src/HOLCF/One.thy
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(*  Title:      HOLCF/One.thy
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    ID:         $Id$
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    Author:     Oscar Slotosch
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The unit domain.
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*)
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header {* The unit domain *}
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theory One
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imports Lift
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begin
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types one = "unit lift"
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constdefs
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  ONE :: "one"
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  "ONE \<equiv> Def ()"
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translations
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  "one" <= (type) "unit lift" 
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text {* Exhaustion and Elimination for type @{typ one} *}
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lemma Exh_one: "t = \<bottom> \<or> t = ONE"
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apply (unfold ONE_def)
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apply (induct t)
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apply simp
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apply simp
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done
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lemma oneE: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = ONE \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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apply (rule Exh_one [THEN disjE])
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apply fast
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apply fast
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done
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lemma dist_less_one [simp]: "\<not> ONE \<sqsubseteq> \<bottom>"
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apply (unfold ONE_def)
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apply simp
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done
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lemma dist_eq_one [simp]: "ONE \<noteq> \<bottom>" "\<bottom> \<noteq> ONE"
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apply (unfold ONE_def)
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apply simp_all
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done
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lemma compact_ONE [simp]: "compact ONE"
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by (rule compact_chfin)
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text {* Case analysis function for type @{typ one} *}
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constdefs
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  one_when :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a"
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  "one_when \<equiv> \<Lambda> a. strictify\<cdot>(\<Lambda> _. a)"
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translations
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  "case x of ONE \<Rightarrow> t" == "one_when\<cdot>t\<cdot>x"
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  "\<Lambda> ONE. t" == "one_when\<cdot>t"
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lemma one_when1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>"
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by (simp add: one_when_def)
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lemma one_when2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t"
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by (simp add: one_when_def)
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lemma one_when3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x"
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by (rule_tac p=x in oneE, simp_all)
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end