(* Title: HOL/HOLCF/IOA/LiveIOA.thy
Author: Olaf Müller
*)
section \<open>Live I/O automata -- specified by temproal formulas\<close>
theory LiveIOA
imports TLS
begin
default_sort type
type_synonym ('a, 's) live_ioa = "('a, 's)ioa \<times> ('a, 's) ioa_temp"
definition validLIOA :: "('a, 's) live_ioa \<Rightarrow> ('a, 's) ioa_temp \<Rightarrow> bool"
where "validLIOA AL P \<longleftrightarrow> validIOA (fst AL) (snd AL \<^bold>\<longrightarrow> P)"
definition WF :: "('a, 's) ioa \<Rightarrow> 'a set \<Rightarrow> ('a, 's) ioa_temp"
where "WF A acts = (\<diamond>\<box>\<langle>\<lambda>(s,a,t). Enabled A acts s\<rangle> \<^bold>\<longrightarrow> \<box>\<diamond>\<langle>xt2 (plift (\<lambda>a. a \<in> acts))\<rangle>)"
definition SF :: "('a, 's) ioa \<Rightarrow> 'a set \<Rightarrow> ('a, 's) ioa_temp"
where "SF A acts = (\<box>\<diamond>\<langle>\<lambda>(s,a,t). Enabled A acts s\<rangle> \<^bold>\<longrightarrow> \<box>\<diamond>\<langle>xt2 (plift (\<lambda>a. a \<in> acts))\<rangle>)"
definition liveexecutions :: "('a, 's) live_ioa \<Rightarrow> ('a, 's) execution set"
where "liveexecutions AP = {exec. exec \<in> executions (fst AP) \<and> (exec \<TTurnstile> snd AP)}"
definition livetraces :: "('a, 's) live_ioa \<Rightarrow> 'a trace set"
where "livetraces AP = {mk_trace (fst AP) \<cdot> (snd ex) |ex. ex \<in> liveexecutions AP}"
definition live_implements :: "('a, 's1) live_ioa \<Rightarrow> ('a, 's2) live_ioa \<Rightarrow> bool"
where "live_implements CL AM \<longleftrightarrow>
inp (fst CL) = inp (fst AM) \<and> out (fst CL) = out (fst AM) \<and>
livetraces CL \<subseteq> livetraces AM"
definition is_live_ref_map :: "('s1 \<Rightarrow> 's2) \<Rightarrow> ('a, 's1) live_ioa \<Rightarrow> ('a, 's2) live_ioa \<Rightarrow> bool"
where "is_live_ref_map f CL AM \<longleftrightarrow>
is_ref_map f (fst CL ) (fst AM) \<and>
(\<forall>exec \<in> executions (fst CL). (exec \<TTurnstile> (snd CL)) \<longrightarrow>
(corresp_ex (fst AM) f exec \<TTurnstile> snd AM))"
lemma live_implements_trans:
"live_implements (A, LA) (B, LB) \<Longrightarrow> live_implements (B, LB) (C, LC) \<Longrightarrow>
live_implements (A, LA) (C, LC)"
by (auto simp: live_implements_def)
subsection \<open>Correctness of live refmap\<close>
lemma live_implements:
"inp C = inp A \<Longrightarrow> out C = out A \<Longrightarrow> is_live_ref_map f (C, M) (A, L)
\<Longrightarrow> live_implements (C, M) (A, L)"
apply (simp add: is_live_ref_map_def live_implements_def livetraces_def liveexecutions_def)
apply auto
apply (rule_tac x = "corresp_ex A f ex" in exI)
apply auto
text \<open>Traces coincide, Lemma 1\<close>
apply (pair ex)
apply (erule lemma_1 [THEN spec, THEN mp])
apply (simp (no_asm) add: externals_def)
apply (auto)[1]
apply (simp add: executions_def reachable.reachable_0)
text \<open>\<open>corresp_ex\<close> is execution, Lemma 2\<close>
apply (pair ex)
apply (simp add: executions_def)
text \<open>start state\<close>
apply (rule conjI)
apply (simp add: is_ref_map_def corresp_ex_def)
text \<open>\<open>is_execution_fragment\<close>\<close>
apply (erule lemma_2 [THEN spec, THEN mp])
apply (simp add: reachable.reachable_0)
done
end