section {* Example 3.8 *}
theory Ex2
imports LCF
begin
axiomatization
P :: "'a => tr" and
F :: "'b => 'b" and
G :: "'a => 'a" and
H :: "'a => 'b => 'b" and
K :: "('a => 'b => 'b) => ('a => 'b => 'b)"
where
F_strict: "F(UU) = UU" and
K: "K = (%h x y. P(x) => y | F(h(G(x),y)))" and
H: "H = FIX(K)"
declare F_strict [simp] K [simp]
lemma example: "ALL x. F(H(x::'a,y::'b)) = H(x,F(y))"
apply (simplesubst H)
apply (induct "K:: ('a=>'b=>'b) => ('a=>'b=>'b)")
apply simp
apply (simp split: COND_cases_iff)
done
end