# File Traces.ML

    (*  Title:      HOLCF/IOA/meta_theory/Traces.ML
        ID:         $Id: Traces.ML,v 1.19 2004/06/21 08:26:10 kleing Exp $
        Author:     Olaf Müller

    Theorems about Executions and Traces of I/O automata in HOLCF.
    *)

    (* global changes to simpset() and claset(), see also TLS.ML *)
    Delsimps (ex_simps @ all_simps);
    Delsimps [split_paired_Ex];
    Addsimps [Let_def];
    claset_ref() := claset() delSWrapper "split_all_tac";

    val exec_rws = [executions_def,is_exec_frag_def];

    (* ----------------------------------------------------------------------------------- *)

    section "recursive equations of operators";

    (* ---------------------------------------------------------------- *)
    (*                               filter_act                         *)
    (* ---------------------------------------------------------------- *)

    Goal  "filter_act$UU = UU";
    by (simp_tac (simpset() addsimps [filter_act_def]) 1);
    qed"filter_act_UU";

    Goal  "filter_act$nil = nil";
    by (simp_tac (simpset() addsimps [filter_act_def]) 1);
    qed"filter_act_nil";

    Goal "filter_act$(x>>xs) = (fst x) >> filter_act$xs";
    by (simp_tac (simpset() addsimps [filter_act_def]) 1);
    qed"filter_act_cons";

    Addsimps [filter_act_UU,filter_act_nil,filter_act_cons];

    (* ---------------------------------------------------------------- *)
    (*                             mk_trace                             *)
    (* ---------------------------------------------------------------- *)

    Goal "mk_trace A$UU=UU";
    by (simp_tac (simpset() addsimps [mk_trace_def]) 1);
    qed"mk_trace_UU";

    Goal "mk_trace A$nil=nil";
    by (simp_tac (simpset() addsimps [mk_trace_def]) 1);
    qed"mk_trace_nil";

    Goal "mk_trace A$(at >> xs) =    \
    \            (if ((fst at):ext A)    \
    \                 then (fst at) >> (mk_trace A$xs) \
    \                 else mk_trace A$xs)";

    by (asm_full_simp_tac (simpset() addsimps [mk_trace_def]) 1);
    qed"mk_trace_cons";

    Addsimps [mk_trace_UU,mk_trace_nil,mk_trace_cons];

    (* ---------------------------------------------------------------- *)
    (*                             is_exec_fragC                             *)
    (* ---------------------------------------------------------------- *)

    Goal "is_exec_fragC A = (LAM ex. (%s. case ex of \
    \      nil => TT \
    \    | x##xs => (flift1 \
    \            (%p. Def ((s,p):trans_of A) andalso (is_exec_fragC A$xs) (snd p)) \
    \             $x) \
    \   ))";
    by (rtac trans 1);
    by (rtac fix_eq2 1);
    by (rtac is_exec_fragC_def 1);
    by (rtac beta_cfun 1);
    by (simp_tac (simpset() addsimps [flift1_def]) 1);
    qed"is_exec_fragC_unfold";

    Goal "(is_exec_fragC A$UU) s=UU";
    by (stac is_exec_fragC_unfold 1);
    by (Simp_tac 1);
    qed"is_exec_fragC_UU";

    Goal "(is_exec_fragC A$nil) s = TT";
    by (stac is_exec_fragC_unfold 1);
    by (Simp_tac 1);
    qed"is_exec_fragC_nil";

    Goal "(is_exec_fragC A$(pr>>xs)) s = \
    \                        (Def ((s,pr):trans_of A) \
    \                andalso (is_exec_fragC A$xs)(snd pr))";
    by (rtac trans 1);
    by (stac is_exec_fragC_unfold 1);
    by (asm_full_simp_tac (simpset() addsimps [Consq_def,flift1_def]) 1);
    by (Simp_tac 1);
    qed"is_exec_fragC_cons";

    Addsimps [is_exec_fragC_UU,is_exec_fragC_nil,is_exec_fragC_cons];

    (* ---------------------------------------------------------------- *)
    (*                        is_exec_frag                              *)
    (* ---------------------------------------------------------------- *)

    Goal "is_exec_frag A (s, UU)";
    by (simp_tac (simpset() addsimps [is_exec_frag_def]) 1);
    qed"is_exec_frag_UU";

    Goal "is_exec_frag A (s, nil)";
    by (simp_tac (simpset() addsimps [is_exec_frag_def]) 1);
    qed"is_exec_frag_nil";

    Goal "is_exec_frag A (s, (a,t)>>ex) = \
    \                               (((s,a,t):trans_of A) & \
    \                               is_exec_frag A (t, ex))";
    by (simp_tac (simpset() addsimps [is_exec_frag_def]) 1);
    qed"is_exec_frag_cons";

    (* Delsimps [is_exec_fragC_UU,is_exec_fragC_nil,is_exec_fragC_cons]; *)
    Addsimps [is_exec_frag_UU,is_exec_frag_nil, is_exec_frag_cons];

    (* ---------------------------------------------------------------------------- *)
                               section "laststate";
    (* ---------------------------------------------------------------------------- *)

    Goal "laststate (s,UU) = s";
    by (simp_tac (simpset() addsimps [laststate_def]) 1);
    qed"laststate_UU";

    Goal "laststate (s,nil) = s";
    by (simp_tac (simpset() addsimps [laststate_def]) 1);
    qed"laststate_nil";

    Goal "!! ex. Finite ex ==> laststate (s,at>>ex) = laststate (snd at,ex)";
    by (simp_tac (simpset() addsimps [laststate_def]) 1);
    by (case_tac "ex=nil" 1);
    by (Asm_simp_tac 1);
    by (Asm_simp_tac 1);
    by (dtac (Finite_Last1 RS mp) 1);
    by (assume_tac 1);
    by (def_tac 1);
    qed"laststate_cons";

    Addsimps [laststate_UU,laststate_nil,laststate_cons];

    Goal "!!ex. Finite ex ==> (! s. ? u. laststate (s,ex)=u)";
    by (Seq_Finite_induct_tac 1);
    qed"exists_laststate";

    (* -------------------------------------------------------------------------------- *)

    section "has_trace, mk_trace";

    (* alternative definition of has_trace tailored for the refinement proof, as it does not
       take the detour of schedules *)

    Goalw  [executions_def,mk_trace_def,has_trace_def,schedules_def,has_schedule_def]
    "has_trace A b = (? ex:executions A. b = mk_trace A$(snd ex))";

    by (safe_tac set_cs);
    (* 1 *)
    by (res_inst_tac[("x","ex")] bexI 1);
    by (stac beta_cfun 1);
    by (cont_tacR 1);
    by (Simp_tac 1);
    by (Asm_simp_tac 1);
    (* 2 *)
    by (res_inst_tac[("x","filter_act$(snd ex)")] bexI 1);
    by (stac beta_cfun 1);
    by (cont_tacR 1);
    by (Simp_tac 1);
    by (safe_tac set_cs);
    by (res_inst_tac[("x","ex")] bexI 1);
    by (REPEAT (Asm_simp_tac 1));
    qed"has_trace_def2";

    (* -------------------------------------------------------------------------------- *)

    section "signatures and executions, schedules";

    (* All executions of A have only actions of A. This is only true because of the
       predicate state_trans (part of the predicate IOA): We have no dependent types.
       For executions of parallel automata this assumption is not needed, as in par_def
       this condition is included once more. (see Lemmas 1.1.1c in CompoExecs for example) *)

    Goal
      "!! A. is_trans_of A ==> \
    \ ! s. is_exec_frag A (s,xs) --> Forall (%a. a:act A) (filter_act$xs)";

    by (pair_induct_tac "xs" [is_exec_frag_def,Forall_def,sforall_def] 1);
    (* main case *)
    ren "ss a t" 1;
    by (safe_tac set_cs);
    by (REPEAT (asm_full_simp_tac (simpset() addsimps [is_trans_of_def]) 1));
    qed"execfrag_in_sig";

    Goal
      "!! A.[|  is_trans_of A; x:executions A |] ==> \
    \ Forall (%a. a:act A) (filter_act$(snd x))";

    by (asm_full_simp_tac (simpset() addsimps [executions_def]) 1);
    by (pair_tac "x" 1);
    by (rtac (execfrag_in_sig RS spec RS mp) 1);
    by Auto_tac;
    qed"exec_in_sig";

    Goalw [schedules_def,has_schedule_def]
      "!! A.[|  is_trans_of A; x:schedules A |] ==> \
    \   Forall (%a. a:act A) x";

    by (fast_tac (claset() addSIs [exec_in_sig]) 1);
    qed"scheds_in_sig";

    (*

    is ok but needs ForallQFilterP which has to been proven first (is trivial also)

    Goalw [traces_def,has_trace_def]
      "!! A.[| x:traces A |] ==> \
    \   Forall (%a. a:act A) x";
     by (safe_tac set_cs );
    by (rtac ForallQFilterP 1);
    by (fast_tac (!claset addSIs [ext_is_act]) 1);
    qed"traces_in_sig";
    *)

    (* -------------------------------------------------------------------------------- *)

    section "executions are prefix closed";

    (* only admissible in y, not if done in x !! *)
    Goal "!x s. is_exec_frag A (s,x) & y<<x  --> is_exec_frag A (s,y)";
    by (pair_induct_tac "y" [is_exec_frag_def] 1);
    by (strip_tac 1);
    by (Seq_case_simp_tac "xa" 1);
    by (pair_tac "a" 1);
    by Auto_tac;
    qed"execfrag_prefixclosed";

    bind_thm ("exec_prefixclosed",conjI RS (execfrag_prefixclosed RS spec RS spec RS mp));

    (* second prefix notion for Finite x *)

    Goal "! y s. is_exec_frag A (s,x@@y) --> is_exec_frag A (s,x)";
    by (pair_induct_tac "x" [is_exec_frag_def] 1);
    by (strip_tac 1);
    by (Seq_case_simp_tac "s" 1);
    by (pair_tac "a" 1);
    by Auto_tac;
    qed_spec_mp"exec_prefix2closed";