‹\\label{section:Infinite_Sum}›\n\ntext ‹In this theory, we introduce the definition of infinite sums, i.e., sums ranging over an\ninfinite, potentially uncountable index set with no particular ordering.\n(This is different from series. Those are sums indexed by natural numbers,\nand the order of the index set matters.)\n\nOur definition is quite standard: $s:=\\sum_{x\\in A} f(x)$ is the limit of finite sums $s_F:=\\sum_{x\\in F} f(x)$ for increasing $F$.\nThat is, $s$ is the limit of the net $s_F$ where $F$ are finite subsets of $A$ ordered by inclusion.\nWe believe that this is the standard definition for such sums.\nSee, e.g., Definition 4.11 in \\<^cite>‹\"conway2013course\"›.\nThis definition is quite general: it is well-defined whenever $f$ takes values in some\ncommutative monoid endowed with a Hausdorff topology.\n(Examples are reals, complex numbers, normed vector spaces, and more.)›\n\ntheory Infinite_Sum\n imports\n Elementary_Topology\n \"HOL-Library.Extended_Nonnegative_Real\"\n \"HOL-Library.Complex_Order\"\nbegin\n\nsubsection ‹Definition and syntax›\n\ndefinition HAS_SUM :: ‹('a ⇒ 'b :: {comm_monoid_add, topological_space}) ⇒ 'a set ⇒ 'b ⇒ bool› \n where has_sum_def: ‹HAS_SUM f A x ≡ (sum f ⤏ x) (finite_subsets_at_top A)›\n\nabbreviation has_sum (infixr \"has'_sum\" 46) where\n \"(f has_sum S) A ≡ HAS_SUM f A S\"\n\ndefinition summable_on :: \"('a ⇒ 'b::{comm_monoid_add, topological_space}) ⇒ 'a set ⇒ bool\" (infixr \"summable'_on\" 46) where\n \"f summable_on A ≡ (∃x. (f has_sum x) A)\"\n\ndefinition infsum :: \"('a ⇒ 'b::{comm_monoid_add,t2_space}) ⇒ 'a set ⇒ 'b\" where\n \"infsum f A = (if f summable_on A then Lim (finite_subsets_at_top A) (sum f) else 0)\"\n\nabbreviation abs_summable_on :: \"('a ⇒ 'b::real_normed_vector) ⇒ 'a set ⇒ bool\" (infixr \"abs'_summable'_on\" 46) where\n \"f abs_summable_on A ≡ (λx. norm (f x)) summable_on A\"\n\nsyntax (ASCII)\n \"_infsum\" :: \"pttrn ⇒ 'a set ⇒ 'b ⇒ 'b::topological_comm_monoid_add\" (\"(3INFSUM (_/:_)./ _)\" [0, 51, 10] 10)\nsyntax\n \"_infsum\" :: \"pttrn ⇒ 'a set ⇒ 'b ⇒ 'b::topological_comm_monoid_add\" (\"(2∑⇩∞(_/∈_)./ _)\" [0, 51, 10] 10)\ntranslations ― ‹Beware of argument permutation!›\n \"∑⇩∞i∈A. b\" ⇌ \"CONST infsum (λi. b) A\"\n\nsyntax (ASCII)\n \"_univinfsum\" :: \"pttrn ⇒ 'a ⇒ 'a\" (\"(3INFSUM _./ _)\" [0, 10] 10)\nsyntax\n \"_univinfsum\" :: \"pttrn ⇒ 'a ⇒ 'a\" (\"(2∑⇩∞_./ _)\" [0, 10] 10)\ntranslations\n \"∑⇩∞x. t\" ⇌ \"CONST infsum (λx. t) (CONST UNIV)\"\n\nsyntax (ASCII)\n \"_qinfsum\" :: \"pttrn ⇒ bool ⇒ 'a ⇒ 'a\" (\"(3INFSUM _ |/ _./ _)\" [0, 0, 10] 10)\nsyntax\n \"_qinfsum\" :: \"pttrn ⇒ bool ⇒ 'a ⇒ 'a\" (\"(2∑⇩∞_ | (_)./ _)\" [0, 0, 10] 10)\ntranslations\n \"∑⇩∞x|P. t\" => \"CONST infsum (λx. t) {x. P}\"\n\nprint_translation ‹\nlet\n fun sum_tr' [Abs (x, Tx, t), Const (@{const_syntax Collect}, _) $ Abs (y, Ty, P)] =\n if x <> y then raise Match\n else\n let\n val x' = Syntax_Trans.mark_bound_body (x, Tx);\n val t' = subst_bound (x', t);\n val P' = subst_bound (x', P);\n in\n Syntax.const @{syntax_const \"_qinfsum\"} $ Syntax_Trans.mark_bound_abs (x, Tx) $ P' $ t'\n end\n | sum_tr' _ = raise Match;\nin [(@{const_syntax infsum}, K sum_tr')] end\n›\n\nsubsection ‹General properties›\n\nlemma infsumI:\n fixes f g :: ‹'a ⇒ 'b::{comm_monoid_add, t2_space}›\n assumes ‹(f has_sum x) A›\n shows ‹infsum f A = x›\n by (metis assms (*‹(f has_sum x) A›*) finite_subsets_at_top_neq_bot (*‹finite_subsets_at_top ?A ≠ bot›*) infsum_def (*‹infsum ?f ?A = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else 0)›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) tendsto_Lim (*‹⟦?net ≠ bot; (?f ⤏ ?l) ?net⟧ ⟹ Lim ?net ?f = ?l›*))\n\nlemma infsum_eqI:\n fixes f g :: ‹'a ⇒ 'b::{comm_monoid_add, t2_space}›\n assumes ‹x = y›\n assumes ‹(f has_sum x) A›\n assumes ‹(g has_sum y) B›\n shows ‹infsum f A = infsum g B›\n using assms (*‹x = y› ‹(f has_sum x) A› ‹(g has_sum y) B›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) by blast\n\nlemma infsum_eqI':\n fixes f g :: ‹'a ⇒ 'b::{comm_monoid_add, t2_space}›\n assumes ‹⋀x. (f has_sum x) A ⟷ (g has_sum x) B›\n shows ‹infsum f A = infsum g B›\n by (metis assms (*‹((f::'a ⇒ 'b) has_sum (?x::'b)) (A::'a set) = ((g::'a ⇒ 'b) has_sum ?x) (B::'a set)›*) infsum_def (*‹infsum (?f::?'a ⇒ ?'b) (?A::?'a set) = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else (0::?'b))›*) infsum_eqI (*‹⟦(?x::?'b) = (?y::?'b); ((?f::?'a ⇒ ?'b) has_sum ?x) (?A::?'a set); ((?g::?'a ⇒ ?'b) has_sum ?y) (?B::?'a set)⟧ ⟹ infsum ?f ?A = infsum ?g ?B›*) summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*))\n\nlemma infsum_not_exists:\n fixes f :: ‹'a ⇒ 'b::{comm_monoid_add, t2_space}›\n assumes ‹¬ f summable_on A›\n shows ‹infsum f A = 0›\n by (simp add: assms (*‹¬ f summable_on A›*) infsum_def (*‹infsum ?f ?A = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else 0)›*))\n\nlemma summable_iff_has_sum_infsum: \"f summable_on A ⟷ (f has_sum (infsum f A)) A\"\n using infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) summable_on_def (*‹(?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) summable_on (?A::?'a::type set) ≡ ∃x::?'b::{comm_monoid_add,topological_space}. (?f has_sum x) ?A›*) by blast\n\nlemma has_sum_infsum[simp]:\n assumes ‹f summable_on S›\n shows ‹(f has_sum (infsum f S)) S›\n using assms (*‹f summable_on S›*) summable_iff_has_sum_infsum (*‹(?f summable_on ?A) = (?f has_sum infsum ?f ?A) ?A›*) by blast\n\nlemma has_sum_cong_neutral:\n fixes f g :: ‹'a ⇒ 'b::{comm_monoid_add, topological_space}›\n assumes ‹⋀x. x∈T-S ⟹ g x = 0›\n assumes ‹⋀x. x∈S-T ⟹ f x = 0›\n assumes ‹⋀x. x∈S∩T ⟹ f x = g x›\n shows \"(f has_sum x) S ⟷ (g has_sum x) T\"\nproof (-)\n (*goal: ‹((f::'a ⇒ 'b) has_sum (x::'b)) (S::'a set) = ((g::'a ⇒ 'b) has_sum x) (T::'a set)›*)\n have \"eventually P (filtermap (sum f) (finite_subsets_at_top S))\n = eventually P (filtermap (sum g) (finite_subsets_at_top T))\" for P\n proof (standard)\n (*goals:\n 1. ‹eventually (P::'b ⇒ bool) (filtermap (sum (f::'a ⇒ 'b)) (finite_subsets_at_top (S::'a set))) ⟹ eventually P (filtermap (sum (g::'a ⇒ 'b)) (finite_subsets_at_top (T::'a set)))›\n 2. ‹eventually (P::'b ⇒ bool) (filtermap (sum (g::'a ⇒ 'b)) (finite_subsets_at_top (T::'a set))) ⟹ eventually P (filtermap (sum (f::'a ⇒ 'b)) (finite_subsets_at_top (S::'a set)))›*)\n assume \"eventually P (filtermap (sum f) (finite_subsets_at_top S))\" (*‹eventually (P::'b ⇒ bool) (filtermap (sum (f::'a ⇒ 'b)) (finite_subsets_at_top (S::'a set)))›*)\n then obtain F0 where \"finite F0\" and \"F0 ⊆ S\" and F0_P: \"⋀F. finite F ⟹ F ⊆ S ⟹ F ⊇ F0 ⟹ P (sum f F)\"\n (*goal: ‹(⋀F0. ⟦finite F0; F0 ⊆ S; ⋀F. ⟦finite F; F ⊆ S; F0 ⊆ F⟧ ⟹ P (sum f F)⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis (no_types, lifting) eventually_filtermap (*‹eventually ?P (filtermap ?f ?F) = (∀⇩F x in ?F. ?P (?f x))›*) eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n define F0' where \"F0' = F0 ∩ T\"\n have [simp]: \"finite F0'\" \"F0' ⊆ T\"\n (*goals:\n 1. ‹finite F0'›\n 2. ‹F0' ⊆ T›\n discuss goal 1*)\n apply (simp add: F0'_def (*‹F0' = F0 ∩ T›*) ‹finite F0›)\n (*discuss goal 2*)\n apply (simp add: F0'_def (*‹F0' = F0 ∩ T›*) ‹finite F0›)\n (*proven 2 subgoals*) .\n have \"P (sum g F)\" if \"finite F\" \"F ⊆ T\" \"F ⊇ F0'\" for F\n proof (-)\n (*goal: ‹P (sum g F)›*)\n have \"P (sum f ((F∩S) ∪ (F0∩S)))\"\n apply (intro F0_P (*‹⟦finite ?F; ?F ⊆ S; F0 ⊆ ?F⟧ ⟹ P (sum f ?F)›*))\n (*goals:\n 1. ‹finite (F ∩ S ∪ F0 ∩ S)›\n 2. ‹F ∩ S ∪ F0 ∩ S ⊆ S›\n 3. ‹F0 ⊆ F ∩ S ∪ F0 ∩ S›\n discuss goal 1*)\n apply ((use ‹F0 ⊆ S› ‹finite F0› that in auto)[1])\n (*discuss goal 2*)\n apply ((use ‹F0 ⊆ S› ‹finite F0› that in auto)[1])\n (*discuss goal 3*)\n apply ((use ‹F0 ⊆ S› ‹finite F0› that in auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹P (sum f (F ∩ S ∪ F0 ∩ S))›*) have \"sum f ((F∩S) ∪ (F0∩S)) = sum g F\"\n apply (intro sum.mono_neutral_cong (*‹⟦finite ?T; finite ?S; ⋀i. i ∈ ?T - ?S ⟹ ?h i = 0; ⋀i. i ∈ ?S - ?T ⟹ ?g i = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?S = sum ?h ?T›*))\n (*goals:\n 1. ‹finite F›\n 2. ‹finite (F ∩ S ∪ F0 ∩ S)›\n 3. ‹⋀i. i ∈ F - (F ∩ S ∪ F0 ∩ S) ⟹ g i = 0›\n 4. ‹⋀i. i ∈ F ∩ S ∪ F0 ∩ S - F ⟹ f i = 0›\n 5. ‹⋀x. x ∈ (F ∩ S ∪ F0 ∩ S) ∩ F ⟹ f x = g x›\n discuss goal 1*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 2*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 3*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 4*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 5*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*proven 5 subgoals*) .\n finally (*calculation: ‹P (sum g F)›*) show \"?thesis\"\n (*goal: ‹(P::'b::{comm_monoid_add,topological_space} ⇒ bool) (sum (g::'a::type ⇒ 'b::{comm_monoid_add,topological_space}) (F::'a::type set))›*) .\n qed\n with ‹F0' ⊆ T› (*‹(F0'::'a::type set) ⊆ (T::'a::type set)›*) ‹finite F0'› (*‹finite F0'›*) show \"eventually P (filtermap (sum g) (finite_subsets_at_top T))\"\n by (metis (no_types, lifting) eventually_filtermap (*‹eventually ?P (filtermap ?f ?F) = (∀⇩F x in ?F. ?P (?f x))›*) eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n next\n (*goal: ‹eventually P (filtermap (sum g) (finite_subsets_at_top T)) ⟹ eventually P (filtermap (sum f) (finite_subsets_at_top S))›*)\n assume \"eventually P (filtermap (sum g) (finite_subsets_at_top T))\" (*‹eventually (P::'b ⇒ bool) (filtermap (sum (g::'a ⇒ 'b)) (finite_subsets_at_top (T::'a set)))›*)\n then obtain F0 where \"finite F0\" and \"F0 ⊆ T\" and F0_P: \"⋀F. finite F ⟹ F ⊆ T ⟹ F ⊇ F0 ⟹ P (sum g F)\"\n (*goal: ‹(⋀F0. ⟦finite F0; F0 ⊆ T; ⋀F. ⟦finite F; F ⊆ T; F0 ⊆ F⟧ ⟹ P (sum g F)⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis (no_types, lifting) eventually_filtermap (*‹eventually (?P::?'a ⇒ bool) (filtermap (?f::?'b ⇒ ?'a) (?F::?'b filter)) = (∀⇩F x::?'b in ?F. ?P (?f x))›*) eventually_finite_subsets_at_top (*‹eventually (?P::?'a set ⇒ bool) (finite_subsets_at_top (?A::?'a set)) = (∃X::?'a set. finite X ∧ X ⊆ ?A ∧ (∀Y::?'a set. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n define F0' where \"F0' = F0 ∩ S\"\n have [simp]: \"finite F0'\" \"F0' ⊆ S\"\n (*goals:\n 1. ‹finite (F0'::'a::type set)›\n 2. ‹(F0'::'a::type set) ⊆ (S::'a::type set)›\n discuss goal 1*)\n apply (simp add: F0'_def (*‹F0' = F0 ∩ S›*) ‹finite F0›)\n (*discuss goal 2*)\n apply (simp add: F0'_def (*‹F0' = F0 ∩ S›*) ‹finite F0›)\n (*proven 2 subgoals*) .\n have \"P (sum f F)\" if \"finite F\" \"F ⊆ S\" \"F ⊇ F0'\" for F\n proof (-)\n (*goal: ‹P (sum f F)›*)\n have \"P (sum g ((F∩T) ∪ (F0∩T)))\"\n apply (intro F0_P (*‹⟦finite (?F::'a::type set); ?F ⊆ (T::'a::type set); (F0::'a::type set) ⊆ ?F⟧ ⟹ (P::'b::{comm_monoid_add,topological_space} ⇒ bool) (sum (g::'a::type ⇒ 'b::{comm_monoid_add,topological_space}) ?F)›*))\n (*goals:\n 1. ‹finite ((F::'a set) ∩ (T::'a set) ∪ (F0::'a set) ∩ T)›\n 2. ‹(F::'a set) ∩ (T::'a set) ∪ (F0::'a set) ∩ T ⊆ T›\n 3. ‹(F0::'a set) ⊆ (F::'a set) ∩ (T::'a set) ∪ F0 ∩ T›\n discuss goal 1*)\n apply ((use ‹F0 ⊆ T› ‹finite F0› that in auto)[1])\n (*discuss goal 2*)\n apply ((use ‹F0 ⊆ T› ‹finite F0› that in auto)[1])\n (*discuss goal 3*)\n apply ((use ‹F0 ⊆ T› ‹finite F0› that in auto)[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹P (sum g (F ∩ T ∪ F0 ∩ T))›*) have \"sum g ((F∩T) ∪ (F0∩T)) = sum f F\"\n apply (intro sum.mono_neutral_cong (*‹⟦finite ?T; finite ?S; ⋀i. i ∈ ?T - ?S ⟹ ?h i = 0; ⋀i. i ∈ ?S - ?T ⟹ ?g i = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?S = sum ?h ?T›*))\n (*goals:\n 1. ‹finite F›\n 2. ‹finite (F ∩ T ∪ F0 ∩ T)›\n 3. ‹⋀i. i ∈ F - (F ∩ T ∪ F0 ∩ T) ⟹ f i = 0›\n 4. ‹⋀i. i ∈ F ∩ T ∪ F0 ∩ T - F ⟹ g i = 0›\n 5. ‹⋀x. x ∈ (F ∩ T ∪ F0 ∩ T) ∩ F ⟹ g x = f x›\n discuss goal 1*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 2*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 3*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 4*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*discuss goal 5*)\n apply ((use that ‹finite F0› F0'_def assms in auto)[1])\n (*proven 5 subgoals*) .\n finally (*calculation: ‹P (sum f F)›*) show \"?thesis\"\n (*goal: ‹P (sum f F)›*) .\n qed\n with ‹F0' ⊆ S› (*‹(F0'::'a set) ⊆ (S::'a set)›*) ‹finite F0'› (*‹finite F0'›*) show \"eventually P (filtermap (sum f) (finite_subsets_at_top S))\"\n by (metis (no_types, lifting) eventually_filtermap (*‹eventually ?P (filtermap ?f ?F) = (∀⇩F x in ?F. ?P (?f x))›*) eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n qed\n then have tendsto_x: \"(sum f ⤏ x) (finite_subsets_at_top S) ⟷ (sum g ⤏ x) (finite_subsets_at_top T)\" for x\n by (simp add: le_filter_def (*‹(?F ≤ ?F') = (∀P. eventually P ?F' ⟶ eventually P ?F)›*) filterlim_def (*‹filterlim ?f ?F2.0 ?F1.0 = (filtermap ?f ?F1.0 ≤ ?F2.0)›*))\n then show \"?thesis\"\n (*goal: ‹(f has_sum x) S = (g has_sum x) T›*)\n by (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\nqed\n\nlemma summable_on_cong_neutral: \n fixes f g :: ‹'a ⇒ 'b::{comm_monoid_add, topological_space}›\n assumes ‹⋀x. x∈T-S ⟹ g x = 0›\n assumes ‹⋀x. x∈S-T ⟹ f x = 0›\n assumes ‹⋀x. x∈S∩T ⟹ f x = g x›\n shows \"f summable_on S ⟷ g summable_on T\"\n using has_sum_cong_neutral[of T S g f, OF assms] (*‹⟦⋀x. x ∈ T - S ⟹ x ∈ T - S; ⋀x. x ∈ S - T ⟹ x ∈ S - T; ⋀x. x ∈ S ∩ T ⟹ x ∈ S ∩ T⟧ ⟹ (f has_sum ?x) S = (g has_sum ?x) T›*) by (simp add: summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma infsum_cong_neutral: \n fixes f g :: ‹'a ⇒ 'b::{comm_monoid_add, t2_space}›\n assumes ‹⋀x. x∈T-S ⟹ g x = 0›\n assumes ‹⋀x. x∈S-T ⟹ f x = 0›\n assumes ‹⋀x. x∈S∩T ⟹ f x = g x›\n shows ‹infsum f S = infsum g T›\n by (smt (verit, best) assms (*‹?x ∈ T - S ⟹ g ?x = 0› ‹?x ∈ S - T ⟹ f ?x = 0› ‹?x ∈ S ∩ T ⟹ f ?x = g ?x›*) has_sum_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f has_sum ?x) ?S = (?g has_sum ?x) ?T›*) infsum_eqI' (*‹(⋀x. (?f has_sum x) ?A = (?g has_sum x) ?B) ⟹ infsum ?f ?A = infsum ?g ?B›*))\n\nlemma has_sum_cong: \n assumes \"⋀x. x∈A ⟹ f x = g x\"\n shows \"(f has_sum x) A ⟷ (g has_sum x) A\"\n using assms (*‹?x ∈ A ⟹ f ?x = g ?x›*) apply (intro has_sum_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f has_sum ?x) ?S = (?g has_sum ?x) ?T›*))\n (*goals:\n 1. ‹⋀x. ⟦⋀x. x ∈ A ⟹ f x = g x; x ∈ A - A⟧ ⟹ g x = 0›\n 2. ‹⋀x. ⟦⋀x. x ∈ A ⟹ f x = g x; x ∈ A - A⟧ ⟹ f x = 0›\n 3. ‹⋀x. ⟦⋀x. x ∈ A ⟹ f x = g x; x ∈ A ∩ A⟧ ⟹ 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\nlemma summable_on_cong:\n assumes \"⋀x. x∈A ⟹ f x = g x\"\n shows \"f summable_on A ⟷ g summable_on A\"\n by (metis assms (*‹?x ∈ A ⟹ f ?x = g ?x›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f has_sum ?x) ?A = (?g has_sum ?x) ?A›*))\n\nlemma infsum_cong:\n assumes \"⋀x. x∈A ⟹ f x = g x\"\n shows \"infsum f A = infsum g A\"\n using assms (*‹?x ∈ A ⟹ f ?x = g ?x›*) infsum_eqI' (*‹(⋀x::?'b. ((?f::?'a ⇒ ?'b) has_sum x) (?A::?'a set) = ((?g::?'a ⇒ ?'b) has_sum x) (?B::?'a set)) ⟹ infsum ?f ?A = infsum ?g ?B›*) has_sum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f has_sum ?x) ?A = (?g has_sum ?x) ?A›*) by blast\n\nlemma summable_on_cofin_subset:\n fixes f :: \"'a ⇒ 'b::topological_ab_group_add\"\n assumes \"f summable_on A\" and [simp]: \"finite F\"\n shows \"f summable_on (A - F)\"\nproof (-)\n (*goal: ‹f summable_on A - F›*)\n from assms(1) (*‹f summable_on A›*) obtain x where lim_f: \"(sum f ⤏ x) (finite_subsets_at_top A)\"\n (*goal: ‹(⋀x::'b. (sum (f::'a ⇒ 'b) ⤏ x) (finite_subsets_at_top (A::'a set)) ⟹ thesis::bool) ⟹ thesis›*)\n unfolding summable_on_def has_sum_def\n (*goal: ‹(⋀x. (sum f ⤏ x) (finite_subsets_at_top A) ⟹ thesis) ⟹ thesis›*)\n by auto\n define F' where \"F' = F∩A\"\n with assms (*‹f summable_on A› ‹finite F›*) have \"finite F'\" and \"A-F = A-F'\"\n apply -\n (*goals:\n 1. ‹⟦f summable_on A; finite F; F' = F ∩ A⟧ ⟹ finite F'›\n 2. ‹⟦f summable_on A; finite F; F' = F ∩ A⟧ ⟹ A - F = A - F'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have \"filtermap ((∪)F') (finite_subsets_at_top (A-F))\n ≤ finite_subsets_at_top A\"\n proof (rule filter_leI (*‹(⋀P. eventually P ?F' ⟹ eventually P ?F) ⟹ ?F ≤ ?F'›*))\n (*goal: ‹⋀P. eventually P (finite_subsets_at_top A) ⟹ eventually P (filtermap ((∪) F') (finite_subsets_at_top (A - F)))›*)\n fix P\n assume \"eventually P (finite_subsets_at_top A)\" (*‹eventually (P::'a set ⇒ bool) (finite_subsets_at_top (A::'a set))›*)\n then obtain X where [simp]: \"finite X\" and XA: \"X ⊆ A\" and P: \"∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ P Y\"\n (*goal: ‹(⋀X::'a::type set. ⟦finite X; X ⊆ (A::'a::type set); ∀Y::'a::type set. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ (P::'a::type set ⇒ bool) Y⟧ ⟹ thesis::bool) ⟹ thesis›*)\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹(⋀X. ⟦finite X; X ⊆ A; ∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ P Y⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n define X' where \"X' = X-F\"\n hence [simp]: \"finite X'\" and [simp]: \"X' ⊆ A-F\"\n using XA (*‹X ⊆ A›*) apply -\n (*goals:\n 1. ‹⟦X' = X - F; X ⊆ A⟧ ⟹ finite X'›\n 2. ‹⟦X' = X - F; X ⊆ A⟧ ⟹ X' ⊆ A - F›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence \"finite Y ∧ X' ⊆ Y ∧ Y ⊆ A - F ⟶ P (F' ∪ Y)\" for Y\n using P (*‹∀Y::'a set. finite Y ∧ (X::'a set) ⊆ Y ∧ Y ⊆ (A::'a set) ⟶ (P::'a set ⇒ bool) Y›*) XA (*‹X ⊆ A›*) unfolding X'_def\n (*goal: ‹finite Y ∧ X - F ⊆ Y ∧ Y ⊆ A - F ⟶ P (F' ∪ Y)›*)\n using F'_def (*‹F' = F ∩ A›*) ‹finite F'› (*‹finite F'›*) by blast\n thus \"eventually P (filtermap ((∪) F') (finite_subsets_at_top (A - F)))\"\n unfolding eventually_filtermap eventually_finite_subsets_at_top\n (*goal: ‹∃X. finite X ∧ X ⊆ A - F ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A - F ⟶ P (F' ∪ Y))›*)\n apply (rule_tac x=X' in exI (*‹?P ?x ⟹ ∃x. ?P x›*))\n (*goal: ‹∃X. finite X ∧ X ⊆ A - F ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A - F ⟶ P (F' ∪ Y))›*)\n by simp\n qed\n with lim_f (*‹(sum f ⤏ x) (finite_subsets_at_top A)›*) have \"(sum f ⤏ x) (filtermap ((∪)F') (finite_subsets_at_top (A-F)))\"\n using tendsto_mono (*‹⟦(?F::?'b filter) ≤ (?F'::?'b filter); ((?f::?'b ⇒ ?'a) ⤏ (?l::?'a)) ?F'⟧ ⟹ (?f ⤏ ?l) ?F›*) by blast\n have \"((λG. sum f (F' ∪ G)) ⤏ x) (finite_subsets_at_top (A - F))\" if \"((sum f ∘ (∪) F') ⤏ x) (finite_subsets_at_top (A - F))\"\n using that (*‹((sum f ∘ (∪) F') ⤏ x) (finite_subsets_at_top (A - F))›*) unfolding o_def\n (*goal: ‹((λG. sum f (F' ∪ G)) ⤏ x) (finite_subsets_at_top (A - F))›*)\n by auto\n hence \"((λG. sum f (F' ∪ G)) ⤏ x) (finite_subsets_at_top (A-F))\"\n using tendsto_compose_filtermap[symmetric] (*‹(?g ⤏ ?T) (filtermap ?f ?F) = ((?g ∘ ?f) ⤏ ?T) ?F›*) by (simp add: ‹(sum f ⤏ x) (filtermap ((∪) F') (finite_subsets_at_top (A - F)))› tendsto_compose_filtermap (*‹((?g ∘ ?f) ⤏ ?T) ?F = (?g ⤏ ?T) (filtermap ?f ?F)›*))\n have \"∀Y. finite Y ∧ Y ⊆ A - F ⟶ sum f (F' ∪ Y) = sum f F' + sum f Y\"\n by (metis Diff_disjoint (*‹?A ∩ (?B - ?A) = {}›*) Int_Diff (*‹?A ∩ ?B - ?C = ?A ∩ (?B - ?C)›*) ‹A - F = A - F'› ‹finite F'› inf.orderE (*‹⟦?a ≤ ?b; ?a = inf ?a ?b ⟹ ?thesis⟧ ⟹ ?thesis›*) sum.union_disjoint (*‹⟦finite ?A; finite ?B; ?A ∩ ?B = {}⟧ ⟹ sum ?g (?A ∪ ?B) = sum ?g ?A + sum ?g ?B›*))\n hence \"∀⇩F x in finite_subsets_at_top (A - F). sum f (F' ∪ x) = sum f F' + sum f x\"\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹∃X. finite X ∧ X ⊆ A - F ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A - F ⟶ sum f (F' ∪ Y) = sum f F' + sum f Y)›*)\n using exI[where x = \"{}\"] (*‹(?P::?'c1 set ⇒ bool) {} ⟹ ∃x::?'c1 set. ?P x›*) by (simp add: ‹⋀P. P {} ⟹ ∃x. P x›)\n hence \"((λG. sum f F' + sum f G) ⤏ x) (finite_subsets_at_top (A-F))\"\n using tendsto_cong[THEN iffD1, rotated] (*‹⟦(?f1 ⤏ ?c1) ?F1; ∀⇩F x in ?F1. ?f1 x = ?g1 x⟧ ⟹ (?g1 ⤏ ?c1) ?F1›*) ‹((λG. sum f (F' ∪ G)) ⤏ x) (finite_subsets_at_top (A - F))› (*‹((λG. sum f (F' ∪ G)) ⤏ x) (finite_subsets_at_top (A - F))›*) by fastforce\n hence \"((λG. sum f F' + sum f G) ⤏ sum f F' + (x-sum f F')) (finite_subsets_at_top (A-F))\"\n by simp\n hence \"(sum f ⤏ x - sum f F') (finite_subsets_at_top (A-F))\"\n using tendsto_add_const_iff (*‹((λx::?'b::type. (?c::?'a::topological_group_add) + (?f::?'b::type ⇒ ?'a::topological_group_add) x) ⤏ ?c + (?d::?'a::topological_group_add)) (?F::?'b::type filter) = (?f ⤏ ?d) ?F›*) by blast\n thus \"f summable_on (A - F)\"\n unfolding summable_on_def has_sum_def\n (*goal: ‹∃x. (sum f ⤏ x) (finite_subsets_at_top (A - F))›*)\n by auto\nqed\n\nlemma\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add}\"\n assumes ‹(f has_sum b) B› and ‹(f has_sum a) A› and AB: \"A ⊆ B\"\n shows has_sum_Diff: \"(f has_sum (b - a)) (B - A)\"\nproof (-)\n (*goal: ‹(f has_sum b - a) (B - A)›*)\n have finite_subsets1: \"finite_subsets_at_top (B - A) ≤ filtermap (λF. F - A) (finite_subsets_at_top B)\"\n proof (rule filter_leI (*‹(⋀P. eventually P ?F' ⟹ eventually P ?F) ⟹ ?F ≤ ?F'›*))\n (*goal: ‹⋀P. eventually P (filtermap (λF. F - A) (finite_subsets_at_top B)) ⟹ eventually P (finite_subsets_at_top (B - A))›*)\n fix P\n assume \"eventually P (filtermap (λF. F - A) (finite_subsets_at_top B))\" (*‹eventually (P::'a set ⇒ bool) (filtermap (λF::'a set. F - (A::'a set)) (finite_subsets_at_top (B::'a set)))›*)\n then obtain X where \"finite X\" and \"X ⊆ B\" and P: \"finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ P (Y - A)\" for Y\n (*goal: ‹(⋀X. ⟦finite X; X ⊆ B; ⋀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ P (Y - A)⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding eventually_filtermap eventually_finite_subsets_at_top\n (*goal: ‹(⋀X. ⟦finite X; X ⊆ B; ⋀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ P (Y - A)⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n hence \"finite (X-A)\" and \"X-A ⊆ B - A\"\n apply -\n (*goals:\n 1. ‹⟦finite X; X ⊆ B; ⋀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ P (Y - A)⟧ ⟹ finite (X - A)›\n 2. ‹⟦finite X; X ⊆ B; ⋀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ P (Y - A)⟧ ⟹ X - A ⊆ B - A›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n moreover have \"finite Y ∧ X-A ⊆ Y ∧ Y ⊆ B - A ⟶ P Y\" for Y\n using P[where Y = \"Y∪X\"] (*‹finite (Y ∪ X) ∧ X ⊆ Y ∪ X ∧ Y ∪ X ⊆ B ⟶ P (Y ∪ X - A)›*) ‹finite X› (*‹finite X›*) ‹X ⊆ B› (*‹X ⊆ B›*) by (metis Diff_subset (*‹?A - ?B ⊆ ?A›*) Int_Diff (*‹?A ∩ ?B - ?C = ?A ∩ (?B - ?C)›*) Un_Diff (*‹?A ∪ ?B - ?C = ?A - ?C ∪ (?B - ?C)›*) finite_Un (*‹finite (?F ∪ ?G) = (finite ?F ∧ finite ?G)›*) inf.orderE (*‹⟦?a ≤ ?b; ?a = inf ?a ?b ⟹ ?thesis⟧ ⟹ ?thesis›*) le_sup_iff (*‹(sup ?x ?y ≤ ?z) = (?x ≤ ?z ∧ ?y ≤ ?z)›*) sup.orderE (*‹⟦?b ≤ ?a; ?a = sup ?a ?b ⟹ ?thesis⟧ ⟹ ?thesis›*) sup_ge2 (*‹?y ≤ sup ?x ?y›*))\n ultimately show \"eventually P (finite_subsets_at_top (B - A))\"\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹∃X. finite X ∧ X ⊆ B - A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B - A ⟶ P Y)›*)\n by meson\n qed\n have finite_subsets2: \"filtermap (λF. F ∩ A) (finite_subsets_at_top B) ≤ finite_subsets_at_top A\"\n apply (rule filter_leI (*‹(⋀P. eventually P ?F' ⟹ eventually P ?F) ⟹ ?F ≤ ?F'›*))\n (*goal: ‹filtermap (λF. F ∩ A) (finite_subsets_at_top B) ≤ finite_subsets_at_top A›*)\n using assms (*‹((f::'a ⇒ 'b) has_sum (b::'b)) (B::'a set)› ‹(f has_sum a) A› ‹A ⊆ B›*) unfolding eventually_filtermap eventually_finite_subsets_at_top\n (*goal: ‹⋀P. ∃X. finite X ∧ X ⊆ A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ P Y) ⟹ ∃X. finite X ∧ X ⊆ B ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ P (Y ∩ A))›*)\n by (metis Int_subset_iff (*‹(?C ⊆ ?A ∩ ?B) = (?C ⊆ ?A ∧ ?C ⊆ ?B)›*) finite_Int (*‹finite ?F ∨ finite ?G ⟹ finite (?F ∩ ?G)›*) inf_le2 (*‹inf ?x ?y ≤ ?y›*) subset_trans (*‹⟦?A ⊆ ?B; ?B ⊆ ?C⟧ ⟹ ?A ⊆ ?C›*))\n from assms(1) (*‹(f has_sum b) B›*) have limB: \"(sum f ⤏ b) (finite_subsets_at_top B)\"\n using has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by auto\n from assms(2) (*‹((f::'a::type ⇒ 'b::topological_ab_group_add) has_sum (a::'b::topological_ab_group_add)) (A::'a::type set)›*) have limA: \"(sum f ⤏ a) (finite_subsets_at_top A)\"\n using has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\n have \"((λF. sum f (F∩A)) ⤏ a) (finite_subsets_at_top B)\"\n proof (subst asm_rl [of \"(λF. sum f (F∩A)) = sum f ∘ (λF. F∩A)\"] (*‹(λF. sum f (F ∩ A)) = sum f ∘ (λF. F ∩ A) ⟹ (λF. sum f (F ∩ A)) = sum f ∘ (λF. F ∩ A)›*))\n (*goals:\n 1. ‹(λF. sum f (F ∩ A)) = sum f ∘ (λF. F ∩ A)›\n 2. ‹((sum f ∘ (λF. F ∩ A)) ⤏ a) (finite_subsets_at_top B)›*)\n show \"(λF. sum f (F ∩ A)) = sum f ∘ (λF. F ∩ A)\"\n unfolding o_def\n (*goal: ‹(λF. sum f (F ∩ A)) = (λx. sum f (x ∩ A))›*)\n by auto\n show \"((sum f ∘ (λF. F ∩ A)) ⤏ a) (finite_subsets_at_top B)\"\n unfolding o_def\n (*goal: ‹((λx::'a::type set. sum (f::'a::type ⇒ 'b::topological_ab_group_add) (x ∩ (A::'a::type set))) ⤏ (a::'b::topological_ab_group_add)) (finite_subsets_at_top (B::'a::type set))›*)\n using tendsto_compose_filtermap (*‹((?g ∘ ?f) ⤏ ?T) ?F = (?g ⤏ ?T) (filtermap ?f ?F)›*) finite_subsets2 (*‹filtermap (λF::'a set. F ∩ (A::'a set)) (finite_subsets_at_top (B::'a set)) ≤ finite_subsets_at_top A›*) limA (*‹(sum f ⤏ a) (finite_subsets_at_top A)›*) tendsto_mono (*‹⟦?F ≤ ?F'; (?f ⤏ ?l) ?F'⟧ ⟹ (?f ⤏ ?l) ?F›*) ‹(λF. sum f (F ∩ A)) = sum f ∘ (λF. F ∩ A)› (*‹(λF. sum f (F ∩ A)) = sum f ∘ (λF. F ∩ A)›*) by fastforce\n qed\n with limB (*‹(sum f ⤏ b) (finite_subsets_at_top B)›*) have \"((λF. sum f F - sum f (F∩A)) ⤏ b - a) (finite_subsets_at_top B)\"\n using tendsto_diff (*‹⟦(?f ⤏ ?a) ?F; (?g ⤏ ?b) ?F⟧ ⟹ ((λx. ?f x - ?g x) ⤏ ?a - ?b) ?F›*) by blast\n have \"sum f X - sum f (X ∩ A) = sum f (X - A)\" if \"finite X\" and \"X ⊆ B\" for X :: \"'a set\"\n using that (*‹finite X› ‹X ⊆ B›*) by (metis add_diff_cancel_left' (*‹?a + ?b - ?a = ?b›*) sum.Int_Diff (*‹finite ?A ⟹ sum ?g ?A = sum ?g (?A ∩ ?B) + sum ?g (?A - ?B)›*))\n hence \"∀⇩F x in finite_subsets_at_top B. sum f x - sum f (x ∩ A) = sum f (x - A)\"\n apply (rule eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*))\n (*goals:\n 1. ‹⋀x. ⟦finite x; x ⊆ B⟧ ⟹ finite x›\n 2. ‹⋀x. ⟦finite x; x ⊆ B⟧ ⟹ x ⊆ B›\n discuss goal 1*)\n apply ((assumption)[1])\n (*discuss goal 2*) .\n (*proven 2 subgoals*)\n hence \"((λF. sum f (F-A)) ⤏ b - a) (finite_subsets_at_top B)\"\n using tendsto_cong[THEN iffD1, rotated] (*‹⟦(?f1 ⤏ ?c1) ?F1; ∀⇩F x in ?F1. ?f1 x = ?g1 x⟧ ⟹ (?g1 ⤏ ?c1) ?F1›*) ‹((λF. sum f F - sum f (F ∩ A)) ⤏ b - a) (finite_subsets_at_top B)› (*‹((λF::'a::type set. sum (f::'a::type ⇒ 'b::topological_ab_group_add) F - sum f (F ∩ (A::'a::type set))) ⤏ (b::'b::topological_ab_group_add) - (a::'b::topological_ab_group_add)) (finite_subsets_at_top (B::'a::type set))›*) by fastforce\n hence \"(sum f ⤏ b - a) (filtermap (λF. F-A) (finite_subsets_at_top B))\"\n apply (subst tendsto_compose_filtermap[symmetric] (*‹(?g ⤏ ?T) (filtermap ?f ?F) = ((?g ∘ ?f) ⤏ ?T) ?F›*))\n (*goal: ‹(sum f ⤏ b - a) (filtermap (λF. F - A) (finite_subsets_at_top B))›*)\n by (simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n thus \"?thesis\"\n (*goal: ‹((f::'a ⇒ 'b) has_sum (b::'b) - (a::'b)) ((B::'a set) - (A::'a set))›*)\n using finite_subsets1 (*‹finite_subsets_at_top (B - A) ≤ filtermap (λF. F - A) (finite_subsets_at_top B)›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) tendsto_mono (*‹⟦?F ≤ ?F'; (?f ⤏ ?l) ?F'⟧ ⟹ (?f ⤏ ?l) ?F›*) by blast\nqed\n\n\nlemma\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add}\"\n assumes \"f summable_on B\" and \"f summable_on A\" and \"A ⊆ B\"\n shows summable_on_Diff: \"f summable_on (B-A)\"\n by (meson assms (*‹f summable_on B› ‹f summable_on A› ‹A ⊆ B›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_Diff (*‹⟦(?f has_sum ?b) ?B; (?f has_sum ?a) ?A; ?A ⊆ ?B⟧ ⟹ (?f has_sum ?b - ?a) (?B - ?A)›*))\n\nlemma\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add,t2_space}\"\n assumes \"f summable_on B\" and \"f summable_on A\" and AB: \"A ⊆ B\"\n shows infsum_Diff: \"infsum f (B - A) = infsum f B - infsum f A\"\n by (metis AB (*‹A ⊆ B›*) assms (*‹f summable_on B› ‹f summable_on A› ‹A ⊆ B›*) has_sum_Diff (*‹⟦(?f has_sum ?b) ?B; (?f has_sum ?a) ?A; ?A ⊆ ?B⟧ ⟹ (?f has_sum ?b - ?a) (?B - ?A)›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma has_sum_mono_neutral:\n fixes f :: \"'a⇒'b::{ordered_comm_monoid_add,linorder_topology}\"\n (* Does this really require a linorder topology? (Instead of order topology.) *)\n assumes ‹(f has_sum a) A› and \"(g has_sum b) B\"\n assumes ‹⋀x. x ∈ A∩B ⟹ f x ≤ g x›\n assumes ‹⋀x. x ∈ A-B ⟹ f x ≤ 0›\n assumes ‹⋀x. x ∈ B-A ⟹ g x ≥ 0›\n shows \"a ≤ b\"\nproof (-)\n (*goal: ‹a ≤ b›*)\n define f' and g' where \"f' x = (if x ∈ A then f x else 0)\" and \"g' x = (if x ∈ B then g x else 0)\" for x\n have [simp]: \"f summable_on A\" \"g summable_on B\"\n using assms(1,2) (*‹((f::'a ⇒ 'b) has_sum (a::'b)) (A::'a set)› ‹(g has_sum b) B›*) summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*) apply -\n (*goals:\n 1. ‹⟦(f has_sum a) A; (g has_sum b) B; ⋀f A. f summable_on A ≡ ∃x. (f has_sum x) A⟧ ⟹ f summable_on A›\n 2. ‹⟦(f has_sum a) A; (g has_sum b) B; ⋀f A. f summable_on A ≡ ∃x. (f has_sum x) A⟧ ⟹ g summable_on B›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have \"(f' has_sum a) (A∪B)\"\n by (smt (verit, best) DiffE (*‹⟦(?c::?'a::type) ∈ (?A::?'a::type set) - (?B::?'a::type set); ⟦?c ∈ ?A; ?c ∉ ?B⟧ ⟹ ?P::bool⟧ ⟹ ?P›*) IntE (*‹⟦(?c::?'a::type) ∈ (?A::?'a::type set) ∩ (?B::?'a::type set); ⟦?c ∈ ?A; ?c ∈ ?B⟧ ⟹ ?P::bool⟧ ⟹ ?P›*) Un_iff (*‹((?c::?'a::type) ∈ (?A::?'a::type set) ∪ (?B::?'a::type set)) = (?c ∈ ?A ∨ ?c ∈ ?B)›*) f'_def (*‹(f'::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) (?x::'a::type) = (if ?x ∈ (A::'a::type set) then (f::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) ?x else (0::'b::{ordered_comm_monoid_add,linorder_topology}))›*) assms( (*‹((f::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) has_sum (a::'b::{ordered_comm_monoid_add,linorder_topology})) (A::'a::type set)›*) 1) has_sum_cong_neutral (*‹⟦⋀x::?'a::type. x ∈ (?T::?'a::type set) - (?S::?'a::type set) ⟹ (?g::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) x = (0::?'b::{comm_monoid_add,topological_space}); ⋀x::?'a::type. x ∈ ?S - ?T ⟹ (?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) x = (0::?'b::{comm_monoid_add,topological_space}); ⋀x::?'a::type. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f has_sum (?x::?'b::{comm_monoid_add,topological_space})) ?S = (?g has_sum ?x) ?T›*))\n then have f'_lim: \"(sum f' ⤏ a) (finite_subsets_at_top (A∪B))\"\n by (meson has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n have \"(g' has_sum b) (A∪B)\"\n by (smt (verit, best) DiffD1 (*‹?c ∈ ?A - ?B ⟹ ?c ∈ ?A›*) DiffD2 (*‹⟦?c ∈ ?A - ?B; ?c ∈ ?B⟧ ⟹ ?P›*) IntE (*‹⟦?c ∈ ?A ∩ ?B; ⟦?c ∈ ?A; ?c ∈ ?B⟧ ⟹ ?P⟧ ⟹ ?P›*) UnCI (*‹(?c ∉ ?B ⟹ ?c ∈ ?A) ⟹ ?c ∈ ?A ∪ ?B›*) g'_def (*‹g' ?x = (if ?x ∈ B then g ?x else 0)›*) assms( (*‹(g has_sum b) B›*) 2) has_sum_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f has_sum ?x) ?S = (?g has_sum ?x) ?T›*))\n then have g'_lim: \"(sum g' ⤏ b) (finite_subsets_at_top (A∪B))\"\n using has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\n have \"⋀X i. ⟦X ⊆ A ∪ B; i ∈ X⟧ ⟹ f' i ≤ g' i\"\n using assms (*‹(f has_sum a) A› ‹(g has_sum b) B› ‹(?x::'a) ∈ (A::'a set) ∩ (B::'a set) ⟹ (f::'a ⇒ 'b) ?x ≤ (g::'a ⇒ 'b) ?x› ‹?x ∈ A - B ⟹ f ?x ≤ 0› ‹?x ∈ B - A ⟹ 0 ≤ g ?x›*) by (auto simp: f'_def (*‹(f'::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) (?x::'a::type) = (if ?x ∈ (A::'a::type set) then (f::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) ?x else (0::'b::{ordered_comm_monoid_add,linorder_topology}))›*) g'_def (*‹(g'::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) (?x::'a::type) = (if ?x ∈ (B::'a::type set) then (g::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) ?x else (0::'b::{ordered_comm_monoid_add,linorder_topology}))›*))\n then have \"∀⇩F x in finite_subsets_at_top (A ∪ B). sum f' x ≤ sum g' x\"\n by (intro eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*) sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*))\n then show \"?thesis\"\n (*goal: ‹a ≤ b›*)\n using f'_lim (*‹(sum (f'::'a ⇒ 'b) ⤏ (a::'b)) (finite_subsets_at_top ((A::'a set) ∪ (B::'a set)))›*) finite_subsets_at_top_neq_bot (*‹finite_subsets_at_top (?A::?'a set) ≠ bot›*) g'_lim (*‹(sum g' ⤏ b) (finite_subsets_at_top (A ∪ B))›*) tendsto_le (*‹⟦?F ≠ bot; (?f ⤏ ?x) ?F; (?g ⤏ ?y) ?F; ∀⇩F x in ?F. ?g x ≤ ?f x⟧ ⟹ ?y ≤ ?x›*) by blast\nqed\n\nlemma infsum_mono_neutral:\n fixes f :: \"'a⇒'b::{ordered_comm_monoid_add,linorder_topology}\"\n assumes \"f summable_on A\" and \"g summable_on B\"\n assumes ‹⋀x. x ∈ A∩B ⟹ f x ≤ g x›\n assumes ‹⋀x. x ∈ A-B ⟹ f x ≤ 0›\n assumes ‹⋀x. x ∈ B-A ⟹ g x ≥ 0›\n shows \"infsum f A ≤ infsum g B\"\n by (smt (verit, best) assms (*‹f summable_on A› ‹g summable_on B› ‹?x ∈ A ∩ B ⟹ f ?x ≤ g ?x› ‹?x ∈ A - B ⟹ f ?x ≤ 0› ‹?x ∈ B - A ⟹ 0 ≤ g ?x›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_mono_neutral (*‹⟦(?f has_sum ?a) ?A; (?g has_sum ?b) ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ ?a ≤ ?b›*))\n\nlemma has_sum_mono:\n fixes f :: \"'a⇒'b::{ordered_comm_monoid_add,linorder_topology}\"\n assumes \"(f has_sum x) A\" and \"(g has_sum y) A\"\n assumes ‹⋀x. x ∈ A ⟹ f x ≤ g x›\n shows \"x ≤ y\"\n using assms (*‹(f has_sum x) A› ‹((g::'a::type ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}) has_sum (y::'b::{ordered_comm_monoid_add,linorder_topology})) (A::'a::type set)› ‹?x ∈ A ⟹ f ?x ≤ g ?x›*) has_sum_mono_neutral (*‹⟦((?f::?'a::type ⇒ ?'b::{ordered_comm_monoid_add,linorder_topology}) has_sum (?a::?'b::{ordered_comm_monoid_add,linorder_topology})) (?A::?'a::type set); ((?g::?'a::type ⇒ ?'b::{ordered_comm_monoid_add,linorder_topology}) has_sum (?b::?'b::{ordered_comm_monoid_add,linorder_topology})) (?B::?'a::type set); ⋀x::?'a::type. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x::?'a::type. x ∈ ?A - ?B ⟹ ?f x ≤ (0::?'b::{ordered_comm_monoid_add,linorder_topology}); ⋀x::?'a::type. x ∈ ?B - ?A ⟹ (0::?'b::{ordered_comm_monoid_add,linorder_topology}) ≤ ?g x⟧ ⟹ ?a ≤ ?b›*) by force\n\nlemma infsum_mono:\n fixes f :: \"'a⇒'b::{ordered_comm_monoid_add,linorder_topology}\"\n assumes \"f summable_on A\" and \"g summable_on A\"\n assumes ‹⋀x. x ∈ A ⟹ f x ≤ g x›\n shows \"infsum f A ≤ infsum g A\"\n by (meson assms (*‹f summable_on A› ‹g summable_on A› ‹?x ∈ A ⟹ f ?x ≤ g ?x›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_mono (*‹⟦(?f has_sum ?x) ?A; (?g has_sum ?y) ?A; ⋀x. x ∈ ?A ⟹ ?f x ≤ ?g x⟧ ⟹ ?x ≤ ?y›*))\n\nlemma has_sum_finite[simp]:\n assumes \"finite F\"\n shows \"(f has_sum (sum f F)) F\"\n using assms (*‹finite F›*) by (auto intro: tendsto_Lim (*‹⟦(?net::?'a filter) ≠ bot; ((?f::?'a ⇒ ?'b) ⤏ (?l::?'b)) ?net⟧ ⟹ Lim ?net ?f = ?l›*) simp: finite_subsets_at_top_finite (*‹finite (?A::?'a set) ⟹ finite_subsets_at_top ?A = principal {?A}›*) infsum_def (*‹infsum (?f::?'a ⇒ ?'b) (?A::?'a set) = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else (0::?'b))›*) has_sum_def (*‹((?f::?'a ⇒ ?'b) has_sum (?x::?'b)) (?A::?'a set) ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) principal_eq_bot_iff (*‹(principal (?X::?'a set) = bot) = (?X = {})›*))\n\nlemma summable_on_finite[simp]:\n fixes f :: ‹'a ⇒ 'b::{comm_monoid_add,topological_space}›\n assumes \"finite F\"\n shows \"f summable_on F\"\n using assms (*‹finite F›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_finite (*‹finite ?F ⟹ (?f has_sum sum ?f ?F) ?F›*) by blast\n\nlemma infsum_finite[simp]:\n assumes \"finite F\"\n shows \"infsum f F = sum f F\"\n by (simp add: assms (*‹finite F›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*))\n\nlemma has_sum_finite_approximation:\n fixes f :: \"'a ⇒ 'b::{comm_monoid_add,metric_space}\"\n assumes \"(f has_sum x) A\" and \"ε > 0\"\n shows \"∃F. finite F ∧ F ⊆ A ∧ dist (sum f F) x ≤ ε\"\nproof (-)\n (*goal: ‹∃F. finite F ∧ F ⊆ A ∧ dist (sum f F) x ≤ ε›*)\n have \"(sum f ⤏ x) (finite_subsets_at_top A)\"\n by (meson assms( (*‹(f has_sum x) A›*) 1) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n hence \"*\": \"∀⇩F F in (finite_subsets_at_top A). dist (sum f F) x < ε\"\n using assms(2) (*‹0 < ε›*) by (rule tendstoD (*‹⟦(?f ⤏ ?l) ?F; 0 < ?e⟧ ⟹ ∀⇩F x in ?F. dist (?f x) ?l < ?e›*))\n thus \"?thesis\"\n (*goal: ‹∃F. finite F ∧ F ⊆ A ∧ dist (sum f F) x ≤ ε›*)\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹∃F::'a::type set. finite F ∧ F ⊆ (A::'a::type set) ∧ dist (sum (f::'a::type ⇒ 'b::{comm_monoid_add,metric_space}) F) (x::'b::{comm_monoid_add,metric_space}) ≤ (ε::real)›*)\n by fastforce\nqed\n\nlemma infsum_finite_approximation:\n fixes f :: \"'a ⇒ 'b::{comm_monoid_add,metric_space}\"\n assumes \"f summable_on A\" and \"ε > 0\"\n shows \"∃F. finite F ∧ F ⊆ A ∧ dist (sum f F) (infsum f A) ≤ ε\"\nproof (-)\n (*goal: ‹∃F. finite F ∧ F ⊆ A ∧ dist (sum f F) (infsum f A) ≤ ε›*)\n from assms (*‹f summable_on A› ‹0 < ε›*) have \"(f has_sum (infsum f A)) A\"\n by (simp add: summable_iff_has_sum_infsum (*‹(?f summable_on ?A) = (?f has_sum infsum ?f ?A) ?A›*))\n from this (*‹(f has_sum infsum f A) A›*) ‹ε > 0› (*‹0 < ε›*) show \"?thesis\"\n (*goal: ‹∃F. finite F ∧ F ⊆ A ∧ dist (sum f F) (infsum f A) ≤ ε›*)\n by (rule has_sum_finite_approximation (*‹⟦((?f::?'a::type ⇒ ?'b::{comm_monoid_add,metric_space}) has_sum (?x::?'b::{comm_monoid_add,metric_space})) (?A::?'a::type set); (0::real) < (?ε::real)⟧ ⟹ ∃F::?'a::type set. finite F ∧ F ⊆ ?A ∧ dist (sum ?f F) ?x ≤ ?ε›*))\nqed\n\nlemma abs_summable_summable:\n fixes f :: ‹'a ⇒ 'b :: banach›\n assumes ‹f abs_summable_on A›\n shows ‹f summable_on A›\nproof (-)\n (*goal: ‹f summable_on A›*)\n from assms (*‹f abs_summable_on A›*) obtain L where lim: \"(sum (λx. norm (f x)) ⤏ L) (finite_subsets_at_top A)\"\n (*goal: ‹(⋀L. (sum (λx. norm (f x)) ⤏ L) (finite_subsets_at_top A) ⟹ thesis) ⟹ thesis›*)\n unfolding has_sum_def summable_on_def\n (*goal: ‹(⋀L. (sum (λx. norm (f x)) ⤏ L) (finite_subsets_at_top A) ⟹ thesis) ⟹ thesis›*)\n by blast\n then have \"*\": \"cauchy_filter (filtermap (sum (λx. norm (f x))) (finite_subsets_at_top A))\"\n by (auto intro!: nhds_imp_cauchy_filter (*‹?F ≤ nhds ?x ⟹ cauchy_filter ?F›*) simp: filterlim_def (*‹filterlim ?f ?F2.0 ?F1.0 = (filtermap ?f ?F1.0 ≤ ?F2.0)›*))\n have \"∃P. eventually P (finite_subsets_at_top A) ∧\n (∀F F'. P F ∧ P F' ⟶ dist (sum f F) (sum f F') < e)\" if \"e>0\" for e\n proof (-)\n (*goal: ‹∃P. eventually P (finite_subsets_at_top A) ∧ (∀F F'. P F ∧ P F' ⟶ dist (sum f F) (sum f F') < e)›*)\n define d and P where \"d = e/4\" and \"P F ⟷ finite F ∧ F ⊆ A ∧ dist (sum (λx. norm (f x)) F) L < d\" for F\n then have \"d > 0\"\n by (simp add: d_def (*‹d = e / 4›*) that (*‹0 < e›*))\n have ev_P: \"eventually P (finite_subsets_at_top A)\"\n using lim (*‹(sum (λx::'a. norm ((f::'a ⇒ 'b) x)) ⤏ (L::real)) (finite_subsets_at_top (A::'a set))›*) by (auto simp add: P_def[abs_def] (*‹P ≡ λF. finite F ∧ F ⊆ A ∧ dist (∑x∈F. norm (f x)) L < d›*) ‹0 < d› eventually_conj_iff (*‹(∀⇩F x in ?F. ?P x ∧ ?Q x) = (eventually ?P ?F ∧ eventually ?Q ?F)›*) eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*) tendsto_iff (*‹(?f ⤏ ?l) ?F = (∀e>0. ∀⇩F x in ?F. dist (?f x) ?l < e)›*))\n moreover have \"dist (sum f F1) (sum f F2) < e\" if \"P F1\" and \"P F2\" for F1 and F2\n proof (-)\n (*goal: ‹dist (sum f F1) (sum f F2) < e›*)\n from ev_P (*‹eventually P (finite_subsets_at_top A)›*) obtain F' where \"finite F'\" and \"F' ⊆ A\" and P_sup_F': \"finite F ∧ F ⊇ F' ∧ F ⊆ A ⟹ P F\" for F\n (*goal: ‹(⋀F'. ⟦finite F'; F' ⊆ A; ⋀F. finite F ∧ F' ⊆ F ∧ F ⊆ A ⟹ P F⟧ ⟹ thesis) ⟹ thesis›*)\n apply atomize_elim\n (*goal: ‹(⋀F'. ⟦finite F'; F' ⊆ A; ⋀F. finite F ∧ F' ⊆ F ∧ F ⊆ A ⟹ P F⟧ ⟹ thesis) ⟹ thesis›*)\n by (simp add: eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n define F where \"F = F' ∪ F1 ∪ F2\"\n have \"finite F\" and \"F ⊆ A\"\n using F_def (*‹(F::'a::type set) = (F'::'a::type set) ∪ (F1::'a::type set) ∪ (F2::'a::type set)›*) P_def[abs_def] (*‹P ≡ λF. finite F ∧ F ⊆ A ∧ dist (∑x∈F. norm (f x)) L < d›*) that (*‹P F1› ‹(P::'a set ⇒ bool) (F2::'a set)›*) ‹finite F'› (*‹finite F'›*) ‹F' ⊆ A› (*‹F' ⊆ A›*) apply -\n (*goals:\n 1. ‹⟦F = F' ∪ F1 ∪ F2; P ≡ λF. finite F ∧ F ⊆ A ∧ dist (∑x∈F. norm (f x)) L < d; P F1; P F2; finite F'; F' ⊆ A⟧ ⟹ finite F›\n 2. ‹⟦F = F' ∪ F1 ∪ F2; P ≡ λF. finite F ∧ F ⊆ A ∧ dist (∑x∈F. norm (f x)) L < d; P F1; P F2; finite F'; 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 have dist_F: \"dist (sum (λx. norm (f x)) F) L < d\"\n by (metis F_def (*‹F = F' ∪ F1 ∪ F2›*) ‹F ⊆ A› P_def (*‹P ?F = (finite ?F ∧ ?F ⊆ A ∧ dist (∑x∈?F. norm (f x)) L < d)›*) P_sup_F' (*‹finite ?F ∧ F' ⊆ ?F ∧ ?F ⊆ A ⟹ P ?F›*) ‹finite F› le_supE (*‹⟦sup ?a ?b ≤ ?x; ⟦?a ≤ ?x; ?b ≤ ?x⟧ ⟹ ?P⟧ ⟹ ?P›*) order_refl (*‹?x ≤ ?x›*))\n have dist_F_subset: \"dist (sum f F) (sum f F') < 2*d\" if F': \"F' ⊆ F\" \"P F'\" for F'\n proof (-)\n (*goal: ‹dist (sum f F) (sum f F') < 2 * d›*)\n have \"dist (sum f F) (sum f F') = norm (sum f (F-F'))\"\n unfolding dist_norm\n (*goal: ‹norm (sum (f::'a ⇒ 'b) (F::'a set) - sum f (F'::'a set)) = norm (sum f (F - F'))›*)\n using ‹finite F› (*‹finite F›*) F' (*‹(F'::'a set) ⊆ (F::'a set)› ‹P F'›*) apply (subst sum_diff (*‹⟦finite ?A; ?B ⊆ ?A⟧ ⟹ sum ?f (?A - ?B) = sum ?f ?A - sum ?f ?B›*))\n (*goals:\n 1. ‹⟦finite F; F' ⊆ F; P F'⟧ ⟹ finite F›\n 2. ‹⟦finite F; F' ⊆ F; P F'⟧ ⟹ F' ⊆ F›\n 3. ‹⟦finite F; F' ⊆ F; P F'⟧ ⟹ norm (sum f F - sum f F') = norm (sum f F - sum f F')›\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: ‹dist (sum f F) (sum f F') = norm (sum f (F - F'))›*) have \"… ≤ norm (∑x∈F-F'. norm (f x))\"\n apply (rule order.trans[OF sum_norm_le[OF order.refl]] (*‹(∑x::?'b1::type∈(?S1::?'b1::type set). norm ((?f1::?'b1::type ⇒ ?'a1::real_normed_vector) x)) ≤ (?c::real) ⟹ norm (sum ?f1 ?S1) ≤ ?c›*))\n (*goal: ‹norm (sum f (F - F')) ≤ norm (∑x∈F - F'. norm (f x))›*)\n by auto\n also (*calculation: ‹dist (sum f F) (sum f F') ≤ norm (∑x∈F - F'. norm (f x))›*) have \"… = dist (∑x∈F. norm (f x)) (∑x∈F'. norm (f x))\"\n unfolding dist_norm\n (*goal: ‹norm (∑x∈F - F'. norm (f x)) = norm ((∑x∈F. norm (f x)) - (∑x∈F'. norm (f x)))›*)\n using ‹finite F› (*‹finite F›*) F' (*‹F' ⊆ F› ‹(P::'a set ⇒ bool) (F'::'a set)›*) apply (subst sum_diff (*‹⟦finite ?A; ?B ⊆ ?A⟧ ⟹ sum ?f (?A - ?B) = sum ?f ?A - sum ?f ?B›*))\n (*goals:\n 1. ‹⟦finite F; F' ⊆ F; P F'⟧ ⟹ finite F›\n 2. ‹⟦finite F; F' ⊆ F; P F'⟧ ⟹ F' ⊆ F›\n 3. ‹⟦finite F; F' ⊆ F; P F'⟧ ⟹ norm ((∑x∈F. norm (f x)) - (∑x∈F'. norm (f x))) = norm ((∑x∈F. norm (f x)) - (∑x∈F'. norm (f 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: ‹dist (sum (f::'a::type ⇒ 'b::banach) (F::'a::type set)) (sum f (F'::'a::type set)) ≤ dist (∑x::'a::type∈F. norm (f x)) (∑x::'a::type∈F'. norm (f x))›*) have \"… < 2 * d\"\n using dist_F (*‹dist (∑x∈F. norm (f x)) L < d›*) F' (*‹F' ⊆ F› ‹P F'›*) unfolding P_def dist_norm real_norm_def\n (*goal: ‹¦(∑x::'a∈(F::'a set). norm ((f::'a ⇒ 'b) x)) - (∑x::'a∈(F'::'a set). norm (f x))¦ < (2::real) * (d::real)›*)\n by linarith\n finally (*calculation: ‹dist (sum f F) (sum f F') < 2 * d›*) show \"dist (sum f F) (sum f F') < 2*d\" .\n qed\n have \"dist (sum f F1) (sum f F2) ≤ dist (sum f F) (sum f F1) + dist (sum f F) (sum f F2)\"\n by (rule dist_triangle3 (*‹dist (?x::?'a) (?y::?'a) ≤ dist (?a::?'a) ?x + dist ?a ?y›*))\n also (*calculation: ‹dist (sum f F1) (sum f F2) ≤ dist (sum f F) (sum f F1) + dist (sum f F) (sum f F2)›*) have \"… < 2 * d + 2 * d\"\n apply (intro add_strict_mono (*‹⟦(?a::?'a::strict_ordered_ab_semigroup_add) < (?b::?'a::strict_ordered_ab_semigroup_add); (?c::?'a::strict_ordered_ab_semigroup_add) < (?d::?'a::strict_ordered_ab_semigroup_add)⟧ ⟹ ?a + ?c < ?b + ?d›*) dist_F_subset (*‹⟦(?F'::'a::type set) ⊆ (F::'a::type set); (P::'a::type set ⇒ bool) ?F'⟧ ⟹ dist (sum (f::'a::type ⇒ 'b::banach) F) (sum f ?F') < (2::real) * (d::real)›*) that (*‹(P::'a::type set ⇒ bool) (F1::'a::type set)› ‹(P::'a::type set ⇒ bool) (F2::'a::type set)›*))\n (*goals:\n 1. ‹F1 ⊆ F›\n 2. ‹F2 ⊆ F›\n discuss goal 1*)\n apply ((auto simp: F_def (*‹F = F' ∪ F1 ∪ F2›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: F_def (*‹F = F' ∪ F1 ∪ F2›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹dist (sum f F1) (sum f F2) < 2 * d + 2 * d›*) have \"… ≤ e\"\n by (auto simp: d_def (*‹d = e / 4›*))\n finally (*calculation: ‹dist (sum f F1) (sum f F2) < e›*) show \"dist (sum f F1) (sum f F2) < e\" .\n qed\n then show \"?thesis\"\n (*goal: ‹∃P::'a::type set ⇒ bool. eventually P (finite_subsets_at_top (A::'a::type set)) ∧ (∀(F::'a::type set) F'::'a::type set. P F ∧ P F' ⟶ dist (sum (f::'a::type ⇒ 'b::banach) F) (sum f F') < (e::real))›*)\n using ev_P (*‹eventually (P::'a::type set ⇒ bool) (finite_subsets_at_top (A::'a::type set))›*) by blast\n qed\n then have \"cauchy_filter (filtermap (sum f) (finite_subsets_at_top A))\"\n by (simp add: cauchy_filter_metric_filtermap (*‹cauchy_filter (filtermap ?f ?F) = (∀e>0. ∃P. eventually P ?F ∧ (∀x y. P x ∧ P y ⟶ dist (?f x) (?f y) < e))›*))\n moreover have \"complete (UNIV::'b set)\"\n by (meson Cauchy_convergent (*‹Cauchy ?X ⟹ convergent ?X›*) UNIV_I (*‹?x ∈ UNIV›*) complete_def (*‹complete ?S = (∀f. (∀n. f n ∈ ?S) ∧ Cauchy f ⟶ (∃l∈?S. f ⇢ l))›*) convergent_def (*‹convergent ?X = (∃L. ?X ⇢ L)›*))\n ultimately obtain L' where \"(sum f ⤏ L') (finite_subsets_at_top A)\"\n (*goal: ‹(⋀L'::'b. (sum (f::'a ⇒ 'b) ⤏ L') (finite_subsets_at_top (A::'a set)) ⟹ thesis::bool) ⟹ thesis›*)\n using complete_uniform[where S = UNIV] (*‹complete UNIV = (∀F≤principal UNIV. F ≠ bot ⟶ cauchy_filter F ⟶ (∃x∈UNIV. F ≤ nhds x))›*) by (force simp add: filterlim_def (*‹filterlim (?f::?'a ⇒ ?'b) (?F2.0::?'b filter) (?F1.0::?'a filter) = (filtermap ?f ?F1.0 ≤ ?F2.0)›*))\n then show \"?thesis\"\n (*goal: ‹(f::'a ⇒ 'b) summable_on (A::'a set)›*)\n using summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\nqed\n\ntext ‹The converse of @{thm [source] abs_summable_summable} does not hold:\n Consider the Hilbert space of square-summable sequences.\n Let $e_i$ denote the sequence with 1 in the $i$th position and 0 elsewhere.\n Let $f(i) := e_i/i$ for $i\\geq1$. We have \\<^term>‹¬ f abs_summable_on UNIV› because $\\lVert f(i)\\rVert=1/i$\n and thus the sum over $\\lVert f(i)\\rVert$ diverges. On the other hand, we have \\<^term>‹f summable_on UNIV›;\n the limit is the sequence with $1/i$ in the $i$th position.\n\n (We have not formalized this separating example here because to the best of our knowledge,\n this Hilbert space has not been formalized in Isabelle/HOL yet.)›\n\nlemma norm_has_sum_bound:\n fixes f :: \"'b ⇒ 'a::real_normed_vector\"\n and A :: \"'b set\"\n assumes \"((λx. norm (f x)) has_sum n) A\"\n assumes \"(f has_sum a) A\"\n shows \"norm a ≤ n\"\nproof (-)\n (*goal: ‹norm a ≤ n›*)\n have \"norm a ≤ n + ε\" if \"ε>0\" for ε\n proof (-)\n (*goal: ‹norm a ≤ n + ε›*)\n have \"∃F. norm (a - sum f F) ≤ ε ∧ finite F ∧ F ⊆ A\"\n using has_sum_finite_approximation[where A = A and f = f and ε = \"ε\"] (*‹⟦(f has_sum ?x) A; 0 < ε⟧ ⟹ ∃F. finite F ∧ F ⊆ A ∧ dist (sum f F) ?x ≤ ε›*) assms (*‹((λx. norm (f x)) has_sum n) A› ‹(f has_sum a) A›*) ‹0 < ε› (*‹0 < ε›*) by (metis dist_commute (*‹dist ?x ?y = dist ?y ?x›*) dist_norm (*‹dist ?x ?y = norm (?x - ?y)›*))\n then obtain F where \"norm (a - sum f F) ≤ ε\" and \"finite F\" and \"F ⊆ A\"\n (*goal: ‹(⋀F. ⟦norm (a - sum f F) ≤ ε; finite F; F ⊆ A⟧ ⟹ thesis) ⟹ thesis›*)\n by (simp add: atomize_elim (*‹(⋀x. ?P x ⟹ ?Q) ≡ (∃x. ?P x ⟹ ?Q)› ‹(⟦?A; ?B⟧ ⟹ ?C) ≡ (?A ∧ ?B ⟹ ?C)› ‹(⟦?A ⟹ ?C; ?B ⟹ ?C⟧ ⟹ ?C) ≡ ((?A ∨ ?B ⟹ ?C) ⟹ ?C)› ‹(⋀B. (?A ⟹ B) ⟹ B) ≡ ?A›*))\n hence \"norm a ≤ norm (sum f F) + ε\"\n by (metis add.commute (*‹?a + ?b = ?b + ?a›*) diff_add_cancel (*‹?a - ?b + ?b = ?a›*) dual_order.refl (*‹?a ≤ ?a›*) norm_triangle_mono (*‹⟦norm ?a ≤ ?r; norm ?b ≤ ?s⟧ ⟹ norm (?a + ?b) ≤ ?r + ?s›*))\n also (*calculation: ‹norm a ≤ norm (sum f F) + ε›*) have \"… ≤ sum (λx. norm (f x)) F + ε\"\n using norm_sum (*‹norm (sum (?f::?'b ⇒ ?'a) (?A::?'b set)) ≤ (∑i::?'b∈?A. norm (?f i))›*) by auto\n also (*calculation: ‹norm a ≤ (∑x∈F. norm (f x)) + ε›*) have \"… ≤ n + ε\"\n proof (intro add_right_mono [OF has_sum_mono_neutral] (*‹⟦(?f1 has_sum ?a) ?A1; (?g1 has_sum ?b) ?B1; ⋀x. x ∈ ?A1 ∩ ?B1 ⟹ ?f1 x ≤ ?g1 x; ⋀x. x ∈ ?A1 - ?B1 ⟹ ?f1 x ≤ 0; ⋀x. x ∈ ?B1 - ?A1 ⟹ 0 ≤ ?g1 x⟧ ⟹ ?a + ?c ≤ ?b + ?c›*))\n (*goals:\n 1. ‹((?f1::?'a1 ⇒ real) has_sum (∑x::'b∈(F::'b set). norm ((f::'b ⇒ 'a) x))) (?A1::?'a1 set)›\n 2. ‹((?g1::?'a1 ⇒ real) has_sum (n::real)) (?B1::?'a1 set)›\n 3. ‹⋀x::?'a1. x ∈ (?A1::?'a1 set) ∩ (?B1::?'a1 set) ⟹ (?f1::?'a1 ⇒ real) x ≤ (?g1::?'a1 ⇒ real) x›\n 4. ‹⋀x::?'a1. x ∈ (?A1::?'a1 set) - (?B1::?'a1 set) ⟹ (?f1::?'a1 ⇒ real) x ≤ (0::real)›\n 5. ‹⋀x::?'a1. x ∈ (?B1::?'a1 set) - (?A1::?'a1 set) ⟹ (0::real) ≤ (?g1::?'a1 ⇒ real) x›*)\n show \"((λx. norm (f x)) has_sum (∑x∈F. norm (f x))) F\"\n by (simp add: ‹finite F›)\n qed (use ‹F ⊆ A› assms in auto)\n (*solves the remaining goals:\n 1. ‹(?g1 has_sum n) ?B1›\n 2. ‹⋀x. x ∈ F ∩ ?B1 ⟹ norm (f x) ≤ ?g1 x›\n 3. ‹⋀x. x ∈ F - ?B1 ⟹ norm (f x) ≤ 0›\n 4. ‹⋀x. x ∈ ?B1 - F ⟹ 0 ≤ ?g1 x›*)\n finally (*calculation: ‹norm (a::'a) ≤ (n::real) + (ε::real)›*) show \"?thesis\"\n (*goal: ‹norm a ≤ n + ε›*)\n by assumption\n qed\n thus \"?thesis\"\n (*goal: ‹norm a ≤ n›*)\n using linordered_field_class.field_le_epsilon (*‹(⋀e. 0 < e ⟹ ?x ≤ ?y + e) ⟹ ?x ≤ ?y›*) by blast\nqed\n\nlemma norm_infsum_bound:\n fixes f :: \"'b ⇒ 'a::real_normed_vector\"\n and A :: \"'b set\"\n assumes \"f abs_summable_on A\"\n shows \"norm (infsum f A) ≤ infsum (λx. norm (f x)) A\"\nproof (cases \"f summable_on A\")\n (*goals:\n 1. ‹f summable_on A ⟹ norm (infsum f A) ≤ (∑⇩∞x∈A. norm (f x))›\n 2. ‹¬ f summable_on A ⟹ norm (infsum f A) ≤ (∑⇩∞x∈A. norm (f x))›*)\n case True (*‹f summable_on A›*)\n have \"((λx. norm (f x)) has_sum (∑⇩∞x∈A. norm (f x))) A\"\n by (simp add: assms (*‹(f::'b::type ⇒ 'a::real_normed_vector) abs_summable_on (A::'b::type set)›*))\n then show \"?thesis\"\n (*goal: ‹norm (infsum f A) ≤ (∑⇩∞x∈A. norm (f x))›*)\n by (metis True (*‹f summable_on A›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) norm_has_sum_bound (*‹⟦((λx. norm (?f x)) has_sum ?n) ?A; (?f has_sum ?a) ?A⟧ ⟹ norm ?a ≤ ?n›*))\nnext\n (*goal: ‹¬ (f::'b::type ⇒ 'a::real_normed_vector) summable_on (A::'b::type set) ⟹ norm (infsum f A) ≤ (∑⇩∞x::'b::type∈A. norm (f x))›*)\n case False (*‹¬ f summable_on A›*)\n obtain t where t_def: \"(sum (λx. norm (f x)) ⤏ t) (finite_subsets_at_top A)\"\n (*goal: ‹(⋀t. (sum (λx. norm (f x)) ⤏ t) (finite_subsets_at_top A) ⟹ thesis) ⟹ thesis›*)\n using assms (*‹f abs_summable_on A›*) unfolding summable_on_def has_sum_def\n (*goal: ‹(⋀t. (sum (λx. norm (f x)) ⤏ t) (finite_subsets_at_top A) ⟹ thesis) ⟹ thesis›*)\n by blast\n have sumpos: \"sum (λx. norm (f x)) X ≥ 0\" for X\n by (simp add: sum_nonneg (*‹(⋀x::?'b::type. x ∈ (?A::?'b::type set) ⟹ (0::?'a::ordered_comm_monoid_add) ≤ (?f::?'b::type ⇒ ?'a::ordered_comm_monoid_add) x) ⟹ (0::?'a::ordered_comm_monoid_add) ≤ sum ?f ?A›*))\n have tgeq0: \"t ≥ 0\"\n proof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹¬ 0 ≤ t ⟹ False›*)\n define S :: \"real set\" where \"S = {s. s < 0}\"\n assume \"¬ 0 ≤ t\" (*‹¬ (0::real) ≤ (t::real)›*)\n hence \"t < 0\"\n by simp\n hence \"t ∈ S\"\n unfolding S_def\n (*goal: ‹t ∈ {s. s < 0}›*)\n by blast\n moreover have \"open S\"\n by (metis S_def (*‹S = {s. s < 0}›*) lessThan_def (*‹{..<?u} ≡ {x. x < ?u}›*) open_real_lessThan (*‹open {..<?a}›*))\n ultimately have \"∀⇩F X in finite_subsets_at_top A. (∑x∈X. norm (f x)) ∈ S\"\n using t_def (*‹(sum (λx::'b. norm ((f::'b ⇒ 'a) x)) ⤏ (t::real)) (finite_subsets_at_top (A::'b set))›*) unfolding tendsto_def\n (*goal: ‹∀⇩F X in finite_subsets_at_top A. (∑x∈X. norm (f x)) ∈ S›*)\n by blast\n hence \"∃X. (∑x∈X. norm (f x)) ∈ S\"\n by (metis (no_types, lifting) eventually_mono (*‹⟦eventually ?P ?F; ⋀x. ?P x ⟹ ?Q x⟧ ⟹ eventually ?Q ?F›*) filterlim_iff (*‹filterlim ?f ?F2.0 ?F1.0 = (∀P. eventually P ?F2.0 ⟶ (∀⇩F x in ?F1.0. P (?f x)))›*) finite_subsets_at_top_neq_bot (*‹finite_subsets_at_top ?A ≠ bot›*) tendsto_Lim (*‹⟦?net ≠ bot; (?f ⤏ ?l) ?net⟧ ⟹ Lim ?net ?f = ?l›*))\n then obtain X where \"(∑x∈X. norm (f x)) ∈ S\"\n (*goal: ‹(⋀X. (∑x∈X. norm (f x)) ∈ S ⟹ thesis) ⟹ thesis›*)\n by blast\n hence \"(∑x∈X. norm (f x)) < 0\"\n unfolding S_def\n (*goal: ‹(∑x∈X. norm (f x)) < 0›*)\n by auto\n thus False\n by (simp add: leD (*‹?y ≤ ?x ⟹ ¬ ?x < ?y›*) sumpos (*‹0 ≤ (∑x∈?X. norm (f x))›*))\n qed\n have \"∃!h. (sum (λx. norm (f x)) ⤏ h) (finite_subsets_at_top A)\"\n using t_def (*‹(sum (λx. norm (f x)) ⤏ t) (finite_subsets_at_top A)›*) finite_subsets_at_top_neq_bot (*‹finite_subsets_at_top ?A ≠ bot›*) tendsto_unique (*‹⟦?F ≠ bot; (?f ⤏ ?a) ?F; (?f ⤏ ?b) ?F⟧ ⟹ ?a = ?b›*) by blast\n hence \"t = (Topological_Spaces.Lim (finite_subsets_at_top A) (sum (λx. norm (f x))))\"\n using t_def (*‹(sum (λx::'b. norm ((f::'b ⇒ 'a) x)) ⤏ (t::real)) (finite_subsets_at_top (A::'b set))›*) unfolding Topological_Spaces.Lim_def\n (*goal: ‹t = (THE l. (sum (λx. norm (f x)) ⤏ l) (finite_subsets_at_top A))›*)\n by (metis the_equality (*‹⟦?P ?a; ⋀x. ?P x ⟹ x = ?a⟧ ⟹ (THE x. ?P x) = ?a›*))\n hence \"Lim (finite_subsets_at_top A) (sum (λx. norm (f x))) ≥ 0\"\n using tgeq0 (*‹0 ≤ t›*) by blast\n thus \"?thesis\"\n (*goal: ‹norm (infsum f A) ≤ (∑⇩∞x∈A. norm (f x))›*)\n unfolding infsum_def\n (*goal: ‹norm (if f summable_on A then Lim (finite_subsets_at_top A) (sum f) else 0) ≤ (if f abs_summable_on A then Lim (finite_subsets_at_top A) (sum (λx. norm (f x))) else 0)›*)\n using False (*‹¬ f summable_on A›*) by auto\nqed\n\nlemma infsum_tendsto:\n assumes ‹f summable_on S›\n shows ‹((λF. sum f F) ⤏ infsum f S) (finite_subsets_at_top S)›\n using assms (*‹f summable_on S›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) has_sum_infsum (*‹(?f::?'a ⇒ ?'b) summable_on (?S::?'a set) ⟹ (?f has_sum infsum ?f ?S) ?S›*) by blast\n\nlemma has_sum_0: \n assumes ‹⋀x. x∈M ⟹ f x = 0›\n shows ‹(f has_sum 0) M›\n by (metis assms (*‹?x ∈ M ⟹ f ?x = 0›*) finite.intros( (*‹finite {}›*) 1) has_sum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f has_sum ?x) ?A = (?g has_sum ?x) ?A›*) has_sum_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f has_sum ?x) ?S = (?g has_sum ?x) ?T›*) has_sum_finite (*‹finite ?F ⟹ (?f has_sum sum ?f ?F) ?F›*) sum.neutral_const (*‹(∑_∈?A. 0) = 0›*))\n\nlemma summable_on_0:\n assumes ‹⋀x. x∈M ⟹ f x = 0›\n shows ‹f summable_on M›\n using assms (*‹?x ∈ M ⟹ f ?x = 0›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_0 (*‹(⋀x::?'a::type. x ∈ (?M::?'a::type set) ⟹ (?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) x = (0::?'b::{comm_monoid_add,topological_space})) ⟹ (?f has_sum (0::?'b::{comm_monoid_add,topological_space})) ?M›*) by blast\n\nlemma infsum_0:\n assumes ‹⋀x. x∈M ⟹ f x = 0›\n shows ‹infsum f M = 0›\n by (metis assms (*‹(?x::'a) ∈ (M::'a set) ⟹ (f::'a ⇒ 'b) ?x = (0::'b)›*) finite_subsets_at_top_neq_bot (*‹finite_subsets_at_top (?A::?'a set) ≠ bot›*) infsum_def (*‹infsum (?f::?'a ⇒ ?'b) (?A::?'a set) = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else (0::?'b))›*) has_sum_0 (*‹(⋀x::?'a. x ∈ (?M::?'a set) ⟹ (?f::?'a ⇒ ?'b) x = (0::?'b)) ⟹ (?f has_sum (0::?'b)) ?M›*) has_sum_def (*‹((?f::?'a ⇒ ?'b) has_sum (?x::?'b)) (?A::?'a set) ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) tendsto_Lim (*‹⟦(?net::?'a filter) ≠ bot; ((?f::?'a ⇒ ?'b) ⤏ (?l::?'b)) ?net⟧ ⟹ Lim ?net ?f = ?l›*))\n\ntext ‹Variants of @{thm [source] infsum_0} etc. suitable as simp-rules›\nlemma infsum_0_simp[simp]: ‹infsum (λ_. 0) M = 0›\n by (simp add: infsum_0 (*‹(⋀x. x ∈ ?M ⟹ ?f x = 0) ⟹ infsum ?f ?M = 0›*))\n\nlemma summable_on_0_simp[simp]: ‹(λ_. 0) summable_on M›\n by (simp add: summable_on_0 (*‹(⋀x::?'a::type. x ∈ (?M::?'a::type set) ⟹ (?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) x = (0::?'b::{comm_monoid_add,topological_space})) ⟹ ?f summable_on ?M›*))\n\nlemma has_sum_0_simp[simp]: ‹((λ_. 0) has_sum 0) M›\n by (simp add: has_sum_0 (*‹(⋀x. x ∈ ?M ⟹ ?f x = 0) ⟹ (?f has_sum 0) ?M›*))\n\n\nlemma has_sum_add:\n fixes f g :: \"'a ⇒ 'b::{topological_comm_monoid_add}\"\n assumes ‹(f has_sum a) A›\n assumes ‹(g has_sum b) A›\n shows ‹((λx. f x + g x) has_sum (a + b)) A›\nproof (-)\n (*goal: ‹((λx. f x + g x) has_sum a + b) A›*)\n from assms (*‹(f has_sum a) A› ‹((g::'a::type ⇒ 'b::topological_comm_monoid_add) has_sum (b::'b::topological_comm_monoid_add)) (A::'a::type set)›*) have lim_f: \"(sum f ⤏ a) (finite_subsets_at_top A)\" and lim_g: \"(sum g ⤏ b) (finite_subsets_at_top A)\"\n apply -\n (*goals:\n 1. ‹⟦(f has_sum a) A; (g has_sum b) A⟧ ⟹ (sum f ⤏ a) (finite_subsets_at_top A)›\n 2. ‹⟦(f has_sum a) A; (g has_sum b) A⟧ ⟹ (sum g ⤏ b) (finite_subsets_at_top A)›\n discuss goal 1*)\n apply (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n (*discuss goal 2*)\n apply (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n (*proven 2 subgoals*) .\n then have lim: \"(sum (λx. f x + g x) ⤏ a + b) (finite_subsets_at_top A)\"\n unfolding sum.distrib\n (*goal: ‹((λA. sum f A + sum g A) ⤏ a + b) (finite_subsets_at_top A)›*)\n by (rule tendsto_add (*‹⟦(?f ⤏ ?a) ?F; (?g ⤏ ?b) ?F⟧ ⟹ ((λx. ?f x + ?g x) ⤏ ?a + ?b) ?F›*))\n then show \"?thesis\"\n (*goal: ‹((λx. f x + g x) has_sum a + b) A›*)\n by (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\nqed\n\nlemma summable_on_add:\n fixes f g :: \"'a ⇒ 'b::{topological_comm_monoid_add}\"\n assumes ‹f summable_on A›\n assumes ‹g summable_on A›\n shows ‹(λx. f x + g x) summable_on A›\n by (metis (full_types) assms (*‹f summable_on A› ‹g summable_on A›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_add (*‹⟦(?f has_sum ?a) ?A; (?g has_sum ?b) ?A⟧ ⟹ ((λx. ?f x + ?g x) has_sum ?a + ?b) ?A›*))\n\nlemma infsum_add:\n fixes f g :: \"'a ⇒ 'b::{topological_comm_monoid_add, t2_space}\"\n assumes ‹f summable_on A›\n assumes ‹g summable_on A›\n shows ‹infsum (λx. f x + g x) A = infsum f A + infsum g A›\nproof (-)\n (*goal: ‹(∑⇩∞x∈A. f x + g x) = infsum f A + infsum g A›*)\n have \"((λx. f x + g x) has_sum (infsum f A + infsum g A)) A\"\n by (simp add: assms (*‹f summable_on A› ‹g summable_on A›*) has_sum_add (*‹⟦(?f has_sum ?a) ?A; (?g has_sum ?b) ?A⟧ ⟹ ((λx. ?f x + ?g x) has_sum ?a + ?b) ?A›*))\n then show \"?thesis\"\n (*goal: ‹(∑⇩∞x∈A. f x + g x) = infsum f A + infsum g A›*)\n using infsumI (*‹((?f::?'a ⇒ ?'b) has_sum (?x::?'b)) (?A::?'a set) ⟹ infsum ?f ?A = ?x›*) by blast\nqed\n\n\nlemma has_sum_Un_disjoint:\n fixes f :: \"'a ⇒ 'b::topological_comm_monoid_add\"\n assumes \"(f has_sum a) A\"\n assumes \"(f has_sum b) B\"\n assumes disj: \"A ∩ B = {}\"\n shows ‹(f has_sum (a + b)) (A ∪ B)›\nproof (-)\n (*goal: ‹((f::'a ⇒ 'b) has_sum (a::'b) + (b::'b)) ((A::'a set) ∪ (B::'a set))›*)\n define fA and fB where \"fA x = (if x ∈ A then f x else 0)\" and \"fB x = (if x ∉ A then f x else 0)\" for x\n have fA: \"(fA has_sum a) (A ∪ B)\"\n by (smt (verit, ccfv_SIG) DiffD1 (*‹?c ∈ ?A - ?B ⟹ ?c ∈ ?A›*) DiffD2 (*‹⟦?c ∈ ?A - ?B; ?c ∈ ?B⟧ ⟹ ?P›*) UnCI (*‹(?c ∉ ?B ⟹ ?c ∈ ?A) ⟹ ?c ∈ ?A ∪ ?B›*) fA_def (*‹fA ?x = (if ?x ∈ A then f ?x else 0)›*) assms( (*‹(f has_sum a) A›*) 1) has_sum_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f has_sum ?x) ?S = (?g has_sum ?x) ?T›*) inf_sup_absorb (*‹inf ?x (sup ?x ?y) = ?x›*))\n have fB: \"(fB has_sum b) (A ∪ B)\"\n by (smt (verit, best) DiffD1 (*‹?c ∈ ?A - ?B ⟹ ?c ∈ ?A›*) DiffD2 (*‹⟦?c ∈ ?A - ?B; ?c ∈ ?B⟧ ⟹ ?P›*) IntE (*‹⟦?c ∈ ?A ∩ ?B; ⟦?c ∈ ?A; ?c ∈ ?B⟧ ⟹ ?P⟧ ⟹ ?P›*) Un_iff (*‹(?c ∈ ?A ∪ ?B) = (?c ∈ ?A ∨ ?c ∈ ?B)›*) fB_def (*‹fB ?x = (if ?x ∉ A then f ?x else 0)›*) assms( (*‹(f has_sum b) B›*) 2) disj (*‹A ∩ B = {}›*) disjoint_iff (*‹(?A ∩ ?B = {}) = (∀x. x ∈ ?A ⟶ x ∉ ?B)›*) has_sum_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f has_sum ?x) ?S = (?g has_sum ?x) ?T›*))\n have fAB: \"f x = fA x + fB x\" for x\n unfolding fA_def fB_def\n (*goal: ‹(f::'a ⇒ 'b) (x::'a) = (if x ∈ (A::'a set) then f x else (0::'b)) + (if x ∉ A then f x else (0::'b))›*)\n by simp\n show \"?thesis\"\n (*goal: ‹((f::'a ⇒ 'b) has_sum (a::'b) + (b::'b)) ((A::'a set) ∪ (B::'a set))›*)\n unfolding fAB\n (*goal: ‹((λx. fA x + fB x) has_sum a + b) (A ∪ B)›*)\n using fA (*‹(fA has_sum a) (A ∪ B)›*) fB (*‹(fB has_sum b) (A ∪ B)›*) by (rule has_sum_add (*‹⟦(?f has_sum ?a) ?A; (?g has_sum ?b) ?A⟧ ⟹ ((λx. ?f x + ?g x) has_sum ?a + ?b) ?A›*))\nqed\n\nlemma summable_on_Un_disjoint:\n fixes f :: \"'a ⇒ 'b::topological_comm_monoid_add\"\n assumes \"f summable_on A\"\n assumes \"f summable_on B\"\n assumes disj: \"A ∩ B = {}\"\n shows ‹f summable_on (A ∪ B)›\n by (meson assms (*‹f summable_on A› ‹f summable_on B› ‹A ∩ B = {}›*) disj (*‹A ∩ B = {}›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_Un_disjoint (*‹⟦(?f has_sum ?a) ?A; (?f has_sum ?b) ?B; ?A ∩ ?B = {}⟧ ⟹ (?f has_sum ?a + ?b) (?A ∪ ?B)›*))\n\nlemma infsum_Un_disjoint:\n fixes f :: \"'a ⇒ 'b::{topological_comm_monoid_add, t2_space}\"\n assumes \"f summable_on A\"\n assumes \"f summable_on B\"\n assumes disj: \"A ∩ B = {}\"\n shows ‹infsum f (A ∪ B) = infsum f A + infsum f B›\n by (intro infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) has_sum_Un_disjoint (*‹⟦(?f has_sum ?a) ?A; (?f has_sum ?b) ?B; ?A ∩ ?B = {}⟧ ⟹ (?f has_sum ?a + ?b) (?A ∪ ?B)›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) assms (*‹f summable_on A› ‹f summable_on B› ‹A ∩ B = {}›*)) \n\nlemma norm_summable_imp_has_sum:\n fixes f :: \"nat ⇒ 'a :: banach\"\n assumes \"summable (λn. norm (f n))\" and \"f sums S\"\n shows \"(f has_sum S) (UNIV :: nat set)\"\n unfolding has_sum_def tendsto_iff eventually_finite_subsets_at_top\n (*goal: ‹∀e>0. ∃X. finite X ∧ X ⊆ UNIV ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ UNIV ⟶ dist (sum f Y) S < e)›*)\n proof (clarsimp)\n (*goal: ‹⋀e. 0 < e ⟹ ∃X. finite X ∧ (∀Y. finite Y ∧ X ⊆ Y ⟶ dist (sum f Y) S < e)›*)\n fix ε :: real\n assume \"ε > 0\" (*‹(0::real) < (ε::real)›*)\n from assms (*‹summable (λn. norm (f n))› ‹f sums S›*) obtain S' where S': \"(λn. norm (f n)) sums S'\"\n (*goal: ‹(⋀S'. (λn. norm (f n)) sums S' ⟹ thesis) ⟹ thesis›*)\n by (auto simp: summable_def (*‹summable ?f = (∃s. ?f sums s)›*))\n with ‹ε > 0› (*‹0 < ε›*) obtain N where N: \"⋀n. n ≥ N ⟹ ¦S' - (∑i<n. norm (f i))¦ < ε\"\n (*goal: ‹(⋀N. (⋀n. N ≤ n ⟹ ¦S' - (∑i<n. norm (f i))¦ < ε) ⟹ thesis) ⟹ thesis›*)\n by (auto simp: tendsto_iff (*‹((?f::?'b ⇒ ?'a) ⤏ (?l::?'a)) (?F::?'b filter) = (∀e>0::real. ∀⇩F x::?'b in ?F. dist (?f x) ?l < e)›*) eventually_at_top_linorder (*‹eventually (?P::?'a ⇒ bool) at_top = (∃N::?'a. ∀n≥N. ?P n)›*) sums_def (*‹(?f::nat ⇒ ?'a) sums (?s::?'a) = (λn::nat. sum ?f {..<n}) ⇢ ?s›*) dist_norm (*‹dist (?x::?'a) (?y::?'a) = norm (?x - ?y)›*) abs_minus_commute (*‹¦(?a::?'a) - (?b::?'a)¦ = ¦?b - ?a¦›*))\n have \"dist (sum f Y) S < ε\" if \"finite Y\" \"{..<N} ⊆ Y\" for Y\n proof (-)\n (*goal: ‹dist (sum (f::nat ⇒ 'a::banach) (Y::nat set)) (S::'a::banach) < (ε::real)›*)\n from that (*‹finite Y› ‹{..<N} ⊆ Y›*) have \"(λn. if n ∈ Y then 0 else f n) sums (S - sum f Y)\"\n apply (intro sums_If_finite_set'[OF ‹f sums S›] (*‹⟦finite ?A; ?S' = S + (∑n∈?A. ?f n - f n)⟧ ⟹ (λn. if n ∈ ?A then ?f n else f n) sums ?S'›*))\n (*goals:\n 1. ‹⟦finite Y; {..<N} ⊆ Y⟧ ⟹ finite Y›\n 2. ‹⟦finite Y; {..<N} ⊆ Y⟧ ⟹ S - sum f Y = S + (∑n∈Y. 0 - f n)›\n discuss goal 1*)\n apply ((auto simp: sum_negf (*‹(∑x∈?A. - ?f x) = - sum ?f ?A›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: sum_negf (*‹(∑x∈?A. - ?f x) = - sum ?f ?A›*))[1])\n (*proven 2 subgoals*) .\n hence \"S - sum f Y = (∑n. if n ∈ Y then 0 else f n)\"\n by (simp add: sums_iff (*‹?f sums ?x = (summable ?f ∧ suminf ?f = ?x)›*))\n also (*calculation: ‹S - sum f Y = (∑n. if n ∈ Y then 0 else f n)›*) have \"norm … ≤ (∑n. norm (if n ∈ Y then 0 else f n))\"\n apply (rule summable_norm[OF summable_comparison_test'[OF assms(1)]] (*‹(⋀n. ?N1 ≤ n ⟹ norm (norm (?f n)) ≤ norm (f n)) ⟹ norm (suminf ?f) ≤ (∑n. norm (?f n))›*))\n (*goal: ‹norm (∑n. if n ∈ Y then 0 else f n) ≤ (∑n. norm (if n ∈ Y then 0 else f n))›*)\n by auto\n also (*calculation: ‹norm (S - sum f Y) ≤ (∑n. norm (if n ∈ Y then 0 else f n))›*) have \"… ≤ (∑n. if n < N then 0 else norm (f n))\"\n using that (*‹finite (Y::nat set)› ‹{..<N::nat} ⊆ (Y::nat set)›*) apply (intro suminf_le (*‹⟦⋀n. ?f n ≤ ?g n; summable ?f; summable ?g⟧ ⟹ suminf ?f ≤ suminf ?g›*) summable_comparison_test'[OF assms(1)] (*‹(⋀n. ?N ≤ n ⟹ norm (?f n) ≤ norm (f n)) ⟹ summable ?f›*))\n (*goals:\n 1. ‹⋀n::nat. ⟦finite (Y::nat set); {..<N::nat} ⊆ Y⟧ ⟹ norm (if n ∈ Y then 0::'a else (f::nat ⇒ 'a) n) ≤ (if n < N then 0::real else norm (f n))›\n 2. ‹⋀n::nat. ⟦finite (Y::nat set); {..<N::nat} ⊆ Y; (?N4::nat) ≤ n⟧ ⟹ norm (norm (if n ∈ Y then 0::'a else (f::nat ⇒ 'a) n)) ≤ norm (f n)›\n 3. ‹⋀n::nat. ⟦finite (Y::nat set); {..<N::nat} ⊆ Y; (?N3::nat) ≤ n⟧ ⟹ norm (if n < N then 0::real else norm ((f::nat ⇒ 'a) n)) ≤ norm (f n)›\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 (S - sum f Y) ≤ (∑n. if n < N then 0 else norm (f n))›*) have \"(λn. if n ∈ {..<N} then 0 else norm (f n)) sums (S' - (∑i<N. norm (f i)))\"\n apply (intro sums_If_finite_set'[OF S'] (*‹⟦finite (?A::nat set); (?S'::real) = (S'::real) + (∑n::nat∈?A. (?f::nat ⇒ real) n - norm ((f::nat ⇒ 'a::banach) n))⟧ ⟹ (λn::nat. if n ∈ ?A then ?f n else norm (f n)) sums ?S'›*))\n (*goals:\n 1. ‹finite {..<N::nat}›\n 2. ‹(S'::real) - (∑i::nat<N::nat. norm ((f::nat ⇒ 'a) i)) = S' + (∑n::nat<N. (0::real) - norm (f n))›\n discuss goal 1*)\n apply ((auto simp: sum_negf (*‹(∑x∈?A. - ?f x) = - sum ?f ?A›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: sum_negf (*‹(∑x∈?A. - ?f x) = - sum ?f ?A›*))[1])\n (*proven 2 subgoals*) .\n hence \"(∑n. if n < N then 0 else norm (f n)) = S' - (∑i<N. norm (f i))\"\n by (simp add: sums_iff (*‹?f sums ?x = (summable ?f ∧ suminf ?f = ?x)›*))\n also (*calculation: ‹norm (S - sum f Y) ≤ S' - (∑i<N. norm (f i))›*) have \"S' - (∑i<N. norm (f i)) ≤ ¦S' - (∑i<N. norm (f i))¦\"\n by simp\n also (*calculation: ‹norm (S - sum f Y) ≤ ¦S' - (∑i<N. norm (f i))¦›*) have \"… < ε\"\n apply (rule N (*‹N ≤ ?n ⟹ ¦S' - (∑i<?n. norm (f i))¦ < ε›*))\n (*goal: ‹¦S' - (∑i<N. norm (f i))¦ < ε›*)\n by auto\n finally (*calculation: ‹norm (S - sum f Y) < ε›*) show \"?thesis\"\n (*goal: ‹dist (sum f Y) S < ε›*)\n by (simp add: dist_norm (*‹dist ?x ?y = norm (?x - ?y)›*) norm_minus_commute (*‹norm (?a - ?b) = norm (?b - ?a)›*))\n qed\n then show \"∃X. finite X ∧ (∀Y. finite Y ∧ X ⊆ Y ⟶ dist (sum f Y) S < ε)\"\n by (meson finite_lessThan (*‹finite {..<?k}›*) subset_UNIV (*‹?A ⊆ UNIV›*))\n qed\n\nlemma norm_summable_imp_summable_on:\n fixes f :: \"nat ⇒ 'a :: banach\"\n assumes \"summable (λn. norm (f n))\"\n shows \"f summable_on UNIV\"\n using norm_summable_imp_has_sum[OF assms, of \"suminf f\"] (*‹f sums suminf f ⟹ (f has_sum suminf f) UNIV›*) assms (*‹summable (λn. norm (f n))›*) by (auto simp: sums_iff (*‹?f sums ?x = (summable ?f ∧ suminf ?f = ?x)›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) dest: summable_norm_cancel (*‹summable (λn. norm (?f n)) ⟹ summable ?f›*))\n\ntext ‹The following lemma indeed needs a complete space (as formalized by the premise \\<^term>‹complete UNIV›).\n The following two counterexamples show this:\n \\begin{itemize}\n \\item Consider the real vector space $V$ of sequences with finite support, and with the $\\ell_2$-norm (sum of squares).\n Let $e_i$ denote the sequence with a $1$ at position $i$.\n Let $f : \\mathbb Z \\to V$ be defined as $f(n) := e_{\\lvert n\\rvert} / n$ (with $f(0) := 0$).\n We have that $\\sum_{n\\in\\mathbb Z} f(n) = 0$ (it even converges absolutely). \n But $\\sum_{n\\in\\mathbb N} f(n)$ does not exist (it would converge against a sequence with infinite support).\n \n \\item Let $f$ be a positive rational valued function such that $\\sum_{x\\in B} f(x)$ is $\\sqrt 2$ and $\\sum_{x\\in A} f(x)$ is 1 (over the reals, with $A\\subseteq B$).\n Then $\\sum_{x\\in B} f(x)$ does not exist over the rationals. But $\\sum_{x\\in A} f(x)$ exists.\n \\end{itemize}\n\n The lemma also requires uniform continuity of the addition. And example of a topological group with continuous \n but not uniformly continuous addition would be the positive reals with the usual multiplication as the addition.\n We do not know whether the lemma would also hold for such topological groups.›\n\nlemma summable_on_subset_aux:\n fixes A B and f :: ‹'a ⇒ 'b::{ab_group_add, uniform_space}›\n assumes ‹complete (UNIV :: 'b set)›\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'b,y). x+y)›\n assumes ‹f summable_on A›\n assumes ‹B ⊆ A›\n shows ‹f summable_on B›\nproof (-)\n (*goal: ‹f summable_on B›*)\n let ?filter_fB = \"filtermap (sum f) (finite_subsets_at_top B)\"\n from ‹f summable_on A› (*‹f summable_on A›*) obtain S where \"(sum f ⤏ S) (finite_subsets_at_top A)\" (is \"(sum f ⤏ S) ?filter_A\")\n (*goal: ‹(⋀S. (sum f ⤏ S) (finite_subsets_at_top A) ⟹ thesis) ⟹ thesis›*)\n using summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\n then have cauchy_fA: \"cauchy_filter (filtermap (sum f) (finite_subsets_at_top A))\" (is \"cauchy_filter ?filter_fA\")\n by (auto intro!: nhds_imp_cauchy_filter (*‹(?F::?'a filter) ≤ nhds (?x::?'a) ⟹ cauchy_filter ?F›*) simp: filterlim_def (*‹filterlim (?f::?'a ⇒ ?'b) (?F2.0::?'b filter) (?F1.0::?'a filter) = (filtermap ?f ?F1.0 ≤ ?F2.0)›*))\n have \"cauchy_filter (filtermap (sum f) (finite_subsets_at_top B))\"\n apply (unfold cauchy_filter_def (*‹cauchy_filter ?F = (?F ×⇩F ?F ≤ uniformity)›*))\n (*goal: ‹cauchy_filter (filtermap (sum f) (finite_subsets_at_top B))›*)\n proof (rule filter_leI (*‹(⋀P. eventually P ?F' ⟹ eventually P ?F) ⟹ ?F ≤ ?F'›*))\n (*goal: ‹⋀P. eventually P uniformity ⟹ eventually P (filtermap (sum f) (finite_subsets_at_top B) ×⇩F filtermap (sum f) (finite_subsets_at_top B))›*)\n fix E :: \"('b×'b) ⇒ bool\"\n assume \"eventually E uniformity\" (*‹eventually (E::'b × 'b ⇒ bool) uniformity›*)\n then obtain E' where \"eventually E' uniformity\" and E'E'E: \"E' (x, y) ⟶ E' (y, z) ⟶ E (x, z)\" for x and y and z\n (*goal: ‹(⋀E'::'b::{ab_group_add,uniform_space} × 'b::{ab_group_add,uniform_space} ⇒ bool. ⟦eventually E' uniformity; ⋀(x::'b::{ab_group_add,uniform_space}) (y::'b::{ab_group_add,uniform_space}) z::'b::{ab_group_add,uniform_space}. E' (x, y) ⟶ E' (y, z) ⟶ (E::'b::{ab_group_add,uniform_space} × 'b::{ab_group_add,uniform_space} ⇒ bool) (x, z)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using uniformity_trans (*‹eventually (?E::?'a × ?'a ⇒ bool) uniformity ⟹ ∃D::?'a × ?'a ⇒ bool. eventually D uniformity ∧ (∀(x::?'a) (y::?'a) z::?'a. D (x, y) ⟶ D (y, z) ⟶ ?E (x, z))›*) by blast\n obtain D where \"eventually D uniformity\" and DE: \"D (x, y) ⟹ E' (x+c, y+c)\" for x and y and c\n (*goal: ‹(⋀D. ⟦eventually D uniformity; ⋀x y c. D (x, y) ⟹ E' (x + c, y + c)⟧ ⟹ thesis) ⟹ thesis›*)\n using plus_cont (*‹uniformly_continuous_on UNIV (λ(x, y). x + y)›*) ‹eventually E' uniformity› (*‹eventually E' uniformity›*) unfolding uniformly_continuous_on_uniformity filterlim_def le_filter_def uniformity_prod_def\n (*goal: ‹(⋀D. ⟦eventually D uniformity; ⋀x y c. D (x, y) ⟹ E' (x + c, y + c)⟧ ⟹ thesis) ⟹ thesis›*)\n by (auto simp: case_prod_beta (*‹(case ?p of (x, xa) ⇒ ?f x xa) = ?f (fst ?p) (snd ?p)›*) eventually_filtermap (*‹eventually ?P (filtermap ?f ?F) = (∀⇩F x in ?F. ?P (?f x))›*) eventually_prod_same (*‹eventually ?P (?F ×⇩F ?F) = (∃Q. eventually Q ?F ∧ (∀x y. Q x ⟶ Q y ⟶ ?P (x, y)))›*) uniformity_refl (*‹eventually ?E uniformity ⟹ ?E (?x, ?x)›*))\n have DE': \"E' (x, y)\" if \"D (x + c, y + c)\" for x and y and c\n using DE[of \"x + c\" \"y + c\" \"-c\"] (*‹D (x + c, y + c) ⟹ E' (x + c + - c, y + c + - c)›*) that (*‹D (x + c, y + c)›*) by simp\n from ‹eventually D uniformity› (*‹eventually D uniformity›*) cauchy_fA (*‹cauchy_filter (filtermap (sum (f::'a::type ⇒ 'b::{ab_group_add,uniform_space})) (finite_subsets_at_top (A::'a::type set)))›*) have \"eventually D (?filter_fA ×⇩F ?filter_fA)\"\n unfolding cauchy_filter_def le_filter_def\n (*goal: ‹eventually D (filtermap (sum f) (finite_subsets_at_top A) ×⇩F filtermap (sum f) (finite_subsets_at_top A))›*)\n by simp\n then obtain P1 and P2 where ev_P1: \"eventually (λF. P1 (sum f F)) ?filter_A\" and ev_P2: \"eventually (λF. P2 (sum f F)) ?filter_A\" and P1P2E: \"P1 x ⟹ P2 y ⟹ D (x, y)\" for x and y\n (*goal: ‹(⋀P1 P2. ⟦∀⇩F F in finite_subsets_at_top A. P1 (sum f F); ∀⇩F F in finite_subsets_at_top A. P2 (sum f F); ⋀x y. ⟦P1 x; P2 y⟧ ⟹ D (x, y)⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding eventually_prod_filter eventually_filtermap\n (*goal: ‹(⋀(P1::'b::{ab_group_add,uniform_space} ⇒ bool) P2::'b::{ab_group_add,uniform_space} ⇒ bool. ⟦∀⇩F F::'a::type set in finite_subsets_at_top (A::'a::type set). P1 (sum (f::'a::type ⇒ 'b::{ab_group_add,uniform_space}) F); ∀⇩F F::'a::type set in finite_subsets_at_top A. P2 (sum f F); ⋀(x::'b::{ab_group_add,uniform_space}) y::'b::{ab_group_add,uniform_space}. ⟦P1 x; P2 y⟧ ⟹ (D::'b::{ab_group_add,uniform_space} × 'b::{ab_group_add,uniform_space} ⇒ bool) (x, y)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n from ev_P1 (*‹∀⇩F F in finite_subsets_at_top A. P1 (sum f F)›*) obtain F1 where F1: \"finite F1\" \"F1 ⊆ A\" \"⋀F. F⊇F1 ⟹ finite F ⟹ F⊆A ⟹ P1 (sum f F)\"\n (*goal: ‹(⋀F1. ⟦finite F1; F1 ⊆ A; ⋀F. ⟦F1 ⊆ F; finite F; F ⊆ A⟧ ⟹ P1 (sum f F)⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n from ev_P2 (*‹∀⇩F F in finite_subsets_at_top A. P2 (sum f F)›*) obtain F2 where F2: \"finite F2\" \"F2 ⊆ A\" \"⋀F. F⊇F2 ⟹ finite F ⟹ F⊆A ⟹ P2 (sum f F)\"\n (*goal: ‹(⋀F2. ⟦finite F2; F2 ⊆ A; ⋀F. ⟦F2 ⊆ F; finite F; F ⊆ A⟧ ⟹ P2 (sum f F)⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n define F0 and F0A and F0B where \"F0 ≡ F1 ∪ F2\" and \"F0A ≡ F0 - B\" and \"F0B ≡ F0 ∩ B\"\n have [simp]: \"finite F0\" \"F0 ⊆ A\"\n using ‹F1 ⊆ A› (*‹F1 ⊆ A›*) ‹F2 ⊆ A› (*‹(F2::'a set) ⊆ (A::'a set)›*) ‹finite F1› (*‹finite F1›*) ‹finite F2› (*‹finite F2›*) unfolding F0_def\n (*goals:\n 1. ‹finite (F1 ∪ F2)›\n 2. ‹F1 ∪ F2 ⊆ A›*)\n apply -\n (*goals:\n 1. ‹⟦F1 ⊆ A; F2 ⊆ A; finite F1; finite F2⟧ ⟹ finite (F1 ∪ F2)›\n 2. ‹⟦F1 ⊆ A; F2 ⊆ A; finite F1; finite F2⟧ ⟹ F1 ∪ F2 ⊆ A›\n discuss goal 1*)\n apply blast\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*) .\n have \"*\": \"E' (sum f F1', sum f F2')\" if \"F1'⊇F0B\" \"F2'⊇F0B\" \"finite F1'\" \"finite F2'\" \"F1'⊆B\" \"F2'⊆B\" for F1' and F2'\n proof (intro DE'[where c = \"sum f F0A\"] (*‹D (?x + sum f F0A, ?y + sum f F0A) ⟹ E' (?x, ?y)›*) P1P2E (*‹⟦P1 ?x; P2 ?y⟧ ⟹ D (?x, ?y)›*))\n (*goals:\n 1. ‹P1 (sum f F1' + sum f F0A)›\n 2. ‹P2 (sum f F2' + sum f F0A)›*)\n have \"P1 (sum f (F1' ∪ F0A))\"\n using that (*‹F0B ⊆ F1'› ‹(F0B::'a::type set) ⊆ (F2'::'a::type set)› ‹finite F1'› ‹finite F2'› ‹F1' ⊆ B› ‹F2' ⊆ B›*) assms (*‹complete UNIV› ‹uniformly_continuous_on UNIV (λ(x, y). x + y)› ‹f summable_on A› ‹B ⊆ A›*) F1(1,2) (*‹finite F1› ‹F1 ⊆ A›*) F2(1,2) (*‹finite (F2::'a::type set)› ‹(F2::'a set) ⊆ (A::'a set)›*) apply (intro F1 (*‹finite F1› ‹F1 ⊆ A› ‹⟦F1 ⊆ ?F; finite ?F; ?F ⊆ A⟧ ⟹ P1 (sum f ?F)›*))\n (*goals:\n 1. ‹⟦F0B ⊆ F1'; F0B ⊆ F2'; finite F1'; finite F2'; F1' ⊆ B; F2' ⊆ B; complete UNIV; uniformly_continuous_on UNIV (λ(x, y). x + y); f summable_on A; B ⊆ A; finite F1; F1 ⊆ A; finite F2; F2 ⊆ A⟧ ⟹ F1 ⊆ F1' ∪ F0A›\n 2. ‹⟦F0B ⊆ F1'; F0B ⊆ F2'; finite F1'; finite F2'; F1' ⊆ B; F2' ⊆ B; complete UNIV; uniformly_continuous_on UNIV (λ(x, y). x + y); f summable_on A; B ⊆ A; finite F1; F1 ⊆ A; finite F2; F2 ⊆ A⟧ ⟹ finite (F1' ∪ F0A)›\n 3. ‹⟦F0B ⊆ F1'; F0B ⊆ F2'; finite F1'; finite F2'; F1' ⊆ B; F2' ⊆ B; complete UNIV; uniformly_continuous_on UNIV (λ(x, y). x + y); f summable_on A; B ⊆ A; finite F1; F1 ⊆ A; finite F2; F2 ⊆ A⟧ ⟹ F1' ∪ F0A ⊆ A›\n discuss goal 1*)\n apply ((auto simp: F0A_def (*‹F0A ≡ F0 - B›*) F0B_def (*‹F0B ≡ F0 ∩ B›*) F0_def (*‹F0 ≡ F1 ∪ F2›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: F0A_def (*‹F0A ≡ F0 - B›*) F0B_def (*‹F0B ≡ F0 ∩ B›*) F0_def (*‹F0 ≡ F1 ∪ F2›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: F0A_def (*‹F0A ≡ F0 - B›*) F0B_def (*‹F0B ≡ F0 ∩ B›*) F0_def (*‹F0 ≡ F1 ∪ F2›*))[1])\n (*proven 3 subgoals*) .\n thus \"P1 (sum f F1' + sum f F0A)\"\n apply (subst (asm) sum.union_disjoint (*‹⟦finite ?A; finite ?B; ?A ∩ ?B = {}⟧ ⟹ sum ?g (?A ∪ ?B) = sum ?g ?A + sum ?g ?B›*))\n (*goals:\n 1. ‹finite (F1'::'a set)›\n 2. ‹finite (F0A::'a set)›\n 3. ‹(F1'::'a set) ∩ (F0A::'a set) = {}›\n 4. ‹(P1::'b ⇒ bool) (sum (f::'a ⇒ 'b) (F1'::'a set) + sum f (F0A::'a set)) ⟹ P1 (sum f F1' + sum f F0A)›\n discuss goal 1*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*discuss goal 2*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*discuss goal 3*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*discuss goal 4*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*proven 4 subgoals*) .\n next\n (*goal: ‹P2 (sum f F2' + sum f F0A)›*)\n have \"P2 (sum f (F2' ∪ F0A))\"\n using that (*‹(F0B::'a set) ⊆ (F1'::'a set)› ‹(F0B::'a::type set) ⊆ (F2'::'a::type set)› ‹finite F1'› ‹finite F2'› ‹F1' ⊆ B› ‹F2' ⊆ B›*) assms (*‹complete UNIV› ‹uniformly_continuous_on UNIV (λ(x, y). x + y)› ‹f summable_on A› ‹B ⊆ A›*) F1(1,2) (*‹finite F1› ‹(F1::'a set) ⊆ (A::'a set)›*) F2(1,2) (*‹finite (F2::'a set)› ‹(F2::'a::type set) ⊆ (A::'a::type set)›*) apply (intro F2 (*‹finite (F2::'a::type set)› ‹(F2::'a::type set) ⊆ (A::'a::type set)› ‹⟦(F2::'a::type set) ⊆ (?F::'a::type set); finite ?F; ?F ⊆ (A::'a::type set)⟧ ⟹ (P2::'b::{ab_group_add,uniform_space} ⇒ bool) (sum (f::'a::type ⇒ 'b::{ab_group_add,uniform_space}) ?F)›*))\n (*goals:\n 1. ‹⟦F0B ⊆ F1'; F0B ⊆ F2'; finite F1'; finite F2'; F1' ⊆ B; F2' ⊆ B; complete UNIV; uniformly_continuous_on UNIV (λ(x, y). x + y); f summable_on A; B ⊆ A; finite F1; F1 ⊆ A; finite F2; F2 ⊆ A⟧ ⟹ F2 ⊆ F2' ∪ F0A›\n 2. ‹⟦F0B ⊆ F1'; F0B ⊆ F2'; finite F1'; finite F2'; F1' ⊆ B; F2' ⊆ B; complete UNIV; uniformly_continuous_on UNIV (λ(x, y). x + y); f summable_on A; B ⊆ A; finite F1; F1 ⊆ A; finite F2; F2 ⊆ A⟧ ⟹ finite (F2' ∪ F0A)›\n 3. ‹⟦F0B ⊆ F1'; F0B ⊆ F2'; finite F1'; finite F2'; F1' ⊆ B; F2' ⊆ B; complete UNIV; uniformly_continuous_on UNIV (λ(x, y). x + y); f summable_on A; B ⊆ A; finite F1; F1 ⊆ A; finite F2; F2 ⊆ A⟧ ⟹ F2' ∪ F0A ⊆ A›\n discuss goal 1*)\n apply ((auto simp: F0A_def (*‹F0A ≡ F0 - B›*) F0B_def (*‹F0B ≡ F0 ∩ B›*) F0_def (*‹F0 ≡ F1 ∪ F2›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: F0A_def (*‹F0A ≡ F0 - B›*) F0B_def (*‹F0B ≡ F0 ∩ B›*) F0_def (*‹F0 ≡ F1 ∪ F2›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: F0A_def (*‹F0A ≡ F0 - B›*) F0B_def (*‹F0B ≡ F0 ∩ B›*) F0_def (*‹F0 ≡ F1 ∪ F2›*))[1])\n (*proven 3 subgoals*) .\n thus \"P2 (sum f F2' + sum f F0A)\"\n apply (subst (asm) sum.union_disjoint (*‹⟦finite (?A::?'b::type set); finite (?B::?'b::type set); ?A ∩ ?B = {}⟧ ⟹ sum (?g::?'b::type ⇒ ?'a::comm_monoid_add) (?A ∪ ?B) = sum ?g ?A + sum ?g ?B›*))\n (*goals:\n 1. ‹finite F2'›\n 2. ‹finite F0A›\n 3. ‹F2' ∩ F0A = {}›\n 4. ‹P2 (sum f F2' + sum f F0A) ⟹ P2 (sum f F2' + sum f F0A)›\n discuss goal 1*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*discuss goal 2*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*discuss goal 3*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*discuss goal 4*)\n apply ((use that in ‹auto simp: F0A_def›)[1])\n (*proven 4 subgoals*) .\n qed\n have \"eventually (λx. E' (x, sum f F0B)) (filtermap (sum f) (finite_subsets_at_top B))\" and \"eventually (λx. E' (sum f F0B, x)) (filtermap (sum f) (finite_subsets_at_top B))\"\n unfolding eventually_filtermap eventually_finite_subsets_at_top\n (*goals:\n 1. ‹∃X. finite X ∧ X ⊆ B ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ E' (sum f Y, sum f F0B))›\n 2. ‹∃X. finite X ∧ X ⊆ B ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ E' (sum f F0B, sum f Y))›*)\n (*goals:\n 1. ‹∃X::'a set. finite X ∧ X ⊆ (B::'a set) ∧ (∀Y::'a set. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ (E'::'b × 'b ⇒ bool) (sum (f::'a ⇒ 'b) Y, sum f (F0B::'a set)))›\n 2. ‹∃X::'a set. finite X ∧ X ⊆ (B::'a set) ∧ (∀Y::'a set. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ (E'::'b × 'b ⇒ bool) (sum (f::'a ⇒ 'b) (F0B::'a set), sum f Y))›\n discuss goal 1*)\n apply (rule exI[of _ F0B] (*‹?P F0B ⟹ ∃x. ?P x›*))\n (*top goal: ‹∃X. finite X ∧ X ⊆ B ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ E' (sum f Y, sum f F0B))› and 1 goal remains*)\n apply (use * in ‹force simp: F0B_def›)\n (*discuss goal 2*)\n apply (rule exI[of _ F0B] (*‹?P F0B ⟹ ∃x. ?P x›*))\n (*goal: ‹∃X::'a set. finite X ∧ X ⊆ (B::'a set) ∧ (∀Y::'a set. finite Y ∧ X ⊆ Y ∧ Y ⊆ B ⟶ (E'::'b × 'b ⇒ bool) (sum (f::'a ⇒ 'b) (F0B::'a set), sum f Y))›*)\n apply (use * in ‹force simp: F0B_def›)\n (*proven 2 subgoals*) .\n then show \"eventually E (?filter_fB ×⇩F ?filter_fB)\"\n unfolding eventually_prod_filter\n (*goal: ‹∃Pf Pg. eventually Pf (filtermap (sum f) (finite_subsets_at_top B)) ∧ eventually Pg (filtermap (sum f) (finite_subsets_at_top B)) ∧ (∀x y. Pf x ⟶ Pg y ⟶ E (x, y))›*)\n using E'E'E (*‹E' (?x, ?y) ⟶ E' (?y, ?z) ⟶ E (?x, ?z)›*) by blast\n qed\n then obtain x where \"?filter_fB ≤ nhds x\"\n (*goal: ‹(⋀x. filtermap (sum f) (finite_subsets_at_top B) ≤ nhds x ⟹ thesis) ⟹ thesis›*)\n using cauchy_filter_complete_converges[of ?filter_fB UNIV] (*‹⟦cauchy_filter (filtermap (sum f) (finite_subsets_at_top B)); complete UNIV; filtermap (sum f) (finite_subsets_at_top B) ≤ principal UNIV; filtermap (sum f) (finite_subsets_at_top B) ≠ bot⟧ ⟹ ∃c. filtermap (sum f) (finite_subsets_at_top B) ≤ nhds c›*) ‹complete (UNIV :: _)› (*‹complete UNIV›*) by (auto simp: filtermap_bot_iff (*‹(filtermap ?f ?F = bot) = (?F = bot)›*))\n then have \"(sum f ⤏ x) (finite_subsets_at_top B)\"\n by (auto simp: filterlim_def (*‹filterlim ?f ?F2.0 ?F1.0 = (filtermap ?f ?F1.0 ≤ ?F2.0)›*))\n then show \"?thesis\"\n (*goal: ‹f summable_on B›*)\n by (auto simp: summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*) has_sum_def (*‹((?f::?'a ⇒ ?'b) has_sum (?x::?'b)) (?A::?'a set) ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\nqed\n\ntext ‹A special case of @{thm [source] summable_on_subset_aux} for Banach spaces with fewer premises.›\n\nlemma summable_on_subset_banach:\n fixes A B and f :: ‹'a ⇒ 'b::banach›\n assumes ‹f summable_on A›\n assumes ‹B ⊆ A›\n shows ‹f summable_on B›\n by (meson Cauchy_convergent (*‹Cauchy ?X ⟹ convergent ?X›*) UNIV_I (*‹?x ∈ UNIV›*) assms (*‹f summable_on A› ‹B ⊆ A›*) complete_def (*‹complete ?S = (∀f. (∀n. f n ∈ ?S) ∧ Cauchy f ⟶ (∃l∈?S. f ⇢ l))›*) convergent_def (*‹convergent ?X = (∃L. ?X ⇢ L)›*) isUCont_plus (*‹isUCont (λ(x, y). x + y)›*) summable_on_subset_aux (*‹⟦complete UNIV; uniformly_continuous_on UNIV (λ(x, y). x + y); ?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*))\n\nlemma has_sum_empty[simp]: ‹(f has_sum 0) {}›\n by (meson ex_in_conv (*‹(∃x. x ∈ ?A) = (?A ≠ {})›*) has_sum_0 (*‹(⋀x. x ∈ ?M ⟹ ?f x = 0) ⟹ (?f has_sum 0) ?M›*))\n\nlemma summable_on_empty[simp]: ‹f summable_on {}›\n by auto\n\nlemma infsum_empty[simp]: ‹infsum f {} = 0›\n by simp\n\nlemma sum_has_sum:\n fixes f :: \"'a ⇒ 'b::topological_comm_monoid_add\"\n assumes ‹finite A›\n assumes ‹⋀a. a ∈ A ⟹ (f has_sum (s a)) (B a)›\n assumes ‹⋀a a'. a∈A ⟹ a'∈A ⟹ a≠a' ⟹ B a ∩ B a' = {}›\n shows ‹(f has_sum (sum s A)) (⋃a∈A. B a)›\n using assms (*‹finite A› ‹?a ∈ A ⟹ (f has_sum s ?a) (B ?a)› ‹⟦?a ∈ A; ?a' ∈ A; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*) proof (induction)\n (*goals:\n 1. ‹⟦⋀a. a ∈ {} ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ {}; a' ∈ {}; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s {}) (⋃ (B ` {}))›\n 2. ‹⋀x F. ⟦finite F; x ∉ F; ⟦⋀a. a ∈ F ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ F; a' ∈ F; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s F) (⋃ (B ` F)); ⋀a. a ∈ insert x F ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ insert x F; a' ∈ insert x F; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s (insert x F)) (⋃ (B ` insert x F))›*)\n case empty (*‹(?a::'c::type) ∈ {} ⟹ ((f::'a::type ⇒ 'b::topological_comm_monoid_add) has_sum (s::'c::type ⇒ 'b::topological_comm_monoid_add) ?a) ((B::'c::type ⇒ 'a::type set) ?a)› ‹⟦?a ∈ {}; ?a' ∈ {}; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*)\n then show \"?case\"\n (*goal: ‹(f has_sum sum s {}) (⋃ (B ` {}))›*)\n by simp\n next\n (*goal: ‹⋀x F. ⟦finite F; x ∉ F; ⟦⋀a. a ∈ F ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ F; a' ∈ F; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s F) (⋃ (B ` F)); ⋀a. a ∈ insert x F ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ insert x F; a' ∈ insert x F; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s (insert x F)) (⋃ (B ` insert x F))›*)\n case (insert x A) (*‹finite A› ‹(x::'c) ∉ (A::'c set)› ‹⟦⋀a. a ∈ A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ A; a' ∈ A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s A) (⋃ (B ` A))› ‹?a ∈ insert x A ⟹ (f has_sum s ?a) (B ?a)› ‹⟦?a ∈ insert x A; ?a' ∈ insert x A; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*)\n have \"(f has_sum (s x)) (B x)\"\n by (simp add: insert.prems (*‹?a ∈ insert x A ⟹ (f has_sum s ?a) (B ?a)› ‹⟦?a ∈ insert x A; ?a' ∈ insert x A; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*))\n moreover have IH: \"(f has_sum (sum s A)) (⋃a∈A. B a)\"\n using insert (*‹finite (A::'c::type set)› ‹x ∉ A› ‹⟦⋀a. a ∈ A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ A; a' ∈ A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s A) (⋃ (B ` A))› ‹?a ∈ insert x A ⟹ (f has_sum s ?a) (B ?a)› ‹⟦?a ∈ insert x A; ?a' ∈ insert x A; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*) by simp\n ultimately have \"(f has_sum (s x + sum s A)) (B x ∪ (⋃a∈A. B a))\"\n using insert (*‹finite (A::'c set)› ‹x ∉ A› ‹⟦⋀a. a ∈ A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ A; a' ∈ A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s A) (⋃ (B ` A))› ‹?a ∈ insert x A ⟹ (f has_sum s ?a) (B ?a)› ‹⟦?a ∈ insert x A; ?a' ∈ insert x A; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*) apply (intro has_sum_Un_disjoint (*‹⟦(?f has_sum ?a) ?A; (?f has_sum ?b) ?B; ?A ∩ ?B = {}⟧ ⟹ (?f has_sum ?a + ?b) (?A ∪ ?B)›*))\n (*goals:\n 1. ‹⟦(f has_sum s x) (B x); (f has_sum sum s A) (⋃ (B ` A)); finite A; x ∉ A; ⟦⋀a. a ∈ A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ A; a' ∈ A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s A) (⋃ (B ` A)); ⋀a. a ∈ insert x A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ insert x A; a' ∈ insert x A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum s x) (B x)›\n 2. ‹⟦(f has_sum s x) (B x); (f has_sum sum s A) (⋃ (B ` A)); finite A; x ∉ A; ⟦⋀a. a ∈ A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ A; a' ∈ A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s A) (⋃ (B ` A)); ⋀a. a ∈ insert x A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ insert x A; a' ∈ insert x A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s A) (⋃ (B ` A))›\n 3. ‹⟦(f has_sum s x) (B x); (f has_sum sum s A) (⋃ (B ` A)); finite A; x ∉ A; ⟦⋀a. a ∈ A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ A; a' ∈ A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum s A) (⋃ (B ` A)); ⋀a. a ∈ insert x A ⟹ (f has_sum s a) (B a); ⋀a a'. ⟦a ∈ insert x A; a' ∈ insert x A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ B x ∩ ⋃ (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 then show \"?case\"\n (*goal: ‹(f has_sum sum s (insert x A)) (⋃ (B ` insert x A))›*)\n using insert.hyps (*‹finite A› ‹(x::'c) ∉ (A::'c set)›*) by auto\n qed\n\n\nlemma summable_on_finite_union_disjoint:\n fixes f :: \"'a ⇒ 'b::topological_comm_monoid_add\"\n assumes finite: ‹finite A›\n assumes conv: ‹⋀a. a ∈ A ⟹ f summable_on (B a)›\n assumes disj: ‹⋀a a'. a∈A ⟹ a'∈A ⟹ a≠a' ⟹ B a ∩ B a' = {}›\n shows ‹f summable_on (⋃a∈A. B a)›\n using sum_has_sum[of A f B] (*‹⟦finite A; ⋀a. a ∈ A ⟹ (f has_sum ?s a) (B a); ⋀a a'. ⟦a ∈ A; a' ∈ A; a ≠ a'⟧ ⟹ B a ∩ B a' = {}⟧ ⟹ (f has_sum sum ?s A) (⋃ (B ` A))›*) assms (*‹finite A› ‹?a ∈ A ⟹ f summable_on B ?a› ‹⟦?a ∈ A; ?a' ∈ A; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*) unfolding summable_on_def\n (*goal: ‹∃x. (f has_sum x) (⋃ (B ` A))›*)\n by metis\n\nlemma sum_infsum:\n fixes f :: \"'a ⇒ 'b::{topological_comm_monoid_add, t2_space}\"\n assumes finite: ‹finite A›\n assumes conv: ‹⋀a. a ∈ A ⟹ f summable_on (B a)›\n assumes disj: ‹⋀a a'. a∈A ⟹ a'∈A ⟹ a≠a' ⟹ B a ∩ B a' = {}›\n shows ‹sum (λa. infsum f (B a)) A = infsum f (⋃a∈A. B a)›\n by (metis (no_types, lifting) assms (*‹finite A› ‹?a ∈ A ⟹ f summable_on B ?a› ‹⟦?a ∈ A; ?a' ∈ A; ?a ≠ ?a'⟧ ⟹ B ?a ∩ B ?a' = {}›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) sum_has_sum (*‹⟦finite ?A; ⋀a. a ∈ ?A ⟹ (?f has_sum ?s a) (?B a); ⋀a a'. ⟦a ∈ ?A; a' ∈ ?A; a ≠ a'⟧ ⟹ ?B a ∩ ?B a' = {}⟧ ⟹ (?f has_sum sum ?s ?A) (⋃ (?B ` ?A))›*))\n\ntext ‹The lemmas ‹infsum_comm_additive_general› and ‹infsum_comm_additive› (and variants) below both state that the infinite sum commutes with\n a continuous additive function. ‹infsum_comm_additive_general› is stated more for more general type classes\n at the expense of a somewhat less compact formulation of the premises.\n E.g., by avoiding the constant \\<^const>‹additive› which introduces an additional sort constraint\n (group instead of monoid). For example, extended reals (\\<^typ>‹ereal›, \\<^typ>‹ennreal›) are not covered\n by ‹infsum_comm_additive›.›\n\n\nlemma has_sum_comm_additive_general: \n fixes f :: ‹'b :: {comm_monoid_add,topological_space} ⇒ 'c :: {comm_monoid_add,topological_space}›\n assumes f_sum: ‹⋀F. finite F ⟹ F ⊆ S ⟹ sum (f ∘ g) F = f (sum g F)›\n ― ‹Not using \\<^const>‹additive› because it would add sort constraint \\<^class>‹ab_group_add››\n assumes cont: ‹f ─x→ f x›\n ― ‹For \\<^class>‹t2_space›, this is equivalent to ‹isCont f x› by @{thm [source] isCont_def}.›\n assumes infsum: ‹(g has_sum x) S›\n shows ‹((f ∘ g) has_sum (f x)) S› \nproof (-)\n (*goal: ‹((f::'b ⇒ 'c) ∘ (g::'a ⇒ 'b) has_sum f (x::'b)) (S::'a set)›*)\n have \"(sum g ⤏ x) (finite_subsets_at_top S)\"\n using infsum (*‹(g has_sum x) S›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\n then have \"((f ∘ sum g) ⤏ f x) (finite_subsets_at_top S)\"\n by (meson cont (*‹f ─x→ f x›*) filterlim_def (*‹filterlim ?f ?F2.0 ?F1.0 = (filtermap ?f ?F1.0 ≤ ?F2.0)›*) tendsto_at_iff_tendsto_nhds (*‹?g ─?l→ ?g ?l = (?g ⤏ ?g ?l) (nhds ?l)›*) tendsto_compose_filtermap (*‹((?g ∘ ?f) ⤏ ?T) ?F = (?g ⤏ ?T) (filtermap ?f ?F)›*) tendsto_mono (*‹⟦?F ≤ ?F'; (?f ⤏ ?l) ?F'⟧ ⟹ (?f ⤏ ?l) ?F›*))\n then have \"(sum (f ∘ g) ⤏ f x) (finite_subsets_at_top S)\"\n using tendsto_cong (*‹∀⇩F x in ?F. ?f x = ?g x ⟹ (?f ⤏ ?c) ?F = (?g ⤏ ?c) ?F›*) f_sum (*‹⟦finite (?F::'a set); ?F ⊆ (S::'a set)⟧ ⟹ sum ((f::'b ⇒ 'c) ∘ (g::'a ⇒ 'b)) ?F = f (sum g ?F)›*) by (simp add: Lim_transform_eventually (*‹⟦(?f ⤏ ?l) ?F; ∀⇩F x in ?F. ?f x = ?g x⟧ ⟹ (?g ⤏ ?l) ?F›*) eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*))\n then show \"((f ∘ g) has_sum (f x)) S\"\n using has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\nqed\n\nlemma summable_on_comm_additive_general:\n fixes f :: ‹'b :: {comm_monoid_add,topological_space} ⇒ 'c :: {comm_monoid_add,topological_space}›\n assumes ‹⋀F. finite F ⟹ F ⊆ S ⟹ sum (f ∘ g) F = f (sum g F)›\n ― ‹Not using \\<^const>‹additive› because it would add sort constraint \\<^class>‹ab_group_add››\n assumes ‹⋀x. (g has_sum x) S ⟹ f ─x→ f x›\n ― ‹For \\<^class>‹t2_space›, this is equivalent to ‹isCont f x› by @{thm [source] isCont_def}.›\n assumes ‹g summable_on S›\n shows ‹(f ∘ g) summable_on S›\n by (meson assms (*‹⟦finite ?F; ?F ⊆ S⟧ ⟹ sum (f ∘ g) ?F = f (sum g ?F)› ‹(g has_sum ?x) S ⟹ f ─?x→ f ?x› ‹g summable_on S›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); ?f ─?x→ ?f ?x; (?g has_sum ?x) ?S⟧ ⟹ (?f ∘ ?g has_sum ?f ?x) ?S›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) infsum_tendsto (*‹?f summable_on ?S ⟹ (sum ?f ⤏ infsum ?f ?S) (finite_subsets_at_top ?S)›*))\n\nlemma infsum_comm_additive_general:\n fixes f :: ‹'b :: {comm_monoid_add,t2_space} ⇒ 'c :: {comm_monoid_add,t2_space}›\n assumes f_sum: ‹⋀F. finite F ⟹ F ⊆ S ⟹ sum (f ∘ g) F = f (sum g F)›\n ― ‹Not using \\<^const>‹additive› because it would add sort constraint \\<^class>‹ab_group_add››\n assumes ‹isCont f (infsum g S)›\n assumes ‹g summable_on S›\n shows ‹infsum (f ∘ g) S = f (infsum g S)›\n using assms (*‹⟦finite ?F; ?F ⊆ S⟧ ⟹ sum (f ∘ g) ?F = f (sum g ?F)› ‹isCont f (infsum g S)› ‹g summable_on S›*) apply (intro infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) has_sum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); ?f ─?x→ ?f ?x; (?g has_sum ?x) ?S⟧ ⟹ (?f ∘ ?g has_sum ?f ?x) ?S›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*))\n (*goals:\n 1. ‹⋀F::'a set. ⟦⋀F::'a set. ⟦finite F; F ⊆ (S::'a set)⟧ ⟹ sum ((f::'b ⇒ 'c) ∘ (g::'a ⇒ 'b)) F = f (sum g F); isCont f (infsum g S); g summable_on S; finite F; F ⊆ S⟧ ⟹ sum (f ∘ g) F = f (sum g F)›\n 2. ‹⟦⋀F::'a set. ⟦finite F; F ⊆ (S::'a set)⟧ ⟹ sum ((f::'b ⇒ 'c) ∘ (g::'a ⇒ 'b)) F = f (sum g F); isCont f (infsum g S); g summable_on S⟧ ⟹ f ─infsum g S→ f (infsum g S)›\n 3. ‹⟦⋀F::'a set. ⟦finite F; F ⊆ (S::'a set)⟧ ⟹ sum ((f::'b ⇒ 'c) ∘ (g::'a ⇒ 'b)) F = f (sum g F); isCont f (infsum g S); g summable_on S⟧ ⟹ g summable_on S›\n discuss goal 1*)\n apply ((auto simp: isCont_def (*‹isCont ?f ?a = ?f ─?a→ ?f ?a›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: isCont_def (*‹isCont ?f ?a = ?f ─?a→ ?f ?a›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: isCont_def (*‹isCont ?f ?a = ?f ─?a→ ?f ?a›*))[1])\n (*proven 3 subgoals*) .\n\nlemma has_sum_comm_additive: \n fixes f :: ‹'b :: {ab_group_add,topological_space} ⇒ 'c :: {ab_group_add,topological_space}›\n assumes ‹additive f›\n assumes ‹f ─x→ f x›\n ― ‹For \\<^class>‹t2_space›, this is equivalent to ‹isCont f x› by @{thm [source] isCont_def}.›\n assumes infsum: ‹(g has_sum x) S›\n shows ‹((f ∘ g) has_sum (f x)) S›\n using assms (*‹additive f› ‹f ─x→ f x› ‹(g has_sum x) S›*) apply (intro has_sum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); ?f ─?x→ ?f ?x; (?g has_sum ?x) ?S⟧ ⟹ (?f ∘ ?g has_sum ?f ?x) ?S›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*))\n (*goals:\n 1. ‹⋀F. ⟦additive f; f ─x→ f x; (g has_sum x) S; finite F; F ⊆ S⟧ ⟹ sum (f ∘ g) F = f (sum g F)›\n 2. ‹⟦additive f; f ─x→ f x; (g has_sum x) S⟧ ⟹ f ─x→ f x›\n 3. ‹⟦additive f; f ─x→ f x; (g has_sum x) S⟧ ⟹ (g has_sum x) S›\n discuss goal 1*)\n apply ((auto simp: isCont_def (*‹isCont ?f ?a = ?f ─?a→ ?f ?a›*) additive.sum (*‹additive ?f ⟹ ?f (sum ?g ?A) = (∑x∈?A. ?f (?g x))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: isCont_def (*‹isCont (?f::?'a ⇒ ?'b) (?a::?'a) = ?f ─?a→ ?f ?a›*) additive.sum (*‹additive (?f::?'a ⇒ ?'b) ⟹ ?f (sum (?g::?'c ⇒ ?'a) (?A::?'c set)) = (∑x::?'c∈?A. ?f (?g x))›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: isCont_def (*‹isCont ?f ?a = ?f ─?a→ ?f ?a›*) additive.sum (*‹additive ?f ⟹ ?f (sum ?g ?A) = (∑x∈?A. ?f (?g x))›*))[1])\n (*proven 3 subgoals*) . \n\nlemma summable_on_comm_additive:\n fixes f :: ‹'b :: {ab_group_add,t2_space} ⇒ 'c :: {ab_group_add,topological_space}›\n assumes ‹additive f›\n assumes ‹isCont f (infsum g S)›\n assumes ‹g summable_on S›\n shows ‹(f ∘ g) summable_on S›\n by (meson assms (*‹additive (f::'b ⇒ 'c)› ‹isCont (f::'b ⇒ 'c) (infsum (g::'a ⇒ 'b) (S::'a set))› ‹(g::'a ⇒ 'b) summable_on (S::'a set)›*) summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*) has_sum_comm_additive (*‹⟦additive (?f::?'b ⇒ ?'c); ?f ─?x::?'b→ ?f ?x; ((?g::?'a ⇒ ?'b) has_sum ?x) (?S::?'a set)⟧ ⟹ (?f ∘ ?g has_sum ?f ?x) ?S›*) has_sum_infsum (*‹(?f::?'a ⇒ ?'b) summable_on (?S::?'a set) ⟹ (?f has_sum infsum ?f ?S) ?S›*) isContD (*‹isCont (?f::?'a ⇒ ?'b) (?x::?'a) ⟹ ?f ─?x→ ?f ?x›*))\n\nlemma infsum_comm_additive:\n fixes f :: ‹'b :: {ab_group_add,t2_space} ⇒ 'c :: {ab_group_add,t2_space}›\n assumes ‹additive f›\n assumes ‹isCont f (infsum g S)›\n assumes ‹g summable_on S›\n shows ‹infsum (f ∘ g) S = f (infsum g S)›\n apply (rule infsum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ infsum (?f ∘ ?g) ?S = ?f (infsum ?g ?S)›*))\n (*goals:\n 1. ‹⋀F::'a set. ⟦finite F; F ⊆ (S::'a set)⟧ ⟹ sum ((f::'b ⇒ 'c) ∘ (g::'a ⇒ 'b)) F = f (sum g F)›\n 2. ‹isCont (f::'b ⇒ 'c) (infsum (g::'a ⇒ 'b) (S::'a set))›\n 3. ‹(g::'a ⇒ 'b) summable_on (S::'a set)›\n discuss goal 1*)\n apply ((auto simp: assms (*‹additive f› ‹isCont f (infsum g S)› ‹g summable_on S›*) additive.sum (*‹additive ?f ⟹ ?f (sum ?g ?A) = (∑x∈?A. ?f (?g x))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: assms (*‹additive f› ‹isCont f (infsum g S)› ‹g summable_on S›*) additive.sum (*‹additive ?f ⟹ ?f (sum ?g ?A) = (∑x∈?A. ?f (?g x))›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: assms (*‹additive f› ‹isCont f (infsum g S)› ‹g summable_on S›*) additive.sum (*‹additive ?f ⟹ ?f (sum ?g ?A) = (∑x∈?A. ?f (?g x))›*))[1])\n (*proven 3 subgoals*) .\n\nlemma nonneg_bdd_above_has_sum:\n fixes f :: ‹'a ⇒ 'b :: {conditionally_complete_linorder, ordered_comm_monoid_add, linorder_topology}›\n assumes ‹⋀x. x∈A ⟹ f x ≥ 0›\n assumes ‹bdd_above (sum f ` {F. F⊆A ∧ finite F})›\n shows ‹(f has_sum (SUP F∈{F. finite F ∧ F⊆A}. sum f F)) A›\nproof (-)\n (*goal: ‹(f has_sum Sup (sum f ` {F. finite F ∧ F ⊆ A})) A›*)\n have \"(sum f ⤏ (SUP F∈{F. finite F ∧ F⊆A}. sum f F)) (finite_subsets_at_top A)\"\n proof (rule order_tendstoI (*‹⟦⋀a::?'a::order_topology. a < (?y::?'a::order_topology) ⟹ ∀⇩F x::?'b::type in ?F::?'b::type filter. a < (?f::?'b::type ⇒ ?'a::order_topology) x; ⋀a::?'a::order_topology. ?y < a ⟹ ∀⇩F x::?'b::type in ?F. ?f x < a⟧ ⟹ (?f ⤏ ?y) ?F›*))\n (*goals:\n 1. ‹⋀a. a < Sup (sum f ` {F. finite F ∧ F ⊆ A}) ⟹ ∀⇩F x in finite_subsets_at_top A. a < sum f x›\n 2. ‹⋀a. Sup (sum f ` {F. finite F ∧ F ⊆ A}) < a ⟹ ∀⇩F x in finite_subsets_at_top A. sum f x < a›*)\n fix a\n assume \"a < (SUP F∈{F. finite F ∧ F⊆A}. sum f F)\" (*‹(a::'b) < Sup (sum (f::'a ⇒ 'b) ` {F::'a set. finite F ∧ F ⊆ (A::'a set)})›*)\n then obtain F where \"a < sum f F\" and \"finite F\" and \"F ⊆ A\"\n (*goal: ‹(⋀F. ⟦a < sum f F; finite F; F ⊆ A⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis (mono_tags, lifting) Collect_cong (*‹(⋀x::?'a. (?P::?'a ⇒ bool) x = (?Q::?'a ⇒ bool) x) ⟹ {x::?'a. ?P x} = {x::?'a. ?Q x}›*) Collect_empty_eq (*‹(Collect (?P::?'a ⇒ bool) = {}) = (∀x::?'a. ¬ ?P x)›*) assms( (*‹bdd_above (sum (f::'a ⇒ 'b) ` {F::'a set. F ⊆ (A::'a set) ∧ finite F})›*) 2) empty_subsetI (*‹{} ⊆ (?A::?'a set)›*) finite.emptyI (*‹finite {}›*) less_cSUP_iff (*‹⟦(?A::?'b set) ≠ {}; bdd_above ((?f::?'b ⇒ ?'a) ` ?A)⟧ ⟹ ((?a::?'a) < Sup (?f ` ?A)) = (∃x::?'b∈?A. ?a < ?f x)›*) mem_Collect_eq (*‹((?a::?'a) ∈ Collect (?P::?'a ⇒ bool)) = ?P ?a›*))\n have \"⋀Y. ⟦finite Y; F ⊆ Y; Y ⊆ A⟧ ⟹ a < sum f Y\"\n by (meson DiffE (*‹⟦?c ∈ ?A - ?B; ⟦?c ∈ ?A; ?c ∉ ?B⟧ ⟹ ?P⟧ ⟹ ?P›*) ‹a < sum f F› assms( (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) 1) less_le_trans (*‹⟦?x < ?y; ?y ≤ ?z⟧ ⟹ ?x < ?z›*) subset_iff (*‹(?A ⊆ ?B) = (∀t. t ∈ ?A ⟶ t ∈ ?B)›*) sum_mono2 (*‹⟦finite ?B; ?A ⊆ ?B; ⋀b. b ∈ ?B - ?A ⟹ 0 ≤ ?f b⟧ ⟹ sum ?f ?A ≤ sum ?f ?B›*))\n then show \"∀⇩F x in finite_subsets_at_top A. a < sum f x\"\n by (metis ‹F ⊆ A› ‹finite F› eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n next\n (*goal: ‹⋀a. Sup (sum f ` {F. finite F ∧ F ⊆ A}) < a ⟹ ∀⇩F x in finite_subsets_at_top A. sum f x < a›*)\n fix a\n assume \"*\": \"(SUP F∈{F. finite F ∧ F⊆A}. sum f F) < a\" (*‹Sup (sum (f::'a ⇒ 'b) ` {F::'a set. finite F ∧ F ⊆ (A::'a set)}) < (a::'b)›*)\n have \"sum f F ≤ (SUP F∈{F. finite F ∧ F⊆A}. sum f F)\" if \"F⊆A\" and \"finite F\" for F\n apply (rule cSUP_upper (*‹⟦(?x::?'b) ∈ (?A::?'b set); bdd_above ((?f::?'b ⇒ ?'a) ` ?A)⟧ ⟹ ?f ?x ≤ Sup (?f ` ?A)›*))\n (*goals:\n 1. ‹F ∈ {F. finite F ∧ F ⊆ A}›\n 2. ‹bdd_above (sum f ` {F. finite F ∧ F ⊆ A})›\n discuss goal 1*)\n apply ((use that assms(2) in ‹auto simp: conj_commute›)[1])\n (*discuss goal 2*)\n apply ((use that assms(2) in ‹auto simp: conj_commute›)[1])\n (*proven 2 subgoals*) .\n then show \"∀⇩F x in finite_subsets_at_top A. sum f x < a\"\n by (metis (no_types, lifting) \"*\" (*‹Sup (sum f ` {F. finite F ∧ F ⊆ A}) < a›*) eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*) order_le_less_trans (*‹⟦?x ≤ ?y; ?y < ?z⟧ ⟹ ?x < ?z›*))\n qed\n then show \"?thesis\"\n (*goal: ‹(f has_sum Sup (sum f ` {F. finite F ∧ F ⊆ A})) A›*)\n using has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\nqed\n\nlemma nonneg_bdd_above_summable_on:\n fixes f :: ‹'a ⇒ 'b :: {conditionally_complete_linorder, ordered_comm_monoid_add, linorder_topology}›\n assumes ‹⋀x. x∈A ⟹ f x ≥ 0›\n assumes ‹bdd_above (sum f ` {F. F⊆A ∧ finite F})›\n shows ‹f summable_on A›\n using assms (*‹(?x::'a::type) ∈ (A::'a::type set) ⟹ (0::'b::{conditionally_complete_linorder,ordered_comm_monoid_add,linorder_topology}) ≤ (f::'a::type ⇒ 'b::{conditionally_complete_linorder,ordered_comm_monoid_add,linorder_topology}) ?x› ‹bdd_above (sum (f::'a::type ⇒ 'b::{conditionally_complete_linorder,ordered_comm_monoid_add,linorder_topology}) ` {F::'a::type set. F ⊆ (A::'a::type set) ∧ finite F})›*) summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*) nonneg_bdd_above_has_sum (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ (?f has_sum Sup (sum ?f ` {F. finite F ∧ F ⊆ ?A})) ?A›*) by blast\n\nlemma nonneg_bdd_above_infsum:\n fixes f :: ‹'a ⇒ 'b :: {conditionally_complete_linorder, ordered_comm_monoid_add, linorder_topology}›\n assumes ‹⋀x. x∈A ⟹ f x ≥ 0›\n assumes ‹bdd_above (sum f ` {F. F⊆A ∧ finite F})›\n shows ‹infsum f A = (SUP F∈{F. finite F ∧ F⊆A}. sum f F)›\n using assms (*‹?x ∈ A ⟹ 0 ≤ f ?x› ‹bdd_above (sum f ` {F. F ⊆ A ∧ finite F})›*) by (auto intro!: infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) nonneg_bdd_above_has_sum (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ (?f has_sum Sup (sum ?f ` {F. finite F ∧ F ⊆ ?A})) ?A›*))\n\nlemma nonneg_has_sum_complete:\n fixes f :: ‹'a ⇒ 'b :: {complete_linorder, ordered_comm_monoid_add, linorder_topology}›\n assumes ‹⋀x. x∈A ⟹ f x ≥ 0›\n shows ‹(f has_sum (SUP F∈{F. finite F ∧ F⊆A}. sum f F)) A›\n using assms (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) nonneg_bdd_above_has_sum (*‹⟦⋀x::?'a::type. x ∈ (?A::?'a::type set) ⟹ (0::?'b::{conditionally_complete_linorder,ordered_comm_monoid_add,linorder_topology}) ≤ (?f::?'a::type ⇒ ?'b::{conditionally_complete_linorder,ordered_comm_monoid_add,linorder_topology}) x; bdd_above (sum ?f ` {F::?'a::type set. F ⊆ ?A ∧ finite F})⟧ ⟹ (?f has_sum Sup (sum ?f ` {F::?'a::type set. finite F ∧ F ⊆ ?A})) ?A›*) by blast\n\nlemma nonneg_summable_on_complete:\n fixes f :: ‹'a ⇒ 'b :: {complete_linorder, ordered_comm_monoid_add, linorder_topology}›\n assumes ‹⋀x. x∈A ⟹ f x ≥ 0›\n shows ‹f summable_on A›\n using assms (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) nonneg_bdd_above_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ ?f summable_on ?A›*) by blast\n\nlemma nonneg_infsum_complete:\n fixes f :: ‹'a ⇒ 'b :: {complete_linorder, ordered_comm_monoid_add, linorder_topology}›\n assumes ‹⋀x. x∈A ⟹ f x ≥ 0›\n shows ‹infsum f A = (SUP F∈{F. finite F ∧ F⊆A}. sum f F)›\n using assms (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) nonneg_bdd_above_infsum (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ infsum ?f ?A = Sup (sum ?f ` {F. finite F ∧ F ⊆ ?A})›*) by blast\n\nlemma has_sum_nonneg:\n fixes f :: \"'a ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}\"\n assumes \"(f has_sum a) M\"\n and \"⋀x. x ∈ M ⟹ 0 ≤ f x\"\n shows \"a ≥ 0\"\n by (metis (no_types, lifting) DiffD1 (*‹?c ∈ ?A - ?B ⟹ ?c ∈ ?A›*) assms (*‹(f has_sum a) M› ‹?x ∈ M ⟹ 0 ≤ f ?x›*) empty_iff (*‹(?c ∈ {}) = False›*) has_sum_0 (*‹(⋀x. x ∈ ?M ⟹ ?f x = 0) ⟹ (?f has_sum 0) ?M›*) has_sum_mono_neutral (*‹⟦(?f has_sum ?a) ?A; (?g has_sum ?b) ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ ?a ≤ ?b›*) order_refl (*‹?x ≤ ?x›*))\n\nlemma infsum_nonneg:\n fixes f :: \"'a ⇒ 'b::{ordered_comm_monoid_add,linorder_topology}\"\n assumes \"⋀x. x ∈ M ⟹ 0 ≤ f x\"\n shows \"infsum f M ≥ 0\" (is \"?lhs ≥ _\")\n by (metis assms (*‹?x ∈ M ⟹ 0 ≤ f ?x›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_nonneg (*‹⟦(?f has_sum ?a) ?M; ⋀x. x ∈ ?M ⟹ 0 ≤ ?f x⟧ ⟹ 0 ≤ ?a›*) infsum_not_exists (*‹¬ ?f summable_on ?A ⟹ infsum ?f ?A = 0›*) linorder_linear (*‹?x ≤ ?y ∨ ?y ≤ ?x›*))\n\nlemma has_sum_mono2:\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}\"\n assumes \"(f has_sum S) A\" \"(f has_sum S') B\" \"A ⊆ B\"\n assumes \"⋀x. x ∈ B - A ⟹ f x ≥ 0\"\n shows \"S ≤ S'\"\n by (metis add_0 (*‹0 + ?a = ?a›*) add_right_mono (*‹?a ≤ ?b ⟹ ?a + ?c ≤ ?b + ?c›*) assms (*‹(f has_sum S) A› ‹(f has_sum S') B› ‹A ⊆ B› ‹?x ∈ B - A ⟹ 0 ≤ f ?x›*) diff_add_cancel (*‹?a - ?b + ?b = ?a›*) has_sum_Diff (*‹⟦(?f has_sum ?b) ?B; (?f has_sum ?a) ?A; ?A ⊆ ?B⟧ ⟹ (?f has_sum ?b - ?a) (?B - ?A)›*) has_sum_nonneg (*‹⟦(?f has_sum ?a) ?M; ⋀x. x ∈ ?M ⟹ 0 ≤ ?f x⟧ ⟹ 0 ≤ ?a›*))\n\nlemma infsum_mono2:\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}\"\n assumes \"f summable_on A\" \"f summable_on B\" \"A ⊆ B\"\n assumes \"⋀x. x ∈ B - A ⟹ f x ≥ 0\"\n shows \"infsum f A ≤ infsum f B\"\n apply (rule has_sum_mono2[OF has_sum_infsum has_sum_infsum] (*‹⟦?f summable_on ?A; ?f summable_on ?B; ?A ⊆ ?B; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?f x⟧ ⟹ infsum ?f ?A ≤ infsum ?f ?B›*))\n (*goals:\n 1. ‹(f::'a ⇒ 'b) summable_on (A::'a set)›\n 2. ‹(f::'a ⇒ 'b) summable_on (B::'a set)›\n 3. ‹(A::'a set) ⊆ (B::'a set)›\n 4. ‹⋀x::'a. x ∈ (B::'a set) - (A::'a set) ⟹ (0::'b) ≤ (f::'a ⇒ 'b) x›\n discuss goal 1*)\n apply ((use assms in auto)[1])\n (*discuss goal 2*)\n apply ((use assms in auto)[1])\n (*discuss goal 3*)\n apply ((use assms in auto)[1])\n (*discuss goal 4*)\n apply ((use assms in auto)[1])\n (*proven 4 subgoals*) .\n\nlemma finite_sum_le_has_sum:\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}\"\n assumes \"(f has_sum S) A\" \"finite B\" \"B ⊆ A\"\n assumes \"⋀x. x ∈ A - B ⟹ f x ≥ 0\"\n shows \"sum f B ≤ S\"\n by (meson assms (*‹(f has_sum S) A› ‹finite B› ‹B ⊆ A› ‹?x ∈ A - B ⟹ 0 ≤ f ?x›*) has_sum_finite (*‹finite ?F ⟹ (?f has_sum sum ?f ?F) ?F›*) has_sum_mono2 (*‹⟦(?f has_sum ?S) ?A; (?f has_sum ?S') ?B; ?A ⊆ ?B; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?f x⟧ ⟹ ?S ≤ ?S'›*))\n\nlemma finite_sum_le_infsum:\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add, ordered_comm_monoid_add,linorder_topology}\"\n assumes \"f summable_on A\" \"finite B\" \"B ⊆ A\"\n assumes \"⋀x. x ∈ A - B ⟹ f x ≥ 0\"\n shows \"sum f B ≤ infsum f A\"\n apply (rule finite_sum_le_has_sum[OF has_sum_infsum] (*‹⟦?f summable_on ?A; finite ?B; ?B ⊆ ?A; ⋀x. x ∈ ?A - ?B ⟹ 0 ≤ ?f x⟧ ⟹ sum ?f ?B ≤ infsum ?f ?A›*))\n (*goals:\n 1. ‹f summable_on A›\n 2. ‹finite B›\n 3. ‹B ⊆ A›\n 4. ‹⋀x. x ∈ A - B ⟹ 0 ≤ f x›\n discuss goal 1*)\n apply ((use assms in auto)[1])\n (*discuss goal 2*)\n apply ((use assms in auto)[1])\n (*discuss goal 3*)\n apply ((use assms in auto)[1])\n (*discuss goal 4*)\n apply ((use assms in auto)[1])\n (*proven 4 subgoals*) .\n\nlemma has_sum_reindex:\n assumes ‹inj_on h A›\n shows ‹(g has_sum x) (h ` A) ⟷ ((g ∘ h) has_sum x) A›\nproof (-)\n (*goal: ‹(g has_sum x) (h ` A) = (g ∘ h has_sum x) A›*)\n have \"(g has_sum x) (h ` A) ⟷ (sum g ⤏ x) (finite_subsets_at_top (h ` A))\"\n by (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n also (*calculation: ‹(g has_sum x) (h ` A) = (sum g ⤏ x) (finite_subsets_at_top (h ` A))›*) have \"… ⟷ ((λF. sum g (h ` F)) ⤏ x) (finite_subsets_at_top A)\"\n by (metis assms (*‹inj_on h A›*) filterlim_filtermap (*‹filterlim ?f ?F1.0 (filtermap ?g ?F2.0) = (LIM x ?F2.0. ?f (?g x) :> ?F1.0)›*) filtermap_image_finite_subsets_at_top (*‹inj_on ?f ?A ⟹ filtermap ((`) ?f) (finite_subsets_at_top ?A) = finite_subsets_at_top (?f ` ?A)›*))\n also (*calculation: ‹(g has_sum x) (h ` A) = ((λF. sum g (h ` F)) ⤏ x) (finite_subsets_at_top A)›*) have \"… ⟷ (sum (g ∘ h) ⤏ x) (finite_subsets_at_top A)\"\n proof (intro tendsto_cong (*‹∀⇩F x in ?F. ?f x = ?g x ⟹ (?f ⤏ ?c) ?F = (?g ⤏ ?c) ?F›*) eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*) sum.reindex (*‹inj_on ?h ?A ⟹ sum ?g (?h ` ?A) = sum (?g ∘ ?h) ?A›*))\n (*goal: ‹⋀X. ⟦finite X; X ⊆ A⟧ ⟹ inj_on h X›*)\n show \"⋀X. ⟦finite X; X ⊆ A⟧ ⟹ inj_on h X\"\n using assms (*‹inj_on h A›*) subset_inj_on (*‹⟦inj_on ?f ?B; ?A ⊆ ?B⟧ ⟹ inj_on ?f ?A›*) by blast\n qed\n also (*calculation: ‹(g has_sum x) (h ` A) = (sum (g ∘ h) ⤏ x) (finite_subsets_at_top A)›*) have \"… ⟷ ((g ∘ h) has_sum x) A\"\n by (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n finally (*calculation: ‹(g has_sum x) (h ` A) = (g ∘ h has_sum x) A›*) show \"?thesis\"\n (*goal: ‹(g has_sum x) (h ` A) = (g ∘ h has_sum x) A›*) .\nqed\n\nlemma summable_on_reindex:\n assumes ‹inj_on h A›\n shows ‹g summable_on (h ` A) ⟷ (g ∘ h) summable_on A›\n by (simp add: assms (*‹inj_on h A›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_reindex (*‹inj_on ?h ?A ⟹ (?g has_sum ?x) (?h ` ?A) = (?g ∘ ?h has_sum ?x) ?A›*))\n\nlemma infsum_reindex:\n assumes ‹inj_on h A›\n shows ‹infsum g (h ` A) = infsum (g ∘ h) A›\n by (metis assms (*‹inj_on h A›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_reindex (*‹inj_on ?h ?A ⟹ (?g has_sum ?x) (?h ` ?A) = (?g ∘ ?h has_sum ?x) ?A›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) infsum_def (*‹infsum ?f ?A = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else 0)›*))\n\nlemma summable_on_reindex_bij_betw:\n assumes \"bij_betw g A B\"\n shows \"(λx. f (g x)) summable_on A ⟷ f summable_on B\"\n by (smt (verit) assms (*‹bij_betw g A B›*) bij_betw_def (*‹bij_betw ?f ?A ?B = (inj_on ?f ?A ∧ ?f ` ?A = ?B)›*) o_apply (*‹(?f ∘ ?g) ?x = ?f (?g ?x)›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*) summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*)) \n\nlemma infsum_reindex_bij_betw:\n assumes \"bij_betw g A B\"\n shows \"infsum (λx. f (g x)) A = infsum f B\"\n by (metis (mono_tags, lifting) assms (*‹bij_betw g A B›*) bij_betw_def (*‹bij_betw ?f ?A ?B = (inj_on ?f ?A ∧ ?f ` ?A = ?B)›*) infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*) infsum_reindex (*‹inj_on ?h ?A ⟹ infsum ?g (?h ` ?A) = infsum (?g ∘ ?h) ?A›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n\nlemma sum_uniformity:\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'b::{uniform_space,comm_monoid_add},y). x+y)›\n assumes EE: ‹eventually E uniformity›\n obtains D where ‹eventually D uniformity› \n and ‹⋀M::'a set. ⋀f f' :: 'a ⇒ 'b. card M ≤ n ∧ (∀m∈M. D (f m, f' m)) ⟹ E (sum f M, sum f' M)›\nproof (atomize_elim, insert EE (*‹eventually (E::'b::{comm_monoid_add,uniform_space} × 'b::{comm_monoid_add,uniform_space} ⇒ bool) uniformity›*), induction n arbitrary: E rule:nat_induct)\n (*goals:\n 1. ‹⋀E. eventually E uniformity ⟹ ∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ 0 ∧ (∀m∈M. D (f m, f' m)) ⟶ E (sum f M, sum f' M))›\n 2. ‹⋀n E. ⟦⋀E. eventually E uniformity ⟹ ∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ n ∧ (∀m∈M. D (f m, f' m)) ⟶ E (sum f M, sum f' M)); eventually E uniformity⟧ ⟹ ∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ Suc n ∧ (∀m∈M. D (f m, f' m)) ⟶ E (sum f M, sum f' M))›*)\n case 0 (*‹eventually E uniformity›*)\n then show \"?case\"\n (*goal: ‹∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ 0 ∧ (∀m∈M. D (f m, f' m)) ⟶ E (sum f M, sum f' M))›*)\n by (metis card_eq_0_iff (*‹(card (?A::?'a::type set) = (0::nat)) = (?A = {} ∨ infinite ?A)›*) equals0D (*‹(?A::?'a::type set) = {} ⟹ (?a::?'a::type) ∉ ?A›*) le_zero_eq (*‹((?n::?'a::canonically_ordered_monoid_add) ≤ (0::?'a::canonically_ordered_monoid_add)) = (?n = (0::?'a::canonically_ordered_monoid_add))›*) sum.infinite (*‹infinite (?A::?'b::type set) ⟹ sum (?g::?'b::type ⇒ ?'a::comm_monoid_add) ?A = (0::?'a::comm_monoid_add)›*) sum.not_neutral_contains_not_neutral (*‹⟦sum (?g::?'b::type ⇒ ?'a::comm_monoid_add) (?A::?'b::type set) ≠ (0::?'a::comm_monoid_add); ⋀a::?'b::type. ⟦a ∈ ?A; ?g a ≠ (0::?'a::comm_monoid_add)⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*) uniformity_refl (*‹eventually (?E::?'a::uniform_space × ?'a::uniform_space ⇒ bool) uniformity ⟹ ?E (?x::?'a::uniform_space, ?x)›*))\nnext\n (*goal: ‹⋀n E. ⟦⋀E. eventually E uniformity ⟹ ∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ n ∧ (∀m∈M. D (f m, f' m)) ⟶ E (sum f M, sum f' M)); eventually E uniformity⟧ ⟹ ∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ Suc n ∧ (∀m∈M. D (f m, f' m)) ⟶ E (sum f M, sum f' M))›*)\n case (Suc n) (*‹eventually ?E uniformity ⟹ ∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ n ∧ (∀m∈M. D (f m, f' m)) ⟶ ?E (sum f M, sum f' M))› ‹eventually E uniformity›*)\n from plus_cont[unfolded uniformly_continuous_on_uniformity filterlim_def le_filter_def, rule_format, OF Suc.prems] (*‹eventually E (filtermap (λ(x, y). (case x of (x, xa) ⇒ x + xa, case y of (x, xa) ⇒ x + xa)) (uniformity_on UNIV))›*) obtain D1 and D2 where \"eventually D1 uniformity\" and \"eventually D2 uniformity\" and D1D2E: \"D1 (x, y) ⟹ D2 (x', y') ⟹ E (x + x', y + y')\" for x and y and x' and y'\n (*goal: ‹(⋀D1 D2. ⟦eventually D1 uniformity; eventually D2 uniformity; ⋀x y x' y'. ⟦D1 (x, y); D2 (x', y')⟧ ⟹ E (x + x', y + y')⟧ ⟹ thesis) ⟹ thesis›*)\n apply atomize_elim\n (*goal: ‹(⋀D1 D2. ⟦eventually D1 uniformity; eventually D2 uniformity; ⋀x y x' y'. ⟦D1 (x, y); D2 (x', y')⟧ ⟹ E (x + x', y + y')⟧ ⟹ thesis) ⟹ thesis›*)\n by (auto simp: eventually_prod_filter (*‹eventually ?P (?F ×⇩F ?G) = (∃Pf Pg. eventually Pf ?F ∧ eventually Pg ?G ∧ (∀x y. Pf x ⟶ Pg y ⟶ ?P (x, y)))›*) case_prod_beta (*‹(case ?p of (x, xa) ⇒ ?f x xa) = ?f (fst ?p) (snd ?p)›*) uniformity_prod_def (*‹uniformity = filtermap (λ((x1, x2), y1, y2). ((x1, y1), x2, y2)) (uniformity ×⇩F uniformity)›*) eventually_filtermap (*‹eventually ?P (filtermap ?f ?F) = (∀⇩F x in ?F. ?P (?f x))›*))\n from Suc.IH[OF ‹eventually D2 uniformity›] (*‹∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ n ∧ (∀m∈M. D (f m, f' m)) ⟶ D2 (sum f M, sum f' M))›*) obtain D3 where \"eventually D3 uniformity\" and D3: \"card M ≤ n ⟹ (∀m∈M. D3 (f m, f' m)) ⟹ D2 (sum f M, sum f' M)\" for M :: \"'a set\" and f and f'\n (*goal: ‹(⋀D3. ⟦eventually D3 uniformity; ⋀M f f'. ⟦card M ≤ n; ∀m∈M. D3 (f m, f' m)⟧ ⟹ D2 (sum f M, sum f' M)⟧ ⟹ thesis) ⟹ thesis›*)\n by metis\n define D where \"D x ≡ D1 x ∧ D3 x\" for x\n have \"eventually D uniformity\"\n using D_def (*‹D ?x ≡ D1 ?x ∧ D3 ?x›*) ‹eventually D1 uniformity› (*‹eventually D1 uniformity›*) ‹eventually D3 uniformity› (*‹eventually D3 uniformity›*) eventually_elim2 (*‹⟦eventually ?P ?F; eventually ?Q ?F; ⋀i. ⟦?P i; ?Q i⟧ ⟹ ?R i⟧ ⟹ eventually ?R ?F›*) by blast\n have \"E (sum f M, sum f' M)\" if \"card M ≤ Suc n\" and DM: \"∀m∈M. D (f m, f' m)\" for M :: \"'a set\" and f and f'\n proof (cases ‹card M = 0›)\n (*goals:\n 1. ‹card M = 0 ⟹ E (sum f M, sum f' M)›\n 2. ‹card M ≠ 0 ⟹ E (sum f M, sum f' M)›*)\n case True (*‹card M = 0›*)\n then show \"?thesis\"\n (*goal: ‹E (sum f M, sum f' M)›*)\n by (metis Suc.prems (*‹eventually (E::'b::{comm_monoid_add,uniform_space} × 'b::{comm_monoid_add,uniform_space} ⇒ bool) uniformity›*) card_eq_0_iff (*‹(card (?A::?'a::type set) = (0::nat)) = (?A = {} ∨ infinite ?A)›*) sum.empty (*‹sum (?g::?'b::type ⇒ ?'a::comm_monoid_add) {} = (0::?'a::comm_monoid_add)›*) sum.infinite (*‹infinite (?A::?'b::type set) ⟹ sum (?g::?'b::type ⇒ ?'a::comm_monoid_add) ?A = (0::?'a::comm_monoid_add)›*) uniformity_refl (*‹eventually (?E::?'a::uniform_space × ?'a::uniform_space ⇒ bool) uniformity ⟹ ?E (?x::?'a::uniform_space, ?x)›*))\n next\n (*goal: ‹card M ≠ 0 ⟹ E (sum f M, sum f' M)›*)\n case False (*‹card M ≠ 0›*)\n with ‹card M ≤ Suc n› (*‹card (M::'a::type set) ≤ Suc (n::nat)›*) obtain N and x where \"card N ≤ n\" and \"x ∉ N\" and \"M = insert x N\"\n (*goal: ‹(⋀N x. ⟦card N ≤ n; x ∉ N; M = insert x N⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis card_Suc_eq (*‹(card ?A = Suc ?k) = (∃b B. ?A = insert b B ∧ b ∉ B ∧ card B = ?k ∧ (?k = 0 ⟶ B = {}))›*) less_Suc_eq_0_disj (*‹(?m < Suc ?n) = (?m = 0 ∨ (∃j. ?m = Suc j ∧ j < ?n))›*) less_Suc_eq_le (*‹(?m < Suc ?n) = (?m ≤ ?n)›*))\n from DM (*‹∀m∈M. D (f m, f' m)›*) have \"⋀m. m∈N ⟹ D (f m, f' m)\"\n using ‹M = insert x N› (*‹M = insert x N›*) by blast\n with D3[OF ‹card N ≤ n›] (*‹∀m::'a∈N::'a set. (D3::'b × 'b ⇒ bool) ((?f::'a ⇒ 'b) m, (?f'::'a ⇒ 'b) m) ⟹ (D2::'b × 'b ⇒ bool) (sum ?f N, sum ?f' N)›*) have D2_N: \"D2 (sum f N, sum f' N)\"\n using D_def (*‹(D::'b × 'b ⇒ bool) (?x::'b × 'b) ≡ (D1::'b × 'b ⇒ bool) ?x ∧ (D3::'b × 'b ⇒ bool) ?x›*) by blast\n from DM (*‹∀m∈M. D (f m, f' m)›*) have \"D (f x, f' x)\"\n using ‹M = insert x N› (*‹M = insert x N›*) by blast\n then have \"D1 (f x, f' x)\"\n by (simp add: D_def (*‹D ?x ≡ D1 ?x ∧ D3 ?x›*))\n with D2_N (*‹D2 (sum f N, sum f' N)›*) have \"E (f x + sum f N, f' x + sum f' N)\"\n using D1D2E (*‹⟦D1 (?x, ?y); D2 (?x', ?y')⟧ ⟹ E (?x + ?x', ?y + ?y')›*) by presburger\n then show \"E (sum f M, sum f' M)\"\n by (metis False (*‹card (M::'a::type set) ≠ (0::nat)›*) ‹M = insert x N› ‹x ∉ N› card.infinite (*‹infinite (?A::?'a::type set) ⟹ card ?A = (0::nat)›*) finite_insert (*‹finite (insert (?a::?'a::type) (?A::?'a::type set)) = finite ?A›*) sum.insert (*‹⟦finite (?A::?'b::type set); (?x::?'b::type) ∉ ?A⟧ ⟹ sum (?g::?'b::type ⇒ ?'a::comm_monoid_add) (insert ?x ?A) = ?g ?x + sum ?g ?A›*))\n qed\n with ‹eventually D uniformity› (*‹eventually D uniformity›*) show \"?case\"\n (*goal: ‹∃D. eventually D uniformity ∧ (∀M f f'. card M ≤ Suc n ∧ (∀m∈M. D (f m, f' m)) ⟶ E (sum f M, sum f' M))›*)\n by auto\nqed\n\nlemma has_sum_Sigma:\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a × 'b ⇒ 'c::{comm_monoid_add,uniform_space}›\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'c,y). x+y)›\n assumes summableAB: \"(f has_sum a) (Sigma A B)\"\n assumes summableB: ‹⋀x. x∈A ⟹ ((λy. f (x, y)) has_sum b x) (B x)›\n shows \"(b has_sum a) A\"\nproof (-)\n (*goal: ‹(b has_sum a) A›*)\n define F and FB and FA where \"F = finite_subsets_at_top (Sigma A B)\" and \"FB x = finite_subsets_at_top (B x)\" and \"FA = finite_subsets_at_top A\" for x\n from summableB (*‹?x ∈ A ⟹ ((λy. f (?x, y)) has_sum b ?x) (B ?x)›*) have sum_b: \"(sum (λy. f (x, y)) ⤏ b x) (FB x)\" if \"x ∈ A\" for x\n using FB_def[abs_def] (*‹FB ≡ λx. finite_subsets_at_top (B x)›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) that (*‹(x::'a) ∈ (A::'a set)›*) by auto\n from summableAB (*‹(f has_sum a) (Sigma A B)›*) have sum_S: \"(sum f ⤏ a) F\"\n using F_def (*‹F = finite_subsets_at_top (Sigma A B)›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) by blast\n have finite_proj: \"finite {b| b. (a,b) ∈ H}\" if \"finite H\" for H :: \"('a×'b) set\" and a\n by (metis (no_types, lifting) finite_imageI (*‹finite ?F ⟹ finite (?h ` ?F)›*) finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) image_eqI (*‹⟦?b = ?f ?x; ?x ∈ ?A⟧ ⟹ ?b ∈ ?f ` ?A›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) snd_conv (*‹snd (?x1.0, ?x2.0) = ?x2.0›*) subsetI (*‹(⋀x. x ∈ ?A ⟹ x ∈ ?B) ⟹ ?A ⊆ ?B›*) that (*‹finite H›*))\n have \"(sum b ⤏ a) FA\"\n proof (rule tendsto_iff_uniformity[THEN iffD2, rule_format] (*‹(⋀E. eventually E uniformity ⟹ ∀⇩F x in ?F. E (?f x, ?l)) ⟹ (?f ⤏ ?l) ?F›*))\n (*goal: ‹⋀E. eventually E uniformity ⟹ ∀⇩F x in FA. E (sum b x, a)›*)\n fix E :: \"('c × 'c) ⇒ bool\"\n assume \"eventually E uniformity\" (*‹eventually (E::'c × 'c ⇒ bool) uniformity›*)\n then obtain D where D_uni: \"eventually D uniformity\" and DDE': \"⋀x y z. D (x, y) ⟹ D (y, z) ⟹ E (x, z)\"\n (*goal: ‹(⋀D::'c × 'c ⇒ bool. ⟦eventually D uniformity; ⋀(x::'c) (y::'c) z::'c. ⟦D (x, y); D (y, z)⟧ ⟹ (E::'c × 'c ⇒ bool) (x, z)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by (metis (no_types, lifting) ‹eventually E uniformity› uniformity_transE (*‹⟦eventually ?E uniformity; ⋀D. ⟦eventually D uniformity; ⋀x y z. ⟦D (x, y); D (y, z)⟧ ⟹ ?E (x, z)⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*))\n from sum_S (*‹(sum f ⤏ a) F›*) obtain G where \"finite G\" and \"G ⊆ Sigma A B\" and G_sum: \"G ⊆ H ⟹ H ⊆ Sigma A B ⟹ finite H ⟹ D (sum f H, a)\" for H\n (*goal: ‹(⋀G::('a::type × 'b::type) set. ⟦finite G; G ⊆ Sigma (A::'a::type set) (B::'a::type ⇒ 'b::type set); ⋀H::('a::type × 'b::type) set. ⟦G ⊆ H; H ⊆ Sigma A B; finite H⟧ ⟹ (D::'c::{comm_monoid_add,uniform_space} × 'c::{comm_monoid_add,uniform_space} ⇒ bool) (sum (f::'a::type × 'b::type ⇒ 'c::{comm_monoid_add,uniform_space}) H, a::'c::{comm_monoid_add,uniform_space})⟧ ⟹ thesis::bool) ⟹ thesis›*)\n unfolding tendsto_iff_uniformity\n (*goal: ‹(⋀G. ⟦finite G; G ⊆ Sigma A B; ⋀H. ⟦G ⊆ H; H ⊆ Sigma A B; finite H⟧ ⟹ D (sum f H, a)⟧ ⟹ thesis) ⟹ thesis›*)\n by (metis (mono_tags, lifting) D_uni (*‹eventually D uniformity›*) F_def (*‹F = finite_subsets_at_top (Sigma A B)›*) eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n have \"finite (fst ` G)\" and \"fst ` G ⊆ A\"\n using ‹finite G› (*‹finite G›*) ‹G ⊆ Sigma A B› (*‹(G::('a × 'b) set) ⊆ Sigma (A::'a set) (B::'a ⇒ 'b set)›*) apply -\n (*goals:\n 1. ‹⟦finite G; G ⊆ Sigma A B⟧ ⟹ finite (fst ` G)›\n 2. ‹⟦finite G; G ⊆ Sigma A B⟧ ⟹ fst ` G ⊆ A›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n thm uniformity_prod_def\n define Ga where \"Ga a = {b. (a,b) ∈ G}\" for a\n have Ga_fin: \"finite (Ga a)\" and Ga_B: \"Ga a ⊆ B a\" for a\n using ‹finite G› (*‹finite G›*) ‹G ⊆ Sigma A B› (*‹G ⊆ Sigma A B›*) finite_proj (*‹finite ?H ⟹ finite {b |b. (?a, b) ∈ ?H}›*) apply -\n (*goals:\n 1. ‹⟦finite (G::('a × 'b) set); G ⊆ Sigma (A::'a set) (B::'a ⇒ 'b set); ⋀(H::('a × 'b) set) a::'a. finite H ⟹ finite {b |b::'b. (a, b) ∈ H}⟧ ⟹ finite ((Ga::'a ⇒ 'b set) (a::'a))›\n 2. ‹⟦finite (G::('a × 'b) set); G ⊆ Sigma (A::'a set) (B::'a ⇒ 'b set); ⋀(H::('a × 'b) set) a::'a. finite H ⟹ finite {b |b::'b. (a, b) ∈ H}⟧ ⟹ (Ga::'a ⇒ 'b set) (a::'a) ⊆ B a›\n discuss goal 1*)\n apply ((auto simp: Ga_def (*‹Ga ?a = {b. (?a, b) ∈ G}›*) finite_proj (*‹finite ?H ⟹ finite {b |b. (?a, b) ∈ ?H}›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: Ga_def (*‹(Ga::'a ⇒ 'b set) (?a::'a) = {b::'b. (?a, b) ∈ (G::('a × 'b) set)}›*) finite_proj (*‹finite (?H::('a × 'b) set) ⟹ finite {b |b::'b. (?a::'a, b) ∈ ?H}›*))[1])\n (*proven 2 subgoals*) .\n have \"E (sum b M, a)\" if \"M ⊇ fst ` G\" and \"finite M\" and \"M ⊆ A\" for M\n proof (-)\n (*goal: ‹E (sum b M, a)›*)\n define FMB where \"FMB = finite_subsets_at_top (Sigma M B)\"\n have \"eventually (λH. D (∑a∈M. b a, ∑(a,b)∈H. f (a,b))) FMB\"\n proof (-)\n (*goal: ‹∀⇩F H in FMB. D (sum b M, ∑(a, b)∈H. f (a, b))›*)\n obtain D' where D'_uni: \"eventually D' uniformity\" and \"card M' ≤ card M ∧ (∀m∈M'. D' (g m, g' m)) ⟹ D (sum g M', sum g' M')\" for M' :: \"'a set\" and g and g'\n (*goal: ‹(⋀D'::'c::{comm_monoid_add,uniform_space} × 'c::{comm_monoid_add,uniform_space} ⇒ bool. ⟦eventually D' uniformity; ⋀(M'::'a::type set) (g::'a::type ⇒ 'c::{comm_monoid_add,uniform_space}) g'::'a::type ⇒ 'c::{comm_monoid_add,uniform_space}. card M' ≤ card (M::'a::type set) ∧ (∀m::'a::type∈M'. D' (g m, g' m)) ⟹ (D::'c::{comm_monoid_add,uniform_space} × 'c::{comm_monoid_add,uniform_space} ⇒ bool) (sum g M', sum g' M')⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using sum_uniformity[OF plus_cont ‹eventually D uniformity›] (*‹(⋀Da. ⟦eventually Da uniformity; ⋀M f f'. card M ≤ ?n ∧ (∀m∈M. Da (f m, f' m)) ⟹ D (sum f M, sum f' M)⟧ ⟹ ?thesis) ⟹ ?thesis›*) by blast\n then have D'_sum_D: \"(∀m∈M. D' (g m, g' m)) ⟹ D (sum g M, sum g' M)\" for g and g'\n by auto\n obtain Ha where \"Ha a ⊇ Ga a\" and Ha_fin: \"finite (Ha a)\" and Ha_B: \"Ha a ⊆ B a\" and D'_sum_Ha: \"Ha a ⊆ L ⟹ L ⊆ B a ⟹ finite L ⟹ D' (b a, sum (λb. f (a,b)) L)\" if \"a ∈ A\" for a and L\n (*goal: ‹(⋀Ha::'a ⇒ 'b set. ⟦⋀a::'a. a ∈ (A::'a set) ⟹ (Ga::'a ⇒ 'b set) a ⊆ Ha a; ⋀a::'a. a ∈ A ⟹ finite (Ha a); ⋀a::'a. a ∈ A ⟹ Ha a ⊆ (B::'a ⇒ 'b set) a; ⋀(a::'a) L::'b set. ⟦a ∈ A; Ha a ⊆ L; L ⊆ B a; finite L⟧ ⟹ (D'::'c × 'c ⇒ bool) ((b::'a ⇒ 'c) a, ∑b::'b∈L. (f::'a × 'b ⇒ 'c) (a, b))⟧ ⟹ thesis::bool) ⟹ thesis›*)\n proof (-)\n (*goal: ‹(⋀Ha. ⟦⋀a. a ∈ A ⟹ Ga a ⊆ Ha a; ⋀a. a ∈ A ⟹ finite (Ha a); ⋀a. a ∈ A ⟹ Ha a ⊆ B a; ⋀a L. ⟦a ∈ A; Ha a ⊆ L; L ⊆ B a; finite L⟧ ⟹ D' (b a, ∑b∈L. f (a, b))⟧ ⟹ thesis) ⟹ thesis›*)\n from sum_b[unfolded tendsto_iff_uniformity, rule_format, OF _ D'_uni [ THEN uniformity_sym ]] (*‹?x ∈ A ⟹ ∀⇩F x in FB ?x. case (∑y∈x. f (?x, y), b ?x) of (x, y) ⇒ D' (y, x)›*) obtain Ha0 where \"finite (Ha0 a)\" and \"Ha0 a ⊆ B a\" and \"Ha0 a ⊆ L ⟹ L ⊆ B a ⟹ finite L ⟹ D' (b a, sum (λb. f (a,b)) L)\" if \"a ∈ A\" for a and L\n (*goal: ‹(⋀Ha0. ⟦⋀a. a ∈ A ⟹ finite (Ha0 a); ⋀a. a ∈ A ⟹ Ha0 a ⊆ B a; ⋀a L. ⟦a ∈ A; Ha0 a ⊆ L; L ⊆ B a; finite L⟧ ⟹ D' (b a, ∑b∈L. f (a, b))⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding FB_def eventually_finite_subsets_at_top\n (*goal: ‹(⋀Ha0. ⟦⋀a. a ∈ A ⟹ finite (Ha0 a); ⋀a. a ∈ A ⟹ Ha0 a ⊆ B a; ⋀a L. ⟦a ∈ A; Ha0 a ⊆ L; L ⊆ B a; finite L⟧ ⟹ D' (b a, ∑b∈L. f (a, b))⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding prod.case\n (*goal: ‹(⋀Ha0. ⟦⋀a. a ∈ A ⟹ finite (Ha0 a); ⋀a. a ∈ A ⟹ Ha0 a ⊆ B a; ⋀a L. ⟦a ∈ A; Ha0 a ⊆ L; L ⊆ B a; finite L⟧ ⟹ D' (b a, ∑b∈L. f (a, b))⟧ ⟹ thesis) ⟹ thesis›*)\n by metis\n moreover define Ha where \"Ha a = Ha0 a ∪ Ga a\" for a\n ultimately show \"?thesis\"\n (*goal: ‹thesis›*)\n using that[where Ha = Ha] (*‹⟦⋀a. a ∈ A ⟹ Ga a ⊆ Ha a; ⋀a. a ∈ A ⟹ finite (Ha a); ⋀a. a ∈ A ⟹ Ha a ⊆ B a; ⋀a L. ⟦a ∈ A; Ha a ⊆ L; L ⊆ B a; finite L⟧ ⟹ D' (b a, ∑b∈L. f (a, b))⟧ ⟹ thesis›*) using Ga_fin (*‹finite (Ga ?a)›*) Ga_B (*‹(Ga::'a::type ⇒ 'b::type set) (?a::'a::type) ⊆ (B::'a::type ⇒ 'b::type set) ?a›*) by auto\n qed\n have \"D (∑a∈M. b a, ∑(a,b)∈H. f (a,b))\" if \"finite H\" and \"H ⊆ Sigma M B\" and \"H ⊇ Sigma M Ha\" for H\n proof (-)\n (*goal: ‹D (sum b M, ∑(a, b)∈H. f (a, b))›*)\n define Ha' where \"Ha' a = {b| b. (a,b) ∈ H}\" for a\n have [simp]: \"finite (Ha' a)\" and [simp]: \"Ha' a ⊇ Ha a\" and [simp]: \"Ha' a ⊆ B a\" if \"a ∈ M\" for a\n unfolding Ha'_def\n (*goals:\n 1. ‹finite {b |b::'b::type. (a::'a::type, b) ∈ (H::('a::type × 'b::type) set)}›\n 2. ‹(Ha::'a::type ⇒ 'b::type set) (a::'a::type) ⊆ {b |b::'b::type. (a, b) ∈ (H::('a::type × 'b::type) set)}›\n 3. ‹{b |b::'b::type. (a::'a::type, b) ∈ (H::('a::type × 'b::type) set)} ⊆ (B::'a::type ⇒ 'b::type set) a›*)\n using ‹finite H› (*‹finite H›*) ‹H ⊆ Sigma M B› (*‹H ⊆ Sigma M B›*) ‹Sigma M Ha ⊆ H› (*‹Sigma M Ha ⊆ H›*) that (*‹(a::'a) ∈ (M::'a set)›*) finite_proj (*‹finite ?H ⟹ finite {b |b. (?a, b) ∈ ?H}›*) apply -\n (*goals:\n 1. ‹⟦finite (H::('a × 'b) set); H ⊆ Sigma (M::'a set) (B::'a ⇒ 'b set); Sigma M (Ha::'a ⇒ 'b set) ⊆ H; (a::'a) ∈ M; ⋀(H::('a × 'b) set) a::'a. finite H ⟹ finite {b |b::'b. (a, b) ∈ H}⟧ ⟹ finite {b |b::'b. (a, b) ∈ H}›\n 2. ‹⟦finite (H::('a × 'b) set); H ⊆ Sigma (M::'a set) (B::'a ⇒ 'b set); Sigma M (Ha::'a ⇒ 'b set) ⊆ H; (a::'a) ∈ M; ⋀(H::('a × 'b) set) a::'a. finite H ⟹ finite {b |b::'b. (a, b) ∈ H}⟧ ⟹ Ha a ⊆ {b |b::'b. (a, b) ∈ H}›\n 3. ‹⟦finite (H::('a × 'b) set); H ⊆ Sigma (M::'a set) (B::'a ⇒ 'b set); Sigma M (Ha::'a ⇒ 'b set) ⊆ H; (a::'a) ∈ M; ⋀(H::('a × 'b) set) a::'a. finite H ⟹ finite {b |b::'b. (a, b) ∈ H}⟧ ⟹ {b |b::'b. (a, b) ∈ H} ⊆ 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 have \"Sigma M Ha' = H\"\n using that (*‹finite H› ‹H ⊆ Sigma M B› ‹Sigma M Ha ⊆ H›*) by (auto simp: Ha'_def (*‹Ha' ?a = {b |b. (?a, b) ∈ H}›*))\n then have \"*\": \"(∑(a,b)∈H. f (a,b)) = (∑a∈M. ∑b∈Ha' a. f (a,b))\"\n by (simp add: ‹finite M› sum.Sigma (*‹⟦finite ?A; ∀x∈?A. finite (?B x)⟧ ⟹ (∑x∈?A. sum (?g x) (?B x)) = (∑(x, y)∈Sigma ?A ?B. ?g x y)›*))\n have \"D' (b a, sum (λb. f (a,b)) (Ha' a))\" if \"a ∈ M\" for a\n using D'_sum_Ha (*‹⟦(?a::'a) ∈ (A::'a set); (Ha::'a ⇒ 'b set) ?a ⊆ (?L::'b set); ?L ⊆ (B::'a ⇒ 'b set) ?a; finite ?L⟧ ⟹ (D'::'c × 'c ⇒ bool) ((b::'a ⇒ 'c) ?a, ∑b::'b∈?L. (f::'a × 'b ⇒ 'c) (?a, b))›*) ‹M ⊆ A› (*‹M ⊆ A›*) that (*‹a ∈ M›*) by auto\n then have \"D (∑a∈M. b a, ∑a∈M. sum (λb. f (a,b)) (Ha' a))\"\n apply (rule_tac D'_sum_D (*‹∀m∈M. D' (?g m, ?g' m) ⟹ D (sum ?g M, sum ?g' M)›*))\n (*goal: ‹D (sum b M, ∑a∈M. ∑b∈Ha' a. f (a, b))›*)\n by auto\n with \"*\" (*‹(∑(a::'a, b::'b)∈(H::('a × 'b) set). (f::'a × 'b ⇒ 'c) (a, b)) = (∑a::'a∈(M::'a set). ∑b::'b∈(Ha'::'a ⇒ 'b set) a. f (a, b))›*) show \"?thesis\"\n (*goal: ‹D (sum b M, ∑(a, b)∈H. f (a, b))›*)\n by auto\n qed\n moreover have \"Sigma M Ha ⊆ Sigma M B\"\n using Ha_B (*‹?a ∈ A ⟹ Ha ?a ⊆ B ?a›*) ‹M ⊆ A› (*‹M ⊆ A›*) by auto\n ultimately show \"?thesis\"\n (*goal: ‹∀⇩F H in FMB. D (sum b M, ∑(a, b)∈H. f (a, b))›*)\n unfolding FMB_def eventually_finite_subsets_at_top\n (*goal: ‹∃X. finite X ∧ X ⊆ Sigma M B ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ Sigma M B ⟶ D (sum b M, ∑(a, b)∈Y. f (a, b)))›*)\n by (metis (no_types, lifting) Ha_fin (*‹(?a::'a) ∈ (A::'a set) ⟹ finite ((Ha::'a ⇒ 'b set) ?a)›*) finite_SigmaI (*‹⟦finite (?A::?'a set); ⋀a::?'a. a ∈ ?A ⟹ finite ((?B::?'a ⇒ ?'b set) a)⟧ ⟹ finite (Sigma ?A ?B)›*) subsetD (*‹⟦(?A::?'a set) ⊆ (?B::?'a set); (?c::?'a) ∈ ?A⟧ ⟹ ?c ∈ ?B›*) that( (*‹finite (M::'a set)›*) 2) that( (*‹(M::'a set) ⊆ (A::'a set)›*) 3))\n qed\n moreover have \"eventually (λH. D (∑(a,b)∈H. f (a,b), a)) FMB\"\n unfolding FMB_def eventually_finite_subsets_at_top\n (*goal: ‹∃X. finite X ∧ X ⊆ Sigma M B ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ Sigma M B ⟶ D (∑(a, b)∈Y. f (a, b), a))›*)\n proof (rule exI[of _ G] (*‹?P G ⟹ ∃x. ?P x›*), safe)\n (*goals:\n 1. ‹finite G›\n 2. ‹⋀a b. (a, b) ∈ G ⟹ a ∈ M›\n 3. ‹⋀a b. (a, b) ∈ G ⟹ b ∈ B a›\n 4. ‹⋀Y. ⟦finite Y; G ⊆ Y; Y ⊆ Sigma M B⟧ ⟹ D (∑(a, b)∈Y. f (a, b), a)›*)\n fix Y\n assume Y: \"finite Y\" \"G ⊆ Y\" \"Y ⊆ Sigma M B\" (*‹finite (Y::('a × 'b) set)› ‹(G::('a × 'b) set) ⊆ (Y::('a × 'b) set)› ‹(Y::('a × 'b) set) ⊆ Sigma (M::'a set) (B::'a ⇒ 'b set)›*)\n thus \"D (∑(a,b)∈Y. f (a, b), a)\"\n using G_sum[of Y] (*‹⟦G ⊆ Y; Y ⊆ Sigma A B; finite Y⟧ ⟹ D (sum f Y, a)›*) Y (*‹finite Y› ‹(G::('a × 'b) set) ⊆ (Y::('a × 'b) set)› ‹Y ⊆ Sigma M B›*) using that(3) (*‹M ⊆ A›*) by fastforce\n qed (use ‹finite G› ‹G ⊆ Sigma A B› that in auto)\n (*solves the remaining goals:\n 1. ‹finite G›\n 2. ‹⋀a b. (a, b) ∈ G ⟹ a ∈ M›\n 3. ‹⋀a b. (a, b) ∈ G ⟹ b ∈ B a›*)\n ultimately have \"∀⇩F x in FMB. E (sum b M, a)\"\n apply eventually_elim\n (*goal: ‹∀⇩F x in FMB. E (sum b M, a)›*)\n by (use DDE' in auto)\n then show \"E (sum b M, a)\"\n using FMB_def (*‹(FMB::('a × 'b) set filter) = finite_subsets_at_top (Sigma (M::'a set) (B::'a ⇒ 'b set))›*) by force\n qed\n then show \"∀⇩F x in FA. E (sum b x, a)\"\n using ‹finite (fst ` G)› (*‹finite (fst ` (G::('a × 'b) set))›*) ‹fst ` G ⊆ A› (*‹fst ` G ⊆ A›*) by (metis (mono_tags, lifting) FA_def (*‹FA = finite_subsets_at_top A›*) eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n qed\n then show \"?thesis\"\n (*goal: ‹(b has_sum a) A›*)\n by (simp add: FA_def (*‹FA = finite_subsets_at_top A›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\nqed\n\nlemma summable_on_Sigma:\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a ⇒ 'b ⇒ 'c::{comm_monoid_add, t2_space, uniform_space}›\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'c,y). x+y)›\n assumes summableAB: \"(λ(x,y). f x y) summable_on (Sigma A B)\"\n assumes summableB: ‹⋀x. x∈A ⟹ (f x) summable_on (B x)›\n shows ‹(λx. infsum (f x) (B x)) summable_on A›\nproof (-)\n (*goal: ‹(λx. infsum (f x) (B x)) summable_on A›*)\n from summableAB (*‹(λ(x, y). f x y) summable_on Sigma A B›*) obtain a where a: \"((λ(x,y). f x y) has_sum a) (Sigma A B)\"\n (*goal: ‹(⋀a. ((λ(x, y). f x y) has_sum a) (Sigma A B) ⟹ thesis) ⟹ thesis›*)\n using has_sum_infsum (*‹(?f::?'a::type ⇒ ?'b::{comm_monoid_add,t2_space}) summable_on (?S::?'a::type set) ⟹ (?f has_sum infsum ?f ?S) ?S›*) by blast\n from summableB (*‹?x ∈ A ⟹ f ?x summable_on B ?x›*) have b: \"⋀x. x∈A ⟹ (f x has_sum infsum (f x) (B x)) (B x)\"\n by (auto intro!: has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*))\n show \"?thesis\"\n (*goal: ‹(λx. infsum (f x) (B x)) summable_on A›*)\n using plus_cont (*‹uniformly_continuous_on UNIV (λ(x::'c::{comm_monoid_add,t2_space,uniform_space}, y::'c::{comm_monoid_add,t2_space,uniform_space}). x + y)›*) a (*‹((λ(x, y). f x y) has_sum a) (Sigma A B)›*) b (*‹?x ∈ A ⟹ (f ?x has_sum infsum (f ?x) (B ?x)) (B ?x)›*) by (smt (verit) has_sum_Sigma[where f=‹λ(x,y). f x y›] (*‹⟦uniformly_continuous_on UNIV (λ(x, y). x + y); ((λ(x, y). f x y) has_sum ?a) (Sigma ?A ?B); ⋀x. x ∈ ?A ⟹ ((λy. case (x, y) of (x, y) ⇒ f x y) has_sum ?b x) (?B x)⟧ ⟹ (?b has_sum ?a) ?A›*) has_sum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f has_sum ?x) ?A = (?g has_sum ?x) ?A›*) old.prod.case (*‹(case (?x1.0, ?x2.0) of (x, xa) ⇒ ?f x xa) = ?f ?x1.0 ?x2.0›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\nqed\n\nlemma infsum_Sigma:\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a × 'b ⇒ 'c::{comm_monoid_add, t2_space, uniform_space}›\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'c,y). x+y)›\n assumes summableAB: \"f summable_on (Sigma A B)\"\n assumes summableB: ‹⋀x. x∈A ⟹ (λy. f (x, y)) summable_on (B x)›\n shows \"infsum f (Sigma A B) = infsum (λx. infsum (λy. f (x, y)) (B x)) A\"\nproof (-)\n (*goal: ‹infsum (f::'a × 'b ⇒ 'c) (Sigma (A::'a set) (B::'a ⇒ 'b set)) = (∑⇩∞x::'a∈A. ∑⇩∞y::'b∈B x. f (x, y))›*)\n from summableAB (*‹(f::'a::type × 'b::type ⇒ 'c::{comm_monoid_add,t2_space,uniform_space}) summable_on Sigma (A::'a::type set) (B::'a::type ⇒ 'b::type set)›*) have a: \"(f has_sum infsum f (Sigma A B)) (Sigma A B)\"\n using has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) by blast\n from summableB (*‹(?x::'a) ∈ (A::'a set) ⟹ (λy::'b. (f::'a × 'b ⇒ 'c) (?x, y)) summable_on (B::'a ⇒ 'b set) ?x›*) have b: \"⋀x. x∈A ⟹ ((λy. f (x, y)) has_sum infsum (λy. f (x, y)) (B x)) (B x)\"\n by (auto intro!: has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*))\n show \"?thesis\"\n (*goal: ‹infsum f (Sigma A B) = (∑⇩∞x∈A. ∑⇩∞y∈B x. f (x, y))›*)\n using plus_cont (*‹uniformly_continuous_on UNIV (λ(x, y). x + y)›*) a (*‹(f has_sum infsum f (Sigma A B)) (Sigma A B)›*) b (*‹?x ∈ A ⟹ ((λy. f (?x, y)) has_sum (∑⇩∞y∈B ?x. f (?x, y))) (B ?x)›*) by (auto intro: infsumI[symmetric] (*‹((?f::?'a::type ⇒ ?'b::{comm_monoid_add,t2_space}) has_sum (?t::?'b::{comm_monoid_add,t2_space})) (?A::?'a::type set) ⟹ ?t = infsum ?f ?A›*) has_sum_Sigma (*‹⟦uniformly_continuous_on UNIV (λ(x::?'c::{comm_monoid_add,uniform_space}, y::?'c::{comm_monoid_add,uniform_space}). x + y); ((?f::?'a::type × ?'b::type ⇒ ?'c::{comm_monoid_add,uniform_space}) has_sum (?a::?'c::{comm_monoid_add,uniform_space})) (Sigma (?A::?'a::type set) (?B::?'a::type ⇒ ?'b::type set)); ⋀x::?'a::type. x ∈ ?A ⟹ ((λy::?'b::type. ?f (x, y)) has_sum (?b::?'a::type ⇒ ?'c::{comm_monoid_add,uniform_space}) x) (?B x)⟧ ⟹ (?b has_sum ?a) ?A›*) simp: summable_on_def (*‹(?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) summable_on (?A::?'a::type set) ≡ ∃x::?'b::{comm_monoid_add,topological_space}. (?f has_sum x) ?A›*))\nqed\n\nlemma infsum_Sigma':\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a ⇒ 'b ⇒ 'c::{comm_monoid_add, t2_space, uniform_space}›\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'c,y). x+y)›\n assumes summableAB: \"(λ(x,y). f x y) summable_on (Sigma A B)\"\n assumes summableB: ‹⋀x. x∈A ⟹ (f x) summable_on (B x)›\n shows ‹infsum (λx. infsum (f x) (B x)) A = infsum (λ(x,y). f x y) (Sigma A B)›\n using infsum_Sigma[of ‹λ(x,y). f x y› A B] (*‹⟦uniformly_continuous_on UNIV (λ(x, y). x + y); (λ(x, y). f x y) summable_on Sigma A B; ⋀x. x ∈ A ⟹ (λy. case (x, y) of (x, y) ⇒ f x y) summable_on B x⟧ ⟹ (∑⇩∞(x, y)∈Sigma A B. f x y) = (∑⇩∞x∈A. ∑⇩∞y∈B x. case (x, y) of (x, y) ⇒ f x y)›*) using assms (*‹uniformly_continuous_on UNIV (λ(x, y). x + y)› ‹(λ(x, y). f x y) summable_on Sigma A B› ‹?x ∈ A ⟹ f ?x summable_on B ?x›*) by auto\n\ntext ‹A special case of @{thm [source] infsum_Sigma} etc. for Banach spaces. It has less premises.›\nlemma\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a ⇒ 'b ⇒ 'c::banach›\n assumes [simp]: \"(λ(x,y). f x y) summable_on (Sigma A B)\"\n shows infsum_Sigma'_banach: ‹infsum (λx. infsum (f x) (B x)) A = infsum (λ(x,y). f x y) (Sigma A B)› (is ?thesis1)\n and summable_on_Sigma_banach: ‹(λx. infsum (f x) (B x)) summable_on A› (is ?thesis2)\nproof (-)\n (*goals:\n 1. ‹(∑⇩∞x::'a::type∈(A::'a::type set). infsum ((f::'a::type ⇒ 'b::type ⇒ 'c::banach) x) ((B::'a::type ⇒ 'b::type set) x)) = (∑⇩∞(x::'a::type, y::'b::type)∈Sigma A B. f x y)›\n 2. ‹(λx::'a::type. infsum ((f::'a::type ⇒ 'b::type ⇒ 'c::banach) x) ((B::'a::type ⇒ 'b::type set) x)) summable_on (A::'a::type set)›*)\n have fsum: \"(f x) summable_on (B x)\" if \"x ∈ A\" for x\n proof (-)\n (*goal: ‹f x summable_on B x›*)\n from assms (*‹(λ(x, y). f x y) summable_on Sigma A B›*) have \"(λ(x,y). f x y) summable_on (Pair x ` B x)\"\n by (meson image_subset_iff (*‹((?f::?'b ⇒ ?'a) ` (?A::?'b set) ⊆ (?B::?'a set)) = (∀x::?'b∈?A. ?f x ∈ ?B)›*) summable_on_subset_banach (*‹⟦(?f::?'a ⇒ ?'b) summable_on (?A::?'a set); (?B::?'a set) ⊆ ?A⟧ ⟹ ?f summable_on ?B›*) mem_Sigma_iff (*‹((?a::?'a, ?b::?'b) ∈ Sigma (?A::?'a set) (?B::?'a ⇒ ?'b set)) = (?a ∈ ?A ∧ ?b ∈ ?B ?a)›*) that (*‹(x::'a) ∈ (A::'a set)›*))\n then have \"((λ(x,y). f x y) ∘ Pair x) summable_on (B x)\"\n by (metis summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*) inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*) prod.inject (*‹((?x1.0, ?x2.0) = (?y1.0, ?y2.0)) = (?x1.0 = ?y1.0 ∧ ?x2.0 = ?y2.0)›*))\n then show \"?thesis\"\n (*goal: ‹f x summable_on B x›*)\n by (auto simp: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n qed\n show \"?thesis1\"\n (*goal: ‹(∑⇩∞x∈A. infsum (f x) (B x)) = (∑⇩∞(x, y)∈Sigma A B. f x y)›*)\n using fsum (*‹?x ∈ A ⟹ f ?x summable_on B ?x›*) assms (*‹(λ(x, y). f x y) summable_on Sigma A B›*) infsum_Sigma' (*‹⟦uniformly_continuous_on UNIV (λ(x::?'c, y::?'c). x + y); (λ(x::?'a, y::?'b). (?f::?'a ⇒ ?'b ⇒ ?'c) x y) summable_on Sigma (?A::?'a set) (?B::?'a ⇒ ?'b set); ⋀x::?'a. x ∈ ?A ⟹ ?f x summable_on ?B x⟧ ⟹ (∑⇩∞x::?'a∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x::?'a, y::?'b)∈Sigma ?A ?B. ?f x y)›*) isUCont_plus (*‹isUCont (λ(x, y). x + y)›*) by blast\n show \"?thesis2\"\n (*goal: ‹(λx::'a. infsum ((f::'a ⇒ 'b ⇒ 'c) x) ((B::'a ⇒ 'b set) x)) summable_on (A::'a set)›*)\n using fsum (*‹?x ∈ A ⟹ f ?x summable_on B ?x›*) assms (*‹(λ(x, y). f x y) summable_on Sigma A B›*) isUCont_plus (*‹isUCont (λ(x, y). x + y)›*) summable_on_Sigma (*‹⟦uniformly_continuous_on UNIV (λ(x, y). x + y); (λ(x, y). ?f x y) summable_on Sigma ?A ?B; ⋀x. x ∈ ?A ⟹ ?f x summable_on ?B x⟧ ⟹ (λx. infsum (?f x) (?B x)) summable_on ?A›*) by blast\nqed\n\nlemma infsum_Sigma_banach:\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a × 'b ⇒ 'c::banach›\n assumes [simp]: \"f summable_on (Sigma A B)\"\n shows ‹infsum (λx. infsum (λy. f (x,y)) (B x)) A = infsum f (Sigma A B)›\n using assms (*‹f summable_on Sigma A B›*) by (simp add: infsum_Sigma'_banach (*‹(λ(x, y). ?f x y) summable_on Sigma ?A ?B ⟹ (∑⇩∞x∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x, y)∈Sigma ?A ?B. ?f x y)›*))\n\nlemma infsum_swap:\n fixes A :: \"'a set\" and B :: \"'b set\"\n fixes f :: \"'a ⇒ 'b ⇒ 'c::{comm_monoid_add,t2_space,uniform_space}\"\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'c,y). x+y)›\n assumes ‹(λ(x, y). f x y) summable_on (A × B)›\n assumes ‹⋀a. a∈A ⟹ (f a) summable_on B›\n assumes ‹⋀b. b∈B ⟹ (λa. f a b) summable_on A›\n shows ‹infsum (λx. infsum (λy. f x y) B) A = infsum (λy. infsum (λx. f x y) A) B›\nproof (-)\n (*goal: ‹(∑⇩∞x::'a∈(A::'a set). infsum ((f::'a ⇒ 'b ⇒ 'c) x) (B::'b set)) = (∑⇩∞y::'b∈B. ∑⇩∞x::'a∈A. f x y)›*)\n have \"(λ(x, y). f y x) ∘ prod.swap summable_on A × B\"\n by (simp add: assms( (*‹(λ(x, y). f x y) summable_on A × B›*) 2) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n then have fyx: \"(λ(x, y). f y x) summable_on (B × A)\"\n by (metis has_sum_reindex (*‹inj_on ?h ?A ⟹ (?g has_sum ?x) (?h ` ?A) = (?g ∘ ?h has_sum ?x) ?A›*) infsum_reindex (*‹inj_on ?h ?A ⟹ infsum ?g (?h ` ?A) = infsum (?g ∘ ?h) ?A›*) inj_swap (*‹inj_on prod.swap ?A›*) product_swap (*‹prod.swap ` (?A × ?B) = ?B × ?A›*) summable_iff_has_sum_infsum (*‹(?f summable_on ?A) = (?f has_sum infsum ?f ?A) ?A›*))\n have \"infsum (λx. infsum (λy. f x y) B) A = infsum (λ(x,y). f x y) (A × B)\"\n using assms (*‹uniformly_continuous_on UNIV (λ(x, y). x + y)› ‹(λ(x, y). f x y) summable_on A × B› ‹?a ∈ A ⟹ f ?a summable_on B› ‹?b ∈ B ⟹ (λa. f a ?b) summable_on A›*) infsum_Sigma' (*‹⟦uniformly_continuous_on UNIV (λ(x, y). x + y); (λ(x, y). ?f x y) summable_on Sigma ?A ?B; ⋀x. x ∈ ?A ⟹ ?f x summable_on ?B x⟧ ⟹ (∑⇩∞x∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x, y)∈Sigma ?A ?B. ?f x y)›*) by blast\n also (*calculation: ‹(∑⇩∞x∈A. infsum (f x) B) = (∑⇩∞(x, y)∈A × B. f x y)›*) have \"… = infsum (λ(x,y). f y x) (B × A)\"\n apply (subst product_swap[symmetric] (*‹?B × ?A = prod.swap ` (?A × ?B)›*))\n (*goal: ‹(∑⇩∞(x, y)∈A × B. f x y) = (∑⇩∞(x, y)∈B × A. f y x)›*)\n apply (subst infsum_reindex (*‹inj_on ?h ?A ⟹ infsum ?g (?h ` ?A) = infsum (?g ∘ ?h) ?A›*))\n (*goal: ‹(∑⇩∞(x, y)∈prod.swap ` (B × A). f x y) = (∑⇩∞(x, y)∈B × A. f y x)›*)\n using assms (*‹uniformly_continuous_on UNIV (λ(x, y). x + y)› ‹(λ(x, y). f x y) summable_on A × B› ‹?a ∈ A ⟹ f ?a summable_on B› ‹?b ∈ B ⟹ (λa. f a ?b) summable_on A›*)\n (*goals:\n 1. ‹inj_on prod.swap (B × A)›\n 2. ‹infsum ((λ(x, y). f x y) ∘ prod.swap) (B × A) = (∑⇩∞(x, y)∈B × A. f y x)›\n discuss goal 1*)\n apply ((auto simp: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(∑⇩∞x::'a∈(A::'a set). infsum ((f::'a ⇒ 'b ⇒ 'c) x) (B::'b set)) = (∑⇩∞(x::'b, y::'a)∈B × A. f y x)›*) have \"… = infsum (λy. infsum (λx. f x y) A) B\"\n by (smt (verit) fyx (*‹(λ(x::'b, y::'a). (f::'a ⇒ 'b ⇒ 'c) y x) summable_on (B::'b set) × (A::'a set)›*) assms( (*‹uniformly_continuous_on UNIV (λ(x::'c, y::'c). x + y)›*) 1) assms( (*‹(?b::'b) ∈ (B::'b set) ⟹ (λa::'a. (f::'a ⇒ 'b ⇒ 'c) a ?b) summable_on (A::'a set)›*) 4) infsum_Sigma' (*‹⟦uniformly_continuous_on UNIV (λ(x::?'c, y::?'c). x + y); (λ(x::?'a, y::?'b). (?f::?'a ⇒ ?'b ⇒ ?'c) x y) summable_on Sigma (?A::?'a set) (?B::?'a ⇒ ?'b set); ⋀x::?'a. x ∈ ?A ⟹ ?f x summable_on ?B x⟧ ⟹ (∑⇩∞x::?'a∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x::?'a, y::?'b)∈Sigma ?A ?B. ?f x y)›*) infsum_cong (*‹(⋀x::?'a. x ∈ (?A::?'a set) ⟹ (?f::?'a ⇒ ?'b) x = (?g::?'a ⇒ ?'b) x) ⟹ infsum ?f ?A = infsum ?g ?A›*))\n finally (*calculation: ‹(∑⇩∞x∈A. infsum (f x) B) = (∑⇩∞y∈B. ∑⇩∞x∈A. f x y)›*) show \"?thesis\"\n (*goal: ‹(∑⇩∞x::'a::type∈(A::'a::type set). infsum ((f::'a::type ⇒ 'b::type ⇒ 'c::{comm_monoid_add,t2_space,uniform_space}) x) (B::'b::type set)) = (∑⇩∞y::'b::type∈B. ∑⇩∞x::'a::type∈A. f x y)›*) .\nqed\n\nlemma infsum_swap_banach:\n fixes A :: \"'a set\" and B :: \"'b set\"\n fixes f :: \"'a ⇒ 'b ⇒ 'c::banach\"\n assumes ‹(λ(x, y). f x y) summable_on (A × B)›\n shows \"infsum (λx. infsum (λy. f x y) B) A = infsum (λy. infsum (λx. f x y) A) B\"\nproof (-)\n (*goal: ‹(∑⇩∞x∈A. infsum (f x) B) = (∑⇩∞y∈B. ∑⇩∞x∈A. f x y)›*)\n have \"§\": \"(λ(x, y). f y x) summable_on (B × A)\"\n by (metis (mono_tags, lifting) assms (*‹(λ(x, y). f x y) summable_on A × B›*) case_swap (*‹(case prod.swap ?p of (y, x) ⇒ ?f x y) = (case ?p of (x, y) ⇒ ?f x y)›*) inj_swap (*‹inj_on prod.swap ?A›*) o_apply (*‹(?f ∘ ?g) ?x = ?f (?g ?x)›*) product_swap (*‹prod.swap ` (?A × ?B) = ?B × ?A›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*) summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*))\n have \"infsum (λx. infsum (λy. f x y) B) A = infsum (λ(x,y). f x y) (A × B)\"\n using assms (*‹(λ(x, y). f x y) summable_on A × B›*) infsum_Sigma'_banach (*‹(λ(x, y). ?f x y) summable_on Sigma ?A ?B ⟹ (∑⇩∞x∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x, y)∈Sigma ?A ?B. ?f x y)›*) by blast\n also (*calculation: ‹(∑⇩∞x∈A. infsum (f x) B) = (∑⇩∞(x, y)∈A × B. f x y)›*) have \"… = infsum (λ(x,y). f y x) (B × A)\"\n apply (subst product_swap[symmetric] (*‹?B × ?A = prod.swap ` (?A × ?B)›*))\n (*goal: ‹(∑⇩∞(x::'a, y::'b)∈(A::'a set) × (B::'b set). (f::'a ⇒ 'b ⇒ 'c) x y) = (∑⇩∞(x::'b, y::'a)∈B × A. f y x)›*)\n apply (subst infsum_reindex (*‹inj_on ?h ?A ⟹ infsum ?g (?h ` ?A) = infsum (?g ∘ ?h) ?A›*))\n (*goal: ‹(∑⇩∞(x, y)∈prod.swap ` (B × A). f x y) = (∑⇩∞(x, y)∈B × A. f y x)›*)\n using assms (*‹(λ(x, y). f x y) summable_on A × B›*)\n (*goals:\n 1. ‹inj_on prod.swap (B × A)›\n 2. ‹infsum ((λ(x, y). f x y) ∘ prod.swap) (B × A) = (∑⇩∞(x, y)∈B × A. f y x)›\n discuss goal 1*)\n apply ((auto simp: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(∑⇩∞x∈A. infsum (f x) B) = (∑⇩∞(x, y)∈B × A. f y x)›*) have \"… = infsum (λy. infsum (λx. f x y) A) B\"\n by (metis (mono_tags, lifting) § (*‹(λ(x, y). f y x) summable_on B × A›*) infsum_Sigma'_banach (*‹(λ(x, y). ?f x y) summable_on Sigma ?A ?B ⟹ (∑⇩∞x∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x, y)∈Sigma ?A ?B. ?f x y)›*) infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*))\n finally (*calculation: ‹(∑⇩∞x∈A. infsum (f x) B) = (∑⇩∞y∈B. ∑⇩∞x∈A. f x y)›*) show \"?thesis\"\n (*goal: ‹(∑⇩∞x∈A. infsum (f x) B) = (∑⇩∞y∈B. ∑⇩∞x∈A. f x y)›*) .\nqed\n\nlemma nonneg_infsum_le_0D:\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add,ordered_ab_group_add,linorder_topology}\"\n assumes \"infsum f A ≤ 0\"\n and abs_sum: \"f summable_on A\"\n and nneg: \"⋀x. x ∈ A ⟹ f x ≥ 0\"\n and \"x ∈ A\"\n shows \"f x = 0\"\nproof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹f x ≠ 0 ⟹ False›*)\n assume \"f x ≠ 0\" (*‹(f::'a ⇒ 'b) (x::'a) ≠ (0::'b)›*)\n have ex: \"f summable_on (A-{x})\"\n apply (rule summable_on_cofin_subset (*‹⟦?f summable_on ?A; finite ?F⟧ ⟹ ?f summable_on ?A - ?F›*))\n (*goals:\n 1. ‹f summable_on A›\n 2. ‹finite {x}›\n discuss goal 1*)\n apply ((use assms in auto)[1])\n (*discuss goal 2*)\n apply ((use assms in auto)[1])\n (*proven 2 subgoals*) .\n have pos: \"infsum f (A - {x}) ≥ 0\"\n apply (rule infsum_nonneg (*‹(⋀x. x ∈ ?M ⟹ 0 ≤ ?f x) ⟹ 0 ≤ infsum ?f ?M›*))\n (*goal: ‹(0::'b) ≤ infsum (f::'a ⇒ 'b) ((A::'a set) - {x::'a})›*)\n by (use nneg in auto)\n have [trans]: \"x ≥ y ⟹ y > z ⟹ x > z\" for x :: 'b and y :: 'b and z :: 'b\n by auto\n have \"infsum f A = infsum f (A-{x}) + infsum f {x}\"\n apply (subst infsum_Un_disjoint[symmetric] (*‹⟦(?f::?'a ⇒ ?'b) summable_on (?A::?'a set); ?f summable_on (?B::?'a set); ?A ∩ ?B = {}⟧ ⟹ infsum ?f ?A + infsum ?f ?B = infsum ?f (?A ∪ ?B)›*))\n (*goals:\n 1. ‹f summable_on A - {x}›\n 2. ‹f summable_on {x}›\n 3. ‹(A - {x}) ∩ {x} = {}›\n 4. ‹infsum f A = infsum f (A - {x} ∪ {x})›\n discuss goal 1*)\n apply ((use assms ex in ‹auto simp: insert_absorb›)[1])\n (*discuss goal 2*)\n apply ((use assms ex in ‹auto simp: insert_absorb›)[1])\n (*discuss goal 3*)\n apply ((use assms ex in ‹auto simp: insert_absorb›)[1])\n (*discuss goal 4*)\n apply ((use assms ex in ‹auto simp: insert_absorb›)[1])\n (*proven 4 subgoals*) .\n also (*calculation: ‹infsum f A = infsum f (A - {x}) + infsum f {x}›*) have \"… ≥ infsum f {x}\" (is \"_ ≥ …\")\n using pos (*‹0 ≤ infsum f (A - {x})›*) apply (rule add_increasing (*‹⟦0 ≤ ?a; ?b ≤ ?c⟧ ⟹ ?b ≤ ?a + ?c›*))\n (*goal: ‹infsum f {x} ≤ infsum f (A - {x}) + infsum f {x}›*)\n by simp\n also (*calculation: ‹infsum f {x} ≤ infsum f A›*) have \"… = f x\" (is \"_ = …\")\n apply (subst infsum_finite (*‹finite ?F ⟹ infsum ?f ?F = sum ?f ?F›*))\n (*goals:\n 1. ‹finite {x}›\n 2. ‹sum f {x} = f x›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹f x ≤ infsum f A›*) have \"… > 0\"\n using ‹f x ≠ 0› (*‹f x ≠ 0›*) assms(4) (*‹x ∈ A›*) nneg (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) by fastforce\n finally (*calculation: ‹0 < infsum f A›*) show False\n using assms (*‹infsum (f::'a ⇒ 'b) (A::'a set) ≤ (0::'b)› ‹(f::'a ⇒ 'b) summable_on (A::'a set)› ‹?x ∈ A ⟹ 0 ≤ f ?x› ‹(x::'a) ∈ (A::'a set)›*) by auto\nqed\n\nlemma nonneg_has_sum_le_0D:\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add,ordered_ab_group_add,linorder_topology}\"\n assumes \"(f has_sum a) A\" ‹a ≤ 0›\n and \"⋀x. x ∈ A ⟹ f x ≥ 0\"\n and \"x ∈ A\"\n shows \"f x = 0\"\n by (metis assms (*‹(f has_sum a) A› ‹a ≤ 0› ‹?x ∈ A ⟹ 0 ≤ f ?x› ‹x ∈ A›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) nonneg_infsum_le_0D (*‹⟦infsum ?f ?A ≤ 0; ?f summable_on ?A; ⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; ?x ∈ ?A⟧ ⟹ ?f ?x = 0›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma has_sum_cmult_left:\n fixes f :: \"'a ⇒ 'b :: {topological_semigroup_mult, semiring_0}\"\n assumes ‹(f has_sum a) A›\n shows \"((λx. f x * c) has_sum (a * c)) A\"\n using assms (*‹(f has_sum a) A›*) tendsto_mult_right (*‹(?f ⤏ ?l) ?F ⟹ ((λx. ?f x * ?c) ⤏ ?l * ?c) ?F›*) by (force simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) sum_distrib_right (*‹sum ?f ?A * ?r = (∑n∈?A. ?f n * ?r)›*))\n\nlemma infsum_cmult_left:\n fixes f :: \"'a ⇒ 'b :: {t2_space, topological_semigroup_mult, semiring_0}\"\n assumes ‹c ≠ 0 ⟹ f summable_on A›\n shows \"infsum (λx. f x * c) A = infsum f A * c\"\n using assms (*‹c ≠ 0 ⟹ f summable_on A›*) has_sum_cmult_left (*‹(?f has_sum ?a) ?A ⟹ ((λx. ?f x * ?c) has_sum ?a * ?c) ?A›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) summable_iff_has_sum_infsum (*‹(?f summable_on ?A) = (?f has_sum infsum ?f ?A) ?A›*) by fastforce\n\nlemma summable_on_cmult_left:\n fixes f :: \"'a ⇒ 'b :: {t2_space, topological_semigroup_mult, semiring_0}\"\n assumes ‹f summable_on A›\n shows \"(λx. f x * c) summable_on A\"\n using assms (*‹f summable_on A›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_cmult_left (*‹(?f has_sum ?a) ?A ⟹ ((λx. ?f x * ?c) has_sum ?a * ?c) ?A›*) by blast\n\nlemma has_sum_cmult_right:\n fixes f :: \"'a ⇒ 'b :: {topological_semigroup_mult, semiring_0}\"\n assumes ‹(f has_sum a) A›\n shows \"((λx. c * f x) has_sum (c * a)) A\"\n using assms (*‹(f has_sum a) A›*) tendsto_mult_left (*‹(?f ⤏ ?l) ?F ⟹ ((λx. ?c * ?f x) ⤏ ?c * ?l) ?F›*) by (force simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) sum_distrib_left (*‹?r * sum ?f ?A = (∑n∈?A. ?r * ?f n)›*))\n\nlemma infsum_cmult_right:\n fixes f :: \"'a ⇒ 'b :: {t2_space, topological_semigroup_mult, semiring_0}\"\n assumes ‹c ≠ 0 ⟹ f summable_on A›\n shows ‹infsum (λx. c * f x) A = c * infsum f A›\n using assms (*‹c ≠ 0 ⟹ f summable_on A›*) has_sum_cmult_right (*‹(?f has_sum ?a) ?A ⟹ ((λx. ?c * ?f x) has_sum ?c * ?a) ?A›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) summable_iff_has_sum_infsum (*‹(?f summable_on ?A) = (?f has_sum infsum ?f ?A) ?A›*) by fastforce\n\nlemma summable_on_cmult_right:\n fixes f :: \"'a ⇒ 'b :: {t2_space, topological_semigroup_mult, semiring_0}\"\n assumes ‹f summable_on A›\n shows \"(λx. c * f x) summable_on A\"\n using assms (*‹f summable_on A›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_cmult_right (*‹(?f has_sum ?a) ?A ⟹ ((λx. ?c * ?f x) has_sum ?c * ?a) ?A›*) by blast\n\nlemma summable_on_cmult_left':\n fixes f :: \"'a ⇒ 'b :: {t2_space, topological_semigroup_mult, division_ring}\"\n assumes ‹c ≠ 0›\n shows \"(λx. f x * c) summable_on A ⟷ f summable_on A\"\nproof (standard)\n (*goals:\n 1. ‹(λx. f x * c) summable_on A ⟹ f summable_on A›\n 2. ‹f summable_on A ⟹ (λx. f x * c) summable_on A›*)\n assume \"f summable_on A\" (*‹(f::'a ⇒ 'b) summable_on (A::'a set)›*)\n then show \"(λx. f x * c) summable_on A\"\n by (rule summable_on_cmult_left (*‹?f summable_on ?A ⟹ (λx. ?f x * ?c) summable_on ?A›*))\nnext\n (*goal: ‹(λx. f x * c) summable_on A ⟹ f summable_on A›*)\n assume \"(λx. f x * c) summable_on A\" (*‹(λx::'a. (f::'a ⇒ 'b) x * (c::'b)) summable_on (A::'a set)›*)\n then have \"(λx. f x * c * inverse c) summable_on A\"\n by (rule summable_on_cmult_left (*‹?f summable_on ?A ⟹ (λx. ?f x * ?c) summable_on ?A›*))\n then show \"f summable_on A\"\n by (smt (verit, del_insts) assms (*‹c ≠ 0›*) divide_inverse (*‹?a / ?b = ?a * inverse ?b›*) nonzero_divide_eq_eq (*‹?c ≠ 0 ⟹ (?b / ?c = ?a) = (?b = ?a * ?c)›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\nqed\n\nlemma summable_on_cmult_right':\n fixes f :: \"'a ⇒ 'b :: {t2_space, topological_semigroup_mult, division_ring}\"\n assumes ‹c ≠ 0›\n shows \"(λx. c * f x) summable_on A ⟷ f summable_on A\"\n by (metis (no_types, lifting) assms (*‹c ≠ 0›*) left_inverse (*‹?a ≠ 0 ⟹ inverse ?a * ?a = 1›*) mult.assoc (*‹?a * ?b * ?c = ?a * (?b * ?c)›*) mult_1 (*‹1 * ?a = ?a›*) summable_on_cmult_right (*‹?f summable_on ?A ⟹ (λx. ?c * ?f x) summable_on ?A›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n\nlemma infsum_cmult_left':\n fixes f :: \"'a ⇒ 'b :: {t2_space, topological_semigroup_mult, division_ring}\"\n shows \"infsum (λx. f x * c) A = infsum f A * c\"\n by (metis (full_types) infsum_cmult_left (*‹(?c ≠ 0 ⟹ ?f summable_on ?A) ⟹ (∑⇩∞x∈?A. ?f x * ?c) = infsum ?f ?A * ?c›*) infsum_not_exists (*‹¬ ?f summable_on ?A ⟹ infsum ?f ?A = 0›*) mult_eq_0_iff (*‹(?a * ?b = 0) = (?a = 0 ∨ ?b = 0)›*) summable_on_cmult_left' (*‹?c ≠ 0 ⟹ ((λx. ?f x * ?c) summable_on ?A) = (?f summable_on ?A)›*))\n\nlemma infsum_cmult_right':\n fixes f :: \"'a ⇒ 'b :: {t2_space,topological_semigroup_mult,division_ring}\"\n shows \"infsum (λx. c * f x) A = c * infsum f A\"\n by (metis (full_types) infsum_cmult_right (*‹(?c ≠ 0 ⟹ ?f summable_on ?A) ⟹ (∑⇩∞x∈?A. ?c * ?f x) = ?c * infsum ?f ?A›*) infsum_not_exists (*‹¬ ?f summable_on ?A ⟹ infsum ?f ?A = 0›*) mult_eq_0_iff (*‹(?a * ?b = 0) = (?a = 0 ∨ ?b = 0)›*) summable_on_cmult_right' (*‹?c ≠ 0 ⟹ ((λx. ?c * ?f x) summable_on ?A) = (?f summable_on ?A)›*))\n\nlemma has_sum_constant[simp]:\n assumes ‹finite F›\n shows ‹((λ_. c) has_sum of_nat (card F) * c) F›\n by (metis assms (*‹finite F›*) has_sum_finite (*‹finite ?F ⟹ (?f has_sum sum ?f ?F) ?F›*) sum_constant (*‹(∑x∈?A. ?y) = of_nat (card ?A) * ?y›*))\n\nlemma infsum_constant[simp]:\n assumes ‹finite F›\n shows ‹infsum (λ_. c) F = of_nat (card F) * c›\n by (simp add: assms (*‹finite F›*))\n\nlemma infsum_diverge_constant:\n ― ‹This probably does not really need all of \\<^class>‹archimedean_field› but Isabelle/HOL\n has no type class such as, e.g., \"archimedean ring\".›\n fixes c :: ‹'a::{archimedean_field, comm_monoid_add, linorder_topology, topological_semigroup_mult}›\n assumes ‹infinite A› and ‹c ≠ 0›\n shows ‹¬ (λ_. c) summable_on A›\nproof (rule notI (*‹(?P ⟹ False) ⟹ ¬ ?P›*))\n (*goal: ‹(λ_::'b::type. c::'a::{archimedean_field,topological_semigroup_mult,linorder_topology}) summable_on (A::'b::type set) ⟹ False›*)\n assume \"(λ_. c) summable_on A\" (*‹(λ_::'b. c::'a) summable_on (A::'b set)›*)\n then have \"(λ_. inverse c * c) summable_on A\"\n by (rule summable_on_cmult_right (*‹?f summable_on ?A ⟹ (λx. ?c * ?f x) summable_on ?A›*))\n then have [simp]: \"(λ_. 1::'a) summable_on A\"\n using assms (*‹infinite (A::'b set)› ‹c ≠ 0›*) by auto\n have \"infsum (λ_. 1) A ≥ d\" for d :: 'a\n proof (-)\n (*goal: ‹d ≤ (∑⇩∞_∈A. 1)›*)\n obtain n :: nat where \"of_nat n ≥ d\"\n (*goal: ‹(⋀n. d ≤ of_nat n ⟹ thesis) ⟹ thesis›*)\n by (meson real_arch_simple (*‹∃n. ?x ≤ of_nat n›*))\n from assms (*‹infinite A› ‹c ≠ 0›*) obtain F where \"F ⊆ A\" and \"finite F\" and \"card F = n\"\n (*goal: ‹(⋀F. ⟦F ⊆ A; finite F; card F = n⟧ ⟹ thesis) ⟹ thesis›*)\n by (meson infinite_arbitrarily_large (*‹infinite ?A ⟹ ∃B. finite B ∧ card B = ?n ∧ B ⊆ ?A›*))\n note ‹d ≤ of_nat n› (*‹(d::'a::{archimedean_field,topological_semigroup_mult,linorder_topology}) ≤ of_nat (n::nat)›*)\n also (*calculation: ‹(d::'a::{archimedean_field,topological_semigroup_mult,linorder_topology}) ≤ of_nat (n::nat)›*) have \"of_nat n = infsum (λ_. 1::'a) F\"\n by (simp add: ‹card F = n› ‹finite F›)\n also (*calculation: ‹d ≤ (∑⇩∞_∈F. 1)›*) have \"… ≤ infsum (λ_. 1::'a) A\"\n apply (rule infsum_mono_neutral (*‹⟦?f summable_on ?A; ?g summable_on ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?B›*))\n (*goal: ‹(∑⇩∞_∈F. 1) ≤ (∑⇩∞_∈A. 1)›*)\n using ‹finite F› (*‹finite F›*) ‹F ⊆ A› (*‹F ⊆ A›*) apply -\n (*goals:\n 1. ‹⟦finite (F::'b set); F ⊆ (A::'b set)⟧ ⟹ (λ_::'b. 1::'a) summable_on F›\n 2. ‹⟦finite (F::'b set); F ⊆ (A::'b set)⟧ ⟹ (λ_::'b. 1::'a) summable_on A›\n 3. ‹⋀x::'b. ⟦x ∈ (F::'b set) ∩ (A::'b set); finite F; F ⊆ A⟧ ⟹ (1::'a) ≤ (1::'a)›\n 4. ‹⋀x::'b. ⟦x ∈ (F::'b set) - (A::'b set); finite F; F ⊆ A⟧ ⟹ (1::'a) ≤ (0::'a)›\n 5. ‹⋀x::'b. ⟦x ∈ (A::'b set) - (F::'b set); finite F; F ⊆ A⟧ ⟹ (0::'a) ≤ (1::'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 finally (*calculation: ‹d ≤ (∑⇩∞_∈A. 1)›*) show \"?thesis\"\n (*goal: ‹d ≤ (∑⇩∞_∈A. 1)›*) .\n qed\n then show False\n by (meson linordered_field_no_ub (*‹∀x::?'a. ∃y::?'a. x < y›*) not_less (*‹(¬ (?x::?'a) < (?y::?'a)) = (?y ≤ ?x)›*))\nqed\n\nlemma has_sum_constant_archimedean[simp]:\n ― ‹This probably does not really need all of \\<^class>‹archimedean_field› but Isabelle/HOL\n has no type class such as, e.g., \"archimedean ring\".›\n fixes c :: ‹'a::{archimedean_field, comm_monoid_add, linorder_topology, topological_semigroup_mult}›\n shows ‹infsum (λ_. c) A = of_nat (card A) * c›\n by (metis infsum_0 (*‹(⋀x. x ∈ ?M ⟹ ?f x = 0) ⟹ infsum ?f ?M = 0›*) infsum_constant (*‹finite ?F ⟹ (∑⇩∞_∈?F. ?c) = of_nat (card ?F) * ?c›*) infsum_diverge_constant (*‹⟦infinite ?A; ?c ≠ 0⟧ ⟹ ¬ (λ_. ?c) summable_on ?A›*) infsum_not_exists (*‹¬ ?f summable_on ?A ⟹ infsum ?f ?A = 0›*) sum.infinite (*‹infinite ?A ⟹ sum ?g ?A = 0›*) sum_constant (*‹(∑x∈?A. ?y) = of_nat (card ?A) * ?y›*))\n\nlemma has_sum_uminus:\n fixes f :: ‹'a ⇒ 'b::topological_ab_group_add›\n shows ‹((λx. - f x) has_sum a) A ⟷ (f has_sum (- a)) A›\n by (auto simp add: sum_negf[abs_def] (*‹sum (λx. - ?f x) ≡ λA. - sum ?f A›*) tendsto_minus_cancel_left (*‹(?f ⤏ - ?y) ?F = ((λx. - ?f x) ⤏ ?y) ?F›*) has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n\nlemma summable_on_uminus:\n fixes f :: ‹'a ⇒ 'b::topological_ab_group_add›\n shows‹(λx. - f x) summable_on A ⟷ f summable_on A›\n by (metis summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*) has_sum_uminus (*‹((λx::?'a. - (?f::?'a ⇒ ?'b) x) has_sum (?a::?'b)) (?A::?'a set) = (?f has_sum - ?a) ?A›*) verit_minus_simplify( (*‹- (- (?b::?'b)) = ?b›*) 4))\n\nlemma infsum_uminus:\n fixes f :: ‹'a ⇒ 'b::{topological_ab_group_add, t2_space}›\n shows ‹infsum (λx. - f x) A = - infsum f A›\n by (metis (full_types) add.inverse_inverse (*‹- (- ?a) = ?a›*) add.inverse_neutral (*‹- 0 = 0›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) infsum_def (*‹infsum ?f ?A = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else 0)›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_uminus (*‹((λx. - ?f x) has_sum ?a) ?A = (?f has_sum - ?a) ?A›*))\n\nlemma has_sum_le_finite_sums:\n fixes a :: ‹'a::{comm_monoid_add,topological_space,linorder_topology}›\n assumes ‹(f has_sum a) A›\n assumes ‹⋀F. finite F ⟹ F ⊆ A ⟹ sum f F ≤ b›\n shows ‹a ≤ b›\n by (metis assms (*‹((f::'b ⇒ 'a) has_sum (a::'a)) (A::'b set)› ‹⟦finite (?F::'b set); ?F ⊆ (A::'b set)⟧ ⟹ sum (f::'b ⇒ 'a) ?F ≤ (b::'a)›*) eventually_finite_subsets_at_top_weakI (*‹(⋀X::?'a set. ⟦finite X; X ⊆ (?A::?'a set)⟧ ⟹ (?P::?'a set ⇒ bool) X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*) finite_subsets_at_top_neq_bot (*‹finite_subsets_at_top (?A::?'a set) ≠ bot›*) has_sum_def (*‹((?f::?'a ⇒ ?'b) has_sum (?x::?'b)) (?A::?'a set) ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) tendsto_upperbound (*‹⟦((?f::?'b ⇒ ?'a) ⤏ (?x::?'a)) (?F::?'b filter); ∀⇩F i::?'b in ?F. ?f i ≤ (?a::?'a); ?F ≠ bot⟧ ⟹ ?x ≤ ?a›*))\n\nlemma infsum_le_finite_sums:\n fixes b :: ‹'a::{comm_monoid_add,topological_space,linorder_topology}›\n assumes ‹f summable_on A›\n assumes ‹⋀F. finite F ⟹ F ⊆ A ⟹ sum f F ≤ b›\n shows ‹infsum f A ≤ b›\n by (meson assms (*‹f summable_on A› ‹⟦finite ?F; ?F ⊆ A⟧ ⟹ sum f ?F ≤ b›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_le_finite_sums (*‹⟦(?f has_sum ?a) ?A; ⋀F. ⟦finite F; F ⊆ ?A⟧ ⟹ sum ?f F ≤ ?b⟧ ⟹ ?a ≤ ?b›*))\n\n\nlemma summable_on_scaleR_left [intro]:\n fixes c :: ‹'a :: real_normed_vector›\n assumes \"c ≠ 0 ⟹ f summable_on A\"\n shows \"(λx. f x *⇩R c) summable_on A\"\nproof (cases ‹c = 0›)\n (*goals:\n 1. ‹c = 0 ⟹ (λx. f x *⇩R c) summable_on A›\n 2. ‹c ≠ 0 ⟹ (λx. f x *⇩R c) summable_on A›*)\n case False (*‹c ≠ 0›*)\n then have \"(λy. y *⇩R c) ∘ f summable_on A\"\n using assms (*‹c ≠ 0 ⟹ f summable_on A›*) by (auto simp add: scaleR_left.additive_axioms (*‹additive (λa. a *⇩R ?x)›*) summable_on_comm_additive (*‹⟦additive ?f; isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ ?f ∘ ?g summable_on ?S›*))\n then show \"?thesis\"\n (*goal: ‹(λx::'b. (f::'b ⇒ real) x *⇩R (c::'a)) summable_on (A::'b set)›*)\n by (metis (mono_tags, lifting) comp_apply (*‹(?f ∘ ?g) ?x = ?f (?g ?x)›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\nqed (auto)\n(*solved the remaining goal: ‹c = 0 ⟹ (λx. f x *⇩R c) summable_on A›*)\n\n\nlemma summable_on_scaleR_right [intro]:\n fixes f :: ‹'a ⇒ 'b :: real_normed_vector›\n assumes \"c ≠ 0 ⟹ f summable_on A\"\n shows \"(λx. c *⇩R f x) summable_on A\"\nproof (cases ‹c = 0›)\n (*goals:\n 1. ‹(c::real) = (0::real) ⟹ (λx::'a. c *⇩R (f::'a ⇒ 'b) x) summable_on (A::'a set)›\n 2. ‹(c::real) ≠ (0::real) ⟹ (λx::'a. c *⇩R (f::'a ⇒ 'b) x) summable_on (A::'a set)›*)\n case False (*‹c ≠ 0›*)\n then have \"(*⇩R) c ∘ f summable_on A\"\n using assms (*‹c ≠ 0 ⟹ f summable_on A›*) by (auto simp add: scaleR_right.additive_axioms (*‹additive ((*⇩R) ?a)›*) summable_on_comm_additive (*‹⟦additive ?f; isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ ?f ∘ ?g summable_on ?S›*))\n then show \"?thesis\"\n (*goal: ‹(λx. c *⇩R f x) summable_on A›*)\n by (metis (mono_tags, lifting) comp_apply (*‹(?f ∘ ?g) ?x = ?f (?g ?x)›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\nqed (auto)\n(*solved the remaining goal: ‹c = 0 ⟹ (λx. c *⇩R f x) summable_on A›*)\n\nlemma infsum_scaleR_left:\n fixes c :: ‹'a :: real_normed_vector›\n assumes \"c ≠ 0 ⟹ f summable_on A\"\n shows \"infsum (λx. f x *⇩R c) A = infsum f A *⇩R c\"\nproof (cases ‹c = 0›)\n (*goals:\n 1. ‹c = 0 ⟹ (∑⇩∞x∈A. f x *⇩R c) = infsum f A *⇩R c›\n 2. ‹c ≠ 0 ⟹ (∑⇩∞x∈A. f x *⇩R c) = infsum f A *⇩R c›*)\n case False (*‹c ≠ 0›*)\n then have \"infsum ((λy. y *⇩R c) ∘ f) A = infsum f A *⇩R c\"\n using assms (*‹c ≠ 0 ⟹ f summable_on A›*) by (auto simp add: scaleR_left.additive_axioms (*‹additive (λa. a *⇩R ?x)›*) infsum_comm_additive (*‹⟦additive ?f; isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ infsum (?f ∘ ?g) ?S = ?f (infsum ?g ?S)›*))\n then show \"?thesis\"\n (*goal: ‹(∑⇩∞x∈A. f x *⇩R c) = infsum f A *⇩R c›*)\n by (metis (mono_tags, lifting) comp_apply (*‹(?f ∘ ?g) ?x = ?f (?g ?x)›*) infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*))\nqed (auto)\n(*solved the remaining goal: ‹c = 0 ⟹ (∑⇩∞x∈A. f x *⇩R c) = infsum f A *⇩R c›*)\n\nlemma infsum_scaleR_right:\n fixes f :: ‹'a ⇒ 'b :: real_normed_vector›\n shows \"infsum (λx. c *⇩R f x) A = c *⇩R infsum f A\"\nproof -\n consider (summable) ‹f summable_on A› | (c0) ‹c = 0› | (not_summable) ‹¬ f summable_on A› ‹c ≠ 0›\n by auto\n then show ?thesis\n proof cases\n case summable\n then have \"infsum ((*⇩R) c ∘ f) A = c *⇩R infsum f A\"\n by (auto simp add: scaleR_right.additive_axioms infsum_comm_additive)\n then show ?thesis\n by (metis (mono_tags, lifting) comp_apply infsum_cong)\n next\n case c0\n then show ?thesis by auto\n next\n case not_summable\n have ‹¬ (λx. c *⇩R f x) summable_on A›\n proof (rule notI)\n assume ‹(λx. c *⇩R f x) summable_on A›\n then have ‹(λx. inverse c *⇩R c *⇩R f x) summable_on A›\n using summable_on_scaleR_right by blast\n with not_summable show False\n by simp\n qed\n then show ?thesis\n by (simp add: infsum_not_exists not_summable(1)) \n qed\nqed\n\n\nlemma infsum_Un_Int:\n fixes f :: \"'a ⇒ 'b::{topological_ab_group_add, t2_space}\"\n assumes \"f summable_on A - B\" \"f summable_on B - A\" ‹f summable_on A ∩ B›\n shows \"infsum f (A ∪ B) = infsum f A + infsum f B - infsum f (A ∩ B)\"\nproof (-)\n (*goal: ‹infsum f (A ∪ B) = infsum f A + infsum f B - infsum f (A ∩ B)›*)\n obtain \"f summable_on A\" \"f summable_on B\"\n (*goal: ‹(⟦f summable_on A; f summable_on B⟧ ⟹ thesis) ⟹ thesis›*)\n using assms (*‹f summable_on A - B› ‹f summable_on B - A› ‹f summable_on A ∩ B›*) by (metis Int_Diff_Un (*‹?A ∩ ?B ∪ (?A - ?B) = ?A›*) Int_Diff_disjoint (*‹?A ∩ ?B ∩ (?A - ?B) = {}›*) inf_commute (*‹inf ?x ?y = inf ?y ?x›*) summable_on_Un_disjoint (*‹⟦?f summable_on ?A; ?f summable_on ?B; ?A ∩ ?B = {}⟧ ⟹ ?f summable_on ?A ∪ ?B›*))\n then have \"infsum f (A ∪ B) = infsum f A + infsum f (B - A)\"\n using assms(2) (*‹f summable_on B - A›*) infsum_Un_disjoint (*‹⟦(?f::?'a ⇒ ?'b) summable_on (?A::?'a set); ?f summable_on (?B::?'a set); ?A ∩ ?B = {}⟧ ⟹ infsum ?f (?A ∪ ?B) = infsum ?f ?A + infsum ?f ?B›*) by fastforce\n moreover have \"infsum f (B - A) = infsum f B - infsum f (A ∩ B)\"\n using assms (*‹f summable_on A - B› ‹f summable_on B - A› ‹f summable_on A ∩ B›*) by (metis Diff_Int2 (*‹(?A::?'a set) ∩ (?C::?'a set) - (?B::?'a set) ∩ ?C = ?A ∩ ?C - ?B›*) Un_Int_eq( (*‹((?S::?'a set) ∪ (?T::?'a set)) ∩ ?T = ?T›*) 2) ‹f summable_on B› inf_le2 (*‹inf (?x::?'a) (?y::?'a) ≤ ?y›*) infsum_Diff (*‹⟦(?f::?'a ⇒ ?'b) summable_on (?B::?'a set); ?f summable_on (?A::?'a set); ?A ⊆ ?B⟧ ⟹ infsum ?f (?B - ?A) = infsum ?f ?B - infsum ?f ?A›*))\n ultimately show \"?thesis\"\n (*goal: ‹infsum f (A ∪ B) = infsum f A + infsum f B - infsum f (A ∩ B)›*)\n by auto\nqed\n\nlemma inj_combinator':\n assumes \"x ∉ F\"\n shows ‹inj_on (λ(g, y). g(x := y)) (Pi⇩E F B × B x)›\nproof (-)\n (*goal: ‹inj_on (λ(g, y). g(x := y)) (Pi⇩E F B × B x)›*)\n have \"inj_on ((λ(y, g). g(x := y)) ∘ prod.swap) (Pi⇩E F B × B x)\"\n using inj_combinator[of x F B] (*‹x ∉ F ⟹ inj_on (λ(y, g). g(x := y)) (B x × Pi⇩E F B)›*) assms (*‹x ∉ F›*) apply (intro comp_inj_on (*‹⟦inj_on (?f::?'a ⇒ ?'b) (?A::?'a set); inj_on (?g::?'b ⇒ ?'c) (?f ` ?A)⟧ ⟹ inj_on (?g ∘ ?f) ?A›*))\n (*goals:\n 1. ‹⟦x ∉ F ⟹ inj_on (λ(y, g). g(x := y)) (B x × Pi⇩E F B); x ∉ F⟧ ⟹ inj_on prod.swap (Pi⇩E F B × B x)›\n 2. ‹⟦x ∉ F ⟹ inj_on (λ(y, g). g(x := y)) (B x × Pi⇩E F B); x ∉ F⟧ ⟹ inj_on (λ(y, g). g(x := y)) (prod.swap ` (Pi⇩E F B × B x))›\n discuss goal 1*)\n apply ((auto simp: product_swap (*‹prod.swap ` (?A × ?B) = ?B × ?A›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: product_swap (*‹prod.swap ` (?A × ?B) = ?B × ?A›*))[1])\n (*proven 2 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹inj_on (λ(g::'a ⇒ 'b, y::'b). g(x::'a := y)) (Pi⇩E (F::'a set) (B::'a ⇒ 'b set) × B x)›*)\n by (simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\nqed\n\nlemma infsum_prod_PiE:\n ― ‹See also ‹infsum_prod_PiE_abs› below with incomparable premises.›\n fixes f :: \"'a ⇒ 'b ⇒ 'c :: {comm_monoid_mult, topological_semigroup_mult, division_ring, banach}\"\n assumes finite: \"finite A\"\n assumes \"⋀x. x ∈ A ⟹ f x summable_on B x\"\n assumes \"(λg. ∏x∈A. f x (g x)) summable_on (PiE A B)\"\n shows \"infsum (λg. ∏x∈A. f x (g x)) (PiE A B) = (∏x∈A. infsum (f x) (B x))\"\nproof (use finite assms(2-) in induction)\n (*goals:\n 1. ‹⟦⋀x. x ∈ {} ⟹ f x summable_on B x; (λg. ∏x∈{}. f x (g x)) summable_on Pi⇩E {} B⟧ ⟹ (∑⇩∞g∈Pi⇩E {} B. ∏x∈{}. f x (g x)) = (∏x∈{}. infsum (f x) (B x))›\n 2. ‹⋀x F. ⟦finite F; x ∉ F; ⟦⋀x. x ∈ F ⟹ f x summable_on B x; (λg. ∏x∈F. f x (g x)) summable_on Pi⇩E F B⟧ ⟹ (∑⇩∞g∈Pi⇩E F B. ∏x∈F. f x (g x)) = (∏x∈F. infsum (f x) (B x)); ⋀xa. xa ∈ insert x F ⟹ f xa summable_on B xa; (λg. ∏x∈insert x F. f x (g x)) summable_on Pi⇩E (insert x F) B⟧ ⟹ (∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = (∏x∈insert x F. infsum (f x) (B x))›*)\n case empty (*‹?x ∈ {} ⟹ f ?x summable_on B ?x› ‹(λg. ∏x∈{}. f x (g x)) summable_on Pi⇩E {} B›*)\n then show \"?case\"\n (*goal: ‹(∑⇩∞g∈Pi⇩E {} B. ∏x∈{}. f x (g x)) = (∏x∈{}. infsum (f x) (B x))›*)\n by auto\nnext\n (*goal: ‹⋀x F. ⟦finite F; x ∉ F; ⟦⋀x. x ∈ F ⟹ f x summable_on B x; (λg. ∏x∈F. f x (g x)) summable_on Pi⇩E F B⟧ ⟹ (∑⇩∞g∈Pi⇩E F B. ∏x∈F. f x (g x)) = (∏x∈F. infsum (f x) (B x)); ⋀xa. xa ∈ insert x F ⟹ f xa summable_on B xa; (λg. ∏x∈insert x F. f x (g x)) summable_on Pi⇩E (insert x F) B⟧ ⟹ (∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = (∏x∈insert x F. infsum (f x) (B x))›*)\n case (insert x F) (*‹finite F› ‹(x::'a::type) ∉ (F::'a::type set)› ‹⟦⋀x. x ∈ F ⟹ f x summable_on B x; (λg. ∏x∈F. f x (g x)) summable_on Pi⇩E F B⟧ ⟹ (∑⇩∞g∈Pi⇩E F B. ∏x∈F. f x (g x)) = (∏x∈F. infsum (f x) (B x))› ‹?x ∈ insert x F ⟹ f ?x summable_on B ?x› ‹(λg. ∏x∈insert x F. f x (g x)) summable_on Pi⇩E (insert x F) B›*)\n have pi: \"Pi⇩E (insert x F) B = (λ(g,y). g(x:=y)) ` (Pi⇩E F B × B x)\"\n unfolding PiE_insert_eq\n (*goal: ‹(λ(y, g). g(x := y)) ` (B x × Pi⇩E F B) = (λ(g, y). g(x := y)) ` (Pi⇩E F B × B x)›*)\n apply (subst swap_product [symmetric] (*‹(?B::?'a::type set) × (?A::?'b::type set) = (λ(i::?'b::type, j::?'a::type). (j, i)) ` (?A × ?B)›*))\n (*goal: ‹(λ(y, g). g(x := y)) ` (B x × Pi⇩E F B) = (λ(g, y). g(x := y)) ` (Pi⇩E F B × B x)›*)\n by (simp add: image_image (*‹(?f::?'b ⇒ ?'a) ` (?g::?'c ⇒ ?'b) ` (?A::?'c set) = (λx::?'c. ?f (?g x)) ` ?A›*) case_prod_unfold (*‹case_prod = (λ(c::?'a ⇒ ?'b ⇒ ?'c) p::?'a × ?'b. c (fst p) (snd p))›*))\n have prod: \"(∏x'∈F. f x' ((p(x:=y)) x')) = (∏x'∈F. f x' (p x'))\" for p and y\n apply (rule prod.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ prod ?g ?A = prod ?h ?B›*))\n (*goals:\n 1. ‹(F::'a::type set) = F›\n 2. ‹⋀xa::'a::type. xa ∈ (F::'a::type set) ⟹ (f::'a::type ⇒ 'b::type ⇒ 'c::{division_ring,comm_monoid_mult,topological_semigroup_mult,banach}) xa (((p::'a::type ⇒ 'b::type)(x::'a::type := y::'b::type)) xa) = f xa (p xa)›\n discuss goal 1*)\n apply ((use insert.hyps in auto)[1])\n (*discuss goal 2*)\n apply ((use insert.hyps in auto)[1])\n (*proven 2 subgoals*) .\n have inj: \"inj_on (λ(g, y). g(x := y)) (Pi⇩E F B × B x)\"\n using ‹x ∉ F› (*‹(x::'a) ∉ (F::'a set)›*) by (rule inj_combinator' (*‹(?x::?'a) ∉ (?F::?'a set) ⟹ inj_on (λ(g::?'a ⇒ ?'b, y::?'b). g(?x := y)) (Pi⇩E ?F (?B::?'a ⇒ ?'b set) × ?B ?x)›*))\n have summable1: \"(λg. ∏x∈insert x F. f x (g x)) summable_on Pi⇩E (insert x F) B\"\n using insert.prems(2) (*‹(λg. ∏x∈insert x F. f x (g x)) summable_on Pi⇩E (insert x F) B›*) .\n also (*calculation: ‹(λg. ∏x∈insert x F. f x (g x)) summable_on Pi⇩E (insert x F) B›*) have \"Pi⇩E (insert x F) B = (λ(g,y). g(x:=y)) ` (Pi⇩E F B × B x)\"\n by (simp only: pi (*‹Pi⇩E (insert x F) B = (λ(g, y). g(x := y)) ` (Pi⇩E F B × B x)›*))\n also (*calculation: ‹(λg::'a ⇒ 'b. ∏x::'a∈insert (x::'a) (F::'a set). (f::'a ⇒ 'b ⇒ 'c) x (g x)) summable_on (λ(g::'a ⇒ 'b, y::'b). g(x := y)) ` (Pi⇩E F (B::'a ⇒ 'b set) × B x)›*) have \"(λg. ∏x∈insert x F. f x (g x)) summable_on … ⟷\n ((λg. ∏x∈insert x F. f x (g x)) ∘ (λ(g,y). g(x:=y))) summable_on (Pi⇩E F B × B x)\"\n using inj (*‹inj_on (λ(g, y). g(x := y)) (Pi⇩E F B × B x)›*) by (rule summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*))\n also (*calculation: ‹(λg. ∏x∈insert x F. f x (g x)) ∘ (λ(g, y). g(x := y)) summable_on Pi⇩E F B × B x›*) have \"(∏z∈F. f z ((g(x := y)) z)) = (∏z∈F. f z (g z))\" for g and y\n using insert.hyps (*‹finite (F::'a set)› ‹x ∉ F›*) apply (intro prod.cong (*‹⟦(?A::?'b set) = (?B::?'b set); ⋀x::?'b. x ∈ ?B ⟹ (?g::?'b ⇒ ?'a) x = (?h::?'b ⇒ ?'a) x⟧ ⟹ prod ?g ?A = prod ?h ?B›*))\n (*goals:\n 1. ‹⟦finite F; x ∉ F⟧ ⟹ F = F›\n 2. ‹⋀xa. ⟦finite F; x ∉ F; xa ∈ F⟧ ⟹ f xa ((g(x := y)) xa) = f xa (g xa)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence \"((λg. ∏x∈insert x F. f x (g x)) ∘ (λ(g,y). g(x:=y))) =\n (λ(p, y). f x y * (∏x'∈F. f x' (p x')))\"\n using insert.hyps (*‹finite F› ‹x ∉ F›*) by (auto simp: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) cong: prod.cong_simp (*‹⟦?A = ?B; ⋀x. x ∈ ?B =simp=> ?g x = ?h x⟧ ⟹ prod ?g ?A = prod ?h ?B›*))\n finally (*calculation: ‹(λa. case a of (p, y) ⇒ f x y * (∏x'∈F. f x' (p x'))) summable_on Pi⇩E F B × B x›*) have summable2: \"(λ(p, y). f x y * (∏x'∈F. f x' (p x'))) summable_on Pi⇩E F B × B x\" .\n then have \"(λp. ∑⇩∞y∈B x. f x y * (∏x'∈F. f x' (p x'))) summable_on Pi⇩E F B\"\n by (rule summable_on_Sigma_banach (*‹(λ(x, y). ?f x y) summable_on Sigma ?A ?B ⟹ (λx. infsum (?f x) (?B x)) summable_on ?A›*))\n then have \"(λp. (∑⇩∞y∈B x. f x y) * (∏x'∈F. f x' (p x'))) summable_on Pi⇩E F B\"\n by (metis (mono_tags, lifting) infsum_cmult_left' (*‹(∑⇩∞x∈?A. ?f x * ?c) = infsum ?f ?A * ?c›*) infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n then have summable3: \"(λp. (∏x'∈F. f x' (p x'))) summable_on Pi⇩E F B\" if \"(∑⇩∞y∈B x. f x y) ≠ 0\"\n using summable_on_cmult_right' (*‹(?c::?'b::{division_ring,topological_semigroup_mult,t2_space}) ≠ (0::?'b::{division_ring,topological_semigroup_mult,t2_space}) ⟹ ((λx::?'a::type. ?c * (?f::?'a::type ⇒ ?'b::{division_ring,topological_semigroup_mult,t2_space}) x) summable_on (?A::?'a::type set)) = (?f summable_on ?A)›*) that (*‹infsum (f x) (B x) ≠ 0›*) by blast\n have \"(∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x))\n = (∑⇩∞(p,y)∈Pi⇩E F B × B x. ∏x'∈insert x F. f x' ((p(x:=y)) x'))\"\n by (smt (verit, ccfv_SIG) comp_apply (*‹(?f ∘ ?g) ?x = ?f (?g ?x)›*) infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*) infsum_reindex (*‹inj_on ?h ?A ⟹ infsum ?g (?h ` ?A) = infsum (?g ∘ ?h) ?A›*) inj (*‹inj_on (λ(g, y). g(x := y)) (Pi⇩E F B × B x)›*) pi (*‹Pi⇩E (insert x F) B = (λ(g, y). g(x := y)) ` (Pi⇩E F B × B x)›*) prod.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ prod ?g ?A = prod ?h ?B›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*))\n also (*calculation: ‹(∑⇩∞g::'a ⇒ 'b∈Pi⇩E (insert x F) (B::'a ⇒ 'b set). ∏x::'a∈insert (x::'a) (F::'a set). (f::'a ⇒ 'b ⇒ 'c) x (g x)) = (∑⇩∞(p::'a ⇒ 'b, y::'b)∈Pi⇩E F B × B x. ∏x'::'a∈insert x F. f x' ((p(x := y)) x'))›*) have \"… = (∑⇩∞(p, y)∈Pi⇩E F B × B x. f x y * (∏x'∈F. f x' ((p(x:=y)) x')))\"\n using insert.hyps (*‹finite F› ‹x ∉ F›*) by auto\n also (*calculation: ‹(∑⇩∞g::'a ⇒ 'b∈Pi⇩E (insert x F) (B::'a ⇒ 'b set). ∏x::'a∈insert (x::'a) (F::'a set). (f::'a ⇒ 'b ⇒ 'c) x (g x)) = (∑⇩∞(p::'a ⇒ 'b, y::'b)∈Pi⇩E F B × B x. f x y * (∏x'::'a∈F. f x' ((p(x := y)) x')))›*) have \"… = (∑⇩∞(p, y)∈Pi⇩E F B × B x. f x y * (∏x'∈F. f x' (p x')))\"\n using prod (*‹(∏x'∈F. f x' ((?p(x := ?y)) x')) = (∏x'∈F. f x' (?p x'))›*) by presburger\n also (*calculation: ‹(∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = (∑⇩∞(p, y)∈Pi⇩E F B × B x. f x y * (∏x'∈F. f x' (p x')))›*) have \"… = (∑⇩∞p∈Pi⇩E F B. ∑⇩∞y∈B x. f x y * (∏x'∈F. f x' (p x')))\"\n using infsum_Sigma'_banach (*‹(λ(x, y). ?f x y) summable_on Sigma ?A ?B ⟹ (∑⇩∞x∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x, y)∈Sigma ?A ?B. ?f x y)›*) summable2 (*‹(λ(p, y). f x y * (∏x'∈F. f x' (p x'))) summable_on Pi⇩E F B × B x›*) by force\n also (*calculation: ‹(∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = (∑⇩∞p∈Pi⇩E F B. ∑⇩∞y∈B x. f x y * (∏x'∈F. f x' (p x')))›*) have \"… = (∑⇩∞y∈B x. f x y) * (∑⇩∞p∈Pi⇩E F B. ∏x'∈F. f x' (p x'))\"\n by (smt (verit) infsum_cmult_left' (*‹(∑⇩∞x∈?A. ?f x * ?c) = infsum ?f ?A * ?c›*) infsum_cmult_right' (*‹(∑⇩∞x∈?A. ?c * ?f x) = ?c * infsum ?f ?A›*) infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*))\n also (*calculation: ‹(∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = infsum (f x) (B x) * (∑⇩∞p∈Pi⇩E F B. ∏x'∈F. f x' (p x'))›*) have \"… = (∏x∈insert x F. infsum (f x) (B x))\"\n using insert (*‹finite F› ‹x ∉ F› ‹⟦⋀x. x ∈ F ⟹ f x summable_on B x; (λg. ∏x∈F. f x (g x)) summable_on Pi⇩E F B⟧ ⟹ (∑⇩∞g∈Pi⇩E F B. ∏x∈F. f x (g x)) = (∏x∈F. infsum (f x) (B x))› ‹?x ∈ insert x F ⟹ f ?x summable_on B ?x› ‹(λg. ∏x∈insert x F. f x (g x)) summable_on Pi⇩E (insert x F) B›*) summable3 (*‹infsum (f x) (B x) ≠ 0 ⟹ (λp. ∏x'∈F. f x' (p x')) summable_on Pi⇩E F B›*) by auto\n finally (*calculation: ‹(∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = (∏x∈insert x F. infsum (f x) (B x))›*) show \"?case\"\n (*goal: ‹(∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = (∏x∈insert x F. infsum (f x) (B x))›*)\n by simp\nqed\n\nlemma infsum_prod_PiE_abs:\n ― ‹See also @{thm [source] infsum_prod_PiE} above with incomparable premises.›\n fixes f :: \"'a ⇒ 'b ⇒ 'c :: {banach, real_normed_div_algebra, comm_semiring_1}\"\n assumes finite: \"finite A\"\n assumes \"⋀x. x ∈ A ⟹ f x abs_summable_on B x\"\n shows \"infsum (λg. ∏x∈A. f x (g x)) (PiE A B) = (∏x∈A. infsum (f x) (B x))\"\nproof (use finite assms(2) in induction)\n (*goals:\n 1. ‹(⋀x. x ∈ {} ⟹ f x abs_summable_on B x) ⟹ (∑⇩∞g∈Pi⇩E {} B. ∏x∈{}. f x (g x)) = (∏x∈{}. infsum (f x) (B x))›\n 2. ‹⋀x F. ⟦finite F; x ∉ F; (⋀x. x ∈ F ⟹ f x abs_summable_on B x) ⟹ (∑⇩∞g∈Pi⇩E F B. ∏x∈F. f x (g x)) = (∏x∈F. infsum (f x) (B x)); ⋀xa. xa ∈ insert x F ⟹ f xa abs_summable_on B xa⟧ ⟹ (∑⇩∞g∈Pi⇩E (insert x F) B. ∏x∈insert x F. f x (g x)) = (∏x∈insert x F. infsum (f x) (B x))›*)\n case empty (*‹?x ∈ {} ⟹ f ?x abs_summable_on B ?x›*)\n then show \"?case\"\n (*goal: ‹(∑⇩∞g∈Pi⇩E {} B. ∏x∈{}. f x (g x)) = (∏x∈{}. infsum (f x) (B x))›*)\n by auto\nnext\n (*goal: ‹⋀(x::'a::type) F::'a::type set. ⟦finite F; x ∉ F; (⋀x::'a::type. x ∈ F ⟹ (f::'a::type ⇒ 'b::type ⇒ 'c::{banach,real_normed_div_algebra,comm_semiring_1}) x abs_summable_on (B::'a::type ⇒ 'b::type set) x) ⟹ (∑⇩∞g::'a::type ⇒ 'b::type∈Pi⇩E F B. ∏x::'a::type∈F. f x (g x)) = (∏x::'a::type∈F. infsum (f x) (B x)); ⋀xa::'a::type. xa ∈ insert x F ⟹ f xa abs_summable_on B xa⟧ ⟹ (∑⇩∞g::'a::type ⇒ 'b::type∈Pi⇩E (insert x F) B. ∏x::'a::type∈insert x F. f x (g x)) = (∏x::'a::type∈insert x F. infsum (f x) (B x))›*)\n case (insert x A) (*‹finite A› ‹x ∉ A› ‹(⋀x. x ∈ A ⟹ f x abs_summable_on B x) ⟹ (∑⇩∞g∈Pi⇩E A B. ∏x∈A. f x (g x)) = (∏x∈A. infsum (f x) (B x))› ‹(?x::'a) ∈ insert (x::'a) (A::'a set) ⟹ (f::'a ⇒ 'b ⇒ 'c) ?x abs_summable_on (B::'a ⇒ 'b set) ?x›*)\n have pi: \"Pi⇩E (insert x F) B = (λ(g,y). g(x:=y)) ` (Pi⇩E F B × B x)\" for x and F and B :: \"'a ⇒ 'b set\"\n unfolding PiE_insert_eq\n (*goal: ‹(λ(y, g). g(x := y)) ` (B x × Pi⇩E F B) = (λ(g, y). g(x := y)) ` (Pi⇩E F B × B x)›*)\n apply (subst swap_product [symmetric] (*‹?B × ?A = (λ(i, j). (j, i)) ` (?A × ?B)›*))\n (*goal: ‹(λ(y, g). g(x := y)) ` (B x × Pi⇩E F B) = (λ(g, y). g(x := y)) ` (Pi⇩E F B × B x)›*)\n by (simp add: image_image (*‹?f ` ?g ` ?A = (λx. ?f (?g x)) ` ?A›*) case_prod_unfold (*‹case_prod = (λc p. c (fst p) (snd p))›*))\n have prod: \"(∏x'∈A. f x' ((p(x:=y)) x')) = (∏x'∈A. f x' (p x'))\" for p and y\n apply (rule prod.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ prod ?g ?A = prod ?h ?B›*))\n (*goals:\n 1. ‹A = A›\n 2. ‹⋀xa. xa ∈ A ⟹ f xa ((p(x := y)) xa) = f xa (p xa)›\n discuss goal 1*)\n apply ((use insert.hyps in auto)[1])\n (*discuss goal 2*)\n apply ((use insert.hyps in auto)[1])\n (*proven 2 subgoals*) .\n have inj: \"inj_on (λ(g, y). g(x := y)) (Pi⇩E A B × B x)\"\n using ‹x ∉ A› (*‹x ∉ A›*) by (rule inj_combinator' (*‹?x ∉ ?F ⟹ inj_on (λ(g, y). g(?x := y)) (Pi⇩E ?F ?B × ?B ?x)›*))\n define s where \"s x = infsum (λy. norm (f x y)) (B x)\" for x\n have \"(∑p∈P. norm (∏x∈F. f x (p x))) ≤ prod s F\" if P: \"P ⊆ Pi⇩E F B\" and [simp]: \"finite P\" \"finite F\" and sum: \"⋀x. x ∈ F ⟹ f x abs_summable_on B x\" for P and F\n proof (-)\n (*goal: ‹(∑p∈P. norm (∏x∈F. f x (p x))) ≤ prod s F›*)\n define B' where \"B' x = {p x| p. p∈P}\" for x\n have fin_B'[simp]: \"finite (B' x)\" for x\n using that (*‹P ⊆ Pi⇩E F B› ‹finite P› ‹finite F› ‹(?x::'a) ∈ (F::'a set) ⟹ (f::'a ⇒ 'b ⇒ 'c) ?x abs_summable_on (B::'a ⇒ 'b set) ?x›*) by (auto simp: B'_def (*‹B' ?x = {p ?x |p. p ∈ P}›*))\n have [simp]: \"finite (Pi⇩E F B')\"\n by (simp add: finite_PiE (*‹⟦finite ?S; ⋀i. i ∈ ?S ⟹ finite (?T i)⟧ ⟹ finite (Pi⇩E ?S ?T)›*))\n have [simp]: \"P ⊆ Pi⇩E F B'\"\n using that (*‹P ⊆ Pi⇩E F B› ‹finite P› ‹finite F› ‹?x ∈ F ⟹ f ?x abs_summable_on B ?x›*) by (auto simp: B'_def (*‹B' ?x = {p ?x |p. p ∈ P}›*))\n have B'B: \"B' x ⊆ B x\" if \"x ∈ F\" for x\n unfolding B'_def\n (*goal: ‹{p (x::'a::type) |p::'a::type ⇒ 'b::type. p ∈ (P::('a::type ⇒ 'b::type) set)} ⊆ (B::'a::type ⇒ 'b::type set) x›*)\n using P (*‹(P::('a ⇒ 'b) set) ⊆ Pi⇩E (F::'a set) (B::'a ⇒ 'b set)›*) that (*‹x ∈ F›*) by auto\n have s_bound: \"(∑y∈B' x. norm (f x y)) ≤ s x\" if \"x ∈ F\" for x\n by (metis B'B (*‹(?x::'a) ∈ (F::'a set) ⟹ (B'::'a ⇒ 'b set) ?x ⊆ (B::'a ⇒ 'b set) ?x›*) fin_B' (*‹finite ((B'::'a ⇒ 'b set) (?x::'a))›*) finite_sum_le_has_sum (*‹⟦((?f::?'a ⇒ ?'b) has_sum (?S::?'b)) (?A::?'a set); finite (?B::?'a set); ?B ⊆ ?A; ⋀x::?'a. x ∈ ?A - ?B ⟹ (0::?'b) ≤ ?f x⟧ ⟹ sum ?f ?B ≤ ?S›*) has_sum_infsum (*‹(?f::?'a ⇒ ?'b) summable_on (?S::?'a set) ⟹ (?f has_sum infsum ?f ?S) ?S›*) norm_ge_zero (*‹(0::real) ≤ norm (?x::?'a)›*) s_def (*‹(s::'a ⇒ real) (?x::'a) = (∑⇩∞y::'b∈(B::'a ⇒ 'b set) ?x. norm ((f::'a ⇒ 'b ⇒ 'c) ?x y))›*) sum (*‹(?x::'a) ∈ (F::'a set) ⟹ (f::'a ⇒ 'b ⇒ 'c) ?x abs_summable_on (B::'a ⇒ 'b set) ?x›*) that (*‹(x::'a) ∈ (F::'a set)›*))\n have \"(∑p∈P. norm (∏x∈F. f x (p x))) ≤ (∑p∈Pi⇩E F B'. norm (∏x∈F. f x (p x)))\"\n by (simp add: sum_mono2 (*‹⟦finite ?B; ?A ⊆ ?B; ⋀b. b ∈ ?B - ?A ⟹ 0 ≤ ?f b⟧ ⟹ sum ?f ?A ≤ sum ?f ?B›*))\n also (*calculation: ‹(∑p∈P. norm (∏x∈F. f x (p x))) ≤ (∑p∈Pi⇩E F B'. norm (∏x∈F. f x (p x)))›*) have \"… = (∑p∈Pi⇩E F B'. ∏x∈F. norm (f x (p x)))\"\n by (simp add: prod_norm (*‹(∏x∈?A. norm (?f x)) = norm (prod ?f ?A)›*))\n also (*calculation: ‹(∑p∈P. norm (∏x∈F. f x (p x))) ≤ (∑p∈Pi⇩E F B'. ∏x∈F. norm (f x (p x)))›*) have \"… = (∏x∈F. ∑y∈B' x. norm (f x y))\"\n proof (use ‹finite F› in induction)\n (*goals:\n 1. ‹(∑p::'a::type ⇒ 'b::type∈Pi⇩E {} (B'::'a::type ⇒ 'b::type set). ∏x::'a::type∈{}. norm ((f::'a::type ⇒ 'b::type ⇒ 'c::{banach,real_normed_div_algebra,comm_semiring_1}) x (p x))) = (∏x::'a::type∈{}. ∑y::'b::type∈B' x. norm (f x y))›\n 2. ‹⋀(x::'a::type) F::'a::type set. ⟦finite F; x ∉ F; (∑p::'a::type ⇒ 'b::type∈Pi⇩E F (B'::'a::type ⇒ 'b::type set). ∏x::'a::type∈F. norm ((f::'a::type ⇒ 'b::type ⇒ 'c::{banach,real_normed_div_algebra,comm_semiring_1}) x (p x))) = (∏x::'a::type∈F. ∑y::'b::type∈B' x. norm (f x y))⟧ ⟹ (∑p::'a::type ⇒ 'b::type∈Pi⇩E (insert x F) B'. ∏x::'a::type∈insert x F. norm (f x (p x))) = (∏x::'a::type∈insert x F. ∑y::'b::type∈B' x. norm (f x y))›*)\n case empty (*no hyothesis introduced yet*)\n then show \"?case\"\n (*goal: ‹(∑p∈Pi⇩E {} B'. ∏x∈{}. norm (f x (p x))) = (∏x∈{}. ∑y∈B' x. norm (f x y))›*)\n by simp\n next\n (*goal: ‹⋀x F. ⟦finite F; x ∉ F; (∑p∈Pi⇩E F B'. ∏x∈F. norm (f x (p x))) = (∏x∈F. ∑y∈B' x. norm (f x y))⟧ ⟹ (∑p∈Pi⇩E (insert x F) B'. ∏x∈insert x F. norm (f x (p x))) = (∏x∈insert x F. ∑y∈B' x. norm (f x y))›*)\n case (insert x F) (*‹finite F› ‹x ∉ F› ‹(∑p∈Pi⇩E F B'. ∏x∈F. norm (f x (p x))) = (∏x∈F. ∑y∈B' x. norm (f x y))›*)\n have inj: \"inj_on (λ(g, y). g(x := y)) (Pi⇩E F B' × B' x)\"\n by (simp add: inj_combinator' (*‹?x ∉ ?F ⟹ inj_on (λ(g, y). g(?x := y)) (Pi⇩E ?F ?B × ?B ?x)›*) insert.hyps (*‹finite F› ‹x ∉ F›*))\n then have \"(∑p∈Pi⇩E (insert x F) B'. ∏x∈insert x F. norm (f x (p x)))\n = (∑(p,y)∈Pi⇩E F B' × B' x. ∏x'∈insert x F. norm (f x' ((p(x := y)) x')))\"\n by (simp add: pi (*‹Pi⇩E (insert (?x::'a) (?F::'a set)) (?B::'a ⇒ 'b set) = (λ(g::'a ⇒ 'b, y::'b). g(?x := y)) ` (Pi⇩E ?F ?B × ?B ?x)›*) sum.reindex (*‹inj_on (?h::?'b ⇒ ?'c) (?A::?'b set) ⟹ sum (?g::?'c ⇒ ?'a) (?h ` ?A) = sum (?g ∘ ?h) ?A›*) case_prod_unfold (*‹case_prod = (λ(c::?'a ⇒ ?'b ⇒ ?'c) p::?'a × ?'b. c (fst p) (snd p))›*))\n also (*calculation: ‹(∑p∈Pi⇩E (insert x F) B'. ∏x∈insert x F. norm (f x (p x))) = (∑(p, y)∈Pi⇩E F B' × B' x. ∏x'∈insert x F. norm (f x' ((p(x := y)) x')))›*) have \"… = (∑(p, y)∈Pi⇩E F B' × B' x. norm (f x y) * (∏x'∈F. norm (f x' (p x'))))\"\n by (smt (verit, del_insts) fun_upd_apply (*‹(?f(?x := ?y)) ?z = (if ?z = ?x then ?y else ?f ?z)›*) insert.hyps (*‹finite F› ‹x ∉ F›*) prod.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ prod ?g ?A = prod ?h ?B›*) prod.insert (*‹⟦finite ?A; ?x ∉ ?A⟧ ⟹ prod ?g (insert ?x ?A) = ?g ?x * prod ?g ?A›*) split_def (*‹case_prod = (λc p. c (fst p) (snd p))›*) sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n also (*calculation: ‹(∑p∈Pi⇩E (insert x F) B'. ∏x∈insert x F. norm (f x (p x))) = (∑(p, y)∈Pi⇩E F B' × B' x. norm (f x y) * (∏x'∈F. norm (f x' (p x'))))›*) have \"… = (∑y∈B' x. norm (f x y)) * (∑p∈Pi⇩E F B'. ∏x'∈F. norm (f x' (p x')))\"\n by (simp add: sum_product (*‹sum ?f ?A * sum ?g ?B = (∑i∈?A. ∑j∈?B. ?f i * ?g j)›*) sum.swap [of _ \"Pi⇩E F B'\"] (*‹(∑i∈?A. sum (?g i) (Pi⇩E F B')) = (∑j∈Pi⇩E F B'. ∑i∈?A. ?g i j)›*) sum.cartesian_product (*‹(∑x∈?A. sum (?g x) ?B) = (∑(x, y)∈?A × ?B. ?g x y)›*))\n also (*calculation: ‹(∑p∈Pi⇩E (insert x F) B'. ∏x∈insert x F. norm (f x (p x))) = (∑y∈B' x. norm (f x y)) * (∑p∈Pi⇩E F B'. ∏x'∈F. norm (f x' (p x')))›*) have \"… = (∏x∈insert x F. ∑y∈B' x. norm (f x y))\"\n using insert (*‹finite F› ‹x ∉ F› ‹(∑p∈Pi⇩E F B'. ∏x∈F. norm (f x (p x))) = (∏x∈F. ∑y∈B' x. norm (f x y))›*) by force\n finally (*calculation: ‹(∑p∈Pi⇩E (insert x F) B'. ∏x∈insert x F. norm (f x (p x))) = (∏x∈insert x F. ∑y∈B' x. norm (f x y))›*) show \"?case\"\n (*goal: ‹(∑p∈Pi⇩E (insert x F) B'. ∏x∈insert x F. norm (f x (p x))) = (∏x∈insert x F. ∑y∈B' x. norm (f x y))›*) .\n qed\n also (*calculation: ‹(∑p::'a::type ⇒ 'b::type∈(P::('a::type ⇒ 'b::type) set). norm (∏x::'a::type∈(F::'a::type set). (f::'a::type ⇒ 'b::type ⇒ 'c::{banach,real_normed_div_algebra,comm_semiring_1}) x (p x))) ≤ (∏x::'a::type∈F. ∑y::'b::type∈(B'::'a::type ⇒ 'b::type set) x. norm (f x y))›*) have \"… ≤ (∏x∈F. s x)\"\n using s_bound (*‹(?x::'a) ∈ (F::'a set) ⟹ (∑y::'b∈(B'::'a ⇒ 'b set) ?x. norm ((f::'a ⇒ 'b ⇒ 'c) ?x y)) ≤ (s::'a ⇒ real) ?x›*) by (simp add: prod_mono (*‹(⋀i. i ∈ ?A ⟹ 0 ≤ ?f i ∧ ?f i ≤ ?g i) ⟹ prod ?f ?A ≤ prod ?g ?A›*) sum_nonneg (*‹(⋀x. x ∈ ?A ⟹ 0 ≤ ?f x) ⟹ 0 ≤ sum ?f ?A›*))\n finally (*calculation: ‹(∑p∈P. norm (∏x∈F. f x (p x))) ≤ prod s F›*) show \"?thesis\"\n (*goal: ‹(∑p∈P. norm (∏x∈F. f x (p x))) ≤ prod s F›*) .\n qed\n then have \"bdd_above\n (sum (λg. norm (∏x∈insert x A. f x (g x))) ` {F. F ⊆ Pi⇩E (insert x A) B ∧ finite F})\"\n using insert.hyps (*‹finite A› ‹x ∉ A›*) insert.prems (*‹?x ∈ insert x A ⟹ f ?x abs_summable_on B ?x›*) apply (intro bdd_aboveI (*‹(⋀x. x ∈ ?A ⟹ x ≤ ?M) ⟹ bdd_above ?A›*))\n (*goal: ‹bdd_above (sum (λg. norm (∏x∈insert x A. f x (g x))) ` {F. F ⊆ Pi⇩E (insert x A) B ∧ finite F})›*)\n by blast\n then have \"(λg. ∏x∈insert x A. f x (g x)) abs_summable_on Pi⇩E (insert x A) B\"\n using nonneg_bdd_above_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ ?f summable_on ?A›*) by (metis (mono_tags, lifting) Collect_cong (*‹(⋀x::?'a::type. (?P::?'a::type ⇒ bool) x = (?Q::?'a::type ⇒ bool) x) ⟹ {x::?'a::type. ?P x} = {x::?'a::type. ?Q x}›*) norm_ge_zero (*‹(0::real) ≤ norm (?x::?'a::real_normed_vector)›*))\n also (*calculation: ‹(λxa. ∏x∈insert x A. f x (xa x)) abs_summable_on Pi⇩E (insert x A) B›*) have \"Pi⇩E (insert x A) B = (λ(g,y). g(x:=y)) ` (Pi⇩E A B × B x)\"\n by (simp only: pi (*‹Pi⇩E (insert ?x ?F) ?B = (λ(g, y). g(?x := y)) ` (Pi⇩E ?F ?B × ?B ?x)›*))\n also (*calculation: ‹(λxa. ∏x∈insert x A. f x (xa x)) abs_summable_on (λ(g, y). g(x := y)) ` (Pi⇩E A B × B x)›*) have \"(λg. ∏x∈insert x A. f x (g x)) abs_summable_on … ⟷\n ((λg. ∏x∈insert x A. f x (g x)) ∘ (λ(g,y). g(x:=y))) abs_summable_on (Pi⇩E A B × B x)\"\n using inj (*‹inj_on (λ(g, y). g(x := y)) (Pi⇩E A B × B x)›*) apply (subst summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*))\n (*goals:\n 1. ‹inj_on (λ(g, y). g(x := y)) (Pi⇩E A B × B x) ⟹ inj_on (λ(g, y). g(x := y)) (Pi⇩E A B × B x)›\n 2. ‹inj_on (λ(g, y). g(x := y)) (Pi⇩E A B × B x) ⟹ ((λxa. norm (∏x∈insert x A. f x (xa x))) ∘ (λ(g, y). g(x := y)) summable_on Pi⇩E A B × B x) = ((λg. ∏x∈insert x A. f x (g x)) ∘ (λ(g, y). g(x := y)) abs_summable_on Pi⇩E A B × B x)›\n discuss goal 1*)\n apply ((auto simp: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹(λg::'a ⇒ 'b. ∏x::'a∈insert (x::'a) (A::'a set). (f::'a ⇒ 'b ⇒ 'c) x (g x)) ∘ (λ(g::'a ⇒ 'b, y::'b). g(x := y)) abs_summable_on Pi⇩E A (B::'a ⇒ 'b set) × B x›*) have \"(∏z∈A. f z ((g(x := y)) z)) = (∏z∈A. f z (g z))\" for g and y\n using insert.hyps (*‹finite A› ‹(x::'a) ∉ (A::'a set)›*) apply (intro prod.cong (*‹⟦(?A::?'b::type set) = (?B::?'b::type set); ⋀x::?'b::type. x ∈ ?B ⟹ (?g::?'b::type ⇒ ?'a::comm_monoid_mult) x = (?h::?'b::type ⇒ ?'a::comm_monoid_mult) x⟧ ⟹ prod ?g ?A = prod ?h ?B›*))\n (*goals:\n 1. ‹⟦finite (A::'a::type set); (x::'a::type) ∉ A⟧ ⟹ A = A›\n 2. ‹⋀xa::'a::type. ⟦finite (A::'a::type set); (x::'a::type) ∉ A; xa ∈ A⟧ ⟹ (f::'a::type ⇒ 'b::type ⇒ 'c::{banach,real_normed_div_algebra,comm_semiring_1}) xa (((g::'a::type ⇒ 'b::type)(x := y::'b::type)) xa) = f xa (g xa)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence \"((λg. ∏x∈insert x A. f x (g x)) ∘ (λ(g,y). g(x:=y))) =\n (λ(p, y). f x y * (∏x'∈A. f x' (p x')))\"\n using insert.hyps (*‹finite A› ‹(x::'a) ∉ (A::'a set)›*) by (auto simp: fun_eq_iff (*‹(?f = ?g) = (∀x. ?f x = ?g x)›*) cong: prod.cong_simp (*‹⟦?A = ?B; ⋀x. x ∈ ?B =simp=> ?g x = ?h x⟧ ⟹ prod ?g ?A = prod ?h ?B›*))\n finally (*calculation: ‹(λxa. case xa of (p, y) ⇒ f x y * (∏x'∈A. f x' (p x'))) abs_summable_on Pi⇩E A B × B x›*) have summable2: \"(λ(p, y). f x y * (∏x'∈A. f x' (p x'))) abs_summable_on Pi⇩E A B × B x\" .\n have \"(∑⇩∞g∈Pi⇩E (insert x A) B. ∏x∈insert x A. f x (g x))\n = (∑⇩∞(p,y)∈Pi⇩E A B × B x. ∏x'∈insert x A. f x' ((p(x:=y)) x'))\"\n using inj (*‹inj_on (λ(g::'a ⇒ 'b, y::'b). g(x::'a := y)) (Pi⇩E (A::'a set) (B::'a ⇒ 'b set) × B x)›*) by (simp add: pi (*‹Pi⇩E (insert (?x::'a) (?F::'a set)) (?B::'a ⇒ 'b set) = (λ(g::'a ⇒ 'b, y::'b). g(?x := y)) ` (Pi⇩E ?F ?B × ?B ?x)›*) infsum_reindex (*‹inj_on (?h::?'a ⇒ ?'b) (?A::?'a set) ⟹ infsum (?g::?'b ⇒ ?'c) (?h ` ?A) = infsum (?g ∘ ?h) ?A›*) o_def (*‹(?f::?'b ⇒ ?'c) ∘ (?g::?'a ⇒ ?'b) = (λx::?'a. ?f (?g x))›*) case_prod_unfold (*‹case_prod = (λ(c::?'a ⇒ ?'b ⇒ ?'c) p::?'a × ?'b. c (fst p) (snd p))›*))\n also (*calculation: ‹(∑⇩∞g∈Pi⇩E (insert x A) B. ∏x∈insert x A. f x (g x)) = (∑⇩∞(p, y)∈Pi⇩E A B × B x. ∏x'∈insert x A. f x' ((p(x := y)) x'))›*) have \"… = (∑⇩∞(p,y) ∈ Pi⇩E A B × B x. f x y * (∏x'∈A. f x' (p x')))\"\n using prod (*‹(∏x'::'a∈(A::'a set). (f::'a ⇒ 'b ⇒ 'c) x' (((?p::'a ⇒ 'b)(x::'a := ?y::'b)) x')) = (∏x'::'a∈A. f x' (?p x'))›*) insert.hyps (*‹finite (A::'a set)› ‹x ∉ A›*) by auto\n also (*calculation: ‹(∑⇩∞g∈Pi⇩E (insert x A) B. ∏x∈insert x A. f x (g x)) = (∑⇩∞(p, y)∈Pi⇩E A B × B x. f x y * (∏x'∈A. f x' (p x')))›*) have \"… = (∑⇩∞p∈Pi⇩E A B. ∑⇩∞y∈B x. f x y * (∏x'∈A. f x' (p x')))\"\n using abs_summable_summable (*‹?f abs_summable_on ?A ⟹ ?f summable_on ?A›*) infsum_Sigma'_banach (*‹(λ(x, y). ?f x y) summable_on Sigma ?A ?B ⟹ (∑⇩∞x∈?A. infsum (?f x) (?B x)) = (∑⇩∞(x, y)∈Sigma ?A ?B. ?f x y)›*) summable2 (*‹(λxa. case xa of (p, y) ⇒ f x y * (∏x'∈A. f x' (p x'))) abs_summable_on Pi⇩E A B × B x›*) by fastforce\n also (*calculation: ‹(∑⇩∞g::'a ⇒ 'b∈Pi⇩E (insert x A) (B::'a ⇒ 'b set). ∏x::'a∈insert (x::'a) (A::'a set). (f::'a ⇒ 'b ⇒ 'c) x (g x)) = (∑⇩∞p::'a ⇒ 'b∈Pi⇩E A B. ∑⇩∞y::'b∈B x. f x y * (∏x'::'a∈A. f x' (p x')))›*) have \"… = (∑⇩∞y∈B x. f x y) * (∑⇩∞p∈Pi⇩E A B. ∏x'∈A. f x' (p x'))\"\n by (smt (verit, best) infsum_cmult_left' (*‹(∑⇩∞x∈?A. ?f x * ?c) = infsum ?f ?A * ?c›*) infsum_cmult_right' (*‹(∑⇩∞x∈?A. ?c * ?f x) = ?c * infsum ?f ?A›*) infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*))\n finally (*calculation: ‹(∑⇩∞g∈Pi⇩E (insert x A) B. ∏x∈insert x A. f x (g x)) = infsum (f x) (B x) * (∑⇩∞p∈Pi⇩E A B. ∏x'∈A. f x' (p x'))›*) show \"?case\"\n (*goal: ‹(∑⇩∞g∈Pi⇩E (insert x A) B. ∏x∈insert x A. f x (g x)) = (∏x∈insert x A. infsum (f x) (B x))›*)\n by (simp add: insert (*‹finite A› ‹x ∉ A› ‹(⋀x. x ∈ A ⟹ f x abs_summable_on B x) ⟹ (∑⇩∞g∈Pi⇩E A B. ∏x∈A. f x (g x)) = (∏x∈A. infsum (f x) (B x))› ‹?x ∈ insert x A ⟹ f ?x abs_summable_on B ?x›*))\nqed\n\n\n\nsubsection ‹Absolute convergence›\n\nlemma abs_summable_countable:\n assumes ‹f abs_summable_on A›\n shows ‹countable {x∈A. f x ≠ 0}›\nproof (-)\n (*goal: ‹countable {x ∈ A. f x ≠ 0}›*)\n have fin: \"finite {x∈A. norm (f x) ≥ t}\" if \"t > 0\" for t\n proof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹infinite {x ∈ A. t ≤ norm (f x)} ⟹ False›*)\n assume \"*\": \"infinite {x ∈ A. t ≤ norm (f x)}\" (*‹infinite {x::'a ∈ A::'a set. (t::real) ≤ norm ((f::'a ⇒ 'b) x)}›*)\n have \"infsum (λx. norm (f x)) A ≥ b\" for b\n proof (-)\n (*goal: ‹b ≤ (∑⇩∞x∈A. norm (f x))›*)\n obtain b' where b': \"of_nat b' ≥ b / t\"\n (*goal: ‹(⋀b'. b / t ≤ real b' ⟹ thesis) ⟹ thesis›*)\n by (meson real_arch_simple (*‹∃n. ?x ≤ of_nat n›*))\n from \"*\" (*‹infinite {x ∈ A. t ≤ norm (f x)}›*) obtain F where cardF: \"card F ≥ b'\" and \"finite F\" and F: \"F ⊆ {x ∈ A. t ≤ norm (f x)}\"\n (*goal: ‹(⋀F. ⟦b' ≤ card F; finite F; F ⊆ {x ∈ A. t ≤ norm (f x)}⟧ ⟹ thesis) ⟹ thesis›*)\n by (meson finite_if_finite_subsets_card_bdd (*‹(⋀G::?'a set. ⟦G ⊆ (?F::?'a set); finite G⟧ ⟹ card G ≤ (?C::nat)) ⟹ finite ?F ∧ card ?F ≤ ?C›*) nle_le (*‹(¬ (?a::?'a) ≤ (?b::?'a)) = (?b ≤ ?a ∧ ?b ≠ ?a)›*))\n have \"b ≤ of_nat b' * t\"\n using b' (*‹b / t ≤ real b'›*) ‹t > 0› (*‹0 < t›*) by (simp add: field_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 69 facts*) split: if_splits (*‹(?P::?'a::type ⇒ bool) (if ?Q::bool then ?x::?'a::type else (?y::?'a::type)) = ((?Q ⟶ ?P ?x) ∧ (¬ ?Q ⟶ ?P ?y))› ‹(?P::?'a::type ⇒ bool) (if ?Q::bool then ?x::?'a::type else (?y::?'a::type)) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n also (*calculation: ‹(b::real) ≤ real (b'::nat) * (t::real)›*) have \"… ≤ of_nat (card F) * t\"\n by (simp add: cardF (*‹b' ≤ card F›*) that (*‹0 < t›*))\n also (*calculation: ‹b ≤ real (card F) * t›*) have \"… = sum (λx. t) F\"\n by simp\n also (*calculation: ‹(b::real) ≤ (∑x::'a∈(F::'a set). (t::real))›*) have \"… ≤ sum (λx. norm (f x)) F\"\n by (metis (mono_tags, lifting) F (*‹F ⊆ {x ∈ A. t ≤ norm (f x)}›*) in_mono (*‹?A ⊆ ?B ⟹ ?x ∈ ?A ⟶ ?x ∈ ?B›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*))\n also (*calculation: ‹b ≤ (∑x∈F. norm (f x))›*) have \"… = infsum (λx. norm (f x)) F\"\n using ‹finite F› (*‹finite F›*) by (rule infsum_finite[symmetric] (*‹finite ?F ⟹ sum ?f ?F = infsum ?f ?F›*))\n also (*calculation: ‹b ≤ (∑⇩∞x∈F. norm (f x))›*) have \"… ≤ infsum (λx. norm (f x)) A\"\n apply (rule infsum_mono_neutral (*‹⟦?f summable_on ?A; ?g summable_on ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?B›*))\n (*goals:\n 1. ‹f abs_summable_on F›\n 2. ‹f abs_summable_on A›\n 3. ‹⋀x. x ∈ F ∩ A ⟹ norm (f x) ≤ norm (f x)›\n 4. ‹⋀x. x ∈ F - A ⟹ norm (f x) ≤ 0›\n 5. ‹⋀x. x ∈ A - F ⟹ 0 ≤ norm (f x)›\n discuss goal 1*)\n apply ((use ‹finite F› assms F in auto)[1])\n (*discuss goal 2*)\n apply ((use ‹finite F› assms F in auto)[1])\n (*discuss goal 3*)\n apply ((use ‹finite F› assms F in auto)[1])\n (*discuss goal 4*)\n apply ((use ‹finite F› assms F in auto)[1])\n (*discuss goal 5*)\n apply ((use ‹finite F› assms F in auto)[1])\n (*proven 5 subgoals*) .\n finally (*calculation: ‹b ≤ (∑⇩∞x∈A. norm (f x))›*) show \"?thesis\"\n (*goal: ‹b ≤ (∑⇩∞x∈A. norm (f x))›*) .\n qed\n then show False\n by (meson gt_ex (*‹∃y. ?x < y›*) linorder_not_less (*‹(¬ ?x < ?y) = (?y ≤ ?x)›*))\n qed\n have \"countable (⋃i∈{1..}. {x∈A. norm (f x) ≥ 1/of_nat i})\"\n apply (rule countable_UN (*‹⟦countable (?I::?'i set); ⋀i::?'i. i ∈ ?I ⟹ countable ((?A::?'i ⇒ ?'a set) i)⟧ ⟹ countable (⋃ (?A ` ?I))›*))\n (*goals:\n 1. ‹countable {1..}›\n 2. ‹⋀i. i ∈ {1..} ⟹ countable {x ∈ A. 1 / real i ≤ norm (f x)}›\n discuss goal 1*)\n apply ((use fin in ‹auto intro!: countable_finite›)[1])\n (*discuss goal 2*)\n apply ((use fin in ‹auto intro!: countable_finite›)[1])\n (*proven 2 subgoals*) .\n also (*calculation: ‹countable (⋃i::nat∈{1::nat..}. {x::'a::type ∈ A::'a::type set. (1::real) / real i ≤ norm ((f::'a::type ⇒ 'b::real_normed_vector) x)})›*) have \"… = {x∈A. f x ≠ 0}\"\n proof (safe)\n (*goals:\n 1. ‹⋀x i. ⟦1 ≤ i; x ∈ A; 1 / real i ≤ norm (f x); f x = 0⟧ ⟹ False›\n 2. ‹⋀x. ⟦x ∈ A; f x ≠ 0⟧ ⟹ x ∈ (⋃i∈{1..}. {x ∈ A. 1 / real i ≤ norm (f x)})›*)\n fix x\n assume x: \"x ∈ A\" \"f x ≠ 0\" (*‹(x::'a) ∈ (A::'a set)› ‹(f::'a ⇒ 'b) (x::'a) ≠ (0::'b)›*)\n define i where \"i = max 1 (nat (ceiling (1 / norm (f x))))\"\n have \"i ≥ 1\"\n by (simp add: i_def (*‹i = max 1 (nat ⌈1 / norm (f x)⌉)›*))\n moreover have \"real i ≥ 1 / norm (f x)\"\n unfolding i_def\n (*goal: ‹1 / norm (f x) ≤ real (max 1 (nat ⌈1 / norm (f x)⌉))›*)\n by linarith\n hence \"1 / real i ≤ norm (f x)\"\n using ‹f x ≠ 0› (*‹f x ≠ 0›*) by (auto simp: divide_simps (*‹inverse ?a = 1 / ?a› ‹?a + ?b / ?z = (if ?z = 0 then ?a else (?a * ?z + ?b) / ?z)› ‹?a / ?z + ?b = (if ?z = 0 then ?b else (?a + ?b * ?z) / ?z)› ‹- (?a / ?z) + ?b = (if ?z = 0 then ?b else (- ?a + ?b * ?z) / ?z)› ‹?a - ?b / ?z = (if ?z = 0 then ?a else (?a * ?z - ?b) / ?z)› ‹?a / ?z - ?b = (if ?z = 0 then - ?b else (?a - ?b * ?z) / ?z)› ‹- (?a / ?z) - ?b = (if ?z = 0 then - ?b else (- ?a - ?b * ?z) / ?z)› ‹(?b / ?c = ?a) = (if ?c ≠ 0 then ?b = ?a * ?c else ?a = 0)› ‹(?a = ?b / ?c) = (if ?c ≠ 0 then ?a * ?c = ?b else ?a = 0)› ‹(- (?b / ?c) = ?a) = (if ?c ≠ 0 then - ?b = ?a * ?c else ?a = 0)› ‹(?a = - (?b / ?c)) = (if ?c ≠ 0 then ?a * ?c = - ?b else ?a = 0)› ‹(?a ≤ ?b / ?c) = (if 0 < ?c then ?a * ?c ≤ ?b else if ?c < 0 then ?b ≤ ?a * ?c else ?a ≤ 0)› and more 13 facts*) mult_ac (*‹?a * ?b * ?c = ?a * (?b * ?c)› ‹?a * ?b = ?b * ?a› ‹?b * (?a * ?c) = ?a * (?b * ?c)›*))\n ultimately show \"x ∈ (⋃i∈{1..}. {x ∈ A. 1 / real i ≤ norm (f x)})\"\n using ‹x ∈ A› (*‹(x::'a::type) ∈ (A::'a::type set)›*) by auto\n qed (auto)\n (*solved the remaining goal: ‹⋀x i. ⟦1 ≤ i; x ∈ A; 1 / real i ≤ norm (f x); f x = 0⟧ ⟹ False›*)\n finally (*calculation: ‹countable {x::'a ∈ A::'a set. (f::'a ⇒ 'b) x ≠ (0::'b)}›*) show \"?thesis\"\n (*goal: ‹countable {x::'a ∈ A::'a set. (f::'a ⇒ 'b) x ≠ (0::'b)}›*) .\nqed\n\n(* Logically belongs in the section about reals, but needed as a dependency here *)\nlemma summable_on_iff_abs_summable_on_real:\n fixes f :: ‹'a ⇒ real›\n shows ‹f summable_on A ⟷ f abs_summable_on A›\nproof (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹f summable_on A ⟹ f abs_summable_on A›\n 2. ‹f abs_summable_on A ⟹ f summable_on A›*)\n assume \"f summable_on A\" (*‹(f::'a ⇒ real) summable_on (A::'a set)›*)\n define n and A⇩p and A⇩n where \"n ≡ λx. norm (f x)\" and \"A⇩p = {x∈A. f x ≥ 0}\" and \"A⇩n = {x∈A. f x < 0}\" for x\n have A: \"A⇩p ∪ A⇩n = A\" \"A⇩p ∩ A⇩n = {}\"\n (*goals:\n 1. ‹A⇩p ∪ A⇩n = A›\n 2. ‹A⇩p ∩ A⇩n = {}›\n discuss goal 1*)\n apply ((auto simp: A⇩p_def (*‹A⇩p = {x ∈ A. 0 ≤ f x}›*) A⇩n_def (*‹A⇩n = {x ∈ A. f x < 0}›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: A⇩p_def (*‹A⇩p = {x ∈ A. 0 ≤ f x}›*) A⇩n_def (*‹A⇩n = {x ∈ A. f x < 0}›*))[1])\n (*proven 2 subgoals*) .\n from ‹f summable_on A› (*‹f summable_on A›*) have \"f summable_on A⇩p\" \"f summable_on A⇩n\"\n using \"A⇩p_def\" (*‹(A⇩p::'a set) = {x::'a ∈ A::'a set. (0::real) ≤ (f::'a ⇒ real) x}›*) \"A⇩n_def\" (*‹A⇩n = {x ∈ A. f x < 0}›*) summable_on_subset_banach (*‹⟦?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*) apply -\n (*goals:\n 1. ‹⟦f summable_on A; A⇩p = {x ∈ A. 0 ≤ f x}; A⇩n = {x ∈ A. f x < 0}; ⋀f A B. ⟦f summable_on A; B ⊆ A⟧ ⟹ f summable_on B⟧ ⟹ f summable_on A⇩p›\n 2. ‹⟦f summable_on A; A⇩p = {x ∈ A. 0 ≤ f x}; A⇩n = {x ∈ A. f x < 0}; ⋀f A B. ⟦f summable_on A; B ⊆ A⟧ ⟹ f summable_on B⟧ ⟹ f summable_on A⇩n›\n discuss goal 1*)\n apply fastforce\n (*discuss goal 2*)\n apply fastforce\n (*proven 2 subgoals*) .\n then have \"n summable_on A⇩p\"\n by (smt (verit) A⇩p_def (*‹A⇩p = {x ∈ A. 0 ≤ f x}›*) n_def (*‹n ≡ λx. norm (f x)›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) real_norm_def (*‹norm ?r = ¦?r¦›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n moreover have \"n summable_on A⇩n\"\n by (smt (verit, best) ‹f summable_on A⇩n› summable_on_uminus (*‹((λx. - ?f x) summable_on ?A) = (?f summable_on ?A)›*) A⇩n_def (*‹A⇩n = {x ∈ A. f x < 0}›*) n_def (*‹n ≡ λx. norm (f x)›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) real_norm_def (*‹norm ?r = ¦?r¦›*))\n ultimately show \"n summable_on A\"\n using A (*‹A⇩p ∪ A⇩n = A› ‹A⇩p ∩ A⇩n = {}›*) summable_on_Un_disjoint (*‹⟦?f summable_on ?A; ?f summable_on ?B; ?A ∩ ?B = {}⟧ ⟹ ?f summable_on ?A ∪ ?B›*) by blast\nnext\n (*goal: ‹f abs_summable_on A ⟹ f summable_on A›*)\n show \"f abs_summable_on A ⟹ f summable_on A\"\n using abs_summable_summable (*‹?f abs_summable_on ?A ⟹ ?f summable_on ?A›*) by blast\nqed\n\nlemma abs_summable_on_Sigma_iff:\n shows \"f abs_summable_on Sigma A B ⟷\n (∀x∈A. (λy. f (x, y)) abs_summable_on B x) ∧\n ((λx. infsum (λy. norm (f (x, y))) (B x)) abs_summable_on A)\"\nproof (intro iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*) conjI (*‹⟦?P; ?Q⟧ ⟹ ?P ∧ ?Q›*) ballI (*‹(⋀x. x ∈ ?A ⟹ ?P x) ⟹ ∀x∈?A. ?P x›*))\n (*goals:\n 1. ‹⋀x. ⟦f abs_summable_on Sigma A B; x ∈ A⟧ ⟹ (λxa. f (x, xa)) abs_summable_on B x›\n 2. ‹f abs_summable_on Sigma A B ⟹ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A›\n 3. ‹(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A ⟹ f abs_summable_on Sigma A B›*)\n assume asm: \"f abs_summable_on Sigma A B\" (*‹(f::'a × 'b ⇒ 'c) abs_summable_on Sigma (A::'a set) (B::'a ⇒ 'b set)›*)\n then have \"(λx. infsum (λy. norm (f (x,y))) (B x)) summable_on A\"\n by (simp add: cond_case_prod_eta (*‹(⋀x y. ?f x y = ?g (x, y)) ⟹ (λ(x, y). ?f x y) = ?g›*) summable_on_Sigma_banach (*‹(λ(x, y). ?f x y) summable_on Sigma ?A ?B ⟹ (λx. infsum (?f x) (?B x)) summable_on ?A›*))\n then show \"(λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A\"\n using summable_on_iff_abs_summable_on_real (*‹(?f summable_on ?A) = (?f abs_summable_on ?A)›*) by force\n show \"(λy. f (x, y)) abs_summable_on B x\" if \"x ∈ A\" for x\n proof (-)\n (*goal: ‹(λxa. f (x, xa)) abs_summable_on B x›*)\n from asm (*‹f abs_summable_on Sigma A B›*) have \"f abs_summable_on Pair x ` B x\"\n by (simp add: image_subset_iff (*‹(?f ` ?A ⊆ ?B) = (∀x∈?A. ?f x ∈ ?B)›*) summable_on_subset_banach (*‹⟦?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*) that (*‹x ∈ A›*))\n then show \"?thesis\"\n (*goal: ‹(λxa::'b. (f::'a × 'b ⇒ 'c) (x::'a, xa)) abs_summable_on (B::'a ⇒ 'b set) x›*)\n by (metis (mono_tags, lifting) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*) summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*) prod.inject (*‹((?x1.0, ?x2.0) = (?y1.0, ?y2.0)) = (?x1.0 = ?y1.0 ∧ ?x2.0 = ?y2.0)›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n qed\nnext\n (*goal: ‹(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A ⟹ f abs_summable_on Sigma A B›*)\n assume asm: \"(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧\n (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A\" (*‹(∀x::'a∈A::'a set. (λxa::'b. (f::'a × 'b ⇒ 'c) (x, xa)) abs_summable_on (B::'a ⇒ 'b set) x) ∧ (λx::'a. ∑⇩∞y::'b∈B x. norm (f (x, y))) abs_summable_on A›*)\n have \"(∑xy∈F. norm (f xy)) ≤ (∑⇩∞x∈A. ∑⇩∞y∈B x. norm (f (x, y)))\" if \"F ⊆ Sigma A B\" and [simp]: \"finite F\" for F\n proof (-)\n (*goal: ‹(∑xy∈F. norm (f xy)) ≤ (∑⇩∞x∈A. ∑⇩∞y∈B x. norm (f (x, y)))›*)\n have [simp]: \"(SIGMA x:fst ` F. {y. (x, y) ∈ F}) = F\"\n by (auto intro!: set_eqI (*‹(⋀x. (x ∈ ?A) = (x ∈ ?B)) ⟹ ?A = ?B›*) simp add: Domain.DomainI (*‹(?a, ?b) ∈ ?r ⟹ ?a ∈ Domain ?r›*) fst_eq_Domain (*‹fst ` ?R = Domain ?R›*))\n have [simp]: \"finite {y. (x, y) ∈ F}\" for x\n by (metis ‹finite F› Range.intros (*‹(?a, ?b) ∈ ?r ⟹ ?b ∈ Range ?r›*) finite_Range (*‹finite ?r ⟹ finite (Range ?r)›*) finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) subsetI (*‹(⋀x. x ∈ ?A ⟹ x ∈ ?B) ⟹ ?A ⊆ ?B›*))\n have \"(∑xy∈F. norm (f xy)) = (∑x∈fst ` F. ∑y∈{y. (x,y)∈F}. norm (f (x,y)))\"\n by (simp add: sum.Sigma (*‹⟦finite ?A; ∀x∈?A. finite (?B x)⟧ ⟹ (∑x∈?A. sum (?g x) (?B x)) = (∑(x, y)∈Sigma ?A ?B. ?g x y)›*))\n also (*calculation: ‹(∑xy∈F. norm (f xy)) = (∑x∈fst ` F. ∑y | (x, y) ∈ F. norm (f (x, y)))›*) have \"… = (∑⇩∞x∈fst ` F. ∑⇩∞y∈{y. (x,y)∈F}. norm (f (x,y)))\"\n by auto\n also (*calculation: ‹(∑xy∈F. norm (f xy)) = (∑⇩∞x∈fst ` F. ∑⇩∞y | (x, y) ∈ F. norm (f (x, y)))›*) have \"… ≤ (∑⇩∞x∈fst ` F. ∑⇩∞y∈B x. norm (f (x,y)))\"\n using asm (*‹(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A›*) that(1) (*‹F ⊆ Sigma A B›*) apply (intro infsum_mono (*‹⟦?f summable_on ?A; ?g summable_on ?A; ⋀x. x ∈ ?A ⟹ ?f x ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?A›*) infsum_mono_neutral (*‹⟦?f summable_on ?A; ?g summable_on ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?B›*))\n (*goals:\n 1. ‹⟦(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A; F ⊆ Sigma A B⟧ ⟹ (λx. ∑⇩∞y | (x, y) ∈ F. norm (f (x, y))) summable_on fst ` F›\n 2. ‹⟦(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A; F ⊆ Sigma A B⟧ ⟹ (λx. ∑⇩∞y∈B x. norm (f (x, y))) summable_on fst ` F›\n 3. ‹⋀x. ⟦(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A; F ⊆ Sigma A B; x ∈ fst ` F⟧ ⟹ (λy. f (x, y)) abs_summable_on {y. (x, y) ∈ F}›\n 4. ‹⋀x. ⟦(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A; F ⊆ Sigma A B; x ∈ fst ` F⟧ ⟹ (λy. f (x, y)) abs_summable_on B x›\n 5. ‹⋀x xa. ⟦(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A; F ⊆ Sigma A B; x ∈ fst ` F; xa ∈ {y. (x, y) ∈ F} ∩ B x⟧ ⟹ norm (f (x, xa)) ≤ norm (f (x, xa))›\n 6. ‹⋀x xa. ⟦(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A; F ⊆ Sigma A B; x ∈ fst ` F; xa ∈ {y. (x, y) ∈ F} - B x⟧ ⟹ norm (f (x, xa)) ≤ 0›\n 7. ‹⋀x xa. ⟦(∀x∈A. (λxa. f (x, xa)) abs_summable_on B x) ∧ (λx. ∑⇩∞y∈B x. norm (f (x, y))) abs_summable_on A; F ⊆ Sigma A B; x ∈ fst ` F; xa ∈ B x - {y. (x, y) ∈ F}⟧ ⟹ 0 ≤ norm (f (x, xa))›\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: ‹(∑xy∈F. norm (f xy)) ≤ (∑⇩∞x∈fst ` F. ∑⇩∞y∈B x. norm (f (x, y)))›*) have \"… ≤ (∑⇩∞x∈A. ∑⇩∞y∈B x. norm (f (x,y)))\"\n apply (rule infsum_mono_neutral (*‹⟦?f summable_on ?A; ?g summable_on ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?B›*))\n (*goals:\n 1. ‹(λx. ∑⇩∞y∈B x. norm (f (x, y))) summable_on fst ` F›\n 2. ‹(λx. ∑⇩∞y∈B x. norm (f (x, y))) summable_on A›\n 3. ‹⋀x. x ∈ fst ` F ∩ A ⟹ (∑⇩∞y∈B x. norm (f (x, y))) ≤ (∑⇩∞y∈B x. norm (f (x, y)))›\n 4. ‹⋀x. x ∈ fst ` F - A ⟹ (∑⇩∞y∈B x. norm (f (x, y))) ≤ 0›\n 5. ‹⋀x. x ∈ A - fst ` F ⟹ 0 ≤ (∑⇩∞y∈B x. norm (f (x, y)))›\n discuss goal 1*)\n apply ((use asm that(1) in ‹auto simp add: infsum_nonneg›)[1])\n (*discuss goal 2*)\n apply ((use asm that(1) in ‹auto simp add: infsum_nonneg›)[1])\n (*discuss goal 3*)\n apply ((use asm that(1) in ‹auto simp add: infsum_nonneg›)[1])\n (*discuss goal 4*)\n apply ((use asm that(1) in ‹auto simp add: infsum_nonneg›)[1])\n (*discuss goal 5*)\n apply ((use asm that(1) in ‹auto simp add: infsum_nonneg›)[1])\n (*proven 5 subgoals*) .\n finally (*calculation: ‹(∑xy∈F. norm (f xy)) ≤ (∑⇩∞x∈A. ∑⇩∞y∈B x. norm (f (x, y)))›*) show \"?thesis\"\n (*goal: ‹(∑xy∈F. norm (f xy)) ≤ (∑⇩∞x∈A. ∑⇩∞y∈B x. norm (f (x, y)))›*) .\n qed\n then show \"f abs_summable_on Sigma A B\"\n apply (intro nonneg_bdd_above_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ ?f summable_on ?A›*))\n (*goals:\n 1. ‹⋀x. ⟦⋀F. ⟦F ⊆ Sigma A B; finite F⟧ ⟹ (∑xy∈F. norm (f xy)) ≤ (∑⇩∞x∈A. ∑⇩∞y∈B x. norm (f (x, y))); x ∈ Sigma A B⟧ ⟹ 0 ≤ norm (f x)›\n 2. ‹(⋀F. ⟦F ⊆ Sigma A B; finite F⟧ ⟹ (∑xy∈F. norm (f xy)) ≤ (∑⇩∞x∈A. ∑⇩∞y∈B x. norm (f (x, y)))) ⟹ bdd_above (sum (λx. norm (f x)) ` {F. F ⊆ Sigma A B ∧ finite F})›\n discuss goal 1*)\n apply ((auto simp: bdd_above_def (*‹bdd_above ?A = (∃M. ∀x∈?A. x ≤ M)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: bdd_above_def (*‹bdd_above ?A = (∃M. ∀x∈?A. x ≤ M)›*))[1])\n (*proven 2 subgoals*) .\nqed\n\nlemma abs_summable_on_comparison_test:\n assumes \"g abs_summable_on A\"\n assumes \"⋀x. x ∈ A ⟹ norm (f x) ≤ norm (g x)\"\n shows \"f abs_summable_on A\"\nproof (rule nonneg_bdd_above_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ ?f summable_on ?A›*))\n (*goals:\n 1. ‹⋀x. x ∈ A ⟹ 0 ≤ norm (f x)›\n 2. ‹bdd_above (sum (λx. norm (f x)) ` {F. F ⊆ A ∧ finite F})›*)\n show \"bdd_above (sum (λx. norm (f x)) ` {F. F ⊆ A ∧ finite F})\"\n proof (rule bdd_aboveI2 (*‹(⋀x::?'b. x ∈ (?A::?'b set) ⟹ (?f::?'b ⇒ ?'a) x ≤ (?M::?'a)) ⟹ bdd_above (?f ` ?A)›*))\n (*goal: ‹⋀x. x ∈ {F. F ⊆ A ∧ finite F} ⟹ (∑x∈x. norm (f x)) ≤ ?M›*)\n fix F\n assume F: \"F ∈ {F. F ⊆ A ∧ finite F}\" (*‹(F::'a set) ∈ {F::'a set. F ⊆ (A::'a set) ∧ finite F}›*)\n have \"sum (λx. norm (f x)) F ≤ sum (λx. norm (g x)) F\"\n using assms (*‹g abs_summable_on A› ‹?x ∈ A ⟹ norm (f ?x) ≤ norm (g ?x)›*) F (*‹F ∈ {F. F ⊆ A ∧ finite F}›*) apply (intro sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*))\n (*goal: ‹(∑x∈F. norm (f x)) ≤ (∑x∈F. norm (g x))›*)\n by auto\n also (*calculation: ‹(∑x∈F. norm (f x)) ≤ (∑x∈F. norm (g x))›*) have \"… = infsum (λx. norm (g x)) F\"\n using F (*‹F ∈ {F. F ⊆ A ∧ finite F}›*) by simp\n also (*calculation: ‹(∑x∈F. norm (f x)) ≤ (∑⇩∞x∈F. norm (g x))›*) have \"… ≤ infsum (λx. norm (g x)) A\"\n by (smt (verit) F (*‹F ∈ {F. F ⊆ A ∧ finite F}›*) assms( (*‹g abs_summable_on A›*) 1) infsum_mono2 (*‹⟦?f summable_on ?A; ?f summable_on ?B; ?A ⊆ ?B; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?f x⟧ ⟹ infsum ?f ?A ≤ infsum ?f ?B›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) norm_ge_zero (*‹0 ≤ norm ?x›*) summable_on_subset_banach (*‹⟦?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*))\n finally (*calculation: ‹(∑x∈F. norm (f x)) ≤ (∑⇩∞x∈A. norm (g x))›*) show \"(∑x∈F. norm (f x)) ≤ (∑⇩∞x∈A. norm (g x))\" .\n qed\nqed (auto)\n(*solved the remaining goal: ‹⋀x. x ∈ A ⟹ 0 ≤ norm (f x)›*)\n\nlemma abs_summable_iff_bdd_above:\n fixes f :: ‹'a ⇒ 'b::real_normed_vector›\n shows ‹f abs_summable_on A ⟷ bdd_above (sum (λx. norm (f x)) ` {F. F⊆A ∧ finite F})›\nproof (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹(f::'a ⇒ 'b) abs_summable_on (A::'a set) ⟹ bdd_above (sum (λx::'a. norm (f x)) ` {F::'a set. F ⊆ A ∧ finite F})›\n 2. ‹bdd_above (sum (λx::'a. norm ((f::'a ⇒ 'b) x)) ` {F::'a set. F ⊆ (A::'a set) ∧ finite F}) ⟹ f abs_summable_on A›*)\n assume \"f abs_summable_on A\" (*‹(f::'a ⇒ 'b) abs_summable_on (A::'a set)›*)\n show \"bdd_above (sum (λx. norm (f x)) ` {F. F ⊆ A ∧ finite F})\"\n proof (rule bdd_aboveI2 (*‹(⋀x. x ∈ ?A ⟹ ?f x ≤ ?M) ⟹ bdd_above (?f ` ?A)›*))\n (*goal: ‹⋀x. x ∈ {F. F ⊆ A ∧ finite F} ⟹ (∑x∈x. norm (f x)) ≤ ?M›*)\n fix F\n assume F: \"F ∈ {F. F ⊆ A ∧ finite F}\" (*‹(F::'a set) ∈ {F::'a set. F ⊆ (A::'a set) ∧ finite F}›*)\n show \"(∑x∈F. norm (f x)) ≤ (∑⇩∞x∈A. norm (f x))\"\n apply (rule finite_sum_le_infsum (*‹⟦?f summable_on ?A; finite ?B; ?B ⊆ ?A; ⋀x. x ∈ ?A - ?B ⟹ 0 ≤ ?f x⟧ ⟹ sum ?f ?B ≤ infsum ?f ?A›*))\n (*goals:\n 1. ‹f abs_summable_on A›\n 2. ‹finite F›\n 3. ‹F ⊆ A›\n 4. ‹⋀x. x ∈ A - F ⟹ 0 ≤ norm (f x)›\n discuss goal 1*)\n apply ((use ‹f abs_summable_on A› F in auto)[1])\n (*discuss goal 2*)\n apply ((use ‹f abs_summable_on A› F in auto)[1])\n (*discuss goal 3*)\n apply ((use ‹f abs_summable_on A› F in auto)[1])\n (*discuss goal 4*)\n apply ((use ‹f abs_summable_on A› F in auto)[1])\n (*proven 4 subgoals*) .\n qed\nnext\n (*goal: ‹bdd_above (sum (λx. norm (f x)) ` {F. F ⊆ A ∧ finite F}) ⟹ f abs_summable_on A›*)\n assume \"bdd_above (sum (λx. norm (f x)) ` {F. F⊆A ∧ finite F})\" (*‹bdd_above (sum (λx::'a. norm ((f::'a ⇒ 'b) x)) ` {F::'a set. F ⊆ (A::'a set) ∧ finite F})›*)\n then show \"f abs_summable_on A\"\n by (simp add: nonneg_bdd_above_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ ?f summable_on ?A›*))\nqed\n\nlemma abs_summable_product:\n fixes x :: \"'a ⇒ 'b::{real_normed_div_algebra,banach,second_countable_topology}\"\n assumes x2_sum: \"(λi. (x i) * (x i)) abs_summable_on A\"\n and y2_sum: \"(λi. (y i) * (y i)) abs_summable_on A\"\n shows \"(λi. x i * y i) abs_summable_on A\"\nproof (rule nonneg_bdd_above_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ ?f summable_on ?A›*))\n (*goals:\n 1. ‹⋀xa. xa ∈ A ⟹ 0 ≤ norm (x xa * y xa)›\n 2. ‹bdd_above (sum (λxa. norm (x xa * y xa)) ` {F. F ⊆ A ∧ finite F})›*)\n show \"bdd_above (sum (λxa. norm (x xa * y xa)) ` {F. F ⊆ A ∧ finite F})\"\n proof (rule bdd_aboveI2 (*‹(⋀x. x ∈ ?A ⟹ ?f x ≤ ?M) ⟹ bdd_above (?f ` ?A)›*))\n (*goal: ‹⋀xa::'a set. xa ∈ {F::'a set. F ⊆ (A::'a set) ∧ finite F} ⟹ (∑xa::'a∈xa. norm ((x::'a ⇒ 'b) xa * (y::'a ⇒ 'b) xa)) ≤ (?M::real)›*)\n fix F\n assume F: \"F ∈ {F. F ⊆ A ∧ finite F}\" (*‹(F::'a set) ∈ {F::'a set. F ⊆ (A::'a set) ∧ finite F}›*)\n then have r1: \"finite F\" and b4: \"F ⊆ A\"\n apply -\n (*goals:\n 1. ‹F ∈ {F. F ⊆ A ∧ finite F} ⟹ finite F›\n 2. ‹F ∈ {F. F ⊆ A ∧ finite F} ⟹ F ⊆ A›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have a1: \"(∑⇩∞i∈F. norm (x i * x i)) ≤ (∑⇩∞i∈A. norm (x i * x i))\"\n by (metis (no_types, lifting) b4 (*‹F ⊆ A›*) infsum_mono2 (*‹⟦?f summable_on ?A; ?f summable_on ?B; ?A ⊆ ?B; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?f x⟧ ⟹ infsum ?f ?A ≤ infsum ?f ?B›*) norm_ge_zero (*‹0 ≤ norm ?x›*) summable_on_subset_banach (*‹⟦?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*) x2_sum (*‹(λxa. x xa * x xa) abs_summable_on A›*))\n have \"norm (x i * y i) ≤ norm (x i * x i) + norm (y i * y i)\" for i\n unfolding norm_mult\n (*goal: ‹norm (x i) * norm (y i) ≤ norm (x i) * norm (x i) + norm (y i) * norm (y i)›*)\n by (smt (verit, best) abs_norm_cancel (*‹¦norm ?a¦ = norm ?a›*) mult_mono (*‹⟦?a ≤ ?b; ?c ≤ ?d; 0 ≤ ?b; 0 ≤ ?c⟧ ⟹ ?a * ?c ≤ ?b * ?d›*) not_sum_squares_lt_zero (*‹¬ ?x * ?x + ?y * ?y < 0›*))\n hence \"(∑i∈F. norm (x i * y i)) ≤ (∑i∈F. norm (x i * x i) + norm (y i * y i))\"\n by (simp add: sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*))\n also (*calculation: ‹(∑i::'a∈(F::'a set). norm ((x::'a ⇒ 'b) i * (y::'a ⇒ 'b) i)) ≤ (∑i::'a∈F. norm (x i * x i) + norm (y i * y i))›*) have \"… = (∑i∈F. norm (x i * x i)) + (∑i∈F. norm (y i * y i))\"\n by (simp add: sum.distrib (*‹(∑x∈?A. ?g x + ?h x) = sum ?g ?A + sum ?h ?A›*))\n also (*calculation: ‹(∑i∈F. norm (x i * y i)) ≤ (∑i∈F. norm (x i * x i)) + (∑i∈F. norm (y i * y i))›*) have \"… = (∑⇩∞i∈F. norm (x i * x i)) + (∑⇩∞i∈F. norm (y i * y i))\"\n by (simp add: ‹finite F›)\n also (*calculation: ‹(∑i::'a∈(F::'a set). norm ((x::'a ⇒ 'b) i * (y::'a ⇒ 'b) i)) ≤ (∑⇩∞i::'a∈F. norm (x i * x i)) + (∑⇩∞i::'a∈F. norm (y i * y i))›*) have \"… ≤ (∑⇩∞i∈A. norm (x i * x i)) + (∑⇩∞i∈A. norm (y i * y i))\"\n using F (*‹F ∈ {F. F ⊆ A ∧ finite F}›*) assms (*‹(λxa. x xa * x xa) abs_summable_on A› ‹(λx. y x * y x) abs_summable_on A›*) apply (intro add_mono (*‹⟦?a ≤ ?b; ?c ≤ ?d⟧ ⟹ ?a + ?c ≤ ?b + ?d›*) infsum_mono2 (*‹⟦?f summable_on ?A; ?f summable_on ?B; ?A ⊆ ?B; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?f x⟧ ⟹ infsum ?f ?A ≤ infsum ?f ?B›*))\n (*goals:\n 1. ‹⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A⟧ ⟹ (λi. x i * x i) abs_summable_on F›\n 2. ‹⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A⟧ ⟹ (λi. x i * x i) abs_summable_on A›\n 3. ‹⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A⟧ ⟹ F ⊆ A›\n 4. ‹⋀xa. ⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A; xa ∈ A - F⟧ ⟹ 0 ≤ norm (x xa * x xa)›\n 5. ‹⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A⟧ ⟹ (λi. y i * y i) abs_summable_on F›\n 6. ‹⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A⟧ ⟹ (λi. y i * y i) abs_summable_on A›\n 7. ‹⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A⟧ ⟹ F ⊆ A›\n 8. ‹⋀xa. ⟦F ∈ {F. F ⊆ A ∧ finite F}; (λxa. x xa * x xa) abs_summable_on A; (λx. y x * y x) abs_summable_on A; xa ∈ A - F⟧ ⟹ 0 ≤ norm (y xa * y xa)›\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 finally (*calculation: ‹(∑i∈F. norm (x i * y i)) ≤ (∑⇩∞i∈A. norm (x i * x i)) + (∑⇩∞i∈A. norm (y i * y i))›*) show \"(∑xa∈F. norm (x xa * y xa)) ≤ (∑⇩∞i∈A. norm (x i * x i)) + (∑⇩∞i∈A. norm (y i * y i))\"\n by simp\n qed\nqed (auto)\n(*solved the remaining goal: ‹⋀xa. xa ∈ A ⟹ 0 ≤ norm (x xa * y xa)›*)\n\nsubsection ‹Extended reals and nats›\n\nlemma summable_on_ennreal[simp]: ‹(f::_ ⇒ ennreal) summable_on S› and summable_on_enat[simp]: ‹(f::_ ⇒ enat) summable_on S›\n \n (*goals:\n 1. ‹(λx::'a::type. ennreal_of_enat ((f::'a::type ⇒ enat) x)) summable_on (S::'a::type set)›\n 2. ‹(f::'a::type ⇒ enat) summable_on (S::'a::type set)›\n discuss goal 1*)\n apply (simp add: nonneg_summable_on_complete (*‹(⋀x. x ∈ ?A ⟹ 0 ≤ ?f x) ⟹ ?f summable_on ?A›*))\n (*discuss goal 2*)\n apply (simp add: nonneg_summable_on_complete (*‹(⋀x. x ∈ ?A ⟹ 0 ≤ ?f x) ⟹ ?f summable_on ?A›*))\n (*proven 2 subgoals*) .\n\nlemma has_sum_superconst_infinite_ennreal:\n fixes f :: ‹'a ⇒ ennreal›\n assumes geqb: ‹⋀x. x ∈ S ⟹ f x ≥ b›\n assumes b: ‹b > 0›\n assumes ‹infinite S›\n shows \"(f has_sum ∞) S\"\nproof (-)\n (*goal: ‹(f has_sum ∞) S›*)\n have \"(sum f ⤏ ∞) (finite_subsets_at_top S)\"\n proof (rule order_tendstoI (*‹⟦⋀a. a < ?y ⟹ ∀⇩F x in ?F. a < ?f x; ⋀a. ?y < a ⟹ ∀⇩F x in ?F. ?f x < a⟧ ⟹ (?f ⤏ ?y) ?F›*))\n (*goals:\n 1. ‹⋀a. a < ∞ ⟹ ∀⇩F x in finite_subsets_at_top S. a < sum f x›\n 2. ‹⋀a. ∞ < a ⟹ ∀⇩F x in finite_subsets_at_top S. sum f x < a›*)\n fix y :: ennreal\n assume \"y < ∞\" (*‹(y::ennreal) < ∞›*)\n then have \"y / b < ∞\" \"y < top\"\n using b (*‹(0::ennreal) < (b::ennreal)›*) ennreal_divide_eq_top_iff (*‹(?a / ?b = top) = (?a ≠ 0 ∧ ?b = 0 ∨ ?a = top ∧ ?b ≠ top)›*) top.not_eq_extremum (*‹((?a::?'a) ≠ top) = (?a < top)›*) apply -\n (*goals:\n 1. ‹⟦y < ∞; 0 < b; ⋀a b. (a / b = top) = (a ≠ 0 ∧ b = 0 ∨ a = top ∧ b ≠ top); ⋀a. (a ≠ top) = (a < top)⟧ ⟹ y / b < ∞›\n 2. ‹⟦y < ∞; 0 < b; ⋀a b. (a / b = top) = (a ≠ 0 ∧ b = 0 ∨ a = top ∧ b ≠ top); ⋀a. (a ≠ top) = (a < top)⟧ ⟹ y < top›\n discuss goal 1*)\n apply force\n (*discuss goal 2*)\n apply force\n (*proven 2 subgoals*) .\n then obtain F where \"finite F\" and \"F ⊆ S\" and cardF: \"card F > y / b\"\n (*goal: ‹(⋀F::'a set. ⟦finite F; F ⊆ (S::'a set); (y::ennreal) / (b::ennreal) < of_nat (card F)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using ‹infinite S› (*‹infinite S›*) by (metis ennreal_Ex_less_of_nat (*‹?x < top ⟹ ∃n. ?x < of_nat n›*) infinite_arbitrarily_large (*‹infinite ?A ⟹ ∃B. finite B ∧ card B = ?n ∧ B ⊆ ?A›*) infinity_ennreal_def (*‹∞ = top›*))\n moreover have \"sum f Y > y\" if \"finite Y\" and \"F ⊆ Y\" and \"Y ⊆ S\" for Y\n proof (-)\n (*goal: ‹y < sum f Y›*)\n have \"y < b * card F\"\n by (metis b (*‹0 < b›*) ‹y < top› cardF (*‹y / b < of_nat (card F)›*) divide_less_ennreal (*‹⟦?b ≠ 0; ?b < top⟧ ⟹ (?a / ?b < ?c) = (?a < ?c * ?b)›*) ennreal_mult_eq_top_iff (*‹(?a * ?b = top) = (?a = top ∧ ?b ≠ 0 ∨ ?b = top ∧ ?a ≠ 0)›*) gr_implies_not_zero (*‹?m < ?n ⟹ ?n ≠ 0›*) mult.commute (*‹?a * ?b = ?b * ?a›*) top.not_eq_extremum (*‹(?a ≠ top) = (?a < top)›*))\n also (*calculation: ‹y < b * of_nat (card F)›*) have \"… ≤ b * card Y\"\n by (meson b (*‹0 < b›*) card_mono (*‹⟦finite ?B; ?A ⊆ ?B⟧ ⟹ card ?A ≤ card ?B›*) less_imp_le (*‹?x < ?y ⟹ ?x ≤ ?y›*) mult_left_mono (*‹⟦?a ≤ ?b; 0 ≤ ?c⟧ ⟹ ?c * ?a ≤ ?c * ?b›*) of_nat_le_iff (*‹(of_nat ?m ≤ of_nat ?n) = (?m ≤ ?n)›*) that (*‹finite Y› ‹F ⊆ Y› ‹Y ⊆ S›*))\n also (*calculation: ‹y < b * of_nat (card Y)›*) have \"… = sum (λ_. b) Y\"\n by (simp add: mult.commute (*‹(?a::?'a) * (?b::?'a) = ?b * ?a›*))\n also (*calculation: ‹y < (∑_∈Y. b)›*) have \"… ≤ sum f Y\"\n using geqb (*‹(?x::'a) ∈ (S::'a set) ⟹ (b::ennreal) ≤ (f::'a ⇒ ennreal) ?x›*) by (meson subset_eq (*‹(?A ⊆ ?B) = (∀x∈?A. x ∈ ?B)›*) sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*) that( (*‹Y ⊆ S›*) 3))\n finally (*calculation: ‹y < sum f Y›*) show \"?thesis\"\n (*goal: ‹y < sum f Y›*) .\n qed\n ultimately show \"∀⇩F x in finite_subsets_at_top S. y < sum f x\"\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹∃X::'a set. finite X ∧ X ⊆ (S::'a set) ∧ (∀Y::'a set. finite Y ∧ X ⊆ Y ∧ Y ⊆ S ⟶ (y::ennreal) < sum (f::'a ⇒ ennreal) Y)›*)\n by auto\n qed (auto)\n (*solved the remaining goal: ‹⋀a. ∞ < a ⟹ ∀⇩F x in finite_subsets_at_top S. sum f x < a›*)\n then show \"?thesis\"\n (*goal: ‹((f::'a ⇒ ennreal) has_sum ∞) (S::'a set)›*)\n by (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\nqed\n\nlemma infsum_superconst_infinite_ennreal:\n fixes f :: ‹'a ⇒ ennreal›\n assumes ‹⋀x. x ∈ S ⟹ f x ≥ b›\n assumes ‹b > 0›\n assumes ‹infinite S›\n shows \"infsum f S = ∞\"\n using assms (*‹?x ∈ S ⟹ b ≤ f ?x› ‹0 < b› ‹infinite S›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) has_sum_superconst_infinite_ennreal (*‹⟦⋀x. x ∈ ?S ⟹ ?b ≤ ?f x; 0 < ?b; infinite ?S⟧ ⟹ (?f has_sum ∞) ?S›*) by blast\n\nlemma infsum_superconst_infinite_ereal:\n fixes f :: ‹'a ⇒ ereal›\n assumes geqb: ‹⋀x. x ∈ S ⟹ f x ≥ b›\n assumes b: ‹b > 0›\n assumes ‹infinite S›\n shows \"infsum f S = ∞\"\nproof (-)\n (*goal: ‹infsum f S = ∞›*)\n obtain b' where b': \"e2ennreal b' = b\" and \"b' > 0\"\n (*goal: ‹(⋀b'. ⟦e2ennreal b' = e2ennreal b; 0 < b'⟧ ⟹ thesis) ⟹ thesis›*)\n using b (*‹0 < b›*) by blast\n have \"0 < e2ennreal b\"\n using b' (*‹e2ennreal b' = e2ennreal b›*) b (*‹(0::ereal) < (b::ereal)›*) by (metis dual_order.refl (*‹(?a::?'a) ≤ ?a›*) enn2ereal_e2ennreal (*‹(0::ereal) ≤ (?x::ereal) ⟹ enn2ereal (e2ennreal ?x) = ?x›*) gr_zeroI (*‹((?n::?'a) = (0::?'a) ⟹ False) ⟹ (0::?'a) < ?n›*) order_less_le (*‹((?x::?'a) < (?y::?'a)) = (?x ≤ ?y ∧ ?x ≠ ?y)›*) zero_ennreal.abs_eq (*‹(0::ennreal) = e2ennreal (0::ereal)›*))\n hence \"*\": \"infsum (e2ennreal ∘ f) S = ∞\"\n using assms (*‹?x ∈ S ⟹ b ≤ f ?x› ‹0 < b› ‹infinite S›*) b' (*‹e2ennreal b' = e2ennreal b›*) apply (intro infsum_superconst_infinite_ennreal[where b=b'] (*‹⟦⋀x. x ∈ ?S ⟹ e2ennreal b' ≤ ?f x; 0 < e2ennreal b'; infinite ?S⟧ ⟹ infsum ?f ?S = ∞›*))\n (*goals:\n 1. ‹⋀x. ⟦0 < e2ennreal b; ⋀x. x ∈ S ⟹ b ≤ f x; 0 < b; infinite S; e2ennreal b' = e2ennreal b; x ∈ S⟧ ⟹ e2ennreal b' ≤ (e2ennreal ∘ f) x›\n 2. ‹⟦0 < e2ennreal b; ⋀x. x ∈ S ⟹ b ≤ f x; 0 < b; infinite S; e2ennreal b' = e2ennreal b⟧ ⟹ 0 < e2ennreal b'›\n 3. ‹⟦0 < e2ennreal b; ⋀x. x ∈ S ⟹ b ≤ f x; 0 < b; infinite S; e2ennreal b' = e2ennreal b⟧ ⟹ infinite S›\n discuss goal 1*)\n apply ((auto intro!: e2ennreal_mono (*‹(?x::ereal) ≤ (?y::ereal) ⟹ e2ennreal ?x ≤ e2ennreal ?y›*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: e2ennreal_mono (*‹?x ≤ ?y ⟹ e2ennreal ?x ≤ e2ennreal ?y›*))[1])\n (*discuss goal 3*)\n apply ((auto intro!: e2ennreal_mono (*‹?x ≤ ?y ⟹ e2ennreal ?x ≤ e2ennreal ?y›*))[1])\n (*proven 3 subgoals*) .\n have \"infsum f S = infsum (enn2ereal ∘ (e2ennreal ∘ f)) S\"\n using geqb (*‹(?x::'a) ∈ (S::'a set) ⟹ (b::ereal) ≤ (f::'a ⇒ ereal) ?x›*) b (*‹0 < b›*) apply (intro infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*))\n (*goal: ‹infsum f S = infsum (enn2ereal ∘ (e2ennreal ∘ f)) S›*)\n by (fastforce simp: enn2ereal_e2ennreal (*‹0 ≤ ?x ⟹ enn2ereal (e2ennreal ?x) = ?x›*))\n also (*calculation: ‹infsum f S = infsum (enn2ereal ∘ (e2ennreal ∘ f)) S›*) have \"… = enn2ereal ∞\"\n using \"*\" (*‹infsum (e2ennreal ∘ f) S = ∞›*) by (simp add: infsum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ infsum (?f ∘ ?g) ?S = ?f (infsum ?g ?S)›*) continuous_at_enn2ereal (*‹continuous (at ?x within ?A) enn2ereal›*) nonneg_summable_on_complete (*‹(⋀x. x ∈ ?A ⟹ 0 ≤ ?f x) ⟹ ?f summable_on ?A›*))\n also (*calculation: ‹infsum f S = enn2ereal ∞›*) have \"… = ∞\"\n by simp\n finally (*calculation: ‹infsum f S = ∞›*) show \"?thesis\"\n (*goal: ‹infsum f S = ∞›*) .\nqed\n\nlemma has_sum_superconst_infinite_ereal:\n fixes f :: ‹'a ⇒ ereal›\n assumes ‹⋀x. x ∈ S ⟹ f x ≥ b›\n assumes ‹b > 0›\n assumes ‹infinite S›\n shows \"(f has_sum ∞) S\"\n by (metis Infty_neq_0( (*‹∞ ≠ 0›*) 1) assms (*‹?x ∈ S ⟹ b ≤ f ?x› ‹0 < b› ‹infinite S›*) infsum_def (*‹infsum ?f ?A = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else 0)›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) infsum_superconst_infinite_ereal (*‹⟦⋀x. x ∈ ?S ⟹ ?b ≤ ?f x; 0 < ?b; infinite ?S⟧ ⟹ infsum ?f ?S = ∞›*))\n\nlemma infsum_superconst_infinite_enat:\n fixes f :: ‹'a ⇒ enat›\n assumes geqb: ‹⋀x. x ∈ S ⟹ f x ≥ b›\n assumes b: ‹b > 0›\n assumes ‹infinite S›\n shows \"infsum f S = ∞\"\nproof (-)\n (*goal: ‹infsum f S = ∞›*)\n have \"ennreal_of_enat (infsum f S) = infsum (ennreal_of_enat ∘ f) S\"\n by (simp flip: infsum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ infsum (?f ∘ ?g) ?S = ?f (infsum ?g ?S)›*))\n also (*calculation: ‹ennreal_of_enat (infsum f S) = infsum (ennreal_of_enat ∘ f) S›*) have \"… = ∞\"\n by (metis assms( (*‹infinite S›*) 3) b (*‹0 < b›*) comp_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) ennreal_of_enat_0 (*‹ennreal_of_enat 0 = 0›*) ennreal_of_enat_le_iff (*‹(ennreal_of_enat ?m ≤ ennreal_of_enat ?n) = (?m ≤ ?n)›*) geqb (*‹?x ∈ S ⟹ b ≤ f ?x›*) infsum_superconst_infinite_ennreal (*‹⟦⋀x. x ∈ ?S ⟹ ?b ≤ ?f x; 0 < ?b; infinite ?S⟧ ⟹ infsum ?f ?S = ∞›*) leD (*‹?y ≤ ?x ⟹ ¬ ?x < ?y›*) leI (*‹¬ ?x < ?y ⟹ ?y ≤ ?x›*))\n also (*calculation: ‹ennreal_of_enat (infsum f S) = ∞›*) have \"… = ennreal_of_enat ∞\"\n by simp\n finally (*calculation: ‹ennreal_of_enat (infsum f S) = ennreal_of_enat ∞›*) show \"?thesis\"\n (*goal: ‹infsum f S = ∞›*)\n by (rule ennreal_of_enat_inj[THEN iffD1] (*‹ennreal_of_enat ?i1 = ennreal_of_enat ?j1 ⟹ ?i1 = ?j1›*))\nqed\n\nlemma has_sum_superconst_infinite_enat:\n fixes f :: ‹'a ⇒ enat›\n assumes ‹⋀x. x ∈ S ⟹ f x ≥ b›\n assumes ‹b > 0›\n assumes ‹infinite S›\n shows \"(f has_sum ∞) S\"\n by (metis assms (*‹?x ∈ S ⟹ b ≤ f ?x› ‹0 < b› ‹infinite S›*) i0_lb (*‹0 ≤ ?n›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) infsum_superconst_infinite_enat (*‹⟦⋀x. x ∈ ?S ⟹ ?b ≤ ?f x; 0 < ?b; infinite ?S⟧ ⟹ infsum ?f ?S = ∞›*) nonneg_summable_on_complete (*‹(⋀x. x ∈ ?A ⟹ 0 ≤ ?f x) ⟹ ?f summable_on ?A›*))\n\ntext ‹This lemma helps to relate a real-valued infsum to a supremum over extended nonnegative reals.›\n\nlemma infsum_nonneg_is_SUPREMUM_ennreal:\n fixes f :: \"'a ⇒ real\"\n assumes summable: \"f summable_on A\"\n and fnn: \"⋀x. x∈A ⟹ f x ≥ 0\"\n shows \"ennreal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. (ennreal (sum f F)))\"\nproof (-)\n (*goal: ‹ennreal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. ennreal (sum f F))›*)\n have \"§\": \"⋀F. ⟦finite F; F ⊆ A⟧ ⟹ sum (ennreal ∘ f) F = ennreal (sum f F)\"\n by (metis (mono_tags, lifting) comp_def (*‹(?f::?'b ⇒ ?'c) ∘ (?g::?'a ⇒ ?'b) = (λx::?'a. ?f (?g x))›*) fnn (*‹(?x::'a) ∈ (A::'a set) ⟹ (0::real) ≤ (f::'a ⇒ real) ?x›*) subsetD (*‹⟦(?A::?'a set) ⊆ (?B::?'a set); (?c::?'a) ∈ ?A⟧ ⟹ ?c ∈ ?B›*) 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›*) sum_ennreal (*‹(⋀i::?'a. i ∈ (?I::?'a set) ⟹ (0::real) ≤ (?f::?'a ⇒ real) i) ⟹ (∑i::?'a∈?I. ennreal (?f i)) = ennreal (sum ?f ?I)›*))\n then have \"ennreal (infsum f A) = infsum (ennreal ∘ f) A\"\n by (simp add: infsum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ infsum (?f ∘ ?g) ?S = ?f (infsum ?g ?S)›*) local.summable (*‹f summable_on A›*))\n also (*calculation: ‹ennreal (infsum f A) = infsum (ennreal ∘ f) A›*) have \"… = (SUP F∈{F. finite F ∧ F ⊆ A}. (ennreal (sum f F)))\"\n by (metis (mono_tags, lifting) § (*‹⟦finite ?F; ?F ⊆ A⟧ ⟹ sum (ennreal ∘ f) ?F = ennreal (sum f ?F)›*) image_cong (*‹⟦?M = ?N; ⋀x. x ∈ ?N ⟹ ?f x = ?g x⟧ ⟹ ?f ` ?M = ?g ` ?N›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) nonneg_infsum_complete (*‹(⋀x. x ∈ ?A ⟹ 0 ≤ ?f x) ⟹ infsum ?f ?A = Sup (sum ?f ` {F. finite F ∧ F ⊆ ?A})›*) zero_le (*‹0 ≤ ?x›*))\n finally (*calculation: ‹ennreal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. ennreal (sum f F))›*) show \"?thesis\"\n (*goal: ‹ennreal (infsum (f::'a::type ⇒ real) (A::'a::type set)) = (SUP F::'a::type set∈{F::'a::type set. finite F ∧ F ⊆ A}. ennreal (sum f F))›*) .\nqed\n\ntext ‹This lemma helps to related a real-valued infsum to a supremum over extended reals.›\n\nlemma infsum_nonneg_is_SUPREMUM_ereal:\n fixes f :: \"'a ⇒ real\"\n assumes summable: \"f summable_on A\"\n and fnn: \"⋀x. x∈A ⟹ f x ≥ 0\"\n shows \"ereal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. (ereal (sum f F)))\"\nproof (-)\n (*goal: ‹ereal (infsum (f::'a ⇒ real) (A::'a set)) = (SUP F::'a set∈{F::'a set. finite F ∧ F ⊆ A}. ereal (sum f F))›*)\n have \"⋀F. ⟦finite F; F ⊆ A⟧ ⟹ sum (ereal ∘ f) F = ereal (sum f F)\"\n by auto\n then have \"ereal (infsum f A) = infsum (ereal ∘ f) A\"\n by (simp add: infsum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ infsum (?f ∘ ?g) ?S = ?f (infsum ?g ?S)›*) local.summable (*‹f summable_on A›*))\n also (*calculation: ‹ereal (infsum (f::'a ⇒ real) (A::'a set)) = infsum (ereal ∘ f) A›*) have \"… = (SUP F∈{F. finite F ∧ F ⊆ A}. (ereal (sum f F)))\"\n apply (subst nonneg_infsum_complete (*‹(⋀x. x ∈ ?A ⟹ 0 ≤ ?f x) ⟹ infsum ?f ?A = Sup (sum ?f ` {F. finite F ∧ F ⊆ ?A})›*))\n (*goals:\n 1. ‹⋀x::'a. x ∈ (A::'a set) ⟹ (0::ereal) ≤ (ereal ∘ (f::'a ⇒ real)) x›\n 2. ‹Sup (sum (ereal ∘ (f::'a ⇒ real)) ` {F::'a set. finite F ∧ F ⊆ (A::'a set)}) = (SUP F::'a set∈{F::'a set. finite F ∧ F ⊆ A}. ereal (sum f F))›\n discuss goal 1*)\n apply (simp add: assms (*‹f summable_on A› ‹?x ∈ A ⟹ 0 ≤ f ?x›*))\n (*discuss goal 2*)\n apply (simp add: assms (*‹f summable_on A› ‹?x ∈ A ⟹ 0 ≤ f ?x›*))\n (*proven 2 subgoals*) .\n finally (*calculation: ‹ereal (infsum (f::'a ⇒ real) (A::'a set)) = (SUP F::'a set∈{F::'a set. finite F ∧ F ⊆ A}. ereal (sum f F))›*) show \"?thesis\"\n (*goal: ‹ereal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. ereal (sum f F))›*) .\nqed\n\n\nsubsection ‹Real numbers›\n\ntext ‹Most lemmas in the general property section already apply to real numbers.\n A few ones that are specific to reals are given here.›\n\nlemma infsum_nonneg_is_SUPREMUM_real:\n fixes f :: \"'a ⇒ real\"\n assumes summable: \"f summable_on A\"\n and fnn: \"⋀x. x∈A ⟹ f x ≥ 0\"\n shows \"infsum f A = (SUP F∈{F. finite F ∧ F ⊆ A}. (sum f F))\"\nproof (-)\n (*goal: ‹infsum f A = Sup (sum f ` {F. finite F ∧ F ⊆ A})›*)\n have \"*\": \"ereal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. (ereal (sum f F)))\"\n using assms (*‹f summable_on A› ‹(?x::'a) ∈ (A::'a set) ⟹ (0::real) ≤ (f::'a ⇒ real) ?x›*) by (rule infsum_nonneg_is_SUPREMUM_ereal (*‹⟦?f summable_on ?A; ⋀x. x ∈ ?A ⟹ 0 ≤ ?f x⟧ ⟹ ereal (infsum ?f ?A) = (SUP F∈{F. finite F ∧ F ⊆ ?A}. ereal (sum ?f F))›*))\n also (*calculation: ‹ereal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. ereal (sum f F))›*) have \"… = ereal (SUP F∈{F. finite F ∧ F ⊆ A}. (sum f F))\"\n by (metis (no_types, lifting) * (*‹ereal (infsum f A) = (SUP F∈{F. finite F ∧ F ⊆ A}. ereal (sum f F))›*) MInfty_neq_ereal( (*‹- ∞ ≠ ereal ?r›*) 2) PInfty_neq_ereal( (*‹∞ ≠ ereal ?r›*) 2) SUP_cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?C x = ?D x⟧ ⟹ Sup (?C ` ?A) = Sup (?D ` ?B)›*) abs_eq_infinity_cases (*‹⟦¦?x¦ = ∞; ?x = ∞ ⟹ ?thesis; ?x = - ∞ ⟹ ?thesis⟧ ⟹ ?thesis›*) ereal_SUP (*‹¦SUP a∈?A. ereal (?f a)¦ ≠ ∞ ⟹ ereal (Sup (?f ` ?A)) = (SUP a∈?A. ereal (?f a))›*))\n finally (*calculation: ‹ereal (infsum f A) = ereal (Sup (sum f ` {F. finite F ∧ F ⊆ A}))›*) show \"?thesis\"\n (*goal: ‹infsum f A = Sup (sum f ` {F. finite F ∧ F ⊆ A})›*)\n by simp\nqed\n\n\nlemma has_sum_nonneg_SUPREMUM_real:\n fixes f :: \"'a ⇒ real\"\n assumes \"f summable_on A\" and \"⋀x. x∈A ⟹ f x ≥ 0\"\n shows \"(f has_sum (SUP F∈{F. finite F ∧ F ⊆ A}. (sum f F))) A\"\n by (metis (mono_tags, lifting) assms (*‹(f::'a ⇒ real) summable_on (A::'a set)› ‹(?x::'a) ∈ (A::'a set) ⟹ (0::real) ≤ (f::'a ⇒ real) ?x›*) has_sum_infsum (*‹(?f::?'a ⇒ ?'b) summable_on (?S::?'a set) ⟹ (?f has_sum infsum ?f ?S) ?S›*) infsum_nonneg_is_SUPREMUM_real (*‹⟦(?f::?'a ⇒ real) summable_on (?A::?'a set); ⋀x::?'a. x ∈ ?A ⟹ (0::real) ≤ ?f x⟧ ⟹ infsum ?f ?A = Sup (sum ?f ` {F::?'a set. finite F ∧ F ⊆ ?A})›*))\n\nlemma summable_countable_real:\n fixes f :: ‹'a ⇒ real›\n assumes ‹f summable_on A›\n shows ‹countable {x∈A. f x ≠ 0}›\n using abs_summable_countable (*‹(?f::?'a::type ⇒ ?'b::real_normed_vector) abs_summable_on (?A::?'a::type set) ⟹ countable {x::?'a::type ∈ ?A. ?f x ≠ (0::?'b::real_normed_vector)}›*) assms (*‹f summable_on A›*) summable_on_iff_abs_summable_on_real (*‹(?f summable_on ?A) = (?f abs_summable_on ?A)›*) by blast\n\nsubsection ‹Complex numbers›\n\nlemma has_sum_cnj_iff[simp]: \n fixes f :: ‹'a ⇒ complex›\n shows ‹((λx. cnj (f x)) has_sum cnj a) M ⟷ (f has_sum a) M›\n by (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*) lim_cnj (*‹((λx. cnj (?f x)) ⤏ cnj ?l) ?F = (?f ⤏ ?l) ?F›*) del: cnj_sum (*‹cnj (sum ?f ?s) = (∑x∈?s. cnj (?f x))›*) add: cnj_sum[symmetric, abs_def, of f] (*‹sum (λx. cnj (f x)) ≡ λs. cnj (sum f s)›*))\n\nlemma summable_on_cnj_iff[simp]:\n \"(λi. cnj (f i)) summable_on A ⟷ f summable_on A\"\n by (metis complex_cnj_cnj (*‹cnj (cnj ?z) = ?z›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*) has_sum_cnj_iff (*‹((λx. cnj (?f x)) has_sum cnj ?a) ?M = (?f has_sum ?a) ?M›*))\n\nlemma infsum_cnj[simp]: ‹infsum (λx. cnj (f x)) M = cnj (infsum f M)›\n by (metis complex_cnj_zero (*‹cnj (0::complex) = (0::complex)›*) infsumI (*‹((?f::?'a::type ⇒ ?'b::{comm_monoid_add,t2_space}) has_sum (?x::?'b::{comm_monoid_add,t2_space})) (?A::?'a::type set) ⟹ infsum ?f ?A = ?x›*) has_sum_cnj_iff (*‹((λx::?'a::type. cnj ((?f::?'a::type ⇒ complex) x)) has_sum cnj (?a::complex)) (?M::?'a::type set) = (?f has_sum ?a) ?M›*) infsum_def (*‹infsum (?f::?'a::type ⇒ ?'b::{comm_monoid_add,t2_space}) (?A::?'a::type set) = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else (0::?'b::{comm_monoid_add,t2_space}))›*) summable_on_cnj_iff (*‹((λi::?'a::type. cnj ((?f::?'a::type ⇒ complex) i)) summable_on (?A::?'a::type set)) = (?f summable_on ?A)›*) has_sum_infsum (*‹(?f::?'a::type ⇒ ?'b::{comm_monoid_add,t2_space}) summable_on (?S::?'a::type set) ⟹ (?f has_sum infsum ?f ?S) ?S›*))\n\nlemma has_sum_Re:\n assumes \"(f has_sum a) M\"\n shows \"((λx. Re (f x)) has_sum Re a) M\"\n using has_sum_comm_additive[where f = Re] (*‹⟦additive Re; Re ─?x→ Re ?x; (?g has_sum ?x) ?S⟧ ⟹ (Re ∘ ?g has_sum Re ?x) ?S›*) using assms (*‹(f has_sum a) M›*) tendsto_Re (*‹((?g::?'c::type ⇒ complex) ⤏ (?a::complex)) (?F::?'c::type filter) ⟹ ((λx::?'c::type. Re (?g x)) ⤏ Re ?a) ?F›*) by (fastforce simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) additive_def (*‹additive ?f ≡ ∀x y. ?f (x + y) = ?f x + ?f y›*))\n\nlemma infsum_Re:\n assumes \"f summable_on M\"\n shows \"infsum (λx. Re (f x)) M = Re (infsum f M)\"\n by (simp add: assms (*‹f summable_on M›*) has_sum_Re (*‹(?f has_sum ?a) ?M ⟹ ((λx. Re (?f x)) has_sum Re ?a) ?M›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*))\n\nlemma summable_on_Re: \n assumes \"f summable_on M\"\n shows \"(λx. Re (f x)) summable_on M\"\n by (metis assms (*‹f summable_on M›*) has_sum_Re (*‹(?f has_sum ?a) ?M ⟹ ((λx. Re (?f x)) has_sum Re ?a) ?M›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma has_sum_Im:\n assumes \"(f has_sum a) M\"\n shows \"((λx. Im (f x)) has_sum Im a) M\"\n using has_sum_comm_additive[where f = Im] (*‹⟦additive Im; Im ─?x::complex→ Im ?x; ((?g::?'a::type ⇒ complex) has_sum ?x) (?S::?'a::type set)⟧ ⟹ (Im ∘ ?g has_sum Im ?x) ?S›*) using assms (*‹(f has_sum a) M›*) tendsto_Im (*‹((?g::?'c ⇒ complex) ⤏ (?a::complex)) (?F::?'c filter) ⟹ ((λx::?'c. Im (?g x)) ⤏ Im ?a) ?F›*) by (fastforce simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*) additive_def (*‹additive ?f ≡ ∀x y. ?f (x + y) = ?f x + ?f y›*))\n\nlemma infsum_Im: \n assumes \"f summable_on M\"\n shows \"infsum (λx. Im (f x)) M = Im (infsum f M)\"\n by (simp add: assms (*‹f summable_on M›*) has_sum_Im (*‹(?f has_sum ?a) ?M ⟹ ((λx. Im (?f x)) has_sum Im ?a) ?M›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*))\n\nlemma summable_on_Im: \n assumes \"f summable_on M\"\n shows \"(λx. Im (f x)) summable_on M\"\n by (metis assms (*‹f summable_on M›*) has_sum_Im (*‹(?f has_sum ?a) ?M ⟹ ((λx. Im (?f x)) has_sum Im ?a) ?M›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma nonneg_infsum_le_0D_complex:\n fixes f :: \"'a ⇒ complex\"\n assumes \"infsum f A ≤ 0\"\n and abs_sum: \"f summable_on A\"\n and nneg: \"⋀x. x ∈ A ⟹ f x ≥ 0\"\n and \"x ∈ A\"\n shows \"f x = 0\"\nproof (-)\n (*goal: ‹(f::'a ⇒ complex) (x::'a) = (0::complex)›*)\n have \"Im (f x) = 0\"\n using assms(4) (*‹x ∈ A›*) less_eq_complex_def (*‹(?x ≤ ?y) = (Re ?x ≤ Re ?y ∧ Im ?x = Im ?y)›*) nneg (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) by auto\n moreover have \"Re (f x) = 0\"\n using assms (*‹infsum f A ≤ 0› ‹f summable_on A› ‹?x ∈ A ⟹ 0 ≤ f ?x› ‹x ∈ A›*) by (auto simp add: summable_on_Re (*‹(?f::?'a::type ⇒ complex) summable_on (?M::?'a::type set) ⟹ (λx::?'a::type. Re (?f x)) summable_on ?M›*) infsum_Re (*‹(?f::?'a::type ⇒ complex) summable_on (?M::?'a::type set) ⟹ (∑⇩∞x::?'a::type∈?M. Re (?f x)) = Re (infsum ?f ?M)›*) less_eq_complex_def (*‹((?x::complex) ≤ (?y::complex)) = (Re ?x ≤ Re ?y ∧ Im ?x = Im ?y)›*) intro: nonneg_infsum_le_0D[where A=A] (*‹⟦infsum (?f::'a::type ⇒ ?'b::{ordered_ab_group_add,topological_ab_group_add,linorder_topology}) (A::'a::type set) ≤ (0::?'b::{ordered_ab_group_add,topological_ab_group_add,linorder_topology}); ?f summable_on A; ⋀x::'a::type. x ∈ A ⟹ (0::?'b::{ordered_ab_group_add,topological_ab_group_add,linorder_topology}) ≤ ?f x; (?x::'a::type) ∈ A⟧ ⟹ ?f ?x = (0::?'b::{ordered_ab_group_add,topological_ab_group_add,linorder_topology})›*))\n ultimately show \"?thesis\"\n (*goal: ‹f x = 0›*)\n by (simp add: complex_eqI (*‹⟦Re (?x::complex) = Re (?y::complex); Im ?x = Im ?y⟧ ⟹ ?x = ?y›*))\nqed\n\nlemma nonneg_has_sum_le_0D_complex:\n fixes f :: \"'a ⇒ complex\"\n assumes \"(f has_sum a) A\" and ‹a ≤ 0›\n and \"⋀x. x ∈ A ⟹ f x ≥ 0\" and \"x ∈ A\"\n shows \"f x = 0\"\n by (metis assms (*‹((f::'a ⇒ complex) has_sum (a::complex)) (A::'a set)› ‹(a::complex) ≤ (0::complex)› ‹(?x::'a) ∈ (A::'a set) ⟹ (0::complex) ≤ (f::'a ⇒ complex) ?x› ‹(x::'a) ∈ (A::'a set)›*) infsumI (*‹((?f::?'a ⇒ ?'b) has_sum (?x::?'b)) (?A::?'a set) ⟹ infsum ?f ?A = ?x›*) nonneg_infsum_le_0D_complex (*‹⟦infsum (?f::?'a ⇒ complex) (?A::?'a set) ≤ (0::complex); ?f summable_on ?A; ⋀x::?'a. x ∈ ?A ⟹ (0::complex) ≤ ?f x; (?x::?'a) ∈ ?A⟧ ⟹ ?f ?x = (0::complex)›*) summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*))\n\ntext ‹The lemma @{thm [source] infsum_mono_neutral} above applies to various linear ordered monoids such as the reals but not to the complex numbers.\n Thus we have a separate corollary for those:›\n\nlemma infsum_mono_neutral_complex:\n fixes f :: \"'a ⇒ complex\"\n assumes [simp]: \"f summable_on A\"\n and [simp]: \"g summable_on B\"\n assumes ‹⋀x. x ∈ A∩B ⟹ f x ≤ g x›\n assumes ‹⋀x. x ∈ A-B ⟹ f x ≤ 0›\n assumes ‹⋀x. x ∈ B-A ⟹ g x ≥ 0›\n shows ‹infsum f A ≤ infsum g B›\nproof (-)\n (*goal: ‹infsum f A ≤ infsum g B›*)\n have \"infsum (λx. Re (f x)) A ≤ infsum (λx. Re (g x)) B\"\n by (smt (verit) assms (*‹(f::'a ⇒ complex) summable_on (A::'a set)› ‹(g::'a ⇒ complex) summable_on (B::'a set)› ‹(?x::'a) ∈ (A::'a set) ∩ (B::'a set) ⟹ (f::'a ⇒ complex) ?x ≤ (g::'a ⇒ complex) ?x› ‹(?x::'a) ∈ (A::'a set) - (B::'a set) ⟹ (f::'a ⇒ complex) ?x ≤ (0::complex)› ‹(?x::'a) ∈ (B::'a set) - (A::'a set) ⟹ (0::complex) ≤ (g::'a ⇒ complex) ?x›*) infsum_cong (*‹(⋀x::?'a. x ∈ (?A::?'a set) ⟹ (?f::?'a ⇒ ?'b) x = (?g::?'a ⇒ ?'b) x) ⟹ infsum ?f ?A = infsum ?g ?A›*) infsum_mono_neutral (*‹⟦(?f::?'a ⇒ ?'b) summable_on (?A::?'a set); (?g::?'a ⇒ ?'b) summable_on (?B::?'a set); ⋀x::?'a. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x::?'a. x ∈ ?A - ?B ⟹ ?f x ≤ (0::?'b); ⋀x::?'a. x ∈ ?B - ?A ⟹ (0::?'b) ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?B›*) less_eq_complex_def (*‹((?x::complex) ≤ (?y::complex)) = (Re ?x ≤ Re ?y ∧ Im ?x = Im ?y)›*) summable_on_Re (*‹(?f::?'a ⇒ complex) summable_on (?M::?'a set) ⟹ (λx::?'a. Re (?f x)) summable_on ?M›*) zero_complex.simps( (*‹Re (0::complex) = (0::real)›*) 1))\n then have Re: \"Re (infsum f A) ≤ Re (infsum g B)\"\n by (metis assms( (*‹f summable_on A› ‹g summable_on B›*) 1-2) infsum_Re (*‹?f summable_on ?M ⟹ (∑⇩∞x∈?M. Re (?f x)) = Re (infsum ?f ?M)›*))\n have \"infsum (λx. Im (f x)) A = infsum (λx. Im (g x)) B\"\n by (smt (verit, best) assms( (*‹?x ∈ A ∩ B ⟹ f ?x ≤ g ?x› ‹?x ∈ A - B ⟹ f ?x ≤ 0› ‹?x ∈ B - A ⟹ 0 ≤ g ?x›*) 3-5) infsum_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ infsum ?f ?S = infsum ?g ?T›*) less_eq_complex_def (*‹(?x ≤ ?y) = (Re ?x ≤ Re ?y ∧ Im ?x = Im ?y)›*) zero_complex.simps( (*‹Im 0 = 0›*) 2))\n then have Im: \"Im (infsum f A) = Im (infsum g B)\"\n by (metis assms( (*‹f summable_on A› ‹g summable_on B›*) 1-2) infsum_Im (*‹?f summable_on ?M ⟹ (∑⇩∞x∈?M. Im (?f x)) = Im (infsum ?f ?M)›*))\n from Re (*‹Re (infsum f A) ≤ Re (infsum g B)›*) Im (*‹Im (infsum f A) = Im (infsum g B)›*) show \"?thesis\"\n (*goal: ‹infsum f A ≤ infsum g B›*)\n by (auto simp: less_eq_complex_def (*‹(?x ≤ ?y) = (Re ?x ≤ Re ?y ∧ Im ?x = Im ?y)›*))\nqed\n\nlemma infsum_mono_complex:\n ― ‹For \\<^typ>‹real›, @{thm [source] infsum_mono} can be used. \n But \\<^typ>‹complex› does not have the right typeclass.›\n fixes f g :: \"'a ⇒ complex\"\n assumes f_sum: \"f summable_on A\" and g_sum: \"g summable_on A\"\n assumes leq: \"⋀x. x ∈ A ⟹ f x ≤ g x\"\n shows \"infsum f A ≤ infsum g A\"\n by (metis DiffE (*‹⟦?c ∈ ?A - ?B; ⟦?c ∈ ?A; ?c ∉ ?B⟧ ⟹ ?P⟧ ⟹ ?P›*) IntD1 (*‹?c ∈ ?A ∩ ?B ⟹ ?c ∈ ?A›*) f_sum (*‹f summable_on A›*) g_sum (*‹g summable_on A›*) infsum_mono_neutral_complex (*‹⟦?f summable_on ?A; ?g summable_on ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?B›*) leq (*‹?x ∈ A ⟹ f ?x ≤ g ?x›*))\n\n\nlemma infsum_nonneg_complex:\n fixes f :: \"'a ⇒ complex\"\n assumes \"f summable_on M\"\n and \"⋀x. x ∈ M ⟹ 0 ≤ f x\"\n shows \"infsum f M ≥ 0\" (is \"?lhs ≥ _\")\n by (metis assms (*‹f summable_on M› ‹?x ∈ M ⟹ 0 ≤ f ?x›*) infsum_0_simp (*‹(∑⇩∞_∈?M. 0) = 0›*) summable_on_0_simp (*‹(λ_. 0) summable_on ?M›*) infsum_mono_complex (*‹⟦?f summable_on ?A; ?g summable_on ?A; ⋀x. x ∈ ?A ⟹ ?f x ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?A›*))\n\nlemma infsum_cmod:\n assumes \"f summable_on M\"\n and fnn: \"⋀x. x ∈ M ⟹ 0 ≤ f x\"\n shows \"infsum (λx. cmod (f x)) M = cmod (infsum f M)\"\nproof (-)\n (*goal: ‹(∑⇩∞x∈M. cmod (f x)) = cmod (infsum f M)›*)\n have \"complex_of_real (infsum (λx. cmod (f x)) M) = infsum (λx. complex_of_real (cmod (f x))) M\"\n proof (rule infsum_comm_additive[symmetric, unfolded o_def] (*‹⟦additive ?f; isCont ?f (infsum ?g ?S); ?g summable_on ?S⟧ ⟹ ?f (infsum ?g ?S) = (∑⇩∞x∈?S. ?f (?g x))›*))\n (*goals:\n 1. ‹additive complex_of_real›\n 2. ‹isCont complex_of_real (∑⇩∞x∈M. cmod (f x))›\n 3. ‹f abs_summable_on M›*)\n have \"(λz. Re (f z)) summable_on M\"\n using assms (*‹(f::'a ⇒ complex) summable_on (M::'a set)› ‹(?x::'a) ∈ (M::'a set) ⟹ (0::complex) ≤ (f::'a ⇒ complex) ?x›*) summable_on_Re (*‹?f summable_on ?M ⟹ (λx. Re (?f x)) summable_on ?M›*) by blast\n also (*calculation: ‹(λz. Re (f z)) summable_on M›*) have \"?this ⟷ f abs_summable_on M\"\n using fnn (*‹?x ∈ M ⟹ 0 ≤ f ?x›*) apply (intro summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n (*goal: ‹((λz. Re (f z)) summable_on M) = (f abs_summable_on M)›*)\n by (auto simp: less_eq_complex_def (*‹(?x ≤ ?y) = (Re ?x ≤ Re ?y ∧ Im ?x = Im ?y)›*) cmod_def (*‹cmod ?z = sqrt ((Re ?z)² + (Im ?z)²)›*))\n finally (*calculation: ‹f abs_summable_on M›*) show \"…\" .\n qed (auto simp: additive_def (*‹additive ?f ≡ ∀x y. ?f (x + y) = ?f x + ?f y›*))\n (*solves the remaining goals:\n 1. ‹additive complex_of_real›\n 2. ‹isCont complex_of_real (∑⇩∞x::'a::type∈(M::'a::type set). cmod ((f::'a::type ⇒ complex) x))›*)\n also (*calculation: ‹complex_of_real (∑⇩∞x∈M. cmod (f x)) = (∑⇩∞x∈M. complex_of_real (cmod (f x)))›*) have \"… = infsum f M\"\n using fnn (*‹?x ∈ M ⟹ 0 ≤ f ?x›*) cmod_eq_Re (*‹Im ?z = 0 ⟹ cmod ?z = ¦Re ?z¦›*) complex_is_Real_iff (*‹(?z ∈ ℝ) = (Im ?z = 0)›*) less_eq_complex_def (*‹(?x ≤ ?y) = (Re ?x ≤ Re ?y ∧ Im ?x = Im ?y)›*) by (force cong: infsum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ infsum ?f ?A = infsum ?g ?A›*))\n finally (*calculation: ‹complex_of_real (∑⇩∞x∈M. cmod (f x)) = infsum f M›*) show \"?thesis\"\n (*goal: ‹(∑⇩∞x∈M. cmod (f x)) = cmod (infsum f M)›*)\n by (metis abs_of_nonneg (*‹0 ≤ ?a ⟹ ¦?a¦ = ?a›*) infsum_def (*‹infsum ?f ?A = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else 0)›*) le_less_trans (*‹⟦?x ≤ ?y; ?y < ?z⟧ ⟹ ?x < ?z›*) norm_ge_zero (*‹0 ≤ norm ?x›*) norm_infsum_bound (*‹?f abs_summable_on ?A ⟹ norm (infsum ?f ?A) ≤ (∑⇩∞x∈?A. norm (?f x))›*) norm_of_real (*‹norm (of_real ?r) = ¦?r¦›*) not_le (*‹(¬ ?x ≤ ?y) = (?y < ?x)›*) order_refl (*‹?x ≤ ?x›*))\nqed\n\n\nlemma summable_on_iff_abs_summable_on_complex:\n fixes f :: ‹'a ⇒ complex›\n shows ‹f summable_on A ⟷ f abs_summable_on A›\nproof (rule iffI (*‹⟦?P ⟹ ?Q; ?Q ⟹ ?P⟧ ⟹ ?P = ?Q›*))\n (*goals:\n 1. ‹(f::'a ⇒ complex) summable_on (A::'a set) ⟹ f abs_summable_on A›\n 2. ‹(f::'a ⇒ complex) abs_summable_on (A::'a set) ⟹ f summable_on A›*)\n assume \"f summable_on A\" (*‹(f::'a ⇒ complex) summable_on (A::'a set)›*)\n define i and r and ni and nr and n where \"i x = Im (f x)\" and \"r x = Re (f x)\" and \"ni x = norm (i x)\" and \"nr x = norm (r x)\" and \"n x = norm (f x)\" for x\n from ‹f summable_on A› (*‹f summable_on A›*) have \"i summable_on A\"\n by (simp add: i_def[abs_def] (*‹i ≡ λx. Im (f x)›*) summable_on_Im (*‹?f summable_on ?M ⟹ (λx. Im (?f x)) summable_on ?M›*))\n then have [simp]: \"ni summable_on A\"\n using ni_def[abs_def] (*‹ni ≡ λx. norm (i x)›*) summable_on_iff_abs_summable_on_real (*‹(?f summable_on ?A) = (?f abs_summable_on ?A)›*) by force\n from ‹f summable_on A› (*‹f summable_on A›*) have \"r summable_on A\"\n by (simp add: r_def[abs_def] (*‹r ≡ λx. Re (f x)›*) summable_on_Re (*‹?f summable_on ?M ⟹ (λx. Re (?f x)) summable_on ?M›*))\n then have [simp]: \"nr summable_on A\"\n by (metis nr_def (*‹nr ?x = norm (r ?x)›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*) summable_on_iff_abs_summable_on_real (*‹(?f summable_on ?A) = (?f abs_summable_on ?A)›*))\n have n_sum: \"n x ≤ nr x + ni x\" for x\n by (simp add: n_def (*‹n ?x = cmod (f ?x)›*) nr_def (*‹nr ?x = norm (r ?x)›*) ni_def (*‹ni ?x = norm (i ?x)›*) r_def (*‹r ?x = Re (f ?x)›*) i_def (*‹i ?x = Im (f ?x)›*) cmod_le (*‹cmod ?z ≤ ¦Re ?z¦ + ¦Im ?z¦›*))\n have \"*\": \"(λx. nr x + ni x) summable_on A\"\n by (simp add: summable_on_add (*‹⟦(?f::?'a ⇒ ?'b) summable_on (?A::?'a set); (?g::?'a ⇒ ?'b) summable_on ?A⟧ ⟹ (λx::?'a. ?f x + ?g x) summable_on ?A›*))\n have \"bdd_above (sum n ` {F. F ⊆ A ∧ finite F})\"\n apply (rule bdd_aboveI[where M=‹infsum (λx. nr x + ni x) A›] (*‹(⋀x. x ∈ ?A ⟹ x ≤ (∑⇩∞x∈A. nr x + ni x)) ⟹ bdd_above ?A›*))\n (*goal: ‹bdd_above (sum n ` {F. F ⊆ A ∧ finite F})›*)\n using \"*\" (*‹(λx. nr x + ni x) summable_on A›*) n_sum (*‹n ?x ≤ nr ?x + ni ?x›*) by (auto simp flip: infsum_finite (*‹finite ?F ⟹ infsum ?f ?F = sum ?f ?F›*) simp: ni_def (*‹ni ?x = norm (i ?x)›*) nr_def (*‹nr ?x = norm (r ?x)›*) intro!: infsum_mono_neutral (*‹⟦?f summable_on ?A; ?g summable_on ?B; ⋀x. x ∈ ?A ∩ ?B ⟹ ?f x ≤ ?g x; ⋀x. x ∈ ?A - ?B ⟹ ?f x ≤ 0; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?g x⟧ ⟹ infsum ?f ?A ≤ infsum ?g ?B›*))\n then show \"n summable_on A\"\n by (simp add: n_def (*‹n ?x = cmod (f ?x)›*) nonneg_bdd_above_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; bdd_above (sum ?f ` {F. F ⊆ ?A ∧ finite F})⟧ ⟹ ?f summable_on ?A›*))\nnext\n (*goal: ‹f abs_summable_on A ⟹ f summable_on A›*)\n show \"f abs_summable_on A ⟹ f summable_on A\"\n using abs_summable_summable (*‹?f abs_summable_on ?A ⟹ ?f summable_on ?A›*) by blast\nqed\n\nlemma summable_countable_complex:\n fixes f :: ‹'a ⇒ complex›\n assumes ‹f summable_on A›\n shows ‹countable {x∈A. f x ≠ 0}›\n using abs_summable_countable (*‹(?f::?'a ⇒ ?'b) abs_summable_on (?A::?'a set) ⟹ countable {x::?'a ∈ ?A. ?f x ≠ (0::?'b)}›*) assms (*‹f summable_on A›*) summable_on_iff_abs_summable_on_complex (*‹((?f::?'a ⇒ complex) summable_on (?A::?'a set)) = (?f abs_summable_on ?A)›*) by blast\n\n\n(* TODO: figure all this out *)\ninductive (in topological_space) convergent_filter :: \"'a filter ⇒ bool\" where\n \"F ≤ nhds x ⟹ convergent_filter F\"\n\nlemma (in topological_space) convergent_filter_iff: \"convergent_filter F ⟷ (∃x. F ≤ nhds x)\"\n by (auto simp: convergent_filter.simps (*‹convergent_filter ?a = (∃F x. ?a = F ∧ F ≤ nhds x)›*))\n\nlemma (in uniform_space) cauchy_filter_mono:\n \"cauchy_filter F ⟹ F' ≤ F ⟹ cauchy_filter F'\"\n unfolding cauchy_filter_def\n (*goal: ‹⟦F ×⇩F F ≤ uniformity; F' ≤ F⟧ ⟹ F' ×⇩F F' ≤ uniformity›*)\n by (meson dual_order.trans (*‹⟦?b ≤ ?a; ?c ≤ ?b⟧ ⟹ ?c ≤ ?a›*) prod_filter_mono (*‹⟦?F ≤ ?F'; ?G ≤ ?G'⟧ ⟹ ?F ×⇩F ?G ≤ ?F' ×⇩F ?G'›*))\n\nlemma (in uniform_space) convergent_filter_cauchy: \n assumes \"convergent_filter F\"\n shows \"cauchy_filter F\"\n using assms (*‹convergent_filter F›*) cauchy_filter_mono (*‹⟦cauchy_filter ?F; ?F' ≤ ?F⟧ ⟹ cauchy_filter ?F'›*) nhds_imp_cauchy_filter[OF order.refl] (*‹cauchy_filter (nhds ?x)›*) by (auto simp: convergent_filter_iff (*‹convergent_filter ?F = (∃x. ?F ≤ nhds x)›*))\n\nlemma (in topological_space) convergent_filter_bot [simp, intro]: \"convergent_filter bot\"\n by (simp add: convergent_filter_iff (*‹convergent_filter ?F = (∃x. ?F ≤ nhds x)›*))\n\n\n\nclass complete_uniform_space = uniform_space +\n assumes cauchy_filter_convergent': \"cauchy_filter (F :: 'a filter) ⟹ F ≠ bot ⟹ convergent_filter F\"\n\nlemma (in complete_uniform_space)\n cauchy_filter_convergent: \"cauchy_filter (F :: 'a filter) ⟹ convergent_filter F\"\n using cauchy_filter_convergent'[of F] (*‹⟦cauchy_filter F; F ≠ bot⟧ ⟹ convergent_filter F›*) apply (cases \"F = bot\")\n (*goals:\n 1. ‹⟦cauchy_filter F; ⟦cauchy_filter F; F ≠ bot⟧ ⟹ convergent_filter F; F = bot⟧ ⟹ convergent_filter F›\n 2. ‹⟦cauchy_filter F; ⟦cauchy_filter F; F ≠ bot⟧ ⟹ convergent_filter F; F ≠ bot⟧ ⟹ convergent_filter F›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma (in complete_uniform_space) convergent_filter_iff_cauchy:\n \"convergent_filter F ⟷ cauchy_filter F\"\n using convergent_filter_cauchy (*‹convergent_filter ?F ⟹ cauchy_filter ?F›*) cauchy_filter_convergent (*‹cauchy_filter ?F ⟹ convergent_filter ?F›*) by blast\n\ndefinition countably_generated_filter :: \"'a filter ⇒ bool\" where\n \"countably_generated_filter F ⟷ (∃U :: nat ⇒ 'a set. F = (INF (n::nat). principal (U n)))\"\n\nlemma countably_generated_filter_has_antimono_basis:\n assumes \"countably_generated_filter F\"\n obtains B :: \"nat ⇒ 'a set\"\n where \"antimono B\" and \"F = (INF n. principal (B n))\" and\n \"⋀P. eventually P F ⟷ (∃i. ∀x∈B i. P x)\"\nproof (-)\n (*goal: ‹(⋀B::nat ⇒ 'a::type set. ⟦monotone (≤) (λ(x::'a::type set) y::'a::type set. y ⊆ x) B; (F::'a::type filter) = (INF n::nat. principal (B n)); ⋀P::'a::type ⇒ bool. eventually P F = (∃i::nat. ∀x::'a::type∈B i. P x)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n from assms (*‹countably_generated_filter F›*) obtain B where B: \"F = (INF (n::nat). principal (B n))\"\n (*goal: ‹(⋀B. F = (INF n. principal (B n)) ⟹ thesis) ⟹ thesis›*)\n unfolding countably_generated_filter_def\n (*goal: ‹(⋀B. F = (INF n. principal (B n)) ⟹ thesis) ⟹ thesis›*)\n by blast\n define B' where \"B' = (λn. ⋂k≤n. B k)\"\n have \"antimono B'\"\n unfolding decseq_def B'_def\n (*goal: ‹∀(m::nat) n::nat. m ≤ n ⟶ ⋂ ((B::nat ⇒ 'a set) ` {..n}) ⊆ ⋂ (B ` {..m})›*)\n by force\n have \"(INF n. principal (B' n)) = (INF n. INF k∈{..n}. principal (B k))\"\n unfolding B'_def\n (*goal: ‹(INF n::nat. principal (⋂ ((B::nat ⇒ 'a::type set) ` {..n}))) = (INF n::nat. INF k::nat∈{..n}. principal (B k))›*)\n apply (intro INF_cong (*‹⟦(?A::?'b set) = (?B::?'b set); ⋀x::?'b. x ∈ ?B ⟹ (?C::?'b ⇒ ?'a) x = (?D::?'b ⇒ ?'a) x⟧ ⟹ Inf (?C ` ?A) = Inf (?D ` ?B)›*) refl (*‹(?t::?'a) = ?t›*) INF_principal_finite [symmetric] (*‹finite (?X::?'a set) ⟹ principal (⋂ ((?f::?'a ⇒ ?'b set) ` ?X)) = (INF x::?'a∈?X. principal (?f x))›*))\n (*goal: ‹(INF n. principal (⋂ (B ` {..n}))) = (INF n. INF k∈{..n}. principal (B k))›*)\n by auto\n also (*calculation: ‹(INF n::nat. principal ((B'::nat ⇒ 'a::type set) n)) = (INF n::nat. INF k::nat∈{..n}. principal ((B::nat ⇒ 'a::type set) k))›*) have \"… = (INF (n::nat). principal (B n))\"\n apply (intro antisym (*‹⟦(?a::?'a) ≤ (?b::?'a); ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹(INF n. INF k∈{..n}. principal (B k)) ≤ (INF n. principal (B n))›\n 2. ‹(INF n. principal (B n)) ≤ (INF n. INF k∈{..n}. principal (B k))›\n discuss goal 1*)\n apply (meson INF_lower (*‹(?i::?'b) ∈ (?A::?'b set) ⟹ Inf ((?f::?'b ⇒ ?'a) ` ?A) ≤ ?f ?i›*) INF_mono (*‹(⋀m::?'b. m ∈ (?B::?'b set) ⟹ ∃n::?'c∈?A::?'c set. (?f::?'c ⇒ ?'a) n ≤ (?g::?'b ⇒ ?'a) m) ⟹ Inf (?f ` ?A) ≤ Inf (?g ` ?B)›*) atMost_iff (*‹((?i::?'a) ∈ {..?k::?'a}) = (?i ≤ ?k)›*) order_refl (*‹(?x::?'a) ≤ ?x›*))\n (*discuss goal 2*)\n apply (meson INF_greatest (*‹(⋀i. i ∈ ?A ⟹ ?u ≤ ?f i) ⟹ ?u ≤ Inf (?f ` ?A)›*) INF_lower (*‹?i ∈ ?A ⟹ Inf (?f ` ?A) ≤ ?f ?i›*) UNIV_I (*‹?x ∈ UNIV›*))\n (*proven 2 subgoals*) .\n also (*calculation: ‹(INF n::nat. principal ((B'::nat ⇒ 'a set) n)) = (INF n::nat. principal ((B::nat ⇒ 'a set) n))›*) have \"… = F\"\n by (simp add: B (*‹F = (INF n. principal (B n))›*))\n finally (*calculation: ‹(INF n. principal (B' n)) = F›*) have F: \"F = (INF n. principal (B' n))\"\n by standard\n moreover have \"eventually P F ⟷ (∃i. eventually P (principal (B' i)))\" for P\n unfolding F\n (*goal: ‹eventually P (INF n. principal (B' n)) = (∃i. eventually P (principal (B' i)))›*)\n using ‹antimono B'› (*‹monotone (≤) (λx y. y ⊆ x) B'›*) apply (subst eventually_INF_base (*‹⟦?B ≠ {}; ⋀a b. ⟦a ∈ ?B; b ∈ ?B⟧ ⟹ ∃x∈?B. ?F x ≤ inf (?F a) (?F b)⟧ ⟹ eventually ?P (Inf (?F ` ?B)) = (∃b∈?B. eventually ?P (?F b))›*))\n (*goals:\n 1. ‹monotone (≤) (λx y. y ⊆ x) B' ⟹ UNIV ≠ {}›\n 2. ‹⋀a b. ⟦monotone (≤) (λx y. y ⊆ x) B'; a ∈ UNIV; b ∈ UNIV⟧ ⟹ ∃x∈UNIV. principal (B' x) ≤ inf (principal (B' a)) (principal (B' b))›\n 3. ‹monotone (≤) (λx y. y ⊆ x) B' ⟹ (∃b∈UNIV. eventually P (principal (B' b))) = (∃i. eventually P (principal (B' i)))›\n discuss goal 1*)\n apply ((auto simp: decseq_def (*‹decseq ?X = (∀m n. m ≤ n ⟶ ?X n ≤ ?X m)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: decseq_def (*‹decseq ?X = (∀m n. m ≤ n ⟶ ?X n ≤ ?X m)›*))[1])\n (*top goal: ‹⋀a b. ⟦monotone (≤) (λx y. y ⊆ x) B'; a ∈ UNIV; b ∈ UNIV⟧ ⟹ ∃x∈UNIV. principal (B' x) ≤ inf (principal (B' a)) (principal (B' b))› and 1 goal remains*)\n apply (meson nat_le_linear (*‹?m ≤ ?n ∨ ?n ≤ ?m›*))\n (*discuss goal 3*)\n apply ((auto simp: decseq_def (*‹decseq (?X::nat ⇒ ?'a) = (∀(m::nat) n::nat. m ≤ n ⟶ ?X n ≤ ?X m)›*))[1])\n (*proven 3 subgoals*) .\n ultimately show \"?thesis\"\n (*goal: ‹thesis›*)\n using ‹antimono B'› (*‹monotone (≤) (λx y. y ⊆ x) B'›*) that[of B'] (*‹⟦monotone (≤) (λx y. y ⊆ x) B'; F = (INF n. principal (B' n)); ⋀P. eventually P F = (∃i. ∀x∈B' i. P x)⟧ ⟹ thesis›*) unfolding eventually_principal\n (*goal: ‹thesis›*)\n by blast\nqed\n\nlemma (in uniform_space) cauchy_filter_iff:\n \"cauchy_filter F ⟷ (∀P. eventually P uniformity ⟶ (∃X. eventually (λx. x ∈ X) F ∧ (∀z∈X×X. P z)))\" \n unfolding cauchy_filter_def le_filter_def\n (*goal: ‹(∀P. eventually P uniformity ⟶ eventually P (F ×⇩F F)) = (∀P. eventually P uniformity ⟶ (∃X. (∀⇩F x in F. x ∈ X) ∧ (∀z∈X × X. P z)))›*)\n apply (auto simp: eventually_prod_same (*‹eventually ?P (?F ×⇩F ?F) = (∃Q. eventually Q ?F ∧ (∀x y. Q x ⟶ Q y ⟶ ?P (x, y)))›*))\n (*goals:\n 1. ‹⋀P. ⟦∀P. eventually P uniformity ⟶ (∃Q. eventually Q F ∧ (∀x. Q x ⟶ (∀y. Q y ⟶ P (x, y)))); eventually P uniformity⟧ ⟹ ∃X. (∀⇩F x in F. x ∈ X) ∧ (∀x∈X. ∀y∈X. P (x, y))›\n 2. ‹⋀P. ⟦∀P. eventually P uniformity ⟶ (∃X. (∀⇩F x in F. x ∈ X) ∧ (∀x∈X. ∀y∈X. P (x, y))); eventually P uniformity⟧ ⟹ ∃Q. eventually Q F ∧ (∀x. Q x ⟶ (∀y. Q y ⟶ P (x, y)))›\n discuss goal 1*)\n apply (metis (full_types) eventually_mono (*‹⟦eventually ?P ?F; ⋀x. ?P x ⟹ ?Q x⟧ ⟹ eventually ?Q ?F›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*))\n (*discuss goal 2*)\n apply blast\n (*proven 2 subgoals*) .\n\nlemma (in uniform_space) controlled_sequences_convergent_imp_complete_aux_sequence:\n fixes U :: \"nat ⇒ ('a × 'a) set\"\n fixes F :: \"'a filter\"\n assumes \"cauchy_filter F\" \"F ≠ bot\"\n assumes \"⋀n. eventually (λz. z ∈ U n) uniformity\"\n obtains g G where\n \"antimono G\" \"⋀n. g n ∈ G n\"\n \"⋀n. eventually (λx. x ∈ G n) F\" \"⋀n. G n × G n ⊆ U n\"\nproof (-)\n (*goal: ‹(⋀G g. ⟦monotone (≤) (λx y. y ⊆ x) G; ⋀n. g n ∈ G n; ⋀n. ∀⇩F x in F. x ∈ G n; ⋀n. G n × G n ⊆ U n⟧ ⟹ thesis) ⟹ thesis›*)\n have \"∃C. eventually (λx. x ∈ C) F ∧ C × C ⊆ U n\" for n\n using assms(1) (*‹cauchy_filter F›*) assms(3)[of n] (*‹∀⇩F z in uniformity. z ∈ U n›*) unfolding cauchy_filter_iff\n (*goal: ‹∃C::'a::type set. (∀⇩F x::'a::type in F::'a::type filter. x ∈ C) ∧ C × C ⊆ (U::nat ⇒ ('a::type × 'a::type) set) (n::nat)›*)\n by blast\n then obtain G where G: \"⋀n. eventually (λx. x ∈ G n) F\" \"⋀n. G n × G n ⊆ U n\"\n (*goal: ‹(⋀G::nat ⇒ 'a set. ⟦⋀n::nat. ∀⇩F x::'a in F::'a filter. x ∈ G n; ⋀n::nat. G n × G n ⊆ (U::nat ⇒ ('a × 'a) set) n⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by metis\n define G' where \"G' = (λn. ⋂k≤n. G k)\"\n have 1: \"eventually (λx. x ∈ G' n) F\" for n\n using G (*‹∀⇩F x in F. x ∈ G ?n› ‹G ?n × G ?n ⊆ U ?n›*) by (auto simp: G'_def (*‹G' = (λn. ⋂ (G ` {..n}))›*) intro: eventually_ball_finite (*‹⟦finite ?A; ∀y∈?A. ∀⇩F x in ?net. ?P x y⟧ ⟹ ∀⇩F x in ?net. ∀y∈?A. ?P x y›*))\n have 2: \"G' n × G' n ⊆ U n\" for n\n using G (*‹∀⇩F x in F. x ∈ G ?n› ‹G ?n × G ?n ⊆ U ?n›*) unfolding G'_def\n (*goal: ‹⋂ (G ` {..n}) × ⋂ (G ` {..n}) ⊆ U n›*)\n by fast\n have 3: \"antimono G'\"\n unfolding G'_def decseq_def\n (*goal: ‹∀m n. m ≤ n ⟶ ⋂ (G ` {..n}) ⊆ ⋂ (G ` {..m})›*)\n by force\n have \"∃g. g ∈ G' n\" for n\n using \"1\" (*‹∀⇩F x::'a::type in F::'a::type filter. x ∈ (G'::nat ⇒ 'a::type set) (?n::nat)›*) assms(2) (*‹F ≠ bot›*) eventually_happens' (*‹⟦?F ≠ bot; eventually ?P ?F⟧ ⟹ ∃x. ?P x›*) by auto\n then obtain g where g: \"⋀n. g n ∈ G' n\"\n (*goal: ‹(⋀g. (⋀n. g n ∈ G' n) ⟹ thesis) ⟹ thesis›*)\n by metis\n from g (*‹g ?n ∈ G' ?n›*) \"1\" (*‹∀⇩F x in F. x ∈ G' ?n›*) \"2\" (*‹G' ?n × G' ?n ⊆ U ?n›*) \"3\" (*‹monotone (≤) (λx y. y ⊆ x) G'›*) that[of G' g] (*‹⟦monotone (≤) (λx y. y ⊆ x) G'; ⋀n. g n ∈ G' n; ⋀n. ∀⇩F x in F. x ∈ G' n; ⋀n. G' n × G' n ⊆ U n⟧ ⟹ thesis›*) show \"?thesis\"\n (*goal: ‹thesis::bool›*)\n by metis\nqed\n\ndefinition lift_filter :: \"('a set ⇒ 'b filter) ⇒ 'a filter ⇒ 'b filter\" where\n \"lift_filter f F = (INF X∈{X. eventually (λx. x ∈ X) F}. f X)\"\n\nlemma lift_filter_top [simp]: \"lift_filter g top = g UNIV\"\nproof (-)\n (*goal: ‹lift_filter g top = g UNIV›*)\n have \"{X. ∀x::'b. x ∈ X} = {UNIV}\"\n by auto\n thus \"?thesis\"\n (*goal: ‹lift_filter g top = g UNIV›*)\n by (simp add: lift_filter_def (*‹lift_filter ?f ?F = Inf (?f ` {X. ∀⇩F x in ?F. x ∈ X})›*))\nqed\n\nlemma eventually_lift_filter_iff:\n assumes \"mono g\"\n shows \"eventually P (lift_filter g F) ⟷ (∃X. eventually (λx. x ∈ X) F ∧ eventually P (g X))\"\n unfolding lift_filter_def\n (*goal: ‹eventually P (Inf (g ` {X. ∀⇩F x in F. x ∈ X})) = (∃X. (∀⇩F x in F. x ∈ X) ∧ eventually P (g X))›*)\n proof (subst eventually_INF_base (*‹⟦?B ≠ {}; ⋀a b. ⟦a ∈ ?B; b ∈ ?B⟧ ⟹ ∃x∈?B. ?F x ≤ inf (?F a) (?F b)⟧ ⟹ eventually ?P (Inf (?F ` ?B)) = (∃b∈?B. eventually ?P (?F b))›*), goal_cases)\n (*goals:\n 1. ‹{X. ∀⇩F x in F. x ∈ X} ≠ {}›\n 2. ‹⋀a b. ⟦a ∈ {X. ∀⇩F x in F. x ∈ X}; b ∈ {X. ∀⇩F x in F. x ∈ X}⟧ ⟹ ∃x∈{X. ∀⇩F x in F. x ∈ X}. g x ≤ inf (g a) (g b)›\n 3. ‹(∃b∈{X. ∀⇩F x in F. x ∈ X}. eventually P (g b)) = (∃X. (∀⇩F x in F. x ∈ X) ∧ eventually P (g X))›*)\n case 1 (*no hyothesis introduced yet*)\n thus \"?case\"\n (*goal: ‹{X::'a::type set. ∀⇩F x::'a::type in F::'a::type filter. x ∈ X} ≠ {}›*)\n by (auto intro: exI[of _ UNIV] (*‹?P UNIV ⟹ ∃x. ?P x›*))\n next\n (*goals:\n 1. ‹⋀a b. ⟦a ∈ {X. ∀⇩F x in F. x ∈ X}; b ∈ {X. ∀⇩F x in F. x ∈ X}⟧ ⟹ ∃x∈{X. ∀⇩F x in F. x ∈ X}. g x ≤ inf (g a) (g b)›\n 2. ‹(∃b∈{X. ∀⇩F x in F. x ∈ X}. eventually P (g b)) = (∃X. (∀⇩F x in F. x ∈ X) ∧ eventually P (g X))›*)\n case (2 X Y) (*‹(X::'a set) ∈ {X::'a set. ∀⇩F x::'a in F::'a filter. x ∈ X}› ‹Y ∈ {X. ∀⇩F x in F. x ∈ X}›*)\n thus \"?case\"\n (*goal: ‹∃x::'a set∈{X::'a set. ∀⇩F x::'a in F::'a filter. x ∈ X}. (g::'a set ⇒ 'b filter) x ≤ inf (g (X::'a set)) (g (Y::'a set))›*)\n by (auto intro!: exI[of _ \"X ∩ Y\"] (*‹?P (X ∩ Y) ⟹ ∃x. ?P x›*) eventually_conj (*‹⟦eventually ?P ?F; eventually ?Q ?F⟧ ⟹ ∀⇩F x in ?F. ?P x ∧ ?Q x›*) monoD[OF assms] (*‹?x ⊆ ?y ⟹ g ?x ≤ g ?y›*))\n qed (auto)\n (*solved the remaining goal: ‹(∃b∈{X. ∀⇩F x in F. x ∈ X}. eventually P (g b)) = (∃X. (∀⇩F x in F. x ∈ X) ∧ eventually P (g X))›*)\n\nlemma lift_filter_le:\n assumes \"eventually (λx. x ∈ X) F\" \"g X ≤ F'\"\n shows \"lift_filter g F ≤ F'\"\n unfolding lift_filter_def\n (*goal: ‹Inf ((g::'a set ⇒ 'b filter) ` {X::'a set. ∀⇩F x::'a in F::'a filter. x ∈ X}) ≤ (F'::'b filter)›*)\n by (metis INF_lower2 (*‹⟦?i ∈ ?A; ?f ?i ≤ ?u⟧ ⟹ Inf (?f ` ?A) ≤ ?u›*) assms (*‹∀⇩F x in F. x ∈ X› ‹g X ≤ F'›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*))\n\ndefinition lift_filter' :: \"('a set ⇒ 'b set) ⇒ 'a filter ⇒ 'b filter\" where\n \"lift_filter' f F = lift_filter (principal ∘ f) F\"\n\nlemma lift_filter'_top [simp]: \"lift_filter' g top = principal (g UNIV)\"\n by (simp add: lift_filter'_def (*‹lift_filter' ?f ?F = lift_filter (principal ∘ ?f) ?F›*))\n\nlemma eventually_lift_filter'_iff:\n assumes \"mono g\"\n shows \"eventually P (lift_filter' g F) ⟷ (∃X. eventually (λx. x ∈ X) F ∧ (∀x∈g X. P x))\"\n unfolding lift_filter'_def\n (*goal: ‹eventually (P::'b::type ⇒ bool) (lift_filter (principal ∘ (g::'a::type set ⇒ 'b::type set)) (F::'a::type filter)) = (∃X::'a::type set. (∀⇩F x::'a::type in F. x ∈ X) ∧ (∀x::'b::type∈g X. P x))›*)\n using assms (*‹mono g›*) apply (subst eventually_lift_filter_iff (*‹mono (?g::?'a set ⇒ ?'b filter) ⟹ eventually (?P::?'b ⇒ bool) (lift_filter ?g (?F::?'a filter)) = (∃X::?'a set. (∀⇩F x::?'a in ?F. x ∈ X) ∧ eventually ?P (?g X))›*))\n (*goals:\n 1. ‹mono (g::'a set ⇒ 'b set) ⟹ mono (principal ∘ g)›\n 2. ‹mono (g::'a set ⇒ 'b set) ⟹ (∃X::'a set. (∀⇩F x::'a in F::'a filter. x ∈ X) ∧ eventually (P::'b ⇒ bool) ((principal ∘ g) X)) = (∃X::'a set. (∀⇩F x::'a in F. x ∈ X) ∧ Ball (g X) P)›\n discuss goal 1*)\n apply ((auto simp: mono_def (*‹mono ?f = (∀x y. x ≤ y ⟶ ?f x ≤ ?f y)›*) eventually_principal (*‹eventually ?P (principal ?S) = (∀x∈?S. ?P x)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: mono_def (*‹mono ?f = (∀x y. x ≤ y ⟶ ?f x ≤ ?f y)›*) eventually_principal (*‹eventually ?P (principal ?S) = (∀x∈?S. ?P x)›*))[1])\n (*proven 2 subgoals*) .\n\nlemma lift_filter'_le:\n assumes \"eventually (λx. x ∈ X) F\" \"principal (g X) ≤ F'\"\n shows \"lift_filter' g F ≤ F'\"\n unfolding lift_filter'_def\n (*goal: ‹lift_filter (principal ∘ g) F ≤ F'›*)\n using assms (*‹∀⇩F x in F. x ∈ X› ‹principal (g X) ≤ F'›*) apply (intro lift_filter_le[where X = X] (*‹⟦∀⇩F x in ?F. x ∈ X; ?g X ≤ ?F'⟧ ⟹ lift_filter ?g ?F ≤ ?F'›*))\n (*goals:\n 1. ‹⟦∀⇩F x::'a in F::'a filter. x ∈ (X::'a set); principal ((g::'a set ⇒ 'b set) X) ≤ (F'::'b filter)⟧ ⟹ ∀⇩F x::'a in F. x ∈ X›\n 2. ‹⟦∀⇩F x::'a in F::'a filter. x ∈ (X::'a set); principal ((g::'a set ⇒ 'b set) X) ≤ (F'::'b filter)⟧ ⟹ (principal ∘ g) X ≤ F'›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\n\nlemma (in uniform_space) comp_uniformity_le_uniformity:\n \"lift_filter' (λX. X O X) uniformity ≤ uniformity\"\n unfolding le_filter_def\n (*goal: ‹∀P::'a × 'a ⇒ bool. eventually P uniformity ⟶ eventually P (lift_filter' (λX::('a × 'a) set. X O X) uniformity)›*)\n proof (safe)\n (*goal: ‹⋀P::'a::type × 'a::type ⇒ bool. eventually P uniformity ⟹ eventually P (lift_filter' (λX::('a::type × 'a::type) set. X O X) uniformity)›*)\n fix P\n assume P: \"eventually P uniformity\" (*‹eventually (P::'a × 'a ⇒ bool) uniformity›*)\n have [simp]: \"mono (λX::('a × 'a) set. X O X)\"\n apply (intro monoI (*‹(⋀x y. x ≤ y ⟹ ?f x ≤ ?f y) ⟹ mono ?f›*))\n (*goal: ‹mono (λX::('a × 'a) set. X O X)›*)\n by auto\n from P (*‹eventually P uniformity›*) obtain P' where P': \"eventually P' uniformity \" \"(⋀x y z. P' (x, y) ⟹ P' (y, z) ⟹ P (x, z))\"\n (*goal: ‹(⋀P'::'a × 'a ⇒ bool. ⟦eventually P' uniformity; ⋀(x::'a) (y::'a) z::'a. ⟦P' (x, y); P' (y, z)⟧ ⟹ (P::'a × 'a ⇒ bool) (x, z)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using uniformity_transE (*‹⟦eventually ?E uniformity; ⋀D. ⟦eventually D uniformity; ⋀x y z. ⟦D (x, y); D (y, z)⟧ ⟹ ?E (x, z)⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) by blast\n show \"eventually P (lift_filter' (λX. X O X) uniformity)\"\n by (auto simp: eventually_lift_filter'_iff (*‹mono (?g::?'a set ⇒ ?'b set) ⟹ eventually (?P::?'b ⇒ bool) (lift_filter' ?g (?F::?'a filter)) = (∃X::?'a set. (∀⇩F x::?'a in ?F. x ∈ X) ∧ (∀x::?'b∈?g X. ?P x))›*) intro!: exI[of _ \"{x. P' x}\"] (*‹(?P::('a × 'a) set ⇒ bool) {x::'a × 'a. (P'::'a × 'a ⇒ bool) x} ⟹ ∃x::('a × 'a) set. ?P x›*) P' (*‹eventually (P'::'a × 'a ⇒ bool) uniformity› ‹⟦(P'::'a × 'a ⇒ bool) (?x::'a, ?y::'a); P' (?y, ?z::'a)⟧ ⟹ (P::'a × 'a ⇒ bool) (?x, ?z)›*))\n qed\n\nlemma (in uniform_space) comp_mem_uniformity_sets:\n assumes \"eventually (λz. z ∈ X) uniformity\"\n obtains Y where \"eventually (λz. z ∈ Y) uniformity\" \"Y O Y ⊆ X\"\nproof (-)\n (*goal: ‹(⋀Y. ⟦∀⇩F z in uniformity. z ∈ Y; Y O Y ⊆ X⟧ ⟹ thesis) ⟹ thesis›*)\n have [simp]: \"mono (λX::('a × 'a) set. X O X)\"\n apply (intro monoI (*‹(⋀x y. x ≤ y ⟹ ?f x ≤ ?f y) ⟹ mono ?f›*))\n (*goal: ‹mono (λX. X O X)›*)\n by auto\n have \"eventually (λz. z ∈ X) (lift_filter' (λX. X O X) uniformity)\"\n using assms (*‹∀⇩F z::'a × 'a in uniformity. z ∈ (X::('a × 'a) set)›*) comp_uniformity_le_uniformity (*‹lift_filter' (λX. X O X) uniformity ≤ uniformity›*) using filter_leD (*‹⟦?F ≤ ?F'; eventually ?P ?F'⟧ ⟹ eventually ?P ?F›*) by blast\n thus \"?thesis\"\n (*goal: ‹thesis›*)\n using that (*‹⟦∀⇩F z in uniformity. z ∈ ?Y; ?Y O ?Y ⊆ X⟧ ⟹ thesis›*) by (auto simp: eventually_lift_filter'_iff (*‹mono ?g ⟹ eventually ?P (lift_filter' ?g ?F) = (∃X. (∀⇩F x in ?F. x ∈ X) ∧ (∀x∈?g X. ?P x))›*))\nqed\n\nlemma (in uniform_space) le_nhds_of_cauchy_adhp_aux:\n assumes \"⋀P. eventually P uniformity ⟹ (∃X. eventually (λy. y ∈ X) F ∧ (∀z∈X × X. P z) ∧ (∃y. P (x, y) ∧ y ∈ X))\"\n shows \"F ≤ nhds x\"\n unfolding le_filter_def\n (*goal: ‹∀P. eventually P (nhds x) ⟶ eventually P F›*)\n proof (safe)\n (*goal: ‹⋀P. eventually P (nhds x) ⟹ eventually P F›*)\n fix P\n assume \"eventually P (nhds x)\" (*‹eventually (P::'a ⇒ bool) (nhds (x::'a))›*)\n hence \"∀⇩F z in uniformity. z ∈ {z. fst z = x ⟶ P (snd z)}\"\n by (simp add: eventually_nhds_uniformity (*‹eventually ?P (nhds ?x) = (∀⇩F (x', y) in uniformity. x' = ?x ⟶ ?P y)›*) case_prod_unfold (*‹case_prod = (λc p. c (fst p) (snd p))›*))\n then obtain Y where Y: \"∀⇩F z in uniformity. z ∈ Y\" \"Y O Y ⊆ {z. fst z = x ⟶ P (snd z)}\"\n (*goal: ‹(⋀Y. ⟦∀⇩F z in uniformity. z ∈ Y; Y O Y ⊆ {z. fst z = x ⟶ P (snd z)}⟧ ⟹ thesis) ⟹ thesis›*)\n using comp_mem_uniformity_sets (*‹⟦∀⇩F z in uniformity. z ∈ ?X; ⋀Y. ⟦∀⇩F z in uniformity. z ∈ Y; Y O Y ⊆ ?X⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) by blast\n obtain X and y where Xy: \"eventually (λy. y ∈ X) F\" \"X×X ⊆ Y\" \"(x, y) ∈ Y\" \"y ∈ X\"\n (*goal: ‹(⋀X y. ⟦∀⇩F y in F. y ∈ X; X × X ⊆ Y; (x, y) ∈ Y; y ∈ X⟧ ⟹ thesis) ⟹ thesis›*)\n using assms[OF Y ( 1 )] (*‹∃X::'a::type set. (∀⇩F y::'a::type in F::'a::type filter. y ∈ X) ∧ (∀z::'a::type × 'a::type∈X × X. z ∈ (Y::('a::type × 'a::type) set)) ∧ (∃y::'a::type. (x::'a::type, y) ∈ Y ∧ y ∈ X)›*) by blast\n have \"*\": \"P x\" if \"x ∈ X\" for x\n using Y(2) (*‹Y O Y ⊆ {z. fst z = x ⟶ P (snd z)}›*) Xy(2-4) (*‹X × X ⊆ Y› ‹(x, y) ∈ Y› ‹y ∈ X›*) that (*‹x ∈ X›*) unfolding relcomp_unfold\n (*goal: ‹P x›*)\n by force\n show \"eventually P F\"\n apply (rule eventually_mono[OF Xy(1)] (*‹(⋀x::'a::type. x ∈ (X::'a::type set) ⟹ (?Q::'a::type ⇒ bool) x) ⟹ eventually ?Q (F::'a::type filter)›*))\n (*goal: ‹eventually (P::'a ⇒ bool) (F::'a filter)›*)\n by (use * in auto)\n qed\n\nlemma (in uniform_space) eventually_uniformity_imp_nhds:\n assumes \"eventually P uniformity\"\n shows \"eventually (λy. P (x, y)) (nhds x)\"\n using assms (*‹eventually (P::'a × 'a ⇒ bool) uniformity›*) unfolding eventually_nhds_uniformity\n (*goal: ‹∀⇩F (x', y) in uniformity. x' = x ⟶ P (x, y)›*)\n apply (elim eventually_mono (*‹⟦eventually (?P::?'a ⇒ bool) (?F::?'a filter); ⋀x::?'a. ?P x ⟹ (?Q::?'a ⇒ bool) x⟧ ⟹ eventually ?Q ?F›*))\n (*goal: ‹∀⇩F (x', y) in uniformity. x' = x ⟶ P (x, y)›*)\n by auto\n\nlemma (in uniform_space) controlled_sequences_convergent_imp_complete_aux:\n fixes U :: \"nat ⇒ ('a × 'a) set\"\n assumes gen: \"countably_generated_filter (uniformity :: ('a × 'a) filter)\"\n assumes U: \"⋀n. eventually (λz. z ∈ U n) uniformity\"\n assumes conv: \"⋀(u :: nat ⇒ 'a). (⋀N m n. N ≤ m ⟹ N ≤ n ⟹ (u m, u n) ∈ U N) ⟹ convergent u\"\n assumes \"cauchy_filter F\"\n shows \"convergent_filter F\"\nproof (cases \"F = bot\")\n (*goals:\n 1. ‹F = bot ⟹ convergent_filter F›\n 2. ‹F ≠ bot ⟹ convergent_filter F›*)\n case False (*‹F ≠ bot›*)\n note F = ‹cauchy_filter F› (*‹cauchy_filter F›*) ‹F ≠ bot› (*‹F ≠ bot›*)\n from gen (*‹countably_generated_filter uniformity›*) obtain B :: \"nat ⇒ ('a × 'a) set\" where B: \"antimono B\" \"uniformity = (INF n. principal (B n))\" \"⋀P. eventually P uniformity ⟷ (∃i. ∀x∈B i. P x)\"\n (*goal: ‹(⋀B::nat ⇒ ('a::type × 'a::type) set. ⟦monotone (≤) (λ(x::('a::type × 'a::type) set) y::('a::type × 'a::type) set. y ⊆ x) B; uniformity = (INF n::nat. principal (B n)); ⋀P::'a::type × 'a::type ⇒ bool. eventually P uniformity = (∃i::nat. ∀x::'a::type × 'a::type∈B i. P x)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using countably_generated_filter_has_antimono_basis (*‹⟦countably_generated_filter (?F::?'a::type filter); ⋀B::nat ⇒ ?'a::type set. ⟦monotone (≤) (λ(x::?'a::type set) y::?'a::type set. y ⊆ x) B; ?F = (INF n::nat. principal (B n)); ⋀P::?'a::type ⇒ bool. eventually P ?F = (∃i::nat. ∀x::?'a::type∈B i. P x)⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*) by blast\n have ev_B: \"eventually (λz. z ∈ B n) uniformity\" for n\n apply (subst B( (*‹eventually ?P uniformity = (∃i. ∀x∈B i. ?P x)›*) 3))\n (*goal: ‹∀⇩F z in uniformity. z ∈ B n›*)\n by auto\n hence ev_B': \"eventually (λz. z ∈ B n ∩ U n) uniformity\" for n\n using U (*‹∀⇩F z::'a × 'a in uniformity. z ∈ (U::nat ⇒ ('a × 'a) set) (?n::nat)›*) by (auto intro: eventually_conj (*‹⟦eventually ?P ?F; eventually ?Q ?F⟧ ⟹ ∀⇩F x in ?F. ?P x ∧ ?Q x›*))\n obtain g and G where gG: \"antimono G\" \"⋀n. g n ∈ G n\" \"⋀n. eventually (λx. x ∈ G n) F\" \"⋀n. G n × G n ⊆ B n ∩ U n\"\n (*goal: ‹(⋀G g. ⟦monotone (≤) (λx y. y ⊆ x) G; ⋀n. g n ∈ G n; ⋀n. ∀⇩F x in F. x ∈ G n; ⋀n. G n × G n ⊆ B n ∩ U n⟧ ⟹ thesis) ⟹ thesis›*)\n using controlled_sequences_convergent_imp_complete_aux_sequence[of F \"λn. B n ∩ U n\", OF F ev_B'] (*‹(⋀g G. ⟦monotone (≤) (λx y. y ⊆ x) G; ⋀n. g n ∈ G n; ⋀n. ∀⇩F x in F. x ∈ G n; ⋀n. G n × G n ⊆ B n ∩ U n⟧ ⟹ ?thesis) ⟹ ?thesis›*) by metis\n have \"convergent g\"\n proof (rule conv (*‹(⋀N m n. ⟦N ≤ m; N ≤ n⟧ ⟹ (?u m, ?u n) ∈ U N) ⟹ convergent ?u›*))\n (*goal: ‹⋀N m n. ⟦N ≤ m; N ≤ n⟧ ⟹ (g m, g n) ∈ U N›*)\n fix N :: nat and m :: nat and n :: nat\n assume mn: \"N ≤ m\" \"N ≤ n\" (*‹(N::nat) ≤ (m::nat)› ‹(N::nat) ≤ (n::nat)›*)\n have \"(g m, g n) ∈ G m × G n\"\n using gG (*‹monotone (≤) (λx y. y ⊆ x) G› ‹g ?n ∈ G ?n› ‹∀⇩F x in F. x ∈ G ?n› ‹G ?n × G ?n ⊆ B ?n ∩ U ?n›*) by auto\n also (*calculation: ‹(g m, g n) ∈ G m × G n›*) from mn (*‹(N::nat) ≤ (m::nat)› ‹N ≤ n›*) have \"… ⊆ G N × G N\"\n apply (intro Sigma_mono (*‹⟦?A ⊆ ?C; ⋀x. x ∈ ?A ⟹ ?B x ⊆ ?D x⟧ ⟹ Sigma ?A ?B ⊆ Sigma ?C ?D›*) gG (*‹monotone (≤) (λx y. y ⊆ x) G› ‹g ?n ∈ G ?n› ‹∀⇩F x in F. x ∈ G ?n› ‹G ?n × G ?n ⊆ B ?n ∩ U ?n›*) antimonoD[OF gG(1)] (*‹?x ≤ ?y ⟹ G ?y ⊆ G ?x›*))\n (*goals:\n 1. ‹⟦N ≤ m; N ≤ n⟧ ⟹ N ≤ m›\n 2. ‹⋀x. ⟦N ≤ m; N ≤ n; x ∈ G m⟧ ⟹ N ≤ n›\n discuss goal 1*)\n apply ((assumption)[1])\n (*discuss goal 2*) .\n (*proven 2 subgoals*)\n also (*calculation: ‹(g m, g n) ∈ G N × G N›*) have \"… ⊆ U N\"\n using gG (*‹monotone (≤) (λx y. y ⊆ x) G› ‹g ?n ∈ G ?n› ‹∀⇩F x in F. x ∈ G ?n› ‹G ?n × G ?n ⊆ B ?n ∩ U ?n›*) by blast\n finally (*calculation: ‹(g m, g n) ∈ U N›*) show \"(g m, g n) ∈ U N\" .\n qed\n then obtain L where G: \"g ⇢ L\"\n (*goal: ‹(⋀L. g ⇢ L ⟹ thesis) ⟹ thesis›*)\n unfolding convergent_def\n (*goal: ‹(⋀L. g ⇢ L ⟹ thesis) ⟹ thesis›*)\n by blast\n have \"F ≤ nhds L\"\n proof (rule le_nhds_of_cauchy_adhp_aux (*‹(⋀P::'a × 'a ⇒ bool. eventually P uniformity ⟹ ∃X::'a set. (∀⇩F y::'a in ?F::'a filter. y ∈ X) ∧ (∀z::'a × 'a∈X × X. P z) ∧ (∃y::'a. P (?x::'a, y) ∧ y ∈ X)) ⟹ ?F ≤ nhds ?x›*))\n (*goal: ‹⋀P::'a × 'a ⇒ bool. eventually P uniformity ⟹ ∃X::'a set. (∀⇩F y::'a in F::'a filter. y ∈ X) ∧ (∀z::'a × 'a∈X × X. P z) ∧ (∃y::'a. P (L::'a, y) ∧ y ∈ X)›*)\n fix P :: \"'a × 'a ⇒ bool\"\n assume P: \"eventually P uniformity\" (*‹eventually (P::'a × 'a ⇒ bool) uniformity›*)\n hence \"eventually (λn. ∀x∈B n. P x) sequentially\"\n using ‹antimono B› (*‹monotone (≤) (λx y. y ⊆ x) B›*) unfolding B(3) eventually_sequentially decseq_def\n (*goal: ‹∃N. ∀n≥N. Ball (B n) P›*)\n by blast\n moreover have \"eventually (λn. P (L, g n)) sequentially\"\n using P (*‹eventually P uniformity›*) eventually_compose_filterlim (*‹⟦eventually ?P ?F; filterlim ?f ?F ?G⟧ ⟹ ∀⇩F x in ?G. ?P (?f x)›*) eventually_uniformity_imp_nhds (*‹eventually ?P uniformity ⟹ ∀⇩F y in nhds ?x. ?P (?x, y)›*) G (*‹g ⇢ L›*) by blast\n ultimately have \"eventually (λn. (∀x∈B n. P x) ∧ P (L, g n)) sequentially\"\n apply eventually_elim\n (*goal: ‹∀⇩F n::nat in sequentially. (∀x::'a × 'a∈(B::nat ⇒ ('a × 'a) set) n. (P::'a × 'a ⇒ bool) x) ∧ P (L::'a, (g::nat ⇒ 'a) n)›*)\n by auto\n then obtain n where \"∀x∈B n. P x\" \"P (L, g n)\"\n (*goal: ‹(⋀n. ⟦∀x∈B n. P x; P (L, g n)⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding eventually_at_top_linorder\n (*goal: ‹(⋀n::nat. ⟦∀x::'a × 'a∈(B::nat ⇒ ('a × 'a) set) n. (P::'a × 'a ⇒ bool) x; P (L::'a, (g::nat ⇒ 'a) n)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by blast\n then show \"∃X. (∀⇩F y in F. y ∈ X) ∧ (∀z∈X × X. P z) ∧ (∃y. P (L, y) ∧ y ∈ X)\"\n using gG (*‹monotone (≤) (λx y. y ⊆ x) G› ‹g ?n ∈ G ?n› ‹∀⇩F x in F. x ∈ G ?n› ‹G ?n × G ?n ⊆ B ?n ∩ U ?n›*) by blast\n qed\n thus \"convergent_filter F\"\n by (auto simp: convergent_filter_iff (*‹convergent_filter ?F = (∃x. ?F ≤ nhds x)›*))\nqed (auto)\n(*solved the remaining goal: ‹F = bot ⟹ convergent_filter F›*)\n\ntheorem (in uniform_space) controlled_sequences_convergent_imp_complete:\n fixes U :: \"nat ⇒ ('a × 'a) set\"\n assumes gen: \"countably_generated_filter (uniformity :: ('a × 'a) filter)\"\n assumes U: \"⋀n. eventually (λz. z ∈ U n) uniformity\"\n assumes conv: \"⋀(u :: nat ⇒ 'a). (⋀N m n. N ≤ m ⟹ N ≤ n ⟹ (u m, u n) ∈ U N) ⟹ convergent u\"\n shows \"class.complete_uniform_space open uniformity\"\n apply unfold_locales\n (*goal: ‹class.complete_uniform_space open uniformity›*)\n by (use assms controlled_sequences_convergent_imp_complete_aux in blast)\n\nlemma filtermap_prod_filter: \"filtermap (map_prod f g) (F ×⇩F G) = filtermap f F ×⇩F filtermap g G\"\nproof (intro antisym (*‹⟦(?a::?'a) ≤ (?b::?'a); ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹filtermap (map_prod f g) (F ×⇩F G) ≤ filtermap f F ×⇩F filtermap g G›\n 2. ‹filtermap f F ×⇩F filtermap g G ≤ filtermap (map_prod f g) (F ×⇩F G)›*)\n show \"filtermap (map_prod f g) (F ×⇩F G) ≤ filtermap f F ×⇩F filtermap g G\"\n by (auto simp: le_filter_def (*‹(?F ≤ ?F') = (∀P. eventually P ?F' ⟶ eventually P ?F)›*) eventually_filtermap (*‹eventually ?P (filtermap ?f ?F) = (∀⇩F x in ?F. ?P (?f x))›*) eventually_prod_filter (*‹eventually ?P (?F ×⇩F ?G) = (∃Pf Pg. eventually Pf ?F ∧ eventually Pg ?G ∧ (∀x y. Pf x ⟶ Pg y ⟶ ?P (x, y)))›*))\nnext\n (*goal: ‹filtermap f F ×⇩F filtermap g G ≤ filtermap (map_prod f g) (F ×⇩F G)›*)\n show \"filtermap f F ×⇩F filtermap g G ≤ filtermap (map_prod f g) (F ×⇩F G)\"\n unfolding le_filter_def\n (*goal: ‹∀P. eventually P (filtermap (map_prod f g) (F ×⇩F G)) ⟶ eventually P (filtermap f F ×⇩F filtermap g G)›*)\n proof (safe)\n (*goal: ‹⋀P. eventually P (filtermap (map_prod f g) (F ×⇩F G)) ⟹ eventually P (filtermap f F ×⇩F filtermap g G)›*)\n fix P\n assume P: \"eventually P (filtermap (map_prod f g) (F ×⇩F G))\" (*‹eventually (P::'a × 'b ⇒ bool) (filtermap (map_prod (f::'c ⇒ 'a) (g::'d ⇒ 'b)) ((F::'c filter) ×⇩F (G::'d filter)))›*)\n then obtain Pf and Pg where \"*\": \"eventually Pf F\" \"eventually Pg G\" \"∀x. Pf x ⟶ (∀y. Pg y ⟶ P (f x, g y))\"\n (*goal: ‹(⋀Pf Pg. ⟦eventually Pf F; eventually Pg G; ∀x. Pf x ⟶ (∀y. Pg y ⟶ P (f x, g y))⟧ ⟹ thesis) ⟹ thesis›*)\n by (auto simp: eventually_filtermap (*‹eventually ?P (filtermap ?f ?F) = (∀⇩F x in ?F. ?P (?f x))›*) eventually_prod_filter (*‹eventually ?P (?F ×⇩F ?G) = (∃Pf Pg. eventually Pf ?F ∧ eventually Pg ?G ∧ (∀x y. Pf x ⟶ Pg y ⟶ ?P (x, y)))›*))\n define Pf' where \"Pf' = (λx. ∃y. x = f y ∧ Pf y)\"\n define Pg' where \"Pg' = (λx. ∃y. x = g y ∧ Pg y)\"\n from \"*\"(1) (*‹eventually Pf F›*) have \"∀⇩F x in F. Pf' (f x)\"\n apply eventually_elim\n (*goal: ‹∀⇩F x in F. Pf' (f x)›*)\n by (auto simp: Pf'_def (*‹(Pf'::'a::type ⇒ bool) = (λx::'a::type. ∃y::'c::type. x = (f::'c::type ⇒ 'a::type) y ∧ (Pf::'c::type ⇒ bool) y)›*))\n moreover from \"*\"(2) (*‹eventually Pg G›*) have \"∀⇩F x in G. Pg' (g x)\"\n apply eventually_elim\n (*goal: ‹∀⇩F x in G. Pg' (g x)›*)\n by (auto simp: Pg'_def (*‹Pg' = (λx. ∃y. x = g y ∧ Pg y)›*))\n moreover have \"(∀x y. Pf' x ⟶ Pg' y ⟶ P (x, y))\"\n using \"*\"(3) (*‹∀x. Pf x ⟶ (∀y. Pg y ⟶ P (f x, g y))›*) by (auto simp: Pf'_def (*‹Pf' = (λx. ∃y. x = f y ∧ Pf y)›*) Pg'_def (*‹Pg' = (λx. ∃y. x = g y ∧ Pg y)›*))\n ultimately show \"eventually P (filtermap f F ×⇩F filtermap g G)\"\n unfolding eventually_prod_filter eventually_filtermap\n (*goal: ‹∃(Pf::'a ⇒ bool) Pg::'b ⇒ bool. (∀⇩F x::'c in F::'c filter. Pf ((f::'c ⇒ 'a) x)) ∧ (∀⇩F x::'d in G::'d filter. Pg ((g::'d ⇒ 'b) x)) ∧ (∀(x::'a) y::'b. Pf x ⟶ Pg y ⟶ (P::'a × 'b ⇒ bool) (x, y))›*)\n by blast\n qed\nqed\n \n\nlemma (in uniform_space) Cauchy_seq_iff_tendsto:\n \"Cauchy f ⟷ filterlim (map_prod f f) uniformity (at_top ×⇩F at_top)\"\n unfolding Cauchy_uniform cauchy_filter_def filterlim_def filtermap_prod_filter\n (*goal: ‹(filtermap (f::nat ⇒ 'a::type) sequentially ×⇩F filtermap f sequentially ≤ uniformity) = (filtermap f sequentially ×⇩F filtermap f sequentially ≤ uniformity)›*)\n by standard\n\ntheorem (in uniform_space) controlled_seq_imp_Cauchy_seq:\n fixes U :: \"nat ⇒ ('a × 'a) set\"\n assumes U: \"⋀P. eventually P uniformity ⟹ (∃n. ∀x∈U n. P x)\"\n assumes controlled: \"⋀N m n. N ≤ m ⟹ N ≤ n ⟹ (f m, f n) ∈ U N\"\n shows \"Cauchy f\"\n unfolding Cauchy_seq_iff_tendsto\n (*goal: ‹filterlim (map_prod f f) uniformity (sequentially ×⇩F sequentially)›*)\n proof (-)\n (*goal: ‹filterlim (map_prod (f::nat ⇒ 'a) f) uniformity (sequentially ×⇩F sequentially)›*)\n show \"filterlim (map_prod f f) uniformity (sequentially ×⇩F sequentially)\"\n unfolding filterlim_def le_filter_def\n (*goal: ‹∀P::'a::type × 'a::type ⇒ bool. eventually P uniformity ⟶ eventually P (filtermap (map_prod (f::nat ⇒ 'a::type) f) (sequentially ×⇩F sequentially))›*)\n proof (safe)\n (*goal: ‹⋀P. eventually P uniformity ⟹ eventually P (filtermap (map_prod f f) (sequentially ×⇩F sequentially))›*)\n fix P :: \"'a × 'a ⇒ bool\"\n assume P: \"eventually P uniformity\" (*‹eventually (P::'a × 'a ⇒ bool) uniformity›*)\n from U[OF this] (*‹∃n. ∀x∈U n. P x›*) obtain N where \"∀x∈U N. P x\"\n (*goal: ‹(⋀N. ∀x∈U N. P x ⟹ thesis) ⟹ thesis›*)\n by blast\n then show \"eventually P (filtermap (map_prod f f) (sequentially ×⇩F sequentially))\"\n unfolding eventually_filtermap eventually_prod_sequentially\n (*goal: ‹∃N. ∀m≥N. ∀n≥N. P (map_prod f f (n, m))›*)\n by (metis controlled (*‹⟦?N ≤ ?m; ?N ≤ ?n⟧ ⟹ (f ?m, f ?n) ∈ U ?N›*) map_prod_simp (*‹map_prod ?f ?g (?a, ?b) = (?f ?a, ?g ?b)›*))\n qed\n qed\n\nlemma (in uniform_space) Cauchy_seq_convergent_imp_complete_aux:\n fixes U :: \"nat ⇒ ('a × 'a) set\"\n assumes gen: \"countably_generated_filter (uniformity :: ('a × 'a) filter)\"\n assumes conv: \"⋀(u :: nat ⇒ 'a). Cauchy u ⟹ convergent u\"\n assumes \"cauchy_filter F\"\n shows \"convergent_filter F\"\nproof (-)\n (*goal: ‹convergent_filter (F::'a filter)›*)\n from gen (*‹countably_generated_filter uniformity›*) obtain B :: \"nat ⇒ ('a × 'a) set\" where B: \"antimono B\" \"uniformity = (INF n. principal (B n))\" \"⋀P. eventually P uniformity ⟷ (∃i. ∀x∈B i. P x)\"\n (*goal: ‹(⋀B. ⟦monotone (≤) (λx y. y ⊆ x) B; uniformity = (INF n. principal (B n)); ⋀P. eventually P uniformity = (∃i. ∀x∈B i. P x)⟧ ⟹ thesis) ⟹ thesis›*)\n using countably_generated_filter_has_antimono_basis (*‹⟦countably_generated_filter ?F; ⋀B. ⟦monotone (≤) (λx y. y ⊆ x) B; ?F = (INF n. principal (B n)); ⋀P. eventually P ?F = (∃i. ∀x∈B i. P x)⟧ ⟹ ?thesis⟧ ⟹ ?thesis›*) by blast\n show \"?thesis\"\n (*goal: ‹convergent_filter F›*)\n proof (rule controlled_sequences_convergent_imp_complete_aux[where U = B] (*‹⟦countably_generated_filter uniformity; ⋀n. ∀⇩F z in uniformity. z ∈ B n; ⋀u. (⋀N m n. ⟦N ≤ m; N ≤ n⟧ ⟹ (u m, u n) ∈ B N) ⟹ convergent u; cauchy_filter ?F⟧ ⟹ convergent_filter ?F›*))\n (*goals:\n 1. ‹countably_generated_filter uniformity›\n 2. ‹⋀n::nat. ∀⇩F z::'a::type × 'a::type in uniformity. z ∈ (B::nat ⇒ ('a::type × 'a::type) set) n›\n 3. ‹⋀u::nat ⇒ 'a::type. (⋀(N::nat) (m::nat) n::nat. ⟦N ≤ m; N ≤ n⟧ ⟹ (u m, u n) ∈ (B::nat ⇒ ('a::type × 'a::type) set) N) ⟹ convergent u›\n 4. ‹cauchy_filter (F::'a::type filter)›*)\n show \"∀⇩F z in uniformity. z ∈ B n\" for n\n unfolding B(3)\n (*goal: ‹∃i. ∀z∈B i. z ∈ B n›*)\n by blast\n next\n (*goals:\n 1. ‹countably_generated_filter uniformity›\n 2. ‹⋀u. (⋀N m n. ⟦N ≤ m; N ≤ n⟧ ⟹ (u m, u n) ∈ B N) ⟹ convergent u›\n 3. ‹cauchy_filter F›*)\n fix f :: \"nat ⇒ 'a\"\n assume f: \"⋀N m n. N ≤ m ⟹ N ≤ n ⟹ (f m, f n) ∈ B N\" (*‹⟦(?N::nat) ≤ (?m::nat); ?N ≤ (?n::nat)⟧ ⟹ ((f::nat ⇒ 'a) ?m, f ?n) ∈ (B::nat ⇒ ('a × 'a) set) ?N›*)\n have \"Cauchy f\"\n using f (*‹⟦?N ≤ ?m; ?N ≤ ?n⟧ ⟹ (f ?m, f ?n) ∈ B ?N›*) B (*‹monotone (≤) (λx y. y ⊆ x) B› ‹uniformity = (INF n. principal (B n))› ‹eventually ?P uniformity = (∃i. ∀x∈B i. ?P x)›*) apply (intro controlled_seq_imp_Cauchy_seq[where U = B] (*‹⟦⋀P::'a × 'a ⇒ bool. eventually P uniformity ⟹ ∃n::nat. ∀x::'a × 'a∈(B::nat ⇒ ('a × 'a) set) n. P x; ⋀(N::nat) (m::nat) n::nat. ⟦N ≤ m; N ≤ n⟧ ⟹ ((?f::nat ⇒ 'a) m, ?f n) ∈ B N⟧ ⟹ Cauchy ?f›*))\n (*goals:\n 1. ‹⋀P. ⟦⋀N m n. ⟦N ≤ m; N ≤ n⟧ ⟹ (f m, f n) ∈ B N; monotone (≤) (λx y. y ⊆ x) B; uniformity = (INF n. principal (B n)); ⋀P. eventually P uniformity = (∃i. ∀x∈B i. P x); eventually P uniformity⟧ ⟹ ∃n. ∀x∈B n. P x›\n 2. ‹⋀N m n. ⟦⋀N m n. ⟦N ≤ m; N ≤ n⟧ ⟹ (f m, f n) ∈ B N; monotone (≤) (λx y. y ⊆ x) B; uniformity = (INF n. principal (B n)); ⋀P. eventually P uniformity = (∃i. ∀x∈B i. P x); N ≤ m; N ≤ n⟧ ⟹ (f m, f n) ∈ B N›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n with conv (*‹Cauchy ?u ⟹ convergent ?u›*) show \"convergent f\"\n by simp\n qed ((fact)+)\n (*solves the remaining goals:\n 1. ‹countably_generated_filter uniformity›\n 2. ‹cauchy_filter F›*)\nqed\n\ntheorem (in uniform_space) Cauchy_seq_convergent_imp_complete:\n fixes U :: \"nat ⇒ ('a × 'a) set\"\n assumes gen: \"countably_generated_filter (uniformity :: ('a × 'a) filter)\"\n assumes conv: \"⋀(u :: nat ⇒ 'a). Cauchy u ⟹ convergent u\"\n shows \"class.complete_uniform_space open uniformity\"\n apply unfold_locales\n (*goal: ‹class.complete_uniform_space open uniformity›*)\n by (use assms Cauchy_seq_convergent_imp_complete_aux in blast)\n\nlemma (in metric_space) countably_generated_uniformity:\n \"countably_generated_filter uniformity\"\nproof (-)\n (*goal: ‹countably_generated_filter uniformity›*)\n have \"(INF e∈{0<..}. principal {(x, y). dist (x::'a) y < e}) =\n (INF n∈UNIV. principal {(x, y). dist x y < 1 / real (Suc n)})\" (is \"?F = ?G\")\n unfolding uniformity_dist\n (*goal: ‹(INF e::real∈{0::real<..}. principal {(x::'a::type, y::'a::type). dist x y < e}) = (INF n::nat. principal {(x::'a::type, y::'a::type). dist x y < (1::real) / real (Suc n)})›*)\n proof (intro antisym (*‹⟦?a ≤ ?b; ?b ≤ ?a⟧ ⟹ ?a = ?b›*))\n (*goals:\n 1. ‹(INF e∈{0<..}. principal {(x, y). dist x y < e}) ≤ (INF n. principal {(x, y). dist x y < 1 / real (Suc n)})›\n 2. ‹(INF n. principal {(x, y). dist x y < 1 / real (Suc n)}) ≤ (INF e∈{0<..}. principal {(x, y). dist x y < e})›*)\n have \"?G = (INF e∈(λn. 1 / real (Suc n)) ` UNIV. principal {(x, y). dist x y < e})\"\n by (simp add: image_image (*‹(?f::?'b::type ⇒ ?'a::type) ` (?g::?'c::type ⇒ ?'b::type) ` (?A::?'c::type set) = (λx::?'c::type. ?f (?g x)) ` ?A›*))\n also (*calculation: ‹(INF n. principal {(x, y). dist x y < 1 / real (Suc n)}) = (INF e∈range (λn. 1 / real (Suc n)). principal {(x, y). dist x y < e})›*) have \"… ≥ ?F\"\n apply (intro INF_superset_mono (*‹⟦?B ⊆ ?A; ⋀x. x ∈ ?B ⟹ ?f x ≤ ?g x⟧ ⟹ Inf (?f ` ?A) ≤ Inf (?g ` ?B)›*))\n (*goals:\n 1. ‹range (λn. 1 / real (Suc n)) ⊆ {0<..}›\n 2. ‹⋀e. e ∈ range (λn. 1 / real (Suc n)) ⟹ principal {(x, y). dist x y < e} ≤ principal {(x, y). dist x y < e}›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n finally (*calculation: ‹(INF e∈{0<..}. principal {a. case a of (x, y) ⇒ dist x y < e}) ≤ (INF n. principal {(x, y). dist x y < 1 / real (Suc n)})›*) show \"?F ≤ ?G\" .\n next\n (*goal: ‹(INF n::nat. principal {(x::'a, y::'a). dist x y < (1::real) / real (Suc n)}) ≤ (INF e::real∈{0::real<..}. principal {(x::'a, y::'a). dist x y < e})›*)\n show \"?G ≤ ?F\"\n unfolding le_filter_def\n (*goal: ‹∀P. eventually P (INF e∈{0<..}. principal {(x, y). dist x y < e}) ⟶ eventually P (INF n. principal {(x, y). dist x y < 1 / real (Suc n)})›*)\n proof (safe)\n (*goal: ‹⋀P::'a::type × 'a::type ⇒ bool. eventually P (INF e::real∈{0::real<..}. principal {(x::'a::type, y::'a::type). dist x y < e}) ⟹ eventually P (INF n::nat. principal {(x::'a::type, y::'a::type). dist x y < (1::real) / real (Suc n)})›*)\n fix P\n assume \"eventually P ?F\" (*‹eventually (P::'a × 'a ⇒ bool) (INF e::real∈{0::real<..}. principal {(x::'a, y::'a). dist x y < e})›*)\n then obtain ε where \"ε\": \"ε > 0\" \"eventually P (principal {(x, y). dist x y < ε})\"\n (*goal: ‹(⋀ε. ⟦0 < ε; eventually P (principal {(x, y). dist x y < ε})⟧ ⟹ thesis) ⟹ thesis›*)\n proof (subst (asm) eventually_INF_base (*‹⟦?B ≠ {}; ⋀a b. ⟦a ∈ ?B; b ∈ ?B⟧ ⟹ ∃x∈?B. ?F x ≤ inf (?F a) (?F b)⟧ ⟹ eventually ?P (Inf (?F ` ?B)) = (∃b∈?B. eventually ?P (?F b))›*), goal_cases)\n (*goals:\n 1. ‹(⋀ε::real. ⟦(0::real) < ε; eventually (P::'a × 'a ⇒ bool) (principal {(x::'a, y::'a). dist x y < ε})⟧ ⟹ thesis::bool) ⟹ {0::real<..} ≠ {}›\n 2. ‹⋀(a::real) b::real. ⟦⋀ε::real. ⟦(0::real) < ε; eventually (P::'a × 'a ⇒ bool) (principal {(x::'a, y::'a). dist x y < ε})⟧ ⟹ thesis::bool; a ∈ {0::real<..}; b ∈ {0::real<..}⟧ ⟹ ∃x::real∈{0::real<..}. principal {(xa::'a, y::'a). dist xa y < x} ≤ inf (principal {(x::'a, y::'a). dist x y < a}) (principal {(x::'a, y::'a). dist x y < b})›\n 3. ‹⟦⋀ε::real. ⟦(0::real) < ε; eventually (P::'a × 'a ⇒ bool) (principal {(x::'a, y::'a). dist x y < ε})⟧ ⟹ thesis::bool; ∃b::real∈{0::real<..}. eventually P (principal {(x::'a, y::'a). dist x y < b})⟧ ⟹ thesis›*)\n case (2 ε1 ε2) (*‹⟦0 < ?ε; eventually P (principal {(x, y). dist x y < ?ε})⟧ ⟹ thesis› ‹ε1 ∈ {0<..}› ‹ε2 ∈ {0<..}›*)\n thus \"?case\"\n (*goal: ‹∃x∈{0<..}. principal {(xa, y). dist xa y < x} ≤ inf (principal {(x, y). dist x y < ε1}) (principal {(x, y). dist x y < ε2})›*)\n apply (intro bexI[of _ \"min ε1 ε2\"] (*‹⟦?P (min ε1 ε2); min ε1 ε2 ∈ ?A⟧ ⟹ ∃x∈?A. ?P x›*))\n (*goals:\n 1. ‹⟦⋀ε. ⟦0 < ε; eventually P (principal {(x, y). dist x y < ε})⟧ ⟹ thesis; ε1 ∈ {0<..}; ε2 ∈ {0<..}⟧ ⟹ principal {(x, y). dist x y < min ε1 ε2} ≤ inf (principal {(x, y). dist x y < ε1}) (principal {(x, y). dist x y < ε2})›\n 2. ‹⟦⋀ε. ⟦0 < ε; eventually P (principal {(x, y). dist x y < ε})⟧ ⟹ thesis; ε1 ∈ {0<..}; ε2 ∈ {0<..}⟧ ⟹ min ε1 ε2 ∈ {0<..}›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n qed (auto)\n (*solves the remaining goals:\n 1. ‹(⋀ε. ⟦0 < ε; eventually P (principal {(x, y). dist x y < ε})⟧ ⟹ thesis) ⟹ {0<..} ≠ {}›\n 2. ‹⟦⋀ε. ⟦0 < ε; eventually P (principal {(x, y). dist x y < ε})⟧ ⟹ thesis; ∃b∈{0<..}. eventually P (principal {(x, y). dist x y < b})⟧ ⟹ thesis›*)\n from ‹ε > 0› (*‹0 < ε›*) obtain n where \"1 / real (Suc n) < ε\"\n (*goal: ‹(⋀n. 1 / real (Suc n) < ε ⟹ thesis) ⟹ thesis›*)\n using nat_approx_posE (*‹⟦0 < ?e; ⋀n. 1 / of_nat (Suc n) < ?e ⟹ ?thesis⟧ ⟹ ?thesis›*) by blast\n then have \"eventually P (principal {(x, y). dist x y < 1 / real (Suc n)})\"\n using \"ε\"(2) (*‹eventually P (principal {(x, y). dist x y < ε})›*) by (auto simp: eventually_principal (*‹eventually (?P::?'a ⇒ bool) (principal (?S::?'a set)) = (∀x::?'a∈?S. ?P x)›*))\n thus \"eventually P ?G\"\n apply (intro eventually_INF1 (*‹⟦(?i::?'a) ∈ (?I::?'a set); eventually (?P::?'b ⇒ bool) ((?F::?'a ⇒ ?'b filter) ?i)⟧ ⟹ eventually ?P (Inf (?F ` ?I))›*))\n (*goals:\n 1. ‹eventually P (principal {(x, y). dist x y < 1 / real (Suc n)}) ⟹ ?n1 ∈ UNIV›\n 2. ‹eventually P (principal {(x, y). dist x y < 1 / real (Suc n)}) ⟹ eventually P (principal {(x, y). dist x y < 1 / real (Suc ?n1)})›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n qed\n qed\n thus \"countably_generated_filter uniformity\"\n unfolding countably_generated_filter_def uniformity_dist\n (*goal: ‹∃U. (INF e∈{0<..}. principal {(x, y). dist x y < e}) = (INF n. principal (U n))›*)\n by fast\nqed\n\nsubclass (in complete_space) complete_uniform_space\nproof (rule Cauchy_seq_convergent_imp_complete)\n show \"convergent f\" if \"Cauchy f\" for f\n using Cauchy_convergent that by blast\nqed (fact countably_generated_uniformity)\n\nlemma (in complete_uniform_space) complete_UNIV_cuspace [intro]: \"complete UNIV\"\n unfolding complete_uniform\n (*goal: ‹∀F≤principal UNIV. F ≠ bot ⟶ cauchy_filter F ⟶ (∃x∈UNIV. F ≤ nhds x)›*)\n using cauchy_filter_convergent (*‹cauchy_filter (?F::'a filter) ⟹ convergent_filter ?F›*) by (auto simp: convergent_filter.simps (*‹convergent_filter ?a = (∃F x. ?a = F ∧ F ≤ nhds x)›*))\n\n\n\nlemma norm_infsum_le:\n assumes \"(f has_sum S) X\"\n assumes \"(g has_sum T) X\"\n assumes \"⋀x. x ∈ X ⟹ norm (f x) ≤ g x\"\n shows \"norm S ≤ T\"\nproof (rule tendsto_le (*‹⟦(?F::?'b filter) ≠ bot; ((?f::?'b ⇒ ?'a) ⤏ (?x::?'a)) ?F; ((?g::?'b ⇒ ?'a) ⤏ (?y::?'a)) ?F; ∀⇩F x::?'b in ?F. ?g x ≤ ?f x⟧ ⟹ ?y ≤ ?x›*))\n (*goals:\n 1. ‹?F ≠ bot›\n 2. ‹(?f ⤏ T) ?F›\n 3. ‹(?g ⤏ norm S) ?F›\n 4. ‹∀⇩F x in ?F. ?g x ≤ ?f x›*)\n show \"((λY. norm (∑x∈Y. f x)) ⤏ norm S) (finite_subsets_at_top X)\"\n using assms(1) (*‹(f has_sum S) X›*) unfolding has_sum_def\n (*goal: ‹((λY. norm (sum f Y)) ⤏ norm S) (finite_subsets_at_top X)›*)\n by (intro tendsto_norm (*‹(?f ⤏ ?a) ?F ⟹ ((λx. norm (?f x)) ⤏ norm ?a) ?F›*))\n show \"((λY. ∑x∈Y. g x) ⤏ T) (finite_subsets_at_top X)\"\n using assms(2) (*‹(g has_sum T) X›*) unfolding has_sum_def\n (*goal: ‹(sum g ⤏ T) (finite_subsets_at_top X)›*) .\n show \"∀⇩F x in finite_subsets_at_top X. norm (sum f x) ≤ (∑x∈x. g x)\"\n by (simp add: assms( (*‹?x ∈ X ⟹ norm (f ?x) ≤ g ?x›*) 3) eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*) subsetD (*‹⟦?A ⊆ ?B; ?c ∈ ?A⟧ ⟹ ?c ∈ ?B›*) sum_norm_le (*‹(⋀x. x ∈ ?S ⟹ norm (?f x) ≤ ?g x) ⟹ norm (sum ?f ?S) ≤ sum ?g ?S›*))\nqed (auto)\n(*solved the remaining goal: ‹finite_subsets_at_top X ≠ bot›*)\n\n(*\nlemma summable_on_Sigma:\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a ⇒ 'b ⇒ 'c::{comm_monoid_add, t2_space, uniform_space}›\n assumes plus_cont: ‹uniformly_continuous_on UNIV (λ(x::'c,y). x+y)›\n assumes summableAB: \"(λ(x,y). f x y) summable_on (Sigma A B)\"\n assumes summableB: ‹⋀x. x∈A ⟹ (f x) summable_on (B x)›\n shows ‹(λx. infsum (f x) (B x)) summable_on A›\n*)\n\nlemma has_sum_imp_summable: \"(f has_sum S) A ⟹ f summable_on A\"\n by (auto simp: summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma has_sum_reindex_bij_betw:\n assumes \"bij_betw g A B\"\n shows \"((λx. f (g x)) has_sum S) A = (f has_sum S) B\"\nproof (-)\n (*goal: ‹((λx. f (g x)) has_sum S) A = (f has_sum S) B›*)\n have \"((λx. f (g x)) has_sum S) A ⟷ (f has_sum S) (g ` A)\"\n apply (subst has_sum_reindex (*‹inj_on (?h::?'a ⇒ ?'b) (?A::?'a set) ⟹ ((?g::?'b ⇒ ?'c) has_sum (?x::?'c)) (?h ` ?A) = (?g ∘ ?h has_sum ?x) ?A›*))\n (*goals:\n 1. ‹inj_on g A›\n 2. ‹((λx. f (g x)) has_sum S) A = (f ∘ g has_sum S) A›\n discuss goal 1*)\n apply ((use assms in ‹auto dest: bij_betw_imp_inj_on simp: o_def›)[1])\n (*discuss goal 2*)\n apply ((use assms in ‹auto dest: bij_betw_imp_inj_on simp: o_def›)[1])\n (*proven 2 subgoals*) .\n then show \"?thesis\"\n (*goal: ‹((λx. f (g x)) has_sum S) A = (f has_sum S) B›*)\n using assms (*‹bij_betw g A B›*) bij_betw_imp_surj_on (*‹bij_betw ?f ?A ?B ⟹ ?f ` ?A = ?B›*) by blast\nqed\n\nlemma has_sum_reindex_bij_witness:\n assumes \"⋀a. a ∈ S ⟹ i (j a) = a\"\n assumes \"⋀a. a ∈ S ⟹ j a ∈ T\"\n assumes \"⋀b. b ∈ T ⟹ j (i b) = b\"\n assumes \"⋀b. b ∈ T ⟹ i b ∈ S\"\n assumes \"⋀a. a ∈ S ⟹ h (j a) = g a\"\n assumes \"s = s'\"\n shows \"(g has_sum s) S = (h has_sum s') T\"\n by (smt (verit, del_insts) assms (*‹?a ∈ S ⟹ i (j ?a) = ?a› ‹?a ∈ S ⟹ j ?a ∈ T› ‹?b ∈ T ⟹ j (i ?b) = ?b› ‹?b ∈ T ⟹ i ?b ∈ S› ‹?a ∈ S ⟹ h (j ?a) = g ?a› ‹s = s'›*) bij_betwI' (*‹⟦⋀x y. ⟦x ∈ ?X; y ∈ ?X⟧ ⟹ (?f x = ?f y) = (x = y); ⋀x. x ∈ ?X ⟹ ?f x ∈ ?Y; ⋀y. y ∈ ?Y ⟹ ∃x∈?X. y = ?f x⟧ ⟹ bij_betw ?f ?X ?Y›*) has_sum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f has_sum ?x) ?A = (?g has_sum ?x) ?A›*) has_sum_reindex_bij_betw (*‹bij_betw ?g ?A ?B ⟹ ((λx. ?f (?g x)) has_sum ?S) ?A = (?f has_sum ?S) ?B›*))\n\n\nlemma has_sum_homomorphism:\n assumes \"(f has_sum S) A\" \"h 0 = 0\" \"⋀a b. h (a + b) = h a + h b\" \"continuous_on UNIV h\"\n shows \"((λx. h (f x)) has_sum (h S)) A\"\nproof (-)\n (*goal: ‹((λx. h (f x)) has_sum h S) A›*)\n have \"sum (h ∘ f) X = h (sum f X)\" for X\n apply (induction X rule: infinite_finite_induct (*‹⟦⋀A::?'a set. infinite A ⟹ (?P::?'a set ⇒ bool) A; ?P {}; ⋀(x::?'a) F::?'a set. ⟦finite F; x ∉ F; ?P F⟧ ⟹ ?P (insert x F)⟧ ⟹ ?P (?A::?'a set)›*))\n (*goals:\n 1. ‹⋀A::'a::type set. infinite A ⟹ sum ((h::'b::{comm_monoid_add,topological_space} ⇒ 'c::{comm_monoid_add,topological_space}) ∘ (f::'a::type ⇒ 'b::{comm_monoid_add,topological_space})) A = h (sum f A)›\n 2. ‹sum ((h::'b::{comm_monoid_add,topological_space} ⇒ 'c::{comm_monoid_add,topological_space}) ∘ (f::'a::type ⇒ 'b::{comm_monoid_add,topological_space})) {} = h (sum f {})›\n 3. ‹⋀(x::'a::type) F::'a::type set. ⟦finite F; x ∉ F; sum ((h::'b::{comm_monoid_add,topological_space} ⇒ 'c::{comm_monoid_add,topological_space}) ∘ (f::'a::type ⇒ 'b::{comm_monoid_add,topological_space})) F = h (sum f F)⟧ ⟹ sum (h ∘ f) (insert x F) = h (sum f (insert x F))›\n discuss goal 1*)\n apply (simp add: assms (*‹(f has_sum S) A› ‹h 0 = 0› ‹h (?a + ?b) = h ?a + h ?b› ‹continuous_on UNIV h›*))\n (*discuss goal 2*)\n apply (simp add: assms (*‹(f has_sum S) A› ‹h 0 = 0› ‹h (?a + ?b) = h ?a + h ?b› ‹continuous_on UNIV h›*))\n (*discuss goal 3*)\n apply (simp add: assms (*‹(f has_sum S) A› ‹h 0 = 0› ‹h (?a + ?b) = h ?a + h ?b› ‹continuous_on UNIV h›*))\n (*proven 3 subgoals*) .\n hence sum_h: \"sum (h ∘ f) = h ∘ sum f\"\n apply (intro ext (*‹(⋀x. ?f x = ?g x) ⟹ ?f = ?g›*))\n (*goal: ‹sum (h ∘ f) = h ∘ sum f›*)\n by auto\n then have \"((h ∘ f) has_sum h S) A\"\n using assms (*‹(f has_sum S) A› ‹h 0 = 0› ‹h (?a + ?b) = h ?a + h ?b› ‹continuous_on UNIV (h::'b::{comm_monoid_add,topological_space} ⇒ 'c::{comm_monoid_add,topological_space})›*) by (metis UNIV_I (*‹?x ∈ UNIV›*) continuous_on_def (*‹continuous_on ?s ?f = (∀x∈?s. (?f ⤏ ?f x) (at x within ?s))›*) has_sum_comm_additive_general (*‹⟦⋀F. ⟦finite F; F ⊆ ?S⟧ ⟹ sum (?f ∘ ?g) F = ?f (sum ?g F); ?f ─?x→ ?f ?x; (?g has_sum ?x) ?S⟧ ⟹ (?f ∘ ?g has_sum ?f ?x) ?S›*) o_apply (*‹(?f ∘ ?g) ?x = ?f (?g ?x)›*))\n thus \"?thesis\"\n (*goal: ‹((λx. h (f x)) has_sum h S) A›*)\n by (simp add: o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\nqed\n\nlemma summable_on_homomorphism:\n assumes \"f summable_on A\" \"h 0 = 0\" \"⋀a b. h (a + b) = h a + h b\" \"continuous_on UNIV h\"\n shows \"(λx. h (f x)) summable_on A\"\nproof (-)\n (*goal: ‹(λx. h (f x)) summable_on A›*)\n from assms(1) (*‹f summable_on A›*) obtain S where \"(f has_sum S) A\"\n (*goal: ‹(⋀S. (f has_sum S) A ⟹ thesis) ⟹ thesis›*)\n by (auto simp: summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n hence \"((λx. h (f x)) has_sum h S) A\"\n apply (rule has_sum_homomorphism (*‹⟦(?f has_sum ?S) ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ ((λx. ?h (?f x)) has_sum ?h ?S) ?A›*))\n (*goals:\n 1. ‹h 0 = 0›\n 2. ‹⋀a b. h (a + b) = h a + h b›\n 3. ‹continuous_on UNIV h›\n discuss goal 1*)\n apply ((use assms in auto)[1])\n (*discuss goal 2*)\n apply ((use assms in auto)[1])\n (*discuss goal 3*)\n apply ((use assms in auto)[1])\n (*proven 3 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹(λx::'a. (h::'b ⇒ 'c) ((f::'a ⇒ 'b) x)) summable_on (A::'a set)›*)\n by (auto simp: summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*))\nqed\n\nlemma infsum_homomorphism_strong:\n fixes h :: \"'a :: {t2_space, topological_comm_monoid_add} ⇒\n 'b :: {t2_space, topological_comm_monoid_add}\"\n assumes \"(λx. h (f x)) summable_on A ⟷ f summable_on A\"\n assumes \"h 0 = 0\" \n assumes \"⋀S. (f has_sum S) A ⟹ ((λx. h (f x)) has_sum (h S)) A\"\n shows \"infsum (λx. h (f x)) A = h (infsum f A)\"\n by (metis assms (*‹((λx. h (f x)) summable_on A) = (f summable_on A)› ‹h 0 = 0› ‹(f has_sum ?S) A ⟹ ((λx. h (f x)) has_sum h ?S) A›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) infsum_not_exists (*‹¬ ?f summable_on ?A ⟹ infsum ?f ?A = 0›*))\n\nlemma has_sum_bounded_linear:\n assumes \"bounded_linear h\" and \"(f has_sum S) A\"\n shows \"((λx. h (f x)) has_sum h S) A\"\nproof (-)\n (*goal: ‹((λx. h (f x)) has_sum h S) A›*)\n interpret bounded_linear h\n by fact\n from assms(2) (*‹(f has_sum S) A›*) show \"?thesis\"\n (*goal: ‹((λx::'c::type. (h::'a::real_normed_vector ⇒ 'b::real_normed_vector) ((f::'c::type ⇒ 'a::real_normed_vector) x)) has_sum h (S::'a::real_normed_vector)) (A::'c::type set)›*)\n apply (rule has_sum_homomorphism (*‹⟦(?f has_sum ?S) ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ ((λx. ?h (?f x)) has_sum ?h ?S) ?A›*))\n (*goals:\n 1. ‹h 0 = 0›\n 2. ‹⋀a b. h (a + b) = h a + h b›\n 3. ‹continuous_on UNIV h›\n discuss goal 1*)\n apply ((auto simp: add (*‹(h::'a ⇒ 'b) ((?b1.0::'a) + (?b2.0::'a)) = h ?b1.0 + h ?b2.0›*) intro!: continuous_on (*‹continuous_on (?s::?'c set) (?g::?'c ⇒ 'a) ⟹ continuous_on ?s (λx::?'c. (h::'a ⇒ 'b) (?g x))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: add (*‹(h::'a ⇒ 'b) ((?b1.0::'a) + (?b2.0::'a)) = h ?b1.0 + h ?b2.0›*) intro!: continuous_on (*‹continuous_on (?s::?'c set) (?g::?'c ⇒ 'a) ⟹ continuous_on ?s (λx::?'c. (h::'a ⇒ 'b) (?g x))›*))[1])\n (*discuss goal 3*)\n apply ((auto simp: add (*‹h (?b1.0 + ?b2.0) = h ?b1.0 + h ?b2.0›*) intro!: continuous_on (*‹continuous_on ?s ?g ⟹ continuous_on ?s (λx. h (?g x))›*))[1])\n (*proven 3 subgoals*) .\nqed\n\nlemma summable_on_bounded_linear:\n assumes \"bounded_linear h\" and \"f summable_on A\"\n shows \"(λx. h (f x)) summable_on A\"\n by (metis assms (*‹bounded_linear h› ‹f summable_on A›*) has_sum_bounded_linear (*‹⟦bounded_linear ?h; (?f has_sum ?S) ?A⟧ ⟹ ((λx. ?h (?f x)) has_sum ?h ?S) ?A›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma summable_on_bounded_linear_iff:\n assumes \"bounded_linear h\" and \"bounded_linear h'\" and \"⋀x. h' (h x) = x\"\n shows \"(λx. h (f x)) summable_on A ⟷ f summable_on A\"\n by (metis (full_types) assms (*‹bounded_linear h› ‹bounded_linear h'› ‹h' (h ?x) = ?x›*) summable_on_bounded_linear (*‹⟦bounded_linear ?h; ?f summable_on ?A⟧ ⟹ (λx. ?h (?f x)) summable_on ?A›*) summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n\nlemma infsum_bounded_linear_strong:\n fixes h :: \"'a :: real_normed_vector ⇒ 'b :: real_normed_vector\"\n assumes \"(λx. h (f x)) summable_on A ⟷ f summable_on A\"\n assumes \"bounded_linear h\"\n shows \"infsum (λx. h (f x)) A = h (infsum f A)\"\nproof (-)\n (*goal: ‹(∑⇩∞x∈A. h (f x)) = h (infsum f A)›*)\n interpret bounded_linear h\n by fact\n show \"?thesis\"\n (*goal: ‹(∑⇩∞x∈A. h (f x)) = h (infsum f A)›*)\n apply (rule infsum_homomorphism_strong (*‹⟦((λx. ?h (?f x)) summable_on ?A) = (?f summable_on ?A); ?h 0 = 0; ⋀S. (?f has_sum S) ?A ⟹ ((λx. ?h (?f x)) has_sum ?h S) ?A⟧ ⟹ (∑⇩∞x∈?A. ?h (?f x)) = ?h (infsum ?f ?A)›*))\n (*goals:\n 1. ‹((λx::'c::type. (h::'a::real_normed_vector ⇒ 'b::real_normed_vector) ((f::'c::type ⇒ 'a::real_normed_vector) x)) summable_on (A::'c::type set)) = (f summable_on A)›\n 2. ‹(h::'a::real_normed_vector ⇒ 'b::real_normed_vector) (0::'a::real_normed_vector) = (0::'b::real_normed_vector)›\n 3. ‹⋀S::'a::real_normed_vector. ((f::'c::type ⇒ 'a::real_normed_vector) has_sum S) (A::'c::type set) ⟹ ((λx::'c::type. (h::'a::real_normed_vector ⇒ 'b::real_normed_vector) (f x)) has_sum h S) A›\n discuss goal 1*)\n apply ((insert assms (*‹((λx. h (f x)) summable_on A) = (f summable_on A)› ‹bounded_linear h›*))[1])\n (*top goal: ‹((λx::'c::type. (h::'a::real_normed_vector ⇒ 'b::real_normed_vector) ((f::'c::type ⇒ 'a::real_normed_vector) x)) summable_on (A::'c::type set)) = (f summable_on A)› and 2 goals remain*)\n apply ((auto intro: add (*‹(h::'a ⇒ 'b) ((?b1.0::'a) + (?b2.0::'a)) = h ?b1.0 + h ?b2.0›*) continuous_on (*‹continuous_on (?s::?'c set) (?g::?'c ⇒ 'a) ⟹ continuous_on ?s (λx::?'c. (h::'a ⇒ 'b) (?g x))›*) has_sum_bounded_linear (*‹⟦bounded_linear (?h::?'a ⇒ ?'b); ((?f::?'c ⇒ ?'a) has_sum (?S::?'a)) (?A::?'c set)⟧ ⟹ ((λx::?'c. ?h (?f x)) has_sum ?h ?S) ?A›*))[1])\n (*discuss goal 2*)\n apply ((insert assms (*‹((λx::'c::type. (h::'a::real_normed_vector ⇒ 'b::real_normed_vector) ((f::'c::type ⇒ 'a::real_normed_vector) x)) summable_on (A::'c::type set)) = (f summable_on A)› ‹bounded_linear (h::'a::real_normed_vector ⇒ 'b::real_normed_vector)›*))[1])\n (*top goal: ‹(h::'a ⇒ 'b) (0::'a) = (0::'b)› and 1 goal remains*)\n apply ((auto intro: add (*‹h (?b1.0 + ?b2.0) = h ?b1.0 + h ?b2.0›*) continuous_on (*‹continuous_on ?s ?g ⟹ continuous_on ?s (λx. h (?g x))›*) has_sum_bounded_linear (*‹⟦bounded_linear ?h; (?f has_sum ?S) ?A⟧ ⟹ ((λx. ?h (?f x)) has_sum ?h ?S) ?A›*))[1])\n (*discuss goal 3*)\n apply ((insert assms (*‹((λx::'c. (h::'a ⇒ 'b) ((f::'c ⇒ 'a) x)) summable_on (A::'c set)) = (f summable_on A)› ‹bounded_linear (h::'a ⇒ 'b)›*))[1])\n (*goal: ‹⋀S. (f has_sum S) A ⟹ ((λx. h (f x)) has_sum h S) A›*)\n apply ((auto intro: add (*‹h (?b1.0 + ?b2.0) = h ?b1.0 + h ?b2.0›*) continuous_on (*‹continuous_on ?s ?g ⟹ continuous_on ?s (λx. h (?g x))›*) has_sum_bounded_linear (*‹⟦bounded_linear ?h; (?f has_sum ?S) ?A⟧ ⟹ ((λx. ?h (?f x)) has_sum ?h ?S) ?A›*))[1])\n (*proven 3 subgoals*) .\nqed\n\nlemma infsum_bounded_linear_strong':\n fixes mult :: \"'c :: zero ⇒ 'a :: real_normed_vector ⇒ 'b :: real_normed_vector\"\n assumes \"c ≠ 0 ⟹ (λx. mult c (f x)) summable_on A ⟷ f summable_on A\"\n assumes \"bounded_linear (mult c)\"\n assumes [simp]: \"⋀x. mult 0 x = 0\"\n shows \"infsum (λx. mult c (f x)) A = mult c (infsum f A)\"\n by (metis assms (*‹(c::'c) ≠ (0::'c) ⟹ ((λx::'d. (mult::'c ⇒ 'a ⇒ 'b) c ((f::'d ⇒ 'a) x)) summable_on (A::'d set)) = (f summable_on A)› ‹bounded_linear ((mult::'c ⇒ 'a ⇒ 'b) (c::'c))› ‹(mult::'c ⇒ 'a ⇒ 'b) (0::'c) (?x::'a) = (0::'b)›*) infsum_0 (*‹(⋀x::?'a. x ∈ (?M::?'a set) ⟹ (?f::?'a ⇒ ?'b) x = (0::?'b)) ⟹ infsum ?f ?M = (0::?'b)›*) infsum_bounded_linear_strong (*‹⟦((λx::?'c. (?h::?'a ⇒ ?'b) ((?f::?'c ⇒ ?'a) x)) summable_on (?A::?'c set)) = (?f summable_on ?A); bounded_linear ?h⟧ ⟹ (∑⇩∞x::?'c∈?A. ?h (?f x)) = ?h (infsum ?f ?A)›*))\n\nlemma has_sum_of_nat: \"(f has_sum S) A ⟹ ((λx. of_nat (f x)) has_sum of_nat S) A\"\n apply (erule has_sum_homomorphism (*‹⟦(?f has_sum ?S) ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ ((λx. ?h (?f x)) has_sum ?h ?S) ?A›*))\n (*goals:\n 1. ‹of_nat (0::nat) = (0::'b)›\n 2. ‹⋀(a::nat) b::nat. of_nat (a + b) = of_nat a + of_nat b›\n 3. ‹continuous_on UNIV of_nat›\n discuss goal 1*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open (?S::?'a set); open (?T::?'a set)⟧ ⟹ open (?S ∪ ?T)› ‹∀x::?'b∈?A::?'b set. open ((?B::?'b ⇒ ?'a set) x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite (?S::?'a set set); ∀T::?'a set∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite (?A::?'b set); ∀x::?'b∈?A. open ((?B::?'b ⇒ ?'a set) x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed (?S::?'a set); closed (?T::?'a set)⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed (?S::?'a set); closed (?T::?'a set)⟧ ⟹ closed (?S ∩ ?T)› ‹∀x::?'b∈?A::?'b set. closed ((?B::?'b ⇒ ?'a set) x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S::?'a set∈?K::?'a set set. closed S ⟹ closed (⋂ ?K)› ‹⟦finite (?S::?'a set set); ∀T::?'a set∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 3*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*proven 3 subgoals*) .\n\nlemma has_sum_of_int: \"(f has_sum S) A ⟹ ((λx. of_int (f x)) has_sum of_int S) A\"\n apply (erule has_sum_homomorphism (*‹⟦(?f has_sum ?S) ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ ((λx. ?h (?f x)) has_sum ?h ?S) ?A›*))\n (*goals:\n 1. ‹of_int (0::int) = (0::'b::{ring_1,topological_space})›\n 2. ‹⋀(a::int) b::int. of_int (a + b) = of_int a + of_int b›\n 3. ‹continuous_on UNIV of_int›\n discuss goal 1*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 3*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*proven 3 subgoals*) .\n\nlemma summable_on_of_nat: \"f summable_on A ⟹ (λx. of_nat (f x)) summable_on A\"\n apply (erule summable_on_homomorphism (*‹⟦?f summable_on ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ (λx. ?h (?f x)) summable_on ?A›*))\n (*goals:\n 1. ‹of_nat 0 = 0›\n 2. ‹⋀a b. of_nat (a + b) = of_nat a + of_nat b›\n 3. ‹continuous_on UNIV of_nat›\n discuss goal 1*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 3*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*proven 3 subgoals*) .\n\nlemma summable_on_of_int: \"f summable_on A ⟹ (λx. of_int (f x)) summable_on A\"\n apply (erule summable_on_homomorphism (*‹⟦?f summable_on ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ (λx. ?h (?f x)) summable_on ?A›*))\n (*goals:\n 1. ‹of_int 0 = 0›\n 2. ‹⋀a b. of_int (a + b) = of_int a + of_int b›\n 3. ‹continuous_on UNIV of_int›\n discuss goal 1*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 2*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*discuss goal 3*)\n apply ((auto intro!: continuous_intros (*‹open {}› ‹⟦open ?S; open ?T⟧ ⟹ open (?S ∪ ?T)› ‹∀x∈?A. open (?B x) ⟹ open (⋃ (?B ` ?A))› ‹⟦finite ?S; ∀T∈?S. open T⟧ ⟹ open (⋂ ?S)› ‹⟦finite ?A; ∀x∈?A. open (?B x)⟧ ⟹ open (⋂ (?B ` ?A))› ‹closed {}› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∪ ?T)› ‹closed UNIV› ‹⟦closed ?S; closed ?T⟧ ⟹ closed (?S ∩ ?T)› ‹∀x∈?A. closed (?B x) ⟹ closed (⋂ (?B ` ?A))› ‹∀S∈?K. closed S ⟹ closed (⋂ ?K)› ‹⟦finite ?S; ∀T∈?S. closed T⟧ ⟹ closed (⋃ ?S)› and more 144 facts*))[1])\n (*proven 3 subgoals*) .\n\nlemma summable_on_discrete_iff:\n fixes f :: \"'a ⇒ 'b :: {discrete_topology, topological_comm_monoid_add, cancel_comm_monoid_add}\"\n shows \"f summable_on A ⟷ finite {x∈A. f x ≠ 0}\"\nproof (standard)\n (*goals:\n 1. ‹f summable_on A ⟹ finite {x ∈ A. f x ≠ 0}›\n 2. ‹finite {x ∈ A. f x ≠ 0} ⟹ f summable_on A›*)\n assume \"*\": \"finite {x∈A. f x ≠ 0}\" (*‹finite {x::'a ∈ A::'a set. (f::'a ⇒ 'b) x ≠ (0::'b)}›*)\n hence \"f summable_on {x∈A. f x ≠ 0}\"\n by (rule summable_on_finite (*‹finite ?F ⟹ ?f summable_on ?F›*))\n then show \"f summable_on A\"\n by (smt (verit) DiffE (*‹⟦(?c::?'a::type) ∈ (?A::?'a::type set) - (?B::?'a::type set); ⟦?c ∈ ?A; ?c ∉ ?B⟧ ⟹ ?P::bool⟧ ⟹ ?P›*) mem_Collect_eq (*‹((?a::?'a::type) ∈ Collect (?P::?'a::type ⇒ bool)) = ?P ?a›*) summable_on_cong_neutral (*‹⟦⋀x::?'a::type. x ∈ (?T::?'a::type set) - (?S::?'a::type set) ⟹ (?g::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) x = (0::?'b::{comm_monoid_add,topological_space}); ⋀x::?'a::type. x ∈ ?S - ?T ⟹ (?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) x = (0::?'b::{comm_monoid_add,topological_space}); ⋀x::?'a::type. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f summable_on ?S) = (?g summable_on ?T)›*))\nnext\n (*goal: ‹(f::'a ⇒ 'b) summable_on (A::'a set) ⟹ finite {x::'a ∈ A. f x ≠ (0::'b)}›*)\n assume \"f summable_on A\" (*‹(f::'a ⇒ 'b) summable_on (A::'a set)›*)\n then obtain S where \"(f has_sum S) A\"\n (*goal: ‹(⋀S. (f has_sum S) A ⟹ thesis) ⟹ thesis›*)\n by (auto simp: summable_on_def (*‹(?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ≡ ∃x::?'b. (?f has_sum x) ?A›*))\n hence \"∀⇩F x in finite_subsets_at_top A. sum f x = S\"\n unfolding has_sum_def tendsto_discrete\n (*goal: ‹∀⇩F x in finite_subsets_at_top A. sum f x = S›*) .\n then obtain X where X: \"finite X\" \"X ⊆ A\" \"⋀Y. finite Y ⟹ X ⊆ Y ⟹ Y ⊆ A ⟹ sum f Y = S\"\n (*goal: ‹(⋀X. ⟦finite X; X ⊆ A; ⋀Y. ⟦finite Y; X ⊆ Y; Y ⊆ A⟧ ⟹ sum f Y = S⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹(⋀X::'a set. ⟦finite X; X ⊆ (A::'a set); ⋀Y::'a set. ⟦finite Y; X ⊆ Y; Y ⊆ A⟧ ⟹ sum (f::'a ⇒ 'b) Y = (S::'b)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by metis\n have \"{x∈A. f x ≠ 0} ⊆ X\"\n proof (standard)\n (*goal: ‹⋀x. x ∈ {x ∈ A. f x ≠ 0} ⟹ x ∈ X›*)\n fix x\n assume x: \"x ∈ {x∈A. f x ≠ 0}\" (*‹(x::'a) ∈ {x::'a ∈ A::'a set. (f::'a ⇒ 'b) x ≠ (0::'b)}›*)\n show \"x ∈ X\"\n proof (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n (*goal: ‹x ∉ X ⟹ False›*)\n assume [simp]: \"x ∉ X\" (*‹(x::'a) ∉ (X::'a set)›*)\n have \"sum f (insert x X) = S\"\n using X (*‹finite (X::'a set)› ‹X ⊆ A› ‹⟦finite ?Y; X ⊆ ?Y; ?Y ⊆ A⟧ ⟹ sum f ?Y = S›*) x (*‹x ∈ {x ∈ A. f x ≠ 0}›*) apply (intro X (*‹finite X› ‹X ⊆ A› ‹⟦finite ?Y; X ⊆ ?Y; ?Y ⊆ A⟧ ⟹ sum f ?Y = S›*))\n (*goals:\n 1. ‹⟦finite X; X ⊆ A; ⋀Y. ⟦finite Y; X ⊆ Y; Y ⊆ A⟧ ⟹ sum f Y = S; x ∈ {x ∈ A. f x ≠ 0}⟧ ⟹ finite (insert x X)›\n 2. ‹⟦finite X; X ⊆ A; ⋀Y. ⟦finite Y; X ⊆ Y; Y ⊆ A⟧ ⟹ sum f Y = S; x ∈ {x ∈ A. f x ≠ 0}⟧ ⟹ X ⊆ insert x X›\n 3. ‹⟦finite X; X ⊆ A; ⋀Y. ⟦finite Y; X ⊆ Y; Y ⊆ A⟧ ⟹ sum f Y = S; x ∈ {x ∈ A. f x ≠ 0}⟧ ⟹ insert x X ⊆ 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 then have \"f x = 0\"\n using X (*‹finite X› ‹X ⊆ A› ‹⟦finite ?Y; X ⊆ ?Y; ?Y ⊆ A⟧ ⟹ sum f ?Y = S›*) by auto\n with x (*‹x ∈ {x ∈ A. f x ≠ 0}›*) show False\n by auto\n qed\n qed\n thus \"finite {x∈A. f x ≠ 0}\"\n using X(1) (*‹finite X›*) finite_subset (*‹⟦?A ⊆ ?B; finite ?B⟧ ⟹ finite ?A›*) by blast\nqed\n\nlemma has_sum_imp_sums: \"(f has_sum S) (UNIV :: nat set) ⟹ f sums S\"\n unfolding sums_def has_sum_def\n (*goal: ‹(sum f ⤏ S) (finite_subsets_at_top UNIV) ⟹ (λn. sum f {..<n}) ⇢ S›*)\n by (rule filterlim_compose[OF _ filterlim_lessThan_at_top] (*‹filterlim ?g ?F3.0 (finite_subsets_at_top UNIV) ⟹ LIM x sequentially. ?g {..<x} :> ?F3.0›*))\n\nlemma summable_on_imp_summable: \"f summable_on (UNIV :: nat set) ⟹ summable f\"\n unfolding summable_on_def summable_def\n (*goal: ‹∃x::'a. ((f::nat ⇒ 'a) has_sum x) UNIV ⟹ Ex ((sums) f)›*)\n by (auto dest: has_sum_imp_sums (*‹(?f has_sum ?S) UNIV ⟹ ?f sums ?S›*))\n\nlemma summable_on_UNIV_nonneg_real_iff:\n assumes \"⋀n. f n ≥ (0 :: real)\"\n shows \"f summable_on UNIV ⟷ summable f\"\n using assms (*‹0 ≤ f ?n›*) by (auto intro: norm_summable_imp_summable_on (*‹summable (λn. norm (?f n)) ⟹ ?f summable_on UNIV›*) summable_on_imp_summable (*‹?f summable_on UNIV ⟹ summable ?f›*))\n\nlemma summable_on_imp_bounded_partial_sums:\n fixes f :: \"_ ⇒ 'a :: {topological_comm_monoid_add, linorder_topology}\"\n assumes f: \"f summable_on A\"\n shows \"∃C. eventually (λX. sum f X ≤ C) (finite_subsets_at_top A)\"\nproof (-)\n (*goal: ‹∃C. ∀⇩F X in finite_subsets_at_top A. sum f X ≤ C›*)\n from assms (*‹f summable_on A›*) obtain S where S: \"(sum f ⤏ S) (finite_subsets_at_top A)\"\n (*goal: ‹(⋀S. (sum f ⤏ S) (finite_subsets_at_top A) ⟹ thesis) ⟹ thesis›*)\n unfolding summable_on_def has_sum_def\n (*goal: ‹(⋀S::'a. (sum (f::'c ⇒ 'a) ⤏ S) (finite_subsets_at_top (A::'c set)) ⟹ thesis::bool) ⟹ thesis›*)\n by blast\n show \"?thesis\"\n (*goal: ‹∃C::'a::{topological_comm_monoid_add,linorder_topology}. ∀⇩F X::'c::type set in finite_subsets_at_top (A::'c::type set). sum (f::'c::type ⇒ 'a::{topological_comm_monoid_add,linorder_topology}) X ≤ C›*)\n proof (cases \"∃C. C > S\")\n (*goals:\n 1. ‹∃C::'a. (S::'a) < C ⟹ ∃C::'a. ∀⇩F X::'c set in finite_subsets_at_top (A::'c set). sum (f::'c ⇒ 'a) X ≤ C›\n 2. ‹∄C::'a. (S::'a) < C ⟹ ∃C::'a. ∀⇩F X::'c set in finite_subsets_at_top (A::'c set). sum (f::'c ⇒ 'a) X ≤ C›*)\n case True (*‹∃C. S < C›*)\n then obtain C where C: \"C > S\"\n (*goal: ‹(⋀C::'a. (S::'a) < C ⟹ thesis::bool) ⟹ thesis›*)\n by blast\n have \"∀⇩F X in finite_subsets_at_top A. sum f X < C\"\n using S (*‹(sum f ⤏ S) (finite_subsets_at_top A)›*) C (*‹S < C›*) by (rule order_tendstoD( (*‹⟦(?f ⤏ ?y) ?F; ?y < ?a⟧ ⟹ ∀⇩F x in ?F. ?f x < ?a›*) 2))\n thus \"?thesis\"\n (*goal: ‹∃C::'a. ∀⇩F X::'c set in finite_subsets_at_top (A::'c set). sum (f::'c ⇒ 'a) X ≤ C›*)\n by (meson eventually_mono (*‹⟦eventually ?P ?F; ⋀x. ?P x ⟹ ?Q x⟧ ⟹ eventually ?Q ?F›*) nless_le (*‹(¬ ?a < ?b) = (¬ ?a ≤ ?b ∨ ?a = ?b)›*))\n next\n (*goal: ‹∄C. S < C ⟹ ∃C. ∀⇩F X in finite_subsets_at_top A. sum f X ≤ C›*)\n case False (*‹∄C. S < C›*)\n thus \"?thesis\"\n (*goal: ‹∃C. ∀⇩F X in finite_subsets_at_top A. sum f X ≤ C›*)\n by (meson not_eventuallyD (*‹¬ eventually ?P ?F ⟹ ∃x. ¬ ?P x›*) not_le_imp_less (*‹¬ ?y ≤ ?x ⟹ ?x < ?y›*))\n qed\nqed\n\nlemma has_sum_mono':\n fixes S S' :: \"'a :: {linorder_topology, ordered_comm_monoid_add, topological_comm_monoid_add}\"\n assumes f: \"(f has_sum S) A\" \"(f has_sum S') B\" \n and AB: \"A ⊆ B\" \"⋀x. x ∈ B - A ⟹ f x ≥ 0\"\n shows \"S ≤ S'\"\n using AB (*‹A ⊆ B› ‹?x ∈ B - A ⟹ 0 ≤ f ?x›*) has_sum_mono_neutral[OF f] (*‹⟦⋀x. x ∈ A ∩ B ⟹ f x ≤ f x; ⋀x. x ∈ A - B ⟹ f x ≤ 0; ⋀x. x ∈ B - A ⟹ 0 ≤ f x⟧ ⟹ S ≤ S'›*) by fastforce\n\n\ncontext\n assumes \"SORT_CONSTRAINT('a :: {topological_comm_monoid_add, order_topology,\n ordered_comm_monoid_add, conditionally_complete_linorder})\"\nbegin\n\ntext ‹\n Any family of non-negative numbers with bounded partial sums is summable, and the sum\n is simply the supremum of the partial sums.\n›\nlemma nonneg_bounded_partial_sums_imp_has_sum_SUP:\n assumes nonneg: \"⋀x. x ∈ A ⟹ f x ≥ (0::'a)\"\n and bound: \"eventually (λX. sum f X ≤ C) (finite_subsets_at_top A)\"\n shows \"(f has_sum (SUP X∈{X. X ⊆ A ∧ finite X}. sum f X)) A\"\nproof (-)\n (*goal: ‹(f has_sum Sup (sum f ` {X. X ⊆ A ∧ finite X})) A›*)\n from bound (*‹∀⇩F X::'b::type set in finite_subsets_at_top (A::'b::type set). sum (f::'b::type ⇒ 'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,order_topology}) X ≤ (C::'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,order_topology})›*) obtain X0 where X0: \"X0 ⊆ A\" \"finite X0\" \"⋀X. X0 ⊆ X ⟹ X ⊆ A ⟹ finite X ⟹ sum f X ≤ C\"\n (*goal: ‹(⋀X0. ⟦X0 ⊆ A; finite X0; ⋀X. ⟦X0 ⊆ X; X ⊆ A; finite X⟧ ⟹ sum f X ≤ C⟧ ⟹ thesis) ⟹ thesis›*)\n by (force simp: eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n have bound': \"sum f X ≤ C\" if \"X ⊆ A\" \"finite X\" for X\n proof (-)\n (*goal: ‹sum f X ≤ C›*)\n have \"sum f X ≤ sum f (X ∪ X0)\"\n using that (*‹(X::'b set) ⊆ (A::'b set)› ‹finite X›*) X0 (*‹X0 ⊆ A› ‹finite X0› ‹⟦X0 ⊆ ?X; ?X ⊆ A; finite ?X⟧ ⟹ sum f ?X ≤ C›*) assms(1) (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) apply (intro sum_mono2 (*‹⟦finite ?B; ?A ⊆ ?B; ⋀b. b ∈ ?B - ?A ⟹ 0 ≤ ?f b⟧ ⟹ sum ?f ?A ≤ sum ?f ?B›*))\n (*goals:\n 1. ‹⟦(X::'b set) ⊆ (A::'b set); finite X; (X0::'b set) ⊆ A; finite X0; ⋀X::'b set. ⟦X0 ⊆ X; X ⊆ A; finite X⟧ ⟹ sum (f::'b ⇒ 'a) X ≤ (C::'a); ⋀x::'b. x ∈ A ⟹ (0::'a) ≤ f x⟧ ⟹ finite (X ∪ X0)›\n 2. ‹⟦(X::'b set) ⊆ (A::'b set); finite X; (X0::'b set) ⊆ A; finite X0; ⋀X::'b set. ⟦X0 ⊆ X; X ⊆ A; finite X⟧ ⟹ sum (f::'b ⇒ 'a) X ≤ (C::'a); ⋀x::'b. x ∈ A ⟹ (0::'a) ≤ f x⟧ ⟹ X ⊆ X ∪ X0›\n 3. ‹⋀b::'b. ⟦(X::'b set) ⊆ (A::'b set); finite X; (X0::'b set) ⊆ A; finite X0; ⋀X::'b set. ⟦X0 ⊆ X; X ⊆ A; finite X⟧ ⟹ sum (f::'b ⇒ 'a) X ≤ (C::'a); ⋀x::'b. x ∈ A ⟹ (0::'a) ≤ f x; b ∈ X ∪ X0 - X⟧ ⟹ (0::'a) ≤ f b›\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: ‹sum f X ≤ sum f (X ∪ X0)›*) have \"… ≤ C\"\n by (simp add: X0 (*‹X0 ⊆ A› ‹finite X0› ‹⟦X0 ⊆ ?X; ?X ⊆ A; finite ?X⟧ ⟹ sum f ?X ≤ C›*) that (*‹X ⊆ A› ‹finite X›*))\n finally (*calculation: ‹sum f X ≤ C›*) show \"?thesis\"\n (*goal: ‹sum f X ≤ C›*) .\n qed\n hence bdd: \"bdd_above (sum f ` {X. X ⊆ A ∧ finite X})\"\n by (auto simp: bdd_above_def (*‹bdd_above ?A = (∃M. ∀x∈?A. x ≤ M)›*))\n show \"?thesis\"\n (*goal: ‹(f has_sum Sup (sum f ` {X. X ⊆ A ∧ finite X})) A›*)\n unfolding has_sum_def\n (*goal: ‹(sum f ⤏ Sup (sum f ` {X. X ⊆ A ∧ finite X})) (finite_subsets_at_top A)›*)\n proof (rule increasing_tendsto (*‹⟦∀⇩F n in ?F. ?f n ≤ ?l; ⋀x. x < ?l ⟹ ∀⇩F n in ?F. x < ?f n⟧ ⟹ (?f ⤏ ?l) ?F›*))\n (*goals:\n 1. ‹∀⇩F n in finite_subsets_at_top A. sum f n ≤ Sup (sum f ` {X. X ⊆ A ∧ finite X})›\n 2. ‹⋀x. x < Sup (sum f ` {X. X ⊆ A ∧ finite X}) ⟹ ∀⇩F n in finite_subsets_at_top A. x < sum f n›*)\n show \"∀⇩F X in finite_subsets_at_top A. sum f X ≤ Sup (sum f ` {X. X ⊆ A ∧ finite X})\"\n apply (intro eventually_finite_subsets_at_top_weakI (*‹(⋀X. ⟦finite X; X ⊆ ?A⟧ ⟹ ?P X) ⟹ eventually ?P (finite_subsets_at_top ?A)›*) cSUP_upper[OF _ bdd] (*‹?x ∈ {X. X ⊆ A ∧ finite X} ⟹ sum f ?x ≤ Sup (sum f ` {X. X ⊆ A ∧ finite X})›*))\n (*goal: ‹∀⇩F X in finite_subsets_at_top A. sum f X ≤ Sup (sum f ` {X. X ⊆ A ∧ finite X})›*)\n by auto\n next\n (*goal: ‹⋀x::'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,order_topology}. x < Sup (sum (f::'b::type ⇒ 'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,order_topology}) ` {X::'b::type set. X ⊆ (A::'b::type set) ∧ finite X}) ⟹ ∀⇩F n::'b::type set in finite_subsets_at_top A. x < sum f n›*)\n fix y\n assume \"y < Sup (sum f ` {X. X ⊆ A ∧ finite X})\" (*‹(y::'a) < Sup (sum (f::'b ⇒ 'a) ` {X::'b set. X ⊆ (A::'b set) ∧ finite X})›*)\n then obtain X where X: \"X ⊆ A\" \"finite X\" \"y < sum f X\"\n (*goal: ‹(⋀X. ⟦X ⊆ A; finite X; y < sum f X⟧ ⟹ thesis) ⟹ thesis›*)\n apply (subst (asm) less_cSUP_iff[OF _ bdd] (*‹{X. X ⊆ A ∧ finite X} ≠ {} ⟹ (?a < Sup (sum f ` {X. X ⊆ A ∧ finite X})) = (∃x∈{X. X ⊆ A ∧ finite X}. ?a < sum f x)›*))\n (*goals:\n 1. ‹(⋀X. ⟦X ⊆ A; finite X; y < sum f X⟧ ⟹ thesis) ⟹ {X. X ⊆ A ∧ finite X} ≠ {}›\n 2. ‹⟦⋀X. ⟦X ⊆ A; finite X; y < sum f X⟧ ⟹ thesis; ∃x∈{X. X ⊆ A ∧ finite X}. y < sum f x⟧ ⟹ thesis›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n from X (*‹(X::'b set) ⊆ (A::'b set)› ‹finite X› ‹y < sum f X›*) have \"eventually (λX'. X ⊆ X' ∧ X' ⊆ A ∧ finite X') (finite_subsets_at_top A)\"\n by (auto simp: eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*))\n thus \"eventually (λX'. y < sum f X') (finite_subsets_at_top A)\"\n proof (eventually_elim)\n (*goal: ‹⋀X'. X ⊆ X' ∧ X' ⊆ A ∧ finite X' ⟹ y < sum f X'›*)\n case (elim X') (*‹X ⊆ X' ∧ X' ⊆ A ∧ finite X'›*)\n note ‹y < sum f X› (*‹y < sum f X›*)\n also (*calculation: ‹y < sum f X›*) have \"sum f X ≤ sum f X'\"\n using nonneg (*‹?x ∈ A ⟹ 0 ≤ f ?x›*) elim (*‹(X::'b set) ⊆ (X'::'b set) ∧ X' ⊆ (A::'b set) ∧ finite X'›*) apply (intro sum_mono2 (*‹⟦finite ?B; ?A ⊆ ?B; ⋀b. b ∈ ?B - ?A ⟹ 0 ≤ ?f b⟧ ⟹ sum ?f ?A ≤ sum ?f ?B›*))\n (*goals:\n 1. ‹⟦⋀x::'b. x ∈ (A::'b set) ⟹ (0::'a) ≤ (f::'b ⇒ 'a) x; (X::'b set) ⊆ (X'::'b set) ∧ X' ⊆ A ∧ finite X'⟧ ⟹ finite X'›\n 2. ‹⟦⋀x::'b. x ∈ (A::'b set) ⟹ (0::'a) ≤ (f::'b ⇒ 'a) x; (X::'b set) ⊆ (X'::'b set) ∧ X' ⊆ A ∧ finite X'⟧ ⟹ X ⊆ X'›\n 3. ‹⋀b::'b. ⟦⋀x::'b. x ∈ (A::'b set) ⟹ (0::'a) ≤ (f::'b ⇒ 'a) x; (X::'b set) ⊆ (X'::'b set) ∧ X' ⊆ A ∧ finite X'; b ∈ X' - X⟧ ⟹ (0::'a) ≤ f b›\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: ‹y < sum f X'›*) show \"?case\"\n (*goal: ‹(y::'a) < sum (f::'b ⇒ 'a) (X'::'b set)›*) .\n qed\n qed\nqed\n\nlemma nonneg_bounded_partial_sums_imp_summable_on:\n assumes nonneg: \"⋀x. x ∈ A ⟹ f x ≥ (0::'a)\"\n and bound: \"eventually (λX. sum f X ≤ C) (finite_subsets_at_top A)\"\n shows \"f summable_on A\"\n using nonneg_bounded_partial_sums_imp_has_sum_SUP[OF assms] (*‹(⋀x. x ∈ A ⟹ x ∈ A) ⟹ (f has_sum Sup (sum f ` {X. X ⊆ A ∧ finite X})) A›*) by (auto simp: summable_on_def (*‹(?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) summable_on (?A::?'a::type set) ≡ ∃x::?'b::{comm_monoid_add,topological_space}. (?f has_sum x) ?A›*))\n\nend\n\ncontext\n assumes \"SORT_CONSTRAINT('a :: {topological_comm_monoid_add, linorder_topology,\n ordered_comm_monoid_add, conditionally_complete_linorder})\"\nbegin\n\nlemma summable_on_comparison_test:\n assumes \"f summable_on A\" and \"⋀x. x ∈ A ⟹ g x ≤ f x\" and \"⋀x. x ∈ A ⟹ (0::'a) ≤ g x\"\n shows \"g summable_on A\"\nproof (-)\n (*goal: ‹g summable_on A›*)\n obtain C where C: \"∀⇩F X in finite_subsets_at_top A. sum f X ≤ C\"\n (*goal: ‹(⋀C. ∀⇩F X in finite_subsets_at_top A. sum f X ≤ C ⟹ thesis) ⟹ thesis›*)\n using assms(1) (*‹f summable_on A›*) summable_on_imp_bounded_partial_sums (*‹?f summable_on ?A ⟹ ∃C. ∀⇩F X in finite_subsets_at_top ?A. sum ?f X ≤ C›*) by blast\n show \"?thesis\"\n (*goal: ‹g summable_on A›*)\n proof (rule nonneg_bounded_partial_sums_imp_summable_on (*‹⟦⋀x::?'b::type. x ∈ (?A::?'b::type set) ⟹ (0::?'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,order_topology}) ≤ (?f::?'b::type ⇒ ?'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,order_topology}) x; ∀⇩F X::?'b::type set in finite_subsets_at_top ?A. sum ?f X ≤ (?C::?'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,order_topology})⟧ ⟹ ?f summable_on ?A›*))\n (*goals:\n 1. ‹⋀x. x ∈ A ⟹ 0 ≤ g x›\n 2. ‹∀⇩F X in finite_subsets_at_top A. sum g X ≤ ?C›*)\n show \"∀⇩F X in finite_subsets_at_top A. sum g X ≤ C\"\n using C (*‹∀⇩F X::'b set in finite_subsets_at_top (A::'b set). sum (f::'b ⇒ 'a) X ≤ (C::'a)›*) assms (*‹f summable_on A› ‹?x ∈ A ⟹ g ?x ≤ f ?x› ‹?x ∈ A ⟹ 0 ≤ g ?x›*) unfolding eventually_finite_subsets_at_top\n (*goal: ‹∃X::'b::type set. finite X ∧ X ⊆ (A::'b::type set) ∧ (∀Y::'b::type set. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ sum (g::'b::type ⇒ 'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,linorder_topology}) Y ≤ (C::'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,linorder_topology}))›*)\n by (smt (verit, ccfv_SIG) order_trans (*‹⟦(?x::?'a::preorder) ≤ (?y::?'a::preorder); ?y ≤ (?z::?'a::preorder)⟧ ⟹ ?x ≤ ?z›*) subsetD (*‹⟦(?A::?'a::type set) ⊆ (?B::?'a::type set); (?c::?'a::type) ∈ ?A⟧ ⟹ ?c ∈ ?B›*) sum_mono (*‹(⋀i::?'b::type. i ∈ (?K::?'b::type set) ⟹ (?f::?'b::type ⇒ ?'a::ordered_comm_monoid_add) i ≤ (?g::?'b::type ⇒ ?'a::ordered_comm_monoid_add) i) ⟹ sum ?f ?K ≤ sum ?g ?K›*))\n qed (use assms in auto)\n (*solved the remaining goal: ‹⋀x. x ∈ A ⟹ 0 ≤ g x›*)\nqed\n\nend\n\n\n\nlemma summable_on_subset:\n fixes f :: \"_ ⇒ 'a :: {uniform_topological_group_add, topological_comm_monoid_add, ab_group_add, complete_uniform_space}\"\n assumes \"f summable_on A\" \"B ⊆ A\"\n shows \"f summable_on B\"\n apply (rule summable_on_subset_aux[OF _ _ assms] (*‹⟦complete UNIV; uniformly_continuous_on UNIV (λ(x::'a::{ab_group_add,complete_uniform_space,topological_comm_monoid_add,uniform_topological_group_add}, y::'a::{ab_group_add,complete_uniform_space,topological_comm_monoid_add,uniform_topological_group_add}). x + y)⟧ ⟹ (f::'c::type ⇒ 'a::{ab_group_add,complete_uniform_space,topological_comm_monoid_add,uniform_topological_group_add}) summable_on (B::'c::type set)›*))\n (*goals:\n 1. ‹complete UNIV›\n 2. ‹uniformly_continuous_on UNIV (λ(x, y). x + y)›\n discuss goal 1*)\n apply ((auto simp: uniformly_continuous_add (*‹uniformly_continuous_on UNIV (λ(x::?'a, y::?'a). x + y)›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: uniformly_continuous_add (*‹uniformly_continuous_on UNIV (λ(x, y). x + y)›*))[1])\n (*proven 2 subgoals*) .\n\nlemma summable_on_union:\n fixes f :: \"_ ⇒ 'a :: {uniform_topological_group_add, topological_comm_monoid_add, ab_group_add, complete_uniform_space}\"\n assumes \"f summable_on A\" \"f summable_on B\"\n shows \"f summable_on (A ∪ B)\"\nproof (-)\n (*goal: ‹(f::'c ⇒ 'a) summable_on (A::'c set) ∪ (B::'c set)›*)\n have \"f summable_on (A ∪ (B - A))\"\n by (meson Diff_disjoint (*‹?A ∩ (?B - ?A) = {}›*) Diff_subset (*‹?A - ?B ⊆ ?A›*) assms (*‹f summable_on A› ‹f summable_on B›*) summable_on_Un_disjoint (*‹⟦?f summable_on ?A; ?f summable_on ?B; ?A ∩ ?B = {}⟧ ⟹ ?f summable_on ?A ∪ ?B›*) summable_on_subset (*‹⟦?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*))\n also (*calculation: ‹f summable_on A ∪ (B - A)›*) have \"A ∪ (B - A) = A ∪ B\"\n by blast\n finally (*calculation: ‹f summable_on A ∪ B›*) show \"?thesis\"\n (*goal: ‹f summable_on A ∪ B›*) .\nqed\n\nlemma summable_on_insert_iff:\n fixes f :: \"_ ⇒ 'a :: {uniform_topological_group_add, topological_comm_monoid_add, ab_group_add, complete_uniform_space}\"\n shows \"f summable_on insert x A ⟷ f summable_on A\"\n using summable_on_union[of f A \"{x}\"] (*‹⟦f summable_on A; f summable_on {x}⟧ ⟹ f summable_on A ∪ {x}›*) by (auto intro: summable_on_subset (*‹⟦?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*))\n\nlemma has_sum_finiteI: \"finite A ⟹ S = sum f A ⟹ (f has_sum S) A\"\n by simp\n\nlemma has_sum_insert:\n fixes f :: \"'a ⇒ 'b :: topological_comm_monoid_add\"\n assumes \"x ∉ A\" and \"(f has_sum S) A\"\n shows \"(f has_sum (f x + S)) (insert x A)\"\nproof (-)\n (*goal: ‹(f has_sum f x + S) (insert x A)›*)\n have \"(f has_sum (f x + S)) ({x} ∪ A)\"\n using assms (*‹x ∉ A› ‹(f has_sum S) A›*) apply (intro has_sum_Un_disjoint (*‹⟦(?f has_sum ?a) ?A; (?f has_sum ?b) ?B; ?A ∩ ?B = {}⟧ ⟹ (?f has_sum ?a + ?b) (?A ∪ ?B)›*))\n (*goals:\n 1. ‹⟦x ∉ A; (f has_sum S) A⟧ ⟹ (f has_sum f x) {x}›\n 2. ‹⟦x ∉ A; (f has_sum S) A⟧ ⟹ (f has_sum S) A›\n 3. ‹⟦x ∉ A; (f has_sum S) A⟧ ⟹ {x} ∩ A = {}›\n discuss goal 1*)\n apply ((auto intro: has_sum_finiteI (*‹⟦finite ?A; ?S = sum ?f ?A⟧ ⟹ (?f has_sum ?S) ?A›*))[1])\n (*discuss goal 2*)\n apply ((auto intro: has_sum_finiteI (*‹⟦finite ?A; ?S = sum ?f ?A⟧ ⟹ (?f has_sum ?S) ?A›*))[1])\n (*discuss goal 3*)\n apply ((auto intro: has_sum_finiteI (*‹⟦finite ?A; ?S = sum ?f ?A⟧ ⟹ (?f has_sum ?S) ?A›*))[1])\n (*proven 3 subgoals*) .\n thus \"?thesis\"\n (*goal: ‹(f has_sum f x + S) (insert x A)›*)\n by simp\nqed\n\nlemma infsum_insert:\n fixes f :: \"_ ⇒ 'a :: {topological_comm_monoid_add, t2_space}\"\n assumes \"f summable_on A\" \"a ∉ A\"\n shows \"infsum f (insert a A) = f a + infsum f A\"\n by (meson assms (*‹f summable_on A› ‹a ∉ A›*) has_sum_insert (*‹⟦?x ∉ ?A; (?f has_sum ?S) ?A⟧ ⟹ (?f has_sum ?f ?x + ?S) (insert ?x ?A)›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) summable_iff_has_sum_infsum (*‹(?f summable_on ?A) = (?f has_sum infsum ?f ?A) ?A›*))\n\nlemma has_sum_SigmaD:\n fixes f :: \"'b × 'c ⇒ 'a :: {topological_comm_monoid_add,t3_space}\"\n assumes sum1: \"(f has_sum S) (Sigma A B)\"\n assumes sum2: \"⋀x. x ∈ A ⟹ ((λy. f (x, y)) has_sum g x) (B x)\"\n shows \"(g has_sum S) A\"\n unfolding has_sum_def tendsto_def eventually_finite_subsets_at_top\n (*goal: ‹∀Sa::'a set. open Sa ⟶ (S::'a) ∈ Sa ⟶ (∃X::'b set. finite X ∧ X ⊆ (A::'b set) ∧ (∀Y::'b set. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ sum (g::'b ⇒ 'a) Y ∈ Sa))›*)\n apply safe\n (*goal: ‹∀Sa. open Sa ⟶ S ∈ Sa ⟶ (∃X. finite X ∧ X ⊆ A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ sum g Y ∈ Sa))›*)\n proof (goal_cases)\n (*goal: ‹⋀Sa::'a set. ⟦open Sa; (S::'a) ∈ Sa⟧ ⟹ ∃X::'b set. finite X ∧ X ⊆ (A::'b set) ∧ (∀Y::'b set. finite Y ∧ X ⊆ Y ∧ Y ⊆ A ⟶ sum (g::'b ⇒ 'a) Y ∈ Sa)›*)\n case (1 X) (*‹open X› ‹S ∈ X›*)\n with nhds_closed[of S X] (*‹⟦S ∈ X; open X⟧ ⟹ ∃A'. S ∈ A' ∧ closed A' ∧ A' ⊆ X ∧ (∀⇩F y in nhds S. y ∈ A')›*) obtain X' where X': \"S ∈ X'\" \"closed X'\" \"X' ⊆ X\" \"eventually (λy. y ∈ X') (nhds S)\"\n (*goal: ‹(⋀X'. ⟦S ∈ X'; closed X'; X' ⊆ X; ∀⇩F y in nhds S. y ∈ X'⟧ ⟹ thesis) ⟹ thesis›*)\n by blast\n from X'(4) (*‹∀⇩F y in nhds S. y ∈ X'›*) obtain X'' where X'': \"S ∈ X''\" \"open X''\" \"X'' ⊆ X'\"\n (*goal: ‹(⋀X''. ⟦S ∈ X''; open X''; X'' ⊆ X'⟧ ⟹ thesis) ⟹ thesis›*)\n by (auto simp: eventually_nhds (*‹eventually ?P (nhds ?a) = (∃S. open S ∧ ?a ∈ S ∧ (∀x∈S. ?P x))›*))\n with sum1 (*‹(f has_sum S) (Sigma A B)›*) obtain Y :: \"('b × 'c) set\" where Y: \"Y ⊆ Sigma A B\" \"finite Y\" \"⋀Z. Y ⊆ Z ⟹ Z ⊆ Sigma A B ⟹ finite Z ⟹ sum f Z ∈ X''\"\n (*goal: ‹(⋀Y. ⟦Y ⊆ Sigma A B; finite Y; ⋀Z. ⟦Y ⊆ Z; Z ⊆ Sigma A B; finite Z⟧ ⟹ sum f Z ∈ X''⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding has_sum_def tendsto_def eventually_finite_subsets_at_top\n (*goal: ‹(⋀Y. ⟦Y ⊆ Sigma A B; finite Y; ⋀Z. ⟦Y ⊆ Z; Z ⊆ Sigma A B; finite Z⟧ ⟹ sum f Z ∈ X''⟧ ⟹ thesis) ⟹ thesis›*)\n by force\n define Y1 :: \"'b set\" where \"Y1 = fst ` Y\"\n from Y (*‹Y ⊆ Sigma A B› ‹finite Y› ‹⟦Y ⊆ ?Z; ?Z ⊆ Sigma A B; finite ?Z⟧ ⟹ sum f ?Z ∈ X''›*) have Y1: \"Y1 ⊆ A\"\n by (auto simp: Y1_def (*‹Y1 = fst ` Y›*))\n define Y2 :: \"'b ⇒ 'c set\" where \"Y2 = (λx. {y. (x, y) ∈ Y})\"\n have Y2: \"finite (Y2 x)\" \"Y2 x ⊆ B x\" if \"x ∈ A\" for x\n using that (*‹(x::'b) ∈ (A::'b set)›*) Y(1,2) (*‹Y ⊆ Sigma A B› ‹finite Y›*) unfolding Y2_def\n (*goals:\n 1. ‹finite {y. (x, y) ∈ Y}›\n 2. ‹{y. (x, y) ∈ Y} ⊆ B x›*)\n apply -\n (*goals:\n 1. ‹⟦x ∈ A; Y ⊆ Sigma A B; finite Y⟧ ⟹ finite {y. (x, y) ∈ Y}›\n 2. ‹⟦x ∈ A; Y ⊆ Sigma A B; finite Y⟧ ⟹ {y. (x, y) ∈ Y} ⊆ B x›\n discuss goal 1*)\n apply (force simp: image_iff (*‹(?z ∈ ?f ` ?A) = (∃x∈?A. ?z = ?f x)›*) intro: finite_subset[of _ \"snd ` Y\"] (*‹⟦?A ⊆ snd ` Y; finite (snd ` Y)⟧ ⟹ finite ?A›*))\n (*discuss goal 2*)\n apply (force simp: image_iff (*‹(?z ∈ ?f ` ?A) = (∃x∈?A. ?z = ?f x)›*) intro: finite_subset[of _ \"snd ` Y\"] (*‹⟦?A ⊆ snd ` Y; finite (snd ` Y)⟧ ⟹ finite ?A›*))\n (*proven 2 subgoals*) .\n show \"?case\"\n (*goal: ‹∃Xa. finite Xa ∧ Xa ⊆ A ∧ (∀Y. finite Y ∧ Xa ⊆ Y ∧ Y ⊆ A ⟶ sum g Y ∈ X)›*)\n proof (rule exI[of _ Y1] (*‹?P Y1 ⟹ ∃x. ?P x›*), safe, goal_cases)\n (*goals:\n 1. ‹finite Y1›\n 2. ‹⋀x. x ∈ Y1 ⟹ x ∈ A›\n 3. ‹⋀Y. ⟦finite Y; Y1 ⊆ Y; Y ⊆ A⟧ ⟹ sum g Y ∈ X›*)\n case (3 Z) (*‹finite (Z::'b::type set)› ‹(Y1::'b set) ⊆ (Z::'b set)› ‹(Z::'b set) ⊆ (A::'b set)›*)\n define H where \"H = (INF x∈Z. filtercomap (λp. p x) (finite_subsets_at_top (B x)))\"\n have \"sum g Z ∈ X'\"\n proof (rule Lim_in_closed_set (*‹⟦closed ?S; ∀⇩F x in ?F. ?f x ∈ ?S; ?F ≠ bot; (?f ⤏ ?l) ?F⟧ ⟹ ?l ∈ ?S›*))\n (*goals:\n 1. ‹closed X'›\n 2. ‹∀⇩F x in ?F. ?f x ∈ X'›\n 3. ‹?F ≠ bot›\n 4. ‹(?f ⤏ sum g Z) ?F›*)\n show \"closed X'\"\n by fact\n next\n (*goals:\n 1. ‹∀⇩F x::?'b in ?F::?'b filter. (?f::?'b ⇒ 'a) x ∈ (X'::'a set)›\n 2. ‹(?F::?'b filter) ≠ bot›\n 3. ‹((?f::?'b ⇒ 'a) ⤏ sum (g::'b ⇒ 'a) (Z::'b set)) (?F::?'b filter)›*)\n show \"((λB'. sum (λx. sum (λy. f (x, y)) (B' x)) Z) ⤏ sum g Z) H\"\n unfolding H_def\n (*goal: ‹((λB'::'b ⇒ 'c set. ∑x::'b∈(Z::'b set). ∑y::'c∈B' x. (f::'b × 'c ⇒ 'a) (x, y)) ⤏ sum (g::'b ⇒ 'a) Z) (INF x::'b∈Z. filtercomap (λp::'b ⇒ 'c set. p x) (finite_subsets_at_top ((B::'b ⇒ 'c set) x)))›*)\n proof (intro tendsto_sum (*‹(⋀i. i ∈ ?I ⟹ (?f i ⤏ ?a i) ?F) ⟹ ((λx. ∑i∈?I. ?f i x) ⤏ sum ?a ?I) ?F›*) filterlim_INF' (*‹⟦?x ∈ ?A; filterlim ?f ?F (?G ?x)⟧ ⟹ filterlim ?f ?F (Inf (?G ` ?A))›*))\n (*goals:\n 1. ‹⋀x::'b. x ∈ (Z::'b set) ⟹ (?x1::'b ⇒ 'b) x ∈ Z›\n 2. ‹⋀x::'b. x ∈ (Z::'b set) ⟹ ((λxa::'b ⇒ 'c set. ∑y::'c∈xa x. (f::'b × 'c ⇒ 'a) (x, y)) ⤏ (g::'b ⇒ 'a) x) (filtercomap (λp::'b ⇒ 'c set. p ((?x1::'b ⇒ 'b) x)) (finite_subsets_at_top ((B::'b ⇒ 'c set) (?x1 x))))›*)\n fix x\n assume x: \"x ∈ Z\" (*‹(x::'b) ∈ (Z::'b set)›*)\n with \"3\" (*‹finite Z› ‹Y1 ⊆ Z› ‹(Z::'b set) ⊆ (A::'b set)›*) have \"x ∈ A\"\n by auto\n from sum2[OF this] (*‹((λy. f (x, y)) has_sum g x) (B x)›*) have \"(sum (λy. f (x, y)) ⤏ g x) (finite_subsets_at_top (B x))\"\n by (simp add: has_sum_def (*‹(?f has_sum ?x) ?A ≡ (sum ?f ⤏ ?x) (finite_subsets_at_top ?A)›*))\n thus \"((λB'. sum (λy. f (x, y)) (B' x)) ⤏ g x)\n (filtercomap (λp. p x) (finite_subsets_at_top (B x)))\"\n by (rule filterlim_compose[OF _ filterlim_filtercomap] (*‹filterlim ?g ?F3.0 ?F2.0 ⟹ LIM x filtercomap ?f ?F2.0. ?g (?f x) :> ?F3.0›*))\n qed (auto)\n (*solved the remaining goal: ‹⋀x. x ∈ Z ⟹ x ∈ Z›*)\n next\n (*goals:\n 1. ‹∀⇩F x in H. (∑xa∈Z. ∑y∈x xa. f (xa, y)) ∈ X'›\n 2. ‹H ≠ bot›*)\n show \"∀⇩F h in H. sum (λx. sum (λy. f (x, y)) (h x)) Z ∈ X'\"\n unfolding H_def\n (*goal: ‹∀⇩F h in INF x∈Z. filtercomap (λp. p x) (finite_subsets_at_top (B x)). (∑x∈Z. ∑y∈h x. f (x, y)) ∈ X'›*)\n proof (subst eventually_INF_finite[OF ‹finite Z›] (*‹eventually ?P (Inf (?F ` Z)) = (∃Q. (∀x∈Z. eventually (Q x) (?F x)) ∧ (∀y. (∀x∈Z. Q x y) ⟶ ?P y))›*), rule exI (*‹?P ?x ⟹ ∃x. ?P x›*), safe)\n (*goals:\n 1. ‹⋀x::'b. x ∈ (Z::'b set) ⟹ eventually ((?Q::'b ⇒ ('b ⇒ 'c set) ⇒ bool) x) (filtercomap (λp::'b ⇒ 'c set. p x) (finite_subsets_at_top ((B::'b ⇒ 'c set) x)))›\n 2. ‹⋀y::'b ⇒ 'c set. ∀x::'b∈Z::'b set. (?Q::'b ⇒ ('b ⇒ 'c set) ⇒ bool) x y ⟹ (∑x::'b∈Z. ∑y::'c∈y x. (f::'b × 'c ⇒ 'a) (x, y)) ∈ (X'::'a set)›*)\n fix x\n assume x: \"x ∈ Z\" (*‹(x::'b) ∈ (Z::'b set)›*)\n hence x': \"x ∈ A\"\n using \"3\" (*‹finite Z› ‹Y1 ⊆ Z› ‹(Z::'b::type set) ⊆ (A::'b::type set)›*) by auto\n show \"eventually (λh. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x)\n (filtercomap (λp. p x) (finite_subsets_at_top (B x)))\"\n using \"3\" (*‹finite Z› ‹(Y1::'b set) ⊆ (Z::'b set)› ‹Z ⊆ A›*) Y2[OF x'] (*‹finite (Y2 x)› ‹Y2 x ⊆ B x›*) apply (intro eventually_filtercomapI (*‹eventually ?P ?F ⟹ ∀⇩F x in filtercomap ?f ?F. ?P (?f x)›*))\n (*goal: ‹∀⇩F h in filtercomap (λp. p x) (finite_subsets_at_top (B x)). finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x›*)\n by (auto simp: eventually_finite_subsets_at_top (*‹eventually ?P (finite_subsets_at_top ?A) = (∃X. finite X ∧ X ⊆ ?A ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ ?A ⟶ ?P Y))›*) intro: exI[of _ \"Y2 x\"] (*‹?P (Y2 x) ⟹ ∃x. ?P x›*))\n next\n (*goal: ‹⋀y::'b::type ⇒ 'c::type set. ∀x::'b::type∈Z::'b::type set. finite (y x) ∧ (Y2::'b::type ⇒ 'c::type set) x ⊆ y x ∧ y x ⊆ (B::'b::type ⇒ 'c::type set) x ⟹ (∑x::'b::type∈Z. ∑y::'c::type∈y x. (f::'b::type × 'c::type ⇒ 'a::{topological_comm_monoid_add,t3_space}) (x, y)) ∈ (X'::'a::{topological_comm_monoid_add,t3_space} set)›*)\n fix h\n assume \"*\": \"∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x\" (*‹∀x::'b∈Z::'b set. finite ((h::'b ⇒ 'c set) x) ∧ (Y2::'b ⇒ 'c set) x ⊆ h x ∧ h x ⊆ (B::'b ⇒ 'c set) x›*)\n hence \"sum (λx. sum (λy. f (x, y)) (h x)) Z = sum f (Sigma Z h)\"\n using ‹finite Z› (*‹finite Z›*) apply (subst sum.Sigma (*‹⟦finite (?A::?'b set); ∀x::?'b∈?A. finite ((?B::?'b ⇒ ?'c set) x)⟧ ⟹ (∑x::?'b∈?A. sum ((?g::?'b ⇒ ?'c ⇒ ?'a) x) (?B x)) = (∑(x::?'b, y::?'c)∈Sigma ?A ?B. ?g x y)›*))\n (*goals:\n 1. ‹⟦∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x; finite Z⟧ ⟹ finite Z›\n 2. ‹⟦∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x; finite Z⟧ ⟹ ∀x∈Z. finite (h x)›\n 3. ‹⟦∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x; finite Z⟧ ⟹ (∑(x, y)∈Sigma Z h. f (x, y)) = sum f (Sigma Z h)›\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: ‹(∑x∈Z. ∑y∈h x. f (x, y)) = sum f (Sigma Z h)›*) have \"… ∈ X''\"\n using \"*\" (*‹∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x›*) \"3\" (*‹finite Z› ‹(Y1::'b::type set) ⊆ (Z::'b::type set)› ‹Z ⊆ A›*) Y(1,2) (*‹(Y::('b::type × 'c::type) set) ⊆ Sigma (A::'b::type set) (B::'b::type ⇒ 'c::type set)› ‹finite Y›*) apply (intro Y (*‹Y ⊆ Sigma A B› ‹finite Y› ‹⟦Y ⊆ ?Z; ?Z ⊆ Sigma A B; finite ?Z⟧ ⟹ sum f ?Z ∈ X''›*))\n (*goals:\n 1. ‹⟦∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x; finite Z; Y1 ⊆ Z; Z ⊆ A; Y ⊆ Sigma A B; finite Y⟧ ⟹ Y ⊆ Sigma Z h›\n 2. ‹⟦∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x; finite Z; Y1 ⊆ Z; Z ⊆ A; Y ⊆ Sigma A B; finite Y⟧ ⟹ Sigma Z h ⊆ Sigma A B›\n 3. ‹⟦∀x∈Z. finite (h x) ∧ Y2 x ⊆ h x ∧ h x ⊆ B x; finite Z; Y1 ⊆ Z; Z ⊆ A; Y ⊆ Sigma A B; finite Y⟧ ⟹ finite (Sigma Z h)›\n discuss goal 1*)\n apply (force simp: Y1_def (*‹Y1 = fst ` Y›*) Y2_def (*‹Y2 = (λx. {y. (x, y) ∈ Y})›*))\n (*discuss goal 2*)\n apply (force simp: Y1_def (*‹(Y1::'b::type set) = fst ` (Y::('b::type × 'c::type) set)›*) Y2_def (*‹(Y2::'b::type ⇒ 'c::type set) = (λx::'b::type. {y::'c::type. (x, y) ∈ (Y::('b::type × 'c::type) set)})›*))\n (*discuss goal 3*)\n apply (force simp: Y1_def (*‹Y1 = fst ` Y›*) Y2_def (*‹Y2 = (λx. {y. (x, y) ∈ Y})›*))\n (*proven 3 subgoals*) .\n also (*calculation: ‹(∑x∈Z. ∑y∈h x. f (x, y)) ∈ X''›*) have \"X'' ⊆ X'\"\n by fact\n finally (*calculation: ‹(∑x::'b∈(Z::'b set). ∑y::'c∈(h::'b ⇒ 'c set) x. (f::'b × 'c ⇒ 'a) (x, y)) ∈ (X'::'a set)›*) show \"sum (λx. sum (λy. f (x, y)) (h x)) Z ∈ X'\" .\n qed\n next\n (*goal: ‹H ≠ bot›*)\n have \"H = (INF x∈SIGMA x:Z. {X. finite X ∧ X ⊆ B x}.\n principal {y. finite (y (fst x)) ∧ snd x ⊆ y (fst x) ∧ y (fst x) ⊆ B (fst x)})\"\n unfolding H_def finite_subsets_at_top_def filtercomap_INF filtercomap_principal\n (*goal: ‹(INF x∈Z. INF b∈{X. finite X ∧ X ⊆ B x}. principal ((λp. p x) -` {Y. finite Y ∧ b ⊆ Y ∧ Y ⊆ B x})) = (INF x∈SIGMA x:Z. {X. finite X ∧ X ⊆ B x}. principal {y. finite (y (fst x)) ∧ snd x ⊆ y (fst x) ∧ y (fst x) ⊆ B (fst x)})›*)\n by (simp add: INF_Sigma (*‹(INF i∈?A. Inf (?f i ` ?B i)) = (INF p∈Sigma ?A ?B. ?f (fst p) (snd p))›*))\n also (*calculation: ‹(H::('b::type ⇒ 'c::type set) filter) = (INF x::'b::type × 'c::type set∈SIGMA x::'b::type:Z::'b::type set. {X::'c::type set. finite X ∧ X ⊆ B x}. principal {y::'b::type ⇒ 'c::type set. finite (y (fst x)) ∧ snd x ⊆ y (fst x) ∧ y (fst x) ⊆ (B::'b::type ⇒ 'c::type set) (fst x)})›*) have \"… ≠ bot\"\n proof (rule INF_filter_not_bot (*‹(⋀X::?'i set. ⟦X ⊆ (?B::?'i set); finite X⟧ ⟹ Inf ((?F::?'i ⇒ ?'a filter) ` X) ≠ bot) ⟹ Inf (?F ` ?B) ≠ bot›*), subst INF_principal_finite (*‹finite ?X ⟹ (INF x∈?X. principal (?f x)) = principal (⋂ (?f ` ?X))›*), goal_cases)\n (*goals:\n 1. ‹⋀X. ⟦X ⊆ (SIGMA x:Z. {X. finite X ∧ X ⊆ B x}); finite X⟧ ⟹ finite X›\n 2. ‹⋀X. ⟦X ⊆ (SIGMA x:Z. {X. finite X ∧ X ⊆ B x}); finite X⟧ ⟹ principal (⋂x∈X. {y. finite (y (fst x)) ∧ snd x ⊆ y (fst x) ∧ y (fst x) ⊆ B (fst x)}) ≠ bot›*)\n case (2 X) (*‹X ⊆ (SIGMA x:Z. {X. finite X ∧ X ⊆ B x})› ‹finite X›*)\n define H' where \"H' = (⋂x∈X. {y. finite (y (fst x)) ∧ snd x ⊆ y (fst x) ∧ y (fst x) ⊆ B (fst x)})\"\n from \"2\" (*‹X ⊆ (SIGMA x:Z. {X. finite X ∧ X ⊆ B x})› ‹finite X›*) have \"(λx. ⋃(y,Y)∈X. if x = y then Y else {}) ∈ H'\"\n by (force 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))›*) simp: H'_def (*‹H' = (⋂x∈X. {y. finite (y (fst x)) ∧ snd x ⊆ y (fst x) ∧ y (fst x) ⊆ B (fst x)})›*))\n hence \"H' ≠ {}\"\n by blast\n thus \"principal H' ≠ bot\"\n by (simp add: principal_eq_bot_iff (*‹(principal ?X = bot) = (?X = {})›*))\n qed\n finally (*calculation: ‹H ≠ bot›*) show \"H ≠ bot\" .\n qed\n also (*calculation: ‹sum (g::'b::type ⇒ 'a::{topological_comm_monoid_add,t3_space}) (Z::'b::type set) ∈ (X'::'a::{topological_comm_monoid_add,t3_space} set)›*) have \"X' ⊆ X\"\n by fact\n finally (*calculation: ‹sum (g::'b ⇒ 'a) (Z::'b set) ∈ (X::'a set)›*) show \"sum g Z ∈ X\" .\n qed (insert Y( (*‹Y ⊆ Sigma A B› ‹finite Y›*) 1,2), auto simp: Y1_def (*‹Y1 = fst ` Y›*))\n (*solves the remaining goals:\n 1. ‹finite (Y1::'b set)›\n 2. ‹⋀x::'b. x ∈ (Y1::'b set) ⟹ x ∈ (A::'b set)›*)\n qed\n\nlemma has_sum_unique:\n fixes f :: \"_ ⇒ 'a :: {topological_comm_monoid_add, t2_space}\"\n assumes \"(f has_sum x) A\" \"(f has_sum y) A\"\n shows \"x = y\"\n using assms (*‹(f has_sum x) A› ‹(f has_sum y) A›*) unfolding has_sum_def\n (*goal: ‹x = y›*)\n using tendsto_unique (*‹⟦?F ≠ bot; (?f ⤏ ?a) ?F; (?f ⤏ ?b) ?F⟧ ⟹ ?a = ?b›*) finite_subsets_at_top_neq_bot (*‹finite_subsets_at_top ?A ≠ bot›*) by blast\n\nlemma has_sum_SigmaI:\n fixes f :: \"_ ⇒ 'a :: {topological_comm_monoid_add, t3_space}\"\n assumes f: \"⋀x. x ∈ A ⟹ ((λy. f (x, y)) has_sum g x) (B x)\"\n assumes g: \"(g has_sum S) A\"\n assumes summable: \"f summable_on Sigma A B\"\n shows \"(f has_sum S) (Sigma A B)\"\n by (metis f (*‹?x ∈ A ⟹ ((λy. f (?x, y)) has_sum g ?x) (B ?x)›*) g (*‹(g has_sum S) A›*) has_sum_SigmaD (*‹⟦(?f has_sum ?S) (Sigma ?A ?B); ⋀x. x ∈ ?A ⟹ ((λy. ?f (x, y)) has_sum ?g x) (?B x)⟧ ⟹ (?g has_sum ?S) ?A›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_unique (*‹⟦(?f has_sum ?x) ?A; (?f has_sum ?y) ?A⟧ ⟹ ?x = ?y›*) local.summable (*‹f summable_on Sigma A B›*))\n\nlemma summable_on_SigmaD1:\n fixes f :: \"_ ⇒ _ ⇒ 'a :: {complete_uniform_space, uniform_topological_group_add, ab_group_add, topological_comm_monoid_add}\"\n assumes f: \"(λ(x,y). f x y) summable_on Sigma A B\"\n assumes x: \"x ∈ A\"\n shows \"f x summable_on B x\"\nproof (-)\n (*goal: ‹f x summable_on B x›*)\n have \"(λ(x,y). f x y) summable_on Sigma {x} B\"\n using f (*‹(λ(x, y). f x y) summable_on Sigma A B›*) apply (rule summable_on_subset (*‹⟦?f summable_on ?A; ?B ⊆ ?A⟧ ⟹ ?f summable_on ?B›*))\n (*goal: ‹(λ(x, y). f x y) summable_on Sigma {x} B›*)\n by (use x in auto)\n also (*calculation: ‹(λ(x::'d, y::'e). (f::'d ⇒ 'e ⇒ 'a) x y) summable_on Sigma {x::'d} (B::'d ⇒ 'e set)›*) have \"?this ⟷ ((λy. f x y) ∘ snd) summable_on Sigma {x} B\"\n apply (intro summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n (*goal: ‹((λ(x, y). f x y) summable_on Sigma {x} B) = (f x ∘ snd summable_on Sigma {x} B)›*)\n by auto\n also (*calculation: ‹f x ∘ snd summable_on Sigma {x} B›*) have \"… ⟷ (λy. f x y) summable_on snd ` Sigma {x} B\"\n apply (intro summable_on_reindex [symmetric] (*‹inj_on (?h::?'a ⇒ ?'b) (?A::?'a set) ⟹ ((?g::?'b ⇒ ?'c) ∘ ?h summable_on ?A) = (?g summable_on ?h ` ?A)›*) 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: ‹((f::'d::type ⇒ 'e::type ⇒ 'a::{ab_group_add,complete_uniform_space,topological_comm_monoid_add,uniform_topological_group_add}) (x::'d::type) ∘ snd summable_on Sigma {x} (B::'d::type ⇒ 'e::type set)) = (f x summable_on snd ` Sigma {x} B)›*)\n by auto\n also (*calculation: ‹f x summable_on snd ` Sigma {x} B›*) have \"snd ` Sigma {x} B = B x\"\n by (force simp: Sigma_def (*‹Sigma ?A ?B ≡ ⋃x∈?A. ⋃y∈?B x. {(x, y)}›*))\n finally (*calculation: ‹f x summable_on B x›*) show \"?thesis\"\n (*goal: ‹f x summable_on B x›*) .\nqed\n\nlemma has_sum_swap:\n \"(f has_sum S) (A × B) ⟷ ((λ(x,y). f (y,x)) has_sum S) (B × A)\"\nproof (-)\n (*goal: ‹((f::'a × 'b ⇒ 'c) has_sum (S::'c)) ((A::'a set) × (B::'b set)) = ((λ(x::'b, y::'a). f (y, x)) has_sum S) (B × A)›*)\n have \"bij_betw (λ(x,y). (y,x)) (B × A) (A × B)\"\n apply (rule bij_betwI[of _ _ _ \"λ(x,y). (y,x)\"] (*‹⟦?f ∈ ?A → ?B; (λ(x, y). (y, x)) ∈ ?B → ?A; ⋀x. x ∈ ?A ⟹ (case ?f x of (x, y) ⇒ (y, x)) = x; ⋀y. y ∈ ?B ⟹ ?f (case y of (x, y) ⇒ (y, x)) = y⟧ ⟹ bij_betw ?f ?A ?B›*))\n (*goals:\n 1. ‹(λ(x, y). (y, x)) ∈ B × A → A × B›\n 2. ‹(λ(x, y). (y, x)) ∈ A × B → B × A›\n 3. ‹⋀x. x ∈ B × A ⟹ (case case x of (x, y) ⇒ (y, x) of (x, y) ⇒ (y, x)) = x›\n 4. ‹⋀y. y ∈ A × B ⟹ (case case y of (x, y) ⇒ (y, x) of (x, y) ⇒ (y, x)) = 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 from has_sum_reindex_bij_betw[OF this, where f = f] (*‹((λx::'b × 'a. (f::'a × 'b ⇒ 'c) (case x of (x::'b, y::'a) ⇒ (y, x))) has_sum (?S::'c)) ((B::'b set) × (A::'a set)) = (f has_sum ?S) (A × B)›*) show \"?thesis\"\n (*goal: ‹((f::'a × 'b ⇒ 'c) has_sum (S::'c)) ((A::'a set) × (B::'b set)) = ((λ(x::'b, y::'a). f (y, x)) has_sum S) (B × A)›*)\n by (simp add: case_prod_unfold (*‹case_prod = (λ(c::?'a ⇒ ?'b ⇒ ?'c) p::?'a × ?'b. c (fst p) (snd p))›*))\nqed\n\n\nlemma summable_on_swap:\n \"f summable_on (A × B) ⟷ (λ(x,y). f (y,x)) summable_on (B × A)\"\n by (metis has_sum_swap (*‹(?f has_sum ?S) (?A × ?B) = ((λ(x, y). ?f (y, x)) has_sum ?S) (?B × ?A)›*) summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n\nlemma has_sum_cmult_right_iff:\n fixes c :: \"'a :: {topological_semigroup_mult, field}\"\n assumes \"c ≠ 0\"\n shows \"((λx. c * f x) has_sum S) A ⟷ (f has_sum (S / c)) A\"\n using has_sum_cmult_right[of f A \"S/c\" c] (*‹(f has_sum S / c) A ⟹ ((λx. c * f x) has_sum c * (S / c)) A›*) has_sum_cmult_right[of \"λx. c * f x\" A S \"inverse c\"] (*‹((λx. c * f x) has_sum S) A ⟹ ((λx. inverse c * (c * f x)) has_sum inverse c * S) A›*) assms (*‹c ≠ 0›*) by (auto simp: field_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 69 facts*))\n\nlemma has_sum_cmult_left_iff:\n fixes c :: \"'a :: {topological_semigroup_mult, field}\"\n assumes \"c ≠ 0\"\n shows \"((λx. f x * c) has_sum S) A ⟷ (f has_sum (S / c)) A\"\n by (smt (verit, best) assms (*‹c ≠ 0›*) has_sum_cmult_right_iff (*‹?c ≠ 0 ⟹ ((λx. ?c * ?f x) has_sum ?S) ?A = (?f has_sum ?S / ?c) ?A›*) has_sum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f has_sum ?x) ?A = (?g has_sum ?x) ?A›*) mult.commute (*‹?a * ?b = ?b * ?a›*))\n\nlemma finite_nonzero_values_imp_summable_on:\n assumes \"finite {x∈X. f x ≠ 0}\"\n shows \"f summable_on X\"\n by (smt (verit, del_insts) Diff_iff (*‹(?c ∈ ?A - ?B) = (?c ∈ ?A ∧ ?c ∉ ?B)›*) assms (*‹finite {x ∈ X. f x ≠ 0}›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) summable_on_cong_neutral (*‹⟦⋀x. x ∈ ?T - ?S ⟹ ?g x = 0; ⋀x. x ∈ ?S - ?T ⟹ ?f x = 0; ⋀x. x ∈ ?S ∩ ?T ⟹ ?f x = ?g x⟧ ⟹ (?f summable_on ?S) = (?g summable_on ?T)›*) summable_on_finite (*‹finite ?F ⟹ ?f summable_on ?F›*))\n\nlemma summable_on_of_int_iff:\n \"(λx::'a. of_int (f x) :: 'b :: real_normed_algebra_1) summable_on A ⟷ f summable_on A\"\nproof (standard)\n (*goals:\n 1. ‹(λx. of_int (f x)) summable_on A ⟹ f summable_on A›\n 2. ‹f summable_on A ⟹ (λx. of_int (f x)) summable_on A›*)\n assume \"f summable_on A\" (*‹(f::'a ⇒ int) summable_on (A::'a set)›*)\n thus \"(λx. of_int (f x)) summable_on A\"\n apply (rule summable_on_homomorphism (*‹⟦?f summable_on ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ (λx. ?h (?f x)) summable_on ?A›*))\n (*goals:\n 1. ‹of_int 0 = 0›\n 2. ‹⋀a b. of_int (a + b) = of_int a + of_int b›\n 3. ‹continuous_on UNIV of_int›\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*) .\nnext\n (*goal: ‹(λx. of_int (f x)) summable_on A ⟹ f summable_on A›*)\n assume \"(λx. of_int (f x) :: 'b) summable_on A\" (*‹(λx::'a. of_int ((f::'a ⇒ int) x)) summable_on (A::'a set)›*)\n then obtain S where \"((λx. of_int (f x) :: 'b) has_sum S) A\"\n (*goal: ‹(⋀S. ((λx. of_int (f x)) has_sum S) A ⟹ thesis) ⟹ thesis›*)\n by (auto simp: summable_on_def (*‹?f summable_on ?A ≡ ∃x. (?f has_sum x) ?A›*))\n hence \"(sum (λx. of_int (f x) :: 'b) ⤏ S) (finite_subsets_at_top A)\"\n unfolding has_sum_def\n (*goal: ‹(sum (λx. of_int (f x)) ⤏ S) (finite_subsets_at_top A)›*) .\n moreover have \"1/2 > (0 :: real)\"\n by auto\n ultimately have \"eventually (λX. dist (sum (λx. of_int (f x) :: 'b) X) S < 1/2)\n (finite_subsets_at_top A)\"\n unfolding tendsto_iff\n (*goal: ‹∀⇩F X in finite_subsets_at_top A. dist (∑x∈X. of_int (f x)) S < 1 / 2›*)\n by blast\n then obtain X where X: \"finite X\" \"X ⊆ A\" \"⋀Y. finite Y ⟹ X ⊆ Y ⟹ Y ⊆ A ⟹ dist (sum (λx. of_int (f x)) Y) S < 1/2\"\n (*goal: ‹(⋀X. ⟦finite X; X ⊆ A; ⋀Y. ⟦finite Y; X ⊆ Y; Y ⊆ A⟧ ⟹ dist (∑x∈Y. of_int (f x)) S < 1 / 2⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹(⋀X::'a set. ⟦finite X; X ⊆ (A::'a set); ⋀Y::'a set. ⟦finite Y; X ⊆ Y; Y ⊆ A⟧ ⟹ dist (∑x::'a∈Y. of_int ((f::'a ⇒ int) x)) (S::'b) < (1::real) / (2::real)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n by metis\n have \"sum f Y = sum f X\" if \"finite Y\" \"X ⊆ Y\" \"Y ⊆ A\" for Y\n proof (-)\n (*goal: ‹sum (f::'a ⇒ int) (Y::'a set) = sum f (X::'a set)›*)\n have \"dist (sum (λx. of_int (f x)) X) S < 1/2\"\n apply (intro X (*‹finite X› ‹X ⊆ A› ‹⟦finite ?Y; X ⊆ ?Y; ?Y ⊆ A⟧ ⟹ dist (∑x∈?Y. of_int (f x)) S < 1 / 2›*))\n (*goal: ‹dist (∑x::'a∈(X::'a set). of_int ((f::'a ⇒ int) x)) (S::'b) < (1::real) / (2::real)›*)\n by auto\n moreover have \"dist (sum (λx. of_int (f x)) Y) S < 1/2\"\n by (intro X (*‹finite X› ‹X ⊆ A› ‹⟦finite ?Y; X ⊆ ?Y; ?Y ⊆ A⟧ ⟹ dist (∑x∈?Y. of_int (f x)) S < 1 / 2›*) that (*‹finite Y› ‹X ⊆ Y› ‹Y ⊆ A›*))\n ultimately have \"dist (sum (λx. of_int (f x)) X) (sum (λx. of_int (f x) :: 'b) Y) <\n 1/2 + 1/2\"\n using dist_triangle_less_add (*‹⟦dist ?x1.0 ?y < ?e1.0; dist ?x2.0 ?y < ?e2.0⟧ ⟹ dist ?x1.0 ?x2.0 < ?e1.0 + ?e2.0›*) by blast\n thus \"?thesis\"\n (*goal: ‹sum (f::'a::type ⇒ int) (Y::'a::type set) = sum f (X::'a::type set)›*)\n by (simp add: dist_norm (*‹dist ?x ?y = norm (?x - ?y)›*) flip: of_int_sum (*‹of_int (sum ?f ?A) = (∑x∈?A. of_int (?f x))›*) of_int_diff (*‹of_int (?w - ?z) = of_int ?w - of_int ?z›*))\n qed\n then have \"{x∈A. f x ≠ 0} ⊆ X\"\n by (smt (verit) X (*‹finite X› ‹X ⊆ A› ‹⟦finite ?Y; X ⊆ ?Y; ?Y ⊆ A⟧ ⟹ dist (∑x∈?Y. of_int (f x)) S < 1 / 2›*) finite_insert (*‹finite (insert ?a ?A) = finite ?A›*) insert_iff (*‹(?a ∈ insert ?b ?A) = (?a = ?b ∨ ?a ∈ ?A)›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) subset_eq (*‹(?A ⊆ ?B) = (∀x∈?A. x ∈ ?B)›*) sum.insert (*‹⟦finite ?A; ?x ∉ ?A⟧ ⟹ sum ?g (insert ?x ?A) = ?g ?x + sum ?g ?A›*))\n with ‹finite X› (*‹finite X›*) have \"finite {x∈A. f x ≠ 0}\"\n using finite_subset (*‹⟦(?A::?'a::type set) ⊆ (?B::?'a::type set); finite ?B⟧ ⟹ finite ?A›*) by blast\n thus \"f summable_on A\"\n by (rule finite_nonzero_values_imp_summable_on (*‹finite {x::?'a::type ∈ ?X::?'a::type set. (?f::?'a::type ⇒ ?'b::{comm_monoid_add,topological_space}) x ≠ (0::?'b::{comm_monoid_add,topological_space})} ⟹ ?f summable_on ?X›*))\nqed\n\nlemma summable_on_of_nat_iff:\n \"(λx::'a. of_nat (f x) :: 'b :: real_normed_algebra_1) summable_on A ⟷ f summable_on A\"\nproof (standard)\n (*goals:\n 1. ‹(λx::'a. of_nat ((f::'a ⇒ nat) x)) summable_on (A::'a set) ⟹ f summable_on A›\n 2. ‹(f::'a ⇒ nat) summable_on (A::'a set) ⟹ (λx::'a. of_nat (f x)) summable_on A›*)\n assume \"f summable_on A\" (*‹(f::'a ⇒ nat) summable_on (A::'a set)›*)\n thus \"(λx. of_nat (f x) :: 'b) summable_on A\"\n apply (rule summable_on_homomorphism (*‹⟦?f summable_on ?A; ?h 0 = 0; ⋀a b. ?h (a + b) = ?h a + ?h b; continuous_on UNIV ?h⟧ ⟹ (λx. ?h (?f x)) summable_on ?A›*))\n (*goals:\n 1. ‹of_nat 0 = 0›\n 2. ‹⋀a b. of_nat (a + b) = of_nat a + of_nat b›\n 3. ‹continuous_on UNIV of_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*) .\nnext\n (*goal: ‹(λx. of_nat (f x)) summable_on A ⟹ f summable_on A›*)\n assume \"(λx. of_nat (f x) :: 'b) summable_on A\" (*‹(λx::'a. of_nat ((f::'a ⇒ nat) x)) summable_on (A::'a set)›*)\n hence \"(λx. of_int (int (f x)) :: 'b) summable_on A\"\n by simp\n also (*calculation: ‹(λx::'a::type. of_int (int ((f::'a::type ⇒ nat) x))) summable_on (A::'a::type set)›*) have \"?this ⟷ (λx. int (f x)) summable_on A\"\n by (rule summable_on_of_int_iff (*‹((λx::?'a. of_int ((?f::?'a ⇒ int) x)) summable_on (?A::?'a set)) = (?f summable_on ?A)›*))\n also (*calculation: ‹(λx. int (f x)) summable_on A›*) have \"… ⟷ f summable_on A\"\n by (simp add: summable_on_discrete_iff (*‹(?f summable_on ?A) = finite {x ∈ ?A. ?f x ≠ 0}›*))\n finally (*calculation: ‹f summable_on A›*) show \"f summable_on A\" .\nqed\n\nlemma infsum_of_nat:\n \"infsum (λx::'a. of_nat (f x) :: 'b :: {real_normed_algebra_1}) A = of_nat (infsum f A)\"\n by (metis has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_of_nat (*‹(?f has_sum ?S) ?A ⟹ ((λx. of_nat (?f x)) has_sum of_nat ?S) ?A›*) infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*) infsum_def (*‹infsum ?f ?A = (if ?f summable_on ?A then Lim (finite_subsets_at_top ?A) (sum ?f) else 0)›*) of_nat_0 (*‹of_nat 0 = 0›*) summable_on_of_nat_iff (*‹((λx. of_nat (?f x)) summable_on ?A) = (?f summable_on ?A)›*))\n\nlemma infsum_of_int:\n \"infsum (λx::'a. of_int (f x) :: 'b :: {real_normed_algebra_1}) A = of_int (infsum f A)\"\n by (metis has_sum_infsum (*‹(?f::?'a ⇒ ?'b) summable_on (?S::?'a set) ⟹ (?f has_sum infsum ?f ?S) ?S›*) has_sum_of_int (*‹((?f::?'a ⇒ int) has_sum (?S::int)) (?A::?'a set) ⟹ ((λx::?'a. of_int (?f x)) has_sum of_int ?S) ?A›*) infsumI (*‹((?f::?'a ⇒ ?'b) has_sum (?x::?'b)) (?A::?'a set) ⟹ infsum ?f ?A = ?x›*) infsum_not_exists (*‹¬ (?f::?'a ⇒ ?'b) summable_on (?A::?'a set) ⟹ infsum ?f ?A = (0::?'b)›*) of_int_0 (*‹of_int (0::int) = (0::?'a)›*) summable_on_of_int_iff (*‹((λx::?'a. of_int ((?f::?'a ⇒ int) x)) summable_on (?A::?'a set)) = (?f summable_on ?A)›*))\n\n\nlemma summable_on_SigmaI:\n fixes f :: \"_ ⇒ 'a :: {linorder_topology, ordered_comm_monoid_add, topological_comm_monoid_add,\n conditionally_complete_linorder}\"\n assumes f: \"⋀x. x ∈ A ⟹ ((λy. f (x, y)) has_sum g x) (B x)\"\n assumes g: \"g summable_on A\"\n assumes f_nonneg: \"⋀x y. x ∈ A ⟹ y ∈ B x ⟹ f (x, y) ≥ (0 :: 'a)\"\n shows \"f summable_on Sigma A B\"\nproof (-)\n (*goal: ‹f summable_on Sigma A B›*)\n have g_nonneg: \"g x ≥ 0\" if \"x ∈ A\" for x\n using f (*‹?x ∈ A ⟹ ((λy. f (?x, y)) has_sum g ?x) (B ?x)›*) apply (rule has_sum_nonneg (*‹⟦(?f has_sum ?a) ?M; ⋀x. x ∈ ?M ⟹ 0 ≤ ?f x⟧ ⟹ 0 ≤ ?a›*))\n (*goals:\n 1. ‹x ∈ A›\n 2. ‹⋀xa. xa ∈ B x ⟹ 0 ≤ f (x, xa)›\n discuss goal 1*)\n apply ((use f_nonneg that in auto)[1])\n (*discuss goal 2*)\n apply ((use f_nonneg that in auto)[1])\n (*proven 2 subgoals*) .\n obtain C where C: \"eventually (λX. sum g X ≤ C) (finite_subsets_at_top A)\"\n (*goal: ‹(⋀C. ∀⇩F X in finite_subsets_at_top A. sum g X ≤ C ⟹ thesis) ⟹ thesis›*)\n using summable_on_imp_bounded_partial_sums[OF g] (*‹∃C. ∀⇩F X in finite_subsets_at_top A. sum g X ≤ C›*) by blast\n have sum_g_le: \"sum g X ≤ C\" if X: \"finite X\" \"X ⊆ A\" for X\n proof (-)\n (*goal: ‹sum g X ≤ C›*)\n from C (*‹∀⇩F X in finite_subsets_at_top A. sum g X ≤ C›*) obtain X' where X': \"finite X'\" \"X' ⊆ A\" \"⋀Y. finite Y ⟹ X' ⊆ Y ⟹ Y ⊆ A ⟹ sum g Y ≤ C\"\n (*goal: ‹(⋀X'. ⟦finite X'; X' ⊆ A; ⋀Y. ⟦finite Y; X' ⊆ Y; Y ⊆ A⟧ ⟹ sum g Y ≤ C⟧ ⟹ thesis) ⟹ thesis›*)\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹(⋀X'. ⟦finite X'; X' ⊆ A; ⋀Y. ⟦finite Y; X' ⊆ Y; Y ⊆ A⟧ ⟹ sum g Y ≤ C⟧ ⟹ thesis) ⟹ thesis›*)\n by metis\n have \"sum g X ≤ sum g (X ∪ X')\"\n using X (*‹finite X› ‹X ⊆ A›*) X' (*‹finite X'› ‹X' ⊆ A› ‹⟦finite ?Y; X' ⊆ ?Y; ?Y ⊆ A⟧ ⟹ sum g ?Y ≤ C›*) apply (intro sum_mono2 (*‹⟦finite ?B; ?A ⊆ ?B; ⋀b. b ∈ ?B - ?A ⟹ 0 ≤ ?f b⟧ ⟹ sum ?f ?A ≤ sum ?f ?B›*) g_nonneg (*‹?x ∈ A ⟹ 0 ≤ g ?x›*))\n (*goals:\n 1. ‹⟦finite (X::'c set); X ⊆ (A::'c set); finite (X'::'c set); X' ⊆ A; ⋀Y::'c set. ⟦finite Y; X' ⊆ Y; Y ⊆ A⟧ ⟹ sum (g::'c ⇒ 'a) Y ≤ (C::'a)⟧ ⟹ finite (X ∪ X')›\n 2. ‹⟦finite (X::'c set); X ⊆ (A::'c set); finite (X'::'c set); X' ⊆ A; ⋀Y::'c set. ⟦finite Y; X' ⊆ Y; Y ⊆ A⟧ ⟹ sum (g::'c ⇒ 'a) Y ≤ (C::'a)⟧ ⟹ X ⊆ X ∪ X'›\n 3. ‹⋀b::'c. ⟦finite (X::'c set); X ⊆ (A::'c set); finite (X'::'c set); X' ⊆ A; ⋀Y::'c set. ⟦finite Y; X' ⊆ Y; Y ⊆ A⟧ ⟹ sum (g::'c ⇒ 'a) Y ≤ (C::'a); b ∈ X ∪ X' - X⟧ ⟹ 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 also (*calculation: ‹sum g X ≤ sum g (X ∪ X')›*) have \"… ≤ C\"\n using X (*‹finite X› ‹X ⊆ A›*) X'(1,2) (*‹finite X'› ‹X' ⊆ A›*) apply (intro X'( (*‹⟦finite ?Y; X' ⊆ ?Y; ?Y ⊆ A⟧ ⟹ sum g ?Y ≤ C›*) 3))\n (*goals:\n 1. ‹⟦finite X; X ⊆ A; finite X'; X' ⊆ A⟧ ⟹ finite (X ∪ X')›\n 2. ‹⟦finite X; X ⊆ A; finite X'; X' ⊆ A⟧ ⟹ X' ⊆ X ∪ X'›\n 3. ‹⟦finite X; X ⊆ A; finite X'; X' ⊆ A⟧ ⟹ X ∪ X' ⊆ 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 finally (*calculation: ‹sum (g::'c ⇒ 'a) (X::'c set) ≤ (C::'a)›*) show \"?thesis\"\n (*goal: ‹sum g X ≤ C›*) .\n qed\n have \"sum f Y ≤ C\" if Y: \"finite Y\" \"Y ⊆ Sigma A B\" for Y\n proof (-)\n (*goal: ‹sum f Y ≤ C›*)\n define Y1 and Y2 where \"Y1 = fst ` Y\" and \"Y2 = (λx. snd ` {z∈Y. fst z = x})\"\n have Y12: \"Y = Sigma Y1 Y2\"\n unfolding Y1_def Y2_def\n (*goal: ‹Y = (SIGMA x:fst ` Y. snd ` {z ∈ Y. fst z = x})›*)\n by force\n have [intro]: \"finite Y1\" \"⋀x. x ∈ Y1 ⟹ finite (Y2 x)\"\n using Y (*‹finite Y› ‹Y ⊆ Sigma A B›*) unfolding Y1_def Y2_def\n (*goals:\n 1. ‹finite (fst ` Y)›\n 2. ‹⋀x. x ∈ fst ` Y ⟹ finite (snd ` {z ∈ Y. fst z = x})›*)\n apply -\n (*goals:\n 1. ‹⟦finite Y; Y ⊆ Sigma A B⟧ ⟹ finite (fst ` Y)›\n 2. ‹⋀x. ⟦x ∈ fst ` Y; finite Y; Y ⊆ Sigma A B⟧ ⟹ finite (snd ` {z ∈ Y. fst z = x})›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n have Y12_subset: \"Y1 ⊆ A\" \"⋀x. Y2 x ⊆ B x\"\n using Y (*‹finite (Y::('c × 'd) set)› ‹Y ⊆ Sigma A B›*) apply -\n (*goals:\n 1. ‹⟦finite Y; Y ⊆ Sigma A B⟧ ⟹ Y1 ⊆ A›\n 2. ‹⋀x. ⟦finite Y; Y ⊆ Sigma A B⟧ ⟹ Y2 x ⊆ B x›\n discuss goal 1*)\n apply ((auto simp: Y1_def (*‹Y1 = fst ` Y›*) Y2_def (*‹Y2 = (λx. snd ` {z ∈ Y. fst z = x})›*))[1])\n (*discuss goal 2*)\n apply ((auto simp: Y1_def (*‹Y1 = fst ` Y›*) Y2_def (*‹Y2 = (λx. snd ` {z ∈ Y. fst z = x})›*))[1])\n (*proven 2 subgoals*) .\n have \"sum f Y = sum f (Sigma Y1 Y2)\"\n by (simp add: Y12 (*‹Y = Sigma Y1 Y2›*))\n also (*calculation: ‹sum f Y = sum f (Sigma Y1 Y2)›*) have \"… = (∑x∈Y1. ∑y∈Y2 x. f (x, y))\"\n apply (subst sum.Sigma (*‹⟦finite (?A::?'b set); ∀x::?'b∈?A. finite ((?B::?'b ⇒ ?'c set) x)⟧ ⟹ (∑x::?'b∈?A. sum ((?g::?'b ⇒ ?'c ⇒ ?'a) x) (?B x)) = (∑(x::?'b, y::?'c)∈Sigma ?A ?B. ?g x y)›*))\n (*goals:\n 1. ‹finite (Y1::'c set)›\n 2. ‹∀x::'c∈Y1::'c set. finite ((Y2::'c ⇒ 'd set) x)›\n 3. ‹sum (f::'c × 'd ⇒ 'a) (Sigma (Y1::'c set) (Y2::'c ⇒ 'd set)) = (∑(x::'c, y::'d)∈Sigma Y1 Y2. f (x, 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 also (*calculation: ‹sum f Y = (∑x∈Y1. ∑y∈Y2 x. f (x, y))›*) have \"… ≤ (∑x∈Y1. g x)\"\n proof (rule sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*))\n (*goal: ‹⋀x. x ∈ Y1 ⟹ (∑y∈Y2 x. f (x, y)) ≤ g x›*)\n fix x\n assume x: \"x ∈ Y1\" (*‹(x::'c) ∈ (Y1::'c set)›*)\n show \"(∑y∈Y2 x. f (x, y)) ≤ g x\"\n proof (rule has_sum_mono' (*‹⟦(?f has_sum ?S) ?A; (?f has_sum ?S') ?B; ?A ⊆ ?B; ⋀x. x ∈ ?B - ?A ⟹ 0 ≤ ?f x⟧ ⟹ ?S ≤ ?S'›*))\n (*goals:\n 1. ‹((?f::?'b ⇒ 'a) has_sum (∑y::'d∈(Y2::'c ⇒ 'd set) x. (f::'c × 'd ⇒ 'a) (x::'c, y))) (?A::?'b set)›\n 2. ‹((?f::?'b ⇒ 'a) has_sum (g::'c ⇒ 'a) (x::'c)) (?B::?'b set)›\n 3. ‹(?A::?'b set) ⊆ (?B::?'b set)›\n 4. ‹⋀x::?'b. x ∈ (?B::?'b set) - (?A::?'b set) ⟹ (0::'a) ≤ (?f::?'b ⇒ 'a) x›*)\n show \"((λy. f (x, y)) has_sum (∑y∈Y2 x. f (x, y))) (Y2 x)\"\n using x (*‹x ∈ Y1›*) apply (intro has_sum_finite (*‹finite ?F ⟹ (?f has_sum sum ?f ?F) ?F›*))\n (*goal: ‹((λy. f (x, y)) has_sum (∑y∈Y2 x. f (x, y))) (Y2 x)›*)\n by auto\n show \"((λy. f (x, y)) has_sum g x) (B x)\"\n apply (rule f (*‹?x ∈ A ⟹ ((λy. f (?x, y)) has_sum g ?x) (B ?x)›*))\n (*goal: ‹((λy. f (x, y)) has_sum g x) (B x)›*)\n by (use x Y12_subset in auto)\n show \"f (x, y) ≥ 0\" if \"y ∈ B x - Y2 x\" for y\n using x (*‹x ∈ Y1›*) that (*‹y ∈ B x - Y2 x›*) Y12_subset (*‹Y1 ⊆ A› ‹Y2 ?x ⊆ B ?x›*) apply (intro f_nonneg (*‹⟦?x ∈ A; ?y ∈ B ?x⟧ ⟹ 0 ≤ f (?x, ?y)›*))\n (*goals:\n 1. ‹⟦x ∈ Y1; y ∈ B x - Y2 x; Y1 ⊆ A; ⋀x. Y2 x ⊆ B x⟧ ⟹ x ∈ A›\n 2. ‹⟦x ∈ Y1; y ∈ B x - Y2 x; Y1 ⊆ A; ⋀x. Y2 x ⊆ B x⟧ ⟹ y ∈ B x›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n qed (use Y12_subset in auto)\n (*solved the remaining goal: ‹Y2 x ⊆ B x›*)\n qed\n also (*calculation: ‹sum f Y ≤ sum g Y1›*) have \"… ≤ C\"\n using Y12_subset (*‹Y1 ⊆ A› ‹Y2 ?x ⊆ B ?x›*) apply (intro sum_g_le (*‹⟦finite ?X; ?X ⊆ A⟧ ⟹ sum g ?X ≤ C›*))\n (*goals:\n 1. ‹⟦Y1 ⊆ A; ⋀x. Y2 x ⊆ B x⟧ ⟹ finite Y1›\n 2. ‹⟦Y1 ⊆ A; ⋀x. Y2 x ⊆ B x⟧ ⟹ Y1 ⊆ A›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n finally (*calculation: ‹sum (f::'c × 'd ⇒ 'a) (Y::('c × 'd) set) ≤ (C::'a)›*) show \"?thesis\"\n (*goal: ‹sum f Y ≤ C›*) .\n qed\n hence \"∀⇩F X in finite_subsets_at_top (Sigma A B). sum f X ≤ C\"\n unfolding eventually_finite_subsets_at_top\n (*goal: ‹∃X. finite X ∧ X ⊆ Sigma A B ∧ (∀Y. finite Y ∧ X ⊆ Y ∧ Y ⊆ Sigma A B ⟶ sum f Y ≤ C)›*)\n by auto\n thus \"?thesis\"\n (*goal: ‹(f::'c × 'd ⇒ 'a) summable_on Sigma (A::'c set) (B::'c ⇒ 'd set)›*)\n by (metis SigmaE (*‹⟦?c ∈ Sigma ?A ?B; ⋀x y. ⟦x ∈ ?A; y ∈ ?B x; ?c = (x, y)⟧ ⟹ ?P⟧ ⟹ ?P›*) f_nonneg (*‹⟦?x ∈ A; ?y ∈ B ?x⟧ ⟹ 0 ≤ f (?x, ?y)›*) nonneg_bounded_partial_sums_imp_summable_on (*‹⟦⋀x. x ∈ ?A ⟹ 0 ≤ ?f x; ∀⇩F X in finite_subsets_at_top ?A. sum ?f X ≤ ?C⟧ ⟹ ?f summable_on ?A›*))\nqed\n\nlemma summable_on_UnionI:\n fixes f :: \"_ ⇒ 'a :: {linorder_topology, ordered_comm_monoid_add, topological_comm_monoid_add,\n conditionally_complete_linorder}\"\n assumes f: \"⋀x. x ∈ A ⟹ (f has_sum g x) (B x)\"\n assumes g: \"g summable_on A\"\n assumes f_nonneg: \"⋀x y. x ∈ A ⟹ y ∈ B x ⟹ f y ≥ (0 :: 'a)\"\n assumes disj: \"disjoint_family_on B A\"\n shows \"f summable_on (⋃x∈A. B x)\"\nproof (-)\n (*goal: ‹f summable_on ⋃ (B ` A)›*)\n have \"f ∘ snd summable_on Sigma A B\"\n using assms (*‹?x ∈ A ⟹ (f has_sum g ?x) (B ?x)› ‹g summable_on A› ‹⟦?x ∈ A; ?y ∈ B ?x⟧ ⟹ 0 ≤ f ?y› ‹disjoint_family_on B A›*) apply (intro summable_on_SigmaI[where g = g] (*‹⟦⋀x. x ∈ ?A ⟹ ((λy. ?f (x, y)) has_sum g x) (?B x); g summable_on ?A; ⋀x y. ⟦x ∈ ?A; y ∈ ?B x⟧ ⟹ 0 ≤ ?f (x, y)⟧ ⟹ ?f summable_on Sigma ?A ?B›*))\n (*goals:\n 1. ‹⋀x::'c. ⟦⋀x::'c. x ∈ (A::'c set) ⟹ ((f::'d ⇒ 'a) has_sum (g::'c ⇒ 'a) x) ((B::'c ⇒ 'd set) x); g summable_on A; ⋀(x::'c) y::'d. ⟦x ∈ A; y ∈ B x⟧ ⟹ (0::'a) ≤ f y; disjoint_family_on B A; x ∈ A⟧ ⟹ ((λy::'d. (f ∘ snd) (x, y)) has_sum g x) (B x)›\n 2. ‹⟦⋀x::'c. x ∈ (A::'c set) ⟹ ((f::'d ⇒ 'a) has_sum (g::'c ⇒ 'a) x) ((B::'c ⇒ 'd set) x); g summable_on A; ⋀(x::'c) y::'d. ⟦x ∈ A; y ∈ B x⟧ ⟹ (0::'a) ≤ f y; disjoint_family_on B A⟧ ⟹ g summable_on A›\n 3. ‹⋀(x::'c) y::'d. ⟦⋀x::'c. x ∈ (A::'c set) ⟹ ((f::'d ⇒ 'a) has_sum (g::'c ⇒ 'a) x) ((B::'c ⇒ 'd set) x); g summable_on A; ⋀(x::'c) y::'d. ⟦x ∈ A; y ∈ B x⟧ ⟹ (0::'a) ≤ f y; disjoint_family_on B A; x ∈ A; y ∈ B x⟧ ⟹ (0::'a) ≤ (f ∘ snd) (x, 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 also (*calculation: ‹f ∘ snd summable_on Sigma A B›*) have \"?this ⟷ f summable_on (snd ` Sigma A B)\"\n using assms (*‹?x ∈ A ⟹ (f has_sum g ?x) (B ?x)› ‹(g::'c ⇒ 'a) summable_on (A::'c set)› ‹⟦?x ∈ A; ?y ∈ B ?x⟧ ⟹ 0 ≤ f ?y› ‹disjoint_family_on B A›*) apply (subst summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*))\n (*goals:\n 1. ‹⟦⋀x::'c. x ∈ (A::'c set) ⟹ ((f::'d ⇒ 'a) has_sum (g::'c ⇒ 'a) x) ((B::'c ⇒ 'd set) x); g summable_on A; ⋀(x::'c) y::'d. ⟦x ∈ A; y ∈ B x⟧ ⟹ (0::'a) ≤ f y; disjoint_family_on B A⟧ ⟹ inj_on snd (Sigma A B)›\n 2. ‹⟦⋀x::'c. x ∈ (A::'c set) ⟹ ((f::'d ⇒ 'a) has_sum (g::'c ⇒ 'a) x) ((B::'c ⇒ 'd set) x); g summable_on A; ⋀(x::'c) y::'d. ⟦x ∈ A; y ∈ B x⟧ ⟹ (0::'a) ≤ f y; disjoint_family_on B A⟧ ⟹ (f ∘ snd summable_on Sigma A B) = (f ∘ snd summable_on Sigma A B)›\n discuss goal 1*)\n apply (force simp: disjoint_family_on_def (*‹disjoint_family_on ?A ?S = (∀m∈?S. ∀n∈?S. m ≠ n ⟶ ?A m ∩ ?A n = {})›*) inj_on_def (*‹inj_on ?f ?A = (∀x∈?A. ∀y∈?A. ?f x = ?f y ⟶ x = y)›*))\n (*discuss goal 2*)\n apply (force simp: disjoint_family_on_def (*‹disjoint_family_on (?A::?'i::type ⇒ ?'a::type set) (?S::?'i::type set) = (∀m::?'i::type∈?S. ∀n::?'i::type∈?S. m ≠ n ⟶ ?A m ∩ ?A n = {})›*) inj_on_def (*‹inj_on (?f::?'a::type ⇒ ?'b::type) (?A::?'a::type set) = (∀x::?'a::type∈?A. ∀y::?'a::type∈?A. ?f x = ?f y ⟶ x = y)›*))\n (*proven 2 subgoals*) .\n also (*calculation: ‹f summable_on snd ` Sigma A B›*) have \"snd ` (Sigma A B) = (⋃x∈A. B x)\"\n by force\n finally (*calculation: ‹f summable_on ⋃ (B ` A)›*) show \"?thesis\"\n (*goal: ‹f summable_on ⋃ (B ` A)›*) .\nqed\n\nlemma summable_on_SigmaD:\n fixes f :: \"'a × 'b ⇒ 'c :: {topological_comm_monoid_add,t3_space}\"\n assumes sum1: \"f summable_on (Sigma A B)\"\n assumes sum2: \"⋀x. x ∈ A ⟹ (λy. f (x, y)) summable_on (B x)\"\n shows \"(λx. infsum (λy. f (x, y)) (B x)) summable_on A\"\n using assms (*‹f summable_on Sigma A B› ‹?x ∈ A ⟹ (λy. f (?x, y)) summable_on B ?x›*) unfolding summable_on_def\n (*goal: ‹∃x. ((λx. ∑⇩∞y∈B x. f (x, y)) has_sum x) A›*)\n by (smt (verit, del_insts) assms (*‹f summable_on Sigma A B› ‹?x ∈ A ⟹ (λy. f (?x, y)) summable_on B ?x›*) has_sum_SigmaD (*‹⟦(?f has_sum ?S) (Sigma ?A ?B); ⋀x. x ∈ ?A ⟹ ((λy. ?f (x, y)) has_sum ?g x) (?B x)⟧ ⟹ (?g has_sum ?S) ?A›*) has_sum_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f has_sum ?x) ?A = (?g has_sum ?x) ?A›*) has_sum_infsum (*‹?f summable_on ?S ⟹ (?f has_sum infsum ?f ?S) ?S›*))\n\nlemma summable_on_UnionD:\n fixes f :: \"'a ⇒ 'c :: {topological_comm_monoid_add,t3_space}\"\n assumes sum1: \"f summable_on (⋃x∈A. B x)\"\n assumes sum2: \"⋀x. x ∈ A ⟹ f summable_on (B x)\"\n assumes disj: \"disjoint_family_on B A\"\n shows \"(λx. infsum f (B x)) summable_on A\"\nproof (-)\n (*goal: ‹(λx. infsum f (B x)) summable_on A›*)\n have \"(⋃x∈A. B x) = snd ` Sigma A B\"\n by (force simp: Sigma_def (*‹Sigma ?A ?B ≡ ⋃x∈?A. ⋃y∈?B x. {(x, y)}›*))\n with sum1 (*‹f summable_on ⋃ (B ` A)›*) have \"f summable_on (snd ` Sigma A B)\"\n by simp\n also (*calculation: ‹f summable_on snd ` Sigma A B›*) have \"?this ⟷ (f ∘ snd) summable_on (Sigma A B)\"\n using disj (*‹disjoint_family_on B A›*) apply (intro summable_on_reindex (*‹inj_on ?h ?A ⟹ (?g summable_on ?h ` ?A) = (?g ∘ ?h summable_on ?A)›*) inj_onI (*‹(⋀x y. ⟦x ∈ ?A; y ∈ ?A; ?f x = ?f y⟧ ⟹ x = y) ⟹ inj_on ?f ?A›*))\n (*goal: ‹(f summable_on snd ` Sigma A B) = (f ∘ snd summable_on Sigma A B)›*)\n by (force simp: disjoint_family_on_def (*‹disjoint_family_on (?A::?'i ⇒ ?'a set) (?S::?'i set) = (∀m::?'i∈?S. ∀n::?'i∈?S. m ≠ n ⟶ ?A m ∩ ?A n = {})›*))\n finally (*calculation: ‹f ∘ snd summable_on Sigma A B›*) show \"(λx. infsum f (B x)) summable_on A\"\n using summable_on_SigmaD[of \"f ∘ snd\" A B] (*‹⟦f ∘ snd summable_on Sigma A B; ⋀x. x ∈ A ⟹ (λy. (f ∘ snd) (x, y)) summable_on B x⟧ ⟹ (λx. ∑⇩∞y∈B x. (f ∘ snd) (x, y)) summable_on A›*) sum2 (*‹(?x::'b) ∈ (A::'b set) ⟹ (f::'a ⇒ 'c) summable_on (B::'b ⇒ 'a set) ?x›*) by simp\nqed\n\nlemma summable_on_Union_iff:\n fixes f :: \"_ ⇒ 'a :: {linorder_topology, ordered_comm_monoid_add, topological_comm_monoid_add,\n conditionally_complete_linorder, t3_space}\"\n assumes f: \"⋀x. x ∈ A ⟹ (f has_sum g x) (B x)\"\n assumes f_nonneg: \"⋀x y. x ∈ A ⟹ y ∈ B x ⟹ f y ≥ 0\"\n assumes disj: \"disjoint_family_on B A\"\n shows \"f summable_on (⋃x∈A. B x) ⟷ g summable_on A\"\nproof (standard)\n (*goals:\n 1. ‹f summable_on ⋃ (B ` A) ⟹ g summable_on A›\n 2. ‹g summable_on A ⟹ f summable_on ⋃ (B ` A)›*)\n assume \"g summable_on A\" (*‹(g::'c ⇒ 'a) summable_on (A::'c set)›*)\n thus \"f summable_on (⋃x∈A. B x)\"\n using summable_on_UnionI[of A f B g] (*‹⟦⋀x::'c::type. x ∈ (A::'c::type set) ⟹ ((f::'d::type ⇒ 'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,linorder_topology,t3_space}) has_sum (g::'c::type ⇒ 'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,linorder_topology,t3_space}) x) ((B::'c::type ⇒ 'd::type set) x); g summable_on A; ⋀(x::'c::type) y::'d::type. ⟦x ∈ A; y ∈ B x⟧ ⟹ (0::'a::{conditionally_complete_linorder,ordered_comm_monoid_add,topological_comm_monoid_add,linorder_topology,t3_space}) ≤ f y; disjoint_family_on B A⟧ ⟹ f summable_on ⋃ (B ` A)›*) assms (*‹?x ∈ A ⟹ (f has_sum g ?x) (B ?x)› ‹⟦?x ∈ A; ?y ∈ B ?x⟧ ⟹ 0 ≤ f ?y› ‹disjoint_family_on B A›*) by auto\nnext\n (*goal: ‹f summable_on ⋃ (B ` A) ⟹ g summable_on A›*)\n assume \"f summable_on (⋃x∈A. B x)\" (*‹(f::'d ⇒ 'a) summable_on ⋃ ((B::'c ⇒ 'd set) ` (A::'c set))›*)\n hence \"(λx. infsum f (B x)) summable_on A\"\n using assms (*‹?x ∈ A ⟹ (f has_sum g ?x) (B ?x)› ‹⟦?x ∈ A; ?y ∈ B ?x⟧ ⟹ 0 ≤ f ?y› ‹disjoint_family_on B A›*) apply (intro summable_on_UnionD (*‹⟦?f summable_on ⋃ (?B ` ?A); ⋀x. x ∈ ?A ⟹ ?f summable_on ?B x; disjoint_family_on ?B ?A⟧ ⟹ (λx. infsum ?f (?B x)) summable_on ?A›*))\n (*goals:\n 1. ‹⟦f summable_on ⋃ (B ` A); ⋀x. x ∈ A ⟹ (f has_sum g x) (B x); ⋀x y. ⟦x ∈ A; y ∈ B x⟧ ⟹ 0 ≤ f y; disjoint_family_on B A⟧ ⟹ f summable_on ⋃ (B ` A)›\n 2. ‹⋀x. ⟦f summable_on ⋃ (B ` A); ⋀x. x ∈ A ⟹ (f has_sum g x) (B x); ⋀x y. ⟦x ∈ A; y ∈ B x⟧ ⟹ 0 ≤ f y; disjoint_family_on B A; x ∈ A⟧ ⟹ f summable_on B x›\n 3. ‹⟦f summable_on ⋃ (B ` A); ⋀x. x ∈ A ⟹ (f has_sum g x) (B x); ⋀x y. ⟦x ∈ A; y ∈ B x⟧ ⟹ 0 ≤ f y; disjoint_family_on B A⟧ ⟹ disjoint_family_on B A›\n discuss goal 1*)\n apply ((auto dest: has_sum_imp_summable (*‹(?f has_sum ?S) ?A ⟹ ?f summable_on ?A›*))[1])\n (*discuss goal 2*)\n apply ((auto dest: has_sum_imp_summable (*‹(?f has_sum ?S) ?A ⟹ ?f summable_on ?A›*))[1])\n (*discuss goal 3*)\n apply ((auto dest: has_sum_imp_summable (*‹(?f has_sum ?S) ?A ⟹ ?f summable_on ?A›*))[1])\n (*proven 3 subgoals*) .\n also (*calculation: ‹(λx. infsum f (B x)) summable_on A›*) have \"?this ⟷ g summable_on A\"\n using assms (*‹?x ∈ A ⟹ (f has_sum g ?x) (B ?x)› ‹⟦?x ∈ A; ?y ∈ B ?x⟧ ⟹ 0 ≤ f ?y› ‹disjoint_family_on B A›*) apply (intro summable_on_cong (*‹(⋀x. x ∈ ?A ⟹ ?f x = ?g x) ⟹ (?f summable_on ?A) = (?g summable_on ?A)›*))\n (*goal: ‹((λx. infsum f (B x)) summable_on A) = (g summable_on A)›*)\n by (auto simp: infsumI (*‹(?f has_sum ?x) ?A ⟹ infsum ?f ?A = ?x›*))\n finally (*calculation: ‹g summable_on A›*) show \"g summable_on A\" .\nqed\n\nlemma has_sum_Sigma':\n fixes A :: \"'a set\" and B :: \"'a ⇒ 'b set\"\n and f :: ‹'a × 'b ⇒ 'c::{comm_monoid_add,uniform_space,uniform_topological_group_add}›\n assumes summableAB: \"(f has_sum a) (Sigma A B)\"\n assumes summableB: ‹⋀x. x∈A ⟹ ((λy. f (x, y)) has_sum (b x)) (B x)›\n shows \"(b has_sum a) A\"\n by (intro has_sum_Sigma[OF _ assms] (*‹⟦uniformly_continuous_on UNIV (λ(x, y). x + y); ⋀x. x ∈ A ⟹ x ∈ A⟧ ⟹ (b has_sum a) A›*) uniformly_continuous_add (*‹uniformly_continuous_on UNIV (λ(x, y). x + y)›*))\n\nlemma abs_summable_on_comparison_test':\n assumes \"g summable_on A\"\n assumes \"⋀x. x ∈ A ⟹ norm (f x) ≤ g x\"\n shows \"(λx. norm (f x)) summable_on A\"\nproof (rule Infinite_Sum.abs_summable_on_comparison_test (*‹⟦?g abs_summable_on ?A; ⋀x. x ∈ ?A ⟹ norm (?f x) ≤ norm (?g x)⟧ ⟹ ?f abs_summable_on ?A›*))\n (*goals:\n 1. ‹?g abs_summable_on A›\n 2. ‹⋀x. x ∈ A ⟹ norm (f x) ≤ norm (?g x)›*)\n have \"g summable_on A ⟷ (λx. norm (g x)) summable_on A\"\n by (metis summable_on_iff_abs_summable_on_real (*‹(?f summable_on ?A) = (?f abs_summable_on ?A)›*))\n with assms (*‹g summable_on A› ‹?x ∈ A ⟹ norm (f ?x) ≤ g ?x›*) show \"(λx. norm (g x)) summable_on A\"\n by blast\nqed (use assms in fastforce)\n(*solved the remaining goal: ‹⋀x. x ∈ A ⟹ norm (f x) ≤ norm (g x)›*)\n\nlemma has_sum_geometric_from_1:\n fixes z :: \"'a :: {real_normed_field, banach}\"\n assumes \"norm z < 1\"\n shows \"((λn. z ^ n) has_sum (z / (1 - z))) {1..}\"\nproof (-)\n (*goal: ‹((^) z has_sum z / (1 - z)) {1..}›*)\n have [simp]: \"z ≠ 1\"\n using assms (*‹norm (z::'a::{banach,real_normed_field}) < (1::real)›*) by auto\n have \"(λn. z ^ Suc n) sums (1 / (1 - z) - 1)\"\n using geometric_sums[of z] (*‹norm z < 1 ⟹ (^) z sums (1 / (1 - z))›*) assms (*‹norm z < 1›*) apply (subst sums_Suc_iff (*‹(λn::nat. (?f::nat ⇒ ?'a::real_normed_vector) (Suc n)) sums (?s::?'a::real_normed_vector) = ?f sums (?s + ?f (0::nat))›*))\n (*goal: ‹(λn. z ^ Suc n) sums (1 / (1 - z) - 1)›*)\n by auto\n also (*calculation: ‹(λn. z ^ Suc n) sums (1 / (1 - z) - 1)›*) have \"1 / (1 - z) - 1 = z / (1 - z)\"\n by (auto simp: field_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 69 facts*))\n finally (*calculation: ‹(λn. z ^ Suc n) sums (z / (1 - z))›*) have \"(λn. z ^ Suc n) sums (z / (1 - z))\" .\n moreover have \"summable (λn. norm (z ^ Suc n))\"\n using assms (*‹norm z < 1›*) apply (subst summable_Suc_iff (*‹summable (λn. ?f (Suc n)) = summable ?f›*))\n (*goal: ‹summable (λn. norm (z ^ Suc n))›*)\n by (auto simp: norm_power (*‹norm (?x ^ ?n) = norm ?x ^ ?n›*) intro!: summable_geometric (*‹norm ?c < 1 ⟹ summable ((^) ?c)›*))\n ultimately have \"((λn. z ^ Suc n) has_sum (z / (1 - z))) UNIV\"\n apply (intro norm_summable_imp_has_sum (*‹⟦summable (λn. norm (?f n)); ?f sums ?S⟧ ⟹ (?f has_sum ?S) UNIV›*))\n (*goals:\n 1. ‹⟦(λn. z ^ Suc n) sums (z / (1 - z)); summable (λn. norm (z ^ Suc n))⟧ ⟹ summable (λn. norm (z ^ Suc n))›\n 2. ‹⟦(λn. z ^ Suc n) sums (z / (1 - z)); summable (λn. norm (z ^ Suc n))⟧ ⟹ (λn. z ^ Suc n) sums (z / (1 - z))›\n discuss goal 1*)\n apply ((assumption)[1])\n (*discuss goal 2*) .\n (*proven 2 subgoals*)\n also (*calculation: ‹((λn. z ^ Suc n) has_sum z / (1 - z)) UNIV›*) have \"?this ⟷ ?thesis\"\n apply (intro has_sum_reindex_bij_witness[of _ \"λn. n-1\" \"λn. n+1\"] (*‹⟦⋀a::?'b1. a ∈ (?S::?'b1 set) ⟹ a + (1::?'b1) - (1::?'b1) = a; ⋀a::?'b1. a ∈ ?S ⟹ a + (1::?'b1) ∈ (?T::?'b1 set); ⋀b::?'b1. b ∈ ?T ⟹ b - (1::?'b1) + (1::?'b1) = b; ⋀b::?'b1. b ∈ ?T ⟹ b - (1::?'b1) ∈ ?S; ⋀a::?'b1. a ∈ ?S ⟹ (?h::?'b1 ⇒ ?'c) (a + (1::?'b1)) = (?g::?'b1 ⇒ ?'c) a; (?s::?'c) = (?s'::?'c)⟧ ⟹ (?g has_sum ?s) ?S = (?h has_sum ?s') ?T›*))\n (*goals:\n 1. ‹⋀a. a ∈ UNIV ⟹ a + 1 - 1 = a›\n 2. ‹⋀a. a ∈ UNIV ⟹ a + 1 ∈ {1..}›\n 3. ‹⋀b. b ∈ {1..} ⟹ b - 1 + 1 = b›\n 4. ‹⋀b. b ∈ {1..} ⟹ b - 1 ∈ UNIV›\n 5. ‹⋀a. a ∈ UNIV ⟹ z ^ (a + 1) = z ^ Suc a›\n 6. ‹z / (1 - z) = z / (1 - z)›\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: ‹((^) z has_sum z / (1 - z)) {1..}›*) show \"?thesis\"\n (*goal: ‹((^) z has_sum z / (1 - z)) {1..}›*) .\nqed \n\nlemma has_sum_divide_const:\n fixes f :: \"'a ⇒ 'b :: {topological_semigroup_mult, field, semiring_0}\"\n shows \"(f has_sum S) A ⟹ ((λx. f x / c) has_sum (S / c)) A\"\n using has_sum_cmult_right[of f A S \"inverse c\"] (*‹(f has_sum S) A ⟹ ((λx. inverse c * f x) has_sum inverse c * S) A›*) by (simp add: field_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::?'a) - (?b::?'a) - (?c::?'a) = ?a - (?b + ?c)› ‹(?a::?'a) + ((?b::?'a) - (?c::?'a)) = ?a + ?b - ?c› ‹((?a::?'a) - (?b::?'a) = (?c::?'a)) = (?a = ?c + ?b)› ‹((?a::?'a) = (?c::?'a) - (?b::?'a)) = (?a + ?b = ?c)› ‹(?a::?'a) - ((?b::?'a) - (?c::?'a)) = ?a + ?c - ?b› ‹(?a::?'a) - (?b::?'a) + (?c::?'a) = ?a + ?c - ?b› and more 69 facts*))\n\nlemma has_sum_uminusI:\n fixes f :: \"'a ⇒ 'b :: {topological_semigroup_mult, ring_1}\"\n shows \"(f has_sum S) A ⟹ ((λx. -f x) has_sum (-S)) A\"\n using has_sum_cmult_right[of f A S \"-1\"] (*‹(f has_sum S) A ⟹ ((λx. - 1 * f x) has_sum - 1 * S) A›*) by simp\n\nend\n\n"},"meta":{"kind":"string","value":"{\n \"path\": \"Isabelle2024/src/HOL/Analysis/Infinite_Sum.thy\",\n \"repo\": \"Isabelle2024\",\n \"sha\": \"bc79361c2879d9ba15f9faa14569ad35506e77beda849fb6f30f4930fa80e951\"\n}"}}},{"rowIdx":95,"cells":{"text":{"kind":"string","value":"(* Title: Miscellaneous results\n Author: Tim Makarios <tjm1983 at gmail.com>, 2012\n Maintainer: Tim Makarios <tjm1983 at gmail.com>\n*)\n\n(*Some of these theorems were moved to the Isabelle repository *)\n\nsection \"Miscellaneous results\"\n\ntheory Miscellany\nimports Metric\nbegin\n\nlemma unordered_pair_element_equality:\n assumes \"{p, q} = {r, s}\" and \"p = r\"\n shows \"q = s\"\n using assms (*‹{p::'a::type, q::'a::type} = {r::'a::type, s::'a::type}› ‹p = r›*) by (auto simp: doubleton_eq_iff (*‹({?a, ?b} = {?c, ?d}) = (?a = ?c ∧ ?b = ?d ∨ ?a = ?d ∧ ?b = ?c)›*))\n\nlemma unordered_pair_equality: \"{p, q} = {q, p}\"\n by auto\n\nlemma cosine_rule:\n fixes a b c :: \"real ^ ('n::finite)\"\n shows \"(norm_dist a c)² =\n (norm_dist a b)² + (norm_dist b c)² + 2 * ((a - b) ∙ (b - c))\"\nproof (-)\n (*goal: ‹(norm_dist a c)² = (norm_dist a b)² + (norm_dist b c)² + 2 * ((a - b) ∙ (b - c))›*)\n have \"(a - b) + (b - c) = a - c\"\n by simp\n with dot_norm[of \"a - b\" \"b - c\"] (*‹((a::(real, 'n) vec) - (b::(real, 'n) vec)) ∙ (b - (c::(real, 'n) vec)) = ((norm (a - b + (b - c)))² - (norm (a - b))² - (norm (b - c))²) / (2::real)›*) have \"(a - b) ∙ (b - c) =\n ((norm (a - c))² - (norm (a - b))² - (norm (b - c))²) / 2\"\n by simp\n thus \"?thesis\"\n (*goal: ‹(norm_dist a c)² = (norm_dist a b)² + (norm_dist b c)² + 2 * ((a - b) ∙ (b - c))›*)\n by simp\nqed\n\nlemma scalar_equiv: \"r *s x = r *⇩R x\"\n by vector\n\nlemma norm_dist_dot: \"(norm_dist x y)² = (x - y) ∙ (x - y)\"\n by (simp add: power2_norm_eq_inner (*‹(norm ?x)² = ?x ∙ ?x›*))\n\ndefinition dep2 :: \"'a::real_vector ⇒ 'a ⇒ bool\" where\n \"dep2 u v ≡ ∃w r s. u = r *⇩R w ∧ v = s *⇩R w\"\n\nlemma real2_eq:\n fixes u v :: \"real^2\"\n assumes \"u$1 = v$1\" and \"u$2 = v$2\"\n shows \"u = v\"\n by (simp add: vec_eq_iff [of u v] (*‹((u::(real, 2) vec) = (v::(real, 2) vec)) = (∀i::2. u $ i = v $ i)›*) forall_2 (*‹(∀i::2. (?P::2 ⇒ bool) i) = (?P (1::2) ∧ ?P (2::2))›*) assms (*‹(u::(real, 2) vec) $ (1::2) = (v::(real, 2) vec) $ (1::2)› ‹(u::(real, 2) vec) $ (2::2) = (v::(real, 2) vec) $ (2::2)›*))\n\ndefinition rotate2 :: \"real^2 ⇒ real^2\" where\n \"rotate2 x ≡ vector [-x$2, x$1]\"\n\ndeclare vector_2 [simp]\n\nlemma rotate2 [simp]:\n \"(rotate2 x)$1 = -x$2\"\n \"(rotate2 x)$2 = x$1\"\n \n (*goals:\n 1. ‹rotate2 x $ 1 = - x $ 2›\n 2. ‹rotate2 x $ 2 = x $ 1›\n discuss goal 1*)\n apply (simp add: rotate2_def (*‹rotate2 ?x ≡ vector [- ?x $ 2, ?x $ 1]›*))\n (*discuss goal 2*)\n apply (simp add: rotate2_def (*‹rotate2 ?x ≡ vector [- ?x $ 2, ?x $ 1]›*))\n (*proven 2 subgoals*) .\n\nlemma rotate2_rotate2 [simp]: \"rotate2 (rotate2 x) = -x\"\nproof (-)\n (*goal: ‹rotate2 (rotate2 x) = - x›*)\n have \"(rotate2 (rotate2 x))$1 = -x$1\" and \"(rotate2 (rotate2 x))$2 = -x$2\"\n (*goals:\n 1. ‹rotate2 (rotate2 x) $ 1 = - x $ 1›\n 2. ‹rotate2 (rotate2 x) $ 2 = - x $ 2›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n with real2_eq (*‹⟦?u $ 1 = ?v $ 1; ?u $ 2 = ?v $ 2⟧ ⟹ ?u = ?v›*) show \"rotate2 (rotate2 x) = -x\"\n by simp\nqed\n\nlemma rotate2_dot [simp]: \"(rotate2 u) ∙ (rotate2 v) = u ∙ v\"\n unfolding inner_vec_def\n (*goal: ‹(∑i::2∈UNIV. rotate2 (u::(real, 2) vec) $ i ∙ rotate2 (v::(real, 2) vec) $ i) = (∑i::2∈UNIV. u $ i ∙ v $ i)›*)\n by (simp add: sum_2 (*‹sum ?f UNIV = ?f 1 + ?f 2›*))\n\nlemma rotate2_scaleR [simp]: \"rotate2 (k *⇩R x) = k *⇩R (rotate2 x)\"\nproof (-)\n (*goal: ‹rotate2 ((k::real) *⇩R (x::(real, 2) vec)) = k *⇩R rotate2 x›*)\n have \"(rotate2 (k *⇩R x))$1 = (k *⇩R (rotate2 x))$1\" and \"(rotate2 (k *⇩R x))$2 = (k *⇩R (rotate2 x))$2\"\n (*goals:\n 1. ‹rotate2 (k *⇩R x) $ 1 = (k *⇩R rotate2 x) $ 1›\n 2. ‹rotate2 (k *⇩R x) $ 2 = (k *⇩R rotate2 x) $ 2›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n with real2_eq (*‹⟦?u $ 1 = ?v $ 1; ?u $ 2 = ?v $ 2⟧ ⟹ ?u = ?v›*) show \"?thesis\"\n (*goal: ‹rotate2 (k *⇩R x) = k *⇩R rotate2 x›*)\n by simp\nqed\n\nlemma rotate2_uminus [simp]: \"rotate2 (-x) = -(rotate2 x)\"\nproof (-)\n (*goal: ‹rotate2 (- x) = - rotate2 x›*)\n from scaleR_minus_left[of 1] (*‹- (1::real) *⇩R (?x::?'a::real_vector) = - ((1::real) *⇩R ?x)›*) have \"-1 *⇩R x = -x\" and \"-1 *⇩R (rotate2 x) = -(rotate2 x)\"\n (*goals:\n 1. ‹- 1 *⇩R x = - x›\n 2. ‹- 1 *⇩R rotate2 x = - rotate2 x›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n with rotate2_scaleR[of \"-1\" x] (*‹rotate2 (- 1 *⇩R x) = - 1 *⇩R rotate2 x›*) show \"?thesis\"\n (*goal: ‹rotate2 (- (x::(real, 2) vec)) = - rotate2 x›*)\n by simp\nqed\n\nlemma rotate2_eq [iff]: \"rotate2 x = rotate2 y ⟷ x = y\"\nproof (standard)\n (*goals:\n 1. ‹rotate2 x = rotate2 y ⟹ x = y›\n 2. ‹x = y ⟹ rotate2 x = rotate2 y›*)\n assume \"x = y\" (*‹(x::(real, 2) vec) = (y::(real, 2) vec)›*)\n thus \"rotate2 x = rotate2 y\"\n by simp\nnext\n (*goal: ‹rotate2 x = rotate2 y ⟹ x = y›*)\n assume \"rotate2 x = rotate2 y\" (*‹rotate2 (x::(real, 2) vec) = rotate2 (y::(real, 2) vec)›*)\n hence \"rotate2 (rotate2 x) = rotate2 (rotate2 y)\"\n by simp\n hence \"-(-x) = -(-y)\"\n by simp\n thus \"x = y\"\n by simp\nqed\n\nlemma dot2_rearrange_1:\n fixes u x :: \"real^2\"\n assumes \"u ∙ x = 0\" and \"x$1 ≠ 0\"\n shows \"u = (u$2 / x$1) *⇩R (rotate2 x)\" (is \"u = ?u'\")\nproof (-)\n (*goal: ‹(u::(real, 2) vec) = (u $ (2::2) / (x::(real, 2) vec) $ (1::2)) *⇩R rotate2 x›*)\n from ‹u ∙ x = 0› (*‹(u::(real, 2) vec) ∙ (x::(real, 2) vec) = (0::real)›*) have \"u$1 * x$1 = -(u$2) * (x$2)\"\n unfolding inner_vec_def\n (*goal: ‹(u::(real, 2) vec) $ (1::2) * (x::(real, 2) vec) $ (1::2) = - u $ (2::2) * x $ (2::2)›*)\n by (simp add: sum_2 (*‹sum ?f UNIV = ?f 1 + ?f 2›*))\n hence \"u$1 * x$1 / x$1 = -u$2 / x$1 * x$2\"\n by simp\n with ‹x$1 ≠ 0› (*‹x $ 1 ≠ 0›*) have \"u$1 = ?u'$1\"\n by simp\n from ‹x$1 ≠ 0› (*‹x $ 1 ≠ 0›*) have \"u$2 = ?u'$2\"\n by simp\n with ‹u$1 = ?u'$1› (*‹u $ 1 = ((u $ 2 / x $ 1) *⇩R rotate2 x) $ 1›*) real2_eq (*‹⟦?u $ 1 = ?v $ 1; ?u $ 2 = ?v $ 2⟧ ⟹ ?u = ?v›*) show \"u = ?u'\"\n by simp\nqed\n\nlemma dot2_rearrange_2:\n fixes u x :: \"real^2\"\n assumes \"u ∙ x = 0\" and \"x$2 ≠ 0\"\n shows \"u = -(u$1 / x$2) *⇩R (rotate2 x)\" (is \"u = ?u'\")\nproof (-)\n (*goal: ‹u = - (u $ 1 / x $ 2) *⇩R rotate2 x›*)\n from assms (*‹(u::(real, 2) vec) ∙ (x::(real, 2) vec) = (0::real)› ‹(x::(real, 2) vec) $ (2::2) ≠ (0::real)›*) dot2_rearrange_1[of \"rotate2 u\" \"rotate2 x\"] (*‹⟦rotate2 u ∙ rotate2 x = 0; rotate2 x $ 1 ≠ 0⟧ ⟹ rotate2 u = (rotate2 u $ 2 / rotate2 x $ 1) *⇩R rotate2 (rotate2 x)›*) have \"rotate2 u = rotate2 ?u'\"\n by simp\n thus \"u = ?u'\"\n by blast\nqed\n\nlemma dot2_rearrange:\n fixes u x :: \"real^2\"\n assumes \"u ∙ x = 0\" and \"x ≠ 0\"\n shows \"∃k. u = k *⇩R (rotate2 x)\"\nproof (cases)\n (*goals:\n 1. ‹?P ⟹ ∃k. u = k *⇩R rotate2 x›\n 2. ‹¬ ?P ⟹ ∃k. u = k *⇩R rotate2 x›*)\n assume \"x$1 = 0\" (*‹(x::(real, 2) vec) $ (1::2) = (0::real)›*)\n with real2_eq[of x 0] (*‹⟦x $ 1 = 0 $ 1; x $ 2 = 0 $ 2⟧ ⟹ x = 0›*) ‹x ≠ 0› (*‹x ≠ 0›*) have \"x$2 ≠ 0\"\n by auto\n with dot2_rearrange_2 (*‹⟦?u ∙ ?x = 0; ?x $ 2 ≠ 0⟧ ⟹ ?u = - (?u $ 1 / ?x $ 2) *⇩R rotate2 ?x›*) ‹u ∙ x = 0› (*‹u ∙ x = 0›*) show \"?thesis\"\n (*goal: ‹∃k. u = k *⇩R rotate2 x›*)\n by blast\nnext\n (*goal: ‹x $ 1 ≠ 0 ⟹ ∃k. u = k *⇩R rotate2 x›*)\n assume \"x$1 ≠ 0\" (*‹(x::(real, 2) vec) $ (1::2) ≠ (0::real)›*)\n with dot2_rearrange_1 (*‹⟦?u ∙ ?x = 0; ?x $ 1 ≠ 0⟧ ⟹ ?u = (?u $ 2 / ?x $ 1) *⇩R rotate2 ?x›*) ‹u ∙ x = 0› (*‹u ∙ x = 0›*) show \"?thesis\"\n (*goal: ‹∃k. u = k *⇩R rotate2 x›*)\n by blast\nqed\n\nlemma real2_orthogonal_dep2:\n fixes u v x :: \"real^2\"\n assumes \"x ≠ 0\" and \"u ∙ x = 0\" and \"v ∙ x = 0\"\n shows \"dep2 u v\"\nsorry\n\nlemma dot_left_diff_distrib:\n fixes u v x :: \"real^'n\"\n shows \"(u - v) ∙ x = (u ∙ x) - (v ∙ x)\"\nproof (-)\n (*goal: ‹((u::(real, 'n) vec) - (v::(real, 'n) vec)) ∙ (x::(real, 'n) vec) = u ∙ x - v ∙ x›*)\n have \"(u ∙ x) - (v ∙ x) = (∑i∈UNIV. u$i * x$i) - (∑i∈UNIV. v$i * x$i)\"\n unfolding inner_vec_def\n (*goal: ‹(∑i∈UNIV. u $ i ∙ x $ i) - (∑i∈UNIV. v $ i ∙ x $ i) = (∑i∈UNIV. u $ i * x $ i) - (∑i∈UNIV. v $ i * x $ i)›*)\n by simp\n also (*calculation: ‹(u::(real, 'n::finite) vec) ∙ (x::(real, 'n::finite) vec) - (v::(real, 'n::finite) vec) ∙ x = (∑i::'n::finite∈UNIV. u $ i * x $ i) - (∑i::'n::finite∈UNIV. v $ i * x $ i)›*) from sum_subtractf[of \"λ i. u$i * x$i\" \"λ i. v$i * x$i\"] (*‹(∑xa∈?A. u $ xa * x $ xa - v $ xa * x $ xa) = (∑xa∈?A. u $ xa * x $ xa) - (∑xa∈?A. v $ xa * x $ xa)›*) have \"… = (∑i∈UNIV. u$i * x$i - v$i * x$i)\"\n by simp\n also (*calculation: ‹u ∙ x - v ∙ x = (∑i∈UNIV. u $ i * x $ i - v $ i * x $ i)›*) from left_diff_distrib[where 'a = real] (*‹((?a::real) - (?b::real)) * (?c::real) = ?a * ?c - ?b * ?c›*) have \"… = (∑i∈UNIV. (u$i - v$i) * x$i)\"\n by simp\n also (*calculation: ‹u ∙ x - v ∙ x = (∑i∈UNIV. (u $ i - v $ i) * x $ i)›*) have \"… = (u - v) ∙ x\"\n unfolding inner_vec_def\n (*goal: ‹(∑i::'n∈UNIV. ((u::(real, 'n) vec) $ i - (v::(real, 'n) vec) $ i) * (x::(real, 'n) vec) $ i) = (∑i::'n∈UNIV. (u - v) $ i ∙ x $ i)›*)\n by simp\n finally (*calculation: ‹u ∙ x - v ∙ x = (u - v) ∙ x›*) show \"?thesis\"\n (*goal: ‹(u - v) ∙ x = u ∙ x - v ∙ x›*)\n by standard\nqed\n\nlemma dot_right_diff_distrib:\n fixes u v x :: \"real^'n\"\n shows \"x ∙ (u - v) = (x ∙ u) - (x ∙ v)\"\nproof (-)\n (*goal: ‹x ∙ (u - v) = x ∙ u - x ∙ v›*)\n from inner_commute (*‹?x ∙ ?y = ?y ∙ ?x›*) have \"x ∙ (u - v) = (u - v) ∙ x\"\n by auto\n also (*calculation: ‹x ∙ (u - v) = (u - v) ∙ x›*) from dot_left_diff_distrib[of u v x] (*‹(u - v) ∙ x = u ∙ x - v ∙ x›*) have \"… = u ∙ x - v ∙ x\" .\n also (*calculation: ‹x ∙ (u - v) = u ∙ x - v ∙ x›*) from inner_commute[of x] (*‹x ∙ ?y = ?y ∙ x›*) have \"… = x ∙ u - x ∙ v\"\n by simp\n finally (*calculation: ‹x ∙ (u - v) = x ∙ u - x ∙ v›*) show \"?thesis\"\n (*goal: ‹x ∙ (u - v) = x ∙ u - x ∙ v›*) .\nqed\n\nlemma am_gm2:\n fixes a b :: real\n assumes \"a ≥ 0\" and \"b ≥ 0\"\n shows \"sqrt (a * b) ≤ (a + b) / 2\"\n and \"sqrt (a * b) = (a + b) / 2 ⟷ a = b\"\nproof (-)\n (*goals:\n 1. ‹sqrt ((a::real) * (b::real)) ≤ (a + b) / (2::real)›\n 2. ‹(sqrt ((a::real) * (b::real)) = (a + b) / (2::real)) = (a = b)›*)\n have \"0 ≤ (a - b) * (a - b)\" and \"0 = (a - b) * (a - b) ⟷ a = b\"\n (*goals:\n 1. ‹0 ≤ (a - b) * (a - b)›\n 2. ‹(0 = (a - b) * (a - b)) = (a = b)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n with right_diff_distrib[of \"a - b\" a b] (*‹(a - b) * (a - b) = (a - b) * a - (a - b) * b›*) left_diff_distrib[of a b] (*‹(a - b) * ?c = a * ?c - b * ?c›*) have \"0 ≤ a * a - 2 * a * b + b * b\" and \"0 = a * a - 2 * a * b + b * b ⟷ a = b\"\n apply -\n (*goals:\n 1. ‹⟦(a - b) * (a - b) = (a - b) * a - (a - b) * b; ⋀c. (a - b) * c = a * c - b * c; 0 ≤ (a - b) * (a - b); (0 = (a - b) * (a - b)) = (a = b)⟧ ⟹ 0 ≤ a * a - 2 * a * b + b * b›\n 2. ‹⟦(a - b) * (a - b) = (a - b) * a - (a - b) * b; ⋀c. (a - b) * c = a * c - b * c; 0 ≤ (a - b) * (a - b); (0 = (a - b) * (a - b)) = (a = b)⟧ ⟹ (0 = a * a - 2 * a * b + b * b) = (a = b)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n hence \"4 * a * b ≤ a * a + 2 * a * b + b * b\" and \"4 * a * b = a * a + 2 * a * b + b * b ⟷ a = b\"\n apply -\n (*goals:\n 1. ‹⟦0 ≤ a * a - 2 * a * b + b * b; (0 = a * a - 2 * a * b + b * b) = (a = b)⟧ ⟹ 4 * a * b ≤ a * a + 2 * a * b + b * b›\n 2. ‹⟦0 ≤ a * a - 2 * a * b + b * b; (0 = a * a - 2 * a * b + b * b) = (a = b)⟧ ⟹ (4 * a * b = a * a + 2 * a * b + b * b) = (a = b)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n with distrib_right[of \"a + b\" a b] (*‹((a::real) + (b::real) + a) * b = (a + b) * b + a * b›*) distrib_left[of a b] (*‹a * (b + ?c) = a * b + a * ?c›*) have \"4 * a * b ≤ (a + b) * (a + b)\" and \"4 * a * b = (a + b) * (a + b) ⟷ a = b\"\n apply -\n (*goals:\n 1. ‹⟦(a + b + a) * b = (a + b) * b + a * b; ⋀c. a * (b + c) = a * b + a * c; 4 * a * b ≤ a * a + 2 * a * b + b * b; (4 * a * b = a * a + 2 * a * b + b * b) = (a = b)⟧ ⟹ 4 * a * b ≤ (a + b) * (a + b)›\n 2. ‹⟦(a + b + a) * b = (a + b) * b + a * b; ⋀c. a * (b + c) = a * b + a * c; 4 * a * b ≤ a * a + 2 * a * b + b * b; (4 * a * b = a * a + 2 * a * b + b * b) = (a = b)⟧ ⟹ (4 * a * b = (a + b) * (a + b)) = (a = b)›\n discuss goal 1*)\n apply (simp add: field_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 77 facts*))\n (*discuss goal 2*)\n apply (simp add: field_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 77 facts*))\n (*proven 2 subgoals*) .\n with real_sqrt_le_mono[of \"4 * a * b\" \"(a + b) * (a + b)\"] (*‹4 * a * b ≤ (a + b) * (a + b) ⟹ sqrt (4 * a * b) ≤ sqrt ((a + b) * (a + b))›*) real_sqrt_eq_iff[of \"4 * a * b\" \"(a + b) * (a + b)\"] (*‹(sqrt (4 * a * b) = sqrt ((a + b) * (a + b))) = (4 * a * b = (a + b) * (a + b))›*) have \"sqrt (4 * a * b) ≤ sqrt ((a + b) * (a + b))\" and \"sqrt (4 * a * b) = sqrt ((a + b) * (a + b)) ⟷ a = b\"\n apply -\n (*goals:\n 1. ‹⟦4 * a * b ≤ (a + b) * (a + b) ⟹ sqrt (4 * a * b) ≤ sqrt ((a + b) * (a + b)); (sqrt (4 * a * b) = sqrt ((a + b) * (a + b))) = (4 * a * b = (a + b) * (a + b)); 4 * a * b ≤ (a + b) * (a + b); (4 * a * b = (a + b) * (a + b)) = (a = b)⟧ ⟹ sqrt (4 * a * b) ≤ sqrt ((a + b) * (a + b))›\n 2. ‹⟦4 * a * b ≤ (a + b) * (a + b) ⟹ sqrt (4 * a * b) ≤ sqrt ((a + b) * (a + b)); (sqrt (4 * a * b) = sqrt ((a + b) * (a + b))) = (4 * a * b = (a + b) * (a + b)); 4 * a * b ≤ (a + b) * (a + b); (4 * a * b = (a + b) * (a + b)) = (a = b)⟧ ⟹ (sqrt (4 * a * b) = sqrt ((a + b) * (a + b))) = (a = b)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n with ‹a ≥ 0› (*‹0 ≤ a›*) ‹b ≥ 0› (*‹0 ≤ b›*) have \"sqrt (4 * a * b) ≤ a + b\" and \"sqrt (4 * a * b) = a + b ⟷ a = b\"\n apply -\n (*goals:\n 1. ‹⟦0 ≤ a; 0 ≤ b; sqrt (4 * a * b) ≤ sqrt ((a + b) * (a + b)); (sqrt (4 * a * b) = sqrt ((a + b) * (a + b))) = (a = b)⟧ ⟹ sqrt (4 * a * b) ≤ a + b›\n 2. ‹⟦0 ≤ a; 0 ≤ b; sqrt (4 * a * b) ≤ sqrt ((a + b) * (a + b)); (sqrt (4 * a * b) = sqrt ((a + b) * (a + b))) = (a = b)⟧ ⟹ (sqrt (4 * a * b) = a + b) = (a = b)›\n discuss goal 1*)\n apply simp\n (*discuss goal 2*)\n apply simp\n (*proven 2 subgoals*) .\n with real_sqrt_abs2[of 2] (*‹sqrt (2 * 2) = ¦2¦›*) real_sqrt_mult[of 4 \"a * b\"] (*‹sqrt (4 * (a * b)) = sqrt 4 * sqrt (a * b)›*) show \"sqrt (a * b) ≤ (a + b) / 2\" and \"sqrt (a * b) = (a + b) / 2 ⟷ a = b\"\n apply -\n (*goals:\n 1. ‹⟦sqrt (2 * 2) = ¦2¦; sqrt (4 * (a * b)) = sqrt 4 * sqrt (a * b); sqrt (4 * a * b) ≤ a + b; (sqrt (4 * a * b) = a + b) = (a = b)⟧ ⟹ sqrt (a * b) ≤ (a + b) / 2›\n 2. ‹⟦sqrt (2 * 2) = ¦2¦; sqrt (4 * (a * b)) = sqrt 4 * sqrt (a * b); sqrt (4 * a * b) ≤ a + b; (sqrt (4 * a * b) = a + b) = (a = b)⟧ ⟹ (sqrt (a * b) = (a + b) / 2) = (a = b)›\n discuss goal 1*)\n apply (simp add: 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 37 facts*))\n (*discuss goal 2*)\n apply (simp add: 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 37 facts*))\n (*proven 2 subgoals*) .\nqed\n\nlemma refl_on_allrel: \"refl_on A (A × A)\"\n unfolding refl_on_def\n (*goal: ‹A × A ⊆ A × A ∧ (∀x∈A. (x, x) ∈ A × A)›*)\n by simp\n\nlemma refl_on_restrict:\n assumes \"refl_on A r\"\n shows \"refl_on (A ∩ B) (r ∩ B × B)\"\nproof (-)\n (*goal: ‹refl_on ((A::'a::type set) ∩ (B::'a::type set)) (Restr (r::('a::type × 'a::type) set) B)›*)\n from ‹refl_on A r› (*‹refl_on (A::'a::type set) (r::('a::type × 'a::type) set)›*) refl_on_allrel[of B] (*‹refl_on B (B × B)›*) refl_on_Int (*‹⟦refl_on ?A ?r; refl_on ?B ?s⟧ ⟹ refl_on (?A ∩ ?B) (?r ∩ ?s)›*) show \"?thesis\"\n (*goal: ‹refl_on (A ∩ B) (Restr r B)›*)\n by auto\nqed\n\nlemma sym_allrel: \"sym (A × A)\"\n unfolding sym_def\n (*goal: ‹∀(x::'a) y::'a. (x, y) ∈ (A::'a set) × A ⟶ (y, x) ∈ A × A›*)\n by simp\n\nlemma sym_restrict:\n assumes \"sym r\"\n shows \"sym (r ∩ A × A)\"\nproof (-)\n (*goal: ‹sym (Restr r A)›*)\n from ‹sym r› (*‹sym r›*) sym_allrel (*‹sym (?A × ?A)›*) sym_Int (*‹⟦sym ?r; sym ?s⟧ ⟹ sym (?r ∩ ?s)›*) show \"?thesis\"\n (*goal: ‹sym (Restr r A)›*)\n by auto\nqed\n\nlemma trans_allrel: \"trans (A × A)\"\n unfolding trans_def\n (*goal: ‹∀x y z. (x, y) ∈ A × A ⟶ (y, z) ∈ A × A ⟶ (x, z) ∈ A × A›*)\n by simp\n\nlemma equiv_Int:\n assumes \"equiv A r\" and \"equiv B s\"\n shows \"equiv (A ∩ B) (r ∩ s)\"\nproof (-)\n (*goal: ‹equiv (A ∩ B) (r ∩ s)›*)\n from assms (*‹equiv (A::'a set) (r::('a × 'a) set)› ‹equiv B s›*) refl_on_Int[of A r B s] (*‹⟦refl_on A r; refl_on B s⟧ ⟹ refl_on (A ∩ B) (r ∩ s)›*) sym_Int (*‹⟦sym ?r; sym ?s⟧ ⟹ sym (?r ∩ ?s)›*) trans_Int (*‹⟦trans ?r; trans ?s⟧ ⟹ trans (?r ∩ ?s)›*) show \"?thesis\"\n (*goal: ‹equiv (A ∩ B) (r ∩ s)›*)\n unfolding equiv_def\n (*goal: ‹refl_on (A ∩ B) (r ∩ s) ∧ sym (r ∩ s) ∧ trans (r ∩ s)›*)\n by auto\nqed\n\nlemma equiv_allrel: \"equiv A (A × A)\"\n unfolding equiv_def\n (*goal: ‹refl_on A (A × A) ∧ sym (A × A) ∧ trans (A × A)›*)\n by (simp add: refl_on_allrel (*‹refl_on ?A (?A × ?A)›*) sym_allrel (*‹sym (?A × ?A)›*) trans_allrel (*‹trans (?A × ?A)›*))\n\nlemma equiv_restrict:\n assumes \"equiv A r\"\n shows \"equiv (A ∩ B) (r ∩ B × B)\"\nproof (-)\n (*goal: ‹equiv (A ∩ B) (Restr r B)›*)\n from ‹equiv A r› (*‹equiv A r›*) equiv_allrel[of B] (*‹equiv B (B × B)›*) equiv_Int (*‹⟦equiv ?A ?r; equiv ?B ?s⟧ ⟹ equiv (?A ∩ ?B) (?r ∩ ?s)›*) show \"?thesis\"\n (*goal: ‹equiv (A ∩ B) (Restr r B)›*)\n by auto\nqed\n\nlemma invertible_times_eq_zero:\n fixes x :: \"real^'n\" and A :: \"real^'n^'n\"\n assumes \"invertible A\" and \"A *v x = 0\"\n shows \"x = 0\"\n using assms (*‹invertible (A::((real, 'n) vec, 'n) vec)› ‹A *v x = 0›*) invertible_def (*‹invertible ?A = (∃A'. ?A ** A' = mat 1 ∧ A' ** ?A = mat 1)›*) matrix_left_invertible_ker (*‹(∃B. B ** ?A = mat 1) = (∀x. ?A *v x = 0 ⟶ x = 0)›*) by blast\n\nlemma times_invertible_eq_zero:\n fixes x :: \"real^'n\" and A :: \"real^'n^'n\"\n assumes \"invertible A\" and \"x v* A = 0\"\n shows \"x = 0\"\n using transpose_invertible (*‹invertible (?A::((real, ?'n::finite) vec, ?'n::finite) vec) ⟹ invertible (Finite_Cartesian_Product.transpose ?A)›*) assms (*‹invertible A› ‹(x::(real, 'n) vec) v* (A::((real, 'n) vec, 'n) vec) = (0::(real, 'n) vec)›*) invertible_times_eq_zero (*‹⟦invertible ?A; ?A *v ?x = 0⟧ ⟹ ?x = 0›*) by fastforce\n\nlemma matrix_id_invertible:\n \"invertible (mat 1 :: ('a::semiring_1)^'n^'n)\"\n by (simp add: invertible_def (*‹invertible ?A = (∃A'. ?A ** A' = mat 1 ∧ A' ** ?A = mat 1)›*))\n\nlemma Image_refl_on_nonempty:\n assumes \"refl_on A r\" and \"x ∈ A\"\n shows \"x ∈ r``{x}\"\nproof (standard)\n (*goal: ‹(x, x) ∈ r›*)\n from ‹refl_on A r› (*‹refl_on A r›*) ‹x ∈ A› (*‹(x::'a::type) ∈ (A::'a::type set)›*) show \"(x, x) ∈ r\"\n unfolding refl_on_def\n (*goal: ‹(x, x) ∈ r›*)\n by simp\nqed\n\nlemma quotient_element_nonempty:\n assumes \"equiv A r\" and \"X ∈ A//r\"\n shows \"∃ x. x ∈ X\"\n using assms (*‹equiv A r› ‹X ∈ A // r›*) in_quotient_imp_non_empty (*‹⟦equiv ?A ?r; ?X ∈ ?A // ?r⟧ ⟹ ?X ≠ {}›*) by fastforce\n\nlemma zero_3: \"(3::3) = 0\"\n by simp\n\nlemma card_suc_ge_insert:\n fixes A and x\n shows \"card A + 1 ≥ card (insert x A)\"\n using card_insert_le_m1 (*‹⟦0 < ?n; card ?y ≤ ?n - 1⟧ ⟹ card (insert ?x ?y) ≤ ?n›*) by fastforce\n\nlemma card_le_UNIV:\n fixes A :: \"('n::finite) set\"\n shows \"card A ≤ CARD('n)\"\n by (simp add: card_mono (*‹⟦finite ?B; ?A ⊆ ?B⟧ ⟹ card ?A ≤ card ?B›*))\n\nlemma partition_Image_element:\n assumes \"equiv A r\" and \"X ∈ A//r\" and \"x ∈ X\"\n shows \"r``{x} = X\"\n by (metis Image_singleton_iff (*‹(?b ∈ ?r `` {?a}) = ((?a, ?b) ∈ ?r)›*) assms (*‹equiv A r› ‹X ∈ A // r› ‹x ∈ X›*) equiv_class_eq_iff (*‹equiv ?A ?r ⟹ ((?x, ?y) ∈ ?r) = (?r `` {?x} = ?r `` {?y} ∧ ?x ∈ ?A ∧ ?y ∈ ?A)›*) quotientE (*‹⟦?X ∈ ?A // ?r; ⋀x. ⟦?X = ?r `` {x}; x ∈ ?A⟧ ⟹ ?P⟧ ⟹ ?P›*))\n\nlemma card_insert_ge: \"card (insert x A) ≥ card A\"\n by (metis card.infinite (*‹infinite (?A::?'a set) ⟹ card ?A = (0::nat)›*) card_insert_le (*‹card (?A::?'a set) ≤ card (insert (?x::?'a) ?A)›*) zero_le (*‹(0::?'a) ≤ (?x::?'a)›*))\n\nlemma choose_1:\n assumes \"card S = 1\"\n shows \"∃ x. S = {x}\"\n using ‹card S = 1› (*‹card S = 1›*) card_eq_SucD[of S 0] (*‹card S = Suc 0 ⟹ ∃b B. S = insert b B ∧ b ∉ B ∧ card B = 0 ∧ (0 = 0 ⟶ B = {})›*) by simp\n\nlemma choose_2:\n assumes \"card S = 2\"\n shows \"∃ x y. S = {x,y}\"\nproof (-)\n (*goal: ‹∃x y. S = {x, y}›*)\n from ‹card S = 2› (*‹card S = 2›*) card_eq_SucD[of S 1] (*‹card S = Suc 1 ⟹ ∃b B. S = insert b B ∧ b ∉ B ∧ card B = 1 ∧ (1 = 0 ⟶ B = {})›*) obtain x and T where \"S = insert x T\" and \"card T = 1\"\n (*goal: ‹(⋀x T. ⟦S = insert x T; card T = 1⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n from ‹card T = 1› (*‹card T = 1›*) choose_1 (*‹card (?S::?'a set) = (1::nat) ⟹ ∃x::?'a. ?S = {x}›*) obtain y where \"T = {y}\"\n (*goal: ‹(⋀y::'a. (T::'a set) = {y} ⟹ thesis::bool) ⟹ thesis›*)\n by auto\n with ‹S = insert x T› (*‹(S::'a::type set) = insert (x::'a::type) (T::'a::type set)›*) have \"S = {x,y}\"\n by simp\n thus \"∃ x y. S = {x,y}\"\n by auto\nqed\n\nlemma choose_3:\n assumes \"card S = 3\"\n shows \"∃ x y z. S = {x,y,z}\"\nproof (-)\n (*goal: ‹∃x y z. S = {x, y, z}›*)\n from ‹card S = 3› (*‹card S = 3›*) card_eq_SucD[of S 2] (*‹card S = Suc 2 ⟹ ∃b B. S = insert b B ∧ b ∉ B ∧ card B = 2 ∧ (2 = 0 ⟶ B = {})›*) obtain x and T where \"S = insert x T\" and \"card T = 2\"\n (*goal: ‹(⋀x T. ⟦S = insert x T; card T = 2⟧ ⟹ thesis) ⟹ thesis›*)\n by auto\n from ‹card T = 2› (*‹card T = 2›*) choose_2[of T] (*‹card T = 2 ⟹ ∃x y. T = {x, y}›*) obtain y and z where \"T = {y,z}\"\n (*goal: ‹(⋀y z. T = {y, z} ⟹ thesis) ⟹ thesis›*)\n by auto\n with ‹S = insert x T› (*‹(S::'a set) = insert (x::'a) (T::'a set)›*) have \"S = {x,y,z}\"\n by simp\n thus \"∃ x y z. S = {x,y,z}\"\n by auto\nqed\n\nlemma card_gt_0_diff_singleton:\n assumes \"card S > 0\" and \"x ∈ S\"\n shows \"card (S - {x}) = card S - 1\"\nproof (-)\n (*goal: ‹card (S - {x}) = card S - 1›*)\n from ‹card S > 0› (*‹0 < card S›*) have \"finite S\"\n by (rule card_ge_0_finite (*‹0 < card ?A ⟹ finite ?A›*))\n with ‹x ∈ S› (*‹(x::'a) ∈ (S::'a set)›*) show \"card (S - {x}) = card S - 1\"\n by (simp add: card_Diff_singleton (*‹?x ∈ ?A ⟹ card (?A - {?x}) = card ?A - 1›*))\nqed\n\nlemma eq_3_or_of_3:\n fixes j :: 4\n shows \"j = 3 ∨ (∃ j'::3. j = of_int (Rep_bit1 j'))\"\nproof (induct j)\n (*goal: ‹⋀z. ⟦0 ≤ z; z < int CARD(4)⟧ ⟹ of_int z = 3 ∨ (∃j'. of_int z = of_int (Rep_bit1 j'))›*)\n fix j_int :: int\n assume \"0 ≤ j_int\" (*‹(0::int) ≤ (j_int::int)›*)\n assume \"j_int < int CARD(4)\" (*‹(j_int::int) < int CARD(4)›*)\n hence \"j_int ≤ 3\"\n by simp\n show \"of_int j_int = (3::4) ∨ (∃ j'::3. of_int j_int = of_int (Rep_bit1 j'))\"\n proof (cases)\n (*goals:\n 1. ‹?P ⟹ of_int j_int = 3 ∨ (∃j'. of_int j_int = of_int (Rep_bit1 j'))›\n 2. ‹¬ ?P ⟹ of_int j_int = 3 ∨ (∃j'. of_int j_int = of_int (Rep_bit1 j'))›*)\n assume \"j_int = 3\" (*‹(j_int::int) = (3::int)›*)\n thus \"of_int j_int = (3::4) ∨ (∃ j'::3. of_int j_int = of_int (Rep_bit1 j'))\"\n by simp\n next\n (*goal: ‹j_int ≠ 3 ⟹ of_int j_int = 3 ∨ (∃j'. of_int j_int = of_int (Rep_bit1 j'))›*)\n assume \"j_int ≠ 3\" (*‹(j_int::int) ≠ (3::int)›*)\n with ‹j_int ≤ 3› (*‹j_int ≤ 3›*) have \"j_int < 3\"\n by simp\n with ‹0 ≤ j_int› (*‹0 ≤ j_int›*) have \"j_int ∈ {0..<3}\"\n by simp\n hence \"Rep_bit1 (Abs_bit1 j_int :: 3) = j_int\"\n by (simp add: bit1.Abs_inverse (*‹?m ∈ {0..<int CARD(?'a bit1)} ⟹ Rep_bit1 (Abs_bit1 ?m) = ?m›*))\n hence \"of_int j_int = of_int (Rep_bit1 (Abs_bit1 j_int :: 3))\"\n by simp\n thus \"of_int j_int = (3::4) ∨ (∃ j'::3. of_int j_int = of_int (Rep_bit1 j'))\"\n by auto\n qed\nqed\n\nlemma sgn_plus:\n fixes x y :: \"'a::linordered_idom\"\n assumes \"sgn x = sgn y\"\n shows \"sgn (x + y) = sgn x\"\n by (simp add: assms (*‹sgn x = sgn y›*) same_sgn_sgn_add (*‹sgn ?b = sgn ?a ⟹ sgn (?a + ?b) = sgn ?a›*))\n\nlemma sgn_div:\n fixes x y :: \"'a::linordered_field\"\n assumes \"y ≠ 0\" and \"sgn x = sgn y\"\n shows \"x / y > 0\"\n using assms (*‹y ≠ 0› ‹sgn x = sgn y›*) sgn_1_pos (*‹(sgn ?a = 1) = (0 < ?a)›*) sgn_eq_0_iff (*‹(sgn ?a = 0) = (?a = 0)›*) by fastforce\n\nlemma abs_plus:\n fixes x y :: \"'a::linordered_idom\"\n assumes \"sgn x = sgn y\"\n shows \"¦x + y¦ = ¦x¦ + ¦y¦\"\n by (simp add: assms (*‹sgn x = sgn y›*) same_sgn_abs_add (*‹sgn ?b = sgn ?a ⟹ ¦?a + ?b¦ = ¦?a¦ + ¦?b¦›*))\n\nlemma sgn_plus_abs:\n fixes x y :: \"'a::linordered_idom\"\n assumes \"¦x¦ > ¦y¦\"\n shows \"sgn (x + y) = sgn x\"\n apply (cases \"x > 0\")\n (*goals:\n 1. ‹0 < x ⟹ sgn (x + y) = sgn x›\n 2. ‹¬ 0 < x ⟹ sgn (x + y) = sgn x›\n discuss goal 1*)\n apply ((use assms in auto)[1])\n (*discuss goal 2*)\n apply ((use assms in auto)[1])\n (*proven 2 subgoals*) .\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Tarskis_Geometry/Miscellany.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"18f5efe3581d5f7d146102856023e81a5ad349cca855c8d87384ef39d55d3117\"\n}"}}},{"rowIdx":96,"cells":{"text":{"kind":"string","value":"theory RM_Sums_Diophantine imports Equation_Setup \"../Diophantine/Register_Machine_Sums\"\n \"../Diophantine/Binary_And\"\n\nbegin\n\ncontext register_machine\nbegin\n\ndefinition sum_ssub_nzero_of_bit_and :: \"polynomial ⇒ nat ⇒ polynomial list ⇒ polynomial list\n ⇒ relation\" \n (\"[_ = ∑S- _ '(_ && _')]\") where\n \"[x = ∑S- d (s && z)] ≡ let x' = push_param x (length p);\n s' = push_param_list s (length p); \n z' = push_param_list z (length p)\n in [∃length p] [∀<length p] (λi. [Param i = s'!i && z'!i])\n [∧] x' [=] ([∑S-] p d Param)\"\n\nlemma sum_ssub_nzero_of_bit_and_dioph[dioph]:\n fixes s z :: \"polynomial list\" and d :: nat and x\n shows \"is_dioph_rel [x = ∑S- d (s && z)]\"\n unfolding sum_ssub_nzero_of_bit_and_def\n (*goal: ‹is_dioph_rel (let x' = push_param x (length p); s' = push_param_list s (length p); z' = push_param_list z (length p) in [∃length p] [∀<length p] λi. [polynomial.Param i = s' ! i && z' ! i] [∧] x' [=] [∑S-] p d polynomial.Param)›*)\n by (auto 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 11 facts*))\n\nlemma sum_rsub_nzero_of_bit_and_eval:\n fixes z s :: \"polynomial list\" and d :: nat and x :: polynomial\n assumes \"length s = Suc m\" \"length z = Suc m\" \"length p > 0\"\n shows \"eval [x = ∑S- d (s && z)] a\n ⟷ peval x a = ∑S- p d (λk. peval (s!k) a && peval (z!k) a)\" (is \"?P ⟷ ?Q\")\nproof (-)\n (*goal: ‹eval [x = ∑S- d (s && z)] a = (peval x a = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) )›*)\n have invariance: \"∀k<length p. y1 k = y2 k ⟹ ∑S- p d y1 = ∑S- p d y2\" for y1 and y2\n unfolding sum_ssub_nzero.simps\n (*goal: ‹∀k<length p. y1 k = y2 k ⟹ (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to (p ! k) then y1 k else 0) = (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to (p ! k) then y2 k else 0)›*)\n apply (intro sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*), simp)\n (*goal: ‹∀k<length p. y1 k = y2 k ⟹ (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to (p ! k) then y1 k else 0) = (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to (p ! k) then y2 k else 0)›*)\n using ‹length p > 0› (*‹(0::nat) < length (p::instruction list)›*) apply auto\n (*goal: ‹⋀x. ⟦∀k<length p. y1 k = y2 k; x ∈ {0..length p - 1}⟧ ⟹ (if issub (p ! x) ∧ d = goes_to (p ! x) then y1 x else 0) = (if issub (p ! x) ∧ d = goes_to (p ! x) then y2 x else 0)›*)\n by (metis Suc_pred (*‹0 < ?n ⟹ Suc (?n - Suc 0) = ?n›*) le_imp_less_Suc (*‹?m ≤ ?n ⟹ ?m < Suc ?n›*) length_greater_0_conv (*‹(0 < length ?xs) = (?xs ≠ [])›*))\n have len_ps: \"length s = length p\"\n using m_def (*‹m ≡ length (p::instruction list) - (1::nat)›*) ‹length s = Suc m› (*‹length s = Suc m›*) ‹length p > 0› (*‹0 < length p›*) by auto\n have len_pz: \"length z = length p\"\n using m_def (*‹m ≡ length (p::instruction list) - (1::nat)›*) ‹length z = Suc m› (*‹length z = Suc m›*) ‹length p > 0› (*‹0 < length p›*) by auto\n show \"?thesis\"\n (*goal: ‹eval [x = ∑S- d (s && z)] a = (peval x a = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) )›*)\n proof (rule)\n (*goals:\n 1. ‹eval [x = ∑S- d (s && z)] a ⟹ peval x a = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ›\n 2. ‹peval x a = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ⟹ eval [x = ∑S- d (s && z)] a›*)\n assume \"?P\" (*‹eval [x::polynomial = ∑S- d::nat (s::polynomial list && z::polynomial list)] (a::nat ⇒ nat)›*)\n thus \"?Q\"\n using sum_ssub_nzero_of_bit_and_def (*‹[?x = ∑S- ?d (?s && ?z)] ≡ let x' = push_param ?x (length p); s' = push_param_list ?s (length p); z' = push_param_list ?z (length p) in [∃length p] [∀<length p] λi. [polynomial.Param i = s' ! i && z' ! i] [∧] x' [=] [∑S-] p ?d polynomial.Param›*) ‹length p > 0› (*‹0 < length p›*) apply (auto simp add: defs (*‹❙0 = polynomial.Const (0::nat)› ‹❙1 = polynomial.Const (1::nat)› ‹(?Q::polynomial) [=] (?R::polynomial) ≡ BINARY (=) ?Q ?R› ‹(?Q::polynomial) [<] (?R::polynomial) ≡ BINARY (<) ?Q ?R› ‹(?Q::polynomial) [≤] (?R::polynomial) ≡ ?Q [<] ?R [∨] ?Q [=] ?R› ‹(?Q::polynomial) [>] (?R::polynomial) ≡ ?R [<] ?Q› ‹(?Q::polynomial) [≥] (?R::polynomial) ≡ ?Q [>] ?R [∨] ?Q [=] ?R› ‹eval (LARY (?R::nat list list ⇒ bool) (?PLL::polynomial list list)) (?a::nat ⇒ nat) = ?R (map (map (λP::polynomial. peval P ?a)) ?PLL)› ‹UNARY (?R::nat ⇒ bool) (?P::polynomial) = NARY (λl::nat list. ?R (l ! (0::nat))) [?P]› ‹BINARY (?R::nat ⇒ nat ⇒ bool) (?P₁::polynomial) (?P₂::polynomial) = NARY (λl::nat list. ?R (l ! (0::nat)) (l ! (1::nat))) [?P₁, ?P₂]› ‹TERNARY (?R::nat ⇒ nat ⇒ nat ⇒ bool) (?P₁::polynomial) (?P₂::polynomial) (?P₃::polynomial) = NARY (λl::nat list. ?R (l ! (0::nat)) (l ! (1::nat)) (l ! (2::nat))) [?P₁, ?P₂, ?P₃]› ‹QUATERNARY (?R::nat ⇒ nat ⇒ nat ⇒ nat ⇒ bool) (?P₁::polynomial) (?P₂::polynomial) (?P₃::polynomial) (?P₄::polynomial) = NARY (λl::nat list. ?R (l ! (0::nat)) (l ! (1::nat)) (l ! (2::nat)) (l ! (3::nat))) [?P₁, ?P₂, ?P₃, ?P₄]› and more 19 facts*) push_push (*‹length (?ns::nat list) = (?n::nat) ⟹ peval (push_param (?P::polynomial) ?n) (push_list (?a::nat ⇒ nat) ?ns) = peval ?P ?a›*))\n (*goal: ‹peval x a = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n using push_push_map_i (*‹⟦length ?ns = ?n; ?i < length ?ls⟧ ⟹ peval (map (λx. push_param x ?n) ?ls ! ?i) (push_list ?a ?ns) = list_eval ?ls ?a ?i›*) apply (simp add: push_param_list_def (*‹push_param_list ?s ?k ≡ map (λx. push_param x ?k) ?s›*) len_ps (*‹length s = length p›*) len_pz (*‹length z = length p›*))\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S- d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, push_param_list s (length ks) ! i, push_param_list z (length ks) ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S-] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = peval (push_param_list s (length ks) ! k) (push_list a ks) && peval (push_param_list z (length ks) ! k) (push_list a ks); peval x a = ∑S- p d push_list a ks ⟧ ⟹ ∑S- p d push_list a ks = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n unfolding list_eval_def\n (*goal: ‹⋀ks::nat list. ⟦⋀(x::polynomial) (d::nat) (s::polynomial list) z::polynomial list. [x = ∑S- d (s && z)] ≡ [∃length ks] [∀<length ks] λi::nat. NARY (λl::nat list. l ! (0::nat) = l ! Suc (0::nat) && l ! (2::nat)) [polynomial.Param i, map (λx::polynomial. push_param x (length ks)) s ! i, map (λx::polynomial. push_param x (length ks)) z ! i] [∧] NARY (λl::nat list. l ! (0::nat) = l ! Suc (0::nat)) [push_param x (length ks), [∑S-] p::instruction list d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list (a::nat ⇒ nat) ks k = map (λx::polynomial. peval x a) (s::polynomial list) ! k && map (λx::polynomial. peval x a) (z::polynomial list) ! k; peval (x::polynomial) a = ∑S- p (d::nat) push_list a ks ; ⋀(ns::nat list) (n::nat) (i::nat) (ls::polynomial list) a::nat ⇒ nat. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = map (λx::polynomial. peval x a) ls ! i⟧ ⟹ ∑S- p d push_list a ks = ∑S- p d (λk::nat. peval (s ! k) a && peval (z ! k) a) ›*)\n apply (auto simp: assms (*‹length s = Suc m› ‹length z = Suc m› ‹0 < length p›*) len_ps (*‹length s = length p›*) len_pz (*‹length z = length p›*) invariance (*‹∀k<length p. ?y1.0 k = ?y2.0 k ⟹ ∑S- p d ?y1.0 = ∑S- p d ?y2.0 ›*))\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S- d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, map (λx. push_param x (length ks)) s ! i, map (λx. push_param x (length ks)) z ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S-] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = map (λx. peval x a) s ! k && map (λx. peval x a) z ! k; peval x a = ∑S- p d push_list a ks ; ⋀ns n i ls a. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = map (λx. peval x a) ls ! i⟧ ⟹ ∑S- p d push_list a ks = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n apply (rule sum_ssub_nzero_cong (*‹∀k. k ≤ length ?p - 1 ∧ issub (?p ! k) ∧ ?l = goes_to (?p ! k) ⟶ ?f k = ?g k ⟹ ∑S- ?p ?l ?f = ∑S- ?p ?l ?g ›*))\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S- d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, map (λx. push_param x (length ks)) s ! i, map (λx. push_param x (length ks)) z ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S-] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = map (λx. peval x a) s ! k && map (λx. peval x a) z ! k; peval x a = ∑S- p d push_list a ks ; ⋀ns n i ls a. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = peval (ls ! i) a⟧ ⟹ ∑S- p d push_list a ks = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n apply auto\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S- d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, map (λx. push_param x (length ks)) s ! i, map (λx. push_param x (length ks)) z ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S-] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = map (λx. peval x a) s ! k && map (λx. peval x a) z ! k; peval x a = ∑S- p d push_list a ks ; ⋀ns n i ls a. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = peval (ls ! i) a⟧ ⟹ ∀k. k ≤ length p - 1 ∧ issub (p ! k) ∧ d = goes_to (p ! k) ⟶ push_list a ks k = peval (s ! k) a && peval (z ! k) a›*)\n by (metis (no_types, lifting) One_nat_def (*‹1 = Suc 0›*) assms( (*‹length s = Suc m›*) 1) assms( (*‹length z = Suc m›*) 2) le_imp_less_Suc (*‹?m ≤ ?n ⟹ ?m < Suc ?n›*) len_ps (*‹length s = length p›*) m_def (*‹m ≡ length p - 1›*) nth_map (*‹?n < length ?xs ⟹ map ?f ?xs ! ?n = ?f (?xs ! ?n)›*))\n next\n (*goal: ‹peval x a = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ⟹ eval [x = ∑S- d (s && z)] a›*)\n assume \"?Q\" (*‹peval (x::polynomial) (a::nat ⇒ nat) = ∑S- (p::instruction list) (d::nat) (λk::nat. peval ((s::polynomial list) ! k) a && peval ((z::polynomial list) ! k) a) ›*)\n thus \"?P\"\n using sum_ssub_nzero_of_bit_and_def (*‹[?x = ∑S- ?d (?s && ?z)] ≡ let x' = push_param ?x (length p); s' = push_param_list ?s (length p); z' = push_param_list ?z (length p) in [∃length p] [∀<length p] λi. [polynomial.Param i = s' ! i && z' ! i] [∧] x' [=] [∑S-] p ?d polynomial.Param›*) ‹length p > 0› (*‹0 < length p›*) apply (auto simp add: defs (*‹❙0 = polynomial.Const 0› ‹❙1 = polynomial.Const 1› ‹?Q [=] ?R ≡ BINARY (=) ?Q ?R› ‹?Q [<] ?R ≡ BINARY (<) ?Q ?R› ‹?Q [≤] ?R ≡ ?Q [<] ?R [∨] ?Q [=] ?R› ‹?Q [>] ?R ≡ ?R [<] ?Q› ‹?Q [≥] ?R ≡ ?Q [>] ?R [∨] ?Q [=] ?R› ‹eval (LARY ?R ?PLL) ?a = ?R (map (map (λP. peval P ?a)) ?PLL)› ‹UNARY ?R ?P = NARY (λl. ?R (l ! 0)) [?P]› ‹BINARY ?R ?P₁ ?P₂ = NARY (λl. ?R (l ! 0) (l ! 1)) [?P₁, ?P₂]› ‹TERNARY ?R ?P₁ ?P₂ ?P₃ = NARY (λl. ?R (l ! 0) (l ! 1) (l ! 2)) [?P₁, ?P₂, ?P₃]› ‹QUATERNARY ?R ?P₁ ?P₂ ?P₃ ?P₄ = NARY (λl. ?R (l ! 0) (l ! 1) (l ! 2) (l ! 3)) [?P₁, ?P₂, ?P₃, ?P₄]› and more 19 facts*) push_push (*‹length ?ns = ?n ⟹ peval (push_param ?P ?n) (push_list ?a ?ns) = peval ?P ?a›*))\n (*goal: ‹eval [x = ∑S- d (s && z)] a›*)\n apply (rule exI[of _ \"map (λk. peval (s ! k) a && peval (z ! k) a) [0..<length p]\"] (*‹?P (map (λk. peval (s ! k) a && peval (z ! k) a) [0..<length p]) ⟹ ∃x. ?P x›*), simp)\n (*goal: ‹⟦peval x a = ∑S- p d (λk. peval (s ! k) a && peval (z ! k) a) ; ⋀x d s z. [x = ∑S- d (s && z)] ≡ [∃length p] [∀<length p] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, push_param_list s (length p) ! i, push_param_list z (length p) ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length p), [∑S-] p d polynomial.Param]; p ≠ []⟧ ⟹ ∃ks. length p = length ks ∧ (∀k<length p. push_list a ks k = peval (push_param_list s (length p) ! k) (push_list a ks) && peval (push_param_list z (length p) ! k) (push_list a ks)) ∧ peval (push_param x (length p)) (push_list a ks) = ∑S- p d push_list a ks ›*)\n using push_push (*‹length ?ns = ?n ⟹ peval (push_param ?P ?n) (push_list ?a ?ns) = peval ?P ?a›*) push_push_map_i (*‹⟦length (?ns::nat list) = (?n::nat); (?i::nat) < length (?ls::polynomial list)⟧ ⟹ peval (map (λx::polynomial. push_param x ?n) ?ls ! ?i) (push_list (?a::nat ⇒ nat) ?ns) = list_eval ?ls ?a ?i›*) by (simp add: push_param_list_def (*‹push_param_list (?s::polynomial list) (?k::nat) ≡ map (λx::polynomial. push_param x ?k) ?s›*) invariance (*‹∀k<length (p::instruction list). (?y1.0::nat ⇒ nat) k = (?y2.0::nat ⇒ nat) k ⟹ ∑S- p (d::nat) ?y1.0 = ∑S- p d ?y2.0 ›*) push_list_eval (*‹(?i::nat) < length (?ns::nat list) ⟹ push_list (?a::nat ⇒ nat) ?ns ?i = ?ns ! ?i›*) len_ps (*‹length (s::polynomial list) = length (p::instruction list)›*) len_pz (*‹length (z::polynomial list) = length (p::instruction list)›*))\n qed\nqed\n\ndefinition sum_ssub_zero_of_bit_and :: \"polynomial ⇒ nat ⇒ polynomial list ⇒ polynomial list\n ⇒ relation\" \n (\"[_ = ∑S0 _ '(_ && _')]\") where\n \"[x = ∑S0 d (s && z)] ≡ let x' = push_param x (length p);\n s' = push_param_list s (length p); \n z' = push_param_list z (length p)\n in [∃length p] [∀<length p] (λi. [Param i = s'!i && z'!i])\n [∧] x' [=] [∑S0] p d Param\"\n\nlemma sum_ssub_zero_of_bit_and_dioph[dioph]:\n fixes s z :: \"polynomial list\" and d :: nat and x\n shows \"is_dioph_rel [x = ∑S0 d (s && z)]\"\n unfolding sum_ssub_zero_of_bit_and_def\n (*goal: ‹is_dioph_rel (let x' = push_param x (length p); s' = push_param_list s (length p); z' = push_param_list z (length p) in [∃length p] [∀<length p] λi. [polynomial.Param i = s' ! i && z' ! i] [∧] x' [=] [∑S0] p d polynomial.Param)›*)\n by (auto 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 12 facts*))\n\nlemma sum_rsub_zero_of_bit_and_eval:\n fixes z s :: \"polynomial list\" and d :: nat and x :: polynomial\n assumes \"length s = Suc m\" \"length z = Suc m\" \"length p > 0\"\n shows \"eval [x = ∑S0 d (s && z)] a\n ⟷ peval x a = ∑S0 p d (λk. peval (s!k) a && peval (z!k) a)\" (is \"?P ⟷ ?Q\")\nproof (-)\n (*goal: ‹eval [x::polynomial = ∑S0 d::nat (s::polynomial list && z::polynomial list)] (a::nat ⇒ nat) = (peval x a = ∑S0 (p::instruction list) d (λk::nat. peval (s ! k) a && peval (z ! k) a) )›*)\n have invariance: \"∀k<length p. y1 k = y2 k ⟹ ∑S0 p d y1 = ∑S0 p d y2\" for y1 and y2\n unfolding sum_ssub_zero.simps\n (*goal: ‹∀k<length p. y1 k = y2 k ⟹ (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to_alt (p ! k) then y1 k else 0) = (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to_alt (p ! k) then y2 k else 0)›*)\n apply (intro sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*), simp)\n (*goal: ‹∀k<length p. y1 k = y2 k ⟹ (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to_alt (p ! k) then y1 k else 0) = (∑k = 0..length p - 1. if issub (p ! k) ∧ d = goes_to_alt (p ! k) then y2 k else 0)›*)\n using ‹length p > 0› (*‹0 < length p›*) apply auto\n (*goal: ‹⋀x. ⟦∀k<length p. y1 k = y2 k; x ∈ {0..length p - 1}⟧ ⟹ (if issub (p ! x) ∧ d = goes_to_alt (p ! x) then y1 x else 0) = (if issub (p ! x) ∧ d = goes_to_alt (p ! x) then y2 x else 0)›*)\n by (metis Suc_pred (*‹0 < ?n ⟹ Suc (?n - Suc 0) = ?n›*) le_imp_less_Suc (*‹?m ≤ ?n ⟹ ?m < Suc ?n›*) length_greater_0_conv (*‹(0 < length ?xs) = (?xs ≠ [])›*))\n have len_ps: \"length s = length p\"\n using m_def (*‹m ≡ length p - 1›*) ‹length s = Suc m› (*‹length (s::polynomial list) = Suc m›*) ‹length p > 0› (*‹0 < length p›*) by auto\n have len_pz: \"length z = length p\"\n using m_def (*‹m ≡ length (p::instruction list) - (1::nat)›*) ‹length z = Suc m› (*‹length z = Suc m›*) ‹length p > 0› (*‹0 < length p›*) by auto\n show \"?thesis\"\n (*goal: ‹eval [x = ∑S0 d (s && z)] a = (peval x a = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) )›*)\n proof (rule)\n (*goals:\n 1. ‹eval [x = ∑S0 d (s && z)] a ⟹ peval x a = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ›\n 2. ‹peval x a = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ⟹ eval [x = ∑S0 d (s && z)] a›*)\n assume \"?P\" (*‹eval [x::polynomial = ∑S0 d::nat (s::polynomial list && z::polynomial list)] (a::nat ⇒ nat)›*)\n thus \"?Q\"\n using sum_ssub_zero_of_bit_and_def (*‹[?x = ∑S0 ?d (?s && ?z)] ≡ let x' = push_param ?x (length p); s' = push_param_list ?s (length p); z' = push_param_list ?z (length p) in [∃length p] [∀<length p] λi. [polynomial.Param i = s' ! i && z' ! i] [∧] x' [=] [∑S0] p ?d polynomial.Param›*) ‹length p > 0› (*‹0 < length p›*) apply (auto simp add: defs (*‹❙0 = polynomial.Const 0› ‹❙1 = polynomial.Const 1› ‹?Q [=] ?R ≡ BINARY (=) ?Q ?R› ‹?Q [<] ?R ≡ BINARY (<) ?Q ?R› ‹?Q [≤] ?R ≡ ?Q [<] ?R [∨] ?Q [=] ?R› ‹?Q [>] ?R ≡ ?R [<] ?Q› ‹?Q [≥] ?R ≡ ?Q [>] ?R [∨] ?Q [=] ?R› ‹eval (LARY ?R ?PLL) ?a = ?R (map (map (λP. peval P ?a)) ?PLL)› ‹UNARY ?R ?P = NARY (λl. ?R (l ! 0)) [?P]› ‹BINARY ?R ?P₁ ?P₂ = NARY (λl. ?R (l ! 0) (l ! 1)) [?P₁, ?P₂]› ‹TERNARY ?R ?P₁ ?P₂ ?P₃ = NARY (λl. ?R (l ! 0) (l ! 1) (l ! 2)) [?P₁, ?P₂, ?P₃]› ‹QUATERNARY ?R ?P₁ ?P₂ ?P₃ ?P₄ = NARY (λl. ?R (l ! 0) (l ! 1) (l ! 2) (l ! 3)) [?P₁, ?P₂, ?P₃, ?P₄]› and more 19 facts*) push_push (*‹length ?ns = ?n ⟹ peval (push_param ?P ?n) (push_list ?a ?ns) = peval ?P ?a›*))\n (*goal: ‹peval x a = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n using push_push_map_i (*‹⟦length ?ns = ?n; ?i < length ?ls⟧ ⟹ peval (map (λx. push_param x ?n) ?ls ! ?i) (push_list ?a ?ns) = list_eval ?ls ?a ?i›*) apply (simp add: push_param_list_def (*‹push_param_list ?s ?k ≡ map (λx. push_param x ?k) ?s›*) len_ps (*‹length s = length p›*) len_pz (*‹length z = length p›*))\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S0 d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, push_param_list s (length ks) ! i, push_param_list z (length ks) ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S0] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = peval (push_param_list s (length ks) ! k) (push_list a ks) && peval (push_param_list z (length ks) ! k) (push_list a ks); peval x a = ∑S0 p d push_list a ks ⟧ ⟹ ∑S0 p d push_list a ks = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n unfolding list_eval_def\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S0 d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, map (λx. push_param x (length ks)) s ! i, map (λx. push_param x (length ks)) z ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S0] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = map (λx. peval x a) s ! k && map (λx. peval x a) z ! k; peval x a = ∑S0 p d push_list a ks ; ⋀ns n i ls a. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = map (λx. peval x a) ls ! i⟧ ⟹ ∑S0 p d push_list a ks = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n apply (auto simp: assms (*‹length s = Suc m› ‹length z = Suc m› ‹0 < length p›*) len_ps (*‹length s = length p›*) len_pz (*‹length z = length p›*) invariance (*‹∀k<length p. ?y1.0 k = ?y2.0 k ⟹ ∑S0 p d ?y1.0 = ∑S0 p d ?y2.0 ›*))\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S0 d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, map (λx. push_param x (length ks)) s ! i, map (λx. push_param x (length ks)) z ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S0] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = map (λx. peval x a) s ! k && map (λx. peval x a) z ! k; peval x a = ∑S0 p d push_list a ks ; ⋀ns n i ls a. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = map (λx. peval x a) ls ! i⟧ ⟹ ∑S0 p d push_list a ks = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n apply (rule sum_ssub_zero_cong (*‹∀k. k ≤ length ?p - 1 ∧ issub (?p ! k) ∧ ?l = goes_to_alt (?p ! k) ⟶ ?f k = ?g k ⟹ ∑S0 ?p ?l ?f = ∑S0 ?p ?l ?g ›*))\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S0 d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, map (λx. push_param x (length ks)) s ! i, map (λx. push_param x (length ks)) z ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S0] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = map (λx. peval x a) s ! k && map (λx. peval x a) z ! k; peval x a = ∑S0 p d push_list a ks ; ⋀ns n i ls a. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = peval (ls ! i) a⟧ ⟹ ∑S0 p d push_list a ks = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ›*)\n apply auto\n (*goal: ‹⋀ks. ⟦⋀x d s z. [x = ∑S0 d (s && z)] ≡ [∃length ks] [∀<length ks] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, map (λx. push_param x (length ks)) s ! i, map (λx. push_param x (length ks)) z ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length ks), [∑S0] p d polynomial.Param]; p ≠ []; length p = length ks; ∀k<length ks. push_list a ks k = map (λx. peval x a) s ! k && map (λx. peval x a) z ! k; peval x a = ∑S0 p d push_list a ks ; ⋀ns n i ls a. ⟦length ns = n; i < length ls⟧ ⟹ peval (push_param (ls ! i) n) (push_list a ns) = peval (ls ! i) a⟧ ⟹ ∀k. k ≤ length p - 1 ∧ issub (p ! k) ∧ d = goes_to_alt (p ! k) ⟶ push_list a ks k = peval (s ! k) a && peval (z ! k) a›*)\n by (metis (no_types, lifting) One_nat_def (*‹(1::nat) = Suc (0::nat)›*) assms( (*‹length (s::polynomial list) = Suc m›*) 1) assms( (*‹length (z::polynomial list) = Suc m›*) 2) le_imp_less_Suc (*‹(?m::nat) ≤ (?n::nat) ⟹ ?m < Suc ?n›*) len_ps (*‹length (s::polynomial list) = length (p::instruction list)›*) m_def (*‹m ≡ length (p::instruction list) - (1::nat)›*) nth_map (*‹(?n::nat) < length (?xs::?'a list) ⟹ map (?f::?'a ⇒ ?'b) ?xs ! ?n = ?f (?xs ! ?n)›*))\n next\n (*goal: ‹peval x a = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ⟹ eval [x = ∑S0 d (s && z)] a›*)\n assume \"?Q\" (*‹peval (x::polynomial) (a::nat ⇒ nat) = ∑S0 (p::instruction list) (d::nat) (λk::nat. peval ((s::polynomial list) ! k) a && peval ((z::polynomial list) ! k) a) ›*)\n thus \"?P\"\n using sum_ssub_zero_of_bit_and_def (*‹[?x = ∑S0 ?d (?s && ?z)] ≡ let x' = push_param ?x (length p); s' = push_param_list ?s (length p); z' = push_param_list ?z (length p) in [∃length p] [∀<length p] λi. [polynomial.Param i = s' ! i && z' ! i] [∧] x' [=] [∑S0] p ?d polynomial.Param›*) ‹length p > 0› (*‹0 < length p›*) apply (auto simp add: defs (*‹❙0 = polynomial.Const 0› ‹❙1 = polynomial.Const 1› ‹?Q [=] ?R ≡ BINARY (=) ?Q ?R› ‹?Q [<] ?R ≡ BINARY (<) ?Q ?R› ‹?Q [≤] ?R ≡ ?Q [<] ?R [∨] ?Q [=] ?R› ‹?Q [>] ?R ≡ ?R [<] ?Q› ‹?Q [≥] ?R ≡ ?Q [>] ?R [∨] ?Q [=] ?R› ‹eval (LARY ?R ?PLL) ?a = ?R (map (map (λP. peval P ?a)) ?PLL)› ‹UNARY ?R ?P = NARY (λl. ?R (l ! 0)) [?P]› ‹BINARY ?R ?P₁ ?P₂ = NARY (λl. ?R (l ! 0) (l ! 1)) [?P₁, ?P₂]› ‹TERNARY ?R ?P₁ ?P₂ ?P₃ = NARY (λl. ?R (l ! 0) (l ! 1) (l ! 2)) [?P₁, ?P₂, ?P₃]› ‹QUATERNARY ?R ?P₁ ?P₂ ?P₃ ?P₄ = NARY (λl. ?R (l ! 0) (l ! 1) (l ! 2) (l ! 3)) [?P₁, ?P₂, ?P₃, ?P₄]› and more 19 facts*) push_push (*‹length ?ns = ?n ⟹ peval (push_param ?P ?n) (push_list ?a ?ns) = peval ?P ?a›*))\n (*goal: ‹eval [x = ∑S0 d (s && z)] a›*)\n apply (rule exI[of _ \"map (λk. peval (s ! k) a && peval (z!k) a) [0..<length p]\"] (*‹(?P::nat list ⇒ bool) (map (λk::nat. peval ((s::polynomial list) ! k) (a::nat ⇒ nat) && peval ((z::polynomial list) ! k) a) [0::nat..<length (p::instruction list)]) ⟹ ∃x::nat list. ?P x›*), simp)\n (*goal: ‹⟦peval x a = ∑S0 p d (λk. peval (s ! k) a && peval (z ! k) a) ; ⋀x d s z. [x = ∑S0 d (s && z)] ≡ [∃length p] [∀<length p] λi. NARY (λl. l ! 0 = l ! Suc 0 && l ! 2) [polynomial.Param i, push_param_list s (length p) ! i, push_param_list z (length p) ! i] [∧] NARY (λl. l ! 0 = l ! Suc 0) [push_param x (length p), [∑S0] p d polynomial.Param]; p ≠ []⟧ ⟹ ∃ks. length p = length ks ∧ (∀k<length p. push_list a ks k = peval (push_param_list s (length p) ! k) (push_list a ks) && peval (push_param_list z (length p) ! k) (push_list a ks)) ∧ peval (push_param x (length p)) (push_list a ks) = ∑S0 p d push_list a ks ›*)\n using push_push (*‹length (?ns::nat list) = (?n::nat) ⟹ peval (push_param (?P::polynomial) ?n) (push_list (?a::nat ⇒ nat) ?ns) = peval ?P ?a›*) push_push_map_i (*‹⟦length ?ns = ?n; ?i < length ?ls⟧ ⟹ peval (map (λx. push_param x ?n) ?ls ! ?i) (push_list ?a ?ns) = list_eval ?ls ?a ?i›*) by (simp add: push_param_list_def (*‹push_param_list ?s ?k ≡ map (λx. push_param x ?k) ?s›*) invariance (*‹∀k<length p. ?y1.0 k = ?y2.0 k ⟹ ∑S0 p d ?y1.0 = ∑S0 p d ?y2.0 ›*) push_list_eval (*‹?i < length ?ns ⟹ push_list ?a ?ns ?i = ?ns ! ?i›*) len_ps (*‹length s = length p›*) len_pz (*‹length z = length p›*))\n qed\nqed\n\nend\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/DPRM_Theorem/Machine_Equations/RM_Sums_Diophantine.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"f3d636e55d659cfa2e5a0371507c37e5fb2a7c95f1037605c302539899fad61d\"\n}"}}},{"rowIdx":97,"cells":{"text":{"kind":"string","value":"(* Title: JinjaThreads/Framework/FWWellform.thy\n Author: Andreas Lochbihler\n*)\n\nsection ‹Wellformedness conditions for the multithreaded state›\n\ntheory FWWellform\nimports\n FWLocking\n FWThread\n FWWait\n FWCondAction\nbegin\n\ntext‹Well-formedness property: Locks are held by real threads›\n\ndefinition\n lock_thread_ok :: \"('l, 't) locks ⇒ ('l, 't,'x) thread_info ⇒ bool\"\nwhere [code del]:\n \"lock_thread_ok ls ts ≡ ∀l t. has_lock (ls $ l) t ⟶ (∃xw. ts t = ⌊xw⌋)\"\n\nlemma lock_thread_ok_code [code]: \n \"lock_thread_ok ls ts = finfun_All ((λl. case l of None ⇒ True | ⌊(t, n)⌋ ⇒ (ts t ≠ None)) ∘$ ls)\"\nby (simp add: lock_thread_ok_def (*‹lock_thread_ok ?ls ?ts ≡ ∀l t. has_lock (?ls $ l) t ⟶ (∃xw. ?ts t = ⌊xw⌋)›*) finfun_All_All (*‹finfun_All ?P = All (($) ?P)›*) has_lock_has_locks_conv (*‹has_lock ?l ?t = (∃n. has_locks ?l ?t = Suc n)›*) has_locks_iff (*‹(has_locks ?l ?t = ?n) = (?l = None ∧ ?n = 0 ∨ (∃n'. ?l = ⌊(?t, n')⌋ ∧ Suc n' = ?n) ∨ (∃t' n'. ?l = ⌊(t', n')⌋ ∧ t' ≠ ?t ∧ ?n = 0))›*) o_def (*‹?f ∘ ?g = (λx. ?f (?g x))›*))\n\nlemma lock_thread_okI:\n \"(⋀l t. has_lock (ls $ l) t ⟹ ∃xw. ts t = ⌊xw⌋) ⟹ lock_thread_ok ls ts\"\nby (auto simp add: lock_thread_ok_def (*‹lock_thread_ok ?ls ?ts ≡ ∀l t. has_lock (?ls $ l) t ⟶ (∃xw. ?ts t = ⌊xw⌋)›*))\n\nlemma lock_thread_okD:\n \"⟦ lock_thread_ok ls ts; has_lock (ls $ l) t ⟧ ⟹ ∃xw. ts t = ⌊xw⌋\"\nby (fastforce simp add: lock_thread_ok_def (*‹lock_thread_ok ?ls ?ts ≡ ∀l t. has_lock (?ls $ l) t ⟶ (∃xw. ?ts t = ⌊xw⌋)›*))\n\nlemma lock_thread_okD':\n \"⟦ lock_thread_ok ls ts; has_locks (ls $ l) t = Suc n ⟧ ⟹ ∃xw. ts t = ⌊xw⌋\"\nby (auto elim: lock_thread_okD[where l=l] (*‹⟦lock_thread_ok ?ls ?ts; has_lock (?ls $ l) ?t⟧ ⟹ ∃xw. ?ts ?t = ⌊xw⌋›*) simp del: split_paired_Ex (*‹(∃x. ?P x) = (∃a b. ?P (a, b))›*))\n\nlemma lock_thread_okE:\n \"⟦ lock_thread_ok ls ts; ∀l t. has_lock (ls $ l) t ⟶ (∃xw. ts t = ⌊xw⌋) ⟹ P ⟧ ⟹ P\"\nby (auto simp add: lock_thread_ok_def (*‹lock_thread_ok ?ls ?ts ≡ ∀l t. has_lock (?ls $ l) t ⟶ (∃xw. ?ts t = ⌊xw⌋)›*) simp del: split_paired_Ex (*‹(∃x. ?P x) = (∃a b. ?P (a, b))›*))\n\nlemma lock_thread_ok_upd:\n \"lock_thread_ok ls ts ⟹ lock_thread_ok ls (ts(t ↦ xw))\"\nby (auto intro!: lock_thread_okI (*‹(⋀l t. has_lock (?ls $ l) t ⟹ ∃xw. ?ts t = ⌊xw⌋) ⟹ lock_thread_ok ?ls ?ts›*) dest: lock_thread_okD (*‹⟦lock_thread_ok ?ls ?ts; has_lock (?ls $ ?l) ?t⟧ ⟹ ∃xw. ?ts ?t = ⌊xw⌋›*))\n\nlemma lock_thread_ok_has_lockE:\n assumes \"lock_thread_ok ls ts\"\n and \"has_lock (ls $ l) t\"\n obtains x ln' where \"ts t = ⌊(x, ln')⌋\"\nusing assms (*‹lock_thread_ok ls ts› ‹has_lock (ls $ l) t›*) by (auto dest!: lock_thread_okD (*‹⟦lock_thread_ok ?ls ?ts; has_lock (?ls $ ?l) ?t⟧ ⟹ ∃xw. ?ts ?t = ⌊xw⌋›*))\n\nlemma redT_updLs_preserves_lock_thread_ok:\n assumes lto: \"lock_thread_ok ls ts\"\n and tst: \"ts t = ⌊xw⌋\"\n shows \"lock_thread_ok (redT_updLs ls t las) ts\"\nproof (rule lock_thread_okI (*‹(⋀l t. has_lock (?ls $ l) t ⟹ ∃xw. ?ts t = ⌊xw⌋) ⟹ lock_thread_ok ?ls ?ts›*))\n (*goal: ‹⋀l ta. has_lock (redT_updLs ls t las $ l) ta ⟹ ∃xw. ts ta = ⌊xw⌋›*)\n fix L and T\n assume ru: \"has_lock (redT_updLs ls t las $ L) T\" (*‹has_lock (redT_updLs (ls::('a, 'b) locks) (t::'b) (las::'a lock_actions) $ (L::'a)) (T::'b)›*)\n show \"∃xw. ts T = ⌊xw⌋\"\n proof (cases \"t = T\")\n (*goals:\n 1. ‹t = T ⟹ ∃xw. ts T = ⌊xw⌋›\n 2. ‹t ≠ T ⟹ ∃xw. ts T = ⌊xw⌋›*)\n case True (*‹(t::'b::type) = (T::'b::type)›*)\n thus \"?thesis\"\n (*goal: ‹∃xw. ts T = ⌊xw⌋›*)\n using tst (*‹(ts::'b ⇒ ('c × 'a ⇒f nat) option) (t::'b) = ⌊xw::'c × 'a ⇒f nat⌋›*) lto (*‹lock_thread_ok (ls::('a, 'b) locks) (ts::'b ⇒ ('c × 'a ⇒f nat) option)›*) by (auto elim: lock_thread_okE (*‹⟦lock_thread_ok ?ls ?ts; ∀l t. has_lock (?ls $ l) t ⟶ (∃xw. ?ts t = ⌊xw⌋) ⟹ ?P⟧ ⟹ ?P›*))\n next\n (*goal: ‹t ≠ T ⟹ ∃xw. ts T = ⌊xw⌋›*)\n case False (*‹t ≠ T›*)\n with ru (*‹has_lock (redT_updLs ls t las $ L) T›*) have \"has_lock (ls $ L) T\"\n by (rule redT_updLs_Some_thread_idD (*‹⟦has_lock (redT_updLs ?ls ?t ?las $ ?l) ?t'; ?t ≠ ?t'⟧ ⟹ has_lock (?ls $ ?l) ?t'›*))\n thus \"?thesis\"\n (*goal: ‹∃xw. ts T = ⌊xw⌋›*)\n using lto (*‹lock_thread_ok ls ts›*) by (auto elim!: lock_thread_okE (*‹⟦lock_thread_ok ?ls ?ts; ∀l t. has_lock (?ls $ l) t ⟶ (∃xw. ?ts t = ⌊xw⌋) ⟹ ?P⟧ ⟹ ?P›*) simp del: split_paired_Ex (*‹(∃x. ?P x) = (∃a b. ?P (a, b))›*))\n qed\nqed\n\nlemma redT_updTs_preserves_lock_thread_ok:\n assumes lto: \"lock_thread_ok ls ts\"\n shows \"lock_thread_ok ls (redT_updTs ts nts)\"\nproof (rule lock_thread_okI (*‹(⋀l t. has_lock (?ls $ l) t ⟹ ∃xw. ?ts t = ⌊xw⌋) ⟹ lock_thread_ok ?ls ?ts›*))\n (*goal: ‹⋀l t. has_lock (ls $ l) t ⟹ ∃xw. redT_updTs ts nts t = ⌊xw⌋›*)\n fix l and t\n assume \"has_lock (ls $ l) t\" (*‹has_lock ((ls::('a, 'b) locks) $ (l::'a)) (t::'b)›*)\n with lto (*‹lock_thread_ok ls ts›*) have \"∃xw. ts t = ⌊xw⌋\"\n by (auto elim!: lock_thread_okE (*‹⟦lock_thread_ok ?ls ?ts; ∀l t. has_lock (?ls $ l) t ⟶ (∃xw. ?ts t = ⌊xw⌋) ⟹ ?P⟧ ⟹ ?P›*) simp del: split_paired_Ex (*‹(∃x. ?P x) = (∃a b. ?P (a, b))›*))\n thus \"∃xw. redT_updTs ts nts t = ⌊xw⌋\"\n by (auto intro: redT_updTs_Some1 (*‹?ts ?t = ⌊?xw⌋ ⟹ ∃xw. redT_updTs ?ts ?tas ?t = ⌊xw⌋›*) simp del: split_paired_Ex (*‹(∃x. ?P x) = (∃a b. ?P (a, b))›*))\nqed\n\nlemma lock_thread_ok_has_lock:\n assumes \"lock_thread_ok ls ts\"\n and \"has_lock (ls $ l) t\"\n obtains xw where \"ts t = ⌊xw⌋\"\nsorry\n\nlemma lock_thread_ok_None_has_locks_0:\n \"⟦ lock_thread_ok ls ts; ts t = None ⟧ ⟹ has_locks (ls $ l) t = 0\"\napply (rule ccontr (*‹(¬ (?P::bool) ⟹ False) ⟹ ?P›*))\n(*goal: ‹⟦lock_thread_ok (ls::('a, 'b) locks) (ts::'b ⇒ ('c × 'a ⇒f nat) option); ts (t::'b) = None⟧ ⟹ has_locks (ls $ (l::'a)) t = (0::nat)›*)\nby (auto dest: lock_thread_okD (*‹⟦lock_thread_ok ?ls ?ts; has_lock (?ls $ ?l) ?t⟧ ⟹ ∃xw. ?ts ?t = ⌊xw⌋›*))\n\nlemma redT_upds_preserves_lock_thread_ok:\n \"⟦lock_thread_ok ls ts; ts t = ⌊xw⌋; thread_oks ts tas⟧\n ⟹ lock_thread_ok (redT_updLs ls t las) ((redT_updTs ts tas)(t ↦ xw'))\"\napply (rule lock_thread_okI (*‹(⋀l t. has_lock (?ls $ l) t ⟹ ∃xw. ?ts t = ⌊xw⌋) ⟹ lock_thread_ok ?ls ?ts›*))\n(*goal: ‹⟦lock_thread_ok ls ts; ts t = ⌊xw⌋; thread_oks ts tas⟧ ⟹ lock_thread_ok (redT_updLs ls t las) ((redT_updTs ts tas)(t ↦ xw'))›*)\napply (clarsimp simp del: split_paired_Ex (*‹(∃x::?'a::type × ?'b::type. (?P::?'a::type × ?'b::type ⇒ bool) x) = (∃(a::?'a::type) b::?'b::type. ?P (a, b))›*))\n(*goal: ‹⋀l ta. ⟦lock_thread_ok ls ts; ts t = ⌊xw⌋; thread_oks ts tas; has_lock (redT_updLs ls t las $ l) ta⟧ ⟹ ∃xw. ((redT_updTs ts tas)(t ↦ xw')) ta = ⌊xw⌋›*)\napply (drule has_lock_upd_locks_implies_has_lock (*‹⟦has_lock (upd_locks (?l::(?'a × nat) option) (?t::?'a) (?Ls::lock_action list)) (?t'::?'a); ?t ≠ ?t'⟧ ⟹ has_lock ?l ?t'›*))\n(*goals:\n 1. ‹⋀(l::'a) ta::'b. ⟦lock_thread_ok (ls::('a, 'b) locks) (ts::'b ⇒ ('c × 'a ⇒f nat) option); ts (t::'b) = ⌊xw::'c × 'a ⇒f nat⌋; thread_oks ts (tas::('b, 'c, 'd) new_thread_action list); ta ≠ t⟧ ⟹ t ≠ ta›\n 2. ‹⋀(l::'a) ta::'b. ⟦lock_thread_ok (ls::('a, 'b) locks) (ts::'b ⇒ ('c × 'a ⇒f nat) option); ts (t::'b) = ⌊xw::'c × 'a ⇒f nat⌋; thread_oks ts (tas::('b, 'c, 'd) new_thread_action list); ta ≠ t; has_lock (ls $ l) ta⟧ ⟹ ∃xw::'c × 'a ⇒f nat. redT_updTs ts tas ta = ⌊xw⌋›\ndiscuss goal 1*)\n apply simp\n(*discuss goal 2*)\n apply (drule lock_thread_okD (*‹⟦lock_thread_ok (?ls::(?'a, ?'b) locks) (?ts::?'b ⇒ (?'c × ?'a ⇒f nat) option); has_lock (?ls $ (?l::?'a)) (?t::?'b)⟧ ⟹ ∃xw::?'c × ?'a ⇒f nat. ?ts ?t = ⌊xw⌋›*))\n (*goals:\n 1. ‹⋀(l::'a::type) ta::'b::type. ⟦(ts::'b::type ⇒ ('c::type × 'a::type ⇒f nat) option) (t::'b::type) = ⌊xw::'c::type × 'a::type ⇒f nat⌋; thread_oks ts (tas::('b::type, 'c::type, 'd::type) new_thread_action list); ta ≠ t; has_lock ((ls::('a::type, 'b::type) locks) $ l) ta⟧ ⟹ has_lock (ls $ ((?l10::'a::type ⇒ 'b::type ⇒ 'a::type) l ta)) ((?t10::'a::type ⇒ 'b::type ⇒ 'b::type) l ta)›\n 2. ‹⋀(l::'a::type) ta::'b::type. ⟦(ts::'b::type ⇒ ('c::type × 'a::type ⇒f nat) option) (t::'b::type) = ⌊xw::'c::type × 'a::type ⇒f nat⌋; thread_oks ts (tas::('b::type, 'c::type, 'd::type) new_thread_action list); ta ≠ t; has_lock ((ls::('a::type, 'b::type) locks) $ l) ta; ∃xw::'c::type × 'a::type ⇒f nat. ts ((?t10::'a::type ⇒ 'b::type ⇒ 'b::type) l ta) = ⌊xw⌋⟧ ⟹ ∃xw::'c::type × 'a::type ⇒f nat. redT_updTs ts tas ta = ⌊xw⌋›\n discuss goal 1*)\n apply assumption\n (*discuss goal 2*)\n apply (erule exE (*‹⟦∃x::?'a::type. (?P::?'a::type ⇒ bool) x; ⋀x::?'a::type. ?P x ⟹ ?Q::bool⟧ ⟹ ?Q›*))\n (*goal: ‹⋀l ta. ⟦ts t = ⌊xw⌋; thread_oks ts tas; ta ≠ t; has_lock (ls $ l) ta; ∃xw. ts ta = ⌊xw⌋⟧ ⟹ ∃xw. redT_updTs ts tas ta = ⌊xw⌋›*)\n by (rule redT_updTs_Some1 (*‹?ts ?t = ⌊?xw⌋ ⟹ ∃xw. redT_updTs ?ts ?tas ?t = ⌊xw⌋›*))\n (*proven 2 subgoals*)\n(*proven 2 subgoals*)\n\nlemma acquire_all_preserves_lock_thread_ok:\n fixes ln\n shows \"⟦ lock_thread_ok ls ts; ts t = ⌊(x, ln)⌋ ⟧ ⟹ lock_thread_ok (acquire_all ls t ln) (ts(t ↦ xw))\"\napply (rule lock_thread_okI (*‹(⋀l t. has_lock (?ls $ l) t ⟹ ∃xw. ?ts t = ⌊xw⌋) ⟹ lock_thread_ok ?ls ?ts›*))\n(*goal: ‹⟦lock_thread_ok ls ts; ts t = ⌊(x, ln)⌋⟧ ⟹ lock_thread_ok (acquire_all ls t ln) (ts(t ↦ xw))›*)\nby (auto dest!: has_lock_acquire_locks_implies_has_lock (*‹⟦has_lock (acquire_locks ?l ?t ?n) ?t'; ?t ≠ ?t'⟧ ⟹ has_lock ?l ?t'›*) dest: lock_thread_okD (*‹⟦lock_thread_ok ?ls ?ts; has_lock (?ls $ ?l) ?t⟧ ⟹ ∃xw. ?ts ?t = ⌊xw⌋›*))\n\ntext ‹Well-formedness condition: Wait sets contain only real threads›\n\ndefinition wset_thread_ok :: \"('w, 't) wait_sets ⇒ ('l, 't, 'x) thread_info ⇒ bool\"\nwhere \"wset_thread_ok ws ts ≡ ∀t. ts t = None ⟶ ws t = None\"\n\nlemma wset_thread_okI:\n \"(⋀t. ts t = None ⟹ ws t = None) ⟹ wset_thread_ok ws ts\"\nby (simp add: wset_thread_ok_def (*‹wset_thread_ok ?ws ?ts ≡ ∀t. ?ts t = None ⟶ ?ws t = None›*))\n\nlemma wset_thread_okD:\n \"⟦ wset_thread_ok ws ts; ts t = None ⟧ ⟹ ws t = None\"\nby (simp add: wset_thread_ok_def (*‹wset_thread_ok ?ws ?ts ≡ ∀t. ?ts t = None ⟶ ?ws t = None›*))\n\nlemma wset_thread_ok_conv_dom:\n \"wset_thread_ok ws ts ⟷ dom ws ⊆ dom ts\"\nby (auto simp add: wset_thread_ok_def (*‹wset_thread_ok ?ws ?ts ≡ ∀t. ?ts t = None ⟶ ?ws t = None›*))\n\nlemma wset_thread_ok_upd:\n \"wset_thread_ok ls ts ⟹ wset_thread_ok ls (ts(t ↦ xw))\"\nby (auto intro!: wset_thread_okI (*‹(⋀t. ?ts t = None ⟹ ?ws t = None) ⟹ wset_thread_ok ?ws ?ts›*) dest: wset_thread_okD (*‹⟦wset_thread_ok ?ws ?ts; ?ts ?t = None⟧ ⟹ ?ws ?t = None›*) split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n\nlemma wset_thread_ok_upd_None:\n \"wset_thread_ok ws ts ⟹ wset_thread_ok (ws(t := None)) (ts(t := None))\"\nby (auto intro!: wset_thread_okI (*‹(⋀t. ?ts t = None ⟹ ?ws t = None) ⟹ wset_thread_ok ?ws ?ts›*) dest: wset_thread_okD (*‹⟦wset_thread_ok ?ws ?ts; ?ts ?t = None⟧ ⟹ ?ws ?t = None›*) split: if_split_asm (*‹?P (if ?Q then ?x else ?y) = (¬ (?Q ∧ ¬ ?P ?x ∨ ¬ ?Q ∧ ¬ ?P ?y))›*))\n\nlemma wset_thread_ok_upd_Some:\n \"wset_thread_ok ws ts ⟹ wset_thread_ok (ws(t := wo)) (ts(t ↦ xln))\"\nby (auto intro!: wset_thread_okI (*‹(⋀t::?'a::type. (?ts::?'a::type ⇒ (?'b::type × ?'c::type ⇒f nat) option) t = None ⟹ (?ws::?'a::type ⇒ ?'d::type wait_set_status option) t = None) ⟹ wset_thread_ok ?ws ?ts›*) dest: wset_thread_okD (*‹⟦wset_thread_ok (?ws::?'a::type ⇒ ?'b::type wait_set_status option) (?ts::?'a::type ⇒ (?'c::type × ?'d::type ⇒f nat) option); ?ts (?t::?'a::type) = None⟧ ⟹ ?ws ?t = None›*) 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\nlemma wset_thread_ok_upd_ws:\n \"⟦ wset_thread_ok ws ts; ts t = ⌊xln⌋ ⟧ ⟹ wset_thread_ok (ws(t := w)) ts\"\nby (auto intro!: wset_thread_okI (*‹(⋀t::?'a. (?ts::?'a ⇒ (?'b × ?'c ⇒f nat) option) t = None ⟹ (?ws::?'a ⇒ ?'d wait_set_status option) t = None) ⟹ wset_thread_ok ?ws ?ts›*) dest: wset_thread_okD (*‹⟦wset_thread_ok (?ws::?'a ⇒ ?'b wait_set_status option) (?ts::?'a ⇒ (?'c × ?'d ⇒f nat) option); ?ts (?t::?'a) = None⟧ ⟹ ?ws ?t = None›*))\n\nlemma wset_thread_ok_NotifyAllI: \n \"wset_thread_ok ws ts ⟹ wset_thread_ok (λt. if ws t = ⌊w t⌋ then ⌊w' t⌋ else ws t) ts\"\nby (simp add: wset_thread_ok_def (*‹wset_thread_ok (?ws::?'t::type ⇒ ?'w::type wait_set_status option) (?ts::?'t::type ⇒ (?'x::type × ?'l::type ⇒f nat) option) ≡ ∀t::?'t::type. ?ts t = None ⟶ ?ws t = None›*))\n\nlemma redT_updTs_preserves_wset_thread_ok:\n assumes wto: \"wset_thread_ok ws ts\"\n shows \"wset_thread_ok ws (redT_updTs ts nts)\"\nproof (rule wset_thread_okI (*‹(⋀t. ?ts t = None ⟹ ?ws t = None) ⟹ wset_thread_ok ?ws ?ts›*))\n (*goal: ‹⋀t. redT_updTs ts nts t = None ⟹ ws t = None›*)\n fix t\n assume \"redT_updTs ts nts t = None\" (*‹redT_updTs (ts::'a ⇒ ('c × 'd ⇒f nat) option) (nts::('a, 'c, 'e) new_thread_action list) (t::'a) = None›*)\n hence \"ts t = None\"\n by (rule redT_updTs_None (*‹redT_updTs ?ts ?tas ?t = None ⟹ ?ts ?t = None›*))\n with wto (*‹wset_thread_ok ws ts›*) show \"ws t = None\"\n by (rule wset_thread_okD (*‹⟦wset_thread_ok ?ws ?ts; ?ts ?t = None⟧ ⟹ ?ws ?t = None›*))\nqed\n\nlemma redT_updW_preserve_wset_thread_ok: \n \"⟦ wset_thread_ok ws ts; redT_updW t ws wa ws'; ts t = ⌊xln⌋ ⟧ ⟹ wset_thread_ok ws' ts\"\nby (fastforce simp add: redT_updW.simps (*‹redT_updW ?t ?ws ?a1.0 ?a2.0 = ((∃t' w. ?a1.0 = Notify w ∧ ?a2.0 = ?ws(t' ↦ PostWS WSNotified) ∧ ?ws t' = ⌊InWS w⌋) ∨ (∃w. ?a1.0 = Notify w ∧ ?a2.0 = ?ws ∧ (∀x. ?ws x ≠ ⌊InWS w⌋)) ∨ (∃w. ?a1.0 = NotifyAll w ∧ ?a2.0 = (λt. if ?ws t = ⌊InWS w⌋ then ⌊PostWS WSNotified⌋ else ?ws t)) ∨ (∃w. ?a1.0 = Suspend w ∧ ?a2.0 = ?ws(?t ↦ InWS w)) ∨ (∃t' w WSInterrupted. ?a1.0 = WakeUp t' ∧ ?a2.0 = ?ws(t' ↦ PostWS WSInterrupted) ∧ ?ws t' = ⌊InWS w⌋) ∨ (∃t'. ?a1.0 = WakeUp t' ∧ ?a2.0 = ?ws ∧ (∀x. ?ws t' ≠ ⌊InWS x⌋)) ∨ ?a1.0 = Notified ∧ ?a2.0 = ?ws(?t := None) ∨ ?a1.0 = WokenUp ∧ ?a2.0 = ?ws(?t := None))›*) intro: wset_thread_okI (*‹(⋀t. ?ts t = None ⟹ ?ws t = None) ⟹ wset_thread_ok ?ws ?ts›*) wset_thread_ok_NotifyAllI (*‹wset_thread_ok ?ws ?ts ⟹ wset_thread_ok (λt. if ?ws t = ⌊?w t⌋ then ⌊?w' t⌋ else ?ws t) ?ts›*) wset_thread_ok_upd_ws (*‹⟦wset_thread_ok ?ws ?ts; ?ts ?t = ⌊?xln⌋⟧ ⟹ wset_thread_ok (?ws(?t := ?w)) ?ts›*) dest: wset_thread_okD (*‹⟦wset_thread_ok ?ws ?ts; ?ts ?t = None⟧ ⟹ ?ws ?t = None›*))\n\nlemma redT_updWs_preserve_wset_thread_ok:\n \"⟦ wset_thread_ok ws ts; redT_updWs t ws was ws'; ts t = ⌊xln⌋ ⟧ ⟹ wset_thread_ok ws' ts\"\nunfolding redT_updWs_def\n(*goal: ‹⟦wset_thread_ok ws ts; rtrancl3p (redT_updW t) ws was ws'; ts t = ⌊xln⌋⟧ ⟹ wset_thread_ok ws' ts›*)\napply (rotate_tac 1)\n(*goal: ‹⟦wset_thread_ok (ws::'a ⇒ 'b wait_set_status option) (ts::'a ⇒ ('c × 'd ⇒f nat) option); rtrancl3p (redT_updW (t::'a)) ws (was::('a, 'b) wait_set_action list) (ws'::'a ⇒ 'b wait_set_status option); ts t = ⌊xln::'c × 'd ⇒f nat⌋⟧ ⟹ wset_thread_ok ws' ts›*)\napply (induct rule: rtrancl3p_converse_induct (*‹⟦rtrancl3p ?r ?a ?bs ?a''; ⋀a. ?P a [] a; ⋀a b a' bs a''. ⟦rtrancl3p ?r a' bs a''; ?r a b a'; ?P a' bs a''⟧ ⟹ ?P a (b # bs) a''⟧ ⟹ ?P ?a ?bs ?a''›*))\n(*goals:\n 1. ‹⋀a. ⟦ts t = ⌊xln⌋; wset_thread_ok a ts⟧ ⟹ wset_thread_ok a ts›\n 2. ‹⋀a b a' bs a''. ⟦rtrancl3p (redT_updW t) a' bs a''; redT_updW t a b a'; ⟦ts t = ⌊xln⌋; wset_thread_ok a' ts⟧ ⟹ wset_thread_ok a'' ts; ts t = ⌊xln⌋; wset_thread_ok a ts⟧ ⟹ wset_thread_ok a'' ts›\ndiscuss goal 1*)\n apply ((auto intro: redT_updW_preserve_wset_thread_ok (*‹⟦wset_thread_ok ?ws ?ts; redT_updW ?t ?ws ?wa ?ws'; ?ts ?t = ⌊?xln⌋⟧ ⟹ wset_thread_ok ?ws' ?ts›*))[1])\n(*discuss goal 2*)\n apply ((auto intro: redT_updW_preserve_wset_thread_ok (*‹⟦wset_thread_ok ?ws ?ts; redT_updW ?t ?ws ?wa ?ws'; ?ts ?t = ⌊?xln⌋⟧ ⟹ wset_thread_ok ?ws' ?ts›*))[1])\n(*proven 2 subgoals*) .\n\ntext ‹Well-formedness condition: Wait sets contain only non-final threads›\n\ncontext final_thread begin\n\ndefinition wset_final_ok :: \"('w, 't) wait_sets ⇒ ('l, 't, 'x) thread_info ⇒ bool\"\nwhere \"wset_final_ok ws ts ⟷ (∀t ∈ dom ws. ∃x ln. ts t = ⌊(x, ln)⌋ ∧ ¬ final x)\"\n\nlemma wset_final_okI:\n \"(⋀t w. ws t = ⌊w⌋ ⟹ ∃x ln. ts t = ⌊(x, ln)⌋ ∧ ¬ final x) ⟹ wset_final_ok ws ts\"\nunfolding wset_final_ok_def\n(*goal: ‹(⋀t w. ws t = ⌊w⌋ ⟹ ∃x ln. ts t = ⌊(x, ln)⌋ ∧ ¬ final x) ⟹ ∀t∈dom ws. ∃x ln. ts t = ⌊(x, ln)⌋ ∧ ¬ final x›*)\nby blast\n\nlemma wset_final_okD:\n \"⟦ wset_final_ok ws ts; ws t = ⌊w⌋ ⟧ ⟹ ∃x ln. ts t = ⌊(x, ln)⌋ ∧ ¬ final x\"\nunfolding wset_final_ok_def\n(*goal: ‹⟦∀t∈dom ws. ∃x ln. ts t = ⌊(x, ln)⌋ ∧ ¬ final x; ws t = ⌊w⌋⟧ ⟹ ∃x ln. ts t = ⌊(x, ln)⌋ ∧ ¬ final x›*)\nby blast\n\nlemma wset_final_okE:\n assumes \"wset_final_ok ws ts\" \"ws t = ⌊w⌋\"\n and \"⋀x ln. ts t = ⌊(x, ln)⌋ ⟹ ¬ final x ⟹ thesis\"\n shows thesis\nusing assms (*‹wset_final_ok (ws::'a::type ⇒ 'b::type wait_set_status option) (ts::'a::type ⇒ ('x::type × 'c::type ⇒f nat) option)› ‹ws t = ⌊w⌋› ‹⟦ts t = ⌊(?x5, ?ln5)⌋; ¬ final ?x5⟧ ⟹ thesis›*) by (blast dest: wset_final_okD (*‹⟦wset_final_ok (?ws::?'a ⇒ ?'b wait_set_status option) (?ts::?'a ⇒ ('x × ?'c ⇒f nat) option); ?ws (?t::?'a) = ⌊?w::?'b wait_set_status⌋⟧ ⟹ ∃(x::'x) ln::?'c ⇒f nat. ?ts ?t = ⌊(x, ln)⌋ ∧ ¬ (final::'x ⇒ bool) x›*))\n\nlemma wset_final_ok_imp_wset_thread_ok:\n \"wset_final_ok ws ts ⟹ wset_thread_ok ws ts\"\napply (rule wset_thread_okI (*‹(⋀t. ?ts t = None ⟹ ?ws t = None) ⟹ wset_thread_ok ?ws ?ts›*))\n(*goal: ‹wset_final_ok (ws::'a ⇒ 'b wait_set_status option) (ts::'a ⇒ ('x × 'c ⇒f nat) option) ⟹ wset_thread_ok ws ts›*)\napply (rule ccontr (*‹(¬ ?P ⟹ False) ⟹ ?P›*))\n(*goal: ‹⋀t. ⟦wset_final_ok ws ts; ts t = None⟧ ⟹ ws t = None›*)\nby (auto elim: wset_final_okE (*‹⟦wset_final_ok (?ws::?'a ⇒ ?'b wait_set_status option) (?ts::?'a ⇒ ('x × ?'c ⇒f nat) option); ?ws (?t::?'a) = ⌊?w::?'b wait_set_status⌋; ⋀(x::'x) ln::?'c ⇒f nat. ⟦?ts ?t = ⌊(x, ln)⌋; ¬ (final::'x ⇒ bool) x⟧ ⟹ ?thesis::bool⟧ ⟹ ?thesis›*))\n\nend\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/JinjaThreads/Framework/FWWellform.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"99efa4d1b22480765e290fc944a853fe9df6e899850543f3ee550b0106c1a9cb\"\n}"}}},{"rowIdx":98,"cells":{"text":{"kind":"string","value":"(*\nAuthors: \n Hanna Lachnitt, TU Wien, lachnitt@student.tuwien.ac.at\n Anthony Bordg, University of Cambridge, apdb3@cam.ac.uk\n*) \n\nsection ‹The Deutsch-Jozsa Algorithm›\n\ntheory Deutsch_Jozsa\nimports\n Deutsch\n More_Tensor\n Binary_Nat\nbegin\n\n\ntext ‹\nGiven a function $f:{0,1}^n \\mapsto {0,1}$, the Deutsch-Jozsa algorithm decides if this function is \nconstant or balanced with a single $f(x)$ circuit to evaluate the function for multiple values of $x$ \nsimultaneously. The algorithm makes use of quantum parallelism and quantum interference.\n\nA constant function with values in {0,1} returns either always 0 or always 1. \nA balanced function is 0 for half of the inputs and 1 for the other half. \n›\n\nlocale bob_fun =\n fixes f:: \"nat ⇒ nat\" and n:: \"nat\"\n assumes dom: \"f ∈ ({(i::nat). i < 2^n} →⇩E {0,1})\"\n assumes dim: \"n ≥ 1\"\n\ncontext bob_fun\nbegin\n\ndefinition const:: \"nat ⇒ bool\" where \n\"const c = (∀x∈{i::nat. i<2^n}. f x = c)\"\n\ndefinition is_const:: bool where \n\"is_const ≡ const 0 ∨ const 1\"\n\ndefinition is_balanced:: bool where\n\"is_balanced ≡ ∃A B ::nat set. A ⊆ {i::nat. i < 2^n} ∧ B ⊆ {i::nat. i < 2^n}\n ∧ card A = 2^(n-1) ∧ card B = 2^(n-1) \n ∧ (∀x∈A. f x = 0) ∧ (∀x∈B. f x = 1)\"\n\nlemma is_balanced_inter: \n fixes A B:: \"nat set\"\n assumes \"∀x ∈ A. f x = 0\" and \"∀x ∈ B. f x = 1\" \n shows \"A ∩ B = {}\" \n using assms (*‹∀x∈A. f x = 0› ‹∀x∈B. f x = 1›*) by auto\n\nlemma is_balanced_union:\n fixes A B:: \"nat set\"\n assumes \"A ⊆ {i::nat. i < 2^n}\" and \"B ⊆ {i::nat. i < 2^n}\" \n and \"card A = 2^(n-1)\" and \"card B = 2^(n-1)\" \n and \"A ∩ B = {}\"\n shows \"A ∪ B = {i::nat. i < 2^n}\"\nproof (-)\n (*goal: ‹A ∪ B = {i. i < 2 ^ n}›*)\n have \"finite A\" and \"finite B\"\n (*goals:\n 1. ‹finite A›\n 2. ‹finite B›\n discuss goal 1*)\n apply (simp add: assms( (*‹card (A::nat set) = (2::nat) ^ ((n::nat) - (1::nat))›*) 3) card_ge_0_finite (*‹(0::nat) < card (?A::?'a::type set) ⟹ finite ?A›*))\n (*discuss goal 2*)\n apply (simp add: assms( (*‹card (B::nat set) = (2::nat) ^ ((n::nat) - (1::nat))›*) 4) card_ge_0_finite (*‹(0::nat) < card (?A::?'a set) ⟹ finite ?A›*))\n (*proven 2 subgoals*) .\n then have \"card(A ∪ B) = 2 * 2^(n-1)\"\n using assms(3-5) (*‹card A = 2 ^ (n - 1)› ‹card (B::nat set) = (2::nat) ^ ((n::nat) - (1::nat))› ‹A ∩ B = {}›*) by (simp add: card_Un_disjoint (*‹⟦finite ?A; finite ?B; ?A ∩ ?B = {}⟧ ⟹ card (?A ∪ ?B) = card ?A + card ?B›*))\n then have \"card(A ∪ B) = 2^n\"\n by (metis Nat.nat.simps( (*‹Suc ?x2.0 ≠ 0›*) 3) One_nat_def (*‹1 = Suc 0›*) dim (*‹1 ≤ n›*) le_0_eq (*‹(?n ≤ 0) = (?n = 0)›*) power_eq_if (*‹?p ^ ?m = (if ?m = 0 then 1 else ?p * ?p ^ (?m - 1))›*))\n moreover have \"… = card({i::nat. i < 2^n})\"\n by simp\n moreover have \"A ∪ B ⊆ {i::nat. i < 2^n}\"\n using assms(1,2) (*‹A ⊆ {i. i < 2 ^ n}› ‹B ⊆ {i. i < 2 ^ n}›*) by simp\n moreover have \"finite ({i::nat. i < 2^n})\"\n by simp\n ultimately show \"?thesis\"\n (*goal: ‹A ∪ B = {i. i < 2 ^ n}›*)\n using card_subset_eq[of \"{i::nat. i < 2^n}\" \"A ∪ B\"] (*‹⟦finite {i. i < 2 ^ n}; A ∪ B ⊆ {i. i < 2 ^ n}; card (A ∪ B) = card {i. i < 2 ^ n}⟧ ⟹ A ∪ B = {i. i < 2 ^ n}›*) by simp\nqed\n\nlemma f_ge_0: \"∀x. f x ≥ 0\" by simp\n\nlemma f_dom_not_zero: \n shows \"f ∈ ({i::nat. n ≥ 1 ∧ i < 2^n} →⇩E {0,1})\" \n using dim (*‹(1::nat) ≤ (n::nat)›*) dom (*‹f ∈ {i. i < 2 ^ n} →⇩E {0, 1}›*) by simp\n\nlemma f_values: \"∀x ∈ {(i::nat). i < 2^n} . f x = 0 ∨ f x = 1\" \n using dom (*‹f ∈ {i. i < 2 ^ n} →⇩E {0, 1}›*) by auto\n\nend (* bob_fun *)\n\ntext ‹The input function has to be constant or balanced.›\n\nlocale jozsa = bob_fun +\n assumes const_or_balanced: \"is_const ∨ is_balanced \"\n\ntext ‹\nIntroduce two customised rules: disjunctions with four disjuncts and induction starting from one \ninstead of zero.\n›\n\n(* To deal with Uf it is often necessary to do a case distinction with four different cases.*)\nlemma disj_four_cases:\n assumes \"A ∨ B ∨ C ∨ D\" and \"A ⟹ P\" and \"B ⟹ P\" and \"C ⟹ P\" and \"D ⟹ P\"\n shows \"P\" \n using assms (*‹A ∨ B ∨ C ∨ D› ‹A ⟹ P› ‹B ⟹ P› ‹C::bool ⟹ P::bool› ‹D::bool ⟹ P::bool›*) by auto\n\ntext ‹The unitary transform @{term U⇩f}.›\n\ndefinition (in jozsa) jozsa_transform:: \"complex Matrix.mat\" (\"U⇩f\") where \n\"U⇩f ≡ Matrix.mat (2^(n+1)) (2^(n+1)) (λ(i,j). \n if i = j then (1-f(i div 2)) else \n if i = j + 1 ∧ odd i then f(i div 2) else\n if i = j - 1 ∧ even i ∧ j≥1 then f(i div 2) else 0)\"\n\nlemma (in jozsa) jozsa_transform_dim [simp]:\n shows \"dim_row U⇩f = 2^(n+1)\" and \"dim_col U⇩f = 2^(n+1)\" \n \n (*goals:\n 1. ‹dim_row U⇩f = 2 ^ (n + 1)›\n 2. ‹dim_col U⇩f = 2 ^ (n + 1)›\n discuss goal 1*)\n apply ((auto simp add: jozsa_transform_def (*‹U⇩f ≡ Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_nat (case x of (i::nat, j::nat) ⇒ if i = j then (1::nat) - (f::nat ⇒ nat) (i div (2::nat)) else if i = j + (1::nat) ∧ odd i then f (i div (2::nat)) else if i = j - (1::nat) ∧ even i ∧ (1::nat) ≤ j then f (i div (2::nat)) else (0::nat)))›*))[1])\n (*discuss goal 2*)\n apply ((auto simp add: jozsa_transform_def (*‹U⇩f ≡ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_nat (case x of (i, j) ⇒ if i = j then 1 - f (i div 2) else if i = j + 1 ∧ odd i then f (i div 2) else if i = j - 1 ∧ even i ∧ 1 ≤ j then f (i div 2) else 0))›*))[1])\n (*proven 2 subgoals*) .\n\nlemma (in jozsa) jozsa_transform_coeff_is_zero [simp]:\n assumes \"i < dim_row U⇩f ∧ j < dim_col U⇩f\"\n shows \"(i≠j ∧ ¬(i=j+1 ∧ odd i) ∧ ¬ (i=j-1 ∧ even i ∧ j≥1)) ⟶ U⇩f $$ (i,j) = 0\"\n using jozsa_transform_def (*‹U⇩f ≡ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_nat (case x of (i, j) ⇒ if i = j then 1 - f (i div 2) else if i = j + 1 ∧ odd i then f (i div 2) else if i = j - 1 ∧ even i ∧ 1 ≤ j then f (i div 2) else 0))›*) assms (*‹(i::nat) < dim_row U⇩f ∧ (j::nat) < dim_col U⇩f›*) by auto\n\nlemma (in jozsa) jozsa_transform_coeff [simp]: \n assumes \"i < dim_row U⇩f ∧ j < dim_col U⇩f\"\n shows \"i = j ⟶ U⇩f $$ (i,j) = 1 - f (i div 2)\"\n and \"i = j + 1 ∧ odd i ⟶ U⇩f $$ (i,j) = f (i div 2)\"\n and \"j ≥ 1 ∧ i = j - 1 ∧ even i ⟶ U⇩f $$ (i,j) = f (i div 2)\" \n using jozsa_transform_def (*‹U⇩f ≡ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_nat (case x of (i, j) ⇒ if i = j then 1 - f (i div 2) else if i = j + 1 ∧ odd i then f (i div 2) else if i = j - 1 ∧ even i ∧ 1 ≤ j then f (i div 2) else 0))›*) assms (*‹i < dim_row U⇩f ∧ j < dim_col U⇩f›*) apply -\n (*goals:\n 1. ‹⟦U⇩f ≡ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_nat (case x of (i, j) ⇒ if i = j then 1 - f (i div 2) else if i = j + 1 ∧ odd i then f (i div 2) else if i = j - 1 ∧ even i ∧ 1 ≤ j then f (i div 2) else 0)); i < dim_row U⇩f ∧ j < dim_col U⇩f⟧ ⟹ i = j ⟶ U⇩f $$ (i, j) = complex_of_nat (1 - f (i div 2))›\n 2. ‹⟦U⇩f ≡ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_nat (case x of (i, j) ⇒ if i = j then 1 - f (i div 2) else if i = j + 1 ∧ odd i then f (i div 2) else if i = j - 1 ∧ even i ∧ 1 ≤ j then f (i div 2) else 0)); i < dim_row U⇩f ∧ j < dim_col U⇩f⟧ ⟹ i = j + 1 ∧ odd i ⟶ U⇩f $$ (i, j) = complex_of_nat (f (i div 2))›\n 3. ‹⟦U⇩f ≡ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_nat (case x of (i, j) ⇒ if i = j then 1 - f (i div 2) else if i = j + 1 ∧ odd i then f (i div 2) else if i = j - 1 ∧ even i ∧ 1 ≤ j then f (i div 2) else 0)); i < dim_row U⇩f ∧ j < dim_col U⇩f⟧ ⟹ 1 ≤ j ∧ i = j - 1 ∧ even i ⟶ U⇩f $$ (i, j) = complex_of_nat (f (i div 2))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*proven 3 subgoals*) .\n\nlemma (in jozsa) U⇩f_mult_without_empty_summands_sum_even:\n fixes i j A\n assumes \"i < dim_row U⇩f\" and \"j < dim_col A\" and \"even i\" and \"dim_col U⇩f = dim_row A\"\n shows \"(∑k∈{0..< dim_row A}. U⇩f $$ (i,k) * A $$ (k,j)) =(∑k∈{i,i+1}. U⇩f $$ (i,k) * A $$ (k,j))\"\nproof (-)\n (*goal: ‹(∑k = 0..<dim_row A. U⇩f $$ (i, k) * A $$ (k, j)) = (∑k∈{i, i + 1}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n have \"(∑k ∈ {0..< 2^(n+1)}. U⇩f $$ (i,k) * A $$ (k,j)) = \n (∑k ∈ {0..<i}. U⇩f $$ (i,k) * A $$ (k,j)) +\n (∑k ∈ {i,i+1}. U⇩f $$ (i,k) * A $$ (k,j)) +\n (∑k ∈ {(i+2)..< 2^(n+1)}. U⇩f $$ (i,k) * A $$ (k,j))\"\n proof (-)\n (*goal: ‹(∑k = 0..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j)) = (∑k = 0..<i. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k∈{i, i + 1}. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k = i + 2..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j))›*)\n have \"{0..< 2^(n+1)} = {0..<i} ∪ {i..< 2^(n+1)} \n ∧ {i..< 2^(n+1)} = {i,i+1} ∪ {(i+2)..<2^(n+1)}\"\n using assms(1-3) (*‹i < dim_row U⇩f› ‹j < dim_col A› ‹even i›*) by auto\n moreover have \"{0..<i} ∩ {i,i+1} = {} \n ∧ {i,i+1} ∩ {(i+2)..< 2^(n+1)} = {} \n ∧ {0..<i} ∩ {(i+2)..< 2^(n+1)} = {}\"\n using assms (*‹i < dim_row U⇩f› ‹j < dim_col A› ‹even i› ‹dim_col U⇩f = dim_row A›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹(∑k = 0..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j)) = (∑k = 0..<i. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k∈{i, i + 1}. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k = i + 2..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j))›*)\n using sum.union_disjoint (*‹⟦finite (?A::?'b set); finite (?B::?'b set); ?A ∩ ?B = {}⟧ ⟹ sum (?g::?'b ⇒ ?'a) (?A ∪ ?B) = sum ?g ?A + sum ?g ?B›*) by (metis (no_types, lifting) finite_Un (*‹finite (?F ∪ ?G) = (finite ?F ∧ finite ?G)›*) finite_atLeastLessThan (*‹finite {?l..<?u}›*) is_num_normalize( (*‹?a + ?b + ?c = ?a + (?b + ?c)›*) 1) ivl_disj_int_two( (*‹{?l..<?m} ∩ {?m..<?u} = {}›*) 3))\n qed\n moreover have \"(∑k ∈ {0..<i}. U⇩f $$ (i,k) * A $$ (k,j)) = 0\"\n proof (-)\n (*goal: ‹(∑k = 0..<i. U⇩f $$ (i, k) * A $$ (k, j)) = 0›*)\n have \"k ∈ {0..<i} ⟶ (i≠k ∧ ¬(i=k+1 ∧ odd i) ∧ ¬ (i=k-1 ∧ even i ∧ k≥1))\" for k\n using assms (*‹(i::nat) < dim_row U⇩f› ‹j < dim_col A› ‹even i› ‹dim_col U⇩f = dim_row (A::complex Matrix.mat)›*) by auto\n then have \"k ∈ {0..<i} ⟶ U⇩f $$ (i,k) = 0\" for k\n using assms(1) (*‹i < dim_row U⇩f›*) by auto\n then show \"?thesis\"\n (*goal: ‹(∑k = 0..<i. U⇩f $$ (i, k) * A $$ (k, j)) = 0›*)\n by simp\n qed\n moreover have \"(∑k ∈ {(i+2)..< 2^(n+1)}. U⇩f $$ (i,k) * A $$ (k,j)) = 0\"\n proof (-)\n (*goal: ‹(∑k = i + 2..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j)) = 0›*)\n have \"k∈{(i+2)..< 2^(n+1)} ⟶ (i≠k ∧ ¬(i=k+1 ∧ odd i) ∧ ¬ (i=k-1 ∧ even i ∧ k≥1))\" for k\n by auto\n then have \"k ∈ {(i+2)..< 2^(n+1)}⟶ U⇩f $$ (i,k) = 0\" for k\n using assms(1) (*‹i < dim_row U⇩f›*) by auto\n then show \"?thesis\"\n (*goal: ‹(∑k::nat = i + (2::nat)..<(2::nat) ^ ((n::nat) + (1::nat)). U⇩f $$ (i::nat, k) * (A::complex Matrix.mat) $$ (k, j::nat)) = (0::complex)›*)\n by simp\n qed\n moreover have \"dim_row A = 2^(n+1)\"\n using assms(4) (*‹dim_col U⇩f = dim_row (A::complex Matrix.mat)›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹(∑k::nat = 0::nat..<dim_row A. U⇩f $$ (i::nat, k) * (A::complex Matrix.mat) $$ (k, j::nat)) = (∑k::nat∈{i, i + (1::nat)}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n by (metis (no_types, lifting) add.left_neutral (*‹0 + ?a = ?a›*) add.right_neutral (*‹?a + 0 = ?a›*))\nqed\n\nlemma (in jozsa) U⇩f_mult_without_empty_summands_even: \n fixes i j A\n assumes \"i < dim_row U⇩f\" and \"j < dim_col A\" and \"even i\" and \"dim_col U⇩f = dim_row A\"\n shows \"(U⇩f * A) $$ (i,j) = (∑k ∈ {i,i+1}. U⇩f $$ (i,k) * A $$ (k,j))\"\nproof (-)\n (*goal: ‹(U⇩f * A) $$ (i, j) = (∑k∈{i, i + 1}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n have \"(U⇩f * A) $$ (i,j) = (∑ k∈{0..< dim_row A}. (U⇩f $$ (i,k)) * (A $$ (k,j)))\"\n using assms(1,2,4) (*‹i < dim_row U⇩f› ‹(j::nat) < dim_col (A::complex Matrix.mat)› ‹dim_col U⇩f = dim_row A›*) index_matrix_prod (*‹⟦?i < dim_row ?A; ?j < dim_col ?B; dim_col ?A = dim_row ?B⟧ ⟹ (?A * ?B) $$ (?i, ?j) = (∑k<dim_row ?B. ?A $$ (?i, k) * ?B $$ (k, ?j))›*) by (simp add: atLeast0LessThan (*‹{0..<?n} = {..<?n}›*))\n then show \"?thesis\"\n (*goal: ‹(U⇩f * A) $$ (i, j) = (∑k∈{i, i + 1}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n using assms (*‹(i::nat) < dim_row U⇩f› ‹(j::nat) < dim_col (A::complex Matrix.mat)› ‹even i› ‹dim_col U⇩f = dim_row A›*) \"U⇩f_mult_without_empty_summands_sum_even\" (*‹⟦?i < dim_row U⇩f; ?j < dim_col ?A; even ?i; dim_col U⇩f = dim_row ?A⟧ ⟹ (∑k = 0..<dim_row ?A. U⇩f $$ (?i, k) * ?A $$ (k, ?j)) = (∑k∈{?i, ?i + 1}. U⇩f $$ (?i, k) * ?A $$ (k, ?j))›*) by simp\nqed\n\nlemma (in jozsa) U⇩f_mult_without_empty_summands_sum_odd:\n fixes i j A\n assumes \"i < dim_row U⇩f\" and \"j < dim_col A\" and \"odd i\" and \"dim_col U⇩f = dim_row A\"\n shows \"(∑k∈{0..< dim_row A}. U⇩f $$ (i,k) * A $$ (k,j)) =(∑k∈{i-1,i}. U⇩f $$ (i,k) * A $$ (k,j))\"\nproof (-)\n (*goal: ‹(∑k = 0..<dim_row A. U⇩f $$ (i, k) * A $$ (k, j)) = (∑k∈{i - 1, i}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n have \"(∑k∈{0..< 2^(n+1)}. U⇩f $$ (i,k) * A $$ (k,j)) = \n (∑k ∈ {0..<i-1}. U⇩f $$ (i,k) * A $$ (k,j)) +\n (∑k ∈ {i-1,i}. U⇩f $$ (i,k) * A $$ (k,j)) +\n (∑k ∈ {i+1..< 2^(n+1)}. U⇩f $$ (i,k) * A $$ (k,j))\"\n proof (-)\n (*goal: ‹(∑k = 0..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j)) = (∑k = 0..<i - 1. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k∈{i - 1, i}. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k = i + 1..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j))›*)\n have \"{0..< 2^(n+1)} = {0..<i-1} ∪ {i-1..< 2^(n+1)} \n ∧ {i-1..< 2^(n+1)} = {i-1,i} ∪ {i+1..<2^(n+1)}\"\n using assms(1-3) (*‹i < dim_row U⇩f› ‹j < dim_col A› ‹odd i›*) by auto\n moreover have \"{0..<i-1} ∩ {i-1,i} = {} \n ∧ {i-1,i} ∩ {i+1..< 2^(n+1)} = {} \n ∧ {0..<i-1} ∩ {i+1..< 2^(n+1)} = {}\"\n using assms (*‹i < dim_row U⇩f› ‹j < dim_col A› ‹odd i› ‹dim_col U⇩f = dim_row A›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹(∑k = 0..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j)) = (∑k = 0..<i - 1. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k∈{i - 1, i}. U⇩f $$ (i, k) * A $$ (k, j)) + (∑k = i + 1..<2 ^ (n + 1). U⇩f $$ (i, k) * A $$ (k, j))›*)\n using sum.union_disjoint (*‹⟦finite ?A; finite ?B; ?A ∩ ?B = {}⟧ ⟹ sum ?g (?A ∪ ?B) = sum ?g ?A + sum ?g ?B›*) by (metis (no_types, lifting) finite_Un (*‹finite ((?F::?'a set) ∪ (?G::?'a set)) = (finite ?F ∧ finite ?G)›*) finite_atLeastLessThan (*‹finite {?l::nat..<?u::nat}›*) is_num_normalize( (*‹(?a::?'a) + (?b::?'a) + (?c::?'a) = ?a + (?b + ?c)›*) 1) ivl_disj_int_two( (*‹{?l::?'a..<?m::?'a} ∩ {?m..<?u::?'a} = {}›*) 3))\n qed\n moreover have \"(∑k ∈ {0..<i-1}. U⇩f $$ (i,k) * A $$ (k,j)) = 0\"\n proof (-)\n (*goal: ‹(∑k::nat = 0::nat..<i - (1::nat). U⇩f $$ (i::nat, k) * (A::complex Matrix.mat) $$ (k, j::nat)) = (0::complex)›*)\n have \"k ∈ {0..<i-1} ⟶ (i≠k ∧ ¬(i=k+1 ∧ odd i) ∧ ¬ (i=k-1 ∧ even i ∧ k≥1))\" for k\n by auto\n then have \"k ∈ {0..<i-1} ⟶ U⇩f $$ (i,k) = 0\" for k\n using assms(1) (*‹i < dim_row U⇩f›*) by auto\n then show \"?thesis\"\n (*goal: ‹(∑k = 0..<i - 1. U⇩f $$ (i, k) * A $$ (k, j)) = 0›*)\n by simp\n qed\n moreover have \"(∑k ∈ {i+1..< 2^(n+1)}. U⇩f $$ (i,k) * A $$ (k,j)) = 0\"\n using assms(3) (*‹odd i›*) by auto\n moreover have \"dim_row A = 2^(n+1)\"\n using assms(4) (*‹dim_col U⇩f = dim_row A›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹(∑k = 0..<dim_row A. U⇩f $$ (i, k) * A $$ (k, j)) = (∑k∈{i - 1, i}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n by (metis (no_types, lifting) add.left_neutral (*‹0 + ?a = ?a›*) add.right_neutral (*‹?a + 0 = ?a›*))\nqed\n\nlemma (in jozsa) U⇩f_mult_without_empty_summands_odd: \n fixes i j A\n assumes \"i < dim_row U⇩f\" and \"j < dim_col A\" and \"odd i\" and \"dim_col U⇩f = dim_row A\"\n shows \"(U⇩f * A) $$ (i,j) = (∑k ∈ {i-1,i}. U⇩f $$ (i,k) * A $$ (k,j)) \"\nproof (-)\n (*goal: ‹(U⇩f * A) $$ (i, j) = (∑k∈{i - 1, i}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n have \"(U⇩f * A) $$ (i,j) = (∑k ∈ {0 ..< dim_row A}. (U⇩f $$ (i,k)) * (A $$ (k,j)))\"\n using assms(1,2,4) (*‹i < dim_row U⇩f› ‹(j::nat) < dim_col (A::complex Matrix.mat)› ‹dim_col U⇩f = dim_row (A::complex Matrix.mat)›*) index_matrix_prod (*‹⟦?i < dim_row ?A; ?j < dim_col ?B; dim_col ?A = dim_row ?B⟧ ⟹ (?A * ?B) $$ (?i, ?j) = (∑k<dim_row ?B. ?A $$ (?i, k) * ?B $$ (k, ?j))›*) by (simp add: atLeast0LessThan (*‹{0..<?n} = {..<?n}›*))\n then show \"?thesis\"\n (*goal: ‹(U⇩f * A) $$ (i, j) = (∑k∈{i - 1, i}. U⇩f $$ (i, k) * A $$ (k, j))›*)\n using assms (*‹i < dim_row U⇩f› ‹j < dim_col A› ‹odd i› ‹dim_col U⇩f = dim_row A›*) \"U⇩f_mult_without_empty_summands_sum_odd\" (*‹⟦?i < dim_row U⇩f; ?j < dim_col ?A; odd ?i; dim_col U⇩f = dim_row ?A⟧ ⟹ (∑k = 0..<dim_row ?A. U⇩f $$ (?i, k) * ?A $$ (k, ?j)) = (∑k∈{?i - 1, ?i}. U⇩f $$ (?i, k) * ?A $$ (k, ?j))›*) by auto\nqed\n\ntext ‹@{term U⇩f} is a gate.›\n\nlemma (in jozsa) transpose_of_jozsa_transform:\n shows \"(U⇩f)⇧t = U⇩f\"\nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹dim_row U⇩f⇧t = dim_row U⇩f›\n 3. ‹dim_col U⇩f⇧t = dim_col U⇩f›*)\n show \"dim_row (U⇩f⇧t) = dim_row U⇩f\"\n by simp\nnext\n (*goals:\n 1. ‹⋀(i::nat) j::nat. ⟦i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹dim_col U⇩f⇧t = dim_col U⇩f›*)\n show \"dim_col (U⇩f⇧t) = dim_col U⇩f\"\n by simp\nnext\n (*goal: ‹⋀i j. ⟦i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row U⇩f\" and a1: \"j < dim_col U⇩f\" (*‹(i::nat) < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f›*)\n then show \"U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)\"\n proof (induct rule: disj_four_cases (*‹⟦?A ∨ ?B ∨ ?C ∨ ?D; ?A ⟹ ?P; ?B ⟹ ?P; ?C ⟹ ?P; ?D ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹?A4 ∨ ?B4 ∨ ?C4 ∨ ?D4›\n 2. ‹⟦?A4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 3. ‹⟦?B4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 4. ‹⟦?C4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 5. ‹⟦?D4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›*)\n show \"i=j ∨ (i=j+1 ∧ odd i) ∨ (i=j-1 ∧ even i ∧ j≥1) ∨ (i≠j ∧ ¬(i=j+1 ∧ odd i) ∧ ¬ (i=j-1 ∧ even i ∧ j≥1))\"\n by linarith\n next\n (*goals:\n 1. ‹⟦i = j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹⟦i = j + 1 ∧ odd i; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 3. ‹⟦i = j - 1 ∧ even i ∧ 1 ≤ j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 4. ‹⟦i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›*)\n assume \"i = j\" (*‹(i::nat) = (j::nat)›*)\n then show \"U⇩f⇧t $$ (i,j) = U⇩f $$ (i,j)\"\n using a0 (*‹i < dim_row U⇩f›*) by simp\n next\n (*goals:\n 1. ‹⟦i = j + 1 ∧ odd i; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹⟦i = j - 1 ∧ even i ∧ 1 ≤ j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 3. ‹⟦i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›*)\n assume \"(i=j+1 ∧ odd i)\" (*‹(i::nat) = (j::nat) + (1::nat) ∧ odd i›*)\n then show \"U⇩f⇧t $$ (i,j) = U⇩f $$ (i,j)\"\n using transpose_mat_def (*‹?A⇧t ≡ Matrix.mat (dim_col ?A) (dim_row ?A) (λ(i, j). ?A $$ (j, i))›*) a0 (*‹i < dim_row U⇩f›*) a1 (*‹(j::nat) < dim_col U⇩f›*) by auto\n next\n (*goals:\n 1. ‹⟦i = j - 1 ∧ even i ∧ 1 ≤ j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹⟦i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›*)\n assume a2: \"(i=j-1 ∧ even i ∧ j≥1)\" (*‹(i::nat) = (j::nat) - (1::nat) ∧ even i ∧ (1::nat) ≤ j›*)\n then have \"U⇩f $$ (i,j) = f (i div 2)\"\n using a0 (*‹(i::nat) < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) jozsa_transform_coeff (*‹(?i::nat) < dim_row U⇩f ∧ (?j::nat) < dim_col U⇩f ⟹ ?i = ?j ⟶ U⇩f $$ (?i, ?j) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (?i div (2::nat)))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j + 1 ∧ odd ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ 1 ≤ ?j ∧ ?i = ?j - 1 ∧ even ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))›*) by auto\n moreover have \"U⇩f $$ (j,i) = f (i div 2)\"\n using a0 (*‹i < dim_row U⇩f›*) a1 (*‹(j::nat) < dim_col U⇩f›*) a2 (*‹i = j - 1 ∧ even i ∧ 1 ≤ j›*) jozsa_transform_coeff (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (1 - f (?i div 2))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j + 1 ∧ odd ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ 1 ≤ ?j ∧ ?i = ?j - 1 ∧ even ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))›*) by (metis add_diff_assoc2 (*‹(?k::nat) ≤ (?j::nat) ⟹ ?j - ?k + (?i::nat) = ?j + ?i - ?k›*) diff_add_inverse2 (*‹(?m::nat) + (?n::nat) - ?n = ?m›*) even_plus_one_iff (*‹even ((?a::?'a) + (1::?'a)) = odd ?a›*) even_succ_div_two (*‹even (?a::?'a) ⟹ (?a + (1::?'a)) div (2::?'a) = ?a div (2::?'a)›*) jozsa_transform_dim (*‹dim_row U⇩f = (2::nat) ^ ((n::nat) + (1::nat))› ‹dim_col U⇩f = (2::nat) ^ ((n::nat) + (1::nat))›*))\n ultimately show \"?thesis\"\n (*goal: ‹U⇩f⇧t $$ (i::nat, j::nat) = U⇩f $$ (i, j)›*)\n using transpose_mat_def (*‹?A⇧t ≡ Matrix.mat (dim_col ?A) (dim_row ?A) (λ(i, j). ?A $$ (j, i))›*) a0 (*‹i < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) by simp\n next\n (*goal: ‹⟦i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧t $$ (i, j) = U⇩f $$ (i, j)›*)\n assume a2: \"(i≠j ∧ ¬(i=j+1 ∧ odd i) ∧ ¬ (i=j-1 ∧ even i ∧ j≥1))\" (*‹(i::nat) ≠ (j::nat) ∧ ¬ (i = j + (1::nat) ∧ odd i) ∧ ¬ (i = j - (1::nat) ∧ even i ∧ (1::nat) ≤ j)›*)\n then have \"(j≠i ∧ ¬(j=i+1 ∧ odd j) ∧ ¬ (j=i-1 ∧ even j ∧ i≥1))\"\n by (metis le_imp_diff_is_add (*‹?i ≤ ?j ⟹ (?j - ?i = ?k) = (?j = ?k + ?i)›*) diff_add_inverse (*‹?n + ?m - ?n = ?m›*) even_plus_one_iff (*‹even (?a + 1) = odd ?a›*) le_add1 (*‹?n ≤ ?n + ?m›*))\n then have \"U⇩f $$ (j,i) = 0\"\n using jozsa_transform_coeff_is_zero (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i ≠ ?j ∧ ¬ (?i = ?j + 1 ∧ odd ?i) ∧ ¬ (?i = ?j - 1 ∧ even ?i ∧ 1 ≤ ?j) ⟶ U⇩f $$ (?i, ?j) = 0›*) a0 (*‹i < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) by auto\n moreover have \"U⇩f $$ (i,j) = 0\"\n using jozsa_transform_coeff_is_zero (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i ≠ ?j ∧ ¬ (?i = ?j + 1 ∧ odd ?i) ∧ ¬ (?i = ?j - 1 ∧ even ?i ∧ 1 ≤ ?j) ⟶ U⇩f $$ (?i, ?j) = 0›*) a0 (*‹i < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) a2 (*‹i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j)›*) by auto\n ultimately show \"U⇩f⇧t $$ (i,j) = U⇩f $$ (i,j)\"\n using transpose_mat_def (*‹?A⇧t ≡ Matrix.mat (dim_col ?A) (dim_row ?A) (λ(i, j). ?A $$ (j, i))›*) a0 (*‹(i::nat) < dim_row U⇩f›*) a1 (*‹(j::nat) < dim_col U⇩f›*) by simp\n qed\nqed\n\nlemma (in jozsa) adjoint_of_jozsa_transform: \n shows \"(U⇩f)⇧† = U⇩f\"\nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹dim_row U⇩f⇧† = dim_row U⇩f›\n 3. ‹dim_col U⇩f⇧† = dim_col U⇩f›*)\n show \"dim_row (U⇩f⇧†) = dim_row U⇩f\"\n by simp\nnext\n (*goals:\n 1. ‹⋀(i::nat) j::nat. ⟦i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹dim_col U⇩f⇧† = dim_col U⇩f›*)\n show \"dim_col (U⇩f⇧†) = dim_col U⇩f\"\n by simp\nnext\n (*goal: ‹⋀i j. ⟦i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row U⇩f\" and a1: \"j < dim_col U⇩f\" (*‹(i::nat) < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f›*)\n then show \"U⇩f⇧† $$ (i,j) = U⇩f $$ (i,j)\"\n proof (induct rule: disj_four_cases (*‹⟦(?A::bool) ∨ (?B::bool) ∨ (?C::bool) ∨ (?D::bool); ?A ⟹ ?P::bool; ?B ⟹ ?P; ?C ⟹ ?P; ?D ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹?A4 ∨ ?B4 ∨ ?C4 ∨ ?D4›\n 2. ‹⟦?A4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 3. ‹⟦?B4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 4. ‹⟦?C4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 5. ‹⟦?D4; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›*)\n show \"i=j ∨ (i=j+1 ∧ odd i) ∨ (i=j-1 ∧ even i ∧ j≥1) ∨ (i≠j ∧ ¬(i=j+1 ∧ odd i) ∧ ¬ (i=j-1 ∧ even i ∧ j≥1))\"\n by linarith\n next\n (*goals:\n 1. ‹⟦i = j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹⟦i = j + 1 ∧ odd i; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 3. ‹⟦i = j - 1 ∧ even i ∧ 1 ≤ j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 4. ‹⟦i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›*)\n assume \"i=j\" (*‹(i::nat) = (j::nat)›*)\n then show \"U⇩f⇧† $$ (i,j) = U⇩f $$ (i,j)\"\n using a0 (*‹i < dim_row U⇩f›*) dagger_def (*‹?M⇧† ≡ Matrix.mat (dim_col ?M) (dim_row ?M) (λ(i, j). cnj (?M $$ (j, i)))›*) by simp\n next\n (*goals:\n 1. ‹⟦(i::nat) = (j::nat) + (1::nat) ∧ odd i; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹⟦(i::nat) = (j::nat) - (1::nat) ∧ even i ∧ (1::nat) ≤ j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 3. ‹⟦(i::nat) ≠ (j::nat) ∧ ¬ (i = j + (1::nat) ∧ odd i) ∧ ¬ (i = j - (1::nat) ∧ even i ∧ (1::nat) ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›*)\n assume \"(i=j+1 ∧ odd i)\" (*‹(i::nat) = (j::nat) + (1::nat) ∧ odd i›*)\n then show \"U⇩f⇧† $$ (i,j) = U⇩f $$ (i,j)\"\n using a0 (*‹(i::nat) < dim_row U⇩f›*) dagger_def (*‹?M⇧† ≡ Matrix.mat (dim_col ?M) (dim_row ?M) (λ(i, j). cnj (?M $$ (j, i)))›*) by auto\n next\n (*goals:\n 1. ‹⟦i = j - 1 ∧ even i ∧ 1 ≤ j; i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›\n 2. ‹⟦i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›*)\n assume a2: \"(i=j-1 ∧ even i ∧ j≥1)\" (*‹(i::nat) = (j::nat) - (1::nat) ∧ even i ∧ (1::nat) ≤ j›*)\n then have \"U⇩f $$ (i,j) = f (i div 2)\"\n using a0 (*‹(i::nat) < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) jozsa_transform_coeff (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (1 - f (?i div 2))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j + 1 ∧ odd ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ 1 ≤ ?j ∧ ?i = ?j - 1 ∧ even ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))›*) by auto\n moreover have \"U⇩f⇧† $$ (j,i) = f (i div 2)\"\n using a1 (*‹j < dim_col U⇩f›*) a2 (*‹i = j - 1 ∧ even i ∧ 1 ≤ j›*) jozsa_transform_coeff (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (1 - f (?i div 2))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j + 1 ∧ odd ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))› ‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ 1 ≤ ?j ∧ ?i = ?j - 1 ∧ even ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))›*) dagger_def (*‹?M⇧† ≡ Matrix.mat (dim_col ?M) (dim_row ?M) (λ(i, j). cnj (?M $$ (j, i)))›*) by auto\n ultimately show \"U⇩f⇧† $$ (i,j) = U⇩f $$ (i,j)\"\n by (metis a0 (*‹i < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) cnj_transpose_is_dagger (*‹?M⇧⋆⇧t = ?M⇧†›*) dim_row_of_dagger (*‹dim_row ?M⇧† = dim_col ?M›*) index_transpose_mat (*‹⟦?i < dim_col ?A; ?j < dim_row ?A⟧ ⟹ ?A⇧t $$ (?i, ?j) = ?A $$ (?j, ?i)› ‹dim_row ?A⇧t = dim_col ?A› ‹dim_col ?A⇧t = dim_row ?A›*) dagger_of_transpose_is_cnj (*‹?M⇧t⇧† = ?M⇧⋆›*) transpose_of_jozsa_transform (*‹U⇩f⇧t = U⇩f›*))\n next\n (*goal: ‹⟦i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j); i < dim_row U⇩f; j < dim_col U⇩f⟧ ⟹ U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)›*)\n assume a2: \"(i≠j ∧ ¬(i=j+1 ∧ odd i) ∧ ¬ (i=j-1 ∧ even i ∧ j≥1))\" (*‹(i::nat) ≠ (j::nat) ∧ ¬ (i = j + (1::nat) ∧ odd i) ∧ ¬ (i = j - (1::nat) ∧ even i ∧ (1::nat) ≤ j)›*)\n then have f0: \"(i≠j ∧ ¬(j=i+1 ∧ odd j) ∧ ¬ (j=i-1 ∧ even j ∧ i≥1))\"\n by (metis le_imp_diff_is_add (*‹?i ≤ ?j ⟹ (?j - ?i = ?k) = (?j = ?k + ?i)›*) diff_add_inverse (*‹?n + ?m - ?n = ?m›*) even_plus_one_iff (*‹even (?a + 1) = odd ?a›*) le_add1 (*‹?n ≤ ?n + ?m›*))\n then have \"U⇩f $$ (j,i) = 0\" and \"cnj 0 = 0\"\n using jozsa_transform_coeff_is_zero (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i ≠ ?j ∧ ¬ (?i = ?j + 1 ∧ odd ?i) ∧ ¬ (?i = ?j - 1 ∧ even ?i ∧ 1 ≤ ?j) ⟶ U⇩f $$ (?i, ?j) = 0›*) a0 (*‹(i::nat) < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) a2 (*‹i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j)›*) apply -\n (*goals:\n 1. ‹⟦i ≠ j ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i); ⋀i j. i < dim_row U⇩f ∧ j < dim_col U⇩f ⟹ i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j) ⟶ U⇩f $$ (i, j) = 0; i < dim_row U⇩f; j < dim_col U⇩f; i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j)⟧ ⟹ U⇩f $$ (j, i) = 0›\n 2. ‹⟦i ≠ j ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i); ⋀i j. i < dim_row U⇩f ∧ j < dim_col U⇩f ⟹ i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j) ⟶ U⇩f $$ (i, j) = 0; i < dim_row U⇩f; j < dim_col U⇩f; i ≠ j ∧ ¬ (i = j + 1 ∧ odd i) ∧ ¬ (i = j - 1 ∧ even i ∧ 1 ≤ j)⟧ ⟹ cnj 0 = 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n then have \"U⇩f⇧† $$ (i,j) = 0\"\n using a0 (*‹i < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) dagger_def (*‹?M⇧† ≡ Matrix.mat (dim_col ?M) (dim_row ?M) (λ(i, j). cnj (?M $$ (j, i)))›*) by simp\n then show \"U⇩f⇧† $$ (i, j) = U⇩f $$ (i, j)\"\n using a0 (*‹i < dim_row U⇩f›*) a1 (*‹j < dim_col U⇩f›*) a2 (*‹(i::nat) ≠ (j::nat) ∧ ¬ (i = j + (1::nat) ∧ odd i) ∧ ¬ (i = j - (1::nat) ∧ even i ∧ (1::nat) ≤ j)›*) jozsa_transform_coeff_is_zero (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i ≠ ?j ∧ ¬ (?i = ?j + 1 ∧ odd ?i) ∧ ¬ (?i = ?j - 1 ∧ even ?i ∧ 1 ≤ ?j) ⟶ U⇩f $$ (?i, ?j) = 0›*) by auto\n qed\nqed\n\nlemma (in jozsa) jozsa_transform_is_unitary_index_even:\n fixes i j:: nat\n assumes \"i < dim_row U⇩f\" and \"j < dim_col U⇩f\" and \"even i\"\n shows \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\nproof (-)\n (*goal: ‹(U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n have \"(U⇩f * U⇩f) $$ (i,j) = (∑k ∈ {i,i+1}. U⇩f $$ (i,k) * U⇩f $$ (k,j)) \"\n using \"U⇩f_mult_without_empty_summands_even\"[of i j U⇩f] (*‹⟦i < dim_row U⇩f; j < dim_col U⇩f; even i; dim_col U⇩f = dim_row U⇩f⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = (∑k∈{i, i + 1}. U⇩f $$ (i, k) * U⇩f $$ (k, j))›*) assms (*‹i < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f› ‹even i›*) by simp\n moreover have \"U⇩f $$ (i,i) * U⇩f $$ (i,j) = (1-f(i div 2)) * U⇩f $$ (i,j)\"\n using assms(1,3) (*‹i < dim_row U⇩f› ‹even i›*) by simp\n moreover have f0: \"U⇩f $$ (i,i+1) * U⇩f $$ (i+1,j) = f(i div 2) * U⇩f $$ (i+1,j)\"\n by (metis One_nat_def (*‹1 = Suc 0›*) Suc_leI (*‹?m < ?n ⟹ Suc ?m ≤ ?n›*) add.right_neutral (*‹?a + 0 = ?a›*) add_Suc_right (*‹?m + Suc ?n = Suc (?m + ?n)›*) assms( (*‹i < dim_row U⇩f›*) 1) assms( (*‹even i›*) 3) diff_add_inverse2 (*‹?m + ?n - ?n = ?m›*) even_add (*‹even (?a + ?b) = (even ?a = even ?b)›*) even_mult_iff (*‹even (?a * ?b) = (even ?a ∨ even ?b)›*) jozsa_transform_coeff( (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ 1 ≤ ?j ∧ ?i = ?j - 1 ∧ even ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))›*) 3) jozsa_transform_dim (*‹dim_row U⇩f = 2 ^ (n + 1)› ‹dim_col U⇩f = 2 ^ (n + 1)›*) le_add2 (*‹?n ≤ ?m + ?n›*) le_eq_less_or_eq (*‹(?m ≤ ?n) = (?m < ?n ∨ ?m = ?n)›*) odd_one (*‹odd 1›*) one_add_one (*‹1 + 1 = 2›*) power.simps( (*‹?a ^ Suc ?n = ?a * ?a ^ ?n›*) 2))\n ultimately have f1: \"(U⇩f * U⇩f) $$ (i,j) = (1-f(i div 2)) * U⇩f $$ (i,j) + f(i div 2) * U⇩f $$ (i+1,j)\"\n by auto\n thus \"?thesis\"\n (*goal: ‹(U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n proof (induct rule: disj_four_cases (*‹⟦?A ∨ ?B ∨ ?C ∨ ?D; ?A ⟹ ?P; ?B ⟹ ?P; ?C ⟹ ?P; ?D ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹(?A2::bool) ∨ (?B2::bool) ∨ (?C2::bool) ∨ (?D2::bool)›\n 2. ‹⟦?A2::bool; (U⇩f * U⇩f) $$ (i::nat, j::nat) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 3. ‹⟦?B2::bool; (U⇩f * U⇩f) $$ (i::nat, j::nat) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 4. ‹⟦?C2::bool; (U⇩f * U⇩f) $$ (i::nat, j::nat) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 5. ‹⟦?D2::bool; (U⇩f * U⇩f) $$ (i::nat, j::nat) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n show \"j=i ∨ (j=i+1 ∧ odd j) ∨ (j=i-1 ∧ even j ∧ i≥1) ∨ (j≠i ∧ ¬(j=i+1 ∧ odd j) ∧ ¬ (j=i-1 ∧ even j ∧ i≥1))\"\n by linarith\n next\n (*goals:\n 1. ‹⟦(j::nat) = (i::nat); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 2. ‹⟦(j::nat) = (i::nat) + (1::nat) ∧ odd j; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 3. ‹⟦(j::nat) = (i::nat) - (1::nat) ∧ even j ∧ (1::nat) ≤ i; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 4. ‹⟦(j::nat) ≠ (i::nat) ∧ ¬ (j = i + (1::nat) ∧ odd j) ∧ ¬ (j = i - (1::nat) ∧ even j ∧ (1::nat) ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume a0: \"j=i\" (*‹(j::nat) = (i::nat)›*)\n then have \"U⇩f $$ (i,j) = (1-f(i div 2))\"\n using assms(1,2) (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f›*) a0 (*‹j = i›*) by simp\n moreover have \"U⇩f $$ (i+1,j) = f(i div 2)\"\n using assms(1,3) (*‹i < dim_row U⇩f› ‹even i›*) a0 (*‹j = i›*) by auto\n ultimately have \"(U⇩f * U⇩f) $$ (i,j) = (1-f(i div 2)) * (1-f(i div 2)) + f(i div 2) * f(i div 2)\"\n using f1 (*‹(U⇩f * U⇩f) $$ (i::nat, j::nat) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)›*) by simp\n moreover have \"(1-f(i div 2)) * (1-f(i div 2)) + f(i div 2) * f(i div 2) = 1\"\n using f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) assms(1) (*‹(i::nat) < dim_row U⇩f›*) by (metis (no_types, lifting) Nat.minus_nat.diff_0 (*‹?m - 0 = ?m›*) diff_add_0 (*‹?n - (?n + ?m) = 0›*) diff_add_inverse (*‹?n + ?m - ?n = ?m›*) jozsa_transform_dim( (*‹dim_row U⇩f = 2 ^ (n + 1)›*) 1) less_power_add_imp_div_less (*‹?i < 2 ^ (?m + ?n) ⟹ ?i div 2 ^ ?n < 2 ^ ?m›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) mult_eq_if (*‹?m * ?n = (if ?m = 0 then 0 else ?n + (?m - 1) * ?n)›*) one_power2 (*‹1² = 1›*) power2_eq_square (*‹?a² = ?a * ?a›*) power_one_right (*‹?a ^ 1 = ?a›*))\n ultimately show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n by (metis assms( (*‹j < dim_col U⇩f›*) 2) a0 (*‹j = i›*) index_one_mat( (*‹⟦?i < ?n; ?j < ?n⟧ ⟹ 1⇩m ?n $$ (?i, ?j) = (if ?i = ?j then 1 else 0)›*) 1) of_nat_1 (*‹of_nat 1 = 1›*))\n next\n (*goals:\n 1. ‹⟦j = i + 1 ∧ odd j; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j) + complex_of_nat (f (i div 2)) * U⇩f $$ (i + 1, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 2. ‹⟦j = i - 1 ∧ even j ∧ 1 ≤ i; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j) + complex_of_nat (f (i div 2)) * U⇩f $$ (i + 1, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 3. ‹⟦j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j) + complex_of_nat (f (i div 2)) * U⇩f $$ (i + 1, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume a0: \"(j=i+1 ∧ odd j)\" (*‹(j::nat) = (i::nat) + (1::nat) ∧ odd j›*)\n then have \"U⇩f $$ (i,j) = f(i div 2)\"\n using assms(1,2) (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f›*) a0 (*‹(j::nat) = (i::nat) + (1::nat) ∧ odd j›*) by simp\n moreover have \"U⇩f $$ (i+1,j) = (1-f(i div 2))\"\n using assms(2,3) (*‹j < dim_col U⇩f› ‹even (i::nat)›*) a0 (*‹(j::nat) = (i::nat) + (1::nat) ∧ odd j›*) by simp\n ultimately have \"(U⇩f * U⇩f) $$ (i,j) = (1-f(i div 2)) * f(i div 2) + f(i div 2) * (1-f(i div 2))\"\n using f0 (*‹U⇩f $$ (i, i + 1) * U⇩f $$ (i + 1, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i + 1, j)›*) f1 (*‹(U⇩f * U⇩f) $$ (i, j) = complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j) + complex_of_nat (f (i div 2)) * U⇩f $$ (i + 1, j)›*) assms (*‹(i::nat) < dim_row U⇩f› ‹j < dim_col U⇩f› ‹even i›*) by simp\n then show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n using assms(1,2) (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f›*) a0 (*‹(j::nat) = (i::nat) + (1::nat) ∧ odd j›*) by auto\n next\n (*goals:\n 1. ‹⟦j = i - 1 ∧ even j ∧ 1 ≤ i; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j) + complex_of_nat (f (i div 2)) * U⇩f $$ (i + 1, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 2. ‹⟦j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j) + complex_of_nat (f (i div 2)) * U⇩f $$ (i + 1, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume \"(j=i-1 ∧ even j ∧ i≥1)\" (*‹(j::nat) = (i::nat) - (1::nat) ∧ even j ∧ (1::nat) ≤ i›*)\n then show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n using assms(3) (*‹even i›*) dvd_diffD1 (*‹⟦(?k::nat) dvd (?m::nat) - (?n::nat); ?k dvd ?m; ?n ≤ ?m⟧ ⟹ ?k dvd ?n›*) odd_one (*‹odd (1::?'a)›*) by blast\n next\n (*goal: ‹⟦(j::nat) ≠ (i::nat) ∧ ¬ (j = i + (1::nat) ∧ odd j) ∧ ¬ (j = i - (1::nat) ∧ even j ∧ (1::nat) ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume a0: \"(j≠i ∧ ¬(j=i+1 ∧ odd j) ∧ ¬ (j=i-1 ∧ even j ∧ i≥1))\" (*‹(j::nat) ≠ (i::nat) ∧ ¬ (j = i + (1::nat) ∧ odd j) ∧ ¬ (j = i - (1::nat) ∧ even j ∧ (1::nat) ≤ i)›*)\n then have \"U⇩f $$ (i,j) = 0\"\n using assms(1,2) (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f›*) by (metis index_transpose_mat( (*‹⟦?i < dim_col ?A; ?j < dim_row ?A⟧ ⟹ ?A⇧t $$ (?i, ?j) = ?A $$ (?j, ?i)›*) 1) jozsa_transform_coeff_is_zero (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i ≠ ?j ∧ ¬ (?i = ?j + 1 ∧ odd ?i) ∧ ¬ (?i = ?j - 1 ∧ even ?i ∧ 1 ≤ ?j) ⟶ U⇩f $$ (?i, ?j) = 0›*) jozsa_transform_dim (*‹dim_row U⇩f = 2 ^ (n + 1)› ‹dim_col U⇩f = 2 ^ (n + 1)›*) transpose_of_jozsa_transform (*‹U⇩f⇧t = U⇩f›*))\n moreover have \"U⇩f $$ (i+1,j) = 0\"\n using assms (*‹i < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f› ‹even i›*) a0 (*‹j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i)›*) by auto\n ultimately have \"(U⇩f * U⇩f) $$ (i,j) = (1-f(i div 2)) * 0 + f(i div 2) * 0\"\n by (simp add: f1 (*‹(U⇩f * U⇩f) $$ (i::nat, j::nat) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i, j) + complex_of_nat (f (i div (2::nat))) * U⇩f $$ (i + (1::nat), j)›*))\n then show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n using a0 (*‹j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i)›*) assms(1,2) (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f›*) by (metis add.left_neutral (*‹0 + ?a = ?a›*) index_one_mat( (*‹⟦?i < ?n; ?j < ?n⟧ ⟹ 1⇩m ?n $$ (?i, ?j) = (if ?i = ?j then 1 else 0)›*) 1) jozsa_transform_dim (*‹dim_row U⇩f = 2 ^ (n + 1)› ‹dim_col U⇩f = 2 ^ (n + 1)›*) mult_0_right (*‹?m * 0 = 0›*) of_nat_0 (*‹of_nat 0 = 0›*))\n qed\nqed\n\nlemma (in jozsa) jozsa_transform_is_unitary_index_odd:\n fixes i j:: nat\n assumes \"i < dim_row U⇩f\" and \"j < dim_col U⇩f\" and \"odd i\"\n shows \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\nproof (-)\n (*goal: ‹(U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n have f0: \"i ≥ 1\"\n using linorder_not_less (*‹(¬ (?x::?'a) < (?y::?'a)) = (?y ≤ ?x)›*) assms(3) (*‹odd i›*) by auto\n have \"(U⇩f * U⇩f) $$ (i,j) = (∑k ∈ {i-1,i}. U⇩f $$ (i,k) * U⇩f $$ (k,j)) \"\n using \"U⇩f_mult_without_empty_summands_odd\"[of i j U⇩f] (*‹⟦i < dim_row U⇩f; j < dim_col U⇩f; odd i; dim_col U⇩f = dim_row U⇩f⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = (∑k∈{i - 1, i}. U⇩f $$ (i, k) * U⇩f $$ (k, j))›*) assms (*‹i < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f› ‹odd i›*) by simp\n moreover have \"(∑k ∈ {i-1,i}. U⇩f $$ (i,k) * U⇩f $$ (k,j)) \n = U⇩f $$ (i,i-1) * U⇩f $$ (i-1,j) + U⇩f $$ (i,i) * U⇩f $$ (i,j)\"\n using f0 (*‹1 ≤ i›*) by simp\n moreover have \"U⇩f $$ (i,i) * U⇩f $$ (i,j) = (1-f(i div 2)) * U⇩f $$ (i,j)\"\n using assms(1,2) (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f›*) by simp\n moreover have f1: \"U⇩f $$ (i,i-1) * U⇩f $$ (i-1,j) = f(i div 2) * U⇩f $$ (i-1,j)\"\n using assms(1) (*‹i < dim_row U⇩f›*) assms(3) (*‹odd i›*) by simp\n ultimately have f2: \"(U⇩f * U⇩f) $$ (i,j) = f(i div 2) * U⇩f $$ (i-1,j) + (1-f(i div 2)) * U⇩f $$ (i,j)\"\n by simp\n then show \"?thesis\"\n (*goal: ‹(U⇩f * U⇩f) $$ (i::nat, j::nat) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n proof (induct rule: disj_four_cases (*‹⟦?A ∨ ?B ∨ ?C ∨ ?D; ?A ⟹ ?P; ?B ⟹ ?P; ?C ⟹ ?P; ?D ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹?A2 ∨ ?B2 ∨ ?C2 ∨ ?D2›\n 2. ‹⟦?A2; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 3. ‹⟦?B2; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 4. ‹⟦?C2; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 5. ‹⟦?D2; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n show \"j=i ∨ (j=i+1 ∧ odd j) ∨ (j=i-1 ∧ even j ∧ i≥1) ∨ (j≠i ∧ ¬(j=i+1 ∧ odd j) ∧ ¬ (j=i-1 ∧ even j ∧ i≥1))\"\n by linarith\n next\n (*goals:\n 1. ‹⟦j = i; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 2. ‹⟦j = i + 1 ∧ odd j; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 3. ‹⟦j = i - 1 ∧ even j ∧ 1 ≤ i; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 4. ‹⟦j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume a0: \"j=i\" (*‹(j::nat) = (i::nat)›*)\n then have \"U⇩f $$ (i,j) = (1-f(i div 2))\"\n using assms(1,2) (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f›*) by simp\n moreover have \"U⇩f $$ (i-1,j) = f(i div 2)\"\n using a0 (*‹j = i›*) assms (*‹(i::nat) < dim_row U⇩f› ‹j < dim_col U⇩f› ‹odd i›*) by (metis index_transpose_mat( (*‹⟦?i < dim_col ?A; ?j < dim_row ?A⟧ ⟹ ?A⇧t $$ (?i, ?j) = ?A $$ (?j, ?i)›*) 1) jozsa_transform_coeff( (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i = ?j + 1 ∧ odd ?i ⟶ U⇩f $$ (?i, ?j) = complex_of_nat (f (?i div 2))›*) 2) less_imp_diff_less (*‹?j < ?k ⟹ ?j - ?n < ?k›*) odd_two_times_div_two_nat (*‹odd ?n ⟹ 2 * (?n div 2) = ?n - 1›*) odd_two_times_div_two_succ (*‹odd ?a ⟹ 2 * (?a div 2) + 1 = ?a›*) transpose_of_jozsa_transform (*‹U⇩f⇧t = U⇩f›*))\n ultimately have \"(U⇩f * U⇩f) $$ (i,j) = f(i div 2) * f(i div 2) + (1-f(i div 2)) * (1-f(i div 2))\"\n using f2 (*‹(U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)›*) by simp\n moreover have \"f(i div 2) * f(i div 2) + (1-f(i div 2)) * (1-f(i div 2)) = 1\"\n using f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) assms(1) (*‹i < dim_row U⇩f›*) by (metis (no_types, lifting) Nat.minus_nat.diff_0 (*‹?m - 0 = ?m›*) diff_add_0 (*‹?n - (?n + ?m) = 0›*) diff_add_inverse (*‹?n + ?m - ?n = ?m›*) jozsa_transform_dim( (*‹dim_row U⇩f = 2 ^ (n + 1)›*) 1) less_power_add_imp_div_less (*‹?i < 2 ^ (?m + ?n) ⟹ ?i div 2 ^ ?n < 2 ^ ?m›*) mem_Collect_eq (*‹(?a ∈ Collect ?P) = ?P ?a›*) mult_eq_if (*‹?m * ?n = (if ?m = 0 then 0 else ?n + (?m - 1) * ?n)›*) one_power2 (*‹1² = 1›*) power2_eq_square (*‹?a² = ?a * ?a›*) power_one_right (*‹?a ^ 1 = ?a›*))\n ultimately show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n by (metis assms( (*‹j < dim_col U⇩f›*) 2) a0 (*‹j = i›*) index_one_mat( (*‹⟦?i < ?n; ?j < ?n⟧ ⟹ 1⇩m ?n $$ (?i, ?j) = (if ?i = ?j then 1 else 0)›*) 1) of_nat_1 (*‹of_nat 1 = 1›*))\n next\n (*goals:\n 1. ‹⟦(j::nat) = (i::nat) + (1::nat) ∧ odd j; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i - (1::nat), j) + complex_of_nat ((1::nat) - f (i div (2::nat))) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 2. ‹⟦(j::nat) = (i::nat) - (1::nat) ∧ even j ∧ (1::nat) ≤ i; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i - (1::nat), j) + complex_of_nat ((1::nat) - f (i div (2::nat))) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 3. ‹⟦(j::nat) ≠ (i::nat) ∧ ¬ (j = i + (1::nat) ∧ odd j) ∧ ¬ (j = i - (1::nat) ∧ even j ∧ (1::nat) ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i - (1::nat), j) + complex_of_nat ((1::nat) - f (i div (2::nat))) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume a0: \"(j=i+1 ∧ odd j)\" (*‹(j::nat) = (i::nat) + (1::nat) ∧ odd j›*)\n then show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n using assms(3) (*‹odd i›*) dvd_diffD1 (*‹⟦?k dvd ?m - ?n; ?k dvd ?m; ?n ≤ ?m⟧ ⟹ ?k dvd ?n›*) odd_one (*‹odd 1›*) even_plus_one_iff (*‹even (?a + 1) = odd ?a›*) by blast\n next\n (*goals:\n 1. ‹⟦(j::nat) = (i::nat) - (1::nat) ∧ even j ∧ (1::nat) ≤ i; (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i - (1::nat), j) + complex_of_nat ((1::nat) - f (i div (2::nat))) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 2. ‹⟦(j::nat) ≠ (i::nat) ∧ ¬ (j = i + (1::nat) ∧ odd j) ∧ ¬ (j = i - (1::nat) ∧ even j ∧ (1::nat) ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat ((f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i - (1::nat), j) + complex_of_nat ((1::nat) - f (i div (2::nat))) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume a0: \"(j=i-1 ∧ even j ∧ i≥1)\" (*‹(j::nat) = (i::nat) - (1::nat) ∧ even j ∧ (1::nat) ≤ i›*)\n then have \"(U⇩f * U⇩f) $$ (i,j) = f(i div 2) * (1-f(i div 2)) + (1-f(i div 2)) * f(i div 2)\"\n using f0 (*‹1 ≤ i›*) f1 (*‹U⇩f $$ (i, i - 1) * U⇩f $$ (i - 1, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j)›*) f2 (*‹(U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)›*) assms (*‹i < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f› ‹odd i›*) by (metis jozsa_transform_coeff( (*‹(?i::nat) < dim_row U⇩f ∧ (?j::nat) < dim_col U⇩f ⟹ ?i = ?j ⟶ U⇩f $$ (?i, ?j) = complex_of_nat ((1::nat) - (f::nat ⇒ nat) (?i div (2::nat)))›*) 1) Groups.ab_semigroup_mult_class.mult.commute (*‹(?a::?'a::ab_semigroup_mult) * (?b::?'a::ab_semigroup_mult) = ?b * ?a›*) even_succ_div_two (*‹even (?a::?'a::linordered_euclidean_semiring) ⟹ (?a + (1::?'a::linordered_euclidean_semiring)) div (2::?'a::linordered_euclidean_semiring) = ?a div (2::?'a::linordered_euclidean_semiring)›*) f2 (*‹(U⇩f * U⇩f) $$ (i::nat, j::nat) = complex_of_nat ((f::nat ⇒ nat) (i div (2::nat))) * U⇩f $$ (i - (1::nat), j) + complex_of_nat ((1::nat) - f (i div (2::nat))) * U⇩f $$ (i, j)›*) jozsa_transform_dim (*‹dim_row U⇩f = (2::nat) ^ ((n::nat) + (1::nat))› ‹dim_col U⇩f = (2::nat) ^ ((n::nat) + (1::nat))›*) odd_two_times_div_two_nat (*‹odd (?n::nat) ⟹ (2::nat) * (?n div (2::nat)) = ?n - (1::nat)›*) odd_two_times_div_two_succ (*‹odd (?a::?'a::linordered_euclidean_semiring) ⟹ (2::?'a::linordered_euclidean_semiring) * (?a div (2::?'a::linordered_euclidean_semiring)) + (1::?'a::linordered_euclidean_semiring) = ?a›*) of_nat_add (*‹of_nat ((?m::nat) + (?n::nat)) = of_nat ?m + of_nat ?n›*) of_nat_mult (*‹of_nat ((?m::nat) * (?n::nat)) = of_nat ?m * of_nat ?n›*))\n then show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n using assms(1) (*‹i < dim_row U⇩f›*) a0 (*‹(j::nat) = (i::nat) - (1::nat) ∧ even j ∧ (1::nat) ≤ i›*) by auto\n next\n (*goal: ‹⟦j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i); (U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›*)\n assume a0: \"j≠i ∧ ¬(j=i+1 ∧ odd j) ∧ ¬ (j=i-1 ∧ even j ∧ i≥1)\" (*‹(j::nat) ≠ (i::nat) ∧ ¬ (j = i + (1::nat) ∧ odd j) ∧ ¬ (j = i - (1::nat) ∧ even j ∧ (1::nat) ≤ i)›*)\n then have \"U⇩f $$ (i,j) = 0\"\n by (metis assms( (*‹(i::nat) < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f›*) 1,2) index_transpose_mat( (*‹⟦(?i::nat) < dim_col (?A::?'a Matrix.mat); (?j::nat) < dim_row ?A⟧ ⟹ ?A⇧t $$ (?i, ?j) = ?A $$ (?j, ?i)›*) 1) jozsa_transform_coeff_is_zero (*‹(?i::nat) < dim_row U⇩f ∧ (?j::nat) < dim_col U⇩f ⟹ ?i ≠ ?j ∧ ¬ (?i = ?j + (1::nat) ∧ odd ?i) ∧ ¬ (?i = ?j - (1::nat) ∧ even ?i ∧ (1::nat) ≤ ?j) ⟶ U⇩f $$ (?i, ?j) = (0::complex)›*) jozsa_transform_dim (*‹dim_row U⇩f = (2::nat) ^ ((n::nat) + (1::nat))› ‹dim_col U⇩f = (2::nat) ^ ((n::nat) + (1::nat))›*) transpose_of_jozsa_transform (*‹U⇩f⇧t = U⇩f›*))\n moreover have \"U⇩f $$ (i-1,j) = 0\"\n using assms (*‹i < dim_row U⇩f› ‹j < dim_col U⇩f› ‹odd (i::nat)›*) a0 (*‹j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i)›*) f0 (*‹(1::nat) ≤ (i::nat)›*) apply auto\n (*goal: ‹U⇩f $$ (i - 1, j) = 0›*)\n by (smt (verit) One_nat_def (*‹1 = Suc 0›*) Suc_n_not_le_n (*‹¬ Suc ?n ≤ ?n›*) add_diff_inverse_nat (*‹¬ ?m < ?n ⟹ ?n + (?m - ?n) = ?m›*) assms( (*‹i < dim_row U⇩f›*) 1) assms( (*‹j < dim_col U⇩f›*) 2) diff_Suc_less (*‹0 < ?n ⟹ ?n - Suc ?i < ?n›*) even_add (*‹even (?a + ?b) = (even ?a = even ?b)›*) jozsa_transform_coeff_is_zero (*‹?i < dim_row U⇩f ∧ ?j < dim_col U⇩f ⟹ ?i ≠ ?j ∧ ¬ (?i = ?j + 1 ∧ odd ?i) ∧ ¬ (?i = ?j - 1 ∧ even ?i ∧ 1 ≤ ?j) ⟶ U⇩f $$ (?i, ?j) = 0›*) jozsa_axioms (*‹jozsa f n›*) less_imp_le (*‹?x < ?y ⟹ ?x ≤ ?y›*) less_le_trans (*‹⟦?x < ?y; ?y ≤ ?z⟧ ⟹ ?x < ?z›*) less_one (*‹(?n < 1) = (?n = 0)›*) odd_one (*‹odd 1›*))\n ultimately have \"(U⇩f * U⇩f) $$ (i,j) = (1-f(i div 2)) * 0 + f(i div 2) * 0\"\n using f2 (*‹(U⇩f * U⇩f) $$ (i, j) = complex_of_nat (f (i div 2)) * U⇩f $$ (i - 1, j) + complex_of_nat (1 - f (i div 2)) * U⇩f $$ (i, j)›*) by simp\n then show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n using a0 (*‹j ≠ i ∧ ¬ (j = i + 1 ∧ odd j) ∧ ¬ (j = i - 1 ∧ even j ∧ 1 ≤ i)›*) assms (*‹i < dim_row U⇩f› ‹(j::nat) < dim_col U⇩f› ‹odd i›*) by (metis add.left_neutral (*‹0 + ?a = ?a›*) index_one_mat( (*‹⟦?i < ?n; ?j < ?n⟧ ⟹ 1⇩m ?n $$ (?i, ?j) = (if ?i = ?j then 1 else 0)›*) 1) jozsa_transform_dim (*‹dim_row U⇩f = 2 ^ (n + 1)› ‹dim_col U⇩f = 2 ^ (n + 1)›*) mult_0_right (*‹?m * 0 = 0›*) of_nat_0 (*‹of_nat 0 = 0›*))\n qed\nqed\n\nlemma (in jozsa) jozsa_transform_is_gate:\n shows \"gate (n+1) U⇩f\"\nproof (standard)\n (*goals:\n 1. ‹dim_row U⇩f = 2 ^ (n + 1)›\n 2. ‹square_mat U⇩f›\n 3. ‹unitary U⇩f›*)\n show \"dim_row U⇩f = 2^(n+1)\"\n by simp\nnext\n (*goals:\n 1. ‹square_mat U⇩f›\n 2. ‹unitary U⇩f›*)\n show \"square_mat U⇩f\"\n by simp\nnext\n (*goal: ‹unitary U⇩f›*)\n show \"unitary U⇩f\"\n proof (-)\n (*goal: ‹unitary U⇩f›*)\n have \"U⇩f * U⇩f = 1⇩m (dim_col U⇩f)\"\n proof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (1⇩m (dim_col U⇩f)); j < dim_col (1⇩m (dim_col U⇩f))⟧ ⟹ (U⇩f * U⇩f) $$ (i, j) = 1⇩m (dim_col U⇩f) $$ (i, j)›\n 2. ‹dim_row (U⇩f * U⇩f) = dim_row (1⇩m (dim_col U⇩f))›\n 3. ‹dim_col (U⇩f * U⇩f) = dim_col (1⇩m (dim_col U⇩f))›*)\n show \"dim_row (U⇩f * U⇩f) = dim_row (1⇩m (dim_col U⇩f))\"\n by simp\n show \"dim_col (U⇩f * U⇩f) = dim_col (1⇩m (dim_col U⇩f))\"\n by simp\n fix i :: nat and j :: nat\n assume \"i < dim_row (1⇩m (dim_col U⇩f))\" and \"j < dim_col (1⇩m (dim_col U⇩f))\" (*‹(i::nat) < dim_row (1⇩m (dim_col U⇩f))› ‹(j::nat) < dim_col (1⇩m (dim_col U⇩f))›*)\n then have \"i < dim_row U⇩f\" and \"j < dim_col U⇩f\"\n apply -\n (*goals:\n 1. ‹⟦i < dim_row (1⇩m (dim_col U⇩f)); j < dim_col (1⇩m (dim_col U⇩f))⟧ ⟹ i < dim_row U⇩f›\n 2. ‹⟦i < dim_row (1⇩m (dim_col U⇩f)); j < dim_col (1⇩m (dim_col U⇩f))⟧ ⟹ j < dim_col U⇩f›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n then show \"(U⇩f * U⇩f) $$ (i,j) = 1⇩m (dim_col U⇩f) $$ (i,j)\"\n using jozsa_transform_is_unitary_index_odd (*‹⟦(?i::nat) < dim_row U⇩f; (?j::nat) < dim_col U⇩f; odd ?i⟧ ⟹ (U⇩f * U⇩f) $$ (?i, ?j) = 1⇩m (dim_col U⇩f) $$ (?i, ?j)›*) jozsa_transform_is_unitary_index_even (*‹⟦?i < dim_row U⇩f; ?j < dim_col U⇩f; even ?i⟧ ⟹ (U⇩f * U⇩f) $$ (?i, ?j) = 1⇩m (dim_col U⇩f) $$ (?i, ?j)›*) by blast\n qed\n thus \"?thesis\"\n (*goal: ‹unitary U⇩f›*)\n by (simp add: adjoint_of_jozsa_transform (*‹U⇩f⇧† = U⇩f›*) unitary_def (*‹unitary (?M::complex Matrix.mat) ≡ ?M⇧† * ?M = 1⇩m (dim_col ?M) ∧ ?M * ?M⇧† = 1⇩m (dim_row ?M)›*))\n qed\nqed\n\ntext ‹N-fold application of the tensor product›\n\nfun iter_tensor:: \"complex Matrix.mat ⇒ nat ⇒ complex Matrix.mat\" (\"_ ⊗⇗_⇖\" 75) where\n \"A ⊗⇗(Suc 0)⇖ = A\" \n| \"A ⊗⇗(Suc k)⇖ = A ⨂ (A ⊗⇗k⇖)\"\n\nlemma one_tensor_is_id [simp]:\n fixes A\n shows \"A ⊗⇗1⇖ = A\"\n using one_mat_def (*‹1⇩m (?n::nat) ≡ Matrix.mat ?n ?n (λ(i::nat, j::nat). if i = j then 1::?'a else (0::?'a))›*) by simp\n\nlemma iter_tensor_suc: \n fixes n\n assumes \"n ≥ 1\"\n shows \" A ⊗⇗(Suc n)⇖ = A ⨂ (A ⊗⇗n⇖)\" \n using assms (*‹1 ≤ n›*) by (metis Deutsch_Jozsa.iter_tensor.simps( (*‹?A ⊗⇗Suc (Suc ?v)⇖ = ?A ⨂ ?A ⊗⇗Suc ?v⇖›*) 2) One_nat_def (*‹1 = Suc 0›*) Suc_le_D (*‹Suc ?n ≤ ?m' ⟹ ∃m. ?m' = Suc m›*))\n\nlemma dim_row_of_iter_tensor [simp]:\n fixes A n\n assumes \"n ≥ 1\"\n shows \"dim_row(A ⊗⇗n⇖) = (dim_row A)^n\"\n using assms (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦?m ≤ ?n; ?P ?m; ⋀n. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹dim_row (A ⊗⇗1⇖) = dim_row A ^ 1›\n 2. ‹⋀n. ⟦1 ≤ n; dim_row (A ⊗⇗n⇖) = dim_row A ^ n⟧ ⟹ dim_row (A ⊗⇗Suc n⇖) = dim_row A ^ Suc n›*)\n show \"dim_row (A ⊗⇗1⇖) = (dim_row A)^1\"\n using one_tensor_is_id (*‹?A ⊗⇗1⇖ = ?A›*) by simp\n fix n :: nat\n assume \"n ≥ 1\" and \"dim_row (A ⊗⇗n⇖) = (dim_row A)^n\" (*‹(1::nat) ≤ (n::nat)› ‹dim_row (A::complex Matrix.mat ⊗⇗n::nat⇖) = dim_row A ^ n›*)\n then show \"dim_row (A ⊗⇗Suc n⇖) = (dim_row A)^Suc n\"\n using iter_tensor_suc (*‹1 ≤ ?n ⟹ ?A ⊗⇗Suc ?n⇖ = ?A ⨂ ?A ⊗⇗?n⇖›*) assms (*‹1 ≤ n›*) dim_row_tensor_mat (*‹dim_row (?A ⨂ ?B) = dim_row ?A * dim_row ?B›*) by simp\n qed\n\nlemma dim_col_of_iter_tensor [simp]:\n fixes A n\n assumes \"n ≥ 1\"\n shows \"dim_col(A ⊗⇗n⇖) = (dim_col A)^n\"\n using assms (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦?m ≤ ?n; ?P ?m; ⋀n. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹dim_col (A ⊗⇗1⇖) = dim_col A ^ 1›\n 2. ‹⋀n. ⟦1 ≤ n; dim_col (A ⊗⇗n⇖) = dim_col A ^ n⟧ ⟹ dim_col (A ⊗⇗Suc n⇖) = dim_col A ^ Suc n›*)\n show \"dim_col (A ⊗⇗1⇖) = (dim_col A)^1\"\n using one_tensor_is_id (*‹?A ⊗⇗1⇖ = ?A›*) by simp\n fix n :: nat\n assume \"n ≥ 1\" and \"dim_col (A ⊗⇗n⇖) = (dim_col A)^n\" (*‹(1::nat) ≤ (n::nat)› ‹dim_col (A::complex Matrix.mat ⊗⇗n::nat⇖) = dim_col A ^ n›*)\n then show \"dim_col (A ⊗⇗Suc n⇖) = (dim_col A)^Suc n\"\n using iter_tensor_suc (*‹1 ≤ ?n ⟹ ?A ⊗⇗Suc ?n⇖ = ?A ⨂ ?A ⊗⇗?n⇖›*) assms (*‹(1::nat) ≤ (n::nat)›*) dim_col_tensor_mat (*‹dim_col (?A ⨂ ?B) = dim_col ?A * dim_col ?B›*) by simp\n qed\n\nlemma iter_tensor_values:\n fixes A n i j\n assumes \"n ≥ 1\" and \"i < dim_row (A ⨂ (A ⊗⇗n⇖))\" and \"j < dim_col (A ⨂ (A ⊗⇗n⇖))\"\n shows \"(A ⊗⇗(Suc n)⇖) $$ (i,j) = (A ⨂ (A ⊗⇗n⇖)) $$ (i,j)\"\n using assms (*‹1 ≤ n› ‹(i::nat) < dim_row ((A::complex Matrix.mat) ⨂ A ⊗⇗n::nat⇖)› ‹(j::nat) < dim_col ((A::complex Matrix.mat) ⨂ A ⊗⇗n::nat⇖)›*) by (metis One_nat_def (*‹1 = Suc 0›*) le_0_eq (*‹(?n ≤ 0) = (?n = 0)›*) not0_implies_Suc (*‹?n ≠ 0 ⟹ ∃m. ?n = Suc m›*) iter_tensor.simps( (*‹?A ⊗⇗Suc (Suc ?v)⇖ = ?A ⨂ ?A ⊗⇗Suc ?v⇖›*) 2))\n\nlemma iter_tensor_mult_distr:\n assumes \"n ≥ 1\" and \"dim_col A = dim_row B\" and \"dim_col A > 0\" and \"dim_col B > 0\"\n shows \"(A ⊗⇗(Suc n)⇖) * (B ⊗⇗(Suc n)⇖) = (A * B) ⨂ ((A ⊗⇗n⇖) * (B ⊗⇗n⇖))\" \nproof (-)\n (*goal: ‹A::complex Matrix.mat ⊗⇗Suc (n::nat)⇖ * B::complex Matrix.mat ⊗⇗Suc n⇖ = A * B ⨂ A ⊗⇗n⇖ * B ⊗⇗n⇖›*)\n have \"(A ⊗⇗(Suc n)⇖) * (B ⊗⇗(Suc n)⇖) = (A ⨂ (A ⊗⇗n⇖)) * (B ⨂ (B ⊗⇗n⇖))\"\n using Suc_le_D (*‹Suc ?n ≤ ?m' ⟹ ∃m. ?m' = Suc m›*) assms(1) (*‹1 ≤ n›*) by fastforce\n then show \"?thesis\"\n (*goal: ‹A::complex Matrix.mat ⊗⇗Suc (n::nat)⇖ * B::complex Matrix.mat ⊗⇗Suc n⇖ = A * B ⨂ A ⊗⇗n⇖ * B ⊗⇗n⇖›*)\n using mult_distr_tensor[of \"A\" \"B\" \"(iter_tensor A n)\" \"(iter_tensor B n)\"] (*‹⟦dim_col A = dim_row B; dim_col (A ⊗⇗n⇖) = dim_row (B ⊗⇗n⇖); 0 < dim_col A; 0 < dim_col B; 0 < dim_col (A ⊗⇗n⇖); 0 < dim_col (B ⊗⇗n⇖)⟧ ⟹ A * B ⨂ A ⊗⇗n⇖ * B ⊗⇗n⇖ = (A ⨂ A ⊗⇗n⇖) * (B ⨂ B ⊗⇗n⇖)›*) assms (*‹1 ≤ n› ‹dim_col A = dim_row B› ‹0 < dim_col A› ‹0 < dim_col B›*) by simp\nqed\n\nlemma index_tensor_mat_with_vec2_row_cond:\n fixes A B:: \"complex Matrix.mat\" and i:: \"nat\" \n assumes \"i < 2 * (dim_row B)\" and \"i ≥ dim_row B\" and \"dim_col B > 0\"\nand \"dim_row A = 2\" and \"dim_col A = 1\"\n shows \"(A ⨂ B) $$ (i,0) = (A $$ (1,0)) * (B $$ (i-dim_row B,0))\"\nproof (-)\n (*goal: ‹(A ⨂ B) $$ (i, 0) = A $$ (1, 0) * B $$ (i - dim_row B, 0)›*)\n have \"(A ⨂ B) $$ (i,0) = A $$ (i div (dim_row B),0) * B $$ (i mod (dim_row B),0)\"\n using assms (*‹i < 2 * dim_row B› ‹dim_row B ≤ i› ‹0 < dim_col B› ‹dim_row (A::complex Matrix.mat) = (2::nat)› ‹dim_col (A::complex Matrix.mat) = (1::nat)›*) index_tensor_mat[of A \"dim_row A\" \"dim_col A\" B \"dim_row B\" \"dim_col B\" i 0] (*‹⟦dim_row A = dim_row A; dim_col A = dim_col A; dim_row B = dim_row B; dim_col B = dim_col B; i < dim_row A * dim_row B; 0 < dim_col A * dim_col B; 0 < dim_col A; 0 < dim_col B⟧ ⟹ (A ⨂ B) $$ (i, 0) = A $$ (i div dim_row B, 0 div dim_col B) * B $$ (i mod dim_row B, 0 mod dim_col B)›*) by simp\n moreover have \"i div (dim_row B) = 1\"\n using assms(1,2,4) (*‹i < 2 * dim_row B› ‹dim_row B ≤ i› ‹dim_row A = 2›*) by simp\n then have \"i mod (dim_row B) = i - (dim_row B)\"\n by (simp add: modulo_nat_def (*‹?m mod ?n = ?m - ?m div ?n * ?n›*))\n ultimately show \"(A ⨂ B) $$ (i,0) = (A $$ (1,0)) * (B $$ (i-dim_row B,0))\"\n by (simp add: ‹i div dim_row B = 1›)\nqed\n\nlemma iter_tensor_of_gate_is_gate:\n fixes A:: \"complex Matrix.mat\" and n m:: \"nat\" \n assumes \"gate m A\" and \"n ≥ 1\" \n shows \"gate (m*n) (A ⊗⇗n⇖)\"\n using assms(2) (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦?m ≤ ?n; ?P ?m; ⋀n. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹gate (m * 1) (A ⊗⇗1⇖)›\n 2. ‹⋀n. ⟦1 ≤ n; gate (m * n) (A ⊗⇗n⇖)⟧ ⟹ gate (m * Suc n) (A ⊗⇗Suc n⇖)›*)\n show \"gate (m * 1) (A ⊗⇗1⇖)\"\n using assms(1) (*‹gate (m::nat) (A::complex Matrix.mat)›*) by simp\n fix n :: nat\n assume \"n ≥ 1\" and IH: \"gate (m * n) (A ⊗⇗n⇖)\" (*‹(1::nat) ≤ (n::nat)› ‹gate ((m::nat) * (n::nat)) (A::complex Matrix.mat ⊗⇗n⇖)›*)\n then have \"A ⊗⇗(Suc n)⇖ = A ⨂ (A ⊗⇗n⇖)\"\n by (simp add: iter_tensor_suc (*‹(1::nat) ≤ (?n::nat) ⟹ ?A::complex Matrix.mat ⊗⇗Suc ?n⇖ = ?A ⨂ ?A ⊗⇗?n⇖›*))\n moreover have \"gate (m*n + m) (A ⊗⇗(Suc n)⇖)\"\n using tensor_gate (*‹⟦gate ?m ?G1.0; gate ?n ?G2.0⟧ ⟹ gate (?m + ?n) (?G1.0 ⨂ ?G2.0)›*) assms(1) (*‹gate m A›*) by (simp add: IH (*‹gate ((m::nat) * (n::nat)) (A::complex Matrix.mat ⊗⇗n⇖)›*) add.commute (*‹(?a::?'a) + (?b::?'a) = ?b + ?a›*) calculation( (*‹A::complex Matrix.mat ⊗⇗Suc (n::nat)⇖ = A ⨂ A ⊗⇗n⇖›*) 1))\n then show \"gate (m*(Suc n)) (A ⊗⇗(Suc n)⇖)\"\n by (simp add: add.commute (*‹?a + ?b = ?b + ?a›*))\n qed\n\nlemma iter_tensor_of_state_is_state:\n fixes A:: \"complex Matrix.mat\" and n m:: \"nat\" \n assumes \"state m A\" and \"n≥1\" \n shows \"state (m*n) (A ⊗⇗n⇖)\"\n using assms(2) (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦?m ≤ ?n; ?P ?m; ⋀n. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹state (m * 1) (A ⊗⇗1⇖)›\n 2. ‹⋀n. ⟦1 ≤ n; state (m * n) (A ⊗⇗n⇖)⟧ ⟹ state (m * Suc n) (A ⊗⇗Suc n⇖)›*)\n show \"state (m * 1) (A ⊗⇗1⇖)\"\n using one_tensor_is_id (*‹?A::complex Matrix.mat ⊗⇗1::nat⇖ = ?A›*) assms(1) (*‹state m A›*) by simp\n fix n :: nat\n assume \"n ≥ 1\" and IH: \"state (m * n) (A ⊗⇗n⇖)\" (*‹(1::nat) ≤ (n::nat)› ‹state ((m::nat) * (n::nat)) (A::complex Matrix.mat ⊗⇗n⇖)›*)\n then have \"A ⊗⇗(Suc n)⇖ = A ⨂ (A ⊗⇗n⇖)\"\n by (simp add: iter_tensor_suc (*‹1 ≤ ?n ⟹ ?A ⊗⇗Suc ?n⇖ = ?A ⨂ ?A ⊗⇗?n⇖›*))\n moreover have \"state (m*n + m) (A ⊗⇗(Suc n)⇖)\"\n using tensor_gate (*‹⟦gate ?m ?G1.0; gate ?n ?G2.0⟧ ⟹ gate (?m + ?n) (?G1.0 ⨂ ?G2.0)›*) assms(1) (*‹state m A›*) by (simp add: IH (*‹state (m * n) (A ⊗⇗n⇖)›*) add.commute (*‹?a + ?b = ?b + ?a›*) calculation (*‹A ⊗⇗Suc n⇖ = A ⨂ A ⊗⇗n⇖›*))\n then show \"state (m*(Suc n)) (A ⊗⇗(Suc n)⇖)\"\n by (simp add: add.commute (*‹?a + ?b = ?b + ?a›*))\n qed\n\ntext ‹\nWe prepare n+1 qubits. The first n qubits in the state $|0\\rangle$, the last one in the state \n$|1\\rangle$.\n›\n\nabbreviation ψ₁₀:: \"nat ⇒ complex Matrix.mat\" where\n\"ψ₁₀ n ≡ Matrix.mat (2^n) 1 (λ(i,j). 1/(sqrt 2)^n)\" \n\nlemma ψ₁₀_values:\n fixes i j n\n assumes \"i < dim_row (ψ₁₀ n)\" and \"j < dim_col (ψ₁₀ n)\" \n shows \"(ψ₁₀ n) $$ (i,j) = 1/(sqrt 2)^n\" \n using assms (*‹i < dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)))› ‹j < dim_col (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)))›*) 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›*) by simp\n\ntext ‹$H^{\\otimes n}$ is applied to $|0\\rangle^{\\otimes n}$.›\n\nlemma H_on_ket_zero: \n shows \"(H * |zero⟩) = ψ₁₀ 1\"\nproof (standard)\n (*goals:\n 1. ‹⋀(i::nat) j::nat. ⟦i < dim_row (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))); j < dim_col (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))))⟧ ⟹ (H * |Deutsch.zero⟩) $$ (i, j) = Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))) $$ (i, j)›\n 2. ‹dim_row (H * |Deutsch.zero⟩) = dim_row (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))))›\n 3. ‹dim_col (H * |Deutsch.zero⟩) = dim_col (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))))›*)\n fix i :: nat and j :: nat\n assume \"i < dim_row (ψ₁₀ 1)\" and \"j < dim_col (ψ₁₀ 1)\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))))› ‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))))›*)\n then have f1: \"i ∈ {0,1} ∧ j = 0\"\n by (simp add: less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*))\n then show \"(H * |zero⟩) $$ (i,j) = (ψ₁₀ 1) $$ (i,j)\"\n by (auto simp add: times_mat_def (*‹(?A::?'a Matrix.mat) * (?B::?'a Matrix.mat) ≡ Matrix.mat (dim_row ?A) (dim_col ?B) (λ(i::nat, j::nat). Matrix.row ?A i ∙ Matrix.col ?B j)›*) scalar_prod_def (*‹(?v::?'a Matrix.vec) ∙ (?w::?'a Matrix.vec) ≡ ∑i::nat = 0::nat..<dim_vec ?w. ?v $ i * ?w $ i›*) H_def (*‹H ≡ complex_of_real ((1::real) / sqrt (2::real)) ⋅⇩m Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i ≠ j then 1::complex else if i = (0::nat) then 1::complex else - (1::complex))›*) ket_vec_def (*‹|?v::complex Matrix.vec⟩ ≡ Matrix.mat (dim_vec ?v) (1::nat) (λ(i::nat, j::nat). ?v $ i)›*))\nnext\n (*goals:\n 1. ‹dim_row (H * |Deutsch.zero⟩) = dim_row (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)))›\n 2. ‹dim_col (H * |Deutsch.zero⟩) = dim_col (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)))›*)\n show \"dim_row (H * |zero⟩) = dim_row (ψ₁₀ 1)\"\n by (simp add: H_def (*‹H ≡ complex_of_real (1 / sqrt 2) ⋅⇩m Matrix.mat 2 2 (λ(i, j). if i ≠ j then 1 else if i = 0 then 1 else - 1)›*))\n show \"dim_col (H * |zero⟩) = dim_col (ψ₁₀ 1)\"\n using H_def (*‹H ≡ complex_of_real (1 / sqrt 2) ⋅⇩m Matrix.mat 2 2 (λ(i, j). if i ≠ j then 1 else if i = 0 then 1 else - 1)›*) by (simp add: ket_vec_def (*‹|?v⟩ ≡ Matrix.mat (dim_vec ?v) 1 (λ(i, j). ?v $ i)›*))\nqed\n\nlemma ψ₁₀_tensor: \n assumes \"n ≥ 1\"\n shows \"(ψ₁₀ 1) ⨂ (ψ₁₀ n) = (ψ₁₀ (Suc n))\"\nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n))); j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))⟧ ⟹ (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j)›\n 2. ‹dim_row (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) = dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))›\n 3. ‹dim_col (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) = dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))›*)\n have \"dim_row (ψ₁₀ 1) * dim_row (ψ₁₀ n) = 2^(Suc n)\"\n by simp\n then show \"dim_row ((ψ₁₀ 1) ⨂ (ψ₁₀ n)) = dim_row (ψ₁₀ (Suc n))\"\n by simp\n have \"dim_col (ψ₁₀ 1) * dim_col (ψ₁₀ n) = 1\"\n by simp\n then show \"dim_col ((ψ₁₀ 1) ⨂ (ψ₁₀ n)) = dim_col (ψ₁₀ (Suc n))\"\n by simp\nnext\n (*goal: ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n))); j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))⟧ ⟹ (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j)›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row (ψ₁₀ (Suc n))\" and a1: \"j < dim_col (ψ₁₀ (Suc n))\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ Suc (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ Suc n)))› ‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ Suc (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ Suc n)))›*)\n then have f0: \"j = 0\" and f1: \"i < 2^(Suc n)\"\n apply -\n (*goals:\n 1. ‹⟦i < dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n))); j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))⟧ ⟹ j = 0›\n 2. ‹⟦i < dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n))); j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))⟧ ⟹ i < 2 ^ Suc n›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n then have f2: \"(ψ₁₀ (Suc n)) $$ (i,j) = 1/(sqrt 2)^(Suc n)\"\n using \"ψ₁₀_values\"[of \"i\" \"(Suc n)\" \"j\"] (*‹⟦i < dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n))); j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))⟧ ⟹ Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (1 / sqrt 2 ^ Suc n)›*) a0 (*‹i < dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))›*) by simp\n show \"((ψ₁₀ 1) ⨂ (ψ₁₀ n)) $$ (i,j) = (ψ₁₀ (Suc n)) $$ (i,j)\"\n proof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹?P ∨ ?Q›\n 2. ‹?P ⟹ (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j)›\n 3. ‹?Q ⟹ (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j)›*)\n show \"i < dim_row (ψ₁₀ n) ∨ i ≥ dim_row (ψ₁₀ n)\"\n by linarith\n next\n (*goals:\n 1. ‹i < dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) ⟹ (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j)›\n 2. ‹dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) ≤ i ⟹ (Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j)›*)\n assume a2: \"i < dim_row (ψ₁₀ n)\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n)))›*)\n then have \"((ψ₁₀ 1) ⨂ (ψ₁₀ n)) $$ (i,j) = (ψ₁₀ 1) $$ (0,0) * (ψ₁₀ n) $$ (i,0)\"\n using index_tensor_mat (*‹⟦dim_row (?A::complex Matrix.mat) = (?rA::nat); dim_col ?A = (?cA::nat); dim_row (?B::complex Matrix.mat) = (?rB::nat); dim_col ?B = (?cB::nat); (?i::nat) < ?rA * ?rB; (?j::nat) < ?cA * ?cB; (0::nat) < ?cA; (0::nat) < ?cB⟧ ⟹ (?A ⨂ ?B) $$ (?i, ?j) = ?A $$ (?i div ?rB, ?j div ?cB) * ?B $$ (?i mod ?rB, ?j mod ?cB)›*) f0 (*‹j = 0›*) assms (*‹1 ≤ n›*) by simp\n also (*calculation: ‹(Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) $$ (0, 0) * Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) $$ (i, 0)›*) have \"... = 1/sqrt(2) * 1/(sqrt(2)^n)\"\n using \"ψ₁₀_values\" (*‹⟦?i < dim_row (Matrix.mat (2 ^ ?n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ ?n))); ?j < dim_col (Matrix.mat (2 ^ ?n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ ?n)))⟧ ⟹ Matrix.mat (2 ^ ?n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ ?n)) $$ (?i, ?j) = complex_of_real (1 / sqrt 2 ^ ?n)›*) a2 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n)))›*) assms (*‹1 ≤ n›*) by simp\n finally (*calculation: ‹(Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) $$ (i, j) = complex_of_real (1 / sqrt 2 * 1 / sqrt 2 ^ n)›*) show \"((ψ₁₀ 1) ⨂ (ψ₁₀ n)) $$ (i,j) = (ψ₁₀ (Suc n)) $$ (i,j)\"\n using f2 (*‹Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (1 / sqrt 2 ^ Suc n)›*) divide_divide_eq_left (*‹?a / ?b / ?c = ?a / (?b * ?c)›*) power_Suc (*‹(?a::?'a) ^ Suc (?n::nat) = ?a * ?a ^ ?n›*) by simp\n next\n (*goal: ‹dim_row (Matrix.mat ((2::nat) ^ (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n))) ≤ (i::nat) ⟹ (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))) ⨂ Matrix.mat ((2::nat) ^ n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n))) $$ (i, j::nat) = Matrix.mat ((2::nat) ^ Suc n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ Suc n)) $$ (i, j)›*)\n assume \"i ≥ dim_row (ψ₁₀ n)\" (*‹dim_row (Matrix.mat ((2::nat) ^ (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n))) ≤ (i::nat)›*)\n then have \"((ψ₁₀ 1) ⨂ (ψ₁₀ n)) $$ (i,0) = ((ψ₁₀ 1) $$ (1, 0)) * ((ψ₁₀ n) $$ ( i -dim_row (ψ₁₀ n),0))\"\n using index_tensor_mat_with_vec2_row_cond[of i \"(ψ₁₀ 1)\" \"(ψ₁₀ n)\"] (*‹⟦(i::nat) < (2::nat) * dim_row (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))); dim_row (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))) ≤ i; (0::nat) < dim_col (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))); dim_row (Matrix.mat ((2::nat) ^ (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n))) = (2::nat); dim_col (Matrix.mat ((2::nat) ^ n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n))) = (1::nat)⟧ ⟹ (Matrix.mat ((2::nat) ^ n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n)) ⨂ Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))) $$ (i, 0::nat) = Matrix.mat ((2::nat) ^ n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n)) $$ (1::nat, 0::nat) * Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))) $$ (i - dim_row (Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))), 0::nat)›*) a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ Suc (n::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))›*) f0 (*‹j = 0›*) by (metis dim_col_mat( (*‹dim_col (Matrix.mat ?nr ?nc ?f) = ?nc›*) 1) dim_row_mat( (*‹dim_row (Matrix.mat ?nr ?nc ?f) = ?nr›*) 1) index_tensor_mat_with_vec2_row_cond (*‹⟦?i < 2 * dim_row ?B; dim_row ?B ≤ ?i; 0 < dim_col ?B; dim_row ?A = 2; dim_col ?A = 1⟧ ⟹ (?A ⨂ ?B) $$ (?i, 0) = ?A $$ (1, 0) * ?B $$ (?i - dim_row ?B, 0)›*) power_Suc (*‹?a ^ Suc ?n = ?a * ?a ^ ?n›*) power_one_right (*‹?a ^ 1 = ?a›*))\n then have \"((ψ₁₀ 1) ⨂ (ψ₁₀ n)) $$ (i,0) = 1/sqrt(2) * 1/(sqrt 2)^n\"\n using \"ψ₁₀_values\"[of \"i -dim_row (ψ₁₀ n)\" \"n\" \"j\"] (*‹⟦i - dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))) < dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))); j < dim_col (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)))⟧ ⟹ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) $$ (i - dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))), j) = complex_of_real (1 / sqrt 2 ^ n)›*) a0 (*‹i < dim_row (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n)))›*) by simp\n then show \"((ψ₁₀ 1) ⨂ (ψ₁₀ n)) $$ (i,j) = (ψ₁₀ (Suc n)) $$ (i,j)\"\n using f0 (*‹j = 0›*) f1 (*‹(i::nat) < (2::nat) ^ Suc (n::nat)›*) divide_divide_eq_left (*‹?a / ?b / ?c = ?a / (?b * ?c)›*) power_Suc (*‹?a ^ Suc ?n = ?a * ?a ^ ?n›*) by simp\n qed\nqed\n\nlemma ψ₁₀_tensor_is_state:\n assumes \"n ≥ 1\"\n shows \"state n ( |zero⟩ ⊗⇗n⇖)\" \n using iter_tensor_of_state_is_state (*‹⟦state ?m ?A; 1 ≤ ?n⟧ ⟹ state (?m * ?n) (?A ⊗⇗?n⇖)›*) ket_zero_is_state (*‹state 1 |Deutsch.zero⟩›*) assms (*‹1 ≤ n›*) by fastforce\n\nlemma iter_tensor_of_H_is_gate:\n assumes \"n ≥ 1\"\n shows \"gate n (H ⊗⇗n⇖)\" \n using iter_tensor_of_gate_is_gate (*‹⟦gate (?m::nat) (?A::complex Matrix.mat); (1::nat) ≤ (?n::nat)⟧ ⟹ gate (?m * ?n) (?A ⊗⇗?n⇖)›*) H_is_gate (*‹gate 1 H›*) assms (*‹1 ≤ n›*) by fastforce\n\nlemma iter_tensor_of_H_on_zero_tensor: \n assumes \"n ≥ 1\"\n shows \"(H ⊗⇗n⇖) * ( |zero⟩ ⊗⇗n⇖) = ψ₁₀ n\"\n using assms (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦?m ≤ ?n; ?P ?m; ⋀n. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹H ⊗⇗1::nat⇖ * |Deutsch.zero⟩ ⊗⇗1::nat⇖ = Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))›\n 2. ‹⋀n::nat. ⟦(1::nat) ≤ n; H ⊗⇗n⇖ * |Deutsch.zero⟩ ⊗⇗n⇖ = Matrix.mat ((2::nat) ^ n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n))⟧ ⟹ H ⊗⇗Suc n⇖ * |Deutsch.zero⟩ ⊗⇗Suc n⇖ = Matrix.mat ((2::nat) ^ Suc n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ Suc n))›*)\n show \"(H ⊗⇗1⇖) * ( |zero⟩ ⊗⇗1⇖) = ψ₁₀ 1\"\n using H_on_ket_zero (*‹H * |Deutsch.zero⟩ = Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))›*) by simp\n next\n (*goal: ‹⋀n. ⟦1 ≤ n; H ⊗⇗n⇖ * |Deutsch.zero⟩ ⊗⇗n⇖ = Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))⟧ ⟹ H ⊗⇗Suc n⇖ * |Deutsch.zero⟩ ⊗⇗Suc n⇖ = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n))›*)\n fix n :: nat\n assume a0: \"n ≥ 1\" and IH: \"(H ⊗⇗n⇖) * ( |zero⟩ ⊗⇗n⇖) = ψ₁₀ n\" (*‹(1::nat) ≤ (n::nat)› ‹H ⊗⇗n::nat⇖ * |Deutsch.zero⟩ ⊗⇗n⇖ = Matrix.mat ((2::nat) ^ n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ n))›*)\n then have \"(H ⊗⇗(Suc n)⇖) * ( |zero⟩ ⊗⇗(Suc n)⇖) = (H * |zero⟩) ⨂ ((H ⊗⇗n⇖) * ( |zero⟩ ⊗⇗n⇖))\"\n using iter_tensor_mult_distr[of \"n\" \"H\" \"|zero⟩\"] (*‹⟦1 ≤ n; dim_col H = dim_row |Deutsch.zero⟩; 0 < dim_col H; 0 < dim_col |Deutsch.zero⟩⟧ ⟹ H ⊗⇗Suc n⇖ * |Deutsch.zero⟩ ⊗⇗Suc n⇖ = H * |Deutsch.zero⟩ ⨂ H ⊗⇗n⇖ * |Deutsch.zero⟩ ⊗⇗n⇖›*) a0 (*‹1 ≤ n›*) ket_vec_def (*‹|?v⟩ ≡ Matrix.mat (dim_vec ?v) 1 (λ(i, j). ?v $ i)›*) H_def (*‹H ≡ complex_of_real (1 / sqrt 2) ⋅⇩m Matrix.mat 2 2 (λ(i, j). if i ≠ j then 1 else if i = 0 then 1 else - 1)›*) by (simp add: H_def (*‹H ≡ complex_of_real (1 / sqrt 2) ⋅⇩m Matrix.mat 2 2 (λ(i, j). if i ≠ j then 1 else if i = 0 then 1 else - 1)›*))\n also (*calculation: ‹H ⊗⇗Suc n⇖ * |Deutsch.zero⟩ ⊗⇗Suc n⇖ = H * |Deutsch.zero⟩ ⨂ H ⊗⇗n⇖ * |Deutsch.zero⟩ ⊗⇗n⇖›*) have \"... = (H * |zero⟩) ⨂ (ψ₁₀ n)\"\n using IH (*‹H ⊗⇗n⇖ * |Deutsch.zero⟩ ⊗⇗n⇖ = Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))›*) by simp\n also (*calculation: ‹H ⊗⇗Suc n⇖ * |Deutsch.zero⟩ ⊗⇗Suc n⇖ = H * |Deutsch.zero⟩ ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))›*) have \"... = (ψ₁₀ 1) ⨂ (ψ₁₀ n)\"\n using H_on_ket_zero (*‹H * |Deutsch.zero⟩ = Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat)))›*) by simp\n also (*calculation: ‹H ⊗⇗Suc n⇖ * |Deutsch.zero⟩ ⊗⇗Suc n⇖ = Matrix.mat (2 ^ 1) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ 1)) ⨂ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))›*) have \"... = (ψ₁₀ (Suc n))\"\n using \"ψ₁₀_tensor\" (*‹(1::nat) ≤ (?n::nat) ⟹ Matrix.mat ((2::nat) ^ (1::nat)) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ (1::nat))) ⨂ Matrix.mat ((2::nat) ^ ?n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ ?n)) = Matrix.mat ((2::nat) ^ Suc ?n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ Suc ?n))›*) a0 (*‹1 ≤ n›*) by simp\n finally (*calculation: ‹H ⊗⇗Suc n⇖ * |Deutsch.zero⟩ ⊗⇗Suc n⇖ = Matrix.mat (2 ^ Suc n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ Suc n))›*) show \"(H ⊗⇗(Suc n)⇖) * ( |zero⟩ ⊗⇗(Suc n)⇖) = (ψ₁₀ (Suc n))\"\n by simp\n qed\n\nlemma ψ₁₀_is_state:\n assumes \"n ≥ 1\"\n shows \"state n (ψ₁₀ n)\"\n using iter_tensor_of_H_is_gate (*‹1 ≤ ?n ⟹ gate ?n (H ⊗⇗?n⇖)›*) \"ψ₁₀_tensor_is_state\" (*‹1 ≤ ?n ⟹ state ?n (|Deutsch.zero⟩ ⊗⇗?n⇖)›*) assms (*‹1 ≤ n›*) gate_on_state_is_state (*‹⟦gate ?n ?A; state ?n ?v⟧ ⟹ state ?n (?A * ?v)›*) iter_tensor_of_H_on_zero_tensor (*‹(1::nat) ≤ (?n::nat) ⟹ H ⊗⇗?n⇖ * |Deutsch.zero⟩ ⊗⇗?n⇖ = Matrix.mat ((2::nat) ^ ?n) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (1::real) / sqrt (2::real) ^ ?n))›*) assms (*‹1 ≤ n›*) by metis\n\nabbreviation ψ₁₁:: \"complex Matrix.mat\" where\n\"ψ₁₁ ≡ Matrix.mat 2 1 (λ(i,j). if i=0 then 1/sqrt(2) else -1/sqrt(2))\"\n\nlemma H_on_ket_one_is_ψ₁₁: \n shows \"(H * |one⟩) = ψ₁₁\"\nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))); j < dim_col (Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)))⟧ ⟹ (H * |Deutsch.one⟩) $$ (i, j) = Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)) $$ (i, j)›\n 2. ‹dim_row (H * |Deutsch.one⟩) = dim_row (Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)))›\n 3. ‹dim_col (H * |Deutsch.one⟩) = dim_col (Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)))›*)\n fix i :: nat and j :: nat\n assume \"i < dim_row ψ₁₁\" and \"j < dim_col ψ₁₁\" (*‹(i::nat) < dim_row (Matrix.mat (2::nat) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i = (0::nat) then (1::real) / sqrt (2::real) else - (1::real) / sqrt (2::real))))› ‹(j::nat) < dim_col (Matrix.mat (2::nat) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i = (0::nat) then (1::real) / sqrt (2::real) else - (1::real) / sqrt (2::real))))›*)\n then have \"i ∈ {0,1} ∧ j = 0\"\n by (simp add: less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*))\n then show \"(H * |one⟩) $$ (i,j) = ψ₁₁ $$ (i,j)\"\n by (auto simp add: times_mat_def (*‹?A * ?B ≡ Matrix.mat (dim_row ?A) (dim_col ?B) (λ(i, j). Matrix.row ?A i ∙ Matrix.col ?B j)›*) scalar_prod_def (*‹?v ∙ ?w ≡ ∑i = 0..<dim_vec ?w. ?v $ i * ?w $ i›*) H_def (*‹H ≡ complex_of_real (1 / sqrt 2) ⋅⇩m Matrix.mat 2 2 (λ(i, j). if i ≠ j then 1 else if i = 0 then 1 else - 1)›*) ket_vec_def (*‹|?v⟩ ≡ Matrix.mat (dim_vec ?v) 1 (λ(i, j). ?v $ i)›*))\nnext\n (*goals:\n 1. ‹dim_row (H * |Deutsch.one⟩) = dim_row (Matrix.mat (2::nat) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i = (0::nat) then (1::real) / sqrt (2::real) else - (1::real) / sqrt (2::real))))›\n 2. ‹dim_col (H * |Deutsch.one⟩) = dim_col (Matrix.mat (2::nat) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i = (0::nat) then (1::real) / sqrt (2::real) else - (1::real) / sqrt (2::real))))›*)\n show \"dim_row (H * |one⟩) = dim_row ψ₁₁\"\n by (simp add: H_def (*‹H ≡ complex_of_real ((1::real) / sqrt (2::real)) ⋅⇩m Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i ≠ j then 1::complex else if i = (0::nat) then 1::complex else - (1::complex))›*))\nnext\n (*goal: ‹dim_col (H * |Deutsch.one⟩) = dim_col (Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)))›*)\n show \"dim_col (H * |one⟩) = dim_col ψ₁₁\"\n by (simp add: H_def (*‹H ≡ complex_of_real ((1::real) / sqrt (2::real)) ⋅⇩m Matrix.mat (2::nat) (2::nat) (λ(i::nat, j::nat). if i ≠ j then 1::complex else if i = (0::nat) then 1::complex else - (1::complex))›*) ket_vec_def (*‹|?v::complex Matrix.vec⟩ ≡ Matrix.mat (dim_vec ?v) (1::nat) (λ(i::nat, j::nat). ?v $ i)›*))\nqed\n\nabbreviation ψ₁:: \"nat ⇒ complex Matrix.mat\" where\n\"ψ₁ n ≡ Matrix.mat (2^(n+1)) 1 (λ(i,j). if even i then 1/(sqrt 2)^(n+1) else -1/(sqrt 2)^(n+1))\"\n\nlemma ψ₁_values_even[simp]:\n fixes i j n\n assumes \"i < dim_row (ψ₁ n)\" and \"j < dim_col (ψ₁ n)\" and \"even i\"\n shows \"(ψ₁ n) $$ (i,j) = 1/(sqrt 2)^(n+1)\" \n using assms (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))› ‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))› ‹even (i::nat)›*) case_prod_conv (*‹(case (?a, ?b) of (c, d) ⇒ ?f c d) = ?f ?a ?b›*) by simp\n\nlemma ψ₁_values_odd [simp]:\n fixes i j n\n assumes \"i < dim_row (ψ₁ n)\" and \"j < dim_col (ψ₁ n)\" and \"odd i\"\n shows \"(ψ₁ n) $$ (i,j) = -1/(sqrt 2)^(n+1)\" \n using assms (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat)))))› ‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))› ‹odd i›*) case_prod_conv (*‹(case (?a, ?b) of (c, d) ⇒ ?f c d) = ?f ?a ?b›*) by simp\n\nlemma \"ψ₁₀_tensor_ψ₁₁_is_ψ₁\":\n assumes \"n ≥ 1\"\n shows \"(ψ₁₀ n) ⨂ ψ₁₁ = ψ₁ n\" \nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))⟧ ⟹ (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) $$ (i, j)›\n 2. ‹dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) = dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›\n 3. ‹dim_col (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) = dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*)\n show \"dim_col ((ψ₁₀ n) ⨂ ψ₁₁) = dim_col (ψ₁ n)\"\n by simp\nnext\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))⟧ ⟹ (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) $$ (i, j)›\n 2. ‹dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) = dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*)\n show \"dim_row ((ψ₁₀ n) ⨂ ψ₁₁) = dim_row (ψ₁ n)\"\n by simp\nnext\n (*goal: ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))⟧ ⟹ (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) $$ (i, j)›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row (ψ₁ n)\" and a1: \"j < dim_col (ψ₁ n)\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat)))))› ‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat)))))›*)\n then have \"i < 2^(n+1)\" and \"j = 0\"\n apply -\n (*goals:\n 1. ‹⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))⟧ ⟹ i < 2 ^ (n + 1)›\n 2. ‹⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))⟧ ⟹ j = 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n then have f0: \"((ψ₁₀ n) ⨂ ψ₁₁) $$ (i,j) = 1/(sqrt 2)^n * ψ₁₁ $$ (i mod 2, j)\"\n using \"ψ₁₀_values\"[of \"i div 2\" n \"j div 1\"] (*‹⟦i div 2 < dim_row (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n))); j div 1 < dim_col (Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)))⟧ ⟹ Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) $$ (i div 2, j div 1) = complex_of_real (1 / sqrt 2 ^ n)›*) a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*) by simp\n show \"((ψ₁₀ n) ⨂ ψ₁₁) $$ (i,j) = (ψ₁ n) $$ (i,j)\"\n using f0 (*‹(Matrix.mat (2 ^ n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))) $$ (i, j) = complex_of_real (1 / sqrt 2 ^ n) * Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)) $$ (i mod 2, j)›*) \"ψ₁_values_even\" (*‹⟦?i < dim_row (Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1)))); ?j < dim_col (Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1)))); even ?i⟧ ⟹ Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1))) $$ (?i, ?j) = complex_of_real (1 / sqrt 2 ^ (?n + 1))›*) \"ψ₁_values_odd\" (*‹⟦?i < dim_row (Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1)))); ?j < dim_col (Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1)))); odd ?i⟧ ⟹ Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1))) $$ (?i, ?j) = complex_of_real (- 1 / sqrt 2 ^ (?n + 1))›*) a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*) by auto\nqed\n\nlemma ψ₁_is_state:\n assumes \"n ≥ 1\"\n shows \"state (n+1) (ψ₁ n)\" \n using assms (*‹1 ≤ n›*) \"ψ₁₀_tensor_ψ₁₁_is_ψ₁\" (*‹1 ≤ ?n ⟹ Matrix.mat (2 ^ ?n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ ?n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)) = Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1)))›*) \"ψ₁₀_is_state\" (*‹1 ≤ ?n ⟹ state ?n (Matrix.mat (2 ^ ?n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ ?n)))›*) H_on_ket_one_is_state (*‹state (1::nat) (H * |Deutsch.one⟩)›*) \"H_on_ket_one_is_ψ₁₁\" (*‹H * |Deutsch.one⟩ = Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))›*) tensor_state (*‹⟦state ?m ?u; state ?n ?v⟧ ⟹ state (?m + ?n) (?u ⨂ ?v)›*) by metis\n\nabbreviation (in jozsa) ψ₂:: \"complex Matrix.mat\" where\n\"ψ₂ ≡ Matrix.mat (2^(n+1)) 1 (λ(i,j). if even i then (-1)^f(i div 2)/(sqrt 2)^(n+1) \n else (-1)^(f(i div 2)+1)/(sqrt 2)^(n+1))\"\n\nlemma (in jozsa) ψ₂_values_even [simp]:\n fixes i j \n assumes \"i < dim_row ψ₂ \" and \"j < dim_col ψ₂\" and \"even i\"\n shows \"ψ₂ $$ (i,j) = (-1)^f(i div 2)/(sqrt 2)^(n+1)\" \n using assms (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))› ‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))› ‹even i›*) case_prod_conv (*‹(case (?a::?'b, ?b::?'c) of (c::?'b, d::?'c) ⇒ (?f::?'b ⇒ ?'c ⇒ ?'a) c d) = ?f ?a ?b›*) by simp\n\nlemma (in jozsa) ψ₂_values_odd [simp]:\n fixes i j \n assumes \"i < dim_row ψ₂\" and \"j < dim_col ψ₂\" and \"odd i\"\n shows \"ψ₂ $$ (i,j) = (-1)^(f(i div 2)+1)/(sqrt 2)^(n+1)\" \n using assms (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))› ‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))› ‹odd i›*) case_prod_conv (*‹(case (?a, ?b) of (c, d) ⇒ ?f c d) = ?f ?a ?b›*) by simp\n\nlemma (in jozsa) ψ₂_values_odd_hidden [simp]:\n assumes \"2*k+1 < dim_row ψ₂\" and \"j < dim_col ψ₂\" \n shows \"ψ₂ $$ (2*k+1,j) = ((-1)^(f((2*k+1) div 2)+1))/(sqrt 2)^(n+1)\" \n using assms (*‹(2::nat) * (k::nat) + (1::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))› ‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) by simp\n\nlemma (in jozsa) snd_rep_of_ψ₂:\n assumes \"i < dim_row ψ₂\"\n shows \"((1-f(i div 2)) + -f(i div 2)) * 1/(sqrt 2)^(n+1) = (-1)^f(i div 2)/(sqrt 2)^(n+1)\"\n and \"(-(1-f(i div 2))+(f(i div 2)))* 1/(sqrt 2)^(n+1) = (-1)^(f(i div 2)+1)/(sqrt 2)^(n+1)\"\nproof (-)\n (*goals:\n 1. ‹real_of_int ((int (1 - f (i div 2)) + - int (f (i div 2))) * 1) / sqrt 2 ^ (n + 1) = (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1)›\n 2. ‹real_of_int ((- int (1 - f (i div 2)) + int (f (i div 2))) * 1) / sqrt 2 ^ (n + 1) = (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)›*)\n have \"i div 2 ∈ {i. i < 2 ^ n}\"\n using assms (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))›*) by auto\n then have \"real (Suc 0 - f (i div 2)) - real (f (i div 2)) = (- 1) ^ f (i div 2)\"\n using assms (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) by auto\n thus \"((1-f(i div 2)) + -f(i div 2)) * 1/(sqrt 2)^(n+1) = (-1)^f(i div 2)/(sqrt 2)^(n+1)\"\n by auto\nnext\n (*goal: ‹real_of_int ((- int (1 - f (i div 2)) + int (f (i div 2))) * 1) / sqrt 2 ^ (n + 1) = (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)›*)\n have \"i div 2 ∈ {i. i < 2^n}\"\n using assms (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) by simp\n then have \"(real (f (i div 2)) - real (Suc 0 - f (i div 2))) / (sqrt 2 ^ (n+1)) =\n - ((- 1) ^ f (i div 2) / (sqrt 2 ^ (n+1)))\"\n using assms (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) by fastforce\n then show \"(-(1-f(i div 2))+(f(i div 2)))* 1/(sqrt 2)^(n+1) = (-1)^(f(i div 2)+1)/(sqrt 2)^(n+1)\"\n by simp\nqed\n\nlemma (in jozsa) jozsa_transform_times_ψ₁_is_ψ₂:\n shows \"U⇩f * (ψ₁ n) = ψ₂\" \nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))⟧ ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j)›\n 2. ‹dim_row (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) = dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›\n 3. ‹dim_col (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) = dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*)\n show \"dim_row (U⇩f * (ψ₁ n)) = dim_row ψ₂\"\n by simp\nnext\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))⟧ ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j)›\n 2. ‹dim_col (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) = dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*)\n show \"dim_col (U⇩f * (ψ₁ n)) = dim_col ψ₂\"\n by simp\nnext\n (*goal: ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))⟧ ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j)›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row ψ₂\" and a1: \"j < dim_col ψ₂\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))› ‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))›*)\n then have f0: \"i ∈ {0..2^(n+1)} ∧ j=0\"\n by simp\n then have f1: \"i < dim_row U⇩f ∧ j < dim_col U⇩f \"\n using a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))›*) by simp\n have f2: \"i < dim_row (ψ₁ n) ∧ j < dim_col (ψ₁ n)\"\n using a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) by simp\n show \"(U⇩f * (ψ₁ n)) $$ (i,j) = ψ₂ $$ (i,j)\"\n proof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹?P ∨ ?Q›\n 2. ‹?P ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j)›\n 3. ‹?Q ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j)›*)\n show \"even i ∨ odd i\"\n by auto\n next\n (*goals:\n 1. ‹even i ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j)›\n 2. ‹odd i ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j)›*)\n assume a2: \"even i\" (*‹even (i::nat)›*)\n then have \"(U⇩f * (ψ₁ n)) $$ (i,j) = (∑k ∈ {i,i+1}. U⇩f $$ (i,k) * (ψ₁ n) $$ (k,j))\"\n using f1 (*‹i < dim_row U⇩f ∧ j < dim_col U⇩f›*) f2 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) ∧ j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*) \"U⇩f_mult_without_empty_summands_even\"[of i j \"(ψ₁ n)\"] (*‹⟦i < dim_row U⇩f; j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))); even i; dim_col U⇩f = dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))⟧ ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = (∑k∈{i, i + 1}. U⇩f $$ (i, k) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) $$ (k, j))›*) by simp\n moreover have \"U⇩f $$ (i,i) * (ψ₁ n) $$ (i,j) = (1-f(i div 2))* 1/(sqrt 2)^(n+1)\"\n using f0 (*‹i ∈ {0..2 ^ (n + 1)} ∧ j = 0›*) f1 (*‹(i::nat) < dim_row U⇩f ∧ (j::nat) < dim_col U⇩f›*) a2 (*‹even (i::nat)›*) by simp\n moreover have \"U⇩f $$ (i,i+1) * (ψ₁ n) $$ (i+1,j) = (-f(i div 2))* 1/(sqrt 2)^(n+1)\"\n using f0 (*‹i ∈ {0..2 ^ (n + 1)} ∧ j = 0›*) f1 (*‹i < dim_row U⇩f ∧ j < dim_col U⇩f›*) a2 (*‹even i›*) by auto\n ultimately have \"(U⇩f * (ψ₁ n)) $$ (i,j) = (1-f(i div 2))* 1/(sqrt 2)^(n+1) + (-f(i div 2))* 1/(sqrt 2)^(n+1)\"\n by simp\n also (*calculation: ‹(U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = complex_of_real (real ((1 - f (i div 2)) * 1) / sqrt 2 ^ (n + 1) + real_of_int (- int (f (i div 2)) * 1) / sqrt 2 ^ (n + 1))›*) have \"... = ((1-f(i div 2))+-f(i div 2)) * 1/(sqrt 2)^(n+1)\"\n using add_divide_distrib (*‹(?a + ?b) / ?c = ?a / ?c + ?b / ?c›*) by (metis (no_types, opaque_lifting) mult.right_neutral (*‹?a * 1 = ?a›*) of_int_add (*‹of_int (?w + ?z) = of_int ?w + of_int ?z›*) of_int_of_nat_eq (*‹of_int (int ?n) = of_nat ?n›*))\n also (*calculation: ‹(U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = complex_of_real (real_of_int ((int (1 - f (i div 2)) + - int (f (i div 2))) * 1) / sqrt 2 ^ (n + 1))›*) have \"... = ψ₂ $$ (i,j)\"\n using a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) a2 (*‹even i›*) \"snd_rep_of_ψ₂\" (*‹(?i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat))))) ⟹ real_of_int ((int ((1::nat) - f (?i div (2::nat))) + - int (f (?i div (2::nat)))) * (1::int)) / sqrt (2::real) ^ (n + (1::nat)) = (- (1::real)) ^ f (?i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat))› ‹(?i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat))))) ⟹ real_of_int ((- int ((1::nat) - f (?i div (2::nat))) + int (f (?i div (2::nat)))) * (1::int)) / sqrt (2::real) ^ (n + (1::nat)) = (- (1::real)) ^ (f (?i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat))›*) by simp\n finally (*calculation: ‹(U⇩f * Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat))))) $$ (i::nat, j::nat) = Matrix.mat ((2::nat) ^ (n + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))) $$ (i, j)›*) show \"(U⇩f * (ψ₁ n)) $$ (i,j) = ψ₂ $$ (i,j)\"\n by simp\n next\n (*goal: ‹odd (i::nat) ⟹ (U⇩f * Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat))))) $$ (i, j::nat) = Matrix.mat ((2::nat) ^ (n + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))) $$ (i, j)›*)\n assume a2: \"odd i\" (*‹odd (i::nat)›*)\n then have f6: \"i≥1\"\n using linorder_not_less (*‹(¬ ?x < ?y) = (?y ≤ ?x)›*) by auto\n have \"(U⇩f * (ψ₁ n)) $$ (i,j) = (∑k ∈ {i-1,i}. U⇩f $$ (i,k) * (ψ₁ n) $$ (k,j))\"\n using f1 (*‹i < dim_row U⇩f ∧ j < dim_col U⇩f›*) f2 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat))))) ∧ (j::nat) < dim_col (Matrix.mat ((2::nat) ^ (n + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat)))))›*) a2 (*‹odd i›*) \"U⇩f_mult_without_empty_summands_odd\"[of i j \"(ψ₁ n)\"] (*‹⟦i < dim_row U⇩f; j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))); odd i; dim_col U⇩f = dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))⟧ ⟹ (U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = (∑k∈{i - 1, i}. U⇩f $$ (i, k) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) $$ (k, j))›*) by (metis dim_row_mat( (*‹dim_row (Matrix.mat ?nr ?nc ?f) = ?nr›*) 1) jozsa_transform_dim( (*‹dim_col U⇩f = 2 ^ (n + 1)›*) 2))\n moreover have \"(∑k ∈ {i-1,i}. U⇩f $$ (i,k) * (ψ₁ n) $$ (k,j)) \n = U⇩f $$ (i,i-1) * (ψ₁ n) $$ (i-1,j) + U⇩f $$ (i,i) * (ψ₁ n) $$ (i,j)\"\n using a2 (*‹odd i›*) f6 (*‹1 ≤ i›*) by simp\n moreover have \"U⇩f $$ (i,i) * (ψ₁ n) $$ (i,j) = (1-f(i div 2))* -1/(sqrt 2)^(n+1)\"\n using f1 (*‹i < dim_row U⇩f ∧ j < dim_col U⇩f›*) f2 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) ∧ j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))›*) a2 (*‹odd i›*) by simp\n moreover have \"U⇩f $$ (i,i-1) * (ψ₁ n) $$ (i-1,j) = f(i div 2)* 1/(sqrt 2)^(n+1)\"\n using a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) a1 (*‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))›*) a2 (*‹odd i›*) by simp\n ultimately have \"(U⇩f * (ψ₁ n)) $$ (i,j) = (1-f(i div 2))* -1/(sqrt 2)^(n+1) +(f(i div 2))* 1/(sqrt 2)^(n+1)\"\n using of_real_add (*‹of_real (?x + ?y) = of_real ?x + of_real ?y›*) by simp\n also (*calculation: ‹(U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))) $$ (i, j) = complex_of_real (real_of_int (int (1 - f (i div 2)) * - 1) / sqrt 2 ^ (n + 1) + real (f (i div 2) * 1) / sqrt 2 ^ (n + 1))›*) have \"... = (-(1-f(i div 2)) + (f(i div 2))) * 1/(sqrt 2)^(n+1)\"\n by (metis (no_types, opaque_lifting) mult.right_neutral (*‹?a * 1 = ?a›*) add_divide_distrib (*‹(?a + ?b) / ?c = ?a / ?c + ?b / ?c›*) mult_minus1_right (*‹?z * - 1 = - ?z›*) of_int_add (*‹of_int (?w + ?z) = of_int ?w + of_int ?z›*) of_int_of_nat_eq (*‹of_int (int ?n) = of_nat ?n›*))\n also (*calculation: ‹(U⇩f * Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat))))) $$ (i::nat, j::nat) = complex_of_real (real_of_int ((- int ((1::nat) - (f::nat ⇒ nat) (i div (2::nat))) + int (f (i div (2::nat)))) * (1::int)) / sqrt (2::real) ^ (n + (1::nat)))›*) have \"... = (-1)^(f(i div 2)+1)/(sqrt 2)^(n+1)\"\n using a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))›*) a1 (*‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))))›*) a2 (*‹odd i›*) \"snd_rep_of_ψ₂\" (*‹?i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) ⟹ real_of_int ((int (1 - f (?i div 2)) + - int (f (?i div 2))) * 1) / sqrt 2 ^ (n + 1) = (- 1) ^ f (?i div 2) / sqrt 2 ^ (n + 1)› ‹(?i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat))))) ⟹ real_of_int ((- int ((1::nat) - f (?i div (2::nat))) + int (f (?i div (2::nat)))) * (1::int)) / sqrt (2::real) ^ (n + (1::nat)) = (- (1::real)) ^ (f (?i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat))›*) by simp\n finally (*calculation: ‹(U⇩f * Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (1::real) / sqrt (2::real) ^ (n + (1::nat)) else - (1::real) / sqrt (2::real) ^ (n + (1::nat))))) $$ (i::nat, j::nat) = complex_of_real ((- (1::real)) ^ ((f::nat ⇒ nat) (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))›*) show \"(U⇩f * (ψ₁ n)) $$ (i,j) = ψ₂ $$ (i,j)\"\n using a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) a2 (*‹odd i›*) by simp\n qed\nqed\n\nlemma (in jozsa) ψ₂_is_state:\n shows \"state (n+1) ψ₂\" \n using \"jozsa_transform_times_ψ₁_is_ψ₂\" (*‹U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))›*) jozsa_transform_is_gate (*‹gate (n + 1) U⇩f›*) \"ψ₁_is_state\" (*‹1 ≤ ?n ⟹ state (?n + 1) (Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1))))›*) dim (*‹1 ≤ n›*) gate_on_state_is_state (*‹⟦gate ?n ?A; state ?n ?v⟧ ⟹ state ?n (?A * ?v)›*) by fastforce\n\ntext ‹@{text \"H^⇩⊗ n\"} is the result of taking the nth tensor product of H›\n\nabbreviation iter_tensor_of_H_rep:: \"nat ⇒ complex Matrix.mat\" (\"H^⇩⊗ _\") where\n\"iter_tensor_of_H_rep n ≡ Matrix.mat (2^n) (2^n) (λ(i,j).(-1)^(i ⋅⇘n⇙ j)/(sqrt 2)^n)\"\n\nlemma tensor_of_H_values [simp]:\n fixes n i j:: nat\n assumes \"i < dim_row (H^⇩⊗ n)\" and \"j < dim_col (H^⇩⊗ n)\"\n shows \"(H^⇩⊗ n) $$ (i,j) = (-1)^(i ⋅⇘n⇙ j)/(sqrt 2)^n\"\n using assms (*‹i < dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)))› ‹j < dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)))›*) by simp\n\nlemma dim_row_of_iter_tensor_of_H [simp]:\n assumes \"n ≥ 1\"\n shows \"1 < dim_row (H^⇩⊗ n)\" \n using assms (*‹1 ≤ n›*) by (metis One_nat_def (*‹1 = Suc 0›*) Suc_1 (*‹Suc 1 = 2›*) dim_row_mat( (*‹dim_row (Matrix.mat ?nr ?nc ?f) = ?nr›*) 1) le_trans (*‹⟦?i ≤ ?j; ?j ≤ ?k⟧ ⟹ ?i ≤ ?k›*) lessI (*‹?n < Suc ?n›*) linorder_not_less (*‹(¬ ?x < ?y) = (?y ≤ ?x)›*) one_less_power (*‹⟦1 < ?a; 0 < ?n⟧ ⟹ 1 < ?a ^ ?n›*))\n\nlemma iter_tensor_of_H_fst_pos:\n fixes n i j:: nat\n assumes \"i < 2^n ∨ j < 2^n\" and \"i < 2^(n+1) ∧ j < 2^(n+1)\"\n shows \"(H^⇩⊗ (Suc n)) $$ (i,j) = 1/sqrt(2) * ((H^⇩⊗ n) $$ (i mod 2^n, j mod 2^n))\"\nproof (-)\n (*goal: ‹Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)›*)\n have \"(H^⇩⊗ (Suc n)) $$ (i,j) = (-1)^(bip i (Suc n) j)/(sqrt 2)^(Suc n)\"\n using assms (*‹i < 2 ^ n ∨ j < 2 ^ n› ‹(i::nat) < (2::nat) ^ ((n::nat) + (1::nat)) ∧ (j::nat) < (2::nat) ^ (n + (1::nat))›*) by simp\n moreover have \"bip i (Suc n) j = bip (i mod 2^n) n (j mod 2^n)\"\n using bitwise_inner_prod_fst_el_0 (*‹?i < 2 ^ ?n ∨ ?j < 2 ^ ?n ⟹ ?i ⋅⇘Suc ?n⇙ ?j = ?i mod 2 ^ ?n ⋅⇘?n⇙ ?j mod 2 ^ ?n›*) assms(1) (*‹i < 2 ^ n ∨ j < 2 ^ n›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)›*)\n using bitwise_inner_prod_def (*‹?i ⋅⇘?n⇙ ?j = (∑k = 0..<?n. bin_rep ?n ?i ! k * bin_rep ?n ?j ! k)›*) by simp\nqed\n\nlemma iter_tensor_of_H_fst_neg:\n fixes n i j:: nat\n assumes \"i ≥ 2^n ∧ j ≥ 2^n\" and \"i < 2^(n+1) ∧ j < 2^(n+1)\"\n shows \"(H^⇩⊗ (Suc n)) $$ (i,j) = -1/sqrt(2) * (H^⇩⊗ n) $$ (i mod 2^n, j mod 2^n)\"\nproof (-)\n (*goal: ‹Matrix.mat ((2::nat) ^ Suc (n::nat)) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)) $$ (i::nat, j::nat) = complex_of_real (- (1::real) / sqrt (2::real)) * Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n)) $$ (i mod (2::nat) ^ n, j mod (2::nat) ^ n)›*)\n have \"(H^⇩⊗ (Suc n)) $$ (i,j) = (-1)^(bip i (n+1) j)/(sqrt 2)^(n+1)\"\n using assms(2) (*‹i < 2 ^ (n + 1) ∧ j < 2 ^ (n + 1)›*) by simp\n moreover have \"bip i (n+1) j = 1 + bip (i mod 2^n) n (j mod 2^n)\"\n using bitwise_inner_prod_fst_el_is_1 (*‹⟦2 ^ ?n ≤ ?i ∧ 2 ^ ?n ≤ ?j; ?i < 2 ^ (?n + 1) ∧ ?j < 2 ^ (?n + 1)⟧ ⟹ ?i ⋅⇘?n + 1⇙ ?j = 1 + ?i mod 2 ^ ?n ⋅⇘?n⇙ ?j mod 2 ^ ?n›*) assms (*‹2 ^ n ≤ i ∧ 2 ^ n ≤ j› ‹i < 2 ^ (n + 1) ∧ j < 2 ^ (n + 1)›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹Matrix.mat ((2::nat) ^ Suc (n::nat)) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)) $$ (i::nat, j::nat) = complex_of_real (- (1::real) / sqrt (2::real)) * Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n)) $$ (i mod (2::nat) ^ n, j mod (2::nat) ^ n)›*)\n by simp\nqed \n\nlemma H_tensor_iter_tensor_of_H: \n fixes n:: nat\n shows \"(H ⨂ H^⇩⊗ n) = H^⇩⊗ (Suc n)\" \nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n))); j < dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))⟧ ⟹ (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j)›\n 2. ‹dim_row (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) = dim_row (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›\n 3. ‹dim_col (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) = dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row (H^⇩⊗ (Suc n))\" and a1: \"j < dim_col (H^⇩⊗ (Suc n))\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ Suc (n::nat)) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))› ‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ Suc (n::nat)) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))›*)\n then have f0: \"i ∈ {0..<2^(n+1)} ∧ j ∈ {0..<2^(n+1)}\"\n by simp\n then have f1: \"(H ⨂ H^⇩⊗ n) $$ (i,j) = H $$ (i div (dim_row (H^⇩⊗ n)),j div (dim_col (H^⇩⊗ n))) \n * (H^⇩⊗ n) $$ (i mod (dim_row (H^⇩⊗ n)),j mod (dim_col (H^⇩⊗ n)))\"\n by (simp add: H_without_scalar_prod (*‹H = Matrix.mat 2 2 (λx. complex_of_real (case x of (i, j) ⇒ if i ≠ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))›*))\n show \"(H ⨂ H^⇩⊗ n) $$ (i,j) = (H^⇩⊗ (Suc n)) $$ (i,j)\"\n proof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹?P ∨ ?Q›\n 2. ‹?P ⟹ (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j)›\n 3. ‹?Q ⟹ (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j)›*)\n show \"(i < 2^n ∨ j < 2^n) ∨ ¬(i < 2^n ∨ j < 2^n)\"\n by auto\n next\n (*goals:\n 1. ‹i < 2 ^ n ∨ j < 2 ^ n ⟹ (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j)›\n 2. ‹¬ (i < 2 ^ n ∨ j < 2 ^ n) ⟹ (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j)›*)\n assume a2: \"(i < 2^n ∨ j < 2^n)\" (*‹(i::nat) < (2::nat) ^ (n::nat) ∨ (j::nat) < (2::nat) ^ n›*)\n then have \"(H^⇩⊗ (Suc n)) $$ (i,j) = 1/sqrt(2) * ((H^⇩⊗ n) $$ (i mod 2^n, j mod 2^n))\"\n using a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ Suc (n::nat)) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*) f0 (*‹i ∈ {0..<2 ^ (n + 1)} ∧ j ∈ {0..<2 ^ (n + 1)}›*) iter_tensor_of_H_fst_pos (*‹⟦(?i::nat) < (2::nat) ^ (?n::nat) ∨ (?j::nat) < (2::nat) ^ ?n; ?i < (2::nat) ^ (?n + (1::nat)) ∧ ?j < (2::nat) ^ (?n + (1::nat))⟧ ⟹ Matrix.mat ((2::nat) ^ Suc ?n) ((2::nat) ^ Suc ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc ?n⇙ j / sqrt (2::real) ^ Suc ?n)) $$ (?i, ?j) = complex_of_real ((1::real) / sqrt (2::real)) * Matrix.mat ((2::nat) ^ ?n) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)) $$ (?i mod (2::nat) ^ ?n, ?j mod (2::nat) ^ ?n)›*) by (metis (mono_tags, lifting) atLeastLessThan_iff (*‹(?i ∈ {?l..<?u}) = (?l ≤ ?i ∧ ?i < ?u)›*))\n moreover have \"H $$ (i div (dim_row (H^⇩⊗ n)),j div (dim_col (H^⇩⊗ n))) = 1/sqrt 2\"\n using a0 (*‹i < dim_row (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*) f0 (*‹i ∈ {0..<2 ^ (n + 1)} ∧ j ∈ {0..<2 ^ (n + 1)}›*) H_without_scalar_prod (*‹H = Matrix.mat 2 2 (λx. complex_of_real (case x of (i, j) ⇒ if i ≠ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))›*) H_values (*‹⟦?i < dim_row H; ?j < dim_col H; ?i ≠ 1 ∨ ?j ≠ 1⟧ ⟹ H $$ (?i, ?j) = complex_of_real (1 / sqrt 2)›*) a2 (*‹i < 2 ^ n ∨ j < 2 ^ n›*) by (metis (no_types, lifting) dim_col_mat( (*‹dim_col (Matrix.mat (?nr::nat) (?nc::nat) (?f::nat × nat ⇒ ?'a)) = ?nc›*) 1) dim_row_mat( (*‹dim_row (Matrix.mat (?nr::nat) (?nc::nat) (?f::nat × nat ⇒ ?'a)) = ?nr›*) 1) div_less (*‹(?m::nat) < (?n::nat) ⟹ ?m div ?n = (0::nat)›*) le_eq_less_or_eq (*‹((?m::nat) ≤ (?n::nat)) = (?m < ?n ∨ ?m = ?n)›*) le_numeral_extra( (*‹¬ (1::?'a) ≤ (0::?'a)›*) 2) less_power_add_imp_div_less (*‹(?i::nat) < (2::nat) ^ ((?m::nat) + (?n::nat)) ⟹ ?i div (2::nat) ^ ?n < (2::nat) ^ ?m›*) plus_1_eq_Suc (*‹(+) (1::nat) = Suc›*) power_one_right (*‹(?a::?'a) ^ (1::nat) = ?a›*))\n ultimately show \"(H ⨂ H^⇩⊗ n) $$ (i,j) = (H^⇩⊗ (Suc n)) $$ (i,j)\"\n using f1 (*‹(H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = H $$ (i div dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))), j div dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)))) * Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) $$ (i mod dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))), j mod dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))))›*) by simp\n next\n (*goal: ‹¬ (i < 2 ^ n ∨ j < 2 ^ n) ⟹ (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j)›*)\n assume a2: \"¬(i < 2^n ∨ j < 2^n)\" (*‹¬ ((i::nat) < (2::nat) ^ (n::nat) ∨ (j::nat) < (2::nat) ^ n)›*)\n then have \"i ≥ 2^n ∧ j ≥ 2^n\"\n by simp\n then have f2: \"(H^⇩⊗ (Suc n)) $$ (i,j) = -1/sqrt(2) * ((H^⇩⊗ n) $$ (i mod 2^n, j mod 2^n))\"\n using a0 (*‹i < dim_row (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*) f0 (*‹i ∈ {0..<2 ^ (n + 1)} ∧ j ∈ {0..<2 ^ (n + 1)}›*) iter_tensor_of_H_fst_neg (*‹⟦(2::nat) ^ (?n::nat) ≤ (?i::nat) ∧ (2::nat) ^ ?n ≤ (?j::nat); ?i < (2::nat) ^ (?n + (1::nat)) ∧ ?j < (2::nat) ^ (?n + (1::nat))⟧ ⟹ Matrix.mat ((2::nat) ^ Suc ?n) ((2::nat) ^ Suc ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc ?n⇙ j / sqrt (2::real) ^ Suc ?n)) $$ (?i, ?j) = complex_of_real (- (1::real) / sqrt (2::real)) * Matrix.mat ((2::nat) ^ ?n) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)) $$ (?i mod (2::nat) ^ ?n, ?j mod (2::nat) ^ ?n)›*) by simp\n have \"i div (dim_row (H^⇩⊗ n)) =1\" and \"j div (dim_row (H^⇩⊗ n)) = 1\"\n using a2 (*‹¬ (i < 2 ^ n ∨ j < 2 ^ n)›*) a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ Suc (n::nat)) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*) apply -\n (*goals:\n 1. ‹⟦¬ ((i::nat) < (2::nat) ^ (n::nat) ∨ (j::nat) < (2::nat) ^ n); i < dim_row (Matrix.mat ((2::nat) ^ Suc n) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n))); j < dim_col (Matrix.mat ((2::nat) ^ Suc n) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))⟧ ⟹ i div dim_row (Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n))) = (1::nat)›\n 2. ‹⟦¬ ((i::nat) < (2::nat) ^ (n::nat) ∨ (j::nat) < (2::nat) ^ n); i < dim_row (Matrix.mat ((2::nat) ^ Suc n) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n))); j < dim_col (Matrix.mat ((2::nat) ^ Suc n) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))⟧ ⟹ j div dim_row (Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n))) = (1::nat)›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n then have \"H $$ (i div (dim_row (H^⇩⊗ n)),j div (dim_col (H^⇩⊗ n))) = -1/sqrt 2\"\n using a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ Suc (n::nat)) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*) f0 (*‹i ∈ {0..<2 ^ (n + 1)} ∧ j ∈ {0..<2 ^ (n + 1)}›*) H_values_right_bottom[of \"i div (dim_row (H^⇩⊗ n))\" \"j div (dim_col (H^⇩⊗ n))\"] (*‹i div dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) = 1 ∧ j div dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) = 1 ⟹ H $$ (i div dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))), j div dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)))) = complex_of_real (- 1 / sqrt 2)›*) a2 (*‹¬ (i < 2 ^ n ∨ j < 2 ^ n)›*) by fastforce\n then show \"(H ⨂ H^⇩⊗ n) $$ (i,j) = (H^⇩⊗ (Suc n)) $$ (i,j)\"\n using f1 (*‹(H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) $$ (i, j) = H $$ (i div dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))), j div dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)))) * Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) $$ (i mod dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))), j mod dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))))›*) f2 (*‹Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)›*) by simp\n qed\nnext\n (*goals:\n 1. ‹dim_row (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) = dim_row (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›\n 2. ‹dim_col (H ⨂ Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))) = dim_col (Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n)))›*)\n show \"dim_row (H ⨂ H^⇩⊗ n) = dim_row (H^⇩⊗ (Suc n))\"\n by (simp add: H_without_scalar_prod (*‹H = Matrix.mat 2 2 (λx. complex_of_real (case x of (i, j) ⇒ if i ≠ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))›*))\nnext\n (*goal: ‹dim_col (H ⨂ Matrix.mat ((2::nat) ^ (n::nat)) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n))) = dim_col (Matrix.mat ((2::nat) ^ Suc n) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n)))›*)\n show \"dim_col (H ⨂ H^⇩⊗ n) = dim_col (H^⇩⊗ (Suc n))\"\n by (simp add: H_without_scalar_prod (*‹H = Matrix.mat 2 2 (λx. complex_of_real (case x of (i, j) ⇒ if i ≠ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))›*))\nqed\n\ntext ‹\nWe prove that @{term \"H^⇩⊗ n\"} is indeed the matrix representation of @{term \"H ⊗⇗n⇖\"}, the iterated \ntensor product of the Hadamard gate H.\n›\n\nlemma one_tensor_of_H_is_H:\n shows \"(H^⇩⊗ 1) = H\"\nproof (rule eq_matI (*‹⟦⋀i j. ⟦i < dim_row ?B; j < dim_col ?B⟧ ⟹ ?A $$ (i, j) = ?B $$ (i, j); dim_row ?A = dim_row ?B; dim_col ?A = dim_col ?B⟧ ⟹ ?A = ?B›*))\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row H; j < dim_col H⟧ ⟹ Matrix.mat (2 ^ 1) (2 ^ 1) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘1⇙ j / sqrt 2 ^ 1)) $$ (i, j) = H $$ (i, j)›\n 2. ‹dim_row (Matrix.mat (2 ^ 1) (2 ^ 1) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘1⇙ j / sqrt 2 ^ 1))) = dim_row H›\n 3. ‹dim_col (Matrix.mat (2 ^ 1) (2 ^ 1) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘1⇙ j / sqrt 2 ^ 1))) = dim_col H›*)\n show \"dim_row (H^⇩⊗ 1) = dim_row H\"\n by (simp add: H_without_scalar_prod (*‹H = Matrix.mat 2 2 (λx. complex_of_real (case x of (i, j) ⇒ if i ≠ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))›*))\n show \"dim_col (H^⇩⊗ 1) = dim_col H\"\n by (simp add: H_without_scalar_prod (*‹H = Matrix.mat (2::nat) (2::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i ≠ j then (1::real) / sqrt (2::real) else if i = (0::nat) then (1::real) / sqrt (2::real) else - ((1::real) / sqrt (2::real))))›*))\nnext\n (*goal: ‹⋀(i::nat) j::nat. ⟦i < dim_row H; j < dim_col H⟧ ⟹ Matrix.mat ((2::nat) ^ (1::nat)) ((2::nat) ^ (1::nat)) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘1::nat⇙ j / sqrt (2::real) ^ (1::nat))) $$ (i, j) = H $$ (i, j)›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row H\" and a1: \"j < dim_col H\" (*‹(i::nat) < dim_row H› ‹(j::nat) < dim_col H›*)\n then show \"(H^⇩⊗ 1) $$ (i,j) = H $$ (i,j)\"\n proof (-)\n (*goal: ‹⟦i < dim_row H; j < dim_col H⟧ ⟹ Matrix.mat (2 ^ 1) (2 ^ 1) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘1⇙ j / sqrt 2 ^ 1)) $$ (i, j) = H $$ (i, j)›*)\n have \"(H^⇩⊗ 1) $$ (0, 0) = 1/sqrt(2)\"\n using bitwise_inner_prod_def (*‹?i ⋅⇘?n⇙ ?j = (∑k = 0..<?n. bin_rep ?n ?i ! k * bin_rep ?n ?j ! k)›*) bin_rep_def (*‹bin_rep ?n ?m = butlast (bin_rep_aux ?n ?m)›*) by simp\n moreover have \"(H^⇩⊗ 1) $$ (0,1) = 1/sqrt(2)\"\n using bitwise_inner_prod_def (*‹?i ⋅⇘?n⇙ ?j = (∑k = 0..<?n. bin_rep ?n ?i ! k * bin_rep ?n ?j ! k)›*) bin_rep_def (*‹bin_rep ?n ?m = butlast (bin_rep_aux ?n ?m)›*) by simp\n moreover have \"(H^⇩⊗ 1) $$ (1,0) = 1/sqrt(2)\"\n using bitwise_inner_prod_def (*‹?i::nat ⋅⇘?n::nat⇙ ?j::nat = (∑k::nat = 0::nat..<?n. bin_rep ?n ?i ! k * bin_rep ?n ?j ! k)›*) bin_rep_def (*‹bin_rep ?n ?m = butlast (bin_rep_aux ?n ?m)›*) by simp\n moreover have \"(H^⇩⊗ 1) $$ (1,1) = -1/sqrt(2)\"\n using bitwise_inner_prod_def (*‹?i ⋅⇘?n⇙ ?j = (∑k = 0..<?n. bin_rep ?n ?i ! k * bin_rep ?n ?j ! k)›*) bin_rep_def (*‹bin_rep (?n::nat) (?m::nat) = butlast (bin_rep_aux ?n ?m)›*) by simp\n ultimately show \"(H^⇩⊗ 1) $$ (i,j) = H $$ (i,j)\"\n using a0 (*‹i < dim_row H›*) a1 (*‹j < dim_col H›*) H_values (*‹⟦?i < dim_row H; ?j < dim_col H; ?i ≠ 1 ∨ ?j ≠ 1⟧ ⟹ H $$ (?i, ?j) = complex_of_real (1 / sqrt 2)›*) H_values_right_bottom (*‹?i = 1 ∧ ?j = 1 ⟹ H $$ (?i, ?j) = complex_of_real (- 1 / sqrt 2)›*) by (metis (no_types, lifting) H_without_scalar_prod (*‹H = Matrix.mat 2 2 (λx. complex_of_real (case x of (i, j) ⇒ if i ≠ j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))›*) One_nat_def (*‹1 = Suc 0›*) dim_col_mat( (*‹dim_col (Matrix.mat ?nr ?nc ?f) = ?nc›*) 1) dim_row_mat( (*‹dim_row (Matrix.mat ?nr ?nc ?f) = ?nr›*) 1) divide_minus_left (*‹- ?a / ?b = - (?a / ?b)›*) less_2_cases (*‹?n < 2 ⟹ ?n = 0 ∨ ?n = Suc 0›*))\n qed\nqed\n\nlemma iter_tensor_of_H_rep_is_correct:\n fixes n:: nat\n assumes \"n ≥ 1\"\n shows \"(H ⊗⇗n⇖) = H^⇩⊗ n\"\n using assms (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦(?m::nat) ≤ (?n::nat); (?P::nat ⇒ bool) ?m; ⋀n::nat. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹H ⊗⇗1::nat⇖ = Matrix.mat ((2::nat) ^ (1::nat)) ((2::nat) ^ (1::nat)) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘1::nat⇙ j / sqrt (2::real) ^ (1::nat)))›\n 2. ‹⋀n::nat. ⟦(1::nat) ≤ n; H ⊗⇗n⇖ = Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n))⟧ ⟹ H ⊗⇗Suc n⇖ = Matrix.mat ((2::nat) ^ Suc n) ((2::nat) ^ Suc n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘Suc n⇙ j / sqrt (2::real) ^ Suc n))›*)\n show \"(H ⊗⇗1⇖) = H^⇩⊗ 1\"\n using one_tensor_is_id (*‹?A ⊗⇗1⇖ = ?A›*) one_tensor_of_H_is_H (*‹Matrix.mat (2 ^ 1) (2 ^ 1) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘1⇙ j / sqrt 2 ^ 1)) = H›*) by simp\n next\n (*goal: ‹⋀n. ⟦1 ≤ n; H ⊗⇗n⇖ = Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n))⟧ ⟹ H ⊗⇗Suc n⇖ = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n))›*)\n fix n :: nat\n assume a0: \"n ≥ 1\" and IH: \"(H ⊗⇗n⇖) = H^⇩⊗ n\" (*‹(1::nat) ≤ (n::nat)› ‹H ⊗⇗n::nat⇖ = Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n))›*)\n then have \"(H ⊗⇗(Suc n)⇖) = H ⨂ (H ⊗⇗n⇖)\"\n using iter_tensor_suc (*‹1 ≤ ?n ⟹ ?A ⊗⇗Suc ?n⇖ = ?A ⨂ ?A ⊗⇗?n⇖›*) Nat.Suc_eq_plus1 (*‹Suc ?n = ?n + 1›*) by metis\n also (*calculation: ‹H ⊗⇗Suc n⇖ = H ⨂ H ⊗⇗n⇖›*) have \"... = H ⨂ (H^⇩⊗ n)\"\n using IH (*‹H ⊗⇗n::nat⇖ = Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n))›*) by simp\n also (*calculation: ‹H ⊗⇗Suc (n::nat)⇖ = H ⨂ Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n))›*) have \"... = H^⇩⊗ (Suc n)\"\n using a0 (*‹1 ≤ n›*) H_tensor_iter_tensor_of_H (*‹H ⨂ Matrix.mat (2 ^ ?n) (2 ^ ?n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘?n⇙ j / sqrt 2 ^ ?n)) = Matrix.mat (2 ^ Suc ?n) (2 ^ Suc ?n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc ?n⇙ j / sqrt 2 ^ Suc ?n))›*) by simp\n finally (*calculation: ‹H ⊗⇗Suc n⇖ = Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘Suc n⇙ j / sqrt 2 ^ Suc n))›*) show \"(H ⊗⇗(Suc n)⇖) = H^⇩⊗ (Suc n)\"\n by simp\n qed\n\ntext ‹@{text \"HId^⇩⊗ 1\"} is the result of taking the tensor product of the nth tensor of H and Id 1 ›\n\nabbreviation tensor_of_H_tensor_Id:: \"nat ⇒ complex Matrix.mat\" (\"HId^⇩⊗ _\") where\n\"tensor_of_H_tensor_Id n ≡ Matrix.mat (2^(n+1)) (2^(n+1)) (λ(i,j).\n if (i mod 2 = j mod 2) then (-1)^((i div 2) ⋅⇘n⇙ (j div 2))/(sqrt 2)^n else 0)\"\n\nlemma mod_2_is_both_even_or_odd:\n \"((even i ∧ even j) ∨ (odd i ∧ odd j)) ⟷ (i mod 2 = j mod 2)\" \n by (metis even_iff_mod_2_eq_zero (*‹even ?a = (?a mod 2 = 0)›*) odd_iff_mod_2_eq_one (*‹odd ?a = (?a mod 2 = 1)›*))\n \nlemma HId_values [simp]:\n assumes \"n ≥ 1\" and \"i < dim_row (HId^⇩⊗ n)\" and \"j < dim_col (HId^⇩⊗ n)\"\n shows \"even i ∧ even j ⟶ (HId^⇩⊗ n) $$ (i,j) = (-1)^((i div 2) ⋅⇘n⇙ (j div 2))/(sqrt 2)^n\"\nand \"odd i ∧ odd j ⟶ (HId^⇩⊗ n) $$ (i,j) = (-1)^((i div 2) ⋅⇘n⇙ (j div 2))/(sqrt 2)^n\"\nand \"(i mod 2 = j mod 2) ⟶ (HId^⇩⊗ n) $$ (i,j) = (-1)^((i div 2) ⋅⇘n⇙ (j div 2))/(sqrt 2)^n\"\nand \"¬(i mod 2 = j mod 2) ⟶ (HId^⇩⊗ n) $$ (i,j) = 0\"\n using assms (*‹(1::nat) ≤ (n::nat)› ‹i < dim_row (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)))› ‹j < dim_col (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)))›*) mod_2_is_both_even_or_odd (*‹(even (?i::?'a::semiring_parity) ∧ even (?j::?'a::semiring_parity) ∨ odd ?i ∧ odd ?j) = (?i mod (2::?'a::semiring_parity) = ?j mod (2::?'a::semiring_parity))›*) apply -\n (*goals:\n 1. ‹⟦1 ≤ n; i < dim_row (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); j < dim_col (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); ⋀i j. (even i ∧ even j ∨ odd i ∧ odd j) = (i mod 2 = j mod 2)⟧ ⟹ even i ∧ even j ⟶ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j) = complex_of_real ((- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n)›\n 2. ‹⟦1 ≤ n; i < dim_row (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); j < dim_col (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); ⋀i j. (even i ∧ even j ∨ odd i ∧ odd j) = (i mod 2 = j mod 2)⟧ ⟹ odd i ∧ odd j ⟶ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j) = complex_of_real ((- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n)›\n 3. ‹⟦1 ≤ n; i < dim_row (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); j < dim_col (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); ⋀i j. (even i ∧ even j ∨ odd i ∧ odd j) = (i mod 2 = j mod 2)⟧ ⟹ i mod 2 = j mod 2 ⟶ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j) = complex_of_real ((- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n)›\n 4. ‹⟦1 ≤ n; i < dim_row (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); j < dim_col (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); ⋀i j. (even i ∧ even j ∨ odd i ∧ odd j) = (i mod 2 = j mod 2)⟧ ⟹ i mod 2 ≠ j mod 2 ⟶ Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j) = 0›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*discuss goal 3*)\n apply ((auto)[1])\n (*discuss goal 4*)\n apply ((auto)[1])\n (*proven 4 subgoals*) .\n\nlemma iter_tensor_of_H_tensor_Id_is_HId:\n shows \"(H^⇩⊗ n) ⨂ Id 1 = HId^⇩⊗ n\"\nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0))); j < dim_col (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)))⟧ ⟹ (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j)›\n 2. ‹dim_row (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) = dim_row (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)))›\n 3. ‹dim_col (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) = dim_col (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)))›*)\n show \"dim_row ((H^⇩⊗ n) ⨂ Id 1) = dim_row (HId^⇩⊗ n)\"\n by (simp add: Quantum.Id_def (*‹Quantum.Id (?n::nat) ≡ 1⇩m ((2::nat) ^ ?n)›*))\n show \"dim_col ((H^⇩⊗ n) ⨂ Id 1) = dim_col (HId^⇩⊗ n)\"\n by (simp add: Quantum.Id_def (*‹Quantum.Id (?n::nat) ≡ 1⇩m ((2::nat) ^ ?n)›*))\nnext\n (*goal: ‹⋀(i::nat) j::nat. ⟦i < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real)))); j < dim_col (Matrix.mat ((2::nat) ^ (n + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))))⟧ ⟹ (Matrix.mat ((2::nat) ^ n) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n)) ⨂ Quantum.Id (1::nat)) $$ (i, j) = Matrix.mat ((2::nat) ^ (n + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))) $$ (i, j)›*)\n fix i :: nat and j :: nat\n assume a0: \"i < dim_row (HId^⇩⊗ n)\" and a1: \"j < dim_col (HId^⇩⊗ n)\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))))› ‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))))›*)\n then have f0: \"i < (2^(n+1)) ∧ j < (2^(n+1))\"\n by simp\n then have \"i < dim_row (H^⇩⊗ n) * dim_row (Id 1) ∧ j < dim_col (H^⇩⊗ n) * dim_col (Id 1)\"\n using Id_def (*‹Quantum.Id ?n ≡ 1⇩m (2 ^ ?n)›*) by simp\n moreover have \"dim_col (H^⇩⊗ n) ≥ 0 ∧ dim_col (Id 1) ≥ 0\"\n using Id_def (*‹Quantum.Id ?n ≡ 1⇩m (2 ^ ?n)›*) by simp\n ultimately have f1: \"((H^⇩⊗ n) ⨂ (Id 1)) $$ (i,j) \n = (H^⇩⊗ n) $$ (i div (dim_row (Id 1)),j div (dim_col (Id 1))) * \n (Id 1) $$ (i mod (dim_row (Id 1)),j mod (dim_col (Id 1)))\"\n by (simp add: Quantum.Id_def (*‹Quantum.Id (?n::nat) ≡ 1⇩m ((2::nat) ^ ?n)›*))\n show \"((H^⇩⊗ n)⨂Id 1) $$ (i,j) = (HId^⇩⊗ n) $$ (i,j)\"\n proof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹?P ∨ ?Q›\n 2. ‹?P ⟹ (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j)›\n 3. ‹?Q ⟹ (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j)›*)\n show \"(i mod 2 = j mod 2) ∨ ¬ (i mod 2 = j mod 2)\"\n by simp\n next\n (*goals:\n 1. ‹(i::nat) mod (2::nat) = (j::nat) mod (2::nat) ⟹ (Matrix.mat ((2::nat) ^ (n::nat)) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n)) ⨂ Quantum.Id (1::nat)) $$ (i, j) = Matrix.mat ((2::nat) ^ (n + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))) $$ (i, j)›\n 2. ‹(i::nat) mod (2::nat) ≠ (j::nat) mod (2::nat) ⟹ (Matrix.mat ((2::nat) ^ (n::nat)) ((2::nat) ^ n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘n⇙ j / sqrt (2::real) ^ n)) ⨂ Quantum.Id (1::nat)) $$ (i, j) = Matrix.mat ((2::nat) ^ (n + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))) $$ (i, j)›*)\n assume a2: \"(i mod 2 = j mod 2)\" (*‹(i::nat) mod (2::nat) = (j::nat) mod (2::nat)›*)\n then have \"(Id 1) $$ (i mod (dim_row (Id 1)),j mod (dim_col (Id 1))) = 1\"\n by (simp add: Quantum.Id_def (*‹Quantum.Id ?n ≡ 1⇩m (2 ^ ?n)›*))\n moreover have \"(H^⇩⊗ n) $$ (i div (dim_row (Id 1)), j div (dim_col (Id 1))) \n = (-1)^((i div (dim_row (Id 1))) ⋅⇘n⇙ (j div (dim_col (Id 1))))/(sqrt 2)^n\"\n using tensor_of_H_values (*‹⟦(?i::nat) < dim_row (Matrix.mat ((2::nat) ^ (?n::nat)) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n))); (?j::nat) < dim_col (Matrix.mat ((2::nat) ^ ?n) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)))⟧ ⟹ Matrix.mat ((2::nat) ^ ?n) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)) $$ (?i, ?j) = complex_of_real ((- (1::real)) ^ ?i ⋅⇘?n⇙ ?j / sqrt (2::real) ^ ?n)›*) Id_def (*‹Quantum.Id ?n ≡ 1⇩m (2 ^ ?n)›*) f0 (*‹i < 2 ^ (n + 1) ∧ j < 2 ^ (n + 1)›*) less_mult_imp_div_less (*‹?m < ?i * ?n ⟹ ?m div ?n < ?i›*) by simp\n ultimately show \"((H^⇩⊗ n) ⨂ Id 1) $$ (i,j) = (HId^⇩⊗ n) $$ (i,j)\"\n using a2 (*‹i mod 2 = j mod 2›*) f0 (*‹i < 2 ^ (n + 1) ∧ j < 2 ^ (n + 1)›*) f1 (*‹(Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) $$ (i, j) = Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) $$ (i div dim_row (Quantum.Id 1), j div dim_col (Quantum.Id 1)) * Quantum.Id 1 $$ (i mod dim_row (Quantum.Id 1), j mod dim_col (Quantum.Id 1))›*) Id_def (*‹Quantum.Id (?n::nat) ≡ 1⇩m ((2::nat) ^ ?n)›*) by simp\n next\n (*goal: ‹i mod 2 ≠ j mod 2 ⟹ (Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, j)›*)\n assume a2: \"¬(i mod 2 = j mod 2)\" (*‹(i::nat) mod (2::nat) ≠ (j::nat) mod (2::nat)›*)\n then have \"(Id 1) $$ (i mod (dim_row (Id 1)),j mod (dim_col (Id 1))) = 0\"\n by (simp add: Quantum.Id_def (*‹Quantum.Id ?n ≡ 1⇩m (2 ^ ?n)›*))\n then show \"((H^⇩⊗ n) ⨂ Id 1) $$ (i,j) = (HId^⇩⊗ n) $$ (i,j)\"\n using a2 (*‹i mod 2 ≠ j mod 2›*) f0 (*‹i < 2 ^ (n + 1) ∧ j < 2 ^ (n + 1)›*) f1 (*‹(Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) $$ (i, j) = Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) $$ (i div dim_row (Quantum.Id 1), j div dim_col (Quantum.Id 1)) * Quantum.Id 1 $$ (i mod dim_row (Quantum.Id 1), j mod dim_col (Quantum.Id 1))›*) by simp\n qed\nqed\n\nlemma HId_is_gate:\n assumes \"n ≥ 1\"\n shows \"gate (n+1) (HId^⇩⊗ n)\" \nproof (-)\n (*goal: ‹gate (n + 1) (Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)))›*)\n have \"(HId^⇩⊗ n) = (H^⇩⊗ n) ⨂ Id 1\"\n using iter_tensor_of_H_tensor_Id_is_HId (*‹Matrix.mat ((2::nat) ^ (?n::nat)) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)) ⨂ Quantum.Id (1::nat) = Matrix.mat ((2::nat) ^ (?n + (1::nat))) ((2::nat) ^ (?n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘?n⇙ j div (2::nat) / sqrt (2::real) ^ ?n else (0::real)))›*) by simp\n moreover have \"gate 1 (Id 1)\"\n using id_is_gate (*‹gate (?n::nat) (Quantum.Id ?n)›*) by simp\n moreover have \"gate n (H^⇩⊗ n)\"\n using H_is_gate (*‹gate 1 H›*) iter_tensor_of_gate_is_gate[of 1 H n] (*‹⟦gate 1 H; 1 ≤ n⟧ ⟹ gate (1 * n) (H ⊗⇗n⇖)›*) assms (*‹1 ≤ n›*) by (simp add: iter_tensor_of_H_rep_is_correct (*‹1 ≤ ?n ⟹ H ⊗⇗?n⇖ = Matrix.mat (2 ^ ?n) (2 ^ ?n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘?n⇙ j / sqrt 2 ^ ?n))›*))\n ultimately show \"gate (n+1) (HId^⇩⊗ n)\"\n using tensor_gate (*‹⟦gate ?m ?G1.0; gate ?n ?G2.0⟧ ⟹ gate (?m + ?n) (?G1.0 ⨂ ?G2.0)›*) by presburger\nqed\n\ntext ‹State @{term \"ψ₃\"} is obtained by the multiplication of @{term \"HId^⇩⊗ n\"} and @{term \"ψ₂\"}›\n\nabbreviation (in jozsa) ψ₃:: \"complex Matrix.mat\" where\n\"ψ₃ ≡ Matrix.mat (2^(n+1)) 1 (λ(i,j). \nif even i \n then (∑k<2^n. (-1)^(f(k) + ((i div 2) ⋅⇘n⇙ k))/((sqrt 2)^n * (sqrt 2)^(n+1))) \n else (∑k<2^n. (-1)^(f(k)+ 1 + ((i div 2) ⋅⇘n⇙ k)) /((sqrt 2)^n * (sqrt 2)^(n+1))))\"\n\nlemma (in jozsa) ψ₃_values:\n assumes \"i < dim_row ψ₃\"\n shows \"odd i ⟶ ψ₃ $$ (i,0) = (∑k<2^n. (-1)^(f(k) + 1 + ((i div 2) ⋅⇘n⇙ k))/((sqrt 2)^n * (sqrt 2)^(n+1)))\"\n using assms (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) by simp\n\nlemma (in jozsa) ψ₃_dim [simp]:\n shows \"1 < dim_row ψ₃\"\n using dim_row_mat(1) (*‹dim_row (Matrix.mat ?nr ?nc ?f) = ?nr›*) nat_neq_iff (*‹(?m ≠ ?n) = (?m < ?n ∨ ?n < ?m)›*) by fastforce\n\nlemma sum_every_odd_summand_is_zero:\n fixes n:: nat \n assumes \"n ≥ 1\"\n shows \"∀f::(nat ⇒ complex).(∀i. i<2^(n+1) ∧ odd i ⟶ f i = 0) ⟶ \n (∑k∈{0..<2^(n+1)}. f k) = (∑k∈{0..<2^n}. f (2*k))\"\n using assms (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦?m ≤ ?n; ?P ?m; ⋀n. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹∀f. (∀i. i < 2 ^ (1 + 1) ∧ odd i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (1 + 1)} = (∑k = 0..<2 ^ 1. f (2 * k))›\n 2. ‹⋀n. ⟦1 ≤ n; ∀f. (∀i. i < 2 ^ (n + 1) ∧ odd i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k))⟧ ⟹ ∀f. (∀i. i < 2 ^ (Suc n + 1) ∧ odd i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (Suc n + 1)} = (∑k = 0..<2 ^ Suc n. f (2 * k))›*)\n show \"∀f::(nat ⇒ complex).(∀i. i<2^(1+1) ∧ odd i ⟶ f i = 0) ⟶\n (∑k∈{0..<2^(1+1)}. f k) = (∑k ∈ {0..<2^1}. f (2*k))\"\n apply (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n (*goal: ‹∀f. (∀i. i < 2 ^ (1 + 1) ∧ odd i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (1 + 1)} = (∑k = 0..<2 ^ 1. f (2 * k))›*)\n proof (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹⋀f. ∀i. i < 2 ^ (1 + 1) ∧ odd i ⟶ f i = 0 ⟹ sum f {0..<2 ^ (1 + 1)} = (∑k = 0..<2 ^ 1. f (2 * k))›*)\n fix f :: \"(nat ⇒ complex)\"\n assume asm: \"(∀i. i<2^(1+1) ∧ odd i ⟶ f i = 0)\" (*‹∀i::nat. i < (2::nat) ^ ((1::nat) + (1::nat)) ∧ odd i ⟶ (f::nat ⇒ complex) i = (0::complex)›*)\n moreover have \"(∑k∈{0..<4}. f k) = f 0 + f 1 + f 2 + f 3\"\n by (simp add: add.commute (*‹?a + ?b = ?b + ?a›*) add.left_commute (*‹?b + (?a + ?c) = ?a + (?b + ?c)›*))\n moreover have \"f 1 = 0\"\n using asm (*‹∀i::nat. i < (2::nat) ^ ((1::nat) + (1::nat)) ∧ odd i ⟶ (f::nat ⇒ complex) i = (0::complex)›*) by simp\n moreover have \"f 3 = 0\"\n using asm (*‹∀i::nat. i < (2::nat) ^ ((1::nat) + (1::nat)) ∧ odd i ⟶ (f::nat ⇒ complex) i = (0::complex)›*) by simp\n moreover have \"(∑k∈{0..<2^1}. f (2*k)) = f 0 + f 2\"\n using add.commute (*‹?a + ?b = ?b + ?a›*) add.left_commute (*‹?b + (?a + ?c) = ?a + (?b + ?c)›*) by simp\n ultimately show \"(∑k∈{0..<2^(1+1)}. f k) = (∑k∈{0..<2^1}. f (2*k))\"\n by simp\n qed\n next\n (*goal: ‹⋀n::nat. ⟦(1::nat) ≤ n; ∀f::nat ⇒ complex. (∀i::nat. i < (2::nat) ^ (n + (1::nat)) ∧ odd i ⟶ f i = (0::complex)) ⟶ sum f {0::nat..<(2::nat) ^ (n + (1::nat))} = (∑k::nat = 0::nat..<(2::nat) ^ n. f ((2::nat) * k))⟧ ⟹ ∀f::nat ⇒ complex. (∀i::nat. i < (2::nat) ^ (Suc n + (1::nat)) ∧ odd i ⟶ f i = (0::complex)) ⟶ sum f {0::nat..<(2::nat) ^ (Suc n + (1::nat))} = (∑k::nat = 0::nat..<(2::nat) ^ Suc n. f ((2::nat) * k))›*)\n fix n :: nat\n assume \"n ≥ 1\" and IH: \"∀f::(nat ⇒complex).(∀i. i<2^(n+1) ∧ odd i ⟶ f i = 0) ⟶\n (∑k∈{0..<2^(n+1)}. f k) = (∑k∈{0..<2^n}. f (2*k))\" (*‹(1::nat) ≤ (n::nat)› ‹∀f::nat ⇒ complex. (∀i::nat. i < (2::nat) ^ ((n::nat) + (1::nat)) ∧ odd i ⟶ f i = (0::complex)) ⟶ sum f {0::nat..<(2::nat) ^ (n + (1::nat))} = (∑k::nat = 0::nat..<(2::nat) ^ n. f ((2::nat) * k))›*)\n show \"∀f::(nat ⇒complex).(∀i. i<2^(Suc n +1) ∧ odd i ⟶ f i = 0) ⟶\n (∑k∈{0..<2^(Suc n +1)}. f k) = (∑k∈{0..< 2^(Suc n)}. f (2*k))\"\n apply (rule allI (*‹(⋀x::?'a. (?P::?'a ⇒ bool) x) ⟹ ∀x::?'a. ?P x›*))\n (*goal: ‹∀f::nat ⇒ complex. (∀i::nat. i < (2::nat) ^ (Suc (n::nat) + (1::nat)) ∧ odd i ⟶ f i = (0::complex)) ⟶ sum f {0::nat..<(2::nat) ^ (Suc n + (1::nat))} = (∑k::nat = 0::nat..<(2::nat) ^ Suc n. f ((2::nat) * k))›*)\n proof (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹⋀f. ∀i. i < 2 ^ (Suc n + 1) ∧ odd i ⟶ f i = 0 ⟹ sum f {0..<2 ^ (Suc n + 1)} = (∑k = 0..<2 ^ Suc n. f (2 * k))›*)\n fix f :: \"nat ⇒ complex\"\n assume asm: \"(∀i. i<2^(Suc n +1) ∧ odd i ⟶ f i = 0)\" (*‹∀i::nat. i < (2::nat) ^ (Suc (n::nat) + (1::nat)) ∧ odd i ⟶ (f::nat ⇒ complex) i = (0::complex)›*)\n have f0: \"(∑k∈{0..<2^(n+1)}. f k) = (∑k∈{0..<2^n}. f (2*k))\"\n using asm (*‹∀i. i < 2 ^ (Suc n + 1) ∧ odd i ⟶ f i = 0›*) IH (*‹∀f. (∀i. i < 2 ^ (n + 1) ∧ odd i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k))›*) by simp\n have f1: \"(∑k∈{0..<2^(n+1)}. (λx. f (x+2^(n+1))) k) = (∑k∈{0..< 2^n}. (λx. f (x+2^(n+1))) (2*k))\"\n using asm (*‹∀i::nat. i < (2::nat) ^ (Suc (n::nat) + (1::nat)) ∧ odd i ⟶ (f::nat ⇒ complex) i = (0::complex)›*) IH (*‹∀f. (∀i. i < 2 ^ (n + 1) ∧ odd i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k))›*) by simp\n have \"(∑k∈{0..<2^(n+2)}. f k) = (∑k∈{0..<2^(n+1)}. f k) + (∑k∈{2^(n+1)..<2^(n+2)}. f k)\"\n by (simp add: sum.atLeastLessThan_concat (*‹⟦?m ≤ ?n; ?n ≤ ?p⟧ ⟹ sum ?g {?m..<?n} + sum ?g {?n..<?p} = sum ?g {?m..<?p}›*))\n also (*calculation: ‹sum (f::nat ⇒ complex) {0::nat..<(2::nat) ^ ((n::nat) + (2::nat))} = sum f {0::nat..<(2::nat) ^ (n + (1::nat))} + sum f {(2::nat) ^ (n + (1::nat))..<(2::nat) ^ (n + (2::nat))}›*) have \"... = (∑k∈{0..<2^n}. f (2*k)) + (∑k∈{2^(n+1)..<2^(n+2)}. f k)\"\n using f0 (*‹sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k))›*) by simp\n also (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ n. f (2 * k)) + sum f {2 ^ (n + 1)..<2 ^ (n + 2)}›*) have \"... = (∑k∈{0..<2^n}. f (2*k)) + (∑k∈{0..<2^(n+1)}. f (k+2^(n+1)))\"\n using sum.shift_bounds_nat_ivl[of \"f\" \"0\" \"2^(n+1)\" \"2^(n+1)\"] (*‹sum f {0 + 2 ^ (n + 1)..<2 ^ (n + 1) + 2 ^ (n + 1)} = (∑i = 0..<2 ^ (n + 1). f (i + 2 ^ (n + 1)))›*) by simp\n also (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ n. f (2 * k)) + (∑k = 0..<2 ^ (n + 1). f (k + 2 ^ (n + 1)))›*) have \"... = (∑k∈{0..<2^n}. f (2*k)) + (∑k∈{0..< 2^n}. (λx. f (x+2^(n+1))) (2*k))\"\n using f1 (*‹(∑k::nat = 0::nat..<(2::nat) ^ (n + (1::nat)). (f::nat ⇒ complex) (k + (2::nat) ^ ((n::nat) + (1::nat)))) = (∑k::nat = 0::nat..<(2::nat) ^ n. f ((2::nat) * k + (2::nat) ^ (n + (1::nat))))›*) by simp\n also (*calculation: ‹sum (f::nat ⇒ complex) {0::nat..<(2::nat) ^ ((n::nat) + (2::nat))} = (∑k::nat = 0::nat..<(2::nat) ^ n. f ((2::nat) * k)) + (∑k::nat = 0::nat..<(2::nat) ^ n. f ((2::nat) * k + (2::nat) ^ (n + (1::nat))))›*) have \"... = (∑k∈{0..<2^n}. f (2*k)) + (∑k∈{2^n..< 2^(n+1)}. f (2 *k))\"\n using sum.shift_bounds_nat_ivl[of \"λx. (f::nat⇒complex) (2*(x-2^n)+2^(n+1))\" \"0\" \"2^n\" \"2^n\"] (*‹(∑x = 0 + 2 ^ n..<2 ^ n + 2 ^ n. f (2 * (x - 2 ^ n) + 2 ^ (n + 1))) = (∑i = 0..<2 ^ n. f (2 * (i + 2 ^ n - 2 ^ n) + 2 ^ (n + 1)))›*) by (simp add: mult_2 (*‹2 * ?z = ?z + ?z›*))\n also (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ n. f (2 * k)) + (∑k = 2 ^ n..<2 ^ (n + 1). f (2 * k))›*) have \"... = (∑k ∈ {0..<2^(n+1)}. f (2*k))\"\n by (metis Suc_eq_plus1 (*‹Suc ?n = ?n + 1›*) lessI (*‹?n < Suc ?n›*) less_imp_le_nat (*‹?m < ?n ⟹ ?m ≤ ?n›*) one_le_numeral (*‹1 ≤ numeral ?n›*) power_increasing (*‹⟦?n ≤ ?N; 1 ≤ ?a⟧ ⟹ ?a ^ ?n ≤ ?a ^ ?N›*) sum.atLeastLessThan_concat (*‹⟦?m ≤ ?n; ?n ≤ ?p⟧ ⟹ sum ?g {?m..<?n} + sum ?g {?n..<?p} = sum ?g {?m..<?p}›*) zero_le (*‹0 ≤ ?x›*))\n finally (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ (n + 1). f (2 * k))›*) show \"(∑k∈{0..<2^((Suc n)+1)}. f k) = (∑k∈{0..< 2^(Suc n)}. f (2*k))\"\n by (metis Suc_eq_plus1 (*‹Suc ?n = ?n + 1›*) add_2_eq_Suc' (*‹?n + 2 = Suc (Suc ?n)›*))\n qed\n qed\n\nlemma sum_every_even_summand_is_zero:\n fixes n:: nat \n assumes \"n ≥ 1\"\n shows \"∀f::(nat ⇒ complex).(∀i. i<2^(n+1) ∧ even i ⟶ f i = 0) ⟶ \n (∑k∈{0..<2^(n+1)}. f k) = (∑k∈{0..< 2^n}. f (2*k+1))\"\n using assms (*‹1 ≤ n›*) proof (rule nat_induct_at_least (*‹⟦(?m::nat) ≤ (?n::nat); (?P::nat ⇒ bool) ?m; ⋀n::nat. ⟦?m ≤ n; ?P n⟧ ⟹ ?P (Suc n)⟧ ⟹ ?P ?n›*))\n (*goals:\n 1. ‹∀f. (∀i. i < 2 ^ (1 + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (1 + 1)} = (∑k = 0..<2 ^ 1. f (2 * k + 1))›\n 2. ‹⋀n. ⟦1 ≤ n; ∀f. (∀i. i < 2 ^ (n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k + 1))⟧ ⟹ ∀f. (∀i. i < 2 ^ (Suc n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (Suc n + 1)} = (∑k = 0..<2 ^ Suc n. f (2 * k + 1))›*)\n show \"∀f::(nat ⇒ complex).(∀i. i<2^(1+1) ∧ even i ⟶ f i = 0) ⟶ \n (∑k∈{0..<2^(1+1)}. f k) = (∑k∈{0..< 2^1}. f (2*k+1))\"\n apply (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n (*goal: ‹∀f. (∀i. i < 2 ^ (1 + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (1 + 1)} = (∑k = 0..<2 ^ 1. f (2 * k + 1))›*)\n proof (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹⋀f. ∀i. i < 2 ^ (1 + 1) ∧ even i ⟶ f i = 0 ⟹ sum f {0..<2 ^ (1 + 1)} = (∑k = 0..<2 ^ 1. f (2 * k + 1))›*)\n fix f :: \"nat ⇒complex\"\n assume asm: \"(∀i. i<2^(1+1) ∧ even i ⟶ f i = 0)\" (*‹∀i::nat. i < (2::nat) ^ ((1::nat) + (1::nat)) ∧ even i ⟶ (f::nat ⇒ complex) i = (0::complex)›*)\n moreover have \"(∑k∈{0..<4}. f k) = f 0 + f 1 + f 2 + f 3\"\n by (simp add: add.commute (*‹?a + ?b = ?b + ?a›*) add.left_commute (*‹?b + (?a + ?c) = ?a + (?b + ?c)›*))\n moreover have \"f 0 = 0\"\n using asm (*‹∀i. i < 2 ^ (1 + 1) ∧ even i ⟶ f i = 0›*) by simp\n moreover have \"f 2 = 0\"\n using asm (*‹∀i. i < 2 ^ (1 + 1) ∧ even i ⟶ f i = 0›*) by simp\n moreover have \"(∑k ∈ {0..< 2^1}. f (2*k+1)) = f 1 + f 3\"\n using add.commute (*‹?a + ?b = ?b + ?a›*) add.left_commute (*‹?b + (?a + ?c) = ?a + (?b + ?c)›*) by simp\n ultimately show \"(∑k∈{0..<2^(1+1)}. f k) = (∑k∈{0..< 2^1}. f (2*k+1))\"\n by simp\n qed\n next\n (*goal: ‹⋀n. ⟦1 ≤ n; ∀f. (∀i. i < 2 ^ (n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k + 1))⟧ ⟹ ∀f. (∀i. i < 2 ^ (Suc n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (Suc n + 1)} = (∑k = 0..<2 ^ Suc n. f (2 * k + 1))›*)\n fix n :: nat\n assume \"n ≥ 1\" and IH: \"∀f::(nat ⇒complex).(∀i. i<2^(n+1) ∧ even i ⟶ f i = 0) ⟶\n (∑k∈{0..<2^(n+1)}. f k) = (∑k∈{0..< 2^n}. f (2*k+1))\" (*‹(1::nat) ≤ (n::nat)› ‹∀f::nat ⇒ complex. (∀i::nat. i < (2::nat) ^ ((n::nat) + (1::nat)) ∧ even i ⟶ f i = (0::complex)) ⟶ sum f {0::nat..<(2::nat) ^ (n + (1::nat))} = (∑k::nat = 0::nat..<(2::nat) ^ n. f ((2::nat) * k + (1::nat)))›*)\n show \"∀f::(nat ⇒complex).(∀i. i<2^((Suc n)+1) ∧ even i ⟶ f i = 0) ⟶\n (∑k∈{0..<2^((Suc n)+1)}. f k) = (∑k∈{0..< 2^(Suc n)}. f (2*k+1))\"\n apply (rule allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n (*goal: ‹∀f. (∀i. i < 2 ^ (Suc n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (Suc n + 1)} = (∑k = 0..<2 ^ Suc n. f (2 * k + 1))›*)\n proof (rule impI (*‹(?P ⟹ ?Q) ⟹ ?P ⟶ ?Q›*))\n (*goal: ‹⋀f. ∀i. i < 2 ^ (Suc n + 1) ∧ even i ⟶ f i = 0 ⟹ sum f {0..<2 ^ (Suc n + 1)} = (∑k = 0..<2 ^ Suc n. f (2 * k + 1))›*)\n fix f :: \"nat ⇒complex\"\n assume asm: \"(∀i. i<2^((Suc n)+1) ∧ even i ⟶ f i = 0)\" (*‹∀i::nat. i < (2::nat) ^ (Suc (n::nat) + (1::nat)) ∧ even i ⟶ (f::nat ⇒ complex) i = (0::complex)›*)\n have f0: \"(∑k ∈{0..<2^(n+1)}. f k) = (∑k ∈ {0..< 2^n}. f (2*k+1))\"\n using asm (*‹∀i::nat. i < (2::nat) ^ (Suc (n::nat) + (1::nat)) ∧ even i ⟶ (f::nat ⇒ complex) i = (0::complex)›*) IH (*‹∀f. (∀i. i < 2 ^ (n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k + 1))›*) by simp\n have f1: \"(∑k∈{0..<2^(n+1)}. (λx. f (x+2^(n+1))) k) \n = (∑k∈{0..< 2^n}. (λx. f (x+2^(n+1))) (2*k+1))\"\n using asm (*‹∀i. i < 2 ^ (Suc n + 1) ∧ even i ⟶ f i = 0›*) IH (*‹∀f. (∀i. i < 2 ^ (n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (n + 1)} = (∑k = 0..<2 ^ n. f (2 * k + 1))›*) by simp\n have \"(∑k∈{0..<2^(n+2)}. f k) \n = (∑k∈{0..<2^(n+1)}. f k) + (∑k∈{2^(n+1)..<2^(n+2)}. f k)\"\n by (simp add: sum.atLeastLessThan_concat (*‹⟦?m ≤ ?n; ?n ≤ ?p⟧ ⟹ sum ?g {?m..<?n} + sum ?g {?n..<?p} = sum ?g {?m..<?p}›*))\n also (*calculation: ‹sum (f::nat ⇒ complex) {0::nat..<(2::nat) ^ ((n::nat) + (2::nat))} = sum f {0::nat..<(2::nat) ^ (n + (1::nat))} + sum f {(2::nat) ^ (n + (1::nat))..<(2::nat) ^ (n + (2::nat))}›*) have \"... = (∑k∈{0..< 2^n}. f (2*k+1)) + (∑k∈{2^(n+1)..<2^(n+2)}. f k)\"\n using f0 (*‹sum (f::nat ⇒ complex) {0::nat..<(2::nat) ^ ((n::nat) + (1::nat))} = (∑k::nat = 0::nat..<(2::nat) ^ n. f ((2::nat) * k + (1::nat)))›*) by simp\n also (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ n. f (2 * k + 1)) + sum f {2 ^ (n + 1)..<2 ^ (n + 2)}›*) have \"... = (∑k∈{0..< 2^n}. f (2*k+1)) + (∑k∈{0..<2^(n+1)}. f (k+(2^(n+1))))\"\n using sum.shift_bounds_nat_ivl[of \"f\" \"0\" \"2^(n+1)\" \"2^(n+1)\"] (*‹sum f {0 + 2 ^ (n + 1)..<2 ^ (n + 1) + 2 ^ (n + 1)} = (∑i = 0..<2 ^ (n + 1). f (i + 2 ^ (n + 1)))›*) by simp\n also (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ n. f (2 * k + 1)) + (∑k = 0..<2 ^ (n + 1). f (k + 2 ^ (n + 1)))›*) have \"... = (∑k∈{0..< 2^n}. f (2*k+1)) + (∑k∈{0..< 2^n}. (λx. f (x+2^(n+1))) (2*k+1))\"\n using f1 (*‹(∑k = 0..<2 ^ (n + 1). f (k + 2 ^ (n + 1))) = (∑k = 0..<2 ^ n. f (2 * k + 1 + 2 ^ (n + 1)))›*) by simp\n also (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ n. f (2 * k + 1)) + (∑k = 0..<2 ^ n. f (2 * k + 1 + 2 ^ (n + 1)))›*) have \"... = (∑k∈{0..< 2^n}. f (2*k+1)) + (∑k∈{2^n..< 2^(n+1)}. f (2 *k+1))\"\n using sum.shift_bounds_nat_ivl[of \"λx. (f::nat⇒complex) (2*(x-2^n)+1+2^(n+1))\" \"0\" \"2^n\" \"2^n\"] (*‹(∑x = 0 + 2 ^ n..<2 ^ n + 2 ^ n. f (2 * (x - 2 ^ n) + 1 + 2 ^ (n + 1))) = (∑i = 0..<2 ^ n. f (2 * (i + 2 ^ n - 2 ^ n) + 1 + 2 ^ (n + 1)))›*) by (simp add: mult_2 (*‹2 * ?z = ?z + ?z›*))\n also (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ n. f (2 * k + 1)) + (∑k = 2 ^ n..<2 ^ (n + 1). f (2 * k + 1))›*) have \"... = (∑k∈{0..< 2^(n+1)}. f (2*k+1))\"\n by (metis Suc_eq_plus1 (*‹Suc (?n::nat) = ?n + (1::nat)›*) lessI (*‹(?n::nat) < Suc ?n›*) less_imp_le_nat (*‹(?m::nat) < (?n::nat) ⟹ ?m ≤ ?n›*) one_le_numeral (*‹(1::?'a) ≤ numeral (?n::num)›*) power_increasing (*‹⟦(?n::nat) ≤ (?N::nat); (1::?'a) ≤ (?a::?'a)⟧ ⟹ ?a ^ ?n ≤ ?a ^ ?N›*) sum.atLeastLessThan_concat (*‹⟦(?m::nat) ≤ (?n::nat); ?n ≤ (?p::nat)⟧ ⟹ sum (?g::nat ⇒ ?'a) {?m..<?n} + sum ?g {?n..<?p} = sum ?g {?m..<?p}›*) zero_le (*‹(0::?'a) ≤ (?x::?'a)›*))\n finally (*calculation: ‹sum f {0..<2 ^ (n + 2)} = (∑k = 0..<2 ^ (n + 1). f (2 * k + 1))›*) show \"(∑k∈{0..<2^((Suc n)+1)}. f k) = (∑k∈{0..< 2^(Suc n)}. f (2*k+1))\"\n by (metis Suc_eq_plus1 (*‹Suc ?n = ?n + 1›*) add_2_eq_Suc' (*‹?n + 2 = Suc (Suc ?n)›*))\n qed\n qed\n\nlemma (in jozsa) iter_tensor_of_H_times_ψ₂_is_ψ₃:\n shows \"((H^⇩⊗ n) ⨂ Id 1) * ψ₂ = ψ₃\"\nproof (standard)\n (*goals:\n 1. ‹⋀i j. ⟦i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))); j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))⟧ ⟹ ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (i, j)›\n 2. ‹dim_row ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) = dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›\n 3. ‹dim_col ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) = dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*)\n fix i and j\n assume a0: \"i < dim_row ψ₃\" and a1: \"j < dim_col ψ₃\" (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))› ‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*)\n then have f0: \"i < (2^(n+1)) ∧ j = 0\"\n by simp\n have f1: \"((HId^⇩⊗ n)* ψ₂) $$ (i,j) = (∑k<(2^(n+1)). ((HId^⇩⊗ n) $$ (i,k)) * (ψ₂ $$ (k,j)))\"\n using a1 (*‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) f0 (*‹i < 2 ^ (n + 1) ∧ j = 0›*) by (simp add: atLeast0LessThan (*‹{0..<?n} = {..<?n}›*))\n show \"(((H^⇩⊗ n) ⨂ Id 1) * ψ₂) $$ (i,j) = ψ₃ $$ (i,j)\"\n proof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹?P ∨ ?Q›\n 2. ‹?P ⟹ ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (i, j)›\n 3. ‹?Q ⟹ ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (i, j)›*)\n show \"even i ∨ odd i\"\n by simp\n next\n (*goals:\n 1. ‹even i ⟹ ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (i, j)›\n 2. ‹odd i ⟹ ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (i, j)›*)\n assume a2: \"even i\" (*‹even (i::nat)›*)\n have \"(¬(i mod 2 = k mod 2) ∧ k<dim_col (HId^⇩⊗ n)) ⟶ ((HId^⇩⊗ n) $$ (i,k)) * (ψ₂ $$ (k,j)) = 0\" for k\n using f0 (*‹i < 2 ^ (n + 1) ∧ j = 0›*) by simp\n then have \"k<(2^(n+1)) ∧ odd k ⟶ ((HId^⇩⊗ n) $$ (i,k)) * (ψ₂ $$ (k,j)) = 0\" for k\n using a2 (*‹even i›*) mod_2_is_both_even_or_odd (*‹(even ?i ∧ even ?j ∨ odd ?i ∧ odd ?j) = (?i mod 2 = ?j mod 2)›*) f0 (*‹i < 2 ^ (n + 1) ∧ j = 0›*) by (metis (no_types, lifting) dim_col_mat( (*‹dim_col (Matrix.mat ?nr ?nc ?f) = ?nc›*) 1))\n then have \"(∑k∈{(0::nat)..<(2^(n+1))}. ((HId^⇩⊗ n) $$ (i,k)) * (ψ₂ $$ (k,j)))\n = (∑k∈{(0::nat)..< (2^n)}. ((HId^⇩⊗ n) $$ (i,2*k)) * (ψ₂ $$ (2*k,j)))\"\n using sum_every_odd_summand_is_zero (*‹1 ≤ ?n ⟹ ∀f. (∀i. i < 2 ^ (?n + 1) ∧ odd i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (?n + 1)} = (∑k = 0..<2 ^ ?n. f (2 * k))›*) dim (*‹1 ≤ n›*) by simp\n moreover have \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k)) * (ψ₂ $$ (2*k,j))) \n = (∑k<2^n.(-1)^((i div 2) ⋅⇘n⇙ k)/(sqrt(2)^n) *((-1)^f(k))/(sqrt(2)^(n+1)))\"\n proof (-)\n (*goal: ‹(∑k<2 ^ n. Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, 2 * k) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (2 * k, j)) = complex_of_real (∑k<2 ^ n. (- 1) ^ i div 2 ⋅⇘n⇙ k / sqrt 2 ^ n * (- 1) ^ f k / sqrt 2 ^ (n + 1))›*)\n have \"(even k ∧ k<dim_row ψ₂) ⟶ (ψ₂ $$ (k,j)) = ((-1)^f(k div 2))/(sqrt(2)^(n+1))\" for k\n using a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) by simp\n then have \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k)) * (ψ₂ $$ (2*k,j))) \n = (∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k)) *((-1)^f((2*k) div 2))/(sqrt(2)^(n+1)))\"\n by simp\n moreover have \"(even k ∧ k<dim_col (HId^⇩⊗ n))\n ⟶ ((HId^⇩⊗ n) $$ (i,k)) = (-1)^ ((i div 2) ⋅⇘n⇙ (k div 2))/(sqrt(2)^n)\" for k\n using a2 (*‹even i›*) a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) by simp\n ultimately have \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k)) * (ψ₂ $$ (2*k,j))) \n = (∑k<2^n. (-1)^((i div 2) ⋅⇘n⇙ ((2*k) div 2))/(sqrt(2)^n) * \n ((-1)^f((2*k) div 2))/(sqrt(2)^(n+1)))\"\n by simp\n then show \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k)) * (ψ₂ $$ (2*k,j))) \n = (∑k<2^n. (-1)^((i div 2) ⋅⇘n⇙ k)/(sqrt(2)^n) *((-1)^f(k))/(sqrt(2)^(n+1)))\"\n by simp\n qed\n ultimately have \"((HId^⇩⊗ n)* ψ₂) $$ (i,j) = (∑k<2^n. (-1)^((i div 2) ⋅⇘n⇙ k)/(sqrt(2)^n) \n * ((-1)^f(k))/(sqrt(2)^(n+1)))\"\n using f1 (*‹(Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))) * Matrix.mat ((2::nat) ^ (n + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat))))) $$ (i::nat, j::nat) = (∑k::nat<(2::nat) ^ (n + (1::nat)). Matrix.mat ((2::nat) ^ (n + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))) $$ (i, k) * Matrix.mat ((2::nat) ^ (n + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ f (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))) $$ (k, j))›*) by (metis atLeast0LessThan (*‹{0..<?n} = {..<?n}›*))\n also (*calculation: ‹(Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = complex_of_real (∑k<2 ^ n. (- 1) ^ i div 2 ⋅⇘n⇙ k / sqrt 2 ^ n * (- 1) ^ f k / sqrt 2 ^ (n + 1))›*) have \"... = (∑k<2^n. (-1)^(f(k)+((i div 2) ⋅⇘n⇙ k))/((sqrt(2)^n)*(sqrt(2)^(n+1))))\"\n by (simp add: power_add (*‹?a ^ (?m + ?n) = ?a ^ ?m * ?a ^ ?n›*) mult.commute (*‹?a * ?b = ?b * ?a›*))\n finally (*calculation: ‹(Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = complex_of_real (∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))›*) have \"((HId^⇩⊗ n)* ψ₂) $$ (i,j) = (∑k<2^n. (-1)^(f(k)+((i div 2) ⋅⇘n⇙ k))/((sqrt(2)^n)*(sqrt(2)^(n+1))))\"\n by simp\n moreover have \"ψ₃ $$ (i,j) = (∑k<2^n. (-1)^(f(k) + ((i div 2) ⋅⇘n⇙ k))/(sqrt(2)^n * sqrt(2)^(n+1)))\"\n using a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) a1 (*‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) a2 (*‹even i›*) by simp\n ultimately show \"(((H^⇩⊗ n) ⨂ Id 1)* ψ₂) $$ (i,j) = ψ₃ $$ (i,j)\"\n using iter_tensor_of_H_tensor_Id_is_HId (*‹Matrix.mat ((2::nat) ^ (?n::nat)) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)) ⨂ Quantum.Id (1::nat) = Matrix.mat ((2::nat) ^ (?n + (1::nat))) ((2::nat) ^ (?n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘?n⇙ j div (2::nat) / sqrt (2::real) ^ ?n else (0::real)))›*) dim (*‹1 ≤ n›*) by simp\n next\n (*goal: ‹odd i ⟹ ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (i, j)›*)\n assume a2: \"odd i\" (*‹odd (i::nat)›*)\n have \"(¬(i mod 2 = k mod 2) ∧ k<dim_col (HId^⇩⊗ n)) ⟶ ((HId^⇩⊗ n) $$ (i,k)) * (ψ₂ $$ (k,j)) = 0\" for k\n using f0 (*‹i < 2 ^ (n + 1) ∧ j = 0›*) by simp\n then have \"k<(2^(n+1)) ∧ even k ⟶ ((HId^⇩⊗ n) $$ (i,k)) * (ψ₂ $$ (k,j)) = 0\" for k\n using a2 (*‹odd i›*) mod_2_is_both_even_or_odd (*‹(even (?i::?'a::semiring_parity) ∧ even (?j::?'a::semiring_parity) ∨ odd ?i ∧ odd ?j) = (?i mod (2::?'a::semiring_parity) = ?j mod (2::?'a::semiring_parity))›*) f0 (*‹i < 2 ^ (n + 1) ∧ j = 0›*) by (metis (no_types, lifting) dim_col_mat( (*‹dim_col (Matrix.mat ?nr ?nc ?f) = ?nc›*) 1))\n then have \"(∑k∈{0..<2^(n+1)}. ((HId^⇩⊗ n) $$ (i,k)) * (ψ₂ $$ (k,j)))\n = (∑k∈{0..<2^n}. ((HId^⇩⊗ n) $$ (i,2*k+1)) * (ψ₂ $$ (2*k+1,j)))\"\n using sum_every_even_summand_is_zero (*‹1 ≤ ?n ⟹ ∀f. (∀i. i < 2 ^ (?n + 1) ∧ even i ⟶ f i = 0) ⟶ sum f {0..<2 ^ (?n + 1)} = (∑k = 0..<2 ^ ?n. f (2 * k + 1))›*) dim (*‹1 ≤ n›*) by simp\n moreover have \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k+1)) * (ψ₂ $$ (2*k+1,j))) \n = (∑ k<2^n. (-1)^((i div 2) ⋅⇘n⇙ k)/(sqrt(2)^n) * ((-1)^(f(k)+1))/(sqrt(2)^(n+1)))\"\n proof (-)\n (*goal: ‹(∑k<2 ^ n. Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, 2 * k + 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (2 * k + 1, j)) = complex_of_real (∑k<2 ^ n. (- 1) ^ i div 2 ⋅⇘n⇙ k / sqrt 2 ^ n * (- 1) ^ (f k + 1) / sqrt 2 ^ (n + 1))›*)\n have \"(odd k ∧ k<dim_row ψ₂) ⟶ (ψ₂ $$ (k,j)) = ((-1)^(f(k div 2)+1))/(sqrt(2)^(n+1))\" for k\n using a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) a2 (*‹odd i›*) by simp\n then have f2: \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k+1)) * (ψ₂ $$ (2*k+1,j))) \n = (∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k+1)) * ((-1)^(f((2*k+1) div 2)+1))/(sqrt(2)^(n+1)))\"\n by simp\n have \"i < dim_row (HId^⇩⊗ n)\"\n using f0 (*‹i < 2 ^ (n + 1) ∧ j = 0›*) a2 (*‹odd i›*) mod_2_is_both_even_or_odd (*‹(even ?i ∧ even ?j ∨ odd ?i ∧ odd ?j) = (?i mod 2 = ?j mod 2)›*) by simp\n then have \"((i mod 2 = k mod 2) ∧ k<dim_col (HId^⇩⊗ n))\n ⟶ ((HId^⇩⊗ n) $$ (i,k)) = (-1)^((i div 2) ⋅⇘n⇙ (k div 2))/(sqrt(2)^n) \" for k\n using a2 (*‹odd (i::nat)›*) a0 (*‹i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) a1 (*‹(j::nat) < dim_col (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) f0 (*‹i < 2 ^ (n + 1) ∧ j = 0›*) dim (*‹(1::nat) ≤ (n::nat)›*) HId_values (*‹⟦1 ≤ ?n; ?i < dim_row (Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0))); ?j < dim_col (Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0)))⟧ ⟹ even ?i ∧ even ?j ⟶ Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0)) $$ (?i, ?j) = complex_of_real ((- 1) ^ ?i div 2 ⋅⇘?n⇙ ?j div 2 / sqrt 2 ^ ?n)› ‹⟦1 ≤ ?n; ?i < dim_row (Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0))); ?j < dim_col (Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0)))⟧ ⟹ odd ?i ∧ odd ?j ⟶ Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0)) $$ (?i, ?j) = complex_of_real ((- 1) ^ ?i div 2 ⋅⇘?n⇙ ?j div 2 / sqrt 2 ^ ?n)› ‹⟦1 ≤ ?n; ?i < dim_row (Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0))); ?j < dim_col (Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0)))⟧ ⟹ ?i mod 2 = ?j mod 2 ⟶ Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0)) $$ (?i, ?j) = complex_of_real ((- 1) ^ ?i div 2 ⋅⇘?n⇙ ?j div 2 / sqrt 2 ^ ?n)› ‹⟦(1::nat) ≤ (?n::nat); (?i::nat) < dim_row (Matrix.mat ((2::nat) ^ (?n + (1::nat))) ((2::nat) ^ (?n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘?n⇙ j div (2::nat) / sqrt (2::real) ^ ?n else (0::real)))); (?j::nat) < dim_col (Matrix.mat ((2::nat) ^ (?n + (1::nat))) ((2::nat) ^ (?n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘?n⇙ j div (2::nat) / sqrt (2::real) ^ ?n else (0::real))))⟧ ⟹ ?i mod (2::nat) ≠ ?j mod (2::nat) ⟶ Matrix.mat ((2::nat) ^ (?n + (1::nat))) ((2::nat) ^ (?n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘?n⇙ j div (2::nat) / sqrt (2::real) ^ ?n else (0::real))) $$ (?i, ?j) = (0::complex)›*) by simp\n moreover have \"odd k ⟶ (i mod 2 = k mod 2)\" for k\n using a2 (*‹odd i›*) mod_2_is_both_even_or_odd (*‹(even ?i ∧ even ?j ∨ odd ?i ∧ odd ?j) = (?i mod 2 = ?j mod 2)›*) by auto\n ultimately have \"(odd k ∧ k<dim_col (HId^⇩⊗ n))\n ⟶ ((HId^⇩⊗ n) $$ (i,k)) = (-1)^((i div 2) ⋅⇘n⇙ (k div 2))/(sqrt(2)^n)\" for k\n by simp\n then have \"k<2^n ⟶ ((HId^⇩⊗ n) $$ (i,2*k+1)) = (-1)^((i div 2) ⋅⇘n⇙ ((2*k+1) div 2))/(sqrt(2)^n) \" for k\n by simp\n then have \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k+1)) * (ψ₂ $$ (2*k+1,j))) \n = (∑k<2^n. (-1)^((i div 2) ⋅⇘n⇙ ((2*k+1) div 2))/(sqrt(2)^n) \n * ((-1)^(f((2*k+1) div 2)+1))/(sqrt(2)^(n+1)))\"\n using f2 (*‹(∑k::nat<(2::nat) ^ n. Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))) $$ (i::nat, (2::nat) * k + (1::nat)) * Matrix.mat ((2::nat) ^ (n + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then (- (1::real)) ^ (f::nat ⇒ nat) (i div (2::nat)) / sqrt (2::real) ^ (n + (1::nat)) else (- (1::real)) ^ (f (i div (2::nat)) + (1::nat)) / sqrt (2::real) ^ (n + (1::nat)))) $$ ((2::nat) * k + (1::nat), j::nat)) = (∑k::nat<(2::nat) ^ n. Matrix.mat ((2::nat) ^ (n + (1::nat))) ((2::nat) ^ (n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘n⇙ j div (2::nat) / sqrt (2::real) ^ n else (0::real))) $$ (i, (2::nat) * k + (1::nat)) * (- (1::complex)) ^ (f (((2::nat) * k + (1::nat)) div (2::nat)) + (1::nat)) / complex_of_real (sqrt (2::real) ^ (n + (1::nat))))›*) by simp\n then show \"(∑k<2^n. ((HId^⇩⊗ n) $$ (i,2*k+1)) * (ψ₂ $$ (2*k+1,j))) \n = (∑k<2^n. (-1)^((i div 2) ⋅⇘n⇙ k)/(sqrt(2)^n) *((-1)^(f(k)+1))/(sqrt(2)^(n+1)))\"\n by simp\n qed\n ultimately have \"((HId^⇩⊗ n)* ψ₂) $$ (i,j) = (∑k<2^n. (-1)^((i div 2) ⋅⇘n⇙ k)/(sqrt(2)^n) \n * ((-1)^(f(k)+1))/(sqrt(2)^(n+1)))\"\n using f1 (*‹(Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = (∑k<2 ^ (n + 1). Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) $$ (i, k) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (k, j))›*) by (metis atLeast0LessThan (*‹{0::nat..<?n::nat} = {..<?n}›*))\n also (*calculation: ‹(Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = complex_of_real (∑k<2 ^ n. (- 1) ^ i div 2 ⋅⇘n⇙ k / sqrt 2 ^ n * (- 1) ^ (f k + 1) / sqrt 2 ^ (n + 1))›*) have \"... = (∑k<2^n. (-1)^(f(k)+1+((i div 2) ⋅⇘n⇙ k))/((sqrt(2)^n)*(sqrt(2)^(n+1))))\"\n by (simp add: mult.commute (*‹?a * ?b = ?b * ?a›*) power_add (*‹?a ^ (?m + ?n) = ?a ^ ?m * ?a ^ ?n›*))\n finally (*calculation: ‹(Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘n⇙ j div 2 / sqrt 2 ^ n else 0)) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) $$ (i, j) = complex_of_real (∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))›*) have \"((HId^⇩⊗ n)* ψ₂) $$ (i,j) \n = (∑k< 2^n. (-1)^(f(k)+1+((i div 2) ⋅⇘n⇙ k))/((sqrt(2)^n)*(sqrt(2)^(n+1))))\"\n by simp\n then show \"(((H^⇩⊗ n) ⨂ Id 1)* ψ₂) $$ (i,j) = ψ₃ $$ (i,j)\"\n using iter_tensor_of_H_tensor_Id_is_HId (*‹Matrix.mat (2 ^ ?n) (2 ^ ?n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘?n⇙ j / sqrt 2 ^ ?n)) ⨂ Quantum.Id 1 = Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0))›*) dim (*‹1 ≤ n›*) a2 (*‹odd (i::nat)›*) a0 (*‹(i::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) a1 (*‹j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) by simp\n qed\nnext\n (*goals:\n 1. ‹dim_row ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) = dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›\n 2. ‹dim_col ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) = dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*)\n show \"dim_row (((H^⇩⊗ n) ⨂ Id 1) * ψ₂) = dim_row ψ₃\"\n using iter_tensor_of_H_tensor_Id_is_HId (*‹Matrix.mat ((2::nat) ^ (?n::nat)) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)) ⨂ Quantum.Id (1::nat) = Matrix.mat ((2::nat) ^ (?n + (1::nat))) ((2::nat) ^ (?n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘?n⇙ j div (2::nat) / sqrt (2::real) ^ ?n else (0::real)))›*) dim (*‹1 ≤ n›*) by simp\nnext\n (*goal: ‹dim_col ((Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))) = dim_col (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*)\n show \"dim_col (((H^⇩⊗ n) ⨂ Id 1)* ψ₂) = dim_col ψ₃\"\n using iter_tensor_of_H_tensor_Id_is_HId (*‹Matrix.mat ((2::nat) ^ (?n::nat)) ((2::nat) ^ ?n) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ (- (1::real)) ^ i ⋅⇘?n⇙ j / sqrt (2::real) ^ ?n)) ⨂ Quantum.Id (1::nat) = Matrix.mat ((2::nat) ^ (?n + (1::nat))) ((2::nat) ^ (?n + (1::nat))) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if i mod (2::nat) = j mod (2::nat) then (- (1::real)) ^ i div (2::nat) ⋅⇘?n⇙ j div (2::nat) / sqrt (2::real) ^ ?n else (0::real)))›*) dim (*‹1 ≤ n›*) by simp\nqed\n\nlemma (in jozsa) ψ₃_is_state:\n shows \"state (n+1) ψ₃\"\nproof (-)\n (*goal: ‹state (n + 1) (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*)\n have \"((H^⇩⊗ n) ⨂ Id 1) * ψ₂ = ψ₃\"\n using \"iter_tensor_of_H_times_ψ₂_is_ψ₃\" (*‹(Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))›*) by simp\n moreover have \"gate (n+1) ((H^⇩⊗ n) ⨂ Id 1)\"\n using iter_tensor_of_H_tensor_Id_is_HId (*‹Matrix.mat (2 ^ ?n) (2 ^ ?n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘?n⇙ j / sqrt 2 ^ ?n)) ⨂ Quantum.Id 1 = Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0))›*) HId_is_gate (*‹1 ≤ ?n ⟹ gate (?n + 1) (Matrix.mat (2 ^ (?n + 1)) (2 ^ (?n + 1)) (λx. complex_of_real (case x of (i, j) ⇒ if i mod 2 = j mod 2 then (- 1) ^ i div 2 ⋅⇘?n⇙ j div 2 / sqrt 2 ^ ?n else 0)))›*) dim (*‹1 ≤ n›*) by simp\n moreover have \"state (n+1) ψ₂\"\n using \"ψ₂_is_state\" (*‹state (n + 1) (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))›*) by simp\n ultimately show \"state (n+1) ψ₃\"\n using gate_on_state_is_state (*‹⟦gate ?n ?A; state ?n ?v⟧ ⟹ state ?n (?A * ?v)›*) dim (*‹(1::nat) ≤ (n::nat)›*) by (metis (no_types, lifting))\nqed\n\ntext ‹\nFinally, all steps are put together. The result depends on the function f. If f is constant\nthe first n qubits are 0, if f is balanced there is at least one qubit in state 1 among the \nfirst n qubits. \nThe algorithm only uses one evaluation of f(x) and will always succeed. \n›\n\ndefinition (in jozsa) jozsa_algo:: \"complex Matrix.mat\" where \n\"jozsa_algo ≡ ((H ⊗⇗n⇖) ⨂ Id 1) * (U⇩f * (((H ⊗⇗n⇖) * ( |zero⟩ ⊗⇗n⇖)) ⨂ (H * |one⟩)))\"\n\nlemma (in jozsa) jozsa_algo_result [simp]: \n shows \"jozsa_algo = ψ₃\" \n using jozsa_algo_def (*‹jozsa_algo ≡ (H ⊗⇗n::nat⇖ ⨂ Quantum.Id (1::nat)) * (U⇩f * (H ⊗⇗n⇖ * |Deutsch.zero⟩ ⊗⇗n⇖ ⨂ H * |Deutsch.one⟩))›*) \"H_on_ket_one_is_ψ₁₁\" (*‹H * |Deutsch.one⟩ = Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))›*) iter_tensor_of_H_on_zero_tensor (*‹1 ≤ ?n ⟹ H ⊗⇗?n⇖ * |Deutsch.zero⟩ ⊗⇗?n⇖ = Matrix.mat (2 ^ ?n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ ?n))›*) \"ψ₁₀_tensor_ψ₁₁_is_ψ₁\" (*‹1 ≤ ?n ⟹ Matrix.mat (2 ^ ?n) 1 (λx. complex_of_real (case x of (i, j) ⇒ 1 / sqrt 2 ^ ?n)) ⨂ Matrix.mat 2 1 (λx. complex_of_real (case x of (i, j) ⇒ if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)) = Matrix.mat (2 ^ (?n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (?n + 1) else - 1 / sqrt 2 ^ (?n + 1)))›*) \"jozsa_transform_times_ψ₁_is_ψ₂\" (*‹U⇩f * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1)))›*) \"iter_tensor_of_H_times_ψ₂_is_ψ₃\" (*‹(Matrix.mat (2 ^ n) (2 ^ n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘n⇙ j / sqrt 2 ^ n)) ⨂ Quantum.Id 1) * Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))›*) dim (*‹1 ≤ n›*) iter_tensor_of_H_rep_is_correct (*‹1 ≤ ?n ⟹ H ⊗⇗?n⇖ = Matrix.mat (2 ^ ?n) (2 ^ ?n) (λx. complex_of_real (case x of (i, j) ⇒ (- 1) ^ i ⋅⇘?n⇙ j / sqrt 2 ^ ?n))›*) by simp\n\nlemma (in jozsa) jozsa_algo_result_is_state: \n shows \"state (n+1) jozsa_algo\" \n using \"ψ₃_is_state\" (*‹state (n + 1) (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) by simp\n\nlemma (in jozsa) prob0_fst_qubits_of_jozsa_algo: \n shows \"(prob0_fst_qubits n jozsa_algo) = (∑j∈{0,1}. (cmod(jozsa_algo $$ (j,0)))²)\"\n using prob0_fst_qubits_eq (*‹prob0_fst_qubits ?n ?v = (cmod (?v $$ (0, 0)))² + (cmod (?v $$ (1, 0)))²›*) by simp\n\ntext ‹General lemmata required to compute probabilities.›\n\nlemma aux_comp_with_sqrt2:\n shows \"(sqrt 2)^n * (sqrt 2)^n = 2^n\"\n by (smt (verit) power_mult_distrib (*‹((?a::?'a) * (?b::?'a)) ^ (?n::nat) = ?a ^ ?n * ?b ^ ?n›*) real_sqrt_mult_self (*‹sqrt (?a::real) * sqrt ?a = ¦?a¦›*))\n\nlemma aux_comp_with_sqrt2_bis [simp]:\n shows \"2^n/(sqrt(2)^n * sqrt(2)^(n+1)) = 1/sqrt 2\"\n using aux_comp_with_sqrt2 (*‹sqrt 2 ^ ?n * sqrt 2 ^ ?n = 2 ^ ?n›*) by (simp add: mult.left_commute (*‹(?b::?'a::ab_semigroup_mult) * ((?a::?'a::ab_semigroup_mult) * (?c::?'a::ab_semigroup_mult)) = ?a * (?b * ?c)›*))\n\nlemma aux_ineq_with_card: \n fixes g:: \"nat ⇒ nat\" and A:: \"nat set\"\n assumes \"finite A\" \n shows \"(∑k∈A. (-1)^(g k)) ≤ card A\" and \"(∑k∈A. (-1)^(g k)) ≥ -card A\" \n \n (*goals:\n 1. ‹(∑k∈A. (- 1) ^ g k) ≤ int (card A)›\n 2. ‹- int (card A) ≤ (∑k∈A. (- 1) ^ g k)›\n discuss goal 1*)\n apply (smt (verit) assms (*‹finite A›*) neg_one_even_power (*‹even ?n ⟹ (- 1) ^ ?n = 1›*) neg_one_odd_power (*‹odd ?n ⟹ (- 1) ^ ?n = - 1›*) card_eq_sum (*‹card ?A = (∑x∈?A. 1)›*) of_nat_1 (*‹of_nat 1 = 1›*) of_nat_sum (*‹of_nat (sum ?f ?A) = (∑x∈?A. of_nat (?f x))›*) sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*))\n (*discuss goal 2*)\n apply (smt (verit) assms (*‹finite A›*) neg_one_even_power (*‹even ?n ⟹ (- 1) ^ ?n = 1›*) neg_one_odd_power (*‹odd ?n ⟹ (- 1) ^ ?n = - 1›*) card_eq_sum (*‹card ?A = (∑x∈?A. 1)›*) of_nat_1 (*‹of_nat 1 = 1›*) of_nat_sum (*‹of_nat (sum ?f ?A) = (∑x∈?A. of_nat (?f x))›*) sum_mono (*‹(⋀i. i ∈ ?K ⟹ ?f i ≤ ?g i) ⟹ sum ?f ?K ≤ sum ?g ?K›*) sum_negf (*‹(∑x∈?A. - ?f x) = - sum ?f ?A›*))\n (*proven 2 subgoals*) .\n\nlemma aux_comp_with_cmod:\n fixes g:: \"nat ⇒ nat\"\n assumes \"(∀x<2^n. g x = 0) ∨ (∀x<2^n. g x = 1)\"\n shows \"(cmod (∑k<2^n. (-1)^(g k)))² = 2^(2*n)\"\nproof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹?P ∨ ?Q›\n 2. ‹?P ⟹ (cmod (∑k<2 ^ n. (- 1) ^ g k))² = 2 ^ (2 * n)›\n 3. ‹?Q ⟹ (cmod (∑k<2 ^ n. (- 1) ^ g k))² = 2 ^ (2 * n)›*)\n show \"(∀x<2^n. g x = 0) ∨ (∀x<2^n. g x = 1)\"\n using assms (*‹(∀x<2 ^ n. g x = 0) ∨ (∀x<2 ^ n. g x = 1)›*) by simp\nnext\n (*goals:\n 1. ‹∀x<2 ^ n. g x = 0 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ g k))² = 2 ^ (2 * n)›\n 2. ‹∀x<2 ^ n. g x = 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ g k))² = 2 ^ (2 * n)›*)\n assume \"∀x<2^n. g x = 0\" (*‹∀x<(2::nat) ^ (n::nat). (g::nat ⇒ nat) x = (0::nat)›*)\n then have \"(cmod (∑k<2^n. (-1)^(g k)))² = (2^n)²\"\n by (simp add: norm_power (*‹norm ((?x::?'a) ^ (?n::nat)) = norm ?x ^ ?n›*))\n then show \"?thesis\"\n (*goal: ‹(cmod (∑k::nat<(2::nat) ^ (n::nat). (- (1::complex)) ^ (g::nat ⇒ nat) k))² = (2::real) ^ ((2::nat) * n)›*)\n by (simp add: power_even_eq (*‹?a ^ (2 * ?n) = (?a ^ ?n)²›*))\nnext\n (*goal: ‹∀x<2 ^ n. g x = 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ g k))² = 2 ^ (2 * n)›*)\n assume \"∀x<2^n. g x = 1\" (*‹∀x<(2::nat) ^ (n::nat). (g::nat ⇒ nat) x = (1::nat)›*)\n then have \"(cmod (∑k<2^n. (-1)^(g k)))² = (2^n)²\"\n by (simp add: norm_power (*‹norm ((?x::?'a::real_normed_div_algebra) ^ (?n::nat)) = norm ?x ^ ?n›*))\n then show \"?thesis\"\n (*goal: ‹(cmod (∑k<2 ^ n. (- 1) ^ g k))² = 2 ^ (2 * n)›*)\n by (simp add: power_even_eq (*‹?a ^ (2 * ?n) = (?a ^ ?n)²›*))\nqed\n\nlemma cmod_less:\n fixes a n:: int\n assumes \"a < n\" and \"a > -n\"\n shows \"cmod a < n\" \n using assms (*‹a < n› ‹- n < a›*) by simp\n\nlemma square_less:\n fixes a n:: real\n assumes \"a < n\" and \"a > -n\" \n shows \"a² < n²\"\n using assms (*‹a < n› ‹- (n::real) < (a::real)›*) by (smt (verit) power2_eq_iff (*‹(?x² = ?y²) = (?x = ?y ∨ ?x = - ?y)›*) power2_minus (*‹(- ?a)² = ?a²›*) power_less_imp_less_base (*‹⟦?a ^ ?n < ?b ^ ?n; 0 ≤ ?b⟧ ⟹ ?a < ?b›*))\n\nlemma cmod_square_real [simp]:\n fixes n:: real\n shows \"(cmod n)² = n²\" \n by simp\n\nlemma aux_comp_sum_divide_cmod:\n fixes n:: nat and g:: \"nat ⇒ int\" and a:: real\n shows \"(cmod(complex_of_real(∑k<n. g k / a)))² = (cmod (∑k<n. g k) / a)²\"\n by (metis cmod_square_real (*‹(cmod (complex_of_real (?n::real)))² = ?n²›*) of_int_sum (*‹of_int (sum (?f::?'b::type ⇒ int) (?A::?'b::type set)) = (∑x::?'b::type∈?A. of_int (?f x))›*) of_real_of_int_eq (*‹of_real (real_of_int (?z::int)) = of_int ?z›*) power_divide (*‹((?a::?'a::field) / (?b::?'a::field)) ^ (?n::nat) = ?a ^ ?n / ?b ^ ?n›*) sum_divide_distrib (*‹sum (?f::?'b::type ⇒ ?'a::field) (?A::?'b::type set) / (?r::?'a::field) = (∑n::?'b::type∈?A. ?f n / ?r)›*))\n\n\ntext ‹\nThe function is constant if and only if the first n qubits are 0. So, if the function is constant, \nthen the probability of measuring 0 for the first n qubits is 1.\n›\n\nlemma (in jozsa) prob0_jozsa_algo_of_const_0:\n assumes \"const 0\"\n shows \"prob0_fst_qubits n jozsa_algo = 1\"\nproof (-)\n (*goal: ‹prob0_fst_qubits n jozsa_algo = 1›*)\n have \"prob0_fst_qubits n jozsa_algo = (∑j∈{0,1}. (cmod(jozsa_algo $$ (j,0)))²)\"\n using prob0_fst_qubits_of_jozsa_algo (*‹prob0_fst_qubits (n::nat) jozsa_algo = (∑j::nat∈{0::nat, 1::nat}. (cmod (jozsa_algo $$ (j, 0::nat)))²)›*) by simp\n moreover have \"(cmod(jozsa_algo $$ (0,0)))² = 1/2\"\n proof (-)\n (*goal: ‹(cmod (jozsa_algo $$ (0, 0)))² = 1 / 2›*)\n have \"k<2^n ⟶ ((0 div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹(?m::nat) < (2::nat) ^ (?n::nat) ⟹ 0::nat ⋅⇘?n⇙ ?m = (0::nat)›*) by simp\n then have \"(cmod(jozsa_algo $$ (0,0)))² = (cmod(∑k::nat<2^n. 1/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using jozsa_algo_result (*‹jozsa_algo = Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat)))))›*) const_def (*‹const ?c = (∀x∈{i. i < 2 ^ n}. f x = ?c)›*) assms (*‹const 0›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. 1 / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = (cmod((2::nat)^n/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_real (real (2 ^ n) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = (cmod(1/(sqrt(2))))²\"\n using aux_comp_with_sqrt2_bis (*‹2 ^ ?n / (sqrt 2 ^ ?n * sqrt 2 ^ (?n + 1)) = 1 / sqrt 2›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (0::nat, 0::nat)))² = (cmod (complex_of_real ((1::real) / sqrt (2::real))))²›*) have \"... = 1/2\"\n by (simp add: norm_divide (*‹norm (?a / ?b) = norm ?a / norm ?b›*) power2_eq_square (*‹?a² = ?a * ?a›*))\n finally (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = 1 / 2›*) show \"?thesis\"\n (*goal: ‹(cmod (jozsa_algo $$ (0, 0)))² = 1 / 2›*)\n by simp\n qed\n moreover have \"(cmod(jozsa_algo $$ (1,0)))² = 1/2\"\n proof (-)\n (*goal: ‹(cmod (jozsa_algo $$ (1, 0)))² = 1 / 2›*)\n have \"k<2^n ⟶ ((1 div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹(?m::nat) < (2::nat) ^ (?n::nat) ⟹ 0::nat ⋅⇘?n⇙ ?m = (0::nat)›*) by simp\n then have \"k<2^n ⟶ f k + 1 + ((1 div 2) ⋅⇘n⇙ k) = 1\" for k :: nat\n using const_def (*‹const ?c = (∀x∈{i. i < 2 ^ n}. f x = ?c)›*) assms (*‹const 0›*) by simp\n moreover have \"(cmod(jozsa_algo $$ (1,0)))² \n = (cmod (∑k::nat<2^n. (-1)^(f k + 1 + ((1 div 2) ⋅⇘n⇙ k))/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using \"ψ₃_dim\" (*‹1 < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) by simp\n ultimately have \"(cmod(jozsa_algo $$ (1,0)))² = (cmod(∑k::nat<2^n. -1/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n by (smt (verit) lessThan_iff (*‹(?i ∈ {..<?k}) = (?i < ?k)›*) power_one_right (*‹?a ^ 1 = ?a›*) sum.cong (*‹⟦?A = ?B; ⋀x. x ∈ ?B ⟹ ?g x = ?h x⟧ ⟹ sum ?g ?A = sum ?h ?B›*))\n also (*calculation: ‹(cmod (jozsa_algo $$ (1, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. - 1 / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = (cmod(-1/(sqrt(2))))²\"\n using aux_comp_with_sqrt2_bis (*‹2 ^ ?n / (sqrt 2 ^ ?n * sqrt 2 ^ (?n + 1)) = 1 / sqrt 2›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (1, 0)))² = (cmod (complex_of_real (- 1 / sqrt 2)))²›*) have \"... = 1/2\"\n by (simp add: norm_divide (*‹norm (?a / ?b) = norm ?a / norm ?b›*) power2_eq_square (*‹?a² = ?a * ?a›*))\n finally (*calculation: ‹(cmod (jozsa_algo $$ (1::nat, 0::nat)))² = (1::real) / (2::real)›*) show \"?thesis\"\n (*goal: ‹(cmod (jozsa_algo $$ (1::nat, 0::nat)))² = (1::real) / (2::real)›*)\n by simp\n qed\n ultimately have \"prob0_fst_qubits n jozsa_algo = 1/2 + 1/2\"\n by simp\n then show \"?thesis\"\n (*goal: ‹prob0_fst_qubits n jozsa_algo = 1›*)\n by simp\nqed\n\nlemma (in jozsa) prob0_jozsa_algo_of_const_1:\n assumes \"const 1\"\n shows \"prob0_fst_qubits n jozsa_algo = 1\"\nproof (-)\n (*goal: ‹prob0_fst_qubits n jozsa_algo = 1›*)\n have \"prob0_fst_qubits n jozsa_algo = (∑j∈{0,1}. (cmod(jozsa_algo $$ (j,0)))²)\"\n using prob0_fst_qubits_of_jozsa_algo (*‹prob0_fst_qubits n jozsa_algo = (∑j∈{0, 1}. (cmod (jozsa_algo $$ (j, 0)))²)›*) by simp\n moreover have \"(cmod(jozsa_algo $$ (0,0)))² = 1/2\"\n proof (-)\n (*goal: ‹(cmod (jozsa_algo $$ (0::nat, 0::nat)))² = (1::real) / (2::real)›*)\n have \"k<2^n ⟶ ((0 div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹?m < 2 ^ ?n ⟹ 0 ⋅⇘?n⇙ ?m = 0›*) by simp\n then have \"(cmod(jozsa_algo $$ (0,0)))² = (cmod(∑k::nat<2^n. 1/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using jozsa_algo_result (*‹jozsa_algo = Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))›*) const_def (*‹const ?c = (∀x∈{i. i < 2 ^ n}. f x = ?c)›*) assms (*‹const 1›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. 1 / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = (cmod((-1)/(sqrt(2))))² \"\n using aux_comp_with_sqrt2_bis (*‹2 ^ ?n / (sqrt 2 ^ ?n * sqrt 2 ^ (?n + 1)) = 1 / sqrt 2›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_real (- 1 / sqrt 2)))²›*) have \"... = 1/2\"\n by (simp add: norm_divide (*‹norm ((?a::?'a) / (?b::?'a)) = norm ?a / norm ?b›*) power2_eq_square (*‹(?a::?'a)² = ?a * ?a›*))\n finally (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = 1 / 2›*) show \"?thesis\"\n (*goal: ‹(cmod (jozsa_algo $$ (0, 0)))² = 1 / 2›*)\n by simp\n qed\n moreover have \"(cmod(jozsa_algo $$ (1,0)))² = 1/2\"\n proof (-)\n (*goal: ‹(cmod (jozsa_algo $$ (1::nat, 0::nat)))² = (1::real) / (2::real)›*)\n have \"k<2^n ⟶ ((1 div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹?m < 2 ^ ?n ⟹ 0 ⋅⇘?n⇙ ?m = 0›*) by simp\n then have \"(∑k::nat<2^n. (-1)^(f k +1 + ((1 div 2) ⋅⇘n⇙ k))/(sqrt(2)^n * sqrt(2)^(n+1)))\n = (∑k::nat<2^n. 1/(sqrt(2)^n * sqrt(2)^(n+1)))\"\n using const_def (*‹const (?c::nat) = (∀x::nat∈{i::nat. i < (2::nat) ^ (n::nat)}. (f::nat ⇒ nat) x = ?c)›*) assms (*‹const (1::nat)›*) by simp\n moreover have \"(cmod(jozsa_algo $$ (1,0)))² \n = (cmod (∑k::nat<2^n. (-1)^(f k + 1 + ((1 div 2) ⋅⇘n⇙ k))/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using \"ψ₃_dim\" (*‹(1::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) by simp\n ultimately have \"(cmod(jozsa_algo $$ (1,0)))² = (cmod(∑k::nat<2^n. 1/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (1, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. 1 / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = (cmod(1/(sqrt(2))))² \"\n using aux_comp_with_sqrt2_bis (*‹2 ^ ?n / (sqrt 2 ^ ?n * sqrt 2 ^ (?n + 1)) = 1 / sqrt 2›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (1::nat, 0::nat)))² = (cmod (complex_of_real ((1::real) / sqrt (2::real))))²›*) have \"... = 1/2\"\n by (simp add: norm_divide (*‹norm (?a / ?b) = norm ?a / norm ?b›*) power2_eq_square (*‹?a² = ?a * ?a›*))\n finally (*calculation: ‹(cmod (jozsa_algo $$ (1, 0)))² = 1 / 2›*) show \"?thesis\"\n (*goal: ‹(cmod (jozsa_algo $$ (1, 0)))² = 1 / 2›*)\n by simp\n qed\n ultimately have \"prob0_fst_qubits n jozsa_algo = 1/2 + 1/2\"\n by simp\n then show \"?thesis\"\n (*goal: ‹prob0_fst_qubits n jozsa_algo = 1›*)\n by simp\nqed\n\ntext ‹If the probability of measuring 0 for the first n qubits is 1, then the function is constant.›\n\nlemma (in jozsa) max_value_of_not_const_less:\n assumes \"¬ const 0\" and \"¬ const 1\"\n shows \"(cmod (∑k::nat<2^n. (-(1::nat))^(f k)))² < (2::nat)^(2*n)\"\nproof (-)\n (*goal: ‹(cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ f k)))² < real (2 ^ (2 * n))›*)\n have \"cmod (∑k::nat<2^n. (-(1::nat))^(f k)) < 2^n\"\n proof (-)\n (*goal: ‹cmod (complex_of_int (∑k::nat<(2::nat) ^ (n::nat). (- int (1::nat)) ^ (f::nat ⇒ nat) k)) < (2::real) ^ n›*)\n have \"(∑k::nat<2^n. (-(1::nat))^(f k)) < 2^n\"\n proof (-)\n (*goal: ‹(∑k<2 ^ n. (- int 1) ^ f k) < 2 ^ n›*)\n obtain x where f0: \"x < 2^n\" and f1: \"f x = 1\"\n (*goal: ‹(⋀x. ⟦x < 2 ^ n; f x = 1⟧ ⟹ thesis) ⟹ thesis›*)\n using assms(1) (*‹¬ const 0›*) const_def (*‹const (?c::nat) = (∀x::nat∈{i::nat. i < (2::nat) ^ (n::nat)}. (f::nat ⇒ nat) x = ?c)›*) f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) by auto\n then have \"(∑k::nat<2^n. (-(1::nat))^(f k)) < (∑k∈{i| i:: nat. i < 2^n}-{x}. (-(1::nat))^(f k))\"\n proof (-)\n (*goal: ‹⟦x < 2 ^ n; f x = 1⟧ ⟹ (∑k<2 ^ n. (- int 1) ^ f k) < (∑k∈{i |i. i < 2 ^ n} - {x}. (- int 1) ^ f k)›*)\n have \"(-(1::nat))^ f x = -1\"\n using f1 (*‹f x = 1›*) by simp\n moreover have \"x∈{i| i::nat. i<2^n}\"\n using f0 (*‹x < 2 ^ n›*) by simp\n moreover have \"finite {i| i::nat. i<2^n}\"\n by simp\n moreover have \"(∑k∈{i| i::nat. i<2^n}. (-(1::nat))^(f k)) < \n(∑k∈{i| i:: nat. i<2^n}-{x}. (-(1::nat))^(f k))\"\n using calculation(1,2,3) (*‹(- int 1) ^ f x = - 1› ‹x ∈ {i |i. i < 2 ^ n}› ‹finite {i |i::nat. i < (2::nat) ^ (n::nat)}›*) sum_diff1 (*‹finite ?A ⟹ sum ?f (?A - {?a}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*) by (simp add: sum_diff1 (*‹finite ?A ⟹ sum ?f (?A - {?a}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*))\n ultimately show \"?thesis\"\n (*goal: ‹(∑k<2 ^ n. (- int 1) ^ f k) < (∑k∈{i |i. i < 2 ^ n} - {x}. (- int 1) ^ f k)›*)\n by (metis Collect_cong (*‹(⋀x. ?P x = ?Q x) ⟹ {x. ?P x} = {x. ?Q x}›*) Collect_mem_eq (*‹{x. x ∈ ?A} = ?A›*) lessThan_iff (*‹(?i ∈ {..<?k}) = (?i < ?k)›*))\n qed\n moreover have \"… ≤ int (2^n - 1)\"\n using aux_ineq_with_card(1)[of \"{i| i:: nat. i<2^n}-{x}\"] (*‹finite ({i |i. i < 2 ^ n} - {x}) ⟹ (∑k∈{i |i. i < 2 ^ n} - {x}. (- 1) ^ ?g k) ≤ int (card ({i |i. i < 2 ^ n} - {x}))›*) f0 (*‹x < 2 ^ n›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹(∑k<2 ^ n. (- int 1) ^ f k) < 2 ^ n›*)\n by (meson diff_le_self (*‹?m - ?n ≤ ?m›*) less_le_trans (*‹⟦?x < ?y; ?y ≤ ?z⟧ ⟹ ?x < ?z›*) of_nat_le_numeral_power_cancel_iff (*‹(of_nat ?x ≤ numeral ?i ^ ?n) = (?x ≤ numeral ?i ^ ?n)›*))\n qed\n moreover have \"(∑k::nat<2^n. (-(1::nat))^(f k)) > - (2^n)\"\n proof (-)\n (*goal: ‹- ((2::int) ^ (n::nat)) < (∑k::nat<(2::nat) ^ n. (- int (1::nat)) ^ (f::nat ⇒ nat) k)›*)\n obtain x where f0: \"x < 2^n\" and f1: \"f x = 0\"\n (*goal: ‹(⋀x. ⟦x < 2 ^ n; f x = 0⟧ ⟹ thesis) ⟹ thesis›*)\n using assms(2) (*‹¬ const 1›*) const_def (*‹const ?c = (∀x∈{i. i < 2 ^ n}. f x = ?c)›*) f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) by auto\n then have \"(∑k::nat<2^n. (-(1::nat))^(f k)) > (∑k∈{i| i:: nat. i < 2^n}-{x}. (-(1::nat))^(f k))\"\n proof (-)\n (*goal: ‹⟦(x::nat) < (2::nat) ^ (n::nat); (f::nat ⇒ nat) x = (0::nat)⟧ ⟹ (∑k::nat∈{i |i::nat. i < (2::nat) ^ n} - {x}. (- int (1::nat)) ^ f k) < (∑k::nat<(2::nat) ^ n. (- int (1::nat)) ^ f k)›*)\n have \"(-(1::nat))^ f x = 1\"\n using f1 (*‹f x = 0›*) by simp\n moreover have \"x∈{i| i::nat. i<2^n}\"\n using f0 (*‹x < 2 ^ n›*) by simp\n moreover have \"finite {i| i::nat. i<2^n}\"\n by simp\n moreover have \"(∑k∈{i| i::nat. i<2^n}. (-(1::nat))^(f k)) > \n(∑k∈{i| i:: nat. i<2^n}-{x}. (-(1::nat))^(f k))\"\n using calculation(1,2,3) (*‹(- int 1) ^ f x = 1› ‹x ∈ {i |i. i < 2 ^ n}› ‹finite {i |i. i < 2 ^ n}›*) sum_diff1 (*‹finite ?A ⟹ sum ?f (?A - {?a}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*) by (simp add: sum_diff1 (*‹finite ?A ⟹ sum ?f (?A - {?a}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*))\n ultimately show \"?thesis\"\n (*goal: ‹(∑k∈{i |i. i < 2 ^ n} - {x}. (- int 1) ^ f k) < (∑k<2 ^ n. (- int 1) ^ f k)›*)\n by (metis Collect_cong (*‹(⋀x. ?P x = ?Q x) ⟹ {x. ?P x} = {x. ?Q x}›*) Collect_mem_eq (*‹{x. x ∈ ?A} = ?A›*) lessThan_iff (*‹(?i ∈ {..<?k}) = (?i < ?k)›*))\n qed\n moreover have \"- int (2^n - 1) ≤ (∑k∈{i| i:: nat. i < 2^n}-{x}. (-(1::nat))^(f k))\"\n using aux_ineq_with_card(2)[of \"{i| i:: nat. i<2^n}-{x}\"] (*‹finite ({i |i. i < 2 ^ n} - {x}) ⟹ - int (card ({i |i. i < 2 ^ n} - {x})) ≤ (∑k∈{i |i. i < 2 ^ n} - {x}. (- 1) ^ ?g k)›*) f0 (*‹x < 2 ^ n›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹- (2 ^ n) < (∑k<2 ^ n. (- int 1) ^ f k)›*)\n by (smt (verit) diff_le_self (*‹?m - ?n ≤ ?m›*) of_nat_1 (*‹of_nat 1 = 1›*) of_nat_add (*‹of_nat (?m + ?n) = of_nat ?m + of_nat ?n›*) of_nat_power_le_of_nat_cancel_iff (*‹(of_nat ?x ≤ of_nat ?b ^ ?w) = (?x ≤ ?b ^ ?w)›*) one_add_one (*‹1 + 1 = 2›*))\n qed\n ultimately show \"?thesis\"\n (*goal: ‹cmod (complex_of_int (∑k::nat<(2::nat) ^ (n::nat). (- int (1::nat)) ^ (f::nat ⇒ nat) k)) < (2::real) ^ n›*)\n using cmod_less (*‹⟦?a < ?n; - ?n < ?a⟧ ⟹ cmod (complex_of_int ?a) < real_of_int ?n›*) of_int_of_nat_eq (*‹of_int (int (?n::nat)) = of_nat ?n›*) of_nat_numeral (*‹of_nat (numeral ?n) = numeral ?n›*) of_nat_power (*‹of_nat (?m ^ ?n) = of_nat ?m ^ ?n›*) by (metis (no_types, lifting))\n qed\n then have \"(cmod (∑k::nat<2^n. (-(1::nat))^(f k)))² < (2^n)²\"\n using square_less (*‹⟦?a < ?n; - ?n < ?a⟧ ⟹ ?a² < ?n²›*) norm_ge_zero (*‹0 ≤ norm ?x›*) by (smt (verit))\n thus \"?thesis\"\n (*goal: ‹(cmod (complex_of_int (∑k::nat<(2::nat) ^ (n::nat). (- int (1::nat)) ^ (f::nat ⇒ nat) k)))² < real ((2::nat) ^ ((2::nat) * n))›*)\n by (simp add: power_even_eq (*‹(?a::?'a) ^ ((2::nat) * (?n::nat)) = (?a ^ ?n)²›*))\nqed\n\nlemma (in jozsa) max_value_of_not_const_less_bis:\n assumes \"¬ const 0\" and \"¬ const 1\"\n shows \"(cmod (∑k::nat<2^n. (-(1::nat))^(f k + 1)))² < (2::nat)^(2*n)\"\nproof (-)\n (*goal: ‹(cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ (f k + 1))))² < real (2 ^ (2 * n))›*)\n have \"cmod (∑k::nat<2^n. (-(1::nat))^(f k + 1)) < 2^n\"\n proof (-)\n (*goal: ‹cmod (complex_of_int (∑k::nat<(2::nat) ^ (n::nat). (- int (1::nat)) ^ ((f::nat ⇒ nat) k + (1::nat)))) < (2::real) ^ n›*)\n have \"(∑k::nat<2^n. (-(1::nat))^(f k + 1)) < 2^n\"\n proof (-)\n (*goal: ‹(∑k<2 ^ n. (- int 1) ^ (f k + 1)) < 2 ^ n›*)\n obtain x where f0: \"x < 2^n\" and f1: \"f x = 0\"\n (*goal: ‹(⋀x. ⟦x < 2 ^ n; f x = 0⟧ ⟹ thesis) ⟹ thesis›*)\n using assms(2) (*‹¬ const 1›*) const_def (*‹const ?c = (∀x∈{i. i < 2 ^ n}. f x = ?c)›*) f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) by auto\n then have \"(∑k::nat<2^n. (-(1::nat))^(f k + 1)) < (∑k∈{i| i:: nat. i < 2^n}-{x}. (-(1::nat))^(f k + 1))\"\n proof (-)\n (*goal: ‹⟦x < 2 ^ n; f x = 0⟧ ⟹ (∑k<2 ^ n. (- int 1) ^ (f k + 1)) < (∑k∈{i |i. i < 2 ^ n} - {x}. (- int 1) ^ (f k + 1))›*)\n have \"(-(1::nat))^ (f x + 1) = -1\"\n using f1 (*‹(f::nat ⇒ nat) (x::nat) = (0::nat)›*) by simp\n moreover have \"x∈{i| i::nat. i<2^n}\"\n using f0 (*‹x < 2 ^ n›*) by simp\n moreover have \"finite {i| i::nat. i<2^n}\"\n by simp\n moreover have \"(∑k∈{i| i::nat. i<2^n}. (-(1::nat))^(f k + 1)) < \n(∑k∈{i| i:: nat. i<2^n}-{x}. (-(1::nat))^(f k + 1))\"\n using calculation(1,2,3) (*‹(- int (1::nat)) ^ ((f::nat ⇒ nat) (x::nat) + (1::nat)) = - (1::int)› ‹x ∈ {i |i. i < 2 ^ n}› ‹finite {i |i. i < 2 ^ n}›*) sum_diff1 (*‹finite ?A ⟹ sum ?f (?A - {?a}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*) by (simp add: sum_diff1 (*‹finite (?A::?'b set) ⟹ sum (?f::?'b ⇒ ?'a) (?A - {?a::?'b}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*))\n ultimately show \"?thesis\"\n (*goal: ‹(∑k<2 ^ n. (- int 1) ^ (f k + 1)) < (∑k∈{i |i. i < 2 ^ n} - {x}. (- int 1) ^ (f k + 1))›*)\n by (metis Collect_cong (*‹(⋀x::?'a::type. (?P::?'a::type ⇒ bool) x = (?Q::?'a::type ⇒ bool) x) ⟹ {x::?'a::type. ?P x} = {x::?'a::type. ?Q x}›*) Collect_mem_eq (*‹{x::?'a::type. x ∈ (?A::?'a::type set)} = ?A›*) lessThan_iff (*‹((?i::?'a::ord) ∈ {..<?k::?'a::ord}) = (?i < ?k)›*))\n qed\n moreover have \"… ≤ int (2^n - 1)\"\n using aux_ineq_with_card(1)[of \"{i| i:: nat. i<2^n}-{x}\" \"λk. f k + 1\"] (*‹finite ({i |i. i < 2 ^ n} - {x}) ⟹ (∑k∈{i |i. i < 2 ^ n} - {x}. (- 1) ^ (f k + 1)) ≤ int (card ({i |i. i < 2 ^ n} - {x}))›*) f0 (*‹x < 2 ^ n›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹(∑k<2 ^ n. (- int 1) ^ (f k + 1)) < 2 ^ n›*)\n by (meson diff_le_self (*‹?m - ?n ≤ ?m›*) less_le_trans (*‹⟦?x < ?y; ?y ≤ ?z⟧ ⟹ ?x < ?z›*) of_nat_le_numeral_power_cancel_iff (*‹(of_nat ?x ≤ numeral ?i ^ ?n) = (?x ≤ numeral ?i ^ ?n)›*))\n qed\n moreover have \"(∑k::nat<2^n. (-(1::nat))^(f k + 1)) > - (2^n)\"\n proof (-)\n (*goal: ‹- (2 ^ n) < (∑k<2 ^ n. (- int 1) ^ (f k + 1))›*)\n obtain x where f0: \"x < 2^n\" and f1: \"f x = 1\"\n (*goal: ‹(⋀x::nat. ⟦x < (2::nat) ^ (n::nat); (f::nat ⇒ nat) x = (1::nat)⟧ ⟹ thesis::bool) ⟹ thesis›*)\n using assms(1) (*‹¬ const 0›*) const_def (*‹const (?c::nat) = (∀x::nat∈{i::nat. i < (2::nat) ^ (n::nat)}. (f::nat ⇒ nat) x = ?c)›*) f_values (*‹∀x∈{i. i < 2 ^ n}. f x = 0 ∨ f x = 1›*) by auto\n then have \"(∑k::nat<2^n. (-(1::nat))^(f k + 1)) > (∑k∈{i| i:: nat. i < 2^n}-{x}. (-(1::nat))^(f k + 1))\"\n proof (-)\n (*goal: ‹⟦x < 2 ^ n; f x = 1⟧ ⟹ (∑k∈{i |i. i < 2 ^ n} - {x}. (- int 1) ^ (f k + 1)) < (∑k<2 ^ n. (- int 1) ^ (f k + 1))›*)\n have \"(-(1::nat))^ (f x + 1) = 1\"\n using f1 (*‹f x = 1›*) by simp\n moreover have \"x∈{i| i::nat. i<2^n}\"\n using f0 (*‹x < 2 ^ n›*) by simp\n moreover have \"finite {i| i::nat. i<2^n}\"\n by simp\n moreover have \"(∑k∈{i| i::nat. i<2^n}. (-(1::nat))^(f k + 1)) > \n(∑k∈{i| i:: nat. i<2^n}-{x}. (-(1::nat))^(f k + 1))\"\n using calculation(1,2,3) (*‹(- int 1) ^ (f x + 1) = 1› ‹(x::nat) ∈ {i |i::nat. i < (2::nat) ^ (n::nat)}› ‹finite {i |i. i < 2 ^ n}›*) sum_diff1 (*‹finite ?A ⟹ sum ?f (?A - {?a}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*) by (simp add: sum_diff1 (*‹finite ?A ⟹ sum ?f (?A - {?a}) = (if ?a ∈ ?A then sum ?f ?A - ?f ?a else sum ?f ?A)›*))\n ultimately show \"?thesis\"\n (*goal: ‹(∑k∈{i |i. i < 2 ^ n} - {x}. (- int 1) ^ (f k + 1)) < (∑k<2 ^ n. (- int 1) ^ (f k + 1))›*)\n by (metis Collect_cong (*‹(⋀x. ?P x = ?Q x) ⟹ {x. ?P x} = {x. ?Q x}›*) Collect_mem_eq (*‹{x. x ∈ ?A} = ?A›*) lessThan_iff (*‹(?i ∈ {..<?k}) = (?i < ?k)›*))\n qed\n moreover have \"- int (2^n - 1) ≤ (∑k∈{i| i:: nat. i < 2^n}-{x}. (-(1::nat))^(f k + 1))\"\n using aux_ineq_with_card(2)[of \"{i| i:: nat. i<2^n}-{x}\" \"λk. f k + 1\"] (*‹finite ({i |i. i < 2 ^ n} - {x}) ⟹ - int (card ({i |i. i < 2 ^ n} - {x})) ≤ (∑k∈{i |i. i < 2 ^ n} - {x}. (- 1) ^ (f k + 1))›*) f0 (*‹(x::nat) < (2::nat) ^ (n::nat)›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹- (2 ^ n) < (∑k<2 ^ n. (- int 1) ^ (f k + 1))›*)\n by (smt (verit) diff_le_self (*‹?m - ?n ≤ ?m›*) of_nat_1 (*‹of_nat 1 = 1›*) of_nat_add (*‹of_nat (?m + ?n) = of_nat ?m + of_nat ?n›*) of_nat_power_le_of_nat_cancel_iff (*‹(of_nat ?x ≤ of_nat ?b ^ ?w) = (?x ≤ ?b ^ ?w)›*) one_add_one (*‹1 + 1 = 2›*))\n qed\n ultimately show \"?thesis\"\n (*goal: ‹cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ (f k + 1))) < 2 ^ n›*)\n using cmod_less (*‹⟦?a < ?n; - ?n < ?a⟧ ⟹ cmod (complex_of_int ?a) < real_of_int ?n›*) of_int_of_nat_eq (*‹of_int (int ?n) = of_nat ?n›*) of_nat_numeral (*‹of_nat (numeral ?n) = numeral ?n›*) of_nat_power (*‹of_nat (?m ^ ?n) = of_nat ?m ^ ?n›*) by (metis (no_types, lifting))\n qed\n then have \"(cmod (∑k::nat<2^n. (-(1::nat))^(f k + 1)))² < (2^n)²\"\n using square_less (*‹⟦?a < ?n; - ?n < ?a⟧ ⟹ ?a² < ?n²›*) norm_ge_zero (*‹0 ≤ norm ?x›*) by (smt (verit))\n thus \"?thesis\"\n (*goal: ‹(cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ (f k + 1))))² < real (2 ^ (2 * n))›*)\n by (simp add: power_even_eq (*‹?a ^ (2 * ?n) = (?a ^ ?n)²›*))\nqed\n\nlemma (in jozsa) f_const_has_max_value: \n assumes \"const 0 ∨ const 1\"\n shows \"(cmod (∑k<(2::nat)^n. (-1)^(f k)))² = (2::nat)^(2*n)\" \n and \"(cmod (∑k<(2::nat)^n. (-1)^(f k + 1)))² = (2::nat)^(2*n)\" \n using aux_comp_with_cmod[of n \"λk. f k\"] (*‹(∀x<2 ^ n. f x = 0) ∨ (∀x<2 ^ n. f x = 1) ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² = 2 ^ (2 * n)›*) aux_comp_with_cmod[of n \"λk. f k + 1\"] (*‹(∀x<2 ^ n. f x + 1 = 0) ∨ (∀x<2 ^ n. f x + 1 = 1) ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² = 2 ^ (2 * n)›*) const_def (*‹const ?c = (∀x∈{i. i < 2 ^ n}. f x = ?c)›*) assms (*‹const (0::nat) ∨ const (1::nat)›*) apply -\n (*goals:\n 1. ‹⟦(∀x<2 ^ n. f x = 0) ∨ (∀x<2 ^ n. f x = 1) ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² = 2 ^ (2 * n); (∀x<2 ^ n. f x + 1 = 0) ∨ (∀x<2 ^ n. f x + 1 = 1) ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² = 2 ^ (2 * n); ⋀c. const c = (∀x∈{i. i < 2 ^ n}. f x = c); const 0 ∨ const 1⟧ ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² = real (2 ^ (2 * n))›\n 2. ‹⟦(∀x<2 ^ n. f x = 0) ∨ (∀x<2 ^ n. f x = 1) ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² = 2 ^ (2 * n); (∀x<2 ^ n. f x + 1 = 0) ∨ (∀x<2 ^ n. f x + 1 = 1) ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² = 2 ^ (2 * n); ⋀c. const c = (∀x∈{i. i < 2 ^ n}. f x = c); const 0 ∨ const 1⟧ ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² = real (2 ^ (2 * n))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma (in jozsa) prob0_fst_qubits_leq:\n shows \"(cmod (∑k<(2::nat)^n. (-1)^(f k)))² ≤ (2::nat)^(2*n)\" \n and \"(cmod (∑k<(2::nat)^n. (-1)^(f k + 1)))² ≤ (2::nat)^(2*n)\" \nproof (-)\n (*goals:\n 1. ‹(cmod (∑k<2 ^ n. (- 1) ^ f k))² ≤ real (2 ^ (2 * n))›\n 2. ‹(cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›*)\n show \"(cmod (∑k<(2::nat)^n. (-1)^(f k)))² ≤ (2::nat)^(2*n)\"\n proof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹(?P::bool) ∨ (?Q::bool)›\n 2. ‹?P::bool ⟹ (cmod (∑k::nat<(2::nat) ^ (n::nat). (- (1::complex)) ^ (f::nat ⇒ nat) k))² ≤ real ((2::nat) ^ ((2::nat) * n))›\n 3. ‹?Q::bool ⟹ (cmod (∑k::nat<(2::nat) ^ (n::nat). (- (1::complex)) ^ (f::nat ⇒ nat) k))² ≤ real ((2::nat) ^ ((2::nat) * n))›*)\n show \"(const 0 ∨ const 1) ∨ (¬ const 0 ∧ ¬ const 1)\"\n by auto\n next\n (*goals:\n 1. ‹const 0 ∨ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² ≤ real (2 ^ (2 * n))›\n 2. ‹¬ const 0 ∧ ¬ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² ≤ real (2 ^ (2 * n))›*)\n assume \"const 0 ∨ const 1\" (*‹const (0::nat) ∨ const (1::nat)›*)\n then show \"(cmod (∑k<(2::nat)^n. (-1)^(f k)))² ≤ (2::nat)^(2*n)\"\n using f_const_has_max_value (*‹const 0 ∨ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² = real (2 ^ (2 * n))› ‹const 0 ∨ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² = real (2 ^ (2 * n))›*) by simp\n next\n (*goal: ‹¬ const (0::nat) ∧ ¬ const (1::nat) ⟹ (cmod (∑k::nat<(2::nat) ^ (n::nat). (- (1::complex)) ^ (f::nat ⇒ nat) k))² ≤ real ((2::nat) ^ ((2::nat) * n))›*)\n assume \"¬ const 0 ∧ ¬ const 1\" (*‹¬ const (0::nat) ∧ ¬ const (1::nat)›*)\n then show \"(cmod (∑k<(2::nat)^n. (-1)^(f k)))² ≤ (2::nat)^(2*n)\"\n using max_value_of_not_const_less (*‹⟦¬ const 0; ¬ const 1⟧ ⟹ (cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ f k)))² < real (2 ^ (2 * n))›*) by simp\n qed\nnext\n (*goal: ‹(cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›*)\n show \"(cmod (∑k<(2::nat)^n. (-1)^(f k + 1)))² ≤ (2::nat)^(2*n)\"\n proof (rule disjE (*‹⟦?P ∨ ?Q; ?P ⟹ ?R; ?Q ⟹ ?R⟧ ⟹ ?R›*))\n (*goals:\n 1. ‹?P ∨ ?Q›\n 2. ‹?P ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›\n 3. ‹?Q ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›*)\n show \"(const 0 ∨ const 1) ∨ (¬ const 0 ∧ ¬ const 1)\"\n by auto\n next\n (*goals:\n 1. ‹const 0 ∨ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›\n 2. ‹¬ const 0 ∧ ¬ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›*)\n assume \"const 0 ∨ const 1\" (*‹const (0::nat) ∨ const (1::nat)›*)\n then show \"(cmod (∑k<(2::nat)^n. (-1)^(f k + 1)))² ≤ (2::nat)^(2*n)\"\n using f_const_has_max_value (*‹const 0 ∨ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ f k))² = real (2 ^ (2 * n))› ‹const 0 ∨ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² = real (2 ^ (2 * n))›*) by simp\n next\n (*goal: ‹¬ const 0 ∧ ¬ const 1 ⟹ (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›*)\n assume \"¬ const 0 ∧ ¬ const 1\" (*‹¬ const (0::nat) ∧ ¬ const (1::nat)›*)\n then show \"(cmod (∑k<(2::nat)^n. (-1)^(f k + 1)))² ≤ (2::nat)^(2*n)\"\n using max_value_of_not_const_less_bis (*‹⟦¬ const 0; ¬ const 1⟧ ⟹ (cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ (f k + 1))))² < real (2 ^ (2 * n))›*) by simp\n qed\nqed\n\nlemma (in jozsa) prob0_jozsa_algo_1_is_const:\n assumes \"prob0_fst_qubits n jozsa_algo = 1\"\n shows \"const 0 ∨ const 1\"\nproof (-)\n (*goal: ‹const (0::nat) ∨ const (1::nat)›*)\n have f0: \"(∑j∈{0,1}. (cmod(jozsa_algo $$ (j,0)))²) = 1\"\n using prob0_fst_qubits_of_jozsa_algo (*‹prob0_fst_qubits n jozsa_algo = (∑j∈{0, 1}. (cmod (jozsa_algo $$ (j, 0)))²)›*) assms (*‹prob0_fst_qubits n jozsa_algo = 1›*) by simp\n have \"k < 2^n⟶((0 div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹?m < 2 ^ ?n ⟹ 0 ⋅⇘?n⇙ ?m = 0›*) by simp\n then have f1: \"(cmod(jozsa_algo $$ (0,0)))² = (cmod(∑k<(2::nat)^n. (-1)^(f k)/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n by simp\n have \"k < 2^n⟶((1 div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹?m < 2 ^ ?n ⟹ 0 ⋅⇘?n⇙ ?m = 0›*) by simp\n moreover have \"(cmod(jozsa_algo $$ (1,0)))² \n = (cmod (∑k<(2::nat)^n. (-1)^(f k+ 1 + ((1 div 2) ⋅⇘n⇙ k))/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using \"ψ₃_dim\" (*‹(1::nat) < dim_row (Matrix.mat ((2::nat) ^ ((n::nat) + (1::nat))) (1::nat) (λx::nat × nat. complex_of_real (case x of (i::nat, j::nat) ⇒ if even i then ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ ((f::nat ⇒ nat) k + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))) else ∑k::nat<(2::nat) ^ n. (- (1::real)) ^ (f k + (1::nat) + i div (2::nat) ⋅⇘n⇙ k) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))))›*) by simp\n ultimately have f2: \"(cmod(jozsa_algo $$ (1,0)))² \n = (cmod (∑k<(2::nat)^n. (-1)^(f k + 1)/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n by simp\n have f3: \"1 = (cmod(∑k::nat<(2::nat)^n.(-1)^(f k)/(sqrt(2)^n * sqrt(2)^(n+1))))²\n + (cmod (∑k::nat<(2::nat)^n. (-1)^(f k + 1)/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using f0 (*‹(∑j∈{0, 1}. (cmod (jozsa_algo $$ (j, 0)))²) = 1›*) f1 (*‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. (- 1) ^ f k / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) f2 (*‹(cmod (jozsa_algo $$ (1, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. (- 1) ^ (f k + 1) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) by simp\n also (*calculation: ‹1 = (cmod (complex_of_real (∑k<2 ^ n. (- 1) ^ f k / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))² + (cmod (complex_of_real (∑k<2 ^ n. (- 1) ^ (f k + 1) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = ((cmod (∑k::nat<(2::nat)^n. (-1)^(f k)) ) /(sqrt(2)^n * sqrt(2)^(n+1)))²\n + ((cmod(∑k::nat<(2::nat)^n. (-1)^(f k + 1))) /(sqrt(2)^n * sqrt(2)^(n+1)))²\"\n using aux_comp_sum_divide_cmod[of \"λk. (-1)^(f k)\" \"(sqrt(2)^n * sqrt(2)^(n+1))\" \"(2::nat)^n\"] (*‹(cmod (complex_of_real (∑k::nat<(2::nat) ^ n. real_of_int ((- (1::int)) ^ (f::nat ⇒ nat) k) / (sqrt (2::real) ^ (n::nat) * sqrt (2::real) ^ (n + (1::nat))))))² = (cmod (complex_of_int (∑k::nat<(2::nat) ^ n. (- (1::int)) ^ f k)) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))²›*) aux_comp_sum_divide_cmod[of \"λk. (-1)^(f k + 1)\" \"(sqrt(2)^n * sqrt(2)^(n+1))\" \"(2::nat)^n\"] (*‹(cmod (complex_of_real (∑k<2 ^ n. real_of_int ((- 1) ^ (f k + 1)) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))² = (cmod (complex_of_int (∑k<2 ^ n. (- 1) ^ (f k + 1))) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))²›*) by simp\n also (*calculation: ‹1 = (cmod (∑k<2 ^ n. (- 1) ^ f k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))² + (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))²›*) have \"... = ((cmod (∑k::nat<(2::nat)^n. (-1)^(f k))))² /((sqrt(2)^n * sqrt(2)^(n+1)))²\n + ((cmod(∑k::nat<(2::nat)^n. (-1)^(f k +1))))² /((sqrt(2)^n * sqrt(2)^(n+1)))²\"\n by (simp add: power_divide (*‹(?a / ?b) ^ ?n = ?a ^ ?n / ?b ^ ?n›*))\n also (*calculation: ‹(1::real) = (cmod (∑k::nat<(2::nat) ^ (n::nat). (- (1::complex)) ^ (f::nat ⇒ nat) k))² / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat)))² + (cmod (∑k::nat<(2::nat) ^ n. (- (1::complex)) ^ (f k + (1::nat))))² / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat)))²›*) have \"... = ((cmod (∑k::nat<(2::nat)^n. (-1)^(f k)) ) )²/(2^(2*n+1))\n + ((cmod(∑k::nat<(2::nat)^n. (-1)^(f k + 1))))² /(2^(2*n+1))\"\n by (smt (verit) left_add_twice (*‹?a + (?a + ?b) = 2 * ?a + ?b›*) power2_eq_square (*‹?a² = ?a * ?a›*) power_add (*‹?a ^ (?m + ?n) = ?a ^ ?m * ?a ^ ?n›*) power_mult_distrib (*‹(?a * ?b) ^ ?n = ?a ^ ?n * ?b ^ ?n›*) real_sqrt_pow2 (*‹0 ≤ ?x ⟹ (sqrt ?x)² = ?x›*))\n also (*calculation: ‹1 = (cmod (∑k<2 ^ n. (- 1) ^ f k))² / 2 ^ (2 * n + 1) + (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² / 2 ^ (2 * n + 1)›*) have \"... = (((cmod (∑k::nat<(2::nat)^n. (-1)^(f k))))² \n + ((cmod(∑k::nat<(2::nat)^n. (-1)^(f k + 1))))²)/(2^(2*n+1)) \"\n by (simp add: add_divide_distrib (*‹(?a + ?b) / ?c = ?a / ?c + ?b / ?c›*))\n finally (*calculation: ‹1 = ((cmod (∑k<2 ^ n. (- 1) ^ f k))² + (cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))²) / 2 ^ (2 * n + 1)›*) have \"((2::nat)^(2*n+1)) = (((cmod (∑k::nat<(2::nat)^n. (-1)^(f k))))² \n + ((cmod(∑k::nat<(2::nat)^n. (-1)^(f k + 1))))²)\"\n by simp\n moreover have \"((2::nat)^(2*n+1)) = 2^(2*n) + 2^(2*n)\"\n by auto\n moreover have \"(cmod (∑k<(2::nat)^n. (-1)^(f k)))² ≤ 2^(2*n)\"\n using prob0_fst_qubits_leq (*‹(cmod (∑k<2 ^ n. (- 1) ^ f k))² ≤ real (2 ^ (2 * n))› ‹(cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›*) by simp\n moreover have \"(cmod (∑k<(2::nat)^n. (-1)^(f k + 1)))² ≤ 2^(2*n)\"\n using prob0_fst_qubits_leq (*‹(cmod (∑k<2 ^ n. (- 1) ^ f k))² ≤ real (2 ^ (2 * n))› ‹(cmod (∑k<2 ^ n. (- 1) ^ (f k + 1)))² ≤ real (2 ^ (2 * n))›*) by simp\n ultimately have \"2^(2*n) = ((cmod (∑k::nat<(2::nat)^n. (-1)^(f k))))²\"\n by simp\n then show \"?thesis\"\n (*goal: ‹const 0 ∨ const 1›*)\n using max_value_of_not_const_less (*‹⟦¬ const (0::nat); ¬ const (1::nat)⟧ ⟹ (cmod (complex_of_int (∑k::nat<(2::nat) ^ (n::nat). (- int (1::nat)) ^ (f::nat ⇒ nat) k)))² < real ((2::nat) ^ ((2::nat) * n))›*) by auto\nqed\n\ntext ‹\nThe function is balanced if and only if at least one qubit among the first n qubits is not zero.\nSo, if the function is balanced then the probability of measuring 0 for the first n qubits is 0.\n›\n\nlemma sum_union_disjoint_finite_set:\n fixes C::\"nat set\" and g::\"nat ⇒ int\"\n assumes \"finite C\"\n shows \"∀A B. A ∩ B = {} ∧ A ∪ B = C ⟶ (∑k∈C. g k) = (∑k∈A. g k) + (∑k∈B. g k)\" \n using assms (*‹finite C›*) sum.union_disjoint (*‹⟦finite ?A; finite ?B; ?A ∩ ?B = {}⟧ ⟹ sum ?g (?A ∪ ?B) = sum ?g ?A + sum ?g ?B›*) by auto\n\nlemma (in jozsa) balanced_pos_and_neg_terms_cancel_out1:\n assumes \"is_balanced\" \n shows \"(∑k<(2::nat)^n. (-(1::nat))^(f k)) = 0\"\nsorry\n\nlemma (in jozsa) balanced_pos_and_neg_terms_cancel_out2:\n assumes \"is_balanced\" \n shows \"(∑k<(2::nat)^n. (-(1::nat))^(f k + 1)) = 0\"\nproof (-)\n (*goal: ‹(∑k<2 ^ n. (- int 1) ^ (f k + 1)) = 0›*)\n have \"⋀A B. A ⊆ {i::nat. i < (2::nat)^n} ∧ B ⊆ {i::nat. i < (2::nat)^n}\n ∧ card A = ((2::nat)^(n-1)) ∧ card B = ((2::nat)^(n-1)) \n ∧ (∀(x::nat)∈A. f x = (0::nat)) ∧ (∀(x::nat)∈B. f x = 1)\n ⟶ (∑k<(2::nat)^n. (-(1::nat))^(f k + 1)) = 0\"\n proof (standard)\n (*goal: ‹⋀A B. A ⊆ {i. i < 2 ^ n} ∧ B ⊆ {i. i < 2 ^ n} ∧ card A = 2 ^ (n - 1) ∧ card B = 2 ^ (n - 1) ∧ (∀x∈A. f x = 0) ∧ (∀x∈B. f x = 1) ⟹ (∑k<2 ^ n. (- int 1) ^ (f k + 1)) = 0›*)\n fix A :: \"nat set\" and B :: \"nat set\"\n assume asm: \"A ⊆ {i::nat. i < (2::nat)^n} ∧ B ⊆ {i::nat. i < (2::nat)^n}\n ∧ card A = ((2::nat)^(n-1)) ∧ card B = ((2::nat)^(n-1)) \n ∧ (∀(x::nat) ∈ A. f x = (0::nat)) ∧ (∀(x::nat) ∈ B. f x = 1)\" (*‹(A::nat set) ⊆ {i::nat. i < (2::nat) ^ (n::nat)} ∧ (B::nat set) ⊆ {i::nat. i < (2::nat) ^ n} ∧ card A = (2::nat) ^ (n - (1::nat)) ∧ card B = (2::nat) ^ (n - (1::nat)) ∧ (∀x::nat∈A. (f::nat ⇒ nat) x = (0::nat)) ∧ (∀x::nat∈B. f x = (1::nat))›*)\n have \"A ∩ B = {}\" and \"{0..<(2::nat)^n} = A ∪ B\"\n using is_balanced_union (*‹⟦?A ⊆ {i. i < 2 ^ n}; ?B ⊆ {i. i < 2 ^ n}; card ?A = 2 ^ (n - 1); card ?B = 2 ^ (n - 1); ?A ∩ ?B = {}⟧ ⟹ ?A ∪ ?B = {i. i < 2 ^ n}›*) is_balanced_inter (*‹⟦∀x∈?A. f x = 0; ∀x∈?B. f x = 1⟧ ⟹ ?A ∩ ?B = {}›*) asm (*‹A ⊆ {i. i < 2 ^ n} ∧ B ⊆ {i. i < 2 ^ n} ∧ card A = 2 ^ (n - 1) ∧ card B = 2 ^ (n - 1) ∧ (∀x∈A. f x = 0) ∧ (∀x∈B. f x = 1)›*) apply -\n (*goals:\n 1. ‹⟦⋀A B. ⟦A ⊆ {i. i < 2 ^ n}; B ⊆ {i. i < 2 ^ n}; card A = 2 ^ (n - 1); card B = 2 ^ (n - 1); A ∩ B = {}⟧ ⟹ A ∪ B = {i. i < 2 ^ n}; ⋀A B. ⟦∀x∈A. f x = 0; ∀x∈B. f x = 1⟧ ⟹ A ∩ B = {}; A ⊆ {i. i < 2 ^ n} ∧ B ⊆ {i. i < 2 ^ n} ∧ card A = 2 ^ (n - 1) ∧ card B = 2 ^ (n - 1) ∧ (∀x∈A. f x = 0) ∧ (∀x∈B. f x = 1)⟧ ⟹ A ∩ B = {}›\n 2. ‹⟦⋀A B. ⟦A ⊆ {i. i < 2 ^ n}; B ⊆ {i. i < 2 ^ n}; card A = 2 ^ (n - 1); card B = 2 ^ (n - 1); A ∩ B = {}⟧ ⟹ A ∪ B = {i. i < 2 ^ n}; ⋀A B. ⟦∀x∈A. f x = 0; ∀x∈B. f x = 1⟧ ⟹ A ∩ B = {}; A ⊆ {i. i < 2 ^ n} ∧ B ⊆ {i. i < 2 ^ n} ∧ card A = 2 ^ (n - 1) ∧ card B = 2 ^ (n - 1) ∧ (∀x∈A. f x = 0) ∧ (∀x∈B. f x = 1)⟧ ⟹ {0..<2 ^ n} = A ∪ B›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n then have \"(∑k∈{0..<(2::nat)^n}. (-(1::nat))^(f k + 1)) =\n (∑k∈A. (-(1::nat))^(f k + 1)) \n + (∑k∈B. (-(1::nat))^(f k + 1))\"\n by (metis finite_atLeastLessThan (*‹finite {?l..<?u}›*) sum_union_disjoint_finite_set (*‹finite ?C ⟹ ∀A B. A ∩ B = {} ∧ A ∪ B = ?C ⟶ sum ?g ?C = sum ?g A + sum ?g B›*))\n moreover have \"(∑k∈A. (-1)^(f k + 1)) = -((2::nat)^(n-1))\"\n using asm (*‹(A::nat set) ⊆ {i::nat. i < (2::nat) ^ (n::nat)} ∧ (B::nat set) ⊆ {i::nat. i < (2::nat) ^ n} ∧ card A = (2::nat) ^ (n - (1::nat)) ∧ card B = (2::nat) ^ (n - (1::nat)) ∧ (∀x::nat∈A. (f::nat ⇒ nat) x = (0::nat)) ∧ (∀x::nat∈B. f x = (1::nat))›*) by simp\n moreover have \"(∑k∈B. (-1)^(f k + 1)) = ((2::nat)^(n-1))\"\n using asm (*‹A ⊆ {i. i < 2 ^ n} ∧ B ⊆ {i. i < 2 ^ n} ∧ card A = 2 ^ (n - 1) ∧ card B = 2 ^ (n - 1) ∧ (∀x∈A. f x = 0) ∧ (∀x∈B. f x = 1)›*) by simp\n ultimately have \"(∑k∈{0..<(2::nat)^n}. (-(1::nat))^(f k + 1)) = 0 \"\n by simp\n then show \"(∑k<(2::nat)^n. (-(1::nat))^(f k + 1)) = 0\"\n by (simp add: lessThan_atLeast0 (*‹{..<?n} = {0..<?n}›*))\n qed\n then show \"(∑k<(2::nat)^n. (-(1::nat))^(f k + 1)) = 0\"\n using assms (*‹is_balanced›*) is_balanced_def (*‹is_balanced ≡ ∃A B. A ⊆ {i. i < 2 ^ n} ∧ B ⊆ {i. i < 2 ^ n} ∧ card A = 2 ^ (n - 1) ∧ card B = 2 ^ (n - 1) ∧ (∀x∈A. f x = 0) ∧ (∀x∈B. f x = 1)›*) by auto\nqed\n\nlemma (in jozsa) prob0_jozsa_algo_of_balanced:\nassumes \"is_balanced\"\n shows \"prob0_fst_qubits n jozsa_algo = 0\"\nproof (-)\n (*goal: ‹prob0_fst_qubits n jozsa_algo = 0›*)\n have \"prob0_fst_qubits n jozsa_algo = (∑j∈{0,1}. (cmod(jozsa_algo $$ (j,0)))²)\"\n using prob0_fst_qubits_of_jozsa_algo (*‹prob0_fst_qubits n jozsa_algo = (∑j∈{0, 1}. (cmod (jozsa_algo $$ (j, 0)))²)›*) by simp\n moreover have \"(cmod(jozsa_algo $$ (0,0)))² = 0\"\n proof (-)\n (*goal: ‹(cmod (jozsa_algo $$ (0, 0)))² = 0›*)\n have \"k < 2^n⟶((1 div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹?m < 2 ^ ?n ⟹ 0 ⋅⇘?n⇙ ?m = 0›*) by simp\n then have \"(cmod(jozsa_algo $$ (0,0)))² = (cmod(∑ k < (2::nat)^n. (-1)^(f k)/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using \"ψ₃_values\" (*‹?i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))) ⟹ odd ?i ⟶ Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))) $$ (?i, 0) = complex_of_real (∑k<2 ^ n. (- 1) ^ (f k + 1 + ?i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. (- 1) ^ f k / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = (cmod(∑k<(2::nat)^n. (-(1::nat))^(f k))/(sqrt(2)^n * sqrt(2)^(n+1)))²\"\n using aux_comp_sum_divide_cmod[of \"λk.(-(1::nat))^(f k)\" \"(sqrt(2)^n * sqrt(2)^(n+1))\" \"2^n\"] (*‹(cmod (complex_of_real (∑k<2 ^ n. real_of_int ((- int 1) ^ f k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))² = (cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ f k)) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))²›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ f k)) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))²›*) have \"... = (cmod ((0::int)/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using balanced_pos_and_neg_terms_cancel_out1 (*‹is_balanced ⟹ (∑k::nat<(2::nat) ^ (n::nat). (- int (1::nat)) ^ (f::nat ⇒ nat) k) = (0::int)›*) assms (*‹is_balanced›*) by (simp add: bob_fun_axioms (*‹bob_fun f n›*))\n also (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = (cmod (complex_of_real (real_of_int 0 / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = 0\"\n by simp\n finally (*calculation: ‹(cmod (jozsa_algo $$ (0, 0)))² = 0›*) show \"?thesis\"\n (*goal: ‹(cmod (jozsa_algo $$ (0, 0)))² = 0›*)\n by simp\n qed\n moreover have \"(cmod(jozsa_algo $$ (1,0)))² = 0\"\n proof (-)\n (*goal: ‹(cmod (jozsa_algo $$ (1::nat, 0::nat)))² = (0::real)›*)\n have \"k < 2^n ⟶ (((1::nat) div 2) ⋅⇘n⇙ k) = 0\" for k :: nat\n using bitwise_inner_prod_with_zero (*‹?m < 2 ^ ?n ⟹ 0 ⋅⇘?n⇙ ?m = 0›*) by auto\n moreover have \"(cmod(jozsa_algo $$ (1,0)))² \n = (cmod (∑k<(2::nat)^n. (-(1::nat))^(f k + (1::nat) + ((1 div 2) ⋅⇘n⇙ k))/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using \"ψ₃_dim\" (*‹1 < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (λx. complex_of_real (case x of (i, j) ⇒ if even i then ∑k<2 ^ n. (- 1) ^ (f k + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else ∑k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 ⋅⇘n⇙ k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))›*) by simp\n ultimately have \"(cmod(jozsa_algo $$ (1,0)))² \n = (cmod(∑k<(2::nat)^n. (-(1::nat))^(f k + (1::nat))/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (1, 0)))² = (cmod (complex_of_real (∑k<2 ^ n. real_of_int ((- int 1) ^ (f k + 1)) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))²›*) have \"... = (cmod(∑k<(2::nat)^n. (-(1::nat))^(f k + 1))/(sqrt(2)^n * sqrt(2)^(n+1)))²\"\n using aux_comp_sum_divide_cmod[of \"λk.(-(1::nat))^(f k + 1)\" \"(sqrt(2)^n * sqrt(2)^(n+1))\" \"2^n\"] (*‹(cmod (complex_of_real (∑k::nat<(2::nat) ^ n. real_of_int ((- int (1::nat)) ^ ((f::nat ⇒ nat) k + (1::nat))) / (sqrt (2::real) ^ (n::nat) * sqrt (2::real) ^ (n + (1::nat))))))² = (cmod (complex_of_int (∑k::nat<(2::nat) ^ n. (- int (1::nat)) ^ (f k + (1::nat)))) / (sqrt (2::real) ^ n * sqrt (2::real) ^ (n + (1::nat))))²›*) by simp\n also (*calculation: ‹(cmod (jozsa_algo $$ (1, 0)))² = (cmod (complex_of_int (∑k<2 ^ n. (- int 1) ^ (f k + 1))) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))²›*) have \"... = (cmod ((0::int)/(sqrt(2)^n * sqrt(2)^(n+1))))²\"\n using balanced_pos_and_neg_terms_cancel_out2 (*‹is_balanced ⟹ (∑k<2 ^ n. (- int 1) ^ (f k + 1)) = 0›*) assms (*‹is_balanced›*) by (simp add: bob_fun_axioms (*‹bob_fun f n›*))\n also (*calculation: ‹(cmod (jozsa_algo $$ (1::nat, 0::nat)))² = (cmod (complex_of_real (real_of_int (0::int) / (sqrt (2::real) ^ (n::nat) * sqrt (2::real) ^ (n + (1::nat))))))²›*) have \"... = 0\"\n by simp\n finally (*calculation: ‹(cmod (jozsa_algo $$ (1, 0)))² = 0›*) show \"?thesis\"\n (*goal: ‹(cmod (jozsa_algo $$ (1::nat, 0::nat)))² = (0::real)›*)\n by simp\n qed\n ultimately have \"prob0_fst_qubits n jozsa_algo = 0 + 0\"\n by simp\n then show \"?thesis\"\n (*goal: ‹prob0_fst_qubits n jozsa_algo = 0›*)\n by simp\nqed\n\ntext ‹If the probability that the first n qubits are 0 is 0, then the function is balanced.›\n\nlemma (in jozsa) balanced_prob0_jozsa_algo:\n assumes \"prob0_fst_qubits n jozsa_algo = 0\"\n shows \"is_balanced\"\nproof (-)\n (*goal: ‹is_balanced›*)\n have \"is_const ∨ is_balanced\"\n using const_or_balanced (*‹is_const ∨ is_balanced›*) by simp\n moreover have \"is_const ⟶ ¬ prob0_fst_qubits n jozsa_algo = 0\"\n using is_const_def (*‹is_const ≡ const (0::nat) ∨ const (1::nat)›*) prob0_jozsa_algo_of_const_0 (*‹const 0 ⟹ prob0_fst_qubits n jozsa_algo = 1›*) prob0_jozsa_algo_of_const_1 (*‹const (1::nat) ⟹ prob0_fst_qubits (n::nat) jozsa_algo = (1::real)›*) by simp\n ultimately show \"?thesis\"\n (*goal: ‹is_balanced›*)\n using assms (*‹prob0_fst_qubits n jozsa_algo = 0›*) by simp\nqed\n\ntext ‹We prove the correctness of the algorithm.›\n\ndefinition (in jozsa) jozsa_algo_eval:: \"real\" where\n\"jozsa_algo_eval ≡ prob0_fst_qubits n jozsa_algo\"\n\ntheorem (in jozsa) jozsa_algo_is_correct:\n shows \"jozsa_algo_eval = 1 ⟷ is_const\" \n and \"jozsa_algo_eval = 0 ⟷ is_balanced\" \n using prob0_jozsa_algo_of_const_1 (*‹const 1 ⟹ prob0_fst_qubits n jozsa_algo = 1›*) prob0_jozsa_algo_of_const_0 (*‹const 0 ⟹ prob0_fst_qubits n jozsa_algo = 1›*) jozsa_algo_eval_def (*‹jozsa_algo_eval ≡ prob0_fst_qubits (n::nat) jozsa_algo›*) prob0_jozsa_algo_1_is_const (*‹prob0_fst_qubits n jozsa_algo = 1 ⟹ const 0 ∨ const 1›*) is_const_def (*‹is_const ≡ const 0 ∨ const 1›*) balanced_prob0_jozsa_algo (*‹prob0_fst_qubits (n::nat) jozsa_algo = (0::real) ⟹ is_balanced›*) prob0_jozsa_algo_of_balanced (*‹is_balanced ⟹ prob0_fst_qubits n jozsa_algo = 0›*) apply -\n (*goals:\n 1. ‹⟦const 1 ⟹ prob0_fst_qubits n jozsa_algo = 1; const 0 ⟹ prob0_fst_qubits n jozsa_algo = 1; jozsa_algo_eval ≡ prob0_fst_qubits n jozsa_algo; prob0_fst_qubits n jozsa_algo = 1 ⟹ const 0 ∨ const 1; is_const ≡ const 0 ∨ const 1; prob0_fst_qubits n jozsa_algo = 0 ⟹ is_balanced; is_balanced ⟹ prob0_fst_qubits n jozsa_algo = 0⟧ ⟹ (jozsa_algo_eval = 1) = is_const›\n 2. ‹⟦const 1 ⟹ prob0_fst_qubits n jozsa_algo = 1; const 0 ⟹ prob0_fst_qubits n jozsa_algo = 1; jozsa_algo_eval ≡ prob0_fst_qubits n jozsa_algo; prob0_fst_qubits n jozsa_algo = 1 ⟹ const 0 ∨ const 1; is_const ≡ const 0 ∨ const 1; prob0_fst_qubits n jozsa_algo = 0 ⟹ is_balanced; is_balanced ⟹ prob0_fst_qubits n jozsa_algo = 0⟧ ⟹ (jozsa_algo_eval = 0) = is_balanced›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nend"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/Isabelle_Marries_Dirac/Deutsch_Jozsa.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"a3a135605aa0c8b6a893c9f6fbfb40e656eda61f597c54b631c8a2c92a92ba77\"\n}"}}},{"rowIdx":99,"cells":{"text":{"kind":"string","value":"(* Title: variants/a_norreqid/Loop_Freedom.thy\n License: BSD 2-Clause. See LICENSE.\n Author: Timothy Bourke, Inria\n*)\n\nsection \"Routing graphs and loop freedom\"\n\ntheory A_Loop_Freedom\nimports A_Aodv_Predicates A_Fresher\nbegin\n\ntext ‹Define the central theorem that relates an invariant over network states to the absence\n of loops in the associate routing graph.›\n\ndefinition\n rt_graph :: \"(ip ⇒ state) ⇒ ip ⇒ ip rel\"\nwhere\n \"rt_graph σ = (λdip.\n {(ip, ip') | ip ip' dsn dsk hops pre.\n ip ≠ dip ∧ rt (σ ip) dip = Some (dsn, dsk, val, hops, ip', pre)})\"\n\ntext ‹Given the state of a network @{term σ}, a routing graph for a given destination\n ip address @{term dip} abstracts the details of routing tables into nodes\n (ip addresses) and vertices (valid routes between ip addresses).›\n\nlemma rt_graphE [elim]:\n fixes n dip ip ip'\n assumes \"(ip, ip') ∈ rt_graph σ dip\"\n shows \"ip ≠ dip ∧ (∃r. rt (σ ip) = r\n ∧ (∃dsn dsk hops pre. r dip = Some (dsn, dsk, val, hops, ip', pre)))\"\n using assms (*‹(ip, ip') ∈ rt_graph σ dip›*) unfolding rt_graph_def\n (*goal: ‹ip ≠ dip ∧ (∃r. rt (σ ip) = r ∧ (∃dsn dsk hops pre. r dip = Some (dsn, dsk, val, hops, ip', pre)))›*)\n by auto\n\nlemma rt_graph_vD [dest]:\n \"⋀ip ip' σ dip. (ip, ip') ∈ rt_graph σ dip ⟹ dip ∈ vD(rt (σ ip))\"\n unfolding rt_graph_def vD_def\n (*goal: ‹⋀ip ip' σ dip. (ip, ip') ∈ {uu_. ∃ip ip' dsn dsk hops pre. uu_ = (ip, ip') ∧ ip ≠ dip ∧ rt (σ ip) dip = Some (dsn, dsk, val, hops, ip', pre)} ⟹ dip ∈ {dip. flag (rt (σ ip)) dip = Some val}›*)\n by auto\n\nlemma rt_graph_vD_trans [dest]:\n \"⋀ip ip' σ dip. (ip, ip') ∈ (rt_graph σ dip)⁺ ⟹ dip ∈ vD(rt (σ ip))\"\n apply (erule converse_tranclE (*‹⟦(?x::?'a::type, ?z::?'a::type) ∈ (?r::(?'a::type × ?'a::type) set)⁺; (?x, ?z) ∈ ?r ⟹ ?P::bool; ⋀y::?'a::type. ⟦(?x, y) ∈ ?r; (y, ?z) ∈ ?r⁺⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀ip ip' σ dip. (ip, ip') ∈ rt_graph σ dip ⟹ dip ∈ vD (rt (σ ip))›\n 2. ‹⋀ip ip' σ dip y. ⟦(ip, y) ∈ rt_graph σ dip; (y, ip') ∈ (rt_graph σ dip)⁺⟧ ⟹ dip ∈ vD (rt (σ ip))›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\nlemma rt_graph_not_dip [dest]:\n \"⋀ip ip' σ dip. (ip, ip') ∈ rt_graph σ dip ⟹ ip ≠ dip\"\n unfolding rt_graph_def\n (*goal: ‹⋀ip ip' σ dip. (ip, ip') ∈ {uu_. ∃ip ip' dsn dsk hops pre. uu_ = (ip, ip') ∧ ip ≠ dip ∧ rt (σ ip) dip = Some (dsn, dsk, val, hops, ip', pre)} ⟹ ip ≠ dip›*)\n by auto\n\nlemma rt_graph_not_dip_trans [dest]:\n \"⋀ip ip' σ dip. (ip, ip') ∈ (rt_graph σ dip)⁺ ⟹ ip ≠ dip\"\n apply (erule converse_tranclE (*‹⟦(?x, ?z) ∈ ?r⁺; (?x, ?z) ∈ ?r ⟹ ?P; ⋀y. ⟦(?x, y) ∈ ?r; (y, ?z) ∈ ?r⁺⟧ ⟹ ?P⟧ ⟹ ?P›*))\n (*goals:\n 1. ‹⋀ip ip' σ dip. (ip, ip') ∈ rt_graph σ dip ⟹ ip ≠ dip›\n 2. ‹⋀ip ip' σ dip y. ⟦(ip, y) ∈ rt_graph σ dip; (y, ip') ∈ (rt_graph σ dip)⁺⟧ ⟹ ip ≠ dip›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n\ntext \"NB: the property below cannot be lifted to the transitive closure\"\n\nlemma rt_graph_nhip_is_nhop [dest]:\n \"⋀ip ip' σ dip. (ip, ip') ∈ rt_graph σ dip ⟹ ip' = the (nhop (rt (σ ip)) dip)\"\n unfolding rt_graph_def\n (*goal: ‹⋀ip ip' σ dip. (ip, ip') ∈ {uu_. ∃ip ip' dsn dsk hops pre. uu_ = (ip, ip') ∧ ip ≠ dip ∧ rt (σ ip) dip = Some (dsn, dsk, val, hops, ip', pre)} ⟹ ip' = the (nhop (rt (σ ip)) dip)›*)\n by auto\n\ntheorem inv_to_loop_freedom:\n assumes \"∀i dip. let nhip = the (nhop (rt (σ i)) dip)\n in dip ∈ vD (rt (σ i)) ∩ vD (rt (σ nhip)) ∧ nhip ≠ dip\n ⟶ (rt (σ i)) ⊏⇘dip⇙ (rt (σ nhip))\"\n shows \"∀dip. irrefl ((rt_graph σ dip)⁺)\"\n using assms (*‹∀i dip. let nhip = the (nhop (rt (σ i)) dip) in dip ∈ vD (rt (σ i)) ∩ vD (rt (σ nhip)) ∧ nhip ≠ dip ⟶ rt (σ i) ⊏⇘dip⇙ rt (σ nhip)›*) proof (intro allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*))\n (*goal: ‹⋀dip. ∀i dip. let nhip = the (nhop (rt (σ i)) dip) in dip ∈ vD (rt (σ i)) ∩ vD (rt (σ nhip)) ∧ nhip ≠ dip ⟶ rt (σ i) ⊏⇘dip⇙ rt (σ nhip) ⟹ irrefl ((rt_graph σ dip)⁺)›*)\n fix σ :: \"ip ⇒ state\" and dip\n assume inv: \"∀ip dip.\n let nhip = the (nhop (rt (σ ip)) dip)\n in dip ∈ vD(rt (σ ip)) ∩ vD(rt (σ nhip)) ∧\n nhip ≠ dip ⟶ rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)\" (*‹∀(ip::nat) dip::nat. let nhip::nat = the (nhop (rt ((σ::nat ⇒ state) ip)) dip) in dip ∈ vD (rt (σ ip)) ∩ vD (rt (σ nhip)) ∧ nhip ≠ dip ⟶ rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)›*)\n {\n fix ip and ip'\n assume \"(ip, ip') ∈ (rt_graph σ dip)⁺\" and \"dip ∈ vD(rt (σ ip'))\" and \"ip' ≠ dip\" (*‹(ip::nat, ip'::nat) ∈ (rt_graph (σ::nat ⇒ state) (dip::nat))⁺› ‹(dip::nat) ∈ vD (rt ((σ::nat ⇒ state) (ip'::nat)))› ‹(ip'::nat) ≠ (dip::nat)›*)\n hence \"rt (σ ip) ⊏⇘dip⇙ rt (σ ip')\"\n proof (induction)\n (*goals:\n 1. ‹⋀y. ⟦(ip, y) ∈ rt_graph σ dip; dip ∈ vD (rt (σ y)); y ≠ dip⟧ ⟹ rt (σ ip) ⊏⇘dip⇙ rt (σ y)›\n 2. ‹⋀y z. ⟦(ip, y) ∈ (rt_graph σ dip)⁺; (y, z) ∈ rt_graph σ dip; ⟦dip ∈ vD (rt (σ y)); y ≠ dip⟧ ⟹ rt (σ ip) ⊏⇘dip⇙ rt (σ y); dip ∈ vD (rt (σ z)); z ≠ dip⟧ ⟹ rt (σ ip) ⊏⇘dip⇙ rt (σ z)›*)\n fix nhip\n assume \"(ip, nhip) ∈ rt_graph σ dip\" and \"dip ∈ vD(rt (σ nhip))\" and \"nhip ≠ dip\" (*‹(ip::nat, nhip::nat) ∈ rt_graph (σ::nat ⇒ state) (dip::nat)› ‹(dip::nat) ∈ vD (rt ((σ::nat ⇒ state) (nhip::nat)))› ‹(nhip::nat) ≠ (dip::nat)›*)\n from ‹(ip, nhip) ∈ rt_graph σ dip› (*‹(ip, nhip) ∈ rt_graph σ dip›*) have \"dip ∈ vD(rt (σ ip))\" and \"nhip = the (nhop (rt (σ ip)) dip)\"\n apply -\n (*goals:\n 1. ‹(ip::nat, nhip::nat) ∈ rt_graph (σ::nat ⇒ state) (dip::nat) ⟹ dip ∈ vD (rt (σ ip))›\n 2. ‹(ip::nat, nhip::nat) ∈ rt_graph (σ::nat ⇒ state) (dip::nat) ⟹ nhip = the (nhop (rt (σ ip)) 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 (σ ip))› (*‹dip ∈ vD (rt (σ ip))›*) ‹dip ∈ vD(rt (σ nhip))› (*‹dip ∈ vD (rt (σ nhip))›*) have \"dip ∈ vD(rt (σ ip)) ∩ vD(rt (σ nhip))\"\n by standard\n with ‹nhip = the (nhop (rt (σ ip)) dip)› (*‹(nhip::nat) = the (nhop (rt ((σ::nat ⇒ state) (ip::nat))) (dip::nat))›*) ‹nhip ≠ dip› (*‹(nhip::nat) ≠ (dip::nat)›*) inv (*‹∀(ip::nat) dip::nat. let nhip::nat = the (nhop (rt ((σ::nat ⇒ state) ip)) dip) in dip ∈ vD (rt (σ ip)) ∩ vD (rt (σ nhip)) ∧ nhip ≠ dip ⟶ rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)›*) show \"rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)\"\n by (clarsimp simp: Let_def (*‹Let ?s ?f ≡ ?f ?s›*))\n next\n (*goal: ‹⋀y z. ⟦(ip, y) ∈ (rt_graph σ dip)⁺; (y, z) ∈ rt_graph σ dip; ⟦dip ∈ vD (rt (σ y)); y ≠ dip⟧ ⟹ rt (σ ip) ⊏⇘dip⇙ rt (σ y); dip ∈ vD (rt (σ z)); z ≠ dip⟧ ⟹ rt (σ ip) ⊏⇘dip⇙ rt (σ z)›*)\n fix nhip and nhip'\n assume \"(ip, nhip) ∈ (rt_graph σ dip)⁺\" and \"(nhip, nhip') ∈ rt_graph σ dip\" and IH: \"⟦ dip ∈ vD(rt (σ nhip)); nhip ≠ dip ⟧ ⟹ rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)\" and \"dip ∈ vD(rt (σ nhip'))\" and \"nhip' ≠ dip\" (*‹(ip::nat, nhip::nat) ∈ (rt_graph (σ::nat ⇒ state) (dip::nat))⁺› ‹(nhip::nat, nhip'::nat) ∈ rt_graph (σ::nat ⇒ state) (dip::nat)› ‹⟦(dip::nat) ∈ vD (rt ((σ::nat ⇒ state) (nhip::nat))); nhip ≠ dip⟧ ⟹ rt (σ (ip::nat)) ⊏⇘dip⇙ rt (σ nhip)› ‹(dip::nat) ∈ vD (rt ((σ::nat ⇒ state) (nhip'::nat)))› ‹(nhip'::nat) ≠ (dip::nat)›*)\n from ‹(nhip, nhip') ∈ rt_graph σ dip› (*‹(nhip, nhip') ∈ rt_graph σ dip›*) have 1: \"dip ∈ vD(rt (σ nhip))\" and 2: \"nhip ≠ dip\" and \"nhip' = the (nhop (rt (σ nhip)) dip)\"\n apply -\n (*goals:\n 1. ‹(nhip, nhip') ∈ rt_graph σ dip ⟹ dip ∈ vD (rt (σ nhip))›\n 2. ‹(nhip, nhip') ∈ rt_graph σ dip ⟹ nhip ≠ dip›\n 3. ‹(nhip, nhip') ∈ rt_graph σ dip ⟹ nhip' = the (nhop (rt (σ nhip)) dip)›\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 \"1\" (*‹dip ∈ vD (rt (σ nhip))›*) \"2\" (*‹(nhip::nat) ≠ (dip::nat)›*) have \"rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)\"\n by (rule IH (*‹⟦dip ∈ vD (rt (σ nhip)); nhip ≠ dip⟧ ⟹ rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)›*))\n also (*calculation: ‹rt ((σ::nat ⇒ state) (ip::nat)) ⊏⇘(dip::nat)⇙ rt (σ (nhip::nat))›*) have \"rt (σ nhip) ⊏⇘dip⇙ rt (σ nhip')\"\n proof (-)\n (*goal: ‹rt ((σ::nat ⇒ state) (nhip::nat)) ⊏⇘(dip::nat)⇙ rt (σ (nhip'::nat))›*)\n from ‹dip ∈ vD(rt (σ nhip))› (*‹dip ∈ vD (rt (σ nhip))›*) ‹dip ∈ vD(rt (σ nhip'))› (*‹dip ∈ vD (rt (σ nhip'))›*) have \"dip ∈ vD(rt (σ nhip)) ∩ vD(rt (σ nhip'))\"\n by standard\n with ‹nhip' ≠ dip› (*‹nhip' ≠ dip›*) ‹nhip' = the (nhop (rt (σ nhip)) dip)› (*‹(nhip'::nat) = the (nhop (rt ((σ::nat ⇒ state) (nhip::nat))) (dip::nat))›*) inv (*‹∀ip dip. let nhip = the (nhop (rt (σ ip)) dip) in dip ∈ vD (rt (σ ip)) ∩ vD (rt (σ nhip)) ∧ nhip ≠ dip ⟶ rt (σ ip) ⊏⇘dip⇙ rt (σ nhip)›*) show \"rt (σ nhip) ⊏⇘dip⇙ rt (σ nhip')\"\n by (clarsimp simp: Let_def (*‹Let (?s::?'a::type) (?f::?'a::type ⇒ ?'b::type) ≡ ?f ?s›*))\n qed\n finally (*calculation: ‹rt (σ ip) ⊏⇘dip⇙ rt (σ nhip')›*) show \"rt (σ ip) ⊏⇘dip⇙ rt (σ nhip')\" .\n qed\n }\n note fresher = this (*‹⟦(?ip2, ?ip'2) ∈ (rt_graph σ dip)⁺; dip ∈ vD (rt (σ ?ip'2)); ?ip'2 ≠ dip⟧ ⟹ rt (σ ?ip2) ⊏⇘dip⇙ rt (σ ?ip'2)›*)\n show \"irrefl ((rt_graph σ dip)⁺)\"\n unfolding irrefl_def\n (*goal: ‹∀a. (a, a) ∉ (rt_graph σ dip)⁺›*)\n proof (intro allI (*‹(⋀x. ?P x) ⟹ ∀x. ?P x›*) notI (*‹(?P ⟹ False) ⟹ ¬ ?P›*))\n (*goal: ‹⋀a. (a, a) ∈ (rt_graph σ dip)⁺ ⟹ False›*)\n fix ip\n assume \"(ip, ip) ∈ (rt_graph σ dip)⁺\" (*‹(ip::nat, ip) ∈ (rt_graph (σ::nat ⇒ state) (dip::nat))⁺›*)\n moreover then have \"dip ∈ vD(rt (σ ip))\" and \"ip ≠ dip\"\n apply -\n (*goals:\n 1. ‹(ip, ip) ∈ (rt_graph σ dip)⁺ ⟹ dip ∈ vD (rt (σ ip))›\n 2. ‹(ip, ip) ∈ (rt_graph σ dip)⁺ ⟹ ip ≠ dip›\n discuss goal 1*)\n apply ((auto)[1])\n (*discuss goal 2*)\n apply ((auto)[1])\n (*proven 2 subgoals*) .\n ultimately have \"rt (σ ip) ⊏⇘dip⇙ rt (σ ip)\"\n by (rule fresher (*‹⟦(?ip2, ?ip'2) ∈ (rt_graph σ dip)⁺; dip ∈ vD (rt (σ ?ip'2)); ?ip'2 ≠ dip⟧ ⟹ rt (σ ?ip2) ⊏⇘dip⇙ rt (σ ?ip'2)›*))\n thus False\n by simp\n qed\n qed\n\nend\n"},"meta":{"kind":"string","value":"{\n \"path\": \"afp-2025-02-12/thys/AODV/variants/a_norreqid/A_Loop_Freedom.thy\",\n \"repo\": \"afp-2025-02-12\",\n \"sha\": \"c6234458e16f884cc7b13d457dae6a2d35c10643028bd8ce549b129f107b0586\"\n}"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":8946,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3NjQ1Mzc0Niwic3ViIjoiL2RhdGFzZXRzL0FOVFBHL2Fubm90YXRlZC1pc2FiZWxsZSIsImV4cCI6MTc3NjQ1NzM0NiwiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.nyCP___PF80EZPzT-uHbmSa0fiy55YJfve6tFoZcZhMtmscn_PLg1YgwdBqnGsN_AYuQTDsA42yx_zA6bFn6Aw","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 |
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(* Title: HOL/Algebra/Coset.thy
Authors: Florian Kammueller, L C Paulson, Stephan Hohe
With additional contributions from Martin Baillon and Paulo EmÃlio de Vilhena.
*)
theory Coset
imports Group
begin
section ‹Cosets and Quotient Groups›
definition
r_coset :: "[_, 'a set, 'a] ⇒ 'a set" (infix... | {
"path": "Isabelle2024/src/HOL/Algebra/Coset.thy",
"repo": "Isabelle2024",
"sha": "1f04909087455c55d370d187d0b01b1e100c01a9117e04493d8d2f7fed044951"
} |
section ‹Standard Rules›
text ‹We define the standard rules here, and prove the relation to standard rules.
This means proving that the graph rules do what they say they do.›
theory StandardRules
imports StandardModels RuleSemanticsConnection
begin
text ‹Definition 16 makes this remark. We don't have a specific... | {
"path": "afp-2025-02-12/thys/Graph_Saturation/StandardRules.thy",
"repo": "afp-2025-02-12",
"sha": "55a9f695c9ff5ce75a09a60e090141048b81eb3dcb37a4123489ca2fa4386e98"
} |
(*
Title: The pi-calculus
Author/Maintainer: Jesper Bengtson (jebe.dk), 2012
*)
theory Strong_Early_Bisim_Pres
imports Strong_Early_Bisim Strong_Early_Sim_Pres
begin
(************* Preservation rules *************)
lemma tauPres:
fixes P :: pi
and Q :: pi
assumes "P ∼ Q"
shows "τ.(P) ∼ τ.(Q)"
... | {
"path": "afp-2025-02-12/thys/Pi_Calculus/Strong_Early_Bisim_Pres.thy",
"repo": "afp-2025-02-12",
"sha": "7f1697498a0eb757420b9da7f7b05ce3f185bd5bb467fa5d6b46dd04a89691cb"
} |
theory Uncertainty_Principle
imports "QHLProver.Complex_Matrix"
begin
section‹Setup›
abbreviation bra_ket ("⟨_|_⟩")
where "⟨u|v⟩ ≡ inner_prod u v"
text‹Fix an n-dimensional normalized quantum state $\psi$.›
locale quantum_state =
fixes n:: nat
and ψ:: "complex Matrix.vec"
assumes dim[simp]: "ψ ∈ carrier_... | {
"path": "afp-2025-02-12/thys/Uncertainty_Principle/Uncertainty_Principle.thy",
"repo": "afp-2025-02-12",
"sha": "998f7f0546983fe39fb851e4506f882cfb8c49675399aae22af7762c1c49f2d6"
} |
(* Title: HOL/HOLCF/Universal.thy
Author: Brian Huffman
*)
section ‹A universal bifinite domain›
theory Universal
imports Bifinite Completion "HOL-Library.Nat_Bijection"
begin
no_notation binomial (infixl "choose" 65)
subsection ‹Basis for universal domain›
subsubsection ‹Basis datatype›
type_synon... | {
"path": "Isabelle2024/src/HOL/HOLCF/Universal.thy",
"repo": "Isabelle2024",
"sha": "5f53c7ae070a6ad0081d98f12d2b1b2185b5629401303604863ca0fbae9ed5ae"
} |
section ‹The Proof System›
theory OG_Hoare imports OG_Tran begin
primrec assertions :: "'a ann_com ⇒ ('a assn) set" where
"assertions (AnnBasic r f) = {r}"
| "assertions (AnnSeq c1 c2) = assertions c1 ∪ assertions c2"
| "assertions (AnnCond1 r b c1 c2) = {r} ∪ assertions c1 ∪ assertions c2"
| "assertions (AnnCond2 ... | {
"path": "Isabelle2024/src/HOL/Hoare_Parallel/OG_Hoare.thy",
"repo": "Isabelle2024",
"sha": "ae755d774145c68a3ba4f9e3793fef9ea7325a233a2b8681c4dc1a2ce83f2cd4"
} |
(* Title: HOL/Library/Code_Target_Nat.thy
Author: Florian Haftmann, TU Muenchen
*)
section ‹Implementation of natural numbers by target-language integers›
theory Code_Target_Nat
imports Code_Abstract_Nat
begin
subsection ‹Implementation for \<^typ>‹nat››
context
includes natural.lifting integer.liftin... | {
"path": "Isabelle2024/src/HOL/Library/Code_Target_Nat.thy",
"repo": "Isabelle2024",
"sha": "33af182eacc76c7b30adaad7813767ac102c56dd4b6d38e8d24f5734b24266c9"
} |
chapter ‹Arithmetic Constructs›
text ‹Less genereric syntax, more committed towards embedding arithmetics›
(*<*)
theory Syntax_Arith imports Syntax
begin
(*>*)
text ‹(An embedding of) the syntax of arithmetic, obtained by adding plus and times›
locale Syntax_Arith_aux =
Syntax_with_Connectives_Rename
var trm fmla... | {
"path": "afp-2025-02-12/thys/Syntax_Independent_Logic/Syntax_Arith.thy",
"repo": "afp-2025-02-12",
"sha": "cae8e04558d2c0fb7cd88ace539d0d963d235512e237b7cfad4448e4525884e6"
} |
section ‹Partial and total correctness›
theory Quantum_Hoare
imports Quantum_Program
begin
context state_sig
begin
definition density_states :: "state set" where
"density_states = {ρ ∈ carrier_mat d d. partial_density_operator ρ}"
lemma denote_density_states:
"ρ ∈ density_states ⟹ well_com S ⟹ denote S ρ ∈ de... | {
"path": "afp-2025-02-12/thys/QHLProver/Quantum_Hoare.thy",
"repo": "afp-2025-02-12",
"sha": "78603246b6f024f4f13129280b404b0dc828d1c3c605819023715021e2481e17"
} |
theory Variable_Substitution
imports
Abstract_Substitution.Substitution
"HOL-Library.FSet"
"HOL-Library.Multiset"
begin
locale finite_set =
fixes set :: "'b ⇒ 'a set"
assumes finite_set [simp]: "⋀b. finite (set b)"
begin
abbreviation finite_set :: "'b ⇒ 'a fset" where
"finite_set b ≡ Abs_fset (se... | {
"path": "afp-2025-02-12/thys/Superposition_Calculus/Variable_Substitution.thy",
"repo": "afp-2025-02-12",
"sha": "f3c144ce197fbdc2ea9ed1a4e6fb70e01e70cba4ad6a219f7e865acc862877be"
} |
(* Title: HOL/Partial_Function.thy
Author: Alexander Krauss, TU Muenchen
*)
section ‹Partial Function Definitions›
theory Partial_Function
imports Complete_Partial_Order Option
keywords "partial_function" :: thy_defn
begin
named_theorems partial_function_mono "monotonicity rules for partial function defi... | {
"path": "Isabelle2024/src/HOL/Partial_Function.thy",
"repo": "Isabelle2024",
"sha": "43cc017289a69f030ac623bb2ebfc5ef62c1bd936fd8f2cea7d9b1ecabf44701"
} |
(* Title: Sturm-Tarski Theorem
Author: Wenda Li <wl302@cam.ac.uk / liwenda1990@hotmail.com>
*)
section "Sturm--Tarski Theorem"
theory Sturm_Tarski
imports Complex_Main PolyMisc "HOL-Computational_Algebra.Field_as_Ring"
begin
subsection‹Misc›
lemma eventually_at_right:
fixes x::"'a::{archimedean_field,l... | {
"path": "afp-2025-02-12/thys/Sturm_Tarski/Sturm_Tarski.thy",
"repo": "afp-2025-02-12",
"sha": "79318959df16a28f3bf45740f94d85ee9ea28a06da08d77975fbccce9fa52556"
} |
(* Author: L C Paulson, University of Cambridge
Author: Amine Chaieb, University of Cambridge
Author: Robert Himmelmann, TU Muenchen
Author: Brian Huffman, Portland State University
*)
chapter ‹Topology›
theory Elementary_Topology
imports
"HOL-Library.Set_Idioms"
"HOL-Library.Disjoint... | {
"path": "Isabelle2024/src/HOL/Analysis/Elementary_Topology.thy",
"repo": "Isabelle2024",
"sha": "cf0598090caa5e200f9f3f5750d4818b493800ae900bfa1a40b2d6423429e39d"
} |
(*******************************************************************************
Project: IsaNet
Author: Tobias Klenze, ETH Zurich <tobias.klenze@inf.ethz.ch>
Christoph Sprenger, ETH Zurich <sprenger@inf.ethz.ch>
Version: JCSPaper.1.0
Isabelle Version: Isabelle2021-1
Copyright (c) 2022 Tobias ... | {
"path": "afp-2025-02-12/thys/IsaNet/Network_Model.thy",
"repo": "afp-2025-02-12",
"sha": "fe90bbaa9b4f5c83f9fa61ae8bf636da2fe3e9574f00d4d34d4abb3b7b60de85"
} |
(* Title: Rtrancl_On.thy
Author: Lars Noschinski, TU München
Author: René Neumann, TU München
*)
theory Rtrancl_On
imports Main
begin
section ‹Reflexive-Transitive Closure on a Domain›
text ‹
In this section we introduce a variant of the reflexive-transitive closure
of a relation which is useful to for... | {
"path": "afp-2025-02-12/thys/Graph_Theory/Rtrancl_On.thy",
"repo": "afp-2025-02-12",
"sha": "e4b20bfc6a16e6043cf17779dfea1ec152893192e2490230a68fda2b21fb7e3b"
} |
theory MutableRefProps
imports MutableRef
begin
inductive_cases
vfun_le_inv[elim!]: "VFun t1 ⊑ VFun t2" and
le_fun_nat_inv[elim!]: "VFun t2 ⊑ VNat x1" and
le_any_nat_inv[elim!]: "v ⊑ VNat n" and
le_nat_any_inv[elim!]: "VNat n ⊑ v" and
le_fun_any_inv[elim!]: "VFun t ⊑ v" and
le_any_fun_inv[elim!]: "v ⊑... | {
"path": "afp-2025-02-12/thys/Decl_Sem_Fun_PL/MutableRefProps.thy",
"repo": "afp-2025-02-12",
"sha": "b43e9f8027708f98fe18de7fdb733097d126ebaeb22008b77c94be6c7c3ab7ee"
} |
(* 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: OrderedMap, algorithms for iterators, min, max, to_sorted_list
*)
sec... | {
"path": "afp-2025-02-12/thys/Collections/ICF/gen_algo/MapGA.thy",
"repo": "afp-2025-02-12",
"sha": "71229f869d22a88fde30dd542f2684b50146fe7b906b1a98d0f113e82766d8ca"
} |
(*****************************************************************************
* Featherweight-OCL --- A Formal Semantics for UML-OCL Version OCL 2.5
* for the OMG Standard.
* http://www.brucker.ch/projects/hol-testgen/
*
* Design_UML.thy --- OCL Contracts and an Example... | {
"path": "afp-2025-02-12/thys/Featherweight_OCL/examples/Employee_Model/Design/Design_UML.thy",
"repo": "afp-2025-02-12",
"sha": "9c61cdf19226f004551b2f608d3b5d6d98b4b33253ce9ab4a536f73f31dabfaa"
} |
(* Title: HOL/Nonstandard_Analysis/CLim.thy
Author: Jacques D. Fleuriot
Copyright: 2001 University of Edinburgh
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)
section ‹Limits, Continuity and Differentiation for Complex Functions›
theory CLim
imports CStar
begin
(*not in sim... | {
"path": "Isabelle2024/src/HOL/Nonstandard_Analysis/CLim.thy",
"repo": "Isabelle2024",
"sha": "d14d4c377245aa4514fd9f1e15b92c3fb93c049142bdc98a48ddabad4cdd02e4"
} |
(* Title: HOL/HOLCF/IOA/Deadlock.thy
Author: Olaf Müller
*)
section ‹Deadlock freedom of I/O Automata›
theory Deadlock
imports RefCorrectness CompoScheds
begin
text ‹Input actions may always be added to a schedule.›
lemma scheds_input_enabled:
"Filter (λx. x ∈ act A) ⋅ sch ∈ schedules A ⟹ a ∈ inp A... | {
"path": "Isabelle2024/src/HOL/HOLCF/IOA/Deadlock.thy",
"repo": "Isabelle2024",
"sha": "c71bd981102550e4b88f0b39380b6444365d7745f56a821ac4d014be9956f704"
} |
subsection‹‹Σ›-AND statements›
theory Sigma_AND imports
Sigma_Protocols
Xor
begin
locale Σ_AND_base = Σ0: Σ_protocols_base init0 response0 check0 Rel0 S0_raw 𝒜ss0 "carrier L" valid_pub0
+ Σ1: Σ_protocols_base init1 response1 check1 Rel1 S1_raw 𝒜ss1 "carrier L" valid_pub1
for init1 :: "'pub1 ⇒ 'witness1 ⇒ (... | {
"path": "afp-2025-02-12/thys/Sigma_Commit_Crypto/Sigma_AND.thy",
"repo": "afp-2025-02-12",
"sha": "36a3ad238f71d32ce4d277c9487083ac6a70f0246609c50ef0fee4c6fbcdcb1f"
} |
(* Title: HOL/UNITY/Union.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Partly from Misra's Chapter 5: Asynchronous Compositions of Programs.
*)
section‹Unions of Programs›
theory Union imports SubstAx FP begin
(*FIXME: conjoin... | {
"path": "Isabelle2024/src/HOL/UNITY/Union.thy",
"repo": "Isabelle2024",
"sha": "ffbad27aa953ca15a4cfcd0d622b9cf683d3afb216e7b3e20d047b89194c38b5"
} |
section ‹Filtermap for Lazy Lists›
theory List_Filtermap
imports Main
begin
text ‹ This theory defines the filtermap operator for lazy lists, proves its basic properties,
and proves coinductive criteria for the equqlity of two filtermapped lazy lsits. ›
subsection ‹ Preliminaries ›
(* Hiding the "filtermap" con... | {
"path": "afp-2025-02-12/thys/More_LazyLists/List_Filtermap.thy",
"repo": "afp-2025-02-12",
"sha": "abf0b4e751a37da5eb72ffe6da749003563ae66e6565b8f5361cf2062a833463"
} |
(* Copyright (C) 2007--2010 Norbert Schirmer
* All rights reserved, DFKI GmbH
*)
(*
header {* Parallel - IMP *}
*)
subsection ‹PIMP›
theory PIMP
imports ReduceStoreBufferSimulation
begin
datatype expr = Const val | Mem bool addr | Tmp sop
| Unop "val ⇒ val" expr
| Binop "val ⇒ val ⇒ v... | {
"path": "afp-2025-02-12/thys/Store_Buffer_Reduction/PIMP.thy",
"repo": "afp-2025-02-12",
"sha": "269b572c9bed25ecdc3e4f477507c963c2b7e2719549b28228bf75fc8f10eb88"
} |
theory Partially_Filled_Array
imports
"Refine_Imperative_HOL.IICF_Array_List"
Array_SBlit
begin
section "Partially Filled Arrays"
text ‹An array that is only partially filled.
The number of actual elements contained is kept in a second element.
This represents a weakened version of the array\_list from IICF... | {
"path": "afp-2025-02-12/thys/BTree/Partially_Filled_Array.thy",
"repo": "afp-2025-02-12",
"sha": "59cb8bdf298762efbafb1dc277ddc0721fdf08536a49967006b58dc9f049aa33"
} |
(*
File: Solovay_Strassen.thy
Authors: Daniel Stüwe, Manuel Eberl
The Solovay--Strassen primality test.
*)
section ‹The Solovay--Strassen Test›
theory Solovay_Strassen_Test
imports
Generalized_Primality_Test
Euler_Witness
begin
definition solovay_strassen_witness :: "nat ⇒ nat ⇒ bool" where
"solova... | {
"path": "afp-2025-02-12/thys/Probabilistic_Prime_Tests/Solovay_Strassen_Test.thy",
"repo": "afp-2025-02-12",
"sha": "85f42f910798b23188aaf4a6ae9cce409efa0df8c4ae22dd2a1a9e2aca8eb3ca"
} |
theory Canton_Transaction_Tree imports
Inclusion_Proof_Construction
begin
section ‹Canton's hierarchical transaction trees›
typedecl view_data
typedecl view_metadata
typedecl common_metadata
typedecl participant_metadata
datatype view = View view_metadata view_data (subviews: "view list")
datatype transaction = T... | {
"path": "afp-2025-02-12/thys/ADS_Functor/Canton_Transaction_Tree.thy",
"repo": "afp-2025-02-12",
"sha": "77fc73202aed09c1f0c347b67304eec260d8e448298f579afb37f6bbb7538284"
} |
(* Author: Xingyuan Zhang, Chunhan Wu, Christian Urban *)
theory Folds
imports "Regular-Sets.Regular_Exp"
begin
section ‹``Summation'' for regular expressions›
text ‹
To obtain equational system out of finite set of equivalence classes, a fold operation
on finite sets ‹folds› is defined. The use of ‹SOME› makes ‹... | {
"path": "afp-2025-02-12/thys/Myhill-Nerode/Folds.thy",
"repo": "afp-2025-02-12",
"sha": "21c5436f8c60593dd180097bb3f2408059032cae269d6428acc96c80614a1ffc"
} |
(* Title: HOL/MicroJava/BV/Opt.thy
Author: Tobias Nipkow
Copyright 2000 TUM
More about options.
*)
section ‹More about Options›
theory Opt
imports
Err
begin
definition le :: "'a ord ⇒ 'a option ord"
where
"le r o₁ o₂ =
(case o₂ of None ⇒ o₁=None | Some y ⇒ (case o₁ of None ⇒ True | Some x ... | {
"path": "afp-2025-02-12/thys/JinjaThreads/DFA/Opt.thy",
"repo": "afp-2025-02-12",
"sha": "3775bae001b432a4c328fd29bf392419280139d1d37237c207dc1edac2a7c32a"
} |
theory Girth_Chromatic_Misc
imports
Main
"HOL-Library.Extended_Real"
begin
section ‹Auxilliary lemmas and setup›
text ‹
This section contains facts about general concepts which are not directly
connected to the proof of the Chromatic-Girth theorem. At some point in time,
most of them could be moved to the I... | {
"path": "afp-2025-02-12/thys/Girth_Chromatic/Girth_Chromatic_Misc.thy",
"repo": "afp-2025-02-12",
"sha": "281c86ec8dd279a4d42f76efebb3381137680f715fd815f515e3ca0cdd584577"
} |
(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
* Copyright (c) 2022 Apple Inc. All rights reserved.
*
* SPDX-License-Identifier: BSD-2-Clause
*)
(*
* Strengthen functions into simpler monads.
*
* Each block of lifting lemmas converts functions in the "L2" monadic
* framework (an exception framework) ... | {
"path": "afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy",
"repo": "afp-2025-02-12",
"sha": "0cc3a077be788b01f011459be69a0d97beab81d1a9d5e66ac0700b46fc755531"
} |
(* Title: Kleene Relation Algebras
Author: Walter Guttmann
Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>
*)
section ‹Kleene Relation Algebras›
text ‹
This theory combines Kleene algebras with Stone relation algebras.
Relation algebras with transitive closure have been studied by \<^... | {
"path": "afp-2025-02-12/thys/Stone_Kleene_Relation_Algebras/Kleene_Relation_Algebras.thy",
"repo": "afp-2025-02-12",
"sha": "a4a8747a27fba43cd6352134873de9e34334e70bac1cbb07f13bcbc78538fce5"
} |
(*
* Copyright 2014, 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)
*)
(*
Miscellaneous library definitions and lemmas.
*)
chapter "Library"
theory Li... | {
"path": "afp-2025-02-12/thys/Planarity_Certificates/l4v/lib/Lib.thy",
"repo": "afp-2025-02-12",
"sha": "7c6784c0102fce15842ee97e76d44c417c89ccd343cd2d0d5bb34f34e30bb2f8"
} |
(*
Author: Wenda Li <wl302@cam.ac.uk / liwenda1990@hotmail.com>
*)
section ‹Some useful lemmas about transcendental functions›
theory Missing_Transcendental imports
Missing_Topology
Missing_Algebraic
begin
subsection ‹Misc›
lemma exp_Arg2pi2pi_multivalue:
assumes "exp (𝗂 * of_real x) = z"
shows "∃k... | {
"path": "afp-2025-02-12/thys/Winding_Number_Eval/Missing_Transcendental.thy",
"repo": "afp-2025-02-12",
"sha": "03a3b318e4baf946c69c7152beaab5776b8756a042f6e84d32ba0c93e40988ce"
} |
(* Title: HOL/UNITY/Simple/Network.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
The Communication Network.
From Misra, "A Logic for Concurrent Programming" (1994), section 5.7.
*)
theory Network imports "../UNITY" begin
(*The st... | {
"path": "Isabelle2024/src/HOL/UNITY/Simple/Network.thy",
"repo": "Isabelle2024",
"sha": "3291aa1187c119983d1a1b83fc143d69ea5e722e0b4afc0ff7cc192d9f77ad9d"
} |
(* Title: Examples/TTS_Foundations/Orders/Type_Simple_Orders.thy
Author: Mihails Milehins
Copyright 2021 (C) Mihails Milehins
*)
section‹Abstract orders on types›
theory Type_Simple_Orders
imports
"../Foundations/FNDS_Definite_Description"
FNDS_Auxiliary
begin
subsection‹Background›
text‹
The resul... | {
"path": "afp-2025-02-12/thys/Types_To_Sets_Extension/Examples/TTS_Foundations/Orders/Type_Simple_Orders.thy",
"repo": "afp-2025-02-12",
"sha": "0cba0b60f48d3461fb80164ed31a18784d9e84f277efc145fb4bb8b80db7f112"
} |
(* Title: AnalyticTestFunction.thy
Authors: Filip Smola and Jacques D. Fleuriot, University of Edinburgh, 2019-2021
*)
theory AnalyticTestFunction
imports HyperdualFunctionExtension "HOL-Decision_Procs.Approximation"
begin
subsection‹Analytic Test Function›
text‹
We investigate the analytic test function u... | {
"path": "afp-2025-02-12/thys/Hyperdual/AnalyticTestFunction.thy",
"repo": "afp-2025-02-12",
"sha": "2625d1b46c1b37d8606629dc6bc5f7769f6d5e31e2bd79f334b8c8ed8b801114"
} |
(*<*)
―‹ ********************************************************************
* Project : HOL-CSP - A Shallow Embedding of CSP in Isabelle/HOL
* Version : 2.0
*
* Author : Burkhart Wolff, Safouan Taha, Lina Ye.
* (Based on HOL-CSP 1.0 by Haykal Tej and Burkhart Wolff)
*... | {
"path": "afp-2025-02-12/thys/HOL-CSP/CopyBuffer.thy",
"repo": "afp-2025-02-12",
"sha": "9eb5813e78c288d7056f66267d7d01b5f512194a3591228a1c3c63565cc66b25"
} |
(* Author: Andreas Lochbihler, ETH Zurich
Author: Joshua Schneider, ETH Zurich
*)
subsection ‹Pointwise arithmetic on infinite binary trees›
theory Cotree_Algebra
imports Cotree
begin
subsubsection ‹Constants and operators›
instantiation tree :: (zero) zero begin
definition [applicative_unfold]: "0 = pure_tree... | {
"path": "afp-2025-02-12/thys/Stern_Brocot/Cotree_Algebra.thy",
"repo": "afp-2025-02-12",
"sha": "e9241385d802f5e6e4d80da5bf3e1f716141ced433d98433d2804343e381cf1c"
} |
(*******************************************************************************
Project: Sumcheck Protocol
Authors: Azucena Garvia Bosshard <zucegb@gmail.com>
Christoph Sprenger, ETH Zurich <sprenger@inf.ethz.ch>
Jonathan Bootle, IBM Research Europe <jbt@zurich.ibm.com>
*******************... | {
"path": "afp-2025-02-12/thys/Sumcheck_Protocol/Instantiation_with_Polynomials/Polynomial_Instantiation.thy",
"repo": "afp-2025-02-12",
"sha": "ae8569b83d90532877a6665fdec613b60b1514685bd80c1442a3836366ae299d"
} |
(* Title: Error_Monad
Author: Christian Sternagel
Author: René Thiemann
*)
section ‹The Sum Type as Error Monad›
theory Error_Monad
imports
"HOL-Library.Monad_Syntax"
Error_Syntax
begin
text ‹Make monad syntax (including do-notation) available for the sum type.›
definition bind :: "'e + 'a ⇒ ('a... | {
"path": "afp-2025-02-12/thys/Certification_Monads/Error_Monad.thy",
"repo": "afp-2025-02-12",
"sha": "9470f51cf98b391d40c6afa636bdb669f43f3b52638d59c60e6d40328d38f842"
} |
section‹Separative notions and proper extensions›
theory Proper_Extension
imports
Names
begin
text‹The key ingredient to obtain a proper extension is to have
a ∗‹separative preorder›:›
locale separative_notion = forcing_notion +
assumes separative: "p∈ℙ ⟹ ∃q∈ℙ. ∃r∈ℙ. q ≼ p ∧ r ≼ p ∧ q ⊥ r"
begin
text‹For se... | {
"path": "afp-2025-02-12/thys/Independence_CH/Proper_Extension.thy",
"repo": "afp-2025-02-12",
"sha": "e75b2f4e2a96e6766dff6b448bcafd0e89ef5c05b862605ec5899532aadf6b99"
} |
theory Asymptotic_Security imports Concrete_Security begin
section ‹Asymptotic security definition›
locale constructive_security_obsf' =
fixes real_resource :: "security ⇒ ('a + 'e, 'b + 'f) resource"
and ideal_resource :: "security ⇒ ('c + 'e, 'd + 'f) resource"
and sim :: "security ⇒ ('a, 'b, 'c, 'd) conv... | {
"path": "afp-2025-02-12/thys/Constructive_Cryptography_CM/Asymptotic_Security.thy",
"repo": "afp-2025-02-12",
"sha": "d32f51650a6f0614b8e853a16727afa8c9b8983586cdf1b6ef594b3a1b53c401"
} |
(* Title: More_Manifolds
Author: Richard Schmoetten <richard.schmoetten@ed.ac.uk>, 2024
Maintainer: Richard Schmoetten <richard.schmoetten@ed.ac.uk>
*)
section ‹Results about Manifolds and Analysis.›
theory More_Manifolds
imports
Types_To_Sets_Extension.VS_Modules
Smooth_Manifolds.Tangent_Spac... | {
"path": "afp-2025-02-12/thys/Lie_Groups/More_Manifolds.thy",
"repo": "afp-2025-02-12",
"sha": "1e573d76ee583d30471a5e0a788d7fd528f5df816c7053fb65f28ac28cad2bb2"
} |
(* Title: Isomorphism Classes of Groups
Author: Jakob von Raumer, Karlsruhe Institute of Technology
Maintainer: Jakob von Raumer <jakob.raumer@student.kit.edu>
*)
theory GroupIsoClasses
imports
"HOL-Algebra.Coset"
begin
section ‹Isomorphism Classes of Groups›
text ‹We construct a quotient type fo... | {
"path": "afp-2025-02-12/thys/Jordan_Hoelder/GroupIsoClasses.thy",
"repo": "afp-2025-02-12",
"sha": "6caa7eff858cc10b7ad0062fa38a0f89ae43a955de5983ba3a95c64b8d48144a"
} |
(* Title: variants/e_all_abcd/OAodv.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke, Inria
*)
section "The `open' AODV model"
theory E_OAodv
imports E_Aodv AWN.OAWN_SOS_Labels AWN.OAWN_Convert
begin
text ‹Definitions for stating and proving global network properties over indivi... | {
"path": "afp-2025-02-12/thys/AODV/variants/e_all_abcd/E_OAodv.thy",
"repo": "afp-2025-02-12",
"sha": "d9bdc5c8b265a338ebb0c4708c69bccbf024ad883132db97b1250883bda2fa0a"
} |
(* Title: Signed (Finite) Multisets
Author: Jasmin Blanchette <jasmin.blanchette at inria.fr>, 2016
Maintainer: Jasmin Blanchette <jasmin.blanchette at inria.fr>
*)
section ‹Signed (Finite) Multisets›
theory Signed_Multiset
imports Multiset_More
abbrevs
"!z" = "⇩z"
begin
unbundle multiset.lift... | {
"path": "afp-2025-02-12/thys/Nested_Multisets_Ordinals/Signed_Multiset.thy",
"repo": "afp-2025-02-12",
"sha": "98fc4666e08c33288fdfdbdcb7d2aa4029cb0ab8974d89c96fbb67a41b2b732f"
} |
(* Author: Tobias Nipkow *)
subsection "Parity Analysis"
theory Abs_Int1_parity
imports Abs_Int1
begin
datatype parity = Even | Odd | Either
text‹Instantiation of class \<^class>‹order› with type \<^typ>‹parity›:›
instantiation parity :: order
begin
text‹First the definition of the interface function ‹≤›. Note th... | {
"path": "Isabelle2024/src/HOL/IMP/Abs_Int1_parity.thy",
"repo": "Isabelle2024",
"sha": "51944ece9ff9fcac35e55fa8f965f08d5e34fc9c379402f35e301690481457e2"
} |
section ‹Symbol sequence operations›
theory Symbol_Ops
imports Two_Four_Symbols
begin
text ‹
While previous sections have focused on ``formatted'' symbol sequences for
numbers and lists, in this section we devise some Turing machines dealing with
``unstructured'' arbitrary symbol sequences. The only ``structure'' t... | {
"path": "afp-2025-02-12/thys/Cook_Levin/Symbol_Ops.thy",
"repo": "afp-2025-02-12",
"sha": "2b6e4782e44b13f2e89689c293016aaa7110892ef1e73ee85295908050291950"
} |
theory Window
imports "HOL-Library.AList" "HOL-Library.Mapping" "HOL-Library.While_Combinator" Timestamp
begin
type_synonym ('a, 'b) mmap = "('a × 'b) list"
(* 'b is a polymorphic input symbol; 'c is a polymorphic DFA state;
'd is a timestamp; 'e is a submonitor state *)
inductive chain_le :: "'d :: timestamp l... | {
"path": "afp-2025-02-12/thys/VYDRA_MDL/Window.thy",
"repo": "afp-2025-02-12",
"sha": "db44879f4a6f0536619e9a10b8cee279d0ffd4d45781576d8e774e16777e55af"
} |
theory Padic_Field_Polynomials
imports Padic_Fields
begin
(**************************************************************************************************)
(**************************************************************************************************)
section‹$p$-adic Univariate Polynomials and Hensel's Lem... | {
"path": "afp-2025-02-12/thys/Padic_Field/Padic_Field_Polynomials.thy",
"repo": "afp-2025-02-12",
"sha": "b26076a426fc9355981e8996579acef39991eb54d29fad14545f7d35c2d42fb4"
} |
(*
* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
* Copyright (c) 2022 Apple Inc. All rights reserved.
*
* SPDX-License-Identifier: BSD-2-Clause
*)
(*
Miscellaneous library definitions and lemmas.
*)
chapter "Misc. Definitions and Lemmas"
theory More_Lib
imports
Introduction_AutoCorres2
"HOL-Libr... | {
"path": "afp-2025-02-12/thys/AutoCorres2/lib/More_Lib.thy",
"repo": "afp-2025-02-12",
"sha": "648d769a0f14f8d5c0088eeb8f95c0451bdcb67d746ded5153457a8034c3b3b8"
} |
section ‹Translating terms into Graphs›
text ‹We define the translation function and its properties.›
theory RuleSemanticsConnection
imports LabeledGraphSemantics RulesAndChains
begin
text ‹Definition 15.›
fun translation :: "'c allegorical_term ⇒ ('c, nat) labeled_graph" where
"translation (A_Lbl l) = LG {(l,0,1)} {... | {
"path": "afp-2025-02-12/thys/Graph_Saturation/RuleSemanticsConnection.thy",
"repo": "afp-2025-02-12",
"sha": "64cb5533c7b4ed1d90f5996833b8edcfac5d1a3c01ff7c3a730e2c28cec57b28"
} |
theory AList_Extra
imports "HOL-Library.AList" List_util
begin
lemma list_all2_rel_prod_updateI:
assumes "list_all2 (rel_prod (=) R) xs ys" and "R xval yval"
shows "list_all2 (rel_prod (=) R) (AList.update k xval xs) (AList.update k yval ys)"
using assms(1,1,2) (*‹list_all2 (rel_prod (=) (R::'b ⇒ 'c ⇒ bool))... | {
"path": "afp-2025-02-12/thys/Interpreter_Optimizations/AList_Extra.thy",
"repo": "afp-2025-02-12",
"sha": "969f4d5792811789df2a8a81739a91908100d90858c2c4cb3c83a7170cf676a4"
} |
section ‹Set Cover›
theory Approx_SC_Hoare
imports
"HOL-Hoare.Hoare_Logic"
Complex_Main (* "HOL-Analysis.Harmonic_Numbers" *)
begin
text ‹This is a formalization of the set cover algorithm and proof
in the book by Kleinberg and Tardos \<^cite>‹"KleinbergT06"›.›
definition harm :: "nat ⇒ 'a :: real_normed_field" ... | {
"path": "afp-2025-02-12/thys/Approximation_Algorithms/Approx_SC_Hoare.thy",
"repo": "afp-2025-02-12",
"sha": "e4d8ce1344a4336cbc7277ec1d991d975d455a6f0c51d11b35ffdb11f497ff55"
} |
section ‹The Same Vote Model›
theory Same_Vote
imports Voting
begin
context quorum_process begin
subsection ‹Model definition›
(******************************************************************************)
text ‹The system... | {
"path": "afp-2025-02-12/thys/Consensus_Refined/Same_Vote.thy",
"repo": "afp-2025-02-12",
"sha": "b0e1b5d73151b7a30c5d6078291c362258d98b4f692e4ef235499453e6efbdc5"
} |
(* Title: JinjaDCI/Compiler/Compiler1.thy
Author: Tobias Nipkow, Susannah Mansky
Copyright TUM 2003, UIUC 2019-20
Based on the Jinja theory Compiler/Compiler1.thy by Tobias Nipkow
*)
section ‹ Compilation Stage 1 ›
theory Compiler1 imports PCompiler J1 Hidden begin
text‹ Replacing variable n... | {
"path": "afp-2025-02-12/thys/JinjaDCI/Compiler/Compiler1.thy",
"repo": "afp-2025-02-12",
"sha": "803c5dd2b4660baf44fadaecf9da862e310d0456c6c38ddd2181245d4389bc63"
} |
(* Title: JinjaThreads/Common/WellForm.thy
Author: Tobias Nipkow, Andreas Lochbihler
Based on the Jinja theory Common/WellForm.thy by Tobias Nipkow
*)
section ‹Generic Well-formedness of programs›
theory WellForm
imports
SystemClasses
ExternalCall
begin
text ‹\noindent This theory defines glob... | {
"path": "afp-2025-02-12/thys/JinjaThreads/Common/WellForm.thy",
"repo": "afp-2025-02-12",
"sha": "c066da78761cfd37e45351a2aabbc9ee5c899072fe1e869912b5b2d35b5331b8"
} |
(* Title: HOL/HOLCF/One.thy
Author: Oscar Slotosch
*)
section ‹The unit domain›
theory One
imports Lift
begin
type_synonym one = "unit lift"
translations
(type) "one" ↽ (type) "unit lift"
definition ONE :: "one"
where "ONE ≡ Def ()"
text ‹Exhaustion and Elimination for type \<^typ>‹one››
lemm... | {
"path": "Isabelle2024/src/HOL/HOLCF/One.thy",
"repo": "Isabelle2024",
"sha": "d0364857dc9132187e84b08a6e455d6cc30c4ab5ffd0cf33c3930bb3b997fef4"
} |
(* Title: Jive Data and Store Model
Author: Norbert Schirmer <schirmer at informatik.tu-muenchen.de> and
Nicole Rauch <rauch at informatik.uni-kl.de>, 2005
Maintainer: Nicole Rauch <rauch at informatik.uni-kl.de>
License: LGPL
*)
section ‹The Direct Subtype Relation of ... | {
"path": "afp-2025-02-12/thys/JiveDataStoreModel/Isa_Counter/DirectSubtypes.thy",
"repo": "afp-2025-02-12",
"sha": "5c10db58f0ef21f5ea7851d144118d181f21b7958604469c6bd0331158f80a9f"
} |
section‹Tag-Based Encodings›
theory T
imports T_G_Prelim
begin
subsection‹The tag translation›
text‹The extension of the function symbols with type tags and type witnesses:›
datatype ('fsym,'tp) efsym = Oldf 'fsym | Tag 'tp | Wit 'tp
context ProblemIkTpart
begin
text‹``TE'' stands for ``tag encoding''›
fun TE_... | {
"path": "afp-2025-02-12/thys/Sort_Encodings/T.thy",
"repo": "afp-2025-02-12",
"sha": "85ba2bcd9cb3e9894237d93346ed729d8d5528d84741f6ffdbaf454163aeede6"
} |
(*<*)
(*
* 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 Local_Invariants_Lemmas
imports
Local_Invariants
begin
declare subst... | {
"path": "afp-2025-02-12/thys/ConcurrentGC/Local_Invariants_Lemmas.thy",
"repo": "afp-2025-02-12",
"sha": "bd2136eabd7116f4a5cb3c62cf92063cf30244585bbd0f1afd07d0519136f5b0"
} |
theory MLSS_Logic
imports Main
begin
section ‹Propositional formulae›
text ‹
This theory contains syntax and semantics of propositional formulae.
›
datatype (atoms: 'a) fm =
is_Atom: Atom 'a | And "'a fm" "'a fm" | Or "'a fm" "'a fm" |
Neg "'a fm"
fun "interp" :: "('model ⇒ 'a ⇒ bool) ⇒ 'model ⇒ 'a fm ⇒ bool... | {
"path": "afp-2025-02-12/thys/MLSS_Decision_Proc/MLSS_Logic.thy",
"repo": "afp-2025-02-12",
"sha": "5164f2593af2e1cd74eb4ec71a29bcd6ae9cd238b68367869274a0df1ecda30f"
} |
(*
File: Master_Theorem_Examples.thy
Author: Manuel Eberl <manuel@pruvisto.org>
Examples for the application of the Master theorem and related proof methods.
*)
section ‹Examples›
theory Master_Theorem_Examples
imports
Complex_Main
Akra_Bazzi_Method
Akra_Bazzi_Approximation
begin
subsection ‹Merge sort... | {
"path": "afp-2025-02-12/thys/Akra_Bazzi/Master_Theorem_Examples.thy",
"repo": "afp-2025-02-12",
"sha": "c32b07d24a0d951e55ca46d62f82044f9f5e5ddbc8dd9f0aa270c9e78871a9b5"
} |
(*
File: Algebraic_Auxiliaries.thy
Authors: Daniel Stüwe
Miscellaneous facts about algebra and number theory
*)
section ‹Auxiliary Material›
theory Algebraic_Auxiliaries
imports
"HOL-Algebra.Algebra"
"HOL-Computational_Algebra.Squarefree"
"HOL-Number_Theory.Number_Theory"
begin
hi... | {
"path": "afp-2025-02-12/thys/Probabilistic_Prime_Tests/Algebraic_Auxiliaries.thy",
"repo": "afp-2025-02-12",
"sha": "7ac0539a36576565393103bde5a966cbedf712ff7e230fa0f942a596e10fd874"
} |
(*<*)
(*
* 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 Phases
imports
Global_Invariants_Lemmas
Local_Invariants_Lemmas
T... | {
"path": "afp-2025-02-12/thys/ConcurrentGC/Phases.thy",
"repo": "afp-2025-02-12",
"sha": "6b41b8c9e0546234c6b0ac66d84c124ae9c939ed9d9fd090bdc5148f19b6aff0"
} |
(*
Title: The Calculus of Communicating Systems
Author/Maintainer: Jesper Bengtson (jebe@itu.dk), 2012
*)
theory Strong_Bisim
imports Strong_Sim
begin
lemma monotonic:
fixes P :: ccs
and A :: "(ccs × ccs) set"
and Q :: ccs
and B :: "(ccs × ccs) set"
assumes "P ↝[A] Q"
and "A ⊆ B"
... | {
"path": "afp-2025-02-12/thys/CCS/Strong_Bisim.thy",
"repo": "afp-2025-02-12",
"sha": "c4dfbeef3e8dd843e71499f205988a41ec0dc6405de4fc61d9c9dc5d1aff9cc0"
} |
(* Title: Map Function on Two Parallel Lists
Author: Anders Schlichtkrull <andschl at dtu.dk>, 2017
Maintainer: Anders Schlichtkrull <andschl at dtu.dk>
*)
section ‹Map Function on Two Parallel Lists›
theory Map2
imports Main
begin
text ‹
This theory defines a map function that applies a (curr... | {
"path": "afp-2025-02-12/thys/Ordered_Resolution_Prover/Map2.thy",
"repo": "afp-2025-02-12",
"sha": "559a1a66db962fd6d0cfc5fbf4348ebf7a18cd04e6a4df59bb173650ceaaf10f"
} |
(*<*)
theory Infinite_Proof_Soundness
imports Finite_Proof_Soundness "HOL-Library.BNF_Corec"
begin
(* Reference: A Generic Cyclic Theorem Prover
by James Brotherston, Nikos Gorogiannis, and Rasmus L. Petersen
*)
(*>*)
section ‹Soundness of Infinite Proof Trees›
context
begin
private definition "num P xs ≡ LEAST n. ... | {
"path": "afp-2025-02-12/thys/Abstract_Soundness/Infinite_Proof_Soundness.thy",
"repo": "afp-2025-02-12",
"sha": "6a84ee2239a43b96a47066d3fb36e91f652cf9088e7c3ef573a86f929ca968f9"
} |
section ‹Result›
theory Result imports Soundness Completeness begin
theorem prover_soundness_completeness:
fixes A B :: ‹fm list›
defines ‹t ≡ prover (A, B)›
shows ‹tfinite t ∧ wf t ⟷ (∀(E :: _ ⇒ tm) F G. sc (E, F, G) (A, B))›
using assms (*‹t ≡ prover (A, B)›*) prover_soundness (*‹⟦tfinite ?t; Prover.wf ?t... | {
"path": "afp-2025-02-12/thys/FOL_Seq_Calc3/Result.thy",
"repo": "afp-2025-02-12",
"sha": "2fb2e3cdf071fd35c5b652f8c0c7706bd1814ea322e242c4996bc069c87a02bb"
} |
section ‹Operational Semantics›
theory RG_Tran
imports RG_Com
begin
subsection ‹Semantics of Component Programs›
subsubsection ‹Environment transitions›
type_synonym 'a conf = "(('a com) option) × 'a"
inductive_set
etran :: "('a conf × 'a conf) set"
and etran' :: "'a conf ⇒ 'a conf ⇒ bool" ("_ -e→ _" [81,81]... | {
"path": "Isabelle2024/src/HOL/Hoare_Parallel/RG_Tran.thy",
"repo": "Isabelle2024",
"sha": "3bca39ddbedfada250c2911e1fc00f4906eea0c9b78c97fc7a3813cd5c7cfdd9"
} |
(*
Author: René Thiemann
License: LGPL
*)
section ‹Show Implemetation for Real Numbers via Rational Numbers›
text ‹We just provide an implementation for show of real numbers where we assume
that real numbers are implemented via rational numbers.›
theory Show_Real_Impl
imports
Show_Real
... | {
"path": "afp-2025-02-12/thys/Show/Show_Real_Impl.thy",
"repo": "afp-2025-02-12",
"sha": "b143d6248421486ac8bbf5bd776f637e555ed6d192e302b557e4e3866485c6ce"
} |
section ‹Denotational semantics of Circus actions›
theory Denotational_Semantics
imports Circus_Actions Var_list
begin
text ‹In this section, we introduce the definitions of Circus actions denotational semantics.
We provide the proof of well-formedness of every action. We also provide proofs concerning
the monotoni... | {
"path": "afp-2025-02-12/thys/Circus/Denotational_Semantics.thy",
"repo": "afp-2025-02-12",
"sha": "6242356afbf5d62f50266e7010a41fb3e2975e1311a071f09c009dd702084723"
} |
(* Title: Tree Automata
Author: Peter Lammich <peter dot lammich at uni-muenster.de>
Maintainer: Peter Lammich <peter dot lammich at uni-muenster.de>
*)
section "Abstract Tree Automata Algorithms"
theory AbsAlgo
imports
Ta
Collections_Examples.Exploration
Collections.CollectionsV1
begin
no... | {
"path": "afp-2025-02-12/thys/Tree-Automata/AbsAlgo.thy",
"repo": "afp-2025-02-12",
"sha": "38ff073ed89b7908d3ee90883458c5bfc121e7c19e6c87c5876ef0a878f251c8"
} |
(* Title: HOL/ex/Birthday_Paradox.thy
Author: Lukas Bulwahn, TU Muenchen, 2007
*)
section ‹A Formulation of the Birthday Paradox›
theory Birthday_Paradox
imports Main "HOL-Library.FuncSet"
begin
section ‹Cardinality›
lemma card_product_dependent:
assumes "finite S"
assumes "∀x ∈ S. finite (T x)"
shows "c... | {
"path": "Isabelle2024/src/HOL/ex/Birthday_Paradox.thy",
"repo": "Isabelle2024",
"sha": "1d737d6f2c7119b9b64efed0180d491c48b94f58b1959fdc0e5ce4a2dc9afd03"
} |
(*
Title: Projections.thy
Author: Jose Divasón <jose.divasonm at unirioja.es>
Author: Jesús Aransay <jesus-maria.aransay at unirioja.es>
*)
section‹Projections›
theory Projections
imports
Miscellaneous_QR
begin
subsection‹Definitions of vector projection and projection of a vector ont... | {
"path": "afp-2025-02-12/thys/QR_Decomposition/Projections.thy",
"repo": "afp-2025-02-12",
"sha": "516dbca6c37c6064fb53f4126f89f8cad31d630b5162b20ce41e98903167dfce"
} |
section‹Shadowed Rules›
theory Shadowed
imports SimpleFw_Semantics
begin
subsection‹Removing Shadowed Rules›
text‹Testing, not executable›
text‹Assumes: @{term "simple_ruleset"}›
fun rmshadow :: "'i::len simple_rule list ⇒ 'i simple_packet set ⇒ 'i simple_rule list" where
"rmshadow [] _ = []" |
"rmshadow ((Simpl... | {
"path": "afp-2025-02-12/thys/Simple_Firewall/Shadowed.thy",
"repo": "afp-2025-02-12",
"sha": "a8a5c7682f3fd797f3276824eb17763e7c694874f04b9e7773fc0cb88ebce948"
} |
section ‹Transition Systems and Trace Theory›
theory Transition_System_Traces
imports
Transition_System_Extensions
Traces
begin
lemma (in transition_system) words_infI_construct[rule_format, intro?]:
assumes "∀ v. v ≤⇩F⇩I w ⟶ path v p"
shows "run w p"
using assms (*‹∀v. v ≤⇩F⇩I w ⟶ path v p›*) a... | {
"path": "afp-2025-02-12/thys/Partial_Order_Reduction/Transition_System_Traces.thy",
"repo": "afp-2025-02-12",
"sha": "720015277cab71aa99a0b940f12f20b283417506013468c45a720847cd427261"
} |
section ‹Tries via Functions›
theory Trie_Fun
imports
Set_Specs
begin
text ‹A trie where each node maps a key to sub-tries via a function.
Nice abstract model. Not efficient because of the function space.›
datatype 'a trie = Nd bool "'a ⇒ 'a trie option"
definition empty :: "'a trie" where
[simp]: "empty = Nd Fal... | {
"path": "Isabelle2024/src/HOL/Data_Structures/Trie_Fun.thy",
"repo": "Isabelle2024",
"sha": "93e76331845cb8a756a2d5576e7fcc04d369ff727bb1197a9b1b1cd448f0a089"
} |
(*
File: Perfect_Fields/Perfect_Fields.thy
Authors: Katharina Kreuzer (TU München)
Manuel Eberl (University of Innsbruck)
Typeclasses for fields of prime characteristic
Frobenius endomorphism on a field type
Fields with a surjective Frobenius endomorphism
Typeclass for perfect fields a... | {
"path": "afp-2025-02-12/thys/Perfect_Fields/Perfect_Fields.thy",
"repo": "afp-2025-02-12",
"sha": "a8e827d4e33309ea70a254db2ec83f05d37263a7321ee0b5bb316c2f3ae1c140"
} |
(* Anders Schlichtkrull & Jørgen Villadsen, DTU Compute, Denmark *)
chapter ‹On Paraconsistency›
text
‹
Paraconsistency concerns inference systems that do not explode given a contradiction.
The Internet Encyclopedia of Philosophy has a survey article on paraconsistent logic.
The following Isabelle theory formalize... | {
"path": "afp-2025-02-12/thys/Paraconsistency/Paraconsistency.thy",
"repo": "afp-2025-02-12",
"sha": "af3973df4369703ac704b83ad1b265db0d488e4ac56346fa2e924c6f670ea5a1"
} |
subsection ‹Consistency›
theory Guards
imports StateModel CommCSL AbstractCommutativity
begin
text ‹A state is "consistent" iff:
1. All its permissions are full
2. Has unique guards iff has shared guard
3. The values in the fractional heaps are "reachable" wrt to the sequence and multiset of actions
4. Has exactly ... | {
"path": "afp-2025-02-12/thys/CommCSL/Guards.thy",
"repo": "afp-2025-02-12",
"sha": "b5c4b39200b75bab584638b34e3ed9c71b424b6e91bdd77603627830f31dc461"
} |
(* Title: Conflict analysis/Constraint Systems
Author: Peter Lammich <peter.lammich@uni-muenster.de>
Maintainer: Peter Lammich <peter.lammich@uni-muenster.de>
*)
section "Constraint Systems"
theory ConstraintSystems
imports Main AcquisitionHistory Normalization
begin
text_raw ‹\label{thy:Constraint... | {
"path": "afp-2025-02-12/thys/Program-Conflict-Analysis/ConstraintSystems.thy",
"repo": "afp-2025-02-12",
"sha": "2e0466eea88b074d00a31640566d764623ed91ac02d131e69ae2237e4940682b"
} |
section "Proof Terms and proof checker"
theory ProofTerm
imports Term Logic Term_Subst SortConstants EqualityProof
begin
(* Move *)
type_synonym tyinst = "(variable × sort) × typ"
type_synonym tinst = "(variable × typ) × term"
datatype proofterm = PAxm "term" "tyinst list"
| PBound nat
| Abst "t... | {
"path": "afp-2025-02-12/thys/Metalogic_ProofChecker/ProofTerm.thy",
"repo": "afp-2025-02-12",
"sha": "3604a7d6832e1ef8eac85815fa69b2bec15e092492bba2832432ac58ea81ee39"
} |
chapter ‹Abstract Formulation of Gödel's Second Incompleteness Theorem›
(*<*)
theory Abstract_Second_Goedel imports Abstract_First_Goedel Derivability_Conditions
begin
(*>*)
text ‹We assume all three derivability conditions, and assumptions
behind Gödel formulas:›
locale Goedel_Second_Assumptions =
HBL1_2_3
var t... | {
"path": "afp-2025-02-12/thys/Goedel_Incompleteness/Abstract_Second_Goedel.thy",
"repo": "afp-2025-02-12",
"sha": "9d5f6afcb43b5f690ad113b1b15c723683f313b60666c115137c62b04a4b952e"
} |
(* Author: Alexander Maletzky *)
section ‹Direct Decompositions and Hilbert Functions›
theory Hilbert_Function
imports
"HOL-Combinatorics.Permutations"
Dube_Prelims
Degree_Section
begin
subsection ‹Direct Decompositions›
text ‹The main reason for defining ‹direct_decomp› in terms of lists rather than sets is ... | {
"path": "afp-2025-02-12/thys/Groebner_Macaulay/Hilbert_Function.thy",
"repo": "afp-2025-02-12",
"sha": "08d7051dabbd35e69891674b559b29d14e5ab930f72c3790d1819808149ebedc"
} |
section ‹Chain-Complete Partial Orders›
theory CCPO_Extensions
imports
"HOL-Library.Complete_Partial_Order2"
ENat_Extensions
Set_Extensions
begin
lemma chain_split[dest]:
assumes "Complete_Partial_Order.chain ord C" "x ∈ C"
shows "C = {y ∈ C. ord x y} ∪ {y ∈ C. ord y x}"
proof (-)
(*goal: ‹C =... | {
"path": "afp-2025-02-12/thys/Partial_Order_Reduction/Extensions/CCPO_Extensions.thy",
"repo": "afp-2025-02-12",
"sha": "75cd235d83dd28fd22c83760c4bc949435c5aa32c65c0db5fb81546204ec4f8d"
} |
section ‹ Scene Spaces ›
theory Scene_Spaces
imports Scenes
begin
subsection ‹ Preliminaries ›
abbreviation foldr_scene :: "'a scene list ⇒ 'a scene" ("⨆⇩S") where
"foldr_scene as ≡ foldr (⊔⇩S) as ⊥⇩S"
lemma pairwise_indep_then_compat [simp]: "pairwise (⨝⇩S) A ⟹ pairwise (##⇩S) A"
by (simp add: pa... | {
"path": "afp-2025-02-12/thys/Optics/Scene_Spaces.thy",
"repo": "afp-2025-02-12",
"sha": "9880d169546d01b2f100285d3a8fb2f71ce3088e17ea772efe7fa2bf40a1d06e"
} |
section ‹Algorithms to compute all complex and real roots of a quartic polynomial›
theory Quartic_Polynomials
imports
Ferraris_Formula
Cubic_Polynomials
begin
text ‹The complex case is straight-forward›
definition solve_depressed_quartic_complex :: "complex ⇒ complex ⇒ complex ⇒ complex list" where
"sol... | {
"path": "afp-2025-02-12/thys/Cubic_Quartic_Equations/Quartic_Polynomials.thy",
"repo": "afp-2025-02-12",
"sha": "547fa83bef46f1f1e74ebcce7710ed44536b58f3548b4c6a82add1c4beeb4359"
} |
theory SM_Datastructures
imports Main
CAVA_Base.CAVA_Base
"../Lib/SOS_Misc_Add"
DFS_Framework.Feedback_Arcs (* TODO: Only for oo-symbol !?*)
begin
lemma [code_unfold]: "{(a,b). (a,b)∈X ∧ P a b} = Set.filter (λ(a,b). P a b) X"
by auto
lemma in_dom_map_code[code_unfold]:
"x∈dom m = (case m x of... | {
"path": "afp-2025-02-12/thys/CAVA_LTL_Modelchecker/SM/Impl/SM_Datastructures.thy",
"repo": "afp-2025-02-12",
"sha": "87d20a73699671313d7491079c892d89bfd41569a84a1096aef0598d388c582a"
} |
(* Title: Omega Algebra
Author: Alasdair Armstrong, Georg Struth, Tjark Weber
Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
Tjark Weber <tjark.weber at it.uu.se>
*)
section ‹Omega Algebras›
theory Omega_Algebra
imports Kleene_Algebra
begin
text ‹
\emph{Omega algebras}~\<^cite>‹... | {
"path": "afp-2025-02-12/thys/Kleene_Algebra/Omega_Algebra.thy",
"repo": "afp-2025-02-12",
"sha": "8c58889d11250d05e1433f3bc8607062c5f830dbc559343d510231b081386474"
} |
section "Kruskal Implementation"
theory Kruskal_Impl
imports Kruskal_Refine Refine_Imperative_HOL.IICF
begin
subsection ‹Refinement III: concrete edges›
text ‹Given a concrete representation of edges and their endpoints as a pair, we refine
Kruskal's algorithm to work on these concrete edges.›
locale Kruskal_conc... | {
"path": "afp-2025-02-12/thys/Kruskal/Kruskal_Impl.thy",
"repo": "afp-2025-02-12",
"sha": "7992eae3c1d5cd874a146b9e867f47bbc7c491eeeaa885f217fa6c636d975b7b"
} |
(*
Author: Alexander Katovsky
*)
section "Category"
theory Category
imports "HOL-Library.FuncSet"
begin
record ('o,'m) Category =
Obj :: "'o set" ("objı" 70)
Mor :: "'m set" ("morı" 70)
Dom :: "'m ⇒ 'o" ("domı _" [80] 70)
Cod :: "'m ⇒ 'o" ("codı _" [80] 70)
Id :: "'o ⇒ 'm" ("idı _" [80] 75)
Comp :: "'... | {
"path": "afp-2025-02-12/thys/Category2/Category.thy",
"repo": "afp-2025-02-12",
"sha": "edbf406349a21f15ea803e694bdb72b2c584285c382301d907ccfa2ff8ed6482"
} |
section ‹The Hadjicostas--Chapman formula›
theory Hadjicostas_Chapman
imports Zeta_Laurent_Expansion
begin
text ‹
In this section, we will derive a formula for the ‹ζ› function that was conjectured by
Hadjicostas~\<^cite>‹"hadjicostas2004"› and proven shortly afterwards by Chapman~\<^cite>‹"chapman2004"›.
The... | {
"path": "afp-2025-02-12/thys/Zeta_Function/Hadjicostas_Chapman.thy",
"repo": "afp-2025-02-12",
"sha": "2950183625215df4c8efb4e4bd9b0a010ad4194faa57cac5014064b9c665e24c"
} |
(*
Title: HOL/Analysis/Infinite_Sum.thy
Author: Dominique Unruh, University of Tartu
Manuel Eberl, University of Innsbruck
A theory of sums over possibly infinite sets.
*)
section ‹Infinite sums›
\<^latex>‹\label{section:Infinite_Sum}›
text ‹In this theory, we introduce the definition of infin... | {
"path": "Isabelle2024/src/HOL/Analysis/Infinite_Sum.thy",
"repo": "Isabelle2024",
"sha": "bc79361c2879d9ba15f9faa14569ad35506e77beda849fb6f30f4930fa80e951"
} |
(* Title: Miscellaneous results
Author: Tim Makarios <tjm1983 at gmail.com>, 2012
Maintainer: Tim Makarios <tjm1983 at gmail.com>
*)
(*Some of these theorems were moved to the Isabelle repository *)
section "Miscellaneous results"
theory Miscellany
imports Metric
begin
lemma unordered_pair_elem... | {
"path": "afp-2025-02-12/thys/Tarskis_Geometry/Miscellany.thy",
"repo": "afp-2025-02-12",
"sha": "18f5efe3581d5f7d146102856023e81a5ad349cca855c8d87384ef39d55d3117"
} |
theory RM_Sums_Diophantine imports Equation_Setup "../Diophantine/Register_Machine_Sums"
"../Diophantine/Binary_And"
begin
context register_machine
begin
definition sum_ssub_nzero_of_bit_and :: "polynomial ⇒ nat ⇒ polynomial list ⇒ polynomial list
... | {
"path": "afp-2025-02-12/thys/DPRM_Theorem/Machine_Equations/RM_Sums_Diophantine.thy",
"repo": "afp-2025-02-12",
"sha": "f3d636e55d659cfa2e5a0371507c37e5fb2a7c95f1037605c302539899fad61d"
} |
(* Title: JinjaThreads/Framework/FWWellform.thy
Author: Andreas Lochbihler
*)
section ‹Wellformedness conditions for the multithreaded state›
theory FWWellform
imports
FWLocking
FWThread
FWWait
FWCondAction
begin
text‹Well-formedness property: Locks are held by real threads›
definition
lock_... | {
"path": "afp-2025-02-12/thys/JinjaThreads/Framework/FWWellform.thy",
"repo": "afp-2025-02-12",
"sha": "99efa4d1b22480765e290fc944a853fe9df6e899850543f3ee550b0106c1a9cb"
} |
(*
Authors:
Hanna Lachnitt, TU Wien, lachnitt@student.tuwien.ac.at
Anthony Bordg, University of Cambridge, apdb3@cam.ac.uk
*)
section ‹The Deutsch-Jozsa Algorithm›
theory Deutsch_Jozsa
imports
Deutsch
More_Tensor
Binary_Nat
begin
text ‹
Given a function $f:{0,1}^n \mapsto {0,1}$, the Deutsch-Jozsa algor... | {
"path": "afp-2025-02-12/thys/Isabelle_Marries_Dirac/Deutsch_Jozsa.thy",
"repo": "afp-2025-02-12",
"sha": "a3a135605aa0c8b6a893c9f6fbfb40e656eda61f597c54b631c8a2c92a92ba77"
} |
(* Title: variants/a_norreqid/Loop_Freedom.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke, Inria
*)
section "Routing graphs and loop freedom"
theory A_Loop_Freedom
imports A_Aodv_Predicates A_Fresher
begin
text ‹Define the central theorem that relates an invariant over network... | {
"path": "afp-2025-02-12/thys/AODV/variants/a_norreqid/A_Loop_Freedom.thy",
"repo": "afp-2025-02-12",
"sha": "c6234458e16f884cc7b13d457dae6a2d35c10643028bd8ce549b129f107b0586"
} |
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