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Weak_Psi_Congruence.thy
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Weak_Psi_Congruence.thy
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(*
Title: Psi-calculi
Based on the AFP entry by Jesper Bengtson (jebe@itu.dk), 2012
*)
theory Weak_Psi_Congruence
imports Weak_Cong_Simulation Weak_Bisimulation
begin
context env begin
definition weakPsiCongruence :: "'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" ("_ \<rhd> _ \<doteq> _" [70, 70, 70] 65)
where
"\<Psi> \<rhd> P \<doteq> Q \<equiv> \<Psi> \<rhd> P \<approx> Q \<and> \<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q \<and> \<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
abbreviation
weakPsiCongNilJudge ("_ \<doteq> _" [70, 70] 65) where "P \<doteq> Q \<equiv> \<one> \<rhd> P \<doteq> Q"
lemma weakPsiCongSym:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<rhd> P \<doteq> Q"
shows "\<Psi> \<rhd> Q \<doteq> P"
using assms
by(auto simp add: weakPsiCongruence_def weakBisimE)
lemma weakPsiCongE:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and \<Psi>' :: 'b
assumes "\<Psi> \<rhd> P \<doteq> Q"
shows "\<Psi> \<rhd> P \<approx> Q"
and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"
and "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
using assms
by(auto simp add: weakPsiCongruence_def)
lemma weakPsiCongI[case_names cWeakBisim cSimLeft cSimRight]:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and \<Psi>' :: 'b
assumes "\<Psi> \<rhd> P \<approx> Q"
and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"
and "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
shows "\<Psi> \<rhd> P \<doteq> Q"
using assms
by(auto simp add: weakPsiCongruence_def)
lemma weakPsiCongSymI[consumes 1, case_names cSym cWeakBisim cSim]:
fixes \<Psi> :: 'b
and P :: "'d::fs_name"
and Q :: 'd
and \<Psi>' :: 'b
assumes "Prop P Q"
and "\<And>P Q. Prop P Q \<Longrightarrow> Prop Q P"
and "\<And>P Q. Prop P Q \<Longrightarrow> \<Psi> \<rhd> (C P) \<approx> (C Q)"
and "\<And>P Q. Prop P Q \<Longrightarrow> \<Psi> \<rhd> (C P) \<leadsto>\<guillemotleft>weakBisim\<guillemotright> (C Q)"
shows "\<Psi> \<rhd> (C P) \<doteq> (C Q)"
using assms
by(rule_tac weakPsiCongI) auto
lemma weakPsiCongSym2[consumes 1, case_names cWeakBisim cSim]:
fixes \<Psi> :: 'b
and \<Psi>' :: 'b
assumes "\<Psi> \<rhd> P \<doteq> Q"
and "\<And>P Q. \<Psi> \<rhd> P \<doteq> Q \<Longrightarrow> \<Psi> \<rhd> (C P) \<approx> (C Q)"
and "\<And>P Q. \<Psi> \<rhd> P \<doteq> Q \<Longrightarrow> \<Psi> \<rhd> (C P) \<leadsto>\<guillemotleft>weakBisim\<guillemotright> (C Q)"
shows "\<Psi> \<rhd> (C P) \<doteq> (C Q)"
using assms
apply(rule_tac weakPsiCongSymI[where C=C])
apply assumption
by(auto simp add: weakPsiCongruence_def dest: weakBisimE)
lemma statEqWeakCong:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and \<Psi>' :: 'b
assumes "\<Psi> \<rhd> P \<doteq> Q"
and "\<Psi> \<simeq> \<Psi>'"
shows "\<Psi>' \<rhd> P \<doteq> Q"
proof -
let ?Prop = "\<lambda>P Q. \<Psi> \<rhd> P \<doteq> Q \<and> \<Psi> \<simeq> \<Psi>'"
from assms have "?Prop P Q" by auto
thus ?thesis
proof(induct rule: weakPsiCongSymI)
case(cSym P Q)
thus ?case by(blast dest: weakPsiCongSym)
next
case(cSim P Q)
from `\<Psi> \<rhd> P \<doteq> Q \<and> \<Psi> \<simeq> \<Psi>'` have "\<Psi> \<rhd> P \<doteq> Q" and "\<Psi> \<simeq> \<Psi>'" by simp+
from `\<Psi> \<rhd> P \<doteq> Q` have "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q" by(rule weakPsiCongE)
with `\<Psi> \<simeq> \<Psi>'` show ?case using statEqWeakBisim
by(rule_tac weakCongSimStatEq) auto
next
case(cWeakBisim P Q)
from `\<Psi> \<rhd> P \<doteq> Q \<and> \<Psi> \<simeq> \<Psi>'`
have "\<Psi> \<rhd> P \<approx> Q" and "\<Psi> \<simeq> \<Psi>'" by(auto dest: weakPsiCongE)
thus ?case by(rule statEqWeakBisim)
qed
qed
lemma weakPsiCongReflexive:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
shows "\<Psi> \<rhd> P \<doteq> P"
by(fastforce intro: weakPsiCongI weakCongSimReflexive weakBisimReflexive)
lemma weakPsiCongClosed:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and p :: "name prm"
assumes "\<Psi> \<rhd> P \<doteq> Q"
shows "(p \<bullet> \<Psi>) \<rhd> (p \<bullet> P) \<doteq> (p \<bullet> Q)"
using assms
proof(induct rule: weakPsiCongSymI)
case(cSym P Q)
thus ?case by(rule weakPsiCongSym)
next
case(cSim P Q)
from `\<Psi> \<rhd> P \<doteq> Q` have "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q" by(rule weakPsiCongE)
thus ?case by(drule_tac p=p in weakCongSimClosed(1)[OF weakBisimEqvt]) (simp add: eqvts)
next
case(cWeakBisim P Q)
from `\<Psi> \<rhd> P \<doteq> Q` have "\<Psi> \<rhd> P \<approx> Q" by(rule weakPsiCongE)
thus ?case by(rule weakBisimClosed)
qed
lemma weakPsiCongTransitive:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
and R :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<rhd> P \<doteq> Q"
and "\<Psi> \<rhd> Q \<doteq> R"
shows "\<Psi> \<rhd> P \<doteq> R"
proof -
from assms have "\<Psi> \<rhd> P \<doteq> Q \<and> \<Psi> \<rhd> Q \<doteq> R" by auto
thus ?thesis
proof(induct rule: weakPsiCongSymI)
case(cSym P R)
thus ?case by(auto dest: weakPsiCongSym)
next
case(cSim P R)
hence "\<Psi> \<rhd> P \<doteq> Q" and "\<Psi> \<rhd> Q \<doteq> R" by auto
moreover from `\<Psi> \<rhd> P \<doteq> Q` have "\<Psi> \<rhd> P \<approx> Q" by(metis weakBisimE weakPsiCongE)
moreover from `\<Psi> \<rhd> P \<doteq> Q` have "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q" by(rule weakPsiCongE)
moreover from `\<Psi> \<rhd> Q \<doteq> R` have "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> R" by(rule weakPsiCongE)
moreover have "{(\<Psi>, P, R) | \<Psi> P R. \<exists>Q. \<Psi> \<rhd> P \<approx> Q \<and> \<Psi> \<rhd> Q \<approx> R} \<subseteq> weakBisim"
by(auto dest: weakBisimTransitive)
ultimately show ?case using weakBisimE(2) by(rule_tac weakCongSimTransitive)
next
case(cWeakBisim P R)
thus ?case by(auto dest: weakBisimTransitive weakPsiCongE)
qed
qed
lemma strongBisimWeakPsiCong:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<rhd> P \<sim> Q"
shows "\<Psi> \<rhd> P \<doteq> Q"
using assms
proof(induct rule: weakPsiCongSymI)
case(cSym P Q)
from `\<Psi> \<rhd> P \<sim> Q` show ?case by(rule bisimE)
next
case(cSim P Q)
from `\<Psi> \<rhd> P \<sim> Q` have "\<Psi> \<rhd> P \<leadsto>[bisim] Q" by(rule bisimE)
thus "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q" using strongBisimWeakBisim
by(rule_tac strongSimWeakCongSim) auto
next
case(cWeakBisim P Q)
thus ?case by(rule strongBisimWeakBisim)
qed
lemma structCongWeakPsiCong:
fixes P :: "('a, 'b, 'c) psi"
and Q :: "('a, 'b, 'c) psi"
assumes "P \<equiv>\<^sub>s Q"
shows "P \<doteq> Q"
using assms
by(metis struct_cong_bisim strongBisimWeakPsiCong)
end
end