ 𝔓 ⟹ ∀i<length ?p. ∃u::nat. ?p ! i ∈ 𝒵 (charslength (take i ?p)) u›*) by blast\n obtain q where q: \"q = drop r p\"\n (*goal: ‹(⋀q. q = drop r p ⟹ thesis) ⟹ thesis›*)\n by blast\n {\n fix j :: nat\n assume j: \"j < length q\" (*‹(j::nat) < length (q::(symbol × character list) list)›*)\n have length_p_q_r: \"length p = length q + r\"\n using r (*‹r ≤ length p›*) q (*‹q = drop r p›*) add.commute (*‹?a + ?b = ?b + ?a›*) diff_add_inverse (*‹?n + ?m - ?n = ?m›*) le_Suc_ex (*‹?k ≤ ?l ⟹ ∃n. ?l = ?k + n›*) length_drop (*‹length (drop ?n ?xs) = length ?xs - ?n›*) by simp\n have j_plus_r_bound: \"j + r < length p\"\n by (simp add: j (*‹j < length q›*) length_p_q_r (*‹length p = length q + r›*))\n with \"p_𝒵\" (*‹∀i<length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u›*) have \"∃u. p ! (j + r) ∈ 𝒵 (charslength (take (j + r) p)) u\"\n by blast\n then obtain u where u: \"p ! (j + r) ∈ 𝒵 (charslength (take (j + r) p)) u\"\n (*goal: ‹(⋀u. p ! (j + r) ∈ 𝒵 (charslength (take (j + r) p)) u ⟹ thesis) ⟹ thesis›*)\n by blast\n have p_at_is_q_at: \"p ! (j + r) = q ! j\"\n by (simp add: add.commute (*‹?a + ?b = ?b + ?a›*) q (*‹q = drop r p›*) r (*‹r ≤ length p›*))\n have \"take (j + r) p = (take r p) @ (take j q)\"\n by (metis add.commute (*‹?a + ?b = ?b + ?a›*) q (*‹q = drop r p›*) take_add (*‹take (?i + ?j) ?xs = take ?i ?xs @ take ?j (drop ?i ?xs)›*))\n with empty (*‹charslength (take (r::nat) (p::(symbol × character list) list)) = (0::nat)›*) have \"charslength (take (j + r) p) = charslength (take j q)\"\n by auto\n with u (*‹p ! (j + r) ∈ 𝒵 (charslength (take (j + r) p)) u›*) p_at_is_q_at (*‹p ! (j + r) = q ! j›*) have \"q ! j ∈ 𝒵 (charslength (take j q)) u\"\n by simp\n then have \"∃ u. q ! j ∈ 𝒵 (charslength (take j q)) u\"\n by auto\n }\n then have \"∀i<length q. ∃u. q ! i ∈ 𝒵 (charslength (take i q)) u\"\n by blast\n from \"characterize_𝒫\"[OF this] (*‹admissible q ⟹ ∃u. q ∈ 𝒫 (charslength q) u›*) have \"∃u. q ∈ 𝒫 (charslength q) u\"\n using admissible (*‹admissible (drop r p)›*) q (*‹q = drop r p›*) by auto\n then show \"?thesis\"\n (*goal: ‹drop r p ∈ 𝔓›*)\n using \"𝔓_covers_𝒫\" (*‹𝒫 ?k ?u ⊆ 𝔓›*) q (*‹q = drop r p›*) by blast\nqed\n \nend\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/LocalLexing/PathLemmas.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"511de6471118ee6864efdbf215b3ef9613a6af385c27e8ad13c3ce84ef47777c\"\n}"}}},{"rowIdx":194,"cells":{"text":{"kind":"string","value":"theory Clause_Could_Propagate\n imports\n Background\n Implicit_Exhaustive_Factorization\nbegin\n\ncontext simulation_SCLFOL_ground_ordered_resolution begin\n\ndefinition clause_could_propagate where\n \"clause_could_propagate Γ C L ⟷ ¬ trail_defined_lit Γ L ∧\n linorder_lit.is_maximal_in_mset C L ∧ trail_false_cls Γ {#K ∈# C. K ≠ L#}\"\n\nlemma trail_false_if_could_have_propagatated:\n \"clause_could_propagate Γ C L ⟹ trail_false_cls ((- L, n) # Γ) C\"\n unfolding clause_could_propagate_def trail_false_cls_def trail_false_lit_def\n (*goal: ‹¬ Background.trail_defined_lit Γ L ∧ ord_res.is_maximal_lit L C ∧ (∀L∈#{#K ∈# C. K ≠ L#}. - L ∈ fst ` set Γ) ⟹ ∀La∈#C. - La ∈ fst ` set ((- L, n) # Γ)›*)\n by auto\n\nlemma atoms_of_trail_lt_atom_of_propagatable_literal:\n assumes\n Γ_lower: \"linorder_trm.is_lower_set (fset (trail_atms Γ)) 𝒜\" and\n C_prop: \"clause_could_propagate Γ C L\" and\n \"atm_of L ∈ 𝒜\"\n shows \"∀A |∈| trail_atms Γ. A ≺⇩t atm_of L\"\nproof (-)\n (*goal: ‹∀A|∈|trail_atms Γ. A ≺⇩t atm_of L›*)\n have \"atm_of L |∉| trail_atms Γ\"\n using C_prop (*‹clause_could_propagate Γ C L›*) unfolding clause_could_propagate_def trail_defined_lit_iff_trail_defined_atm\n (*goal: ‹atm_of L |∉| trail_atms Γ›*)\n by argo\n then show \"?thesis\"\n (*goal: ‹∀A::'f::type gterm|∈|trail_atms (Γ::('f::type gterm literal × 'a::type option) list). A ≺⇩t atm_of (L::'f::type gterm literal)›*)\n using \"Γ_lower\" (*‹linorder_trm.is_lower_set (fset (trail_atms Γ)) 𝒜›*) ‹atm_of L ∈ 𝒜› (*‹atm_of L ∈ 𝒜›*) by (metis linorder_trm.is_lower_set_iff (*‹linorder_trm.is_lower_set ?L ?X = (?L ⊆ ?X ∧ (∀l∈?L. ∀x∈?X. x ≺⇩t l ⟶ x ∈ ?L))›*) linorder_trm.neqE (*‹⟦?x ≠ ?y; ?x ≺⇩t ?y ⟹ ?R; ?y ≺⇩t ?x ⟹ ?R⟧ ⟹ ?R›*))\nqed\n\nlemma trail_false_cls_filter_mset_iff:\n \"trail_false_cls Γ {#Ka ∈# C. Ka ≠ K#} ⟷ (∀L∈#C. L ≠ K ⟶ trail_false_lit Γ L)\"\n unfolding trail_false_cls_def\n (*goal: ‹Multiset.Ball {#Ka ∈# C. Ka ≠ K#} (Background.trail_false_lit Γ) = (∀L∈#C. L ≠ K ⟶ Background.trail_false_lit Γ L)›*)\n by auto\n\nlemma clause_could_propagate_iff: \"clause_could_propagate Γ C K ⟷\n ¬ trail_defined_lit Γ K ∧ ord_res.is_maximal_lit K C ∧ (∀L∈#C. L ≠ K ⟶ trail_false_lit Γ L)\"\n unfolding clause_could_propagate_def trail_false_cls_filter_mset_iff\n (*goal: ‹(¬ Background.trail_defined_lit Γ K ∧ ord_res.is_maximal_lit K C ∧ (∀L∈#C. L ≠ K ⟶ Background.trail_false_lit Γ L)) = (¬ Background.trail_defined_lit Γ K ∧ ord_res.is_maximal_lit K C ∧ (∀L∈#C. L ≠ K ⟶ Background.trail_false_lit Γ L))›*)\n by standard\n\nlemma clause_could_propagate_efac: \"clause_could_propagate Γ (efac C) = clause_could_propagate Γ C\"\nproof (rule ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹⋀x::'f gterm literal. clause_could_propagate (Γ::('f gterm literal × 'a option) list) (efac (C::'f gterm literal multiset)) x = clause_could_propagate Γ C x›*)\n fix L\n show \"clause_could_propagate Γ (efac C) L ⟷ clause_could_propagate Γ C L\"\n unfolding clause_could_propagate_def\n (*goal: ‹(¬ Background.trail_defined_lit Γ L ∧ ord_res.is_maximal_lit L (efac C) ∧ Background.trail_false_cls Γ {#K ∈# efac C. K ≠ L#}) = (¬ Background.trail_defined_lit Γ L ∧ ord_res.is_maximal_lit L C ∧ Background.trail_false_cls Γ {#K ∈# C. K ≠ L#})›*)\n by (metis (mono_tags, lifting) ex1_efac_eq_factoring_chain (*‹∃!C'. efac ?C = C' ∧ ord_res.ground_factoring⇧*⇧* ?C C' ∧ (∄C''. ord_res.ground_factoring C' C'')›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) ord_res.ground_factorings_preserves_maximal_literal (*‹ord_res.ground_factoring⇧*⇧* ?P ?C ⟹ ord_res.is_maximal_lit ?L ?P = ord_res.is_maximal_lit ?L ?C›*) set_mset_efac (*‹set_mset (efac ?C) = set_mset ?C›*) set_mset_filter (*‹set_mset (filter_mset ?P ?M) = {a. a ∈# ?M ∧ ?P a}›*) trail_false_cls_def (*‹Background.trail_false_cls ?Γ ?C = (∀L∈#?C. Background.trail_false_lit ?Γ L)›*))\nqed\n\nlemma bex_clause_could_propagate_simp:\n fixes ℱ N Γ L\n shows \"fBex (iefac ℱ |`| N) (λC. clause_could_propagate Γ C L) ⟷\n fBex N (λC. clause_could_propagate Γ C L)\"\n sketch (rule iffI; elim bexE)\n proof (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*); elim bexE (*‹⟦∃x∈?A. ?P x; ⋀x. ⟦x ∈ ?A; ?P x⟧ ⟹ ?Q⟧ ⟹ ?Q›*))\n (*goals:\n 1. ‹⋀C. ⟦C |∈| iefac ℱ |`| N; clause_could_propagate Γ C L⟧ ⟹ ∃C|∈|N. clause_could_propagate Γ C L›\n 2. ‹⋀C. ⟦C |∈| N; clause_could_propagate Γ C L⟧ ⟹ ∃C|∈|iefac ℱ |`| N. clause_could_propagate Γ C L›*)\n fix C :: \"'f gclause\"\n assume \"C |∈| iefac ℱ |`| N\" and \"clause_could_propagate Γ C L\" (*‹(C::'f gterm literal multiset) |∈| iefac (ℱ::'f gterm literal multiset fset) |`| (N::'f gterm literal multiset fset)› ‹clause_could_propagate (Γ::('f gterm literal × 'a option) list) (C::'f gterm literal multiset) (L::'f gterm literal)›*)\n thus \"∃C |∈| N. clause_could_propagate Γ C L\"\n by (metis clause_could_propagate_efac (*‹clause_could_propagate (?Γ::('f gterm literal × ?'a option) list) (efac (?C::'f gterm literal multiset)) = clause_could_propagate ?Γ ?C›*) fimageE (*‹⟦(?b::?'a) |∈| (?f::?'b ⇒ ?'a) |`| (?A::?'b fset); ⋀x::?'b. ⟦?b = ?f x; x |∈| ?A⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*) iefac_def (*‹iefac (?ℱ::'f gterm literal multiset fset) (?C::'f gterm literal multiset) = (if ?C |∈| ?ℱ then efac ?C else ?C)›*))\n next\n (*goal: ‹⋀C. ⟦C |∈| N; clause_could_propagate Γ C L⟧ ⟹ ∃C|∈|iefac ℱ |`| N. clause_could_propagate Γ C L›*)\n fix C :: \"'f gclause\"\n assume \"C |∈| N\" and \"clause_could_propagate Γ C L\" (*‹(C::'f gterm literal multiset) |∈| (N::'f gterm literal multiset fset)› ‹clause_could_propagate (Γ::('f gterm literal × 'a option) list) (C::'f gterm literal multiset) (L::'f gterm literal)›*)\n thus \"∃C|∈|iefac ℱ |`| N. clause_could_propagate Γ C L\"\n by (metis clause_could_propagate_efac (*‹clause_could_propagate ?Γ (efac ?C) = clause_could_propagate ?Γ ?C›*) fimageI (*‹?x |∈| ?A ⟹ ?f ?x |∈| ?f |`| ?A›*) iefac_def (*‹iefac ?ℱ ?C = (if ?C |∈| ?ℱ then efac ?C else ?C)›*))\n qed\n\nend\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/SCL_Simulates_Ground_Resolution/Clause_Could_Propagate.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"c136afa916bf15069dc50b1b93ece57ff8dbbcdc737c2b62db2d3e6307baecc3\"\n}"}}},{"rowIdx":195,"cells":{"text":{"kind":"string","value":"(* Title: HOL/HOLCF/IOA/Abstraction.thy\n Author: Olaf Müller\n*)\n\nsection ‹Abstraction Theory -- tailored for I/O automata›\n\ntheory Abstraction\nimports LiveIOA\nbegin\n\ndefault_sort type\n\ndefinition cex_abs :: \"('s1 ⇒ 's2) ⇒ ('a, 's1) execution ⇒ ('a, 's2) execution\"\n where \"cex_abs f ex = (f (fst ex), Map (λ(a, t). (a, f t)) ⋅ (snd ex))\"\n\ntext ‹equals cex_abs on Sequences -- after ex2seq application›\ndefinition cex_absSeq ::\n \"('s1 ⇒ 's2) ⇒ ('a option, 's1) transition Seq ⇒ ('a option, 's2)transition Seq\"\n where \"cex_absSeq f = (λs. Map (λ(s, a, t). (f s, a, f t)) ⋅ s)\"\n\ndefinition is_abstraction :: \"('s1 ⇒ 's2) ⇒ ('a, 's1) ioa ⇒ ('a, 's2) ioa ⇒ bool\"\n where \"is_abstraction f C A ⟷\n ((∀s ∈ starts_of C. f s ∈ starts_of A) ∧\n (∀s t a. reachable C s ∧ s ─a─C→ t ⟶ f s ─a─A→ f t))\"\n\ndefinition weakeningIOA :: \"('a, 's2) ioa ⇒ ('a, 's1) ioa ⇒ ('s1 ⇒ 's2) ⇒ bool\"\n where \"weakeningIOA A C h ⟷ (∀ex. ex ∈ executions C ⟶ cex_abs h ex ∈ executions A)\"\n\ndefinition temp_strengthening :: \"('a, 's2) ioa_temp ⇒ ('a, 's1) ioa_temp ⇒ ('s1 ⇒ 's2) ⇒ bool\"\n where \"temp_strengthening Q P h ⟷ (∀ex. (cex_abs h ex ⊫ Q) ⟶ (ex ⊫ P))\"\n\ndefinition temp_weakening :: \"('a, 's2) ioa_temp ⇒ ('a, 's1) ioa_temp ⇒ ('s1 ⇒ 's2) ⇒ bool\"\n where \"temp_weakening Q P h ⟷ temp_strengthening (❙¬ Q) (❙¬ P) h\"\n\ndefinition state_strengthening :: \"('a, 's2) step_pred ⇒ ('a, 's1) step_pred ⇒ ('s1 ⇒ 's2) ⇒ bool\"\n where \"state_strengthening Q P h ⟷ (∀s t a. Q (h s, a, h t) ⟶ P (s, a, t))\"\n\ndefinition state_weakening :: \"('a, 's2) step_pred ⇒ ('a, 's1) step_pred ⇒ ('s1 ⇒ 's2) ⇒ bool\"\n where \"state_weakening Q P h ⟷ state_strengthening (❙¬ Q) (❙¬ P) h\"\n\ndefinition is_live_abstraction :: \"('s1 ⇒ 's2) ⇒ ('a, 's1) live_ioa ⇒ ('a, 's2) live_ioa ⇒ bool\"\n where \"is_live_abstraction h CL AM ⟷\n is_abstraction h (fst CL) (fst AM) ∧ temp_weakening (snd AM) (snd CL) h\"\n\n\n(* thm about ex2seq which is not provable by induction as ex2seq is not continuous *)\naxiomatization where\n ex2seq_abs_cex: \"ex2seq (cex_abs h ex) = cex_absSeq h (ex2seq ex)\"\n\n(* analog to the proved thm strength_Box - proof skipped as trivial *)\naxiomatization where\n weak_Box: \"temp_weakening P Q h ⟹ temp_weakening (□P) (□Q) h\"\n\n(* analog to the proved thm strength_Next - proof skipped as trivial *)\naxiomatization where\n weak_Next: \"temp_weakening P Q h ⟹ temp_weakening (○P) (○Q) h\"\n\n\nsubsection ‹‹cex_abs››\n\nlemma cex_abs_UU [simp]: \"cex_abs f (s, UU) = (f s, UU)\"\n by (simp add: cex_abs_def (*‹cex_abs ?f ?ex = (?f (fst ?ex), Map (λ(a, t). (a, ?f t))⋅(snd ?ex))›*))\n\nlemma cex_abs_nil [simp]: \"cex_abs f (s,nil) = (f s, nil)\"\n by (simp add: cex_abs_def (*‹cex_abs ?f ?ex = (?f (fst ?ex), Map (λ(a, t). (a, ?f t))⋅(snd ?ex))›*))\n\nlemma cex_abs_cons [simp]:\n \"cex_abs f (s, (a, t) ↝ ex) = (f s, (a, f t) ↝ (snd (cex_abs f (t, ex))))\"\n by (simp add: cex_abs_def (*‹cex_abs ?f ?ex = (?f (fst ?ex), Map (λ(a, t). (a, ?f t))⋅(snd ?ex))›*))\n\n\nsubsection ‹Lemmas›\n\nlemma temp_weakening_def2: \"temp_weakening Q P h ⟷ (∀ex. (ex ⊫ P) ⟶ (cex_abs h ex ⊫ Q))\"\n apply (simp add: temp_weakening_def (*‹temp_weakening ?Q ?P ?h = temp_strengthening (❙¬ ?Q) (❙¬ ?P) ?h›*) temp_strengthening_def (*‹temp_strengthening ?Q ?P ?h = (∀ex. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*) NOT_def (*‹(❙¬ ?P) ?s = (¬ ?P ?s)›*) temp_sat_def (*‹(?ex ⊫ ?P) = (ex2seq ?ex ⊨ ?P)›*) satisfies_def (*‹(?s ⊨ ?P) = ?P ?s›*))\n (*goal: ‹temp_weakening Q P h = (∀ex. (ex ⊫ P) ⟶ (cex_abs h ex ⊫ Q))›*)\n by auto\n\nlemma state_weakening_def2: \"state_weakening Q P h ⟷ (∀s t a. P (s, a, t) ⟶ Q (h s, a, h t))\"\n apply (simp add: state_weakening_def (*‹state_weakening ?Q ?P ?h = state_strengthening (❙¬ ?Q) (❙¬ ?P) ?h›*) state_strengthening_def (*‹state_strengthening ?Q ?P ?h = (∀s t a. ?Q (?h s, a, ?h t) ⟶ ?P (s, a, t))›*) NOT_def (*‹(❙¬ ?P) ?s = (¬ ?P ?s)›*))\n (*goal: ‹state_weakening Q P h = (∀s t a. P (s, a, t) ⟶ Q (h s, a, h t))›*)\n by auto\n\n\nsubsection ‹Abstraction Rules for Properties›\n\nlemma exec_frag_abstraction [rule_format]:\n \"is_abstraction h C A ⟹\n ∀s. reachable C s ∧ is_exec_frag C (s, xs) ⟶ is_exec_frag A (cex_abs h (s, xs))\"\n apply (simp add: cex_abs_def (*‹cex_abs (?f::?'s1.0::type ⇒ ?'s2.0::type) (?ex::?'s1.0::type × (?'a::type × ?'s1.0::type) lift seq) = (?f (fst ?ex), Map (λ(a::?'a::type, t::?'s1.0::type). (a, ?f t))⋅(snd ?ex))›*))\n (*goal: ‹is_abstraction h C A ⟹ ∀s. reachable C s ∧ is_exec_frag C (s, xs) ⟶ is_exec_frag A (cex_abs h (s, xs))›*)\n apply (pair_induct xs simp: is_exec_frag_def)\n (*goal: ‹is_abstraction h C A ⟹ ∀s. reachable C s ∧ is_exec_frag C (s, xs) ⟶ is_exec_frag A (h s, Map (λ(a, t). (a, h t))⋅xs)›*)\n txt ‹main case›\n by (auto dest: reachable.reachable_n (*‹⟦reachable ?C ?s; ?s ─?a─?C→ ?t⟧ ⟹ reachable ?C ?t›*) simp add: is_abstraction_def (*‹is_abstraction ?f ?C ?A = ((∀s∈starts_of ?C. ?f s ∈ starts_of ?A) ∧ (∀s t a. reachable ?C s ∧ s ─a─?C→ t ⟶ ?f s ─a─?A→ ?f t))›*))\n\nlemma abs_is_weakening: \"is_abstraction h C A ⟹ weakeningIOA A C h\"\n apply (simp add: weakeningIOA_def (*‹weakeningIOA ?A ?C ?h = (∀ex. ex ∈ executions ?C ⟶ cex_abs ?h ex ∈ executions ?A)›*))\n (*goal: ‹is_abstraction h C A ⟹ weakeningIOA A C h›*)\n apply auto\n (*goal: ‹is_abstraction h C A ⟹ ∀a b. (a, b) ∈ executions C ⟶ cex_abs h (a, b) ∈ executions A›*)\n apply (simp add: executions_def (*‹executions ?ioa = {e. fst e ∈ starts_of ?ioa ∧ is_exec_frag ?ioa e}›*))\n (*goal: ‹⋀(a::'a::type) b::('c::type × 'a::type) lift seq. ⟦is_abstraction (h::'a::type ⇒ 'b::type) (C::('c::type set × 'c::type set × 'c::type set) × 'a::type set × ('a::type × 'c::type × 'a::type) set × 'c::type set set × 'c::type set set) (A::('c::type set × 'c::type set × 'c::type set) × 'b::type set × ('b::type × 'c::type × 'b::type) set × 'c::type set set × 'c::type set set); (a, b) ∈ executions C⟧ ⟹ cex_abs h (a, b) ∈ executions A›*)\n txt ‹start state›\n apply (rule conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*))\n (*goal: ‹⋀a b. ⟦is_abstraction h C A; a ∈ starts_of C ∧ is_exec_frag C (a, b)⟧ ⟹ fst (cex_abs h (a, b)) ∈ starts_of A ∧ is_exec_frag A (cex_abs h (a, b))›*)\n apply (simp add: is_abstraction_def (*‹is_abstraction (?f::?'s1.0 ⇒ ?'s2.0) (?C::(?'a set × ?'a set × ?'a set) × ?'s1.0 set × (?'s1.0 × ?'a × ?'s1.0) set × ?'a set set × ?'a set set) (?A::(?'a set × ?'a set × ?'a set) × ?'s2.0 set × (?'s2.0 × ?'a × ?'s2.0) set × ?'a set set × ?'a set set) = ((∀s::?'s1.0∈starts_of ?C. ?f s ∈ starts_of ?A) ∧ (∀(s::?'s1.0) (t::?'s1.0) a::?'a. reachable ?C s ∧ s ─a─?C→ t ⟶ ?f s ─a─?A→ ?f t))›*) cex_abs_def (*‹cex_abs (?f::?'s1.0 ⇒ ?'s2.0) (?ex::?'s1.0 × (?'a × ?'s1.0) lift seq) = (?f (fst ?ex), Map (λ(a::?'a, t::?'s1.0). (a, ?f t))⋅(snd ?ex))›*))\n (*top goal: ‹⋀a b. ⟦is_abstraction h C A; a ∈ starts_of C ∧ is_exec_frag C (a, b)⟧ ⟹ fst (cex_abs h (a, b)) ∈ starts_of A› and 1 goal remains*)\n txt ‹is-execution-fragment›\n apply (erule exec_frag_abstraction (*‹⟦is_abstraction ?h ?C ?A; reachable ?C ?s ∧ is_exec_frag ?C (?s, ?xs)⟧ ⟹ is_exec_frag ?A (cex_abs ?h (?s, ?xs))›*))\n (*goal: ‹⋀a b. ⟦is_abstraction h C A; a ∈ starts_of C ∧ is_exec_frag C (a, b)⟧ ⟹ is_exec_frag A (cex_abs h (a, b))›*)\n by (simp add: reachable.reachable_0 (*‹?s ∈ starts_of ?C ⟹ reachable ?C ?s›*))\n\n\nlemma AbsRuleT1:\n \"is_abstraction h C A ⟹ validIOA A Q ⟹ temp_strengthening Q P h ⟹ validIOA C P\"\n apply (drule abs_is_weakening (*‹is_abstraction (?h::?'a ⇒ ?'b) (?C::(?'c set × ?'c set × ?'c set) × ?'a set × (?'a × ?'c × ?'a) set × ?'c set set × ?'c set set) (?A::(?'c set × ?'c set × ?'c set) × ?'b set × (?'b × ?'c × ?'b) set × ?'c set set × ?'c set set) ⟹ weakeningIOA ?A ?C ?h›*))\n (*goal: ‹⟦is_abstraction h C A; validIOA A Q; temp_strengthening Q P h⟧ ⟹ validIOA C P›*)\n apply (simp add: weakeningIOA_def (*‹weakeningIOA ?A ?C ?h = (∀ex. ex ∈ executions ?C ⟶ cex_abs ?h ex ∈ executions ?A)›*) validIOA_def (*‹validIOA ?A ?P = (∀ex∈executions ?A. ex ⊫ ?P)›*) temp_strengthening_def (*‹temp_strengthening ?Q ?P ?h = (∀ex. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n (*goal: ‹⟦validIOA A Q; temp_strengthening Q P h; weakeningIOA A C h⟧ ⟹ validIOA C P›*)\n by (auto simp add: split_paired_all (*‹(⋀x. PROP ?P x) ≡ (⋀a b. PROP ?P (a, b))›*))\n\n\nlemma AbsRuleT2:\n \"is_live_abstraction h (C, L) (A, M) ⟹ validLIOA (A, M) Q ⟹ temp_strengthening Q P h\n ⟹ validLIOA (C, L) P\"\n apply (unfold is_live_abstraction_def (*‹is_live_abstraction (?h::?'s1.0 ⇒ ?'s2.0) (?CL::((?'a set × ?'a set × ?'a set) × ?'s1.0 set × (?'s1.0 × ?'a × ?'s1.0) set × ?'a set set × ?'a set set) × ((?'s1.0 × ?'a option × ?'s1.0) lift seq ⇒ bool)) (?AM::((?'a set × ?'a set × ?'a set) × ?'s2.0 set × (?'s2.0 × ?'a × ?'s2.0) set × ?'a set set × ?'a set set) × ((?'s2.0 × ?'a option × ?'s2.0) lift seq ⇒ bool)) = (is_abstraction ?h (fst ?CL) (fst ?AM) ∧ temp_weakening (snd ?AM) (snd ?CL) ?h)›*))\n (*goal: ‹⟦is_live_abstraction h (C, L) (A, M); validLIOA (A, M) Q; temp_strengthening Q P h⟧ ⟹ validLIOA (C, L) P›*)\n apply auto\n (*goal: ‹⟦is_abstraction h (fst (C, L)) (fst (A, M)) ∧ temp_weakening (snd (A, M)) (snd (C, L)) h; validLIOA (A, M) Q; temp_strengthening Q P h⟧ ⟹ validLIOA (C, L) P›*)\n apply (drule abs_is_weakening (*‹is_abstraction ?h ?C ?A ⟹ weakeningIOA ?A ?C ?h›*))\n (*goal: ‹⟦validLIOA (A, M) Q; temp_strengthening Q P h; is_abstraction h C A; temp_weakening M L h⟧ ⟹ validLIOA (C, L) P›*)\n apply (simp add: weakeningIOA_def (*‹weakeningIOA ?A ?C ?h = (∀ex. ex ∈ executions ?C ⟶ cex_abs ?h ex ∈ executions ?A)›*) temp_weakening_def2 (*‹temp_weakening ?Q ?P ?h = (∀ex. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*) validLIOA_def (*‹validLIOA ?AL ?P = validIOA (fst ?AL) (snd ?AL ❙⟶ ?P)›*) validIOA_def (*‹validIOA ?A ?P = (∀ex∈executions ?A. ex ⊫ ?P)›*) temp_strengthening_def (*‹temp_strengthening ?Q ?P ?h = (∀ex. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n (*goal: ‹⟦validLIOA (A::('c set × 'c set × 'c set) × 'b set × ('b × 'c × 'b) set × 'c set set × 'c set set, M::('b × 'c option × 'b) lift seq ⇒ bool) (Q::('b × 'c option × 'b) lift seq ⇒ bool); temp_strengthening Q (P::('a × 'c option × 'a) lift seq ⇒ bool) (h::'a ⇒ 'b); temp_weakening M (L::('a × 'c option × 'a) lift seq ⇒ bool) h; weakeningIOA A (C::('c set × 'c set × 'c set) × 'a set × ('a × 'c × 'a) set × 'c set set × 'c set set) h⟧ ⟹ validLIOA (C, L) P›*)\n by (auto simp add: split_paired_all (*‹(⋀x::?'a × ?'b. PROP (?P::?'a × ?'b ⇒ prop) x) ≡ (⋀(a::?'a) b::?'b. PROP ?P (a, b))›*))\n\n\nlemma AbsRuleTImprove:\n \"is_live_abstraction h (C, L) (A, M) ⟹ validLIOA (A,M) (H1 ❙⟶ Q) ⟹\n temp_strengthening Q P h ⟹ temp_weakening H1 H2 h ⟹ validLIOA (C, L) H2 ⟹\n validLIOA (C, L) P\"\n apply (unfold is_live_abstraction_def (*‹is_live_abstraction ?h ?CL ?AM = (is_abstraction ?h (fst ?CL) (fst ?AM) ∧ temp_weakening (snd ?AM) (snd ?CL) ?h)›*))\n (*goal: ‹⟦is_live_abstraction h (C, L) (A, M); validLIOA (A, M) (H1 ❙⟶ Q); temp_strengthening Q P h; temp_weakening H1 H2 h; validLIOA (C, L) H2⟧ ⟹ validLIOA (C, L) P›*)\n apply auto\n (*goal: ‹⟦is_abstraction h (fst (C, L)) (fst (A, M)) ∧ temp_weakening (snd (A, M)) (snd (C, L)) h; validLIOA (A, M) (H1 ❙⟶ Q); temp_strengthening Q P h; temp_weakening H1 H2 h; validLIOA (C, L) H2⟧ ⟹ validLIOA (C, L) P›*)\n apply (drule abs_is_weakening (*‹is_abstraction ?h ?C ?A ⟹ weakeningIOA ?A ?C ?h›*))\n (*goal: ‹⟦validLIOA (A, M) (H1 ❙⟶ Q); temp_strengthening Q P h; temp_weakening H1 H2 h; validLIOA (C, L) H2; is_abstraction h C A; temp_weakening M L h⟧ ⟹ validLIOA (C, L) P›*)\n apply (simp add: weakeningIOA_def (*‹weakeningIOA ?A ?C ?h = (∀ex. ex ∈ executions ?C ⟶ cex_abs ?h ex ∈ executions ?A)›*) temp_weakening_def2 (*‹temp_weakening ?Q ?P ?h = (∀ex. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*) validLIOA_def (*‹validLIOA ?AL ?P = validIOA (fst ?AL) (snd ?AL ❙⟶ ?P)›*) validIOA_def (*‹validIOA ?A ?P = (∀ex∈executions ?A. ex ⊫ ?P)›*) temp_strengthening_def (*‹temp_strengthening ?Q ?P ?h = (∀ex. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n (*goal: ‹⟦validLIOA (A, M) (H1 ❙⟶ Q); temp_strengthening Q P h; temp_weakening H1 H2 h; validLIOA (C, L) H2; temp_weakening M L h; weakeningIOA A C h⟧ ⟹ validLIOA (C, L) P›*)\n by (auto simp add: split_paired_all (*‹(⋀x. PROP ?P x) ≡ (⋀a b. PROP ?P (a, b))›*))\n\n\nsubsection ‹Correctness of safe abstraction›\n\nlemma abstraction_is_ref_map: \"is_abstraction h C A ⟹ is_ref_map h C A\"\n apply (auto simp: is_abstraction_def (*‹is_abstraction (?f::?'s1.0 ⇒ ?'s2.0) (?C::(?'a set × ?'a set × ?'a set) × ?'s1.0 set × (?'s1.0 × ?'a × ?'s1.0) set × ?'a set set × ?'a set set) (?A::(?'a set × ?'a set × ?'a set) × ?'s2.0 set × (?'s2.0 × ?'a × ?'s2.0) set × ?'a set set × ?'a set set) = ((∀s::?'s1.0∈starts_of ?C. ?f s ∈ starts_of ?A) ∧ (∀(s::?'s1.0) (t::?'s1.0) a::?'a. reachable ?C s ∧ s ─a─?C→ t ⟶ ?f s ─a─?A→ ?f t))›*) is_ref_map_def (*‹is_ref_map (?f::?'s1.0 ⇒ ?'s2.0) (?C::(?'a set × ?'a set × ?'a set) × ?'s1.0 set × (?'s1.0 × ?'a × ?'s1.0) set × ?'a set set × ?'a set set) (?A::(?'a set × ?'a set × ?'a set) × ?'s2.0 set × (?'s2.0 × ?'a × ?'s2.0) set × ?'a set set × ?'a set set) = ((∀s::?'s1.0∈starts_of ?C. ?f s ∈ starts_of ?A) ∧ (∀(s::?'s1.0) (t::?'s1.0) a::?'a. reachable ?C s ∧ s ─a─?C→ t ⟶ (∃ex::(?'a × ?'s2.0) lift seq. move ?A ex (?f s) a (?f t))))›*))\n (*goal: ‹is_abstraction h C A ⟹ is_ref_map h C A›*)\n apply (rule_tac x = \"(a,h t) ↝nil\" in exI (*‹(?P::?'a::type ⇒ bool) (?x::?'a::type) ⟹ ∃x::?'a::type. ?P x›*))\n (*goal: ‹⋀s t a. ⟦∀s∈starts_of C. h s ∈ starts_of A; ∀s t a. reachable C s ∧ s ─a─C→ t ⟶ h s ─a─A→ h t; reachable C s; s ─a─C→ t⟧ ⟹ ∃ex. move A ex (h s) a (h t)›*)\n by (simp add: move_def (*‹move ?ioa ?ex ?s ?a ?t = (is_exec_frag ?ioa (?s, ?ex) ∧ Finite ?ex ∧ laststate (?s, ?ex) = ?t ∧ mk_trace ?ioa⋅?ex = (if ?a ∈ ext ?ioa then [?a!] else nil))›*))\n\nlemma abs_safety: \"inp C = inp A ⟹ out C = out A ⟹ is_abstraction h C A ⟹ C =<| A\"\n apply (simp add: ioa_implements_def (*‹((?ioa1.0::(?'a set × ?'a set × ?'a set) × ?'s1.0 set × (?'s1.0 × ?'a × ?'s1.0) set × ?'a set set × ?'a set set) =<| (?ioa2.0::(?'a set × ?'a set × ?'a set) × ?'s2.0 set × (?'s2.0 × ?'a × ?'s2.0) set × ?'a set set × ?'a set set)) = ((inp ?ioa1.0 = inp ?ioa2.0 ∧ out ?ioa1.0 = out ?ioa2.0) ∧ traces ?ioa1.0 ⊆ traces ?ioa2.0)›*))\n (*goal: ‹⟦inp C = inp A; out C = out A; is_abstraction h C A⟧ ⟹ C =<| A›*)\n apply (rule trace_inclusion (*‹⟦ext ?C = ext ?A; is_ref_map ?f ?C ?A⟧ ⟹ traces ?C ⊆ traces ?A›*))\n (*goals:\n 1. ‹⟦inp C = inp A; out C = out A; is_abstraction h C A⟧ ⟹ ext C = ext A›\n 2. ‹⟦inp C = inp A; out C = out A; is_abstraction h C A⟧ ⟹ is_ref_map ?f1 C A›\n discuss goal 1*)\n apply (simp (no_asm) add: externals_def (*‹externals ?asig = inputs ?asig ∪ outputs ?asig›*))\n (*top goal: ‹⟦inp C = inp A; out C = out A; is_abstraction h C A⟧ ⟹ ext C = ext A› and 1 goal remains*)\n apply auto\n (*discuss goal 2*)\n apply (erule abstraction_is_ref_map (*‹is_abstraction ?h ?C ?A ⟹ is_ref_map ?h ?C ?A›*))\n (*proven 2 subgoals*) .\n\n\nsubsection ‹Correctness of life abstraction›\n\ntext ‹\n Reduces to ‹Filter (Map fst x) = Filter (Map fst (Map (λ(a, t). (a, x)) x)›,\n that is to special Map Lemma.\n›\nlemma traces_coincide_abs:\n \"ext C = ext A ⟹ mk_trace C ⋅ xs = mk_trace A ⋅ (snd (cex_abs f (s, xs)))\"\n apply (unfold cex_abs_def (*‹cex_abs ?f ?ex = (?f (fst ?ex), Map (λ(a, t). (a, ?f t))⋅(snd ?ex))›*) mk_trace_def (*‹mk_trace ?ioa = (Λ tr. Filter (λa. a ∈ ext ?ioa)⋅(filter_act⋅tr))›*) filter_act_def (*‹filter_act = Map fst›*))\n (*goal: ‹ext C = ext A ⟹ mk_trace C⋅xs = mk_trace A⋅(snd (cex_abs f (s, xs)))›*)\n apply simp\n (*goal: ‹ext C = ext A ⟹ (Λ tr. Filter (λa. a ∈ ext C)⋅(Map fst⋅tr))⋅xs = (Λ tr. Filter (λa. a ∈ ext A)⋅(Map fst⋅tr))⋅(snd (f (fst (s, xs)), Map (λ(a, t). (a, f t))⋅(snd (s, xs))))›*)\n by (pair_induct xs)\n\n\ntext ‹\n Does not work with ‹abstraction_is_ref_map› as proof of ‹abs_safety›, because\n ‹is_live_abstraction› includes ‹temp_strengthening› which is necessarily based\n on ‹cex_abs› and not on ‹corresp_ex›. Thus, the proof is redone in a more specific\n way for ‹cex_abs›.\n›\nlemma abs_liveness:\n \"inp C = inp A ⟹ out C = out A ⟹ is_live_abstraction h (C, M) (A, L) ⟹\n live_implements (C, M) (A, L)\"\n apply (simp add: is_live_abstraction_def (*‹is_live_abstraction ?h ?CL ?AM = (is_abstraction ?h (fst ?CL) (fst ?AM) ∧ temp_weakening (snd ?AM) (snd ?CL) ?h)›*) live_implements_def (*‹live_implements ?CL ?AM = (inp (fst ?CL) = inp (fst ?AM) ∧ out (fst ?CL) = out (fst ?AM) ∧ livetraces ?CL ⊆ livetraces ?AM)›*) livetraces_def (*‹livetraces ?AP = {mk_trace (fst ?AP)⋅(snd ex) |ex. ex ∈ liveexecutions ?AP}›*) liveexecutions_def (*‹liveexecutions ?AP = {exec ∈ executions (fst ?AP). exec ⊫ snd ?AP}›*))\n (*goal: ‹⟦inp C = inp A; out C = out A; is_live_abstraction h (C, M) (A, L)⟧ ⟹ live_implements (C, M) (A, L)›*)\n apply auto\n (*goal: ‹⟦inp C = inp A; out C = out A; is_abstraction h C A ∧ temp_weakening L M h⟧ ⟹ {mk_trace C⋅(snd ex) |ex. ex ∈ executions C ∧ (ex ⊫ M)} ⊆ {mk_trace A⋅(snd ex) |ex. ex ∈ executions A ∧ (ex ⊫ L)}›*)\n apply (rule_tac x = \"cex_abs h ex\" in exI (*‹?P ?x ⟹ ∃x. ?P x›*))\n (*goal: ‹⋀ex::'b × ('a × 'b) lift seq. ⟦inp (C::('a set × 'a set × 'a set) × 'b set × ('b × 'a × 'b) set × 'a set set × 'a set set) = inp (A::('a set × 'a set × 'a set) × 'c set × ('c × 'a × 'c) set × 'a set set × 'a set set); out C = out A; is_abstraction (h::'b ⇒ 'c) C A; temp_weakening (L::('c × 'a option × 'c) lift seq ⇒ bool) (M::('b × 'a option × 'b) lift seq ⇒ bool) h; ex ∈ executions C; ex ⊫ M⟧ ⟹ ∃exa::'c × ('a × 'c) lift seq. mk_trace C⋅(snd ex) = mk_trace A⋅(snd exa) ∧ exa ∈ executions A ∧ (exa ⊫ L)›*)\n apply auto\n (*goal: ‹⋀ex. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; ex ∈ executions C; ex ⊫ M⟧ ⟹ mk_trace C⋅(snd ex) = mk_trace A⋅(snd (cex_abs h ex)) ∧ cex_abs h ex ∈ executions A ∧ (cex_abs h ex ⊫ L)›*)\n text ‹Traces coincide›\n apply (pair ex)\n (*top goal: ‹⋀ex. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; ex ∈ executions C; ex ⊫ M⟧ ⟹ mk_trace C⋅(snd ex) = mk_trace A⋅(snd (cex_abs h ex))› and 2 goals remain*)\n apply (rule traces_coincide_abs (*‹ext ?C = ext ?A ⟹ mk_trace ?C⋅?xs = mk_trace ?A⋅(snd (cex_abs ?f (?s, ?xs)))›*))\n (*top goal: ‹⋀x1 x2. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; (x1, x2) ∈ executions C; (x1, x2) ⊫ M⟧ ⟹ mk_trace C⋅x2 = mk_trace A⋅(snd (cex_abs h (x1, x2)))› and 2 goals remain*)\n apply (simp (no_asm) add: externals_def (*‹externals ?asig = inputs ?asig ∪ outputs ?asig›*))\n (*top goal: ‹⋀(x1::'b) x2::('a × 'b) lift seq. ⟦inp (C::('a set × 'a set × 'a set) × 'b set × ('b × 'a × 'b) set × 'a set set × 'a set set) = inp (A::('a set × 'a set × 'a set) × 'c set × ('c × 'a × 'c) set × 'a set set × 'a set set); out C = out A; is_abstraction (h::'b ⇒ 'c) C A; temp_weakening (L::('c × 'a option × 'c) lift seq ⇒ bool) (M::('b × 'a option × 'b) lift seq ⇒ bool) h; (x1, x2) ∈ executions C; (x1, x2) ⊫ M⟧ ⟹ ext C = ext A› and 2 goals remain*)\n apply ((auto)[1])\n (*top goal: ‹⋀x1 x2. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; (x1, x2) ∈ executions C; (x1, x2) ⊫ M⟧ ⟹ inp C ∪ out C = inp A ∪ out A› and 2 goals remain*)\n text ‹‹cex_abs› is execution›\n apply (pair ex)\n (*top goal: ‹⋀ex::'b × ('a × 'b) lift seq. ⟦inp (C::('a set × 'a set × 'a set) × 'b set × ('b × 'a × 'b) set × 'a set set × 'a set set) = inp (A::('a set × 'a set × 'a set) × 'c set × ('c × 'a × 'c) set × 'a set set × 'a set set); out C = out A; is_abstraction (h::'b ⇒ 'c) C A; temp_weakening (L::('c × 'a option × 'c) lift seq ⇒ bool) (M::('b × 'a option × 'b) lift seq ⇒ bool) h; ex ∈ executions C; ex ⊫ M⟧ ⟹ cex_abs h ex ∈ executions A› and 1 goal remains*)\n apply (simp add: executions_def (*‹executions ?ioa = {e. fst e ∈ starts_of ?ioa ∧ is_exec_frag ?ioa e}›*))\n (*top goal: ‹⋀x1 x2. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; (x1, x2) ∈ executions C; (x1, x2) ⊫ M⟧ ⟹ cex_abs h (x1, x2) ∈ executions A› and 1 goal remains*)\n text ‹start state›\n apply (rule conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*top goal: ‹⋀(x1::'b) x2::('a × 'b) lift seq. ⟦inp (C::('a set × 'a set × 'a set) × 'b set × ('b × 'a × 'b) set × 'a set set × 'a set set) = inp (A::('a set × 'a set × 'a set) × 'c set × ('c × 'a × 'c) set × 'a set set × 'a set set); out C = out A; is_abstraction (h::'b ⇒ 'c) C A; temp_weakening (L::('c × 'a option × 'c) lift seq ⇒ bool) (M::('b × 'a option × 'b) lift seq ⇒ bool) h; x1 ∈ starts_of C ∧ is_exec_frag C (x1, x2); (x1, x2) ⊫ M⟧ ⟹ fst (cex_abs h (x1, x2)) ∈ starts_of A ∧ is_exec_frag A (cex_abs h (x1, x2))› and 1 goal remains*)\n apply (simp add: is_abstraction_def (*‹is_abstraction ?f ?C ?A = ((∀s∈starts_of ?C. ?f s ∈ starts_of ?A) ∧ (∀s t a. reachable ?C s ∧ s ─a─?C→ t ⟶ ?f s ─a─?A→ ?f t))›*) cex_abs_def (*‹cex_abs ?f ?ex = (?f (fst ?ex), Map (λ(a, t). (a, ?f t))⋅(snd ?ex))›*))\n (*top goal: ‹⋀x1 x2. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; x1 ∈ starts_of C ∧ is_exec_frag C (x1, x2); (x1, x2) ⊫ M⟧ ⟹ fst (cex_abs h (x1, x2)) ∈ starts_of A› and 2 goals remain*)\n text ‹‹is_execution_fragment››\n apply (erule exec_frag_abstraction (*‹⟦is_abstraction ?h ?C ?A; reachable ?C ?s ∧ is_exec_frag ?C (?s, ?xs)⟧ ⟹ is_exec_frag ?A (cex_abs ?h (?s, ?xs))›*))\n (*top goal: ‹⋀x1 x2. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; x1 ∈ starts_of C ∧ is_exec_frag C (x1, x2); (x1, x2) ⊫ M⟧ ⟹ is_exec_frag A (cex_abs h (x1, x2))› and 1 goal remains*)\n apply (simp add: reachable.reachable_0 (*‹?s ∈ starts_of ?C ⟹ reachable ?C ?s›*))\n (*top goal: ‹⋀x1 x2. ⟦inp C = inp A; out C = out A; temp_weakening L M h; x1 ∈ starts_of C ∧ is_exec_frag C (x1, x2); (x1, x2) ⊫ M⟧ ⟹ reachable C x1 ∧ is_exec_frag C (x1, x2)› and 1 goal remains*)\n text ‹Liveness›\n apply (simp add: temp_weakening_def2 (*‹temp_weakening (?Q::(?'a × ?'b option × ?'a) lift seq ⇒ bool) (?P::(?'c × ?'b option × ?'c) lift seq ⇒ bool) (?h::?'c ⇒ ?'a) = (∀ex::?'c × (?'b × ?'c) lift seq. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*))\n (*goal: ‹⋀ex. ⟦inp C = inp A; out C = out A; is_abstraction h C A; temp_weakening L M h; ex ∈ executions C; ex ⊫ M⟧ ⟹ cex_abs h ex ⊫ L›*)\n by (pair ex)\n\n\nsubsection ‹Abstraction Rules for Automata›\n\nlemma AbsRuleA1:\n \"inp C = inp A ⟹ out C = out A ⟹ inp Q = inp P ⟹ out Q = out P ⟹\n is_abstraction h1 C A ⟹ A =<| Q ⟹ is_abstraction h2 Q P ⟹ C =<| P\"\n apply (drule abs_safety (*‹⟦inp ?C = inp ?A; out ?C = out ?A; is_abstraction ?h ?C ?A⟧ ⟹ ?C =<| ?A›*))\n (*goals:\n 1. ‹⟦out C = out A; inp Q = inp P; out Q = out P; is_abstraction h1 C A; A =<| Q; is_abstraction h2 Q P⟧ ⟹ out C = out A›\n 2. ‹⟦out C = out A; inp Q = inp P; out Q = out P; is_abstraction h1 C A; A =<| Q; is_abstraction h2 Q P⟧ ⟹ is_abstraction ?h C A›\n 3. ‹⟦out C = out A; inp Q = inp P; out Q = out P; is_abstraction h1 C A; A =<| Q; is_abstraction h2 Q P; C =<| A⟧ ⟹ C =<| P›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply (drule abs_safety (*‹⟦inp ?C = inp ?A; out ?C = out ?A; is_abstraction ?h ?C ?A⟧ ⟹ ?C =<| ?A›*))\n (*goals:\n 1. ‹⟦out (C::('a::type set × 'a::type set × 'a::type set) × 'b::type set × ('b::type × 'a::type × 'b::type) set × 'a::type set set × 'a::type set set) = out (A::('a::type set × 'a::type set × 'a::type set) × 'c::type set × ('c::type × 'a::type × 'c::type) set × 'a::type set set × 'a::type set set); out (Q::('a::type set × 'a::type set × 'a::type set) × 'd::type set × ('d::type × 'a::type × 'd::type) set × 'a::type set set × 'a::type set set) = out (P::('a::type set × 'a::type set × 'a::type set) × 'e::type set × ('e::type × 'a::type × 'e::type) set × 'a::type set set × 'a::type set set); is_abstraction (h1::'b::type ⇒ 'c::type) C A; A =<| Q; is_abstraction (h2::'d::type ⇒ 'e::type) Q P; C =<| A⟧ ⟹ out Q = out P›\n 2. ‹⟦out (C::('a::type set × 'a::type set × 'a::type set) × 'b::type set × ('b::type × 'a::type × 'b::type) set × 'a::type set set × 'a::type set set) = out (A::('a::type set × 'a::type set × 'a::type set) × 'c::type set × ('c::type × 'a::type × 'c::type) set × 'a::type set set × 'a::type set set); out (Q::('a::type set × 'a::type set × 'a::type set) × 'd::type set × ('d::type × 'a::type × 'd::type) set × 'a::type set set × 'a::type set set) = out (P::('a::type set × 'a::type set × 'a::type set) × 'e::type set × ('e::type × 'a::type × 'e::type) set × 'a::type set set × 'a::type set set); is_abstraction (h1::'b::type ⇒ 'c::type) C A; A =<| Q; is_abstraction (h2::'d::type ⇒ 'e::type) Q P; C =<| A⟧ ⟹ is_abstraction (?h4::'d::type ⇒ 'e::type) Q P›\n 3. ‹⟦out (C::('a::type set × 'a::type set × 'a::type set) × 'b::type set × ('b::type × 'a::type × 'b::type) set × 'a::type set set × 'a::type set set) = out (A::('a::type set × 'a::type set × 'a::type set) × 'c::type set × ('c::type × 'a::type × 'c::type) set × 'a::type set set × 'a::type set set); out (Q::('a::type set × 'a::type set × 'a::type set) × 'd::type set × ('d::type × 'a::type × 'd::type) set × 'a::type set set × 'a::type set set) = out (P::('a::type set × 'a::type set × 'a::type set) × 'e::type set × ('e::type × 'a::type × 'e::type) set × 'a::type set set × 'a::type set set); is_abstraction (h1::'b::type ⇒ 'c::type) C A; A =<| Q; is_abstraction (h2::'d::type ⇒ 'e::type) Q P; C =<| A; Q =<| P⟧ ⟹ C =<| P›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply (erule implements_trans (*‹⟦?A =<| ?B; ?B =<| ?C⟧ ⟹ ?A =<| ?C›*))\n (*goal: ‹⟦out C = out A; out Q = out P; is_abstraction h1 C A; A =<| Q; is_abstraction h2 Q P; C =<| A; Q =<| P⟧ ⟹ C =<| P›*)\n apply (erule implements_trans (*‹⟦?A =<| ?B; ?B =<| ?C⟧ ⟹ ?A =<| ?C›*))\n (*goal: ‹⟦out C = out A; out Q = out P; is_abstraction h1 C A; A =<| Q; is_abstraction h2 Q P; Q =<| P⟧ ⟹ A =<| P›*)\n apply assumption\n (*proven 3 subgoals*)\n (*proven 3 subgoals*) .\n\nlemma AbsRuleA2:\n \"inp C = inp A ⟹ out C = out A ⟹ inp Q = inp P ⟹ out Q = out P ⟹\n is_live_abstraction h1 (C, LC) (A, LA) ⟹ live_implements (A, LA) (Q, LQ) ⟹\n is_live_abstraction h2 (Q, LQ) (P, LP) ⟹ live_implements (C, LC) (P, LP)\"\n apply (drule abs_liveness (*‹⟦inp (?C::(?'a set × ?'a set × ?'a set) × ?'b set × (?'b × ?'a × ?'b) set × ?'a set set × ?'a set set) = inp (?A::(?'a set × ?'a set × ?'a set) × ?'c set × (?'c × ?'a × ?'c) set × ?'a set set × ?'a set set); out ?C = out ?A; is_live_abstraction (?h::?'b ⇒ ?'c) (?C, ?M::(?'b × ?'a option × ?'b) lift seq ⇒ bool) (?A, ?L::(?'c × ?'a option × ?'c) lift seq ⇒ bool)⟧ ⟹ live_implements (?C, ?M) (?A, ?L)›*))\n (*goals:\n 1. ‹⟦out (C::('a::type set × 'a::type set × 'a::type set) × 'b::type set × ('b::type × 'a::type × 'b::type) set × 'a::type set set × 'a::type set set) = out (A::('a::type set × 'a::type set × 'a::type set) × 'c::type set × ('c::type × 'a::type × 'c::type) set × 'a::type set set × 'a::type set set); inp (Q::('a::type set × 'a::type set × 'a::type set) × 'd::type set × ('d::type × 'a::type × 'd::type) set × 'a::type set set × 'a::type set set) = inp (P::('a::type set × 'a::type set × 'a::type set) × 'e::type set × ('e::type × 'a::type × 'e::type) set × 'a::type set set × 'a::type set set); out Q = out P; is_live_abstraction (h1::'b::type ⇒ 'c::type) (C, LC::('b::type × 'a::type option × 'b::type) lift seq ⇒ bool) (A, LA::('c::type × 'a::type option × 'c::type) lift seq ⇒ bool); live_implements (A, LA) (Q, LQ::('d::type × 'a::type option × 'd::type) lift seq ⇒ bool); is_live_abstraction (h2::'d::type ⇒ 'e::type) (Q, LQ) (P, LP::('e::type × 'a::type option × 'e::type) lift seq ⇒ bool)⟧ ⟹ out C = out A›\n 2. ‹⟦out (C::('a::type set × 'a::type set × 'a::type set) × 'b::type set × ('b::type × 'a::type × 'b::type) set × 'a::type set set × 'a::type set set) = out (A::('a::type set × 'a::type set × 'a::type set) × 'c::type set × ('c::type × 'a::type × 'c::type) set × 'a::type set set × 'a::type set set); inp (Q::('a::type set × 'a::type set × 'a::type set) × 'd::type set × ('d::type × 'a::type × 'd::type) set × 'a::type set set × 'a::type set set) = inp (P::('a::type set × 'a::type set × 'a::type set) × 'e::type set × ('e::type × 'a::type × 'e::type) set × 'a::type set set × 'a::type set set); out Q = out P; is_live_abstraction (h1::'b::type ⇒ 'c::type) (C, LC::('b::type × 'a::type option × 'b::type) lift seq ⇒ bool) (A, LA::('c::type × 'a::type option × 'c::type) lift seq ⇒ bool); live_implements (A, LA) (Q, LQ::('d::type × 'a::type option × 'd::type) lift seq ⇒ bool); is_live_abstraction (h2::'d::type ⇒ 'e::type) (Q, LQ) (P, LP::('e::type × 'a::type option × 'e::type) lift seq ⇒ bool)⟧ ⟹ is_live_abstraction (?h::'b::type ⇒ 'c::type) (C, ?M::('b::type × 'a::type option × 'b::type) lift seq ⇒ bool) (A, ?L::('c::type × 'a::type option × 'c::type) lift seq ⇒ bool)›\n 3. ‹⟦out (C::('a::type set × 'a::type set × 'a::type set) × 'b::type set × ('b::type × 'a::type × 'b::type) set × 'a::type set set × 'a::type set set) = out (A::('a::type set × 'a::type set × 'a::type set) × 'c::type set × ('c::type × 'a::type × 'c::type) set × 'a::type set set × 'a::type set set); inp (Q::('a::type set × 'a::type set × 'a::type set) × 'd::type set × ('d::type × 'a::type × 'd::type) set × 'a::type set set × 'a::type set set) = inp (P::('a::type set × 'a::type set × 'a::type set) × 'e::type set × ('e::type × 'a::type × 'e::type) set × 'a::type set set × 'a::type set set); out Q = out P; is_live_abstraction (h1::'b::type ⇒ 'c::type) (C, LC::('b::type × 'a::type option × 'b::type) lift seq ⇒ bool) (A, LA::('c::type × 'a::type option × 'c::type) lift seq ⇒ bool); live_implements (A, LA) (Q, LQ::('d::type × 'a::type option × 'd::type) lift seq ⇒ bool); is_live_abstraction (h2::'d::type ⇒ 'e::type) (Q, LQ) (P, LP::('e::type × 'a::type option × 'e::type) lift seq ⇒ bool); live_implements (C, ?M::('b::type × 'a::type option × 'b::type) lift seq ⇒ bool) (A, ?L::('c::type × 'a::type option × 'c::type) lift seq ⇒ bool)⟧ ⟹ live_implements (C, LC) (P, LP)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply (drule abs_liveness (*‹⟦inp (?C::(?'a set × ?'a set × ?'a set) × ?'b set × (?'b × ?'a × ?'b) set × ?'a set set × ?'a set set) = inp (?A::(?'a set × ?'a set × ?'a set) × ?'c set × (?'c × ?'a × ?'c) set × ?'a set set × ?'a set set); out ?C = out ?A; is_live_abstraction (?h::?'b ⇒ ?'c) (?C, ?M::(?'b × ?'a option × ?'b) lift seq ⇒ bool) (?A, ?L::(?'c × ?'a option × ?'c) lift seq ⇒ bool)⟧ ⟹ live_implements (?C, ?M) (?A, ?L)›*))\n (*goals:\n 1. ‹⟦out C = out A; out Q = out P; is_live_abstraction h1 (C, LC) (A, LA); live_implements (A, LA) (Q, LQ); is_live_abstraction h2 (Q, LQ) (P, LP); live_implements (C, LC) (A, LA)⟧ ⟹ out Q = out P›\n 2. ‹⟦out C = out A; out Q = out P; is_live_abstraction h1 (C, LC) (A, LA); live_implements (A, LA) (Q, LQ); is_live_abstraction h2 (Q, LQ) (P, LP); live_implements (C, LC) (A, LA)⟧ ⟹ is_live_abstraction ?h4 (Q, ?M4) (P, ?L4)›\n 3. ‹⟦out C = out A; out Q = out P; is_live_abstraction h1 (C, LC) (A, LA); live_implements (A, LA) (Q, LQ); is_live_abstraction h2 (Q, LQ) (P, LP); live_implements (C, LC) (A, LA); live_implements (Q, ?M4) (P, ?L4)⟧ ⟹ live_implements (C, LC) (P, LP)›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply (erule live_implements_trans (*‹⟦live_implements (?A::(?'a set × ?'a set × ?'a set) × ?'b set × (?'b × ?'a × ?'b) set × ?'a set set × ?'a set set, ?LA::(?'b × ?'a option × ?'b) lift seq ⇒ bool) (?B::(?'a set × ?'a set × ?'a set) × ?'c set × (?'c × ?'a × ?'c) set × ?'a set set × ?'a set set, ?LB::(?'c × ?'a option × ?'c) lift seq ⇒ bool); live_implements (?B, ?LB) (?C::(?'a set × ?'a set × ?'a set) × ?'d set × (?'d × ?'a × ?'d) set × ?'a set set × ?'a set set, ?LC::(?'d × ?'a option × ?'d) lift seq ⇒ bool)⟧ ⟹ live_implements (?A, ?LA) (?C, ?LC)›*))\n (*goal: ‹⟦out C = out A; out Q = out P; is_live_abstraction h1 (C, LC) (A, LA); live_implements (A, LA) (Q, LQ); is_live_abstraction h2 (Q, LQ) (P, LP); live_implements (C, LC) (A, LA); live_implements (Q, LQ) (P, LP)⟧ ⟹ live_implements (C, LC) (P, LP)›*)\n apply (erule live_implements_trans (*‹⟦live_implements (?A::(?'a set × ?'a set × ?'a set) × ?'b set × (?'b × ?'a × ?'b) set × ?'a set set × ?'a set set, ?LA::(?'b × ?'a option × ?'b) lift seq ⇒ bool) (?B::(?'a set × ?'a set × ?'a set) × ?'c set × (?'c × ?'a × ?'c) set × ?'a set set × ?'a set set, ?LB::(?'c × ?'a option × ?'c) lift seq ⇒ bool); live_implements (?B, ?LB) (?C::(?'a set × ?'a set × ?'a set) × ?'d set × (?'d × ?'a × ?'d) set × ?'a set set × ?'a set set, ?LC::(?'d × ?'a option × ?'d) lift seq ⇒ bool)⟧ ⟹ live_implements (?A, ?LA) (?C, ?LC)›*))\n (*goal: ‹⟦out (C::('a::type set × 'a::type set × 'a::type set) × 'b::type set × ('b::type × 'a::type × 'b::type) set × 'a::type set set × 'a::type set set) = out (A::('a::type set × 'a::type set × 'a::type set) × 'c::type set × ('c::type × 'a::type × 'c::type) set × 'a::type set set × 'a::type set set); out (Q::('a::type set × 'a::type set × 'a::type set) × 'd::type set × ('d::type × 'a::type × 'd::type) set × 'a::type set set × 'a::type set set) = out (P::('a::type set × 'a::type set × 'a::type set) × 'e::type set × ('e::type × 'a::type × 'e::type) set × 'a::type set set × 'a::type set set); is_live_abstraction (h1::'b::type ⇒ 'c::type) (C, LC::('b::type × 'a::type option × 'b::type) lift seq ⇒ bool) (A, LA::('c::type × 'a::type option × 'c::type) lift seq ⇒ bool); live_implements (A, LA) (Q, LQ::('d::type × 'a::type option × 'd::type) lift seq ⇒ bool); is_live_abstraction (h2::'d::type ⇒ 'e::type) (Q, LQ) (P, LP::('e::type × 'a::type option × 'e::type) lift seq ⇒ bool); live_implements (Q, LQ) (P, LP)⟧ ⟹ live_implements (A, LA) (P, LP)›*)\n apply assumption\n (*proven 3 subgoals*)\n (*proven 3 subgoals*) .\n\n\ndeclare split_paired_All [simp del]\n\n\nsubsection ‹Localizing Temporal Strengthenings and Weakenings›\n\nlemma strength_AND:\n \"temp_strengthening P1 Q1 h ⟹ temp_strengthening P2 Q2 h ⟹\n temp_strengthening (P1 ❙∧ P2) (Q1 ❙∧ Q2) h\"\n by (auto simp: temp_strengthening_def (*‹temp_strengthening (?Q::(?'s2.0 × ?'a option × ?'s2.0) lift seq ⇒ bool) (?P::(?'s1.0 × ?'a option × ?'s1.0) lift seq ⇒ bool) (?h::?'s1.0 ⇒ ?'s2.0) = (∀ex::?'s1.0 × (?'a × ?'s1.0) lift seq. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n\nlemma strength_OR:\n \"temp_strengthening P1 Q1 h ⟹ temp_strengthening P2 Q2 h ⟹\n temp_strengthening (P1 ❙∨ P2) (Q1 ❙∨ Q2) h\"\n by (auto simp: temp_strengthening_def (*‹temp_strengthening (?Q::(?'s2.0 × ?'a option × ?'s2.0) lift seq ⇒ bool) (?P::(?'s1.0 × ?'a option × ?'s1.0) lift seq ⇒ bool) (?h::?'s1.0 ⇒ ?'s2.0) = (∀ex::?'s1.0 × (?'a × ?'s1.0) lift seq. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n\nlemma strength_NOT: \"temp_weakening P Q h ⟹ temp_strengthening (❙¬ P) (❙¬ Q) h\"\n by (auto simp: temp_weakening_def2 (*‹temp_weakening ?Q ?P ?h = (∀ex. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*) temp_strengthening_def (*‹temp_strengthening ?Q ?P ?h = (∀ex. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n\nlemma strength_IMPLIES:\n \"temp_weakening P1 Q1 h ⟹ temp_strengthening P2 Q2 h ⟹\n temp_strengthening (P1 ❙⟶ P2) (Q1 ❙⟶ Q2) h\"\n by (simp add: temp_weakening_def2 (*‹temp_weakening ?Q ?P ?h = (∀ex. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*) temp_strengthening_def (*‹temp_strengthening ?Q ?P ?h = (∀ex. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n\n\nlemma weak_AND:\n \"temp_weakening P1 Q1 h ⟹ temp_weakening P2 Q2 h ⟹\n temp_weakening (P1 ❙∧ P2) (Q1 ❙∧ Q2) h\"\n by (simp add: temp_weakening_def2 (*‹temp_weakening ?Q ?P ?h = (∀ex. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*))\n\nlemma weak_OR:\n \"temp_weakening P1 Q1 h ⟹ temp_weakening P2 Q2 h ⟹\n temp_weakening (P1 ❙∨ P2) (Q1 ❙∨ Q2) h\"\n by (simp add: temp_weakening_def2 (*‹temp_weakening ?Q ?P ?h = (∀ex. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*))\n\nlemma weak_NOT:\n \"temp_strengthening P Q h ⟹ temp_weakening (❙¬ P) (❙¬ Q) h\"\n by (auto simp add: temp_weakening_def2 (*‹temp_weakening (?Q::(?'a::type × ?'b::type option × ?'a::type) lift seq ⇒ bool) (?P::(?'c::type × ?'b::type option × ?'c::type) lift seq ⇒ bool) (?h::?'c::type ⇒ ?'a::type) = (∀ex::?'c::type × (?'b::type × ?'c::type) lift seq. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*) temp_strengthening_def (*‹temp_strengthening (?Q::(?'s2.0::type × ?'a::type option × ?'s2.0::type) lift seq ⇒ bool) (?P::(?'s1.0::type × ?'a::type option × ?'s1.0::type) lift seq ⇒ bool) (?h::?'s1.0::type ⇒ ?'s2.0::type) = (∀ex::?'s1.0::type × (?'a::type × ?'s1.0::type) lift seq. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n\nlemma weak_IMPLIES:\n \"temp_strengthening P1 Q1 h ⟹ temp_weakening P2 Q2 h ⟹\n temp_weakening (P1 ❙⟶ P2) (Q1 ❙⟶ Q2) h\"\n by (simp add: temp_weakening_def2 (*‹temp_weakening (?Q::(?'a::type × ?'b::type option × ?'a::type) lift seq ⇒ bool) (?P::(?'c::type × ?'b::type option × ?'c::type) lift seq ⇒ bool) (?h::?'c::type ⇒ ?'a::type) = (∀ex::?'c::type × (?'b::type × ?'c::type) lift seq. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*) temp_strengthening_def (*‹temp_strengthening (?Q::(?'s2.0::type × ?'a::type option × ?'s2.0::type) lift seq ⇒ bool) (?P::(?'s1.0::type × ?'a::type option × ?'s1.0::type) lift seq ⇒ bool) (?h::?'s1.0::type ⇒ ?'s2.0::type) = (∀ex::?'s1.0::type × (?'a::type × ?'s1.0::type) lift seq. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*))\n\n\nsubsubsection ‹Box›\n\n(* FIXME: should be same as nil_is_Conc2 when all nils are turned to right side! *)\nlemma UU_is_Conc: \"(UU = x @@ y) = (((x::'a Seq)= UU) | (x = nil ∧ y = UU))\"\n by (Seq_case_simp x)\n\nlemma ex2seqConc [rule_format]:\n \"Finite s1 ⟶ (∀ex. (s ≠ nil ∧ s ≠ UU ∧ ex2seq ex = s1 @@ s) ⟶ (∃ex'. s = ex2seq ex'))\"\n apply (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹Finite s1 ⟶ (∀ex. s ≠ nil ∧ s ≠ ⊥ ∧ ex2seq ex = s1 @@ s ⟶ (∃ex'. s = ex2seq ex'))›*)\n apply Seq_Finite_induct\n (*goal: ‹Finite s1 ⟹ ∀ex. s ≠ nil ∧ s ≠ ⊥ ∧ ex2seq ex = s1 @@ s ⟶ (∃ex'. s = ex2seq ex')›*)\n apply blast\n (*top goal: ‹∀ex. s ≠ nil ∧ s ≠ ⊥ ∧ ex2seq ex = s ⟶ (∃ex'. s = ex2seq ex')› and 1 goal remains*)\n text ‹main case›\n apply clarify\n (*goal: ‹⋀a sa. ⟦Finite sa; ∀ex. s ≠ nil ∧ s ≠ ⊥ ∧ ex2seq ex = sa @@ s ⟶ (∃ex'. s = ex2seq ex')⟧ ⟹ ∀ex. s ≠ nil ∧ s ≠ ⊥ ∧ ex2seq ex = (a↝sa) @@ s ⟶ (∃ex'. s = ex2seq ex')›*)\n apply (pair ex)\n (*goal: ‹⋀a sa ex. ⟦Finite sa; ∀ex. s ≠ nil ∧ s ≠ ⊥ ∧ ex2seq ex = sa @@ s ⟶ (∃ex'. s = ex2seq ex'); s ≠ nil; s ≠ ⊥; ex2seq ex = (a↝sa) @@ s⟧ ⟹ ∃ex'. s = ex2seq ex'›*)\n apply (Seq_case_simp x2)\n (*goal: ‹⋀a sa x1 x2. ⟦Finite sa; ∀ex. ex2seq ex = sa @@ s ⟶ (∃ex'. s = ex2seq ex'); s ≠ nil; s ≠ ⊥; ex2seq (x1, x2) = a↝sa @@ s⟧ ⟹ ∃ex'. s = ex2seq ex'›*)\n text ‹‹UU› case›\n apply (simp add: nil_is_Conc (*‹(nil = (?x::?'a lift seq) @@ (?y::?'a lift seq)) = (?x = nil ∧ ?y = nil)›*))\n (*top goal: ‹⋀sa. ⟦Finite sa; ∀ex. ex2seq ex = nil ⟶ (∃ex'. s = ex2seq ex'); s ≠ nil; s ≠ ⊥; nil = sa @@ s⟧ ⟹ ∃ex'. s = ex2seq ex'› and 2 goals remain*)\n text ‹‹nil› case›\n apply (simp add: nil_is_Conc (*‹(nil = (?x::?'a::type lift seq) @@ (?y::?'a::type lift seq)) = (?x = nil ∧ ?y = nil)›*))\n (*top goal: ‹⋀sa::('a × 'b option × 'a) lift seq. ⟦Finite sa; ∀ex::'a × ('b × 'a) lift seq. ex2seq ex = nil ⟶ (∃ex'::'a × ('b × 'a) lift seq. (s::('a × 'b option × 'a) lift seq) = ex2seq ex'); s ≠ nil; s ≠ ⊥; nil = sa @@ s⟧ ⟹ ∃ex'::'a × ('b × 'a) lift seq. s = ex2seq ex'› and 1 goal remains*)\n text ‹cons case›\n by (pair aa)\n\n(* important property of ex2seq: can be shiftet, as defined \"pointwise\" *)\nlemma ex2seq_tsuffix: \"tsuffix s (ex2seq ex) ⟹ ∃ex'. s = (ex2seq ex')\"\n apply (unfold tsuffix_def (*‹tsuffix ?s2.0 ?s = (?s2.0 ≠ nil ∧ ?s2.0 ≠ ⊥ ∧ suffix ?s2.0 ?s)›*) suffix_def (*‹suffix ?s2.0 ?s = (∃s1. Finite s1 ∧ ?s = s1 @@ ?s2.0)›*))\n (*goal: ‹tsuffix s (ex2seq ex) ⟹ ∃ex'. s = ex2seq ex'›*)\n apply auto\n (*goal: ‹(s::('a × 'b option × 'a) lift seq) ≠ nil ∧ s ≠ ⊥ ∧ (∃s1::('a × 'b option × 'a) lift seq. Finite s1 ∧ ex2seq (ex::'a × ('b × 'a) lift seq) = s1 @@ s) ⟹ ∃ex'::'a × ('b × 'a) lift seq. s = ex2seq ex'›*)\n apply (drule ex2seqConc (*‹⟦Finite ?s1.0; ?s ≠ nil ∧ ?s ≠ ⊥ ∧ ex2seq ?ex = ?s1.0 @@ ?s⟧ ⟹ ∃ex'. ?s = ex2seq ex'›*))\n (*goals:\n 1. ‹⋀s1. ⟦s ≠ nil; s ≠ ⊥; ex2seq ex = s1 @@ s⟧ ⟹ ?s4 s1 ≠ nil ∧ ?s4 s1 ≠ ⊥ ∧ ex2seq (?ex4 s1) = s1 @@ ?s4 s1›\n 2. ‹⋀s1. ⟦s ≠ nil; s ≠ ⊥; ex2seq ex = s1 @@ s; ∃ex'. ?s4 s1 = ex2seq ex'⟧ ⟹ ∃ex'. s = ex2seq ex'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\n\n(*important property of cex_absSeq: As it is a 1to1 correspondence,\n properties carry over *)\nlemma cex_absSeq_tsuffix: \"tsuffix s t ⟹ tsuffix (cex_absSeq h s) (cex_absSeq h t)\"\n apply (unfold tsuffix_def (*‹tsuffix ?s2.0 ?s = (?s2.0 ≠ nil ∧ ?s2.0 ≠ ⊥ ∧ suffix ?s2.0 ?s)›*) suffix_def (*‹suffix ?s2.0 ?s = (∃s1. Finite s1 ∧ ?s = s1 @@ ?s2.0)›*) cex_absSeq_def (*‹cex_absSeq ?f = Rep_cfun (Map (λ(s, a, t). (?f s, a, ?f t)))›*))\n (*goal: ‹tsuffix s t ⟹ tsuffix (cex_absSeq h s) (cex_absSeq h t)›*)\n apply auto\n (*goals:\n 1. ‹⋀s1. ⟦s ≠ nil; s ≠ ⊥; Finite s1; t = s1 @@ s; Map (λ(s, a, t). (h s, a, h t))⋅s = nil⟧ ⟹ False›\n 2. ‹⋀s1. ⟦s ≠ nil; s ≠ ⊥; Finite s1; t = s1 @@ s; Map (λ(s, a, t). (h s, a, h t))⋅s = ⊥⟧ ⟹ False›\n 3. ‹⋀s1. ⟦s ≠ nil; s ≠ ⊥; Finite s1; t = s1 @@ s⟧ ⟹ ∃s1a. Finite s1a ∧ Map (λ(s, a, t). (h s, a, h t))⋅(s1 @@ s) = s1a @@ Map (λ(s, a, t). (h s, a, h t))⋅s›\n discuss goal 1*)\n apply (simp add: Mapnil (*‹(Map ?f⋅?s = nil) = (?s = nil)›*))\n (*discuss goal 2*)\n apply (simp add: MapUU (*‹(Map ?f⋅?s = ⊥) = (?s = ⊥)›*))\n (*discuss goal 3*)\n apply (rule_tac x = \"Map (% (s,a,t) . (h s,a, h t))⋅s1\" in exI (*‹?P ?x ⟹ ∃x. ?P x›*))\n (*goal: ‹⋀s1. ⟦s ≠ nil; s ≠ ⊥; Finite s1; t = s1 @@ s⟧ ⟹ ∃s1a. Finite s1a ∧ Map (λ(s, a, t). (h s, a, h t))⋅(s1 @@ s) = s1a @@ Map (λ(s, a, t). (h s, a, h t))⋅s›*)\n apply (simp add: Map2Finite (*‹Finite (Map ?f⋅?s) = Finite ?s›*) MapConc (*‹Map ?f⋅(?x @@ ?y) = Map ?f⋅?x @@ Map ?f⋅?y›*))\n (*proven 3 subgoals*) .\n\n\nlemma strength_Box: \"temp_strengthening P Q h ⟹ temp_strengthening (□P) (□Q) h\"\n apply (unfold temp_strengthening_def (*‹temp_strengthening ?Q ?P ?h = (∀ex. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*) state_strengthening_def (*‹state_strengthening ?Q ?P ?h = (∀s t a. ?Q (?h s, a, ?h t) ⟶ ?P (s, a, t))›*) temp_sat_def (*‹(?ex ⊫ ?P) = (ex2seq ?ex ⊨ ?P)›*) satisfies_def (*‹(?s ⊨ ?P) = ?P ?s›*) Box_def (*‹(□?P) ?s = (∀s2. tsuffix s2 ?s ⟶ ?P s2)›*))\n (*goal: ‹temp_strengthening P Q h ⟹ temp_strengthening (□P) (□Q) h›*)\n apply clarify\n (*goal: ‹∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex) ⟹ ∀ex. (∀s2. tsuffix s2 (ex2seq (cex_abs h ex)) ⟶ P s2) ⟶ (∀s2. tsuffix s2 (ex2seq ex) ⟶ Q s2)›*)\n apply (frule ex2seq_tsuffix (*‹tsuffix ?s (ex2seq ?ex) ⟹ ∃ex'. ?s = ex2seq ex'›*))\n (*goal: ‹⋀ex s2. ⟦∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex); ∀s2. tsuffix s2 (ex2seq (cex_abs h ex)) ⟶ P s2; tsuffix s2 (ex2seq ex)⟧ ⟹ Q s2›*)\n apply clarify\n (*goal: ‹⋀ex s2. ⟦∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex); ∀s2. tsuffix s2 (ex2seq (cex_abs h ex)) ⟶ P s2; tsuffix s2 (ex2seq ex); ∃ex'. s2 = ex2seq ex'⟧ ⟹ Q s2›*)\n apply (drule_tac h = \"h\" in cex_absSeq_tsuffix (*‹tsuffix ?s ?t ⟹ tsuffix (cex_absSeq ?h ?s) (cex_absSeq ?h ?t)›*))\n (*goal: ‹⋀(ex::'c × ('b × 'c) lift seq) (s2::('c × 'b option × 'c) lift seq) ex'::'c × ('b × 'c) lift seq. ⟦∀ex::'c × ('b × 'c) lift seq. (P::('a × 'b option × 'a) lift seq ⇒ bool) (ex2seq (cex_abs (h::'c ⇒ 'a) ex)) ⟶ (Q::('c × 'b option × 'c) lift seq ⇒ bool) (ex2seq ex); ∀s2::('a × 'b option × 'a) lift seq. tsuffix s2 (ex2seq (cex_abs h ex)) ⟶ P s2; tsuffix (ex2seq ex') (ex2seq ex)⟧ ⟹ Q (ex2seq ex')›*)\n by (simp add: ex2seq_abs_cex (*‹ex2seq (cex_abs ?h ?ex) = cex_absSeq ?h (ex2seq ?ex)›*))\n\n\nsubsubsection ‹Init›\n\nlemma strength_Init:\n \"state_strengthening P Q h ⟹ temp_strengthening (Init P) (Init Q) h\"\n apply (unfold temp_strengthening_def (*‹temp_strengthening (?Q::(?'s2.0 × ?'a option × ?'s2.0) lift seq ⇒ bool) (?P::(?'s1.0 × ?'a option × ?'s1.0) lift seq ⇒ bool) (?h::?'s1.0 ⇒ ?'s2.0) = (∀ex::?'s1.0 × (?'a × ?'s1.0) lift seq. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*) state_strengthening_def (*‹state_strengthening (?Q::?'s2.0 × ?'a option × ?'s2.0 ⇒ bool) (?P::?'s1.0 × ?'a option × ?'s1.0 ⇒ bool) (?h::?'s1.0 ⇒ ?'s2.0) = (∀(s::?'s1.0) (t::?'s1.0) a::?'a option. ?Q (?h s, a, ?h t) ⟶ ?P (s, a, t))›*) temp_sat_def (*‹((?ex::?'s × (?'a × ?'s) lift seq) ⊫ (?P::(?'s × ?'a option × ?'s) lift seq ⇒ bool)) = (ex2seq ?ex ⊨ ?P)›*) satisfies_def (*‹((?s::?'a) ⊨ (?P::?'a ⇒ bool)) = ?P ?s›*) Init_def (*‹⟨?P::?'a ⇒ bool⟩ (?s::?'a lift seq) = ?P (unlift (HD⋅?s))›*) unlift_def (*‹unlift (?x::?'a lift) = (case ?x of Def (y::?'a) ⇒ y)›*))\n (*goal: ‹state_strengthening P Q h ⟹ temp_strengthening ⟨P⟩ ⟨Q⟩ h›*)\n apply auto\n (*goal: ‹∀s t a. P (h s, a, h t) ⟶ Q (s, a, t) ⟹ ∀ex. P (case HD⋅(ex2seq (cex_abs h ex)) of Def y ⇒ y) ⟶ Q (case HD⋅(ex2seq ex) of Def y ⇒ y)›*)\n apply (pair ex)\n (*goal: ‹⋀ex. ⟦∀s t a. P (h s, a, h t) ⟶ Q (s, a, t); P (case HD⋅(ex2seq (cex_abs h ex)) of Def y ⇒ y)⟧ ⟹ Q (case HD⋅(ex2seq ex) of Def y ⇒ y)›*)\n apply (Seq_case_simp x2)\n (*goal: ‹⋀x1 x2. ⟦∀s t a. P (h s, a, h t) ⟶ Q (s, a, t); P (case HD⋅(ex2seq (cex_abs h (x1, x2))) of Def y ⇒ y)⟧ ⟹ Q (case HD⋅(ex2seq (x1, x2)) of Def y ⇒ y)›*)\n by (pair a)\n\n\nsubsubsection ‹Next›\n\nlemma TL_ex2seq_UU: \"TL ⋅ (ex2seq (cex_abs h ex)) = UU ⟷ TL ⋅ (ex2seq ex) = UU\"\n apply (pair ex)\n (*goal: ‹(TL⋅(ex2seq (cex_abs h ex)) = ⊥) = (TL⋅(ex2seq ex) = ⊥)›*)\n apply (Seq_case_simp x2)\n (*goal: ‹⋀x1 x2. ex = (x1, x2) ⟹ (TL⋅(ex2seq (cex_abs h (x1, x2))) = ⊥) = (TL⋅(ex2seq (x1, x2)) = ⊥)›*)\n apply (pair a)\n (*goal: ‹⋀x1 a s. ex = (x1, a↝s) ⟹ (TL⋅(ex2seq (cex_abs h (x1, a↝s))) = ⊥) = (TL⋅(ex2seq (x1, a↝s)) = ⊥)›*)\n apply (Seq_case_simp s)\n (*goal: ‹⋀(x1::'c) (s::('b × 'c) lift seq) (x1a::'b) x2::'c. (ex::'c × ('b × 'c) lift seq) = (x1, (x1a, x2)↝s) ⟹ (ex2seq ((h::'c ⇒ 'a) x2, snd (cex_abs h (x2, s))) = ⊥) = (ex2seq (x2, s) = ⊥)›*)\n by (pair a)\n\nlemma TL_ex2seq_nil: \"TL ⋅ (ex2seq (cex_abs h ex)) = nil ⟷ TL ⋅ (ex2seq ex) = nil\"\n apply (pair ex)\n (*goal: ‹(TL⋅(ex2seq (cex_abs h ex)) = nil) = (TL⋅(ex2seq ex) = nil)›*)\n apply (Seq_case_simp x2)\n (*goal: ‹⋀x1 x2. ex = (x1, x2) ⟹ (TL⋅(ex2seq (cex_abs h (x1, x2))) = nil) = (TL⋅(ex2seq (x1, x2)) = nil)›*)\n apply (pair a)\n (*goal: ‹⋀(x1::'c::type) (a::'b::type × 'c::type) s::('b::type × 'c::type) lift seq. (ex::'c::type × ('b::type × 'c::type) lift seq) = (x1, a↝s) ⟹ (TL⋅(ex2seq (cex_abs (h::'c::type ⇒ 'a::type) (x1, a↝s))) = nil) = (TL⋅(ex2seq (x1, a↝s)) = nil)›*)\n apply (Seq_case_simp s)\n (*goal: ‹⋀x1 s x1a x2. ex = (x1, (x1a, x2)↝s) ⟹ (ex2seq (h x2, snd (cex_abs h (x2, s))) = nil) = (ex2seq (x2, s) = nil)›*)\n by (pair a)\n\n(*important property of cex_absSeq: As it is a 1to1 correspondence,\n properties carry over*)\nlemma cex_absSeq_TL: \"cex_absSeq h (TL ⋅ s) = TL ⋅ (cex_absSeq h s)\"\n by (simp add: MapTL (*‹Map ?f⋅(TL⋅?s) = TL⋅(Map ?f⋅?s)›*) cex_absSeq_def (*‹cex_absSeq ?f = Rep_cfun (Map (λ(s, a, t). (?f s, a, ?f t)))›*))\n\n(* important property of ex2seq: can be shiftet, as defined \"pointwise\" *)\nlemma TLex2seq: \"snd ex ≠ UU ⟹ snd ex ≠ nil ⟹ ∃ex'. TL⋅(ex2seq ex) = ex2seq ex'\"\n apply (pair ex)\n (*goal: ‹⟦snd ex ≠ ⊥; snd ex ≠ nil⟧ ⟹ ∃ex'. TL⋅(ex2seq ex) = ex2seq ex'›*)\n apply (Seq_case_simp x2)\n (*goal: ‹⋀x1 x2. ⟦x2 ≠ ⊥; x2 ≠ nil; ex = (x1, x2)⟧ ⟹ ∃ex'. TL⋅(ex2seq (x1, x2)) = ex2seq ex'›*)\n apply (pair a)\n (*goal: ‹⋀x1 a s. ⟦a↝s ≠ ⊥; a↝s ≠ nil; ex = (x1, a↝s)⟧ ⟹ ∃ex'. TL⋅(ex2seq (x1, a↝s)) = ex2seq ex'›*)\n by auto\n\nlemma ex2seqnilTL: \"TL ⋅ (ex2seq ex) ≠ nil ⟷ snd ex ≠ nil ∧ snd ex ≠ UU\"\n apply (pair ex)\n (*goal: ‹(TL⋅(ex2seq ex) ≠ nil) = (snd ex ≠ nil ∧ snd ex ≠ ⊥)›*)\n apply (Seq_case_simp x2)\n (*goal: ‹⋀(x1::'a::type) x2::('b::type × 'a::type) lift seq. (ex::'a::type × ('b::type × 'a::type) lift seq) = (x1, x2) ⟹ (TL⋅(ex2seq (x1, x2)) ≠ nil) = (x2 ≠ nil ∧ x2 ≠ ⊥)›*)\n apply (pair a)\n (*goal: ‹⋀x1 a s. ex = (x1, a↝s) ⟹ TL⋅(ex2seq (x1, a↝s)) ≠ nil›*)\n apply (Seq_case_simp s)\n (*goal: ‹⋀x1 s x1a x2. ex = (x1, (x1a, x2)↝s) ⟹ ex2seq (x2, s) ≠ nil›*)\n by (pair a)\n\nlemma strength_Next: \"temp_strengthening P Q h ⟹ temp_strengthening (○P) (○Q) h\"\n apply (unfold temp_strengthening_def (*‹temp_strengthening (?Q::(?'s2.0 × ?'a option × ?'s2.0) lift seq ⇒ bool) (?P::(?'s1.0 × ?'a option × ?'s1.0) lift seq ⇒ bool) (?h::?'s1.0 ⇒ ?'s2.0) = (∀ex::?'s1.0 × (?'a × ?'s1.0) lift seq. (cex_abs ?h ex ⊫ ?Q) ⟶ (ex ⊫ ?P))›*) state_strengthening_def (*‹state_strengthening (?Q::?'s2.0 × ?'a option × ?'s2.0 ⇒ bool) (?P::?'s1.0 × ?'a option × ?'s1.0 ⇒ bool) (?h::?'s1.0 ⇒ ?'s2.0) = (∀(s::?'s1.0) (t::?'s1.0) a::?'a option. ?Q (?h s, a, ?h t) ⟶ ?P (s, a, t))›*) temp_sat_def (*‹((?ex::?'s × (?'a × ?'s) lift seq) ⊫ (?P::(?'s × ?'a option × ?'s) lift seq ⇒ bool)) = (ex2seq ?ex ⊨ ?P)›*) satisfies_def (*‹((?s::?'a) ⊨ (?P::?'a ⇒ bool)) = ?P ?s›*) Next_def (*‹(○(?P::?'a lift seq ⇒ bool)) (?s::?'a lift seq) = (if TL⋅?s = ⊥ ∨ TL⋅?s = nil then ?P ?s else ?P (TL⋅?s))›*))\n (*goal: ‹temp_strengthening P Q h ⟹ temp_strengthening (○P) (○Q) h›*)\n apply simp\n (*goal: ‹∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex) ⟹ ∀ex. (if TL⋅(ex2seq (cex_abs h ex)) = ⊥ ∨ TL⋅(ex2seq (cex_abs h ex)) = nil then P (ex2seq (cex_abs h ex)) else P (TL⋅(ex2seq (cex_abs h ex)))) ⟶ (if TL⋅(ex2seq ex) = ⊥ ∨ TL⋅(ex2seq ex) = nil then Q (ex2seq ex) else Q (TL⋅(ex2seq ex)))›*)\n apply auto\n (*goal: ‹∀ex::'c::type × ('b::type × 'c::type) lift seq. (P::('a::type × 'b::type option × 'a::type) lift seq ⇒ bool) (ex2seq (cex_abs (h::'c::type ⇒ 'a::type) ex)) ⟶ (Q::('c::type × 'b::type option × 'c::type) lift seq ⇒ bool) (ex2seq ex) ⟹ ∀ex::'c::type × ('b::type × 'c::type) lift seq. (TL⋅(ex2seq (cex_abs h ex)) = ⊥ ⟶ TL⋅(ex2seq ex) ≠ ⊥ ∧ TL⋅(ex2seq ex) ≠ nil ⟶ P (ex2seq (cex_abs h ex)) ⟶ Q (TL⋅(ex2seq ex))) ∧ (TL⋅(ex2seq (cex_abs h ex)) = nil ⟶ TL⋅(ex2seq ex) ≠ ⊥ ∧ TL⋅(ex2seq ex) ≠ nil ⟶ P (ex2seq (cex_abs h ex)) ⟶ Q (TL⋅(ex2seq ex))) ∧ (TL⋅(ex2seq (cex_abs h ex)) ≠ ⊥ ∧ TL⋅(ex2seq (cex_abs h ex)) ≠ nil ⟶ (TL⋅(ex2seq ex) = ⊥ ⟶ P (TL⋅(ex2seq (cex_abs h ex))) ⟶ Q (ex2seq ex)) ∧ (TL⋅(ex2seq ex) = nil ⟶ P (TL⋅(ex2seq (cex_abs h ex))) ⟶ Q (ex2seq ex)) ∧ (TL⋅(ex2seq ex) ≠ ⊥ ∧ TL⋅(ex2seq ex) ≠ nil ⟶ P (TL⋅(ex2seq (cex_abs h ex))) ⟶ Q (TL⋅(ex2seq ex))))›*)\n apply (simp add: TL_ex2seq_nil (*‹(TL⋅(ex2seq (cex_abs (?h::?'c ⇒ ?'a) (?ex::?'c × (?'b × ?'c) lift seq))) = nil) = (TL⋅(ex2seq ?ex) = nil)›*) TL_ex2seq_UU (*‹(TL⋅(ex2seq (cex_abs (?h::?'c ⇒ ?'a) (?ex::?'c × (?'b × ?'c) lift seq))) = ⊥) = (TL⋅(ex2seq ?ex) = ⊥)›*))\n (*top goal: ‹⋀ex. ⟦∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex); TL⋅(ex2seq (cex_abs h ex)) = ⊥; TL⋅(ex2seq ex) ≠ ⊥; TL⋅(ex2seq ex) ≠ nil; P (ex2seq (cex_abs h ex))⟧ ⟹ Q (TL⋅(ex2seq ex))› and 4 goals remain*)\n apply (simp add: TL_ex2seq_nil (*‹(TL⋅(ex2seq (cex_abs ?h ?ex)) = nil) = (TL⋅(ex2seq ?ex) = nil)›*) TL_ex2seq_UU (*‹(TL⋅(ex2seq (cex_abs ?h ?ex)) = ⊥) = (TL⋅(ex2seq ?ex) = ⊥)›*))\n (*top goal: ‹⋀ex. ⟦∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex); TL⋅(ex2seq (cex_abs h ex)) = nil; TL⋅(ex2seq ex) ≠ ⊥; TL⋅(ex2seq ex) ≠ nil; P (ex2seq (cex_abs h ex))⟧ ⟹ Q (TL⋅(ex2seq ex))› and 3 goals remain*)\n apply (simp add: TL_ex2seq_nil (*‹(TL⋅(ex2seq (cex_abs (?h::?'c::type ⇒ ?'a::type) (?ex::?'c::type × (?'b::type × ?'c::type) lift seq))) = nil) = (TL⋅(ex2seq ?ex) = nil)›*) TL_ex2seq_UU (*‹(TL⋅(ex2seq (cex_abs (?h::?'c::type ⇒ ?'a::type) (?ex::?'c::type × (?'b::type × ?'c::type) lift seq))) = ⊥) = (TL⋅(ex2seq ?ex) = ⊥)›*))\n (*top goal: ‹⋀ex. ⟦∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex); TL⋅(ex2seq (cex_abs h ex)) ≠ ⊥; TL⋅(ex2seq (cex_abs h ex)) ≠ nil; TL⋅(ex2seq ex) = ⊥; P (TL⋅(ex2seq (cex_abs h ex)))⟧ ⟹ Q (ex2seq ex)› and 2 goals remain*)\n apply (simp add: TL_ex2seq_nil (*‹(TL⋅(ex2seq (cex_abs ?h ?ex)) = nil) = (TL⋅(ex2seq ?ex) = nil)›*) TL_ex2seq_UU (*‹(TL⋅(ex2seq (cex_abs ?h ?ex)) = ⊥) = (TL⋅(ex2seq ?ex) = ⊥)›*))\n (*top goal: ‹⋀ex. ⟦∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex); TL⋅(ex2seq (cex_abs h ex)) ≠ ⊥; TL⋅(ex2seq (cex_abs h ex)) ≠ nil; TL⋅(ex2seq ex) = nil; P (TL⋅(ex2seq (cex_abs h ex)))⟧ ⟹ Q (ex2seq ex)› and 1 goal remains*)\n text ‹cons case›\n apply (simp add: TL_ex2seq_nil (*‹(TL⋅(ex2seq (cex_abs ?h ?ex)) = nil) = (TL⋅(ex2seq ?ex) = nil)›*) TL_ex2seq_UU (*‹(TL⋅(ex2seq (cex_abs ?h ?ex)) = ⊥) = (TL⋅(ex2seq ?ex) = ⊥)›*) ex2seq_abs_cex (*‹ex2seq (cex_abs ?h ?ex) = cex_absSeq ?h (ex2seq ?ex)›*) cex_absSeq_TL [symmetric] (*‹TL⋅(cex_absSeq ?h ?s) = cex_absSeq ?h (TL⋅?s)›*) ex2seqnilTL (*‹(TL⋅(ex2seq ?ex) ≠ nil) = (snd ?ex ≠ nil ∧ snd ?ex ≠ ⊥)›*))\n (*goal: ‹⋀ex. ⟦∀ex. P (ex2seq (cex_abs h ex)) ⟶ Q (ex2seq ex); TL⋅(ex2seq (cex_abs h ex)) ≠ ⊥; TL⋅(ex2seq (cex_abs h ex)) ≠ nil; TL⋅(ex2seq ex) ≠ ⊥; TL⋅(ex2seq ex) ≠ nil; P (TL⋅(ex2seq (cex_abs h ex)))⟧ ⟹ Q (TL⋅(ex2seq ex))›*)\n apply (erule conjE (*‹⟦?P ∧ ?Q; ⟦?P; ?Q⟧ ⟹ ?R⟧ ⟹ ?R›*))\n (*goal: ‹⋀ex. ⟦∀ex. P (cex_absSeq h (ex2seq ex)) ⟶ Q (ex2seq ex); cex_absSeq h (TL⋅(ex2seq ex)) ≠ ⊥; cex_absSeq h (TL⋅(ex2seq ex)) ≠ nil; TL⋅(ex2seq ex) ≠ ⊥; snd ex ≠ nil ∧ snd ex ≠ ⊥; P (cex_absSeq h (TL⋅(ex2seq ex)))⟧ ⟹ Q (TL⋅(ex2seq ex))›*)\n apply (drule TLex2seq (*‹⟦snd ?ex ≠ ⊥; snd ?ex ≠ nil⟧ ⟹ ∃ex'. TL⋅(ex2seq ?ex) = ex2seq ex'›*))\n (*goals:\n 1. ‹⋀ex. ⟦∀ex. P (cex_absSeq h (ex2seq ex)) ⟶ Q (ex2seq ex); cex_absSeq h (TL⋅(ex2seq ex)) ≠ ⊥; cex_absSeq h (TL⋅(ex2seq ex)) ≠ nil; TL⋅(ex2seq ex) ≠ ⊥; P (cex_absSeq h (TL⋅(ex2seq ex))); snd ex ≠ nil⟧ ⟹ snd ex ≠ nil›\n 2. ‹⋀ex. ⟦∀ex. P (cex_absSeq h (ex2seq ex)) ⟶ Q (ex2seq ex); cex_absSeq h (TL⋅(ex2seq ex)) ≠ ⊥; cex_absSeq h (TL⋅(ex2seq ex)) ≠ nil; TL⋅(ex2seq ex) ≠ ⊥; P (cex_absSeq h (TL⋅(ex2seq ex))); snd ex ≠ nil; ∃ex'. TL⋅(ex2seq ex) = ex2seq ex'⟧ ⟹ Q (TL⋅(ex2seq ex))›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply auto\n (*proven 2 subgoals*) .\n\n\ntext ‹Localizing Temporal Weakenings - 2›\n\nlemma weak_Init: \"state_weakening P Q h ⟹ temp_weakening (Init P) (Init Q) h\"\n apply (simp add: temp_weakening_def2 (*‹temp_weakening ?Q ?P ?h = (∀ex. (ex ⊫ ?P) ⟶ (cex_abs ?h ex ⊫ ?Q))›*) state_weakening_def2 (*‹state_weakening ?Q ?P ?h = (∀s t a. ?P (s, a, t) ⟶ ?Q (?h s, a, ?h t))›*) temp_sat_def (*‹(?ex ⊫ ?P) = (ex2seq ?ex ⊨ ?P)›*) satisfies_def (*‹(?s ⊨ ?P) = ?P ?s›*) Init_def (*‹⟨?P⟩ ?s = ?P (unlift (HD⋅?s))›*) unlift_def (*‹unlift ?x = (case ?x of Def y ⇒ y)›*))\n (*goal: ‹state_weakening (P::'a × 'b option × 'a ⇒ bool) (Q::'c × 'b option × 'c ⇒ bool) (h::'c ⇒ 'a) ⟹ temp_weakening ⟨P⟩ ⟨Q⟩ h›*)\n apply auto\n (*goal: ‹∀s t a. Q (s, a, t) ⟶ P (h s, a, h t) ⟹ ∀ex. Q (case HD⋅(ex2seq ex) of Def y ⇒ y) ⟶ P (case HD⋅(ex2seq (cex_abs h ex)) of Def y ⇒ y)›*)\n apply (pair ex)\n (*goal: ‹⋀ex::'c × ('b × 'c) lift seq. ⟦∀(s::'c) (t::'c) a::'b option. (Q::'c × 'b option × 'c ⇒ bool) (s, a, t) ⟶ (P::'a × 'b option × 'a ⇒ bool) ((h::'c ⇒ 'a) s, a, h t); Q (case HD⋅(ex2seq ex) of Def (y::'c × 'b option × 'c) ⇒ y)⟧ ⟹ P (case HD⋅(ex2seq (cex_abs h ex)) of Def (y::'a × 'b option × 'a) ⇒ y)›*)\n apply (Seq_case_simp x2)\n (*goal: ‹⋀x1 x2. ⟦∀s t a. Q (s, a, t) ⟶ P (h s, a, h t); Q (case HD⋅(ex2seq (x1, x2)) of Def y ⇒ y)⟧ ⟹ P (case HD⋅(ex2seq (cex_abs h (x1, x2))) of Def y ⇒ y)›*)\n by (pair a)\n\n\ntext ‹Localizing Temproal Strengthenings - 3›\n\nlemma strength_Diamond: \"temp_strengthening P Q h ⟹ temp_strengthening (◇P) (◇Q) h\"\n apply (unfold Diamond_def (*‹◇(?P::?'a lift seq ⇒ bool) = (❙¬ □(❙¬ ?P))›*))\n (*goal: ‹temp_strengthening P Q h ⟹ temp_strengthening (◇P) (◇Q) h›*)\n apply (rule strength_NOT (*‹temp_weakening ?P ?Q ?h ⟹ temp_strengthening (❙¬ ?P) (❙¬ ?Q) ?h›*))\n (*goal: ‹temp_strengthening P Q h ⟹ temp_strengthening (❙¬ □(❙¬ P)) (❙¬ □(❙¬ Q)) h›*)\n apply (rule weak_Box (*‹temp_weakening ?P ?Q ?h ⟹ temp_weakening (□?P) (□?Q) ?h›*))\n (*goal: ‹temp_strengthening P Q h ⟹ temp_weakening (□(❙¬ P)) (□(❙¬ Q)) h›*)\n by (erule weak_NOT (*‹temp_strengthening ?P ?Q ?h ⟹ temp_weakening (❙¬ ?P) (❙¬ ?Q) ?h›*))\n\nlemma strength_Leadsto:\n \"temp_weakening P1 P2 h ⟹ temp_strengthening Q1 Q2 h ⟹\n temp_strengthening (P1 ↝ Q1) (P2 ↝ Q2) h\"\n apply (unfold Leadsto_def (*‹(?P ↝ ?Q) = □(?P ❙⟶ ◇?Q)›*))\n (*goal: ‹⟦temp_weakening P1 P2 h; temp_strengthening Q1 Q2 h⟧ ⟹ temp_strengthening (P1 ↝ Q1) (P2 ↝ Q2) h›*)\n apply (rule strength_Box (*‹temp_strengthening ?P ?Q ?h ⟹ temp_strengthening (□?P) (□?Q) ?h›*))\n (*goal: ‹⟦temp_weakening P1 P2 h; temp_strengthening Q1 Q2 h⟧ ⟹ temp_strengthening (□(P1 ❙⟶ ◇Q1)) (□(P2 ❙⟶ ◇Q2)) h›*)\n apply (erule strength_IMPLIES (*‹⟦temp_weakening (?P1.0::(?'a × ?'b option × ?'a) lift seq ⇒ bool) (?Q1.0::(?'c × ?'b option × ?'c) lift seq ⇒ bool) (?h::?'c ⇒ ?'a); temp_strengthening (?P2.0::(?'a × ?'b option × ?'a) lift seq ⇒ bool) (?Q2.0::(?'c × ?'b option × ?'c) lift seq ⇒ bool) ?h⟧ ⟹ temp_strengthening (?P1.0 ❙⟶ ?P2.0) (?Q1.0 ❙⟶ ?Q2.0) ?h›*))\n (*goal: ‹⟦temp_weakening (P1::('a × 'b option × 'a) lift seq ⇒ bool) (P2::('c × 'b option × 'c) lift seq ⇒ bool) (h::'c ⇒ 'a); temp_strengthening (Q1::('a × 'b option × 'a) lift seq ⇒ bool) (Q2::('c × 'b option × 'c) lift seq ⇒ bool) h⟧ ⟹ temp_strengthening (P1 ❙⟶ ◇Q1) (P2 ❙⟶ ◇Q2) h›*)\n by (erule strength_Diamond (*‹temp_strengthening ?P ?Q ?h ⟹ temp_strengthening (◇?P) (◇?Q) ?h›*))\n\n\ntext ‹Localizing Temporal Weakenings - 3›\n\nlemma weak_Diamond: \"temp_weakening P Q h ⟹ temp_weakening (◇P) (◇Q) h\"\n apply (unfold Diamond_def (*‹◇?P = (❙¬ □(❙¬ ?P))›*))\n (*goal: ‹temp_weakening P Q h ⟹ temp_weakening (◇P) (◇Q) h›*)\n apply (rule weak_NOT (*‹temp_strengthening ?P ?Q ?h ⟹ temp_weakening (❙¬ ?P) (❙¬ ?Q) ?h›*))\n (*goal: ‹temp_weakening P Q h ⟹ temp_weakening (❙¬ □(❙¬ P)) (❙¬ □(❙¬ Q)) h›*)\n apply (rule strength_Box (*‹temp_strengthening ?P ?Q ?h ⟹ temp_strengthening (□?P) (□?Q) ?h›*))\n (*goal: ‹temp_weakening P Q h ⟹ temp_strengthening (□(❙¬ P)) (□(❙¬ Q)) h›*)\n by (erule strength_NOT (*‹temp_weakening ?P ?Q ?h ⟹ temp_strengthening (❙¬ ?P) (❙¬ ?Q) ?h›*))\n\nlemma weak_Leadsto:\n \"temp_strengthening P1 P2 h ⟹ temp_weakening Q1 Q2 h ⟹\n temp_weakening (P1 ↝ Q1) (P2 ↝ Q2) h\"\n apply (unfold Leadsto_def (*‹(?P ↝ ?Q) = □(?P ❙⟶ ◇?Q)›*))\n (*goal: ‹⟦temp_strengthening P1 P2 h; temp_weakening Q1 Q2 h⟧ ⟹ temp_weakening (P1 ↝ Q1) (P2 ↝ Q2) h›*)\n apply (rule weak_Box (*‹temp_weakening ?P ?Q ?h ⟹ temp_weakening (□?P) (□?Q) ?h›*))\n (*goal: ‹⟦temp_strengthening P1 P2 h; temp_weakening Q1 Q2 h⟧ ⟹ temp_weakening (□(P1 ❙⟶ ◇Q1)) (□(P2 ❙⟶ ◇Q2)) h›*)\n apply (erule weak_IMPLIES (*‹⟦temp_strengthening ?P1.0 ?Q1.0 ?h; temp_weakening ?P2.0 ?Q2.0 ?h⟧ ⟹ temp_weakening (?P1.0 ❙⟶ ?P2.0) (?Q1.0 ❙⟶ ?Q2.0) ?h›*))\n (*goal: ‹⟦temp_strengthening (P1::('a × 'b option × 'a) lift seq ⇒ bool) (P2::('c × 'b option × 'c) lift seq ⇒ bool) (h::'c ⇒ 'a); temp_weakening (Q1::('a × 'b option × 'a) lift seq ⇒ bool) (Q2::('c × 'b option × 'c) lift seq ⇒ bool) h⟧ ⟹ temp_weakening (P1 ❙⟶ ◇Q1) (P2 ❙⟶ ◇Q2) h›*)\n by (erule weak_Diamond (*‹temp_weakening ?P ?Q ?h ⟹ temp_weakening (◇?P) (◇?Q) ?h›*))\n\nlemma weak_WF:\n \"(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s)\n ⟹ temp_weakening (WF A acts) (WF C acts) h\"\n apply (unfold WF_def (*‹WF ?A ?acts = (◇□⟨λ(s, a, t). Enabled ?A ?acts s⟩ ❙⟶ □◇⟨xt2 (plift (λa. a ∈ ?acts))⟩)›*))\n (*goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_weakening (WF A acts) (WF C acts) h›*)\n apply (rule weak_IMPLIES (*‹⟦temp_strengthening ?P1.0 ?Q1.0 ?h; temp_weakening ?P2.0 ?Q2.0 ?h⟧ ⟹ temp_weakening (?P1.0 ❙⟶ ?P2.0) (?Q1.0 ❙⟶ ?Q2.0) ?h›*))\n (*goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_weakening (◇□⟨λ(s, a, t). Enabled A acts s⟩ ❙⟶ □◇⟨xt2 (plift (λa. a ∈ acts))⟩) (◇□⟨λ(s, a, t). Enabled C acts s⟩ ❙⟶ □◇⟨xt2 (plift (λa. a ∈ acts))⟩) h›*)\n apply (rule strength_Diamond (*‹temp_strengthening ?P ?Q ?h ⟹ temp_strengthening (◇?P) (◇?Q) ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_strengthening (◇□⟨λ(s, a, t). Enabled A acts s⟩) (◇□⟨λ(s, a, t). Enabled C acts s⟩) h› and 1 goal remains*)\n apply (rule strength_Box (*‹temp_strengthening ?P ?Q ?h ⟹ temp_strengthening (□?P) (□?Q) ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_strengthening (□⟨λ(s, a, t). Enabled A acts s⟩) (□⟨λ(s, a, t). Enabled C acts s⟩) h› and 1 goal remains*)\n apply (rule strength_Init (*‹state_strengthening ?P ?Q ?h ⟹ temp_strengthening ⟨?P⟩ ⟨?Q⟩ ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_strengthening ⟨λ(s, a, t). Enabled A acts s⟩ ⟨λ(s, a, t). Enabled C acts s⟩ h› and 1 goal remains*)\n apply (rule_tac [2] weak_Box (*‹temp_weakening ?P ?Q ?h ⟹ temp_weakening (□?P) (□?Q) ?h›*))\n (*top goal: ‹(⋀s::'a. Enabled (A::('b set × 'b set × 'b set) × 'c set × ('c × 'b × 'c) set × 'b set set × 'b set set) (acts::'b set) ((h::'a ⇒ 'c) s) ⟹ Enabled (C::('b set × 'b set × 'b set) × 'a set × ('a × 'b × 'a) set × 'b set set × 'b set set) acts s) ⟹ state_strengthening (λ(s::'c, a::'b option, t::'c). Enabled A acts s) (λ(s::'a, a::'b option, t::'a). Enabled C acts s) h› and 1 goal remains*)\n apply (rule_tac [2] weak_Diamond (*‹temp_weakening ?P ?Q ?h ⟹ temp_weakening (◇?P) (◇?Q) ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ state_strengthening (λ(s, a, t). Enabled A acts s) (λ(s, a, t). Enabled C acts s) h› and 1 goal remains*)\n apply (rule_tac [2] weak_Init (*‹state_weakening (?P::?'a × ?'b option × ?'a ⇒ bool) (?Q::?'c × ?'b option × ?'c ⇒ bool) (?h::?'c ⇒ ?'a) ⟹ temp_weakening ⟨?P⟩ ⟨?Q⟩ ?h›*))\n (*top goal: ‹(⋀s::'a::type. Enabled (A::('b::type set × 'b::type set × 'b::type set) × 'c::type set × ('c::type × 'b::type × 'c::type) set × 'b::type set set × 'b::type set set) (acts::'b::type set) ((h::'a::type ⇒ 'c::type) s) ⟹ Enabled (C::('b::type set × 'b::type set × 'b::type set) × 'a::type set × ('a::type × 'b::type × 'a::type) set × 'b::type set set × 'b::type set set) acts s) ⟹ state_strengthening (λ(s::'c::type, a::'b::type option, t::'c::type). Enabled A acts s) (λ(s::'a::type, a::'b::type option, t::'a::type). Enabled C acts s) h› and 1 goal remains*)\n by (auto simp add: state_weakening_def (*‹state_weakening ?Q ?P ?h = state_strengthening (❙¬ ?Q) (❙¬ ?P) ?h›*) state_strengthening_def (*‹state_strengthening ?Q ?P ?h = (∀s t a. ?Q (?h s, a, ?h t) ⟶ ?P (s, a, t))›*) xt2_def (*‹xt2 ?P ?tr = ?P (fst (snd ?tr))›*) plift_def (*‹plift ?P = option_lift ?P False›*) option_lift_def (*‹option_lift ?f ?s ?y = (case ?y of None ⇒ ?s | Some x ⇒ ?f x)›*) NOT_def (*‹(❙¬ ?P) ?s = (¬ ?P ?s)›*))\n\nlemma weak_SF:\n \"(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s)\n ⟹ temp_weakening (SF A acts) (SF C acts) h\"\n apply (unfold SF_def (*‹SF ?A ?acts = (□◇⟨λ(s, a, t). Enabled ?A ?acts s⟩ ❙⟶ □◇⟨xt2 (plift (λa. a ∈ ?acts))⟩)›*))\n (*goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_weakening (SF A acts) (SF C acts) h›*)\n apply (rule weak_IMPLIES (*‹⟦temp_strengthening ?P1.0 ?Q1.0 ?h; temp_weakening ?P2.0 ?Q2.0 ?h⟧ ⟹ temp_weakening (?P1.0 ❙⟶ ?P2.0) (?Q1.0 ❙⟶ ?Q2.0) ?h›*))\n (*goal: ‹(⋀s::'a. Enabled (A::('b set × 'b set × 'b set) × 'c set × ('c × 'b × 'c) set × 'b set set × 'b set set) (acts::'b set) ((h::'a ⇒ 'c) s) ⟹ Enabled (C::('b set × 'b set × 'b set) × 'a set × ('a × 'b × 'a) set × 'b set set × 'b set set) acts s) ⟹ temp_weakening (□◇⟨λ(s::'c, a::'b option, t::'c). Enabled A acts s⟩ ❙⟶ □◇⟨xt2 (plift (λa::'b. a ∈ acts))⟩) (□◇⟨λ(s::'a, a::'b option, t::'a). Enabled C acts s⟩ ❙⟶ □◇⟨xt2 (plift (λa::'b. a ∈ acts))⟩) h›*)\n apply (rule strength_Box (*‹temp_strengthening (?P::(?'a × ?'b option × ?'a) lift seq ⇒ bool) (?Q::(?'c × ?'b option × ?'c) lift seq ⇒ bool) (?h::?'c ⇒ ?'a) ⟹ temp_strengthening (□?P) (□?Q) ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_strengthening (□◇⟨λ(s, a, t). Enabled A acts s⟩) (□◇⟨λ(s, a, t). Enabled C acts s⟩) h› and 1 goal remains*)\n apply (rule strength_Diamond (*‹temp_strengthening ?P ?Q ?h ⟹ temp_strengthening (◇?P) (◇?Q) ?h›*))\n (*top goal: ‹(⋀s::'a::type. Enabled (A::('b::type set × 'b::type set × 'b::type set) × 'c::type set × ('c::type × 'b::type × 'c::type) set × 'b::type set set × 'b::type set set) (acts::'b::type set) ((h::'a::type ⇒ 'c::type) s) ⟹ Enabled (C::('b::type set × 'b::type set × 'b::type set) × 'a::type set × ('a::type × 'b::type × 'a::type) set × 'b::type set set × 'b::type set set) acts s) ⟹ temp_strengthening (◇⟨λ(s::'c::type, a::'b::type option, t::'c::type). Enabled A acts s⟩) (◇⟨λ(s::'a::type, a::'b::type option, t::'a::type). Enabled C acts s⟩) h› and 1 goal remains*)\n apply (rule strength_Init (*‹state_strengthening ?P ?Q ?h ⟹ temp_strengthening ⟨?P⟩ ⟨?Q⟩ ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ temp_strengthening ⟨λ(s, a, t). Enabled A acts s⟩ ⟨λ(s, a, t). Enabled C acts s⟩ h› and 1 goal remains*)\n apply (rule_tac [2] weak_Box (*‹temp_weakening (?P::(?'a × ?'b option × ?'a) lift seq ⇒ bool) (?Q::(?'c × ?'b option × ?'c) lift seq ⇒ bool) (?h::?'c ⇒ ?'a) ⟹ temp_weakening (□?P) (□?Q) ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ state_strengthening (λ(s, a, t). Enabled A acts s) (λ(s, a, t). Enabled C acts s) h› and 1 goal remains*)\n apply (rule_tac [2] weak_Diamond (*‹temp_weakening ?P ?Q ?h ⟹ temp_weakening (◇?P) (◇?Q) ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ state_strengthening (λ(s, a, t). Enabled A acts s) (λ(s, a, t). Enabled C acts s) h› and 1 goal remains*)\n apply (rule_tac [2] weak_Init (*‹state_weakening ?P ?Q ?h ⟹ temp_weakening ⟨?P⟩ ⟨?Q⟩ ?h›*))\n (*top goal: ‹(⋀s. Enabled A acts (h s) ⟹ Enabled C acts s) ⟹ state_strengthening (λ(s, a, t). Enabled A acts s) (λ(s, a, t). Enabled C acts s) h› and 1 goal remains*)\n by (auto simp add: state_weakening_def (*‹state_weakening ?Q ?P ?h = state_strengthening (❙¬ ?Q) (❙¬ ?P) ?h›*) state_strengthening_def (*‹state_strengthening ?Q ?P ?h = (∀s t a. ?Q (?h s, a, ?h t) ⟶ ?P (s, a, t))›*) xt2_def (*‹xt2 ?P ?tr = ?P (fst (snd ?tr))›*) plift_def (*‹plift ?P = option_lift ?P False›*) option_lift_def (*‹option_lift ?f ?s ?y = (case ?y of None ⇒ ?s | Some x ⇒ ?f x)›*) NOT_def (*‹(❙¬ ?P) ?s = (¬ ?P ?s)›*))\n\n\nlemmas weak_strength_lemmas =\n weak_OR weak_AND weak_NOT weak_IMPLIES weak_Box weak_Next weak_Init\n weak_Diamond weak_Leadsto strength_OR strength_AND strength_NOT\n strength_IMPLIES strength_Box strength_Next strength_Init\n strength_Diamond strength_Leadsto weak_WF weak_SF\n\nML ‹\nfun abstraction_tac ctxt =\n SELECT_GOAL (auto_tac\n (ctxt addSIs @{thms weak_strength_lemmas}\n addsimps [@{thm state_strengthening_def}, @{thm state_weakening_def}]))\n›\n\nmethod_setup abstraction = ‹Scan.succeed (SIMPLE_METHOD' o abstraction_tac)›\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/HOLCF/IOA/Abstraction.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"668ec7d78200263e343e1729ae5efdcc611b8aac8e4cb21c5c255827e02ffb85\"\n}"}}},{"rowIdx":196,"cells":{"text":{"kind":"string","value":"section ‹Nested DFS›\ntheory Nested_DFS\nimports DFS_Find_Path\nbegin\ntext ‹Nested DFS is a standard method for Buchi-Automaton emptiness check.›\n\nsubsection ‹Auxiliary Lemmas›\n\nlemma closed_restrict_aux:\n assumes CL: \"E``F ⊆ F ∪ S\"\n assumes NR: \"E⇧*``U ∩ S = {}\"\n assumes SS: \"U ⊆ F\"\n shows \"E⇧*``U ⊆ F\"\n ― ‹Auxiliary lemma to show that nodes reachable from a finished node must \n be finished if, additionally, no stack node is reachable›\nproof (clarify)\n (*goal: ‹⋀x xa. ⟦(xa, x) ∈ E⇧*; xa ∈ U⟧ ⟹ x ∈ F›*)\n fix u and v\n assume A: \"(u,v)∈E⇧*\" \"u∈U\" (*‹(u::'a, v::'a) ∈ (E::('a × 'a) set)⇧*› ‹(u::'a) ∈ (U::'a set)›*)\n hence M: \"E⇧*``{u} ∩ S = {}\" \"u∈F\"\n using NR (*‹E⇧* `` U ∩ S = {}›*) SS (*‹(U::'a set) ⊆ (F::'a set)›*) apply -\n (*goals:\n 1. ‹⟦(u, v) ∈ E⇧*; u ∈ U; E⇧* `` U ∩ S = {}; U ⊆ F⟧ ⟹ E⇧* `` {u} ∩ S = {}›\n 2. ‹⟦(u, v) ∈ E⇧*; u ∈ U; E⇧* `` U ∩ S = {}; U ⊆ F⟧ ⟹ u ∈ F›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*) .\n from A(1) (*‹(u, v) ∈ E⇧*›*) M (*‹E⇧* `` {u} ∩ S = {}› ‹(u::'a) ∈ (F::'a set)›*) show \"v∈F\"\n apply (induct rule: converse_rtrancl_induct (*‹⟦(?a, ?b) ∈ ?r⇧*; ?P ?b; ⋀y z. ⟦(y, z) ∈ ?r; (z, ?b) ∈ ?r⇧*; ?P z⟧ ⟹ ?P y⟧ ⟹ ?P ?a›*))\n (*goal: ‹v ∈ F›*)\n using CL (*‹E `` F ⊆ F ∪ S›*) apply -\n (*goals:\n 1. ‹⟦E⇧* `` {v} ∩ S = {}; v ∈ F; E `` F ⊆ F ∪ S⟧ ⟹ v ∈ F›\n 2. ‹⋀y z. ⟦(y, z) ∈ E; (z, v) ∈ E⇧*; ⟦E⇧* `` {z} ∩ S = {}; z ∈ F⟧ ⟹ v ∈ F; E⇧* `` {y} ∩ S = {}; y ∈ F; E `` F ⊆ F ∪ S⟧ ⟹ v ∈ F›\n discuss goal 1*)\n apply ((auto dest: rtrancl_Image_advance_ss (*‹(?u, ?v) ∈ ?E ⟹ ?E⇧* `` {?v} ⊆ ?E⇧* `` {?u}›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: rtrancl_Image_advance_ss (*‹(?u, ?v) ∈ ?E ⟹ ?E⇧* `` {?v} ⊆ ?E⇧* `` {?u}›*))[1])\n (*proven 2 subgoals*) .\nqed\n\n\nsubsection ‹Instantiation of the Framework›\nrecord 'v blue_dfs_state = \"'v state\" +\n lasso :: \"('v list × 'v list) option\" (* pr × pl *)\n red :: \"'v set\"\n\ntype_synonym 'v blue_dfs_param = \"('v, ('v,unit) blue_dfs_state_ext) parameterization\"\n\nlemma lasso_more_cong[cong]:\"state.more s = state.more s' ⟹ lasso s = lasso s'\"\n apply (cases s)\n (*goal: ‹state.more (s::('a, 'b) blue_dfs_state_scheme) = state.more (s'::('a, 'b) blue_dfs_state_scheme) ⟹ lasso s = lasso s'›*)\n apply (cases s')\n (*goal: ‹⋀counter discovered finished pending stack tree_edges back_edges cross_edges lassoa red morea. ⟦state.more s = state.more s'; s = ⦇counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, lasso = lassoa, red = red, … = morea⦈⟧ ⟹ lasso s = lasso s'›*)\n by simp\nlemma red_more_cong[cong]: \"state.more s = state.more s' ⟹ red s = red s'\"\n apply (cases s)\n (*goal: ‹state.more s = state.more s' ⟹ red s = red s'›*)\n apply (cases s')\n (*goal: ‹⋀counter discovered finished pending stack tree_edges back_edges cross_edges lasso reda morea. ⟦state.more s = state.more s'; s = ⦇counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, lasso = lasso, red = reda, … = morea⦈⟧ ⟹ red s = red s'›*)\n by simp\n\nlemma [simp]: \"s⦇ state.more := ⦇ lasso = foo, red = bar ⦈ ⦈ = s ⦇ lasso := foo, red := bar ⦈\"\n apply (cases s)\n (*goal: ‹s⦇state.more := ⦇lasso = foo, red = bar⦈⦈ = s⦇lasso := foo, red := bar⦈›*)\n by simp\n\n\nabbreviation \"dropWhileNot v ≡ dropWhile ((≠) v)\"\nabbreviation \"takeWhileNot v ≡ takeWhile ((≠) v)\"\n\nlocale BlueDFS_defs = graph_defs G\n for G :: \"('v, 'more) graph_rec_scheme\" +\n fixes accpt :: \"'v ⇒ bool\"\nbegin\n\n definition \"blue s ≡ dom (finished s) - red s\"\n definition \"cyan s ≡ set (stack s)\"\n definition \"white s ≡ - dom (discovered s)\"\n\n abbreviation \"red_dfs R ss x ≡ find_path1_restr_spec (G ⦇ g_V0 := {x} ⦈) ss R\"\n\n definition mk_blue_witness \n :: \"'v blue_dfs_state ⇒ 'v fpr_result ⇒ ('v,unit) blue_dfs_state_ext\"\n where\n \"mk_blue_witness s redS ≡ case redS of\n Inl R' ⇒ ⦇ lasso = None, red = (R' ⌦‹∪ red s›) ⦈\n | Inr (vs, v) ⇒ let rs = rev (stack s) in \n ⦇ lasso = Some (rs, vs@dropWhileNot v rs), red = red s⦈\"\n\n definition run_red_dfs \n :: \"'v ⇒ 'v blue_dfs_state ⇒ ('v,unit) blue_dfs_state_ext nres\" \n where\n \"run_red_dfs u s ≡ case lasso s of None ⇒ do {\n redS ← red_dfs (red s) (λx. x = u ∨ x ∈ cyan s) u;\n RETURN (mk_blue_witness s redS)\n }\n | _ ⇒ NOOP s\"\n\n text ‹ Schwoon-Esparza extension ›\n definition \"se_back_edge u v s ≡ case lasso s of\n None ⇒ \n ― ‹it's a back edge, so ‹u› and ‹v› are both on stack›\n ― ‹we differentiate whether ‹u› or ‹v› is the 'culprit'›\n ― ‹to generate a better counter example›\n if accpt u then\n let rs = rev (tl (stack s));\n ur = rs;\n ul = u#dropWhileNot v rs\n in RETURN ⦇lasso = Some (ur,ul), red = red s⦈\n else if accpt v then\n let rs = rev (stack s);\n vr = takeWhileNot v rs;\n vl = dropWhileNot v rs\n in RETURN ⦇lasso = Some (vr,vl), red = red s⦈\n else NOOP s\n | _ ⇒ NOOP s\"\n\n definition blue_dfs_params :: \"'v blue_dfs_param\"\n where \"blue_dfs_params = ⦇\n on_init = RETURN ⦇ lasso = None, red = {} ⦈,\n on_new_root = λv0 s. NOOP s,\n on_discover = λu v s. NOOP s,\n on_finish = λu s. if accpt u then run_red_dfs u s else NOOP s,\n on_back_edge = se_back_edge,\n on_cross_edge = λu v s. NOOP s,\n is_break = λs. lasso s ≠ None ⦈\"\n \n schematic_goal blue_dfs_params_simps[simp]:\n \"on_init blue_dfs_params = ?OI\"\n \"on_new_root blue_dfs_params = ?ONR\"\n \"on_discover blue_dfs_params = ?OD\"\n \"on_finish blue_dfs_params = ?OF\"\n \"on_back_edge blue_dfs_params = ?OBE\"\n \"on_cross_edge blue_dfs_params = ?OCE\"\n \"is_break blue_dfs_params = ?IB\"\n unfolding blue_dfs_params_def gen_parameterization.simps\n (*goals:\n 1. ‹RETURN ⦇lasso = None, red = {}⦈ = ?OI›\n 2. ‹(λv0. NOOP) = ?ONR›\n 3. ‹(λu v. NOOP) = ?OD›\n 4. ‹(λu s. if accpt u then run_red_dfs u s else NOOP s) = ?OF›\n 5. ‹se_back_edge = ?OBE›\n 6. ‹(λu v. NOOP) = ?OCE›\n 7. ‹(λs. lasso s ≠ None) = ?IB›*)\n (*goals:\n 1. ‹RETURN ⦇lasso = None, red = {}⦈ = ?OI›\n 2. ‹(λv0. NOOP) = ?ONR›\n 3. ‹(λu v. NOOP) = ?OD›\n 4. ‹(λu s. if accpt u then run_red_dfs u s else NOOP s) = ?OF›\n 5. ‹se_back_edge = ?OBE›\n 6. ‹(λu v. NOOP) = ?OCE›\n 7. ‹(λs. lasso s ≠ None) = ?IB›\n discuss goal 1*)\n apply (rule refl (*‹(?t::?'a) = ?t›*))\n (*discuss goal 2*)\n apply (rule refl (*‹(?t::?'a) = ?t›*))\n (*discuss goal 3*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 4*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 5*)\n apply (rule refl (*‹(?t::?'a) = ?t›*))\n (*discuss goal 6*)\n apply (rule refl (*‹?t = ?t›*))\n (*discuss goal 7*)\n apply (rule refl (*‹?t = ?t›*))\n (*proven 7 subgoals*) .\n\n\n sublocale param_DFS_defs G blue_dfs_params \n by unfold_locales\n\nend\n\nlocale BlueDFS = BlueDFS_defs G accpt + param_DFS G blue_dfs_params\n for G :: \"('v, 'more) graph_rec_scheme\" and accpt :: \"'v ⇒ bool\"\n\nlemma BlueDFSI: \n assumes \"fb_graph G\" \n shows \"BlueDFS G\"\nproof (-)\n (*goal: ‹BlueDFS G›*)\n interpret fb_graph G\n by fact\n show \"?thesis\"\n (*goal: ‹BlueDFS G›*)\n by unfold_locales\nqed\n\nlocale BlueDFS_invar = BlueDFS +\n DFS_invar where param = blue_dfs_params\n\ncontext BlueDFS_defs begin\n\nlemma BlueDFS_invar_eq[simp]:\n shows \"DFS_invar G blue_dfs_params s ⟷ BlueDFS_invar G accpt s\" \nproof (standard)\n (*goals:\n 1. ‹DFS_invar G blue_dfs_params s ⟹ BlueDFS_invar G accpt s›\n 2. ‹BlueDFS_invar G accpt s ⟹ DFS_invar G blue_dfs_params s›*)\n assume \"DFS_invar G blue_dfs_params s\" (*‹DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params (s::'v blue_dfs_state)›*)\n interpret DFS_invar G blue_dfs_params s\n by fact\n show \"BlueDFS_invar G accpt s\"\n by unfold_locales\nnext\n (*goal: ‹BlueDFS_invar G accpt s ⟹ DFS_invar G blue_dfs_params s›*)\n assume \"BlueDFS_invar G accpt s\" (*‹BlueDFS_invar (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool) (s::'v blue_dfs_state)›*)\n then interpret BlueDFS_invar G accpt s .\n show \"DFS_invar G blue_dfs_params s\"\n by unfold_locales\nqed\n\nend\n\nsubsection ‹Correctness Proof›\ncontext BlueDFS begin\n\n definition \"blue_basic_invar s ≡ \n case lasso s of\n None ⇒ restr_invar E (red s) (λx. x∈set (stack s)) \n ∧ red s ⊆ dom (finished s)\n | Some l ⇒ True\"\n\n lemma (in BlueDFS_invar) red_DFS_precond_aux:\n assumes BI: \"blue_basic_invar s\"\n assumes [simp]: \"lasso s = None\"\n assumes SNE: \"stack s ≠ []\"\n shows\n \"fb_graph (G ⦇ g_V0 := {hd (stack s)} ⦈)\"\n and \"fb_graph (G ⦇ g_E := E ∩ UNIV × - red s, g_V0 := {hd (stack s)} ⦈)\" \n and \"restr_invar E (red s) (λx. x ∈ set (stack s))\"\n using stack_reachable (*‹set (stack s) ⊆ reachable›*) ‹stack s ≠ []› (*‹stack s ≠ []›*) apply ((rule_tac fb_graph_subset (*‹⟦g_V (?G'::('a, ?'a) graph_rec_scheme) = V; g_E ?G' ⊆ E; finite (g_V0 ?G'); g_V0 ?G' ⊆ reachable⟧ ⟹ fb_graph ?G'›*), auto)[1])\n (*top goal: ‹fb_graph (G⦇g_V0 := {hd (stack s)}⦈)› and 2 goals remain*)\n using stack_reachable (*‹set (stack s) ⊆ reachable›*) ‹stack s ≠ []› (*‹stack s ≠ []›*) apply ((rule_tac fb_graph_subset (*‹⟦g_V (?G'::('a, ?'a) graph_rec_scheme) = V; g_E ?G' ⊆ E; finite (g_V0 ?G'); g_V0 ?G' ⊆ reachable⟧ ⟹ fb_graph ?G'›*), auto)[1])\n (*top goal: ‹fb_graph ((G::('a, 'b) graph_rec_scheme)⦇g_E := E ∩ UNIV × - red s, g_V0 := {hd (stack (s::'a blue_dfs_state))}⦈)› and 1 goal remains*)\n using BI (*‹blue_basic_invar s›*) by (simp add: blue_basic_invar_def (*‹blue_basic_invar (?s::('a, ?'a) blue_dfs_state_scheme) ≡ case lasso ?s of None ⇒ restr_invar E (red ?s) (λx::'a. x ∈ set (stack ?s)) ∧ red ?s ⊆ dom (finished ?s) | Some (l::'a list × 'a list) ⇒ True›*))\n\n lemma (in BlueDFS_invar) red_dfs_pres_bbi:\n assumes BI: \"blue_basic_invar s\"\n assumes [simp]: \"lasso s = None\" and SNE: \"stack s ≠ []\"\n assumes \"pending s `` {hd (stack s)} = {}\"\n shows \"run_red_dfs (hd (stack s)) (finish (hd (stack s)) s) ≤⇩n\n SPEC (λe. \n DFS_invar G blue_dfs_params (finish (hd (stack s)) s⦇state.more := e⦈) \n ⟶ blue_basic_invar (finish (hd (stack s)) s⦇state.more := e⦈))\"\n proof -\n have [simp]: \"(λx. x = hd (stack s) ∨ x ∈ cyan (finish (hd (stack s)) s)) = \n (λx. x∈set (stack s))\"\n using ‹stack s ≠ []›\n unfolding finish_def cyan_def by (auto simp: neq_Nil_conv)\n\n show ?thesis\n unfolding run_red_dfs_def\n apply simp\n apply (refine_vcg)\n apply simp\n proof -\n fix fp1\n define s' where \"s' = finish (hd (stack s)) s\"\n assume FP_spec: \n \"find_path1_restr_pred (G ⦇ g_V0 := {hd (stack s)} ⦈) (λx. x ∈ set (stack s)) (red s) fp1\"\n assume \"BlueDFS_invar G accpt (s'⦇state.more := mk_blue_witness s' fp1⦈)\" \n then interpret i: BlueDFS_invar G accpt \"(s'⦇state.more := mk_blue_witness s' fp1⦈)\"\n by simp\n \n have [simp]: \n \"red s' = red s\" \n \"discovered s' = discovered s\" \n \"dom (finished s') = insert (hd (stack s)) (dom (finished s))\"\n unfolding s'_def finish_def by auto\n\n {\n fix R'\n assume [simp]: \"fp1 = Inl R'\"\n from FP_spec[unfolded find_path1_restr_pred_def, simplified] \n have \n R'FMT: \"R' = red s ∪ E⁺ `` {hd (stack s)}\" \n and RI: \"restr_invar E R' (λx. x ∈ set (stack s))\" \n by auto\n\n from BI have \"red s ⊆ dom (finished s)\"\n unfolding blue_basic_invar_def by auto\n also have \"E⁺ `` {hd (stack s)} ⊆ dom (finished s)\"\n proof (intro subsetI, elim ImageE, simp)\n fix v\n assume \"(hd (stack s),v)∈E⁺\"\n\n then obtain u where \"(hd (stack s),u)∈E\" and \"(u,v)∈E⇧*\" \n by (auto simp: trancl_unfold_left)\n\n from RI have NR: \"E⁺ `` {hd (stack s)} ∩ set (stack s) = {}\"\n unfolding restr_invar_def by (auto simp: R'FMT)\n\n with ‹(hd (stack s),u)∈E› have \"u∉set (stack s)\" by auto\n with i.finished_closed[simplified] ‹(hd (stack s),u)∈E› \n have UID: \"u∈dom (finished s)\" by (auto simp: stack_set_def)\n\n from NR ‹(hd (stack s),u)∈E› have NR': \"E⇧*``{u} ∩ set (stack s) = {}\"\n by (auto simp: trancl_unfold_left)\n\n have CL: \"E `` dom (finished s) ⊆ dom (finished s) ∪ set (stack s)\"\n using finished_closed discovered_eq_finished_un_stack\n by simp\n\n from closed_restrict_aux[OF CL NR'] UID \n have \"E⇧* `` {u} ⊆ dom (finished s)\" by simp\n with ‹(u,v)∈E⇧*› show \"v ∈ dom (finished s)\" by auto\n qed\n finally (sup_least) \n have \"R' ⊆ dom (finished s) ∧ red s ⊆ dom (finished s)\" \n by (simp add: R'FMT)\n } note aux1 = this\n\n show \"blue_basic_invar (s'⦇state.more := mk_blue_witness s' fp1⦈)\"\n unfolding blue_basic_invar_def mk_blue_witness_def\n apply (simp split: option.splits sum.splits)\n apply (intro allI conjI impI)\n\n using FP_spec SNE\n apply (auto \n simp: s'_def blue_basic_invar_def find_path1_restr_pred_def\n simp: restr_invar_def\n simp: neq_Nil_conv) []\n\n apply (auto dest!: aux1) []\n done\n qed\n qed\n\n lemma blue_basic_invar: \"is_invar blue_basic_invar\"\n proof (induct rule: establish_invarI (*‹⟦on_init blue_dfs_params ≤⇩n SPEC (λx. ?I (empty_state x)); ⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ on_finish blue_dfs_params u s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈))⟧ ⟹ is_invar ?I›*))\n (*goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx. blue_basic_invar (empty_state x))›\n 2. ‹⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›\n 3. ‹⋀s s' u. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ on_finish blue_dfs_params u s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›\n 4. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›\n 5. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›\n 6. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›*)\n case (finish s) (*‹DFS_invar G blue_dfs_params s› ‹blue_basic_invar (s::'v blue_dfs_state)› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u_ = hd (stack s)› ‹pending s `` {u_} = {}› ‹(s'_::'v blue_dfs_state) = finish (u_::'v) (s::'v blue_dfs_state)›*)\n then interpret BlueDFS_invar where s = s\n by simp\n have [simp]: \"(λx. x = hd (stack s) ∨ x ∈ cyan (finish (hd (stack s)) s)) = \n (λx. x∈set (stack s))\"\n using ‹stack s ≠ []› (*‹stack s ≠ []›*) unfolding finish_def cyan_def\n (*goal: ‹(λx::'v. x = hd (stack (s::'v blue_dfs_state)) ∨ x ∈ set (stack (let f::'v ⇒ nat option = (finished s)(hd (stack s) ↦ counter s); c::nat = counter s + (1::nat); st::'v list = tl (stack s) in s⦇finished := f, counter := c, stack := st⦈))) = (λx::'v. x ∈ set (stack s))›*)\n by (auto simp: neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))\n from finish (*‹DFS_invar G blue_dfs_params s› ‹blue_basic_invar s› ‹cond (s::'v blue_dfs_state)› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u_ = hd (stack s)› ‹pending s `` {u_} = {}› ‹s'_ = finish u_ s›*) show \"?case\"\n (*goal: ‹on_finish blue_dfs_params u_ s'_ ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'_⦇state.more := x⦈) ⟶ blue_basic_invar (s'_⦇state.more := x⦈))›*)\n apply simp\n (*goal: ‹on_finish blue_dfs_params u_ s'_ ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'_⦇state.more := x⦈) ⟶ blue_basic_invar (s'_⦇state.more := x⦈))›*)\n apply (intro conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*))\n (*goals:\n 1. ‹⟦BlueDFS_invar G accpt s; blue_basic_invar s; cond s; lasso s = None; stack s ≠ []; u_ = hd (stack s); pending s `` {hd (stack s)} = {}; s'_ = finish (hd (stack s)) s; accpt (hd (stack s))⟧ ⟹ run_red_dfs (hd (stack s)) (finish (hd (stack s)) s) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (finish (hd (stack s)) s⦇state.more := x⦈) ⟶ blue_basic_invar (finish (hd (stack s)) s⦇state.more := x⦈))›\n 2. ‹⟦BlueDFS_invar G accpt s; blue_basic_invar s; cond s; lasso s = None; stack s ≠ []; u_ = hd (stack s); pending s `` {hd (stack s)} = {}; s'_ = finish (hd (stack s)) s; ¬ accpt (hd (stack s)); BlueDFS_invar G accpt (finish (hd (stack s)) s⦇state.more := state.more s⦈)⟧ ⟹ blue_basic_invar (finish (hd (stack s)) s⦇state.more := state.more s⦈)›\n discuss goal 1*)\n apply (rule leof_trans[OF red_dfs_pres_bbi] (*‹⟦blue_basic_invar (s::'v blue_dfs_state); lasso s = None; stack s ≠ []; pending s `` {hd (stack s)} = {}; SPEC (λe::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params (finish (hd (stack s)) s⦇state.more := e⦈) ⟶ blue_basic_invar (finish (hd (stack s)) s⦇state.more := e⦈)) ≤⇩n (?c::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈ nres)⟧ ⟹ run_red_dfs (hd (stack s)) (finish (hd (stack s)) s) ≤⇩n ?c›*))\n (*goals:\n 1. ‹⟦BlueDFS_invar G accpt s; blue_basic_invar s; cond s; lasso s = None; stack s ≠ []; u_ = hd (stack s); pending s `` {hd (stack s)} = {}; s'_ = finish (hd (stack s)) s; accpt (hd (stack s))⟧ ⟹ blue_basic_invar s›\n 2. ‹⟦BlueDFS_invar G accpt s; blue_basic_invar s; cond s; lasso s = None; stack s ≠ []; u_ = hd (stack s); pending s `` {hd (stack s)} = {}; s'_ = finish (hd (stack s)) s; accpt (hd (stack s))⟧ ⟹ lasso s = None›\n 3. ‹⟦BlueDFS_invar G accpt s; blue_basic_invar s; cond s; lasso s = None; stack s ≠ []; u_ = hd (stack s); pending s `` {hd (stack s)} = {}; s'_ = finish (hd (stack s)) s; accpt (hd (stack s))⟧ ⟹ stack s ≠ []›\n 4. ‹⟦BlueDFS_invar G accpt s; blue_basic_invar s; cond s; lasso s = None; stack s ≠ []; u_ = hd (stack s); pending s `` {hd (stack s)} = {}; s'_ = finish (hd (stack s)) s; accpt (hd (stack s))⟧ ⟹ pending s `` {hd (stack s)} = {}›\n 5. ‹⟦BlueDFS_invar G accpt s; blue_basic_invar s; cond s; lasso s = None; stack s ≠ []; u_ = hd (stack s); pending s `` {hd (stack s)} = {}; s'_ = finish (hd (stack s)) s; accpt (hd (stack s))⟧ ⟹ SPEC (λe. DFS_invar G blue_dfs_params (finish (hd (stack s)) s⦇state.more := e⦈) ⟶ blue_basic_invar (finish (hd (stack s)) s⦇state.more := e⦈)) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (finish (hd (stack s)) s⦇state.more := x⦈) ⟶ blue_basic_invar (finish (hd (stack s)) s⦇state.more := x⦈))›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply assumption\n (*discuss goal 3*)\n apply assumption\n (*discuss goal 4*)\n apply assumption\n (*discuss goal 5*)\n apply simp\n (*proven 5 subgoals*)\n (*discuss goal 2*)\n apply (auto simp: restr_invar_def (*‹restr_invar ?E ?R ?P ≡ ?E `` ?R ⊆ ?R ∧ ?R ∩ Collect ?P = {}›*) blue_basic_invar_def (*‹blue_basic_invar ?s ≡ case lasso ?s of None ⇒ restr_invar E (red ?s) (λx. x ∈ set (stack ?s)) ∧ red ?s ⊆ dom (finished ?s) | Some l ⇒ True›*) neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))\n (*proven 2 subgoals*) .\n qed (auto simp: blue_basic_invar_def (*‹blue_basic_invar ?s ≡ case lasso ?s of None ⇒ restr_invar E (red ?s) (λx. x ∈ set (stack ?s)) ∧ red ?s ⊆ dom (finished ?s) | Some l ⇒ True›*) cond_def (*‹cond ?s = ((V0 ⊆ {v. is_discovered v ?s} ⟶ ¬ is_empty_stack ?s) ∧ ¬ is_break blue_dfs_params ?s)›*) se_back_edge_def (*‹se_back_edge ?u ?v ?s ≡ case lasso ?s of None ⇒ if accpt ?u then let rs = rev (tl (stack ?s)); ur = rs; ul = ?u # dropWhileNot ?v rs in RETURN ⦇lasso = Some (ur, ul), red = red ?s⦈ else if accpt ?v then let rs = rev (stack ?s); vr = takeWhileNot ?v rs; vl = dropWhileNot ?v rs in RETURN ⦇lasso = Some (vr, vl), red = red ?s⦈ else NOOP ?s | Some a ⇒ NOOP ?s›*) simp: restr_invar_def (*‹restr_invar ?E ?R ?P ≡ ?E `` ?R ⊆ ?R ∧ ?R ∩ Collect ?P = {}›*) empty_state_def (*‹empty_state ?e ≡ ⦇counter = 0, discovered = λx. None, finished = λx. None, pending = {}, stack = [], tree_edges = {}, back_edges = {}, cross_edges = {}, … = ?e⦈›*) pred_defs (*‹is_discovered ?u ?s ≡ ?u ∈ dom (discovered ?s)› ‹is_finished ?u ?s ≡ ?u ∈ dom (finished ?s)› ‹is_empty_stack ?s ≡ stack ?s = []›*) simp: DFS_invar.discovered_eq_finished_un_stack (*‹DFS_invar ?G ?param ?s ⟹ dom (discovered ?s) = dom (finished ?s) ∪ set (stack ?s)›*) simp del: BlueDFS_invar_eq (*‹DFS_invar G blue_dfs_params ?s = BlueDFS_invar G accpt ?s›*) split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n (*solves the remaining goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx. blue_basic_invar (empty_state x))›\n 2. ‹⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›\n 3. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›\n 4. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›\n 5. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; blue_basic_invar s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ blue_basic_invar (s'⦇state.more := x⦈))›*)\n\n lemmas (in BlueDFS_invar) s_blue_basic_invar \n = blue_basic_invar[THEN make_invar_thm]\n\n lemmas (in BlueDFS_invar) red_DFS_precond \n = red_DFS_precond_aux[OF s_blue_basic_invar]\n\n sublocale DFS G blue_dfs_params\n apply unfold_locales\n (*goal: ‹DFS G blue_dfs_params›*)\n apply (clarsimp_all simp: se_back_edge_def run_red_dfs_def refine_pw_simps pre_on_defs split: option.splits)\n (*top goal: ‹nofail (on_init blue_dfs_params)› and 5 goals remain*)\n unfolding nofail_SPEC_iff\n (*goal: ‹⋀sa. ⟦BlueDFS_invar G accpt sa; cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; accpt (hd (stack sa))⟧ ⟹ red_dfs (red sa) (λx. x = hd (stack sa) ∨ x ∈ cyan (finish (hd (stack sa)) sa)) (hd (stack sa)) ≤ SPEC (λ_. True)›*)\n apply refine_vcg\n (*goals:\n 1. ‹⋀sa. ⟦BlueDFS_invar G accpt sa; cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; accpt (hd (stack sa))⟧ ⟹ fb_graph (G⦇g_V0 := {hd (stack sa)}⦈)›\n 2. ‹⋀sa. ⟦BlueDFS_invar G accpt sa; cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; accpt (hd (stack sa))⟧ ⟹ restr_invar (g_E (G⦇g_V0 := {hd (stack sa)}⦈)) (red sa) (λx. x = hd (stack sa) ∨ x ∈ cyan (finish (hd (stack sa)) sa))›\n 3. ‹⋀sa x. ⟦BlueDFS_invar G accpt sa; cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; accpt (hd (stack sa)); fb_graph (G⦇g_V0 := {hd (stack sa)}⦈) ∧ restr_invar (g_E (G⦇g_V0 := {hd (stack sa)}⦈)) (red sa) (λx. x = hd (stack sa) ∨ x ∈ cyan (finish (hd (stack sa)) sa)); find_path1_restr_pred (G⦇g_V0 := {hd (stack sa)}⦈) (λx. x = hd (stack sa) ∨ x ∈ cyan (finish (hd (stack sa)) sa)) (red sa) x⟧ ⟹ True›\n discuss goal 1*)\n apply (erule BlueDFS_invar.red_DFS_precond (*‹⟦BlueDFS_invar ?G ?accpt ?s; lasso ?s = None; stack ?s ≠ []⟧ ⟹ fb_graph (?G⦇g_V0 := {hd (stack ?s)}⦈)› ‹⟦BlueDFS_invar ?G ?accpt ?s; lasso ?s = None; stack ?s ≠ []⟧ ⟹ fb_graph (?G⦇g_E := g_E ?G ∩ UNIV × - red ?s, g_V0 := {hd (stack ?s)}⦈)› ‹⟦BlueDFS_invar ?G ?accpt ?s; lasso ?s = None; stack ?s ≠ []⟧ ⟹ restr_invar (g_E ?G) (red ?s) (λx. x ∈ set (stack ?s))›*))\n (*goals:\n 1. ‹⋀sa. ⟦cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; accpt (hd (stack sa))⟧ ⟹ lasso sa = None›\n 2. ‹⋀sa. ⟦cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; accpt (hd (stack sa))⟧ ⟹ stack sa ≠ []›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (simp add: cyan_def (*‹cyan ?s ≡ set (stack ?s)›*) finish_def (*‹finish ?u ?s ≡ let f = (finished ?s)(?u ↦ counter ?s); c = counter ?s + 1; st = tl (stack ?s) in ?s⦇finished := f, counter := c, stack := st⦈›*))\n (*top goal: ‹⋀sa. ⟦BlueDFS_invar G accpt sa; cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; accpt (hd (stack sa))⟧ ⟹ restr_invar (g_E (G⦇g_V0 := {hd (stack sa)}⦈)) (red sa) (λx. x = hd (stack sa) ∨ x ∈ cyan (finish (hd (stack sa)) sa))› and 1 goal remains*)\n apply (erule BlueDFS_invar.red_DFS_precond (*‹⟦BlueDFS_invar ?G ?accpt ?s; lasso ?s = None; stack ?s ≠ []⟧ ⟹ fb_graph (?G⦇g_V0 := {hd (stack ?s)}⦈)› ‹⟦BlueDFS_invar ?G ?accpt ?s; lasso ?s = None; stack ?s ≠ []⟧ ⟹ fb_graph (?G⦇g_E := g_E ?G ∩ UNIV × - red ?s, g_V0 := {hd (stack ?s)}⦈)› ‹⟦BlueDFS_invar ?G ?accpt ?s; lasso ?s = None; stack ?s ≠ []⟧ ⟹ restr_invar (g_E ?G) (red ?s) (λx. x ∈ set (stack ?s))›*))\n (*goals:\n 1. ‹⋀sa::'v blue_dfs_state. ⟦cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; (accpt::'v ⇒ bool) (hd (stack sa))⟧ ⟹ lasso sa = None›\n 2. ‹⋀sa::'v blue_dfs_state. ⟦cond sa; stack sa ≠ []; pending sa `` {hd (stack sa)} = {}; lasso sa = None; (accpt::'v ⇒ bool) (hd (stack sa))⟧ ⟹ stack sa ≠ []›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*)\n (*discuss goal 3*)\n apply (rule TrueI (*‹True›*))\n (*proven 3 subgoals*) .\n\nend\n\ncontext BlueDFS_invar\nbegin\n\n context assumes [simp]: \"lasso s = None\"\n begin\n lemma red_closed:\n \"E `` red s ⊆ red s\"\n using s_blue_basic_invar (*‹blue_basic_invar s›*) unfolding blue_basic_invar_def restr_invar_def\n (*goal: ‹E `` red (s::'a blue_dfs_state) ⊆ red s›*)\n by simp\n \n lemma red_stack_disjoint:\n \"set (stack s) ∩ red s = {}\"\n using s_blue_basic_invar (*‹blue_basic_invar s›*) unfolding blue_basic_invar_def restr_invar_def\n (*goal: ‹set (stack (s::'a blue_dfs_state)) ∩ red s = {}›*)\n by auto\n \n lemma red_finished: \"red s ⊆ dom (finished s)\"\n using s_blue_basic_invar (*‹blue_basic_invar s›*) unfolding blue_basic_invar_def\n (*goal: ‹red (s::'a blue_dfs_state) ⊆ dom (finished s)›*)\n by auto\n\n (* Play of Colors *)\n lemma all_nodes_colored: \"white s ∪ blue s ∪ cyan s ∪ red s = UNIV \" \n unfolding white_def blue_def cyan_def\n (*goal: ‹- dom (discovered s) ∪ (dom (finished s) - red s) ∪ set (stack s) ∪ red s = UNIV›*)\n by (auto simp: stack_set_def (*‹set (stack (s::'a blue_dfs_state)) = dom (discovered s) - dom (finished s)›*))\n\n lemma colors_disjoint:\n \"white s ∩ (blue s ∪ cyan s ∪ red s) = {}\"\n \"blue s ∩ (white s ∪ cyan s ∪ red s) = {}\"\n \"cyan s ∩ (white s ∪ blue s ∪ red s) = {}\"\n \"red s ∩ (white s ∪ blue s ∪ cyan s) = {}\"\n unfolding white_def blue_def cyan_def\n (*goals:\n 1. ‹- dom (discovered s) ∩ (dom (finished s) - red s ∪ set (stack s) ∪ red s) = {}›\n 2. ‹(dom (finished s) - red s) ∩ (- dom (discovered s) ∪ set (stack s) ∪ red s) = {}›\n 3. ‹set (stack s) ∩ (- dom (discovered s) ∪ (dom (finished s) - red s) ∪ red s) = {}›\n 4. ‹red s ∩ (- dom (discovered s) ∪ (dom (finished s) - red s) ∪ set (stack s)) = {}›*)\n using finished_discovered (*‹dom (finished (s::'a blue_dfs_state)) ⊆ dom (discovered s)›*) red_finished (*‹red s ⊆ dom (finished s)›*) unfolding stack_set_def\n (*goals:\n 1. ‹- dom (discovered s) ∩ (dom (finished s) - red s ∪ (dom (discovered s) - dom (finished s)) ∪ red s) = {}›\n 2. ‹(dom (finished s) - red s) ∩ (- dom (discovered s) ∪ (dom (discovered s) - dom (finished s)) ∪ red s) = {}›\n 3. ‹(dom (discovered s) - dom (finished s)) ∩ (- dom (discovered s) ∪ (dom (finished s) - red s) ∪ red s) = {}›\n 4. ‹red s ∩ (- dom (discovered s) ∪ (dom (finished s) - red s) ∪ (dom (discovered s) - dom (finished s))) = {}›*)\n apply -\n (*goals:\n 1. ‹⟦dom (finished s) ⊆ dom (discovered s); red s ⊆ dom (finished s)⟧ ⟹ - dom (discovered s) ∩ (dom (finished s) - red s ∪ (dom (discovered s) - dom (finished s)) ∪ red s) = {}›\n 2. ‹⟦dom (finished s) ⊆ dom (discovered s); red s ⊆ dom (finished s)⟧ ⟹ (dom (finished s) - red s) ∩ (- dom (discovered s) ∪ (dom (discovered s) - dom (finished s)) ∪ red s) = {}›\n 3. ‹⟦dom (finished s) ⊆ dom (discovered s); red s ⊆ dom (finished s)⟧ ⟹ (dom (discovered s) - dom (finished s)) ∩ (- dom (discovered s) ∪ (dom (finished s) - red s) ∪ red s) = {}›\n 4. ‹⟦dom (finished s) ⊆ dom (discovered s); red s ⊆ dom (finished s)⟧ ⟹ red s ∩ (- dom (discovered s) ∪ (dom (finished s) - red s) ∪ (dom (discovered s) - dom (finished s))) = {}›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*discuss goal 4*)\n apply blast\n (*proven 4 subgoals*) .\n \n end\n\n lemma (in BlueDFS) i_no_accpt_cyle_in_finish:\n \"is_invar (λs. lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x,x) ∉ E⁺))\"\n proof (induct rule: establish_invarI (*‹⟦on_init blue_dfs_params ≤⇩n SPEC (λx. ?I (empty_state x)); ⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ on_finish blue_dfs_params u s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈))⟧ ⟹ is_invar ?I›*))\n (*goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. lasso (empty_state x) = None ⟶ (∀xa::'v. (accpt::'v ⇒ bool) xa ∧ xa ∈ dom (finished (empty_state x)) ⟶ (xa, xa) ∉ E⁺))›\n 2. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) v0::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso s = None ⟶ (∀x::'v. (accpt::'v ⇒ bool) x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa::'v. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 3. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) u::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso s = None ⟶ (∀x::'v. (accpt::'v ⇒ bool) x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ on_finish blue_dfs_params u s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa::'v. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 4. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) (u::'v) v::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso s = None ⟶ (∀x::'v. (accpt::'v ⇒ bool) x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa::'v. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 5. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) (u::'v) v::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso s = None ⟶ (∀x::'v. (accpt::'v ⇒ bool) x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa::'v. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 6. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) (u::'v) v::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso s = None ⟶ (∀x::'v. (accpt::'v ⇒ bool) x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa::'v. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›*)\n case (finish s s' u) (*‹DFS_invar G blue_dfs_params s› ‹lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺)› ‹cond (s::'v blue_dfs_state)› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*)\n then interpret BlueDFS_invar where s = s\n by simp\n let ?onstack = \"λx. x∈set (stack s)\"\n let ?rE = \"rel_restrict E (red s)\"\n from finish (*‹DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params (s::'v blue_dfs_state)› ‹lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺)› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) obtain sh and st where [simp]: \"stack s = sh#st\"\n (*goal: ‹(⋀(sh::'v) st::'v list. stack (s::'v blue_dfs_state) = sh # st ⟹ thesis::bool) ⟹ thesis›*)\n by (auto simp: neq_Nil_conv (*‹((?xs::?'a list) ≠ []) = (∃(y::?'a) ys::?'a list. ?xs = y # ys)›*))\n have 1: \"g_E (G ⦇ g_V0 := {hd (stack s)} ⦈) = E\"\n by simp\n {\n fix R' :: \"'v set\"\n let ?R' = \"R' ⌦‹∪ red s›\"\n let ?s = \"s'⦇ lasso := None, red := ?R'⦈\"\n assume \"⋀v. (hd (stack s), v) ∈ ?rE⁺ ⟹ ¬ ?onstack v\" and accpt: \"accpt u\" and NL[simp]: \"lasso s = None\" (*‹(hd (stack (s::'v blue_dfs_state)), ?v8::'v) ∈ (rel_restrict E (red s))⁺ ⟹ ?v8 ∉ set (stack s)› ‹(accpt::'v ⇒ bool) (u::'v)› ‹lasso (s::'v blue_dfs_state) = None›*)\n hence no_hd_cycle: \"(hd (stack s), hd (stack s)) ∉ ?rE⁺\"\n by auto\n from finish (*‹DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params (s::'v blue_dfs_state)› ‹lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺)› ‹cond s› ‹¬ is_break blue_dfs_params (s::'v blue_dfs_state)› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) have \"stack s ≠ []\"\n by simp\n from hd_in_set[OF this] (*‹hd (stack s) ∈ set (stack s)›*) have \"hd (stack s) ∉ red s\"\n using red_stack_disjoint (*‹lasso s = None ⟹ set (stack s) ∩ red s = {}›*) by auto\n hence \"(hd (stack s),hd (stack s)) ∉ E⁺\"\n using no_hd_cycle (*‹(hd (stack s), hd (stack s)) ∉ (rel_restrict E (red s))⁺›*) rel_restrict_tranclI (*‹⟦(?x, ?y) ∈ ?E⁺; ?x ∉ ?R; ?y ∉ ?R; ?E `` ?R ⊆ ?R⟧ ⟹ (?x, ?y) ∈ (rel_restrict ?E ?R)⁺›*) red_closed[OF NL] (*‹E `` red s ⊆ red s›*) by metis\n with accpt (*‹accpt u›*) finish (*‹DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params (s::'v blue_dfs_state)› ‹lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺)› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) have \"∀x. accpt x ∧ x ∈ dom (finished ?s) ⟶ (x,x) ∉ E⁺\"\n by auto\n }\n with finish (*‹DFS_invar G blue_dfs_params s› ‹lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺)› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) have \"red_dfs (red s) ?onstack (hd (stack s))\n ≤ SPEC (λx. ∀R. x = Inl R ⟶\n DFS_invar G blue_dfs_params (lasso_update Map.empty s'⦇red := R ⌦‹∪ red s›⦈) ⟶\n (∀x. accpt x ∧ x∈dom (finished s') ⟶ (x, x) ∉ E⁺))\"\n apply -\n (*goal: ‹red_dfs (red s) (λx. x ∈ set (stack s)) (hd (stack s)) ≤ SPEC (λx. ∀R. x = Inl R ⟶ DFS_invar G blue_dfs_params (s'⦇lasso := None, red := R⦈) ⟶ (∀x. accpt x ∧ x ∈ dom (finished s') ⟶ (x, x) ∉ E⁺))›*)\n apply (rule find_path1_restr_spec_rule (*‹⟦fb_graph (?G::(?'a, ?'b) graph_rec_scheme) ∧ restr_invar (g_E ?G) (?R::?'a set) (?P::?'a ⇒ bool); fb_graph ?G ∧ restr_invar (g_E ?G) ?R ?P ⟹ SPEC (find_path1_restr_pred ?G ?P ?R) ≤ (?m::(?'a set + ?'a list × ?'a) nres)⟧ ⟹ find_path1_restr_spec ?G ?P ?R ≤ ?m› ‹(fb_graph (?G::(?'a, ?'b) graph_rec_scheme) ∧ restr_invar (g_E ?G) (?R::?'a set) (?P::?'a ⇒ bool) ⟹ SPEC (find_path1_restr_pred ?G ?P ?R) ≤⇩n (?m::(?'a set + ?'a list × ?'a) nres)) ⟹ find_path1_restr_spec ?G ?P ?R ≤⇩n ?m›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goal: ‹⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s; ⋀R'. ⟦⋀v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺⟧ ⟹ red_dfs (red s) (λx. x ∈ set (stack s)) (hd (stack s)) ≤ SPEC (λx. ∀R. x = Inl R ⟶ DFS_invar G blue_dfs_params (s'⦇lasso := None, red := R⦈) ⟶ (∀x. accpt x ∧ x ∈ dom (finished s') ⟶ (x, x) ∉ E⁺))›*)\n apply ((rule red_DFS_precond (*‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_E := E ∩ UNIV × - red s, g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ restr_invar E (red s) (λx. x ∈ set (stack s))›*), simp_all)[1])\n (*top goal: ‹⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params (s::'v blue_dfs_state); lasso s = None ⟶ (∀x::'v. (accpt::'v ⇒ bool) x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u::'v) = hd (stack s); pending s `` {u} = {}; (s'::'v blue_dfs_state) = finish u s; ⋀R'::'v set. ⟦⋀v::'v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x::'v. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺⟧ ⟹ fb_graph (G⦇g_V0 := {hd (stack s)}⦈)› and 2 goals remain*)\n unfolding \"1\"\n (*goals:\n 1. ‹⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s; ⋀R'. ⟦⋀v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺⟧ ⟹ restr_invar E (red s) (λx. x ∈ set (stack s))›\n 2. ‹⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s; ⋀R'. ⟦⋀v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺; fb_graph (G⦇g_V0 := {hd (stack s)}⦈) ∧ restr_invar E (red s) (λx. x ∈ set (stack s))⟧ ⟹ SPEC (find_path1_restr_pred (G⦇g_V0 := {hd (stack s)}⦈) (λx. x ∈ set (stack s)) (red s)) ≤ SPEC (λx. ∀R. x = Inl R ⟶ DFS_invar G blue_dfs_params (s'⦇lasso := None, red := R⦈) ⟶ (∀x. accpt x ∧ x ∈ dom (finished s') ⟶ (x, x) ∉ E⁺))›*)\n (*goals:\n 1. ‹⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s; ⋀R'. ⟦⋀v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺⟧ ⟹ restr_invar E (red s) (λx. x ∈ set (stack s))›\n 2. ‹⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s; ⋀R'. ⟦⋀v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺; fb_graph (G⦇g_V0 := {hd (stack s)}⦈) ∧ restr_invar E (red s) (λx. x ∈ set (stack s))⟧ ⟹ SPEC (find_path1_restr_pred (G⦇g_V0 := {hd (stack s)}⦈) (λx. x ∈ set (stack s)) (red s)) ≤ SPEC (λx. ∀R. x = Inl R ⟶ DFS_invar G blue_dfs_params (s'⦇lasso := None, red := R⦈) ⟶ (∀x. accpt x ∧ x ∈ dom (finished s') ⟶ (x, x) ∉ E⁺))›\n discuss goal 1*)\n apply (rule red_DFS_precond (*‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_E := E ∩ UNIV × - red s, g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ restr_invar E (red s) (λx. x ∈ set (stack s))›*))\n (*goals:\n 1. ‹⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s; ⋀R'. ⟦⋀v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺⟧ ⟹ lasso s = None›\n 2. ‹⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s; ⋀R'. ⟦⋀v. (hd (stack s), v) ∈ (rel_restrict E (red s))⁺ ⟹ v ∉ set (stack s); accpt u; lasso s = None⟧ ⟹ ∀x. accpt x ∧ x ∈ dom (finished (s'⦇lasso := None, red := R'⦈)) ⟶ (x, x) ∉ E⁺⟧ ⟹ stack s ≠ []›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply (auto simp: find_path1_restr_pred_def (*‹find_path1_restr_pred ?G ?P ?R ≡ λr. case r of Inl R' ⇒ R' = ?R ∪ (g_E ?G)⁺ `` g_V0 ?G ∧ restr_invar (g_E ?G) R' ?P | Inr (vs, v) ⇒ ?P v ∧ vs ≠ [] ∧ (∃v0∈g_V0 ?G. path (g_E ?G ∩ UNIV × - ?R) v0 vs v)›*) restr_invar_def (*‹restr_invar ?E ?R ?P ≡ ?E `` ?R ⊆ ?R ∧ ?R ∩ Collect ?P = {}›*))\n (*proven 2 subgoals*) .\n note aux = leof_trans[OF this [ simplified , THEN leof_lift ]] (*‹SPEC (λx. ∀R. x = Inl R ⟶ BlueDFS_invar G accpt (s'⦇lasso := None, red := R⦈) ⟶ (∀x. accpt x ∧ x ∈ dom (finished s') ⟶ (x, x) ∉ E⁺)) ≤⇩n ?c ⟹ red_dfs (red s) (λx. x = sh ∨ x ∈ set st) sh ≤⇩n ?c›*)\n note [refine_vcg del] = find_path1_restr_spec_rule (*‹⟦fb_graph (?G::(?'a, ?'b) graph_rec_scheme) ∧ restr_invar (g_E ?G) (?R::?'a::type set) (?P::?'a::type ⇒ bool); fb_graph ?G ∧ restr_invar (g_E ?G) ?R ?P ⟹ SPEC (find_path1_restr_pred ?G ?P ?R) ≤ (?m::(?'a::type set + ?'a::type list × ?'a::type) nres)⟧ ⟹ find_path1_restr_spec ?G ?P ?R ≤ ?m› ‹(fb_graph ?G ∧ restr_invar (g_E ?G) ?R ?P ⟹ SPEC (find_path1_restr_pred ?G ?P ?R) ≤⇩n ?m) ⟹ find_path1_restr_spec ?G ?P ?R ≤⇩n ?m›*)\n from finish (*‹DFS_invar G blue_dfs_params s› ‹lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺)› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) show \"?case\"\n (*goal: ‹on_finish blue_dfs_params (u::'v) (s'::'v blue_dfs_state) ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa::'v. (accpt::'v ⇒ bool) xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›*)\n apply simp\n (*goal: ‹on_finish blue_dfs_params u s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›*)\n apply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹⟦BlueDFS_invar G accpt s; ∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺; cond s; lasso s = None; u = sh; pending s `` {sh} = {}; s' = finish sh s⟧ ⟹ (accpt sh ⟶ run_red_dfs sh (finish sh s) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (finish sh s⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺))) ∧ (¬ accpt sh ⟶ BlueDFS_invar G accpt (finish sh s⦇state.more := state.more s⦈) ⟶ (∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺))›*)\n unfolding run_red_dfs_def mk_blue_witness_def cyan_def\n (*goals:\n 1. ‹⟦BlueDFS_invar G accpt s; ∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺; cond s; lasso s = None; u = sh; pending s `` {sh} = {}; s' = finish sh s; accpt sh⟧ ⟹ (case lasso (finish sh s) of None ⇒ red_dfs (red (finish sh s)) (λx. x = sh ∨ x ∈ set (stack (finish sh s))) sh ⤜ (λredS. RETURN (case redS of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack (finish sh s)) in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red (finish sh s)⦈)) | Some a ⇒ NOOP (finish sh s)) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (finish sh s⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺))›\n 2. ‹⟦BlueDFS_invar G accpt s; ∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺; cond s; lasso s = None; u = sh; pending s `` {sh} = {}; s' = finish sh s; ¬ accpt sh; BlueDFS_invar G accpt (finish sh s⦇state.more := state.more s⦈)⟧ ⟹ ∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺›*)\n (*goals:\n 1. ‹⟦BlueDFS_invar G accpt s; ∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺; cond s; lasso s = None; u = sh; pending s `` {sh} = {}; s' = finish sh s; accpt sh⟧ ⟹ (case lasso (finish sh s) of None ⇒ red_dfs (red (finish sh s)) (λx. x = sh ∨ x ∈ set (stack (finish sh s))) sh ⤜ (λredS. RETURN (case redS of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack (finish sh s)) in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red (finish sh s)⦈)) | Some a ⇒ NOOP (finish sh s)) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (finish sh s⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺))›\n 2. ‹⟦BlueDFS_invar G accpt s; ∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺; cond s; lasso s = None; u = sh; pending s `` {sh} = {}; s' = finish sh s; ¬ accpt sh; BlueDFS_invar G accpt (finish sh s⦇state.more := state.more s⦈)⟧ ⟹ ∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺›\n discuss goal 1*)\n apply clarsimp\n (*top goal: ‹⟦BlueDFS_invar G accpt s; ∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺; cond s; lasso s = None; u = sh; pending s `` {sh} = {}; s' = finish sh s; accpt sh⟧ ⟹ (case lasso (finish sh s) of None ⇒ red_dfs (red (finish sh s)) (λx. x = sh ∨ x ∈ set (stack (finish sh s))) sh ⤜ (λredS. RETURN (case redS of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack (finish sh s)) in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red (finish sh s)⦈)) | Some a ⇒ NOOP (finish sh s)) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (finish sh s⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺))› and 1 goal remains*)\n apply (refine_vcg aux)\n (*top goal: ‹⟦BlueDFS_invar G accpt s; ∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺; cond s; lasso s = None; u = sh; pending s `` {sh} = {}; accpt sh; s' = finish sh s⟧ ⟹ red_dfs (red s) (λx. x = sh ∨ x ∈ set st) sh ⤜ (λredS. RETURN (case redS of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack (finish sh s)) in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red (finish sh s)⦈)) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (finish sh s⦇state.more := x⦈) ⟶ lasso (finish sh s⦇state.more := x⦈) = None ⟶ (∀x. accpt x ∧ (x = sh ∨ x ∈ dom (finished s)) ⟶ (x, x) ∉ E⁺))› and 1 goal remains*)\n apply ((auto split: sum.splits (*‹(?P::?'c ⇒ bool) (case ?sum::?'a + ?'b of Inl (x::?'a) ⇒ (?f1.0::?'a ⇒ ?'c) x | Inr (x::?'b) ⇒ (?f2.0::?'b ⇒ ?'c) x) = ((∀x1::?'a. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2::?'b. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'c ⇒ bool) (case ?sum::?'a + ?'b of Inl (x::?'a) ⇒ (?f1.0::?'a ⇒ ?'c) x | Inr (x::?'b) ⇒ (?f2.0::?'b ⇒ ?'c) x) = (¬ ((∃x1::?'a. ?sum = Inl x1 ∧ ¬ ?P (?f1.0 x1)) ∨ (∃x2::?'b. ?sum = Inr x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n (*discuss goal 2*)\n apply ((auto split: sum.splits (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = (¬ ((∃x1. ?sum = Inl x1 ∧ ¬ ?P (?f1.0 x1)) ∨ (∃x2. ?sum = Inr x2 ∧ ¬ ?P (?f2.0 x2))))›*))[1])\n (*proven 2 subgoals*) .\n next\n (*goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx. lasso (empty_state x) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (empty_state x)) ⟶ (xa, xa) ∉ E⁺))›\n 2. ‹⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 3. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 4. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 5. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›*)\n case back_edge (*‹DFS_invar G blue_dfs_params s_› ‹lasso (s_::'v blue_dfs_state) = None ⟶ (∀x::'v::type. (accpt::'v::type ⇒ bool) x ∧ x ∈ dom (finished s_) ⟶ (x, x) ∉ E⁺)› ‹cond s_› ‹¬ is_break blue_dfs_params s_› ‹stack (s_::'v blue_dfs_state) ≠ []› ‹(u_, v_) ∈ pending s_› ‹u_ = hd (stack s_)› ‹v_ ∈ dom (discovered s_)› ‹v_ ∉ dom (finished s_)› ‹s'_ = back_edge u_ v_ (s_⦇pending := pending s_ - {(u_, v_)}⦈)›*)\n thus \"?case\"\n (*goal: ‹on_back_edge blue_dfs_params u_ v_ s'_ ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'_⦇state.more := x⦈) ⟶ lasso (s'_⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'_⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›*)\n by (simp add: se_back_edge_def (*‹se_back_edge ?u ?v ?s ≡ case lasso ?s of None ⇒ if accpt ?u then let rs = rev (tl (stack ?s)); ur = rs; ul = ?u # dropWhileNot ?v rs in RETURN ⦇lasso = Some (ur, ul), red = red ?s⦈ else if accpt ?v then let rs = rev (stack ?s); vr = takeWhileNot ?v rs; vl = dropWhileNot ?v rs in RETURN ⦇lasso = Some (vr, vl), red = red ?s⦈ else NOOP ?s | Some a ⇒ NOOP ?s›*) split: option.split (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))›*))\n qed (simp_all)\n (*solves the remaining goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx. lasso (empty_state x) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (empty_state x)) ⟶ (xa, xa) ∉ E⁺))›\n 2. ‹⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 3. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›\n 4. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso s = None ⟶ (∀x. accpt x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺); cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso (s'⦇state.more := x⦈) = None ⟶ (∀xa. accpt xa ∧ xa ∈ dom (finished (s'⦇state.more := x⦈)) ⟶ (xa, xa) ∉ E⁺))›*)\n\n lemma no_accpt_cycle_in_finish:\n \"⟦lasso s = None; accpt v; v ∈ dom (finished s)⟧ ⟹ (v,v) ∉ E⁺\"\n using i_no_accpt_cyle_in_finish[THEN make_invar_thm] (*‹lasso (s::'a blue_dfs_state) = None ⟶ (∀x::'a. (accpt::'a ⇒ bool) x ∧ x ∈ dom (finished s) ⟶ (x, x) ∉ E⁺)›*) by blast\n\nend\n\ncontext BlueDFS\nbegin\n definition lasso_inv where\n \"lasso_inv s ≡ ∀pr pl. lasso s = Some (pr,pl) ⟶ \n pl ≠ []\n ∧ (∃v0∈V0. path E v0 pr (hd pl)) \n ∧ accpt (hd pl) \n ∧ path E (hd pl) pl (hd pl)\"\n\n lemma (in BlueDFS_invar) se_back_edge_lasso_inv:\n assumes b_inv: \"lasso_inv s\"\n and ne: \"stack s ≠ []\"\n and R: \"lasso s = None\"\n and p:\"(hd (stack s), v) ∈ pending s\"\n and v: \"v ∈ dom (discovered s)\" \"v ∉ dom (finished s)\"\n and s': \"s' = back_edge (hd (stack s)) v (s⦇pending := pending s - {(u,v)}⦈)\"\n shows \"se_back_edge (hd (stack s)) v s'\n ≤ SPEC (λe. DFS_invar G blue_dfs_params (s'⦇state.more := e⦈) ⟶\n lasso_inv (s'⦇state.more := e⦈))\"\n proof (-)\n (*goal: ‹se_back_edge (hd (stack s)) v s' ≤ SPEC (λe. DFS_invar G blue_dfs_params (s'⦇state.more := e⦈) ⟶ lasso_inv (s'⦇state.more := e⦈))›*)\n from v (*‹v ∈ dom (discovered s)› ‹v ∉ dom (finished s)›*) stack_set_def (*‹set (stack (s::'a blue_dfs_state)) = dom (discovered s) - dom (finished s)›*) have v_in: \"v ∈ set (stack s)\"\n by simp\n from p (*‹(hd (stack (s::'a blue_dfs_state)), v::'a) ∈ pending s›*) have uv_edg: \"(hd (stack s), v) ∈ E\"\n by (auto dest: pendingD (*‹(?u, ?v) ∈ pending s ⟹ (?u, ?v) ∈ E ∧ ?u ∈ dom (discovered s)›*))\n {\n assume accpt: \"accpt (hd (stack s))\" (*‹(accpt::'a ⇒ bool) (hd (stack (s::'a blue_dfs_state)))›*)\n let ?ur = \"rev (tl (stack s))\"\n let ?ul = \"hd (stack s)#dropWhileNot v (rev (tl (stack s)))\"\n let ?s = \"s'⦇lasso := Some (?ur, ?ul), red := red s⦈\"\n assume \"DFS_invar G blue_dfs_params ?s\" (*‹DFS_invar (G::('a, 'b) graph_rec_scheme) blue_dfs_params ((s'::'a blue_dfs_state)⦇lasso := Some (rev (tl (stack s)), hd (stack s) # dropWhileNot (v::'a) (rev (tl (stack s)))), red := red (s::'a blue_dfs_state)⦈)›*)\n have [simp]: \"stack ?s = stack s\"\n by (simp add: s' (*‹s' = back_edge (hd (stack s)) v (s⦇pending := pending s - {(u, v)}⦈)›*))\n have hd_ul[simp]: \"hd ?ul = hd (stack s)\"\n by simp\n have \"?ul ≠ []\"\n by simp\n moreover have P: \"∃v0∈V0. path E v0 ?ur (hd ?ul)\"\n using stack_is_path[OF ne] (*‹∃v0∈V0. path E v0 (rev (tl (stack s))) (hd (stack s))›*) by auto\n moreover from accpt (*‹accpt (hd (stack s))›*) have \"accpt (hd ?ul)\"\n by simp\n moreover have \"path E (hd ?ul) ?ul (hd ?ul)\"\n proof (cases \"v = hd (stack s)\")\n (*goals:\n 1. ‹v = hd (stack s) ⟹ path E (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s))))) (hd (stack s) # dropWhileNot v (rev (tl (stack s)))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›\n 2. ‹v ≠ hd (stack s) ⟹ path E (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s))))) (hd (stack s) # dropWhileNot v (rev (tl (stack s)))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*)\n case True (*‹v = hd (stack s)›*)\n with distinct_hd_tl (*‹⟦distinct ?xs; ?x = hd ?xs⟧ ⟹ ?x ∉ set (tl ?xs)›*) stack_distinct (*‹distinct (stack (s::'a blue_dfs_state))›*) have ul: \"?ul = [hd (stack s)]\"\n by force\n from True (*‹v = hd (stack s)›*) uv_edg (*‹(hd (stack s), v) ∈ E›*) show \"?thesis\"\n (*goal: ‹path E (hd (hd (stack (s::'a blue_dfs_state)) # dropWhileNot (v::'a) (rev (tl (stack s))))) (hd (stack s) # dropWhileNot v (rev (tl (stack s)))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*)\n apply (subst ul (*‹hd (stack s) # dropWhileNot v (rev (tl (stack s))) = [hd (stack s)]›*))\n (*goal: ‹path E (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s))))) (hd (stack s) # dropWhileNot v (rev (tl (stack s)))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*)\n apply (subst ul (*‹hd (stack (s::'a blue_dfs_state)) # dropWhileNot (v::'a) (rev (tl (stack s))) = [hd (stack s)]›*))\n (*goal: ‹⟦v = hd (stack s); (hd (stack s), v) ∈ E⟧ ⟹ path E (hd [hd (stack s)]) (hd (stack s) # dropWhileNot v (rev (tl (stack s)))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*)\n apply (subst ul (*‹hd (stack s) # dropWhileNot v (rev (tl (stack s))) = [hd (stack s)]›*))\n (*goal: ‹⟦(v::'a) = hd (stack (s::'a blue_dfs_state)); (hd (stack s), v) ∈ E⟧ ⟹ path E (hd [hd (stack s)]) [hd (stack s)] (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*)\n by (simp add: path1 (*‹(?u, ?v) ∈ ?E ⟹ path ?E ?u [?u] ?v›*))\n next\n (*goal: ‹(v::'a::type) ≠ hd (stack (s::'a blue_dfs_state)) ⟹ path E (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s))))) (hd (stack s) # dropWhileNot v (rev (tl (stack s)))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*)\n case False (*‹v ≠ hd (stack s)›*)\n with v_in (*‹v ∈ set (stack s)›*) ne (*‹stack s ≠ []›*) have \"v ∈ set ?ur\"\n by (auto simp add: neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))\n with P (*‹∃v0∈V0. path E v0 (rev (tl (stack s))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*) show \"?thesis\"\n (*goal: ‹path E (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s))))) (hd (stack s) # dropWhileNot v (rev (tl (stack s)))) (hd (hd (stack s) # dropWhileNot v (rev (tl (stack s)))))›*)\n by (fastforce intro: path_prepend (*‹⟦(?u, ?v) ∈ ?E; path ?E ?v ?l ?w⟧ ⟹ path ?E ?u (?u # ?l) ?w›*) dropWhileNot_path[where p=\"?ur\"] (*‹⟦rev (tl (stack s)) ≠ []; path ?E ?w (rev (tl (stack s))) ?x; ?v ∈ set (rev (tl (stack s))); dropWhileNot ?v (rev (tl (stack s))) = ?c⟧ ⟹ path ?E ?v ?c ?x›*) uv_edg (*‹(hd (stack s), v) ∈ E›*))\n qed\n ultimately have \"lasso_inv ?s\"\n by (simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n }\n moreover {\n assume accpt: \"accpt v\" (*‹(accpt::'a ⇒ bool) (v::'a)›*)\n let ?vr = \"takeWhileNot v (rev (stack s))\"\n let ?vl = \"dropWhileNot v (rev (stack s))\"\n let ?s = \"s'⦇lasso := Some(?vr, ?vl), red := red s⦈\"\n assume \"DFS_invar G blue_dfs_params ?s\" (*‹DFS_invar (G::('a, 'b) graph_rec_scheme) blue_dfs_params ((s'::'a blue_dfs_state)⦇lasso := Some (takeWhileNot (v::'a) (rev (stack s)), dropWhileNot v (rev (stack s))), red := red (s::'a blue_dfs_state)⦈)›*)\n have [simp]: \"stack ?s = stack s\"\n by (simp add: s' (*‹s' = back_edge (hd (stack s)) v (s⦇pending := pending s - {(u, v)}⦈)›*))\n from ne (*‹stack s ≠ []›*) v_in (*‹v ∈ set (stack s)›*) have hd_vl[simp]: \"hd ?vl = v\"\n apply (induct (\"stack s\") rule: rev_nonempty_induct)\n (*goals:\n 1. ‹⋀x. v ∈ set [x] ⟹ hd (dropWhileNot v (rev [x])) = v›\n 2. ‹⋀x xs. ⟦xs ≠ []; v ∈ set xs ⟹ hd (dropWhileNot v (rev xs)) = v; v ∈ set (xs @ [x])⟧ ⟹ hd (dropWhileNot v (rev (xs @ [x]))) = v›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from v_in (*‹v ∈ set (stack s)›*) have \"?vl ≠ []\"\n by simp\n moreover from hd_succ_stack_is_path[OF ne] (*‹?v ∈ E `` {hd (stack s)} ⟹ ∃v0∈V0. path E v0 (rev (stack s)) ?v›*) uv_edg (*‹(hd (stack s), v) ∈ E›*) have P: \"∃v0∈V0. path E v0 (rev (stack s)) v\"\n by auto\n with ne (*‹stack s ≠ []›*) v_in (*‹v ∈ set (stack s)›*) have \"∃v0∈V0. path E v0 ?vr (hd ?vl)\"\n by (force intro: takeWhileNot_path (*‹⟦?p ≠ []; path ?E ?w ?p ?x; ?v ∈ set ?p; takeWhileNot ?v ?p = ?c⟧ ⟹ path ?E ?w ?c ?v›*))\n moreover from accpt (*‹accpt v›*) have \"accpt (hd ?vl)\"\n by simp\n moreover from P (*‹∃v0::'a∈V0. path E v0 (rev (stack (s::'a blue_dfs_state))) (v::'a)›*) ne (*‹stack s ≠ []›*) v_in (*‹v ∈ set (stack s)›*) have \"path E (hd ?vl) ?vl (hd ?vl)\"\n by (force intro: dropWhileNot_path (*‹⟦?p ≠ []; path ?E ?w ?p ?x; ?v ∈ set ?p; dropWhileNot ?v ?p = ?c⟧ ⟹ path ?E ?v ?c ?x›*))\n ultimately have \"lasso_inv ?s\"\n by (simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n }\n moreover {\n assume \"¬ accpt (hd (stack s))\" \"¬ accpt v\" (*‹¬ (accpt::'a ⇒ bool) (hd (stack (s::'a blue_dfs_state)))› ‹¬ (accpt::'a ⇒ bool) (v::'a)›*)\n let ?s = \"s'⦇state.more := state.more s'⦈\"\n assume \"DFS_invar G blue_dfs_params ?s\" (*‹DFS_invar (G::('a, 'b) graph_rec_scheme) blue_dfs_params (s'⦇state.more := state.more (s'::'a blue_dfs_state)⦈)›*)\n from assms (*‹lasso_inv s› ‹stack s ≠ []› ‹lasso s = None› ‹(hd (stack s), v) ∈ pending s› ‹v ∈ dom (discovered s)› ‹v ∉ dom (finished s)› ‹s' = back_edge (hd (stack s)) v (s⦇pending := pending s - {(u, v)}⦈)›*) have \"lasso_inv ?s\"\n by (auto simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n }\n ultimately show \"?thesis\"\n (*goal: ‹se_back_edge (hd (stack (s::'a blue_dfs_state))) (v::'a::type) (s'::'a blue_dfs_state) ≤ SPEC (λe::⦇lasso :: ('a list × 'a list) option, red :: 'a set⦈. DFS_invar (G::('a, 'b) graph_rec_scheme) blue_dfs_params (s'⦇state.more := e⦈) ⟶ lasso_inv (s'⦇state.more := e⦈))›*)\n using R (*‹lasso s = None›*) s' (*‹s' = back_edge (hd (stack s)) v (s⦇pending := pending s - {(u, v)}⦈)›*) unfolding se_back_edge_def\n (*goal: ‹(case lasso s' of None ⇒ if accpt (hd (stack s)) then let rs = rev (tl (stack s')); ur = rs; ul = hd (stack s) # dropWhileNot v rs in RETURN ⦇lasso = Some (ur, ul), red = red s'⦈ else if accpt v then let rs = rev (stack s'); vr = takeWhileNot v rs; vl = dropWhileNot v rs in RETURN ⦇lasso = Some (vr, vl), red = red s'⦈ else NOOP s' | Some a ⇒ NOOP s') ≤ SPEC (λe. DFS_invar G blue_dfs_params (s'⦇state.more := e⦈) ⟶ lasso_inv (s'⦇state.more := e⦈))›*)\n by (auto split: option.splits (*‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹?P (case ?option of None ⇒ ?f1.0 | Some x ⇒ ?f2.0 x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n qed\n\n\n lemma lasso_inv:\n \"is_invar lasso_inv\"\n proof (induct rule: establish_invarI (*‹⟦on_init blue_dfs_params ≤⇩n SPEC (λx. ?I (empty_state x)); ⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ on_finish blue_dfs_params u s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈)); ⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; ?I s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ ?I (s'⦇state.more := x⦈))⟧ ⟹ is_invar ?I›*))\n (*goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. lasso_inv (empty_state x))›\n 2. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) v0::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 3. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) u::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ on_finish blue_dfs_params u s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 4. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) (u::'v) v::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 5. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) (u::'v) v::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 6. ‹⋀(s::'v blue_dfs_state) (s'::'v blue_dfs_state) (u::'v) v::'v. ⟦DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›*)\n case (finish s s' u) (*‹DFS_invar G blue_dfs_params s› ‹lasso_inv s› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*)\n then interpret BlueDFS_invar where s = s\n by simp\n let ?onstack = \"λx. x ∈ set (stack s)\"\n let ?rE = \"rel_restrict E (red s)\"\n let ?revs = \"rev (tl (stack s))\"\n note ne = ‹stack s ≠ []› (*‹stack s ≠ []›*)\n note [simp] = ‹u=hd (stack s)› (*‹u = hd (stack s)›*)\n from finish (*‹DFS_invar G blue_dfs_params s› ‹lasso_inv s› ‹cond (s::'v blue_dfs_state)› ‹¬ is_break blue_dfs_params (s::'v blue_dfs_state)› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) have [simp]: \"⋀x. x = hd (stack s) ∨ x ∈ set (stack s') ⟷ x∈set (stack s)\" \"red s' = red s\" \"lasso s' = lasso s\"\n apply -\n (*goals:\n 1. ‹⋀x. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ (x = hd (stack s) ∨ x ∈ set (stack s')) = (x ∈ set (stack s))›\n 2. ‹⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ red s' = red s›\n 3. ‹⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ lasso s' = lasso s›\n discuss goal 1*)\n apply ((auto simp: neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))[1])\n (*proven 3 subgoals*) .\n {\n fix v and vs\n let ?cyc = \"vs @ dropWhileNot v ?revs\"\n let ?s = \"s'⦇lasso := Some (?revs, ?cyc), red := red s⦈\"\n assume \"DFS_invar G blue_dfs_params ?s\" and vs: \"vs ≠ []\" \"path ?rE (hd (stack s)) vs v\" and v: \"?onstack v\" and accpt: \"accpt (hd (stack s))\" (*‹DFS_invar (G::('v, 'more) graph_rec_scheme) blue_dfs_params ((s'::'v blue_dfs_state)⦇lasso := Some (rev (tl (stack s)), (vs::'v list) @ dropWhileNot (v::'v) (rev (tl (stack s)))), red := red (s::'v blue_dfs_state)⦈)› ‹(vs::'v list) ≠ []› ‹path (rel_restrict E (red (s::'v blue_dfs_state))) (hd (stack s)) (vs::'v list) (v::'v)› ‹(v::'v) ∈ set (stack (s::'v blue_dfs_state))› ‹(accpt::'v ⇒ bool) (hd (stack (s::'v blue_dfs_state)))›*)\n from vs (*‹(vs::'v::type list) ≠ []› ‹path (rel_restrict E (red (s::'v blue_dfs_state))) (hd (stack s)) (vs::'v::type list) (v::'v::type)›*) have P: \"path E (hd (stack s)) vs v\"\n by (metis path_mono (*‹⟦?E ⊆ ?E'; path ?E ?u ?p ?v⟧ ⟹ path ?E' ?u ?p ?v›*) rel_restrict_sub (*‹rel_restrict ?R ?A ⊆ ?R›*))\n have hds[simp]: \"hd vs = hd (stack s)\" \"hd ?cyc = hd (stack s)\"\n using vs (*‹vs ≠ []› ‹path (rel_restrict E (red s)) (hd (stack s)) vs v›*) path_hd (*‹⟦(?p::?'a list) ≠ []; path (?E::(?'a × ?'a) set) (?v::?'a) ?p (?w::?'a)⟧ ⟹ hd ?p = ?v›*) apply -\n (*goals:\n 1. ‹⟦vs ≠ []; path (rel_restrict E (red s)) (hd (stack s)) vs v; ⋀p E v w. ⟦p ≠ []; path E v p w⟧ ⟹ hd p = v⟧ ⟹ hd vs = hd (stack s)›\n 2. ‹⟦vs ≠ []; path (rel_restrict E (red s)) (hd (stack s)) vs v; ⋀p E v w. ⟦p ≠ []; path E v p w⟧ ⟹ hd p = v⟧ ⟹ hd (vs @ dropWhileNot v (rev (tl (stack s)))) = hd (stack s)›\n discuss goal 1*)\n apply ((assumption)[1])\n (*discuss goal 2*)\n by simp\n (*proven 2 subgoals*)\n from vs (*‹(vs::'v list) ≠ []› ‹path (rel_restrict E (red s)) (hd (stack s)) vs v›*) have \"?cyc ≠ []\"\n by simp\n moreover have P0: \"∃v0∈V0. path E v0 ?revs (hd ?cyc)\"\n using stack_is_path[OF ne] (*‹∃v0∈V0. path E v0 (rev (tl (stack s))) (hd (stack s))›*) by auto\n moreover from accpt (*‹accpt (hd (stack s))›*) have \"accpt (hd ?cyc)\"\n by simp\n moreover have \"path E (hd ?cyc) ?cyc (hd ?cyc)\"\n proof (cases \"tl (stack s) = []\")\n (*goals:\n 1. ‹tl (stack s) = [] ⟹ path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›\n 2. ‹tl (stack s) ≠ [] ⟹ path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n case True (*‹tl (stack s) = []›*)\n with ne (*‹stack s ≠ []›*) last_stack_in_V0 (*‹stack s ≠ [] ⟹ last (stack s) ∈ V0›*) obtain v0 where \"v0 ∈ V0\" and [simp]: \"stack s = [v0]\"\n (*goal: ‹(⋀v0. ⟦v0 ∈ V0; stack s = [v0]⟧ ⟹ thesis) ⟹ thesis›*)\n by (auto simp: neq_Nil_conv (*‹((?xs::?'a list) ≠ []) = (∃(y::?'a) ys::?'a list. ?xs = y # ys)›*))\n with v (*‹v ∈ set (stack s)›*) True (*‹tl (stack s) = []›*) finish (*‹DFS_invar G blue_dfs_params s› ‹lasso_inv s› ‹cond (s::'v blue_dfs_state)› ‹¬ is_break blue_dfs_params (s::'v blue_dfs_state)› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) have [simp]: \"v = v0\"\n by simp\n from True (*‹tl (stack (s::'v blue_dfs_state)) = []›*) P (*‹path E (hd (stack s)) vs v›*) show \"?thesis\"\n (*goal: ‹path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n by simp\n next\n (*goal: ‹tl (stack s) ≠ [] ⟹ path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n case False (*‹tl (stack s) ≠ []›*)\n note tl_ne = this (*‹tl (stack s) ≠ []›*)\n show \"?thesis\"\n (*goal: ‹path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n proof (cases \"v = hd (stack s)\")\n (*goals:\n 1. ‹(v::'v) = hd (stack (s::'v blue_dfs_state)) ⟹ path E (hd ((vs::'v list) @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›\n 2. ‹(v::'v) ≠ hd (stack (s::'v blue_dfs_state)) ⟹ path E (hd ((vs::'v list) @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n case True (*‹v = hd (stack s)›*)\n hence \"v ∉ set ?revs\"\n using ne (*‹stack s ≠ []›*) stack_distinct (*‹distinct (stack s)›*) by (auto simp: neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))\n hence \"?cyc = vs\"\n by fastforce\n with P (*‹path E (hd (stack s)) vs v›*) True (*‹(v::'v) = hd (stack (s::'v blue_dfs_state))›*) show \"?thesis\"\n (*goal: ‹path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n by (simp del: dropWhile_eq_Nil_conv (*‹(dropWhile ?P ?xs = []) = (∀x∈set ?xs. ?P x)›*))\n next\n (*goal: ‹v ≠ hd (stack s) ⟹ path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n case False (*‹v ≠ hd (stack s)›*)\n with finish (*‹DFS_invar G blue_dfs_params s› ‹lasso_inv s› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹u = hd (stack s)› ‹pending s `` {u} = {}› ‹s' = finish u s›*) v (*‹v ∈ set (stack s)›*) have \"v ∈ set ?revs\"\n by (auto simp: neq_Nil_conv (*‹(?xs ≠ []) = (∃y ys. ?xs = y # ys)›*))\n with tl_ne (*‹tl (stack s) ≠ []›*) False (*‹v ≠ hd (stack s)›*) P0 (*‹∃v0::'v∈V0. path E v0 (rev (tl (stack (s::'v blue_dfs_state)))) (hd ((vs::'v list) @ dropWhileNot (v::'v) (rev (tl (stack s)))))›*) show \"?thesis\"\n (*goal: ‹path E (hd (vs @ dropWhileNot v (rev (tl (stack s))))) (vs @ dropWhileNot v (rev (tl (stack s)))) (hd (vs @ dropWhileNot v (rev (tl (stack s)))))›*)\n by (force intro: path_conc[OF P] (*‹path E v ?lb ?w ⟹ path E (hd (stack s)) (vs @ ?lb) ?w›*) dropWhileNot_path[where p=\"?revs\"] (*‹⟦rev (tl (stack s)) ≠ []; path ?E ?w (rev (tl (stack s))) ?x; ?v ∈ set (rev (tl (stack s))); dropWhileNot ?v (rev (tl (stack s))) = ?c⟧ ⟹ path ?E ?v ?c ?x›*))\n qed\n qed\n ultimately have \"lasso_inv ?s\"\n by (simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n }\n hence \"accpt (hd (stack s)) ⟶ lasso s = None ⟶\n red_dfs (red s) ?onstack (hd (stack s)) ≤ SPEC (λrs. ∀vs v. \n rs = Inr (vs,v) ⟶\n DFS_invar G blue_dfs_params (s'⦇lasso := Some (?revs, vs @ dropWhileNot v ?revs), red:= red s⦈) ⟶\n lasso_inv (s'⦇lasso := Some (?revs, vs @ dropWhileNot v ?revs), red:=red s⦈))\"\n apply clarsimp\n (*goal: ‹accpt (hd (stack s)) ⟶ lasso s = None ⟶ red_dfs (red s) (λx. x ∈ set (stack s)) (hd (stack s)) ≤ SPEC (λrs. ∀vs v. rs = Inr (vs, v) ⟶ DFS_invar G blue_dfs_params (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈) ⟶ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈))›*)\n apply (rule find_path1_restr_spec_rule (*‹⟦fb_graph ?G ∧ restr_invar (g_E ?G) ?R ?P; fb_graph ?G ∧ restr_invar (g_E ?G) ?R ?P ⟹ SPEC (find_path1_restr_pred ?G ?P ?R) ≤ ?m⟧ ⟹ find_path1_restr_spec ?G ?P ?R ≤ ?m› ‹(fb_graph ?G ∧ restr_invar (g_E ?G) ?R ?P ⟹ SPEC (find_path1_restr_pred ?G ?P ?R) ≤⇩n ?m) ⟹ find_path1_restr_spec ?G ?P ?R ≤⇩n ?m›*), intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*))\n (*goal: ‹⟦⋀vs v. ⟦BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); vs ≠ []; path (rel_restrict E (red s)) (hd (stack s)) vs v; v ∈ set (stack s)⟧ ⟹ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); accpt (hd (stack s)); lasso s = None⟧ ⟹ red_dfs (red s) (λx. x ∈ set (stack s)) (hd (stack s)) ≤ SPEC (λrs. ∀vs v. rs = Inr (vs, v) ⟶ BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈) ⟶ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈))›*)\n apply ((rule red_DFS_precond (*‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_E := E ∩ UNIV × - red s, g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ restr_invar E (red s) (λx. x ∈ set (stack s))›*), simp_all add: ne)[1])\n (*top goal: ‹⟦⋀vs v. ⟦BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); vs ≠ []; path (rel_restrict E (red s)) (hd (stack s)) vs v; v ∈ set (stack s)⟧ ⟹ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); accpt (hd (stack s)); lasso s = None⟧ ⟹ fb_graph (G⦇g_V0 := {hd (stack s)}⦈)› and 2 goals remain*)\n apply ((simp, rule red_DFS_precond (*‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ fb_graph (G⦇g_E := E ∩ UNIV × - red s, g_V0 := {hd (stack s)}⦈)› ‹⟦lasso s = None; stack s ≠ []⟧ ⟹ restr_invar E (red s) (λx. x ∈ set (stack s))›*), simp_all add: ne)[1])\n (*top goal: ‹⟦⋀vs v. ⟦BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); vs ≠ []; path (rel_restrict E (red s)) (hd (stack s)) vs v; v ∈ set (stack s)⟧ ⟹ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); accpt (hd (stack s)); lasso s = None⟧ ⟹ restr_invar (g_E (G⦇g_V0 := {hd (stack s)}⦈)) (red s) (λx. x ∈ set (stack s))› and 1 goal remains*)\n using red_stack_disjoint (*‹lasso s = None ⟹ set (stack s) ∩ red s = {}›*) ne (*‹stack s ≠ []›*) apply clarsimp\n (*goal: ‹⟦⋀vs v. ⟦BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); vs ≠ []; path (rel_restrict E (red s)) (hd (stack s)) vs v; v ∈ set (stack s)⟧ ⟹ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈); accpt (hd (stack s)); lasso s = None; fb_graph (G⦇g_V0 := {hd (stack s)}⦈) ∧ restr_invar (g_E (G⦇g_V0 := {hd (stack s)}⦈)) (red s) (λx. x ∈ set (stack s))⟧ ⟹ SPEC (find_path1_restr_pred (G⦇g_V0 := {hd (stack s)}⦈) (λx. x ∈ set (stack s)) (red s)) ≤ SPEC (λrs. ∀vs v. rs = Inr (vs, v) ⟶ BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈) ⟶ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈))›*)\n apply rprems\n (*goals:\n 1. ‹⋀vs v. ⟦accpt (hd (stack s)); lasso s = None; set (stack s) ∩ red s = {}; stack s ≠ []; find_path1_restr_pred (G⦇g_V0 := {hd (stack s)}⦈) (λx. x ∈ set (stack s)) (red s) (Inr (vs, v)); fb_graph (G⦇g_V0 := {hd (stack s)}⦈); restr_invar E (red s) (λx. x ∈ set (stack s)); BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈)⟧ ⟹ BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈)›\n 2. ‹⋀vs v. ⟦accpt (hd (stack s)); lasso s = None; set (stack s) ∩ red s = {}; stack s ≠ []; find_path1_restr_pred (G⦇g_V0 := {hd (stack s)}⦈) (λx. x ∈ set (stack s)) (red s) (Inr (vs, v)); fb_graph (G⦇g_V0 := {hd (stack s)}⦈); restr_invar E (red s) (λx. x ∈ set (stack s)); BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈)⟧ ⟹ vs ≠ []›\n 3. ‹⋀vs v. ⟦accpt (hd (stack s)); lasso s = None; set (stack s) ∩ red s = {}; stack s ≠ []; find_path1_restr_pred (G⦇g_V0 := {hd (stack s)}⦈) (λx. x ∈ set (stack s)) (red s) (Inr (vs, v)); fb_graph (G⦇g_V0 := {hd (stack s)}⦈); restr_invar E (red s) (λx. x ∈ set (stack s)); BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈)⟧ ⟹ path (rel_restrict E (red s)) (hd (stack s)) vs v›\n 4. ‹⋀vs v. ⟦accpt (hd (stack s)); lasso s = None; set (stack s) ∩ red s = {}; stack s ≠ []; find_path1_restr_pred (G⦇g_V0 := {hd (stack s)}⦈) (λx. x ∈ set (stack s)) (red s) (Inr (vs, v)); fb_graph (G⦇g_V0 := {hd (stack s)}⦈); restr_invar E (red s) (λx. x ∈ set (stack s)); BlueDFS_invar G accpt (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈)⟧ ⟹ v ∈ set (stack s)›\n discuss goal 1*)\n apply (simp add: find_path1_restr_pred_def (*‹find_path1_restr_pred (?G::(?'a, ?'b) graph_rec_scheme) (?P::?'a ⇒ bool) (?R::?'a set) ≡ λr::?'a set + ?'a list × ?'a. case r of Inl (R'::?'a set) ⇒ R' = ?R ∪ (g_E ?G)⁺ `` g_V0 ?G ∧ restr_invar (g_E ?G) R' ?P | Inr (vs::?'a list, v::?'a) ⇒ ?P v ∧ vs ≠ [] ∧ (∃v0::?'a∈g_V0 ?G. path (g_E ?G ∩ UNIV × - ?R) v0 vs v)›*) restr_invar_def (*‹restr_invar (?E::(?'a × ?'a) set) (?R::?'a set) (?P::?'a ⇒ bool) ≡ ?E `` ?R ⊆ ?R ∧ ?R ∩ Collect ?P = {}›*))\n (*discuss goal 2*)\n apply (simp add: find_path1_restr_pred_def (*‹find_path1_restr_pred ?G ?P ?R ≡ λr. case r of Inl R' ⇒ R' = ?R ∪ (g_E ?G)⁺ `` g_V0 ?G ∧ restr_invar (g_E ?G) R' ?P | Inr (vs, v) ⇒ ?P v ∧ vs ≠ [] ∧ (∃v0∈g_V0 ?G. path (g_E ?G ∩ UNIV × - ?R) v0 vs v)›*) restr_invar_def (*‹restr_invar ?E ?R ?P ≡ ?E `` ?R ⊆ ?R ∧ ?R ∩ Collect ?P = {}›*))\n (*discuss goal 3*)\n apply (simp add: find_path1_restr_pred_def (*‹find_path1_restr_pred (?G::(?'a, ?'b) graph_rec_scheme) (?P::?'a ⇒ bool) (?R::?'a set) ≡ λr::?'a set + ?'a list × ?'a. case r of Inl (R'::?'a set) ⇒ R' = ?R ∪ (g_E ?G)⁺ `` g_V0 ?G ∧ restr_invar (g_E ?G) R' ?P | Inr (vs::?'a list, v::?'a) ⇒ ?P v ∧ vs ≠ [] ∧ (∃v0::?'a∈g_V0 ?G. path (g_E ?G ∩ UNIV × - ?R) v0 vs v)›*) restr_invar_def (*‹restr_invar (?E::(?'a × ?'a) set) (?R::?'a set) (?P::?'a ⇒ bool) ≡ ?E `` ?R ⊆ ?R ∧ ?R ∩ Collect ?P = {}›*))\n (*top goal: ‹⋀(vs::'v list) v::'v. ⟦(accpt::'v ⇒ bool) (hd (stack (s::'v blue_dfs_state))); lasso s = None; set (stack s) ∩ red s = {}; stack s ≠ []; find_path1_restr_pred ((G::('v, 'more) graph_rec_scheme)⦇g_V0 := {hd (stack s)}⦈) (λx::'v. x ∈ set (stack s)) (red s) (Inr (vs, v)); fb_graph (G⦇g_V0 := {hd (stack s)}⦈); restr_invar E (red s) (λx::'v. x ∈ set (stack s)); BlueDFS_invar G accpt ((s'::'v blue_dfs_state)⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈)⟧ ⟹ path (rel_restrict E (red s)) (hd (stack s)) vs v› and 1 goal remains*)\n apply (fastforce intro: path_restrict_tl (*‹⟦?u ∉ ?R; path (?E ∩ UNIV × - ?R) ?u ?p ?v⟧ ⟹ path (rel_restrict ?E ?R) ?u ?p ?v›*) rel_restrictI (*‹⟦?x ∉ ?R; ?y ∉ ?R; (?x, ?y) ∈ ?E⟧ ⟹ (?x, ?y) ∈ rel_restrict ?E ?R›*))\n (*discuss goal 4*)\n apply (simp add: find_path1_restr_pred_def (*‹find_path1_restr_pred ?G ?P ?R ≡ λr. case r of Inl R' ⇒ R' = ?R ∪ (g_E ?G)⁺ `` g_V0 ?G ∧ restr_invar (g_E ?G) R' ?P | Inr (vs, v) ⇒ ?P v ∧ vs ≠ [] ∧ (∃v0∈g_V0 ?G. path (g_E ?G ∩ UNIV × - ?R) v0 vs v)›*) restr_invar_def (*‹restr_invar ?E ?R ?P ≡ ?E `` ?R ⊆ ?R ∧ ?R ∩ Collect ?P = {}›*))\n (*proven 4 subgoals*) .\n note aux1 = this[rule_format, THEN leof_lift] (*‹⟦accpt (hd (stack s)); lasso s = None⟧ ⟹ red_dfs (red s) (λx. x ∈ set (stack s)) (hd (stack s)) ≤⇩n SPEC (λrs. ∀vs v. rs = Inr (vs, v) ⟶ DFS_invar G blue_dfs_params (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈) ⟶ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈))›*)\n show \"?case\"\n (*goal: ‹on_finish blue_dfs_params u s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›*)\n apply simp\n (*goal: ‹on_finish blue_dfs_params u s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›*)\n unfolding run_red_dfs_def mk_blue_witness_def cyan_def\n (*goal: ‹(accpt (hd (stack s)) ⟶ (case lasso s' of None ⇒ red_dfs (red s') (λx. x = hd (stack s) ∨ x ∈ set (stack s')) (hd (stack s)) ⤜ (λredS. RETURN (case redS of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack s') in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red s'⦈)) | Some a ⇒ NOOP s') ≤⇩n SPEC (λx. BlueDFS_invar G accpt (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))) ∧ (¬ accpt (hd (stack s)) ⟶ BlueDFS_invar G accpt s' ⟶ lasso_inv s')›*)\n apply (simp add: run_red_dfs_def (*‹run_red_dfs (?u::'v) (?s::'v blue_dfs_state) ≡ case lasso ?s of None ⇒ red_dfs (red ?s) (λx::'v. x = ?u ∨ x ∈ cyan ?s) ?u ⤜ (λredS::'v set + 'v list × 'v. RETURN (mk_blue_witness ?s redS)) | Some (a::'v list × 'v list) ⇒ NOOP ?s›*) mk_blue_witness_def (*‹mk_blue_witness (?s::'v blue_dfs_state) (?redS::'v set + 'v list × 'v) ≡ case ?redS of Inl (R'::'v set) ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs::'v list, v::'v) ⇒ let rs::'v list = rev (stack ?s) in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red ?s⦈›*) cyan_def (*‹cyan (?s::(?'a, ?'b) state_scheme) ≡ set (stack ?s)›*) split: option.splits (*‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = ((?option = None ⟶ ?P ?f1.0) ∧ (∀x2::?'a. ?option = Some x2 ⟶ ?P (?f2.0 x2)))› ‹(?P::?'b ⇒ bool) (case ?option::?'a option of None ⇒ ?f1.0::?'b | Some (x::?'a) ⇒ (?f2.0::?'a ⇒ ?'b) x) = (¬ (?option = None ∧ ¬ ?P ?f1.0 ∨ (∃x2::?'a. ?option = Some x2 ∧ ¬ ?P (?f2.0 x2))))›*))\n (*goal: ‹(accpt (hd (stack s)) ⟶ (case lasso s' of None ⇒ red_dfs (red s') (λx. x = hd (stack s) ∨ x ∈ set (stack s')) (hd (stack s)) ⤜ (λredS. RETURN (case redS of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack s') in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red s'⦈)) | Some a ⇒ NOOP s') ≤⇩n SPEC (λx. BlueDFS_invar G accpt (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))) ∧ (¬ accpt (hd (stack s)) ⟶ BlueDFS_invar G accpt s' ⟶ lasso_inv s')›*)\n apply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹(lasso (s::'v blue_dfs_state) = None ⟶ ((accpt::'v ⇒ bool) (hd (stack s)) ⟶ red_dfs (red s) (λx::'v. x ∈ set (stack s)) (hd (stack s)) ⤜ (λredS::'v set + 'v list × 'v. RETURN (case redS of Inl (R'::'v set) ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs::'v list, v::'v) ⇒ let rs::'v list = rev (stack (s'::'v blue_dfs_state)) in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red s'⦈)) ≤⇩n SPEC (λx::⦇lasso :: ('v list × 'v list) option, red :: 'v set⦈. BlueDFS_invar (G::('v, 'more) graph_rec_scheme) accpt (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))) ∧ (¬ accpt (hd (stack s)) ⟶ BlueDFS_invar G accpt s' ⟶ lasso_inv s')) ∧ ((∃(a::'v list) b::'v list. lasso s = Some (a, b)) ⟶ BlueDFS_invar G accpt s' ⟶ lasso_inv s')›*)\n apply (refine_vcg leof_trans[OF aux1])\n (*top goal: ‹⟦lasso s = None; accpt (hd (stack s))⟧ ⟹ red_dfs (red s) (λx. x ∈ set (stack s)) (hd (stack s)) ⤜ (λredS. RETURN (case redS of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack s') in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red s'⦈)) ≤⇩n SPEC (λx. BlueDFS_invar G accpt (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))› and 2 goals remain*)\n using finish (*‹DFS_invar G blue_dfs_params s› ‹lasso_inv s› ‹cond (s::'v blue_dfs_state)› ‹¬ is_break blue_dfs_params (s::'v blue_dfs_state)› ‹stack s ≠ []› ‹(u::'v) = hd (stack (s::'v blue_dfs_state))› ‹pending s `` {u} = {}› ‹s' = finish u s›*) apply -\n (*goals:\n 1. ‹⋀x. ⟦lasso s = None; accpt (hd (stack s)); x ∈ {rs. ∀vs v. rs = Inr (vs, v) ⟶ DFS_invar G blue_dfs_params (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈) ⟶ lasso_inv (s'⦇lasso := Some (rev (tl (stack s)), vs @ dropWhileNot v (rev (tl (stack s)))), red := red s⦈)}; BlueDFS_invar G accpt (s'⦇state.more := case x of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack s') in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red s'⦈⦈); DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ lasso_inv (s'⦇state.more := case x of Inl R' ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs, v) ⇒ let rs = rev (stack s') in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red s'⦈⦈)›\n 2. ‹⟦lasso s = None; ¬ accpt (hd (stack s)); BlueDFS_invar G accpt s'; DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ lasso_inv s'›\n 3. ‹⟦∃a b. lasso s = Some (a, b); BlueDFS_invar G accpt s'; DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s⟧ ⟹ lasso_inv s'›\n discuss goal 1*)\n apply ((auto simp add: lasso_inv_def (*‹lasso_inv (?s::('v, ?'a) blue_dfs_state_scheme) ≡ ∀(pr::'v list) pl::'v list. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0::'v∈V0. path E v0 pr (hd pl)) ∧ (accpt::'v ⇒ bool) (hd pl) ∧ path E (hd pl) pl (hd pl)›*) split: sum.split (*‹(?P::?'c ⇒ bool) (case ?sum::?'a + ?'b of Inl (x::?'a) ⇒ (?f1.0::?'a ⇒ ?'c) x | Inr (x::?'b) ⇒ (?f2.0::?'b ⇒ ?'c) x) = ((∀x1::?'a. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2::?'b. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*) split: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))[1])\n (*discuss goal 3*)\n apply ((auto simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*) split: sum.split (*‹?P (case ?sum of Inl x ⇒ ?f1.0 x | Inr x ⇒ ?f2.0 x) = ((∀x1. ?sum = Inl x1 ⟶ ?P (?f1.0 x1)) ∧ (∀x2. ?sum = Inr x2 ⟶ ?P (?f2.0 x2)))›*))[1])\n (*proven 3 subgoals*) .\n next\n (*goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx. lasso_inv (empty_state x))›\n 2. ‹⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 3. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 4. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∉ dom (finished s); s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 5. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›*)\n case (back_edge s s' u v) (*‹DFS_invar G blue_dfs_params s› ‹lasso_inv s› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹(u, v) ∈ pending s› ‹u = hd (stack s)› ‹v ∈ dom (discovered s)› ‹v ∉ dom (finished s)› ‹s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)›*)\n then interpret BlueDFS_invar where s = s\n by simp\n from back_edge (*‹DFS_invar G blue_dfs_params s› ‹lasso_inv s› ‹cond s› ‹¬ is_break blue_dfs_params s› ‹stack s ≠ []› ‹(u::'v, v::'v) ∈ pending (s::'v blue_dfs_state)› ‹u = hd (stack s)› ‹v ∈ dom (discovered s)› ‹(v::'v) ∉ dom (finished (s::'v blue_dfs_state))› ‹s' = back_edge u v (s⦇pending := pending s - {(u, v)}⦈)›*) se_back_edge_lasso_inv[THEN leof_lift] (*‹⟦lasso_inv s; stack s ≠ []; lasso s = None; (hd (stack s), ?v1) ∈ pending s; ?v1 ∈ dom (discovered s); ?v1 ∉ dom (finished s); ?s'1 = back_edge (hd (stack s)) ?v1 (s⦇pending := pending s - {(?u1, ?v1)}⦈)⟧ ⟹ se_back_edge (hd (stack s)) ?v1 ?s'1 ≤⇩n SPEC (λe. DFS_invar G blue_dfs_params (?s'1⦇state.more := e⦈) ⟶ lasso_inv (?s'1⦇state.more := e⦈))›*) show \"?case\"\n (*goal: ‹on_back_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›*)\n by auto\n qed (simp_all add: lasso_inv_def empty_state_def)\n (*solves the remaining goals:\n 1. ‹on_init blue_dfs_params ≤⇩n SPEC (λx. lasso_inv (empty_state x))›\n 2. ‹⋀s s' v0. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s = []; v0 ∈ V0; v0 ∉ dom (discovered s); s' = new_root v0 s⟧ ⟹ on_new_root blue_dfs_params v0 s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 3. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∈ dom (discovered s); v ∈ dom (finished s); s' = cross_edge u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_cross_edge blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›\n 4. ‹⋀s s' u v. ⟦DFS_invar G blue_dfs_params s; lasso_inv s; cond s; ¬ is_break blue_dfs_params s; stack s ≠ []; (u, v) ∈ pending s; u = hd (stack s); v ∉ dom (discovered s); s' = discover u v (s⦇pending := pending s - {(u, v)}⦈)⟧ ⟹ on_discover blue_dfs_params u v s' ≤⇩n SPEC (λx. DFS_invar G blue_dfs_params (s'⦇state.more := x⦈) ⟶ lasso_inv (s'⦇state.more := x⦈))›*)\nend\n\ncontext BlueDFS_invar\nbegin\n lemmas s_lasso_inv = lasso_inv[THEN make_invar_thm]\n\n lemma \n assumes \"lasso s = Some (pr,pl)\"\n shows loop_nonempty: \"pl ≠ []\"\n and accpt_loop: \"accpt (hd pl)\"\n and loop_is_path: \"path E (hd pl) pl (hd pl)\"\n and loop_reachable: \"∃v0∈V0. path E v0 pr (hd pl)\"\n using assms (*‹lasso s = Some (pr, pl)›*) s_lasso_inv (*‹lasso_inv s›*) apply -\n (*goals:\n 1. ‹⟦lasso s = Some (pr, pl); lasso_inv s⟧ ⟹ pl ≠ []›\n 2. ‹⟦lasso s = Some (pr, pl); lasso_inv s⟧ ⟹ accpt (hd pl)›\n 3. ‹⟦lasso s = Some (pr, pl); lasso_inv s⟧ ⟹ path E (hd pl) pl (hd pl)›\n 4. ‹⟦lasso s = Some (pr, pl); lasso_inv s⟧ ⟹ ∃v0∈V0. path E v0 pr (hd pl)›\n discuss goal 1*)\n apply (simp add: lasso_inv_def (*‹lasso_inv (?s::('a, ?'a) blue_dfs_state_scheme) ≡ ∀(pr::'a list) pl::'a list. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0::'a∈V0. path E v0 pr (hd pl)) ∧ (accpt::'a ⇒ bool) (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n (*discuss goal 2*)\n apply (simp add: lasso_inv_def (*‹lasso_inv (?s::('a, ?'a) blue_dfs_state_scheme) ≡ ∀(pr::'a::type list) pl::'a::type list. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0::'a::type∈V0. path E v0 pr (hd pl)) ∧ (accpt::'a::type ⇒ bool) (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n (*discuss goal 3*)\n apply (simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n (*discuss goal 4*)\n apply (simp add: lasso_inv_def (*‹lasso_inv ?s ≡ ∀pr pl. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0∈V0. path E v0 pr (hd pl)) ∧ accpt (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n (*proven 4 subgoals*) .\n\n lemma blue_dfs_correct:\n assumes NC: \"¬ cond s\"\n shows \"case lasso s of\n None ⇒ ¬(∃v0∈V0. ∃v. (v0,v) ∈ E⇧* ∧ accpt v ∧ (v,v) ∈ E⁺)\n | Some (pr,pl) ⇒ (∃v0∈V0. ∃v. \n path E v0 pr v ∧ accpt v ∧ pl≠[] ∧ path E v pl v)\"\n proof (cases \"lasso s\")\n (*goals:\n 1. ‹lasso s = None ⟹ case lasso s of None ⇒ ¬ (∃v0∈V0. ∃v. (v0, v) ∈ E⇧* ∧ accpt v ∧ (v, v) ∈ E⁺) | Some (pr, pl) ⇒ ∃v0∈V0. ∃v. path E v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path E v pl v›\n 2. ‹⋀a. lasso s = Some a ⟹ case lasso s of None ⇒ ¬ (∃v0∈V0. ∃v. (v0, v) ∈ E⇧* ∧ accpt v ∧ (v, v) ∈ E⁺) | Some (pr, pl) ⇒ ∃v0∈V0. ∃v. path E v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path E v pl v›*)\n case None (*‹lasso s = None›*)\n moreover {\n fix v and v0\n assume \"v0 ∈ V0\" \"(v0,v) ∈ E⇧*\" \"accpt v\" \"(v,v) ∈ E⁺\" (*‹(v0::'a) ∈ V0› ‹(v0::'a, v::'a) ∈ E⇧*› ‹(accpt::'a ⇒ bool) (v::'a)› ‹(v::'a, v) ∈ E⁺›*)\n moreover hence \"v ∈ reachable\"\n by auto\n with nc_finished_eq_reachable (*‹⟦¬ cond s; ¬ is_break blue_dfs_params s⟧ ⟹ dom (finished s) = reachable›*) NC (*‹¬ cond (s::'a blue_dfs_state)›*) None (*‹lasso s = None›*) have \"v ∈ dom (finished s)\"\n by simp\n moreover note no_accpt_cycle_in_finish (*‹⟦lasso s = None; accpt ?v; ?v ∈ dom (finished s)⟧ ⟹ (?v, ?v) ∉ E⁺›*) None (*‹lasso s = None›*)\n ultimately have False\n by blast\n }\n ultimately show \"?thesis\"\n (*goal: ‹case lasso (s::'a blue_dfs_state) of None ⇒ ¬ (∃v0::'a∈V0. ∃v::'a. (v0, v) ∈ E⇧* ∧ (accpt::'a ⇒ bool) v ∧ (v, v) ∈ E⁺) | Some (pr::'a list, pl::'a list) ⇒ ∃v0::'a∈V0. ∃v::'a. path E v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path E v pl v›*)\n by auto\n next\n (*goal: ‹⋀a. lasso s = Some a ⟹ case lasso s of None ⇒ ¬ (∃v0∈V0. ∃v. (v0, v) ∈ E⇧* ∧ accpt v ∧ (v, v) ∈ E⁺) | Some (pr, pl) ⇒ ∃v0∈V0. ∃v. path E v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path E v pl v›*)\n case (Some prpl) (*‹lasso s = Some prpl›*)\n with s_lasso_inv (*‹lasso_inv s›*) show \"?thesis\"\n (*goal: ‹case lasso (s::'a blue_dfs_state) of None ⇒ ¬ (∃v0::'a∈V0. ∃v::'a. (v0, v) ∈ E⇧* ∧ (accpt::'a ⇒ bool) v ∧ (v, v) ∈ E⁺) | Some (pr::'a list, pl::'a list) ⇒ ∃v0::'a∈V0. ∃v::'a. path E v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path E v pl v›*)\n apply (cases prpl)\n (*goal: ‹case lasso s of None ⇒ ¬ (∃v0∈V0. ∃v. (v0, v) ∈ E⇧* ∧ accpt v ∧ (v, v) ∈ E⁺) | Some (pr, pl) ⇒ ∃v0∈V0. ∃v. path E v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path E v pl v›*)\n by (auto intro: path_is_rtrancl (*‹path (?E::(?'a × ?'a) set) (?u::?'a) (?l::?'a list) (?v::?'a) ⟹ (?u, ?v) ∈ ?E⇧*›*) path_is_trancl (*‹⟦path (?E::(?'a × ?'a) set) (?u::?'a) (?l::?'a list) (?v::?'a); ?l ≠ []⟧ ⟹ (?u, ?v) ∈ ?E⁺›*) simp: lasso_inv_def (*‹lasso_inv (?s::('a, ?'a) blue_dfs_state_scheme) ≡ ∀(pr::'a list) pl::'a list. lasso ?s = Some (pr, pl) ⟶ pl ≠ [] ∧ (∃v0::'a∈V0. path E v0 pr (hd pl)) ∧ (accpt::'a ⇒ bool) (hd pl) ∧ path E (hd pl) pl (hd pl)›*))\n qed\n\nend\n\nsubsection ‹Interface›\n\ninterpretation BlueDFS_defs for G accpt .\n\ndefinition \"nested_dfs_spec G accpt ≡ λr. case r of\n None ⇒ ¬(∃v0∈g_V0 G. ∃v. (v0,v) ∈ (g_E G)⇧* ∧ accpt v ∧ (v,v) ∈ (g_E G)⁺)\n| Some (pr,pl) ⇒ (∃v0∈g_V0 G. ∃v. \n path (g_E G) v0 pr v ∧ accpt v ∧ pl≠[] ∧ path (g_E G) v pl v)\"\n\ndefinition \"nested_dfs G accpt ≡ do {\n ASSERT (fb_graph G);\n s ← it_dfs TYPE('a) G accpt;\n RETURN (lasso s)\n}\"\n\ntheorem nested_dfs_correct: \n assumes \"fb_graph G\"\n shows \"nested_dfs G accpt ≤ SPEC (nested_dfs_spec G accpt)\"\nproof (-)\n (*goal: ‹nested_dfs (G::('a, 'b) graph_rec_scheme) (accpt::'a::type ⇒ bool) ≤ SPEC (nested_dfs_spec G accpt)›*)\n interpret fb_graph G\n by fact\n interpret BlueDFS G accpt\n by unfold_locales\n show \"?thesis\"\n (*goal: ‹nested_dfs G accpt ≤ SPEC (nested_dfs_spec G accpt)›*)\n unfolding nested_dfs_def\n (*goal: ‹ASSERT (fb_graph G) ⤜ (λ_. param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt) ⤜ (λs. RETURN (lasso s))) ≤ SPEC (nested_dfs_spec G accpt)›*)\n apply (refine_rcg refine_vcg)\n (*goal: ‹ASSERT (fb_graph G) ⤜ (λ_. param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt) ⤜ (λs. RETURN (lasso s))) ≤ SPEC (nested_dfs_spec G accpt)›*)\n apply fact\n (*top goal: ‹fb_graph G› and 1 goal remains*)\n apply (rule weaken_SPEC[OF it_dfs_correct] (*‹(⋀x. DFS_invar G (BlueDFS_defs.blue_dfs_params G accpt) x ∧ ¬ param_DFS_defs.cond G (BlueDFS_defs.blue_dfs_params G accpt) x ⟹ ?Ψ x) ⟹ param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt) ≤ SPEC ?Ψ›*))\n (*goal: ‹fb_graph G ⟹ param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt) ≤ SPEC (λs. RETURN (lasso s) ≤ SPEC (nested_dfs_spec G accpt))›*)\n proof (clarsimp)\n (*goal: ‹⋀x::'a blue_dfs_state. ⟦fb_graph (G::('a, 'b) graph_rec_scheme); BlueDFS_invar G (accpt::'a::type ⇒ bool) x; ¬ param_DFS_defs.cond G (BlueDFS_defs.blue_dfs_params G accpt) x⟧ ⟹ nested_dfs_spec G accpt (lasso x)›*)\n fix s\n assume \"BlueDFS_invar G accpt s\" (*‹BlueDFS_invar (G::('a, 'b) graph_rec_scheme) (accpt::'a ⇒ bool) (s::'a blue_dfs_state)›*)\n then interpret BlueDFS_invar G accpt s .\n assume \"¬cond TYPE('b) G accpt s\" (*‹¬ param_DFS_defs.cond (G::('a, 'b) graph_rec_scheme) (BlueDFS_defs.blue_dfs_params G (accpt::'a ⇒ bool)) (s::'a blue_dfs_state)›*)\n from blue_dfs_correct[OF this] (*‹case lasso s of None ⇒ ¬ (∃v0∈g_V0 G. ∃v. (v0, v) ∈ (g_E G)⇧* ∧ accpt v ∧ (v, v) ∈ (g_E G)⁺) | Some (pr, pl) ⇒ ∃v0∈g_V0 G. ∃v. path (g_E G) v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path (g_E G) v pl v›*) show \"nested_dfs_spec G accpt (lasso s)\"\n unfolding nested_dfs_spec_def\n (*goal: ‹case lasso s of None ⇒ ¬ (∃v0∈g_V0 G. ∃v. (v0, v) ∈ (g_E G)⇧* ∧ accpt v ∧ (v, v) ∈ (g_E G)⁺) | Some (pr, pl) ⇒ ∃v0∈g_V0 G. ∃v. path (g_E G) v0 pr v ∧ accpt v ∧ pl ≠ [] ∧ path (g_E G) v pl v›*)\n by simp\n qed\nqed\n\nsubsection ‹Implementation›\n\nrecord 'v bdfs_state_impl = \"'v simple_state\" +\n lasso_impl :: \"('v list × 'v list) option\"\n red_impl :: \"'v set\"\n\ndefinition \"bdfs_erel ≡ {(⦇lasso_impl=li,red_impl=ri⦈,⦇lasso=l, red=r⦈) \n | li ri l r. li=l ∧ ri=r}\"\n\nabbreviation \"bdfs_rel ≡ ⟨bdfs_erel⟩simple_state_rel\"\n\ndefinition mk_blue_witness_impl\n :: \"'v bdfs_state_impl ⇒ 'v fpr_result ⇒ ('v,unit) bdfs_state_impl_ext\"\n where\n \"mk_blue_witness_impl s redS ≡ \n case redS of\n Inl R' ⇒ ⦇ lasso_impl = None, red_impl = (R' ⌦‹∪ red_impl s›) ⦈\n | Inr (vs, v) ⇒ let \n rs = rev (map fst (CAST (ss_stack s))) \n in ⦇ \n lasso_impl = Some (rs, vs@dropWhileNot v rs), \n red_impl = red_impl s⦈\"\n\nlemma mk_blue_witness_impl[refine]:\n \"⟦ (si,s)∈bdfs_rel; (ri,r)∈⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel ⟧ \n ⟹ (mk_blue_witness_impl si ri, mk_blue_witness s r)∈bdfs_erel\"\n unfolding mk_blue_witness_impl_def mk_blue_witness_def\n (*goal: ‹⟦(si::'a bdfs_state_impl, s::'a blue_dfs_state) ∈ bdfs_rel; (ri::'a set + 'a list × 'a, r::'a set + 'a list × 'a) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel⟧ ⟹ (case ri of Inl (R'::'a set) ⇒ ⦇lasso_impl = None, red_impl = R'⦈ | Inr (vs::'a list, v::'a) ⇒ let rs::'a list = rev (map fst (CAST (ss_stack si))) in ⦇lasso_impl = Some (rs, vs @ dropWhileNot v rs), red_impl = red_impl si⦈, case r of Inl (R'::'a set) ⇒ ⦇lasso = None, red = R'⦈ | Inr (vs::'a list, v::'a) ⇒ let rs::'a list = rev (stack s) in ⦇lasso = Some (rs, vs @ dropWhileNot v rs), red = red s⦈) ∈ bdfs_erel›*)\n apply ((parametricity)[1])\n (*goals:\n 1. ‹⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel⟧ ⟹ (λR'. ⦇lasso_impl = None, red_impl = R'⦈, λR'. ⦇lasso = None, red = R'⦈) ∈ Id → bdfs_erel›\n 2. ‹⋀a a' aa a'a. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; (a, a') ∈ ⟨Id⟩list_rel; (aa, a'a) ∈ Id⟧ ⟹ (map fst (CAST (ss_stack si)), stack s) ∈ ⟨?R51 a a' aa a'a⟩list_rel›\n 3. ‹⋀a a' aa a'a. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; (a, a') ∈ ⟨Id⟩list_rel; (aa, a'a) ∈ Id⟧ ⟹ (λrs. ⦇lasso_impl = Some (rs, a @ dropWhileNot aa rs), red_impl = red_impl si⦈, λrs. ⦇lasso = Some (rs, a' @ dropWhileNot a'a rs), red = red s⦈) ∈ ⟨?R51 a a' aa a'a⟩list_rel → bdfs_erel›\n discuss goal 1*)\n apply ((cases si)[1])\n (*top goal: ‹⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel⟧ ⟹ (λR'. ⦇lasso_impl = None, red_impl = R'⦈, λR'. ⦇lasso = None, red = R'⦈) ∈ Id → bdfs_erel› and 2 goals remain*)\n apply ((cases s)[1])\n (*top goal: ‹⋀ss_stack on_stack visited lasso_impl red_impl. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; si = ⦇ss_stack = ss_stack, on_stack = on_stack, visited = visited, lasso_impl = lasso_impl, red_impl = red_impl⦈⟧ ⟹ (λR'. ⦇lasso_impl = None, red_impl = R'⦈, λR'. ⦇lasso = None, red = R'⦈) ∈ Id → bdfs_erel› and 2 goals remain*)\n apply ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))[1])\n (*discuss goal 2*)\n apply (rule introR[where R=\"⟨Id⟩list_rel\"] (*‹(?a, ?a') ∈ ⟨Id⟩list_rel ⟹ (?a, ?a') ∈ ⟨Id⟩list_rel›*))\n (*top goal: ‹⋀a a' aa a'a. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; (a, a') ∈ ⟨Id⟩list_rel; (aa, a'a) ∈ Id⟧ ⟹ (map fst (CAST (ss_stack si)), stack s) ∈ ⟨?R51 a a' aa a'a⟩list_rel› and 1 goal remains*)\n apply ((cases si)[1])\n (*top goal: ‹⋀a a' aa a'a. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; (a, a') ∈ ⟨Id⟩list_rel; (aa, a'a) ∈ Id⟧ ⟹ (map fst (CAST (ss_stack si)), stack s) ∈ ⟨Id⟩list_rel› and 1 goal remains*)\n apply ((cases s)[1])\n (*top goal: ‹⋀a a' aa a'a ss_stacka on_stack visited lasso_impl red_impl. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; (a, a') ∈ ⟨Id⟩list_rel; (aa, a'a) ∈ Id; si = ⦇ss_stack = ss_stacka, on_stack = on_stack, visited = visited, lasso_impl = lasso_impl, red_impl = red_impl⦈⟧ ⟹ (map fst (CAST (ss_stack si)), stack s) ∈ ⟨Id⟩list_rel› and 1 goal remains*)\n apply ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*) comp_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*discuss goal 3*)\n apply ((cases si)[1])\n (*goal: ‹⋀a a' aa a'a. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; (a, a') ∈ ⟨Id⟩list_rel; (aa, a'a) ∈ Id⟧ ⟹ (λrs. ⦇lasso_impl = Some (rs, a @ dropWhileNot aa rs), red_impl = red_impl si⦈, λrs. ⦇lasso = Some (rs, a' @ dropWhileNot a'a rs), red = red s⦈) ∈ ⟨Id⟩list_rel → bdfs_erel›*)\n apply ((cases s)[1])\n (*goal: ‹⋀a a' aa a'a ss_stack on_stack visited lasso_impl red_impla. ⟦(si, s) ∈ bdfs_rel; (ri, r) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel; (a, a') ∈ ⟨Id⟩list_rel; (aa, a'a) ∈ Id; si = ⦇ss_stack = ss_stack, on_stack = on_stack, visited = visited, lasso_impl = lasso_impl, red_impl = red_impla⦈⟧ ⟹ (λrs. ⦇lasso_impl = Some (rs, a @ dropWhileNot aa rs), red_impl = red_impl si⦈, λrs. ⦇lasso = Some (rs, a' @ dropWhileNot a'a rs), red = red s⦈) ∈ ⟨Id⟩list_rel → bdfs_erel›*)\n apply ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))[1])\n (*proven 3 subgoals*) .\n\ndefinition \"cyan_impl s ≡ on_stack s\"\nlemma cyan_impl[refine]: \"⟦(si,s)∈bdfs_rel⟧ ⟹ (cyan_impl si, cyan s)∈Id\"\n unfolding cyan_impl_def cyan_def\n (*goal: ‹(si::⦇ss_stack :: ('a × 'a set) list, on_stack :: 'a set, visited :: 'a set, lasso_impl :: ('b list × 'b list) option, red_impl :: 'b set⦈, s::⦇counter :: nat, discovered :: 'a ⇒ nat option, finished :: 'a ⇒ nat option, pending :: ('a × 'a) set, stack :: 'a list, tree_edges :: ('a × 'a) set, back_edges :: ('a × 'a) set, cross_edges :: ('a × 'a) set, lasso :: ('b list × 'b list) option, red :: 'b set⦈) ∈ bdfs_rel ⟹ (on_stack si, set (stack s)) ∈ Id›*)\n by (auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |(li::(?'a list × ?'a list) option) (ri::?'a set) (l::(?'a list × ?'a list) option) r::?'a set. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel::(?'b × ?'c) set⟩simple_state_rel ≡ {(s::(?'a, ?'b) simple_state_scheme, s'::(?'a, ?'c) state_scheme). ss_stack s = map (λu::?'a. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))\n\ndefinition run_red_dfs_impl \n :: \"('v, 'more) graph_rec_scheme ⇒ 'v ⇒ 'v bdfs_state_impl ⇒ ('v,unit) bdfs_state_impl_ext nres\" \n where\n \"run_red_dfs_impl G u s ≡ case lasso_impl s of None ⇒ do {\n redS ← red_dfs TYPE('more) G (red_impl s) (λx. x = u ∨ x ∈ cyan_impl s) u;\n RETURN (mk_blue_witness_impl s redS)\n }\n | _ ⇒ RETURN (simple_state.more s)\"\n\n lemma run_red_dfs_impl[refine]: \"⟦(Gi,G)∈Id; (ui,u)∈Id; (si,s)∈bdfs_rel⟧ \n ⟹ run_red_dfs_impl Gi ui si ≤⇓bdfs_erel (run_red_dfs TYPE('a) G u s)\"\n unfolding run_red_dfs_impl_def run_red_dfs_def\n (*goal: ‹⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel⟧ ⟹ (case lasso_impl si of None ⇒ find_path1_restr_spec (Gi⦇g_V0 := {ui}⦈) (λx. x = ui ∨ x ∈ cyan_impl si) (red_impl si) ⤜ (λredS. RETURN (mk_blue_witness_impl si redS)) | Some a ⇒ RETURN (simple_state.more si)) ≤ ⇓ bdfs_erel (case lasso s of None ⇒ find_path1_restr_spec (G⦇g_V0 := {u}⦈) (λx. x = u ∨ x ∈ cyan s) (red s) ⤜ (λredS. RETURN (mk_blue_witness s redS)) | Some a ⇒ NOOP s)›*)\n apply ((refine_rcg)[1])\n (*goals:\n 1. ‹⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel⟧ ⟹ (lasso_impl si, lasso s) ∈ ⟨?Ra1⟩option_rel›\n 2. ‹⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None⟧ ⟹ find_path1_restr_spec (Gi⦇g_V0 := {ui}⦈) (λx. x = ui ∨ x ∈ cyan_impl si) (red_impl si) ≤ ⇓ ?R'2 (find_path1_restr_spec (G⦇g_V0 := {u}⦈) (λx. x = u ∨ x ∈ cyan s) (red s))›\n 3. ‹⋀redS redSa. ⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; (redS, redSa) ∈ ?R'2⟧ ⟹ (redS, redSa) ∈ ⟨Id, ⟨Id⟩list_rel ×⇩r Id⟩sum_rel›\n 4. ‹⋀x x'. ⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = Some x; lasso s = Some x'; (x, x') ∈ ?Ra1⟧ ⟹ (simple_state.more si, state.more s) ∈ bdfs_erel›\n discuss goal 1*)\n apply ((refine_dref_type)[1])\n (*top goal: ‹⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel⟧ ⟹ (lasso_impl si, lasso s) ∈ ⟨?Ra1⟩option_rel› and 3 goals remain*)\n apply ((cases si)[1])\n (*top goal: ‹⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel⟧ ⟹ (lasso_impl si, lasso s) ∈ ⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel› and 3 goals remain*)\n apply ((cases s)[1])\n (*top goal: ‹⋀ss_stack on_stack visited lasso_impla red_impl. ⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel; si = ⦇ss_stack = ss_stack, on_stack = on_stack, visited = visited, lasso_impl = lasso_impla, red_impl = red_impl⦈⟧ ⟹ (lasso_impl si, lasso s) ∈ ⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel› and 3 goals remain*)\n apply ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |(li::(?'a list × ?'a list) option) (ri::?'a set) (l::(?'a list × ?'a list) option) r::?'a set. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel::(?'b × ?'c) set⟩simple_state_rel ≡ {(s::(?'a, ?'b) simple_state_scheme, s'::(?'a, ?'c) state_scheme). ss_stack s = map (λu::?'a. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))[1])\n (*discuss goal 2*)\n apply ((refine_dref_type)[1])\n (*top goal: ‹⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None⟧ ⟹ find_path1_restr_spec (Gi⦇g_V0 := {ui}⦈) (λx. x = ui ∨ x ∈ cyan_impl si) (red_impl si) ≤ ⇓ ?R'2 (find_path1_restr_spec (G⦇g_V0 := {u}⦈) (λx. x = u ∨ x ∈ cyan s) (red s))› and 2 goals remain*)\n apply ((cases si)[1])\n (*top goal: ‹⟦(Gi::('b, 'a) graph_rec_scheme, G::('b, 'a) graph_rec_scheme) ∈ Id; (ui::'b::type, u::'b::type) ∈ Id; (si::'b bdfs_state_impl, s::'b blue_dfs_state) ∈ bdfs_rel; lasso_impl si = None; lasso s = None⟧ ⟹ find_path1_restr_spec (Gi⦇g_V0 := {ui}⦈) (λx::'b::type. x = ui ∨ x ∈ cyan_impl si) (red_impl si) ≤ ⇓ Id (find_path1_restr_spec (G⦇g_V0 := {u}⦈) (λx::'b::type. x = u ∨ x ∈ cyan s) (red s))› and 2 goals remain*)\n apply ((cases s)[1])\n (*top goal: ‹⋀ss_stack on_stack visited lasso_impla red_impla. ⟦(Gi, G) ∈ Id; (ui, u) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; si = ⦇ss_stack = ss_stack, on_stack = on_stack, visited = visited, lasso_impl = lasso_impla, red_impl = red_impla⦈⟧ ⟹ find_path1_restr_spec (Gi⦇g_V0 := {ui}⦈) (λx. x = ui ∨ x ∈ cyan_impl si) (red_impl si) ≤ ⇓ Id (find_path1_restr_spec (G⦇g_V0 := {u}⦈) (λx. x = u ∨ x ∈ cyan s) (red s))› and 2 goals remain*)\n apply ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |(li::(?'a list × ?'a list) option) (ri::?'a set) (l::(?'a list × ?'a list) option) r::?'a set. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel::(?'b × ?'c) set⟩simple_state_rel ≡ {(s::(?'a, ?'b) simple_state_scheme, s'::(?'a, ?'c) state_scheme). ss_stack s = map (λu::?'a. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*) cyan_impl_def (*‹cyan_impl (?s::(?'a, ?'b) simple_state_scheme) ≡ on_stack ?s›*) cyan_def (*‹cyan (?s::(?'a, ?'b) state_scheme) ≡ set (stack ?s)›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))[1])\n (*discuss goal 4*)\n apply ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))[1])\n (*proven 4 subgoals*) .\n\n definition \"se_back_edge_impl accpt u v s ≡ case lasso_impl s of\n None ⇒ \n if accpt u then\n let rs = rev (map fst (tl (CAST (ss_stack s))));\n ur = rs;\n ul = u#dropWhileNot v rs\n in RETURN ⦇lasso_impl = Some (ur,ul), red_impl = red_impl s⦈\n else if accpt v then\n let rs = rev (map fst (CAST (ss_stack s)));\n vr = takeWhileNot v rs;\n vl = dropWhileNot v rs\n in RETURN ⦇lasso_impl = Some (vr,vl), red_impl = red_impl s⦈\n else RETURN (simple_state.more s)\n | _ ⇒ RETURN (simple_state.more s)\"\n\n lemma se_back_edge_impl[refine]: \"⟦ (accpti,accpt)∈Id; (ui,u)∈Id; (vi,v)∈Id; (si,s)∈bdfs_rel ⟧ \n ⟹ se_back_edge_impl accpt ui vi si ≤⇓bdfs_erel (se_back_edge accpt u v s)\"\n unfolding se_back_edge_impl_def se_back_edge_def\n (*goal: ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel⟧ ⟹ (case lasso_impl si of None ⇒ if accpt ui then let rs = rev (map fst (tl (CAST (ss_stack si)))); ur = rs; ul = ui # dropWhileNot vi rs in RETURN ⦇lasso_impl = Some (ur, ul), red_impl = red_impl si⦈ else if accpt vi then let rs = rev (map fst (CAST (ss_stack si))); vr = takeWhileNot vi rs; vl = dropWhileNot vi rs in RETURN ⦇lasso_impl = Some (vr, vl), red_impl = red_impl si⦈ else RETURN (simple_state.more si) | Some a ⇒ RETURN (simple_state.more si)) ≤ ⇓ bdfs_erel (case lasso s of None ⇒ if accpt u then let rs = rev (tl (stack s)); ur = rs; ul = u # dropWhileNot v rs in RETURN ⦇lasso = Some (ur, ul), red = red s⦈ else if accpt v then let rs = rev (stack s); vr = takeWhileNot v rs; vl = dropWhileNot v rs in RETURN ⦇lasso = Some (vr, vl), red = red s⦈ else NOOP s | Some a ⇒ NOOP s)›*)\n apply ((refine_rcg)[1])\n (*goals:\n 1. ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel⟧ ⟹ (lasso_impl si, lasso s) ∈ ⟨?Ra1⟩option_rel›\n 2. ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None⟧ ⟹ accpt ui = accpt u›\n 3. ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; accpt ui; accpt u⟧ ⟹ (⦇lasso_impl = Some (rev (map fst (tl (CAST (ss_stack si)))), ui # dropWhileNot vi (rev (map fst (tl (CAST (ss_stack si)))))), red_impl = red_impl si⦈, ⦇lasso = Some (rev (tl (stack s)), u # dropWhileNot v (rev (tl (stack s)))), red = red s⦈) ∈ bdfs_erel›\n 4. ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; ¬ accpt ui; ¬ accpt u⟧ ⟹ accpt vi = accpt v›\n 5. ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; ¬ accpt ui; ¬ accpt u; accpt vi; accpt v⟧ ⟹ (⦇lasso_impl = Some (takeWhileNot vi (rev (map fst (CAST (ss_stack si)))), dropWhileNot vi (rev (map fst (CAST (ss_stack si))))), red_impl = red_impl si⦈, ⦇lasso = Some (takeWhileNot v (rev (stack s)), dropWhileNot v (rev (stack s))), red = red s⦈) ∈ bdfs_erel›\n 6. ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; ¬ accpt ui; ¬ accpt u; ¬ accpt vi; ¬ accpt v⟧ ⟹ (simple_state.more si, state.more s) ∈ bdfs_erel›\n 7. ‹⋀x x'. ⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = Some x; lasso s = Some x'; (x, x') ∈ ?Ra1⟧ ⟹ (simple_state.more si, state.more s) ∈ bdfs_erel›\n discuss goal 1*)\n apply ((refine_dref_type)[1])\n (*top goal: ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel⟧ ⟹ (lasso_impl si, lasso s) ∈ ⟨?Ra1⟩option_rel› and 6 goals remain*)\n apply simp\n (*top goal: ‹⟦(accpti::'a ⇒ bool, accpt::'a ⇒ bool) ∈ Id; (ui::'a, u::'a) ∈ Id; (vi::'a, v::'a) ∈ Id; (si::'a bdfs_state_impl, s::'a blue_dfs_state) ∈ bdfs_rel⟧ ⟹ (lasso_impl si, lasso s) ∈ ⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel› and 6 goals remain*)\n apply (simp add: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))\n (*top goal: ‹⟦accpti = accpt; ui = u; vi = v; (si, s) ∈ bdfs_rel⟧ ⟹ lasso_impl si = lasso s› and 6 goals remain*)\n apply ((cases si)[1])\n (*top goal: ‹⟦accpti = accpt; ui = u; vi = v; ss_stack si = map (λu. (u, pending s `` {u})) (stack s) ∧ on_stack si = set (stack s) ∧ visited si = dom (discovered s) ∧ dom (finished s) = dom (discovered s) - set (stack s) ∧ set (stack s) ⊆ dom (discovered s) ∧ (∃li ri. simple_state.more si = ⦇lasso_impl = li, red_impl = ri⦈ ∧ state.more s = ⦇lasso = li, red = ri⦈)⟧ ⟹ lasso_impl si = lasso s› and 6 goals remain*)\n apply ((cases s)[1])\n (*top goal: ‹⋀ss_stacka on_stacka visiteda lasso_impla red_impl. ⟦accpti = accpt; ui = u; vi = v; ss_stack si = map (λu. (u, pending s `` {u})) (stack s) ∧ on_stack si = set (stack s) ∧ visited si = dom (discovered s) ∧ dom (finished s) = dom (discovered s) - set (stack s) ∧ set (stack s) ⊆ dom (discovered s) ∧ (∃li ri. simple_state.more si = ⦇lasso_impl = li, red_impl = ri⦈ ∧ state.more s = ⦇lasso = li, red = ri⦈); si = ⦇ss_stack = ss_stacka, on_stack = on_stacka, visited = visiteda, lasso_impl = lasso_impla, red_impl = red_impl⦈⟧ ⟹ lasso_impl si = lasso s› and 6 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply simp\n (*discuss goal 3*)\n apply simp\n (*top goal: ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; accpt ui; accpt u⟧ ⟹ (⦇lasso_impl = Some (rev (map fst (tl (CAST (ss_stack si)))), ui # dropWhileNot vi (rev (map fst (tl (CAST (ss_stack si)))))), red_impl = red_impl si⦈, ⦇lasso = Some (rev (tl (stack s)), u # dropWhileNot v (rev (tl (stack s)))), red = red s⦈) ∈ bdfs_erel› and 4 goals remain*)\n apply (simp add: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))\n (*top goal: ‹⟦accpti = accpt; ui = u; vi = v; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; accpt u⟧ ⟹ (⦇lasso_impl = Some (rev (map fst (tl (ss_stack si))), u # dropWhileNot v (rev (map fst (tl (ss_stack si))))), red_impl = red_impl si⦈, ⦇lasso = Some (rev (tl (stack s)), u # dropWhileNot v (rev (tl (stack s)))), red = red s⦈) ∈ bdfs_erel› and 4 goals remain*)\n apply ((cases si)[1])\n (*top goal: ‹⟦accpti = accpt; ui = u; vi = v; ss_stack si = map (λu. (u, pending s `` {u})) (stack s) ∧ on_stack si = set (stack s) ∧ visited si = dom (discovered s) ∧ dom (finished s) = dom (discovered s) - set (stack s) ∧ set (stack s) ⊆ dom (discovered s) ∧ (∃li ri. simple_state.more si = ⦇lasso_impl = li, red_impl = ri⦈ ∧ state.more s = ⦇lasso = li, red = ri⦈); lasso_impl si = None; lasso s = None; accpt u⟧ ⟹ tl (stack s) = map fst (tl (map (λu. (u, pending s `` {u})) (stack s))) ∧ dropWhileNot v (rev (tl (stack s))) = dropWhileNot v (rev (map fst (tl (map (λu. (u, pending s `` {u})) (stack s))))) ∧ red s = red_impl si› and 4 goals remain*)\n apply ((cases s)[1])\n (*top goal: ‹⋀(ss_stacka::('a × 'a set) list) (on_stacka::'a set) (visiteda::'a set) (lasso_impla::('a list × 'a list) option) red_impla::'a set. ⟦(accpti::'a ⇒ bool) = (accpt::'a ⇒ bool); (ui::'a) = (u::'a); (vi::'a) = (v::'a); ss_stack (si::'a bdfs_state_impl) = map (λu::'a. (u, pending (s::'a blue_dfs_state) `` {u})) (stack s) ∧ on_stack si = set (stack s) ∧ visited si = dom (discovered s) ∧ dom (finished s) = dom (discovered s) - set (stack s) ∧ set (stack s) ⊆ dom (discovered s) ∧ (∃(li::('a list × 'a list) option) ri::'a set. simple_state.more si = ⦇lasso_impl = li, red_impl = ri⦈ ∧ state.more s = ⦇lasso = li, red = ri⦈); lasso_impl si = None; lasso s = None; accpt u; si = ⦇ss_stack = ss_stacka, on_stack = on_stacka, visited = visiteda, lasso_impl = lasso_impla, red_impl = red_impla⦈⟧ ⟹ tl (stack s) = map fst (tl (map (λu::'a. (u, pending s `` {u})) (stack s))) ∧ dropWhileNot v (rev (tl (stack s))) = dropWhileNot v (rev (map fst (tl (map (λu::'a. (u, pending s `` {u})) (stack s))))) ∧ red s = red_impl si› and 4 goals remain*)\n apply ((auto simp: map_tl (*‹map (?f::?'b ⇒ ?'a) (tl (?xs::?'b list)) = tl (map ?f ?xs)›*) comp_def (*‹(?f::?'b ⇒ ?'c) ∘ (?g::?'a ⇒ ?'b) = (λx::?'a. ?f (?g x))›*))[1])\n (*discuss goal 4*)\n apply simp\n (*discuss goal 5*)\n apply simp\n (*top goal: ‹⟦(accpti::'a::type ⇒ bool, accpt::'a::type ⇒ bool) ∈ Id; (ui::'a::type, u::'a::type) ∈ Id; (vi::'a::type, v::'a::type) ∈ Id; (si::'a bdfs_state_impl, s::'a blue_dfs_state) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; ¬ accpt ui; ¬ accpt u; accpt vi; accpt v⟧ ⟹ (⦇lasso_impl = Some (takeWhileNot vi (rev (map fst (CAST (ss_stack si)))), dropWhileNot vi (rev (map fst (CAST (ss_stack si))))), red_impl = red_impl si⦈, ⦇lasso = Some (takeWhileNot v (rev (stack s)), dropWhileNot v (rev (stack s))), red = red s⦈) ∈ bdfs_erel› and 2 goals remain*)\n apply (simp add: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))\n (*top goal: ‹⟦accpti = accpt; ui = u; vi = v; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; ¬ accpt u; accpt v⟧ ⟹ (⦇lasso_impl = Some (takeWhileNot v (rev (map fst (ss_stack si))), dropWhileNot v (rev (map fst (ss_stack si)))), red_impl = red_impl si⦈, ⦇lasso = Some (takeWhileNot v (rev (stack s)), dropWhileNot v (rev (stack s))), red = red s⦈) ∈ bdfs_erel› and 2 goals remain*)\n apply ((cases si)[1])\n (*top goal: ‹⟦(accpti::'a::type ⇒ bool) = (accpt::'a::type ⇒ bool); (ui::'a::type) = (u::'a::type); (vi::'a::type) = (v::'a::type); ss_stack (si::'a bdfs_state_impl) = map (λu::'a::type. (u, pending (s::'a blue_dfs_state) `` {u})) (stack s) ∧ on_stack si = set (stack s) ∧ visited si = dom (discovered s) ∧ dom (finished s) = dom (discovered s) - set (stack s) ∧ set (stack s) ⊆ dom (discovered s) ∧ (∃(li::('a::type list × 'a::type list) option) ri::'a::type set. simple_state.more si = ⦇lasso_impl = li, red_impl = ri⦈ ∧ state.more s = ⦇lasso = li, red = ri⦈); lasso_impl si = None; lasso s = None; ¬ accpt u; accpt v⟧ ⟹ takeWhileNot v (rev (stack s)) = takeWhileNot v (rev (map (fst ∘ (λu::'a::type. (u, pending s `` {u}))) (stack s))) ∧ dropWhileNot v (rev (stack s)) = dropWhileNot v (rev (map (fst ∘ (λu::'a::type. (u, pending s `` {u}))) (stack s))) ∧ red s = red_impl si› and 2 goals remain*)\n apply ((cases s)[1])\n (*top goal: ‹⋀ss_stacka on_stacka visiteda lasso_impla red_impla. ⟦accpti = accpt; ui = u; vi = v; ss_stack si = map (λu. (u, pending s `` {u})) (stack s) ∧ on_stack si = set (stack s) ∧ visited si = dom (discovered s) ∧ dom (finished s) = dom (discovered s) - set (stack s) ∧ set (stack s) ⊆ dom (discovered s) ∧ (∃li ri. simple_state.more si = ⦇lasso_impl = li, red_impl = ri⦈ ∧ state.more s = ⦇lasso = li, red = ri⦈); lasso_impl si = None; lasso s = None; ¬ accpt u; accpt v; si = ⦇ss_stack = ss_stacka, on_stack = on_stacka, visited = visiteda, lasso_impl = lasso_impla, red_impl = red_impla⦈⟧ ⟹ takeWhileNot v (rev (stack s)) = takeWhileNot v (rev (map (fst ∘ (λu. (u, pending s `` {u}))) (stack s))) ∧ dropWhileNot v (rev (stack s)) = dropWhileNot v (rev (map (fst ∘ (λu. (u, pending s `` {u}))) (stack s))) ∧ red s = red_impl si› and 2 goals remain*)\n apply ((auto simp: comp_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*discuss goal 6*)\n apply simp\n (*top goal: ‹⟦(accpti, accpt) ∈ Id; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; lasso_impl si = None; lasso s = None; ¬ accpt ui; ¬ accpt u; ¬ accpt vi; ¬ accpt v⟧ ⟹ (simple_state.more si, state.more s) ∈ bdfs_erel› and 1 goal remains*)\n apply (simp add: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))\n (*discuss goal 7*)\n apply simp\n (*goal: ‹⋀(x::'a list × 'a list) x'::'a list × 'a list. ⟦(accpti::'a ⇒ bool, accpt::'a ⇒ bool) ∈ Id; (ui::'a, u::'a) ∈ Id; (vi::'a, v::'a) ∈ Id; (si::'a bdfs_state_impl, s::'a blue_dfs_state) ∈ bdfs_rel; lasso_impl si = Some x; lasso s = Some x'; (x, x') ∈ ⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟧ ⟹ (simple_state.more si, state.more s) ∈ bdfs_erel›*)\n apply (simp add: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |(li::(?'a list × ?'a list) option) (ri::?'a set) (l::(?'a list × ?'a list) option) r::?'a set. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel::(?'b × ?'c) set⟩simple_state_rel ≡ {(s::(?'a, ?'b) simple_state_scheme, s'::(?'a, ?'c) state_scheme). ss_stack s = map (λu::?'a. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))\n (*proven 7 subgoals*) .\n\n\n lemma NOOP_impl: \"(si, s) ∈ bdfs_rel \n ⟹ RETURN (simple_state.more si) ≤ ⇓ bdfs_erel (NOOP s)\"\n apply (simp add: pw_le_iff (*‹(?S ≤ ?S') = (nofail ?S' ⟶ nofail ?S ∧ (∀x. inres ?S x ⟶ inres ?S' x))›*) refine_pw_simps (*‹inres (filter_nb ?b ?I) ?x = (nofail ?I ⟶ inres ?I ?x ∧ ¬ ?b ?x)› ‹nofail (filter_nb ?b ?I) = nofail ?I› ‹inres (filter_ASSUME ?c ?m) ?x = (nofail ?m ⟶ inres ?m ?x ∧ ?c ?x)› ‹nofail (filter_ASSUME ?c ?m) = nofail ?m› ‹inres (SELECT ?P) ?r = ((?r = None ⟶ (∀x. ¬ ?P x)) ∧ (∀x. ?r = Some x ⟶ ?P x))› ‹nofail (SELECT ?P)› ‹inres (OBTAIN ?P) ?x = ((∀x. ¬ ?P x) ∨ ?P ?x)› ‹nofail (OBTAIN ?P) = (∃x. ?P x)› ‹inres (ASSUME ?Φ) ?x = ?Φ› ‹nofail (ASSUME ?Φ)› ‹inres (ASSERT ?Φ) ?x› ‹nofail (ASSERT ?Φ) = ?Φ› and more 30 facts*))\n (*goal: ‹(si, s) ∈ bdfs_rel ⟹ RETURN (simple_state.more si) ≤ ⇓ bdfs_erel (NOOP s)›*)\n by (auto simp: simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))\n \n\n definition bdfs_params_impl \n :: \"('v, 'more) graph_rec_scheme ⇒ ('v ⇒ bool) ⇒ ('v,'v bdfs_state_impl,('v,unit)bdfs_state_impl_ext) gen_parameterization\"\n where \"bdfs_params_impl G accpt ≡ ⦇\n on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈,\n on_new_root = λv0 s. RETURN (simple_state.more s),\n on_discover = λu v s. RETURN (simple_state.more s),\n on_finish = λu s. \n if accpt u then run_red_dfs_impl G u s else RETURN (simple_state.more s),\n on_back_edge = se_back_edge_impl accpt,\n on_cross_edge = λu v s. RETURN (simple_state.more s),\n is_break = λs. lasso_impl s ≠ None ⦈\"\n\n lemmas bdfs_params_impl_simps[simp, DFS_code_unfold] = \n gen_parameterization.simps[mk_record_simp, OF bdfs_params_impl_def] \n\n\n interpretation impl: simple_impl_defs G \"bdfs_params_impl G accpt\" \"blue_dfs_params TYPE('a) G accpt\"\n for G accpt .\n\ncontext BlueDFS begin\n\n sublocale impl: simple_impl G blue_dfs_params \"bdfs_params_impl G accpt\" bdfs_erel\n apply ((unfold_locales)[1])\n (*goals:\n 1. ‹(simple_state.more_update, state.more_update) ∈ (bdfs_erel → bdfs_erel) → bdfs_rel → bdfs_rel›\n 2. ‹on_init (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool)) ≤ ⇓ bdfs_erel (on_init local.blue_dfs_params)›\n 3. ‹(is_break (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool)), is_break local.blue_dfs_params) ∈ bdfs_rel → bool_rel›\n 4. ‹⋀(v0::'v) (s::'v blue_dfs_state) (v0i::'v) (si::'v bdfs_state_impl) (si'::'v bdfs_state_impl) s'::'v blue_dfs_state. ⟦local.gen_dfs.pre_new_root v0 s; (v0i, v0) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_new_root local.gbs v0 s) s'⟧ ⟹ on_new_root (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool)) v0i si' ≤ ⇓ bdfs_erel (on_new_root local.blue_dfs_params v0 s')›\n 5. ‹⋀(v::'v) (s0::'v blue_dfs_state) (s::'v blue_dfs_state) (vi::'v) (si::'v bdfs_state_impl) (si'::'v bdfs_state_impl) s'::'v blue_dfs_state. ⟦local.gen_dfs.pre_finish v s0 s; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_finish local.gbs v s) s'⟧ ⟹ on_finish (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool)) vi si' ≤ ⇓ bdfs_erel (on_finish local.blue_dfs_params v s')›\n 6. ‹⋀(u::'v) (v::'v) (s0::'v blue_dfs_state) (s::'v blue_dfs_state) (ui::'v) (vi::'v) (si::'v bdfs_state_impl) (si'::'v bdfs_state_impl) s'::'v blue_dfs_state. ⟦local.gen_dfs.pre_cross_edge u v s0 s; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_cross_edge local.gbs u v s) s'⟧ ⟹ on_cross_edge (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool)) ui vi si' ≤ ⇓ bdfs_erel (on_cross_edge local.blue_dfs_params u v s')›\n 7. ‹⋀(u::'v) (v::'v) (s0::'v blue_dfs_state) (s::'v blue_dfs_state) (ui::'v) (vi::'v) (si::'v bdfs_state_impl) (si'::'v bdfs_state_impl) s'::'v blue_dfs_state. ⟦local.gen_dfs.pre_back_edge u v s0 s; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_back_edge local.gbs u v s) s'⟧ ⟹ on_back_edge (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool)) ui vi si' ≤ ⇓ bdfs_erel (on_back_edge local.blue_dfs_params u v s')›\n 8. ‹⋀(u::'v) (v::'v) (s0::'v blue_dfs_state) (s::'v blue_dfs_state) (ui::'v) (vi::'v) (si::'v bdfs_state_impl) (si'::'v bdfs_state_impl) s'::'v blue_dfs_state. ⟦local.gen_dfs.pre_discover u v s0 s; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_discover local.gbs u v s) s'⟧ ⟹ on_discover (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v ⇒ bool)) ui vi si' ≤ ⇓ bdfs_erel (on_discover local.blue_dfs_params u v s')›\n discuss goal 1*)\n apply ((parametricity)[1])\n (*discuss goal 2*)\n apply (simp add: bdfs_params_impl_def (*‹bdfs_params_impl (?G::(?'v, ?'more) graph_rec_scheme) (?accpt::?'v ⇒ bool) ≡ ⦇on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈, on_new_root = λ(v0::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_discover = λ(u::?'v) (v::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_finish = λ(u::?'v) s::?'v bdfs_state_impl. if ?accpt u then run_red_dfs_impl ?G u s else RETURN (simple_state.more s), on_back_edge = se_back_edge_impl ?accpt, on_cross_edge = λ(u::?'v) (v::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), is_break = λs::?'v bdfs_state_impl. lasso_impl s ≠ None⦈›*) run_red_dfs_impl (*‹⟦(?Gi::(?'b, ?'a) graph_rec_scheme, ?G::(?'b, ?'a) graph_rec_scheme) ∈ Id; (?ui::?'b, ?u::?'b) ∈ Id; (?si::?'b bdfs_state_impl, ?s::?'b blue_dfs_state) ∈ bdfs_rel⟧ ⟹ run_red_dfs_impl ?Gi ?ui ?si ≤ ⇓ bdfs_erel (BlueDFS_defs.run_red_dfs ?G ?u ?s)›*) se_back_edge_impl (*‹⟦(?accpti::?'a ⇒ bool, ?accpt::?'a ⇒ bool) ∈ Id; (?ui::?'a, ?u::?'a) ∈ Id; (?vi::?'a, ?v::?'a) ∈ Id; (?si::?'a bdfs_state_impl, ?s::?'a blue_dfs_state) ∈ bdfs_rel⟧ ⟹ se_back_edge_impl ?accpt ?ui ?vi ?si ≤ ⇓ bdfs_erel (Nested_DFS.se_back_edge ?accpt ?u ?v ?s)›*) NOOP_impl (*‹(?si::⦇ss_stack :: (?'a × ?'a set) list, on_stack :: ?'a set, visited :: ?'a set, lasso_impl :: (?'b list × ?'b list) option, red_impl :: ?'b set⦈, ?s::⦇counter :: nat, discovered :: ?'a ⇒ nat option, finished :: ?'a ⇒ nat option, pending :: (?'a × ?'a) set, stack :: ?'a list, tree_edges :: (?'a × ?'a) set, back_edges :: (?'a × ?'a) set, cross_edges :: (?'a × ?'a) set, lasso :: (?'b list × ?'b list) option, red :: ?'b set⦈) ∈ bdfs_rel ⟹ RETURN (simple_state.more ?si) ≤ ⇓ bdfs_erel (NOOP ?s)›*))\n (*top goal: ‹on_init (bdfs_params_impl G accpt) ≤ ⇓ bdfs_erel (on_init local.blue_dfs_params)› and 6 goals remain*)\n apply ((clarsimp_all simp: pw_le_iff refine_pw_simps bdfs_erel_def simple_state_rel_def)[1])\n (*discuss goal 3*)\n apply (simp add: bdfs_params_impl_def (*‹bdfs_params_impl ?G ?accpt ≡ ⦇on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈, on_new_root = λv0 s. RETURN (simple_state.more s), on_discover = λu v s. RETURN (simple_state.more s), on_finish = λu s. if ?accpt u then run_red_dfs_impl ?G u s else RETURN (simple_state.more s), on_back_edge = se_back_edge_impl ?accpt, on_cross_edge = λu v s. RETURN (simple_state.more s), is_break = λs. lasso_impl s ≠ None⦈›*) run_red_dfs_impl (*‹⟦(?Gi, ?G) ∈ Id; (?ui, ?u) ∈ Id; (?si, ?s) ∈ bdfs_rel⟧ ⟹ run_red_dfs_impl ?Gi ?ui ?si ≤ ⇓ bdfs_erel (BlueDFS_defs.run_red_dfs ?G ?u ?s)›*) se_back_edge_impl (*‹⟦(?accpti, ?accpt) ∈ Id; (?ui, ?u) ∈ Id; (?vi, ?v) ∈ Id; (?si, ?s) ∈ bdfs_rel⟧ ⟹ se_back_edge_impl ?accpt ?ui ?vi ?si ≤ ⇓ bdfs_erel (Nested_DFS.se_back_edge ?accpt ?u ?v ?s)›*) NOOP_impl (*‹(?si, ?s) ∈ bdfs_rel ⟹ RETURN (simple_state.more ?si) ≤ ⇓ bdfs_erel (NOOP ?s)›*))\n (*top goal: ‹(is_break (bdfs_params_impl G accpt), is_break local.blue_dfs_params) ∈ bdfs_rel → bool_rel› and 5 goals remain*)\n apply ((clarsimp_all simp: pw_le_iff refine_pw_simps bdfs_erel_def simple_state_rel_def)[1])\n (*top goal: ‹(λs. ∃a b. lasso_impl s = Some (a, b), λs. ∃a b. lasso s = Some (a, b)) ∈ bdfs_rel → bool_rel› and 5 goals remain*)\n apply ((rename_tac si s x y)[1])\n (*top goal: ‹⋀(a::'v bdfs_state_impl) (a'::'v blue_dfs_state) (li::('v list × 'v list) option) ri::'v set. ⟦ss_stack a = map (λu::'v. (u, pending a' `` {u})) (stack a'); on_stack a = set (stack a'); visited a = dom (discovered a'); dom (finished a') = dom (discovered a') - set (stack a'); set (stack a') ⊆ dom (discovered a'); simple_state.more a = ⦇lasso_impl = li, red_impl = ri⦈; state.more a' = ⦇lasso = li, red = ri⦈⟧ ⟹ (∃(aa::'v list) b::'v list. lasso_impl a = Some (aa, b)) = (∃(a::'v list) b::'v list. lasso a' = Some (a, b))› and 5 goals remain*)\n apply ((case_tac si)[1])\n (*top goal: ‹⋀si s x y. ⟦ss_stack si = map (λu. (u, pending s `` {u})) (stack s); on_stack si = set (stack s); visited si = dom (discovered s); dom (finished s) = dom (discovered s) - set (stack s); set (stack s) ⊆ dom (discovered s); simple_state.more si = ⦇lasso_impl = x, red_impl = y⦈; state.more s = ⦇lasso = x, red = y⦈⟧ ⟹ (∃a b. lasso_impl si = Some (a, b)) = (∃a b. lasso s = Some (a, b))› and 5 goals remain*)\n apply ((case_tac s)[1])\n (*top goal: ‹⋀si s x y ss_stacka on_stacka visiteda lasso_impla red_impl. ⟦ss_stack si = map (λu. (u, pending s `` {u})) (stack s); on_stack si = set (stack s); visited si = dom (discovered s); dom (finished s) = dom (discovered s) - set (stack s); set (stack s) ⊆ dom (discovered s); simple_state.more si = ⦇lasso_impl = x, red_impl = y⦈; state.more s = ⦇lasso = x, red = y⦈; si = ⦇ss_stack = ss_stacka, on_stack = on_stacka, visited = visiteda, lasso_impl = lasso_impla, red_impl = red_impl⦈⟧ ⟹ (∃a b. lasso_impl si = Some (a, b)) = (∃a b. lasso s = Some (a, b))› and 5 goals remain*)\n apply ((auto simp add: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))[1])\n (*discuss goal 4*)\n apply (simp add: bdfs_params_impl_def (*‹bdfs_params_impl (?G::(?'v, ?'more) graph_rec_scheme) (?accpt::?'v ⇒ bool) ≡ ⦇on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈, on_new_root = λ(v0::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_discover = λ(u::?'v) (v::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_finish = λ(u::?'v) s::?'v bdfs_state_impl. if ?accpt u then run_red_dfs_impl ?G u s else RETURN (simple_state.more s), on_back_edge = se_back_edge_impl ?accpt, on_cross_edge = λ(u::?'v) (v::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), is_break = λs::?'v bdfs_state_impl. lasso_impl s ≠ None⦈›*) run_red_dfs_impl (*‹⟦(?Gi::(?'b, ?'a) graph_rec_scheme, ?G::(?'b, ?'a) graph_rec_scheme) ∈ Id; (?ui::?'b, ?u::?'b) ∈ Id; (?si::?'b bdfs_state_impl, ?s::?'b blue_dfs_state) ∈ bdfs_rel⟧ ⟹ run_red_dfs_impl ?Gi ?ui ?si ≤ ⇓ bdfs_erel (BlueDFS_defs.run_red_dfs ?G ?u ?s)›*) se_back_edge_impl (*‹⟦(?accpti::?'a ⇒ bool, ?accpt::?'a ⇒ bool) ∈ Id; (?ui::?'a, ?u::?'a) ∈ Id; (?vi::?'a, ?v::?'a) ∈ Id; (?si::?'a bdfs_state_impl, ?s::?'a blue_dfs_state) ∈ bdfs_rel⟧ ⟹ se_back_edge_impl ?accpt ?ui ?vi ?si ≤ ⇓ bdfs_erel (Nested_DFS.se_back_edge ?accpt ?u ?v ?s)›*) NOOP_impl (*‹(?si::⦇ss_stack :: (?'a × ?'a set) list, on_stack :: ?'a set, visited :: ?'a set, lasso_impl :: (?'b list × ?'b list) option, red_impl :: ?'b set⦈, ?s::⦇counter :: nat, discovered :: ?'a ⇒ nat option, finished :: ?'a ⇒ nat option, pending :: (?'a × ?'a) set, stack :: ?'a list, tree_edges :: (?'a × ?'a) set, back_edges :: (?'a × ?'a) set, cross_edges :: (?'a × ?'a) set, lasso :: (?'b list × ?'b list) option, red :: ?'b set⦈) ∈ bdfs_rel ⟹ RETURN (simple_state.more ?si) ≤ ⇓ bdfs_erel (NOOP ?s)›*))\n (*top goal: ‹⋀v0 s v0i si si' s'. ⟦local.gen_dfs.pre_new_root v0 s; (v0i, v0) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_new_root local.gbs v0 s) s'⟧ ⟹ on_new_root (bdfs_params_impl G accpt) v0i si' ≤ ⇓ bdfs_erel (on_new_root local.blue_dfs_params v0 s')› and 4 goals remain*)\n apply ((clarsimp_all simp: pw_le_iff refine_pw_simps bdfs_erel_def simple_state_rel_def)[1])\n (*discuss goal 5*)\n apply (simp add: bdfs_params_impl_def (*‹bdfs_params_impl (?G::(?'v, ?'more) graph_rec_scheme) (?accpt::?'v ⇒ bool) ≡ ⦇on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈, on_new_root = λ(v0::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_discover = λ(u::?'v) (v::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_finish = λ(u::?'v) s::?'v bdfs_state_impl. if ?accpt u then run_red_dfs_impl ?G u s else RETURN (simple_state.more s), on_back_edge = se_back_edge_impl ?accpt, on_cross_edge = λ(u::?'v) (v::?'v) s::?'v bdfs_state_impl. RETURN (simple_state.more s), is_break = λs::?'v bdfs_state_impl. lasso_impl s ≠ None⦈›*) run_red_dfs_impl (*‹⟦(?Gi::(?'b, ?'a) graph_rec_scheme, ?G::(?'b, ?'a) graph_rec_scheme) ∈ Id; (?ui::?'b, ?u::?'b) ∈ Id; (?si::?'b bdfs_state_impl, ?s::?'b blue_dfs_state) ∈ bdfs_rel⟧ ⟹ run_red_dfs_impl ?Gi ?ui ?si ≤ ⇓ bdfs_erel (BlueDFS_defs.run_red_dfs ?G ?u ?s)›*) se_back_edge_impl (*‹⟦(?accpti::?'a ⇒ bool, ?accpt::?'a ⇒ bool) ∈ Id; (?ui::?'a, ?u::?'a) ∈ Id; (?vi::?'a, ?v::?'a) ∈ Id; (?si::?'a bdfs_state_impl, ?s::?'a blue_dfs_state) ∈ bdfs_rel⟧ ⟹ se_back_edge_impl ?accpt ?ui ?vi ?si ≤ ⇓ bdfs_erel (Nested_DFS.se_back_edge ?accpt ?u ?v ?s)›*) NOOP_impl (*‹(?si::⦇ss_stack :: (?'a × ?'a set) list, on_stack :: ?'a set, visited :: ?'a set, lasso_impl :: (?'b list × ?'b list) option, red_impl :: ?'b set⦈, ?s::⦇counter :: nat, discovered :: ?'a ⇒ nat option, finished :: ?'a ⇒ nat option, pending :: (?'a × ?'a) set, stack :: ?'a list, tree_edges :: (?'a × ?'a) set, back_edges :: (?'a × ?'a) set, cross_edges :: (?'a × ?'a) set, lasso :: (?'b list × ?'b list) option, red :: ?'b set⦈) ∈ bdfs_rel ⟹ RETURN (simple_state.more ?si) ≤ ⇓ bdfs_erel (NOOP ?s)›*))\n (*top goal: ‹⋀(v::'v::type) (s0::'v blue_dfs_state) (s::'v blue_dfs_state) (vi::'v::type) (si::'v bdfs_state_impl) (si'::'v bdfs_state_impl) s'::'v blue_dfs_state. ⟦local.gen_dfs.pre_finish v s0 s; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_finish local.gbs v s) s'⟧ ⟹ on_finish (bdfs_params_impl (G::('v, 'more) graph_rec_scheme) (accpt::'v::type ⇒ bool)) vi si' ≤ ⇓ bdfs_erel (on_finish local.blue_dfs_params v s')› and 3 goals remain*)\n apply ((clarsimp_all simp: pw_le_iff refine_pw_simps bdfs_erel_def simple_state_rel_def)[1])\n (*discuss goal 6*)\n apply (simp add: bdfs_params_impl_def (*‹bdfs_params_impl ?G ?accpt ≡ ⦇on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈, on_new_root = λv0 s. RETURN (simple_state.more s), on_discover = λu v s. RETURN (simple_state.more s), on_finish = λu s. if ?accpt u then run_red_dfs_impl ?G u s else RETURN (simple_state.more s), on_back_edge = se_back_edge_impl ?accpt, on_cross_edge = λu v s. RETURN (simple_state.more s), is_break = λs. lasso_impl s ≠ None⦈›*) run_red_dfs_impl (*‹⟦(?Gi, ?G) ∈ Id; (?ui, ?u) ∈ Id; (?si, ?s) ∈ bdfs_rel⟧ ⟹ run_red_dfs_impl ?Gi ?ui ?si ≤ ⇓ bdfs_erel (BlueDFS_defs.run_red_dfs ?G ?u ?s)›*) se_back_edge_impl (*‹⟦(?accpti, ?accpt) ∈ Id; (?ui, ?u) ∈ Id; (?vi, ?v) ∈ Id; (?si, ?s) ∈ bdfs_rel⟧ ⟹ se_back_edge_impl ?accpt ?ui ?vi ?si ≤ ⇓ bdfs_erel (Nested_DFS.se_back_edge ?accpt ?u ?v ?s)›*) NOOP_impl (*‹(?si, ?s) ∈ bdfs_rel ⟹ RETURN (simple_state.more ?si) ≤ ⇓ bdfs_erel (NOOP ?s)›*))\n (*top goal: ‹⋀u v s0 s ui vi si si' s'. ⟦local.gen_dfs.pre_cross_edge u v s0 s; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_cross_edge local.gbs u v s) s'⟧ ⟹ on_cross_edge (bdfs_params_impl G accpt) ui vi si' ≤ ⇓ bdfs_erel (on_cross_edge local.blue_dfs_params u v s')› and 2 goals remain*)\n apply ((clarsimp_all simp: pw_le_iff refine_pw_simps bdfs_erel_def simple_state_rel_def)[1])\n (*discuss goal 7*)\n apply (simp add: bdfs_params_impl_def (*‹bdfs_params_impl ?G ?accpt ≡ ⦇on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈, on_new_root = λv0 s. RETURN (simple_state.more s), on_discover = λu v s. RETURN (simple_state.more s), on_finish = λu s. if ?accpt u then run_red_dfs_impl ?G u s else RETURN (simple_state.more s), on_back_edge = se_back_edge_impl ?accpt, on_cross_edge = λu v s. RETURN (simple_state.more s), is_break = λs. lasso_impl s ≠ None⦈›*) run_red_dfs_impl (*‹⟦(?Gi, ?G) ∈ Id; (?ui, ?u) ∈ Id; (?si, ?s) ∈ bdfs_rel⟧ ⟹ run_red_dfs_impl ?Gi ?ui ?si ≤ ⇓ bdfs_erel (BlueDFS_defs.run_red_dfs ?G ?u ?s)›*) se_back_edge_impl (*‹⟦(?accpti, ?accpt) ∈ Id; (?ui, ?u) ∈ Id; (?vi, ?v) ∈ Id; (?si, ?s) ∈ bdfs_rel⟧ ⟹ se_back_edge_impl ?accpt ?ui ?vi ?si ≤ ⇓ bdfs_erel (Nested_DFS.se_back_edge ?accpt ?u ?v ?s)›*) NOOP_impl (*‹(?si, ?s) ∈ bdfs_rel ⟹ RETURN (simple_state.more ?si) ≤ ⇓ bdfs_erel (NOOP ?s)›*))\n (*discuss goal 8*)\n apply (simp add: bdfs_params_impl_def (*‹bdfs_params_impl (?G::(?'v, ?'more) graph_rec_scheme) (?accpt::?'v::type ⇒ bool) ≡ ⦇on_init = RETURN ⦇lasso_impl = None, red_impl = {}⦈, on_new_root = λ(v0::?'v::type) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_discover = λ(u::?'v::type) (v::?'v::type) s::?'v bdfs_state_impl. RETURN (simple_state.more s), on_finish = λ(u::?'v::type) s::?'v bdfs_state_impl. if ?accpt u then run_red_dfs_impl ?G u s else RETURN (simple_state.more s), on_back_edge = se_back_edge_impl ?accpt, on_cross_edge = λ(u::?'v::type) (v::?'v::type) s::?'v bdfs_state_impl. RETURN (simple_state.more s), is_break = λs::?'v bdfs_state_impl. lasso_impl s ≠ None⦈›*) run_red_dfs_impl (*‹⟦(?Gi::(?'b, ?'a) graph_rec_scheme, ?G::(?'b, ?'a) graph_rec_scheme) ∈ Id; (?ui::?'b::type, ?u::?'b::type) ∈ Id; (?si::?'b bdfs_state_impl, ?s::?'b blue_dfs_state) ∈ bdfs_rel⟧ ⟹ run_red_dfs_impl ?Gi ?ui ?si ≤ ⇓ bdfs_erel (BlueDFS_defs.run_red_dfs ?G ?u ?s)›*) se_back_edge_impl (*‹⟦(?accpti::?'a::type ⇒ bool, ?accpt::?'a::type ⇒ bool) ∈ Id; (?ui::?'a::type, ?u::?'a::type) ∈ Id; (?vi::?'a::type, ?v::?'a::type) ∈ Id; (?si::?'a bdfs_state_impl, ?s::?'a blue_dfs_state) ∈ bdfs_rel⟧ ⟹ se_back_edge_impl ?accpt ?ui ?vi ?si ≤ ⇓ bdfs_erel (Nested_DFS.se_back_edge ?accpt ?u ?v ?s)›*) NOOP_impl (*‹(?si::⦇ss_stack :: (?'a × ?'a set) list, on_stack :: ?'a set, visited :: ?'a set, lasso_impl :: (?'b list × ?'b list) option, red_impl :: ?'b set⦈, ?s::⦇counter :: nat, discovered :: ?'a ⇒ nat option, finished :: ?'a ⇒ nat option, pending :: (?'a × ?'a) set, stack :: ?'a list, tree_edges :: (?'a × ?'a) set, back_edges :: (?'a × ?'a) set, cross_edges :: (?'a × ?'a) set, lasso :: (?'b list × ?'b list) option, red :: ?'b set⦈) ∈ bdfs_rel ⟹ RETURN (simple_state.more ?si) ≤ ⇓ bdfs_erel (NOOP ?s)›*))\n (*goal: ‹⋀u v s0 s ui vi si si' s'. ⟦local.gen_dfs.pre_discover u v s0 s; (ui, u) ∈ Id; (vi, v) ∈ Id; (si, s) ∈ bdfs_rel; (si', s') ∈ bdfs_rel; nf_inres (gbs_discover local.gbs u v s) s'⟧ ⟹ on_discover (bdfs_params_impl G accpt) ui vi si' ≤ ⇓ bdfs_erel (on_discover local.blue_dfs_params u v s')›*)\n apply ((clarsimp_all simp: pw_le_iff refine_pw_simps bdfs_erel_def simple_state_rel_def)[1])\n (*proven 8 subgoals*) .\n\n lemmas impl = impl.simple_tailrec_refine\nend\n\ndefinition \"nested_dfs_impl G accpt ≡ do {\n ASSERT (fb_graph G);\n s ← impl.tailrec_impl TYPE('a) G accpt;\n RETURN (lasso_impl s)\n}\"\n\nlemma nested_dfs_impl[refine]: \n assumes \"(Gi,G)∈Id\"\n assumes \"(accpti,accpt)∈Id\"\n shows \"nested_dfs_impl Gi accpti ≤⇓(⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel) \n (nested_dfs G accpt)\"\n using assms (*‹(Gi::('a, 'b) graph_rec_scheme, G::('a, 'b) graph_rec_scheme) ∈ Id› ‹(accpti, accpt) ∈ Id›*) unfolding nested_dfs_impl_def nested_dfs_def\n (*goal: ‹ASSERT (fb_graph Gi) ⤜ (λ_. tailrec_impl_defs.tailrec_impl Gi (gen_param_dfs_defs.gds (pre_simple_impl.gbs Gi) (bdfs_params_impl Gi accpti) simple_state.more_update) ⤜ (λs. RETURN (lasso_impl s))) ≤ ⇓ (⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel) (ASSERT (fb_graph G) ⤜ (λ_. param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt) ⤜ (λs. RETURN (lasso s))))›*)\n apply refine_rcg\n (*goal: ‹ASSERT (fb_graph (Gi::('a, 'b) graph_rec_scheme)) ⤜ (λ_::unit. tailrec_impl_defs.tailrec_impl Gi (gen_param_dfs_defs.gds (pre_simple_impl.gbs Gi) (bdfs_params_impl Gi (accpti::'a ⇒ bool)) simple_state.more_update) ⤜ (λs::'a bdfs_state_impl. RETURN (lasso_impl s))) ≤ ⇓ (⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel) (ASSERT (fb_graph (G::('a, 'b) graph_rec_scheme)) ⤜ (λ_::unit. param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G (accpt::'a ⇒ bool)) ⤜ (λs::'a blue_dfs_state. RETURN (lasso s))))›*)\n apply simp_all\n (*top goal: ‹⟦(Gi, G) ∈ Id; (accpti, accpt) ∈ Id; fb_graph G⟧ ⟹ fb_graph Gi› and 2 goals remain*)\n apply (rule intro_prgR[where R=bdfs_rel] (*‹(?c::⦇ss_stack :: (?'d1 × ?'d1 set) list, on_stack :: ?'d1 set, visited :: ?'d1 set, lasso_impl :: (?'c1 list × ?'c1 list) option, red_impl :: ?'c1 set⦈ nres) ≤ ⇓ bdfs_rel (?a::⦇counter :: nat, discovered :: ?'d1 ⇒ nat option, finished :: ?'d1 ⇒ nat option, pending :: (?'d1 × ?'d1) set, stack :: ?'d1 list, tree_edges :: (?'d1 × ?'d1) set, back_edges :: (?'d1 × ?'d1) set, cross_edges :: (?'d1 × ?'d1) set, lasso :: (?'c1 list × ?'c1 list) option, red :: ?'c1 set⦈ nres) ⟹ ?c ≤ ⇓ bdfs_rel ?a›*))\n (*top goal: ‹⟦Gi = G; accpti = accpt; fb_graph G⟧ ⟹ tailrec_impl_defs.tailrec_impl G (gen_param_dfs_defs.gds (pre_simple_impl.gbs G) (bdfs_params_impl G accpt) simple_state.more_update) ≤ ⇓ ?R'4 (param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt))› and 1 goal remains*)\n defer 1\n (*top goal: ‹⋀s sa. ⟦Gi = G; accpti = accpt; fb_graph G; (s, sa) ∈ bdfs_rel⟧ ⟹ lasso_impl s = lasso sa› and 1 goal remains*)\n apply (rename_tac si s)\n (*top goal: ‹⋀s sa. ⟦Gi = G; accpti = accpt; fb_graph G; (s, sa) ∈ bdfs_rel⟧ ⟹ lasso_impl s = lasso sa› and 1 goal remains*)\n apply (case_tac si, case_tac s)\n (*top goal: ‹⋀si s. ⟦Gi = G; accpti = accpt; fb_graph G; (si, s) ∈ bdfs_rel⟧ ⟹ lasso_impl si = lasso s› and 1 goal remains*)\n proof ((auto simp: bdfs_erel_def (*‹bdfs_erel ≡ {(⦇lasso_impl = li, red_impl = ri⦈, ⦇lasso = l, red = r⦈) |li ri l r. li = l ∧ ri = r}›*) simple_state_rel_def (*‹⟨?erel⟩simple_state_rel ≡ {(s, s'). ss_stack s = map (λu. (u, pending s' `` {u})) (stack s') ∧ on_stack s = set (stack s') ∧ visited s = dom (discovered s') ∧ dom (finished s') = dom (discovered s') - set (stack s') ∧ set (stack s') ⊆ dom (discovered s') ∧ (simple_state.more s, state.more s') ∈ ?erel}›*))[1])\n (*goal: ‹⟦(Gi::('a, 'b) graph_rec_scheme) = (G::('a, 'b) graph_rec_scheme); (accpti::'a::type ⇒ bool) = (accpt::'a::type ⇒ bool); fb_graph G⟧ ⟹ tailrec_impl_defs.tailrec_impl G (gen_param_dfs_defs.gds (pre_simple_impl.gbs G) (bdfs_params_impl G accpt) simple_state.more_update) ≤ ⇓ bdfs_rel (param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt))›*)\n assume \"fb_graph G\" (*‹fb_graph (G::('a, 'b) graph_rec_scheme)›*)\n then interpret fb_graph G .\n interpret BlueDFS G\n by unfold_locales\n from impl (*‹tailrec_impl_defs.tailrec_impl (G::('a, 'b) graph_rec_scheme) (gen_param_dfs_defs.gds (pre_simple_impl.gbs G) (bdfs_params_impl G (accpt::'a::type ⇒ bool)) simple_state.more_update) ≤ ⇓ bdfs_rel (param_DFS_defs.it_dfs G (BlueDFS_defs.blue_dfs_params G accpt))›*) show \"impl.tailrec_impl TYPE('b) G accpt ≤⇓bdfs_rel (it_dfs TYPE('b) G accpt)\" .\n qed \n\nsubsection ‹Synthesis of Executable Code›\n(* Straightforward autoref implementation *)\nrecord ('v,'si,'nsi)bdfs_state_impl' = \"('si,'nsi)simple_state_impl\" +\n lasso_impl' :: \"('v list × 'v list) option\"\n red_impl' :: 'nsi\n\ndefinition [to_relAPP]: \"bdfs_state_erel' Vi ≡ {\n (⦇lasso_impl' = li, red_impl'=ri⦈,⦇lasso_impl = l, red_impl = r⦈) | li ri l r.\n (li,l)∈⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel ∧ (ri,r)∈⟨Vi⟩dflt_ahs_rel}\"\n\nconsts \n i_bdfs_state_ext :: \"interface ⇒ interface\"\n\nlemmas [autoref_rel_intf] = REL_INTFI[of bdfs_state_erel' i_bdfs_state_ext]\n\nlemma [autoref_rules]:\n fixes ns_rel vis_rel Vi\n defines \"R ≡ ⟨ns_rel,vis_rel,⟨Vi⟩bdfs_state_erel'⟩ss_impl_rel\"\n shows \n \"(bdfs_state_impl'_ext, bdfs_state_impl_ext) \n ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel → ⟨Vi⟩dflt_ahs_rel → unit_rel → ⟨Vi⟩bdfs_state_erel'\"\n \"(lasso_impl', lasso_impl) ∈ R → ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel\"\n \"(red_impl', red_impl) ∈ R → ⟨Vi⟩dflt_ahs_rel\"\n unfolding bdfs_state_erel'_def ss_impl_rel_def R_def\n (*goals:\n 1. ‹(bdfs_state_impl'_ext, bdfs_state_impl_ext) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel → ⟨Vi⟩ahs_rel bounded_hashcode_nat → unit_rel → {(⦇lasso_impl' = li, red_impl' = ri⦈, ⦇lasso_impl = l, red_impl = r⦈) |li ri l r. (li, l) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel ∧ (ri, r) ∈ ⟨Vi⟩ahs_rel bounded_hashcode_nat}›\n 2. ‹(lasso_impl', lasso_impl) ∈ {(⦇ss_stack_impl = si, ss_on_stack_impl = osi, ss_visited_impl = visi, … = mi⦈, ⦇ss_stack = s, on_stack = os, visited = vis, … = m⦈) |si osi visi mi s os vis m. (si, s) ∈ ns_rel ∧ (osi, os) ∈ vis_rel ∧ (visi, vis) ∈ vis_rel ∧ (mi, m) ∈ {(⦇lasso_impl' = li, red_impl' = ri⦈, ⦇lasso_impl = l, red_impl = r⦈) |li ri l r. (li, l) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel ∧ (ri, r) ∈ ⟨Vi⟩ahs_rel bounded_hashcode_nat}} → ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel›\n 3. ‹(red_impl', red_impl) ∈ {(⦇ss_stack_impl = si, ss_on_stack_impl = osi, ss_visited_impl = visi, … = mi⦈, ⦇ss_stack = s, on_stack = os, visited = vis, … = m⦈) |si osi visi mi s os vis m. (si, s) ∈ ns_rel ∧ (osi, os) ∈ vis_rel ∧ (visi, vis) ∈ vis_rel ∧ (mi, m) ∈ {(⦇lasso_impl' = li, red_impl' = ri⦈, ⦇lasso_impl = l, red_impl = r⦈) |li ri l r. (li, l) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel ∧ (ri, r) ∈ ⟨Vi⟩ahs_rel bounded_hashcode_nat}} → ⟨Vi⟩ahs_rel bounded_hashcode_nat›*)\n (*goals:\n 1. ‹(bdfs_state_impl'_ext, bdfs_state_impl_ext) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel → ⟨Vi⟩ahs_rel bounded_hashcode_nat → unit_rel → {(⦇lasso_impl' = li, red_impl' = ri⦈, ⦇lasso_impl = l, red_impl = r⦈) |li ri l r. (li, l) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel ∧ (ri, r) ∈ ⟨Vi⟩ahs_rel bounded_hashcode_nat}›\n 2. ‹(lasso_impl', lasso_impl) ∈ {(⦇ss_stack_impl = si, ss_on_stack_impl = osi, ss_visited_impl = visi, … = mi⦈, ⦇ss_stack = s, on_stack = os, visited = vis, … = m⦈) |si osi visi mi s os vis m. (si, s) ∈ ns_rel ∧ (osi, os) ∈ vis_rel ∧ (visi, vis) ∈ vis_rel ∧ (mi, m) ∈ {(⦇lasso_impl' = li, red_impl' = ri⦈, ⦇lasso_impl = l, red_impl = r⦈) |li ri l r. (li, l) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel ∧ (ri, r) ∈ ⟨Vi⟩ahs_rel bounded_hashcode_nat}} → ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel›\n 3. ‹(red_impl', red_impl) ∈ {(⦇ss_stack_impl = si, ss_on_stack_impl = osi, ss_visited_impl = visi, … = mi⦈, ⦇ss_stack = s, on_stack = os, visited = vis, … = m⦈) |si osi visi mi s os vis m. (si, s) ∈ ns_rel ∧ (osi, os) ∈ vis_rel ∧ (visi, vis) ∈ vis_rel ∧ (mi, m) ∈ {(⦇lasso_impl' = li, red_impl' = ri⦈, ⦇lasso_impl = l, red_impl = r⦈) |li ri l r. (li, l) ∈ ⟨⟨Vi⟩list_rel ×⇩r ⟨Vi⟩list_rel⟩option_rel ∧ (ri, r) ∈ ⟨Vi⟩ahs_rel bounded_hashcode_nat}} → ⟨Vi⟩ahs_rel bounded_hashcode_nat›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n\nschematic_goal nested_dfs_code:\n assumes Vid: \"V = (Id :: ('v::hashable × 'v) set)\"\n assumes [unfolded Vid, autoref_rules]:\n \"(Gi, G) ∈ ⟨Rm, V⟩g_impl_rel_ext\"\n \"(accpti,accpt) ∈ (V → bool_rel)\"\n notes [unfolded Vid, autoref_tyrel] = \n TYRELI[where R=\"⟨V⟩dflt_ahs_rel\"]\n TYRELI[where R=\"⟨V⟩ras_rel\"]\n shows \"(nres_of ?c, nested_dfs_impl G accpt) \n ∈ ⟨⟨⟨V⟩list_rel ×⇩r ⟨V⟩list_rel⟩option_rel⟩nres_rel\"\n unfolding nested_dfs_impl_def[abs_def] Vid \n se_back_edge_impl_def run_red_dfs_impl_def mk_blue_witness_impl_def\n cyan_impl_def\n DFS_code_unfold\n (*apply (subst aux1)*)\n using [[autoref_trace_failed_id]]\n apply (autoref_monadic (trace))\n done\n\nconcrete_definition nested_dfs_code uses nested_dfs_code\n\nexport_code nested_dfs_code checking SML\n\nsubsection ‹Conclusion›\ntext ‹\n We have implemented an efficiently executable nested DFS algorithm.\n The following theorem declares this implementation to the Autoref tool,\n such that it uses it to synthesize efficient code for @{const nested_dfs}.\n Note that you will need the lemma @{thm [source] nested_dfs_correct} to link\n @{const nested_dfs} to an abstract specification, which is usually done in \n a previous refinement step.\n›\ntheorem nested_dfs_autoref[autoref_rules]:\n assumes \"PREFER_id V\"\n shows \"(λ G accpt. nres_of (nested_dfs_code G accpt),nested_dfs) ∈\n ⟨Rm, V⟩g_impl_rel_ext → (V → bool_rel) →\n ⟨⟨⟨V⟩list_rel ×⇩r ⟨V⟩list_rel⟩option_rel⟩nres_rel\"\nproof (-)\n (*goal: ‹(λG accpt. nres_of (nested_dfs_code G accpt), nested_dfs) ∈ ⟨Rm, V⟩g_impl_rel_ext → (V → bool_rel) → ⟨⟨⟨V⟩list_rel ×⇩r ⟨V⟩list_rel⟩option_rel⟩nres_rel›*)\n from assms (*‹PREFER_id V›*) have Vid: \"V=Id\"\n by simp\n note nested_dfs_code.refine[OF Vid, param_fo, THEN nres_relD] (*‹⟦(?Gi1, ?G1) ∈ ⟨?Rm1, V⟩g_impl_rel_ext; (?accpti1, ?accpt1) ∈ V → bool_rel⟧ ⟹ nres_of (nested_dfs_code ?Gi1 ?accpti1) ≤ ⇓ (⟨⟨V⟩list_rel ×⇩r ⟨V⟩list_rel⟩option_rel) (nested_dfs_impl ?G1 ?accpt1)›*)\n also (*calculation: ‹⟦(?Gi1, ?G1) ∈ ⟨?Rm1, V⟩g_impl_rel_ext; (?accpti1, ?accpt1) ∈ V → bool_rel⟧ ⟹ nres_of (nested_dfs_code ?Gi1 ?accpti1) ≤ ⇓ (⟨⟨V⟩list_rel ×⇩r ⟨V⟩list_rel⟩option_rel) (nested_dfs_impl ?G1 ?accpt1)›*) note nested_dfs_impl (*‹⟦(?Gi, ?G) ∈ Id; (?accpti, ?accpt) ∈ Id⟧ ⟹ nested_dfs_impl ?Gi ?accpti ≤ ⇓ (⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel) (nested_dfs ?G ?accpt)›*)\n finally (*calculation: ‹⟦(?Gi3, ?Gi1) ∈ ⟨?Rm3, V⟩g_impl_rel_ext; (?accpti3, ?accpti1) ∈ V → bool_rel; (?Gi1, ?G1) ∈ Id; (?accpti1, ?accpt1) ∈ Id⟧ ⟹ nres_of (nested_dfs_code ?Gi3 ?accpti3) ≤ ⇓ (⟨⟨V⟩list_rel ×⇩r ⟨V⟩list_rel⟩option_rel) (⇓ (⟨⟨Id⟩list_rel ×⇩r ⟨Id⟩list_rel⟩option_rel) (nested_dfs ?G1 ?accpt1))›*) show \"?thesis\"\n (*goal: ‹(λ(G::('a ⇒ bool, 'a ⇒ 'a list, 'a list, 'b) gen_g_impl_scheme) accpt::'a ⇒ bool. nres_of (nested_dfs_code G accpt), nested_dfs) ∈ ⟨Rm::('b × 'c) set, V::('a × 'a) set⟩g_impl_rel_ext → (V → bool_rel) → ⟨⟨⟨V⟩list_rel ×⇩r ⟨V⟩list_rel⟩option_rel⟩nres_rel›*)\n by (fastforce intro: nres_relI (*‹?c ≤ ⇓ ?R ?a ⟹ (?c, ?a) ∈ ⟨?R⟩nres_rel›*))\nqed\n\n\nend\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/DFS_Framework/Examples/Nested_DFS.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"1270ce5f6cc9552a05906d357636b1a577cbdf0532b255921996a9ea8c6045e0\"\n}"}}},{"rowIdx":197,"cells":{"text":{"kind":"string","value":"(*\n Author: Jose Divasón\n Email: jose.divason@unirioja.es\n*)\n\nsection ‹Elementary divisor rings›\n\ntheory Elementary_Divisor_Rings\n imports \n SNF_Algorithm\n Rings2_Extended\nbegin\n\ntext ‹This theory contains the definition of elementary divisor rings and Hermite rings, as \nwell as the corresponding relation between both concepts. \nIt also includes a complete characterization\nfor elementary divisor rings, by means of an \\emph{if and only if}-statement.\n\nThe results presented here follows the article ``Some remarks about elementary divisor rings''\nby Leonard Gillman and Melvin Henriksen.›\n\nsubsection ‹Previous definitions and basic properties of Hermite ring›\n\ndefinition \"admits_triangular_reduction A = \n (∃U::'a::comm_ring_1 mat. U ∈ carrier_mat (dim_col A) (dim_col A) \n ∧ invertible_mat U ∧ lower_triangular (A*U))\"\n\nclass Hermite_ring =\n assumes \"∀(A::'a::comm_ring_1 mat). admits_triangular_reduction A\"\n\nlemma admits_triangular_reduction_intro:\n assumes \"invertible_mat (U::'a::comm_ring_1 mat)\" \n and \"U ∈ carrier_mat (dim_col A) (dim_col A)\"\n and \"lower_triangular (A*U)\"\n shows \"admits_triangular_reduction A\" \n using assms (*‹invertible_mat U› ‹U ∈ carrier_mat (dim_col A) (dim_col A)› ‹lower_triangular (A * U)›*) unfolding admits_triangular_reduction_def\n (*goal: ‹∃U::'a::comm_ring_1 mat. U ∈ carrier_mat (dim_col (A::'a::comm_ring_1 mat)) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n by auto\n\nlemma OFCLASS_Hermite_ring_def:\n \"OFCLASS('a::comm_ring_1, Hermite_ring_class) \n ≡ (⋀(A::'a::comm_ring_1 mat). admits_triangular_reduction A)\"\nproof (standard)\n (*goals:\n 1. ‹⋀A. OFCLASS('a, Hermite_ring_class) ⟹ admits_triangular_reduction A›\n 2. ‹(⋀A. admits_triangular_reduction A) ⟹ OFCLASS('a, Hermite_ring_class)›*)\n fix A :: \"'a mat\"\n assume H: \"OFCLASS('a::comm_ring_1, Hermite_ring_class)\" (*‹OFCLASS('a, Hermite_ring_class)›*)\n have \"∀A. admits_triangular_reduction (A::'a mat)\"\n using conjunctionD2[OF H [ unfolded Hermite_ring_class_def class.Hermite_ring_def ]] (*‹∀A::'a mat. admits_triangular_reduction A›*) by auto\n thus \"admits_triangular_reduction A\"\n by auto\nnext\n (*goal: ‹(⋀A. admits_triangular_reduction A) ⟹ OFCLASS('a, Hermite_ring_class)›*)\n assume i: \"(⋀A::'a mat. admits_triangular_reduction A)\" (*‹admits_triangular_reduction (?A::'a mat)›*)\n show \"OFCLASS('a, Hermite_ring_class)\"\n proof (standard)\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n show \"∀A::'a mat. admits_triangular_reduction A\"\n using i (*‹admits_triangular_reduction ?A›*) by auto\n qed\nqed\n\n\ndefinition admits_diagonal_reduction::\"'a::comm_ring_1 mat ⇒ bool\"\n where \"admits_diagonal_reduction A = (∃P Q. P ∈ carrier_mat (dim_row A) (dim_row A) ∧\n Q ∈ carrier_mat (dim_col A) (dim_col A) \n ∧ invertible_mat P ∧ invertible_mat Q \n ∧ Smith_normal_form_mat (P * A * Q))\"\n\nlemma admits_diagonal_reduction_intro:\n assumes \"P ∈ carrier_mat (dim_row A) (dim_row A)\"\n and \"Q ∈ carrier_mat (dim_col A) (dim_col A)\" \n and \"invertible_mat P\" and \"invertible_mat Q \"\n and \"Smith_normal_form_mat (P * A * Q)\"\n shows \"admits_diagonal_reduction A\" using assms (*‹P ∈ carrier_mat (dim_row A) (dim_row A)› ‹(Q::'a::comm_ring_1 mat) ∈ carrier_mat (dim_col (A::'a::comm_ring_1 mat)) (dim_col A)› ‹invertible_mat P› ‹invertible_mat Q› ‹Smith_normal_form_mat (P * A * Q)›*) unfolding admits_diagonal_reduction_def\n (*goal: ‹∃P Q. P ∈ carrier_mat (dim_row A) (dim_row A) ∧ Q ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat P ∧ invertible_mat Q ∧ Smith_normal_form_mat (P * A * Q)›*)\n by fast\n\n(*Lemmas for equivalence between admits_diagonal_reduction and is_SNF \n via the existence of an algorithm*)\n\nlemma admits_diagonal_reduction_imp_exists_algorithm_is_SNF:\n assumes \"A ∈ carrier_mat m n\"\n and \"admits_diagonal_reduction A\"\nshows \"∃algorithm. is_SNF A (algorithm A)\" \n using assms (*‹A ∈ carrier_mat m n› ‹admits_diagonal_reduction (A::'a mat)›*) unfolding is_SNF_def admits_diagonal_reduction_def\n (*goal: ‹∃algorithm. case algorithm A of (P, S, Q) ⇒ P ∈ carrier_mat (dim_row A) (dim_row A) ∧ Q ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat P ∧ invertible_mat Q ∧ Smith_normal_form_mat S ∧ S = P * A * Q›*)\n by auto\n\nlemma exists_algorithm_is_SNF_imp_admits_diagonal_reduction:\n assumes \"A ∈ carrier_mat m n\"\n and \"∃algorithm. is_SNF A (algorithm A)\"\n shows \"admits_diagonal_reduction A\"\n using assms (*‹A ∈ carrier_mat m n› ‹∃algorithm. is_SNF A (algorithm A)›*) unfolding is_SNF_def admits_diagonal_reduction_def\n (*goal: ‹∃P Q. P ∈ carrier_mat (dim_row A) (dim_row A) ∧ Q ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat P ∧ invertible_mat Q ∧ Smith_normal_form_mat (P * A * Q)›*)\n by auto\n\nlemma admits_diagonal_reduction_eq_exists_algorithm_is_SNF:\n assumes A: \"A ∈ carrier_mat m n\"\n shows \"admits_diagonal_reduction A = (∃algorithm. is_SNF A (algorithm A))\"\n using admits_diagonal_reduction_imp_exists_algorithm_is_SNF[OF A] (*‹admits_diagonal_reduction A ⟹ ∃algorithm. is_SNF A (algorithm A)›*) using exists_algorithm_is_SNF_imp_admits_diagonal_reduction[OF A] (*‹∃algorithm. is_SNF A (algorithm A) ⟹ admits_diagonal_reduction A›*) by auto\n\n\nlemma admits_diagonal_reduction_imp_exists_algorithm_is_SNF_all:\n assumes \"(∀(A::'a::comm_ring_1 mat) ∈ carrier_mat m n. admits_diagonal_reduction A)\" \n shows\" (∃algorithm. ∀(A::'a mat) ∈ carrier_mat m n. is_SNF A (algorithm A))\"\nsorry\n\nlemma exists_algorithm_is_SNF_imp_admits_diagonal_reduction_all:\n assumes \"(∃algorithm. ∀(A::'a mat) ∈ carrier_mat m n. is_SNF A (algorithm A))\"\n shows \"(∀(A::'a::comm_ring_1 mat) ∈ carrier_mat m n. admits_diagonal_reduction A)\"\n using assms (*‹∃algorithm. ∀A∈carrier_mat m n. is_SNF A (algorithm A)›*) exists_algorithm_is_SNF_imp_admits_diagonal_reduction (*‹⟦?A ∈ carrier_mat ?m ?n; ∃algorithm. is_SNF ?A (algorithm ?A)⟧ ⟹ admits_diagonal_reduction ?A›*) by blast \n\n \nlemma admits_diagonal_reduction_eq_exists_algorithm_is_SNF_all:\n shows \"(∀(A::'a::comm_ring_1 mat) ∈ carrier_mat m n. admits_diagonal_reduction A) \n = (∃algorithm. ∀(A::'a mat) ∈ carrier_mat m n. is_SNF A (algorithm A))\"\n using exists_algorithm_is_SNF_imp_admits_diagonal_reduction_all (*‹∃algorithm. ∀A∈carrier_mat ?m ?n. is_SNF A (algorithm A) ⟹ ∀A∈carrier_mat ?m ?n. admits_diagonal_reduction A›*) using admits_diagonal_reduction_imp_exists_algorithm_is_SNF_all (*‹∀A∈carrier_mat ?m ?n. admits_diagonal_reduction A ⟹ ∃algorithm. ∀A∈carrier_mat ?m ?n. is_SNF A (algorithm A)›*) by auto\n\n\nsubsection ‹The class that represents elementary divisor rings›\n\nclass elementary_divisor_ring =\n assumes \"∀(A::'a::comm_ring_1 mat). admits_diagonal_reduction A\"\n\n\nlemma dim_row_mat_diag[simp]: \"dim_row (mat_diag n f) = n\" and \n dim_col_mat_diag[simp]: \"dim_col (mat_diag n f) = n\" \n using mat_diag_dim (*‹mat_diag ?n ?f ∈ carrier_mat ?n ?n›*) unfolding carrier_mat_def\n (*goals:\n 1. ‹dim_row (mat_diag n f) = n›\n 2. ‹dim_col (mat_diag n f) = n›*)\n apply -\n (*goals:\n 1. ‹(⋀n f. mat_diag n f ∈ {m. dim_row m = n ∧ dim_col m = n}) ⟹ dim_row (mat_diag n f) = n›\n 2. ‹(⋀n f. mat_diag n f ∈ {m. dim_row m = n ∧ dim_col m = n}) ⟹ dim_col (mat_diag n f) = n›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\n\nsubsection ‹Hermite ring implies B\\'ezout ring›\n\n(*HERMITE ⟹ BEZOUT*)\n\ntext ‹To prove this fact, we make use of the alternative definition for B\\'ezout rings:\neach finitely generated ideal is principal›\n\nlemma Hermite_ring_imp_Bezout_ring:\n assumes H: \"OFCLASS('a::comm_ring_1, Hermite_ring_class)\"\n shows \" OFCLASS('a::comm_ring_1, bezout_ring_class)\"\napply (rule all_fin_gen_ideals_are_principal_imp_bezout (*‹∀I. finitely_generated_ideal I ⟶ principal_ideal I ⟹ OFCLASS(?'a, bezout_ring_class)›*))\n(*goal: ‹OFCLASS('a, bezout_ring_class)›*)\napply rule\n(*goal: ‹∀I. finitely_generated_ideal I ⟶ principal_ideal I›*)\nproof (rule)\n (*goal: ‹⋀I. finitely_generated_ideal I ⟹ principal_ideal I›*)\n fix I :: \"'a set\"\n assume fin: \"finitely_generated_ideal I\" (*‹finitely_generated_ideal (I::'a set)›*)\n obtain S where ig_S: \"ideal_generated S = I\" and fin_S: \"finite S\"\n (*goal: ‹(⋀S. ⟦ideal_generated S = I; finite S⟧ ⟹ thesis) ⟹ thesis›*)\n using fin (*‹finitely_generated_ideal I›*) unfolding finitely_generated_ideal_def\n (*goal: ‹(⋀S::'a set. ⟦ideal_generated S = (I::'a set); finite S⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n obtain xs where set_xs: \"set xs = S\" and d: \"distinct xs\"\n (*goal: ‹(⋀xs. ⟦set xs = S; distinct xs⟧ ⟹ thesis) ⟹ thesis›*)\n using finite_distinct_list[OF fin_S] (*‹∃xs. set xs = S ∧ distinct xs›*) by blast\n hence length_eq_card: \"length xs = card S\"\n using distinct_card (*‹distinct ?xs ⟹ card (set ?xs) = length ?xs›*) by force\n define n where \"n = card S\"\n define A where \"A = mat_of_rows n [vec_of_list xs]\"\n have A[simp]: \"A ∈ carrier_mat 1 n\"\n unfolding A_def\n (*goal: ‹mat_of_rows n [vec_of_list xs] ∈ carrier_mat 1 n›*)\n using mat_of_rows_carrier (*‹mat_of_rows ?n ?vs ∈ carrier_mat (length ?vs) ?n› ‹dim_row (mat_of_rows ?n ?vs) = length ?vs› ‹dim_col (mat_of_rows ?n ?vs) = ?n›*) by auto\n have \"∀(A::'a::comm_ring_1 mat). admits_triangular_reduction A\"\n using H (*‹OFCLASS('a, Hermite_ring_class)›*) unfolding OFCLASS_Hermite_ring_def\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n by auto\n from this (*‹∀A. admits_triangular_reduction A›*) obtain Q where inv_Q: \"invertible_mat Q\" and t_AQ: \"lower_triangular (A*Q)\" and Q[simp]: \"Q ∈ carrier_mat n n\"\n (*goal: ‹(⋀Q. ⟦invertible_mat Q; lower_triangular (A * Q); Q ∈ carrier_mat n n⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding admits_triangular_reduction_def\n (*goal: ‹(⋀Q::'a mat. ⟦invertible_mat Q; lower_triangular ((A::'a mat) * Q); Q ∈ carrier_mat (n::nat) n⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using A (*‹A ∈ carrier_mat 1 n›*) by auto\n have AQ[simp]: \"A * Q ∈ carrier_mat 1 n\"\n using A (*‹A ∈ carrier_mat 1 n›*) Q (*‹Q ∈ carrier_mat n n›*) by auto\n show \"principal_ideal I\"\n proof (cases \"xs=[]\")\n (*goals:\n 1. ‹(xs::'a::comm_ring_1 list) = [] ⟹ principal_ideal (I::'a::comm_ring_1 set)›\n 2. ‹(xs::'a::comm_ring_1 list) ≠ [] ⟹ principal_ideal (I::'a::comm_ring_1 set)›*)\n case True (*‹xs = []›*)\n then show \"?thesis\"\n (*goal: ‹principal_ideal I›*)\n by (metis empty_set (*‹{} = set []›*) ideal_generated_0 (*‹ideal_generated {0} = {0}›*) ideal_generated_empty (*‹ideal_generated {} = {0}›*) ig_S (*‹ideal_generated S = I›*) principal_ideal_def (*‹principal_ideal ?S = (∃a. ideal_generated {a} = ?S)›*) set_xs (*‹set xs = S›*))\n next\n (*goal: ‹xs ≠ [] ⟹ principal_ideal I›*)\n case False (*‹(xs::'a list) ≠ []›*)\n have a: \"0 < dim_row A\"\n using A (*‹A ∈ carrier_mat 1 n›*) by auto\n have \"0 < length xs\"\n using False (*‹xs ≠ []›*) by auto\n hence b: \"0 < dim_col A\"\n using A (*‹A ∈ carrier_mat 1 n›*) n_def (*‹n = card S›*) length_eq_card (*‹length xs = card S›*) by auto\n have q0: \"0 < dim_col Q\"\n by (metis A (*‹(A::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*) Q (*‹(Q::'a mat) ∈ carrier_mat (n::nat) n›*) b (*‹(0::nat) < dim_col (A::'a mat)›*) carrier_matD( (*‹(?A::?'a mat) ∈ carrier_mat (?nr::nat) (?nc::nat) ⟹ dim_col ?A = ?nc›*) 2))\n have n0: \"0<n\"\n using ‹0 < length xs› (*‹0 < length xs›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) by linarith\n define d where \"d = (A*Q) $$ (0,0)\"\n let ?h = \"(λx. THE i. xs ! i = x ∧ i<n)\"\n let ?u = \"λi. xs ! i\"\n have bij: \"bij_betw ?h (set xs) {0..<n}\"\n proof (rule bij_betw_imageI (*‹⟦inj_on ?f ?A; ?f ` ?A = ?B⟧ ⟹ bij_betw ?f ?A ?B›*))\n (*goals:\n 1. ‹inj_on (λx::'a::comm_ring_1. THE i::nat. (xs::'a::comm_ring_1 list) ! i = x ∧ i < (n::nat)) (set xs)›\n 2. ‹(λx::'a::comm_ring_1. THE i::nat. (xs::'a::comm_ring_1 list) ! i = x ∧ i < (n::nat)) ` set xs = {0::nat..<n}›*)\n show \"inj_on ?h (set xs)\"\n proof (-)\n (*goal: ‹inj_on (λx. THE i. xs ! i = x ∧ i < n) (set xs)›*)\n have \"x=y\" if x: \"x ∈ set xs\" and y: \"y ∈ set xs\" and xy: \"(THE i. xs ! i = x ∧ i < n) = (THE i. xs ! i = y ∧ i < n)\" for x and y\n proof (-)\n (*goal: ‹x = y›*)\n let ?i = \"(THE i. xs ! i = x ∧ i < n)\"\n let ?j = \"(THE i. xs ! i = y ∧ i < n)\"\n obtain i where xs_i: \"xs ! i = x ∧ i < n\"\n (*goal: ‹(⋀i. xs ! i = x ∧ i < n ⟹ thesis) ⟹ thesis›*)\n using x (*‹x ∈ set xs›*) by (metis in_set_conv_nth (*‹((?x::?'a) ∈ set (?xs::?'a list)) = (∃i<length ?xs. ?xs ! i = ?x)›*) length_eq_card (*‹length (xs::'a list) = card (S::'a set)›*) n_def (*‹(n::nat) = card (S::'a set)›*))\n from this (*‹(xs::'a list) ! (i::nat) = (x::'a) ∧ i < (n::nat)›*) have 1: \"xs ! ?i = x ∧ ?i < n\"\n apply (rule theI (*‹⟦(?P::?'a ⇒ bool) (?a::?'a); ⋀x::?'a. ?P x ⟹ x = ?a⟧ ⟹ ?P (THE x::?'a. ?P x)›*))\n (*goal: ‹xs ! (THE i. xs ! i = x ∧ i < n) = x ∧ (THE i. xs ! i = x ∧ i < n) < n›*)\n apply (insert d (*‹distinct xs›*) xs_i (*‹xs ! i = x ∧ i < n›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) nth_eq_iff_index_eq (*‹⟦distinct ?xs; ?i < length ?xs; ?j < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*))\n (*goal: ‹⋀ia::nat. (xs::'a list) ! ia = (x::'a) ∧ ia < (n::nat) ⟹ ia = (i::nat)›*)\n by fastforce\n obtain j where xs_j: \"xs ! j = y ∧ j < n\"\n (*goal: ‹(⋀j. xs ! j = y ∧ j < n ⟹ thesis) ⟹ thesis›*)\n using y (*‹(y::'a) ∈ set (xs::'a list)›*) by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*))\n from this (*‹xs ! j = y ∧ j < n›*) have 2: \"xs ! ?j = y ∧ ?j < n\"\n apply (rule theI (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ?P (THE x. ?P x)›*))\n (*goal: ‹xs ! (THE i. xs ! i = y ∧ i < n) = y ∧ (THE i. xs ! i = y ∧ i < n) < n›*)\n apply (insert d (*‹distinct xs›*) xs_j (*‹xs ! j = y ∧ j < n›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) nth_eq_iff_index_eq (*‹⟦distinct ?xs; ?i < length ?xs; ?j < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*))\n (*goal: ‹⋀i. xs ! i = y ∧ i < n ⟹ i = j›*)\n by fastforce\n show \"?thesis\"\n (*goal: ‹(x::'a) = (y::'a)›*)\n using \"1\" (*‹xs ! (THE i. xs ! i = x ∧ i < n) = x ∧ (THE i. xs ! i = x ∧ i < n) < n›*) \"2\" (*‹xs ! (THE i. xs ! i = y ∧ i < n) = y ∧ (THE i. xs ! i = y ∧ i < n) < n›*) d (*‹distinct xs›*) xy (*‹(THE i. xs ! i = x ∧ i < n) = (THE i. xs ! i = y ∧ i < n)›*) by argo\n qed\n thus \"?thesis\"\n (*goal: ‹inj_on (λx::'a::comm_ring_1. THE i::nat. (xs::'a::comm_ring_1 list) ! i = x ∧ i < (n::nat)) (set xs)›*)\n unfolding inj_on_def\n (*goal: ‹∀x∈set xs. ∀y∈set xs. (THE i. xs ! i = x ∧ i < n) = (THE i. xs ! i = y ∧ i < n) ⟶ x = y›*)\n by auto\n qed\n show \"(λx. THE i. xs ! i = x ∧ i < n) ` set xs = {0..<n}\"\n proof (auto)\n (*goals:\n 1. ‹⋀xa. xa ∈ set xs ⟹ (THE i. xs ! i = xa ∧ i < n) < n›\n 2. ‹⋀x. x < n ⟹ x ∈ (λx. THE i. xs ! i = x ∧ i < n) ` set xs›*)\n fix xa\n assume xa: \"xa ∈ set xs\" (*‹(xa::'a) ∈ set (xs::'a list)›*)\n let ?i = \"(THE i. xs ! i = xa ∧ i < n)\"\n obtain i where xs_i: \"xs ! i = xa ∧ i < n\"\n (*goal: ‹(⋀i::nat. (xs::'a list) ! i = (xa::'a) ∧ i < (n::nat) ⟹ thesis::bool) ⟹ thesis›*)\n using xa (*‹xa ∈ set xs›*) by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*))\n from this (*‹xs ! i = xa ∧ i < n›*) have 1: \"xs ! ?i = xa ∧ ?i < n\"\n apply (rule theI (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ?P (THE x. ?P x)›*))\n (*goal: ‹xs ! (THE i. xs ! i = xa ∧ i < n) = xa ∧ (THE i. xs ! i = xa ∧ i < n) < n›*)\n apply (insert d (*‹distinct xs›*) xs_i (*‹xs ! i = xa ∧ i < n›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) nth_eq_iff_index_eq (*‹⟦distinct ?xs; ?i < length ?xs; ?j < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*))\n (*goal: ‹⋀ia. xs ! ia = xa ∧ ia < n ⟹ ia = i›*)\n by fastforce\n thus \"(THE i. xs ! i = xa ∧ i < n) < n\"\n by simp\n next\n (*goal: ‹⋀x. x < n ⟹ x ∈ (λx. THE i. xs ! i = x ∧ i < n) ` set xs›*)\n fix x\n assume x: \"x<n\" (*‹(x::nat) < (n::nat)›*)\n have \"∃xa∈set xs. x = (THE i. xs ! i = xa ∧ i < n)\"\n apply (rule bexI[of _ \"xs ! x\"] (*‹⟦?P (xs ! x); xs ! x ∈ ?A⟧ ⟹ ∃x∈?A. ?P x›*))\n (*goals:\n 1. ‹x = (THE i. xs ! i = xs ! x ∧ i < n)›\n 2. ‹xs ! x ∈ set xs›\n discuss goal 1*)\n apply (rule the_equality[symmetric] (*‹⟦?P ?t; ⋀x. ?P x ⟹ x = ?t⟧ ⟹ ?t = (THE x. ?P x)›*))\n (*goals:\n 1. ‹xs ! x = xs ! x ∧ x < n›\n 2. ‹⋀i. xs ! i = xs ! x ∧ i < n ⟹ i = x›\n discuss goal 1*)\n apply ((insert x (*‹x < n›*) d (*‹distinct xs›*))[1])\n (*top goal: ‹xs ! x = xs ! x ∧ x < n› and 2 goals remain*)\n apply ((auto simp add: length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) nth_eq_iff_index_eq (*‹⟦distinct ?xs; ?i < length ?xs; ?j < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*))[1])\n (*discuss goal 2*)\n apply ((insert x (*‹x < n›*) d (*‹distinct xs›*))[1])\n (*top goal: ‹⋀i. xs ! i = xs ! x ∧ i < n ⟹ i = x› and 1 goal remains*)\n apply ((auto simp add: length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) nth_eq_iff_index_eq (*‹⟦distinct ?xs; ?i < length ?xs; ?j < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*))[1])\n (*proven 2 subgoals*)\n (*discuss goal 2*)\n apply ((insert x (*‹x < n›*) d (*‹distinct xs›*))[1])\n (*goal: ‹xs ! x ∈ set xs›*)\n apply ((auto simp add: length_eq_card (*‹length (xs::'a list) = card (S::'a set)›*) n_def (*‹(n::nat) = card (S::'a set)›*) nth_eq_iff_index_eq (*‹⟦distinct (?xs::?'a list); (?i::nat) < length ?xs; (?j::nat) < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*))[1])\n (*proven 2 subgoals*) .\n thus \"x ∈ (λx. THE i. xs ! i = x ∧ i < n) ` set xs\"\n unfolding image_def\n (*goal: ‹x ∈ {y. ∃x∈set xs. y = (THE i. xs ! i = x ∧ i < n)}›*)\n by auto\n qed\n qed\n have i: \"ideal_generated {d} = ideal_generated S\"\n proof (-)\n (*goal: ‹ideal_generated {d} = ideal_generated S›*)\n have ideal_S_explicit: \"ideal_generated S = {y. ∃f. (∑i∈S. f i * i) = y}\"\n unfolding ideal_explicit2[OF fin_S]\n (*goal: ‹{y. ∃f. (∑i∈S. f i * i) = y} = {y. ∃f. (∑i∈S. f i * i) = y}›*)\n by simp\n have \"ideal_generated {d} ⊆ ideal_generated S\"\n apply (rule ideal_generated_subset2 (*‹∀b∈?B. b ∈ ideal_generated ?A ⟹ ideal_generated ?B ⊆ ideal_generated ?A›*))\n (*goal: ‹ideal_generated {d::'a::comm_ring_1} ⊆ ideal_generated (S::'a::comm_ring_1 set)›*)\n proof (auto simp add: ideal_S_explicit (*‹ideal_generated S = {y. ∃f. (∑i∈S. f i * i) = y}›*))\n (*goal: ‹∃f. (∑i∈S. f i * i) = d›*)\n have n: \"dim_vec (col Q 0) = n\"\n using Q (*‹Q ∈ carrier_mat n n›*) n_def (*‹(n::nat) = card (S::'a set)›*) by auto\n have aux: \"Matrix.row A 0 $v i = xs ! i\" if i: \"i<n\" for i\n proof (-)\n (*goal: ‹Matrix.row (A::'a::comm_ring_1 mat) (0::nat) $v (i::nat) = (xs::'a::comm_ring_1 list) ! i›*)\n have i2: \"i < dim_col A\"\n by (simp add: A_def (*‹A = mat_of_rows n [vec_of_list xs]›*) i (*‹i < n›*))\n have \"Matrix.row A 0 $v i = A $$ (0,i)\"\n apply (rule index_row( (*‹⟦?i < dim_row ?A; ?j < dim_col ?A⟧ ⟹ Matrix.row ?A ?i $v ?j = ?A $$ (?i, ?j)›*) 1))\n (*goals:\n 1. ‹0 < dim_row A›\n 2. ‹i < dim_col A›\n discuss goal 1*)\n apply ((auto simp add: a (*‹0 < dim_row A›*) b (*‹0 < dim_col A›*) i2 (*‹i < dim_col A›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: a (*‹0 < dim_row A›*) b (*‹0 < dim_col A›*) i2 (*‹i < dim_col A›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹Matrix.row (A::'a::comm_ring_1 mat) (0::nat) $v (i::nat) = A $$ (0::nat, i)›*) have \"... = [vec_of_list xs] ! 0 $v i\"\n unfolding A_def\n (*goal: ‹mat_of_rows (n::nat) [vec_of_list (xs::'a::comm_ring_1 list)] $$ (0::nat, i::nat) = [vec_of_list xs] ! (0::nat) $v i›*)\n apply (rule mat_of_rows_index (*‹⟦(?i::nat) < length (?rs::?'a Matrix.vec list); (?j::nat) < (?n::nat)⟧ ⟹ mat_of_rows ?n ?rs $$ (?i, ?j) = ?rs ! ?i $v ?j›*))\n (*goals:\n 1. ‹0 < length [vec_of_list xs]›\n 2. ‹i < n›\n discuss goal 1*)\n apply ((auto simp add: i (*‹(i::nat) < (n::nat)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: i (*‹(i::nat) < (n::nat)›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹Matrix.row A 0 $v i = [vec_of_list xs] ! 0 $v i›*) have \"... = xs ! i\"\n by (simp add: vec_of_list_index (*‹vec_of_list ?xs $v ?j = ?xs ! ?j›*))\n finally (*calculation: ‹Matrix.row A 0 $v i = xs ! i›*) show \"?thesis\"\n (*goal: ‹Matrix.row (A::'a mat) (0::nat) $v (i::nat) = (xs::'a list) ! i›*) .\n qed\n let ?f = \"λx. let i = (THE i. xs ! i = x ∧ i < n) in col Q 0 $v i\"\n let ?g = \"(λi. xs ! i * col Q 0 $v i)\"\n have \"d = (A*Q) $$ (0,0)\"\n unfolding d_def\n (*goal: ‹(A * Q) $$ (0, 0) = (A * Q) $$ (0, 0)›*)\n by simp\n also (*calculation: ‹d = (A * Q) $$ (0, 0)›*) have \"... = Matrix.row A 0 ∙ col Q 0\"\n by (rule index_mult_mat(1)[OF a q0] (*‹(A * Q) $$ (0, 0) = Matrix.row A 0 ∙ col Q 0›*))\n also (*calculation: ‹d = Matrix.row A 0 ∙ col Q 0›*) have \"... = (∑i = 0..<dim_vec (col Q 0). Matrix.row A 0 $v i * col Q 0 $v i)\"\n unfolding scalar_prod_def\n (*goal: ‹(∑i = 0..<dim_vec (col Q 0). Matrix.row A 0 $v i * col Q 0 $v i) = (∑i = 0..<dim_vec (col Q 0). Matrix.row A 0 $v i * col Q 0 $v i)›*)\n by simp\n also (*calculation: ‹d = (∑i = 0..<dim_vec (col Q 0). Matrix.row A 0 $v i * col Q 0 $v i)›*) have \"... = (∑i = 0..<n. Matrix.row A 0 $v i * col Q 0 $v i)\"\n unfolding n\n (*goal: ‹(∑i::nat = 0::nat..<n::nat. Matrix.row (A::'a mat) (0::nat) $v i * col (Q::'a mat) (0::nat) $v i) = (∑i::nat = 0::nat..<n. Matrix.row A (0::nat) $v i * col Q (0::nat) $v i)›*)\n by auto\n also (*calculation: ‹d = (∑i = 0..<n. Matrix.row A 0 $v i * col Q 0 $v i)›*) have \"... = (∑i = 0..<n. xs ! i * col Q 0 $v i)\"\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0..<n} = {0..<n}›\n 2. ‹⋀x. x ∈ {0..<n} ⟹ Matrix.row A 0 $v x * col Q 0 $v x = xs ! x * col Q 0 $v x›\n discuss goal 1*)\n apply ((auto simp add: aux (*‹?i < n ⟹ Matrix.row A 0 $v ?i = xs ! ?i›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: aux (*‹?i < n ⟹ Matrix.row A 0 $v ?i = xs ! ?i›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹d = (∑i = 0..<n. xs ! i * col Q 0 $v i)›*) have \"... = (∑x ∈ set xs. ?g (?h x))\"\n by (rule sum.reindex_bij_betw[symmetric, OF bij] (*‹sum ?g {0..<n} = (∑x∈set xs. ?g (THE i. xs ! i = x ∧ i < n))›*))\n also (*calculation: ‹d = (∑x∈set xs. xs ! (THE i. xs ! i = x ∧ i < n) * col Q 0 $v (THE i. xs ! i = x ∧ i < n))›*) have \"... = (∑x ∈ set xs. ?f x * x)\"\n proof (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*), auto simp add: Let_def (*‹Let ?s ?f ≡ ?f ?s›*))\n (*goal: ‹⋀x. x ∈ set xs ⟹ xs ! (THE i. xs ! i = x ∧ i < n) * col Q 0 $v (THE i. xs ! i = x ∧ i < n) = col Q 0 $v (THE i. xs ! i = x ∧ i < n) * x›*)\n fix x\n assume x: \"x ∈ set xs\" (*‹(x::'a) ∈ set (xs::'a list)›*)\n let ?i = \"(THE i. xs ! i = x ∧ i < n)\"\n obtain i where xs_i: \"xs ! i = x ∧ i < n\"\n (*goal: ‹(⋀i. xs ! i = x ∧ i < n ⟹ thesis) ⟹ thesis›*)\n by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*) x (*‹x ∈ set xs›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*))\n from this (*‹xs ! i = x ∧ i < n›*) have \"xs ! ?i = x ∧ ?i < n\"\n apply (rule theI (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ ?P (THE x. ?P x)›*))\n (*goal: ‹(xs::'a::comm_ring_1 list) ! (THE i::nat. xs ! i = (x::'a::comm_ring_1) ∧ i < (n::nat)) = x ∧ (THE i::nat. xs ! i = x ∧ i < n) < n›*)\n apply (insert d (*‹distinct xs›*) xs_i (*‹xs ! i = x ∧ i < n›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) nth_eq_iff_index_eq (*‹⟦distinct ?xs; ?i < length ?xs; ?j < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*))\n (*goal: ‹⋀ia. xs ! ia = x ∧ ia < n ⟹ ia = i›*)\n by fastforce\n thus \"xs ! ?i * col Q 0 $v ?i = col Q 0 $v ?i * x\"\n by auto\n qed\n also (*calculation: ‹d = (∑x∈set xs. Let (THE i. xs ! i = x ∧ i < n) (($v) (col Q 0)) * x)›*) have \"... = (∑x ∈ S. ?f x * x)\"\n using set_xs (*‹set (xs::'a list) = (S::'a set)›*) by auto\n finally (*calculation: ‹(d::'a::comm_ring_1) = (∑x::'a::comm_ring_1∈(S::'a::comm_ring_1 set). Let (THE i::nat. (xs::'a::comm_ring_1 list) ! i = x ∧ i < (n::nat)) (($v) (col (Q::'a::comm_ring_1 mat) (0::nat))) * x)›*) show \"∃f. (∑i∈S. f i * i) = d\"\n by auto\n qed\n moreover have \"ideal_generated S ⊆ ideal_generated {d}\"\n proof (standard)\n (*goal: ‹⋀x. x ∈ ideal_generated S ⟹ x ∈ ideal_generated {d}›*)\n fix x\n assume x: \"x ∈ ideal_generated S\" (*‹(x::'a) ∈ ideal_generated (S::'a set)›*)\n thm Matrix.diag_mat_def\n hence x_xs: \"x ∈ ideal_generated (set xs)\"\n by (simp add: set_xs (*‹set (xs::'a list) = (S::'a set)›*))\n from this (*‹x ∈ ideal_generated (set xs)›*) obtain f where f: \"(∑i∈(set xs). f i * i) = x\"\n (*goal: ‹(⋀f. (∑i∈set xs. f i * i) = x ⟹ thesis) ⟹ thesis›*)\n using x (*‹x ∈ ideal_generated S›*) ideal_explicit2 (*‹finite ?S ⟹ ideal_generated ?S = {y. ∃f. (∑i∈?S. f i * i) = y}›*) by auto\n define B where \"B = Matrix.vec n (λi. f (A $$ (0,i)))\"\n have B: \"B ∈ carrier_vec n\"\n unfolding B_def\n (*goal: ‹Matrix.vec n (λi. f (A $$ (0, i))) ∈ carrier_vec n›*)\n by auto\n have \"(A *⇩v B) $v 0 = Matrix.row A 0 ∙ B\"\n by (rule index_mult_mat_vec[OF a] (*‹(A *⇩v ?v) $v 0 = Matrix.row A 0 ∙ ?v›*))\n also (*calculation: ‹(A *⇩v B) $v 0 = Matrix.row A 0 ∙ B›*) have \"... = sum (λi. f (A $$ (0,i)) * A $$ (0,i)) {0..<n}\"\n unfolding B_def Matrix.row_def scalar_prod_def\n (*goal: ‹(∑i::nat = 0::nat..<dim_vec (Matrix.vec n (λi::nat. f (A $$ (0::nat, i)))). Matrix.vec (dim_col (A::'a mat)) (λj::nat. A $$ (0::nat, j)) $v i * Matrix.vec (n::nat) (λi::nat. (f::'a ⇒ 'a) (A $$ (0::nat, i))) $v i) = (∑i::nat = 0::nat..<n. f (A $$ (0::nat, i)) * A $$ (0::nat, i))›*)\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0..<dim_vec (Matrix.vec n (λi. f (A $$ (0, i))))} = {0..<n}›\n 2. ‹⋀x. x ∈ {0..<n} ⟹ Matrix.vec (dim_col A) (λj. A $$ (0, j)) $v x * Matrix.vec n (λi. f (A $$ (0, i))) $v x = f (A $$ (0, x)) * A $$ (0, x)›\n discuss goal 1*)\n apply ((auto simp add: A_def (*‹A = mat_of_rows n [vec_of_list xs]›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: A_def (*‹A = mat_of_rows n [vec_of_list xs]›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(A *⇩v B) $v 0 = (∑i = 0..<n. f (A $$ (0, i)) * A $$ (0, i))›*) have \"... = sum (λi. f i * i) (set xs)\"\n proof (rule sum.reindex_bij_betw (*‹bij_betw ?h ?S ?T ⟹ (∑x∈?S. ?g (?h x)) = sum ?g ?T›*))\n (*goal: ‹bij_betw (λx::nat. (A::'a mat) $$ (0::nat, x)) {0::nat..<n::nat} (set (xs::'a list))›*)\n have 1: \"inj_on (λx. A $$ (0, x)) {0..<n}\"\n apply (unfold inj_on_def (*‹inj_on (?f::?'a ⇒ ?'b) (?A::?'a set) = (∀x::?'a∈?A. ∀y::?'a∈?A. ?f x = ?f y ⟶ x = y)›*))\n (*goal: ‹inj_on (λx. A $$ (0, x)) {0..<n}›*)\n proof (auto)\n (*goal: ‹⋀(x::nat) y::nat. ⟦x < (n::nat); y < n; (A::'a mat) $$ (0::nat, x) = A $$ (0::nat, y)⟧ ⟹ x = y›*)\n fix x and y\n assume x: \"x < n\" and y: \"y < n\" and xy: \"A $$ (0, x) = A $$ (0, y)\" (*‹(x::nat) < (n::nat)› ‹(y::nat) < (n::nat)› ‹(A::'a mat) $$ (0::nat, x::nat) = A $$ (0::nat, y::nat)›*)\n have \"A $$ (0,x) = [vec_of_list xs] ! 0 $v x\"\n unfolding A_def\n (*goal: ‹mat_of_rows n [vec_of_list xs] $$ (0, x) = [vec_of_list xs] ! 0 $v x›*)\n apply (rule mat_of_rows_index (*‹⟦(?i::nat) < length (?rs::?'a::type Matrix.vec list); (?j::nat) < (?n::nat)⟧ ⟹ mat_of_rows ?n ?rs $$ (?i, ?j) = ?rs ! ?i $v ?j›*))\n (*goals:\n 1. ‹0 < length [vec_of_list xs]›\n 2. ‹x < n›\n discuss goal 1*)\n apply ((insert x (*‹(x::nat) < (n::nat)›*) y (*‹(y::nat) < (n::nat)›*))[1])\n (*top goal: ‹0 < length [vec_of_list xs]› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert x (*‹x < n›*) y (*‹y < n›*))[1])\n (*goal: ‹x < n›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹A $$ (0, x) = [vec_of_list xs] ! 0 $v x›*) have \"... = xs ! x\"\n using x (*‹x < n›*) by (simp add: vec_of_list_index (*‹vec_of_list (?xs::?'a list) $v (?j::nat) = ?xs ! ?j›*))\n finally (*calculation: ‹A $$ (0, x) = xs ! x›*) have 1: \"A $$ (0,x) = xs ! x\" .\n have \"A $$ (0,y) = [vec_of_list xs] ! 0 $v y\"\n unfolding A_def\n (*goal: ‹mat_of_rows n [vec_of_list xs] $$ (0, y) = [vec_of_list xs] ! 0 $v y›*)\n apply (rule mat_of_rows_index (*‹⟦?i < length ?rs; ?j < ?n⟧ ⟹ mat_of_rows ?n ?rs $$ (?i, ?j) = ?rs ! ?i $v ?j›*))\n (*goals:\n 1. ‹0 < length [vec_of_list xs]›\n 2. ‹y < n›\n discuss goal 1*)\n apply ((insert x (*‹x < n›*) y (*‹y < n›*))[1])\n (*top goal: ‹0 < length [vec_of_list xs]› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert x (*‹x < n›*) y (*‹y < n›*))[1])\n (*goal: ‹y < n›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(A::'a mat) $$ (0::nat, y::nat) = [vec_of_list (xs::'a list)] ! (0::nat) $v y›*) have \"... = xs ! y\"\n using y (*‹y < n›*) by (simp add: vec_of_list_index (*‹vec_of_list ?xs $v ?j = ?xs ! ?j›*))\n finally (*calculation: ‹A $$ (0, y) = xs ! y›*) have 2: \"A $$ (0,y) = xs ! y\" .\n show \"x = y\"\n using \"1\" (*‹A $$ (0, x) = xs ! x›*) \"2\" (*‹A $$ (0, y) = xs ! y›*) x (*‹x < n›*) y (*‹y < n›*) d (*‹distinct xs›*) length_eq_card (*‹length (xs::'a list) = card (S::'a set)›*) n_def (*‹n = card S›*) nth_eq_iff_index_eq (*‹⟦distinct ?xs; ?i < length ?xs; ?j < length ?xs⟧ ⟹ (?xs ! ?i = ?xs ! ?j) = (?i = ?j)›*) xy (*‹(A::'a::comm_ring_1 mat) $$ (0::nat, x::nat) = A $$ (0::nat, y::nat)›*) by fastforce\n qed\n have 2: \"A $$ (0, xa) ∈ set xs\" if xa: \"xa < n\" for xa\n proof (-)\n (*goal: ‹A $$ (0, xa) ∈ set xs›*)\n have \"A $$ (0,xa) = [vec_of_list xs] ! 0 $v xa\"\n unfolding A_def\n (*goal: ‹mat_of_rows n [vec_of_list xs] $$ (0, xa) = [vec_of_list xs] ! 0 $v xa›*)\n apply (rule mat_of_rows_index (*‹⟦?i < length ?rs; ?j < ?n⟧ ⟹ mat_of_rows ?n ?rs $$ (?i, ?j) = ?rs ! ?i $v ?j›*))\n (*goals:\n 1. ‹0 < length [vec_of_list xs]›\n 2. ‹xa < n›\n discuss goal 1*)\n apply ((insert xa (*‹xa < n›*))[1])\n (*top goal: ‹0 < length [vec_of_list xs]› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert xa (*‹(xa::nat) < (n::nat)›*))[1])\n (*goal: ‹xa < n›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹A $$ (0, xa) = [vec_of_list xs] ! 0 $v xa›*) have \"... = xs ! xa\"\n using xa (*‹xa < n›*) by (simp add: vec_of_list_index (*‹vec_of_list (?xs::?'a list) $v (?j::nat) = ?xs ! ?j›*))\n finally (*calculation: ‹(A::'a::comm_ring_1 mat) $$ (0::nat, xa::nat) = (xs::'a::comm_ring_1 list) ! xa›*) show \"?thesis\"\n (*goal: ‹A $$ (0, xa) ∈ set xs›*)\n using xa (*‹xa < n›*) by (simp add: length_eq_card (*‹length (xs::'a list) = card (S::'a set)›*) n_def (*‹(n::nat) = card (S::'a set)›*))\n qed\n have 3: \"x ∈ (λx. A $$ (0, x)) ` {0..<n}\" if x: \"x∈ set xs\" for x\n proof (-)\n (*goal: ‹(x::'a) ∈ (λx::nat. (A::'a mat) $$ (0::nat, x)) ` {0::nat..<n::nat}›*)\n obtain i where xs: \"xs ! i = x ∧ i < n\"\n (*goal: ‹(⋀i. xs ! i = x ∧ i < n ⟹ thesis) ⟹ thesis›*)\n by (metis in_set_conv_nth (*‹(?x ∈ set ?xs) = (∃i<length ?xs. ?xs ! i = ?x)›*) length_eq_card (*‹length xs = card S›*) n_def (*‹n = card S›*) x (*‹x ∈ set xs›*))\n have \"A $$ (0,i) = [vec_of_list xs] ! 0 $v i\"\n unfolding A_def\n (*goal: ‹mat_of_rows n [vec_of_list xs] $$ (0, i) = [vec_of_list xs] ! 0 $v i›*)\n apply (rule mat_of_rows_index (*‹⟦?i < length ?rs; ?j < ?n⟧ ⟹ mat_of_rows ?n ?rs $$ (?i, ?j) = ?rs ! ?i $v ?j›*))\n (*goals:\n 1. ‹0 < length [vec_of_list xs]›\n 2. ‹i < n›\n discuss goal 1*)\n apply ((insert xs (*‹(xs::'a list) ! (i::nat) = (x::'a) ∧ i < (n::nat)›*))[1])\n (*top goal: ‹0 < length [vec_of_list xs]› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert xs (*‹xs ! i = x ∧ i < n›*))[1])\n (*goal: ‹i < n›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹A $$ (0, i) = [vec_of_list xs] ! 0 $v i›*) have \"... = xs ! i\"\n using xs (*‹xs ! i = x ∧ i < n›*) by (simp add: vec_of_list_index (*‹vec_of_list ?xs $v ?j = ?xs ! ?j›*))\n finally (*calculation: ‹A $$ (0, i) = xs ! i›*) show \"?thesis\"\n (*goal: ‹x ∈ (λx. A $$ (0, x)) ` {0..<n}›*)\n using xs (*‹xs ! i = x ∧ i < n›*) unfolding image_def\n (*goal: ‹x ∈ {y. ∃x∈{0..<n}. y = A $$ (0, x)}›*)\n by auto\n qed\n show \"bij_betw (λx. A $$ (0, x)) {0..<n} (set xs)\"\n using \"1\" (*‹inj_on (λx. A $$ (0, x)) {0..<n}›*) \"2\" (*‹?xa < n ⟹ A $$ (0, ?xa) ∈ set xs›*) \"3\" (*‹?x ∈ set xs ⟹ ?x ∈ (λx. A $$ (0, x)) ` {0..<n}›*) unfolding bij_betw_def\n (*goal: ‹inj_on (λx. A $$ (0, x)) {0..<n} ∧ (λx. A $$ (0, x)) ` {0..<n} = set xs›*)\n by auto\n qed\n finally (*calculation: ‹(A *⇩v B) $v 0 = (∑i∈set xs. f i * i)›*) have AB00_sum: \"(A *⇩v B) $v 0 = sum (λi. f i * i) (set xs)\"\n by auto\n hence AB_00_x: \"(A *⇩v B) $v 0 = x\"\n using f (*‹(∑i∈set xs. f i * i) = x›*) by auto\n obtain Q' where QQ': \"inverts_mat Q Q'\" and Q'Q: \"inverts_mat Q' Q\" and Q': \"Q' ∈ carrier_mat n n\"\n (*goal: ‹(⋀Q'. ⟦inverts_mat Q Q'; inverts_mat Q' Q; Q' ∈ carrier_mat n n⟧ ⟹ thesis) ⟹ thesis›*)\n apply (rule obtain_inverse_matrix[OF Q inv_Q] (*‹(⋀B. ⟦inverts_mat Q B; inverts_mat B Q; B ∈ carrier_mat n n⟧ ⟹ ?thesis) ⟹ ?thesis›*))\n (*goal: ‹(⋀Q'. ⟦inverts_mat Q Q'; inverts_mat Q' Q; Q' ∈ carrier_mat n n⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n have eq: \"A = (A*Q)*Q'\"\n using QQ' (*‹inverts_mat Q Q'›*) unfolding inverts_mat_def\n (*goal: ‹(A::'a mat) = A * (Q::'a mat) * (Q'::'a mat)›*)\n by (metis A (*‹(A::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*) Q (*‹(Q::'a mat) ∈ carrier_mat (n::nat) n›*) Q' (*‹(Q'::'a mat) ∈ carrier_mat (n::nat) n›*) assoc_mult_mat (*‹⟦(?A::?'a mat) ∈ carrier_mat (?n₁::nat) (?n₂::nat); (?B::?'a mat) ∈ carrier_mat ?n₂ (?n₃::nat); (?C::?'a mat) ∈ carrier_mat ?n₃ (?n₄::nat)⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) carrier_matD( (*‹(?A::?'a mat) ∈ carrier_mat (?nr::nat) (?nc::nat) ⟹ dim_row ?A = ?nr›*) 1) right_mult_one_mat (*‹(?A::?'a mat) ∈ carrier_mat (?nr::nat) (?nc::nat) ⟹ ?A * 1⇩m ?nc = ?A›*))\n let ?g = \"λi. Matrix.row (A * Q) 0 $v i * (Matrix.row Q' i ∙ B)\"\n have sum0: \"(∑i = 1..<n. ?g i) = 0\"\n apply (rule sum.neutral (*‹∀x∈?A. ?g x = 0 ⟹ sum ?g ?A = 0›*))\n (*goal: ‹(∑i = 1..<n. Matrix.row (A * Q) 0 $v i * (Matrix.row Q' i ∙ B)) = 0›*)\n proof (rule)\n (*goal: ‹⋀x. x ∈ {1..<n} ⟹ Matrix.row (A * Q) 0 $v x * (Matrix.row Q' x ∙ B) = 0›*)\n fix x\n assume x: \"x ∈ {1..<n}\" (*‹(x::nat) ∈ {1::nat..<n::nat}›*)\n hence \"Matrix.row (A * Q) 0 $v x = 0\"\n using t_AQ (*‹lower_triangular (A * Q)›*) unfolding lower_triangular_def\n (*goal: ‹Matrix.row (A * Q) 0 $v x = 0›*)\n apply -\n (*goal: ‹Matrix.row (A * Q) 0 $v x = 0›*)\n by (msorry)\n thus \"Matrix.row (A * Q) 0 $v x * (Matrix.row Q' x ∙ B) = 0\"\n by simp\n qed\n have set_rw: \"{0..<n} - {0} = {1..<n}\"\n by (simp add: atLeast0LessThan (*‹{0..<?n} = {..<?n}›*) atLeast1_lessThan_eq_remove0 (*‹{Suc 0..<?n} = {..<?n} - {0}›*))\n have mat_rw: \"(A*Q*Q')*⇩v B = A*Q*⇩v(Q' *⇩v B)\"\n apply (rule assoc_mult_mat_vec (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?v ∈ carrier_vec ?n₃⟧ ⟹ ?A * ?B *⇩v ?v = ?A *⇩v (?B *⇩v ?v)›*))\n (*goals:\n 1. ‹(A::'a mat) * (Q::'a mat) ∈ carrier_mat (?n₁::nat) (?n₂::nat)›\n 2. ‹(Q'::'a mat) ∈ carrier_mat (?n₂::nat) (?n₃::nat)›\n 3. ‹(B::'a Matrix.vec) ∈ carrier_vec (?n₃::nat)›\n discuss goal 1*)\n apply ((insert Q (*‹Q ∈ carrier_mat n n›*) Q' (*‹Q' ∈ carrier_mat n n›*) B (*‹B ∈ carrier_vec n›*) AQ (*‹A * Q ∈ carrier_mat 1 n›*))[1])\n (*top goal: ‹A * Q ∈ carrier_mat ?n₁ ?n₂› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert Q (*‹(Q::'a mat) ∈ carrier_mat (n::nat) n›*) Q' (*‹(Q'::'a mat) ∈ carrier_mat (n::nat) n›*) B (*‹(B::'a Matrix.vec) ∈ carrier_vec (n::nat)›*) AQ (*‹(A::'a mat) * (Q::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*))[1])\n (*top goal: ‹Q' ∈ carrier_mat n ?n₃› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert Q (*‹(Q::'a mat) ∈ carrier_mat (n::nat) n›*) Q' (*‹(Q'::'a mat) ∈ carrier_mat (n::nat) n›*) B (*‹(B::'a Matrix.vec) ∈ carrier_vec (n::nat)›*) AQ (*‹(A::'a mat) * (Q::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*))[1])\n (*goal: ‹B ∈ carrier_vec n›*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n from eq (*‹A = A * Q * Q'›*) have \"A *⇩vB = (A*Q)*⇩v(Q'*⇩v B)\"\n using mat_rw (*‹A * Q * Q' *⇩v B = A * Q *⇩v (Q' *⇩v B)›*) by auto\n from this (*‹A *⇩v B = A * Q *⇩v (Q' *⇩v B)›*) have \"(A *⇩v B) $v 0 = (A * Q *⇩v (Q' *⇩v B)) $v 0\"\n by auto\n also (*calculation: ‹(A *⇩v B) $v 0 = (A * Q *⇩v (Q' *⇩v B)) $v 0›*) have \"... = Matrix.row (A*Q) 0 ∙ (Q' *⇩v B)\"\n apply (rule index_mult_mat_vec (*‹?i < dim_row ?A ⟹ (?A *⇩v ?v) $v ?i = Matrix.row ?A ?i ∙ ?v›*))\n (*goal: ‹(A * Q *⇩v (Q' *⇩v B)) $v 0 = Matrix.row (A * Q) 0 ∙ (Q' *⇩v B)›*)\n apply (insert a (*‹0 < dim_row A›*) B_def (*‹B = Matrix.vec n (λi. f (A $$ (0, i)))›*) n0 (*‹0 < n›*))\n (*goal: ‹0 < dim_row (A * Q)›*)\n by auto\n also (*calculation: ‹(A *⇩v B) $v 0 = Matrix.row (A * Q) 0 ∙ (Q' *⇩v B)›*) have \"... = (∑i = 0..<n. ?g i)\"\n using Q' (*‹(Q'::'a mat) ∈ carrier_mat (n::nat) n›*) by (auto simp add: scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $v i * ?w $v i›*))\n also (*calculation: ‹(A *⇩v B) $v 0 = (∑i = 0..<n. Matrix.row (A * Q) 0 $v i * (Matrix.row Q' i ∙ B))›*) have \"... = ?g 0 + (∑i ∈ {0..<n} - {0}. ?g i)\"\n by (metis (no_types, lifting) Q (*‹Q ∈ carrier_mat n n›*) atLeast0LessThan (*‹{0..<?n} = {..<?n}›*) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_col ?A = ?nc›*) 2) finite_atLeastLessThan (*‹finite {?l..<?u}›*) lessThan_iff (*‹(?i ∈ {..<?k}) = (?i < ?k)›*) q0 (*‹0 < dim_col Q›*) sum.remove (*‹⟦finite ?A; ?x ∈ ?A⟧ ⟹ sum ?g ?A = ?g ?x + sum ?g (?A - {?x})›*))\n also (*calculation: ‹(A *⇩v B) $v 0 = Matrix.row (A * Q) 0 $v 0 * (Matrix.row Q' 0 ∙ B) + (∑i∈{0..<n} - {0}. Matrix.row (A * Q) 0 $v i * (Matrix.row Q' i ∙ B))›*) have \"... = ?g 0 + (∑i = 1..<n. ?g i)\"\n using set_rw (*‹{0..<n} - {0} = {1..<n}›*) by simp\n also (*calculation: ‹(A *⇩v B) $v 0 = Matrix.row (A * Q) 0 $v 0 * (Matrix.row Q' 0 ∙ B) + (∑i = 1..<n. Matrix.row (A * Q) 0 $v i * (Matrix.row Q' i ∙ B))›*) have \"... = ?g 0\"\n using sum0 (*‹(∑i::nat = 1::nat..<n::nat. Matrix.row ((A::'a mat) * (Q::'a mat)) (0::nat) $v i * (Matrix.row (Q'::'a mat) i ∙ (B::'a Matrix.vec))) = (0::'a)›*) by auto\n also (*calculation: ‹(A *⇩v B) $v 0 = Matrix.row (A * Q) 0 $v 0 * (Matrix.row Q' 0 ∙ B)›*) have \"... = d * (Matrix.row Q' 0 ∙ B)\"\n by (simp add: a (*‹0 < dim_row A›*) d_def (*‹d = (A * Q) $$ (0, 0)›*) q0 (*‹0 < dim_col Q›*))\n finally (*calculation: ‹(A *⇩v B) $v 0 = d * (Matrix.row Q' 0 ∙ B)›*) show \"x ∈ ideal_generated {d}\"\n using AB_00_x (*‹(A *⇩v B) $v 0 = x›*) unfolding ideal_generated_singleton\n (*goal: ‹(x::'a) ∈ {k * (d::'a) |k::'a. k ∈ UNIV}›*)\n using mult.commute (*‹?a * ?b = ?b * ?a›*) by auto\n qed\n ultimately show \"?thesis\"\n (*goal: ‹ideal_generated {d} = ideal_generated S›*)\n by auto\n qed\n thus \"principal_ideal I\"\n unfolding principal_ideal_def ig_S\n (*goal: ‹∃a::'a. ideal_generated {a} = (I::'a set)›*)\n by blast\n qed\nqed\n\nsubsection ‹Elementary divisor ring implies Hermite ring›\n\ncontext\n assumes \"SORT_CONSTRAINT('a::comm_ring_1)\"\nbegin\n\n\nlemma triangularizable_m0:\nassumes A: \"A ∈ carrier_mat m 0\" \nshows \"∃U. U ∈ carrier_mat 0 0 ∧ invertible_mat U ∧ lower_triangular (A * U)\"\n using A (*‹(A::'b mat) ∈ carrier_mat (m::nat) (0::nat)›*) unfolding lower_triangular_def carrier_mat_def invertible_mat_def inverts_mat_def\n (*goal: ‹∃U. U ∈ {m. dim_row m = 0 ∧ dim_col m = 0} ∧ (square_mat U ∧ (∃B. U * B = 1⇩m (dim_row U) ∧ B * U = 1⇩m (dim_row B))) ∧ (∀i j. i < j ∧ i < dim_row (A * U) ∧ j < dim_col (A * U) ⟶ (A * U) $$ (i, j) = 0)›*)\n apply auto\n (*goal: ‹∃U. U ∈ {m. dim_row m = 0 ∧ dim_col m = 0} ∧ (square_mat U ∧ (∃B. U * B = 1⇩m (dim_row U) ∧ B * U = 1⇩m (dim_row B))) ∧ (∀i j. i < j ∧ i < dim_row (A * U) ∧ j < dim_col (A * U) ⟶ (A * U) $$ (i, j) = 0)›*)\n by (metis gr_implies_not0 (*‹?m < ?n ⟹ ?n ≠ 0›*) index_one_mat( (*‹dim_row (1⇩m ?n) = ?n›*) 2) index_one_mat( (*‹dim_col (1⇩m ?n) = ?n›*) 3) right_mult_one_mat' (*‹dim_col ?A = ?n ⟹ ?A * 1⇩m ?n = ?A›*))\n\nlemma triangularizable_0n:\nassumes A: \"A ∈ carrier_mat 0 n\" \nshows \"∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)\"\n using A (*‹A ∈ carrier_mat 0 n›*) unfolding lower_triangular_def carrier_mat_def invertible_mat_def inverts_mat_def\n (*goal: ‹∃U. U ∈ {m. dim_row m = n ∧ dim_col m = n} ∧ (square_mat U ∧ (∃B. U * B = 1⇩m (dim_row U) ∧ B * U = 1⇩m (dim_row B))) ∧ (∀i j. i < j ∧ i < dim_row (A * U) ∧ j < dim_col (A * U) ⟶ (A * U) $$ (i, j) = 0)›*)\n apply auto\n (*goal: ‹∃U. U ∈ {m. dim_row m = n ∧ dim_col m = n} ∧ (square_mat U ∧ (∃B. U * B = 1⇩m (dim_row U) ∧ B * U = 1⇩m (dim_row B))) ∧ (∀i j. i < j ∧ i < dim_row (A * U) ∧ j < dim_col (A * U) ⟶ (A * U) $$ (i, j) = 0)›*)\n by (metis index_one_mat( (*‹dim_row (1⇩m ?n) = ?n›*) 2) index_one_mat( (*‹dim_col (1⇩m ?n) = ?n›*) 3) right_mult_one_mat' (*‹dim_col ?A = ?n ⟹ ?A * 1⇩m ?n = ?A›*))\n\n\n(*To show this, we have to prove that P is a matrix of one element, which is a unit.*)\nlemma diagonal_imp_triangular_1x2:\n assumes A: \"A ∈ carrier_mat 1 2\" and d: \"admits_diagonal_reduction (A::'a mat)\"\n shows \"admits_triangular_reduction A\"\nproof (-)\n (*goal: ‹admits_triangular_reduction A›*)\n obtain P and Q where P: \"P ∈ carrier_mat (dim_row A) (dim_row A)\" and Q: \"Q ∈ carrier_mat (dim_col A) (dim_col A)\" and inv_P: \"invertible_mat P\" and inv_Q: \"invertible_mat Q\" and SNF: \"Smith_normal_form_mat (P * A * Q)\"\n (*goal: ‹(⋀P Q. ⟦P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A); invertible_mat P; invertible_mat Q; Smith_normal_form_mat (P * A * Q)⟧ ⟹ thesis) ⟹ thesis›*)\n using d (*‹admits_diagonal_reduction A›*) unfolding admits_diagonal_reduction_def\n (*goal: ‹(⋀P Q. ⟦P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A); invertible_mat P; invertible_mat Q; Smith_normal_form_mat (P * A * Q)⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n have \"(P * A * Q) = P * (A * Q)\"\n using P (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) Q (*‹Q ∈ carrier_mat (dim_col A) (dim_col A)›*) assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) by blast\n also (*calculation: ‹(P::'a mat) * (A::'a mat) * (Q::'a mat) = P * (A * Q)›*) have \"... = P $$ (0,0) ⋅⇩m (A * Q)\"\n apply (rule smult_mat_mat_one_element (*‹⟦?A ∈ carrier_mat 1 1; ?B ∈ carrier_mat 1 ?n⟧ ⟹ ?A * ?B = ?A $$ (0, 0) ⋅⇩m ?B›*))\n (*goals:\n 1. ‹P ∈ carrier_mat 1 1›\n 2. ‹A * Q ∈ carrier_mat 1 ?n›\n discuss goal 1*)\n apply ((insert P (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) A (*‹A ∈ carrier_mat 1 2›*) Q (*‹Q ∈ carrier_mat (dim_col A) (dim_col A)›*))[1])\n (*top goal: ‹(P::'a mat) ∈ carrier_mat (1::nat) (1::nat)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert P (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) A (*‹A ∈ carrier_mat 1 2›*) Q (*‹Q ∈ carrier_mat (dim_col A) (dim_col A)›*))[1])\n (*goal: ‹A * Q ∈ carrier_mat 1 ?n›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹P * A * Q = P $$ (0, 0) ⋅⇩m (A * Q)›*) have \"... = A * (P $$ (0,0) ⋅⇩m Q)\"\n using Q (*‹Q ∈ carrier_mat (dim_col A) (dim_col A)›*) by auto\n finally (*calculation: ‹P * A * Q = A * (P $$ (0, 0) ⋅⇩m Q)›*) have eq: \"(P * A * Q) = A * (P $$ (0,0) ⋅⇩m Q)\" .\n have inv: \"invertible_mat (P $$ (0,0) ⋅⇩m Q)\"\n proof (-)\n (*goal: ‹invertible_mat (P $$ (0, 0) ⋅⇩m Q)›*)\n have d: \"Determinant.det P = P $$ (0, 0)\"\n apply (rule determinant_one_element (*‹?A ∈ carrier_mat 1 1 ⟹ Determinant.det ?A = ?A $$ (0, 0)›*))\n (*goal: ‹Determinant.det P = P $$ (0, 0)›*)\n apply (insert P (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) A (*‹A ∈ carrier_mat 1 2›*))\n (*goal: ‹(P::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (1::nat)›*)\n by auto\n from this (*‹Determinant.det P = P $$ (0, 0)›*) have P_dvd_1: \"P $$ (0, 0) dvd 1\"\n using invertible_iff_is_unit_JNF[OF P] (*‹invertible_mat P = (Determinant.det P dvd 1)›*) using inv_P (*‹invertible_mat P›*) by auto\n have Q_dvd_1: \"Determinant.det Q dvd 1\"\n using inv_Q (*‹invertible_mat Q›*) invertible_iff_is_unit_JNF[OF Q] (*‹invertible_mat Q = (Determinant.det Q dvd 1)›*) by simp\n have \"Determinant.det (P $$ (0, 0) ⋅⇩m Q) = P $$ (0, 0) ^ dim_col Q * Determinant.det Q\"\n unfolding det_smult\n (*goal: ‹P $$ (0, 0) ^ dim_col Q * Determinant.det Q = P $$ (0, 0) ^ dim_col Q * Determinant.det Q›*)\n by auto\n also (*calculation: ‹Determinant.det (P $$ (0, 0) ⋅⇩m Q) = P $$ (0, 0) ^ dim_col Q * Determinant.det Q›*) have \"... dvd 1\"\n using P_dvd_1 (*‹P $$ (0, 0) dvd 1›*) Q_dvd_1 (*‹Determinant.det Q dvd 1›*) unfolding is_unit_mult_iff\n (*goal: ‹P $$ (0, 0) ^ dim_col Q dvd 1 ∧ Determinant.det Q dvd 1›*)\n by (metis dvdE (*‹⟦?b dvd ?a; ⋀k. ?a = ?b * k ⟹ ?P⟧ ⟹ ?P›*) dvd_mult_left (*‹?a * ?b dvd ?c ⟹ ?a dvd ?c›*) one_dvd (*‹1 dvd ?a›*) power_mult_distrib (*‹(?a * ?b) ^ ?n = ?a ^ ?n * ?b ^ ?n›*) power_one (*‹1 ^ ?n = 1›*))\n finally (*calculation: ‹Determinant.det (P $$ (0, 0) ⋅⇩m Q) dvd 1›*) have det: \"(Determinant.det (P $$ (0, 0) ⋅⇩m Q) dvd 1)\" .\n have PQ: \"P $$ (0,0) ⋅⇩m Q ∈ carrier_mat 2 2\"\n using A (*‹A ∈ carrier_mat 1 2›*) P (*‹(P::'a::comm_ring_1 mat) ∈ carrier_mat (dim_row (A::'a::comm_ring_1 mat)) (dim_row A)›*) Q (*‹Q ∈ carrier_mat (dim_col A) (dim_col A)›*) by auto\n show \"?thesis\"\n (*goal: ‹invertible_mat (P $$ (0, 0) ⋅⇩m Q)›*)\n using invertible_iff_is_unit_JNF[OF PQ] (*‹invertible_mat (P $$ (0, 0) ⋅⇩m Q) = (Determinant.det (P $$ (0, 0) ⋅⇩m Q) dvd 1)›*) det (*‹Determinant.det ((P::'a mat) $$ (0::nat, 0::nat) ⋅⇩m (Q::'a mat)) dvd (1::'a)›*) by auto\n qed\n moreover have \"lower_triangular (A * (P $$ (0,0) ⋅⇩m Q))\"\n unfolding lower_triangular_def\n (*goal: ‹∀i j. i < j ∧ i < dim_row (A * (P $$ (0, 0) ⋅⇩m Q)) ∧ j < dim_col (A * (P $$ (0, 0) ⋅⇩m Q)) ⟶ (A * (P $$ (0, 0) ⋅⇩m Q)) $$ (i, j) = 0›*)\n using SNF (*‹Smith_normal_form_mat (P * A * Q)›*) eq (*‹P * A * Q = A * (P $$ (0, 0) ⋅⇩m Q)›*) unfolding Smith_normal_form_mat_def isDiagonal_mat_def\n (*goal: ‹∀i j. i < j ∧ i < dim_row (A * (P $$ (0, 0) ⋅⇩m Q)) ∧ j < dim_col (A * (P $$ (0, 0) ⋅⇩m Q)) ⟶ (A * (P $$ (0, 0) ⋅⇩m Q)) $$ (i, j) = 0›*)\n by auto\n moreover have \"(P $$ (0,0) ⋅⇩m Q) ∈ carrier_mat (dim_col A) (dim_col A)\"\n using P (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) Q (*‹Q ∈ carrier_mat (dim_col A) (dim_col A)›*) A (*‹A ∈ carrier_mat 1 2›*) by auto\n ultimately show \"?thesis\"\n (*goal: ‹admits_triangular_reduction A›*)\n unfolding admits_triangular_reduction_def\n (*goal: ‹∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n by auto\nqed\n\nlemma triangular_imp_diagonal_1x2:\nassumes A: \"A ∈ carrier_mat 1 2\" and t: \"admits_triangular_reduction (A::'a mat)\"\nshows \"admits_diagonal_reduction A\"\nproof (-)\n (*goal: ‹admits_diagonal_reduction (A::'a mat)›*)\n obtain U where U: \"U ∈ carrier_mat (dim_col A) (dim_col A)\" and inv_U: \"invertible_mat U\" and AU: \"lower_triangular (A * U)\"\n (*goal: ‹(⋀U. ⟦U ∈ carrier_mat (dim_col A) (dim_col A); invertible_mat U; lower_triangular (A * U)⟧ ⟹ thesis) ⟹ thesis›*)\n using t (*‹admits_triangular_reduction A›*) unfolding admits_triangular_reduction_def\n (*goal: ‹(⋀U. ⟦U ∈ carrier_mat (dim_col A) (dim_col A); invertible_mat U; lower_triangular (A * U)⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n have SNF_AU: \"Smith_normal_form_mat (A * U)\"\n using AU (*‹lower_triangular (A * U)›*) A (*‹(A::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (2::nat)›*) unfolding Smith_normal_form_mat_def lower_triangular_def isDiagonal_mat_def\n (*goal: ‹(∀a. a + 1 < min (dim_row (A * U)) (dim_col (A * U)) ⟶ (A * U) $$ (a, a) dvd (A * U) $$ (a + 1, a + 1)) ∧ (∀i j. i ≠ j ∧ i < dim_row (A * U) ∧ j < dim_col (A * U) ⟶ (A * U) $$ (i, j) = 0)›*)\n by auto\n have \"A * U = (1⇩m 1) * A * U\"\n using A (*‹A ∈ carrier_mat 1 2›*) by auto\n hence SNF: \"Smith_normal_form_mat ((1⇩m 1) * A * U)\"\n using SNF_AU (*‹Smith_normal_form_mat (A * U)›*) by auto\n moreover have \"invertible_mat (1⇩m 1)\"\n using invertible_mat_def (*‹invertible_mat (?A::?'a mat) ≡ square_mat ?A ∧ (∃B::?'a mat. inverts_mat ?A B ∧ inverts_mat B ?A)›*) inverts_mat_def (*‹inverts_mat ?A ?B ≡ ?A * ?B = 1⇩m (dim_row ?A)›*) by fastforce\n ultimately show \"?thesis\"\n (*goal: ‹admits_diagonal_reduction A›*)\n using inv_U (*‹invertible_mat U›*) unfolding admits_diagonal_reduction_def\n (*goal: ‹∃(P::'a mat) Q::'a mat. P ∈ carrier_mat (dim_row (A::'a mat)) (dim_row A) ∧ Q ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat P ∧ invertible_mat Q ∧ Smith_normal_form_mat (P * A * Q)›*)\n by (smt (verit) U (*‹(U::'a::comm_ring_1 mat) ∈ carrier_mat (dim_col (A::'a::comm_ring_1 mat)) (dim_col A)›*) assms( (*‹(A::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (2::nat)›*) 1) carrier_matD( (*‹(?A::?'a::type mat) ∈ carrier_mat (?nr::nat) (?nc::nat) ⟹ dim_row ?A = ?nr›*) 1) one_carrier_mat (*‹1⇩m (?n::nat) ∈ carrier_mat ?n ?n›*))\nqed\n\n\nlemma triangular_eq_diagonal_1x2:\n \"(∀A∈carrier_mat 1 2. admits_triangular_reduction (A::'a mat)) \n = (∀A∈carrier_mat 1 2. admits_diagonal_reduction (A::'a mat))\"\n using triangular_imp_diagonal_1x2 (*‹⟦?A ∈ carrier_mat 1 2; admits_triangular_reduction ?A⟧ ⟹ admits_diagonal_reduction ?A›*) diagonal_imp_triangular_1x2 (*‹⟦?A ∈ carrier_mat 1 2; admits_diagonal_reduction ?A⟧ ⟹ admits_triangular_reduction ?A›*) by auto\n\n\nlemma admits_triangular_mat_1x1:\n assumes A: \"A ∈ carrier_mat 1 1\"\n shows \"admits_triangular_reduction (A::'a mat)\"\n apply (rule admits_triangular_reduction_intro[of \"1⇩m 1\"] (*‹⟦invertible_mat (1⇩m (1::nat)); 1⇩m (1::nat) ∈ carrier_mat (dim_col (?A::?'b1 mat)) (dim_col ?A); lower_triangular (?A * 1⇩m (1::nat))⟧ ⟹ admits_triangular_reduction ?A›*))\n (*goals:\n 1. ‹invertible_mat (1⇩m (1::nat))›\n 2. ‹1⇩m (1::nat) ∈ carrier_mat (dim_col (A::'a mat)) (dim_col A)›\n 3. ‹lower_triangular ((A::'a mat) * 1⇩m (1::nat))›\n discuss goal 1*)\n apply ((insert A (*‹(A::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (1::nat)›*))[1])\n (*top goal: ‹invertible_mat (1⇩m 1)› and 2 goals remain*)\n apply ((auto simp add: admits_triangular_reduction_def (*‹admits_triangular_reduction ?A = (∃U. U ∈ carrier_mat (dim_col ?A) (dim_col ?A) ∧ invertible_mat U ∧ lower_triangular (?A * U))›*) lower_triangular_def (*‹lower_triangular ?A = (∀i j. i < j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 2*)\n apply ((insert A (*‹A ∈ carrier_mat 1 1›*))[1])\n (*top goal: ‹1⇩m (1::nat) ∈ carrier_mat (dim_col (A::'a::comm_ring_1 mat)) (dim_col A)› and 1 goal remains*)\n apply ((auto simp add: admits_triangular_reduction_def (*‹admits_triangular_reduction ?A = (∃U. U ∈ carrier_mat (dim_col ?A) (dim_col ?A) ∧ invertible_mat U ∧ lower_triangular (?A * U))›*) lower_triangular_def (*‹lower_triangular ?A = (∀i j. i < j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 3*)\n apply ((insert A (*‹A ∈ carrier_mat 1 1›*))[1])\n (*goal: ‹lower_triangular (A * 1⇩m 1)›*)\n apply ((auto simp add: admits_triangular_reduction_def (*‹admits_triangular_reduction ?A = (∃U. U ∈ carrier_mat (dim_col ?A) (dim_col ?A) ∧ invertible_mat U ∧ lower_triangular (?A * U))›*) lower_triangular_def (*‹lower_triangular ?A = (∀i j. i < j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*proven 3 subgoals*) .\n\n\nlemma admits_diagonal_mat_1x1:\n assumes A: \"A ∈ carrier_mat 1 1\"\n shows \"admits_diagonal_reduction (A::'a mat)\"\n apply (rule admits_diagonal_reduction_intro[of \"(1⇩m 1)\" _ \"(1⇩m 1)\"] (*‹⟦1⇩m 1 ∈ carrier_mat (dim_row ?A) (dim_row ?A); 1⇩m 1 ∈ carrier_mat (dim_col ?A) (dim_col ?A); invertible_mat (1⇩m 1); invertible_mat (1⇩m 1); Smith_normal_form_mat (1⇩m 1 * ?A * 1⇩m 1)⟧ ⟹ admits_diagonal_reduction ?A›*))\n (*goals:\n 1. ‹1⇩m 1 ∈ carrier_mat (dim_row A) (dim_row A)›\n 2. ‹1⇩m 1 ∈ carrier_mat (dim_col A) (dim_col A)›\n 3. ‹invertible_mat (1⇩m 1)›\n 4. ‹invertible_mat (1⇩m 1)›\n 5. ‹Smith_normal_form_mat (1⇩m 1 * A * 1⇩m 1)›\n discuss goal 1*)\n apply ((insert A (*‹A ∈ carrier_mat 1 1›*))[1])\n (*top goal: ‹1⇩m 1 ∈ carrier_mat (dim_row A) (dim_row A)› and 4 goals remain*)\n apply ((auto simp add: Smith_normal_form_mat_def (*‹Smith_normal_form_mat ?A = ((∀a. a + 1 < min (dim_row ?A) (dim_col ?A) ⟶ ?A $$ (a, a) dvd ?A $$ (a + 1, a + 1)) ∧ isDiagonal_mat ?A)›*) isDiagonal_mat_def (*‹isDiagonal_mat ?A = (∀i j. i ≠ j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 2*)\n apply ((insert A (*‹A ∈ carrier_mat 1 1›*))[1])\n (*top goal: ‹1⇩m (1::nat) ∈ carrier_mat (dim_col (A::'a mat)) (dim_col A)› and 3 goals remain*)\n apply ((auto simp add: Smith_normal_form_mat_def (*‹Smith_normal_form_mat ?A = ((∀a. a + 1 < min (dim_row ?A) (dim_col ?A) ⟶ ?A $$ (a, a) dvd ?A $$ (a + 1, a + 1)) ∧ isDiagonal_mat ?A)›*) isDiagonal_mat_def (*‹isDiagonal_mat ?A = (∀i j. i ≠ j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 3*)\n apply ((insert A (*‹(A::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (1::nat)›*))[1])\n (*top goal: ‹invertible_mat (1⇩m 1)› and 2 goals remain*)\n apply ((auto simp add: Smith_normal_form_mat_def (*‹Smith_normal_form_mat ?A = ((∀a. a + 1 < min (dim_row ?A) (dim_col ?A) ⟶ ?A $$ (a, a) dvd ?A $$ (a + 1, a + 1)) ∧ isDiagonal_mat ?A)›*) isDiagonal_mat_def (*‹isDiagonal_mat ?A = (∀i j. i ≠ j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 4*)\n apply ((insert A (*‹A ∈ carrier_mat 1 1›*))[1])\n (*top goal: ‹invertible_mat (1⇩m 1)› and 1 goal remains*)\n apply ((auto simp add: Smith_normal_form_mat_def (*‹Smith_normal_form_mat ?A = ((∀a. a + 1 < min (dim_row ?A) (dim_col ?A) ⟶ ?A $$ (a, a) dvd ?A $$ (a + 1, a + 1)) ∧ isDiagonal_mat ?A)›*) isDiagonal_mat_def (*‹isDiagonal_mat ?A = (∀i j. i ≠ j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 5*)\n apply ((insert A (*‹A ∈ carrier_mat 1 1›*))[1])\n (*goal: ‹Smith_normal_form_mat (1⇩m 1 * A * 1⇩m 1)›*)\n apply ((auto simp add: Smith_normal_form_mat_def (*‹Smith_normal_form_mat ?A = ((∀a. a + 1 < min (dim_row ?A) (dim_col ?A) ⟶ ?A $$ (a, a) dvd ?A $$ (a + 1, a + 1)) ∧ isDiagonal_mat ?A)›*) isDiagonal_mat_def (*‹isDiagonal_mat ?A = (∀i j. i ≠ j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*proven 5 subgoals*) .\n\n\nlemma admits_diagonal_imp_admits_triangular_1xn:\n assumes a: \"∀A∈carrier_mat 1 2. admits_diagonal_reduction (A::'a mat)\"\n shows \"∀A∈carrier_mat 1 n. admits_triangular_reduction (A::'a mat)\"\nproof (standard)\n (*goal: ‹⋀A. A ∈ carrier_mat 1 n ⟹ admits_triangular_reduction A›*)\n fix A :: \"'a mat\"\n assume A: \"A ∈ carrier_mat 1 n\" (*‹(A::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*)\n have \"∃U. U ∈ carrier_mat (dim_col A) (dim_col A) \n ∧ invertible_mat U ∧ lower_triangular (A * U)\"\n using A (*‹(A::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*) proof (induct n arbitrary: A rule: less_induct)\n (*goal: ‹⋀(x::nat) A::'a mat. ⟦⋀(y::nat) A::'a mat. ⟦y < x; A ∈ carrier_mat (1::nat) y⟧ ⟹ ∃U::'a mat. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U); A ∈ carrier_mat (1::nat) x⟧ ⟹ ∃U::'a mat. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case (less n) (*‹⟦?y < n; ?A ∈ carrier_mat 1 ?y⟧ ⟹ ∃U. U ∈ carrier_mat (dim_col ?A) (dim_col ?A) ∧ invertible_mat U ∧ lower_triangular (?A * U)› ‹A ∈ carrier_mat 1 n›*)\n note A = less.prems(1) (*‹(A::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*)\n show \"?case\"\n (*goal: ‹∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n proof (cases \"n=0\")\n (*goals:\n 1. ‹n = 0 ⟹ ∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›\n 2. ‹n ≠ 0 ⟹ ∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case True (*‹n = 0›*)\n then show \"?thesis\"\n (*goal: ‹∃U::'a mat. U ∈ carrier_mat (dim_col (A::'a mat)) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n using triangularizable_m0 (*‹?A ∈ carrier_mat ?m 0 ⟹ ∃U. U ∈ carrier_mat 0 0 ∧ invertible_mat U ∧ lower_triangular (?A * U)›*) triangularizable_0n (*‹?A ∈ carrier_mat 0 ?n ⟹ ∃U. U ∈ carrier_mat ?n ?n ∧ invertible_mat U ∧ lower_triangular (?A * U)›*) less.prems (*‹A ∈ carrier_mat 1 n›*) by auto\n next\n (*goal: ‹n ≠ 0 ⟹ ∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case False (*‹n ≠ 0›*)\n note nm_not_0 = False (*‹n ≠ 0›*)\n from this (*‹n ≠ 0›*) have n_not_0: \"n ≠ 0\"\n by auto\n show \"?thesis\"\n (*goal: ‹∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n proof (cases \"n>2\")\n (*goals:\n 1. ‹2 < n ⟹ ∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›\n 2. ‹¬ 2 < n ⟹ ∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case False (*‹¬ 2 < n›*)\n note n_less_2 = False (*‹¬ 2 < n›*)\n show \"?thesis\"\n (*goal: ‹∃U::'a mat. U ∈ carrier_mat (dim_col (A::'a mat)) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n using admits_triangular_mat_1x1 (*‹?A ∈ carrier_mat 1 1 ⟹ admits_triangular_reduction ?A›*) a (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A›*) diagonal_imp_triangular_1x2 (*‹⟦?A ∈ carrier_mat 1 2; admits_diagonal_reduction ?A⟧ ⟹ admits_triangular_reduction ?A›*) unfolding admits_triangular_reduction_def\n (*goal: ‹∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n by (metis (full_types) admits_triangular_mat_1x1 (*‹(?A::'a mat) ∈ carrier_mat (1::nat) (1::nat) ⟹ admits_triangular_reduction ?A›*) Suc_1 (*‹Suc (1::nat) = (2::nat)›*) admits_triangular_reduction_def (*‹admits_triangular_reduction (?A::?'a mat) = (∃U::?'a mat. U ∈ carrier_mat (dim_col ?A) (dim_col ?A) ∧ invertible_mat U ∧ lower_triangular (?A * U))›*) less( (*‹(A::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*) 2) less_Suc_eq (*‹((?m::nat) < Suc (?n::nat)) = (?m < ?n ∨ ?m = ?n)›*) less_one (*‹((?n::nat) < (1::nat)) = (?n = (0::nat))›*) linorder_neqE_nat (*‹⟦(?x::nat) ≠ (?y::nat); ?x < ?y ⟹ ?R::bool; ?y < ?x ⟹ ?R⟧ ⟹ ?R›*) n_less_2 (*‹¬ (2::nat) < (n::nat)›*) nm_not_0 (*‹(n::nat) ≠ (0::nat)›*) triangular_eq_diagonal_1x2 (*‹(∀A::'a mat∈carrier_mat (1::nat) (2::nat). admits_triangular_reduction A) = (∀A::'a mat∈carrier_mat (1::nat) (2::nat). admits_diagonal_reduction A)›*))\n next\n (*goal: ‹2 < n ⟹ ∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case True (*‹2 < n›*)\n note n_ge_2 = True (*‹2 < n›*)\n let ?B = \"mat_of_row (vec_last (Matrix.row A 0) (n - 1))\"\n have \"∃V. V∈ carrier_mat (dim_col ?B) (dim_col ?B) \n ∧ invertible_mat V ∧ lower_triangular (?B * V)\"\n proof (rule less.hyps (*‹⟦?y < n; ?A ∈ carrier_mat 1 ?y⟧ ⟹ ∃U. U ∈ carrier_mat (dim_col ?A) (dim_col ?A) ∧ invertible_mat U ∧ lower_triangular (?A * U)›*))\n (*goals:\n 1. ‹?y < n›\n 2. ‹mat_of_row (vec_last (Matrix.row A 0) (n - 1)) ∈ carrier_mat 1 ?y›*)\n show \"n-1 < n\"\n using n_not_0 (*‹n ≠ 0›*) by auto\n show \"mat_of_row (vec_last (Matrix.row A 0) (n - 1)) ∈ carrier_mat 1 (n - 1)\"\n using A (*‹A ∈ carrier_mat 1 n›*) by simp\n qed\n from this (*‹∃V::'a mat. V ∈ carrier_mat (dim_col (mat_of_row (vec_last (Matrix.row (A::'a mat) (0::nat)) ((n::nat) - (1::nat))))) (dim_col (mat_of_row (vec_last (Matrix.row A (0::nat)) (n - (1::nat))))) ∧ invertible_mat V ∧ lower_triangular (mat_of_row (vec_last (Matrix.row A (0::nat)) (n - (1::nat))) * V)›*) obtain V where inv_V: \"invertible_mat V\" and BV: \"lower_triangular (?B * V)\" and V': \"V ∈ carrier_mat (dim_col ?B) (dim_col ?B)\"\n (*goal: ‹(⋀V. ⟦invertible_mat V; lower_triangular (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V); V ∈ carrier_mat (dim_col (mat_of_row (vec_last (Matrix.row A 0) (n - 1)))) (dim_col (mat_of_row (vec_last (Matrix.row A 0) (n - 1))))⟧ ⟹ thesis) ⟹ thesis›*)\n by fast\n have V: \"V ∈ carrier_mat (n-1) (n-1)\"\n using V' (*‹(V::'a mat) ∈ carrier_mat (dim_col (mat_of_row (vec_last (Matrix.row (A::'a mat) (0::nat)) ((n::nat) - (1::nat))))) (dim_col (mat_of_row (vec_last (Matrix.row A (0::nat)) (n - (1::nat)))))›*) by auto\n have BV_0: \"∀j ∈ {1..<n-1}. (?B * V) $$ (0,j) = 0\"\n apply rule\n (*goal: ‹∀j∈{1..<n - 1}. (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, j) = 0›*)\n apply (rule lower_triangular_index[OF BV] (*‹⟦?i < ?j; ?i < dim_row (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V); ?j < dim_col (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V)⟧ ⟹ (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (?i, ?j) = 0›*))\n (*goals:\n 1. ‹⋀j. j ∈ {1..<n - 1} ⟹ 0 < j›\n 2. ‹⋀j. j ∈ {1..<n - 1} ⟹ 0 < dim_row (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V)›\n 3. ‹⋀j. j ∈ {1..<n - 1} ⟹ j < dim_col (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V)›\n discuss goal 1*)\n apply ((insert V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*top goal: ‹⋀j::nat. j ∈ {1::nat..<(n::nat) - (1::nat)} ⟹ (0::nat) < j› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert V (*‹(V::'a::comm_ring_1 mat) ∈ carrier_mat ((n::nat) - (1::nat)) (n - (1::nat))›*))[1])\n (*top goal: ‹⋀j. j ∈ {1..<n - 1} ⟹ 0 < dim_row (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert V (*‹(V::'a::comm_ring_1 mat) ∈ carrier_mat ((n::nat) - (1::nat)) (n - (1::nat))›*))[1])\n (*goal: ‹⋀j::nat. j ∈ {1::nat..<(n::nat) - (1::nat)} ⟹ j < dim_col (mat_of_row (vec_last (Matrix.row (A::'a mat) (0::nat)) (n - (1::nat))) * (V::'a mat))›*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n define b where \"b = (?B * V) $$ (0,0)\"\n define a where \"a = A $$ (0,0)\"\n define ab :: \"'a mat\" where \"ab = Matrix.mat 1 2 (λ(i,j). if i=0 ∧ j=0 then a else b)\"\n have ab[simp]: \"ab ∈ carrier_mat 1 2\"\n unfolding ab_def\n (*goal: ‹Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) ∈ carrier_mat 1 2›*)\n by simp\n hence \"admits_diagonal_reduction ab\"\n using a (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A›*) by auto\n hence \"admits_triangular_reduction ab\"\n using diagonal_imp_triangular_1x2[OF ab] (*‹admits_diagonal_reduction (ab::'a mat) ⟹ admits_triangular_reduction ab›*) by auto\n from this (*‹admits_triangular_reduction ab›*) obtain W where inv_W: \"invertible_mat W\" and ab_W: \"lower_triangular (ab * W)\" and W: \"W ∈ carrier_mat 2 2\"\n (*goal: ‹(⋀W. ⟦invertible_mat W; lower_triangular (ab * W); W ∈ carrier_mat 2 2⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding admits_triangular_reduction_def\n (*goal: ‹(⋀W::'a mat. ⟦invertible_mat W; lower_triangular ((ab::'a mat) * W); W ∈ carrier_mat (2::nat) (2::nat)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using ab (*‹ab ∈ carrier_mat 1 2›*) by auto\n have id_n2_carrier[simp]: \"1⇩m (n-2) ∈ carrier_mat (n-2) (n-2)\"\n by auto\n define U where \"U = (four_block_mat (1⇩m 1) (0⇩m 1 (n-1)) (0⇩m (n-1) 1) V) * \n (four_block_mat W (0⇩m 2 (n-2)) (0⇩m (n-2) 2) (1⇩m (n-2)))\"\n let ?U1 = \"four_block_mat (1⇩m 1) (0⇩m 1 (n-1)) (0⇩m (n-1) 1) V\"\n let ?U2 = \"four_block_mat W (0⇩m 2 (n-2)) (0⇩m (n-2) 2) (1⇩m (n-2))\"\n have U1[simp]: \"?U1 ∈carrier_mat n n\"\n using four_block_carrier_mat[OF _ V] (*‹(?A::'a mat) ∈ carrier_mat (?nr1.0::nat) (?nc1.0::nat) ⟹ four_block_mat ?A (?B::'a mat) (?C::'a mat) (V::'a mat) ∈ carrier_mat (?nr1.0 + ((n::nat) - (1::nat))) (?nc1.0 + (n - (1::nat)))›*) nm_not_0 (*‹n ≠ 0›*) by fastforce\n have U2[simp]: \"?U2 ∈carrier_mat n n\"\n using four_block_carrier_mat[OF W id_n2_carrier] (*‹four_block_mat W ?B ?C (1⇩m (n - 2)) ∈ carrier_mat (2 + (n - 2)) (2 + (n - 2))›*) by (metis True (*‹2 < n›*) add_diff_inverse_nat (*‹¬ ?m < ?n ⟹ ?n + (?m - ?n) = ?m›*) less_imp_add_positive (*‹?i < ?j ⟹ ∃k>0. ?i + k = ?j›*) not_add_less1 (*‹¬ ?i + ?j < ?i›*))\n have U[simp]: \"U ∈ carrier_mat n n\"\n unfolding U_def\n (*goal: ‹four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) ∈ carrier_mat n n›*)\n using U1 (*‹four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V ∈ carrier_mat n n›*) U2 (*‹four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) ∈ carrier_mat n n›*) by auto\n moreover have inv_U: \"invertible_mat U\"\n proof (-)\n (*goal: ‹invertible_mat U›*)\n have \"invertible_mat ?U1\"\n by (metis U1 (*‹four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V ∈ carrier_mat n n›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*) det_four_block_mat_lower_left_zero_col (*‹⟦?A1.0 ∈ carrier_mat 1 1; ?A2.0 ∈ carrier_mat 1 ?n; ?A3.0 = 0⇩m ?n 1; ?A4.0 ∈ carrier_mat ?n ?n⟧ ⟹ Determinant.det (four_block_mat ?A1.0 ?A2.0 ?A3.0 ?A4.0) = Determinant.det ?A1.0 * Determinant.det ?A4.0›*) det_one (*‹Determinant.det (1⇩m ?n) = 1›*) inv_V (*‹invertible_mat V›*) invertible_iff_is_unit_JNF (*‹?A ∈ carrier_mat ?n ?n ⟹ invertible_mat ?A = (Determinant.det ?A dvd 1)›*) more_arith_simps( (*‹1 * ?a = ?a›*) 5) one_carrier_mat (*‹1⇩m ?n ∈ carrier_mat ?n ?n›*) zero_carrier_mat (*‹0⇩m ?nr ?nc ∈ carrier_mat ?nr ?nc›*))\n moreover have \"invertible_mat ?U2\"\n proof (-)\n (*goal: ‹invertible_mat (four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)))›*)\n have \"Determinant.det ?U2 = Determinant.det W\"\n apply (rule det_four_block_mat_lower_right_id (*‹⟦(?A::?'a mat) ∈ carrier_mat (?m::nat) ?m; (?B::?'a mat) = 0⇩m ?m ((?n::nat) - ?m); (?C::?'a mat) = 0⇩m (?n - ?m) ?m; (?D::?'a mat) = 1⇩m (?n - ?m); ?m < ?n⟧ ⟹ Determinant.det (four_block_mat ?A ?B ?C ?D) = Determinant.det ?A›*))\n (*goals:\n 1. ‹W ∈ carrier_mat ?m ?m›\n 2. ‹0⇩m 2 (n - 2) = 0⇩m ?m (?n - ?m)›\n 3. ‹0⇩m (n - 2) 2 = 0⇩m (?n - ?m) ?m›\n 4. ‹1⇩m (n - 2) = 1⇩m (?n - ?m)›\n 5. ‹?m < ?n›\n discuss goal 1*)\n apply ((insert less.prems (*‹(A::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (n::nat)›*) W (*‹(W::'a::comm_ring_1 mat) ∈ carrier_mat (2::nat) (2::nat)›*) n_ge_2 (*‹(2::nat) < (n::nat)›*))[1])\n (*top goal: ‹W ∈ carrier_mat ?m ?m› and 4 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert less.prems (*‹A ∈ carrier_mat 1 n›*) W (*‹W ∈ carrier_mat 2 2›*) n_ge_2 (*‹2 < n›*))[1])\n (*top goal: ‹0⇩m 2 (n - 2) = 0⇩m 2 (?n - 2)› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert less.prems (*‹A ∈ carrier_mat 1 n›*) W (*‹W ∈ carrier_mat 2 2›*) n_ge_2 (*‹2 < n›*))[1])\n (*top goal: ‹0⇩m (n - 2) 2 = 0⇩m (n - 2) 2› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert less.prems (*‹A ∈ carrier_mat 1 n›*) W (*‹W ∈ carrier_mat 2 2›*) n_ge_2 (*‹2 < n›*))[1])\n (*top goal: ‹1⇩m (n - 2) = 1⇩m (n - 2)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((insert less.prems (*‹A ∈ carrier_mat 1 n›*) W (*‹W ∈ carrier_mat 2 2›*) n_ge_2 (*‹2 < n›*))[1])\n (*goal: ‹2 < n›*)\n apply ((auto)[1])\n (*proven 5 subgoals*) .\n also (*calculation: ‹Determinant.det (four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2))) = Determinant.det W›*) have \" ... dvd 1\"\n using W (*‹W ∈ carrier_mat 2 2›*) inv_W (*‹invertible_mat (W::'a::comm_ring_1 mat)›*) invertible_iff_is_unit_JNF (*‹?A ∈ carrier_mat ?n ?n ⟹ invertible_mat ?A = (Determinant.det ?A dvd 1)›*) by auto\n finally (*calculation: ‹Determinant.det (four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2))) dvd 1›*) show \"?thesis\"\n (*goal: ‹invertible_mat (four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)))›*)\n using invertible_iff_is_unit_JNF[OF U2] (*‹invertible_mat (four_block_mat (W::'a mat) (0⇩m (2::nat) ((n::nat) - (2::nat))) (0⇩m (n - (2::nat)) (2::nat)) (1⇩m (n - (2::nat)))) = (Determinant.det (four_block_mat W (0⇩m (2::nat) (n - (2::nat))) (0⇩m (n - (2::nat)) (2::nat)) (1⇩m (n - (2::nat)))) dvd (1::'a))›*) by auto\n qed\n ultimately show \"?thesis\"\n (*goal: ‹invertible_mat U›*)\n using U1 (*‹four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V ∈ carrier_mat n n›*) U2 (*‹four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) ∈ carrier_mat n n›*) U_def (*‹U = four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2))›*) invertible_mult_JNF (*‹⟦?A ∈ carrier_mat ?n ?n; ?B ∈ carrier_mat ?n ?n; invertible_mat ?A; invertible_mat ?B⟧ ⟹ invertible_mat (?A * ?B)›*) by blast\n qed\n moreover have \"lower_triangular (A*U)\"\n proof (-)\n (*goal: ‹lower_triangular (A * U)›*)\n let ?A = \"Matrix.mat 1 n (λ(i,j). if j = 0 then a else if j=1 then b else 0)\"\n let ?T = \"Matrix.mat 1 n (λ(i,j). if j = 0 then (ab*W) $$ (0,0) else 0)\"\n have \"A*?U1 = ?A\"\n proof (rule eq_matI (*‹⟦⋀i j. ⟦i < dim_row ?B; j < dim_col ?B⟧ ⟹ ?A $$ (i, j) = ?B $$ (i, j); dim_row ?A = dim_row ?B; dim_col ?A = dim_col ?B⟧ ⟹ ?A = ?B›*))\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0)); j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))⟧ ⟹ (A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) $$ (i, j) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›\n 2. ‹dim_row (A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) = dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›\n 3. ‹dim_col (A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) = dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*)\n fix i and j\n assume i: \"i<dim_row ?A\" and j: \"j<dim_col ?A\" (*‹(i::nat) < dim_row (Matrix.mat (1::nat) (n::nat) (λ(i::nat, j::nat). if j = (0::nat) then a::'a else if j = (1::nat) then b::'a else (0::'a)))› ‹(j::nat) < dim_col (Matrix.mat (1::nat) (n::nat) (λ(i::nat, j::nat). if j = (0::nat) then a::'a else if j = (1::nat) then b::'a else (0::'a)))›*)\n have i0: \"i=0\"\n using i (*‹i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) by auto\n let ?f = \"λ i. A $$ (0, i) * \n (if i = 0 then if j < 1 then 1⇩m (1) $$ (i, j) else 0⇩m (1) (n - 1) $$ (i, j - 1)\n else if j < 1 then 0⇩m (n - 1) (1) $$ (i - 1, j) else V $$ (i - 1, j - 1))\"\n have \"(A*?U1) $$ (i,j) = Matrix.row A i ∙ col ?U1 j\"\n apply (rule index_mult_mat (*‹⟦?i < dim_row ?A; ?j < dim_col ?B⟧ ⟹ (?A * ?B) $$ (?i, ?j) = Matrix.row ?A ?i ∙ col ?B ?j› ‹dim_row (?A * ?B) = dim_row ?A› ‹dim_col (?A * ?B) = dim_col ?B›*))\n (*goals:\n 1. ‹i < dim_row A›\n 2. ‹j < dim_col (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V)›\n discuss goal 1*)\n apply ((insert i (*‹i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) A (*‹A ∈ carrier_mat 1 n›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*top goal: ‹i < dim_row A› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert i (*‹i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) A (*‹A ∈ carrier_mat 1 n›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*goal: ‹j < dim_col (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V)›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) $$ (i, j) = Matrix.row A i ∙ col (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) j›*) have \"... = (∑i = 0..<n. ?f i)\"\n using i (*‹i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) A (*‹(A::'a mat) ∈ carrier_mat (1::nat) (n::nat)›*) V (*‹(V::'a::comm_ring_1 mat) ∈ carrier_mat ((n::nat) - (1::nat)) (n - (1::nat))›*) unfolding scalar_prod_def\n (*goal: ‹(∑ia = 0..<dim_vec (col (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) j). Matrix.row A i $v ia * col (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) j $v ia) = (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1)))›*)\n apply auto\n (*goal: ‹(∑ia = 0..<dim_vec (col (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) j). Matrix.row A i $v ia * col (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) j $v ia) = (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1)))›*)\n apply (unfold index_one_mat (*‹⟦?i < ?n; ?j < ?n⟧ ⟹ 1⇩m ?n $$ (?i, ?j) = (if ?i = ?j then 1 else 0)› ‹dim_row (1⇩m ?n) = ?n› ‹dim_col (1⇩m ?n) = ?n›*))\n (*goal: ‹⟦i = 0; j < n; A ∈ carrier_mat (Suc 0) n; V ∈ carrier_mat (n - Suc 0) (n - Suc 0)⟧ ⟹ (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < dim_col (1⇩m (Suc 0)) then 1⇩m (Suc 0) $$ (i, j) else 0⇩m (Suc 0) (n - Suc 0) $$ (i, j - dim_col (1⇩m (Suc 0))) else if j < dim_col (1⇩m (Suc 0)) then 0⇩m (n - Suc 0) (Suc 0) $$ (i - dim_row (1⇩m (Suc 0)), j) else V $$ (i - dim_row (1⇩m (Suc 0)), j - dim_col (1⇩m (Suc 0))))) = (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1)))›*)\n apply (insert One_nat_def (*‹1 = Suc 0›*))\n (*goal: ‹⟦i = 0; j < n; A ∈ carrier_mat (Suc 0) n; V ∈ carrier_mat (n - Suc 0) (n - Suc 0)⟧ ⟹ (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < Suc 0 then 1⇩m (Suc 0) $$ (i, j) else 0⇩m (Suc 0) (n - Suc 0) $$ (i, j - Suc 0) else if j < Suc 0 then 0⇩m (n - Suc 0) (Suc 0) $$ (i - Suc 0, j) else V $$ (i - Suc 0, j - Suc 0))) = (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1)))›*)\n by presburger\n also (*calculation: ‹((A::'a mat) * four_block_mat (1⇩m (1::nat)) (0⇩m (1::nat) ((n::nat) - (1::nat))) (0⇩m (n - (1::nat)) (1::nat)) (V::'a mat)) $$ (i::nat, j::nat) = (∑i::nat = 0::nat..<n. A $$ (0::nat, i) * (if i = (0::nat) then if j < (1::nat) then 1⇩m (1::nat) $$ (i, j) else 0⇩m (1::nat) (n - (1::nat)) $$ (i, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ (i - (1::nat), j) else V $$ (i - (1::nat), j - (1::nat))))›*) have \"... = ?A $$ (i,j)\"\n proof (cases \"j=0\")\n (*goals:\n 1. ‹j = 0 ⟹ (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›\n 2. ‹j ≠ 0 ⟹ (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*)\n case True (*‹(j::nat) = (0::nat)›*)\n have rw0: \"sum ?f {1..<n} = 0\"\n apply (rule sum.neutral (*‹∀x∈?A. ?g x = 0 ⟹ sum ?g ?A = 0›*))\n (*goal: ‹(∑i = 1..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = 0›*)\n apply (insert True (*‹j = 0›*))\n (*goal: ‹∀x∈{1..<n}. A $$ (0, x) * (if x = 0 then if j < 1 then 1⇩m 1 $$ (x, j) else 0⇩m 1 (n - 1) $$ (x, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (x - 1, j) else V $$ (x - 1, j - 1)) = 0›*)\n by auto\n have set_rw: \"{0..<n} = insert 0 {1..<n}\"\n using n_ge_2 (*‹2 < n›*) by auto\n hence \"sum ?f {0..<n} = ?f 0 + sum ?f {1..<n}\"\n by auto\n also (*calculation: ‹(∑i::nat = 0::nat..<n. (A::'a::comm_ring_1 mat) $$ (0::nat, i) * (if i = (0::nat) then if (j::nat) < (1::nat) then 1⇩m (1::nat) $$ (i, j) else 0⇩m (1::nat) ((n::nat) - (1::nat)) $$ (i, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ (i - (1::nat), j) else (V::'a::comm_ring_1 mat) $$ (i - (1::nat), j - (1::nat)))) = A $$ (0::nat, 0::nat) * (if (0::nat) = (0::nat) then if j < (1::nat) then 1⇩m (1::nat) $$ (0::nat, j) else 0⇩m (1::nat) (n - (1::nat)) $$ (0::nat, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ ((0::nat) - (1::nat), j) else V $$ ((0::nat) - (1::nat), j - (1::nat))) + (∑i::nat = 1::nat..<n. A $$ (0::nat, i) * (if i = (0::nat) then if j < (1::nat) then 1⇩m (1::nat) $$ (i, j) else 0⇩m (1::nat) (n - (1::nat)) $$ (i, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ (i - (1::nat), j) else V $$ (i - (1::nat), j - (1::nat))))›*) have \"... = ?f 0\"\n unfolding rw0\n (*goal: ‹A $$ (0, 0) * (if 0 = 0 then if j < 1 then 1⇩m 1 $$ (0, j) else 0⇩m 1 (n - 1) $$ (0, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (0 - 1, j) else V $$ (0 - 1, j - 1)) + 0 = A $$ (0, 0) * (if 0 = 0 then if j < 1 then 1⇩m 1 $$ (0, j) else 0⇩m 1 (n - 1) $$ (0, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (0 - 1, j) else V $$ (0 - 1, j - 1))›*)\n by simp\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = A $$ (0, 0) * (if 0 = 0 then if j < 1 then 1⇩m 1 $$ (0, j) else 0⇩m 1 (n - 1) $$ (0, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (0 - 1, j) else V $$ (0 - 1, j - 1))›*) have \"... = a\"\n using True (*‹j = 0›*) unfolding a_def\n (*goal: ‹A $$ (0, 0) * (if 0 = 0 then if j < 1 then 1⇩m 1 $$ (0, j) else 0⇩m 1 (n - 1) $$ (0, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (0 - 1, j) else V $$ (0 - 1, j - 1)) = A $$ (0, 0)›*)\n by simp\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = a›*) have \"... = ?A $$ (i,j)\"\n using True (*‹j = 0›*) i (*‹i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) by auto\n finally (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*) show \"?thesis\"\n (*goal: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*) .\n next\n (*goal: ‹j ≠ 0 ⟹ (∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*)\n case False (*‹j ≠ 0›*)\n note j_not_0 = False (*‹j ≠ 0›*)\n have rw_simp: \"Matrix.row (mat_of_row (vec_last (Matrix.row A 0) (n - 1))) 0 \n = (vec_last (Matrix.row A 0) (n - 1))\"\n unfolding Matrix.row_def\n (*goal: ‹Matrix.vec (dim_col (mat_of_row (vec_last (Matrix.vec (dim_col (A::'a::comm_ring_1 mat)) (λj::nat. A $$ (0::nat, j))) ((n::nat) - (1::nat))))) (λj::nat. mat_of_row (vec_last (Matrix.vec (dim_col A) (λj::nat. A $$ (0::nat, j))) (n - (1::nat))) $$ (0::nat, j)) = vec_last (Matrix.vec (dim_col A) (λj::nat. A $$ (0::nat, j))) (n - (1::nat))›*)\n by auto\n let ?g = \"λi. A $$ (0, i) * V $$ (i - 1, j - 1)\"\n let ?h = \"λi. A $$ (0, i+1) * V $$ (i, j - 1)\"\n have f0: \"?f 0 = 0\"\n using j_not_0 (*‹(j::nat) ≠ (0::nat)›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) by auto\n have set_rw2: \"(λi. i+1)`{0..<n-1} = {1..<n}\"\n unfolding image_def\n (*goal: ‹{y. ∃x∈{0..<n - 1}. y = x + 1} = {1..<n}›*)\n using Suc_le_D (*‹Suc ?n ≤ ?m' ⟹ ∃m. ?m' = Suc m›*) by fastforce\n have set_rw: \"{0..<n} = insert 0 {1..<n}\"\n using n_ge_2 (*‹2 < n›*) by auto\n hence \"sum ?f {0..<n} = ?f 0 + sum ?f {1..<n}\"\n by auto\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = A $$ (0, 0) * (if 0 = 0 then if j < 1 then 1⇩m 1 $$ (0, j) else 0⇩m 1 (n - 1) $$ (0, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (0 - 1, j) else V $$ (0 - 1, j - 1)) + (∑i = 1..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1)))›*) have \"... = sum ?f {1..<n}\"\n using f0 (*‹A $$ (0, 0) * (if 0 = 0 then if j < 1 then 1⇩m 1 $$ (0, j) else 0⇩m 1 (n - 1) $$ (0, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (0 - 1, j) else V $$ (0 - 1, j - 1)) = 0›*) by simp\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = (∑i = 1..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1)))›*) have \"... = sum ?g {1..<n}\"\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{1::nat..<n::nat} = {1::nat..<n}›\n 2. ‹⋀x::nat. x ∈ {1::nat..<n::nat} ⟹ (A::'a::comm_ring_1 mat) $$ (0::nat, x) * (if x = (0::nat) then if (j::nat) < (1::nat) then 1⇩m (1::nat) $$ (x, j) else 0⇩m (1::nat) (n - (1::nat)) $$ (x, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ (x - (1::nat), j) else (V::'a::comm_ring_1 mat) $$ (x - (1::nat), j - (1::nat))) = A $$ (0::nat, x) * V $$ (x - (1::nat), j - (1::nat))›\n discuss goal 1*)\n apply ((insert j_not_0 (*‹j ≠ 0›*))[1])\n (*top goal: ‹{1..<n} = {1..<n}› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert j_not_0 (*‹j ≠ 0›*))[1])\n (*goal: ‹⋀x::nat. x ∈ {1::nat..<n::nat} ⟹ (A::'a::comm_ring_1 mat) $$ (0::nat, x) * (if x = (0::nat) then if (j::nat) < (1::nat) then 1⇩m (1::nat) $$ (x, j) else 0⇩m (1::nat) (n - (1::nat)) $$ (x, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ (x - (1::nat), j) else (V::'a::comm_ring_1 mat) $$ (x - (1::nat), j - (1::nat))) = A $$ (0::nat, x) * V $$ (x - (1::nat), j - (1::nat))›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(∑i::nat = 0::nat..<n. (A::'a mat) $$ (0::nat, i) * (if i = (0::nat) then if (j::nat) < (1::nat) then 1⇩m (1::nat) $$ (i, j) else 0⇩m (1::nat) ((n::nat) - (1::nat)) $$ (i, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ (i - (1::nat), j) else (V::'a mat) $$ (i - (1::nat), j - (1::nat)))) = (∑i::nat = 1::nat..<n. A $$ (0::nat, i) * V $$ (i - (1::nat), j - (1::nat)))›*) have \"... = sum ?g ((λi. i+1)`{0..<n-1})\"\n using set_rw2 (*‹(λi::nat. i + (1::nat)) ` {0::nat..<(n::nat) - (1::nat)} = {1::nat..<n}›*) by simp\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = (∑i∈(λi. i + 1) ` {0..<n - 1}. A $$ (0, i) * V $$ (i - 1, j - 1))›*) have \"... = sum (?g ∘ (λi. i+1)) {0..<n-1}\"\n apply (rule sum.reindex (*‹inj_on ?h ?A ⟹ sum ?g (?h ` ?A) = sum (?g ∘ ?h) ?A›*))\n (*goal: ‹(∑i∈(λi. i + 1) ` {0..<n - 1}. A $$ (0, i) * V $$ (i - 1, j - 1)) = sum ((λi. A $$ (0, i) * V $$ (i - 1, j - 1)) ∘ (λi. i + 1)) {0..<n - 1}›*)\n apply (unfold inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*))\n (*goal: ‹inj_on (λi. i + 1) {0..<n - 1}›*)\n by auto\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = sum ((λi. A $$ (0, i) * V $$ (i - 1, j - 1)) ∘ (λi. i + 1)) {0..<n - 1}›*) have \"... = sum ?h {0..<n-1}\"\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0..<n - 1} = {0..<n - 1}›\n 2. ‹⋀x. x ∈ {0..<n - 1} ⟹ ((λi. A $$ (0, i) * V $$ (i - 1, j - 1)) ∘ (λi. i + 1)) x = A $$ (0, x + 1) * V $$ (x, j - 1)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = (∑i = 0..<n - 1. A $$ (0, i + 1) * V $$ (i, j - 1))›*) have \"... = Matrix.row ?B 0 ∙ col V (j-1)\"\n unfolding scalar_prod_def\n (*goal: ‹(∑i = 0..<n - 1. A $$ (0, i + 1) * V $$ (i, j - 1)) = (∑i = 0..<dim_vec (col V (j - 1)). Matrix.row (mat_of_row (vec_last (Matrix.row A 0) (n - 1))) 0 $v i * col V (j - 1) $v i)›*)\n proof (rule sum.cong (*‹⟦(?A::?'b set) = (?B::?'b set); ⋀x::?'b. x ∈ ?B ⟹ (?g::?'b ⇒ ?'a) x = (?h::?'b ⇒ ?'a) x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0..<n - 1} = {0..<dim_vec (col V (j - 1))}›\n 2. ‹⋀x. x ∈ {0..<dim_vec (col V (j - 1))} ⟹ A $$ (0, x + 1) * V $$ (x, j - 1) = Matrix.row (mat_of_row (vec_last (Matrix.row A 0) (n - 1))) 0 $v x * col V (j - 1) $v x›*)\n fix x\n assume x: \"x ∈ {0..<dim_vec (col V (j - 1))}\" (*‹(x::nat) ∈ {0::nat..<dim_vec (col (V::'a mat) ((j::nat) - (1::nat)))}›*)\n have \"Matrix.row ?B 0 $v x = ?B $$ (0,x)\"\n apply (rule index_row (*‹⟦?i < dim_row ?A; ?j < dim_col ?A⟧ ⟹ Matrix.row ?A ?i $v ?j = ?A $$ (?i, ?j)› ‹dim_vec (Matrix.row ?A ?i) = dim_col ?A›*))\n (*goals:\n 1. ‹0 < dim_row (mat_of_row (vec_last (Matrix.row A 0) (n - 1)))›\n 2. ‹x < dim_col (mat_of_row (vec_last (Matrix.row A 0) (n - 1)))›\n discuss goal 1*)\n apply ((insert x (*‹x ∈ {0..<dim_vec (col V (j - 1))}›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*top goal: ‹(0::nat) < dim_row (mat_of_row (vec_last (Matrix.row (A::'a mat) (0::nat)) ((n::nat) - (1::nat))))› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert x (*‹x ∈ {0..<dim_vec (col V (j - 1))}›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*goal: ‹x < dim_col (mat_of_row (vec_last (Matrix.row A 0) (n - 1)))›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹Matrix.row (mat_of_row (vec_last (Matrix.row A 0) (n - 1))) 0 $v x = mat_of_row (vec_last (Matrix.row A 0) (n - 1)) $$ (0, x)›*) have \"... = (vec_last (Matrix.row A 0) (n - 1)) $v x\"\n apply (rule mat_of_row_index (*‹?x < dim_vec ?y ⟹ mat_of_row ?y $$ (0, ?x) = ?y $v ?x›*))\n (*goal: ‹mat_of_row (vec_last (Matrix.row A 0) (n - 1)) $$ (0, x) = vec_last (Matrix.row A 0) (n - 1) $v x›*)\n apply (insert x (*‹x ∈ {0..<dim_vec (col V (j - 1))}›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*))\n (*goal: ‹x < dim_vec (vec_last (Matrix.row A 0) (n - 1))›*)\n by auto\n also (*calculation: ‹Matrix.row (mat_of_row (vec_last (Matrix.row A 0) (n - 1))) 0 $v x = vec_last (Matrix.row A 0) (n - 1) $v x›*) have \"... = A $$ (0, x + 1)\"\n by (smt (verit) Suc_less_eq (*‹(Suc ?m < Suc ?n) = (?m < ?n)›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*) add.right_neutral (*‹?a + 0 = ?a›*) add_Suc_right (*‹?m + Suc ?n = Suc (?m + ?n)›*) add_diff_cancel_right' (*‹?a + ?b - ?b = ?a›*) add_diff_inverse_nat (*‹¬ ?m < ?n ⟹ ?n + (?m - ?n) = ?m›*) atLeastLessThan_iff (*‹(?i ∈ {?l..<?u}) = (?l ≤ ?i ∧ ?i < ?u)›*) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_row ?A = ?nr›*) 1) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_col ?A = ?nc›*) 2) dim_col (*‹dim_vec (col ?A ?i) = dim_row ?A›*) index_row( (*‹⟦?i < dim_row ?A; ?j < dim_col ?A⟧ ⟹ Matrix.row ?A ?i $v ?j = ?A $$ (?i, ?j)›*) 1) index_row( (*‹dim_vec (Matrix.row ?A ?i) = dim_col ?A›*) 2) index_vec (*‹?i < ?n ⟹ Matrix.vec ?n ?f $v ?i = ?f ?i›*) less.prems (*‹A ∈ carrier_mat 1 n›*) less_Suc0 (*‹(?n < Suc 0) = (?n = 0)›*) n_not_0 (*‹n ≠ 0›*) plus_1_eq_Suc (*‹(+) 1 = Suc›*) vec_last_def (*‹vec_last ?v ?n ≡ Matrix.vec ?n (λi. ?v $v (dim_vec ?v - ?n + i))›*) x (*‹x ∈ {0..<dim_vec (col V (j - 1))}›*))\n finally (*calculation: ‹Matrix.row (mat_of_row (vec_last (Matrix.row A 0) (n - 1))) 0 $v x = A $$ (0, x + 1)›*) have \"Matrix.row ?B 0 $v x = A $$ (0, x + 1)\" .\n moreover have \"col V (j - 1) $v x = V $$ (x, j - 1)\"\n using V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) x (*‹x ∈ {0..<dim_vec (col V (j - 1))}›*) by auto\n ultimately show \"A $$ (0, x + 1) * V $$ (x, j - 1) \n = Matrix.row ?B 0 $v x * col V (j - 1) $v x\"\n by simp\n qed (insert V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*) j_not_0 (*‹j ≠ 0›*), auto)\n (*solved the remaining goal: ‹{0..<n - 1} = {0..<dim_vec (col V (j - 1))}›*)\n also (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.row (mat_of_row (vec_last (Matrix.row A 0) (n - 1))) 0 ∙ col V (j - 1)›*) have \"... = (?B*V) $$ (0,j-1)\"\n apply (rule index_mult_mat[symmetric] (*‹⟦?i < dim_row ?A; ?j < dim_col ?B⟧ ⟹ Matrix.row ?A ?i ∙ col ?B ?j = (?A * ?B) $$ (?i, ?j)› ‹dim_row ?A = dim_row (?A * ?B)› ‹dim_col ?B = dim_col (?A * ?B)›*))\n (*goals:\n 1. ‹0 < dim_row (mat_of_row (vec_last (Matrix.row A 0) (n - 1)))›\n 2. ‹j - 1 < dim_col V›\n discuss goal 1*)\n apply ((insert V (*‹(V::'a mat) ∈ carrier_mat ((n::nat) - (1::nat)) (n - (1::nat))›*) j (*‹(j::nat) < dim_col (Matrix.mat (1::nat) (n::nat) (λ(i::nat, j::nat). if j = (0::nat) then a::'a else if j = (1::nat) then b::'a else (0::'a)))›*) False (*‹(j::nat) ≠ (0::nat)›*))[1])\n (*top goal: ‹0 < dim_row (mat_of_row (vec_last (Matrix.row A 0) (n - 1)))› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) False (*‹j ≠ 0›*))[1])\n (*goal: ‹j - 1 < dim_col V›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(∑i::nat = 0::nat..<n. (A::'a::comm_ring_1 mat) $$ (0::nat, i) * (if i = (0::nat) then if (j::nat) < (1::nat) then 1⇩m (1::nat) $$ (i, j) else 0⇩m (1::nat) ((n::nat) - (1::nat)) $$ (i, j - (1::nat)) else if j < (1::nat) then 0⇩m (n - (1::nat)) (1::nat) $$ (i - (1::nat), j) else (V::'a::comm_ring_1 mat) $$ (i - (1::nat), j - (1::nat)))) = (mat_of_row (vec_last (Matrix.row A (0::nat)) (n - (1::nat))) * V) $$ (0::nat, j - (1::nat))›*) have \"... = ?A $$ (i, j)\"\n apply (cases \"j=1\")\n (*goals:\n 1. ‹j = 1 ⟹ (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, j - 1) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›\n 2. ‹j ≠ 1 ⟹ (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, j - 1) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›\n discuss goal 1*)\n apply ((insert False (*‹j ≠ 0›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) i0 (*‹i = 0›*) BV_0 (*‹∀j∈{1..<n - 1}. (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, j) = 0›*) b_def (*‹b = (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, 0)›*))[1])\n (*top goal: ‹j = 1 ⟹ (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, j - 1) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)› and 1 goal remains*)\n apply ((auto simp add: Suc_leI (*‹?m < ?n ⟹ Suc ?m ≤ ?n›*))[1])\n (*discuss goal 2*)\n apply ((insert False (*‹j ≠ 0›*) V (*‹V ∈ carrier_mat (n - 1) (n - 1)›*) j (*‹j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*) i0 (*‹i = 0›*) BV_0 (*‹∀j∈{1..<n - 1}. (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, j) = 0›*) b_def (*‹b = (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, 0)›*))[1])\n (*goal: ‹j ≠ 1 ⟹ (mat_of_row (vec_last (Matrix.row A 0) (n - 1)) * V) $$ (0, j - 1) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*)\n apply ((auto simp add: Suc_leI (*‹?m < ?n ⟹ Suc ?m ≤ ?n›*))[1])\n (*proven 2 subgoals*) .\n finally (*calculation: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*) show \"?thesis\"\n (*goal: ‹(∑i = 0..<n. A $$ (0, i) * (if i = 0 then if j < 1 then 1⇩m 1 $$ (i, j) else 0⇩m 1 (n - 1) $$ (i, j - 1) else if j < 1 then 0⇩m (n - 1) 1 $$ (i - 1, j) else V $$ (i - 1, j - 1))) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*) .\n qed\n finally (*calculation: ‹(A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) $$ (i, j) = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j)›*) show \"(A*?U1) $$ (i,j) = ?A $$ (i,j)\" .\n next\n (*goals:\n 1. ‹dim_row (A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) = dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›\n 2. ‹dim_col (A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V) = dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0))›*)\n show \"dim_row (A*?U1) = dim_row ?A\"\n using A (*‹A ∈ carrier_mat 1 n›*) by auto\n show \"dim_col (A*?U1) = dim_col ?A\"\n using U1 (*‹four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V ∈ carrier_mat n n›*) by auto\n qed\n also (*calculation: ‹A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V = Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0)›*) have \"... * ?U2 = ?T\"\n proof (-)\n (*goal: ‹Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) = Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0)›*)\n let ?A1.0 = ab\n let ?B1.0 = \"Matrix.mat 1 (n-2) (λ(i,j). 0)\"\n let ?C1.0 = \"Matrix.mat 0 2 (λ(i,j). 0)\"\n let ?D1.0 = \"Matrix.mat 0 (n-2) (λ(i,j). 0)\"\n let ?B2.0 = \"(0⇩m 2 (n - 2))\"\n let ?C2.0 = \"(0⇩m (n - 2) 2)\"\n let ?D2.0 = \"1⇩m (n - 2)\"\n have A_eq: \"?A = four_block_mat ?A1.0 ?B1.0 ?C1.0 ?D1.0\"\n apply (rule eq_matI (*‹⟦⋀i j. ⟦i < dim_row ?B; j < dim_col ?B⟧ ⟹ ?A $$ (i, j) = ?B $$ (i, j); dim_row ?A = dim_row ?B; dim_col ?A = dim_col ?B⟧ ⟹ ?A = ?B›*))\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0))); j < dim_col (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)))⟧ ⟹ Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j) = four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)) $$ (i, j)›\n 2. ‹dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0)) = dim_row (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)))›\n 3. ‹dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0)) = dim_col (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)))›\n discuss goal 1*)\n apply ((insert ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*) n_ge_2 (*‹2 < n›*))[1])\n (*top goal: ‹⋀i j. ⟦i < dim_row (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0))); j < dim_col (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)))⟧ ⟹ Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) $$ (i, j) = four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)) $$ (i, j)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert ab_def (*‹(ab::'a mat) = Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a else (b::'a))›*) n_ge_2 (*‹(2::nat) < (n::nat)›*))[1])\n (*top goal: ‹dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0)) = dim_row (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)))› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*) n_ge_2 (*‹2 < n›*))[1])\n (*goal: ‹dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0)) = dim_col (four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)))›*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n hence \"?A * ?U2 = four_block_mat ?A1.0 ?B1.0 ?C1.0 ?D1.0 * ?U2\"\n by simp\n also (*calculation: ‹Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) = four_block_mat ab (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)) * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2))›*) have \"... = four_block_mat (?A1.0 * W + ?B1.0 * ?C2.0) \n (?A1.0 * ?B2.0 + ?B1.0 * ?D2.0) (?C1.0 * W + ?D1.0 * ?C2.0) \n (?C1.0 * ?B2.0 + ?D1.0 * ?D2.0)\"\n by (rule mult_four_block_mat (*‹⟦?A1.0 ∈ carrier_mat ?nr1.0 ?n1.0; ?B1.0 ∈ carrier_mat ?nr1.0 ?n2.0; ?C1.0 ∈ carrier_mat ?nr2.0 ?n1.0; ?D1.0 ∈ carrier_mat ?nr2.0 ?n2.0; ?A2.0 ∈ carrier_mat ?n1.0 ?nc1.0; ?B2.0 ∈ carrier_mat ?n1.0 ?nc2.0; ?C2.0 ∈ carrier_mat ?n2.0 ?nc1.0; ?D2.0 ∈ carrier_mat ?n2.0 ?nc2.0⟧ ⟹ four_block_mat ?A1.0 ?B1.0 ?C1.0 ?D1.0 * four_block_mat ?A2.0 ?B2.0 ?C2.0 ?D2.0 = four_block_mat (?A1.0 * ?A2.0 + ?B1.0 * ?C2.0) (?A1.0 * ?B2.0 + ?B1.0 * ?D2.0) (?C1.0 * ?A2.0 + ?D1.0 * ?C2.0) (?C1.0 * ?B2.0 + ?D1.0 * ?D2.0)›*), auto simp add: W (*‹W ∈ carrier_mat 2 2›*) ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*))\n also (*calculation: ‹Matrix.mat (1::nat) (n::nat) (λ(i::nat, j::nat). if j = (0::nat) then a::'a else if j = (1::nat) then b::'a else (0::'a)) * four_block_mat (W::'a mat) (0⇩m (2::nat) (n - (2::nat))) (0⇩m (n - (2::nat)) (2::nat)) (1⇩m (n - (2::nat))) = four_block_mat ((ab::'a mat) * W + Matrix.mat (1::nat) (n - (2::nat)) (λ(i::nat, j::nat). 0::'a) * 0⇩m (n - (2::nat)) (2::nat)) (ab * 0⇩m (2::nat) (n - (2::nat)) + Matrix.mat (1::nat) (n - (2::nat)) (λ(i::nat, j::nat). 0::'a) * 1⇩m (n - (2::nat))) (Matrix.mat (0::nat) (2::nat) (λ(i::nat, j::nat). 0::'a) * W + Matrix.mat (0::nat) (n - (2::nat)) (λ(i::nat, j::nat). 0::'a) * 0⇩m (n - (2::nat)) (2::nat)) (Matrix.mat (0::nat) (2::nat) (λ(i::nat, j::nat). 0::'a) * 0⇩m (2::nat) (n - (2::nat)) + Matrix.mat (0::nat) (n - (2::nat)) (λ(i::nat, j::nat). 0::'a) * 1⇩m (n - (2::nat)))›*) have \"... = four_block_mat (?A1.0 * W) (?B1.0) (?C1.0) (?D1.0)\"\n apply (rule cong_four_block_mat (*‹⟦?A1.0 = ?B1.0; ?A2.0 = ?B2.0; ?A3.0 = ?B3.0; ?A4.0 = ?B4.0⟧ ⟹ four_block_mat ?A1.0 ?A2.0 ?A3.0 ?A4.0 = four_block_mat ?B1.0 ?B2.0 ?B3.0 ?B4.0›*))\n (*goals:\n 1. ‹ab * W + Matrix.mat 1 (n - 2) (λ(i, j). 0) * 0⇩m (n - 2) 2 = ab * W›\n 2. ‹ab * 0⇩m 2 (n - 2) + Matrix.mat 1 (n - 2) (λ(i, j). 0) * 1⇩m (n - 2) = Matrix.mat 1 (n - 2) (λ(i, j). 0)›\n 3. ‹Matrix.mat 0 2 (λ(i, j). 0) * W + Matrix.mat 0 (n - 2) (λ(i, j). 0) * 0⇩m (n - 2) 2 = Matrix.mat 0 2 (λ(i, j). 0)›\n 4. ‹Matrix.mat 0 2 (λ(i, j). 0) * 0⇩m 2 (n - 2) + Matrix.mat 0 (n - 2) (λ(i, j). 0) * 1⇩m (n - 2) = Matrix.mat 0 (n - 2) (λ(i, j). 0)›\n discuss goal 1*)\n apply ((insert W (*‹W ∈ carrier_mat 2 2›*) ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*))[1])\n (*top goal: ‹ab * W + Matrix.mat 1 (n - 2) (λ(i, j). 0) * 0⇩m (n - 2) 2 = ab * W› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert W (*‹W ∈ carrier_mat 2 2›*) ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*))[1])\n (*top goal: ‹ab * 0⇩m 2 (n - 2) + Matrix.mat 1 (n - 2) (λ(i, j). 0) * 1⇩m (n - 2) = Matrix.mat 1 (n - 2) (λ(i, j). 0)› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert W (*‹W ∈ carrier_mat 2 2›*) ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*))[1])\n (*top goal: ‹Matrix.mat 0 2 (λ(i, j). 0) * W + Matrix.mat 0 (n - 2) (λ(i, j). 0) * 0⇩m (n - 2) 2 = Matrix.mat 0 2 (λ(i, j). 0)› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert W (*‹(W::'a mat) ∈ carrier_mat (2::nat) (2::nat)›*) ab_def (*‹(ab::'a mat) = Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a else (b::'a))›*))[1])\n (*goal: ‹Matrix.mat 0 2 (λ(i, j). 0) * 0⇩m 2 (n - 2) + Matrix.mat 0 (n - 2) (λ(i, j). 0) * 1⇩m (n - 2) = Matrix.mat 0 (n - 2) (λ(i, j). 0)›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n also (*calculation: ‹Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) = four_block_mat (ab * W) (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0))›*) have \"... = ?T\"\n apply (rule eq_matI (*‹⟦⋀(i::nat) j::nat. ⟦i < dim_row (?B::?'a mat); j < dim_col ?B⟧ ⟹ (?A::?'a mat) $$ (i, j) = ?B $$ (i, j); dim_row ?A = dim_row ?B; dim_col ?A = dim_col ?B⟧ ⟹ ?A = ?B›*))\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0)); j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0))⟧ ⟹ four_block_mat (ab * W) (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)) $$ (i, j) = Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0) $$ (i, j)›\n 2. ‹dim_row (four_block_mat (ab * W) (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0))) = dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0))›\n 3. ‹dim_col (four_block_mat (ab * W) (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0))) = dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0))›\n discuss goal 1*)\n apply ((insert W (*‹W ∈ carrier_mat 2 2›*) n_ge_2 (*‹2 < n›*) ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*) ab_W (*‹lower_triangular (ab * W)›*))[1])\n (*top goal: ‹⋀i j. ⟦i < dim_row (Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0)); j < dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0))⟧ ⟹ four_block_mat (ab * W) (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0)) $$ (i, j) = Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0) $$ (i, j)› and 2 goals remain*)\n apply ((auto simp add: lower_triangular_def (*‹lower_triangular ?A = (∀i j. i < j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 2*)\n apply ((insert W (*‹W ∈ carrier_mat 2 2›*) n_ge_2 (*‹2 < n›*) ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*) ab_W (*‹lower_triangular (ab * W)›*))[1])\n (*top goal: ‹dim_row (four_block_mat ((ab::'a mat) * (W::'a mat)) (Matrix.mat (1::nat) ((n::nat) - (2::nat)) (λ(i::nat, j::nat). 0::'a)) (Matrix.mat (0::nat) (2::nat) (λ(i::nat, j::nat). 0::'a)) (Matrix.mat (0::nat) (n - (2::nat)) (λ(i::nat, j::nat). 0::'a))) = dim_row (Matrix.mat (1::nat) n (λ(i::nat, j::nat). if j = (0::nat) then (ab * W) $$ (0::nat, 0::nat) else (0::'a)))› and 1 goal remains*)\n apply ((auto simp add: lower_triangular_def (*‹lower_triangular ?A = (∀i j. i < j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*discuss goal 3*)\n apply ((insert W (*‹W ∈ carrier_mat 2 2›*) n_ge_2 (*‹2 < n›*) ab_def (*‹ab = Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)›*) ab_W (*‹lower_triangular (ab * W)›*))[1])\n (*goal: ‹dim_col (four_block_mat (ab * W) (Matrix.mat 1 (n - 2) (λ(i, j). 0)) (Matrix.mat 0 2 (λ(i, j). 0)) (Matrix.mat 0 (n - 2) (λ(i, j). 0))) = dim_col (Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0))›*)\n apply ((auto simp add: lower_triangular_def (*‹lower_triangular ?A = (∀i j. i < j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*))[1])\n (*proven 3 subgoals*) .\n finally (*calculation: ‹Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) = Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0)›*) show \"?thesis\"\n (*goal: ‹Matrix.mat 1 n (λ(i, j). if j = 0 then a else if j = 1 then b else 0) * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) = Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0)›*) .\n qed\n finally (*calculation: ‹A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) = Matrix.mat 1 n (λa. case a of (i, j) ⇒ if j = 0 then (ab * W) $$ (0, 0) else 0)›*) have \"A * U = ?T\"\n using assoc_mult_mat[OF _ U1 U2] (*‹?A ∈ carrier_mat ?n₁ n ⟹ ?A * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)) = ?A * (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2)))›*) less.prems (*‹A ∈ carrier_mat 1 n›*) unfolding U_def\n (*goal: ‹A * (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) V * four_block_mat W (0⇩m 2 (n - 2)) (0⇩m (n - 2) 2) (1⇩m (n - 2))) = Matrix.mat 1 n (λ(i, j). if j = 0 then (ab * W) $$ (0, 0) else 0)›*)\n by auto\n moreover have \"lower_triangular ?T\"\n unfolding lower_triangular_def\n (*goal: ‹∀(i::nat) j::nat. i < j ∧ i < dim_row (Matrix.mat (1::nat) (n::nat) (λ(i::nat, j::nat). if j = (0::nat) then ((ab::'a mat) * (W::'a mat)) $$ (0::nat, 0::nat) else (0::'a))) ∧ j < dim_col (Matrix.mat (1::nat) n (λ(i::nat, j::nat). if j = (0::nat) then (ab * W) $$ (0::nat, 0::nat) else (0::'a))) ⟶ Matrix.mat (1::nat) n (λ(i::nat, j::nat). if j = (0::nat) then (ab * W) $$ (0::nat, 0::nat) else (0::'a)) $$ (i, j) = (0::'a)›*)\n by simp\n ultimately show \"?thesis\"\n (*goal: ‹lower_triangular (A * U)›*)\n by simp\n qed\n ultimately show \"?thesis\"\n (*goal: ‹∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n using A (*‹A ∈ carrier_mat 1 n›*) U (*‹U ∈ carrier_mat n n›*) by blast\n qed\n qed\n qed\n from this (*‹∃U::'a mat. U ∈ carrier_mat (dim_col (A::'a mat)) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*) show \"admits_triangular_reduction A\"\n unfolding admits_triangular_reduction_def\n (*goal: ‹∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n by simp\nqed\n\nlemma admits_diagonal_imp_admits_triangular:\n assumes a: \"∀A∈carrier_mat 1 2. admits_diagonal_reduction (A::'a mat)\"\n shows \"∀A. admits_triangular_reduction (A::'a mat)\"\nproof (standard)\n (*goal: ‹⋀A. admits_triangular_reduction A›*)\n fix A :: \"'a mat\"\n obtain m and n where A: \"A ∈ carrier_mat m n\"\n (*goal: ‹(⋀m n. A ∈ carrier_mat m n ⟹ thesis) ⟹ thesis›*)\n by auto\n have \"∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)\"\n using A (*‹(A::'a::comm_ring_1 mat) ∈ carrier_mat (m::nat) (n::nat)›*) proof (induct n arbitrary: m A rule: less_induct)\n (*goal: ‹⋀x m A. ⟦⋀y m A. ⟦y < x; A ∈ carrier_mat m y⟧ ⟹ ∃U. U ∈ carrier_mat y y ∧ invertible_mat U ∧ lower_triangular (A * U); A ∈ carrier_mat m x⟧ ⟹ ∃U. U ∈ carrier_mat x x ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case (less n) (*‹⟦(?y::nat) < (n::nat); (?A::'a mat) ∈ carrier_mat (?m::nat) ?y⟧ ⟹ ∃U::'a mat. U ∈ carrier_mat ?y ?y ∧ invertible_mat U ∧ lower_triangular (?A * U)› ‹A ∈ carrier_mat m n›*)\n note A = less.prems(1) (*‹A ∈ carrier_mat m n›*)\n show \"?case\"\n (*goal: ‹∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n proof (cases \"n=0 ∨ m=0\")\n (*goals:\n 1. ‹n = 0 ∨ m = 0 ⟹ ∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›\n 2. ‹¬ (n = 0 ∨ m = 0) ⟹ ∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case True (*‹(n::nat) = (0::nat) ∨ (m::nat) = (0::nat)›*)\n then show \"?thesis\"\n (*goal: ‹∃U::'a::comm_ring_1 mat. U ∈ carrier_mat (n::nat) n ∧ invertible_mat U ∧ lower_triangular ((A::'a::comm_ring_1 mat) * U)›*)\n using triangularizable_m0 (*‹?A ∈ carrier_mat ?m 0 ⟹ ∃U. U ∈ carrier_mat 0 0 ∧ invertible_mat U ∧ lower_triangular (?A * U)›*) triangularizable_0n (*‹?A ∈ carrier_mat 0 ?n ⟹ ∃U. U ∈ carrier_mat ?n ?n ∧ invertible_mat U ∧ lower_triangular (?A * U)›*) less.prems (*‹A ∈ carrier_mat m n›*) by auto\n next\n (*goal: ‹¬ (n = 0 ∨ m = 0) ⟹ ∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case False (*‹¬ (n = 0 ∨ m = 0)›*)\n note nm_not_0 = False (*‹¬ (n = 0 ∨ m = 0)›*)\n from this (*‹¬ (n = 0 ∨ m = 0)›*) have m_not_0: \"m ≠ 0\" and n_not_0: \"n ≠ 0\"\n apply -\n (*goals:\n 1. ‹¬ (n = 0 ∨ m = 0) ⟹ m ≠ 0›\n 2. ‹¬ (n = 0 ∨ m = 0) ⟹ n ≠ 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n show \"?thesis\"\n (*goal: ‹∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n proof (cases \"m = 1\")\n (*goals:\n 1. ‹m = 1 ⟹ ∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›\n 2. ‹m ≠ 1 ⟹ ∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case True (*‹m = 1›*)\n note m1 = True (*‹m = 1›*)\n show \"?thesis\"\n (*goal: ‹∃U::'a mat. U ∈ carrier_mat (n::nat) n ∧ invertible_mat U ∧ lower_triangular ((A::'a mat) * U)›*)\n using admits_diagonal_imp_admits_triangular_1xn (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A ⟹ ∀A∈carrier_mat 1 ?n. admits_triangular_reduction A›*) A (*‹A ∈ carrier_mat m n›*) m1 (*‹m = 1›*) a (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A›*) unfolding admits_triangular_reduction_def\n (*goal: ‹∃U::'a mat. U ∈ carrier_mat (n::nat) n ∧ invertible_mat U ∧ lower_triangular ((A::'a mat) * U)›*)\n by blast\n next\n (*goal: ‹(m::nat) ≠ (1::nat) ⟹ ∃U::'a mat. U ∈ carrier_mat (n::nat) n ∧ invertible_mat U ∧ lower_triangular ((A::'a mat) * U)›*)\n case False (*‹m ≠ 1›*)\n note m_not_1 = False (*‹(m::nat) ≠ (1::nat)›*)\n show \"?thesis\"\n (*goal: ‹∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n proof (cases \"n=1\")\n (*goals:\n 1. ‹n = 1 ⟹ ∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›\n 2. ‹n ≠ 1 ⟹ ∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n case True (*‹n = 1›*)\n thus \"?thesis\"\n (*goal: ‹∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n using invertible_mat_zero (*‹?A ∈ carrier_mat 0 0 ⟹ invertible_mat ?A›*) lower_triangular_def (*‹lower_triangular ?A = (∀i j. i < j ∧ i < dim_row ?A ∧ j < dim_col ?A ⟶ ?A $$ (i, j) = 0)›*) by (metis carrier_matD( (*‹(?A::?'a mat) ∈ carrier_mat (?nr::nat) (?nc::nat) ⟹ dim_col ?A = ?nc›*) 2) det_one (*‹Determinant.det (1⇩m (?n::nat)) = (1::?'a)›*) gr_implies_not0 (*‹(?m::nat) < (?n::nat) ⟹ ?n ≠ (0::nat)›*) invertible_iff_is_unit_JNF (*‹(?A::?'a mat) ∈ carrier_mat (?n::nat) ?n ⟹ invertible_mat ?A = (Determinant.det ?A dvd (1::?'a))›*) less( (*‹(A::'a mat) ∈ carrier_mat (m::nat) (n::nat)›*) 2) less_one (*‹((?n::nat) < (1::nat)) = (?n = (0::nat))›*) one_carrier_mat (*‹1⇩m (?n::nat) ∈ carrier_mat ?n ?n›*) right_mult_one_mat' (*‹dim_col (?A::?'a mat) = (?n::nat) ⟹ ?A * 1⇩m ?n = ?A›*))\n next\n (*goal: ‹(n::nat) ≠ (1::nat) ⟹ ∃U::'a mat. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular ((A::'a mat) * U)›*)\n case False (*‹n ≠ 1›*)\n note n_not_1 = False (*‹n ≠ 1›*)\n let ?first_row = \"mat_of_row (Matrix.row A 0)\"\n have first_row: \"?first_row ∈ carrier_mat 1 n\"\n using less.prems (*‹A ∈ carrier_mat m n›*) by auto\n have m1: \"m>1\"\n using m_not_1 (*‹m ≠ 1›*) m_not_0 (*‹m ≠ 0›*) by linarith\n have n1: \"n>1\"\n using n_not_1 (*‹n ≠ 1›*) n_not_0 (*‹(n::nat) ≠ (0::nat)›*) by linarith\n obtain V where lt_first_row_V: \"lower_triangular (?first_row * V)\" and inv_V: \"invertible_mat V\" and V: \"V ∈ carrier_mat n n\"\n (*goal: ‹(⋀V::'a mat. ⟦lower_triangular (mat_of_row (Matrix.row (A::'a mat) (0::nat)) * V); invertible_mat V; V ∈ carrier_mat (n::nat) n⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using admits_diagonal_imp_admits_triangular_1xn (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A ⟹ ∀A∈carrier_mat 1 ?n. admits_triangular_reduction A›*) a (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A›*) first_row (*‹mat_of_row (Matrix.row A 0) ∈ carrier_mat 1 n›*) unfolding admits_triangular_reduction_def\n (*goal: ‹(⋀V. ⟦lower_triangular (mat_of_row (Matrix.row A 0) * V); invertible_mat V; V ∈ carrier_mat n n⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n have AV: \"A*V ∈ carrier_mat m n\"\n using V (*‹V ∈ carrier_mat n n›*) less (*‹⟦?y < n; ?A ∈ carrier_mat ?m ?y⟧ ⟹ ∃U. U ∈ carrier_mat ?y ?y ∧ invertible_mat U ∧ lower_triangular (?A * U)› ‹A ∈ carrier_mat m n›*) by auto\n have dim_row_AV: \"dim_row (A * V) = 1 + (m-1)\"\n using m1 (*‹1 < m›*) AV (*‹A * V ∈ carrier_mat m n›*) by auto\n have dim_col_AV: \"dim_col (A * V) = 1 + (n-1)\"\n using n1 (*‹1 < n›*) AV (*‹(A::'a mat) * (V::'a mat) ∈ carrier_mat (m::nat) (n::nat)›*) by fastforce\n have reduced_first_row: \"Matrix.row (?first_row * V) 0 = Matrix.row (A * V) 0\"\n by (rule mult_eq_first_row (*‹⟦?A ∈ carrier_mat 1 ?n; ?B ∈ carrier_mat ?m ?n; ?m ≠ 0; Matrix.row ?A 0 = Matrix.row ?B 0⟧ ⟹ Matrix.row (?A * ?V) 0 = Matrix.row (?B * ?V) 0›*), insert first_row (*‹mat_of_row (Matrix.row A 0) ∈ carrier_mat 1 n›*) m1 (*‹1 < m›*) less.prems (*‹A ∈ carrier_mat m n›*), auto)\n obtain a and zero and B and C where split: \"split_block (A*V) 1 1 = (a, zero, B, C)\"\n (*goal: ‹(⋀a zero B C. split_block (A * V) 1 1 = (a, zero, B, C) ⟹ thesis) ⟹ thesis›*)\n using prod_cases4 (*‹(⋀(a::?'a) (b::?'b) (c::?'c) d::?'d. (?y::?'a × ?'b × ?'c × ?'d) = (a, b, c, d) ⟹ ?thesis::bool) ⟹ ?thesis›*) by blast\n have a: \"a ∈ carrier_mat 1 1\" and zero: \"zero ∈ carrier_mat 1 (n-1)\" and B: \"B ∈ carrier_mat (m-1) 1\" and C: \"C ∈ carrier_mat (m-1) (n-1)\"\n (*goals:\n 1. ‹a ∈ carrier_mat 1 1›\n 2. ‹zero ∈ carrier_mat 1 (n - 1)›\n 3. ‹B ∈ carrier_mat (m - 1) 1›\n 4. ‹C ∈ carrier_mat (m - 1) (n - 1)›\n discuss goal 1*)\n apply (rule split_block[OF split dim_row_AV dim_col_AV] (*‹a ∈ carrier_mat 1 1› ‹zero ∈ carrier_mat 1 (n - 1)› ‹B ∈ carrier_mat (m - 1) 1› ‹C ∈ carrier_mat (m - 1) (n - 1)› ‹A * V = four_block_mat a zero B C›*))\n (*discuss goal 2*)\n apply (rule split_block[OF split dim_row_AV dim_col_AV] (*‹a ∈ carrier_mat 1 1› ‹zero ∈ carrier_mat 1 (n - 1)› ‹B ∈ carrier_mat (m - 1) 1› ‹C ∈ carrier_mat (m - 1) (n - 1)› ‹A * V = four_block_mat a zero B C›*))\n (*discuss goal 3*)\n apply (rule split_block[OF split dim_row_AV dim_col_AV] (*‹a ∈ carrier_mat 1 1› ‹zero ∈ carrier_mat 1 (n - 1)› ‹B ∈ carrier_mat (m - 1) 1› ‹C ∈ carrier_mat (m - 1) (n - 1)› ‹A * V = four_block_mat a zero B C›*))\n (*discuss goal 4*)\n apply (rule split_block[OF split dim_row_AV dim_col_AV] (*‹a ∈ carrier_mat 1 1› ‹zero ∈ carrier_mat 1 (n - 1)› ‹B ∈ carrier_mat (m - 1) 1› ‹C ∈ carrier_mat (m - 1) (n - 1)› ‹A * V = four_block_mat a zero B C›*))\n (*proven 4 subgoals*) .\n have AV_block: \"A*V = four_block_mat a zero B C\"\n by (rule split_block[OF split dim_row_AV dim_col_AV] (*‹a ∈ carrier_mat 1 1› ‹zero ∈ carrier_mat 1 (n - 1)› ‹B ∈ carrier_mat (m - 1) 1› ‹C ∈ carrier_mat (m - 1) (n - 1)› ‹A * V = four_block_mat a zero B C›*))\n have \"∃W. W∈ carrier_mat (n-1) (n-1) ∧ invertible_mat W ∧ lower_triangular (C*W)\"\n apply (rule less.hyps (*‹⟦?y < n; ?A ∈ carrier_mat ?m ?y⟧ ⟹ ∃U. U ∈ carrier_mat ?y ?y ∧ invertible_mat U ∧ lower_triangular (?A * U)›*))\n (*goals:\n 1. ‹n - 1 < n›\n 2. ‹C ∈ carrier_mat ?m (n - 1)›\n discuss goal 1*)\n apply ((insert n1 (*‹(1::nat) < (n::nat)›*) C (*‹(C::'a mat) ∈ carrier_mat ((m::nat) - (1::nat)) ((n::nat) - (1::nat))›*))[1])\n (*top goal: ‹(n::nat) - (1::nat) < n› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert n1 (*‹1 < n›*) C (*‹C ∈ carrier_mat (m - 1) (n - 1)›*))[1])\n (*goal: ‹C ∈ carrier_mat ?m (n - 1)›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from this (*‹∃W. W ∈ carrier_mat (n - 1) (n - 1) ∧ invertible_mat W ∧ lower_triangular (C * W)›*) obtain W where inv_W: \"invertible_mat W\" and lt_CW: \"lower_triangular (C*W)\" and W: \"W ∈ carrier_mat (n-1) (n-1)\"\n (*goal: ‹(⋀W. ⟦invertible_mat W; lower_triangular (C * W); W ∈ carrier_mat (n - 1) (n - 1)⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n let ?W2 = \"four_block_mat (1⇩m 1) (0⇩m 1 (n-1)) (0⇩m (n-1) 1) W\"\n have W2: \"?W2 ∈ carrier_mat n n\"\n using V (*‹V ∈ carrier_mat n n›*) W (*‹W ∈ carrier_mat (n - 1) (n - 1)›*) dim_col_AV (*‹dim_col (A * V) = 1 + (n - 1)›*) by auto\n have \"Determinant.det ?W2 = Determinant.det (1⇩m 1) * Determinant.det W\"\n apply (rule det_four_block_mat_lower_left_zero_col[OF _ _ _ W] (*‹⟦?A1.0 ∈ carrier_mat 1 1; ?A2.0 ∈ carrier_mat 1 (n - 1); ?A3.0 = 0⇩m (n - 1) 1⟧ ⟹ Determinant.det (four_block_mat ?A1.0 ?A2.0 ?A3.0 W) = Determinant.det ?A1.0 * Determinant.det W›*))\n (*goals:\n 1. ‹1⇩m 1 ∈ carrier_mat 1 1›\n 2. ‹0⇩m 1 (n - 1) ∈ carrier_mat 1 (n - 1)›\n 3. ‹0⇩m (n - 1) 1 = 0⇩m (n - 1) 1›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n hence det_W2: \"Determinant.det ?W2 = Determinant.det W\"\n by auto\n hence inv_W2: \"invertible_mat ?W2\"\n by (metis W (*‹W ∈ carrier_mat (n - 1) (n - 1)›*) four_block_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr1.0 ?nc1.0; ?D ∈ carrier_mat ?nr2.0 ?nc2.0⟧ ⟹ four_block_mat ?A ?B ?C ?D ∈ carrier_mat (?nr1.0 + ?nr2.0) (?nc1.0 + ?nc2.0)›*) inv_W (*‹invertible_mat W›*) invertible_iff_is_unit_JNF (*‹?A ∈ carrier_mat ?n ?n ⟹ invertible_mat ?A = (Determinant.det ?A dvd 1)›*) one_carrier_mat (*‹1⇩m ?n ∈ carrier_mat ?n ?n›*))\n have inv_V_W2: \"invertible_mat (V * ?W2)\"\n using inv_W2 (*‹invertible_mat (four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) W)›*) inv_V (*‹invertible_mat V›*) V (*‹V ∈ carrier_mat n n›*) W2 (*‹four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) W ∈ carrier_mat n n›*) invertible_mult_JNF (*‹⟦?A ∈ carrier_mat ?n ?n; ?B ∈ carrier_mat ?n ?n; invertible_mat ?A; invertible_mat ?B⟧ ⟹ invertible_mat (?A * ?B)›*) by blast\n have \"lower_triangular (A*V*?W2)\"\n proof (-)\n (*goal: ‹lower_triangular (A * V * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) W)›*)\n let ?T = \"(four_block_mat a (0⇩m 1 (n-1)) B (C * W))\"\n have zero_eq: \"zero = 0⇩m 1 (n-1)\"\n proof (rule eq_matI (*‹⟦⋀i j. ⟦i < dim_row ?B; j < dim_col ?B⟧ ⟹ ?A $$ (i, j) = ?B $$ (i, j); dim_row ?A = dim_row ?B; dim_col ?A = dim_col ?B⟧ ⟹ ?A = ?B›*))\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (0⇩m 1 (n - 1)); j < dim_col (0⇩m 1 (n - 1))⟧ ⟹ zero $$ (i, j) = 0⇩m 1 (n - 1) $$ (i, j)›\n 2. ‹dim_row zero = dim_row (0⇩m 1 (n - 1))›\n 3. ‹dim_col zero = dim_col (0⇩m 1 (n - 1))›*)\n show 1: \"dim_row zero = dim_row (0⇩m 1 (n - 1))\" and 2: \"dim_col zero = dim_col (0⇩m 1 (n - 1))\"\n using zero (*‹zero ∈ carrier_mat 1 (n - 1)›*) apply -\n (*goals:\n 1. ‹zero ∈ carrier_mat 1 (n - 1) ⟹ dim_row zero = dim_row (0⇩m 1 (n - 1))›\n 2. ‹zero ∈ carrier_mat 1 (n - 1) ⟹ dim_col zero = dim_col (0⇩m 1 (n - 1))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n fix i and j\n assume i: \"i < dim_row (0⇩m 1 (n - 1))\" and j: \"j < dim_col (0⇩m 1 (n - 1))\" (*‹(i::nat) < dim_row (0⇩m (1::nat) ((n::nat) - (1::nat)))› ‹(j::nat) < dim_col (0⇩m (1::nat) ((n::nat) - (1::nat)))›*)\n have i0: \"i=0\"\n using i (*‹i < dim_row (0⇩m 1 (n - 1))›*) by auto\n have \"0 = Matrix.row (?first_row * V) 0 $v (j+1)\"\n using lt_first_row_V (*‹lower_triangular (mat_of_row (Matrix.row A 0) * V)›*) j (*‹j < dim_col (0⇩m 1 (n - 1))›*) unfolding lower_triangular_def\n (*goal: ‹0 = Matrix.row (mat_of_row (Matrix.row A 0) * V) 0 $v (j + 1)›*)\n by (metis Suc_eq_plus1 (*‹Suc ?n = ?n + 1›*) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_col ?A = ?nc›*) 2) index_mult_mat( (*‹dim_row (?A * ?B) = dim_row ?A› ‹dim_col (?A * ?B) = dim_col ?B›*) 2,3) index_row( (*‹⟦?i < dim_row ?A; ?j < dim_col ?A⟧ ⟹ Matrix.row ?A ?i $v ?j = ?A $$ (?i, ?j)›*) 1) less_diff_conv (*‹(?i < ?j - ?k) = (?i + ?k < ?j)›*) mat_of_row_dim( (*‹dim_row (mat_of_row ?y) = 1›*) 1) zero (*‹zero ∈ carrier_mat 1 (n - 1)›*) zero_less_Suc (*‹0 < Suc ?n›*) zero_less_one_class.zero_less_one (*‹0 < 1›*) V (*‹V ∈ carrier_mat n n›*) 2 (*‹dim_col zero = dim_col (0⇩m 1 (n - 1))›*))\n also (*calculation: ‹0 = Matrix.row (mat_of_row (Matrix.row A 0) * V) 0 $v (j + 1)›*) have \"... = Matrix.row (A*V) 0 $v (j+1)\"\n by (simp add: reduced_first_row (*‹Matrix.row (mat_of_row (Matrix.row (A::'a mat) (0::nat)) * (V::'a mat)) (0::nat) = Matrix.row (A * V) (0::nat)›*))\n also (*calculation: ‹0 = Matrix.row (A * V) 0 $v (j + 1)›*) have \"... = (A*V) $$ (i, j+1)\"\n using V (*‹V ∈ carrier_mat n n›*) dim_row_AV (*‹dim_row (A * V) = 1 + (m - 1)›*) i0 (*‹i = 0›*) j (*‹j < dim_col (0⇩m 1 (n - 1))›*) by auto\n also (*calculation: ‹0 = (A * V) $$ (i, j + 1)›*) have \"... = four_block_mat a zero B C $$ (i, j+1)\"\n by (simp add: AV_block (*‹A * V = four_block_mat a zero B C›*))\n also (*calculation: ‹0 = four_block_mat a zero B C $$ (i, j + 1)›*) have \"... = (if i < dim_row a then if (j+1) < dim_col a \n then a $$ (i, (j+1)) else zero $$ (i, (j+1) - dim_col a) else if (j+1) < dim_col a\n then B $$ (i - dim_row a, (j+1)) else C $$ (i - dim_row a, (j+1) - dim_col a))\"\n apply (rule index_mat_four_block (*‹⟦?i < dim_row ?A + dim_row ?D; ?j < dim_col ?A + dim_col ?D⟧ ⟹ four_block_mat ?A ?B ?C ?D $$ (?i, ?j) = (if ?i < dim_row ?A then if ?j < dim_col ?A then ?A $$ (?i, ?j) else ?B $$ (?i, ?j - dim_col ?A) else if ?j < dim_col ?A then ?C $$ (?i - dim_row ?A, ?j) else ?D $$ (?i - dim_row ?A, ?j - dim_col ?A))› ‹dim_row (four_block_mat ?A ?B ?C ?D) = dim_row ?A + dim_row ?D› ‹dim_col (four_block_mat ?A ?B ?C ?D) = dim_col ?A + dim_col ?D›*))\n (*goals:\n 1. ‹i < dim_row a + dim_row C›\n 2. ‹j + 1 < dim_col a + dim_col C›\n discuss goal 1*)\n apply ((insert a (*‹a ∈ carrier_mat 1 1›*) zero (*‹zero ∈ carrier_mat 1 (n - 1)›*) i (*‹i < dim_row (0⇩m 1 (n - 1))›*) j (*‹j < dim_col (0⇩m 1 (n - 1))›*) C (*‹C ∈ carrier_mat (m - 1) (n - 1)›*))[1])\n (*top goal: ‹i < dim_row a + dim_row C› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert a (*‹a ∈ carrier_mat 1 1›*) zero (*‹zero ∈ carrier_mat 1 (n - 1)›*) i (*‹i < dim_row (0⇩m 1 (n - 1))›*) j (*‹j < dim_col (0⇩m 1 (n - 1))›*) C (*‹C ∈ carrier_mat (m - 1) (n - 1)›*))[1])\n (*goal: ‹(j::nat) + (1::nat) < dim_col (a::'a::comm_ring_1 mat) + dim_col (C::'a::comm_ring_1 mat)›*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹0 = (if i < dim_row a then if j + 1 < dim_col a then a $$ (i, j + 1) else zero $$ (i, j + 1 - dim_col a) else if j + 1 < dim_col a then B $$ (i - dim_row a, j + 1) else C $$ (i - dim_row a, j + 1 - dim_col a))›*) have \"... = zero $$ (i, (j+1) - dim_col a)\"\n using a (*‹a ∈ carrier_mat 1 1›*) zero (*‹(zero::'a mat) ∈ carrier_mat (1::nat) ((n::nat) - (1::nat))›*) i (*‹i < dim_row (0⇩m 1 (n - 1))›*) j (*‹(j::nat) < dim_col (0⇩m (1::nat) ((n::nat) - (1::nat)))›*) C (*‹C ∈ carrier_mat (m - 1) (n - 1)›*) by auto\n also (*calculation: ‹0 = zero $$ (i, j + 1 - dim_col a)›*) have \"... = zero $$ (i, j)\"\n using a (*‹a ∈ carrier_mat 1 1›*) i (*‹i < dim_row (0⇩m 1 (n - 1))›*) by auto\n finally (*calculation: ‹0 = zero $$ (i, j)›*) show \"zero $$ (i, j) = 0⇩m 1 (n - 1) $$ (i, j)\"\n using i (*‹i < dim_row (0⇩m 1 (n - 1))›*) j (*‹j < dim_col (0⇩m 1 (n - 1))›*) by auto\n qed\n have rw1: \"a * (1⇩m 1) + zero * (0⇩m (n-1) 1) = a\"\n using a (*‹a ∈ carrier_mat 1 1›*) zero (*‹zero ∈ carrier_mat 1 (n - 1)›*) by auto\n have rw2: \"a * (0⇩m 1 (n-1)) + zero * W = 0⇩m 1 (n-1)\"\n using a (*‹a ∈ carrier_mat 1 1›*) zero (*‹zero ∈ carrier_mat 1 (n - 1)›*) zero_eq (*‹zero = 0⇩m 1 (n - 1)›*) W (*‹W ∈ carrier_mat (n - 1) (n - 1)›*) by auto\n have rw3: \"B * (1⇩m 1) + C * (0⇩m (n-1) 1) = B\"\n using B (*‹B ∈ carrier_mat (m - 1) 1›*) C (*‹C ∈ carrier_mat (m - 1) (n - 1)›*) by auto\n have rw4: \"B * (0⇩m 1 (n-1)) + C * W = C * W\"\n using B (*‹B ∈ carrier_mat (m - 1) 1›*) C (*‹C ∈ carrier_mat (m - 1) (n - 1)›*) W (*‹(W::'a mat) ∈ carrier_mat ((n::nat) - (1::nat)) (n - (1::nat))›*) by auto\n have \"A*V = four_block_mat a zero B C\"\n by (rule AV_block (*‹(A::'a::comm_ring_1 mat) * (V::'a::comm_ring_1 mat) = four_block_mat (a::'a::comm_ring_1 mat) (zero::'a::comm_ring_1 mat) (B::'a::comm_ring_1 mat) (C::'a::comm_ring_1 mat)›*))\n also (*calculation: ‹A * V = four_block_mat a zero B C›*) have \"... * ?W2 = four_block_mat (a * (1⇩m 1) + zero * (0⇩m (n-1) 1)) \n (a * (0⇩m 1 (n-1)) + zero * W) (B * (1⇩m 1) + C * (0⇩m (n-1) 1))\n (B * (0⇩m 1 (n-1)) + C * W)\"\n apply (rule mult_four_block_mat[OF a zero B C] (*‹⟦?A2.0 ∈ carrier_mat 1 ?nc1.0; ?B2.0 ∈ carrier_mat 1 ?nc2.0; ?C2.0 ∈ carrier_mat (n - 1) ?nc1.0; ?D2.0 ∈ carrier_mat (n - 1) ?nc2.0⟧ ⟹ four_block_mat a zero B C * four_block_mat ?A2.0 ?B2.0 ?C2.0 ?D2.0 = four_block_mat (a * ?A2.0 + zero * ?C2.0) (a * ?B2.0 + zero * ?D2.0) (B * ?A2.0 + C * ?C2.0) (B * ?B2.0 + C * ?D2.0)›*))\n (*goals:\n 1. ‹1⇩m (1::nat) ∈ carrier_mat (1::nat) (?nc1.0::nat)›\n 2. ‹0⇩m (1::nat) ((n::nat) - (1::nat)) ∈ carrier_mat (1::nat) (?nc2.0::nat)›\n 3. ‹0⇩m ((n::nat) - (1::nat)) (1::nat) ∈ carrier_mat (n - (1::nat)) (?nc1.0::nat)›\n 4. ‹(W::'a mat) ∈ carrier_mat ((n::nat) - (1::nat)) (?nc2.0::nat)›\n discuss goal 1*)\n apply ((insert W (*‹W ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*top goal: ‹1⇩m (1::nat) ∈ carrier_mat (1::nat) (?nc1.0::nat)› and 3 goals remain*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((insert W (*‹W ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*top goal: ‹0⇩m 1 (n - 1) ∈ carrier_mat 1 ?nc2.0› and 2 goals remain*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((insert W (*‹W ∈ carrier_mat (n - 1) (n - 1)›*))[1])\n (*top goal: ‹0⇩m (n - 1) 1 ∈ carrier_mat (n - 1) (dim_col (1⇩m (Suc 0)))› and 1 goal remains*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((insert W (*‹(W::'a::comm_ring_1 mat) ∈ carrier_mat ((n::nat) - (1::nat)) (n - (1::nat))›*))[1])\n (*goal: ‹W ∈ carrier_mat (n - 1) (dim_col (0⇩m (Suc 0) (n - Suc 0)))›*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n also (*calculation: ‹A * V * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) W = four_block_mat (a * 1⇩m 1 + zero * 0⇩m (n - 1) 1) (a * 0⇩m 1 (n - 1) + zero * W) (B * 1⇩m 1 + C * 0⇩m (n - 1) 1) (B * 0⇩m 1 (n - 1) + C * W)›*) have \"... = ?T\"\n using rw1 (*‹a * 1⇩m 1 + zero * 0⇩m (n - 1) 1 = a›*) rw2 (*‹a * 0⇩m 1 (n - 1) + zero * W = 0⇩m 1 (n - 1)›*) rw3 (*‹B * 1⇩m 1 + C * 0⇩m (n - 1) 1 = B›*) rw4 (*‹B * 0⇩m 1 (n - 1) + C * W = C * W›*) by simp\n finally (*calculation: ‹A * V * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) W = four_block_mat a (0⇩m 1 (n - 1)) B (C * W)›*) have AVW2: \"A*V * ?W2 = ...\" .\n moreover have \"lower_triangular ?T\"\n using lt_CW (*‹lower_triangular (C * W)›*) unfolding lower_triangular_def\n (*goal: ‹∀(i::nat) j::nat. i < j ∧ i < dim_row (four_block_mat (a::'a mat) (0⇩m (1::nat) ((n::nat) - (1::nat))) (B::'a mat) ((C::'a mat) * (W::'a mat))) ∧ j < dim_col (four_block_mat a (0⇩m (1::nat) (n - (1::nat))) B (C * W)) ⟶ four_block_mat a (0⇩m (1::nat) (n - (1::nat))) B (C * W) $$ (i, j) = (0::'a)›*)\n using a (*‹(a::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (1::nat)›*) zero (*‹(zero::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) ((n::nat) - (1::nat))›*) B (*‹B ∈ carrier_mat (m - 1) 1›*) C (*‹C ∈ carrier_mat (m - 1) (n - 1)›*) W (*‹W ∈ carrier_mat (n - 1) (n - 1)›*) apply auto\n (*goal: ‹∀i j. i < j ∧ i < dim_row (four_block_mat a (0⇩m 1 (n - 1)) B (C * W)) ∧ j < dim_col (four_block_mat a (0⇩m 1 (n - 1)) B (C * W)) ⟶ four_block_mat a (0⇩m 1 (n - 1)) B (C * W) $$ (i, j) = 0›*)\n by (metis (full_types) Suc_less_eq (*‹(Suc ?m < Suc ?n) = (?m < ?n)›*) Suc_pred (*‹0 < ?n ⟹ Suc (?n - Suc 0) = ?n›*) basic_trans_rules( (*‹⟦?x < ?y; ?y < ?z⟧ ⟹ ?x < ?z›*) 19))\n ultimately show \"?thesis\"\n (*goal: ‹lower_triangular ((A::'a mat) * (V::'a mat) * four_block_mat (1⇩m (1::nat)) (0⇩m (1::nat) ((n::nat) - (1::nat))) (0⇩m (n - (1::nat)) (1::nat)) (W::'a mat))›*)\n by simp\n qed\n then show \"?thesis\"\n (*goal: ‹∃U. U ∈ carrier_mat n n ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n using inv_V_W2 (*‹invertible_mat (V * four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) W)›*) V (*‹V ∈ carrier_mat n n›*) W2 (*‹four_block_mat (1⇩m 1) (0⇩m 1 (n - 1)) (0⇩m (n - 1) 1) W ∈ carrier_mat n n›*) less.prems (*‹A ∈ carrier_mat m n›*) by (smt (verit) assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*))\n qed\n qed\n qed\n qed\n thus \"admits_triangular_reduction A\"\n using A (*‹A ∈ carrier_mat m n›*) unfolding admits_triangular_reduction_def\n (*goal: ‹∃U::'a mat. U ∈ carrier_mat (dim_col (A::'a mat)) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n by simp\nqed\n\ncorollary admits_diagonal_imp_admits_triangular':\n assumes a: \"∀A. admits_diagonal_reduction (A::'a mat)\"\n shows \"∀A. admits_triangular_reduction (A::'a mat)\"\n using admits_diagonal_imp_admits_triangular (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A ⟹ ∀A. admits_triangular_reduction A›*) assms (*‹∀A. admits_diagonal_reduction A›*) by blast\n\n\nlemma admits_triangular_reduction_1x2:\n assumes \"∀A::'a mat. A ∈ carrier_mat 1 2 ⟶ admits_triangular_reduction A\"\n shows \"∀C::'a mat. admits_triangular_reduction C\" \n using admits_diagonal_imp_admits_triangular (*‹∀A∈carrier_mat 1 2. admits_diagonal_reduction A ⟹ ∀A. admits_triangular_reduction A›*) assms (*‹∀A::'a::comm_ring_1 mat. A ∈ carrier_mat (1::nat) (2::nat) ⟶ admits_triangular_reduction A›*) triangular_eq_diagonal_1x2 (*‹(∀A∈carrier_mat 1 2. admits_triangular_reduction A) = (∀A∈carrier_mat 1 2. admits_diagonal_reduction A)›*) by auto \n\n \nlemma Hermite_ring_OFCLASS:\n assumes \"∀A ∈ carrier_mat 1 2. admits_triangular_reduction (A::'a mat)\"\n shows \"OFCLASS('a, Hermite_ring_class)\"\nproof (standard)\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n show \"∀A::'a mat. admits_triangular_reduction A\"\n by (rule admits_diagonal_imp_admits_triangular[OF assms[unfolded triangular_eq_diagonal_1x2]] (*‹∀A::'a::comm_ring_1 mat. admits_triangular_reduction A›*))\nqed\n\nlemma Hermite_ring_OFCLASS':\n assumes \"∀A ∈ carrier_mat 1 2.admits_diagonal_reduction (A::'a mat)\"\n shows \"OFCLASS('a, Hermite_ring_class)\"\nproof (standard)\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n show \"∀A::'a mat. admits_triangular_reduction A\"\n by (rule admits_diagonal_imp_admits_triangular[OF assms] (*‹∀A. admits_triangular_reduction A›*))\nqed\n\nlemma theorem3_part1:\n assumes T: \"(∀a b::'a. ∃ a1 b1 d. a = a1*d ∧ b = b1*d \n ∧ ideal_generated {a1,b1} = ideal_generated {1})\"\n shows \"∀A::'a mat. admits_triangular_reduction A\"\napply (rule admits_triangular_reduction_1x2 (*‹∀A. A ∈ carrier_mat 1 2 ⟶ admits_triangular_reduction A ⟹ ∀C. admits_triangular_reduction C›*))\n(*goal: ‹∀A::'a mat. admits_triangular_reduction A›*)\napply (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n(*goal: ‹∀A. A ∈ carrier_mat 1 2 ⟶ admits_triangular_reduction A›*)\nproof (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹⋀A. A ∈ carrier_mat 1 2 ⟹ admits_triangular_reduction A›*)\n fix A :: \"'a mat\"\n assume A: \"A ∈ carrier_mat 1 2\" (*‹(A::'a mat) ∈ carrier_mat (1::nat) (2::nat)›*)\n let ?a = \"A $$ (0,0)\"\n let ?b = \"A $$ (0,1)\"\n obtain a1 and b1 and d where a: \"?a = a1*d\" and b: \"?b = b1*d\" and i: \"ideal_generated {a1,b1} = ideal_generated {1}\"\n (*goal: ‹(⋀a1 d b1. ⟦A $$ (0, 0) = a1 * d; A $$ (0, 1) = b1 * d; ideal_generated {a1, b1} = ideal_generated {1}⟧ ⟹ thesis) ⟹ thesis›*)\n using T (*‹∀(a::'a) b::'a. ∃(a1::'a) (b1::'a) d::'a. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1::'a}›*) by blast\n obtain s and t where sa1tb1: \"s*a1+t*b1=1\"\n (*goal: ‹(⋀s t. s * a1 + t * b1 = 1 ⟹ thesis) ⟹ thesis›*)\n using ideal_generated_pair_exists_pq1[OF i [ simplified ]] (*‹∃p q. p * a1 + q * b1 = 1›*) by blast\n let ?Q = \"Matrix.mat 2 2 (λ(i,j). if i = 0 ∧ j = 0 then s else\n if i = 0 ∧ j = 1 then -b1 else\n if i = 1 ∧ j = 0 then t else a1)\"\n have Q: \"?Q ∈ carrier_mat 2 2\"\n by auto\n have det_Q: \"Determinant.det ?Q = 1\"\n unfolding det_2[OF Q]\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1) $$ (0, 0) * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1) $$ (1, 1) - Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1) $$ (0, 1) * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1) $$ (1, 0) = 1›*)\n using sa1tb1 (*‹s * a1 + t * b1 = 1›*) by (simp add: mult.commute (*‹?a * ?b = ?b * ?a›*))\n hence inv_Q: \"invertible_mat ?Q\"\n using invertible_iff_is_unit_JNF[OF Q] (*‹invertible_mat (Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then s::'a else if i = (0::nat) ∧ j = (1::nat) then - (b1::'a) else if i = (1::nat) ∧ j = (0::nat) then t::'a else (a1::'a))) = (Determinant.det (Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then s else if i = (0::nat) ∧ j = (1::nat) then - b1 else if i = (1::nat) ∧ j = (0::nat) then t else a1)) dvd (1::'a))›*) by auto\n have lower_AQ: \"lower_triangular (A*?Q)\"\n proof (-)\n (*goal: ‹lower_triangular (A * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1))›*)\n have \"Matrix.row A 0 $v Suc 0 * a1 = Matrix.row A 0 $v 0 * b1\" if j2: \"j<2\" and j0: \"0<j\" for j\n by (metis A (*‹(A::'a mat) ∈ carrier_mat (1::nat) (2::nat)›*) One_nat_def (*‹(1::nat) = Suc (0::nat)›*) a (*‹(A::'a mat) $$ (0::nat, 0::nat) = (a1::'a) * (d::'a)›*) b (*‹(A::'a mat) $$ (0::nat, 1::nat) = (b1::'a) * (d::'a)›*) carrier_matD( (*‹(?A::?'a mat) ∈ carrier_mat (?nr::nat) (?nc::nat) ⟹ dim_row ?A = ?nr›*) 1) carrier_matD( (*‹(?A::?'a mat) ∈ carrier_mat (?nr::nat) (?nc::nat) ⟹ dim_col ?A = ?nc›*) 2) index_row( (*‹⟦(?i::nat) < dim_row (?A::?'a mat); (?j::nat) < dim_col ?A⟧ ⟹ Matrix.row ?A ?i $v ?j = ?A $$ (?i, ?j)›*) 1) lessI (*‹(?n::nat) < Suc ?n›*) more_arith_simps( (*‹(?a::?'a) * (?b::?'a) * (?c::?'a) = ?a * (?b * ?c)›*) 11) mult.commute (*‹(?a::?'a) * (?b::?'a) = ?b * ?a›*) numeral_2_eq_2 (*‹(2::nat) = Suc (Suc (0::nat))›*) pos2 (*‹(0::nat) < (2::nat)›*))\n thus \"?thesis\"\n (*goal: ‹lower_triangular (A * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1))›*)\n unfolding lower_triangular_def\n (*goal: ‹∀(i::nat) j::nat. i < j ∧ i < dim_row ((A::'a mat) * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then s::'a else if i = (0::nat) ∧ j = (1::nat) then - (b1::'a) else if i = (1::nat) ∧ j = (0::nat) then t::'a else (a1::'a))) ∧ j < dim_col (A * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then s else if i = (0::nat) ∧ j = (1::nat) then - b1 else if i = (1::nat) ∧ j = (0::nat) then t else a1)) ⟶ (A * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then s else if i = (0::nat) ∧ j = (1::nat) then - b1 else if i = (1::nat) ∧ j = (0::nat) then t else a1)) $$ (i, j) = (0::'a)›*)\n using A (*‹(A::'a mat) ∈ carrier_mat (1::nat) (2::nat)›*) by (auto simp add: scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $v i * ?w $v i›*) sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*))\n qed\n show \"admits_triangular_reduction A\"\n unfolding admits_triangular_reduction_def\n (*goal: ‹∃U. U ∈ carrier_mat (dim_col A) (dim_col A) ∧ invertible_mat U ∧ lower_triangular (A * U)›*)\n using lower_AQ (*‹lower_triangular (A * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1))›*) inv_Q (*‹invertible_mat (Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then s::'a else if i = (0::nat) ∧ j = (1::nat) then - (b1::'a) else if i = (1::nat) ∧ j = (0::nat) then t::'a else (a1::'a)))›*) Q (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then s else if i = 0 ∧ j = 1 then - b1 else if i = 1 ∧ j = 0 then t else a1) ∈ carrier_mat 2 2›*) A (*‹(A::'a::comm_ring_1 mat) ∈ carrier_mat (1::nat) (2::nat)›*) by force\nqed\n\n\nlemma theorem3_part2:\n assumes 1: \"∀A::'a mat. admits_triangular_reduction A\"\n shows \"∀a b::'a. ∃ a1 b1 d. a = a1*d ∧ b = b1*d ∧ ideal_generated {a1,b1} = ideal_generated {1}\"\napply (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n(*goal: ‹∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*)\nproof (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n (*goal: ‹⋀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*)\n fix a :: 'a and b :: 'a\n let ?A = \"Matrix.mat 1 2 (λ(i,j). if i = 0 ∧ j = 0 then a else b)\"\n obtain Q where AQ: \"lower_triangular (?A*Q)\" and inv_Q: \"invertible_mat Q\" and Q: \"Q ∈ carrier_mat 2 2\"\n (*goal: ‹(⋀Q::'a mat. ⟦lower_triangular (Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a else (b::'a)) * Q); invertible_mat Q; Q ∈ carrier_mat (2::nat) (2::nat)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using \"1\" (*‹∀A. admits_triangular_reduction A›*) unfolding admits_triangular_reduction_def\n (*goal: ‹(⋀Q. ⟦lower_triangular (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q); invertible_mat Q; Q ∈ carrier_mat 2 2⟧ ⟹ thesis) ⟹ thesis›*)\n by fastforce\n hence [simp]: \"dim_col Q = 2\" and [simp]: \"dim_row Q = 2\"\n apply -\n (*goals:\n 1. ‹⟦lower_triangular (Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a else (b::'a)) * (Q::'a mat)); invertible_mat Q; Q ∈ carrier_mat (2::nat) (2::nat)⟧ ⟹ dim_col Q = (2::nat)›\n 2. ‹⟦lower_triangular (Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a else (b::'a)) * (Q::'a mat)); invertible_mat Q; Q ∈ carrier_mat (2::nat) (2::nat)⟧ ⟹ dim_row Q = (2::nat)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n let ?s = \"Q $$ (0,0)\"\n let ?t = \"Q $$ (1,0)\"\n let ?a1 = \"Q $$ (1,1)\"\n let ?b1 = \"-(Q $$ (0,1))\"\n let ?d = \"(?A*Q) $$ (0,0)\"\n have ab1_ba1: \"a*?b1 = b*?a1\"\n proof (-)\n (*goal: ‹a * - Q $$ (0, 1) = b * Q $$ (1, 1)›*)\n have \"(?A*Q) $$ (0,1) = (∑i = 0..<2. (if i = 0 then a else b) * Q $$ (i, Suc 0))\"\n unfolding times_mat_def col_def scalar_prod_def\n (*goal: ‹Matrix.mat (dim_row (Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a else (b::'a)))) (dim_col (Q::'a mat)) (λ(i::nat, j::nat). ∑ia::nat = 0::nat..<dim_vec (Matrix.vec (dim_row Q) (λi::nat. Q $$ (i, j))). Matrix.row (Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a else b)) i $v ia * Matrix.vec (dim_row Q) (λi::nat. Q $$ (i, j)) $v ia) $$ (0::nat, 1::nat) = (∑i::nat = 0::nat..<2::nat. (if i = (0::nat) then a else b) * Q $$ (i, Suc (0::nat)))›*)\n by auto\n also (*calculation: ‹(Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 1) = (∑i = 0..<2. (if i = 0 then a else b) * Q $$ (i, Suc 0))›*) have \"... = (∑i ∈ {0,1}. (if i = 0 then a else b) * Q $$ (i, Suc 0))\"\n apply (rule sum.cong (*‹⟦(?A::?'b::type set) = (?B::?'b::type set); ⋀x::?'b::type. x ∈ ?B ⟹ (?g::?'b::type ⇒ ?'a::comm_monoid_add) x = (?h::?'b::type ⇒ ?'a::comm_monoid_add) x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0::nat..<2::nat} = {0::nat, 1::nat}›\n 2. ‹⋀x::nat. x ∈ {0::nat, 1::nat} ⟹ (if x = (0::nat) then a::'a::comm_ring_1 else (b::'a::comm_ring_1)) * (Q::'a::comm_ring_1 mat) $$ (x, Suc (0::nat)) = (if x = (0::nat) then a else b) * Q $$ (x, Suc (0::nat))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 1) = (∑i∈{0, 1}. (if i = 0 then a else b) * Q $$ (i, Suc 0))›*) have \"... = - a*?b1 + b*?a1\"\n by auto\n finally (*calculation: ‹(Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 1) = - a * - Q $$ (0, 1) + b * Q $$ (1, 1)›*) have \"(?A*Q) $$ (0,1) = - a*?b1 + b*?a1\"\n by simp\n moreover have \"(?A*Q) $$ (0,1) = 0\"\n using AQ (*‹lower_triangular (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q)›*) unfolding lower_triangular_def\n (*goal: ‹(Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a::comm_ring_1 else (b::'a::comm_ring_1)) * (Q::'a::comm_ring_1 mat)) $$ (0::nat, 1::nat) = (0::'a::comm_ring_1)›*)\n by auto\n ultimately show \"?thesis\"\n (*goal: ‹a * - Q $$ (0, 1) = b * Q $$ (1, 1)›*)\n by (metis add_left_cancel (*‹(?a + ?b = ?a + ?c) = (?b = ?c)›*) more_arith_simps( (*‹- ?a + ?a = 0›*) 3) more_arith_simps( (*‹- ?a * ?b = - (?a * ?b)›*) 7))\n qed\n have sa_tb_d: \"?s*a+?t*b = ?d\"\n proof (-)\n (*goal: ‹Q $$ (0, 0) * a + Q $$ (1, 0) * b = (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0)›*)\n have \"?d = (∑i = 0..<2. (if i = 0 then a else b) * Q $$ (i, 0))\"\n unfolding times_mat_def col_def scalar_prod_def\n (*goal: ‹Matrix.mat (dim_row (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b))) (dim_col Q) (λ(i, j). ∑ia = 0..<dim_vec (Matrix.vec (dim_row Q) (λi. Q $$ (i, j))). Matrix.row (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b)) i $v ia * Matrix.vec (dim_row Q) (λi. Q $$ (i, j)) $v ia) $$ (0, 0) = (∑i = 0..<2. (if i = 0 then a else b) * Q $$ (i, 0))›*)\n by auto\n also (*calculation: ‹(Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0) = (∑i = 0..<2. (if i = 0 then a else b) * Q $$ (i, 0))›*) have \"... = (∑i ∈ {0,1}. (if i = 0 then a else b) * Q $$ (i, 0))\"\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0::nat..<2::nat} = {0::nat, 1::nat}›\n 2. ‹⋀x::nat. x ∈ {0::nat, 1::nat} ⟹ (if x = (0::nat) then a::'a::comm_ring_1 else (b::'a::comm_ring_1)) * (Q::'a::comm_ring_1 mat) $$ (x, 0::nat) = (if x = (0::nat) then a else b) * Q $$ (x, 0::nat)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a else (b::'a)) * (Q::'a mat)) $$ (0::nat, 0::nat) = (∑i::nat∈{0::nat, 1::nat}. (if i = (0::nat) then a else b) * Q $$ (i, 0::nat))›*) have \"... = ?s*a+?t*b\"\n by auto\n finally (*calculation: ‹(Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a::'a::comm_ring_1 else (b::'a::comm_ring_1)) * (Q::'a::comm_ring_1 mat)) $$ (0::nat, 0::nat) = Q $$ (0::nat, 0::nat) * a + Q $$ (1::nat, 0::nat) * b›*) show \"?thesis\"\n (*goal: ‹Q $$ (0, 0) * a + Q $$ (1, 0) * b = (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0)›*)\n by simp\n qed\n have det_Q_dvd_1: \"(Determinant.det Q dvd 1)\"\n using invertible_iff_is_unit_JNF[OF Q] (*‹invertible_mat Q = (Determinant.det Q dvd 1)›*) inv_Q (*‹invertible_mat Q›*) by auto\n moreover have det_Q_eq: \"Determinant.det Q = ?s*?a1 + ?t*?b1\"\n unfolding det_2[OF Q]\n (*goal: ‹Q $$ (0, 0) * Q $$ (1, 1) - Q $$ (0, 1) * Q $$ (1, 0) = Q $$ (0, 0) * Q $$ (1, 1) + Q $$ (1, 0) * - Q $$ (0, 1)›*)\n by simp\n ultimately have \"?s*?a1 + ?t*?b1 dvd 1\"\n by auto\n from this (*‹Q $$ (0, 0) * Q $$ (1, 1) + Q $$ (1, 0) * - Q $$ (0, 1) dvd 1›*) obtain u where u_eq: \"?s*?a1 + ?t*?b1 = u\" and u: \"u dvd 1\"\n (*goal: ‹(⋀u::'a. ⟦(Q::'a mat) $$ (0::nat, 0::nat) * Q $$ (1::nat, 1::nat) + Q $$ (1::nat, 0::nat) * - Q $$ (0::nat, 1::nat) = u; u dvd (1::'a)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n hence eq1: \"?s*?a1*a + ?t*?b1*a = u*a\"\n by (metis ring_class.ring_distribs( (*‹((?a::?'a::ring) + (?b::?'a::ring)) * (?c::?'a::ring) = ?a * ?c + ?b * ?c›*) 2))\n hence \"?s*?a1*a + ?t*?a1*b = u*a\"\n by (metis (no_types, lifting) ab1_ba1 (*‹a * - Q $$ (0, 1) = b * Q $$ (1, 1)›*) mult.assoc (*‹?a * ?b * ?c = ?a * (?b * ?c)›*) mult.commute (*‹?a * ?b = ?b * ?a›*))\n hence a1d_ua: \"?a1*?d=u*a\"\n by (smt (verit) Groups.mult_ac( (*‹?a * ?b = ?b * ?a›*) 2) distrib_left (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c›*) more_arith_simps( (*‹?a * ?b * ?c = ?a * (?b * ?c)›*) 11) sa_tb_d (*‹Q $$ (0, 0) * a + Q $$ (1, 0) * b = (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0)›*))\n hence b1d_ub: \"?b1*?d=u*b\"\n by (smt (verit) Groups.mult_ac( (*‹?a * ?b = ?b * ?a›*) 2) Groups.mult_ac( (*‹?b * (?a * ?c) = ?a * (?b * ?c)›*) 3) ab1_ba1 (*‹a * - Q $$ (0, 1) = b * Q $$ (1, 1)›*) distrib_right (*‹(?a + ?b) * ?c = ?a * ?c + ?b * ?c›*) sa_tb_d (*‹Q $$ (0, 0) * a + Q $$ (1, 0) * b = (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0)›*) u_eq (*‹Q $$ (0, 0) * Q $$ (1, 1) + Q $$ (1, 0) * - Q $$ (0, 1) = u›*))\n obtain inv_u where inv_u: \"inv_u * u = 1\"\n (*goal: ‹(⋀inv_u::'a. inv_u * (u::'a) = (1::'a) ⟹ thesis::bool) ⟹ thesis›*)\n using u (*‹u dvd 1›*) unfolding dvd_def\n (*goal: ‹(⋀inv_u. inv_u * u = 1 ⟹ thesis) ⟹ thesis›*)\n by (metis mult.commute (*‹(?a::?'a) * (?b::?'a) = ?b * ?a›*))\n hence inv_u_dvd_1: \"inv_u dvd 1\"\n unfolding dvd_def\n (*goal: ‹∃k. 1 = inv_u * k›*)\n by auto\n have cond1: \"(inv_u*?b1)*?d = b\"\n using b1d_ub (*‹- Q $$ (0, 1) * (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0) = u * b›*) inv_u (*‹inv_u * u = 1›*) by (metis (no_types, lifting) Groups.mult_ac( (*‹?b * (?a * ?c) = ?a * (?b * ?c)›*) 3) more_arith_simps( (*‹?a * ?b * ?c = ?a * (?b * ?c)›*) 11) more_arith_simps( (*‹?a * 1 = ?a›*) 6))\n have cond2: \"(inv_u*?a1)*?d = a\"\n using a1d_ua (*‹Q $$ (1, 1) * (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0) = u * a›*) inv_u (*‹inv_u * u = 1›*) by (metis (no_types, lifting) Groups.mult_ac( (*‹?b * (?a * ?c) = ?a * (?b * ?c)›*) 3) more_arith_simps( (*‹?a * ?b * ?c = ?a * (?b * ?c)›*) 11) more_arith_simps( (*‹?a * 1 = ?a›*) 6))\n have \"ideal_generated {inv_u*?a1, inv_u*?b1} = ideal_generated {?a1,?b1}\"\n by (rule ideal_generated_mult_unit2[OF inv_u_dvd_1] (*‹ideal_generated {inv_u * ?a, inv_u * ?b} = ideal_generated {?a, ?b}›*))\n also (*calculation: ‹ideal_generated {inv_u * Q $$ (1, 1), inv_u * - Q $$ (0, 1)} = ideal_generated {Q $$ (1, 1), - Q $$ (0, 1)}›*) have \"... = UNIV\"\n using ideal_generated_pair_UNIV[OF u_eq u] (*‹ideal_generated {(Q::'a mat) $$ (1::nat, 1::nat), - Q $$ (0::nat, 1::nat)} = UNIV›*) by simp\n finally (*calculation: ‹ideal_generated {inv_u * Q $$ (1, 1), inv_u * - Q $$ (0, 1)} = UNIV›*) have cond3: \"ideal_generated {inv_u*?a1, inv_u*?b1} = ideal_generated {1}\"\n by auto\n show \"∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}\"\n apply (rule exI[of _ \"inv_u*?a1\"] (*‹(?P::'a ⇒ bool) ((inv_u::'a) * (Q::'a mat) $$ (1::nat, 1::nat)) ⟹ ∃x::'a. ?P x›*))\n (*goal: ‹∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*)\n apply (rule exI[of _ \"inv_u*?b1\"] (*‹?P (inv_u * - Q $$ (0, 1)) ⟹ ∃x. ?P x›*))\n (*goal: ‹∃b1 d. a = inv_u * Q $$ (1, 1) * d ∧ b = b1 * d ∧ ideal_generated {inv_u * Q $$ (1, 1), b1} = ideal_generated {1}›*)\n apply (rule exI[of _ ?d] (*‹?P ((Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0)) ⟹ ∃x. ?P x›*))\n (*goal: ‹∃d. a = inv_u * Q $$ (1, 1) * d ∧ b = inv_u * - Q $$ (0, 1) * d ∧ ideal_generated {inv_u * Q $$ (1, 1), inv_u * - Q $$ (0, 1)} = ideal_generated {1}›*)\n apply (insert cond1 (*‹inv_u * - Q $$ (0, 1) * (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0) = b›*) cond2 (*‹inv_u * Q $$ (1, 1) * (Matrix.mat 1 2 (λ(i, j). if i = 0 ∧ j = 0 then a else b) * Q) $$ (0, 0) = a›*) cond3 (*‹ideal_generated {inv_u * Q $$ (1, 1), inv_u * - Q $$ (0, 1)} = ideal_generated {1}›*))\n (*goal: ‹(a::'a) = (inv_u::'a) * (Q::'a mat) $$ (1::nat, 1::nat) * (Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a else (b::'a)) * Q) $$ (0::nat, 0::nat) ∧ b = inv_u * - Q $$ (0::nat, 1::nat) * (Matrix.mat (1::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a else b) * Q) $$ (0::nat, 0::nat) ∧ ideal_generated {inv_u * Q $$ (1::nat, 1::nat), inv_u * - Q $$ (0::nat, 1::nat)} = ideal_generated {1::'a}›*)\n by auto\nqed\n \n\ntheorem theorem3:\n shows \"(∀A::'a mat. admits_triangular_reduction A)\n = (∀a b::'a. ∃ a1 b1 d. a = a1*d ∧ b = b1*d ∧ ideal_generated {a1,b1} = ideal_generated {1})\"\n using theorem3_part1 (*‹∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1} ⟹ ∀A. admits_triangular_reduction A›*) theorem3_part2 (*‹∀A. admits_triangular_reduction A ⟹ ∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*) by auto\n\nend\n\n\n\ncontext comm_ring_1\nbegin\n\n\nlemma lemma4_prev:\n assumes a: \"a = a1*d\" and b: \"b = b1*d\"\n and i: \"ideal_generated {a1,b1} = ideal_generated {1}\"\nshows \"ideal_generated {a,b} = ideal_generated {d}\"\nproof (-)\n (*goal: ‹ideal_generated {a::'a, b::'a} = ideal_generated {d::'a}›*)\n have 1: \"∃k. p * (a1 * d) + q * (b1 * d) = k * d\" for p and q\n by (metis (full_types) local.distrib_right (*‹(?a + ?b) * ?c = ?a * ?c + ?b * ?c›*) local.mult.semigroup_axioms (*‹semigroup (*)›*) semigroup.assoc (*‹semigroup ?f ⟹ ?f (?f ?a ?b) ?c = ?f ?a (?f ?b ?c)›*))\n have \"ideal_generated {a,b} ⊆ ideal_generated {d}\"\n proof (-)\n (*goal: ‹ideal_generated {a, b} ⊆ ideal_generated {d}›*)\n have \"ideal_generated {a,b} = {p*a+q*b | p q. True}\"\n using ideal_generated_pair (*‹ideal_generated {?a, ?b} = {p * ?a + q * ?b |p q. True}›*) by auto\n also (*calculation: ‹ideal_generated {a, b} = {p * a + q * b |p q. True}›*) have \"... = {p*(a1*d)+q*(b1*d) | p q. True}\"\n using a (*‹a = a1 * d›*) b (*‹(b::'a::type) = (b1::'a::type) * (d::'a::type)›*) by auto\n also (*calculation: ‹ideal_generated {a, b} = {p * (a1 * d) + q * (b1 * d) |p q. True}›*) have \"... ⊆ {k*d|k. True}\"\n using \"1\" (*‹∃k::'a. (?p1::'a) * ((a1::'a) * (d::'a)) + (?q1::'a) * ((b1::'a) * d) = k * d›*) by auto\n finally (*calculation: ‹ideal_generated {a, b} ⊆ {k * d |k. True}›*) show \"?thesis\"\n (*goal: ‹ideal_generated {a, b} ⊆ ideal_generated {d}›*)\n by (simp add: a (*‹a = a1 * d›*) b (*‹b = b1 * d›*) local.dvd_ideal_generated_singleton' (*‹?b dvd ?a ⟹ ideal_generated {?a} ⊆ ideal_generated {?b}›*) local.ideal_generated_subset2 (*‹⟦ideal_generated {?a} ⊆ ideal_generated {?c}; ideal_generated {?b} ⊆ ideal_generated {?c}⟧ ⟹ ideal_generated {?a, ?b} ⊆ ideal_generated {?c}›*))\n qed\n moreover have \"ideal_generated{d} ⊆ ideal_generated {a,b}\"\n proof (rule ideal_generated_singleton_subset (*‹⟦(?d::'a) ∈ ideal_generated (?S::'a set); finite ?S⟧ ⟹ ideal_generated {?d} ⊆ ideal_generated ?S›*))\n (*goals:\n 1. ‹d ∈ ideal_generated {a, b}›\n 2. ‹finite {a, b}›*)\n obtain p and q where \"p*a1+q*b1 = 1\"\n (*goal: ‹(⋀p q. p * a1 + q * b1 = 1 ⟹ thesis) ⟹ thesis›*)\n using ideal_generated_pair_exists_UNIV (*‹(ideal_generated {?a, ?b} = ideal_generated {1}) = (∃p q. p * ?a + q * ?b = 1)›*) i (*‹ideal_generated {a1, b1} = ideal_generated {1}›*) by auto\n hence \"d = p * (a1 * d) + q * (b1 * d)\"\n by (metis local.mult_ac( (*‹?b * (?a * ?c) = ?a * (?b * ?c)›*) 3) local.ring_distribs( (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c›*) 1) local.semiring_normalization_rules( (*‹?a * 1 = ?a›*) 12))\n also (*calculation: ‹d = p * (a1 * d) + q * (b1 * d)›*) have \"... ∈ {p*(a1*d)+q*(b1*d) | p q. True}\"\n by auto\n also (*calculation: ‹d ∈ {p * (a1 * d) + q * (b1 * d) |p q. True}›*) have \"... = ideal_generated {a,b}\"\n unfolding ideal_generated_pair a b\n (*goal: ‹{p * (a1 * d) + q * (b1 * d) |p q. True} = {p * (a1 * d) + q * (b1 * d) |p q. True}›*)\n by auto\n finally (*calculation: ‹d ∈ ideal_generated {a, b}›*) show \"d ∈ ideal_generated {a,b}\"\n by simp\n qed (simp)\n (*solved the remaining goal: ‹finite {a, b}›*)\n ultimately show \"?thesis\"\n (*goal: ‹ideal_generated {a::'a, b::'a} = ideal_generated {d::'a}›*)\n by simp\nqed\n\n\nlemma lemma4:\n assumes a: \"a = a1*d\" and b: \"b = b1*d\"\n and i: \"ideal_generated {a1,b1} = ideal_generated {1}\"\n and i2: \"ideal_generated {a,b} = ideal_generated {d'}\"\n shows \"∃a1' b1'. a = a1' * d' ∧ b = b1' * d' \n ∧ ideal_generated {a1',b1'} = ideal_generated {1}\"\nproof (-)\n (*goal: ‹∃a1' b1'. a = a1' * d' ∧ b = b1' * d' ∧ ideal_generated {a1', b1'} = ideal_generated {1}›*)\n have i3: \"ideal_generated {a,b} = ideal_generated {d}\"\n using lemma4_prev (*‹⟦?a = ?a1.0 * ?d; ?b = ?b1.0 * ?d; ideal_generated {?a1.0, ?b1.0} = ideal_generated {1}⟧ ⟹ ideal_generated {?a, ?b} = ideal_generated {?d}›*) assms (*‹a = a1 * d› ‹b = b1 * d› ‹ideal_generated {a1, b1} = ideal_generated {1}› ‹ideal_generated {a::'a::type, b::'a::type} = ideal_generated {d'::'a::type}›*) by auto\n have d_dvd_d': \"d dvd d'\"\n by (metis a (*‹a = a1 * d›*) b (*‹b = b1 * d›*) i2 (*‹ideal_generated {a, b} = ideal_generated {d'}›*) dvd_ideal_generated_singleton (*‹ideal_generated {?a} ⊆ ideal_generated {?b} ⟹ ?b dvd ?a›*) dvd_ideal_generated_singleton' (*‹?b dvd ?a ⟹ ideal_generated {?a} ⊆ ideal_generated {?b}›*) dvd_triv_right (*‹?a dvd ?b * ?a›*) ideal_generated_subset2 (*‹⟦ideal_generated {?a} ⊆ ideal_generated {?c}; ideal_generated {?b} ⊆ ideal_generated {?c}⟧ ⟹ ideal_generated {?a, ?b} ⊆ ideal_generated {?c}›*))\n have d'_dvd_d: \"d' dvd d\"\n using i3 (*‹ideal_generated {a, b} = ideal_generated {d}›*) i2 (*‹ideal_generated {a, b} = ideal_generated {d'}›*) local.dvd_ideal_generated_singleton (*‹ideal_generated {?a} ⊆ ideal_generated {?b} ⟹ ?b dvd ?a›*) by auto\n obtain k and l where d: \"d = k*d'\" and d': \"d' = l*d\"\n (*goal: ‹(⋀k l. ⟦d = k * d'; d' = l * d⟧ ⟹ thesis) ⟹ thesis›*)\n using d_dvd_d' (*‹d dvd d'›*) d'_dvd_d (*‹d' dvd d›*) mult_ac (*‹?a * ?b * ?c = ?a * (?b * ?c)› ‹?a * ?b = ?b * ?a› ‹?b * (?a * ?c) = ?a * (?b * ?c)›*) unfolding dvd_def\n (*goal: ‹(⋀k l. ⟦d = k * d'; d' = l * d⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain s and t where sa1_tb1: \"s*a1 + t*b1 = 1\"\n (*goal: ‹(⋀s t. s * a1 + t * b1 = 1 ⟹ thesis) ⟹ thesis›*)\n using i (*‹ideal_generated {a1, b1} = ideal_generated {1}›*) ideal_generated_pair_exists_UNIV[of a1 b1] (*‹(ideal_generated {a1, b1} = ideal_generated {1}) = (∃p q. p * a1 + q * b1 = 1)›*) by auto\n let ?a1' = \"k * l * t - t + a1 * k\"\n let ?b1' = \"s - k * l * s + b1 * k\"\n have 1: \"?a1'*d'=a\"\n by (metis a (*‹a = a1 * d›*) d (*‹d = k * d'›*) d' (*‹d' = l * d›*) add_ac( (*‹?a + ?b = ?b + ?a›*) 2) add_diff_cancel (*‹?a + ?b - ?b = ?a›*) add_diff_eq (*‹?a + (?b - ?c) = ?a + ?b - ?c›*) mult_ac( (*‹?a * ?b = ?b * ?a›*) 2) ring_distribs( (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c› ‹?a * (?b - ?c) = ?a * ?b - ?a * ?c›*) 1,4) semiring_normalization_rules( (*‹?lx * (?rx * ?ry) = ?lx * ?rx * ?ry›*) 18))\n have 2: \"?b1'*d' = b\"\n by (metis (no_types, opaque_lifting) b (*‹b = b1 * d›*) d (*‹d = k * d'›*) d' (*‹d' = l * d›*) add_ac( (*‹?a + ?b = ?b + ?a›*) 2) add_diff_cancel (*‹?a + ?b - ?b = ?a›*) add_diff_eq (*‹?a + (?b - ?c) = ?a + ?b - ?c›*) mult_ac( (*‹?a * ?b = ?b * ?a›*) 2) mult_ac( (*‹?b * (?a * ?c) = ?a * (?b * ?c)›*) 3) ring_distribs( (*‹(?a + ?b) * ?c = ?a * ?c + ?b * ?c› ‹?a * (?b - ?c) = ?a * ?b - ?a * ?c›*) 2,4) semiring_normalization_rules( (*‹?lx * (?rx * ?ry) = ?lx * ?rx * ?ry›*) 18))\n have \"(s*l-b1)*?a1' + (t*l+a1)*?b1' = 1\"\n proof (-)\n (*goal: ‹(s * l - b1) * (k * l * t - t + a1 * k) + (t * l + a1) * (s - k * l * s + b1 * k) = 1›*)\n have aux_rw1: \"s * l * k * l * t = t * l * k * l * s\" and aux_rw2: \"s * l * t=t * l * s\" and aux_rw3: \"b1 * a1 * k=a1 * b1 * k\" and aux_rw4: \"t * l * b1 * k=b1 * k * l * t\" and aux_rw5: \"s * l * a1 * k=a1 * k * l * s\"\n using mult.commute (*‹(?a::'a) * (?b::'a) = ?b * ?a›*) mult.assoc (*‹?a * ?b * ?c = ?a * (?b * ?c)›*)\n (*goals:\n 1. ‹s * l * k * l * t = t * l * k * l * s›\n 2. ‹s * l * t = t * l * s›\n 3. ‹b1 * a1 * k = a1 * b1 * k›\n 4. ‹t * l * b1 * k = b1 * k * l * t›\n 5. ‹s * l * a1 * k = a1 * k * l * s›\n discuss goal 1*)\n apply ((msorry)[1])\n (*discuss goal 2*)\n apply ((msorry)[1])\n (*discuss goal 3*)\n apply ((msorry)[1])\n (*discuss goal 4*)\n apply ((msorry)[1])\n (*discuss goal 5*)\n apply ((msorry)[1])\n (*proven 5 subgoals*) .\n note aux_rw = aux_rw1 (*‹s * l * k * l * t = t * l * k * l * s›*) aux_rw2 (*‹s * l * t = t * l * s›*) aux_rw3 (*‹b1 * a1 * k = a1 * b1 * k›*) aux_rw4 (*‹t * l * b1 * k = b1 * k * l * t›*) aux_rw5 (*‹s * l * a1 * k = a1 * k * l * s›*)\n have \"(s*l-b1)*?a1' + (t*l+a1)*?b1' = s*l*?a1' - b1*?a1' + t*l*?b1'+a1*?b1'\"\n using local.add_ac(1) (*‹?a + ?b + ?c = ?a + (?b + ?c)›*) local.left_diff_distrib' (*‹(?b - ?c) * ?a = ?b * ?a - ?c * ?a›*) local.ring_distribs(2) (*‹(?a + ?b) * ?c = ?a * ?c + ?b * ?c›*) by auto\n also (*calculation: ‹(s * l - b1) * (k * l * t - t + a1 * k) + (t * l + a1) * (s - k * l * s + b1 * k) = s * l * (k * l * t - t + a1 * k) - b1 * (k * l * t - t + a1 * k) + t * l * (s - k * l * s + b1 * k) + a1 * (s - k * l * s + b1 * k)›*) have \"... = s * l * k * l*t - s * l * t + s * l * a1 * k-b1 * k * l * t + b1 * t-b1 * a1 * k\n + t * l * s-t * l * k * l * s + t * l * b1 * k + a1 * s - a1 * k * l * s + a1 * b1 * k\"\n by (smt (verit) local.add_diff_eq (*‹?a + (?b - ?c) = ?a + ?b - ?c›*) local.diff_add_eq (*‹?a - ?b + ?c = ?a + ?c - ?b›*) local.diff_diff_eq2 (*‹?a - (?b - ?c) = ?a + ?c - ?b›*) local.mult_ac( (*‹?a * ?b * ?c = ?a * (?b * ?c)›*) 1) local.ring_distribs( (*‹?a * (?b - ?c) = ?a * ?b - ?a * ?c›*) 4))\n also (*calculation: ‹(s * l - b1) * (k * l * t - t + a1 * k) + (t * l + a1) * (s - k * l * s + b1 * k) = s * l * k * l * t - s * l * t + s * l * a1 * k - b1 * k * l * t + b1 * t - b1 * a1 * k + t * l * s - t * l * k * l * s + t * l * b1 * k + a1 * s - a1 * k * l * s + a1 * b1 * k›*) have \"... = a1 * s + b1 * t\"\n unfolding aux_rw\n (*goal: ‹t * l * k * l * s - t * l * s + a1 * k * l * s - b1 * k * l * t + b1 * t - a1 * b1 * k + t * l * s - t * l * k * l * s + b1 * k * l * t + a1 * s - a1 * k * l * s + a1 * b1 * k = a1 * s + b1 * t›*)\n by (smt (verit, ccfv_SIG) local.add_diff_cancel_left' (*‹(?a::'a::type) + (?b::'a::type) - ?a = ?b›*) local.diff_add_eq (*‹(?a::'a::type) - (?b::'a::type) + (?c::'a::type) = ?a + ?c - ?b›*) local.eq_diff_eq (*‹((?a::'a::type) = (?c::'a::type) - (?b::'a::type)) = (?a + ?b = ?c)›*))\n also (*calculation: ‹((s::'a) * (l::'a) - (b1::'a)) * ((k::'a) * l * (t::'a) - t + (a1::'a) * k) + (t * l + a1) * (s - k * l * s + b1 * k) = a1 * s + b1 * t›*) have \"... = 1\"\n using sa1_tb1 (*‹s * a1 + t * b1 = 1›*) mult.commute (*‹?a * ?b = ?b * ?a›*) by auto\n finally (*calculation: ‹(s * l - b1) * (k * l * t - t + a1 * k) + (t * l + a1) * (s - k * l * s + b1 * k) = 1›*) show \"?thesis\"\n (*goal: ‹(s * l - b1) * (k * l * t - t + a1 * k) + (t * l + a1) * (s - k * l * s + b1 * k) = 1›*)\n by simp\n qed\n hence \"ideal_generated {?a1',?b1'} = ideal_generated {1}\"\n using ideal_generated_pair_exists_UNIV[of ?a1' ?b1'] (*‹(ideal_generated {k * l * t - t + a1 * k, s - k * l * s + b1 * k} = ideal_generated {1}) = (∃p q. p * (k * l * t - t + a1 * k) + q * (s - k * l * s + b1 * k) = 1)›*) by auto\n thus \"?thesis\"\n (*goal: ‹∃a1' b1'. a = a1' * d' ∧ b = b1' * d' ∧ ideal_generated {a1', b1'} = ideal_generated {1}›*)\n using \"1\" (*‹((k::'a) * (l::'a) * (t::'a) - t + (a1::'a) * k) * (d'::'a) = (a::'a)›*) \"2\" (*‹(s - k * l * s + b1 * k) * d' = b›*) by auto\nqed\n\n\n(*In the article, this is a corollary. But here, this needs more work.*)\nlemma corollary5:\n assumes T: \"∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d \n ∧ ideal_generated {a1, b1} = ideal_generated {1::'a}\"\n and i2: \"ideal_generated {a,b,c} = ideal_generated {d}\"\n shows \"∃ a1 b1 c1. a = a1 * d ∧ b = b1 * d ∧ c = c1 * d \n ∧ ideal_generated {a1,b1,c1} = ideal_generated {1}\"\nproof (-)\n (*goal: ‹∃a1 b1 c1. a = a1 * d ∧ b = b1 * d ∧ c = c1 * d ∧ ideal_generated {a1, b1, c1} = ideal_generated {1}›*)\n have da: \"d dvd a\"\n using ideal_generated_singleton_dvd[OF i2] (*‹(?x::'a::type) ∈ {a::'a::type, b::'a::type, c::'a::type} ⟹ (d::'a::type) dvd ?x›*) by auto\n have db: \"d dvd b\"\n using ideal_generated_singleton_dvd[OF i2] (*‹(?x::'a) ∈ {a::'a, b::'a, c::'a} ⟹ (d::'a) dvd ?x›*) by auto\n have dc: \"d dvd c\"\n using ideal_generated_singleton_dvd[OF i2] (*‹?x ∈ {a, b, c} ⟹ d dvd ?x›*) by auto\n from this (*‹d dvd c›*) obtain c1' where c: \"c = c1' * d\"\n (*goal: ‹(⋀c1'. c = c1' * d ⟹ thesis) ⟹ thesis›*)\n using dvd_def (*‹(?b dvd ?a) = (∃k. ?a = ?b * k)›*) mult_ac(2) (*‹?a * ?b = ?b * ?a›*) by auto\n obtain a1 and b1 and d' where a: \"a = a1 * d'\" and b: \"b = b1 * d' \" and i: \"ideal_generated {a1, b1} = ideal_generated {1::'a}\"\n (*goal: ‹(⋀(a1::'a) (d'::'a) b1::'a. ⟦(a::'a) = a1 * d'; (b::'a) = b1 * d'; ideal_generated {a1, b1} = ideal_generated {1::'a}⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using T (*‹∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*) by blast\n have i_ab_d': \"ideal_generated {a, b} = ideal_generated {d'}\"\n by (simp add: a (*‹a = a1 * d'›*) b (*‹b = b1 * d'›*) i (*‹ideal_generated {a1, b1} = ideal_generated {1}›*) lemma4_prev (*‹⟦?a = ?a1.0 * ?d; ?b = ?b1.0 * ?d; ideal_generated {?a1.0, ?b1.0} = ideal_generated {1}⟧ ⟹ ideal_generated {?a, ?b} = ideal_generated {?d}›*))\n have i2: \"ideal_generated {d', c} = ideal_generated {d}\"\n by (rule ideal_generated_triple_pair_rewrite[OF i2 i_ab_d'] (*‹ideal_generated {d', c} = ideal_generated {d}›*))\n obtain u and v and dp where d'1: \"d' = u * dp\" and d'2: \"c = v * dp\" and xy: \"ideal_generated{u,v}=ideal_generated{1}\"\n (*goal: ‹(⋀u dp v. ⟦d' = u * dp; c = v * dp; ideal_generated {u, v} = ideal_generated {1}⟧ ⟹ thesis) ⟹ thesis›*)\n using T (*‹∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*) by blast\n have \"∃a1' b1'. d' = a1' * d ∧ c = b1' * d ∧ ideal_generated {a1', b1'} = ideal_generated {1}\"\n by (rule lemma4[OF d'1 d'2 xy i2] (*‹∃(a1'::'a) b1'::'a. (d'::'a) = a1' * (d::'a) ∧ (c::'a) = b1' * d ∧ ideal_generated {a1', b1'} = ideal_generated {1::'a}›*))\n from this (*‹∃a1' b1'. d' = a1' * d ∧ c = b1' * d ∧ ideal_generated {a1', b1'} = ideal_generated {1}›*) obtain a1' and c1 where d'_a1: \"d' = a1' * d\" and c: \"c = c1 * d\" and i3: \"ideal_generated {a1', c1} = ideal_generated {1}\"\n (*goal: ‹(⋀a1' c1. ⟦d' = a1' * d; c = c1 * d; ideal_generated {a1', c1} = ideal_generated {1}⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n have r1: \"a = a1 * a1' * d\"\n by (simp add: d'_a1 (*‹d' = a1' * d›*) a (*‹a = a1 * d'›*) local.semiring_normalization_rules( (*‹?lx * (?rx * ?ry) = ?lx * ?rx * ?ry›*) 18))\n have r2: \"b = b1 * a1' * d\"\n by (simp add: d'_a1 (*‹d' = a1' * d›*) b (*‹b = b1 * d'›*) local.semiring_normalization_rules( (*‹?lx * (?rx * ?ry) = ?lx * ?rx * ?ry›*) 18))\n have i4: \"ideal_generated {a1 * a1',b1 * a1', c1} = ideal_generated {1}\"\n proof (-)\n (*goal: ‹ideal_generated {a1 * a1', b1 * a1', c1} = ideal_generated {1}›*)\n obtain p and q where 1: \"p * a1' + q * c1 = 1\"\n (*goal: ‹(⋀(p::'a::type) q::'a::type. p * (a1'::'a::type) + q * (c1::'a::type) = (1::'a::type) ⟹ thesis::bool) ⟹ thesis›*)\n using i3 (*‹ideal_generated {a1', c1} = ideal_generated {1}›*) unfolding ideal_generated_pair_exists_UNIV\n (*goal: ‹(⋀p q. p * a1' + q * c1 = 1 ⟹ thesis) ⟹ thesis›*)\n by auto\n obtain x and y where 2: \"x*a1 + y*b1 = p\"\n (*goal: ‹(⋀x y. x * a1 + y * b1 = p ⟹ thesis) ⟹ thesis›*)\n using ideal_generated_UNIV_obtain_pair[OF i] (*‹∃p q. p * a1 + q * b1 = ?d›*) by blast\n have \"1 = (x*a1 + y*b1) * a1' + q * c1\"\n using \"1\" (*‹p * a1' + q * c1 = 1›*) \"2\" (*‹(x::'a) * (a1::'a) + (y::'a) * (b1::'a) = (p::'a)›*) by auto\n also (*calculation: ‹1 = (x * a1 + y * b1) * a1' + q * c1›*) have \"... = x*a1*a1' + y*b1*a1' + q * c1\"\n by (simp add: local.ring_distribs( (*‹(?a + ?b) * ?c = ?a * ?c + ?b * ?c›*) 2))\n finally (*calculation: ‹(1::'a::type) = (x::'a::type) * (a1::'a::type) * (a1'::'a::type) + (y::'a::type) * (b1::'a::type) * a1' + (q::'a::type) * (c1::'a::type)›*) have \"1 = x*a1*a1' + y*b1*a1' + q * c1\" .\n hence \"1 ∈ ideal_generated {a1 * a1', b1 * a1', c1}\"\n using ideal_explicit2[of \"{a1 * a1', b1 * a1', c1}\"] (*‹finite {a1 * a1', b1 * a1', c1} ⟹ ideal_generated {a1 * a1', b1 * a1', c1} = {y. ∃f. (∑i∈{a1 * a1', b1 * a1', c1}. f i * i) = y}›*) sum_three_elements' (*‹∃f. (∑i∈{?a, ?b, ?c}. f i * i) = ?x * ?a + ?y * ?b + ?z * ?c›*) by (simp add: mult_assoc (*‹?a * ?b * ?c = ?a * (?b * ?c)›*))\n hence \"ideal_generated {1} ⊆ ideal_generated {a1 * a1',b1 * a1', c1}\"\n apply (rule ideal_generated_singleton_subset (*‹⟦(?d::'a) ∈ ideal_generated (?S::'a set); finite ?S⟧ ⟹ ideal_generated {?d} ⊆ ideal_generated ?S›*))\n (*goal: ‹ideal_generated {1} ⊆ ideal_generated {a1 * a1', b1 * a1', c1}›*)\n by auto\n thus \"?thesis\"\n (*goal: ‹ideal_generated {(a1::'a) * (a1'::'a), (b1::'a) * a1', c1::'a} = ideal_generated {1::'a}›*)\n by auto\n qed\n show \"?thesis\"\n (*goal: ‹∃a1 b1 c1. a = a1 * d ∧ b = b1 * d ∧ c = c1 * d ∧ ideal_generated {a1, b1, c1} = ideal_generated {1}›*)\n using r1 (*‹(a::'a) = (a1::'a) * (a1'::'a) * (d::'a)›*) r2 (*‹b = b1 * a1' * d›*) i4 (*‹ideal_generated {a1 * a1', b1 * a1', c1} = ideal_generated {1}›*) c (*‹c = c1 * d›*) by auto\nqed\n\n\nend\n\ncontext\n assumes \"SORT_CONSTRAINT('a::comm_ring_1)\"\nbegin\n\nlemma OFCLASS_elementary_divisor_ring_imp_class:\n assumes \"OFCLASS('a::comm_ring_1, elementary_divisor_ring_class)\"\n shows \" class.elementary_divisor_ring TYPE('a)\" \n by (rule conjunctionD2[OF assms[unfolded elementary_divisor_ring_class_def]] (*‹class.elementary_divisor_ring TYPE('a)›*))\n\n\n(*ELEMENTARY DIVISOR RING ⟹ HERMITE*)\ncorollary Elementary_divisor_ring_imp_Hermite_ring:\n assumes \"OFCLASS('a::comm_ring_1, elementary_divisor_ring_class) \"\n shows \"OFCLASS('a::comm_ring_1, Hermite_ring_class)\"\nproof (standard)\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n have \"∀A::'a mat. admits_diagonal_reduction A\"\n using OFCLASS_elementary_divisor_ring_imp_class[OF assms] (*‹class.elementary_divisor_ring TYPE('a)›*) unfolding class.elementary_divisor_ring_def\n (*goal: ‹∀A::'a::comm_ring_1 mat. admits_diagonal_reduction A›*)\n by auto\n thus \"∀A::'a mat. admits_triangular_reduction A\"\n using admits_diagonal_imp_admits_triangular (*‹Ball (carrier_mat (1::nat) (2::nat)) admits_diagonal_reduction ⟹ All admits_triangular_reduction›*) by auto\nqed\n\n(*ELEMENTARY DIVISOR RING ⟹ BEZOUT*)\ncorollary Elementary_divisor_ring_imp_Bezout_ring:\n assumes \"OFCLASS('a::comm_ring_1, elementary_divisor_ring_class) \"\n shows \"OFCLASS('a::comm_ring_1, bezout_ring_class)\"\n apply (rule Hermite_ring_imp_Bezout_ring (*‹OFCLASS(?'a, Hermite_ring_class) ⟹ OFCLASS(?'a, bezout_ring_class)›*))\n (*goal: ‹OFCLASS('a, bezout_ring_class)›*)\n by (rule Elementary_divisor_ring_imp_Hermite_ring[OF assms] (*‹OFCLASS('a, Hermite_ring_class)›*))\n\nsubsection ‹Characterization of Elementary divisor rings›\n\nlemma necessity_D': \n assumes edr: \"(∀(A::'a mat). admits_diagonal_reduction A)\"\n shows \"∀a b c::'a. ideal_generated {a,b,c} = ideal_generated{1} \n ⟶ (∃p q. ideal_generated {p*a,p*b+q*c} = ideal_generated {1})\"\napply (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n(*goal: ‹∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*)\napply (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n(*goal: ‹⋀a. ∀b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*)\napply (rule allI (*‹(⋀x::?'a. (?P::?'a ⇒ bool) x) ⟹ ∀x::?'a. ?P x›*))\n(*goal: ‹⋀a b. ∀c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*)\nproof (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹⋀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟹ ∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}›*)\n fix a :: 'a and b :: 'a and c :: 'a\n assume i: \"ideal_generated {a,b,c} = ideal_generated{1}\" (*‹ideal_generated {a::'a, b::'a, c::'a} = ideal_generated {1::'a}›*)\n define A where \"A = Matrix.mat 2 2 (λ(i,j). if i = 0 ∧ j = 0 then a else\n if i = 0 ∧ j = 1 then b else\n if i = 1 ∧ j = 0 then 0 else c)\"\n have A: \"A ∈ carrier_mat 2 2\"\n unfolding A_def\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a else if i = 0 ∧ j = 1 then b else if i = 1 ∧ j = 0 then 0 else c) ∈ carrier_mat 2 2›*)\n by auto\n obtain P and Q where P: \"P ∈ carrier_mat (dim_row A) (dim_row A)\" and Q: \"Q ∈ carrier_mat (dim_col A) (dim_col A)\" and inv_P: \"invertible_mat P\" and inv_Q: \"invertible_mat Q\" and SNF_PAQ: \"Smith_normal_form_mat (P * A * Q)\"\n (*goal: ‹(⋀P Q. ⟦P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A); invertible_mat P; invertible_mat Q; Smith_normal_form_mat (P * A * Q)⟧ ⟹ thesis) ⟹ thesis›*)\n using edr (*‹∀A. admits_diagonal_reduction A›*) unfolding admits_diagonal_reduction_def\n (*goal: ‹(⋀(P::'a mat) Q::'a mat. ⟦P ∈ carrier_mat (dim_row (A::'a mat)) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A); invertible_mat P; invertible_mat Q; Smith_normal_form_mat (P * A * Q)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by blast\n have [simp]: \"dim_row P = 2\" and [simp]: \"dim_col P = 2 \" and [simp]: \"dim_row Q = 2\" and [simp]: \"dim_col Q = 2\" and [simp]: \"dim_col A = 2\" and [simp]: \"dim_row A = 2\"\n using A (*‹A ∈ carrier_mat 2 2›*) P (*‹P ∈ carrier_mat (dim_row A) (dim_row A)›*) Q (*‹(Q::'a mat) ∈ carrier_mat (dim_col (A::'a mat)) (dim_col A)›*) apply -\n (*goals:\n 1. ‹⟦A ∈ carrier_mat 2 2; P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A)⟧ ⟹ dim_row P = 2›\n 2. ‹⟦A ∈ carrier_mat 2 2; P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A)⟧ ⟹ dim_col P = 2›\n 3. ‹⟦A ∈ carrier_mat 2 2; P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A)⟧ ⟹ dim_row Q = 2›\n 4. ‹⟦A ∈ carrier_mat 2 2; P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A)⟧ ⟹ dim_col Q = 2›\n 5. ‹⟦A ∈ carrier_mat 2 2; P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A)⟧ ⟹ dim_col A = 2›\n 6. ‹⟦A ∈ carrier_mat 2 2; P ∈ carrier_mat (dim_row A) (dim_row A); Q ∈ carrier_mat (dim_col A) (dim_col A)⟧ ⟹ dim_row A = 2›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*discuss goal 5*)\n apply ((auto)[1])\n (*discuss goal 6*)\n apply ((auto)[1])\n (*proven 6 subgoals*) .\n define u where \"u = (P*A*Q) $$ (0,0)\"\n define p where \"p = P $$ (0,0)\"\n define q where \"q = P $$ (0,1)\"\n define x where \"x = Q $$ (0,0)\"\n define y where \"y = Q $$ (1,0)\"\n have eq: \"p*a*x + p*b*y + q*c*y = u\"\n proof (-)\n (*goal: ‹p * a * x + p * b * y + q * c * y = u›*)\n have rw1: \"(∑ia = 0..<2. P $$ (0, ia) * A $$ (ia, x)) * Q $$ (x, 0) \n = (∑ia∈{0, 1}. P $$ (0, ia) * A $$ (ia, x)) * Q $$ (x, 0)\" for x\n apply (unfold sum_distrib_right (*‹sum ?f ?A * ?r = (∑n∈?A. ?f n * ?r)›*))\n (*goal: ‹(∑ia = 0..<2. P $$ (0, ia) * A $$ (ia, x)) * Q $$ (x, 0) = (∑ia∈{0, 1}. P $$ (0, ia) * A $$ (ia, x)) * Q $$ (x, 0)›*)\n apply (rule sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n (*goals:\n 1. ‹{0..<2} = {0, 1}›\n 2. ‹⋀x. x ∈ {0, 1} ⟹ P $$ (0, x) * A $$ (x, x) * Q $$ (x, 0) = P $$ (0, x) * A $$ (x, x) * Q $$ (x, 0)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have \"u = (∑i = 0..<2. (∑ia = 0..<2. P $$ (0, ia) * A $$ (ia, i)) * Q $$ (i, 0))\"\n unfolding u_def p_def q_def x_def y_def\n (*goal: ‹(P * A * Q) $$ (0, 0) = (∑i = 0..<2. (∑ia = 0..<2. P $$ (0, ia) * A $$ (ia, i)) * Q $$ (i, 0))›*)\n unfolding times_mat_def scalar_prod_def\n (*goal: ‹Matrix.mat (dim_row (Matrix.mat (dim_row P) (dim_col A) (λ(i, j). ∑ia = 0..<dim_vec (col A j). Matrix.row P i $v ia * col A j $v ia))) (dim_col Q) (λ(i, j). ∑ia = 0..<dim_vec (col Q j). Matrix.row (Matrix.mat (dim_row P) (dim_col A) (λ(i, j). ∑ia = 0..<dim_vec (col A j). Matrix.row P i $v ia * col A j $v ia)) i $v ia * col Q j $v ia) $$ (0, 0) = (∑i = 0..<2. (∑ia = 0..<2. P $$ (0, ia) * A $$ (ia, i)) * Q $$ (i, 0))›*)\n by auto\n also (*calculation: ‹u = (∑i = 0..<2. (∑ia = 0..<2. P $$ (0, ia) * A $$ (ia, i)) * Q $$ (i, 0))›*) have \"... = (∑i ∈{0,1}. (∑ia ∈ {0,1}. P $$ (0, ia) * A $$ (ia, i)) * Q $$ (i, 0))\"\n apply (rule sum.cong[OF _ rw1] (*‹?A = ?B ⟹ (∑x∈?A. (∑ia = 0..<2. P $$ (0, ia) * A $$ (ia, ?x1 x)) * Q $$ (?x1 x, 0)) = (∑x∈?B. (∑ia∈{0, 1}. P $$ (0, ia) * A $$ (ia, ?x1 x)) * Q $$ (?x1 x, 0))›*))\n (*goal: ‹(∑i::nat = 0::nat..<2::nat. (∑ia::nat = 0::nat..<2::nat. (P::'a mat) $$ (0::nat, ia) * (A::'a mat) $$ (ia, i)) * (Q::'a mat) $$ (i, 0::nat)) = (∑i::nat∈{0::nat, 1::nat}. (∑ia::nat∈{0::nat, 1::nat}. P $$ (0::nat, ia) * A $$ (ia, i)) * Q $$ (i, 0::nat))›*)\n by auto\n also (*calculation: ‹u = (∑i∈{0, 1}. (∑ia∈{0, 1}. P $$ (0, ia) * A $$ (ia, i)) * Q $$ (i, 0))›*) have \"... = p*a*x + p*b*y+q*c*y\"\n unfolding u_def p_def q_def x_def y_def A_def\n (*goal: ‹(∑i∈{0, 1}. (∑ia∈{0, 1}. P $$ (0, ia) * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a else if i = 0 ∧ j = 1 then b else if i = 1 ∧ j = 0 then 0 else c) $$ (ia, i)) * Q $$ (i, 0)) = P $$ (0, 0) * a * Q $$ (0, 0) + P $$ (0, 0) * b * Q $$ (1, 0) + P $$ (0, 1) * c * Q $$ (1, 0)›*)\n using ring_class.ring_distribs(2) (*‹(?a + ?b) * ?c = ?a * ?c + ?b * ?c›*) by auto\n finally (*calculation: ‹(u::'a::comm_ring_1) = (p::'a::comm_ring_1) * (a::'a::comm_ring_1) * (x::'a::comm_ring_1) + p * (b::'a::comm_ring_1) * (y::'a::comm_ring_1) + (q::'a::comm_ring_1) * (c::'a::comm_ring_1) * y›*) show \"?thesis\"\n (*goal: ‹(p::'a::comm_ring_1) * (a::'a::comm_ring_1) * (x::'a::comm_ring_1) + p * (b::'a::comm_ring_1) * (y::'a::comm_ring_1) + (q::'a::comm_ring_1) * (c::'a::comm_ring_1) * y = (u::'a::comm_ring_1)›*)\n by standard\n qed\n have u_dvd_1: \"u dvd 1\"\n proof (rule ideal_generated_dvd2[OF i] (*‹⟦finite {a, b, c}; ∀x∈{a, b, c}. ?d' dvd x⟧ ⟹ ?d' dvd 1›*))\n (*goals:\n 1. ‹finite {a, b, c}›\n 2. ‹∀x∈{a, b, c}. u dvd x›*)\n define D where \"D = (P*A*Q)\"\n obtain P' where P'[simp]: \"P' ∈ carrier_mat 2 2\" and inv_P: \"inverts_mat P' P\"\n (*goal: ‹(⋀P'. ⟦P' ∈ carrier_mat 2 2; inverts_mat P' P⟧ ⟹ thesis) ⟹ thesis›*)\n using inv_P (*‹invertible_mat P›*) obtain_inverse_matrix[OF P inv_P] (*‹(⋀B::'a::comm_ring_1 mat. ⟦inverts_mat (P::'a::comm_ring_1 mat) B; inverts_mat B P; B ∈ carrier_mat (dim_row (A::'a::comm_ring_1 mat)) (dim_row A)⟧ ⟹ ?thesis::bool) ⟹ ?thesis›*) by (metis ‹dim_row A = 2›)\n obtain Q' where [simp]: \"Q' ∈ carrier_mat 2 2\" and inv_Q: \"inverts_mat Q Q'\"\n (*goal: ‹(⋀Q'. ⟦Q' ∈ carrier_mat 2 2; inverts_mat Q Q'⟧ ⟹ thesis) ⟹ thesis›*)\n using inv_Q (*‹invertible_mat (Q::'a mat)›*) obtain_inverse_matrix[OF Q inv_Q] (*‹(⋀B. ⟦inverts_mat Q B; inverts_mat B Q; B ∈ carrier_mat (dim_col A) (dim_col A)⟧ ⟹ ?thesis) ⟹ ?thesis›*) by (metis ‹dim_col A = 2›)\n have D[simp]: \"D ∈ carrier_mat 2 2\"\n unfolding D_def\n (*goal: ‹P * A * Q ∈ carrier_mat 2 2›*)\n by auto\n have e: \"P' * D * Q' = A\"\n unfolding D_def\n (*goal: ‹P' * (P * A * Q) * Q' = A›*)\n by (rule inv_P'PAQQ'[OF _ _ inv_P inv_Q] (*‹⟦?A ∈ carrier_mat ?n ?n; P ∈ carrier_mat ?n ?n; Q ∈ carrier_mat ?n ?n; P' ∈ carrier_mat ?n ?n; Q' ∈ carrier_mat ?n ?n⟧ ⟹ P' * (P * ?A * Q) * Q' = ?A›*), auto)\n have [simp]: \"(P' * D) ∈ carrier_mat 2 2\"\n using D (*‹(D::'a mat) ∈ carrier_mat (2::nat) (2::nat)›*) P' (*‹P' ∈ carrier_mat 2 2›*) mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*) by blast\n have D_01: \"D $$ (0, 1) = 0\"\n using D_def (*‹D = P * A * Q›*) SNF_PAQ (*‹Smith_normal_form_mat (P * A * Q)›*) unfolding Smith_normal_form_mat_def isDiagonal_mat_def\n (*goal: ‹(D::'a::comm_ring_1 mat) $$ (0::nat, 1::nat) = (0::'a::comm_ring_1)›*)\n by force\n have D_10: \"D $$ (1, 0) = 0\"\n using D_def (*‹(D::'a mat) = (P::'a mat) * (A::'a mat) * (Q::'a mat)›*) SNF_PAQ (*‹Smith_normal_form_mat (P * A * Q)›*) unfolding Smith_normal_form_mat_def isDiagonal_mat_def\n (*goal: ‹(D::'a mat) $$ (1::nat, 0::nat) = (0::'a)›*)\n by force\n have \"D $$ (0,0) dvd D $$ (1, 1)\"\n using D_def (*‹D = P * A * Q›*) SNF_PAQ (*‹Smith_normal_form_mat (P * A * Q)›*) unfolding Smith_normal_form_mat_def\n (*goal: ‹D $$ (0, 0) dvd D $$ (1, 1)›*)\n by auto\n from this (*‹D $$ (0, 0) dvd D $$ (1, 1)›*) obtain k where D11: \"D $$ (1, 1) = D $$ (0,0) * k\"\n (*goal: ‹(⋀k. D $$ (1, 1) = D $$ (0, 0) * k ⟹ thesis) ⟹ thesis›*)\n unfolding dvd_def\n (*goal: ‹(⋀k. D $$ (1, 1) = D $$ (0, 0) * k ⟹ thesis) ⟹ thesis›*)\n by blast\n have P'D_00: \"(P' * D) $$ (0, 0) = P' $$ (0, 0) * D $$ (0, 0)\"\n using mat_mult2_00[of P' D] (*‹⟦P' ∈ carrier_mat 2 2; D ∈ carrier_mat 2 2; ?A = P' * D⟧ ⟹ ?A $$ (0, 0) = P' $$ (0, 0) * D $$ (0, 0) + P' $$ (0, 1) * D $$ (1, 0)›*) D_10 (*‹D $$ (1, 0) = 0›*) by auto\n have P'D_01: \"(P' * D) $$ (0, 1) = P' $$ (0, 1) * D $$ (1, 1)\"\n using mat_mult2_01[of P' D] (*‹⟦P' ∈ carrier_mat 2 2; D ∈ carrier_mat 2 2; ?A = P' * D⟧ ⟹ ?A $$ (0, 1) = P' $$ (0, 0) * D $$ (0, 1) + P' $$ (0, 1) * D $$ (1, 1)›*) D_01 (*‹D $$ (0, 1) = 0›*) by auto\n have P'D_10: \"(P' * D) $$ (1, 0) = P' $$ (1, 0) * D $$ (0, 0)\"\n using mat_mult2_10[of P' D] (*‹⟦P' ∈ carrier_mat 2 2; D ∈ carrier_mat 2 2; ?A = P' * D⟧ ⟹ ?A $$ (1, 0) = P' $$ (1, 0) * D $$ (0, 0) + P' $$ (1, 1) * D $$ (1, 0)›*) D_10 (*‹D $$ (1, 0) = 0›*) by auto\n have P'D_11: \"(P' * D) $$ (1, 1) = P' $$ (1, 1) * D $$ (1, 1)\"\n using mat_mult2_11[of P' D] (*‹⟦P' ∈ carrier_mat 2 2; D ∈ carrier_mat 2 2; ?A = P' * D⟧ ⟹ ?A $$ (1, 1) = P' $$ (1, 0) * D $$ (0, 1) + P' $$ (1, 1) * D $$ (1, 1)›*) D_01 (*‹D $$ (0, 1) = 0›*) by auto\n have \"a = (P' * D * Q') $$ (0,0)\"\n using e (*‹P' * D * Q' = A›*) A_def (*‹A = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a else if i = 0 ∧ j = 1 then b else if i = 1 ∧ j = 0 then 0 else c)›*) by auto\n also (*calculation: ‹a = (P' * D * Q') $$ (0, 0)›*) have \"... = (P' * D) $$ (0, 0) * Q' $$ (0, 0) + (P' * D) $$ (0, 1) * Q' $$ (1, 0)\"\n apply (rule mat_mult2_00 (*‹⟦?U ∈ carrier_mat 2 2; ?V ∈ carrier_mat 2 2; ?A = ?U * ?V⟧ ⟹ ?A $$ (0, 0) = ?U $$ (0, 0) * ?V $$ (0, 0) + ?U $$ (0, 1) * ?V $$ (1, 0)›*))\n (*goals:\n 1. ‹P' * D ∈ carrier_mat 2 2›\n 2. ‹Q' ∈ carrier_mat 2 2›\n 3. ‹P' * D * Q' = P' * D * Q'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹a = (P' * D) $$ (0, 0) * Q' $$ (0, 0) + (P' * D) $$ (0, 1) * Q' $$ (1, 0)›*) have \"... = P' $$ (0, 0) * D $$ (0, 0) * Q' $$ (0, 0) \n + P' $$ (0, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 0)\"\n unfolding P'D_00 P'D_01 D11\n (*goal: ‹(P'::'a mat) $$ (0::nat, 0::nat) * (D::'a mat) $$ (0::nat, 0::nat) * (Q'::'a mat) $$ (0::nat, 0::nat) + P' $$ (0::nat, 1::nat) * (D $$ (0::nat, 0::nat) * (k::'a)) * Q' $$ (1::nat, 0::nat) = P' $$ (0::nat, 0::nat) * D $$ (0::nat, 0::nat) * Q' $$ (0::nat, 0::nat) + P' $$ (0::nat, 1::nat) * (D $$ (0::nat, 0::nat) * k) * Q' $$ (1::nat, 0::nat)›*)\n by standard\n also (*calculation: ‹(a::'a) = (P'::'a mat) $$ (0::nat, 0::nat) * (D::'a mat) $$ (0::nat, 0::nat) * (Q'::'a mat) $$ (0::nat, 0::nat) + P' $$ (0::nat, 1::nat) * (D $$ (0::nat, 0::nat) * (k::'a)) * Q' $$ (1::nat, 0::nat)›*) have \"... = D $$ (0, 0) * (P' $$ (0, 0) * Q' $$ (0, 0) \n + P' $$ (0, 1) * k * Q' $$ (1, 0))\"\n by (simp add: distrib_left (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c›*))\n finally (*calculation: ‹a = D $$ (0, 0) * (P' $$ (0, 0) * Q' $$ (0, 0) + P' $$ (0, 1) * k * Q' $$ (1, 0))›*) have u_dvd_a: \"u dvd a\"\n unfolding u_def D_def dvd_def\n (*goal: ‹∃k. a = (P * A * Q) $$ (0, 0) * k›*)\n by auto\n have \"b = (P' * D * Q') $$ (0,1)\"\n using e (*‹P' * D * Q' = A›*) A_def (*‹A = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a else if i = 0 ∧ j = 1 then b else if i = 1 ∧ j = 0 then 0 else c)›*) by auto\n also (*calculation: ‹b = (P' * D * Q') $$ (0, 1)›*) have \"... = (P' * D) $$ (0, 0) * Q' $$ (0, 1) + (P' * D) $$ (0, 1) * Q' $$ (1, 1)\"\n apply (rule mat_mult2_01 (*‹⟦?U ∈ carrier_mat 2 2; ?V ∈ carrier_mat 2 2; ?A = ?U * ?V⟧ ⟹ ?A $$ (0, 1) = ?U $$ (0, 0) * ?V $$ (0, 1) + ?U $$ (0, 1) * ?V $$ (1, 1)›*))\n (*goals:\n 1. ‹(P'::'a::comm_ring_1 mat) * (D::'a::comm_ring_1 mat) ∈ carrier_mat (2::nat) (2::nat)›\n 2. ‹(Q'::'a::comm_ring_1 mat) ∈ carrier_mat (2::nat) (2::nat)›\n 3. ‹(P'::'a::comm_ring_1 mat) * (D::'a::comm_ring_1 mat) * (Q'::'a::comm_ring_1 mat) = P' * D * Q'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹b = (P' * D) $$ (0, 0) * Q' $$ (0, 1) + (P' * D) $$ (0, 1) * Q' $$ (1, 1)›*) have \"... = P' $$ (0, 0) * D $$ (0, 0) * Q' $$ (0, 1) +\n P' $$ (0, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1)\"\n unfolding P'D_00 P'D_01 D11\n (*goal: ‹P' $$ (0, 0) * D $$ (0, 0) * Q' $$ (0, 1) + P' $$ (0, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1) = P' $$ (0, 0) * D $$ (0, 0) * Q' $$ (0, 1) + P' $$ (0, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1)›*)\n by standard\n also (*calculation: ‹b = P' $$ (0, 0) * D $$ (0, 0) * Q' $$ (0, 1) + P' $$ (0, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1)›*) have \"... = D $$ (0, 0) * (P' $$ (0, 0) * Q' $$ (0, 1) +\n P' $$ (0, 1) * k * Q' $$ (1, 1))\"\n by (simp add: distrib_left (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c›*))\n finally (*calculation: ‹b = D $$ (0, 0) * (P' $$ (0, 0) * Q' $$ (0, 1) + P' $$ (0, 1) * k * Q' $$ (1, 1))›*) have u_dvd_b: \"u dvd b\"\n unfolding u_def D_def dvd_def\n (*goal: ‹∃k. b = (P * A * Q) $$ (0, 0) * k›*)\n by auto\n have \"c = (P' * D * Q') $$ (1,1)\"\n using e (*‹P' * D * Q' = A›*) A_def (*‹A = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a else if i = 0 ∧ j = 1 then b else if i = 1 ∧ j = 0 then 0 else c)›*) by auto\n also (*calculation: ‹(c::'a) = ((P'::'a mat) * (D::'a mat) * (Q'::'a mat)) $$ (1::nat, 1::nat)›*) have \"... = (P' * D) $$ (1, 0) * Q' $$ (0, 1) + (P' * D) $$ (1, 1) * Q' $$ (1, 1)\"\n apply (rule mat_mult2_11 (*‹⟦?U ∈ carrier_mat 2 2; ?V ∈ carrier_mat 2 2; ?A = ?U * ?V⟧ ⟹ ?A $$ (1, 1) = ?U $$ (1, 0) * ?V $$ (0, 1) + ?U $$ (1, 1) * ?V $$ (1, 1)›*))\n (*goals:\n 1. ‹P' * D ∈ carrier_mat 2 2›\n 2. ‹Q' ∈ carrier_mat 2 2›\n 3. ‹P' * D * Q' = P' * D * Q'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹c = (P' * D) $$ (1, 0) * Q' $$ (0, 1) + (P' * D) $$ (1, 1) * Q' $$ (1, 1)›*) have \"... = P' $$ (1, 0) * D $$ (0, 0) * Q' $$ (0, 1) \n + P' $$ (1, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1)\"\n unfolding P'D_11 P'D_10 D11\n (*goal: ‹P' $$ (1, 0) * D $$ (0, 0) * Q' $$ (0, 1) + P' $$ (1, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1) = P' $$ (1, 0) * D $$ (0, 0) * Q' $$ (0, 1) + P' $$ (1, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1)›*)\n by standard\n also (*calculation: ‹c = P' $$ (1, 0) * D $$ (0, 0) * Q' $$ (0, 1) + P' $$ (1, 1) * (D $$ (0, 0) * k) * Q' $$ (1, 1)›*) have \"... = D $$ (0, 0) * (P' $$ (1, 0) * Q' $$ (0, 1) \n + P' $$ (1, 1) * k * Q' $$ (1, 1))\"\n by (simp add: distrib_left (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c›*))\n finally (*calculation: ‹(c::'a) = (D::'a mat) $$ (0::nat, 0::nat) * ((P'::'a mat) $$ (1::nat, 0::nat) * (Q'::'a mat) $$ (0::nat, 1::nat) + P' $$ (1::nat, 1::nat) * (k::'a) * Q' $$ (1::nat, 1::nat))›*) have u_dvd_c: \"u dvd c\"\n unfolding u_def D_def dvd_def\n (*goal: ‹∃k. c = (P * A * Q) $$ (0, 0) * k›*)\n by auto\n show \"∀x∈{a,b,c}. u dvd x\"\n using u_dvd_a (*‹u dvd a›*) u_dvd_b (*‹(u::'a) dvd (b::'a)›*) u_dvd_c (*‹u dvd c›*) by auto\n qed (simp)\n (*solved the remaining goal: ‹finite {a, b, c}›*)\n have \"ideal_generated {p*a,p*b+q*c} = ideal_generated {1}\"\n by (metis (no_types, lifting) eq (*‹p * a * x + p * b * y + q * c * y = u›*) add.assoc (*‹?a + ?b + ?c = ?a + (?b + ?c)›*) ideal_generated_1 (*‹ideal_generated {1} = UNIV›*) ideal_generated_pair_UNIV (*‹⟦?s * ?a + ?t * ?b = ?u; ?u dvd 1⟧ ⟹ ideal_generated {?a, ?b} = UNIV›*) mult.commute (*‹?a * ?b = ?b * ?a›*) semiring_normalization_rules( (*‹?x * (?y + ?z) = ?x * ?y + ?x * ?z›*) 34) u_dvd_1 (*‹u dvd 1›*))\n from this (*‹ideal_generated {p * a, p * b + q * c} = ideal_generated {1}›*) show \"∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}\"\n by auto\nqed\n\nlemma necessity:\n assumes \"(∀(A::'a mat). admits_diagonal_reduction A)\"\n shows \"(∀(A::'a mat). admits_triangular_reduction A)\"\n and \"∀a b c::'a. ideal_generated{a,b,c} = ideal_generated{1} \n ⟶ (∃p q. ideal_generated {p*a,p*b+q*c} = ideal_generated {1})\"\n using necessity_D' (*‹∀A. admits_diagonal_reduction A ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*) admits_diagonal_imp_admits_triangular (*‹Ball (carrier_mat 1 2) admits_diagonal_reduction ⟹ All admits_triangular_reduction›*) assms (*‹∀A. admits_diagonal_reduction A›*) apply -\n (*goals:\n 1. ‹⟦∀A. admits_diagonal_reduction A ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}); Ball (carrier_mat 1 2) admits_diagonal_reduction ⟹ All admits_triangular_reduction; ∀A. admits_diagonal_reduction A⟧ ⟹ ∀A. admits_triangular_reduction A›\n 2. ‹⟦∀A. admits_diagonal_reduction A ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}); Ball (carrier_mat 1 2) admits_diagonal_reduction ⟹ All admits_triangular_reduction; ∀A. admits_diagonal_reduction A⟧ ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*) .\n\ntext ‹In the article, the authors change the notation and assume $(a,b,c) = (1)$. However,\nwe have to provide here the complete prove. To to this, I obtained a $D$ matrix such that\n$A' = A*D$ and $D$ is a diagonal matrix with $d$ in the diagonal. Proving that $D$ is \nleft and right commutative, I can follow the reasoning in the article›\n\nlemma sufficiency:\n assumes hermite_ring: \"(∀(A::'a mat). admits_triangular_reduction A)\"\n and D': \"∀a b c::'a. ideal_generated{a,b,c} = ideal_generated{1} \n ⟶ (∃p q. ideal_generated {p*a,p*b+q*c} = ideal_generated {1})\"\n shows \"(∀(A::'a mat). admits_diagonal_reduction A)\"\nproof (-)\n (*goal: ‹∀A. admits_diagonal_reduction A›*)\n have admits_1x2: \"∀(A::'a mat) ∈ carrier_mat 1 2. admits_diagonal_reduction A\"\n using hermite_ring (*‹∀A. admits_triangular_reduction A›*) triangular_eq_diagonal_1x2 (*‹Ball (carrier_mat 1 2) admits_triangular_reduction = Ball (carrier_mat 1 2) admits_diagonal_reduction›*) by blast\n have admits_2x2: \"∀(A::'a mat) ∈ carrier_mat 2 2. admits_diagonal_reduction A\"\n proof (standard)\n (*goal: ‹⋀A. A ∈ carrier_mat 2 2 ⟹ admits_diagonal_reduction A›*)\n fix B :: \"'a mat\"\n assume B: \"B ∈ carrier_mat 2 2\" (*‹(B::'a mat) ∈ carrier_mat (2::nat) (2::nat)›*)\n obtain U where BU: \"lower_triangular (B*U)\" and inv_U: \"invertible_mat U\" and U: \"U ∈ carrier_mat 2 2\"\n (*goal: ‹(⋀U. ⟦lower_triangular (B * U); invertible_mat U; U ∈ carrier_mat 2 2⟧ ⟹ thesis) ⟹ thesis›*)\n using hermite_ring (*‹∀A::'a mat. admits_triangular_reduction A›*) unfolding admits_triangular_reduction_def\n (*goal: ‹(⋀U. ⟦lower_triangular (B * U); invertible_mat U; U ∈ carrier_mat 2 2⟧ ⟹ thesis) ⟹ thesis›*)\n using B (*‹B ∈ carrier_mat 2 2›*) by fastforce\n define A where \"A = B*U\"\n define a where \"a = A $$ (0,0)\"\n define b where \"b = A $$ (1,0)\"\n define c where \"c = A $$ (1,1)\"\n have A: \"A ∈ carrier_mat 2 2\"\n using U (*‹U ∈ carrier_mat 2 2›*) B (*‹B ∈ carrier_mat 2 2›*) A_def (*‹(A::'a mat) = (B::'a mat) * (U::'a mat)›*) by auto\n have A_01: \"A$$(0,1) = 0\"\n using BU (*‹lower_triangular (B * U)›*) U (*‹(U::'a::comm_ring_1 mat) ∈ carrier_mat (2::nat) (2::nat)›*) B (*‹(B::'a mat) ∈ carrier_mat (2::nat) (2::nat)›*) unfolding lower_triangular_def A_def\n (*goal: ‹((B::'a mat) * (U::'a mat)) $$ (0::nat, 1::nat) = (0::'a)›*)\n by auto\n obtain d :: 'a where i: \"ideal_generated {a,b,c} = ideal_generated {d}\"\n (*goal: ‹(⋀d::'a. ideal_generated {a::'a, b::'a, c::'a} = ideal_generated {d} ⟹ thesis::bool) ⟹ thesis›*)\n proof (-)\n (*goal: ‹(⋀d. ideal_generated {a, b, c} = ideal_generated {d} ⟹ thesis) ⟹ thesis›*)\n have \"OFCLASS('a, bezout_ring_class)\"\n apply (rule Hermite_ring_imp_Bezout_ring (*‹OFCLASS(?'a, Hermite_ring_class) ⟹ OFCLASS(?'a, bezout_ring_class)›*))\n (*goal: ‹OFCLASS('a::comm_ring_1, bezout_ring_class)›*)\n apply (insert OFCLASS_Hermite_ring_def[where ?'a='a] (*‹OFCLASS('a, Hermite_ring_class) ≡ (⋀A. admits_triangular_reduction A)›*) hermite_ring (*‹∀A. admits_triangular_reduction A›*))\n (*goal: ‹OFCLASS('a, Hermite_ring_class)›*)\n by auto\n hence \"class.bezout_ring (*) (1::'a) (+) 0 (-) uminus\"\n using OFCLASS_bezout_ring_imp_class_bezout_ring[where ?'a = 'a] (*‹OFCLASS('a, bezout_ring_class) ⟹ class.bezout_ring (*) 1 (+) 0 (-) uminus›*) by auto\n hence \"(∀I::'a::comm_ring_1 set. finitely_generated_ideal I ⟶ principal_ideal I)\"\n using bezout_ring_iff_fin_gen_principal_ideal2 (*‹(∀I. finitely_generated_ideal I ⟶ principal_ideal I) = class.bezout_ring (*) 1 (+) 0 (-) uminus›*) by auto\n moreover have \"finitely_generated_ideal (ideal_generated {a,b,c})\"\n unfolding finitely_generated_ideal_def\n (*goal: ‹ideal (ideal_generated {a, b, c}) ∧ (∃S. finite S ∧ ideal_generated S = ideal_generated {a, b, c})›*)\n using ideal_ideal_generated (*‹ideal (ideal_generated ?S)›*) by force\n ultimately have \"principal_ideal (ideal_generated {a,b,c})\"\n by auto\n thus \"?thesis\"\n (*goal: ‹thesis::bool›*)\n using that (*‹ideal_generated {a, b, c} = ideal_generated {?d} ⟹ thesis›*) unfolding principal_ideal_def\n (*goal: ‹thesis›*)\n by auto\n qed\n have d_dvd_a: \"d dvd a\" and d_dvd_b: \"d dvd b\" and d_dvd_c: \"d dvd c\"\n using i (*‹ideal_generated {a, b, c} = ideal_generated {d}›*) ideal_generated_singleton_dvd (*‹⟦ideal_generated ?S = ideal_generated {?d}; ?x ∈ ?S⟧ ⟹ ?d dvd ?x›*) apply -\n (*goals:\n 1. ‹⟦ideal_generated {a, b, c} = ideal_generated {d}; ⋀S d x. ⟦ideal_generated S = ideal_generated {d}; x ∈ S⟧ ⟹ d dvd x⟧ ⟹ d dvd a›\n 2. ‹⟦ideal_generated {a, b, c} = ideal_generated {d}; ⋀S d x. ⟦ideal_generated S = ideal_generated {d}; x ∈ S⟧ ⟹ d dvd x⟧ ⟹ d dvd b›\n 3. ‹⟦ideal_generated {a, b, c} = ideal_generated {d}; ⋀S d x. ⟦ideal_generated S = ideal_generated {d}; x ∈ S⟧ ⟹ d dvd x⟧ ⟹ d dvd c›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*discuss goal 3*)\n apply blast\n (*proven 3 subgoals*) .\n obtain a1 and b1 and c1 where a1: \"a = a1 * d\" and b1: \"b = b1 * d\" and c1: \"c = c1 * d\" and i2: \"ideal_generated {a1,b1,c1} = ideal_generated {1}\"\n (*goal: ‹(⋀(a1::'a) (b1::'a) c1::'a. ⟦(a::'a) = a1 * (d::'a); (b::'a) = b1 * d; (c::'a) = c1 * d; ideal_generated {a1, b1, c1} = ideal_generated {1::'a}⟧ ⟹ thesis::bool) ⟹ thesis›*)\n proof (-)\n (*goal: ‹(⋀(a1::'a::comm_ring_1) (b1::'a::comm_ring_1) c1::'a::comm_ring_1. ⟦(a::'a::comm_ring_1) = a1 * (d::'a::comm_ring_1); (b::'a::comm_ring_1) = b1 * d; (c::'a::comm_ring_1) = c1 * d; ideal_generated {a1, b1, c1} = ideal_generated {1::'a::comm_ring_1}⟧ ⟹ thesis::bool) ⟹ thesis›*)\n have T: \"∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d \n ∧ ideal_generated {a1, b1} = ideal_generated {1::'a}\"\n by (rule theorem3_part2[OF hermite_ring] (*‹∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*))\n from this (*‹∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*) obtain a1' and b1' and d' where 1: \"a = a1' * d'\" and 2: \"b = b1' * d'\" and 3: \"ideal_generated {a1', b1'} = ideal_generated {1::'a}\"\n (*goal: ‹(⋀a1' d' b1'. ⟦a = a1' * d'; b = b1' * d'; ideal_generated {a1', b1'} = ideal_generated {1}⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n have \"∃a1 b1 c1. a = a1 * d ∧ b = b1 * d ∧ c = c1 * d \n ∧ ideal_generated {a1, b1, c1} = ideal_generated {1}\"\n by (rule corollary5[OF T i] (*‹∃(a1::'a::comm_ring_1) (b1::'a::comm_ring_1) c1::'a::comm_ring_1. (a::'a::comm_ring_1) = a1 * (d::'a::comm_ring_1) ∧ (b::'a::comm_ring_1) = b1 * d ∧ (c::'a::comm_ring_1) = c1 * d ∧ ideal_generated {a1, b1, c1} = ideal_generated {1::'a::comm_ring_1}›*))\n from this (*‹∃a1 b1 c1. a = a1 * d ∧ b = b1 * d ∧ c = c1 * d ∧ ideal_generated {a1, b1, c1} = ideal_generated {1}›*) show \"?thesis\"\n (*goal: ‹thesis::bool›*)\n using that (*‹⟦a = ?a1.0 * d; b = ?b1.0 * d; c = ?c1.0 * d; ideal_generated {?a1.0, ?b1.0, ?c1.0} = ideal_generated {1}⟧ ⟹ thesis›*) by auto\n qed\n define D where \"D = d ⋅⇩m (1⇩m 2)\"\n define A' where \"A' = Matrix.mat 2 2 (λ(i,j). if i = 0 ∧ j = 0 then a1 else\n if i = 1 ∧ j = 0 then b1 else\n if i = 0 ∧ j = 1 then 0 else c1)\"\n have D: \"D ∈ carrier_mat 2 2\" and A': \"A'∈ carrier_mat 2 2\"\n unfolding A'_def D_def\n (*goals:\n 1. ‹d ⋅⇩m 1⇩m 2 ∈ carrier_mat 2 2›\n 2. ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1) ∈ carrier_mat 2 2›*)\n (*goals:\n 1. ‹(d::'a) ⋅⇩m 1⇩m (2::nat) ∈ carrier_mat (2::nat) (2::nat)›\n 2. ‹Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a1::'a else if i = (1::nat) ∧ j = (0::nat) then b1::'a else if i = (0::nat) ∧ j = (1::nat) then 0::'a else (c1::'a)) ∈ carrier_mat (2::nat) (2::nat)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have A_A'D: \"A = A' * D\"\n apply (rule eq_matI (*‹⟦⋀i j. ⟦i < dim_row ?B; j < dim_col ?B⟧ ⟹ ?A $$ (i, j) = ?B $$ (i, j); dim_row ?A = dim_row ?B; dim_col ?A = dim_col ?B⟧ ⟹ ?A = ?B›*))\n (*goals:\n 1. ‹⋀(i::nat) j::nat. ⟦i < dim_row ((A'::'a mat) * (D::'a mat)); j < dim_col (A' * D)⟧ ⟹ (A::'a mat) $$ (i, j) = (A' * D) $$ (i, j)›\n 2. ‹dim_row (A::'a mat) = dim_row ((A'::'a mat) * (D::'a mat))›\n 3. ‹dim_col (A::'a mat) = dim_col ((A'::'a mat) * (D::'a mat))›\n discuss goal 1*)\n apply ((insert D (*‹D ∈ carrier_mat 2 2›*) A' (*‹A' ∈ carrier_mat 2 2›*) A (*‹A ∈ carrier_mat 2 2›*) a1 (*‹a = a1 * d›*) b1 (*‹b = b1 * d›*) c1 (*‹c = c1 * d›*) A_01 (*‹A $$ (0, 1) = 0›*) sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*) a_def (*‹a = A $$ (0, 0)›*) b_def (*‹b = A $$ (1, 0)›*) c_def (*‹c = A $$ (1, 1)›*))[1])\n (*top goal: ‹⋀(i::nat) j::nat. ⟦i < dim_row ((A'::'a::comm_ring_1 mat) * (D::'a::comm_ring_1 mat)); j < dim_col (A' * D)⟧ ⟹ (A::'a::comm_ring_1 mat) $$ (i, j) = (A' * D) $$ (i, j)› and 2 goals remain*)\n apply ((unfold scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $v i * ?w $v i›*) Matrix.row_def (*‹Matrix.row ?A ?i = Matrix.vec (dim_col ?A) (λj. ?A $$ (?i, j))›*) col_def (*‹col ?A ?j = Matrix.vec (dim_row ?A) (λi. ?A $$ (i, ?j))›*) D_def (*‹D = d ⋅⇩m 1⇩m 2›*) A'_def (*‹A' = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)›*))[1])\n (*top goal: ‹⋀i j. ⟦i < dim_row (A' * D); j < dim_col (A' * D); D ∈ carrier_mat 2 2; A' ∈ carrier_mat 2 2; A ∈ carrier_mat 2 2; a = a1 * d; b = b1 * d; c = c1 * d; A $$ (0, 1) = 0; ⋀f. sum f {0..<2} = sum f {0, 1}; a = A $$ (0, 0); b = A $$ (1, 0); c = A $$ (1, 1)⟧ ⟹ A $$ (i, j) = (A' * D) $$ (i, j)› and 2 goals remain*)\n apply ((auto simp add: sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*) less_Suc_eq (*‹(?m < Suc ?n) = (?m < ?n ∨ ?m = ?n)›*) numerals( (*‹2 = Suc (Suc 0)›*) 2))[1])\n (*discuss goal 2*)\n apply ((insert D (*‹D ∈ carrier_mat 2 2›*) A' (*‹A' ∈ carrier_mat 2 2›*) A (*‹A ∈ carrier_mat 2 2›*) a1 (*‹a = a1 * d›*) b1 (*‹b = b1 * d›*) c1 (*‹c = c1 * d›*) A_01 (*‹A $$ (0, 1) = 0›*) sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*) a_def (*‹a = A $$ (0, 0)›*) b_def (*‹b = A $$ (1, 0)›*) c_def (*‹c = A $$ (1, 1)›*))[1])\n (*top goal: ‹dim_row A = dim_row (A' * D)› and 1 goal remains*)\n apply ((unfold scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $v i * ?w $v i›*) Matrix.row_def (*‹Matrix.row ?A ?i = Matrix.vec (dim_col ?A) (λj. ?A $$ (?i, j))›*) col_def (*‹col ?A ?j = Matrix.vec (dim_row ?A) (λi. ?A $$ (i, ?j))›*) D_def (*‹D = d ⋅⇩m 1⇩m 2›*) A'_def (*‹A' = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)›*))[1])\n (*top goal: ‹⟦D ∈ carrier_mat 2 2; A' ∈ carrier_mat 2 2; A ∈ carrier_mat 2 2; a = a1 * d; b = b1 * d; c = c1 * d; A $$ (0, 1) = 0; ⋀f. sum f {0..<2} = sum f {0, 1}; a = A $$ (0, 0); b = A $$ (1, 0); c = A $$ (1, 1)⟧ ⟹ dim_row A = dim_row (A' * D)› and 1 goal remains*)\n apply ((auto simp add: sum_two_rw (*‹sum (?f::nat ⇒ ?'a) {0::nat..<2::nat} = sum ?f {0::nat, 1::nat}›*) less_Suc_eq (*‹((?m::nat) < Suc (?n::nat)) = (?m < ?n ∨ ?m = ?n)›*) numerals( (*‹(2::nat) = Suc (Suc (0::nat))›*) 2))[1])\n (*discuss goal 3*)\n apply ((insert D (*‹D ∈ carrier_mat 2 2›*) A' (*‹A' ∈ carrier_mat 2 2›*) A (*‹A ∈ carrier_mat 2 2›*) a1 (*‹a = a1 * d›*) b1 (*‹b = b1 * d›*) c1 (*‹c = c1 * d›*) A_01 (*‹A $$ (0, 1) = 0›*) sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*) a_def (*‹a = A $$ (0, 0)›*) b_def (*‹b = A $$ (1, 0)›*) c_def (*‹c = A $$ (1, 1)›*))[1])\n (*goal: ‹dim_col A = dim_col (A' * D)›*)\n apply ((unfold scalar_prod_def (*‹(?v::?'a::semiring_0 Matrix.vec) ∙ (?w::?'a::semiring_0 Matrix.vec) ≡ ∑i::nat = 0::nat..<dim_vec ?w. ?v $v i * ?w $v i›*) Matrix.row_def (*‹Matrix.row (?A::?'a::type mat) (?i::nat) = Matrix.vec (dim_col ?A) (λj::nat. ?A $$ (?i, j))›*) col_def (*‹col (?A::?'a::type mat) (?j::nat) = Matrix.vec (dim_row ?A) (λi::nat. ?A $$ (i, ?j))›*) D_def (*‹(D::'a::comm_ring_1 mat) = (d::'a::comm_ring_1) ⋅⇩m 1⇩m (2::nat)›*) A'_def (*‹(A'::'a::comm_ring_1 mat) = Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then a1::'a::comm_ring_1 else if i = (1::nat) ∧ j = (0::nat) then b1::'a::comm_ring_1 else if i = (0::nat) ∧ j = (1::nat) then 0::'a::comm_ring_1 else (c1::'a::comm_ring_1))›*))[1])\n (*goal: ‹⟦D ∈ carrier_mat 2 2; A' ∈ carrier_mat 2 2; A ∈ carrier_mat 2 2; a = a1 * d; b = b1 * d; c = c1 * d; A $$ (0, 1) = 0; ⋀f. sum f {0..<2} = sum f {0, 1}; a = A $$ (0, 0); b = A $$ (1, 0); c = A $$ (1, 1)⟧ ⟹ dim_col A = dim_col (A' * D)›*)\n apply ((auto simp add: sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*) less_Suc_eq (*‹(?m < Suc ?n) = (?m < ?n ∨ ?m = ?n)›*) numerals( (*‹2 = Suc (Suc 0)›*) 2))[1])\n (*proven 3 subgoals*) .\n have \"1∈ ideal_generated{a1,b1,c1}\"\n using i2 (*‹ideal_generated {a1, b1, c1} = ideal_generated {1}›*) by (simp add: ideal_generated_in (*‹?a ∈ ?A ⟹ ?a ∈ ideal_generated ?A›*))\n from this (*‹1 ∈ ideal_generated {a1, b1, c1}›*) obtain f where d: \"(∑i∈{a1,b1,c1}. f i * i) = 1\"\n (*goal: ‹(⋀f. (∑i∈{a1, b1, c1}. f i * i) = 1 ⟹ thesis) ⟹ thesis›*)\n using ideal_explicit2[of \"{a1,b1,c1}\"] (*‹finite {a1::'a, b1::'a, c1::'a} ⟹ ideal_generated {a1, b1, c1} = {y::'a. ∃f::'a ⇒ 'a. (∑i::'a∈{a1, b1, c1}. f i * i) = y}›*) by auto\n from this (*‹(∑i::'a∈{a1::'a, b1::'a, c1::'a}. (f::'a ⇒ 'a) i * i) = (1::'a)›*) obtain x and y and z where \"x*a1+y*b1+z*c1 = 1\"\n (*goal: ‹(⋀x y z. x * a1 + y * b1 + z * c1 = 1 ⟹ thesis) ⟹ thesis›*)\n using sum_three_elements[of _ a1 b1 c1] (*‹∃x y z. (∑i∈{a1, b1, c1}. ?f i * i) = x * a1 + y * b1 + z * c1›*) by metis\n hence xa1_yb1_zc1_dvd_1: \"x * a1 + y * b1 + z * c1 dvd 1\"\n by auto\n obtain p and q where i3: \"ideal_generated {p*a1,p*b1+q*c1} = ideal_generated {1}\"\n (*goal: ‹(⋀p q. ideal_generated {p * a1, p * b1 + q * c1} = ideal_generated {1} ⟹ thesis) ⟹ thesis›*)\n using D' (*‹∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*) i2 (*‹ideal_generated {a1, b1, c1} = ideal_generated {1}›*) by blast\n have \"ideal_generated {p,q} = UNIV\"\n proof (-)\n (*goal: ‹ideal_generated {p, q} = UNIV›*)\n obtain X and Y where e: \"X*p*a1 + Y*(p*b1+q*c1) = 1\"\n (*goal: ‹(⋀X Y. X * p * a1 + Y * (p * b1 + q * c1) = 1 ⟹ thesis) ⟹ thesis›*)\n by (metis i3 (*‹ideal_generated {p * a1, p * b1 + q * c1} = ideal_generated {1}›*) ideal_generated_1 (*‹ideal_generated {1} = UNIV›*) ideal_generated_pair_exists_UNIV (*‹(ideal_generated {?a, ?b} = ideal_generated {1}) = (∃p q. p * ?a + q * ?b = 1)›*) mult.assoc (*‹?a * ?b * ?c = ?a * (?b * ?c)›*))\n have \"X*p*a1 + Y*(p*b1+q*c1) = X*p*a1 + Y*p*b1+Y*q*c1\"\n by (simp add: add.assoc (*‹?a + ?b + ?c = ?a + (?b + ?c)›*) mult.assoc (*‹?a * ?b * ?c = ?a * (?b * ?c)›*) semiring_normalization_rules( (*‹?x * (?y + ?z) = ?x * ?y + ?x * ?z›*) 34))\n also (*calculation: ‹X * p * a1 + Y * (p * b1 + q * c1) = X * p * a1 + Y * p * b1 + Y * q * c1›*) have \"... = (X*a1+Y*b1) * p + (Y * c1) * q\"\n by (simp add: mult.commute (*‹?a * ?b = ?b * ?a›*) ring_class.ring_distribs (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c› ‹(?a + ?b) * ?c = ?a * ?c + ?b * ?c› ‹(?a - ?b) * ?c = ?a * ?c - ?b * ?c› ‹?a * (?b - ?c) = ?a * ?b - ?a * ?c›*))\n finally (*calculation: ‹X * p * a1 + Y * (p * b1 + q * c1) = (X * a1 + Y * b1) * p + Y * c1 * q›*) have \"(X*a1+Y*b1) * p + Y * c1 * q = 1\"\n using e (*‹X * p * a1 + Y * (p * b1 + q * c1) = 1›*) by simp\n from this (*‹(X * a1 + Y * b1) * p + Y * c1 * q = 1›*) show \"?thesis\"\n (*goal: ‹ideal_generated {p::'a::comm_ring_1, q::'a::comm_ring_1} = UNIV›*)\n apply (rule ideal_generated_pair_UNIV (*‹⟦?s * ?a + ?t * ?b = ?u; ?u dvd 1⟧ ⟹ ideal_generated {?a, ?b} = UNIV›*))\n (*goal: ‹ideal_generated {p::'a, q::'a} = UNIV›*)\n by simp\n qed\n from this (*‹ideal_generated {p, q} = UNIV›*) obtain u and v where pu_qv_1: \"p*u - q * v = 1\"\n (*goal: ‹(⋀u v. p * u - q * v = 1 ⟹ thesis) ⟹ thesis›*)\n by (metis Groups.mult_ac( (*‹?a * ?b = ?b * ?a›*) 2) diff_minus_eq_add (*‹?a - - ?b = ?a + ?b›*) ideal_generated_1 (*‹ideal_generated {1} = UNIV›*) ideal_generated_pair_exists_UNIV (*‹(ideal_generated {?a, ?b} = ideal_generated {1}) = (∃p q. p * ?a + q * ?b = 1)›*) mult_minus_left (*‹- ?a * ?b = - (?a * ?b)›*))\n let ?P = \"Matrix.mat 2 2 (λ(i,j). if i = 0 ∧ j = 0 then p else\n if i = 1 ∧ j = 0 then q else\n if i = 0 ∧ j = 1 then v else u)\"\n have P: \"?P ∈ carrier_mat 2 2\"\n by auto\n have \"Determinant.det ?P = 1\"\n using pu_qv_1 (*‹p * u - q * v = 1›*) unfolding det_2[OF P]\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) $$ (0, 0) * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) $$ (1, 1) - Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) $$ (0, 1) * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) $$ (1, 0) = 1›*)\n by (simp add: mult.commute (*‹?a * ?b = ?b * ?a›*))\n hence inv_P: \"invertible_mat ?P\"\n by (metis (no_types, lifting) P (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) ∈ carrier_mat 2 2›*) dvd_refl (*‹?a dvd ?a›*) invertible_iff_is_unit_JNF (*‹?A ∈ carrier_mat ?n ?n ⟹ invertible_mat ?A = (Determinant.det ?A dvd 1)›*))\n define S1 where \"S1 = A'*?P\"\n have S1: \"S1 ∈ carrier_mat 2 2\"\n using A' (*‹A' ∈ carrier_mat 2 2›*) P (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) ∈ carrier_mat 2 2›*) S1_def (*‹S1 = A' * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)›*) mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*) by blast\n have S1_00: \"S1 $$(0,0) = p*a1\" and S1_01: \"S1 $$(1,0) = p*b1+q*c1\"\n unfolding S1_def times_mat_def scalar_prod_def\n (*goals:\n 1. ‹Matrix.mat (dim_row A') (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row A' i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (0, 0) = p * a1›\n 2. ‹Matrix.mat (dim_row A') (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row A' i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (1, 0) = p * b1 + q * c1›*)\n using A' (*‹A' ∈ carrier_mat 2 2›*) P (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) ∈ carrier_mat 2 2›*) BU (*‹lower_triangular (B * U)›*) U (*‹U ∈ carrier_mat 2 2›*) B (*‹B ∈ carrier_mat 2 2›*) unfolding A'_def upper_triangular_def\n (*goals:\n 1. ‹Matrix.mat (dim_row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1))) (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)) i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (0, 0) = p * a1›\n 2. ‹Matrix.mat (dim_row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1))) (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)) i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (1, 0) = p * b1 + q * c1›*)\n (*goals:\n 1. ‹Matrix.mat (dim_row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1))) (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)) i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (0, 0) = p * a1›\n 2. ‹Matrix.mat (dim_row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1))) (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)) i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (1, 0) = p * b1 + q * c1›\n discuss goal 1*)\n apply ((auto)[1])\n (*top goal: ‹Matrix.mat (dim_row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1))) (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)) i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (0, 0) = p * a1› and 1 goal remains*)\n apply ((unfold sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*))[1])\n (*top goal: ‹⟦lower_triangular (B * U); U ∈ carrier_mat 2 2; B ∈ carrier_mat 2 2⟧ ⟹ (∑i = 0..<2. (if i = 0 then a1 else if 0 = 1 ∧ i = 0 then b1 else if 0 = 0 ∧ i = 1 then 0 else c1) * (if i = 0 then p else if i = 1 ∧ 0 = 0 then q else if i = 0 ∧ 0 = 1 then v else u)) = p * a1› and 1 goal remains*)\n apply ((auto simp add: A'_def (*‹A' = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)›*) a_def (*‹a = A $$ (0, 0)›*) b_def (*‹b = A $$ (1, 0)›*) c_def (*‹c = A $$ (1, 1)›*))[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*goal: ‹Matrix.mat (dim_row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1))) (dim_col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))) (λ(i, j). ∑ia = 0..<dim_vec (col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j). Matrix.row (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)) i $v ia * col (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u)) j $v ia) $$ (1, 0) = p * b1 + q * c1›*)\n apply ((unfold sum_two_rw (*‹sum ?f {0..<2} = sum ?f {0, 1}›*))[1])\n (*goal: ‹(∑i = 0..<2. (if i = 0 then b1 else if Suc 0 = 0 ∧ i = 1 then 0 else c1) * (if i = 0 then p else if i = 1 ∧ 0 = 0 then q else if i = 0 ∧ 0 = 1 then v else u)) = p * b1 + q * c1›*)\n apply ((auto simp add: A'_def (*‹A' = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then a1 else if i = 1 ∧ j = 0 then b1 else if i = 0 ∧ j = 1 then 0 else c1)›*) a_def (*‹a = A $$ (0, 0)›*) b_def (*‹b = A $$ (1, 0)›*) c_def (*‹c = A $$ (1, 1)›*))[1])\n (*proven 2 subgoals*) .\n obtain q00 and q01 where q00_q01: \"p*a1*q00 + (p*b1+q*c1)*q01 = 1\"\n (*goal: ‹(⋀(q00::'a::comm_ring_1) q01::'a::comm_ring_1. (p::'a::comm_ring_1) * (a1::'a::comm_ring_1) * q00 + (p * (b1::'a::comm_ring_1) + (q::'a::comm_ring_1) * (c1::'a::comm_ring_1)) * q01 = (1::'a::comm_ring_1) ⟹ thesis::bool) ⟹ thesis›*)\n using i3 (*‹ideal_generated {p * a1, p * b1 + q * c1} = ideal_generated {1}›*) by (metis ideal_generated_1 (*‹ideal_generated {1} = UNIV›*) ideal_generated_pair_exists_pq1 (*‹ideal_generated {?a, ?b} = UNIV ⟹ ∃p q. p * ?a + q * ?b = 1›*) mult.commute (*‹?a * ?b = ?b * ?a›*))\n define q10 where \"q10 = - (p*b1+q*c1)\"\n define q11 where \"q11 = p*a1\"\n have q10_q11: \"p*a1*q10 + (p*b1+q*c1)*q11 = 0\"\n unfolding q10_def q11_def\n (*goal: ‹(p::'a::comm_ring_1) * (a1::'a::comm_ring_1) * - (p * (b1::'a::comm_ring_1) + (q::'a::comm_ring_1) * (c1::'a::comm_ring_1)) + (p * b1 + q * c1) * (p * a1) = (0::'a::comm_ring_1)›*)\n by (auto simp add: Rings.ring_distribs( (*‹?a * (?b + ?c) = ?a * ?b + ?a * ?c›*) 1) Rings.ring_distribs( (*‹?a * (?b - ?c) = ?a * ?b - ?a * ?c›*) 4) semiring_normalization_rules( (*‹?a * ?b = ?b * ?a›*) 7))\n let ?Q = \"Matrix.mat 2 2 (λ(i,j). if i = 0 ∧ j = 0 then q00 else\n if i = 1 ∧ j = 0 then q10 else\n if i = 0 ∧ j = 1 then q01 else q11)\"\n have Q: \"?Q ∈ carrier_mat 2 2\"\n by auto\n have \"Determinant.det ?Q = 1\"\n using q00_q01 (*‹(p::'a) * (a1::'a) * (q00::'a) + (p * (b1::'a) + (q::'a) * (c1::'a)) * (q01::'a) = (1::'a)›*) unfolding det_2[OF Q]\n (*goal: ‹Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then q10::'a else if i = (0::nat) ∧ j = (1::nat) then q01::'a else (q11::'a)) $$ (0::nat, 0::nat) * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then q10 else if i = (0::nat) ∧ j = (1::nat) then q01 else q11) $$ (1::nat, 1::nat) - Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then q10 else if i = (0::nat) ∧ j = (1::nat) then q01 else q11) $$ (0::nat, 1::nat) * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then q10 else if i = (0::nat) ∧ j = (1::nat) then q01 else q11) $$ (1::nat, 0::nat) = (1::'a)›*)\n unfolding q10_def q11_def\n (*goal: ‹Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then - ((p::'a) * (b1::'a) + (q::'a) * (c1::'a)) else if i = (0::nat) ∧ j = (1::nat) then q01::'a else p * (a1::'a)) $$ (0::nat, 0::nat) * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then - (p * b1 + q * c1) else if i = (0::nat) ∧ j = (1::nat) then q01 else p * a1) $$ (1::nat, 1::nat) - Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then - (p * b1 + q * c1) else if i = (0::nat) ∧ j = (1::nat) then q01 else p * a1) $$ (0::nat, 1::nat) * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then - (p * b1 + q * c1) else if i = (0::nat) ∧ j = (1::nat) then q01 else p * a1) $$ (1::nat, 0::nat) = (1::'a)›*)\n apply auto\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then - (p * b1 + q * c1) else if i = 0 ∧ j = 1 then q01 else p * a1) $$ (0, 0) * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then - (p * b1 + q * c1) else if i = 0 ∧ j = 1 then q01 else p * a1) $$ (1, 1) - Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then - (p * b1 + q * c1) else if i = 0 ∧ j = 1 then q01 else p * a1) $$ (0, 1) * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then - (p * b1 + q * c1) else if i = 0 ∧ j = 1 then q01 else p * a1) $$ (1, 0) = 1›*)\n by (metis (no_types, lifting) add_uminus_conv_diff (*‹?a + - ?b = ?a - ?b›*) diff_minus_eq_add (*‹?a - - ?b = ?a + ?b›*) more_arith_simps( (*‹- ?a * ?b = - (?a * ?b)›*) 7) more_arith_simps( (*‹- (?a + ?b) = - ?a + - ?b›*) 9) mult.commute (*‹?a * ?b = ?b * ?a›*))\n hence inv_Q: \"invertible_mat ?Q\"\n by (smt (verit) Q (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) ∈ carrier_mat 2 2›*) dvd_refl (*‹?a dvd ?a›*) invertible_iff_is_unit_JNF (*‹?A ∈ carrier_mat ?n ?n ⟹ invertible_mat ?A = (Determinant.det ?A dvd 1)›*))\n define S2 where \"S2 = ?Q * S1 \"\n have S2: \"S2 ∈ carrier_mat 2 2\"\n using A' (*‹(A'::'a mat) ∈ carrier_mat (2::nat) (2::nat)›*) P (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) ∈ carrier_mat 2 2›*) S2_def (*‹S2 = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * S1›*) S1 (*‹(S1::'a mat) ∈ carrier_mat (2::nat) (2::nat)›*) Q (*‹Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then q10::'a else if i = (0::nat) ∧ j = (1::nat) then q01::'a else (q11::'a)) ∈ carrier_mat (2::nat) (2::nat)›*) mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*) by blast\n have S2_00: \"S2 $$ (0,0) = 1\"\n unfolding mat_mult2_00[OF Q S1 S2_def]\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) $$ (0, 0) * S1 $$ (0, 0) + Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) $$ (0, 1) * S1 $$ (1, 0) = 1›*)\n using q00_q01 (*‹p * a1 * q00 + (p * b1 + q * c1) * q01 = 1›*) unfolding S1_00 S1_01\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) $$ (0, 0) * (p * a1) + Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) $$ (0, 1) * (p * b1 + q * c1) = 1›*)\n by (simp add: mult.commute (*‹(?a::?'a) * (?b::?'a) = ?b * ?a›*))\n have S2_10: \"S2 $$ (1,0) = 0\"\n unfolding mat_mult2_10[OF Q S1 S2_def]\n (*goal: ‹Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then q10::'a else if i = (0::nat) ∧ j = (1::nat) then q01::'a else (q11::'a)) $$ (1::nat, 0::nat) * (S1::'a mat) $$ (0::nat, 0::nat) + Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then q10 else if i = (0::nat) ∧ j = (1::nat) then q01 else q11) $$ (1::nat, 1::nat) * S1 $$ (1::nat, 0::nat) = (0::'a)›*)\n using q10_q11 (*‹p * a1 * q10 + (p * b1 + q * c1) * q11 = 0›*) unfolding S1_00 S1_01\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) $$ (1, 0) * (p * a1) + Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) $$ (1, 1) * (p * b1 + q * c1) = 0›*)\n by (simp add: Groups.mult_ac( (*‹?a * ?b = ?b * ?a›*) 2))\n let ?P1 = \"(addrow_mat 2 (- (S2$$(0,1))) 0 1)\"\n have P1: \"?P1 ∈ carrier_mat 2 2\"\n by auto\n have inv_P1: \"invertible_mat ?P1\"\n by (metis addrow_mat_carrier (*‹addrow_mat ?n ?a ?k ?l ∈ carrier_mat ?n ?n›*) arithmetic_simps( (*‹1 * ?a = ?a›*) 78) det_addrow_mat (*‹?k ≠ ?l ⟹ Determinant.det (addrow_mat ?n ?a ?k ?l) = 1›*) dvd_def (*‹(?b dvd ?a) = (∃k. ?a = ?b * k)›*) invertible_iff_is_unit_JNF (*‹?A ∈ carrier_mat ?n ?n ⟹ invertible_mat ?A = (Determinant.det ?A dvd 1)›*) numeral_One (*‹Numeral1 = 1›*) zero_neq_numeral (*‹0 ≠ numeral ?n›*))\n define S3 where \"S3 = S2 * ?P1\"\n have P1_P_A': \" A' *?P *?P1 ∈ carrier_mat 2 2\"\n using P1 (*‹addrow_mat 2 (- S2 $$ (0, 1)) 0 1 ∈ carrier_mat 2 2›*) P (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) ∈ carrier_mat 2 2›*) A' (*‹A' ∈ carrier_mat 2 2›*) mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*) by auto\n have S3: \"S3 ∈ carrier_mat 2 2\"\n using P1 (*‹addrow_mat 2 (- S2 $$ (0, 1)) 0 1 ∈ carrier_mat 2 2›*) S2 (*‹S2 ∈ carrier_mat 2 2›*) S3_def (*‹S3 = S2 * addrow_mat 2 (- S2 $$ (0, 1)) 0 1›*) mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*) by blast\n have S3_00: \"S3 $$ (0,0) = 1\"\n using S2_00 (*‹S2 $$ (0, 0) = 1›*) unfolding mat_mult2_00[OF S2 P1 S3_def]\n (*goal: ‹S2 $$ (0, 0) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1 $$ (0, 0) + S2 $$ (0, 1) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1 $$ (1, 0) = 1›*)\n by auto\n moreover have S3_01: \"S3 $$ (0,1) = 0\"\n using S2_00 (*‹S2 $$ (0, 0) = 1›*) unfolding mat_mult2_01[OF S2 P1 S3_def]\n (*goal: ‹S2 $$ (0, 0) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1 $$ (0, 1) + S2 $$ (0, 1) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1 $$ (1, 1) = 0›*)\n by auto\n moreover have S3_10: \"S3 $$ (1,0) = 0\"\n using S2_10 (*‹S2 $$ (1, 0) = 0›*) unfolding mat_mult2_10[OF S2 P1 S3_def]\n (*goal: ‹(S2::'a mat) $$ (1::nat, 0::nat) * addrow_mat (2::nat) (- S2 $$ (0::nat, 1::nat)) (0::nat) (1::nat) $$ (0::nat, 0::nat) + S2 $$ (1::nat, 1::nat) * addrow_mat (2::nat) (- S2 $$ (0::nat, 1::nat)) (0::nat) (1::nat) $$ (1::nat, 0::nat) = (0::'a)›*)\n by auto\n ultimately have SNF_S3: \"Smith_normal_form_mat S3\"\n using S3 (*‹S3 ∈ carrier_mat 2 2›*) unfolding Smith_normal_form_mat_def isDiagonal_mat_def\n (*goal: ‹(∀a. a + 1 < min (dim_row S3) (dim_col S3) ⟶ S3 $$ (a, a) dvd S3 $$ (a + 1, a + 1)) ∧ (∀i j. i ≠ j ∧ i < dim_row S3 ∧ j < dim_col S3 ⟶ S3 $$ (i, j) = 0)›*)\n using less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*) by auto\n hence SNF_S3_D: \"Smith_normal_form_mat (S3*D)\"\n using D_def (*‹D = d ⋅⇩m 1⇩m 2›*) S3 (*‹S3 ∈ carrier_mat 2 2›*) SNF_preserved_multiples_identity (*‹⟦(?S::?'a mat) ∈ carrier_mat (?m::nat) (?n::nat); Smith_normal_form_mat ?S⟧ ⟹ Smith_normal_form_mat (?S * ((?k::?'a) ⋅⇩m 1⇩m ?n))›*) by blast\n have \"S3 * D = ?Q * A' * ?P * ?P1 * D\"\n using S1_def (*‹(S1::'a mat) = (A'::'a mat) * Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then p::'a else if i = (1::nat) ∧ j = (0::nat) then q::'a else if i = (0::nat) ∧ j = (1::nat) then v::'a else (u::'a))›*) S2_def (*‹S2 = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * S1›*) S3_def (*‹S3 = S2 * addrow_mat 2 (- S2 $$ (0, 1)) 0 1›*) by (smt (verit) A' (*‹A' ∈ carrier_mat 2 2›*) P (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) ∈ carrier_mat 2 2›*) Q (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) ∈ carrier_mat 2 2›*) S1 (*‹S1 ∈ carrier_mat 2 2›*) addrow_mat_carrier (*‹addrow_mat ?n ?a ?k ?l ∈ carrier_mat ?n ?n›*) assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*))\n also (*calculation: ‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * A' * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1 * D›*) have \"... = ?Q * A' * ?P * (?P1 * D)\"\n by (meson A' (*‹A' ∈ carrier_mat 2 2›*) D (*‹D ∈ carrier_mat 2 2›*) addrow_mat_carrier (*‹addrow_mat ?n ?a ?k ?l ∈ carrier_mat ?n ?n›*) assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) mat_carrier (*‹Matrix.mat ?nr ?nc ?f ∈ carrier_mat ?nr ?nc›*) mult_carrier_mat (*‹⟦?A ∈ carrier_mat ?nr ?n; ?B ∈ carrier_mat ?n ?nc⟧ ⟹ ?A * ?B ∈ carrier_mat ?nr ?nc›*))\n also (*calculation: ‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * A' * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * (addrow_mat 2 (- S2 $$ (0, 1)) 0 1 * D)›*) have \"... = ?Q * A' * ?P * (D * ?P1)\"\n using commute_multiples_identity[OF P1] (*‹addrow_mat (2::nat) (- (S2::'a mat) $$ (0::nat, 1::nat)) (0::nat) (1::nat) * ((?k::'a) ⋅⇩m 1⇩m (2::nat)) = ?k ⋅⇩m 1⇩m (2::nat) * addrow_mat (2::nat) (- S2 $$ (0::nat, 1::nat)) (0::nat) (1::nat)›*) unfolding D_def\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * A' * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * (addrow_mat 2 (- S2 $$ (0, 1)) 0 1 * (d ⋅⇩m 1⇩m 2)) = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * A' * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * (d ⋅⇩m 1⇩m 2 * addrow_mat 2 (- S2 $$ (0, 1)) 0 1)›*)\n by auto\n also (*calculation: ‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * A' * Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * (D * addrow_mat 2 (- S2 $$ (0, 1)) 0 1)›*) have \"... = ?Q * A' * (?P * (D * ?P1))\"\n by (smt (verit) A' (*‹A' ∈ carrier_mat 2 2›*) D (*‹D ∈ carrier_mat 2 2›*) assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_row ?A = ?nr›*) 1) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_col ?A = ?nc›*) 2) mat_carrier (*‹Matrix.mat ?nr ?nc ?f ∈ carrier_mat ?nr ?nc›*) times_mat_def (*‹?A * ?B ≡ Matrix.mat (dim_row ?A) (dim_col ?B) (λ(i, j). Matrix.row ?A i ∙ col ?B j)›*))\n also (*calculation: ‹(S3::'a mat) * (D::'a mat) = Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then q10::'a else if i = (0::nat) ∧ j = (1::nat) then q01::'a else (q11::'a)) * (A'::'a mat) * (Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then p::'a else if i = (1::nat) ∧ j = (0::nat) then q::'a else if i = (0::nat) ∧ j = (1::nat) then v::'a else (u::'a)) * (D * addrow_mat (2::nat) (- (S2::'a mat) $$ (0::nat, 1::nat)) (0::nat) (1::nat)))›*) have \"... = ?Q * A' * (D * (?P * ?P1))\"\n by (smt (verit) D (*‹(D::'a mat) ∈ carrier_mat (2::nat) (2::nat)›*) D_def (*‹(D::'a mat) = (d::'a) ⋅⇩m 1⇩m (2::nat)›*) P (*‹Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then p::'a else if i = (1::nat) ∧ j = (0::nat) then q::'a else if i = (0::nat) ∧ j = (1::nat) then v::'a else (u::'a)) ∈ carrier_mat (2::nat) (2::nat)›*) P1 (*‹addrow_mat (2::nat) (- (S2::'a mat) $$ (0::nat, 1::nat)) (0::nat) (1::nat) ∈ carrier_mat (2::nat) (2::nat)›*) assoc_mult_mat (*‹⟦(?A::?'a mat) ∈ carrier_mat (?n₁::nat) (?n₂::nat); (?B::?'a mat) ∈ carrier_mat ?n₂ (?n₃::nat); (?C::?'a mat) ∈ carrier_mat ?n₃ (?n₄::nat)⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) commute_multiples_identity (*‹(?A::?'a mat) ∈ carrier_mat (?n::nat) ?n ⟹ ?A * ((?k::?'a) ⋅⇩m 1⇩m ?n) = ?k ⋅⇩m 1⇩m ?n * ?A›*))\n also (*calculation: ‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * A' * (D * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1))›*) have \"... = ?Q * (A' * D) * (?P * ?P1)\"\n by (smt (verit) A' (*‹A' ∈ carrier_mat 2 2›*) D (*‹D ∈ carrier_mat 2 2›*) assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_row ?A = ?nr›*) 1) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_col ?A = ?nc›*) 2) mat_carrier (*‹Matrix.mat ?nr ?nc ?f ∈ carrier_mat ?nr ?nc›*) times_mat_def (*‹?A * ?B ≡ Matrix.mat (dim_row ?A) (dim_col ?B) (λ(i, j). Matrix.row ?A i ∙ col ?B j)›*))\n also (*calculation: ‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * (A' * D) * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1)›*) have \"... = ?Q * A * (?P * ?P1)\"\n unfolding A_A'D\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * (A' * D) * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1) = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * (A' * D) * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1)›*)\n by auto\n also (*calculation: ‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * A * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1)›*) have \"... = ?Q * B * (U * (?P * ?P1))\"\n unfolding A_def\n (*goal: ‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * (B * U) * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1) = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * B * (U * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1))›*)\n by (smt (verit) B (*‹B ∈ carrier_mat 2 2›*) U (*‹U ∈ carrier_mat 2 2›*) assoc_mult_mat (*‹⟦?A ∈ carrier_mat ?n₁ ?n₂; ?B ∈ carrier_mat ?n₂ ?n₃; ?C ∈ carrier_mat ?n₃ ?n₄⟧ ⟹ ?A * ?B * ?C = ?A * (?B * ?C)›*) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_row ?A = ?nr›*) 1) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_col ?A = ?nc›*) 2) mat_carrier (*‹Matrix.mat ?nr ?nc ?f ∈ carrier_mat ?nr ?nc›*) times_mat_def (*‹?A * ?B ≡ Matrix.mat (dim_row ?A) (dim_col ?B) (λ(i, j). Matrix.row ?A i ∙ col ?B j)›*))\n finally (*calculation: ‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * B * (U * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1))›*) have S3_D_rw: \"S3 * D = ?Q * B * (U * (?P * ?P1))\" .\n show \"admits_diagonal_reduction B\"\n proof (rule admits_diagonal_reduction_intro[OF _ _ inv_Q] (*‹⟦Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then q10::'a else if i = (0::nat) ∧ j = (1::nat) then q01::'a else (q11::'a)) ∈ carrier_mat (dim_row (?A::'a mat)) (dim_row ?A); (?Q::'a mat) ∈ carrier_mat (dim_col ?A) (dim_col ?A); invertible_mat ?Q; Smith_normal_form_mat (Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00 else if i = (1::nat) ∧ j = (0::nat) then q10 else if i = (0::nat) ∧ j = (1::nat) then q01 else q11) * ?A * ?Q)⟧ ⟹ admits_diagonal_reduction ?A›*))\n (*goals:\n 1. ‹Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then q10::'a else if i = (0::nat) ∧ j = (1::nat) then q01::'a else (q11::'a)) ∈ carrier_mat (dim_row (B::'a mat)) (dim_row B)›\n 2. ‹(?Q::'a mat) ∈ carrier_mat (dim_col (B::'a mat)) (dim_col B)›\n 3. ‹invertible_mat (?Q::'a mat)›\n 4. ‹Smith_normal_form_mat (Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i = (0::nat) ∧ j = (0::nat) then q00::'a else if i = (1::nat) ∧ j = (0::nat) then q10::'a else if i = (0::nat) ∧ j = (1::nat) then q01::'a else (q11::'a)) * (B::'a mat) * (?Q::'a mat))›*)\n show \"(U* (?P * ?P1)) ∈ carrier_mat (dim_col B) (dim_col B)\"\n using B (*‹B ∈ carrier_mat 2 2›*) U (*‹U ∈ carrier_mat 2 2›*) by auto\n show \"?Q ∈ carrier_mat (dim_row B) (dim_row B)\"\n using Q (*‹Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) ∈ carrier_mat 2 2›*) B (*‹B ∈ carrier_mat 2 2›*) by auto\n show \"invertible_mat (U * (?P * ?P1))\"\n by (metis (no_types, lifting) P1 (*‹addrow_mat 2 (- S2 $$ (0, 1)) 0 1 ∈ carrier_mat 2 2›*) U (*‹U ∈ carrier_mat 2 2›*) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_row ?A = ?nr›*) 1) carrier_matD( (*‹?A ∈ carrier_mat ?nr ?nc ⟹ dim_col ?A = ?nc›*) 2) inv_P (*‹invertible_mat (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u))›*) inv_P1 (*‹invertible_mat (addrow_mat 2 (- S2 $$ (0, 1)) 0 1)›*) inv_U (*‹invertible_mat U›*) invertible_mult_JNF (*‹⟦?A ∈ carrier_mat ?n ?n; ?B ∈ carrier_mat ?n ?n; invertible_mat ?A; invertible_mat ?B⟧ ⟹ invertible_mat (?A * ?B)›*) mat_carrier (*‹Matrix.mat ?nr ?nc ?f ∈ carrier_mat ?nr ?nc›*) times_mat_def (*‹?A * ?B ≡ Matrix.mat (dim_row ?A) (dim_col ?B) (λ(i, j). Matrix.row ?A i ∙ col ?B j)›*))\n show \"Smith_normal_form_mat (?Q * B *(U* (?P * ?P1)))\"\n using SNF_S3_D (*‹Smith_normal_form_mat (S3 * D)›*) S3_D_rw (*‹S3 * D = Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then q00 else if i = 1 ∧ j = 0 then q10 else if i = 0 ∧ j = 1 then q01 else q11) * B * (U * (Matrix.mat 2 2 (λ(i, j). if i = 0 ∧ j = 0 then p else if i = 1 ∧ j = 0 then q else if i = 0 ∧ j = 1 then v else u) * addrow_mat 2 (- S2 $$ (0, 1)) 0 1))›*) by simp\n qed\n qed\n obtain Smith_1x2 where Smith_1x2: \"∀(A::'a mat)∈carrier_mat 1 2. is_SNF A (Smith_1x2 A)\"\n (*goal: ‹(⋀Smith_1x2. ∀A∈carrier_mat 1 2. is_SNF A (Smith_1x2 A) ⟹ thesis) ⟹ thesis›*)\n using admits_diagonal_reduction_imp_exists_algorithm_is_SNF_all[OF admits_1x2] (*‹∃algorithm::'a mat ⇒ 'a mat × 'a mat × 'a mat. ∀A::'a mat∈carrier_mat (1::nat) (2::nat). is_SNF A (algorithm A)›*) by auto\n from this (*‹∀A∈carrier_mat 1 2. is_SNF A (Smith_1x2 A)›*) obtain Smith_1x2' where Smith_1x2': \"∀(A::'a mat)∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1x2' A)\"\n (*goal: ‹(⋀Smith_1x2'. ∀A∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1x2' A) ⟹ thesis) ⟹ thesis›*)\n using Smith_1xn_two_matrices_all[OF Smith_1x2] (*‹∃Smith_1xn'. ∀A∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1xn' A)›*) by auto\n obtain Smith_2x2 where Smith_2x2: \"∀(A::'a mat)∈carrier_mat 2 2. is_SNF A (Smith_2x2 A)\"\n (*goal: ‹(⋀Smith_2x2::'a::comm_ring_1 mat ⇒ 'a::comm_ring_1 mat × 'a::comm_ring_1 mat × 'a::comm_ring_1 mat. ∀A::'a::comm_ring_1 mat∈carrier_mat (2::nat) (2::nat). is_SNF A (Smith_2x2 A) ⟹ thesis::bool) ⟹ thesis›*)\n using admits_diagonal_reduction_imp_exists_algorithm_is_SNF_all[OF admits_2x2] (*‹∃algorithm. ∀A∈carrier_mat 2 2. is_SNF A (algorithm A)›*) by auto\n have d: \"is_div_op (λa b. (SOME k. k * b = a))\"\n using div_op_SOME (*‹is_div_op (λ(a::?'a) b::?'a. SOME k::?'a. k * b = a)›*) by auto\n interpret Smith_Impl Smith_1x2' Smith_2x2 \"(λa b. (SOME k. k * b = a))\"\n using Smith_1x2' (*‹∀A∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1x2' A)›*) Smith_2x2 (*‹∀A∈carrier_mat 2 2. is_SNF A (Smith_2x2 A)›*) d (*‹is_div_op (λa b. SOME k. k * b = a)›*) apply unfold_locales\n (*goals:\n 1. ‹⟦∀A∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1x2' A); ∀A∈carrier_mat 2 2. is_SNF A (Smith_2x2 A); is_div_op (λa b. SOME k. k * b = a)⟧ ⟹ ∀A∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1x2' A)›\n 2. ‹⟦∀A∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1x2' A); ∀A∈carrier_mat 2 2. is_SNF A (Smith_2x2 A); is_div_op (λa b. SOME k. k * b = a)⟧ ⟹ ∀A∈carrier_mat 2 2. is_SNF A (Smith_2x2 A)›\n 3. ‹⟦∀A∈carrier_mat 1 2. is_SNF A (1⇩m 1, Smith_1x2' A); ∀A∈carrier_mat 2 2. is_SNF A (Smith_2x2 A); is_div_op (λa b. SOME k. k * b = a)⟧ ⟹ is_div_op (λa b. SOME k. k * b = a)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n show \"?thesis\"\n (*goal: ‹∀A. admits_diagonal_reduction A›*)\n using is_SNF_Smith_mxn (*‹?A ∈ carrier_mat ?m ?n ⟹ is_SNF ?A (Smith_mxn ?A)›*) by (meson admits_diagonal_reduction_eq_exists_algorithm_is_SNF (*‹?A ∈ carrier_mat ?m ?n ⟹ admits_diagonal_reduction ?A = (∃algorithm. is_SNF ?A (algorithm ?A))›*) carrier_mat_triv (*‹?m ∈ carrier_mat (dim_row ?m) (dim_col ?m)›*))\nqed\n\nsubsection ‹Final theorem›\n\n(* Characterization of elementary divisor rings (theorem 6)*)\n\ntheorem edr_characterization:\n \"(∀(A::'a mat). admits_diagonal_reduction A) = ((∀(A::'a mat). admits_triangular_reduction A) \n ∧ (∀a b c::'a. ideal_generated{a,b,c} = ideal_generated{1} \n ⟶ (∃p q. ideal_generated {p*a,p*b+q*c} = ideal_generated {1})))\"\n using necessity (*‹∀A. admits_diagonal_reduction A ⟹ ∀A. admits_triangular_reduction A› ‹∀A. admits_diagonal_reduction A ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*) sufficiency (*‹⟦∀A. admits_triangular_reduction A; ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})⟧ ⟹ ∀A. admits_diagonal_reduction A›*) by blast\n\n\ncorollary OFCLASS_edr_characterization:\n\"OFCLASS('a, elementary_divisor_ring_class) ≡ (OFCLASS('a, Hermite_ring_class) \n &&& (∀a b c::'a. ideal_generated{a,b,c} = ideal_generated{1} \n ⟶ (∃p q. ideal_generated {p*a,p*b+q*c} = ideal_generated {1})))\" (is \"?lhs ≡ ?rhs\")\nproof (standard)\n (*goals:\n 1. ‹OFCLASS('a, elementary_divisor_ring_class) ⟹ OFCLASS('a, Hermite_ring_class) &&& ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›\n 2. ‹OFCLASS('a, Hermite_ring_class) &&& ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}) ⟹ OFCLASS('a, elementary_divisor_ring_class)›*)\n assume 1: \"OFCLASS('a, elementary_divisor_ring_class)\" (*‹OFCLASS('a, elementary_divisor_ring_class)›*)\n hence admits_diagonal: \"∀A::'a mat. admits_diagonal_reduction A\"\n using conjunctionD2[OF 1 [ unfolded elementary_divisor_ring_class_def ]] (*‹class.elementary_divisor_ring TYPE('a)›*) unfolding class.elementary_divisor_ring_def\n (*goal: ‹∀A. admits_diagonal_reduction A›*)\n by auto\n have \"∀A::'a mat. admits_triangular_reduction A\"\n by (simp add: admits_diagonal (*‹∀A. admits_diagonal_reduction A›*) necessity( (*‹∀A. admits_diagonal_reduction A ⟹ ∀A. admits_triangular_reduction A›*) 1))\n hence OFCLASS_Hermite: \"OFCLASS('a, Hermite_ring_class)\"\n apply intro_classes\n (*goal: ‹OFCLASS('a, Hermite_ring_class)›*)\n by simp\n moreover have \"∀a b c::'a. ideal_generated {a, b, c} = ideal_generated {1} \n ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})\"\n using admits_diagonal (*‹∀A. admits_diagonal_reduction A›*) necessity(2) (*‹∀A::'a::comm_ring_1 mat. admits_diagonal_reduction A ⟹ ∀(a::'a::comm_ring_1) (b::'a::comm_ring_1) c::'a::comm_ring_1. ideal_generated {a, b, c} = ideal_generated {1::'a::comm_ring_1} ⟶ (∃(p::'a::comm_ring_1) q::'a::comm_ring_1. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a::comm_ring_1})›*) by blast\n ultimately show \"OFCLASS('a, Hermite_ring_class) &&& \n ∀a b c::'a. ideal_generated {a, b, c} = ideal_generated {1} \n ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})\"\n apply -\n (*goals:\n 1. ‹⟦OFCLASS('a, Hermite_ring_class); ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})⟧ ⟹ OFCLASS('a, Hermite_ring_class)›\n 2. ‹⟦OFCLASS('a, Hermite_ring_class); ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})⟧ ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\nnext\n (*goal: ‹OFCLASS('a, Hermite_ring_class) &&& ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}) ⟹ OFCLASS('a, elementary_divisor_ring_class)›*)\n assume 1: \"OFCLASS('a, Hermite_ring_class) &&&\n ∀a b c::'a. ideal_generated {a, b, c} = ideal_generated {1} ⟶\n (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})\" (*‹OFCLASS('a, Hermite_ring_class) &&& ∀(a::'a) (b::'a) c::'a. ideal_generated {a, b, c} = ideal_generated {1::'a} ⟶ (∃(p::'a) q::'a. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})›*)\n have H: \"OFCLASS('a, Hermite_ring_class)\" and 2: \"∀a b c::'a. ideal_generated {a, b, c} = ideal_generated {1} ⟶\n (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})\"\n using conjunctionD1[OF 1] (*‹OFCLASS('a, Hermite_ring_class)›*) conjunctionD2[OF 1] (*‹∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*) apply -\n (*goals:\n 1. ‹⟦OFCLASS('a, Hermite_ring_class); ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})⟧ ⟹ OFCLASS('a, Hermite_ring_class)›\n 2. ‹⟦OFCLASS('a, Hermite_ring_class); ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})⟧ ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have \"∀A::'a mat. admits_triangular_reduction A\"\n using H (*‹OFCLASS('a, Hermite_ring_class)›*) unfolding OFCLASS_Hermite_ring_def\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n by auto\n hence a: \"∀A::'a mat. admits_diagonal_reduction A\"\n using \"2\" (*‹∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*) sufficiency (*‹⟦∀A. admits_triangular_reduction A; ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})⟧ ⟹ ∀A. admits_diagonal_reduction A›*) by blast\n show \"OFCLASS('a, elementary_divisor_ring_class)\"\n apply intro_classes\n (*goal: ‹OFCLASS('a, elementary_divisor_ring_class)›*)\n by (simp add: a (*‹∀A. admits_diagonal_reduction A›*))\nqed\n\ncorollary edr_characterization_class:\n\"class.elementary_divisor_ring TYPE('a) \n = (class.Hermite_ring TYPE('a) \n ∧ (∀a b c::'a. ideal_generated{a,b,c} = ideal_generated{1} \n⟶ (∃p q. ideal_generated {p*a,p*b+q*c} = ideal_generated {1})))\" (is \"?lhs = (?H ∧ ?D')\")\nproof (standard)\n (*goals:\n 1. ‹class.elementary_divisor_ring TYPE('a) ⟹ class.Hermite_ring TYPE('a) ∧ (∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}))›\n 2. ‹class.Hermite_ring TYPE('a) ∧ (∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})) ⟹ class.elementary_divisor_ring TYPE('a)›*)\n assume 1: \"?lhs\" (*‹class.elementary_divisor_ring TYPE('a)›*)\n hence admits_diagonal: \"∀A::'a mat. admits_diagonal_reduction A\"\n unfolding class.elementary_divisor_ring_def\n (*goal: ‹∀A. admits_diagonal_reduction A›*) .\n have admits_triangular: \"∀A::'a mat. admits_triangular_reduction A\"\n using \"1\" (*‹class.elementary_divisor_ring TYPE('a)›*) necessity(1) (*‹∀A. admits_diagonal_reduction A ⟹ ∀A. admits_triangular_reduction A›*) unfolding class.elementary_divisor_ring_def\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n by blast\n hence \"?H\"\n unfolding class.Hermite_ring_def\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n by auto\n moreover have \"?D'\"\n using admits_diagonal (*‹∀A::'a mat. admits_diagonal_reduction A›*) necessity(2) (*‹∀A. admits_diagonal_reduction A ⟹ ∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})›*) by blast\n ultimately show \"(?H ∧ ?D')\"\n by simp\nnext\n (*goal: ‹class.Hermite_ring TYPE('a) ∧ (∀(a::'a) (b::'a) c::'a. ideal_generated {a, b, c} = ideal_generated {1::'a} ⟶ (∃(p::'a) q::'a. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})) ⟹ class.elementary_divisor_ring TYPE('a)›*)\n assume HD': \"(?H ∧ ?D')\" (*‹class.Hermite_ring TYPE('a) ∧ (∀(a::'a) (b::'a) c::'a. ideal_generated {a, b, c} = ideal_generated {1::'a} ⟶ (∃(p::'a) q::'a. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a}))›*)\n hence admits_triangular: \"∀A::'a mat. admits_triangular_reduction A\"\n unfolding class.Hermite_ring_def\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n by auto\n hence admits_diagonal: \"∀A::'a mat. admits_diagonal_reduction A\"\n using edr_characterization (*‹(∀A. admits_diagonal_reduction A) = ((∀A. admits_triangular_reduction A) ∧ (∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})))›*) HD' (*‹class.Hermite_ring TYPE('a) ∧ (∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}))›*) by auto\n thus \"?lhs\"\n unfolding class.elementary_divisor_ring_def\n (*goal: ‹∀A. admits_diagonal_reduction A›*)\n by auto\nqed\n\n\ncorollary edr_iff_T_D':\n shows \"class.elementary_divisor_ring TYPE('a) = (\n (∀a b::'a. ∃ a1 b1 d. a = a1*d ∧ b = b1*d ∧ ideal_generated {a1,b1} = ideal_generated {1})\n ∧ (∀a b c::'a. ideal_generated{a,b,c} = ideal_generated{1} \n ⟶ (∃p q. ideal_generated {p*a,p*b+q*c} = ideal_generated {1}))\n )\" (is \"?lhs = (?T ∧ ?D')\")\nproof (standard)\n (*goals:\n 1. ‹class.elementary_divisor_ring TYPE('a::comm_ring_1) ⟹ (∀(a::'a::comm_ring_1) b::'a::comm_ring_1. ∃(a1::'a::comm_ring_1) (b1::'a::comm_ring_1) d::'a::comm_ring_1. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1::'a::comm_ring_1}) ∧ (∀(a::'a::comm_ring_1) (b::'a::comm_ring_1) c::'a::comm_ring_1. ideal_generated {a, b, c} = ideal_generated {1::'a::comm_ring_1} ⟶ (∃(p::'a::comm_ring_1) q::'a::comm_ring_1. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a::comm_ring_1}))›\n 2. ‹(∀(a::'a::comm_ring_1) b::'a::comm_ring_1. ∃(a1::'a::comm_ring_1) (b1::'a::comm_ring_1) d::'a::comm_ring_1. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1::'a::comm_ring_1}) ∧ (∀(a::'a::comm_ring_1) (b::'a::comm_ring_1) c::'a::comm_ring_1. ideal_generated {a, b, c} = ideal_generated {1::'a::comm_ring_1} ⟶ (∃(p::'a::comm_ring_1) q::'a::comm_ring_1. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a::comm_ring_1})) ⟹ class.elementary_divisor_ring TYPE('a::comm_ring_1)›*)\n assume 1: \"?lhs\" (*‹class.elementary_divisor_ring TYPE('a)›*)\n hence \"∀A::'a mat. admits_triangular_reduction A\"\n unfolding class.elementary_divisor_ring_def\n (*goal: ‹∀A. admits_triangular_reduction A›*)\n using necessity(1) (*‹∀A. admits_diagonal_reduction A ⟹ ∀A. admits_triangular_reduction A›*) by blast\n hence \"?T\"\n using theorem3_part2 (*‹All admits_triangular_reduction ⟹ ∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}›*) by simp\n moreover have \"?D'\"\n using \"1\" (*‹class.elementary_divisor_ring TYPE('a)›*) unfolding edr_characterization_class\n (*goal: ‹∀(a::'a) (b::'a) c::'a. ideal_generated {a, b, c} = ideal_generated {1::'a} ⟶ (∃(p::'a) q::'a. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})›*)\n by auto\n ultimately show \"(?T ∧ ?D')\"\n by simp\nnext\n (*goal: ‹(∀(a::'a::comm_ring_1) b::'a::comm_ring_1. ∃(a1::'a::comm_ring_1) (b1::'a::comm_ring_1) d::'a::comm_ring_1. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1::'a::comm_ring_1}) ∧ (∀(a::'a::comm_ring_1) (b::'a::comm_ring_1) c::'a::comm_ring_1. ideal_generated {a, b, c} = ideal_generated {1::'a::comm_ring_1} ⟶ (∃(p::'a::comm_ring_1) q::'a::comm_ring_1. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a::comm_ring_1})) ⟹ class.elementary_divisor_ring TYPE('a::comm_ring_1)›*)\n assume TD': \"(?T ∧ ?D')\" (*‹(∀(a::'a) b::'a. ∃(a1::'a) (b1::'a) d::'a. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1::'a}) ∧ (∀(a::'a) (b::'a) c::'a. ideal_generated {a, b, c} = ideal_generated {1::'a} ⟶ (∃(p::'a) q::'a. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a}))›*)\n hence \"class.Hermite_ring TYPE('a)\"\n unfolding class.Hermite_ring_def\n (*goal: ‹∀A::'a mat. admits_triangular_reduction A›*)\n using theorem3_part1 (*‹∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1} ⟹ All admits_triangular_reduction›*) TD' (*‹(∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}) ∧ (∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}))›*) by auto\n thus \"?lhs\"\n using edr_characterization_class (*‹class.elementary_divisor_ring TYPE('a) = (class.Hermite_ring TYPE('a) ∧ (∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1})))›*) TD' (*‹(∀a b. ∃a1 b1 d. a = a1 * d ∧ b = b1 * d ∧ ideal_generated {a1, b1} = ideal_generated {1}) ∧ (∀a b c. ideal_generated {a, b, c} = ideal_generated {1} ⟶ (∃p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1}))›*) by auto\nqed\n\nend\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Smith_Normal_Form/Elementary_Divisor_Rings.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"e6bde3141d3a3614dbea7bb386cfbdce553bd74fc92c112a1a5b04de3349930a\"\n}"}}},{"rowIdx":198,"cells":{"text":{"kind":"string","value":"(*****************************************************************************\n * Featherweight-OCL --- A Formal Semantics for UML-OCL Version OCL 2.5\n * for the OMG Standard.\n * http://www.brucker.ch/projects/hol-testgen/\n *\n * UML_Void.thy --- Library definitions.\n * This file is part of HOL-TestGen.\n *\n * Copyright (c) 2012-2015 Université Paris-Saclay, Univ. Paris-Sud, France\n * 2013-2015 IRT SystemX, France\n *\n * All rights reserved.\n *\n * Redistribution and use in source and binary forms, with or without\n * modification, are permitted provided that the following conditions are\n * met:\n *\n * * Redistributions of source code must retain the above copyright\n * notice, this list of conditions and the following disclaimer.\n *\n * * Redistributions in binary form must reproduce the above\n * copyright notice, this list of conditions and the following\n * disclaimer in the documentation and/or other materials provided\n * with the distribution.\n *\n * * Neither the name of the copyright holders nor the names of its\n * contributors may be used to endorse or promote products derived\n * from this software without specific prior written permission.\n *\n * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS\n * \"AS IS\" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT\n * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR\n * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT\n * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,\n * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT\n * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,\n * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY\n * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT\n * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE\n * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.\n ******************************************************************************)\n\ntheory UML_Void\nimports \"../UML_PropertyProfiles\"\nbegin\n\nsection‹Basic Type Void: Operations›\n\n(* For technical reasons, the type does not contain to the null-class yet. *)\ntext ‹This \\emph{minimal} OCL type contains only two elements:\n@{term \"invalid\"} and @{term \"null\"}.\n@{term \"Void\"} could initially be defined as @{typ \"unit option option\"},\nhowever the cardinal of this type is more than two, so it would have the cost to consider\n ‹Some None› and ‹Some (Some ())› seemingly everywhere.›\n \nsubsection‹Fundamental Properties on Voids: Strict Equality›\n\nsubsubsection‹Definition›\n\ninstantiation Void⇩b⇩a⇩s⇩e :: bot\nbegin\n definition bot_Void_def: \"(bot_class.bot :: Void⇩b⇩a⇩s⇩e) ≡ Abs_Void⇩b⇩a⇩s⇩e None\"\n\n instance proof show \"∃x:: Void⇩b⇩a⇩s⇩e. x ≠ bot\"\n apply(rule_tac x=\"Abs_Void⇩b⇩a⇩s⇩e ⌊None⌋\" in exI)\n apply(simp add:bot_Void_def, subst Abs_Void⇩b⇩a⇩s⇩e_inject)\n apply(simp_all add: null_option_def bot_option_def)\n done\n qed\nend\n\ninstantiation Void⇩b⇩a⇩s⇩e :: null\nbegin\n definition null_Void_def: \"(null::Void⇩b⇩a⇩s⇩e) ≡ Abs_Void⇩b⇩a⇩s⇩e ⌊ None ⌋\"\n\n instance proof show \"(null:: Void⇩b⇩a⇩s⇩e) ≠ bot\"\n apply(simp add:null_Void_def bot_Void_def, subst Abs_Void⇩b⇩a⇩s⇩e_inject)\n apply(simp_all add: null_option_def bot_option_def)\n done\n qed\nend\n\n\ntext‹The last basic operation belonging to the fundamental infrastructure\nof a value-type in OCL is the weak equality, which is defined similar\nto the @{typ \"('𝔄)Void\"}-case as strict extension of the strong equality:›\noverloading StrictRefEq ≡ \"StrictRefEq :: [('𝔄)Void,('𝔄)Void] ⇒ ('𝔄)Boolean\"\nbegin\n definition StrictRefEq⇩V⇩o⇩i⇩d[code_unfold] :\n \"(x::('𝔄)Void) ≐ y ≡ λ τ. if (υ x) τ = true τ ∧ (υ y) τ = true τ\n then (x ≜ y) τ\n else invalid τ\"\nend\n\ntext‹Property proof in terms of @{term \"profile_bin⇩S⇩t⇩r⇩o⇩n⇩g⇩E⇩q_⇩v_⇩v\"}›\ninterpretation StrictRefEq⇩V⇩o⇩i⇩d : profile_bin⇩S⇩t⇩r⇩o⇩n⇩g⇩E⇩q_⇩v_⇩v \"λ x y. (x::('𝔄)Void) ≐ y\" \n apply unfold_locales\n (*goal: ‹profile_bin⇩S⇩t⇩r⇩o⇩n⇩g⇩E⇩q_⇩v_⇩v (≐)›*)\n by (auto simp: StrictRefEq⇩V⇩o⇩i⇩d (*‹?x ≐ ?y ≡ λτ. if (υ ?x) τ = true τ ∧ (υ ?y) τ = true τ then (?x ≜ ?y) τ else invalid τ›*))\n \n \nsubsection‹Basic Void Constants›\n\n\nsubsection‹Validity and Definedness Properties›\n\nlemma \"δ(null::('𝔄)Void) = false\" by simp\nlemma \"υ(null::('𝔄)Void) = true\" sorry\n\nlemma [simp,code_unfold]: \"δ (λ_. Abs_Void⇩b⇩a⇩s⇩e None) = false\"\napply (simp add:defined_def (*‹δ ?X ≡ λτ. if ?X τ = ⊥ τ ∨ ?X τ = null τ then false τ else true τ›*) true_def (*‹true ≡ λτ. ⌊⌊True⌋⌋›*) bot_fun_def (*‹⊥ ≡ λx. ⊥›*) bot_option_def (*‹⊥ ≡ ⊥›*))\n(*goal: ‹δ (λ_. Abs_Void⇩b⇩a⇩s⇩e ⊥) = false›*)\napply (rule ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n(*goal: ‹(λτ. if Abs_Void⇩b⇩a⇩s⇩e ⊥ = ⊥ ∨ Abs_Void⇩b⇩a⇩s⇩e ⊥ = null τ then false τ else true τ) = false›*)\napply (simp split:)\n(*goal: ‹⋀τ. (if Abs_Void⇩b⇩a⇩s⇩e ⊥ = ⊥ ∨ Abs_Void⇩b⇩a⇩s⇩e ⊥ = null τ then false τ else true τ) = false τ›*)\napply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n(*goal: ‹⋀τ. Abs_Void⇩b⇩a⇩s⇩e ⊥ ≠ ⊥ ∧ Abs_Void⇩b⇩a⇩s⇩e ⊥ ≠ null τ ⟶ true τ = false τ›*)\nby (simp add: bot_Void_def (*‹⊥ ≡ Abs_Void⇩b⇩a⇩s⇩e ⊥›*))\n\nlemma [simp,code_unfold]: \"υ (λ_. Abs_Void⇩b⇩a⇩s⇩e None) = false\"\napply (simp add:valid_def (*‹υ ?X ≡ λτ. if ?X τ = ⊥ τ then false τ else true τ›*) true_def (*‹true ≡ λτ. ⌊⌊True⌋⌋›*) bot_fun_def (*‹⊥ ≡ λx. ⊥›*) bot_option_def (*‹⊥ ≡ ⊥›*))\n(*goal: ‹υ (λ_. Abs_Void⇩b⇩a⇩s⇩e ⊥) = false›*)\napply (rule ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n(*goal: ‹(λτ::'a state × 'a state. if Abs_Void⇩b⇩a⇩s⇩e ⊥ = ⊥ then false τ else true τ) = false›*)\napply (simp split:)\n(*goal: ‹⋀τ. (if Abs_Void⇩b⇩a⇩s⇩e ⊥ = ⊥ then false τ else true τ) = false τ›*)\napply (intro conjI (*‹⟦?P::bool; ?Q::bool⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P::bool ⟹ ?Q::bool) ⟹ ?P ⟶ ?Q›*))\n(*goal: ‹⋀τ. Abs_Void⇩b⇩a⇩s⇩e ⊥ ≠ ⊥ ⟶ true τ = false τ›*)\nby (simp add: bot_Void_def (*‹⊥ ≡ Abs_Void⇩b⇩a⇩s⇩e ⊥›*))\n\nlemma [simp,code_unfold]: \"δ (λ_. Abs_Void⇩b⇩a⇩s⇩e ⌊None⌋) = false\"\napply (simp add:defined_def (*‹δ ?X ≡ λτ. if ?X τ = ⊥ τ ∨ ?X τ = null τ then false τ else true τ›*) true_def (*‹true ≡ λτ. ⌊⌊True⌋⌋›*) bot_fun_def (*‹⊥ ≡ λx. ⊥›*) bot_option_def (*‹⊥ ≡ ⊥›*) null_fun_def (*‹null ≡ λx. null›*) null_option_def (*‹null ≡ ⌊⊥⌋›*))\n(*goal: ‹δ (λ_. Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋) = false›*)\napply (rule ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n(*goal: ‹(λτ. if Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ = ⊥ ∨ Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ = null then false τ else true τ) = false›*)\napply (simp split:)\n(*goal: ‹⋀τ. (if Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ = ⊥ ∨ Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ = null then false τ else true τ) = false τ›*)\napply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n(*goal: ‹⋀τ. Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ ≠ ⊥ ∧ Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ ≠ null ⟶ true τ = false τ›*)\nby (simp add: null_Void_def (*‹null ≡ Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋›*))\n\nlemma [simp,code_unfold]: \"υ (λ_. Abs_Void⇩b⇩a⇩s⇩e ⌊None⌋) = true\"\napply (simp add:valid_def (*‹υ ?X ≡ λτ. if ?X τ = ⊥ τ then false τ else true τ›*) true_def (*‹true ≡ λτ. ⌊⌊True⌋⌋›*) bot_fun_def (*‹⊥ ≡ λx. ⊥›*) bot_option_def (*‹⊥ ≡ ⊥›*))\n(*goal: ‹υ (λ_. Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋) = true›*)\napply (rule ext (*‹(⋀x::?'a::type. (?f::?'a::type ⇒ ?'b::type) x = (?g::?'a::type ⇒ ?'b::type) x) ⟹ ?f = ?g›*))\n(*goal: ‹(λτ. if Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ = ⊥ then false τ else true τ) = (λτ. ⌊⌊True⌋⌋)›*)\napply (simp split:)\n(*goal: ‹⋀τ::'a state × 'a state. (if Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ = ⊥ then false τ else true τ) = ⌊⌊True⌋⌋›*)\napply (intro conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n(*goals:\n 1. ‹⋀τ. Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ = ⊥ ⟹ false τ = ⌊⌊True⌋⌋›\n 2. ‹⋀τ. Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋ ≠ ⊥ ⟹ true τ = ⌊⌊True⌋⌋›\ndiscuss goal 1*)\n apply (metis null_Void_def (*‹null ≡ Abs_Void⇩b⇩a⇩s⇩e ⌊⊥⌋›*) null_is_valid (*‹null ≠ ⊥›*))\n(*discuss goal 2*)\n apply (simp add: true_def (*‹true ≡ λτ. ⌊⌊True⌋⌋›*))\n(*proven 2 subgoals*) .\n\n\nsubsection‹Test Statements›\n\nAssert \"τ ⊨ ((null::('𝔄)Void) ≐ null)\"\n\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Featherweight_OCL/basic_types/UML_Void.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"0eea4d973c15ca92e10bbe703b2b06d5a0a8cef577439acf2be926171cb8af3a\"\n}"}}},{"rowIdx":199,"cells":{"text":{"kind":"string","value":"section ‹Lazy evaluation within the logic›\ntheory Lazy_Eval\nimports\n Complex_Main\nbegin\n\ntext ‹ \n This is infrastructure to lazily evaluate an expression (typically something corecursive)\n within the logic by simple rewriting. A signature of supported (co-)datatype constructures \n upon which pattern matching is allowed and a list of function equations that are used in\n rewriting must be provided.\n\n One can then e.\\,g.\\ determine whether a given pattern matches a given expression. To do this,\n the expression will be rewritten using the given function equations until enough constructors\n have been exposed to decide whether the pattern matches.\n\n This infrastructure was developed specifically for evaluating Multiseries expressions, but\n can, in principle, be used for other purposes as well.\n›\n\nlemma meta_eq_TrueE: \"PROP P ≡ Trueprop True ⟹ PROP P\" by simp\n\ndatatype cmp_result = LT | EQ | GT\n\ndefinition COMPARE :: \"real ⇒ real ⇒ cmp_result\" where\n \"COMPARE x y = (if x < y then LT else if x > y then GT else EQ)\"\n\nlemma COMPARE_intros: \n \"x < y ⟹ COMPARE x y ≡ LT\" \"x > y ⟹ COMPARE x y ≡ GT\" \"x = y ⟹ COMPARE x y ≡ EQ\"\n \n (*goals:\n 1. ‹(x::real) < (y::real) ⟹ COMPARE x y ≡ LT›\n 2. ‹(y::real) < (x::real) ⟹ COMPARE x y ≡ GT›\n 3. ‹(x::real) = (y::real) ⟹ COMPARE x y ≡ EQ›\n discuss goal 1*)\n apply (simp add: COMPARE_def (*‹COMPARE ?x ?y = (if ?x < ?y then LT else if ?y < ?x then GT else EQ)›*))\n (*discuss goal 2*)\n apply (simp add: COMPARE_def (*‹COMPARE ?x ?y = (if ?x < ?y then LT else if ?y < ?x then GT else EQ)›*))\n (*discuss goal 3*)\n apply (simp add: COMPARE_def (*‹COMPARE ?x ?y = (if ?x < ?y then LT else if ?y < ?x then GT else EQ)›*))\n (*proven 3 subgoals*) .\n \nprimrec CMP_BRANCH :: \"cmp_result ⇒ 'a ⇒ 'a ⇒ 'a ⇒ 'a\" where\n \"CMP_BRANCH LT x y z = x\"\n| \"CMP_BRANCH EQ x y z = y\"\n| \"CMP_BRANCH GT x y z = z\" \n\nML_file ‹lazy_eval.ML›\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/Real_Asymp/Lazy_Eval.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"dc10676a7fc95cf9c19415abd61be9711d33db7bcdd3bdb755ad2d7bff22c5d6\"\n}"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":1,"numItemsPerPage":100,"numTotalItems":8946,"offset":100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3Njc2NDEwNSwic3ViIjoiL2RhdGFzZXRzL0FOVFBHL2Fubm90YXRlZC1pc2FiZWxsZSIsImV4cCI6MTc3Njc2NzcwNSwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.e9RCh7YbTbT9JFAPN-XkwbbDXrQvNiMI_RoPvei34fD99PutmjTwxRkezFI3kgkg6myG_JBwN0T-dkdV0m9bDw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":8946,"numBytesParquet":250274661}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
text stringlengths 157 7.64M | meta dict |
|---|---|
(* Title: HOL/Quotient_Examples/Quotient_Int.thy
Author: Cezary Kaliszyk
Author: Christian Urban
Integers based on Quotients, based on an older version by Larry
Paulson.
*)
theory Quotient_Int
imports "HOL-Library.Quotient_Product"
begin
fun
intrel :: "(nat × nat) ⇒ (nat × nat) ⇒ bool" (infix... | {
"path": "Isabelle2024/src/HOL/Quotient_Examples/Quotient_Int.thy",
"repo": "Isabelle2024",
"sha": "724945a60cf2db7973eb7a575c272a2bceac715b5d1e648fac6158f5c87c087e"
} |
(* Title: QuasiBorel.thy
Author: Michikazu Hirata, Yasuhiko Minamide Tokyo Institute of Technology
*)
section ‹Quasi-Borel Spaces›
theory QuasiBorel
imports "HOL-Probability.Probability"
begin
subsection ‹ Definitions ›
subsubsection ‹ Quasi-Borel Spaces›
definition qbs_closed1 :: "(real ⇒ 'a) set ⇒ bool"
... | {
"path": "afp-2025-02-12/thys/S_Finite_Measure_Monad/QuasiBorel.thy",
"repo": "afp-2025-02-12",
"sha": "6ada33666b2df996beaf2a35918f0bc851c99e24434b2d0c24244c0237eea6db"
} |
subsection ‹Relation between the Hoare logics in big-O style›
theory DiscussionO
imports SepLogK_Hoare QuantK_Hoare Nielson_Hoare
begin
(* here we compare quantitative Hoare logic with constant with Nielson's Hoare logic *)
subsubsection ‹Relation Nielson to quantHoare›
definition emN :: "qassn ⇒ Nielson_Hoar... | {
"path": "afp-2025-02-12/thys/Hoare_Time/DiscussionO.thy",
"repo": "afp-2025-02-12",
"sha": "9c2c8744d239fd87c213be1c40b54943da105e7647466342211048606b4dcece"
} |
section ‹Internally Vertex-Disjoint Paths›
theory DisjointPaths imports Separations begin
text ‹
Menger's Theorem talks about internally vertex-disjoint @{term v0}-@{term v1}-paths. Let us
define this concept.
›
locale DisjointPaths = v0_v1_Digraph +
fixes paths :: "'a Walk set"
assumes paths:
"⋀xs. xs ... | {
"path": "afp-2025-02-12/thys/Menger/DisjointPaths.thy",
"repo": "afp-2025-02-12",
"sha": "f04bc25c0af48809dbc438106a2a6840365d44875cd593fa40a3d0354b262724"
} |
section ‹Follow map›
theory Follow_Map
imports First_Map
begin
type_synonym ('n, 't) follow_map = "('n, 't lookahead list) fmap"
fun updateFo :: "'n set ⇒ ('n, 't) first_map ⇒ 'n ⇒ ('n, 't) rhs ⇒ ('n, 't) follow_map
⇒ ('n, 't) follow_map" where
"updateFo nu fi lx [] fo = fo"
| "updateFo nu fi lx ((T _) # gamma... | {
"path": "afp-2025-02-12/thys/LL1_Parser/Follow_Map.thy",
"repo": "afp-2025-02-12",
"sha": "faa07bdd8c882ffaa746292c73c24a2bac8bc6a249bce78138d4afcf2be8db08"
} |
theory Higher_Projective_Space_Axioms
imports Main
begin
(* Author: Anthony Bordg, University of Cambridge, apdb3@cam.ac.uk .*)
text ‹
Contents:
▪ We introduce the types of 'point and 'line and an incidence relation between them.
▪ A set of axioms for higher projective spaces, i.e. we allow models of dimension ‹>› ... | {
"path": "afp-2025-02-12/thys/Projective_Geometry/Higher_Projective_Space_Axioms.thy",
"repo": "afp-2025-02-12",
"sha": "c0bced8a43c8e4ff8b0801d0c92a81f41fba561528eafc117a614eec041955ac"
} |
(* Title: HOL/Library/Function_Algebras.thy
Author: Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
*)
section ‹Pointwise instantiation of functions to algebra type classes›
theory Function_Algebras
imports Main
begin
text ‹Pointwise operations›
instantiation "fun" :: (type, plus) plus
begin
... | {
"path": "Isabelle2024/src/HOL/Library/Function_Algebras.thy",
"repo": "Isabelle2024",
"sha": "7a96287649c89cd00cc82d5a1b620bcf95a5b13408a43276208bdf9025189eb4"
} |
(* Title: HOL/UNITY/Constrains.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Weak safety relations: restricted to the set of reachable states.
*)
section‹Weak Safety›
theory Constrains imports UNITY begin
(*Initial states and p... | {
"path": "Isabelle2024/src/HOL/UNITY/Constrains.thy",
"repo": "Isabelle2024",
"sha": "623a4c4c959352a7ece78e1789b0553ca4b901e05bdfa2ae0338f4767c7c27f0"
} |
theory Laurent_Convergence
imports "HOL-Computational_Algebra.Formal_Laurent_Series" "HOL-Library.Landau_Symbols"
Residue_Theorem
begin
instance fps :: (semiring_char_0) semiring_char_0
proof
show "inj (of_nat :: nat ⇒ 'a fps)"
proof
fix m n :: nat
assume "of_nat m = (of_nat n :: 'a fps)"
... | {
"path": "Isabelle2024/src/HOL/Complex_Analysis/Laurent_Convergence.thy",
"repo": "Isabelle2024",
"sha": "067c6fe0d54891d8fcecd25ffc01fbf7db07bf964e1197b57ca0c104876646a1"
} |
(*
Author:
Mnacho Echenim, Université Grenoble Alpes
*)
theory Linear_Algebra_Complements imports
"Isabelle_Marries_Dirac.Tensor"
"Isabelle_Marries_Dirac.More_Tensor"
"QHLProver.Gates"
"HOL-Types_To_Sets.Group_On_With"
"HOL-Probability.Probability"
begin
hide_const(open) S
section ‹Preliminaries›
... | {
"path": "afp-2025-02-12/thys/Projective_Measurements/Linear_Algebra_Complements.thy",
"repo": "afp-2025-02-12",
"sha": "b0187b543ea40264bae66965243b4b655d38b8e49c1ca89a1004667740c3effb"
} |
(* Title: Isabelle Collections Library
Author: Peter Lammich <peter dot lammich at uni-muenster.de>
Maintainer: Peter Lammich <peter dot lammich at uni-muenster.de>
*)
(*
Changes since submission on 2009-11-26:
2009-12-10: OrderedSet, implemented iterators, min, max, to_sorted_list
*)
sectio... | {
"path": "afp-2025-02-12/thys/Collections/ICF/impl/RBTSetImpl.thy",
"repo": "afp-2025-02-12",
"sha": "b93aab6a00fcd721ad52b13e2252273277d6a371e4cf4184d76a7ab76dba858b"
} |
(* Title: HOL/Computational_Algebra/Group_Closure.thy
Author: Johannes Hoelzl, TU Muenchen
Author: Florian Haftmann, TU Muenchen
*)
theory Group_Closure
imports
Main
begin
context ab_group_add
begin
inductive_set group_closure :: "'a set ⇒ 'a set" for S
where base: "s ∈ insert 0 S ⟹ s ∈ gro... | {
"path": "Isabelle2024/src/HOL/Computational_Algebra/Group_Closure.thy",
"repo": "Isabelle2024",
"sha": "517667880270759fa9233d4459d9fd75c93ec49e5866e49777911f3e074b8d0e"
} |
theory Access_Matrix_Embeddings
imports Semantics_Embeddings
"Primitive_Matchers/No_Spoof"
Simple_Firewall.Service_Matrix
begin
section‹Applying the Access Matrix to the Bigstep Semantics›
text‹
If the real iptables firewall (@{const iptables_bigstep}) accepts a packet, we have a corresponding
edge ... | {
"path": "afp-2025-02-12/thys/Iptables_Semantics/Access_Matrix_Embeddings.thy",
"repo": "afp-2025-02-12",
"sha": "7acd0b1ee3ec739bd3b6dfa2f41f2459990b2ee92abe46db09b72f7e10486aa5"
} |
(* Title: Quantales
Author: Georg Struth
Maintainer: Georg Struth <g.struth@sheffield.ac.uk>
Contributions by Brijesh Dongol, Victor Gomes, Ian Hayes
*)
section ‹Quantales›
theory Quantales
imports
"Order_Lattice_Props.Closure_Operators"
Kleene_Algebra.Dioid
begin
subsection ‹Families of Proto-Qu... | {
"path": "afp-2025-02-12/thys/Quantales/Quantales.thy",
"repo": "afp-2025-02-12",
"sha": "0f27c6843c3b0b99948596121b4d7a5323c0aa20b32f824cc02f56592aead627"
} |
(*
Mike Stannett
Feb 2023
*)
section ‹ Affine ›
text ‹This theory defines affine transformations and established their
key properties.›
theory Affine
imports Translations LinearMaps
begin
class Affine = Translations + LinearMaps
begin
abbreviation affine :: "('a Point ⇒ 'a Point) ⇒ bool"
where "affin... | {
"path": "afp-2025-02-12/thys/No_FTL_observers_Gen_Rel/Affine.thy",
"repo": "afp-2025-02-12",
"sha": "fd8410c0fdfb3f02f621ffc9ba45bb59dd1b0eaf0b4e5b7d6739b0eb77f05bf1"
} |
theory Confluent_Quotient imports
Confluence
begin
text ‹Functors with finite setters preserve wide intersection for any equivalence relation that respects the mapper.›
lemma Inter_finite_subset:
assumes "∀A ∈ 𝒜. finite A"
shows "∃ℬ⊆𝒜. finite ℬ ∧ (⋂ℬ) = (⋂𝒜)"
proof (cases "𝒜 = {}")
(*goals:
1. ‹(𝒜::'... | {
"path": "Isabelle2024/src/HOL/Library/Confluent_Quotient.thy",
"repo": "Isabelle2024",
"sha": "c612c0040f2d6efb760ee028f8bd32fc8574bab46fc76fe4461e8cced0f4dd12"
} |
section Substitution
text ‹This theory introduces the substitution operation using a locale, and provides two models.›
theory Substitution
imports Renaming
begin
subsection Definition
locale substitution =
fixes subst :: "('r,'l,'v) expr ⇒ 'v ⇒ ('r,'l,'v) expr ⇒ ('r,'l,'v) expr"
assumes
renaming_distr_su... | {
"path": "afp-2025-02-12/thys/Concurrent_Revisions/Substitution.thy",
"repo": "afp-2025-02-12",
"sha": "a4a6f1b8cdef2080a04e136d45c21bc8b08e36c89d76d288f0bbf7ec74cda7a6"
} |
(* Title: JinjaThreads/Framework/FWProgressAux.thy
Author: Andreas Lochbihler
*)
section ‹Auxiliary definitions for the progress theorem for the multithreaded semantics›
theory FWProgressAux
imports
FWSemantics
begin
abbreviation collect_waits :: "('l,'t,'x,'m,'w,'o) thread_action ⇒ ('l + 't + 't) se... | {
"path": "afp-2025-02-12/thys/JinjaThreads/Framework/FWProgressAux.thy",
"repo": "afp-2025-02-12",
"sha": "659c3936b966e70db033f09ec0e1b480be5cdb8ac85558ecf25b36f67c12896f"
} |
section ‹Nondeterministic Büchi Automata Combinations›
theory NBA_Combine
imports NBA NGBA
begin
global_interpretation degeneralization: automaton_degeneralization_run
ngba ngba.alphabet ngba.initial ngba.transition ngba.accepting "λ P w r p. gen infs P (p ## r)"
nba nba.alphabet nba.initial nba.transition... | {
"path": "afp-2025-02-12/thys/Transition_Systems_and_Automata/Automata/NBA/NBA_Combine.thy",
"repo": "afp-2025-02-12",
"sha": "ce911ea8c833e364f6b05ade9a99820ace5864a87da37e9c1986b4c2c93efbc1"
} |
(*File: VDM_OBJ.thy*)
(*Authors: Lennart Beringer and Martin Hofmann, LMU Munich 2008*)
theory VDM_OBJ imports OBJ begin
subsection‹Program logic›
text‹\label{sec:ObjLogic}Apart from the addition of proof rules for the three new
instructions, this section is essentially identical to Section
\ref{sec:VDM}.›
subsubse... | {
"path": "afp-2025-02-12/thys/SIFPL/VDM_OBJ.thy",
"repo": "afp-2025-02-12",
"sha": "e7a821d6a618825685ba537d8718e0ed9ff8aec3c76e48f2247b04b27350b82c"
} |
section ‹Coproducts›
theory Coproduct
imports Equivalence
begin
(* We define our own (ETCS) case_bool later, so we need to hide the HOL version. *)
hide_const case_bool
text ‹The axiomatization below corresponds to Axiom 7 (Coproducts) in Halvorson.›
axiomatization
coprod :: "cset ⇒ cset ⇒ cset" (infixr "∐" 65) ... | {
"path": "afp-2025-02-12/thys/Category_Set/Coproduct.thy",
"repo": "afp-2025-02-12",
"sha": "331f3b3dc464ef3c1344919233fb8792a328fe84ed1e8c41806c59dbf4278e74"
} |
(* Title: JinjaThreads/Common/SemiType.thy
Author: Tobias Nipkow, Gerwin Klein, Andreas Lochbihler
*)
section ‹The Jinja Type System as a Semilattice›
theory SemiType
imports
WellForm
"../DFA/Semilattices"
begin
inductive_set
widen1 :: "'a prog ⇒ (ty × ty) set"
and widen1_syntax :: "'a prog ⇒ t... | {
"path": "afp-2025-02-12/thys/JinjaThreads/Common/SemiType.thy",
"repo": "afp-2025-02-12",
"sha": "18f1118a34632fb6ce3f98d382c6e41cd1e74b7cf745a65098b178b87908254d"
} |
theory SubRel
imports Graph
begin
section ‹Graphs equipped with a subsumption relation›
text ‹In this section, we define subsumption relations and the notion of sub\hyp{}paths in
rooted graphs equipped with such relations. Sub-paths are defined in the same way than in
\verb?Graph.thy?: first we define the consiste... | {
"path": "afp-2025-02-12/thys/InfPathElimination/SubRel.thy",
"repo": "afp-2025-02-12",
"sha": "6fa6c8b129a8249e719ae9699c37865fa71826d0d787e5809511665097f1ed0b"
} |
theory LinuxRouter_OpenFlow_Translation
imports IP_Addresses.CIDR_Split
Automatic_Refinement.Misc (*TODO@Peter: rename and make available at better place :)*)
Simple_Firewall.Generic_SimpleFw
Semantics_OpenFlow
OpenFlow_Matches
OpenFlow_Action
Routing.Linux_Router
"Pure-ex.Guess"
begin
hide_const Misc.uncurry
... | {
"path": "afp-2025-02-12/thys/LOFT/LinuxRouter_OpenFlow_Translation.thy",
"repo": "afp-2025-02-12",
"sha": "c552d86c7efe89f18ec5dacf53b786499787e3025f490dc69a7544e91f25f164"
} |
(* Title: HOL/MicroJava/BV/Kildall.thy
Author: Tobias Nipkow, Gerwin Klein
Copyright 2000 TUM
Kildall's algorithm.
*)
section ‹Kildall's Algorithm \label{sec:Kildall}›
theory Kildall_1
imports SemilatAlg
begin
primrec merges :: "'s binop ⇒ (nat × 's) list ⇒ 's list ⇒ 's list"
where
"merges f [... | {
"path": "afp-2025-02-12/thys/Jinja/DFA/Kildall_1.thy",
"repo": "afp-2025-02-12",
"sha": "cc883177d03e65aa358f1a6250cb89f10ebda2385ef83d10a9d05e6310f0d5f9"
} |
(* Title: HOL/Auth/Guard/Analz.thy
Author: Frederic Blanqui, University of Cambridge Computer Laboratory
Copyright 2001 University of Cambridge
*)
section‹Decomposition of Analz into two parts›
theory Analz imports Extensions begin
text‹decomposition of \<^term>‹analz› into two parts:
\<^... | {
"path": "Isabelle2024/src/HOL/Auth/Guard/Analz.thy",
"repo": "Isabelle2024",
"sha": "f02e310e8b8e65345e81edc47dd0fdc0fb2d66acb6a11780019c0626d10bed39"
} |
(* author: R. Thiemann *)
section ‹The Sunflower Lemma›
text ‹We formalize the proof of the sunflower lemma of Erd\H{o}s and Rado~\<^cite>‹"erdos_rado"›,
as it is presented in the textbook~\<^cite>‹‹Chapter~6› in "book"›.
We further integrate Exercise 6.2 from the textbook,
which provides a lower bound on the exis... | {
"path": "afp-2025-02-12/thys/Sunflowers/Erdos_Rado_Sunflower.thy",
"repo": "afp-2025-02-12",
"sha": "e5221b961479cc392cd31709db3951eae89edd7953cb78c4e5615ab3acc0d3f9"
} |
section ‹Secure composition: Encrypt then MAC›
theory Secure_Channel imports
One_Time_Pad
Message_Authentication_Code
begin
context begin
interpretation INSEC: insec_channel .
interpretation MAC: macode "rnd η" "mac η" for η .
interpretation AUTH: auth_channel .
interpretation CIPHER: cipher "key η" "enc η" "... | {
"path": "afp-2025-02-12/thys/Constructive_Cryptography/Examples/Secure_Channel/Secure_Channel.thy",
"repo": "afp-2025-02-12",
"sha": "4fa815ba29fdd8c2021b3ba5356dc838e406b39821a157bfe3b7b7b44556ee43"
} |
(*
Author: Salomon Sickert
License: BSD
*)
section ‹Mojmir Automata›
theory Mojmir
imports Main Semi_Mojmir
begin
subsection ‹Definitions›
locale mojmir_def = semi_mojmir_def +
fixes
― ‹Final States›
F :: "'b set"
begin
definition token_succeeds :: "nat ⇒ bool"
where
"token_succeeds ... | {
"path": "afp-2025-02-12/thys/LTL_to_DRA/Mojmir.thy",
"repo": "afp-2025-02-12",
"sha": "6060d8c5e7634a8418031e13d2935678c3a7959bd71daa652e10efbc9a3cf23f"
} |
(* Title: JinjaThreads/MM/DRF_J.thy
Author: Andreas Lochbihler
*)
section ‹JMM Instantiation for J›
theory DRF_J
imports
JMM_Common
JMM_J
"../J/ProgressThreaded"
SC_Legal
begin
primrec ka :: "'addr expr ⇒ 'addr set"
and kas :: "'addr expr list ⇒ 'addr set"
where
"ka (new C) = {}"
| "ka (ne... | {
"path": "afp-2025-02-12/thys/JinjaThreads/MM/DRF_J.thy",
"repo": "afp-2025-02-12",
"sha": "25e30634d98be12927a3a8e2ee875d1390dc9e88eacd7313fbd5d7d85c327848"
} |
(*<*)
―‹ ********************************************************************
* Project : HOL-CSPM - Architectural operators for HOL-CSP
*
* Author : Benoît Ballenghien, Safouan Taha, Burkhart Wolff
*
* This file : Multi-deterministic choice
*
* Copyright (c) 2023 Université Paris-Saclay... | {
"path": "afp-2025-02-12/thys/HOL-CSPM/MultiDet.thy",
"repo": "afp-2025-02-12",
"sha": "d22f995c62252dc764839726c4a5c1ffbc97ba87f94f47b4568454b5ce269a84"
} |
theory FIPS186_4_Curves
imports "Efficient_SEC1"
begin
text ‹In FIPS 186-4, NIST defines several recommended curves for government use. In this theory,
we define the NIST recommended curves over prime fields, that is, all the curves that are labelled
with "P-" Note that all these curves have cofactor h=1 and coef... | {
"path": "afp-2025-02-12/thys/Crypto_Standards/FIPS186_4_Curves.thy",
"repo": "afp-2025-02-12",
"sha": "104c916e7ccfbdd42273311461d515afc02e442bc508508c4737a18de30e217b"
} |
(* Title: Semirings
Author: Walter Guttmann
Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>
*)
section ‹Semirings›
text ‹
This theory develops a hierarchy of idempotent semirings.
All kinds of semiring considered here are bounded semilattices, but many lack additional properties typic... | {
"path": "afp-2025-02-12/thys/Stone_Relation_Algebras/Semirings.thy",
"repo": "afp-2025-02-12",
"sha": "0c0580ed614845060fb4c175e74f1c138caeb9abdc271b300b6ea790340c5e22"
} |
(***********************************************************************************
* Copyright (c) 2016-2018 The University of Sheffield, UK
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are me... | {
"path": "afp-2025-02-12/thys/Core_DOM/common/classes/ObjectClass.thy",
"repo": "afp-2025-02-12",
"sha": "8f6ca38a8bddb50d8959053a640f5db68038b22944329b7045a1f07b0ad04433"
} |
(* Title: Minimality.thy
Author: Sebastian Ullrich
*)
section ‹Minimality›
text ‹We show that every reducible CFG without trivial \pf s is minimal, recreating the proof in~\<^cite>‹"braun13cc"›.
The original proof is inlined as prose text.›
theory Minimality
imports SSA_CFG Serial_Rel
begin
context ... | {
"path": "afp-2025-02-12/thys/Formal_SSA/Minimality.thy",
"repo": "afp-2025-02-12",
"sha": "31d117925994f96122f1f1ced58b3f3dbdccb2317ab6a2041d335e9e1d53745a"
} |
theory Planar_Subdivision
imports
Graph_Genus
Reachablen
Permutations_2
begin
section ‹Combinatorial Planarity and Subdivisions›
locale subdiv1_contr = subdiv_step +
fixes HM
assumes H_map: "digraph_map H HM"
assumes edge_rev_conv: "edge_rev HM = rev_H"
sublocale subdiv1_contr ⊆ H: digraph_map H HM
rew... | {
"path": "afp-2025-02-12/thys/Planarity_Certificates/Planarity/Planar_Subdivision.thy",
"repo": "afp-2025-02-12",
"sha": "782343838efeeef53397712ce5b83e3ae3456aa0b6affe2b07d0d0fc7e836d17"
} |
(* Title: HOL/Library/Nonpos_Ints.thy
Author: Manuel Eberl, TU München
*)
section ‹Non-negative, non-positive integers and reals›
theory Nonpos_Ints
imports Complex_Main
begin
subsection‹Non-positive integers›
text ‹
The set of non-positive integers on a ring. (in analogy to the set of non-negative
in... | {
"path": "Isabelle2024/src/HOL/Library/Nonpos_Ints.thy",
"repo": "Isabelle2024",
"sha": "fc875974ccfa1edd9da0fcc1355e3a495f6630997f5c8138e16df34285dafb92"
} |
section‹Scottâs variant›
subsection‹ScottVariantHOML.thy (Figure 12 of \cite{J75})›
text‹Verification of Scottâs variant of Gödelâs ontological argument. Actualist
quantifiers (avoiding existential import) are used for quantification over entities, otherwise possibilist quantifiers
are used.›
theory ScottVaria... | {
"path": "afp-2025-02-12/thys/Notes_On_Goedels_Ontological_Argument/ScottVariantHOML.thy",
"repo": "afp-2025-02-12",
"sha": "3eb5c92cc61ad2c31ddc4f63ef9d73740f85fc626313858a551218c3c2ea0288"
} |
section ‹Bounding the Size of @{term Y}›
theory Bounding_Y imports Red_Steps
begin
text ‹yet another telescope variant, with weaker promises but a different conclusion;
as written it holds even if @{term"n=0"}›
lemma prod_lessThan_telescope_mult:
fixes f::"nat ⇒ 'a::field"
assumes "⋀i. i<n ⟹ f i ≠ 0"
shows "... | {
"path": "afp-2025-02-12/thys/Diagonal_Ramsey/Bounding_Y.thy",
"repo": "afp-2025-02-12",
"sha": "f5191884c6a52c4ad93c0edfecf91449c45800a38893c28ec5b8fff68236771d"
} |
(* Title: Fast Fourier Transform
Author: Clemens Ballarin <ballarin at in.tum.de>, started 12 April 2005
Maintainer: Clemens Ballarin <ballarin at in.tum.de>
*)
theory FFT
imports Complex_Main
begin
text ‹We formalise a functional implementation of the FFT algorithm
over the complex numbers, and i... | {
"path": "afp-2025-02-12/thys/FFT/FFT.thy",
"repo": "afp-2025-02-12",
"sha": "fe1215231af7306746fd9a15697c541504c4d22ec8583a885e7a606cd60c93dc"
} |
(* Title: HOL/Algebra/Galois_Connection.thy
Author: Alasdair Armstrong and Simon Foster
Copyright: Alasdair Armstrong and Simon Foster
*)
theory Galois_Connection
imports Complete_Lattice
begin
section ‹Galois connections›
subsection ‹Definition and basic properties›
record ('a, 'b, 'c, 'd) gal... | {
"path": "Isabelle2024/src/HOL/Algebra/Galois_Connection.thy",
"repo": "Isabelle2024",
"sha": "99ed2c8ab3ffec7417ce48a4839fe5365a81147f2b1dc1db3dcefca8fc3d54b4"
} |
(***********************************************************************************
* Copyright (c) 2019-2022 Achim D. Brucker
*
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* * Redi... | {
"path": "afp-2025-02-12/thys/Nano_JSON/Example.thy",
"repo": "afp-2025-02-12",
"sha": "24a87d24d2d7453822a4af9bbe727373715c91206129997358d43585898b57e4"
} |
(*
* Copyright (c) 2022 Apple Inc. All rights reserved.
*
* SPDX-License-Identifier: BSD-2-Clause
*)
theory explosion imports
"AutoCorres2_Main.AutoCorres_Main"
begin
install_C_file "explosion.c"
text ‹The sequence of unitialised struct variables caused autocorres to explode in performace.›
autocorres [] "... | {
"path": "afp-2025-02-12/thys/AutoCorres2/tests/proof-tests/explosion.thy",
"repo": "afp-2025-02-12",
"sha": "7d7b2121b752d43e964add298b9b59e7a25772633cd4f311359954e54e3fdc46"
} |
(*
Mike Stannett
Feb 2023
*)
section ‹ Translations ›
text ‹This theory describes translation maps. ›
theory Translations
imports Functions
begin
class Translations = Functions
begin
(*
These functions are all single-valued, so we define them
as having type ('a Point ⇒ 'a Point). We can use the
asFu... | {
"path": "afp-2025-02-12/thys/No_FTL_observers_Gen_Rel/Translations.thy",
"repo": "afp-2025-02-12",
"sha": "e77407606e02479d8e5fff4b14c026d6c90348e3065c25a12dfa2f1e3dea8998"
} |
theory Bexp
imports Aexp
begin
section ‹Boolean Expressions›
text ‹We proceed as in \verb?Aexp.thy?.›
subsection‹Basic definitions›
subsubsection ‹The $\mathit{bexp}$ type-synonym›
text ‹We represent boolean expressions, their set of variables and the notion of freshness of a
variable in the same way than for ari... | {
"path": "afp-2025-02-12/thys/InfPathElimination/Bexp.thy",
"repo": "afp-2025-02-12",
"sha": "84e320f2b0ff3bfce9800f6f3f0c48d6fd3e58d9e0ca52be7228d504c5ed372d"
} |
(*******************************************************************************
Project: Refining Authenticated Key Agreement with Strong Adversaries
Module: sklvl3_asymmetric.thy (Isabelle/HOL 2016-1)
ID: $Id: sklvl3_asymmetric.thy 133183 2017-01-31 13:55:43Z csprenge $
Author: Joseph Lallemand, INRI... | {
"path": "afp-2025-02-12/thys/Key_Agreement_Strong_Adversaries/sklvl3_asymmetric.thy",
"repo": "afp-2025-02-12",
"sha": "8c3fd51adc0b620ea1deaafa604fadc232e827bcf3225656fcc070aa4c31d743"
} |
(* Title: Jinja/J/SmallProgress.thy
Author: Tobias Nipkow
Copyright 2003 Technische Universitaet Muenchen
*)
section ‹Progress of Small Step Semantics›
theory Progress
imports Equivalence WellTypeRT DefAss "../Common/Conform"
begin
lemma final_addrE:
"⟦ P,E,h ⊢ e : Class C; final e;
⋀a. e ... | {
"path": "afp-2025-02-12/thys/Jinja/J/Progress.thy",
"repo": "afp-2025-02-12",
"sha": "763e9530c3e0a4dd16ea32528b54e7052253e53a07ddab21979d7d53c3142a98"
} |
(*<*)
theory IVSubst
imports Syntax
begin
(*>*)
chapter ‹Immutable Variable Substitution›
text ‹Substitution involving immutable variables. We define a class and instances for all
of the term forms›
section ‹Class›
class has_subst_v = fs +
fixes subst_v :: "'a::fs ⇒ x ⇒ v ⇒ 'a::fs" ("_[_::=_]⇩v" [1000,50,5... | {
"path": "afp-2025-02-12/thys/MiniSail/IVSubst.thy",
"repo": "afp-2025-02-12",
"sha": "d8c0cb0bd2b647f2575563f314e8ee4041234a9f8b8135e0366eabd1d4a10cb2"
} |
(* Title: variants/d_fwdrreqs/Aodv_Data.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke, Inria
*)
section "Predicates and functions used in the AODV model"
theory D_Aodv_Data
imports D_Fwdrreqs
begin
subsection "Sequence Numbers"
text ‹Sequence numbers approximate the relative... | {
"path": "afp-2025-02-12/thys/AODV/variants/d_fwdrreqs/D_Aodv_Data.thy",
"repo": "afp-2025-02-12",
"sha": "4cd0bd4f1775252ecd1644968440bc76d20520ec3c3ab74a42f99dd7a7eca54d"
} |
(* Copyright 2021 (C) Mihails Milehins *)
section‹Initial and terminal objects as limits and colimits›
theory CZH_UCAT_Limit_IT
imports
CZH_UCAT_Limit
CZH_Elementary_Categories.CZH_ECAT_Comma
begin
subsection‹Initial and terminal objects as limits/colimits of an empty diagram›
subsubsection‹Definition ... | {
"path": "afp-2025-02-12/thys/CZH_Universal_Constructions/czh_ucategories/CZH_UCAT_Limit_IT.thy",
"repo": "afp-2025-02-12",
"sha": "3ab73f87b0360396be50d8630c3bb55314962b907de42343f80485fa249f1745"
} |
(*
Author: Salomon Sickert
Author: Benedikt Seidl
Author: Alexander Schimpf (original entry: CAVA/LTL.thy)
Author: Stephan Merz (original entry: Stuttering_Equivalence/PLTL.thy)
License: BSD
*)
section ‹Linear Temporal Logic›
theory LTL
imports
Main "HOL-Library.Omega_Words_Fun"
begin
... | {
"path": "afp-2025-02-12/thys/LTL/LTL.thy",
"repo": "afp-2025-02-12",
"sha": "43f7e16beae168f43354e662bf56c26ca00814faeb6d778ac08ee11042c4f72f"
} |
theory Post_COMPOSE2
imports
"Post_ISSUER"
"Post_RECEIVER"
"BD_Security_Compositional.Composing_Security"
begin
subsection ‹Confidentiality for the (binary) issuer-receiver composition›
type_synonym ttrans = "(state, act, out) trans"
type_synonym value1 = Post_ISSUER.value type_synonym value2 = Post_RECEIVER.v... | {
"path": "afp-2025-02-12/thys/CoSMeDis/Post_Confidentiality/Post_COMPOSE2.thy",
"repo": "afp-2025-02-12",
"sha": "e840644ea5ca13293720ba7c4bd5189672ea53eb4432e676df56aabdde72c1f6"
} |
(* Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com> *)
section ‹Injections from A to B up to a Permutation of A›
theory Twelvefold_Way_Entry5
imports
Equiv_Relations_on_Functions
begin
subsection ‹Definition of Bijections›
definition subset_of :: "'a set ⇒ ('a ⇒ 'b) set ⇒ 'b set"
where
"subset_of A F = univ... | {
"path": "afp-2025-02-12/thys/Twelvefold_Way/Twelvefold_Way_Entry5.thy",
"repo": "afp-2025-02-12",
"sha": "cbba1cf7ac9948639e040a1ae68c90e71291076d49f6f712c7b390204ccb4be8"
} |
theory Key
imports
"../Fused_Resource"
begin
section ‹Key specification›
locale ideal_key =
fixes valid_keys :: "'key set"
begin
subsection ‹Data-types for Parties, State, Events, Input, and Output›
datatype party = Alice | Bob
type_synonym s_shell = "party set"
datatype 'key' s_kernel = PState_Store | St... | {
"path": "afp-2025-02-12/thys/Constructive_Cryptography_CM/Specifications/Key.thy",
"repo": "afp-2025-02-12",
"sha": "49321e2c7b0c8ec56b1cc761d02802d392a4bd71af60041136f6baab5a0dfba0"
} |
(*
* Copyright 2016, Data61, CSIRO
*
* This software may be distributed and modified according to the terms of
* the BSD 2-Clause license. Note that NO WARRANTY is provided.
* See "LICENSE_BSD2.txt" for details.
*
* @TAG(DATA61_BSD)
*)
section ‹The COMPLX Syntax›
theory Language
imports Main
begin
subsection... | {
"path": "afp-2025-02-12/thys/Complx/Language.thy",
"repo": "afp-2025-02-12",
"sha": "a19b24d158cca83e921a98f40c557d8c2d70dd500e803e59151f18981b46a915"
} |
(* The definitions for a challenge suggested by Adam Chlipala *)
theory Compile
imports "HOL-Nominal.Nominal"
begin
atom_decl name
nominal_datatype data =
DNat
| DProd "data" "data"
| DSum "data" "data"
nominal_datatype ty =
Data "data"
| Arrow "ty" "ty" ("_→_" [100,100] 100)
nominal_datatype trm ... | {
"path": "Isabelle2024/src/HOL/Nominal/Examples/Compile.thy",
"repo": "Isabelle2024",
"sha": "c215a05ae71813549b4b3df1d8d5db9a9145e24414f6fb221208852fb7ae9cc2"
} |
section ‹Characteristic of Rings\label{sec:ring_char}›
theory Ring_Characteristic
imports
"Finite_Fields_Factorization_Ext"
"HOL-Algebra.IntRing"
"HOL-Algebra.Embedded_Algebras"
begin
locale finite_field = field +
assumes finite_carrier: "finite (carrier R)"
begin
lemma finite_field_min_order:
"ord... | {
"path": "afp-2025-02-12/thys/Finite_Fields/Ring_Characteristic.thy",
"repo": "afp-2025-02-12",
"sha": "20f12a2b661260ec420b2d7d092093d37eddeedaba1f25ac16bbd87e777bc09a"
} |
section ‹Zeckendorf's Theorem›
theory Zeckendorf
imports
Main
"HOL-Number_Theory.Number_Theory"
begin
subsection ‹Definitions›
text ‹
Formulate auxiliary definitions. An increasing sequence is a predicate of a function $f$
together with a set $I$. $f$ is an increasing sequence on $I$, if $f(x)+1 < f(x+1)$ ... | {
"path": "afp-2025-02-12/thys/Zeckendorf/Zeckendorf.thy",
"repo": "afp-2025-02-12",
"sha": "2eb81b20926ee2f654efb524698512c9a896d70fa7f4c965eb04e60d0087ee61"
} |
✐‹creator "Kevin Kappelmann"›
section ‹Transport For Functions›
subsection ‹Basic Setup›
theory Transport_Functions_Base
imports
Monotone_Function_Relator
Transport_Base
begin
paragraph ‹Summary›
text ‹Basic setup for closure proofs. We introduce locales for the syntax,
the dependent relator, the non-depende... | {
"path": "afp-2025-02-12/thys/Transport/Transport/Functions/Transport_Functions_Base.thy",
"repo": "afp-2025-02-12",
"sha": "7f5fd31aadb52ff63e82d905d5ae0497436cf086782514d6fb9e71de9c42ed9d"
} |
(* Title: statecharts/DataSpace/Update.thy
Author: Steffen Helke, Software Engineering Group
Copyright 2010 Technische Universitaet Berlin
*)
section ‹Update-Functions on Data Spaces›
theory Update
imports Data
begin
subsection ‹Total update-functions›
definition
Update :: "(('d data) => ('d d... | {
"path": "afp-2025-02-12/thys/Statecharts/Update.thy",
"repo": "afp-2025-02-12",
"sha": "f71a6947463fd50f6b244c9ab3b616b62c5bd991ef2618ad402421a9cf145b39"
} |
section ‹Storjohann's basis reduction algorithm (concrete implementation)›
text ‹We refine the abstract algorithm into a more efficient executable one.›
theory Storjohann_Impl
imports
Storjohann
begin
subsection ‹Implementation›
text ‹We basically store four components:
▪ The $f$-basis (as list, all values... | {
"path": "afp-2025-02-12/thys/Modular_arithmetic_LLL_and_HNF_algorithms/Storjohann_Impl.thy",
"repo": "afp-2025-02-12",
"sha": "be0a330414c18019c786e8357934f847f3e0cd86cdd6e377c8105ce523c0b771"
} |
(* Author: Tobias Nipkow *)
subsection "Verification Conditions for Total Correctness"
subsubsection "The Standard Approach"
theory VCG_Total_EX
imports Hoare_Total_EX
begin
text‹Annotated commands: commands where loops are annotated with
invariants.›
datatype acom =
Askip ("SKIP") |
Aassign v... | {
"path": "Isabelle2024/src/HOL/IMP/VCG_Total_EX.thy",
"repo": "Isabelle2024",
"sha": "35cfbd01b401d7793fdebb4f1e2c0f954b6330d260c33f4246ad8c60ca55a1ea"
} |
section‹The Axiom of Extensionality in $M[G]$›
theory Extensionality_Axiom
imports
Names
begin
context forcing_data
begin
lemma extensionality_in_MG : "extensionality(##(M[G]))"
proof (-)
(*goal: ‹extensionality(##M[G])›*)
{
fix x and y and z
assume asms: "x∈M[G]" "y∈M[G]" "(∀w∈M[G] . w ∈ x ⟷ w ∈ y)" ... | {
"path": "afp-2025-02-12/thys/Forcing/Extensionality_Axiom.thy",
"repo": "afp-2025-02-12",
"sha": "ffd787f00e433070124962757c017a12eab1d6e478259bf8b66dd5f8395aaa1b"
} |
(* Title: OAWN_SOS.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke
*)
section "Open semantics of the Algebra of Wireless Networks"
theory OAWN_SOS
imports TransitionSystems AWN
begin
text ‹
These are variants of the SOS rules that work against a mixed global/local context, wh... | {
"path": "afp-2025-02-12/thys/AWN/OAWN_SOS.thy",
"repo": "afp-2025-02-12",
"sha": "787eddf803c9f46cfa87049d3f61a960ed4303c9c8689da3f2ee41ab262ed144"
} |
(* Title: RBT_Mapping_Exts.thy
Author: Denis Lohner, Sebastian Ullrich
*)
theory RBT_Mapping_Exts
imports
Mapping_Exts
"HOL-Library.RBT_Mapping"
"HOL-Library.RBT_Set"
begin
lemma restrict_mapping_code [code]:
"restrict_mapping f (RBT_Set.Set r) = RBT_Mapping.Mapping (RBT.map (λa _. f a) r)"
... | {
"path": "afp-2025-02-12/thys/Formal_SSA/RBT_Mapping_Exts.thy",
"repo": "afp-2025-02-12",
"sha": "46c7b144ae33b2806e873c368fa56f38ac8690ee726828b1d9bc1c8a7372ef05"
} |
(* Title: The Optimized Graceful Recursive Path Order for Lambda-Free Higher-Order Terms
Author: Jasmin Blanchette <jasmin.blanchette at inria.fr>, 2016
Maintainer: Jasmin Blanchette <jasmin.blanchette at inria.fr>
*)
section ‹The Optimized Graceful Recursive Path Order for Lambda-Free Higher-Orde... | {
"path": "afp-2025-02-12/thys/Lambda_Free_RPOs/Lambda_Free_RPO_Optim.thy",
"repo": "afp-2025-02-12",
"sha": "2b83bff277b2a507568849a83e05062f43f9d7cf85601d3f7a1806c347a88a66"
} |
(* Title: HOL/UNITY/ProgressSets.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2003 University of Cambridge
Progress Sets. From
David Meier and Beverly Sanders,
Composing Leads-to Properties
Theoretical Computer Science 243:1-2 (2000), 339-361.
... | {
"path": "Isabelle2024/src/HOL/UNITY/ProgressSets.thy",
"repo": "Isabelle2024",
"sha": "5d4f04e37f6e967b0f16b7c6d2056d04c7020784a38acbc97c72b6536562cfe1"
} |
(* Author: Tobias Nipkow *)
section "Tries (List Version)"
theory Tries
imports Trie
begin
text‹This is a specialization of tries where values are lists.›
type_synonym ('k,'v)tries = "('k,'v list)trie"
definition lookup_tries :: "('k,'v)tries ⇒ 'k list ⇒ 'v list" where
"lookup_tries t ks = (case lookup_trie t ks... | {
"path": "afp-2025-02-12/thys/Trie/Tries.thy",
"repo": "afp-2025-02-12",
"sha": "9d42301d5219b3a4ce584f64c749e7482d429206ffa61c512b6c3ce430d82d2a"
} |
theory Fused_Resource imports
Fold_Spmf
begin
(* TODO: move these somewhere *)
context includes ℐ.lifting begin
lift_definition eℐ :: "('a, 'b) ℐ ⇒ ('a, 'b × 'c) ℐ" is "λℐ x. ℐ x × UNIV" .
lemma outs_ℐ_eℐ[simp]: "outs_ℐ (eℐ ℐ) = outs_ℐ ℐ"
apply transfer
(*goal: ‹outs_ℐ (eℐ ℐ) = outs_ℐ ℐ›*)
by simp
lemma re... | {
"path": "afp-2025-02-12/thys/Constructive_Cryptography_CM/Fused_Resource.thy",
"repo": "afp-2025-02-12",
"sha": "b3c7ebc96745729e90a16276e18f756f8e3c7e01d1a872c0f9a971d9c6523926"
} |
(* Title: HOL/TLA/TLA.thy
Author: Stephan Merz
Copyright: 1998 University of Munich
*)
section ‹The temporal level of TLA›
theory TLA
imports Init
begin
consts
(** abstract syntax **)
Box :: "('w::world) form ⇒ temporal"
Dmd :: "('w::world) form ⇒ temporal"
leadsto :: "['w... | {
"path": "Isabelle2024/src/HOL/TLA/TLA.thy",
"repo": "Isabelle2024",
"sha": "c9be2e8ab3779532f7f7bbda3f76ad9a5bd2a5a839e70af5d7b5bb5863001810"
} |
section ‹ Key algorithms for WEST ›
theory WEST_Algorithms
imports Mission_Time_LTL.MLTL_Properties
begin
subsection ‹Custom Types›
datatype WEST_bit = Zero | One | S
type_synonym state = "nat set"
type_synonym trace = "nat set list"
type_synonym state_regex = "WEST_bit list"
type_synonym trace_regex = "WEST_bit l... | {
"path": "afp-2025-02-12/thys/Mission_Time_LTL_to_Regular_Expression/WEST_Algorithms.thy",
"repo": "afp-2025-02-12",
"sha": "bc00eb032d5931517ae0922b03d5ce3fef1593da049f1a4817d803e079a38bcc"
} |
(* Title: thys/Rec_Ex.thy
Author: Franz Regensburger (FABR) 03/2022
*)
section ‹Examples for Recursive Functions based on Rec\_def›
theory Rec_Ex
imports Rec_Def
begin
definition plus_2 :: "recf"
where
"plus_2 = (Cn 1 s [s])"
lemma "rec_exec plus_2 [0] = Suc (Suc 0)"
unfolding plus_2_def
(*goal... | {
"path": "afp-2025-02-12/thys/Universal_Turing_Machine/Rec_Ex.thy",
"repo": "afp-2025-02-12",
"sha": "3c144c4459806b0aac3a4a44eedb1b0a749b93f8143dc042c67cc9251bccc3e8"
} |
(*******************************************************************************
Project: Development of Security Protocols by Refinement
Module: Key_establish/m3_kerberos4.thy (Isabelle/HOL 2016-1)
ID: $Id: m3_kerberos4.thy 132890 2016-12-24 10:25:57Z csprenge $
Authors Ivano Somaini, ETH Zurich <soma... | {
"path": "afp-2025-02-12/thys/Security_Protocol_Refinement/Key_establish/m3_kerberos4.thy",
"repo": "afp-2025-02-12",
"sha": "cc5c60af1a6b222168e01518c1ebf504ff3bdc1c6dc0addbeb832e8a18c987bc"
} |
(*
Title: Code_Cayley_Hamilton.thy
Author: Jose Divasón <jose.divasonm at unirioja.es>
Author: Jesús Aransay <jesus-maria.aransay at unirioja.es>
*)
section‹Code Cayley Hamilton›
theory Code_Cayley_Hamilton
imports
"HOL-Computational_Algebra.Polynomial"
Cayley_Hamilton_Compatible
... | {
"path": "afp-2025-02-12/thys/Echelon_Form/Code_Cayley_Hamilton.thy",
"repo": "afp-2025-02-12",
"sha": "8d855fb052e7861ee0eefb95b9aad3cb33fbe43c789aa25233aae000623377a0"
} |
(*<*)
(*
* Copyright 2015, NICTA
*
* This software may be distributed and modified according to the terms of
* the BSD 2-Clause license. Note that NO WARRANTY is provided.
* See "LICENSE_BSD2.txt" for details.
*
* @TAG(NICTA_BSD)
*)
theory Global_Invariants_Lemmas
imports
Global_Invariants
begin
declare sub... | {
"path": "afp-2025-02-12/thys/ConcurrentGC/Global_Invariants_Lemmas.thy",
"repo": "afp-2025-02-12",
"sha": "4ea1e5e6692fa2030b4eb70521c06ddf447e61b364d6fa64a18cf5ec294ed573"
} |
theory Preliminary_Library
imports Main "HOL-Library.Word" "Word_Lib.Word_Lib_Sumo" "HOL-Library.Countable"
begin
section ‹Preliminary Theories›
text ‹In this subsection, we provide the type and value system used by the CHERI-C Memory Model.
We also provide proofs for the conversion between large words (i.e. b... | {
"path": "afp-2025-02-12/thys/CHERI-C_Memory_Model/Preliminary_Library.thy",
"repo": "afp-2025-02-12",
"sha": "55cf8e557dfaa50709c34261001ee6ed8aec56b5d004da0e2b1daeb1506adde1"
} |
(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
* Copyright (c) 2022 Apple Inc. All rights reserved.
*
* SPDX-License-Identifier: BSD-2-Clause
*)
theory globals_fn
imports "AutoCorres2.CTranslation"
begin
install_C_file "globals_fn.c"
print_locale globals_fn_global_addresses
context globals_fn_simpl
be... | {
"path": "afp-2025-02-12/thys/AutoCorres2/tests/c-parser/globals_fn.thy",
"repo": "afp-2025-02-12",
"sha": "984dfbcabafc5efc6359f71af19030e24272ffcd7d2f0012761076b833aea042"
} |
(* Title: Right group actions
Author: Tim Makarios <tjm1983 at gmail.com>, 2012
Maintainer: Tim Makarios <tjm1983 at gmail.com>
*)
(* After Isabelle 2012, this may be moved to ~~/src/HOL/Algebra *)
section "Right group actions"
theory Action
imports "HOL-Algebra.Group"
begin
locale action = g... | {
"path": "afp-2025-02-12/thys/Tarskis_Geometry/Action.thy",
"repo": "afp-2025-02-12",
"sha": "c75ff23224eb19c3e6b9c15003666208d1b709c7d4bfd550cb6bcb4b3a46c361"
} |
(* Title: HOL/Statespace/DistinctTreeProver.thy
Author: Norbert Schirmer, TU Muenchen
*)
section ‹Distinctness of Names in a Binary Tree \label{sec:DistinctTreeProver}›
theory DistinctTreeProver
imports Main
begin
text ‹A state space manages a set of (abstract) names and assumes
that the names are dis... | {
"path": "Isabelle2024/src/HOL/Statespace/DistinctTreeProver.thy",
"repo": "Isabelle2024",
"sha": "0c52313faa5cbb9f00a0a692731277a8deccc2b21838a37e2b53ecc4a7c98e57"
} |
(*
Authors: Jose Divasón
Sebastiaan Joosten
René Thiemann
Akihisa Yamada
License: BSD
*)
section ‹List representation›
theory List_Representation
imports Main
begin
lemma rev_take_Suc: assumes j: "j < length xs"
shows "rev (take (Suc j) xs) = xs ! j... | {
"path": "afp-2025-02-12/thys/LLL_Basis_Reduction/List_Representation.thy",
"repo": "afp-2025-02-12",
"sha": "714b9040bff54d3349330f77da605f811175cf05637d4b92f720d698f3d0d918"
} |
subsection ‹Digit function is Diophantine›
theory Digit_Function
imports Exponential_Relation "Digit_Expansions.Bits_Digits"
begin
definition digit ("[ _ = Digit _ _ _]" [999] 1000)
where "[D = Digit AA K BASE] ≡ (QUATERNARY (λd a k b. b > 1
∧ d = nth_digit a k b) D AA K BASE)"... | {
"path": "afp-2025-02-12/thys/DPRM_Theorem/Diophantine/Digit_Function.thy",
"repo": "afp-2025-02-12",
"sha": "0910ea9c8ad34eb4ad10e8185df932eff86748beaf6628e81f8c24c6f458c450"
} |
section ‹Semantics of SM›
theory SM_Semantics
imports SM_State SM_Cfg Gen_Scheduler
begin
section ‹Evaluation of Expressions›
subsection ‹Basic operations›
text ‹Attention: Silently overflows, using 2's complement.
This should match Java-semantics. In C, signed overflow is undefined!›
context
includes... | {
"path": "afp-2025-02-12/thys/CAVA_LTL_Modelchecker/SM/RefPoint/SM_Semantics.thy",
"repo": "afp-2025-02-12",
"sha": "d3543cf221b283ad9e9482ff5caac4f5f53f819aa92a5f9db691fe0816e16993"
} |
(* Title: Prim's Minimum Spanning Tree Algorithm
Author: Walter Guttmann
Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>
*)
section ‹Prim's Minimum Spanning Tree Algorithm›
text ‹
In this theory we prove total correctness of Prim's minimum spanning tree algorithm.
The proof has the sa... | {
"path": "afp-2025-02-12/thys/Relational_Minimum_Spanning_Trees/Prim.thy",
"repo": "afp-2025-02-12",
"sha": "601c8d58cc563cdb74514b4b08523b5a4650ef7c863860bee2ee39e935af0156"
} |
(*<*)
theory Event_Data
imports
"HOL-Library.Char_ord"
Code_Double
Deriving.Derive
begin
(*>*)
section ‹Event parameters›
definition div_to_zero :: "integer ⇒ integer ⇒ integer" where
"div_to_zero x y = (let z = fst (Code_Numeral.divmod_abs x y) in
if (x < 0) ≠ (y < 0) then - z else z)"
definitio... | {
"path": "afp-2025-02-12/thys/MFODL_Monitor_Optimized/Event_Data.thy",
"repo": "afp-2025-02-12",
"sha": "2823f1fdcd57d42f007fe66fe4378addb2f70e8e7c7788720fdd294d237442ae"
} |
(* Author: Alexander Maletzky *)
section ‹Macaulay Matrices›
theory Macaulay_Matrix
imports More_MPoly_Type_Class Jordan_Normal_Form.Gauss_Jordan_Elimination
begin
text ‹We build upon vectors and matrices represented by dimension and characteristic function, because
later on we need to quantify the dimensions of... | {
"path": "afp-2025-02-12/thys/Groebner_Bases/Macaulay_Matrix.thy",
"repo": "afp-2025-02-12",
"sha": "ae88c6316a514b4fadf51fcfe680a1a1468b2bfee64fb17f9583d27f9ffdcc02"
} |
(*
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Tobias Nipkow, TUM
*)
theory Tilings imports Main begin
section‹Inductive Tiling›
inductive_set
tiling :: "'a set set ⇒ 'a set set"
for A :: "'a set set" where
empty [simp, intro]: "{} ∈ tiling A" |
Un [simp, intro]:... | {
"path": "afp-2025-02-12/thys/FunWithTilings/Tilings.thy",
"repo": "afp-2025-02-12",
"sha": "f3243f87e3dd4d519b177dad828ad35fbe31e080d674d75e963f1e2189c11e3b"
} |
theory Valid_List
imports Main "../util/List_Util"
begin
section ‹Valid List›
definition
valid_list :: "('a :: {linorder, order_bot}) list ⇒ bool"
where
"valid_list s = (length s > 0 ∧ (∀i < length s - 1. s ! i ≠ bot) ∧ last s = bot)"
lemma valid_list_ex_def:
fixes s ::"('a :: {linorder, order_bot}) list"
... | {
"path": "afp-2025-02-12/thys/SuffixArray/order/Valid_List.thy",
"repo": "afp-2025-02-12",
"sha": "0f3e229f371c2d3a44523088be1b2536311acc1e1dd001efd3dd6ee0509a7ca5"
} |
section ‹Preliminary results›
theory Concentration_Inequalities_Preliminary
imports Lp.Lp
begin
text ‹Version of Cauchy-Schwartz for the Lebesgue integral:›
lemma cauchy_schwartz:
fixes f g :: "_ ⇒ real"
assumes "f ∈ borel_measurable M" "g ∈ borel_measurable M"
assumes "integrable M (λx. (f x) ^2)" "integrabl... | {
"path": "afp-2025-02-12/thys/Concentration_Inequalities/Concentration_Inequalities_Preliminary.thy",
"repo": "afp-2025-02-12",
"sha": "a05f62e524f21cfcc7ddeeb8b6b99905b128eca5029059a145f7725768952400"
} |
(* Author: Dmitriy Traytel *)
section ‹Normalization of WS1S Formulas›
(*<*)
theory WS1S_Normalization
imports WS1S
begin
(*>*)
fun nNot where
"nNot (FNot φ) = φ"
| "nNot (FAnd φ1 φ2) = FOr (nNot φ1) (nNot φ2)"
| "nNot (FOr φ1 φ2) = FAnd (nNot φ1) (nNot φ2)"
| "nNot φ = FNot φ"
primrec norm where
"norm (FQ a m)... | {
"path": "afp-2025-02-12/thys/MSO_Regex_Equivalence/WS1S_Normalization.thy",
"repo": "afp-2025-02-12",
"sha": "deee4b7c960f5c82bfad81a1a4ccde6f8bbe13eb607dbc47c1ccc1b4ca31a515"
} |
(* Title: Subset Boolean Algebras
Authors: Walter Guttmann, Bernhard Möller
Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>
*)
theory Subset_Boolean_Algebras
imports Stone_Algebras.P_Algebras
begin
section ‹Boolean Algebras›
text ‹
We show that Isabelle/HOL's ‹boolean_algebra› clas... | {
"path": "afp-2025-02-12/thys/Subset_Boolean_Algebras/Subset_Boolean_Algebras.thy",
"repo": "afp-2025-02-12",
"sha": "04676f97d3db76c970437d6377d3988001df2eb5dd769fc2b89e355a1bad961a"
} |
theory Aleph_Relative
imports
CardinalArith_Relative
begin
definition
HAleph :: "[i,i] ⇒ i" where
"HAleph(i,r) ≡ if(¬(Ord(i)),i,if(i=0, nat, if(¬Limit(i) ∧ i≠0,
csucc(r`( ⋃ i )),
⋃j∈i. r`j)))"
reldb_add functional "Limit" "Limit"
relationalize "... | {
"path": "afp-2025-02-12/thys/Transitive_Models/Aleph_Relative.thy",
"repo": "afp-2025-02-12",
"sha": "f927061b2269ec16878b11ffc3d6ed603a16de06f6b0c68d09f96c8521f7c926"
} |
(*
* Copyright (c) 2022 Apple Inc. All rights reserved.
*
* SPDX-License-Identifier: BSD-2-Clause
*)
section ‹Pointers into Structures in Split Heap \label{sec:open_struct}›
theory open_struct
imports "AutoCorres"
begin
subsection ‹Overview›
text ‹
Heap lifting in AutoCorres transforms the monolithic byte-lev... | {
"path": "afp-2025-02-12/thys/AutoCorres2/doc/open_struct.thy",
"repo": "afp-2025-02-12",
"sha": "efdf615dc432b2089e73483df9535042e0ae4612b42e1bb4c942b00d1e8c9832"
} |
(* Title: variants/c_gtobcast/Fresher.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke, Inria
Author: Peter Höfner, NICTA
*)
section "Quality relations between routes"
theory C_Fresher
imports C_Aodv_Data
begin
subsection "Net sequence numbers"
su... | {
"path": "afp-2025-02-12/thys/AODV/variants/c_gtobcast/C_Fresher.thy",
"repo": "afp-2025-02-12",
"sha": "f085a0e01f9ff9156e6505b35d720a9ed42661a632b5e138aca34964ed3bfb6f"
} |
theory PathLemmas
imports TheoremD14
begin
context LocalLexing begin
lemma characterize_𝒫:
"(∀ i < length p. ∃u. p ! i ∈ 𝒵 (charslength (take i p)) u) ⟹ admissible p ⟹
∃ u. p ∈ 𝒫 (charslength p) u"
proof (induct p rule: rev_induct (*‹⟦?P []; ⋀x xs. ?P xs ⟹ ?P (xs @ [x])⟧ ⟹ ?P ?xs›*))
(*goals:
1. ‹⟦∀i<len... | {
"path": "afp-2025-02-12/thys/LocalLexing/PathLemmas.thy",
"repo": "afp-2025-02-12",
"sha": "511de6471118ee6864efdbf215b3ef9613a6af385c27e8ad13c3ce84ef47777c"
} |
theory Clause_Could_Propagate
imports
Background
Implicit_Exhaustive_Factorization
begin
context simulation_SCLFOL_ground_ordered_resolution begin
definition clause_could_propagate where
"clause_could_propagate Γ C L ⟷ ¬ trail_defined_lit Γ L ∧
linorder_lit.is_maximal_in_mset C L ∧ trail_false_cls Γ {... | {
"path": "afp-2025-02-12/thys/SCL_Simulates_Ground_Resolution/Clause_Could_Propagate.thy",
"repo": "afp-2025-02-12",
"sha": "c136afa916bf15069dc50b1b93ece57ff8dbbcdc737c2b62db2d3e6307baecc3"
} |
(* Title: HOL/HOLCF/IOA/Abstraction.thy
Author: Olaf Müller
*)
section ‹Abstraction Theory -- tailored for I/O automata›
theory Abstraction
imports LiveIOA
begin
default_sort type
definition cex_abs :: "('s1 ⇒ 's2) ⇒ ('a, 's1) execution ⇒ ('a, 's2) execution"
where "cex_abs f ex = (f (fst ex), Map ... | {
"path": "Isabelle2024/src/HOL/HOLCF/IOA/Abstraction.thy",
"repo": "Isabelle2024",
"sha": "668ec7d78200263e343e1729ae5efdcc611b8aac8e4cb21c5c255827e02ffb85"
} |
section ‹Nested DFS›
theory Nested_DFS
imports DFS_Find_Path
begin
text ‹Nested DFS is a standard method for Buchi-Automaton emptiness check.›
subsection ‹Auxiliary Lemmas›
lemma closed_restrict_aux:
assumes CL: "E``F ⊆ F ∪ S"
assumes NR: "E⇧*``U ∩ S = {}"
assumes SS: "U ⊆ F"
shows "E⇧*``U ⊆ F"
― ‹Auxiliary... | {
"path": "afp-2025-02-12/thys/DFS_Framework/Examples/Nested_DFS.thy",
"repo": "afp-2025-02-12",
"sha": "1270ce5f6cc9552a05906d357636b1a577cbdf0532b255921996a9ea8c6045e0"
} |
(*
Author: Jose Divasón
Email: jose.divason@unirioja.es
*)
section ‹Elementary divisor rings›
theory Elementary_Divisor_Rings
imports
SNF_Algorithm
Rings2_Extended
begin
text ‹This theory contains the definition of elementary divisor rings and Hermite rings, as
well as the corresponding r... | {
"path": "afp-2025-02-12/thys/Smith_Normal_Form/Elementary_Divisor_Rings.thy",
"repo": "afp-2025-02-12",
"sha": "e6bde3141d3a3614dbea7bb386cfbdce553bd74fc92c112a1a5b04de3349930a"
} |
(*****************************************************************************
* Featherweight-OCL --- A Formal Semantics for UML-OCL Version OCL 2.5
* for the OMG Standard.
* http://www.brucker.ch/projects/hol-testgen/
*
* UML_Void.thy --- Library definitions.
* This f... | {
"path": "afp-2025-02-12/thys/Featherweight_OCL/basic_types/UML_Void.thy",
"repo": "afp-2025-02-12",
"sha": "0eea4d973c15ca92e10bbe703b2b06d5a0a8cef577439acf2be926171cb8af3a"
} |
section ‹Lazy evaluation within the logic›
theory Lazy_Eval
imports
Complex_Main
begin
text ‹
This is infrastructure to lazily evaluate an expression (typically something corecursive)
within the logic by simple rewriting. A signature of supported (co-)datatype constructures
upon which pattern matching is al... | {
"path": "Isabelle2024/src/HOL/Real_Asymp/Lazy_Eval.thy",
"repo": "Isabelle2024",
"sha": "dc10676a7fc95cf9c19415abd61be9711d33db7bcdd3bdb755ad2d7bff22c5d6"
} |
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