Quelle LiveIOA.thy Sprache: Isabelle

(*  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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