src/LCF/ex.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/LCF/ex.ML	Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,126 @@
+(*  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";