# File Natural.ML

    (*  Title:      HOL/IMPP/Natural.ML
        ID:         $Id: Natural.ML,v 1.5 2005/09/17 18:14:31 wenzelm Exp $
        Author:     David von Oheimb, TUM
        Copyright   1999 TUM
    *)

    AddIs evalc.intros;
    AddIs evaln.intros;

    AddSEs evalc_elim_cases;
    AddSEs evaln_elim_cases;

    (* evaluation of com is deterministic *)
    Goal "<c,s> -c-> t ==> (!u. <c,s> -c-> u --> u=t)";
    by (etac evalc.induct 1);
    by (thin_tac "<?c,s1> -c-> s2" 8);
    (*blast_tac needs Unify.search_bound := 40*)
    by (ALLGOALS (best_tac (claset() addEs [evalc_WHILE_case])));
    qed_spec_mp "com_det";

    Goal "<c,s> -n-> t ==> <c,s> -c-> t";
    by (etac evaln.induct 1);
    by (ALLGOALS (resolve_tac evalc.intros THEN_ALL_NEW atac));
    qed "evaln_evalc";

    Goal "[| Suc n <= m'; (!!m. n <= m ==> P (Suc m)) |] ==> P m'";
    by (cut_facts_tac (premises()) 1);
    by (ftac Suc_le_D 1);
    by (Clarify_tac 1);
    by (eresolve_tac (premises()) 1);
    qed "Suc_le_D_lemma";

    Goal "<c,s> -n-> t ==> !m. n<=m --> <c,s> -m-> t";
    by (etac evaln.induct 1);
    by (ALLGOALS (EVERY'[strip_tac,TRY o etac Suc_le_D_lemma, REPEAT o smp_tac 1]));
    by (ALLGOALS (resolve_tac evaln.intros THEN_ALL_NEW atac));
    qed_spec_mp "evaln_nonstrict";

    Goal "<c,s> -n-> s' ==> <c,s> -Suc n-> s'";
    by (etac evaln_nonstrict 1);
    by Auto_tac;
    qed "evaln_Suc";

    Goal "[| <c1,s1> -n1-> t1;  <c2,s2> -n2-> t2 |] ==> \
    \   ? n. <c1,s1> -n -> t1 & <c2,s2> -n -> t2";
    by (cut_facts_tac [read_instantiate [("m","n1"),("n","n2")] nat_le_linear] 1);
    by (blast_tac (claset() addDs [evaln_nonstrict]) 1);
    qed "evaln_max2";

    Goal "<c,s> -c-> t ==> ? n. <c,s> -n-> t";
    by (etac evalc.induct 1);
    by (ALLGOALS (REPEAT o etac exE));
    by (TRYALL(EVERY'[datac evaln_max2 1, REPEAT o eresolve_tac [exE, conjE]]));
    by (ALLGOALS (rtac exI THEN' resolve_tac evaln.intros THEN_ALL_NEW atac));
    qed "evalc_evaln";

    Goal "<c,s> -c-> t = (? n. <c,s> -n-> t)";
    by (fast_tac (claset() addEs [evalc_evaln, evaln_evalc]) 1);
    qed "eval_eq";