fact
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| filename
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| symbolic_name
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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))
obtain rs' where 4:"s2 = s1 \<star> rs'" using assms
by (auto simp add:less_eq_def)
have "s2 = \<bottom> \<star> ([r]@(rs@rs'))" using 3 4
by (metis exec_append)
hence "s1 = s2" using 3
by (metis single_use)
with 2 have False by auto }
thus ?thesis by blast
qed
interpretation RDR \<delta> \<gamma> \<bottom>
proof (unfold_locales)
fix s r
assume "contains s r"
show "s \<bullet> r = s"
proof -
obtain rs where "s = \<bottom> \<star> rs" and "rs \<noteq> []"
using \<open>contains s r\<close>
by (auto simp add:contains_def, force)
thus ?thesis
by (metis \<delta>.simps(2) rev_exhaust single_use)
qed
|
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 Init_def
Abort_def Reco_def
declare if_split_asm [split]
|
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 < id2" and "a = Invoke i p c"
| (Switch1) p c av
where "Abort p c av s s' \<and> Init p c av t t'"
and "a = Switch id1 p c av"
| (Switch2) p c av
where "s = s' \<and> Abort p c av t t'"
and "a = Switch id2 p c av"
| (Response1) i p ou
where "Resp p ou s s'\<and> t = t'"
and "i < id1" and "a = Response i p ou"
| (Response2) i p ou
where "Resp p ou t t' \<and> s = s'"
and "id1 \<le> i \<and> i < id2" and "a = Response i p ou"
| (Lin1) "Lin s s' \<and> t = t'" and "a = Linearize 0"
| (Lin2) "Lin t t' \<and> s = s'" and "a = Linearize id1"
| (Reco2) "Reco t t' \<and> s = s'" and "a = Recover id1"
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
|
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 \<noteq> Aborted then status s1 p else status s2 p),
dstate = (if dstate s2 = \<bottom> then dstate s1 else dstate s2),
initialized = True\<rparr>"
|
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 = Pending \<and> contains (dstate s1) (pending s1 p)))
\<longrightarrow> (\<exists> rs . dstate s2 = dstate s1 \<star> rs \<and> (\<forall> r \<in> set rs . fst r \<noteq> p)))
\<and> ((status s1 p = Pending \<and> \<not> contains (dstate s1) (pending s1 p))
\<longrightarrow> (\<exists> rs . dstate s2 = dstate s1 \<star> rs \<and> (\<forall> r \<in> set rs .
fst r = p \<longrightarrow> r = pending s1 p)))))"
|
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>composition\<longrightarrow> (t1,t2)"
show "P1 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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>composition\<longrightarrow> (t1,t2)"
show "P2 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P16 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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>composition\<longrightarrow> (t1,t2)"
show "P3 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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>composition\<longrightarrow> (t1,t2)"
show "P4 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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>composition\<longrightarrow> (t1,t2)"
show "P5 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P13 (t1,t2)" using trans and hyp
by (cases rule:trans_elim, auto)
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
and reach: "reachable (composition) (s1,s2)"
from reach have P16:"P16 (s1,s2)" using P16_invariant and ids
by (metis IOA.invariant_def)
show "P20 (t1,t2)" using trans and hyp and P16
by (cases rule:trans_elim, auto simp add:safeInits_def safeAborts_def
initAborts_def uninitAborts_def bot)
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P18 (t1,t2)" using trans and hyp
by (cases rule:trans_elim, auto)
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P14 (t1,t2)" using trans and hyp
by (cases rule:trans_elim, auto simp add:safeInits_def)
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
and reach: "reachable (composition) (s1,s2)"
from reach have P5:"P5 (s1,s2)" using P5_invariant and ids
by (metis IOA.invariant_def)
show "P15 (t1,t2)" using trans and hyp and P5
by (cases rule:trans_elim,
auto simp add:less_eq_def safeAborts_def initAborts_def)
qed
|
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>composition\<longrightarrow> (t1,t2)"
show "P6 (t1,t2)" using trans and hyp
by (cases rule:trans_elim, force+)
qed
|
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>composition\<longrightarrow> (t1,t2)"
show "P7 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P10 (t1,t2)" using trans and hyp
by (cases rule:trans_elim) auto
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
show "P11 (t1,t2)" using trans and hyp
by (cases rule:trans_elim, force+)
qed
|
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>composition\<longrightarrow> (t1,t2)"
and reach: "reachable (composition) (s1,s2)"
from reach have P5:"P5 (s1,s2)" using P5_invariant and ids
by (metis IOA.invariant_def)
from reach have P1:"P1 (s1,s2)" using P1_invariant and ids
by (metis IOA.invariant_def)
from reach have P11:"P11 (s1,s2)" using P11_invariant and ids
by (metis IOA.invariant_def)
show "P8 (t1,t2)" using trans and hyp
proof (cases rule:trans_elim)
case (Invoke1 i p c)
assume "P8 (s1,s2)"
have "pendingSeqs s1 \<subseteq> pendingSeqs t1"
proof -
have "pending t1 = (pending s1)(p := (p,c))"
and "status t1 = (status s1)(p := Pending)"
and "status s1 p = Ready"
using Invoke1(1) by auto
hence "pendingReqs s1 \<subseteq> pendingReqs t1" by (simp add:pendingReqs_def) force
thus ?thesis by (auto simp add:pendingSeqs_def)
qed
moreover have "initVals t2 = initVals s2" and "dstate t1 = dstate s1"
|
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 ?thesis using 2 finite_subset by (auto simp add:initSets_def)
qed
have 4:"\<forall> av \<in> ivs . \<exists> rs \<in> pendingSeqs s1 . av = dstate s1 \<star> rs"
proof -
have P8:"P8 (s1,s2)" using P8_invariant 1
by (metis IOA.invariant_def)
thus ?thesis using 2 by (auto simp add:initSets_def)
qed
show "\<exists> rs \<in> pendingSeqs s1 . \<Sqinter>ivs = dstate s1 \<star> rs"
using 3 4 glb_common_set by (simp add:pendingSeqs_def, metis)
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
and reach: "reachable (composition) (s1,s2)"
from reach have P13:"P13 (s1,s2)" using P13_invariant
by (metis IOA.invariant_def)
from reach have P14:"P14 (s1,s2)" using P14_invariant
by (metis IOA.invariant_def)
show "P12 (t1,t2)" using trans and hyp
proof (cases rule:trans_elim)
case (Invoke1 i p c)
assume "P12 (s1,s2)"
thus "P12 (t1,t2)" using Invoke1(1) by auto
next
case Lin1
assume "P12 (s1,s2)"
thus "P12 (t1,t2)" using Lin1(1) by auto
next
case (Response1 i p ou)
assume "P12 (s1,s2)"
thus "P12 (t1,t2)" using Response1(1) by auto
next
case (Switch1 p c av)
assume ih:"P12 (s1,s2)"
|
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)
moreover
have P11:"P11 (s1,s2)" using P11_invariant 1
by (metis IOA.invariant_def)
moreover
ultimately show "P19 (s1,s2)" by auto
qed
|
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 (s1,s2)" using P13_invariant 1
by (metis IOA.invariant_def)
have P14:"P14 (s1,s2)" using P14_invariant 1
by (metis IOA.invariant_def)
have P11:"P11 (s1,s2)" using P11_invariant 1
by (metis IOA.invariant_def)
have "initialized s2 \<Longrightarrow> dstate s1 \<preceq> \<Sqinter>(abortVals s1)"
using P13 P15 P14 P11 boundedI by simp
thus "P9 (s1,s2)" using P12 P11 by simp (metis trans)
qed
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
and reach:"reachable (composition) (s1,s2)"
show "P17 (t1,t2)" using trans and hyp
proof (cases rule:trans_elim)
case (Invoke1 i p c)
assume "P17 (s1,s2)"
thus "P17 (t1,t2)" using Invoke1(1) by fastforce
next
case (Response1 i p ou)
assume "P17 (s1,s2)"
thus "P17 (t1,t2)" using Response1(1) by auto
next
case (Switch1 p c av)
assume "P17 (s1,s2)"
thus "P17 (t1,t2)" using Switch1(1) by auto
next
case (Invoke2 i p c)
assume "P17 (s1,s2)"
thus "P17 (t1,t2)" using Invoke2(1) by force
next
case (Response2 i p ou)
assume "P17 (s1,s2)"
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
and reach: "reachable (composition) (s1,s2)"
show "P21 (t1,t2)"
proof (cases "initialized t2")
case True
moreover
have P12:"P12 (t1,t2)" using P12_invariant reach trans
by (metis invariant_def reachable_n)
moreover
have P11:"P11 (t1,t2)" using P11_invariant reach trans
by (metis IOA.invariant_def reachable_n)
moreover
have P20:"P20 (t1,t2)" using P20_invariant reach trans
by (metis IOA.invariant_def reachable_n)
ultimately show "P21 (t1,t2)" by simp
(metis pre_RDR.trans)
next
case False
show "P21 (t1,t2)" using trans
proof (cases rule:trans_elim)
case (Switch2 p c av)
obtain av where "abortVals t2 = abortVals s2 \<union> {av}"
|
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 "dstate (f (s1,s2)) = dstate s2"
proof (cases "dstate s2 = \<bottom>")
case False
thus ?thesis by auto
next
case True
show "dstate (f (s1,s2)) = dstate s2"
proof -
have "dstate s1 \<preceq> dstate s2"
using \<open>initialized s2\<close> and \<open>P9 (s1,s2)\<close>
by auto
hence "dstate s1 = dstate s2" using True
by (metis antisym bot)
thus ?thesis by auto
qed
qed
qed
qed
|
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,s2))"
proof -
{ fix r
assume 3:"r \<in> pendingReqs s1"
have 4:"status s1 (fst r) = Pending \<or> status s1 (fst r) = Aborted"
and 5:"pending s1 (fst r) = r"
proof -
have "P1 (s1,s2)" using 1 P1_invariant
by (metis invariant_def)
thus "status s1 (fst r) = Pending \<or> status s1 (fst r) = Aborted"
and "pending s1 (fst r) = r"
using 3 by (auto simp add:pendingReqs_def)
qed
have "r \<in> pendingReqs (f (s1,s2))" using 4
proof
assume "status s1 (fst r) = Pending"
with 5 show ?thesis by (auto simp add:pendingReqs_def)
(metis SLin_status.distinct(9))
next
assume 6:"status s1 (fst r) = Aborted"
have 7:"pending s1 (fst r) = pending s2 (fst r)
|
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\<midarrow>composition\<longrightarrow> (t1,t2)"
and reach:"reachable composition (s1,s2)"
show "P26 (t1,t2)" using trans and hyp
proof (cases rule:trans_elim)
case Lin2
hence 1:"dstate s2 \<preceq> dstate t2"
by auto (metis less_eq_def)
have 2:"t2 = s2\<lparr>dstate := dstate t2\<rparr>" and 3:"s1 = t1"
using Lin2(1) by auto
show ?thesis
proof (simp, clarify)
fix p
assume 4:"status t1 p = Aborted"
and 5:"\<not> contains (dstate t2) (pending t1 p)"
have 6:"status s1 p = Aborted" using 3 4 by auto
have 7:"pending s1 p = pending t1 p" using 3 by simp
have 8:"\<not> contains (dstate s2) (pending s1 p)"
using 1 5 7
by simp (metis contains_star less_eq_def)
have 11:"status s2 p \<in> {Pending,Aborted}"
and 9:"pending s1 p = pending s2 p" using hyp 6 8 by auto
show "(status t2 p = Pending \<or> status t2 p = Aborted)
|
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> \<Sqinter>ivs"
obtain rs' where 4:"dstate s2 \<star> rs' = \<Sqinter>ivs"
and 5:"rs' \<in> pendingSeqs s1" and 6:"\<forall> r \<in> set rs' . \<not> contains (dstate s2) r"
proof -
have 5:"dstate s1 \<preceq> dstate s2"
proof -
have P9:"P9 (s1,s2)" using P9_invariant reach
by (simp add:invariant_def)
thus ?thesis using 2 by auto
qed
obtain rs where 6:"\<Sqinter>ivs = dstate s1 \<star> rs" and 7:"rs \<in> pendingSeqs s1"
proof -
have P8a:"P8a (s1,s2)" using P8a_invariant reach
by (simp add:invariant_def)
thus ?thesis using that 1 by auto
qed
have "\<exists> rs' . dstate s2 \<star> rs' = \<Sqinter> ivs \<and> rs' \<in> pendingSeqs s1"
using 3 5 6 7 consistency[of "dstate s1" "dstate s2" "\<Sqinter>ivs" rs]
by (force simp add:pendingSeqs_def)
with this obtain rs' where "\<Sqinter>ivs = dstate s2 \<star> rs'"
and "rs' \<in> pendingSeqs s1" by metis
|
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 id2)"
\<comment> \<open>Then we show that output signatures match\<close>
using ids by auto
moreover
have "traces (composition) \<subseteq> traces (ioa 0 id2)"
\<comment> \<open>Finally we show trace inclusion\<close>
proof -
have "ext (composition) = ext (ioa 0 id2)"
\<comment> \<open>First we show that they have the same external signature\<close>
using same_input_sig and same_output_sig by simp
moreover
have "is_ref_map f (composition) (ioa 0 id2)"
\<comment> \<open>Then we show that @{text f_comp} is a refinement mapping\<close>
proof (auto simp only:is_ref_map_def)
fix s1 s2
assume 1:"(s1,s2) \<in> ioa.start (composition)"
show "f (s1,s2) \<in> ioa.start (ioa 0 id2)"
proof -
have 2:"ioa.start (ioa 0 id2) = start (0::nat)" by simp
have 3:"ioa.start (composition)
= start (0::nat) \<times> start id1" by fastforce
show ?thesis
using 1 2 3 by simp
|
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 (snd tr))}\<rparr>"
|
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 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" using assms(2) by simp
qed }
thus ?thesis by (simp add:invariant_def)
qed
|
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). int (memb fam i) \<rparr>"
|
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)
\<and> (if a \<in> act A
then fst s \<midarrow>a\<midarrow>A\<longrightarrow> fst t
else fst s = fst t)
\<and> (if a \<in> act B
then snd s \<midarrow>a\<midarrow>B\<longrightarrow> snd t
else snd s = snd t) }\<rparr>"
|
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> act (memb i))
\<and> (\<forall> i\<in>ids .
if a \<in> act (memb i)
then s i \<midarrow>a\<midarrow>(memb i)\<longrightarrow> s' i
else s i = (s' i)) } \<rparr>"
lemmas asig_simps = hide_asig_def is_asig_def locals_def externals_def actions_def
hide_def compatible_def asig_comp_def
lemmas ioa_simps = rename_def rename_set_def is_trans_def is_ioa_def par_def
|
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_exec_frag_of A (s, []) = True"
|
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 (\<lambda> a. a \<in> ext A) sch"
by (auto simp add:traces_def is_trace_of_def)
from 1 obtain e where 3:"is_exec_of A e" and 4:"sch = filter_act (snd e)"
by (auto simp add:is_schedule_of_def)
from 4 and 2 have "trace (ioa.asig A) e = t"
by (simp add:trace_def schedule_def)
with 3 show ?thesis by fast
qed }
moreover
{ fix e
assume "is_exec_of A e"
hence "trace (ioa.asig A) e \<in> traces A"
by (force simp add:trace_def schedule_def traces_def
is_trace_of_def is_schedule_of_def is_exec_of_def) }
ultimately show ?thesis by blast
qed
lemmas trace_simps = traces_def is_trace_of_def is_schedule_of_def filter_act_def is_exec_of_def
trace_def schedule_def
|
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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