(* Title: LCF/ex.ML ID: $Id$ Author: Tobias Nipkow Copyright 1991 University of Cambridge Some examples from Lawrence Paulson's book Logic and Computation. *) LCF_build_completed; (*Cause examples to fail if LCF did*) proof_timing := true; (*** Section 10.4 ***) val ex_thy = extend_theory thy "Ex 10.4" ([], [], [], [], [(["P"], "'a => tr"), (["G","H"], "'a => 'a"), (["K"], "('a => 'a) => ('a => 'a)") ], None) [ ("P_strict", "P(UU) = UU"), ("K", "K = (%h x. P(x) => x | h(h(G(x))))"), ("H", "H = FIX(K)") ]; val ax = get_axiom ex_thy; val P_strict = ax"P_strict"; val K = ax"K"; val H = ax"H"; val ex_ss = LCF_ss addsimps [P_strict,K]; val H_unfold = prove_goal ex_thy "H = K(H)" (fn _ => [stac H 1, rtac (FIX_eq RS sym) 1]); val H_strict = prove_goal ex_thy "H(UU)=UU" (fn _ => [stac H_unfold 1, simp_tac ex_ss 1]); val ex_ss = ex_ss addsimps [H_strict]; goal ex_thy "ALL x. H(FIX(K,x)) = FIX(K,x)"; by(induct_tac "K" 1); by(simp_tac ex_ss 1); by(simp_tac (ex_ss setloop (split_tac [COND_cases_iff])) 1); by(strip_tac 1); by(stac H_unfold 1); by(asm_simp_tac ex_ss 1); val H_idemp_lemma = topthm(); val H_idemp = rewrite_rule [mk_meta_eq (H RS sym)] H_idemp_lemma; (*** Example 3.8 ***) val ex_thy = extend_theory thy "Ex 3.8" ([], [], [], [], [(["P"], "'a => tr"), (["F","G"], "'a => 'a"), (["H"], "'a => 'b => 'b"), (["K"], "('a => 'b => 'b) => ('a => 'b => 'b)") ], None) [ ("F_strict", "F(UU) = UU"), ("K", "K = (%h x y. P(x) => y | F(h(G(x),y)))"), ("H", "H = FIX(K)") ]; val ax = get_axiom ex_thy; val F_strict = ax"F_strict"; val K = ax"K"; val H = ax"H"; val ex_ss = LCF_ss addsimps [F_strict,K]; goal ex_thy "ALL x. F(H(x::'a,y::'b)) = H(x,F(y))"; by(stac H 1); by(induct_tac "K::('a=>'b=>'b)=>('a=>'b=>'b)" 1); by(simp_tac ex_ss 1); by(asm_simp_tac (ex_ss setloop (split_tac [COND_cases_iff])) 1); result(); (*** Addition with fixpoint of successor ***) val ex_thy = extend_theory thy "fix ex" ([], [], [], [], [(["s"], "'a => 'a"), (["p"], "'a => 'a => 'a") ], None) [ ("p_strict", "p(UU) = UU"), ("p_s", "p(s(x),y) = s(p(x,y))") ]; val ax = get_axiom ex_thy; val p_strict = ax"p_strict"; val p_s = ax"p_s"; val ex_ss = LCF_ss addsimps [p_strict,p_s]; goal ex_thy "p(FIX(s),y) = FIX(s)"; by(induct_tac "s" 1); by(simp_tac ex_ss 1); by(simp_tac ex_ss 1); result(); (*** Prefixpoints ***) val asms = goal thy "[| f(p) << p; !!q. f(q) << q ==> p << q |] ==> FIX(f)=p"; by(rewtac eq_def); by (rtac conjI 1); by(induct_tac "f" 1); by (rtac minimal 1); by(strip_tac 1); by (rtac less_trans 1); by (resolve_tac asms 2); by (etac less_ap_term 1); by (resolve_tac asms 1); by (rtac (FIX_eq RS eq_imp_less1) 1); result(); maketest"END: file for LCF examples";