author | haftmann |
Mon, 18 Jul 2011 21:49:39 +0200 | |
changeset 43900 | 7162691e740b |
parent 43899 | 60ef6abb2f92 |
child 43901 | 3ab6c30d256d |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *) |
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header {* Complete lattices, with special focus on sets *} |
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theory Complete_Lattice |
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imports Set |
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begin |
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notation |
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less_eq (infix "\<sqsubseteq>" 50) and |
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less (infix "\<sqsubset>" 50) and |
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inf (infixl "\<sqinter>" 70) and |
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sup (infixl "\<squnion>" 65) and |
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top ("\<top>") and |
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bot ("\<bottom>") |
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subsection {* Syntactic infimum and supremum operations *} |
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class Inf = |
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fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) |
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class Sup = |
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fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) |
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subsection {* Abstract complete lattices *} |
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class complete_lattice = bounded_lattice + Inf + Sup + |
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assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" |
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and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" |
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assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" |
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and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" |
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begin |
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lemma dual_complete_lattice: |
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"class.complete_lattice Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>" |
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by (auto intro!: class.complete_lattice.intro dual_bounded_lattice) |
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(unfold_locales, (fact bot_least top_greatest |
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Sup_upper Sup_least Inf_lower Inf_greatest)+) |
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}" |
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
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lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}" |
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
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lemma Inf_empty [simp]: |
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"\<Sqinter>{} = \<top>" |
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by (auto intro: antisym Inf_greatest) |
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lemma Sup_empty [simp]: |
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"\<Squnion>{} = \<bottom>" |
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by (auto intro: antisym Sup_least) |
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lemma Inf_UNIV [simp]: |
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"\<Sqinter>UNIV = \<bottom>" |
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by (simp add: Sup_Inf Sup_empty [symmetric]) |
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lemma Sup_UNIV [simp]: |
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"\<Squnion>UNIV = \<top>" |
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by (simp add: Inf_Sup Inf_empty [symmetric]) |
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lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" |
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by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) |
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lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" |
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by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) |
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lemma Inf_singleton [simp]: |
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"\<Sqinter>{a} = a" |
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by (auto intro: antisym Inf_lower Inf_greatest) |
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lemma Sup_singleton [simp]: |
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"\<Squnion>{a} = a" |
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by (auto intro: antisym Sup_upper Sup_least) |
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lemma Inf_binary: |
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"\<Sqinter>{a, b} = a \<sqinter> b" |
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by (simp add: Inf_insert) |
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lemma Sup_binary: |
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"\<Squnion>{a, b} = a \<squnion> b" |
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by (simp add: Sup_insert) |
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lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)" |
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by (auto intro: Inf_greatest dest: Inf_lower) |
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lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)" |
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by (auto intro: Sup_least dest: Sup_upper) |
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lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B" |
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by (auto intro: Inf_greatest Inf_lower) |
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lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B" |
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by (auto intro: Sup_least Sup_upper) |
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lemma Inf_mono: |
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assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b" |
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shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B" |
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proof (rule Inf_greatest) |
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fix b assume "b \<in> B" |
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with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast |
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from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower) |
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with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto |
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qed |
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lemma Sup_mono: |
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assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b" |
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shows "\<Squnion>A \<sqsubseteq> \<Squnion>B" |
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proof (rule Sup_least) |
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fix a assume "a \<in> A" |
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with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast |
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from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper) |
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with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto |
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qed |
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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A" |
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using Sup_upper [of u A] by auto |
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v" |
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using Inf_lower [of u A] by auto |
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lemma Inf_less_eq: |
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assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u" |
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and "A \<noteq> {}" |
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shows "\<Sqinter>A \<sqsubseteq> u" |
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proof - |
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from `A \<noteq> {}` obtain v where "v \<in> A" by blast |
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moreover with assms have "v \<sqsubseteq> u" by blast |
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ultimately show ?thesis by (rule Inf_lower2) |
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qed |
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lemma less_eq_Sup: |
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assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v" |
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and "A \<noteq> {}" |
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shows "u \<sqsubseteq> \<Squnion>A" |
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proof - |
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from `A \<noteq> {}` obtain v where "v \<in> A" by blast |
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moreover with assms have "u \<sqsubseteq> v" by blast |
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ultimately show ?thesis by (rule Sup_upper2) |
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qed |
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lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)" |
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by (auto intro: Inf_greatest Inf_lower) |
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lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B " |
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by (auto intro: Sup_least Sup_upper) |
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lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" |
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by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) |
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lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B" |
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by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2) |
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lemma Inf_top_conv [no_atp]: |
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"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
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"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
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proof - |
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show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
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proof |
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assume "\<forall>x\<in>A. x = \<top>" |
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then have "A = {} \<or> A = {\<top>}" by auto |
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then show "\<Sqinter>A = \<top>" by auto |
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next |
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assume "\<Sqinter>A = \<top>" |
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show "\<forall>x\<in>A. x = \<top>" |
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proof (rule ccontr) |
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assume "\<not> (\<forall>x\<in>A. x = \<top>)" |
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then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast |
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then obtain B where "A = insert x B" by blast |
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with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert) |
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qed |
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qed |
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then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto |
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qed |
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lemma Sup_bot_conv [no_atp]: |
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"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P) |
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"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q) |
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proof - |
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interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom> |
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by (fact dual_complete_lattice) |
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from dual.Inf_top_conv show ?P and ?Q by simp_all |
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qed |
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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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INF_def: "INFI A f = \<Sqinter> (f ` A)" |
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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SUP_def: "SUPR A f = \<Squnion> (f ` A)" |
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text {* |
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Note: must use names @{const INFI} and @{const SUPR} here instead of |
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@{text INF} and @{text SUP} to allow the following syntax coexist |
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with the plain constant names. |
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*} |
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end |
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syntax |
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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10) |
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) |
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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) |
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) |
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syntax (xsymbols) |
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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) |
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) |
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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) |
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) |
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translations |
41082 | 213 |
"INF x y. B" == "INF x. INF y. B" |
214 |
"INF x. B" == "CONST INFI CONST UNIV (%x. B)" |
|
215 |
"INF x. B" == "INF x:CONST UNIV. B" |
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216 |
"INF x:A. B" == "CONST INFI A (%x. B)" |
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"SUP x y. B" == "SUP x. SUP y. B" |
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"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" |
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"SUP x. B" == "SUP x:CONST UNIV. B" |
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"SUP x:A. B" == "CONST SUPR A (%x. B)" |
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|
35115 | 222 |
print_translation {* |
42284 | 223 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, |
224 |
Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] |
|
35115 | 225 |
*} -- {* to avoid eta-contraction of body *} |
11979 | 226 |
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context complete_lattice |
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begin |
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229 |
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lemma INF_empty: "(\<Sqinter>x\<in>{}. f x) = \<top>" |
43872 | 231 |
by (simp add: INF_def) |
41971 | 232 |
|
43870 | 233 |
lemma SUP_empty: "(\<Squnion>x\<in>{}. f x) = \<bottom>" |
43872 | 234 |
by (simp add: SUP_def) |
43870 | 235 |
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lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f" |
43872 | 237 |
by (simp add: INF_def Inf_insert) |
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238 |
|
43870 | 239 |
lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f" |
43872 | 240 |
by (simp add: SUP_def Sup_insert) |
43870 | 241 |
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lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i" |
43872 | 243 |
by (auto simp add: INF_def intro: Inf_lower) |
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244 |
|
43872 | 245 |
lemma le_SUP_I: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)" |
246 |
by (auto simp add: SUP_def intro: Sup_upper) |
|
43870 | 247 |
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lemma INF_leI2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u" |
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using INF_leI [of i A f] by auto |
41971 | 250 |
|
43872 | 251 |
lemma le_SUP_I2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)" |
252 |
using le_SUP_I [of i A f] by auto |
|
43870 | 253 |
|
43872 | 254 |
lemma le_INF_I: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)" |
255 |
by (auto simp add: INF_def intro: Inf_greatest) |
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256 |
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lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u" |
43872 | 258 |
by (auto simp add: SUP_def intro: Sup_least) |
43870 | 259 |
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lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> f i)" |
43872 | 261 |
by (auto simp add: INF_def le_Inf_iff) |
35629 | 262 |
|
43870 | 263 |
lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. f i \<sqsubseteq> u)" |
43872 | 264 |
by (auto simp add: SUP_def Sup_le_iff) |
43870 | 265 |
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lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f" |
43872 | 267 |
by (auto intro: antisym INF_leI le_INF_I) |
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268 |
|
43870 | 269 |
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f" |
43872 | 270 |
by (auto intro: antisym SUP_leI le_SUP_I) |
43870 | 271 |
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lemma INF_top: "(\<Sqinter>x\<in>A. \<top>) = \<top>" |
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by (cases "A = {}") (simp_all add: INF_empty) |
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lemma SUP_bot: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>" |
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by (cases "A = {}") (simp_all add: SUP_empty) |
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277 |
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lemma INF_cong: |
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279 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)" |
43872 | 280 |
by (simp add: INF_def image_def) |
38705 | 281 |
|
43870 | 282 |
lemma SUP_cong: |
283 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)" |
|
43872 | 284 |
by (simp add: SUP_def image_def) |
43870 | 285 |
|
38705 | 286 |
lemma INF_mono: |
43753 | 287 |
"(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)" |
43872 | 288 |
by (force intro!: Inf_mono simp: INF_def) |
38705 | 289 |
|
43870 | 290 |
lemma SUP_mono: |
291 |
"(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)" |
|
43872 | 292 |
by (force intro!: Sup_mono simp: SUP_def) |
43870 | 293 |
|
43899 | 294 |
lemma INF_superset_mono: |
295 |
"B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f" |
|
296 |
by (rule INF_mono) auto |
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lemma SUP_subset_mono: |
43870 | 299 |
"A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f" |
43899 | 300 |
by (rule SUP_mono) auto |
43870 | 301 |
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lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)" |
43872 | 303 |
by (iprover intro: INF_leI le_INF_I order_trans antisym) |
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|
43870 | 305 |
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)" |
43872 | 306 |
by (iprover intro: SUP_leI le_SUP_I order_trans antisym) |
43870 | 307 |
|
43871 | 308 |
lemma INF_absorb: |
43868 | 309 |
assumes "k \<in> I" |
310 |
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)" |
|
311 |
proof - |
|
312 |
from assms obtain J where "I = insert k J" by blast |
|
313 |
then show ?thesis by (simp add: INF_insert) |
|
314 |
qed |
|
315 |
||
43871 | 316 |
lemma SUP_absorb: |
317 |
assumes "k \<in> I" |
|
318 |
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)" |
|
319 |
proof - |
|
320 |
from assms obtain J where "I = insert k J" by blast |
|
321 |
then show ?thesis by (simp add: SUP_insert) |
|
322 |
qed |
|
323 |
||
324 |
lemma INF_union: |
|
43868 | 325 |
"(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)" |
43872 | 326 |
by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INF_I INF_leI) |
43868 | 327 |
|
43871 | 328 |
lemma SUP_union: |
329 |
"(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)" |
|
43872 | 330 |
by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_leI le_SUP_I) |
43871 | 331 |
|
332 |
lemma INF_constant: |
|
43868 | 333 |
"(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)" |
334 |
by (simp add: INF_empty) |
|
335 |
||
43871 | 336 |
lemma SUP_constant: |
337 |
"(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)" |
|
338 |
by (simp add: SUP_empty) |
|
339 |
||
340 |
lemma INF_eq: |
|
43868 | 341 |
"(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})" |
43872 | 342 |
by (simp add: INF_def image_def) |
43868 | 343 |
|
43871 | 344 |
lemma SUP_eq: |
345 |
"(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})" |
|
43872 | 346 |
by (simp add: SUP_def image_def) |
43871 | 347 |
|
348 |
lemma INF_top_conv: |
|
43868 | 349 |
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
350 |
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
|
43872 | 351 |
by (auto simp add: INF_def Inf_top_conv) |
43868 | 352 |
|
43871 | 353 |
lemma SUP_bot_conv: |
354 |
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
|
355 |
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
|
43872 | 356 |
by (auto simp add: SUP_def Sup_bot_conv) |
43868 | 357 |
|
43873 | 358 |
lemma INF_UNIV_range: |
43871 | 359 |
"(\<Sqinter>x. f x) = \<Sqinter>range f" |
43872 | 360 |
by (fact INF_def) |
43871 | 361 |
|
43873 | 362 |
lemma SUP_UNIV_range: |
43871 | 363 |
"(\<Squnion>x. f x) = \<Squnion>range f" |
43872 | 364 |
by (fact SUP_def) |
43871 | 365 |
|
43873 | 366 |
lemma INF_UNIV_bool_expand: |
43868 | 367 |
"(\<Sqinter>b. A b) = A True \<sqinter> A False" |
368 |
by (simp add: UNIV_bool INF_empty INF_insert inf_commute) |
|
369 |
||
43873 | 370 |
lemma SUP_UNIV_bool_expand: |
43871 | 371 |
"(\<Squnion>b. A b) = A True \<squnion> A False" |
372 |
by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute) |
|
373 |
||
43899 | 374 |
lemma INF_mono': |
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375 |
"B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)" |
43868 | 376 |
-- {* The last inclusion is POSITIVE! *} |
43899 | 377 |
by (rule INF_mono) auto |
43868 | 378 |
|
43899 | 379 |
lemma SUP_mono': |
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380 |
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)" |
43871 | 381 |
-- {* The last inclusion is POSITIVE! *} |
382 |
by (blast intro: SUP_mono dest: subsetD) |
|
383 |
||
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end |
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41082 | 386 |
lemma Inf_less_iff: |
387 |
fixes a :: "'a\<Colon>{complete_lattice,linorder}" |
|
43753 | 388 |
shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)" |
43754 | 389 |
unfolding not_le [symmetric] le_Inf_iff by auto |
41082 | 390 |
|
43871 | 391 |
lemma less_Sup_iff: |
392 |
fixes a :: "'a\<Colon>{complete_lattice,linorder}" |
|
393 |
shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)" |
|
394 |
unfolding not_le [symmetric] Sup_le_iff by auto |
|
395 |
||
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396 |
lemma INF_less_iff: |
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397 |
fixes a :: "'a::{complete_lattice,linorder}" |
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398 |
shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)" |
43872 | 399 |
unfolding INF_def Inf_less_iff by auto |
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400 |
|
40872 | 401 |
lemma less_SUP_iff: |
402 |
fixes a :: "'a::{complete_lattice,linorder}" |
|
43753 | 403 |
shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)" |
43872 | 404 |
unfolding SUP_def less_Sup_iff by auto |
40872 | 405 |
|
43873 | 406 |
class complete_boolean_algebra = boolean_algebra + complete_lattice |
407 |
begin |
|
408 |
||
409 |
lemma uminus_Inf: |
|
410 |
"- (\<Sqinter>A) = \<Squnion>(uminus ` A)" |
|
411 |
proof (rule antisym) |
|
412 |
show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)" |
|
413 |
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp |
|
414 |
show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A" |
|
415 |
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto |
|
416 |
qed |
|
417 |
||
418 |
lemma uminus_Sup: |
|
419 |
"- (\<Squnion>A) = \<Sqinter>(uminus ` A)" |
|
420 |
proof - |
|
421 |
have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf) |
|
422 |
then show ?thesis by simp |
|
423 |
qed |
|
424 |
||
425 |
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)" |
|
426 |
by (simp add: INF_def SUP_def uminus_Inf image_image) |
|
427 |
||
428 |
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)" |
|
429 |
by (simp add: INF_def SUP_def uminus_Sup image_image) |
|
430 |
||
431 |
end |
|
432 |
||
433 |
||
32139 | 434 |
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *} |
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|
43873 | 436 |
instantiation bool :: complete_boolean_algebra |
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437 |
begin |
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438 |
|
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definition |
41080 | 440 |
"\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" |
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441 |
|
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442 |
definition |
41080 | 443 |
"\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" |
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444 |
|
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changeset
|
445 |
instance proof |
43852 | 446 |
qed (auto simp add: Inf_bool_def Sup_bool_def) |
32077
3698947146b2
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haftmann
parents:
32064
diff
changeset
|
447 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
448 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
449 |
|
43873 | 450 |
lemma INF_bool_eq [simp]: |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
451 |
"INFI = Ball" |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
452 |
proof (rule ext)+ |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
453 |
fix A :: "'a set" |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
454 |
fix P :: "'a \<Rightarrow> bool" |
43753 | 455 |
show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)" |
43872 | 456 |
by (auto simp add: Ball_def INF_def Inf_bool_def) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
457 |
qed |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
458 |
|
43873 | 459 |
lemma SUP_bool_eq [simp]: |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
460 |
"SUPR = Bex" |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
461 |
proof (rule ext)+ |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
462 |
fix A :: "'a set" |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
463 |
fix P :: "'a \<Rightarrow> bool" |
43753 | 464 |
show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)" |
43872 | 465 |
by (auto simp add: Bex_def SUP_def Sup_bool_def) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
466 |
qed |
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
467 |
|
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
468 |
instantiation "fun" :: (type, complete_lattice) complete_lattice |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
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changeset
|
469 |
begin |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
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changeset
|
470 |
|
3698947146b2
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changeset
|
471 |
definition |
41080 | 472 |
"\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" |
473 |
||
474 |
lemma Inf_apply: |
|
475 |
"(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}" |
|
476 |
by (simp add: Inf_fun_def) |
|
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
477 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
478 |
definition |
41080 | 479 |
"\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" |
480 |
||
481 |
lemma Sup_apply: |
|
482 |
"(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}" |
|
483 |
by (simp add: Sup_fun_def) |
|
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
484 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
485 |
instance proof |
41080 | 486 |
qed (auto simp add: le_fun_def Inf_apply Sup_apply |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
487 |
intro: Inf_lower Sup_upper Inf_greatest Sup_least) |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
488 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
489 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
490 |
|
43873 | 491 |
lemma INF_apply: |
41080 | 492 |
"(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" |
43872 | 493 |
by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply) |
38705 | 494 |
|
43873 | 495 |
lemma SUP_apply: |
41080 | 496 |
"(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" |
43872 | 497 |
by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
498 |
|
43873 | 499 |
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra .. |
500 |
||
32077
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haftmann
parents:
32064
diff
changeset
|
501 |
|
41082 | 502 |
subsection {* Inter *} |
503 |
||
504 |
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where |
|
505 |
"Inter S \<equiv> \<Sqinter>S" |
|
506 |
||
507 |
notation (xsymbols) |
|
508 |
Inter ("\<Inter>_" [90] 90) |
|
509 |
||
510 |
lemma Inter_eq: |
|
511 |
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" |
|
512 |
proof (rule set_eqI) |
|
513 |
fix x |
|
514 |
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" |
|
515 |
by auto |
|
516 |
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" |
|
517 |
by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def) |
|
518 |
qed |
|
519 |
||
43741 | 520 |
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" |
41082 | 521 |
by (unfold Inter_eq) blast |
522 |
||
43741 | 523 |
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" |
41082 | 524 |
by (simp add: Inter_eq) |
525 |
||
526 |
text {* |
|
527 |
\medskip A ``destruct'' rule -- every @{term X} in @{term C} |
|
43741 | 528 |
contains @{term A} as an element, but @{prop "A \<in> X"} can hold when |
529 |
@{prop "X \<in> C"} does not! This rule is analogous to @{text spec}. |
|
41082 | 530 |
*} |
531 |
||
43741 | 532 |
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" |
41082 | 533 |
by auto |
534 |
||
43741 | 535 |
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" |
41082 | 536 |
-- {* ``Classical'' elimination rule -- does not require proving |
43741 | 537 |
@{prop "X \<in> C"}. *} |
41082 | 538 |
by (unfold Inter_eq) blast |
539 |
||
43741 | 540 |
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 541 |
by (fact Inf_lower) |
542 |
||
41082 | 543 |
lemma Inter_subset: |
43755 | 544 |
"(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 545 |
by (fact Inf_less_eq) |
41082 | 546 |
|
43755 | 547 |
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A" |
43740 | 548 |
by (fact Inf_greatest) |
41082 | 549 |
|
550 |
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}" |
|
43739 | 551 |
by (fact Inf_binary [symmetric]) |
41082 | 552 |
|
553 |
lemma Inter_empty [simp]: "\<Inter>{} = UNIV" |
|
554 |
by (fact Inf_empty) |
|
555 |
||
556 |
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}" |
|
43739 | 557 |
by (fact Inf_UNIV) |
41082 | 558 |
|
559 |
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B" |
|
43739 | 560 |
by (fact Inf_insert) |
41082 | 561 |
|
562 |
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" |
|
43899 | 563 |
by (fact less_eq_Inf_inter) |
41082 | 564 |
|
565 |
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" |
|
43756 | 566 |
by (fact Inf_union_distrib) |
567 |
||
43868 | 568 |
lemma Inter_UNIV_conv [simp, no_atp]: |
43741 | 569 |
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
570 |
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
|
43801 | 571 |
by (fact Inf_top_conv)+ |
41082 | 572 |
|
43741 | 573 |
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" |
43899 | 574 |
by (fact Inf_superset_mono) |
41082 | 575 |
|
576 |
||
577 |
subsection {* Intersections of families *} |
|
578 |
||
579 |
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where |
|
580 |
"INTER \<equiv> INFI" |
|
581 |
||
43872 | 582 |
text {* |
583 |
Note: must use name @{const INTER} here instead of @{text INT} |
|
584 |
to allow the following syntax coexist with the plain constant name. |
|
585 |
*} |
|
586 |
||
41082 | 587 |
syntax |
588 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) |
|
589 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) |
|
590 |
||
591 |
syntax (xsymbols) |
|
592 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) |
|
593 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) |
|
594 |
||
595 |
syntax (latex output) |
|
596 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
|
597 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) |
|
598 |
||
599 |
translations |
|
600 |
"INT x y. B" == "INT x. INT y. B" |
|
601 |
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" |
|
602 |
"INT x. B" == "INT x:CONST UNIV. B" |
|
603 |
"INT x:A. B" == "CONST INTER A (%x. B)" |
|
604 |
||
605 |
print_translation {* |
|
42284 | 606 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] |
41082 | 607 |
*} -- {* to avoid eta-contraction of body *} |
608 |
||
609 |
lemma INTER_eq_Inter_image: |
|
610 |
"(\<Inter>x\<in>A. B x) = \<Inter>(B`A)" |
|
43872 | 611 |
by (fact INF_def) |
41082 | 612 |
|
613 |
lemma Inter_def: |
|
614 |
"\<Inter>S = (\<Inter>x\<in>S. x)" |
|
615 |
by (simp add: INTER_eq_Inter_image image_def) |
|
616 |
||
617 |
lemma INTER_def: |
|
618 |
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" |
|
619 |
by (auto simp add: INTER_eq_Inter_image Inter_eq) |
|
620 |
||
621 |
lemma Inter_image_eq [simp]: |
|
622 |
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" |
|
43872 | 623 |
by (rule sym) (fact INF_def) |
41082 | 624 |
|
43817 | 625 |
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)" |
41082 | 626 |
by (unfold INTER_def) blast |
627 |
||
43817 | 628 |
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)" |
41082 | 629 |
by (unfold INTER_def) blast |
630 |
||
43852 | 631 |
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a" |
41082 | 632 |
by auto |
633 |
||
43852 | 634 |
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" |
635 |
-- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *} |
|
41082 | 636 |
by (unfold INTER_def) blast |
637 |
||
638 |
lemma INT_cong [cong]: |
|
43854 | 639 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
640 |
by (fact INF_cong) |
41082 | 641 |
|
642 |
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" |
|
643 |
by blast |
|
644 |
||
645 |
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" |
|
646 |
by blast |
|
647 |
||
43817 | 648 |
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a" |
41082 | 649 |
by (fact INF_leI) |
650 |
||
43817 | 651 |
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)" |
43872 | 652 |
by (fact le_INF_I) |
41082 | 653 |
|
654 |
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV" |
|
43872 | 655 |
by (fact INF_empty) |
43854 | 656 |
|
43817 | 657 |
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" |
43872 | 658 |
by (fact INF_absorb) |
41082 | 659 |
|
43854 | 660 |
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)" |
41082 | 661 |
by (fact le_INF_iff) |
662 |
||
663 |
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
664 |
by (fact INF_insert) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
665 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
666 |
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
667 |
by (fact INF_union) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
668 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
669 |
lemma INT_insert_distrib: |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
670 |
"u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
671 |
by blast |
43854 | 672 |
|
41082 | 673 |
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
674 |
by (fact INF_constant) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
675 |
|
41082 | 676 |
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})" |
677 |
-- {* Look: it has an \emph{existential} quantifier *} |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
678 |
by (fact INF_eq) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
679 |
|
43854 | 680 |
lemma INTER_UNIV_conv [simp]: |
43817 | 681 |
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" |
682 |
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
683 |
by (fact INF_top_conv)+ |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
684 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
685 |
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False" |
43873 | 686 |
by (fact INF_UNIV_bool_expand) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
687 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
688 |
lemma INT_anti_mono: |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
689 |
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
690 |
-- {* The last inclusion is POSITIVE! *} |
43899 | 691 |
by (fact INF_mono') |
41082 | 692 |
|
693 |
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" |
|
694 |
by blast |
|
695 |
||
43817 | 696 |
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)" |
41082 | 697 |
by blast |
698 |
||
699 |
||
32139 | 700 |
subsection {* Union *} |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
701 |
|
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset
|
702 |
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where |
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset
|
703 |
"Union S \<equiv> \<Squnion>S" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
704 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
705 |
notation (xsymbols) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
706 |
Union ("\<Union>_" [90] 90) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
707 |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
708 |
lemma Union_eq: |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
709 |
"\<Union>A = {x. \<exists>B \<in> A. x \<in> B}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
38705
diff
changeset
|
710 |
proof (rule set_eqI) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
711 |
fix x |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
712 |
have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
713 |
by auto |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
714 |
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" |
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset
|
715 |
by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
716 |
qed |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
717 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
718 |
lemma Union_iff [simp, no_atp]: |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
719 |
"A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)" |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
720 |
by (unfold Union_eq) blast |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
721 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
722 |
lemma UnionI [intro]: |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
723 |
"X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C" |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
724 |
-- {* The order of the premises presupposes that @{term C} is rigid; |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
725 |
@{term A} may be flexible. *} |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
726 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
727 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
728 |
lemma UnionE [elim!]: |
43817 | 729 |
"A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
730 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
731 |
|
43817 | 732 |
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
733 |
by (iprover intro: subsetI UnionI) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
734 |
|
43817 | 735 |
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
736 |
by (iprover intro: subsetI elim: UnionE dest: subsetD) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
737 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
738 |
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
739 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
740 |
|
43817 | 741 |
lemma Union_empty [simp]: "\<Union>{} = {}" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
742 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
743 |
|
43817 | 744 |
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
745 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
746 |
|
43817 | 747 |
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
748 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
749 |
|
43817 | 750 |
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
751 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
752 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
753 |
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
754 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
755 |
|
43817 | 756 |
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
757 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
758 |
|
43817 | 759 |
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
760 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
761 |
|
43817 | 762 |
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
763 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
764 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
765 |
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
766 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
767 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
768 |
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
769 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
770 |
|
43817 | 771 |
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
772 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
773 |
|
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
774 |
|
32139 | 775 |
subsection {* Unions of families *} |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
776 |
|
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset
|
777 |
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where |
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset
|
778 |
"UNION \<equiv> SUPR" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
779 |
|
43872 | 780 |
text {* |
781 |
Note: must use name @{const UNION} here instead of @{text UN} |
|
782 |
to allow the following syntax coexist with the plain constant name. |
|
783 |
*} |
|
784 |
||
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
785 |
syntax |
35115 | 786 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) |
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset
|
787 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 0, 10] 10) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
788 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
789 |
syntax (xsymbols) |
35115 | 790 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) |
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset
|
791 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
792 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
793 |
syntax (latex output) |
35115 | 794 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset
|
795 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
796 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
797 |
translations |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
798 |
"UN x y. B" == "UN x. UN y. B" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
799 |
"UN x. B" == "CONST UNION CONST UNIV (%x. B)" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
800 |
"UN x. B" == "UN x:CONST UNIV. B" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
801 |
"UN x:A. B" == "CONST UNION A (%x. B)" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
802 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
803 |
text {* |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
804 |
Note the difference between ordinary xsymbol syntax of indexed |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
805 |
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
806 |
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
807 |
former does not make the index expression a subscript of the |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
808 |
union/intersection symbol because this leads to problems with nested |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
809 |
subscripts in Proof General. |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
810 |
*} |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
811 |
|
35115 | 812 |
print_translation {* |
42284 | 813 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] |
35115 | 814 |
*} -- {* to avoid eta-contraction of body *} |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
815 |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
816 |
lemma UNION_eq_Union_image: |
43817 | 817 |
"(\<Union>x\<in>A. B x) = \<Union>(B ` A)" |
43872 | 818 |
by (fact SUP_def) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
819 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
820 |
lemma Union_def: |
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
821 |
"\<Union>S = (\<Union>x\<in>S. x)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
822 |
by (simp add: UNION_eq_Union_image image_def) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
823 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
824 |
lemma UNION_def [no_atp]: |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
825 |
"(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" |
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
826 |
by (auto simp add: UNION_eq_Union_image Union_eq) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
827 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
828 |
lemma Union_image_eq [simp]: |
43817 | 829 |
"\<Union>(B ` A) = (\<Union>x\<in>A. B x)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
830 |
by (rule sym) (fact UNION_eq_Union_image) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
831 |
|
43852 | 832 |
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)" |
11979 | 833 |
by (unfold UNION_def) blast |
834 |
||
43852 | 835 |
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)" |
11979 | 836 |
-- {* The order of the premises presupposes that @{term A} is rigid; |
837 |
@{term b} may be flexible. *} |
|
838 |
by auto |
|
839 |
||
43852 | 840 |
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R" |
11979 | 841 |
by (unfold UNION_def) blast |
923 | 842 |
|
11979 | 843 |
lemma UN_cong [cong]: |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
844 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)" |
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
845 |
by (fact SUP_cong) |
11979 | 846 |
|
29691 | 847 |
lemma strong_UN_cong: |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
848 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)" |
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
849 |
by (unfold simp_implies_def) (fact UN_cong) |
29691 | 850 |
|
43817 | 851 |
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
852 |
by blast |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
853 |
|
43817 | 854 |
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)" |
43872 | 855 |
by (fact le_SUP_I) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
856 |
|
43817 | 857 |
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
858 |
by (fact SUP_leI) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
859 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
860 |
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
861 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
862 |
|
43817 | 863 |
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
864 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
865 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
866 |
lemma UN_empty [simp, no_atp]: "(\<Union>x\<in>{}. B x) = {}" |
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
867 |
by (fact SUP_empty) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
868 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
869 |
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
870 |
by (fact SUP_bot) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
871 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
872 |
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
873 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
874 |
|
43817 | 875 |
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
876 |
by (fact SUP_absorb) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
877 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
878 |
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
879 |
by (fact SUP_insert) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
880 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
881 |
lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
882 |
by (fact SUP_union) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
883 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
884 |
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)" -- "FIXME generalize" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
885 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
886 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
887 |
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" |
35629 | 888 |
by (fact SUP_le_iff) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
889 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
890 |
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
891 |
by (fact SUP_constant) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
892 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
893 |
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
894 |
by (fact SUP_eq) |
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
895 |
|
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
896 |
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" -- "FIXME generalize" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
897 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
898 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
899 |
lemma UNION_empty_conv[simp]: |
43817 | 900 |
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
901 |
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
902 |
by (fact SUP_bot_conv)+ |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
903 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
904 |
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
905 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
906 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
907 |
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
908 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
909 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
910 |
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
911 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
912 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
913 |
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
914 |
by (auto simp add: split_if_mem2) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
915 |
|
43817 | 916 |
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
917 |
by (fact SUP_UNIV_bool_expand) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
918 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
919 |
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
920 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
921 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
922 |
lemma UN_mono: |
43817 | 923 |
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
924 |
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
925 |
by (fact SUP_mono') |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
926 |
|
43817 | 927 |
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
928 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
929 |
|
43817 | 930 |
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
931 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
932 |
|
43817 | 933 |
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
934 |
-- {* NOT suitable for rewriting *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
935 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
936 |
|
43817 | 937 |
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)" |
938 |
by blast |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
939 |
|
11979 | 940 |
|
32139 | 941 |
subsection {* Distributive laws *} |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
942 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
943 |
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
944 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
945 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
946 |
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
947 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
948 |
|
43817 | 949 |
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
950 |
-- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
951 |
-- {* Union of a family of unions *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
952 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
953 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
954 |
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
955 |
-- {* Equivalent version *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
956 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
957 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
958 |
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
959 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
960 |
|
43817 | 961 |
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
962 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
963 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
964 |
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
965 |
-- {* Equivalent version *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
966 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
967 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
968 |
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
969 |
-- {* Halmos, Naive Set Theory, page 35. *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
970 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
971 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
972 |
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
973 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
974 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
975 |
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
976 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
977 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
978 |
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
979 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
980 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
981 |
|
32139 | 982 |
subsection {* Complement *} |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
983 |
|
43873 | 984 |
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)" |
985 |
by (fact uminus_INF) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
986 |
|
43873 | 987 |
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)" |
988 |
by (fact uminus_SUP) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
989 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
990 |
|
32139 | 991 |
subsection {* Miniscoping and maxiscoping *} |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
992 |
|
13860 | 993 |
text {* \medskip Miniscoping: pushing in quantifiers and big Unions |
994 |
and Intersections. *} |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
995 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
996 |
lemma UN_simps [simp]: |
43817 | 997 |
"\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))" |
43852 | 998 |
"\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))" |
999 |
"\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))" |
|
1000 |
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter>B)" |
|
1001 |
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))" |
|
1002 |
"\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)" |
|
1003 |
"\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))" |
|
1004 |
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)" |
|
1005 |
"\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)" |
|
43831 | 1006 |
"\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1007 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1008 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1009 |
lemma INT_simps [simp]: |
43831 | 1010 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter>B)" |
1011 |
"\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))" |
|
43852 | 1012 |
"\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)" |
1013 |
"\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))" |
|
43817 | 1014 |
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)" |
43852 | 1015 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)" |
1016 |
"\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))" |
|
1017 |
"\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)" |
|
1018 |
"\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)" |
|
1019 |
"\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1020 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1021 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
1022 |
lemma ball_simps [simp,no_atp]: |
43852 | 1023 |
"\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)" |
1024 |
"\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))" |
|
1025 |
"\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))" |
|
1026 |
"\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)" |
|
1027 |
"\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True" |
|
1028 |
"\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)" |
|
1029 |
"\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))" |
|
1030 |
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)" |
|
1031 |
"\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)" |
|
1032 |
"\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)" |
|
1033 |
"\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))" |
|
1034 |
"\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1035 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1036 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
1037 |
lemma bex_simps [simp,no_atp]: |
43852 | 1038 |
"\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)" |
1039 |
"\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))" |
|
1040 |
"\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False" |
|
1041 |
"\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)" |
|
1042 |
"\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))" |
|
1043 |
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)" |
|
1044 |
"\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)" |
|
1045 |
"\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)" |
|
1046 |
"\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))" |
|
1047 |
"\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1048 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1049 |
|
13860 | 1050 |
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} |
1051 |
||
1052 |
lemma UN_extend_simps: |
|
43817 | 1053 |
"\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))" |
43852 | 1054 |
"\<And>A B C. (\<Union>x\<in>C. A x) \<union> B = (if C={} then B else (\<Union>x\<in>C. A x \<union> B))" |
1055 |
"\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))" |
|
1056 |
"\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)" |
|
1057 |
"\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)" |
|
43817 | 1058 |
"\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)" |
1059 |
"\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)" |
|
43852 | 1060 |
"\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)" |
1061 |
"\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)" |
|
43831 | 1062 |
"\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)" |
13860 | 1063 |
by auto |
1064 |
||
1065 |
lemma INT_extend_simps: |
|
43852 | 1066 |
"\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))" |
1067 |
"\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))" |
|
1068 |
"\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))" |
|
1069 |
"\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))" |
|
43817 | 1070 |
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))" |
43852 | 1071 |
"\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)" |
1072 |
"\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)" |
|
1073 |
"\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)" |
|
1074 |
"\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)" |
|
1075 |
"\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)" |
|
13860 | 1076 |
by auto |
1077 |
||
1078 |
||
43872 | 1079 |
text {* Legacy names *} |
1080 |
||
1081 |
lemmas (in complete_lattice) INFI_def = INF_def |
|
1082 |
lemmas (in complete_lattice) SUPR_def = SUP_def |
|
1083 |
lemmas (in complete_lattice) le_SUPI = le_SUP_I |
|
1084 |
lemmas (in complete_lattice) le_SUPI2 = le_SUP_I2 |
|
1085 |
lemmas (in complete_lattice) le_INFI = le_INF_I |
|
43899 | 1086 |
lemmas (in complete_lattice) INF_subset = INF_superset_mono |
43873 | 1087 |
lemmas INFI_apply = INF_apply |
1088 |
lemmas SUPR_apply = SUP_apply |
|
43872 | 1089 |
|
1090 |
text {* Finally *} |
|
1091 |
||
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1092 |
no_notation |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1093 |
less_eq (infix "\<sqsubseteq>" 50) and |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1094 |
less (infix "\<sqsubset>" 50) and |
41082 | 1095 |
bot ("\<bottom>") and |
1096 |
top ("\<top>") and |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1097 |
inf (infixl "\<sqinter>" 70) and |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1098 |
sup (infixl "\<squnion>" 65) and |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1099 |
Inf ("\<Sqinter>_" [900] 900) and |
41082 | 1100 |
Sup ("\<Squnion>_" [900] 900) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1101 |
|
41080 | 1102 |
no_syntax (xsymbols) |
41082 | 1103 |
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) |
1104 |
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) |
|
41080 | 1105 |
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) |
1106 |
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) |
|
1107 |
||
30596 | 1108 |
lemmas mem_simps = |
1109 |
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff |
|
1110 |
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff |
|
1111 |
-- {* Each of these has ALREADY been added @{text "[simp]"} above. *} |
|
21669 | 1112 |
|
11979 | 1113 |
end |