(*  Title:      HOLCF/One.thy

     Author:     Oscar Slotosch

 *)

 header {* The unit domain *}

 theory One

 imports Lift

 begin

 types one = "unit lift"

 translations

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

 definition

   ONE :: "one"

 where

   "ONE == Def ()"

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

 lemma Exh_one: "t = \<bottom> \<or> t = ONE"

 unfolding ONE_def by (induct t) simp_all

 lemma oneE: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = ONE \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

 unfolding ONE_def by (induct p) simp_all

 lemma one_induct: "\<lbrakk>P \<bottom>; P ONE\<rbrakk> \<Longrightarrow> P x"

 by (cases x rule: oneE) simp_all

 lemma dist_below_one [simp]: "\<not> ONE \<sqsubseteq> \<bottom>"

 unfolding ONE_def by simp

 lemma below_ONE [simp]: "x \<sqsubseteq> ONE"

 by (induct x rule: one_induct) simp_all

 lemma ONE_below_iff [simp]: "ONE \<sqsubseteq> x \<longleftrightarrow> x = ONE"

 by (induct x rule: one_induct) simp_all

 lemma ONE_defined [simp]: "ONE \<noteq> \<bottom>"

 unfolding ONE_def by simp

 lemma one_neq_iffs [simp]:

   "x \<noteq> ONE \<longleftrightarrow> x = \<bottom>"

   "ONE \<noteq> x \<longleftrightarrow> x = \<bottom>"

   "x \<noteq> \<bottom> \<longleftrightarrow> x = ONE"

   "\<bottom> \<noteq> x \<longleftrightarrow> x = ONE"

 by (induct x rule: one_induct) simp_all

 lemma compact_ONE: "compact ONE"

 by (rule compact_chfin)

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

 definition

   one_when :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" where

   "one_when = (\<Lambda> a. strictify\<cdot>(\<Lambda> _. a))"

 translations

   "case x of XCONST ONE \<Rightarrow> t" == "CONST one_when\<cdot>t\<cdot>x"

   "\<Lambda> (XCONST ONE). t" == "CONST one_when\<cdot>t"

 lemma one_when1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>"

 by (simp add: one_when_def)

 lemma one_when2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t"

 by (simp add: one_when_def)

 lemma one_when3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x"

 by (induct x rule: one_induct) simp_all

 end