Theory Ex2

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theory Ex2
imports LCF
begin

(* $Id: Ex2.thy,v 1.3 2006/06/01 19:14:06 wenzelm Exp $ *)

header {* Example 3.8 *}

theory Ex2
imports LCF
begin

consts
  P     :: "'a => tr"
  F     :: "'a => 'a"
  G     :: "'a => 'a"
  H     :: "'a => 'b => 'b"
  K     :: "('a => 'b => 'b) => ('a => 'b => 'b)"

axioms
  F_strict:     "F(UU) = UU"
  K:            "K = (%h x y. P(x) => y | F(h(G(x),y)))"
  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 (tactic {* induct_tac "K:: ('a=>'b=>'b) => ('a=>'b=>'b)" 1 *})
  apply (simp (no_asm))
  apply (simp (no_asm_simp) split: COND_cases_iff)
  done

end

lemma example:

  x. F(H(x, y)) = H(x, F(y))