(* $Id$ *)
header {* Section 10.4 *}
theory Ex1
imports LCF
begin
consts
P :: "'a => tr"
G :: "'a => 'a"
H :: "'a => 'a"
K :: "('a => 'a) => ('a => 'a)"
axioms
P_strict: "P(UU) = UU"
K: "K = (%h x. P(x) => x | h(h(G(x))))"
H: "H = FIX(K)"
declare P_strict [simp] K [simp]
lemma H_unfold: "H = K(H)"
apply (simplesubst H)
apply (rule FIX_eq [symmetric])
done
lemma H_strict [simp]: "H(UU)=UU"
apply (simplesubst H_unfold)
apply simp
done
lemma H_idemp_lemma: "ALL x. H(FIX(K,x)) = FIX(K,x)"
apply (tactic {* induct_tac "K" 1 *})
apply (simp (no_asm))
apply (simp (no_asm) split: COND_cases_iff)
apply (intro strip)
apply (subst H_unfold)
apply (simp (no_asm_simp))
done
lemma H_idemp: "ALL x. H(H(x)) = H(x)"
apply (rule H_idemp_lemma [folded H])
done
end