# File LiveIOA.ML

    (*  Title:      HOLCF/IOA/meta_theory/LiveIOA.ML
        ID:         $Id: LiveIOA.ML,v 1.10 2005/09/02 15:24:01 wenzelm Exp $
        Author:     Olaf Müller
    *)

    Delsimps [split_paired_Ex];

    Goalw [live_implements_def]
    "!!LC. [| 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 "[| 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";