fact stringlengths 1 2.11k | type stringclasses 27
values | library stringclasses 888
values | imports listlengths 0 81 | filename stringlengths 9 106 | symbolic_name stringlengths 1 113 | docstring stringlengths 0 1.34k ⌀ |
|---|---|---|---|---|---|---|
last_statewhere
"last_state (s,[]) = s"
| "last_state (s,ps#p) = snd p" | fun | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | last_state | null |
last_state_reachable:
fixes A e
assumes "is_exec_of A e"
shows "reachable A (last_state e)" using assms
proof -
have "is_exec_of A e \<Longrightarrow> reachable A (last_state e)"
proof (induction "snd e" arbitrary: e)
case Nil
from Nil.prems have 1:"fst e \<in> start A" by (simp add:is_exec_of_def)
... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | last_state_reachable | null |
trans_from_last_state:
assumes "is_exec_frag_of A e" and "(last_state e)\<midarrow>a\<midarrow>A\<longrightarrow>s'"
shows "is_exec_frag_of A (cons_exec e (a,s'))"
using assms by (cases "(A, fst e, snd e)" rule:is_exec_frag_of.cases, auto simp add:cons_exec_def) | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | trans_from_last_state | null |
exec_frag_prefix:
fixes A p ps
assumes "is_exec_frag_of A (cons_exec e p)"
shows "is_exec_frag_of A e"
using assms by (cases "(A, fst e, snd e)" rule:is_exec_frag_of.cases, auto simp add:cons_exec_def) | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | exec_frag_prefix | null |
trace_same_ext:
fixes A B e
assumes "ext A = ext B"
shows "trace (ioa.asig A) e = trace (ioa.asig B) e"
using assms by (auto simp add:trace_def) | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | trace_same_ext | null |
trace_append_is_append_trace:
fixes e e' sig
shows "trace sig (append_exec e' e) = trace sig e' @ trace sig e"
by (simp add:append_exec_def trace_def schedule_def filter_act_def) | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | trace_append_is_append_trace | null |
append_exec_frags_is_exec_frag:
fixes e e' A as
assumes "is_exec_frag_of A e" and "last_state e = fst e'"
and "is_exec_frag_of A e'"
shows "is_exec_frag_of A (append_exec e e')"
proof -
from assms show ?thesis
proof (induct "(fst e',snd e')" arbitrary:e' rule:is_exec_frag_of.induct)
case (3 A)
from ... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | append_exec_frags_is_exec_frag | null |
last_state_of_append:
fixes e e'
assumes "fst e' = last_state e"
shows "last_state (append_exec e e') = last_state e'"
using assms by (cases e' rule:last_state.cases, auto simp add:append_exec_def) | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/IOA.thy | last_state_of_append | null |
pre_RDR= Sequences +
fixes \<delta>::"'a \<Rightarrow> ('b \<times> 'c) \<Rightarrow> 'a" (infix "\<bullet>" 65)
and \<gamma>::"'a \<Rightarrow> ('b \<times> 'c) \<Rightarrow> 'd"
and bot::'a ("\<bottom>")
begin | locale | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | pre_RDR | null |
exec::"'a \<Rightarrow> ('b\<times>'c)list \<Rightarrow> 'a" (infix "\<star>" 65) where
"exec s Nil = s"
| "exec s (rs#r) = (exec s rs) \<bullet> r" | fun | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | exec | null |
less_eq(infix "\<preceq>" 50) where
"less_eq s s' \<equiv> \<exists> rs . s' = (s\<star>rs)" | definition | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | less_eq | null |
less(infix "\<prec>" 50) where
"less s s' \<equiv> less_eq s s' \<and> s \<noteq> s'" | definition | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | less | null |
is_lbwhere
"is_lb s s1 s2 \<equiv> s \<preceq> s2 \<and> s \<preceq> s1" | definition | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | is_lb | null |
is_glbwhere
"is_glb s s1 s2 \<equiv> is_lb s s1 s2 \<and> (\<forall> s' . is_lb s' s1 s2 \<longrightarrow> s' \<preceq> s)" | definition | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | is_glb | null |
containswhere
"contains s r \<equiv> \<exists> rs . r \<in> set rs \<and> s = (\<bottom> \<star> rs)" | definition | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | contains | null |
inf(infix "\<sqinter>" 65) where
"inf s1 s2 \<equiv> THE s . is_glb s s1 s2" | definition | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | inf | null |
exec_cons:
"s \<star> (rs # r)= (s \<star> rs) \<bullet> r" by simp | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | exec_cons | Useful Lemmas in the pre-RDR locale |
exec_append:
"(s \<star> rs) \<star> rs' = s \<star> (rs@rs')"
proof (induct rs')
show "(s \<star> rs) \<star> [] = s \<star> (rs@[])" by simp
next
fix rs' r
assume ih:"(s \<star> rs) \<star> rs' = s \<star> (rs@rs')"
thus "(s \<star> rs) \<star> (rs'#r) = s \<star> (rs @ (rs'#r))"
by (metis append_Co... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | exec_append | Useful Lemmas in the pre-RDR locale |
trans:
assumes "s1 \<preceq> s2" and "s2 \<preceq> s3"
shows "s1 \<preceq> s3" using assms
by (auto simp add:less_eq_def, metis exec_append) | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | trans | null |
contains_star:
fixes s r rs
assumes "contains s r"
shows "contains (s \<star> rs) r"
proof (induct rs)
case Nil
show "contains (s \<star> []) r" using assms by auto
next
case (Cons r' rs)
with this obtain rs' where 1:"s \<star> rs = \<bottom> \<star> rs'" and 2:"r \<in> set rs'"
by (auto simp add:cont... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | contains_star | null |
preceq_star: "s \<star> (rs#r) \<preceq> s' \<Longrightarrow> s \<star> rs \<preceq> s'"
by (metis pre_RDR.exec.simps(1) pre_RDR.exec.simps(2) pre_RDR.less_eq_def trans) | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | preceq_star | null |
RDR= pre_RDR +
assumes idem1:"contains s r \<Longrightarrow> s \<bullet> r = s"
and idem2:"\<And> s r r' . fst r \<noteq> fst r' \<Longrightarrow> \<gamma> s r = \<gamma> ((s \<bullet> r) \<bullet> r') r"
and antisym:"\<And> s1 s2 . s1 \<preceq> s2 \<and> s2 \<preceq> s1 \<Longrightarrow> s1 = s2"
and glb_exist... | locale | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | RDR | The RDR locale |
inf_glb:"is_glb (s1 \<sqinter> s2) s1 s2"
proof -
{ fix s s'
assume "is_glb s s1 s2" and "is_glb s' s1 s2"
hence "s = s'" using antisym by (auto simp add:is_glb_def is_lb_def) }
from this and glb_exists show ?thesis
by (auto simp add:inf_def, metis (lifting) theI')
qed
sublocale ordering less_eq les... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | inf_glb | null |
idem_star:
fixes r s rs
assumes "contains s r"
shows "s \<star> rs = s \<star> (filter (\<lambda> x . x \<noteq> r) rs)"
proof (induct rs)
case Nil
show "s \<star> [] = s \<star> (filter (\<lambda> x . x \<noteq> r) [])"
using assms by auto
next
case (Cons r' rs)
have 1:"contains (s \<star> rs) r" using ass... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | idem_star | Some useful lemmas |
idem_star2:
fixes s rs'
shows "\<exists> rs' . s \<star> rs = s \<star> rs' \<and> set rs' \<subseteq> set rs
\<and> (\<forall> r \<in> set rs' . \<not> contains s r)"
proof (induct rs)
case Nil
thus "\<exists> rs' . s \<star> [] = s \<star> rs' \<and> set rs' \<subseteq> set []
\<and> (\<forall> r \<in... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | idem_star2 | null |
idem2_star:
assumes "contains s r"
and "\<And> r' . r' \<in> set rs \<Longrightarrow> fst r' \<noteq> fst r"
shows "\<gamma> s r = \<gamma> (s \<star> rs) r" using assms
proof (induct rs)
case Nil
show "\<gamma> s r = \<gamma> (s \<star> []) r" by simp
next
case (Cons r' rs)
thus "\<gamma> s r = \<gamma> (s \<s... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | idem2_star | null |
glb_common:
fixes s1 s2 s rs1 rs2
assumes "s1 = s \<star> rs1" and "s2 = s \<star> rs2"
shows "\<exists> rs . s1 \<sqinter> s2 = s \<star> rs \<and> set rs \<subseteq> set rs1 \<union> set rs2"
proof -
have 1:"s \<preceq> s1" and 2:"s \<preceq> s2" using assms by (auto simp add:less_eq_def)
hence 3:"s \<preceq> s1 ... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | glb_common | null |
glb_common_set:
fixes ss s0 rset
assumes "finite ss" and "ss \<noteq> {}"
and "\<And> s . s \<in> ss \<Longrightarrow> \<exists> rs . s = s0 \<star> rs \<and> set rs \<subseteq> rset"
shows "\<exists> rs . \<Sqinter> ss = s0 \<star> rs \<and> set rs \<subseteq> rset"
using assms
proof (induct ss rule:finite_ne_induct)... | lemma | Abortable_Linearizable_Modules | [
"Main",
"Sequences"
] | Abortable_Linearizable_Modules/RDR.thy | glb_common_set | null |
Sequencesbegin | locale | Abortable_Linearizable_Modules | [
"Main"
] | Abortable_Linearizable_Modules/Sequences.thy | Sequences | Sequences as Lists |
Append(infixl "#" 65)
where "Append xs x \<equiv> Cons x xs"
no_notation append (infixr "@" 65) | abbreviation | Abortable_Linearizable_Modules | [
"Main"
] | Abortable_Linearizable_Modules/Sequences.thy | Append | null |
Concat(infixl "@" 65)
where "Concat xs ys \<equiv> append ys xs" | abbreviation | Abortable_Linearizable_Modules | [
"Main"
] | Abortable_Linearizable_Modules/Sequences.thy | Concat | null |
refineswhere
"refines e s a t A f \<equiv> fst e = f s \<and> last_state e = f t \<and> is_exec_frag_of A e
\<and> (let tr = trace (ioa.asig A) e in
if a \<in> ext A then tr = [a] else tr = [])" | definition | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | refines | null |
is_ref_map:: "('s1 \<Rightarrow> 's2) \<Rightarrow> ('s1,'a)ioa \<Rightarrow> ('s2,'a)ioa \<Rightarrow> bool" where
"is_ref_map f B A \<equiv>
(\<forall> s \<in> start B . f s \<in> start A) \<and> (\<forall> s t a. reachable B s \<and> s \<midarrow>a\<midarrow>B\<longrightarrow> t
\<longrightarrow> (\<exis... | definition | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | is_ref_map | null |
is_forward_sim:: "('s1 \<Rightarrow> ('s2 set)) \<Rightarrow> ('s1,'a)ioa \<Rightarrow> ('s2,'a)ioa \<Rightarrow> bool" where
"is_forward_sim f B A \<equiv>
(\<forall> s \<in> start B . f s \<inter> start A \<noteq> {})
\<and> (\<forall> s s' t a. s' \<in> f s \<and> s \<midarrow>a\<midarrow>B\<longrightarrow> ... | definition | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | is_forward_sim | null |
is_backward_sim:: "('s1 \<Rightarrow> ('s2 set)) \<Rightarrow> ('s1,'a)ioa \<Rightarrow> ('s2,'a)ioa \<Rightarrow> bool" where
"is_backward_sim f B A \<equiv>
(\<forall> s . f s \<noteq> {}) \<comment> \<open>Quantifying over reachable states would suffice\<close>
\<and> (\<forall> s \<in> start B . f s \<subse... | definition | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | is_backward_sim | null |
step_eq_traces:
fixes e_B' A e e_A' a t
defines "e_A \<equiv> append_exec e_A' e" and "e_B \<equiv> cons_exec e_B' (a,t)"
and "tr \<equiv> trace (ioa.asig A) e"
assumes 1:"trace (ioa.asig A) e_A' = trace (ioa.asig A) e_B'"
and 2:"if a \<in> ext A then tr = [a] else tr = []"
shows "trace (ioa.asig A) e_A = t... | lemma | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | step_eq_traces | A series of lemmas that will be useful in the soundness proofs |
exec_inc_imp_trace_inc:
fixes A B
assumes "ext B = ext A"
and "\<And> e_B . is_exec_of B e_B
\<Longrightarrow> \<exists> e_A . is_exec_of A e_A \<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B"
shows "traces B \<subseteq> traces A"
proof -
{ fix t
assume "t \<in> traces B"
with this obtain e... | lemma | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | exec_inc_imp_trace_inc | null |
ref_map_execs:
fixes A::"('sA,'a)ioa" and B::"('sB,'a)ioa" and f::"'sB \<Rightarrow> 'sA" and e_B
assumes "is_ref_map f B A" and "is_exec_of B e_B"
shows "\<exists> e_A . is_exec_of A e_A
\<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B"
proof -
note assms(2)
hence "\<exists> e_A . is_exec_of A e_... | lemma | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | ref_map_execs | Soundness of Refinement Mappings |
ref_map_soundness:
fixes A::"('sA,'a)ioa" and B::"('sB,'a)ioa" and f::"'sB \<Rightarrow> 'sA"
assumes "is_ref_map f B A" and "ext A = ext B"
shows "traces B \<subseteq> traces A"
using assms ref_map_execs exec_inc_imp_trace_inc by metis | theorem | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | ref_map_soundness | null |
forward_sim_execs:
fixes A::"('sA,'a)ioa" and B::"('sB,'a)ioa" and f::"'sB \<Rightarrow> 'sA set" and e_B
assumes "is_forward_sim f B A" and "is_exec_of B e_B"
shows "\<exists> e_A . is_exec_of A e_A
\<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B"
proof -
note assms(2)
hence "\<exists> e_A . is_... | lemma | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | forward_sim_execs | Soundness of Forward Simulations |
forward_sim_soundness:
fixes A::"('sA,'a)ioa" and B::"('sB,'a)ioa" and f::"'sB \<Rightarrow> 'sA set"
assumes "is_forward_sim f B A" and "ext A = ext B"
shows "traces B \<subseteq> traces A"
using assms forward_sim_execs exec_inc_imp_trace_inc by metis | theorem | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | forward_sim_soundness | null |
backward_sim_execs:
fixes A::"('sA,'a)ioa" and B::"('sB,'a)ioa" and f::"'sB \<Rightarrow> 'sA set" and e_B
assumes "is_backward_sim f B A" and "is_exec_of B e_B"
shows "\<exists> e_A . is_exec_of A e_A
\<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B"
proof -
note assms(2)
hence "\<forall> s \<in> ... | lemma | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | backward_sim_execs | Soundness of Backward Simulations |
backward_sim_soundness:
fixes A::"('sA,'a)ioa" and B::"('sB,'a)ioa" and f::"'sB \<Rightarrow> 'sA set"
assumes "is_backward_sim f B A" and "ext A = ext B"
shows "traces B \<subseteq> traces A"
using assms backward_sim_execs exec_inc_imp_trace_inc by metis | theorem | Abortable_Linearizable_Modules | [
"IOA"
] | Abortable_Linearizable_Modules/Simulations.thy | backward_sim_soundness | null |
SLin_status= Sleep | Pending | Ready | Aborted | datatype | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | SLin_status | null |
SLin= RDR + IOA
begin | locale | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | SLin | null |
asig:: "nat \<Rightarrow> nat \<Rightarrow> ('a,'b,'c,'d)SLin_action signature"
\<comment> \<open>The first instance has number 0\<close>
where
"asig i j \<equiv> \<lparr>
inputs = {act . \<exists> p c iv i' .
(i \<le> i' \<and> i' < j \<and> act = Invoke i' p c) \<or> (i > 0 \<and> act = Switch i p c i... | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | asig | null |
pendingReqs:: "('a,'b,'c)SLin_state \<Rightarrow> ('b\<times>'c) set"
where
"pendingReqs s \<equiv> {r . \<exists> p .
r = pending s p
\<and> status s p \<in> {Pending, Aborted}}" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | pendingReqs | null |
Inv:: "'b \<Rightarrow> 'c
\<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> bool"
where
"Inv p c s s' \<equiv>
status s p = Ready
\<and> s' = s\<lparr>pending := (pending s)(p := (p,c)),
status := (status s)(p := Pending)\<rparr>" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | Inv | null |
pendingSeqswhere
"pendingSeqs s \<equiv> {rs . set rs \<subseteq> pendingReqs s}" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | pendingSeqs | null |
Lin:: "('a,'b,'c)SLin_state \<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> bool"
where
"Lin s s' \<equiv> \<exists> rs \<in> pendingSeqs s .
initialized s
\<and> (\<forall> av \<in> abortVals s . (dstate s) \<star> rs \<preceq> av)
\<and> s' = s\<lparr>dstate := (dstate s) \<star> rs\<rparr>" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | Lin | null |
initSetswhere
"initSets s \<equiv> {ivs . ivs \<noteq> {} \<and> ivs \<subseteq> initVals s}" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | initSets | null |
safeInitswhere
"safeInits s \<equiv> if initVals s = {} then {}
else {d . \<exists> ivs \<in> initSets s . \<exists> rs \<in> pendingSeqs s .
d = \<Sqinter>ivs \<star> rs \<and> (\<forall> av \<in> abortVals s . d \<preceq> av)}" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | safeInits | null |
initAbortswhere
"initAborts s \<equiv> { d .dstate s \<preceq> d
\<and> ((\<exists> rs \<in> pendingSeqs s . d = dstate s \<star> rs)
\<or> (\<exists> ivs \<in> initSets s . dstate s \<preceq> \<Sqinter>ivs
\<and> (\<exists> rs \<in> pendingSeqs s . d = \<Sqinter>ivs \<star> rs))) }" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | initAborts | null |
uninitAbortswhere
"uninitAborts s \<equiv> { d .
\<exists> ivs \<in> initSets s . \<exists> rs \<in> pendingSeqs s .
d = \<Sqinter>ivs \<star> rs }" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | uninitAborts | null |
safeAborts::"('a,'b,'c)SLin_state \<Rightarrow> 'a set" where
"safeAborts s \<equiv> if initialized s then initAborts s
else uninitAborts s" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | safeAborts | null |
Reco:: "('a,'b,'c)SLin_state \<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> bool"
where
"Reco s s' \<equiv>
(\<exists> p . status s p \<noteq> Sleep)
\<and> \<not> initialized s
\<and> (\<exists> d \<in> safeInits s .
s' = s\<lparr>dstate := d, initialized := True\<rparr>)" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | Reco | null |
Resp:: "'b \<Rightarrow> 'd \<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> bool"
where
"Resp p ou s s' \<equiv>
status s p = Pending
\<and> initialized s
\<and> contains (dstate s) (pending s p)
\<and> ou = \<gamma> (dstate s) (pending s p)
\<and> s' ... | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | Resp | null |
Init:: "'b \<Rightarrow> 'c \<Rightarrow> 'a
\<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> bool"
where
"Init p c iv s s' \<equiv>
status s p = Sleep
\<and> s' = s \<lparr>initVals := {iv} \<union> (initVals s),
status := (status s)(p := Pending),
pending... | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | Init | null |
Abort:: "'b \<Rightarrow> 'c \<Rightarrow> 'a
\<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> ('a,'b,'c)SLin_state \<Rightarrow> bool"
where
"Abort p c av s s' \<equiv>
status s p = Pending \<and> pending s p = (p,c)
\<and> av \<in> safeAborts s
\<and> s' = s\<lparr>status := (status s)(p := Aborted... | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | Abort | null |
transwhere
"trans i j \<equiv> { (s,a,s') . case a of
Invoke i' p c \<Rightarrow> i \<le> i' \<and> i < j \<and> Inv p c s s'
| Response i' p ou \<Rightarrow> i \<le> i' \<and> i < j \<and> Resp p ou s s'
| Switch i' p c v \<Rightarrow> (i > 0 \<and> i' = i \<and> Init p c v s s')
\<or> (i' = j \<and> Abort p c v... | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | trans | null |
startwhere
"start i \<equiv> { s .
\<forall> p . status s p = (if i > 0 then Sleep else Ready)
\<and> dstate s = \<bottom>
\<and> (if i > 0 then \<not> initialized s else initialized s)
\<and> initVals s = {}
\<and> abortVals s = {}}" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | start | null |
ioawhere
"ioa i j \<equiv>
\<lparr>ioa.asig = asig i j ,
start = start i,
trans = trans i j\<rparr>" | definition | Abortable_Linearizable_Modules | [
"IOA",
"RDR"
] | Abortable_Linearizable_Modules/SLin.thy | ioa | null |
trancl_mono_set:
"r \<subseteq> s \<Longrightarrow> r\<^sup>+ \<subseteq> s\<^sup>+"
by (blast intro: trancl_mono) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | trancl_mono_set | FIXME: move |
relpow_mono:
fixes r :: "'a rel"
assumes "r \<subseteq> r'" shows "r ^^ n \<subseteq> r' ^^ n"
using assms by (induct n) auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | relpow_mono | FIXME: move |
refl_inv_image:
"refl R \<Longrightarrow> refl (inv_image R f)"
by (simp add: inv_image_def refl_on_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | refl_inv_image | null |
join:: "'a rel \<Rightarrow> 'a rel" ("(_\<^sup>\<down>)" [1000] 999) where
"A\<^sup>\<down> = A\<^sup>* O (A\<inverse>)\<^sup>*" | definition | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | join | Definitions |
meet:: "'a rel \<Rightarrow> 'a rel" ("(_\<^sup>\<up>)" [1000] 999) where
"A\<^sup>\<up> = (A\<inverse>)\<^sup>* O A\<^sup>*" | definition | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | meet | null |
symcl:: "'a rel \<Rightarrow> 'a rel" ("(_\<^sup>\<leftrightarrow>)" [1000] 999) where
"A\<^sup>\<leftrightarrow> \<equiv> A \<union> A\<inverse>" | abbreviation | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | symcl | The \emph{symmetric closure} of a relation allows steps in both directions. |
conversion:: "'a rel \<Rightarrow> 'a rel" ("(_\<^sup>\<leftrightarrow>\<^sup>*)" [1000] 999) where
"A\<^sup>\<leftrightarrow>\<^sup>* = (A\<^sup>\<leftrightarrow>)\<^sup>*" | definition | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion | The \emph{symmetric closure} of a relation allows steps in both directions. A \emph{conversion} is a (possibly empty) sequence of steps in the symmetric closure. |
NF:: "'a rel \<Rightarrow> 'a set" where
"NF A = {a. A `` {a} = {}}" | definition | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | NF | null |
normalizability:: "'a rel \<Rightarrow> 'a rel" ("(_\<^sup>!)" [1000] 999) where
"A\<^sup>! = {(a, b). (a, b) \<in> A\<^sup>* \<and> b \<in> NF A}"
notation (ASCII)
symcl ("(_^<->)" [1000] 999) and
conversion ("(_^<->*)" [1000] 999) and
normalizability ("(_^!)" [1000] 999) | definition | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | normalizability | null |
symcl_converse:
"(A\<^sup>\<leftrightarrow>)\<inverse> = A\<^sup>\<leftrightarrow>" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | symcl_converse | null |
symcl_Un: "(A \<union> B)\<^sup>\<leftrightarrow> = A\<^sup>\<leftrightarrow> \<union> B\<^sup>\<leftrightarrow>" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | symcl_Un | null |
no_step:
assumes "A `` {a} = {}" shows "a \<in> NF A"
using assms by (auto simp: NF_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | no_step | null |
joinI:
"(a, c) \<in> A\<^sup>* \<Longrightarrow> (b, c) \<in> A\<^sup>* \<Longrightarrow> (a, b) \<in> A\<^sup>\<down>"
by (auto simp: join_def rtrancl_converse) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | joinI | null |
joinI_left:
"(a, b) \<in> A\<^sup>* \<Longrightarrow> (a, b) \<in> A\<^sup>\<down>"
by (auto simp: join_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | joinI_left | null |
joinI_right: "(b, a) \<in> A\<^sup>* \<Longrightarrow> (a, b) \<in> A\<^sup>\<down>"
by (rule joinI) auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | joinI_right | null |
joinE:
assumes "(a, b) \<in> A\<^sup>\<down>"
obtains c where "(a, c) \<in> A\<^sup>*" and "(b, c) \<in> A\<^sup>*"
using assms by (auto simp: join_def rtrancl_converse) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | joinE | null |
joinD:
"(a, b) \<in> A\<^sup>\<down> \<Longrightarrow> \<exists>c. (a, c) \<in> A\<^sup>* \<and> (b, c) \<in> A\<^sup>*"
by (blast elim: joinE) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | joinD | null |
meetI:
"(a, b) \<in> A\<^sup>* \<Longrightarrow> (a, c) \<in> A\<^sup>* \<Longrightarrow> (b, c) \<in> A\<^sup>\<up>"
by (auto simp: meet_def rtrancl_converse) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | meetI | null |
meetE:
assumes "(b, c) \<in> A\<^sup>\<up>"
obtains a where "(a, b) \<in> A\<^sup>*" and "(a, c) \<in> A\<^sup>*"
using assms by (auto simp: meet_def rtrancl_converse) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | meetE | null |
meetD: "(b, c) \<in> A\<^sup>\<up> \<Longrightarrow> \<exists>a. (a, b) \<in> A\<^sup>* \<and> (a, c) \<in> A\<^sup>*"
by (blast elim: meetE) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | meetD | null |
conversionI: "(a, b) \<in> (A\<^sup>\<leftrightarrow>)\<^sup>* \<Longrightarrow> (a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
by (simp add: conversion_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversionI | null |
conversion_refl[simp]: "(a, a) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
by (simp add: conversion_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_refl | null |
conversionI':
assumes "(a, b) \<in> A\<^sup>*" shows "(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
using assms
proof (induct)
case base then show ?case by simp
next
case (step b c)
then have "(b, c) \<in> A\<^sup>\<leftrightarrow>" by simp
with \<open>(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*\<close> ... | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversionI' | null |
rtrancl_comp_trancl_conv:
"r\<^sup>* O r = r\<^sup>+" by regexp | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | rtrancl_comp_trancl_conv | (a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>* |
trancl_o_refl_is_trancl:
"r\<^sup>+ O r\<^sup>= = r\<^sup>+" by regexp | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | trancl_o_refl_is_trancl | null |
conversionE:
"(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>* \<Longrightarrow> ((a, b) \<in> (A\<^sup>\<leftrightarrow>)\<^sup>* \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: conversion_def) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversionE | null |
conversion_trans:
"trans (A\<^sup>\<leftrightarrow>\<^sup>*)"
unfolding trans_def
proof (intro allI impI)
fix a b c assume "(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*" and "(b, c) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
then show "(a, c) \<in> A\<^sup>\<leftrightarrow>\<^sup>*" unfolding conversion_def
p... | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_trans | null |
conversion_sym:
"sym (A\<^sup>\<leftrightarrow>\<^sup>*)"
unfolding sym_def
proof (intro allI impI)
fix a b assume "(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*" then show "(b, a) \<in> A\<^sup>\<leftrightarrow>\<^sup>*" unfolding conversion_def
proof (induct)
case base then show ?case by simp
next
... | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_sym | null |
conversion_inv:
"(x, y) \<in> R\<^sup>\<leftrightarrow>\<^sup>* \<longleftrightarrow> (y, x) \<in> R\<^sup>\<leftrightarrow>\<^sup>*"
by (auto simp: conversion_def)
(metis (full_types) rtrancl_converseD symcl_converse)+ | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_inv | (c, b) \<in> A\<^sup>\<leftrightarrow> |
conversion_converse[simp]:
"(A\<^sup>\<leftrightarrow>\<^sup>*)\<inverse> = A\<^sup>\<leftrightarrow>\<^sup>*"
by (metis conversion_sym sym_conv_converse_eq) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_converse | null |
conversion_rtrancl[simp]:
"(A\<^sup>\<leftrightarrow>\<^sup>*)\<^sup>* = A\<^sup>\<leftrightarrow>\<^sup>*"
by (metis conversion_def rtrancl_idemp) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | conversion_rtrancl | null |
rtrancl_join_join:
assumes "(a, b) \<in> A\<^sup>*" and "(b, c) \<in> A\<^sup>\<down>" shows "(a, c) \<in> A\<^sup>\<down>"
proof -
from \<open>(b, c) \<in> A\<^sup>\<down>\<close> obtain b' where "(b, b') \<in> A\<^sup>*" and "(b', c) \<in> (A\<inverse>)\<^sup>*"
unfolding join_def by blast
with \<open>(a, b... | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | rtrancl_join_join | null |
join_rtrancl_join:
assumes "(a, b) \<in> A\<^sup>\<down>" and "(c, b) \<in> A\<^sup>*" shows "(a, c) \<in> A\<^sup>\<down>"
proof -
from \<open>(c, b) \<in> A\<^sup>*\<close> have "(b, c) \<in> (A\<inverse>)\<^sup>*" unfolding rtrancl_converse by simp
from \<open>(a, b) \<in> A\<^sup>\<down>\<close> obtain a' whe... | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | join_rtrancl_join | (a, b) \<in> A\<^sup>* (b', c) \<in> (A\<inverse>)\<^sup>* |
NF_I: "(\<And>b. (a, b) \<notin> A) \<Longrightarrow> a \<in> NF A" by (auto intro: no_step) | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | NF_I | (b, c) \<in> (A\<inverse>)\<^sup>* (a, a') \<in> A\<^sup>* |
NF_E: "a \<in> NF A \<Longrightarrow> ((a, b) \<notin> A \<Longrightarrow> P) \<Longrightarrow> P" by (auto simp: NF_def)
declare NF_I [intro]
declare NF_E [elim] | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | NF_E | (a, a') \<in> A\<^sup>* |
NF_no_step: "a \<in> NF A \<Longrightarrow> \<forall>b. (a, b) \<notin> A" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | NF_no_step | null |
NF_anti_mono:
assumes "A \<subseteq> B" shows "NF B \<subseteq> NF A"
using assms by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | NF_anti_mono | null |
NF_iff_no_step: "a \<in> NF A = (\<forall>b. (a, b) \<notin> A)" by auto | lemma | Abstract-Rewriting | [
"HOL-Library.Infinite_Set",
"Regular-Sets.Regexp_Method",
"Seq"
] | Abstract-Rewriting/Abstract_Rewriting.thy | NF_iff_no_step | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.