Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
statement stringlengths 1 113k | proof stringlengths 0 694k | type stringclasses 26
values | symbolic_name stringlengths 1 113 | library stringlengths 2 65 | filename stringlengths 14 111 | imports listlengths 0 0 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
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Consensus | locale | Consensus | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Consensus.thy | [] | [] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
\<delta>::"val option \<Rightarrow> (proc \<times> val) \<Rightarrow> val option" (infix "\<bullet>" 65) | where
"\<delta> None r = Some (snd r)"
| "\<delta> (Some v) r = Some v" | fun | delta | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Consensus.thy | [] | [
"Some",
"bullet",
"delta",
"option",
"proc",
"snd",
"times",
"val",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
\<gamma>::"val option \<Rightarrow> (proc \<times> val) \<Rightarrow> val" | where
"\<gamma> None r = snd r"
| "\<gamma> (Some v) r = v" | fun | gamma | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Consensus.thy | [] | [
"Some",
"gamma",
"option",
"proc",
"snd",
"times",
"val",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
single_use:
fixes r rs
shows "\<bottom> \<star> ([r]@rs) = Some (snd r)" | proof (induct rs)
case Nil
thus ?case by simp
next
case (Cons r rs)
thus ?case by auto
qed | lemma | single_use | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Consensus.thy | [] | [
"Cons",
"Nil",
"Some",
"auto",
"bottom",
"by",
"fixes",
"induct",
"next",
"proof",
"rs",
"shows",
"simp",
"snd",
"star"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
bot: "\<exists> rs . s = \<bottom> \<star> rs" | proof (cases s)
case None
hence "s = \<bottom> \<star> []" by auto
thus ?thesis by blast
next
case (Some v)
obtain r where "\<bottom> \<star> [r] = Some v" by force
thus ?thesis using Some by metis
qed | lemma | bot | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Consensus.thy | [] | [
"Some",
"auto",
"blast",
"bottom",
"by",
"cases",
"exists",
"force",
"metis",
"next",
"proof",
"rs",
"star",
"using",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
prec_eq_None_or_equal:
fixes s1 s2
assumes "s1 \<preceq> s2"
shows "s1 = None \<or> s1 = s2" | using assms single_use
proof -
{ assume 1:"s1 \<noteq> None" and 2:"s1 \<noteq> s2"
obtain r rs where 3:"s1 = \<bottom> \<star> ([r]@rs)" using bot using 1
by (metis append_butlast_last_id pre_RDR.exec.simps(1))
obtain rs' where 4:"s2 = s1 \<star> rs'" using assms
by (auto simp add:less_eq_def... | lemma | prec_eq_None_or_equal | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Consensus.thy | [] | [
"False",
"add",
"and",
"assume",
"assumes",
"auto",
"blast",
"bot",
"bottom",
"by",
"exec_append",
"fixes",
"have",
"metis",
"noteq",
"or",
"proof",
"rs",
"rs'",
"s1",
"s2",
"shows",
"simp",
"single_use",
"star",
"using",
"where",
"with"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
Idempotence = SLin +
fixes id1 id2 :: nat
assumes id1:"0 < id1" and id2:"id1 < id2" | locale | Idempotence | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"SLin",
"and",
"assumes",
"fixes",
"nat"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
composition | where
"composition \<equiv>
hide ((ioa 0 id1) \<parallel> (ioa id1 id2))
{act . \<exists>p c av . act = Switch id1 p c av }"
lemmas comp_simps = hide_def composition_def ioa_def par2_def is_trans_def
start_def actions_def asig_def trans_def
lemmas trans_defs = Inv_def Lin_def Resp_def Init_def
... | definition | composition | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"act",
"av",
"equiv",
"exists",
"hide",
"ioa",
"lemmas",
"parallel",
"trans_def",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
trans_elim:
fixes s t a s' t' P
assumes "(s,t) \<midarrow>a\<midarrow>composition\<longrightarrow> (s',t')"
obtains
(Invoke1) i p c
where "Inv p c s s' \<and> t = t'"
and "i < id1" and "a = Invoke i p c"
| (Invoke2) i p c
where "Inv p c t t' \<and> s = s'"
and "id1 \<le> i \<and> i... | proof %invisible (cases a)
case (Invoke i p c)
with assms have
"(Inv p c s s' \<and> t = t' \<and> i < id1)
\<or> (Inv p c t t' \<and> s = s' \<and> id1 \<le> i \<and> i < id2)" by auto
thus ?thesis using Invoke that by blast
next
case (Response i p ou)
with assms have
"(Resp p ou s s' \<and> t... | lemma | trans_elim | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"Abort",
"Init",
"Inv",
"Lin",
"Reco",
"Resp",
"and",
"assumes",
"auto",
"av",
"blast",
"by",
"cases",
"composition",
"fixes",
"have",
"ids",
"invisible",
"le",
"metis",
"next",
"obtains",
"or",
"proof",
"s'",
"t'",
"that",
"using",
"where",
"with"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
f :: "(('a,'b,'c)SLin_state * ('a,'b,'c)SLin_state) \<Rightarrow> ('a,'b,'c)SLin_state" | where
"f (s1, s2) =
\<lparr>pending = \<lambda> p. (if status s1 p \<noteq> Aborted then pending s1 p else pending s2 p),
initVals = {},
abortVals = abortVals s2,
status = \<lambda> p. (if status s1 p \<noteq> Aborted then status s1 p else status s2 p),
dstate = (if dstate s2 = \<bottom> ... | fun | f | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"SLin_state",
"True",
"bottom",
"dstate",
"if",
"lambda",
"noteq",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P1 | where
"P1 (s1,s2) = (\<forall> p . status s1 p \<in> {Pending, Aborted}
\<longrightarrow> fst (pending s1 p) = p)" | fun | P1 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"forall",
"fst",
"in",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P2 | where
"P2 (s1,s2) = (\<forall> p . status s2 p \<noteq> Sleep \<longrightarrow> fst (pending s2 p) = p)" | fun | P2 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"forall",
"fst",
"noteq",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P3 | where
"P3 (s1,s2) = (\<forall> p . (status s2 p = Ready \<longrightarrow> initialized s2))" | fun | P3 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"forall",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P4 | where
"P4 (s1,s2) = ((\<forall> p . status s2 p = Sleep) = (initVals s2 = {}))" | fun | P4 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"forall",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P5 | where
"P5 (s1,s2) = (\<forall> p . status s1 p \<noteq> Sleep \<and> initialized s1 \<and> initVals s1 = {})" | fun | P5 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"and",
"forall",
"noteq",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P6 | where
"P6 (s1,s2) = (\<forall> p . (status s1 p \<noteq> Aborted) = (status s2 p = Sleep))" | fun | P6 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"forall",
"noteq",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P7 | where
"P7 (s1,s2) = (\<forall> c . status s1 c = Aborted \<and> \<not> initialized s2
\<longrightarrow> (pending s2 c = pending s1 c \<and> status s2 c \<in> {Pending, Aborted}))" | fun | P7 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"and",
"forall",
"in",
"not",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P8 | where
"P8 (s1,s2) = (\<forall> iv \<in> initVals s2 . \<exists> rs \<in> pendingSeqs s1 .
iv = dstate s1 \<star> rs)" | fun | P8 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"dstate",
"exists",
"forall",
"in",
"iv",
"pendingSeqs",
"rs",
"s1",
"s2",
"star",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P8a | where
"P8a (s1,s2) = (\<forall> ivs \<in> initSets s2 . \<exists> rs \<in> pendingSeqs s1 .
\<Sqinter>ivs = dstate s1 \<star> rs)" | fun | P8a | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"dstate",
"exists",
"forall",
"in",
"initSets",
"pendingSeqs",
"rs",
"s1",
"s2",
"star",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P9 | where
"P9 (s1,s2) = (initialized s2 \<longrightarrow> dstate s1 \<preceq> dstate s2)" | fun | P9 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"dstate",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P10 | where
"P10 (s1,s2) = ((\<not> initialized s2) \<longrightarrow> (dstate s2 = \<bottom>))" | fun | P10 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"bottom",
"dstate",
"not",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P11 | where
"P11 (s1,s2) = (initVals s2 = abortVals s1)" | fun | P11 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P12 | where
"P12 (s1,s2) = (initialized s2 \<longrightarrow> \<Sqinter> (initVals s2) \<preceq> dstate s2)" | fun | P12 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"dstate",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P13 | where
"P13 (s1,s2) = (finite (initVals s2)
\<and> finite (abortVals s1) \<and> finite (abortVals s2))" | fun | P13 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"and",
"finite",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P14 | where
"P14 (s1,s2) = (initialized s2 \<longrightarrow> initVals s2 \<noteq> {})" | fun | P14 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"noteq",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P15 | where
"P15 (s1,s2) = (\<forall> av \<in> abortVals s1 . dstate s1 \<preceq> av)" | fun | P15 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"av",
"dstate",
"forall",
"in",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P16 | where
"P16 (s1,s2) = (dstate s2 \<noteq> \<bottom> \<longrightarrow> initialized s2)" | fun | P16 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"bottom",
"dstate",
"noteq",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P17 | where
\<comment> \<open>For the Response1 case of the refinement proof, in case a response
is produced in the first instance and the second instance is already
initialized\<close>
"P17 (s1,s2) = (initialized s2
\<longrightarrow> (\<forall> p .
(... | fun | P17 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"and",
"contains",
"dstate",
"exists",
"forall",
"fst",
"in",
"not",
"noteq",
"or",
"rs",
"s1",
"s2",
"set",
"star",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P18 | where
"P18 (s1,s2) = (abortVals s2 \<noteq> {} \<longrightarrow> (\<exists> p . status s2 p \<noteq> Sleep))" | fun | P18 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"exists",
"noteq",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P19 | where
"P19 (s1,s2) = (abortVals s2 \<noteq> {} \<longrightarrow> abortVals s1 \<noteq> {})" | fun | P19 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"noteq",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P20 | where
"P20 (s1,s2) = (\<forall> av \<in> abortVals s2 . dstate s2 \<preceq> av)" | fun | P20 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"av",
"dstate",
"forall",
"in",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P21 | where
"P21 (s1,s2) = (\<forall> av \<in> abortVals s2 . \<Sqinter>(abortVals s1) \<preceq> av)" | fun | P21 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"av",
"forall",
"in",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P22 | where
"P22 (s1,s2) = (initialized s2 \<longrightarrow> dstate (f (s1,s2)) = dstate s2)" | fun | P22 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"dstate",
"s1",
"s2",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P23 | where
"P23 (s1,s2) = ((\<not> initialized s2) \<longrightarrow>
pendingSeqs s1 \<subseteq> pendingSeqs (f (s1,s2)))" | fun | P23 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"not",
"pendingSeqs",
"s1",
"s2",
"subseteq",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P25 | where
"P25 (s1,s2) = (\<forall> ivs . (ivs \<in> initSets s2 \<and> initialized s2
\<and> dstate s2 \<preceq> \<Sqinter>ivs)
\<longrightarrow> (\<exists> rs' \<in> pendingSeqs (f (s1,s2)) . \<Sqinter>ivs = dstate s2 \<star> rs'))" | fun | P25 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"and",
"dstate",
"exists",
"forall",
"in",
"initSets",
"pendingSeqs",
"rs'",
"s1",
"s2",
"star",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P26 | where
"P26 (s1,s2) = (\<forall> p . (status s1 p = Aborted
\<and> \<not> contains (dstate s2) (pending s1 p))
\<longrightarrow> (status s2 p \<in> {Pending,Aborted}
\<and> pending s1 p = pending s2 p))" | fun | P26 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"and",
"contains",
"dstate",
"forall",
"in",
"not",
"s1",
"s2",
"status",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P1_invariant:
shows "invariant (composition) P1" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P1 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P1 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P1 ... | lemma | P1_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P1",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P2_invariant:
shows "invariant (composition) P2" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P2 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P2 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P2 ... | lemma | P2_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P2",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P16_invariant:
shows "invariant (composition) P16" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P16 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P16 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P... | lemma | P16_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P16",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P3_invariant:
shows "invariant (composition) P3" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P3 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P3 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P3 ... | lemma | P3_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P3",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P4_invariant:
shows "invariant (composition) P4" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P4 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P4 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P4 ... | lemma | P4_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P4",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P5_invariant:
shows "invariant (composition) P5" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P5 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P5 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P5 ... | lemma | P5_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P5",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P13_invariant:
shows "invariant (composition) P13" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P13 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P13 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P... | lemma | P13_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P13",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P20_invariant:
shows "invariant (composition) P20" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P20 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P20 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
and rea... | lemma | P20_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P16",
"P16_invariant",
"P20",
"add",
"and",
"assume",
"auto",
"bot",
"by",
"cases",
"composition",
"fix",
"from",
"have",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"metis",
"next",
"proof",
"reach",
"reachable",
"rule",
"s1",
"s2",
"show",
"shows",
"si... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P18_invariant:
shows "invariant (composition) P18" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P18 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P18 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P... | lemma | P18_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P18",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P14_invariant:
shows "invariant (composition) P14" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P14 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P14 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P... | lemma | P14_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P14",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P15_invariant:
shows "invariant (composition) P15" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P15 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P15 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
and rea... | lemma | P15_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P15",
"P5",
"P5_invariant",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"from",
"have",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"metis",
"next",
"proof",
"reach",
"reachable",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"sim... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P6_invariant:
shows "invariant (composition) P6" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P6 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P6 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P6 ... | lemma | P6_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P6",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"force",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P7_invariant:
shows "invariant (composition) P7" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P7 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P7 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P7 ... | lemma | P7_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P7",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P10_invariant:
shows "invariant (composition) P10" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P10 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P10 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P... | lemma | P10_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P10",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P11_invariant:
shows "invariant (composition) P11" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P11 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P11 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P... | lemma | P11_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P11",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"composition",
"fix",
"force",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"next",
"proof",
"rule",
"s1",
"s2",
"show",
"shows",
"simp",
"simp_all",
"t1",
"t2",
"trans",
"trans_elim",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P8_invariant:
shows "invariant (composition) P8" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P8 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P8 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
and reach... | lemma | P8_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"And",
"P1",
"P11",
"P11_invariant",
"P1_invariant",
"P5",
"P5_invariant",
"P8",
"add",
"and",
"assume",
"auto",
"av",
"ballI",
"by",
"cases",
"close",
"composition",
"consistency",
"contains",
"define",
"dstate",
"exists",
"filter",
"fix",
"force",
"from",
"hav... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P8a_invariant:
shows "invariant (composition) P8a" | proof (auto simp:invariant_def)
fix s1 s2 ivs
assume 1:"reachable (composition) (s1,s2)"
and 2:"ivs \<in> initSets s2"
have 3:"finite ivs \<and> ivs \<noteq> {}"
proof -
have "P13 (s1,s2)" using P13_invariant 1
by (metis IOA.invariant_def)
thus ?thesis using 2 finite_subset by (auto simp add:ini... | lemma | P8a_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P13",
"P13_invariant",
"P8",
"P8_invariant",
"P8a",
"add",
"and",
"assume",
"auto",
"av",
"by",
"composition",
"dstate",
"exists",
"finite",
"finite_subset",
"fix",
"forall",
"glb_common_set",
"have",
"in",
"initSets",
"invariant",
"metis",
"noteq",
"pendingSeqs",
... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P12_invariant:
shows "invariant (composition) P12" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P12 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P12 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
and rea... | lemma | P12_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P12",
"P13",
"P13_invariant",
"P14",
"P14_invariant",
"add",
"and",
"assume",
"auto",
"av",
"by",
"cases",
"commute",
"composition",
"dstate",
"fix",
"force",
"from",
"have",
"hyp",
"ids",
"ih",
"in",
"insert",
"invariant",
"invariantI",
"metis",
"next",
"not... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P19_invariant:
shows "invariant (composition) P19" | proof (auto simp only:invariant_def)
fix s1 s2
assume 1:"reachable (composition) (s1,s2)"
have P4:"P4 (s1,s2)" using P4_invariant 1
by (simp add:invariant_def)
moreover
have P18:"P18 (s1,s2)" using P18_invariant 1
by (metis IOA.invariant_def)
moreover
have P11:"P11 (s1,s2)" using P11_invariant 1
... | lemma | P19_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P11",
"P11_invariant",
"P18",
"P18_invariant",
"P19",
"P4",
"P4_invariant",
"add",
"assume",
"auto",
"by",
"composition",
"fix",
"have",
"invariant",
"metis",
"proof",
"reachable",
"s1",
"s2",
"show",
"shows",
"simp",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P9_invariant:
shows "invariant (composition) P9" | proof (auto simp only:invariant_def)
fix s1 s2
assume 1:"reachable (composition) (s1,s2)"
have P12:"P12 (s1,s2)" using P12_invariant 1
by (simp add:invariant_def)
have P15:"P15 (s1,s2)" using P15_invariant 1
by (metis IOA.invariant_def)
have P13:"P13 (s1,s2)" using P13_invariant 1
by (metis IOA.in... | lemma | P9_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P11",
"P11_invariant",
"P12",
"P12_invariant",
"P13",
"P13_invariant",
"P14",
"P14_invariant",
"P15",
"P15_invariant",
"P9",
"add",
"assume",
"auto",
"boundedI",
"by",
"composition",
"dstate",
"fix",
"have",
"invariant",
"metis",
"proof",
"reachable",
"s1",
"s2",
... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P17_invariant:
shows "invariant (composition) P17" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P17 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P17 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
and rea... | lemma | P17_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"And",
"False",
"P1",
"P17",
"P1_invariant",
"P2",
"P2_invariant",
"P6",
"P6_invariant",
"P8a",
"P8a_invariant",
"P9",
"P9_invariant",
"True",
"add",
"and",
"assume",
"auto",
"av",
"ballI",
"blast",
"by",
"cases",
"close",
"composition",
"consistency",
"contains",... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P21_invariant:
shows "invariant (composition) P21" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P21 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P21 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
and rea... | lemma | P21_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"False",
"P11",
"P11_invariant",
"P12",
"P12_invariant",
"P13",
"P13_invariant",
"P19",
"P19_invariant",
"P20",
"P20_invariant",
"P21",
"True",
"add",
"and",
"assume",
"auto",
"av",
"by",
"cases",
"composition",
"fix",
"have",
"hyp",
"ids",
"in",
"invariant",
"i... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P22_invariant:
shows "invariant (composition) P22" | proof (auto simp only:invariant_def)
fix s1 s2
assume 1:"reachable (composition) (s1,s2)"
have P9:"P9 (s1,s2)" using P9_invariant 1
by (simp add:invariant_def)
show "P22 (s1,s2)"
proof (simp only:P22.simps, rule impI)
assume "initialized s2"
show "dstate (f (s1,s2)) = dstate s2"
proof (cases "... | lemma | P22_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"False",
"P22",
"P9",
"P9_invariant",
"True",
"add",
"and",
"antisym",
"assume",
"auto",
"bot",
"bottom",
"by",
"cases",
"close",
"composition",
"dstate",
"fix",
"have",
"impI",
"invariant",
"metis",
"next",
"open",
"proof",
"reachable",
"rule",
"s1",
"s2",
... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P23_invariant:
shows "invariant (composition) P23" | proof (auto simp only:invariant_def)
fix s1 s2
assume 1:"reachable (composition) (s1,s2)"
show "P23 (s1,s2)"
proof (simp only:P23.simps, clarify)
fix rs
assume 2:"\<not>initialized s2" and 3:"rs\<in>pendingSeqs s1"
show "rs\<in> pendingSeqs (f (s1,s2))"
proof -
{ fix r
assume 3:"r ... | lemma | P23_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P1",
"P1_invariant",
"P23",
"P7",
"P7_invariant",
"add",
"and",
"assume",
"auto",
"by",
"clarify",
"composition",
"fix",
"fst",
"have",
"in",
"invariant",
"metis",
"next",
"not",
"or",
"pendingReqs",
"pendingSeqs",
"proof",
"reachable",
"rs",
"s1",
"s2",
"sho... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P26_invariant:
shows "invariant (composition) P26" | proof (rule invariantI, simp_all only:split_paired_all)
fix s1 s2
assume "(s1,s2) \<in> ioa.start (composition)"
thus "P26 (s1,s2)" using ids by (auto simp add:comp_simps)
next
fix s1 s2 t1 t2 a
assume hyp: "P26 (s1,s2)" and trans:"(s1,s2) \<midarrow>a\<midarrow>composition\<longrightarrow> (t1,t2)"
and rea... | lemma | P26_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P26",
"P7",
"P7_invariant",
"add",
"and",
"assume",
"auto",
"by",
"cases",
"clarify",
"composition",
"contains",
"contains_star",
"dstate",
"fix",
"force",
"from",
"have",
"hyp",
"ids",
"in",
"invariant",
"invariantI",
"metis",
"next",
"not",
"or",
"proof",
"... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
P25_invariant:
shows "invariant (composition) P25" | proof (auto simp only:invariant_def)
fix s1 s2
assume reach:"reachable (composition) (s1,s2)"
show "P25 (s1,s2)"
proof (simp only:P25.simps, clarify)
fix ivs
assume 1:"ivs \<in> initSets s2" and 2:"initialized s2"
and 3:"dstate s2 \<preceq> \<Sqinter>ivs"
obtain rs' where 4:"dstate s2 \<star> ... | lemma | P25_invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"P25",
"P26",
"P26_invariant",
"P8a",
"P8a_invariant",
"P9",
"P9_invariant",
"add",
"and",
"assume",
"auto",
"by",
"clarify",
"close",
"composition",
"consistency",
"contains",
"dstate",
"exists",
"fix",
"forall",
"force",
"from",
"have",
"idem_star2",
"in",
"init... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
idempotence:
shows "((composition) =<| (ioa 0 id2))" | proof -
have same_input_sig:"inp (composition) = inp (ioa 0 id2)"
\<comment> \<open>First we show that both automata have the same input and output signature\<close>
using ids by auto
moreover
have same_output_sig:"out (composition) = out (ioa 0 id2)"
\<comment> \<open>Then we show that output si... | theorem | idempotence | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/Idempotence.thy | [] | [
"False",
"Inv",
"P1",
"P10",
"P10_invariant",
"P13",
"P13_invariant",
"P15",
"P15_invariant",
"P16",
"P16_invariant",
"P17",
"P17_invariant",
"P19",
"P19_invariant",
"P1_invariant",
"P21",
"P21_invariant",
"P22",
"P22_invariant",
"P23",
"P23_invariant",
"P25",
"P25_inva... | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
IOA = Sequences | locale | IOA | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"Sequences"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
'a signature =
inputs::"'a set"
outputs::"'a set"
internals::"'a set" | record | signature | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"inputs",
"outputs",
"set"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
actions :: "'a signature \<Rightarrow> 'a set" | where
"actions asig \<equiv> inputs asig \<union> outputs asig \<union> internals asig" | definition | actions | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"inputs",
"outputs",
"set",
"signature",
"union",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
externals :: "'a signature \<Rightarrow> 'a set" | where
"externals asig \<equiv> inputs asig \<union> outputs asig" | definition | externals | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"inputs",
"outputs",
"set",
"signature",
"union",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
locals :: "'a signature \<Rightarrow> 'a set" | where
"locals asig \<equiv> internals asig \<union> outputs asig" | definition | locals | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"outputs",
"set",
"signature",
"union",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
is_asig :: "'a signature \<Rightarrow> bool" | where
"is_asig triple \<equiv>
inputs triple \<inter> outputs triple = {} \<and>
outputs triple \<inter> internals triple = {} \<and>
inputs triple \<inter> internals triple = {}" | definition | is_asig | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"and",
"bool",
"equiv",
"inputs",
"inter",
"outputs",
"signature",
"triple",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
internal_inter_external:
assumes "is_asig sig"
shows "internals sig \<inter> externals sig = {}" | using assms by (auto simp add:internals_def externals_def is_asig_def) | lemma | internal_inter_external | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"add",
"assumes",
"auto",
"by",
"externals",
"inter",
"is_asig",
"shows",
"sig",
"simp",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
hide_asig | where
"hide_asig asig actns \<equiv>
\<lparr>inputs = inputs asig - actns, outputs = outputs asig - actns,
internals = internals asig \<union>actns\<rparr>" | definition | hide_asig | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"inputs",
"outputs",
"union",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
('s,'a) transition = "'s \<times> 'a \<times> 's" | type_synonym | transition | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"times"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
('s,'a) ioa =
asig::"'a signature"
start::"'s set"
trans::"('s,'a)transition set" | record | ioa | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"set",
"signature",
"start",
"trans",
"transition"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
"act A \<equiv> actions (asig A)" | abbreviation | act | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"actions",
"asig",
"equiv"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
"ext A \<equiv> externals (asig A)" | abbreviation | ext | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"externals"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
int | where "int A \<equiv> internals (asig A)" | abbreviation | int | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
"inp A \<equiv> inputs (asig A)" | abbreviation | inp | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"inputs"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
"out A \<equiv> outputs (asig A)" | abbreviation | out | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"outputs"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
"local A \<equiv> locals (asig A)" | abbreviation | local | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"locals"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
is_ioa::"('s,'a) ioa \<Rightarrow> bool" | where
"is_ioa A \<equiv> is_asig (asig A)
\<and> (\<forall> triple \<in> trans A . (fst o snd) triple \<in> act A)" | definition | is_ioa | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"act",
"and",
"asig",
"bool",
"equiv",
"forall",
"fst",
"in",
"ioa",
"is_asig",
"snd",
"trans",
"triple",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
hide | where
"hide A actns \<equiv> A\<lparr>asig := hide_asig (asig A) actns\<rparr>" | definition | hide | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"equiv",
"hide_asig",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
is_trans::"'s \<Rightarrow> 'a \<Rightarrow> ('s,'a)ioa \<Rightarrow> 's \<Rightarrow> bool" | where
"is_trans s1 a A s2 \<equiv> (s1,a,s2) \<in> trans A" | definition | is_trans | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"bool",
"equiv",
"in",
"ioa",
"s1",
"s2",
"trans",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
rename_set | where
"rename_set A ren \<equiv> {b. \<exists> x \<in> A . ren b = Some x}" | definition | rename_set | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"Some",
"equiv",
"exists",
"in",
"ren",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
rename | where
"rename A ren \<equiv>
\<lparr>asig = \<lparr>inputs = rename_set (inp A) ren,
outputs = rename_set (out A) ren,
internals = rename_set (int A) ren\<rparr>,
start = start A,
trans = {tr. \<exists> x . ren (fst (snd tr)) = Some x \<and> (fst tr) \<midarrow>x\<midarrow>A\<longrightarrow> (snd (snd... | definition | rename | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"Some",
"and",
"asig",
"equiv",
"exists",
"fst",
"inp",
"inputs",
"int",
"out",
"outputs",
"ren",
"rename_set",
"snd",
"start",
"tr",
"trans",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
reachable :: "('s,'a) ioa \<Rightarrow> 's \<Rightarrow> bool"
for A :: "('s,'a) ioa" | where
reachable_0: "s \<in> start A \<Longrightarrow> reachable A s"
| reachable_n: "\<lbrakk> reachable A s; s \<midarrow>a\<midarrow>A\<longrightarrow> t \<rbrakk> \<Longrightarrow> reachable A t" | inductive | reachable | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"bool",
"for",
"in",
"ioa",
"start",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
invariant | where
"invariant A P \<equiv> (\<forall> s . reachable A s \<longrightarrow> P(s))" | definition | invariant | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"equiv",
"forall",
"reachable",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
invariantI:
fixes A P
assumes "\<And> s . s \<in> start A \<Longrightarrow> P s"
and "\<And> s t a . \<lbrakk>reachable A s; P s; s \<midarrow>a\<midarrow>A\<longrightarrow> t\<rbrakk> \<Longrightarrow> P t"
shows "invariant A P" | proof -
{ fix s
assume "reachable A s"
hence "P s"
proof (induct rule:reachable.induct)
fix s
assume "s \<in> start A"
thus "P s" using assms(1) by simp
next
fix a s t
assume "reachable A s" and "P s" and " s \<midarrow>a\<midarrow>A\<longrightarrow> t"
thus "P t" u... | theorem | invariantI | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"And",
"add",
"and",
"assume",
"assumes",
"by",
"fix",
"fixes",
"in",
"induct",
"invariant",
"next",
"proof",
"reachable",
"rule",
"shows",
"simp",
"start",
"using"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
('id, 'a) family =
ids :: "'id set"
memb :: "'id \<Rightarrow> 'a" | record | family | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"ids",
"memb",
"set"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
is_ioa_fam | where
"is_ioa_fam fam \<equiv> \<forall> i \<in> ids fam . is_ioa (memb fam i)" | definition | is_ioa_fam | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"equiv",
"forall",
"ids",
"in",
"is_ioa",
"memb",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
compatible2 | where
"compatible2 A B \<equiv>
out A \<inter> out B = {} \<and>
int A \<inter> act B = {} \<and>
int B \<inter> act A = {}" | definition | compatible2 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"act",
"and",
"equiv",
"int",
"inter",
"out",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
compatible::"('id, ('s,'a)ioa) family \<Rightarrow> bool" | where
"compatible fam \<equiv> finite (ids fam) \<and>
(\<forall> i \<in> ids fam . \<forall> j \<in> ids fam . i \<noteq> j \<longrightarrow>
compatible2 (memb fam i) (memb fam j))" | definition | compatible | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"and",
"bool",
"compatible2",
"equiv",
"family",
"finite",
"forall",
"ids",
"in",
"ioa",
"memb",
"noteq",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
asig_comp2 | where
"asig_comp2 A B \<equiv>
\<lparr>inputs = (inputs A \<union> inputs B) - (outputs A \<union> outputs B),
outputs = outputs A \<union> outputs B,
internals = internals A \<union> internals B\<rparr>" | definition | asig_comp2 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"equiv",
"inputs",
"outputs",
"union",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
asig_comp::"('id, ('s,'a)ioa) family \<Rightarrow> 'a signature" | where
"asig_comp fam \<equiv>
\<lparr> inputs = \<Union>i\<in>(ids fam). inp (memb fam i)
- (\<Union>i\<in>(ids fam). out (memb fam i)),
outputs = \<Union>i\<in>(ids fam). out (memb fam i),
internals = \<Union>i\<in>(ids fam). int (memb fam i) \<rparr>" | definition | asig_comp | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"Union",
"equiv",
"family",
"ids",
"in",
"inp",
"inputs",
"int",
"ioa",
"memb",
"out",
"outputs",
"signature",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
par2 (infixr "\<parallel>" 10) | where
"A \<parallel> B \<equiv>
\<lparr>asig = asig_comp2 (asig A) (asig B),
start = {pr. fst pr \<in> start A \<and> snd pr \<in> start B},
trans = {tr.
let s = fst tr; a = fst (snd tr); t = snd (snd tr)
in (a \<in> act A \<or> a \<in> act B)
\<and> (if a \<in> act A
... | definition | par2 | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"act",
"and",
"asig",
"asig_comp2",
"equiv",
"fst",
"if",
"in",
"let",
"or",
"parallel",
"pr",
"snd",
"start",
"tr",
"trans",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
par::"('id, ('s,'a)ioa) family \<Rightarrow> ('id \<Rightarrow> 's,'a)ioa" | where
"par fam \<equiv> let ids = ids fam; memb = memb fam in
\<lparr> asig = asig_comp fam,
start = {s . \<forall> i\<in>ids . s i \<in> start (memb i)},
trans = { (s, a, s') .
(\<exists> i\<in>ids . a \<in> act (memb i))
\<and> (\<forall> i\<in>ids .
if a \<... | definition | par | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"act",
"and",
"asig",
"asig_comp",
"equiv",
"exists",
"family",
"forall",
"ids",
"if",
"in",
"ioa",
"lemmas",
"let",
"memb",
"s'",
"start",
"trans",
"where"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
('s,'a)pairs = "('a\<times>'s) list" | type_synonym | pairs | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"list",
"times"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
('s,'a)execution = "'s \<times> ('s,'a)pairs" | type_synonym | execution | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"pairs",
"times"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
'a trace | = "'a list" | type_synonym | trace | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"list"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | |
('s,'a)execution_module =
execs::"('s,'a)execution set"
asig::"'a signature" | record | execution_module | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"execs",
"execution",
"set",
"signature"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 | ||
'a trace_module =
traces::"'a trace set"
asig::"'a signature" | record | trace_module | Abortable_Linearizable_Modules | thys/Abortable_Linearizable_Modules/IOA.thy | [] | [
"asig",
"set",
"signature",
"trace",
"traces"
] | https://github.com/isabelle-prover/mirror-afp-2024 | 1549ea5b788b34986e3aef5613a0dce379aaee86 |
End of preview. Expand in Data Studio
Isabelle-AFP
Structured dataset from the Isabelle Archive of Formal Proofs (AFP) - the largest repository of formal proofs in Isabelle/HOL.
Source
- Repository: https://github.com/isabelle-prover/mirror-afp-2024
- Commit:
1549ea5b788b34986e3aef5613a0dce379aaee86 - Files: 9105
- License: bsd-3-clause
Schema
| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 352,334
- With proof: 323,649 (91.9%)
- With docstring: 0 (0.0%)
- Libraries: 1296
By type
| Type | Count |
|---|---|
| lemma | 250,920 |
| definition | 46,866 |
| abbreviation | 10,106 |
| fun | 9,053 |
| theorem | 6,371 |
| locale | 5,869 |
| corollary | 3,425 |
| type_synonym | 2,936 |
| lift_definition | 2,546 |
| primrec | 2,219 |
| datatype | 1,806 |
| instantiation | 1,777 |
| class | 1,459 |
| inductive | 1,369 |
| proposition | 1,074 |
| instance | 895 |
| schematic_goal | 735 |
| function | 582 |
| consts | 574 |
| inductive_set | 458 |
| record | 446 |
| axiomatization | 255 |
| partial_function | 213 |
| primcorec | 187 |
| termination | 98 |
| coinductive | 95 |
Example
\<delta>::"val option \<Rightarrow> (proc \<times> val) \<Rightarrow> val option" (infix "\<bullet>" 65)
where
"\<delta> None r = Some (snd r)"
| "\<delta> (Some v) r = Some v"
- type: fun | symbolic_name:
delta| thys/Abortable_Linearizable_Modules/Consensus.thy
Use
Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
Citation
@misc{isabelle_afp_dataset,
title = {Isabelle-AFP},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/isabelle-prover/mirror-afp-2024, commit 1549ea5b788b},
url = {https://huggingface.co/datasets/phanerozoic/Isabelle-AFP}
}
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