Theory One

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theory One
imports Lift
begin

(*  Title:      HOLCF/One.thy
    ID:         $Id: One.thy,v 1.18 2007/10/21 12:21:48 wenzelm Exp $
    Author:     Oscar Slotosch

The unit domain.
*)

header {* The unit domain *}

theory One
imports Lift
begin

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

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

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

lemma compact_ONE [simp]: "compact ONE"
by (rule compact_chfin)

text {* Case analysis function for type @{typ one} *}

definition
  one_when :: "'a::pcpo -> one -> 'a" where
  "one_when = (Λ a. strictify·(Λ _. a))"

translations
  "case x of CONST ONE => t" == "CONST one_when·t·x"
  "Λ (CONST ONE). t" == "CONST one_when·t"

lemma one_when1 [simp]: "(case ⊥ of ONE => t) = ⊥"
by (simp add: one_when_def)

lemma one_when2 [simp]: "(case ONE of ONE => t) = t"
by (simp add: one_when_def)

lemma one_when3 [simp]: "(case x of ONE => ONE) = x"
by (rule_tac p=x in oneE, simp_all)

end

lemma Exh_one:

  t = UUt = ONE

lemma oneE:

  [| p = UU ==> Q; p = ONE ==> Q |] ==> Q

lemma dist_less_one:

  ¬ ONE << UU

lemma dist_eq_one:

  ONE  UU
  UU  ONE

lemma compact_ONE:

  compact ONE

lemma one_when1:

  (case UU of ONE => t) = UU

lemma one_when2:

  (case ONE of ONE => t) = t

lemma one_when3:

  (case x of ONE => ONE) = x