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(* Title: Qmsg_Lifting.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke
*)
section "Lifting rules for parallel compositions with QMSG"
theory Qmsg_Lifting
imports Qmsg OAWN_SOS Inv_Cterms OAWN_Invariants
begin
lemma oseq_no_change_on_send:
fixes \<sigma> s a \<sigma>' s'
assumes "((\<sigma>, s), a, (\<sigma>', s')) \<in> oseqp_sos \<Gamma> i"
shows "case a of
broadcast m \<Rightarrow> \<sigma>' i = \<sigma> i
| groupcast ips m \<Rightarrow> \<sigma>' i = \<sigma> i
| unicast ips m \<Rightarrow> \<sigma>' i = \<sigma> i
| \<not>unicast ips \<Rightarrow> \<sigma>' i = \<sigma> i
| send m \<Rightarrow> \<sigma>' i = \<sigma> i
| deliver m \<Rightarrow> \<sigma>' i = \<sigma> i
| _ \<Rightarrow> True"
using assms by induction simp_all
lemma qmsg_no_change_on_send_or_receive:
fixes \<sigma> s a \<sigma>' s'
assumes "((\<sigma>, s), a, (\<sigma>', s')) \<in> oparp_sos i (oseqp_sos \<Gamma> i) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"
and "a \<noteq> \<tau>"
shows "\<sigma>' i = \<sigma> i"
proof -
from assms(1) obtain p q p' q'
where "((\<sigma>, (p, q)), a, (\<sigma>', (p', q'))) \<in> oparp_sos i (oseqp_sos \<Gamma> i) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"
by (cases s, cases s', simp)
thus ?thesis
proof
assume "((\<sigma>, p), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
and "\<And>m. a \<noteq> receive m"
with \<open>a \<noteq> \<tau>\<close> show "\<sigma>' i = \<sigma> i"
by - (drule oseq_no_change_on_send, cases a, auto)
next
assume "(q, a, q') \<in> seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
and "\<sigma>' i = \<sigma> i"
thus "\<sigma>' i = \<sigma> i" by simp
next
assume "a = \<tau>" with \<open>a \<noteq> \<tau>\<close> show ?thesis by auto
qed
qed
lemma qmsg_msgs_not_empty:
"qmsg \<TTurnstile> onl \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G (\<lambda>(msgs, l). l = ()-:1 \<longrightarrow> msgs \<noteq> [])"
by inv_cterms
lemma qmsg_send_from_queue:
"qmsg \<TTurnstile>\<^sub>A (\<lambda>((msgs, q), a, _). sendmsg (\<lambda>m. m\<in>set msgs) a)"
proof -
have "qmsg \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G (\<lambda>((msgs, _), a, _). sendmsg (\<lambda>m. m\<in>set msgs) a)"
by (inv_cterms inv add: onl_invariant_sterms [OF qmsg_wf qmsg_msgs_not_empty])
thus ?thesis
by (rule step_invariant_weakenE) (auto dest!: onllD)
qed
lemma qmsg_queue_contents:
"qmsg \<TTurnstile>\<^sub>A (\<lambda>((msgs, q), a, (msgs', q')). case a of
receive m \<Rightarrow> set msgs' \<subseteq> set (msgs @ [m])
| _ \<Rightarrow> set msgs' \<subseteq> set msgs)"
proof -
have "qmsg \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G (\<lambda>((msgs, q), a, (msgs', q')).
case a of
receive m \<Rightarrow> set msgs' \<subseteq> set (msgs @ [m])
| _ \<Rightarrow> set msgs' \<subseteq> set msgs)"
by (inv_cterms) (clarsimp simp add: in_set_tl)+
thus ?thesis
by (rule step_invariant_weakenE) (auto dest!: onllD)
qed
lemma qmsg_send_receive_or_tau:
"qmsg \<TTurnstile>\<^sub>A (\<lambda>(_, a, _). \<exists>m. a = send m \<or> a = receive m \<or> a = \<tau>)"
proof -
have "qmsg \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G (\<lambda>(_, a, _). \<exists>m. a = send m \<or> a = receive m \<or> a = \<tau>)"
by inv_cterms
thus ?thesis
by rule (auto dest!: onllD)
qed
lemma par_qmsg_oreachable:
assumes "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg) (otherwith S {i} (orecvmsg R)) (other U {i})"
(is "_ \<in> oreachable _ ?owS _")
and pinv: "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>)
globala (\<lambda>(\<sigma>, _, \<sigma>'). U (\<sigma> i) (\<sigma>' i))"
and ustutter: "\<And>\<xi>. U \<xi> \<xi>"
and sgivesu: "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'"
and upreservesq: "\<And>\<sigma> \<sigma>' m. \<lbrakk> \<forall>j. U (\<sigma> j) (\<sigma>' j); R \<sigma> m \<rbrakk> \<Longrightarrow> R \<sigma>' m"
shows "(\<sigma>, fst \<zeta>) \<in> oreachable A ?owS (other U {i})
\<and> snd \<zeta> \<in> reachable qmsg (recvmsg (R \<sigma>))
\<and> (\<forall>m\<in>set (fst (snd \<zeta>)). R \<sigma> m)"
using assms(1) proof (induction rule: oreachable_pair_induct)
fix \<sigma> pq
assume "(\<sigma>, pq) \<in> init (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg)"
then obtain p ms q where "pq = (p, (ms, q))"
and "(\<sigma>, p) \<in> init A"
and "(ms, q) \<in> init qmsg"
by (clarsimp simp del: \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_simps)
from this(2) have "(\<sigma>, p) \<in> oreachable A ?owS (other U {i})" ..
moreover from \<open>(ms, q) \<in> init qmsg\<close> have "(ms, q) \<in> reachable qmsg (recvmsg (R \<sigma>))" ..
moreover from \<open>(ms, q) \<in> init qmsg\<close> have "ms = []"
unfolding \<sigma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_def by simp
ultimately show "(\<sigma>, fst pq) \<in> oreachable A ?owS (other U {i})
\<and> snd pq \<in> reachable qmsg (recvmsg (R \<sigma>))
\<and> (\<forall>m\<in>set (fst (snd pq)). R \<sigma> m)"
using \<open>pq = (p, (ms, q))\<close> by simp
next
note \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_simps [simp del]
case (other \<sigma> pq \<sigma>')
hence "(\<sigma>, fst pq) \<in> oreachable A ?owS (other U {i})"
and "other U {i} \<sigma> \<sigma>'"
and qr: "snd pq \<in> reachable qmsg (recvmsg (R \<sigma>))"
and "\<forall>m\<in>set (fst (snd pq)). R \<sigma> m"
by simp_all
from \<open>other U {i} \<sigma> \<sigma>'\<close> and ustutter have "\<forall>j. U (\<sigma> j) (\<sigma>' j)"
by (clarsimp elim!: otherE) metis
from \<open>other U {i} \<sigma> \<sigma>'\<close>
and \<open>(\<sigma>, fst pq) \<in> oreachable A ?owS (other U {i})\<close>
have "(\<sigma>', fst pq) \<in> oreachable A ?owS (other U {i})"
by - (rule oreachable_other')
moreover have "\<forall>m\<in>set (fst (snd pq)). R \<sigma>' m"
proof
fix m assume "m \<in> set (fst (snd pq))"
with \<open>\<forall>m\<in>set (fst (snd pq)). R \<sigma> m\<close> have "R \<sigma> m" ..
with \<open>\<forall>j. U (\<sigma> j) (\<sigma>' j)\<close> show "R \<sigma>' m" by (rule upreservesq)
qed
moreover from qr have "snd pq \<in> reachable qmsg (recvmsg (R \<sigma>'))"
proof
fix a
assume "recvmsg (R \<sigma>) a"
thus "recvmsg (R \<sigma>') a"
proof (rule recvmsgE [where R=R])
fix m assume "R \<sigma> m"
with \<open>\<forall>j. U (\<sigma> j) (\<sigma>' j)\<close> show "R \<sigma>' m" by (rule upreservesq)
qed
qed
ultimately show ?case using qr by simp
next
case (local \<sigma> pq \<sigma>' pq' a)
obtain p ms q p' ms' q' where "pq = (p, (ms, q))"
and "pq' = (p', (ms', q'))"
by (cases pq, cases pq') metis
with local.hyps local.IH
have pqtr: "((\<sigma>, (p, (ms, q))), a, (\<sigma>', (p', (ms', q'))))
\<in> oparp_sos i (trans A) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"
and por: "(\<sigma>, p) \<in> oreachable A ?owS (other U {i})"
and qr: "(ms, q) \<in> reachable qmsg (recvmsg (R \<sigma>))"
and "\<forall>m\<in>set ms. R \<sigma> m"
and "?owS \<sigma> \<sigma>' a"
by (simp_all del: \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_simps)
from \<open>?owS \<sigma> \<sigma>' a\<close> have "\<forall>j. j\<noteq>i \<longrightarrow> S (\<sigma> j) (\<sigma>' j)"
by (clarsimp dest!: otherwith_syncD)
with sgivesu have "\<forall>j. j\<noteq>i \<longrightarrow> U (\<sigma> j) (\<sigma>' j)" by simp
from \<open>?owS \<sigma> \<sigma>' a\<close> have "orecvmsg R \<sigma> a" by (rule otherwithE)
hence "recvmsg (R \<sigma>) a" ..
from pqtr have "(\<sigma>', p') \<in> oreachable A ?owS (other U {i})
\<and> (ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>'))
\<and> (\<forall>m\<in>set ms'. R \<sigma>' m)"
proof
assume "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "\<And>m. a \<noteq> receive m"
and "(ms', q') = (ms, q)"
from this(1) have ptr: "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A" by simp
with pinv por and \<open>?owS \<sigma> \<sigma>' a\<close> have "U (\<sigma> i) (\<sigma>' i)"
by (auto dest!: ostep_invariantD)
with \<open>\<forall>j. j\<noteq>i \<longrightarrow> U (\<sigma> j) (\<sigma>' j)\<close> have "\<forall>j. U (\<sigma> j) (\<sigma>' j)" by auto
hence recvmsg': "\<And>a. recvmsg (R \<sigma>) a \<Longrightarrow> recvmsg (R \<sigma>') a"
by (auto elim!: recvmsgE [where R=R] upreservesq)
from por ptr \<open>?owS \<sigma> \<sigma>' a\<close> have "(\<sigma>', p') \<in> oreachable A ?owS (other U {i})"
by - (rule oreachable_local')
moreover have "(ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>'))"
proof -
from qr and \<open>(ms', q') = (ms, q)\<close>
have "(ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>))" by simp
thus ?thesis by (rule reachable_weakenE) (erule recvmsg')
qed
moreover have "\<forall>m\<in>set ms'. R \<sigma>' m"
proof
fix m
assume "m\<in>set ms'"
with \<open>(ms', q') = (ms, q)\<close> have "m\<in>set ms" by simp
with \<open>\<forall>m\<in>set ms. R \<sigma> m\<close> have "R \<sigma> m" ..
with \<open>\<forall>j. U (\<sigma> j) (\<sigma>' j)\<close> show "R \<sigma>' m"
by (rule upreservesq)
qed
ultimately show
"(\<sigma>', p') \<in> oreachable A ?owS (other U {i})
\<and> (ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>'))
\<and> (\<forall>m\<in>set ms'. R \<sigma>' m)" by simp_all
next
assume qtr: "((ms, q), a, (ms', q')) \<in> seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
and "\<And>m. a \<noteq> send m"
and "p' = p"
and "\<sigma>' i = \<sigma> i"
from this(4) and \<open>\<And>\<xi>. U \<xi> \<xi>\<close> have "U (\<sigma> i) (\<sigma>' i)" by simp
with \<open>\<forall>j. j\<noteq>i \<longrightarrow> U (\<sigma> j) (\<sigma>' j)\<close> have "\<forall>j. U (\<sigma> j) (\<sigma>' j)" by auto
hence recvmsg': "\<And>a. recvmsg (R \<sigma>) a \<Longrightarrow> recvmsg (R \<sigma>') a"
by (auto elim!: recvmsgE [where R=R] upreservesq)
from qtr have tqtr: "((ms, q), a, (ms', q')) \<in> trans qmsg" by simp
from \<open>\<forall>j. U (\<sigma> j) (\<sigma>' j)\<close> and \<open>\<sigma>' i = \<sigma> i\<close> have "other U {i} \<sigma> \<sigma>'" by auto
with por and \<open>p' = p\<close>
have "(\<sigma>', p') \<in> oreachable A ?owS (other U {i})"
by (auto dest: oreachable_other)
moreover have "(ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>'))"
proof (rule reachable_weakenE [where P="recvmsg (R \<sigma>)"])
from qr tqtr \<open>recvmsg (R \<sigma>) a\<close> show "(ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>))" ..
qed (rule recvmsg')
moreover have "\<forall>m\<in>set ms'. R \<sigma>' m"
proof
fix m
assume "m \<in> set ms'"
moreover have "case a of receive m \<Rightarrow> set ms' \<subseteq> set (ms @ [m]) | _ \<Rightarrow> set ms' \<subseteq> set ms"
proof -
from qr have "(ms, q) \<in> reachable qmsg TT" ..
thus ?thesis using tqtr
by (auto dest!: step_invariantD [OF qmsg_queue_contents])
qed
ultimately have "R \<sigma> m" using \<open>\<forall>m\<in>set ms. R \<sigma> m\<close> and \<open>orecvmsg R \<sigma> a\<close>
by (cases a) auto
with \<open>\<forall>j. U (\<sigma> j) (\<sigma>' j)\<close> show "R \<sigma>' m"
by (rule upreservesq)
qed
ultimately show "(\<sigma>', p') \<in> oreachable A ?owS (other U {i})
\<and> (ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>'))
\<and> (\<forall>m\<in>set ms'. R \<sigma>' m)" by simp
next
fix m
assume "a = \<tau>"
and "((\<sigma>, p), receive m, (\<sigma>', p')) \<in> trans A"
and "((ms, q), send m, (ms', q')) \<in> seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
from this(2-3)
have ptr: "((\<sigma>, p), receive m, (\<sigma>', p')) \<in> trans A"
and qtr: "((ms, q), send m, (ms', q')) \<in> trans qmsg" by simp_all
from qr have "(ms, q) \<in> reachable qmsg TT" ..
with qtr have "m \<in> set ms"
by (auto dest!: step_invariantD [OF qmsg_send_from_queue])
with \<open>\<forall>m\<in>set ms. R \<sigma> m\<close> have "R \<sigma> m" ..
hence "orecvmsg R \<sigma> (receive m)" by simp
with \<open>\<forall>j. j\<noteq>i \<longrightarrow> S (\<sigma> j) (\<sigma>' j)\<close> have "?owS \<sigma> \<sigma>' (receive m)"
by (auto intro!: otherwithI)
with pinv por ptr have "U (\<sigma> i) (\<sigma>' i)"
by (auto dest!: ostep_invariantD)
with \<open>\<forall>j. j\<noteq>i \<longrightarrow> U (\<sigma> j) (\<sigma>' j)\<close> have "\<forall>j. U (\<sigma> j) (\<sigma>' j)" by auto
hence recvmsg': "\<And>a. recvmsg (R \<sigma>) a \<Longrightarrow> recvmsg (R \<sigma>') a"
by (auto elim!: recvmsgE [where R=R] upreservesq)
from por ptr have "(\<sigma>', p') \<in> oreachable A ?owS (other U {i})"
using \<open>?owS \<sigma> \<sigma>' (receive m)\<close> by - (erule(1) oreachable_local, simp)
moreover have "(ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>'))"
proof (rule reachable_weakenE [where P="recvmsg (R \<sigma>)"])
have "recvmsg (R \<sigma>) (send m)" by simp
with qr qtr show "(ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>))" ..
qed (rule recvmsg')
moreover have "\<forall>m\<in>set ms'. R \<sigma>' m"
proof
fix m
assume "m \<in> set ms'"
moreover have "set ms' \<subseteq> set ms"
proof -
from qr have "(ms, q) \<in> reachable qmsg TT" ..
thus ?thesis using qtr
by (auto dest!: step_invariantD [OF qmsg_queue_contents])
qed
ultimately have "R \<sigma> m" using \<open>\<forall>m\<in>set ms. R \<sigma> m\<close> by auto
with \<open>\<forall>j. U (\<sigma> j) (\<sigma>' j)\<close> show "R \<sigma>' m"
by (rule upreservesq)
qed
ultimately show "(\<sigma>', p') \<in> oreachable A ?owS (other U {i})
\<and> (ms', q') \<in> reachable qmsg (recvmsg (R \<sigma>'))
\<and> (\<forall>m\<in>set ms'. R \<sigma>' m)" by simp
qed
with \<open>pq = (p, (ms, q))\<close> and \<open>pq' = (p', (ms', q'))\<close> show ?case
by (simp_all del: \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_simps)
qed
lemma par_qmsg_oreachable_statelessassm:
assumes "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg)
(\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>) (other (\<lambda>_ _. True) {i})"
and ustutter: "\<And>\<xi>. U \<xi> \<xi>"
shows "(\<sigma>, fst \<zeta>) \<in> oreachable A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>) (other (\<lambda>_ _. True) {i})
\<and> snd \<zeta> \<in> reachable qmsg (recvmsg R)
\<and> (\<forall>m\<in>set (fst (snd \<zeta>)). R m)"
proof -
from assms(1)
have "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg)
(otherwith (\<lambda>_ _. True) {i} (orecvmsg (\<lambda>_. R)))
(other (\<lambda>_ _. True) {i})" by auto
moreover
have "A \<Turnstile>\<^sub>A (otherwith (\<lambda>_ _. True) {i} (orecvmsg (\<lambda>_. R)),
other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, _, \<sigma>'). True)"
by auto
ultimately
obtain "(\<sigma>, fst \<zeta>) \<in> oreachable A
(otherwith (\<lambda>_ _. True) {i} (orecvmsg (\<lambda>_. R))) (other (\<lambda>_ _. True) {i})"
and *: "snd \<zeta> \<in> reachable qmsg (recvmsg R)"
and **: "(\<forall>m\<in>set (fst (snd \<zeta>)). R m)"
by (auto dest!: par_qmsg_oreachable)
from this(1)
have "(\<sigma>, fst \<zeta>) \<in> oreachable A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>) (other (\<lambda>_ _. True) {i})"
by rule auto
thus ?thesis using * ** by simp
qed
lemma lift_into_qmsg:
assumes "A \<Turnstile> (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) global P"
and "\<And>\<xi>. U \<xi> \<xi>"
and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'"
and "\<And>\<sigma> \<sigma>' m. \<lbrakk> \<forall>j. U (\<sigma> j) (\<sigma>' j); R \<sigma> m \<rbrakk> \<Longrightarrow> R \<sigma>' m"
and "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>)
globala (\<lambda>(\<sigma>, _, \<sigma>'). U (\<sigma> i) (\<sigma>' i))"
shows "A \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile> (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) global P"
proof (rule oinvariant_oreachableI)
fix \<sigma> \<zeta>
assume "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg) (otherwith S {i} (orecvmsg R)) (other U {i})"
then obtain s where "(\<sigma>, s) \<in> oreachable A (otherwith S {i} (orecvmsg R)) (other U {i})"
by (auto dest!: par_qmsg_oreachable [OF _ assms(5,2-4)])
with assms(1) show "global P (\<sigma>, \<zeta>)"
by (auto dest: oinvariant_weakenD [OF assms(1)])
qed
lemma lift_step_into_qmsg:
assumes inv: "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala P"
and ustutter: "\<And>\<xi>. U \<xi> \<xi>"
and sgivesu: "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'"
and upreservesq: "\<And>\<sigma> \<sigma>' m. \<lbrakk> \<forall>j. U (\<sigma> j) (\<sigma>' j); R \<sigma> m \<rbrakk> \<Longrightarrow> R \<sigma>' m"
and self_sync: "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>)
globala (\<lambda>(\<sigma>, _, \<sigma>'). U (\<sigma> i) (\<sigma>' i))"
and recv_stutter: "\<And>\<sigma> \<sigma>' m. \<lbrakk> \<forall>j. U (\<sigma> j) (\<sigma>' j); \<sigma>' i = \<sigma> i \<rbrakk> \<Longrightarrow> P (\<sigma>, receive m, \<sigma>')"
and receive_right: "\<And>\<sigma> \<sigma>' m. P (\<sigma>, receive m, \<sigma>') \<Longrightarrow> P (\<sigma>, \<tau>, \<sigma>')"
shows "A \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala P"
(is "_ \<Turnstile>\<^sub>A (?owS, ?U \<rightarrow>) _")
proof (rule ostep_invariantI)
fix \<sigma> \<zeta> a \<sigma>' \<zeta>'
assume or: "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg) ?owS ?U"
and otr: "((\<sigma>, \<zeta>), a, (\<sigma>', \<zeta>')) \<in> trans (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg)"
and "?owS \<sigma> \<sigma>' a"
from this(2) have "((\<sigma>, \<zeta>), a, (\<sigma>', \<zeta>')) \<in> oparp_sos i (trans A) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"
by simp
then obtain s msgs q s' msgs' q'
where "\<zeta> = (s, (msgs, q))" "\<zeta>' = (s', (msgs', q'))"
and "((\<sigma>, (s, (msgs, q))), a, (\<sigma>', (s', (msgs', q'))))
\<in> oparp_sos i (trans A) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"
by (metis prod_cases3)
from this(1-2) and or
obtain "(\<sigma>, s) \<in> oreachable A ?owS ?U"
"(msgs, q) \<in> reachable qmsg (recvmsg (R \<sigma>))"
"(\<forall>m\<in>set msgs. R \<sigma> m)"
by (auto dest: par_qmsg_oreachable [OF _ self_sync ustutter sgivesu]
elim!: upreservesq)
from otr \<open>\<zeta> = (s, (msgs, q))\<close> \<open>\<zeta>' = (s', (msgs', q'))\<close>
have "((\<sigma>, (s, (msgs, q))), a, (\<sigma>', (s', (msgs', q'))))
\<in> oparp_sos i (trans A) (seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G)"
by simp
hence "globala P ((\<sigma>, s), a, (\<sigma>', s'))"
proof
assume "((\<sigma>, s), a, (\<sigma>', s')) \<in> trans A"
with \<open>(\<sigma>, s) \<in> oreachable A ?owS ?U\<close>
show "globala P ((\<sigma>, s), a, (\<sigma>', s'))"
using \<open>?owS \<sigma> \<sigma>' a\<close> by (rule ostep_invariantD [OF inv])
next
assume "((msgs, q), a, (msgs', q')) \<in> seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
and "\<And>m. a \<noteq> send m"
and "\<sigma>' i = \<sigma> i"
from this(3) and ustutter have "U (\<sigma> i) (\<sigma>' i)" by simp
with \<open>?owS \<sigma> \<sigma>' a\<close> and sgivesu have "\<forall>j. U (\<sigma> j) (\<sigma>' j)"
by (clarsimp dest!: otherwith_syncD) metis
moreover have "(\<exists>m. a = receive m) \<or> (a = \<tau>)"
proof -
from \<open>(msgs, q) \<in> reachable qmsg (recvmsg (R \<sigma>))\<close>
have "(msgs, q) \<in> reachable qmsg TT" ..
moreover from \<open>((msgs, q), a, (msgs', q')) \<in> seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G\<close>
have "((msgs, q), a, (msgs', q')) \<in> trans qmsg" by simp
ultimately show ?thesis
using \<open>\<And>m. a \<noteq> send m\<close>
by (auto dest!: step_invariantD [OF qmsg_send_receive_or_tau])
qed
ultimately show "globala P ((\<sigma>, s), a, (\<sigma>', s'))"
using \<open>\<sigma>' i = \<sigma> i\<close>
by simp (metis receive_right recv_stutter step_seq_tau)
next
fix m
assume "a = \<tau>"
and "((\<sigma>, s), receive m, (\<sigma>', s')) \<in> trans A"
and "((msgs, q), send m, (msgs', q')) \<in> seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G"
from \<open>(msgs, q) \<in> reachable qmsg (recvmsg (R \<sigma>))\<close>
have "(msgs, q) \<in> reachable qmsg TT" ..
moreover from \<open>((msgs, q), send m, (msgs', q')) \<in> seqp_sos \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G\<close>
have "((msgs, q), send m, (msgs', q')) \<in> trans qmsg" by simp
ultimately have "m\<in>set msgs"
by (auto dest!: step_invariantD [OF qmsg_send_from_queue])
with \<open>\<forall>m\<in>set msgs. R \<sigma> m\<close> have "R \<sigma> m" ..
with \<open>?owS \<sigma> \<sigma>' a\<close> have "?owS \<sigma> \<sigma>' (receive m)"
by (auto dest!: otherwith_syncD)
with \<open>((\<sigma>, s), receive m, (\<sigma>', s')) \<in> trans A\<close>
have "globala P ((\<sigma>, s), receive m, (\<sigma>', s'))"
using \<open>(\<sigma>, s) \<in> oreachable A ?owS ?U\<close>
by - (rule ostep_invariantD [OF inv])
hence "P (\<sigma>, receive m, \<sigma>')" by simp
hence "P (\<sigma>, \<tau>, \<sigma>')" by (rule receive_right)
with \<open>a = \<tau>\<close> show "globala P ((\<sigma>, s), a, (\<sigma>', s'))" by simp
qed
with \<open>\<zeta> = (s, (msgs, q))\<close> and \<open>\<zeta>' = (s', (msgs', q'))\<close> show "globala P ((\<sigma>, \<zeta>), a, (\<sigma>', \<zeta>'))"
by simp
qed
lemma lift_step_into_qmsg_statelessassm:
assumes "A \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala P"
and "\<And>\<sigma> \<sigma>' m. \<sigma>' i = \<sigma> i \<Longrightarrow> P (\<sigma>, receive m, \<sigma>')"
and "\<And>\<sigma> \<sigma>' m. P (\<sigma>, receive m, \<sigma>') \<Longrightarrow> P (\<sigma>, \<tau>, \<sigma>')"
shows "A \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala P"
proof -
from assms(1) have *: "A \<Turnstile>\<^sub>A (otherwith (\<lambda>_ _. True) {i} (orecvmsg (\<lambda>_. R)),
other (\<lambda>_ _. True) {i} \<rightarrow>) globala P"
by rule auto
hence "A \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A
(otherwith (\<lambda>_ _. True) {i} (orecvmsg (\<lambda>_. R)), other (\<lambda>_ _. True) {i} \<rightarrow>) globala P"
by (rule lift_step_into_qmsg)
(auto elim!: assms(2-3) simp del: step_seq_tau)
thus ?thesis by rule auto
qed
end
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