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 ⌀ |
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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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