src/HOL/IMP/Expr.ML

Converted proofs to use default clasets.

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

Arithmetic expressions and Boolean expressions.
Not used in the rest of the language, but included for completeness.
*)

open Expr;

val evala_elim_cases = map (evala.mk_cases aexp.simps)
   ["(N(n),sigma) -a-> i", "(X(x),sigma) -a-> i",
    "(Op1 f e,sigma) -a-> i", "(Op2 f a1 a2,sigma)  -a-> i"
   ];

val evalb_elim_cases = map (evalb.mk_cases bexp.simps)
   ["(true,sigma) -b-> x", "(false,sigma) -b-> x",
    "(ROp f a0 a1,sigma) -b-> x", "(noti(b),sigma) -b-> x",
    "(b0 andi b1,sigma) -b-> x", "(b0 ori b1,sigma) -b-> x"
   ];

val evalb_simps = map (fn s => prove_goal Expr.thy s
    (fn _ => [fast_tac (HOL_cs addSIs evalb.intrs addSEs evalb_elim_cases) 1]))
  ["((true,sigma) -b-> w) = (w=True)",
   "((false,sigma) -b-> w) = (w=False)",
   "((ROp f a0 a1,sigma) -b-> w) = \
\   (? m. (a0,sigma) -a-> m & (? n. (a1,sigma) -a-> n & w = f m n))",
   "((noti(b),sigma) -b-> w) = (? x. (b,sigma) -b-> x & w = (~x))",
   "((b0 andi b1,sigma) -b-> w) = \
\   (? x. (b0,sigma) -b-> x & (? y. (b1,sigma) -b-> y & w = (x&y)))",
   "((b0 ori b1,sigma) -b-> w) = \
\   (? x. (b0,sigma) -b-> x & (? y. (b1,sigma) -b-> y & w = (x|y)))"];

goal Expr.thy "!n. ((a,s) -a-> n) = (n = A a s)";
by (aexp.induct_tac "a" 1);                               (* struct. ind. *)
by (ALLGOALS Simp_tac);                                   (* rewr. Den.   *)
by (TRYALL (fast_tac (!claset addSIs (evala.intrs@prems)
                             addSEs evala_elim_cases)));
qed_spec_mp "aexp_iff";

goal Expr.thy "!w. ((b,s) -b-> w) = (w = B b s)";
by (bexp.induct_tac "b" 1);
by (ALLGOALS(asm_simp_tac (!simpset addcongs [conj_cong]
                                    addsimps (aexp_iff::evalb_simps))));
qed_spec_mp "bexp_iff";