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Old_Simulation.thy
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theory Old_Simulation
imports Old_Semantics "NewPsi.Simulation"
begin
context old_psi begin
definition
"old_simulation" :: "'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow>
('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set \<Rightarrow>
('a, 'b, 'c) psi \<Rightarrow> bool" ("_ \<rhd> _ \<leadsto>[_]\<^sub>O _" [80, 80, 80, 80] 80)
where
"\<Psi> \<rhd> P \<leadsto>[Rel]\<^sub>O Q \<equiv> \<forall>\<alpha> Q'. \<Psi> \<rhd> Q \<longmapsto>\<^sub>O\<alpha> \<prec> Q' \<longrightarrow> bn \<alpha> \<sharp>* \<Psi> \<longrightarrow> bn \<alpha> \<sharp>* P \<longrightarrow> (\<exists>P'. \<Psi> \<rhd> P \<longmapsto>\<^sub>O\<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel)"
abbreviation
old_simulation_nil_judge ("_ \<leadsto>[_]\<^sub>O _" [80, 80, 80] 80) where "P \<leadsto>[Rel]\<^sub>O Q \<equiv> S_bottom' \<rhd> P \<leadsto>[Rel]\<^sub>O Q"
lemma old_simulation_is_new:
shows "\<Psi> \<rhd> P \<leadsto>[Rel]\<^sub>O Q = \<Psi> \<rhd> P \<leadsto>[Rel] Q"
unfolding old_simulation_def simulation_def
by(metis old_semantics_complete old_semantics_sound)
lemma old_simE:
fixes F :: 'b
and P :: "('a, 'b, 'c) psi"
and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"
and Q :: "('a, 'b, 'c) psi"
assumes "\<Psi> \<rhd> P \<leadsto>[Rel]\<^sub>O Q"
shows "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto>\<^sub>O\<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<longmapsto>\<^sub>O\<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel"
using assms
by(auto simp add: old_simulation_def)
lemma old_monotonic:
fixes \<Psi> :: 'b
and P :: "('a, 'b, 'c) psi"
and A :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"
and Q :: "('a, 'b, 'c) psi"
and B :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"
assumes "\<Psi> \<rhd> P \<leadsto>[A]\<^sub>O Q"
and "A \<subseteq> B"
shows "\<Psi> \<rhd> P \<leadsto>[B]\<^sub>O Q"
using assms
by(simp (no_asm) add: old_simulation_def) (auto dest: old_simE)
end
end