(* Title: HOLCF/IOA/meta_theory/LiveIOA.ML
ID: $Id$
Author: Olaf M"uller
Copyright 1997 TU Muenchen
Live I/O Automata
*)
Delsimps [split_paired_Ex];
Goalw [live_implements_def]
"!! A. [| live_implements (A,LA) (B,LB); live_implements (B,LB) (C,LC) |] \
\ ==> live_implements (A,LA) (C,LC)";
by Auto_tac;
qed"live_implements_trans";
section "Correctness of live refmap";
(* ---------------------------------------------------------------- *)
(* Correctness of live refmap *)
(* ---------------------------------------------------------------- *)
Goal "!! f. [| inp(C)=inp(A); out(C)=out(A); \
\ is_live_ref_map f (C,M) (A,L) |] \
\ ==> live_implements (C,M) (A,L)";
by (asm_full_simp_tac (simpset() addsimps [is_live_ref_map_def, live_implements_def,
livetraces_def,liveexecutions_def]) 1);
by (safe_tac set_cs);
by (res_inst_tac[("x","corresp_ex A f ex")] exI 1);
by (safe_tac set_cs);
(* Traces coincide, Lemma 1 *)
by (pair_tac "ex" 1);
by (etac (lemma_1 RS spec RS mp) 1);
by (simp_tac (simpset() addsimps [externals_def])1);
by (SELECT_GOAL (auto_tac (claset(),simpset()))1);
by (asm_full_simp_tac (simpset() addsimps [executions_def,reachable.reachable_0]) 1);
(* corresp_ex is execution, Lemma 2 *)
by (pair_tac "ex" 1);
by (asm_full_simp_tac (simpset() addsimps [executions_def]) 1);
(* start state *)
by (rtac conjI 1);
by (asm_full_simp_tac (simpset() addsimps [is_ref_map_def,corresp_ex_def]) 1);
(* is-execution-fragment *)
by (etac (lemma_2 RS spec RS mp) 1);
by (asm_full_simp_tac (simpset() addsimps [reachable.reachable_0]) 1);
(* Liveness *)
by Auto_tac;
qed"live_implements";
(*
(* Classical Fairness and Fairness by Temporal Formula coincide *)
Goal "!! ex. Finite (snd ex) ==> \
\ (ex |== WF A acts) = (~ Enabled A acts (laststate ex))";
In 3 steps:
1) []<>P and <>[]P mean both P (Last`s)
2) Transform this to executions and laststate
3) plift is used to show that plift (laststate ex) : acts == False.
*)