1(* Title: HOL/HOLCF/One.thy 2 Author: Oscar Slotosch 3*) 4 5section \<open>The unit domain\<close> 6 7theory One 8 imports Lift 9begin 10 11type_synonym one = "unit lift" 12 13translations 14 (type) "one" \<leftharpoondown> (type) "unit lift" 15 16definition ONE :: "one" 17 where "ONE \<equiv> Def ()" 18 19text \<open>Exhaustion and Elimination for type \<^typ>\<open>one\<close>\<close> 20 21lemma Exh_one: "t = \<bottom> \<or> t = ONE" 22 by (induct t) (simp_all add: ONE_def) 23 24lemma oneE [case_names bottom ONE]: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = ONE \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 25 by (induct p) (simp_all add: ONE_def) 26 27lemma one_induct [case_names bottom ONE]: "P \<bottom> \<Longrightarrow> P ONE \<Longrightarrow> P x" 28 by (cases x rule: oneE) simp_all 29 30lemma dist_below_one [simp]: "ONE \<notsqsubseteq> \<bottom>" 31 by (simp add: ONE_def) 32 33lemma below_ONE [simp]: "x \<sqsubseteq> ONE" 34 by (induct x rule: one_induct) simp_all 35 36lemma ONE_below_iff [simp]: "ONE \<sqsubseteq> x \<longleftrightarrow> x = ONE" 37 by (induct x rule: one_induct) simp_all 38 39lemma ONE_defined [simp]: "ONE \<noteq> \<bottom>" 40 by (simp add: ONE_def) 41 42lemma one_neq_iffs [simp]: 43 "x \<noteq> ONE \<longleftrightarrow> x = \<bottom>" 44 "ONE \<noteq> x \<longleftrightarrow> x = \<bottom>" 45 "x \<noteq> \<bottom> \<longleftrightarrow> x = ONE" 46 "\<bottom> \<noteq> x \<longleftrightarrow> x = ONE" 47 by (induct x rule: one_induct) simp_all 48 49lemma compact_ONE: "compact ONE" 50 by (rule compact_chfin) 51 52text \<open>Case analysis function for type \<^typ>\<open>one\<close>\<close> 53 54definition one_case :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" 55 where "one_case = (\<Lambda> a x. seq\<cdot>x\<cdot>a)" 56 57translations 58 "case x of XCONST ONE \<Rightarrow> t" \<rightleftharpoons> "CONST one_case\<cdot>t\<cdot>x" 59 "case x of XCONST ONE :: 'a \<Rightarrow> t" \<rightharpoonup> "CONST one_case\<cdot>t\<cdot>x" 60 "\<Lambda> (XCONST ONE). t" \<rightleftharpoons> "CONST one_case\<cdot>t" 61 62lemma one_case1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>" 63 by (simp add: one_case_def) 64 65lemma one_case2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t" 66 by (simp add: one_case_def) 67 68lemma one_case3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x" 69 by (induct x rule: one_induct) simp_all 70 71end 72