src/HOL/IMP/Denotation.ML

induct -> lfp_induct

(*  Title:      HOL/IMP/Denotation.ML
    ID:         $Id$
    Author:     Heiko Loetzbeyer & Robert Sandner & Tobias Nipkow, TUM
    Copyright   1994 TUM
*)

(**** mono (Gamma(b,c)) ****)

qed_goalw "Gamma_mono" Denotation.thy [mono_def,Gamma_def]
        "mono (Gamma b c)"
     (fn _ => [Fast_tac 1]);


Goal "C(WHILE b DO c) = C(IF b THEN c;WHILE b DO c ELSE SKIP)";
by (Simp_tac 1);
by (EVERY[stac (Gamma_mono RS lfp_unfold) 1,
          stac (Gamma_def RS meta_eq_to_obj_eq RS fun_cong) 1,
          Simp_tac 1,
          IF_UNSOLVED no_tac]);
qed "C_While_If";


(* Operational Semantics implies Denotational Semantics *)

Goal "<c,s> -c-> t ==> (s,t) : C(c)";

(* start with rule induction *)
by (etac evalc.induct 1);
by Auto_tac;
(* while *)
by (rewtac Gamma_def);
by (stac (rewrite_rule [Gamma_def] (Gamma_mono RS lfp_unfold)) 1);
by (Fast_tac 1);
by (stac (rewrite_rule [Gamma_def] (Gamma_mono RS lfp_unfold)) 1);
by (Fast_tac 1);

qed "com1";

(* Denotational Semantics implies Operational Semantics *)

Goal "!s t. (s,t):C(c) --> <c,s> -c-> t";
by (induct_tac "c" 1);

by (ALLGOALS Full_simp_tac);
by (ALLGOALS (TRY o Fast_tac));

(* while *)
by (strip_tac 1);
by (etac (Gamma_mono RSN (2,lfp_induct)) 1);
by (rewtac Gamma_def);  
by (Fast_tac 1);

qed_spec_mp "com2";


(**** Proof of Equivalence ****)

Goal "(s,t) : C(c)  =  (<c,s> -c-> t)";
by (fast_tac (claset() addEs [com2] addDs [com1]) 1);
qed "denotational_is_natural";