(* Title: LCF/ex/ex.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Some examples from Lawrence Paulson's book Logic and Computation.
*)
(*** Section 10.4 ***)
val ex_thy =
thy
|> Theory.add_consts
[("P", "'a => tr", NoSyn),
("G", "'a => 'a", NoSyn),
("H", "'a => 'a", NoSyn),
("K", "('a => 'a) => ('a => 'a)", NoSyn)]
|> PureThy.add_store_axioms
[("P_strict", "P(UU) = UU"),
("K", "K = (%h x. P(x) => x | h(h(G(x))))"),
("H", "H = FIX(K)")]
|> Theory.add_name "Ex 10.4";
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 =
thy
|> Theory.add_consts
[("P", "'a => tr", NoSyn),
("F", "'a => 'a", NoSyn),
("G", "'a => 'a", NoSyn),
("H", "'a => 'b => 'b", NoSyn),
("K", "('a => 'b => 'b) => ('a => 'b => 'b)", NoSyn)]
|> PureThy.add_store_axioms
[("F_strict", "F(UU) = UU"),
("K", "K = (%h x y. P(x) => y | F(h(G(x),y)))"),
("H", "H = FIX(K)")]
|> Theory.add_name "Ex 3.8";
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 =
thy
|> Theory.add_consts
[("s", "'a => 'a", NoSyn),
("p", "'a => 'a => 'a", NoSyn)]
|> PureThy.add_store_axioms
[("p_strict", "p(UU) = UU"),
("p_s", "p(s(x),y) = s(p(x,y))")]
|> Theory.add_name "fix ex";
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();