Theory One

(*  Title:      HOLCF/One.thy
    ID:         $Id: One.thy,v 1.14 2005/07/07 18:41:12 huffman Exp $
    Author:     Oscar Slotosch

The unit domain.
*)

header {* The unit domain *}

theory One
imports Lift
begin

types one = "unit lift"

constdefs
  ONE :: "one"
  "ONE ≡ Def ()"

translations
  "one" <= (type) "unit lift" 

text {* Exhaustion and Elimination for type @{typ one} *}

lemma Exh_one: "t = ⊥ ∨ t = ONE"
apply (unfold ONE_def)
apply (induct t)
apply simp
apply simp
done

lemma oneE: "[|p = ⊥ ==> Q; p = ONE ==> Q|] ==> Q"
apply (rule Exh_one [THEN disjE])
apply fast
apply fast
done

lemma dist_less_one [simp]: "¬ ONE \<sqsubseteq> ⊥"
apply (unfold ONE_def)
apply simp
done

lemma dist_eq_one [simp]: "ONE ≠ ⊥" "⊥ ≠ ONE"
apply (unfold ONE_def)
apply simp_all
done

end