(* Title: HOLCF/IOA/meta_theory/LiveIOA.thy ID: $Id: LiveIOA.thy,v 1.10 2007/10/21 14:27:43 wenzelm Exp $ Author: Olaf Müller *) header {* Live I/O automata -- specified by temproal formulas *} theory LiveIOA imports TLS begin defaultsort type types ('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) .--> P)" definition WF :: "('a,'s)ioa => 'a set => ('a,'s)ioa_temp" where "WF A acts = (<> [] <%(s,a,t) . Enabled A acts s> .--> [] <> <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> .--> [] <> <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)) & (out (fst CL) = out (fst AM)) & livetraces CL <= 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) & (! exec : executions (fst CL). (exec |== (snd CL)) --> ((corresp_ex (fst AM) f exec) |== (snd AM))))" declare split_paired_Ex [simp del] lemma live_implements_trans: "!!LC. [| live_implements (A,LA) (B,LB); live_implements (B,LB) (C,LC) |] ==> live_implements (A,LA) (C,LC)" apply (unfold live_implements_def) apply auto done subsection "Correctness of live refmap" lemma live_implements: "[| inp(C)=inp(A); out(C)=out(A); is_live_ref_map f (C,M) (A,L) |] ==> live_implements (C,M) (A,L)" apply (simp add: is_live_ref_map_def live_implements_def livetraces_def liveexecutions_def) apply (tactic "safe_tac set_cs") apply (rule_tac x = "corresp_ex A f ex" in exI) apply (tactic "safe_tac set_cs") (* Traces coincide, Lemma 1 *) apply (tactic {* pair_tac "ex" 1 *}) 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) (* corresp_ex is execution, Lemma 2 *) apply (tactic {* pair_tac "ex" 1 *}) apply (simp add: executions_def) (* start state *) apply (rule conjI) apply (simp add: is_ref_map_def corresp_ex_def) (* is-execution-fragment *) apply (erule lemma_2 [THEN spec, THEN mp]) apply (simp add: reachable.reachable_0) (* Liveness *) apply auto done end
lemma live_implements_trans:
[| live_implements (A, LA) (B, LB); live_implements (B, LB) (C, LC) |]
==> live_implements (A, LA) (C, LC)
lemma live_implements:
[| inp C = inp A; out C = out A; is_live_ref_map f (C, M) (A, L) |]
==> live_implements (C, M) (A, L)