Theory Ex2

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 = (λh x y. P(x) ⇒ y | F(h(G(x),y)))" and
  H:            "H = FIX(K)"

declare F_strict [simp] K [simp]

lemma example: "∀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