# File Ex1.ML

    simpset_ref() := LCF_ss;

    Addsimps [P_strict,K];

    val H_unfold = prove_goal (the_context ()) "H = K(H)"
     (fn _ => [stac H 1, rtac (FIX_eq RS sym) 1]);

    bind_thm ("H_unfold", H_unfold);

    val H_strict = prove_goal (the_context ()) "H(UU)=UU"
     (fn _ => [stac H_unfold 1, Simp_tac 1]);

    bind_thm ("H_strict", H_strict);
    Addsimps [H_strict];

    Goal "ALL x. H(FIX(K,x)) = FIX(K,x)";
    by (induct_tac "K" 1);
    by (Simp_tac 1);
    by (simp_tac (simpset() setloop (split_tac [COND_cases_iff])) 1);
    by (strip_tac 1);
    by (stac H_unfold 1);
    by (Asm_simp_tac 1);
    qed "H_idemp_lemma";

    val H_idemp = rewrite_rule [mk_meta_eq (H RS sym)] H_idemp_lemma;
    bind_thm ("H_idemp", H_idemp);