author | paulson <lp15@cam.ac.uk> |
Sun, 14 Aug 2022 23:51:47 +0100 | |
changeset 75864 | 3842556b757c |
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permissions | -rw-r--r-- |
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(* Title: HOL/HOLCF/One.thy |
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Author: Oscar Slotosch |
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*) |
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section \<open>The unit domain\<close> |
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theory One |
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imports Lift |
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begin |
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type_synonym one = "unit lift" |
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translations |
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(type) "one" \<leftharpoondown> (type) "unit lift" |
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definition ONE :: "one" |
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where "ONE \<equiv> Def ()" |
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text \<open>Exhaustion and Elimination for type \<^typ>\<open>one\<close>\<close> |
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lemma Exh_one: "t = \<bottom> \<or> t = ONE" |
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by (induct t) (simp_all add: ONE_def) |
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lemma oneE [case_names bottom ONE]: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = ONE \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" |
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by (induct p) (simp_all add: ONE_def) |
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lemma one_induct [case_names bottom ONE]: "P \<bottom> \<Longrightarrow> P ONE \<Longrightarrow> P x" |
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by (cases x rule: oneE) simp_all |
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lemma dist_below_one [simp]: "ONE \<notsqsubseteq> \<bottom>" |
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by (simp add: ONE_def) |
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lemma below_ONE [simp]: "x \<sqsubseteq> ONE" |
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by (induct x rule: one_induct) simp_all |
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lemma ONE_below_iff [simp]: "ONE \<sqsubseteq> x \<longleftrightarrow> x = ONE" |
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by (induct x rule: one_induct) simp_all |
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lemma ONE_defined [simp]: "ONE \<noteq> \<bottom>" |
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by (simp add: ONE_def) |
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lemma one_neq_iffs [simp]: |
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"x \<noteq> ONE \<longleftrightarrow> x = \<bottom>" |
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"ONE \<noteq> x \<longleftrightarrow> x = \<bottom>" |
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"x \<noteq> \<bottom> \<longleftrightarrow> x = ONE" |
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"\<bottom> \<noteq> x \<longleftrightarrow> x = ONE" |
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by (induct x rule: one_induct) simp_all |
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lemma compact_ONE: "compact ONE" |
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by (rule compact_chfin) |
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text \<open>Case analysis function for type \<^typ>\<open>one\<close>\<close> |
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definition one_case :: "'a::pcpo \<rightarrow> one \<rightarrow> 'a" |
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where "one_case = (\<Lambda> a x. seq\<cdot>x\<cdot>a)" |
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translations |
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"case x of XCONST ONE \<Rightarrow> t" \<rightleftharpoons> "CONST one_case\<cdot>t\<cdot>x" |
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"case x of XCONST ONE :: 'a \<Rightarrow> t" \<rightharpoonup> "CONST one_case\<cdot>t\<cdot>x" |
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"\<Lambda> (XCONST ONE). t" \<rightleftharpoons> "CONST one_case\<cdot>t" |
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lemma one_case1 [simp]: "(case \<bottom> of ONE \<Rightarrow> t) = \<bottom>" |
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by (simp add: one_case_def) |
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lemma one_case2 [simp]: "(case ONE of ONE \<Rightarrow> t) = t" |
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by (simp add: one_case_def) |
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lemma one_case3 [simp]: "(case x of ONE \<Rightarrow> ONE) = x" |
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by (induct x rule: one_induct) simp_all |
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end |