Datasets:

phanerozoic
/

Isabelle-AFP

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 ⌀
Consensus\<comment> \<open>To avoid name clashes\<close> begin	locale	Abortable_Linearizable_Modules	[ "RDR" ]	Abortable_Linearizable_Modules/Consensus.thy	Consensus	null
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	Abortable_Linearizable_Modules	[ "RDR" ]	Abortable_Linearizable_Modules/Consensus.thy	single_use	null
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	Abortable_Linearizable_Modules	[ "RDR" ]	Abortable_Linearizable_Modules/Consensus.thy	bot	null
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)) ...	lemma	Abortable_Linearizable_Modules	[ "RDR" ]	Abortable_Linearizable_Modules/Consensus.thy	prec_eq_None_or_equal	null
Idempotence= SLin + fixes id1 id2 :: nat assumes id1:"0 < id1" and id2:"id1 < id2" begin lemmas ids = id1 id2	locale	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	Idempotence	Idempotence of the SLin I/O automaton
compositionwhere "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 Ini...	definition	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	composition	null
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 ...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	trans_elim	null
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 \<not...	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	f	Definition of the Refinement Mapping
P1where "P1 (s1,s2) = (\<forall> p . status s1 p \<in> {Pending, Aborted} \<longrightarrow> fst (pending s1 p) = p)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P1	Invariants
P2where "P2 (s1,s2) = (\<forall> p . status s2 p \<noteq> Sleep \<longrightarrow> fst (pending s2 p) = p)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P2	null
P3where "P3 (s1,s2) = (\<forall> p . (status s2 p = Ready \<longrightarrow> initialized s2))"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P3	null
P4where "P4 (s1,s2) = ((\<forall> p . status s2 p = Sleep) = (initVals s2 = {}))"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P4	Used to prove P19 only
P5where "P5 (s1,s2) = (\<forall> p . status s1 p \<noteq> Sleep \<and> initialized s1 \<and> initVals s1 = {})"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P5	Used to prove P19 only
P6where "P6 (s1,s2) = (\<forall> p . (status s1 p \<noteq> Aborted) = (status s2 p = Sleep))"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P6	null
P7where "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	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P7	null
P8where "P8 (s1,s2) = (\<forall> iv \<in> initVals s2 . \<exists> rs \<in> pendingSeqs s1 . iv = dstate s1 \<star> rs)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P8	Only used in the proof of P8a
P8awhere "P8a (s1,s2) = (\<forall> ivs \<in> initSets s2 . \<exists> rs \<in> pendingSeqs s1 . \<Sqinter>ivs = dstate s1 \<star> rs)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P8a	Only used in the proof of P8a
P9where "P9 (s1,s2) = (initialized s2 \<longrightarrow> dstate s1 \<preceq> dstate s2)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P9	null
P10where "P10 (s1,s2) = ((\<not> initialized s2) \<longrightarrow> (dstate s2 = \<bottom>))"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P10	null
P11where "P11 (s1,s2) = (initVals s2 = abortVals s1)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P11	null
P12where "P12 (s1,s2) = (initialized s2 \<longrightarrow> \<Sqinter> (initVals s2) \<preceq> dstate s2)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P12	null
P13where "P13 (s1,s2) = (finite (initVals s2) \<and> finite (abortVals s1) \<and> finite (abortVals s2))"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P13	null
P14where "P14 (s1,s2) = (initialized s2 \<longrightarrow> initVals s2 \<noteq> {})"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P14	null
P15where "P15 (s1,s2) = (\<forall> av \<in> abortVals s1 . dstate s1 \<preceq> av)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P15	null
P16where "P16 (s1,s2) = (dstate s2 \<noteq> \<bottom> \<longrightarrow> initialized s2)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P16	null
P17where \<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 . ((status s1 p = Ready \<or> (status s1 p = Pendin...	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P17	null
P18where "P18 (s1,s2) = (abortVals s2 \<noteq> {} \<longrightarrow> (\<exists> p . status s2 p \<noteq> Sleep))"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P18	Only used for proving P19
P19where "P19 (s1,s2) = (abortVals s2 \<noteq> {} \<longrightarrow> abortVals s1 \<noteq> {})"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P19	Only used for proving P19
P20where "P20 (s1,s2) = (\<forall> av \<in> abortVals s2 . dstate s2 \<preceq> av)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P20	null
P21where "P21 (s1,s2) = (\<forall> av \<in> abortVals s2 . \<Sqinter>(abortVals s1) \<preceq> av)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P21	null
P22where "P22 (s1,s2) = (initialized s2 \<longrightarrow> dstate (f (s1,s2)) = dstate s2)"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P22	null
P23where "P23 (s1,s2) = ((\<not> initialized s2) \<longrightarrow> pendingSeqs s1 \<subseteq> pendingSeqs (f (s1,s2)))"	fun	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P23	null
P25where "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	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P25	null
P26where "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	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P26	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P1_invariant	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P2_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P16_invariant	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P3_invariant	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P4_invariant	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P5_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P13_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P20_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P18_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P14_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P15_invariant	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P6_invariant	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P7_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P10_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P11_invariant	null
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>...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P8_invariant	null
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 ...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P8a_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P12_invariant	null
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) moreov...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P19_invariant	null
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 (s...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P9_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P17_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P21_invariant	null
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 ...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P22_invariant	null
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,s...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P23_invariant	null
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\<midar...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P26_invariant	null
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> \<Sqin...	lemma	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	P25_invariant	null
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 id...	theorem	Abortable_Linearizable_Modules	[ "SLin", "Simulations" ]	Abortable_Linearizable_Modules/Idempotence.thy	idempotence	null
IOA= Sequences	locale	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	IOA	This theory is inspired and draws material from the IOA theory of Nipkow and Müller
'asignature = inputs::"'a set" outputs::"'a set" internals::"'a set" context IOA begin	record	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	'a	This theory is inspired and draws material from the IOA theory of Nipkow and Müller
actions:: "'a signature \<Rightarrow> 'a set" where "actions asig \<equiv> inputs asig \<union> outputs asig \<union> internals asig"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	actions	Signatures
externals:: "'a signature \<Rightarrow> 'a set" where "externals asig \<equiv> inputs asig \<union> outputs asig"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	externals	Signatures
locals:: "'a signature \<Rightarrow> 'a set" where "locals asig \<equiv> internals asig \<union> outputs asig"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	locals	null
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	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_asig	null
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	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	internal_inter_external	null
hide_asigwhere "hide_asig asig actns \<equiv> \<lparr>inputs = inputs asig - actns, outputs = outputs asig - actns, internals = internals asig \<union>actns\<rparr>"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	hide_asig	null
intwhere "int A \<equiv> internals (asig A)"	abbreviation	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	int	null
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	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_ioa	null
hidewhere "hide A actns \<equiv> A\<lparr>asig := hide_asig (asig A) actns\<rparr>"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	hide	null
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" notation is_trans ("_ \<midarrow>_\<midarrow>_\<longrightarrow> _" [81,81,81,81] 100)	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_trans	null
rename_setwhere "rename_set A ren \<equiv> {b. \<exists> x \<in> A . ren b = Some x}"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	rename_set	null
renamewhere "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 ...	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	rename	null
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	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	reachable	Reachable states and invariants
invariantwhere "invariant A P \<equiv> (\<forall> s . reachable A s \<longrightarrow> P(s))"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	invariant	null
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 rul...	theorem	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	invariantI	null
is_ioa_famwhere "is_ioa_fam fam \<equiv> \<forall> i \<in> ids fam . is_ioa (memb fam i)"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_ioa_fam	null
compatible2where "compatible2 A B \<equiv> out A \<inter> out B = {} \<and> int A \<inter> act B = {} \<and> int B \<inter> act A = {}"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	compatible2	null
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	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	compatible	null
asig_comp2where "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	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	asig_comp2	null
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). in...	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	asig_comp	null
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) ...	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	par2	null
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> ac...	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	par	null
'atrace = "'a list"	type_synonym	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	'a	null
'atrace_module = traces::"'a trace set" asig::"'a signature" context IOA begin	record	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	'a	null
is_exec_frag_of::"('s,'a)ioa \<Rightarrow> ('s,'a)execution \<Rightarrow> bool" where "is_exec_frag_of A (s,(ps#p')#p) = (snd p' \<midarrow>fst p\<midarrow>A\<longrightarrow> snd p \<and> is_exec_frag_of A (s, (ps#p')))" \| "is_exec_frag_of A (s, [p]) = s \<midarrow>fst p\<midarrow>A\<longrightarrow> snd p" \| "is_...	fun	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_exec_frag_of	null
is_exec_of::"('s,'a)ioa \<Rightarrow> ('s,'a)execution \<Rightarrow> bool" where "is_exec_of A e \<equiv> fst e \<in> start A \<and> is_exec_frag_of A e"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_exec_of	null
filter_actwhere "filter_act \<equiv> map fst"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	filter_act	null
schedulewhere "schedule \<equiv> filter_act o snd"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	schedule	null
tracewhere "trace sig \<equiv> filter (\<lambda> a . a \<in> externals sig) o schedule"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	trace	null
is_schedule_ofwhere "is_schedule_of A sch \<equiv> (\<exists> e . is_exec_of A e \<and> sch = filter_act (snd e))"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_schedule_of	null
is_trace_ofwhere "is_trace_of A tr \<equiv> (\<exists> sch . is_schedule_of A sch \<and> tr = filter (\<lambda> a. a \<in> ext A) sch)"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	is_trace_of	null
traceswhere "traces A \<equiv> {tr. is_trace_of A tr}"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	traces	null
traces_alt: shows "traces A = {tr . \<exists> e . is_exec_of A e \<and> tr = trace (ioa.asig A) e}" proof - { fix t assume a:"t \<in> traces A" have "\<exists> e . is_exec_of A e \<and> trace (ioa.asig A) e = t" proof - from a obtain sch where 1:"is_schedule_of A sch" and 2:"t = filter...	lemma	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	traces_alt	null
proj_trace::"'a trace \<Rightarrow> ('a signature) \<Rightarrow> 'a trace" (infixr "\<bar>" 12) where "proj_trace t sig \<equiv> filter (\<lambda> a . a \<in> actions sig) t"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	proj_trace	null
ioa_implements:: "('s1,'a)ioa \<Rightarrow> ('s2,'a)ioa \<Rightarrow> bool" (infixr "=<\|" 12) where "A =<\| B \<equiv> inp A = inp B \<and> out A = out B \<and> traces A \<subseteq> traces B"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	ioa_implements	null
cons_execwhere "cons_exec e p \<equiv> (fst e, (snd e)#p)"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	cons_exec	Operations on Executions
append_execwhere "append_exec e e' \<equiv> (fst e, (snd e)@(snd e'))"	definition	Abortable_Linearizable_Modules	[ "Main", "Sequences" ]	Abortable_Linearizable_Modules/IOA.thy	append_exec	Operations on Executions

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Isabelle-AFP

Structured dataset from the Isabelle Archive of Formal Proofs (AFP) - the largest repository of formal proofs in Isabelle/HOL.

Schema

Column	Type	Description
fact	string	Declaration body
type	string	lemma, definition, theorem, fun, locale, datatype, etc.
library	string	AFP entry name (project)
imports	list	Theory imports
filename	string	Source file path
symbolic_name	string	Declaration identifier
docstring	string	Documentation (20% coverage)

Statistics

By Type (Top 15)

Type	Count
lemma	228,980
definition	35,811
fun	10,758
abbreviation	6,005
theorem	5,892
locale	5,773
corollary	3,185
type_synonym	2,369
primrec	2,111
instance	2,087
instantiation	1,743
datatype	1,527
class	1,491
inductive	1,317
proposition	922

Top AFP Entries

Entry	Count
Crypto_Standards	10,367
AutoCorres2	8,760
JinjaThreads	4,937
Cook_Levin	3,008
ConcurrentHOL	2,989

About AFP

The Archive of Formal Proofs is a collection of proof libraries for Isabelle/HOL, maintained by the Isabelle community. It contains formalized mathematics, verified algorithms, and program verification.