(* 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 \ ('a, 's) ioa_temp"
definition validLIOA :: "('a, 's) live_ioa \ ('a, 's) ioa_temp \ bool"
where "validLIOA AL P \ validIOA (fst AL) (snd AL \<^bold>\ P)"
definition WF :: "('a, 's) ioa \ 'a set \ ('a, 's) ioa_temp"
where "WF A acts = (\\\\(s,a,t). Enabled A acts s\ \<^bold>\ \\\xt2 (plift (\a. a \ acts))\)"
definition SF :: "('a, 's) ioa \ 'a set \ ('a, 's) ioa_temp"
where "SF A acts = (\\\\(s,a,t). Enabled A acts s\ \<^bold>\ \\\xt2 (plift (\a. a \ acts))\)"
definition liveexecutions :: "('a, 's) live_ioa \ ('a, 's) execution set"
where "liveexecutions AP = {exec. exec \ executions (fst AP) \ (exec \ snd AP)}"
definition livetraces :: "('a, 's) live_ioa \ 'a trace set"
where "livetraces AP = {mk_trace (fst AP) \ (snd ex) |ex. ex \ liveexecutions AP}"
definition live_implements :: "('a, 's1) live_ioa \ ('a, 's2) live_ioa \ bool"
where "live_implements CL AM \
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 \ 's2) \ ('a, 's1) live_ioa \ ('a, 's2) live_ioa \ bool"
where "is_live_ref_map f CL AM \
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) \ live_implements (B, LB) (C, LC) \
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 \ out C = out A \ 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
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