(* Title: HOLCF/Dnat.thy
ID: $Id$
Author: Franz Regensburger
Theory for the domain of natural numbers dnat = one ++ dnat
*)
theory Dnat imports HOLCF begin
domain dnat = dzero | dsucc (dpred :: dnat)
definition
iterator :: "dnat -> ('a -> 'a) -> 'a -> 'a" where
"iterator = fix $ (LAM h n f x.
case n of dzero => x
| dsucc $ m => f $ (h $ m $ f $ x))"
text {*
\medskip Expand fixed point properties.
*}
lemma iterator_def2:
"iterator = (LAM n f x. case n of dzero => x | dsucc$m => f$(iterator$m$f$x))"
apply (rule trans)
apply (rule fix_eq2)
apply (rule iterator_def [THEN eq_reflection])
apply (rule beta_cfun)
apply simp
done
text {* \medskip Recursive properties. *}
lemma iterator1: "iterator $ UU $ f $ x = UU"
apply (subst iterator_def2)
apply (simp add: dnat.rews)
done
lemma iterator2: "iterator $ dzero $ f $ x = x"
apply (subst iterator_def2)
apply (simp add: dnat.rews)
done
lemma iterator3: "n ~= UU ==> iterator $ (dsucc $ n) $ f $ x = f $ (iterator $ n $ f $ x)"
apply (rule trans)
apply (subst iterator_def2)
apply (simp add: dnat.rews)
apply (rule refl)
done
lemmas iterator_rews = iterator1 iterator2 iterator3
lemma dnat_flat: "ALL x y::dnat. x<<y --> x=UU | x=y"
apply (rule allI)
apply (induct_tac x rule: dnat.ind)
apply fast
apply (rule allI)
apply (rule_tac x = y in dnat.casedist)
apply simp
apply simp
apply (simp add: dnat.dist_les)
apply (rule allI)
apply (rule_tac x = y in dnat.casedist)
apply (fast intro!: UU_I)
apply (thin_tac "ALL y. d << y --> d = UU | d = y")
apply (simp add: dnat.dist_les)
apply (simp (no_asm_simp) add: dnat.rews dnat.injects dnat.inverts)
apply (drule_tac x="da" in spec)
apply simp
done
end