(* Title: HOL/HOLCF/One.thy
Author: Oscar Slotosch
*)
section \<open>The unit domain\<close>
theory One
imports Lift
begin
type_synonym one = "unit lift"
translations
(type) "one" \<leftharpoondown> (type) "unit lift"
definition ONE :: "one"
where "ONE \<equiv> Def ()"
text \<open>Exhaustion and Elimination for type @{typ one}\<close>
lemma Exh_one: "t = \<bottom> \<or> t = ONE"
by (induct t) (simp_all add: ONE_def)
lemma oneE [case_names bottom ONE]: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = ONE \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
by (induct p) (simp_all add: ONE_def)
lemma one_induct [case_names bottom ONE]: "P \<bottom> \<Longrightarrow> P ONE \<Longrightarrow> P x"
by (cases x rule: oneE) simp_all
lemma dist_below_one [simp]: "ONE \<notsqsubseteq> \<bottom>"
by (simp add: ONE_def)
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>"
by (simp add: ONE_def)
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 \<open>Case analysis function for type @{typ one}\<close>
definition one_case :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a"
where "one_case = (\<Lambda> a x. seq\<cdot>x\<cdot>a)"
translations
"case x of XCONST ONE \<Rightarrow> t" \<rightleftharpoons> "CONST one_case\<cdot>t\<cdot>x"
"case x of XCONST ONE :: 'a \<Rightarrow> t" \<rightharpoonup> "CONST one_case\<cdot>t\<cdot>x"
"\<Lambda> (XCONST ONE). t" \<rightleftharpoons> "CONST one_case\<cdot>t"
lemma one_case1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>"
by (simp add: one_case_def)
lemma one_case2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t"
by (simp add: one_case_def)
lemma one_case3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x"
by (induct x rule: one_induct) simp_all
end