src/HOL/Complete_Lattice.thy
author haftmann
Thu Aug 04 19:29:52 2011 +0200 (2011-08-04)
changeset 44032 cb768f4ceaf9
parent 44029 ce4e3090f01a
child 44039 c3d0dac940fc
permissions -rw-r--r--
solving duality problem for complete_distrib_lattice; tuned
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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
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  "INF x y. B"   == "INF x. INF y. B"
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  "INF x. B"     == "CONST INFI CONST UNIV (%x. B)"
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  "INF x. B"     == "INF x:CONST UNIV. B"
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  "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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print_translation {*
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  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
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    Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
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*} -- {* to avoid eta-contraction of body *}
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context complete_lattice
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begin
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lemma INF_empty: "(\<Sqinter>x\<in>{}. f x) = \<top>"
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  by (simp add: INF_def)
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lemma SUP_empty: "(\<Squnion>x\<in>{}. f x) = \<bottom>"
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  by (simp add: SUP_def)
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lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
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  by (simp add: INF_def Inf_insert)
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lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
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  by (simp add: SUP_def Sup_insert)
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lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
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  by (auto simp add: INF_def intro: Inf_lower)
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lemma le_SUP_I: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  by (auto simp add: SUP_def intro: Sup_upper)
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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
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lemma le_SUP_I2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
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  using le_SUP_I [of i A f] by auto
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lemma le_INF_I: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
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  by (auto simp add: INF_def intro: Inf_greatest)
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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"
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  by (auto simp add: SUP_def intro: Sup_least)
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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)"
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  by (auto simp add: INF_def le_Inf_iff)
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lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. f i \<sqsubseteq> u)"
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  by (auto simp add: SUP_def Sup_le_iff)
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lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
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  by (auto intro: antisym INF_leI le_INF_I)
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lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
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  by (auto intro: antisym SUP_leI le_SUP_I)
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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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lemma INF_cong:
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  "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)"
haftmann@43872
   280
  by (simp add: INF_def image_def)
hoelzl@38705
   281
haftmann@43870
   282
lemma SUP_cong:
haftmann@43870
   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)"
haftmann@43872
   284
  by (simp add: SUP_def image_def)
haftmann@43870
   285
hoelzl@38705
   286
lemma INF_mono:
haftmann@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)"
haftmann@43872
   288
  by (force intro!: Inf_mono simp: INF_def)
hoelzl@38705
   289
haftmann@43870
   290
lemma SUP_mono:
haftmann@43870
   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)"
haftmann@43872
   292
  by (force intro!: Sup_mono simp: SUP_def)
haftmann@43870
   293
haftmann@43899
   294
lemma INF_superset_mono:
haftmann@43940
   295
  "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)"
haftmann@43940
   296
  -- {* The last inclusion is POSITIVE! *}
haftmann@43940
   297
  by (blast intro: INF_mono dest: subsetD)
haftmann@43865
   298
haftmann@43900
   299
lemma SUP_subset_mono:
haftmann@43940
   300
  "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)"
haftmann@43940
   301
  by (blast intro: SUP_mono dest: subsetD)
haftmann@43870
   302
haftmann@43865
   303
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)"
haftmann@43872
   304
  by (iprover intro: INF_leI le_INF_I order_trans antisym)
haftmann@43865
   305
haftmann@43870
   306
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)"
haftmann@43872
   307
  by (iprover intro: SUP_leI le_SUP_I order_trans antisym)
haftmann@43870
   308
haftmann@43871
   309
lemma INF_absorb:
haftmann@43868
   310
  assumes "k \<in> I"
haftmann@43868
   311
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   312
proof -
haftmann@43868
   313
  from assms obtain J where "I = insert k J" by blast
haftmann@43868
   314
  then show ?thesis by (simp add: INF_insert)
haftmann@43868
   315
qed
haftmann@43868
   316
haftmann@43871
   317
lemma SUP_absorb:
haftmann@43871
   318
  assumes "k \<in> I"
haftmann@43871
   319
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   320
proof -
haftmann@43871
   321
  from assms obtain J where "I = insert k J" by blast
haftmann@43871
   322
  then show ?thesis by (simp add: SUP_insert)
haftmann@43871
   323
qed
haftmann@43871
   324
haftmann@43871
   325
lemma INF_union:
haftmann@43868
   326
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@43872
   327
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INF_I INF_leI)
haftmann@43868
   328
haftmann@43871
   329
lemma SUP_union:
haftmann@43871
   330
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@43872
   331
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_leI le_SUP_I)
haftmann@43871
   332
haftmann@43871
   333
lemma INF_constant:
haftmann@43868
   334
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
haftmann@43868
   335
  by (simp add: INF_empty)
haftmann@43868
   336
haftmann@43871
   337
lemma SUP_constant:
haftmann@43871
   338
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
haftmann@43871
   339
  by (simp add: SUP_empty)
haftmann@43871
   340
haftmann@43871
   341
lemma INF_eq:
haftmann@43868
   342
  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43872
   343
  by (simp add: INF_def image_def)
haftmann@43868
   344
haftmann@43871
   345
lemma SUP_eq:
haftmann@43871
   346
  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43872
   347
  by (simp add: SUP_def image_def)
haftmann@43871
   348
haftmann@43871
   349
lemma INF_top_conv:
haftmann@43868
   350
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@43868
   351
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@43872
   352
  by (auto simp add: INF_def Inf_top_conv)
haftmann@43868
   353
haftmann@43871
   354
lemma SUP_bot_conv:
haftmann@43871
   355
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@43871
   356
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@43872
   357
  by (auto simp add: SUP_def Sup_bot_conv)
haftmann@43868
   358
haftmann@43943
   359
lemma less_INF_D:
haftmann@43943
   360
  assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i"
haftmann@43943
   361
proof -
haftmann@43943
   362
  note `y < (\<Sqinter>i\<in>A. f i)`
haftmann@43943
   363
  also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
haftmann@43943
   364
    by (rule INF_leI)
haftmann@43943
   365
  finally show "y < f i" .
haftmann@43943
   366
qed
haftmann@43943
   367
haftmann@43943
   368
lemma SUP_lessD:
haftmann@43943
   369
  assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
haftmann@43943
   370
proof -
haftmann@43943
   371
  have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
haftmann@43943
   372
    by (rule le_SUP_I)
haftmann@43943
   373
  also note `(\<Squnion>i\<in>A. f i) < y`
haftmann@43943
   374
  finally show "f i < y" .
haftmann@43943
   375
qed
haftmann@43943
   376
haftmann@43873
   377
lemma INF_UNIV_range:
haftmann@43871
   378
  "(\<Sqinter>x. f x) = \<Sqinter>range f"
haftmann@43872
   379
  by (fact INF_def)
haftmann@43871
   380
haftmann@43873
   381
lemma SUP_UNIV_range:
haftmann@43871
   382
  "(\<Squnion>x. f x) = \<Squnion>range f"
haftmann@43872
   383
  by (fact SUP_def)
haftmann@43871
   384
haftmann@43873
   385
lemma INF_UNIV_bool_expand:
haftmann@43868
   386
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@43868
   387
  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
haftmann@43868
   388
haftmann@43873
   389
lemma SUP_UNIV_bool_expand:
haftmann@43871
   390
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@43871
   391
  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
haftmann@43871
   392
haftmann@44032
   393
lemma INF_foundation_dual [no_atp]:
haftmann@44032
   394
  "complete_lattice.SUPR Inf = INFI"
haftmann@44032
   395
proof (rule ext)+
haftmann@44032
   396
  interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
haftmann@44032
   397
    by (fact dual_complete_lattice)
haftmann@44032
   398
  fix f :: "'b \<Rightarrow> 'a" and A
haftmann@44032
   399
  show "complete_lattice.SUPR Inf A f = (\<Sqinter>a\<in>A. f a)"
haftmann@44032
   400
    by (simp only: dual.SUP_def INF_def)
haftmann@44032
   401
qed
haftmann@44032
   402
haftmann@44032
   403
lemma SUP_foundation_dual [no_atp]:
haftmann@44032
   404
  "complete_lattice.INFI Sup = SUPR"
haftmann@44032
   405
proof (rule ext)+
haftmann@44032
   406
  interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
haftmann@44032
   407
    by (fact dual_complete_lattice)
haftmann@44032
   408
  fix f :: "'b \<Rightarrow> 'a" and A
haftmann@44032
   409
  show "complete_lattice.INFI Sup A f = (\<Squnion>a\<in>A. f a)"
haftmann@44032
   410
    by (simp only: dual.INF_def SUP_def)
haftmann@44032
   411
qed
haftmann@44032
   412
haftmann@32077
   413
end
haftmann@32077
   414
haftmann@44024
   415
class complete_distrib_lattice = complete_lattice +
haftmann@44024
   416
  assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)"
haftmann@44024
   417
  assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)"
haftmann@44024
   418
begin
haftmann@44024
   419
haftmann@44032
   420
lemma dual_complete_distrib_lattice:
haftmann@44024
   421
  "class.complete_distrib_lattice Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
haftmann@44024
   422
  apply (rule class.complete_distrib_lattice.intro)
haftmann@44024
   423
  apply (fact dual_complete_lattice)
haftmann@44024
   424
  apply (rule class.complete_distrib_lattice_axioms.intro)
haftmann@44032
   425
  apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
haftmann@44032
   426
  done
haftmann@44024
   427
haftmann@44024
   428
subclass distrib_lattice proof -- {* Question: is it sufficient to include @{class distrib_lattice}
haftmann@44029
   429
  and prove @{text inf_Sup} and @{text sup_Inf} from that? *}
haftmann@44024
   430
  fix a b c
haftmann@44024
   431
  from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
haftmann@44024
   432
  then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def Inf_binary)
haftmann@44024
   433
qed
haftmann@44024
   434
haftmann@44024
   435
end
haftmann@44024
   436
haftmann@44032
   437
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
haftmann@43873
   438
begin
haftmann@43873
   439
haftmann@43943
   440
lemma dual_complete_boolean_algebra:
haftmann@43943
   441
  "class.complete_boolean_algebra Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
haftmann@44032
   442
  by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
haftmann@43943
   443
haftmann@43873
   444
lemma uminus_Inf:
haftmann@43873
   445
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   446
proof (rule antisym)
haftmann@43873
   447
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   448
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   449
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   450
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   451
qed
haftmann@43873
   452
haftmann@43873
   453
lemma uminus_Sup:
haftmann@43873
   454
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   455
proof -
haftmann@43873
   456
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
haftmann@43873
   457
  then show ?thesis by simp
haftmann@43873
   458
qed
haftmann@43873
   459
  
haftmann@43873
   460
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@43873
   461
  by (simp add: INF_def SUP_def uminus_Inf image_image)
haftmann@43873
   462
haftmann@43873
   463
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@43873
   464
  by (simp add: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   465
haftmann@43873
   466
end
haftmann@43873
   467
haftmann@43940
   468
class complete_linorder = linorder + complete_lattice
haftmann@43940
   469
begin
haftmann@43940
   470
haftmann@43943
   471
lemma dual_complete_linorder:
haftmann@43943
   472
  "class.complete_linorder Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
haftmann@43943
   473
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
haftmann@43943
   474
haftmann@43940
   475
lemma Inf_less_iff:
haftmann@43940
   476
  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43940
   477
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@43940
   478
haftmann@43940
   479
lemma less_Sup_iff:
haftmann@43940
   480
  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43940
   481
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43940
   482
haftmann@43940
   483
lemma INF_less_iff:
haftmann@43940
   484
  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@43940
   485
  unfolding INF_def Inf_less_iff by auto
haftmann@43940
   486
haftmann@43940
   487
lemma less_SUP_iff:
haftmann@43940
   488
  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@43940
   489
  unfolding SUP_def less_Sup_iff by auto
haftmann@43940
   490
haftmann@43943
   491
lemma Sup_eq_top_iff:
haftmann@43943
   492
  "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
haftmann@43943
   493
proof
haftmann@43943
   494
  assume *: "\<Squnion>A = \<top>"
haftmann@43943
   495
  show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
haftmann@43943
   496
  proof (intro allI impI)
haftmann@43943
   497
    fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i"
haftmann@43943
   498
      unfolding less_Sup_iff by auto
haftmann@43943
   499
  qed
haftmann@43943
   500
next
haftmann@43943
   501
  assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i"
haftmann@43943
   502
  show "\<Squnion>A = \<top>"
haftmann@43943
   503
  proof (rule ccontr)
haftmann@43943
   504
    assume "\<Squnion>A \<noteq> \<top>"
haftmann@43943
   505
    with top_greatest [of "\<Squnion>A"]
haftmann@43943
   506
    have "\<Squnion>A < \<top>" unfolding le_less by auto
haftmann@43943
   507
    then have "\<Squnion>A < \<Squnion>A"
haftmann@43943
   508
      using * unfolding less_Sup_iff by auto
haftmann@43943
   509
    then show False by auto
haftmann@43943
   510
  qed
haftmann@43943
   511
qed
haftmann@43943
   512
haftmann@43943
   513
lemma Inf_eq_bot_iff:
haftmann@43943
   514
  "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
haftmann@43943
   515
proof -
haftmann@43943
   516
  interpret dual: complete_linorder Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
haftmann@43943
   517
    by (fact dual_complete_linorder)
haftmann@43943
   518
  from dual.Sup_eq_top_iff show ?thesis .
haftmann@43943
   519
qed
haftmann@43943
   520
haftmann@43967
   521
lemma INF_eq_bot_iff:
haftmann@43967
   522
  "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
haftmann@43967
   523
  unfolding INF_def Inf_eq_bot_iff by auto
haftmann@43967
   524
haftmann@43967
   525
lemma SUP_eq_top_iff:
haftmann@43967
   526
  "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
haftmann@43967
   527
  unfolding SUP_def Sup_eq_top_iff by auto
haftmann@43967
   528
haftmann@43940
   529
end
haftmann@43940
   530
haftmann@43873
   531
haftmann@32139
   532
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077
   533
haftmann@44024
   534
instantiation bool :: complete_lattice
haftmann@32077
   535
begin
haftmann@32077
   536
haftmann@32077
   537
definition
haftmann@41080
   538
  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@32077
   539
haftmann@32077
   540
definition
haftmann@41080
   541
  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@32077
   542
haftmann@32077
   543
instance proof
haftmann@43852
   544
qed (auto simp add: Inf_bool_def Sup_bool_def)
haftmann@32077
   545
haftmann@32077
   546
end
haftmann@32077
   547
haftmann@43873
   548
lemma INF_bool_eq [simp]:
haftmann@32120
   549
  "INFI = Ball"
haftmann@32120
   550
proof (rule ext)+
haftmann@32120
   551
  fix A :: "'a set"
haftmann@32120
   552
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   553
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
haftmann@43872
   554
    by (auto simp add: Ball_def INF_def Inf_bool_def)
haftmann@32120
   555
qed
haftmann@32120
   556
haftmann@43873
   557
lemma SUP_bool_eq [simp]:
haftmann@32120
   558
  "SUPR = Bex"
haftmann@32120
   559
proof (rule ext)+
haftmann@32120
   560
  fix A :: "'a set"
haftmann@32120
   561
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   562
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@43872
   563
    by (auto simp add: Bex_def SUP_def Sup_bool_def)
haftmann@32120
   564
qed
haftmann@32120
   565
haftmann@44032
   566
instance bool :: complete_boolean_algebra proof
haftmann@44024
   567
qed (auto simp add: Inf_bool_def Sup_bool_def)
haftmann@44024
   568
haftmann@32077
   569
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   570
begin
haftmann@32077
   571
haftmann@32077
   572
definition
haftmann@44024
   573
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
haftmann@41080
   574
haftmann@41080
   575
lemma Inf_apply:
haftmann@44024
   576
  "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
haftmann@41080
   577
  by (simp add: Inf_fun_def)
haftmann@32077
   578
haftmann@32077
   579
definition
haftmann@44024
   580
  "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
haftmann@41080
   581
haftmann@41080
   582
lemma Sup_apply:
haftmann@44024
   583
  "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)"
haftmann@41080
   584
  by (simp add: Sup_fun_def)
haftmann@32077
   585
haftmann@32077
   586
instance proof
haftmann@44024
   587
qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_leI le_SUP_I le_INF_I SUP_leI)
haftmann@32077
   588
haftmann@32077
   589
end
haftmann@32077
   590
haftmann@43873
   591
lemma INF_apply:
haftmann@41080
   592
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@43872
   593
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
hoelzl@38705
   594
haftmann@43873
   595
lemma SUP_apply:
haftmann@41080
   596
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@43872
   597
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
haftmann@32077
   598
haftmann@44024
   599
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof
haftmann@44024
   600
qed (auto simp add: inf_apply sup_apply Inf_apply Sup_apply INF_def SUP_def inf_Sup sup_Inf image_image)
haftmann@44024
   601
haftmann@43873
   602
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   603
haftmann@32077
   604
haftmann@41082
   605
subsection {* Inter *}
haftmann@41082
   606
haftmann@41082
   607
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   608
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   609
  
haftmann@41082
   610
notation (xsymbols)
haftmann@41082
   611
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   612
haftmann@41082
   613
lemma Inter_eq:
haftmann@41082
   614
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   615
proof (rule set_eqI)
haftmann@41082
   616
  fix x
haftmann@41082
   617
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
haftmann@41082
   618
    by auto
haftmann@41082
   619
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   620
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082
   621
qed
haftmann@41082
   622
haftmann@43741
   623
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   624
  by (unfold Inter_eq) blast
haftmann@41082
   625
haftmann@43741
   626
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   627
  by (simp add: Inter_eq)
haftmann@41082
   628
haftmann@41082
   629
text {*
haftmann@41082
   630
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   631
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   632
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   633
*}
haftmann@41082
   634
haftmann@43741
   635
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   636
  by auto
haftmann@41082
   637
haftmann@43741
   638
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   639
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   640
    @{prop "X \<in> C"}. *}
haftmann@41082
   641
  by (unfold Inter_eq) blast
haftmann@41082
   642
haftmann@43741
   643
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   644
  by (fact Inf_lower)
haftmann@43740
   645
haftmann@41082
   646
lemma Inter_subset:
haftmann@43755
   647
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   648
  by (fact Inf_less_eq)
haftmann@41082
   649
haftmann@43755
   650
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   651
  by (fact Inf_greatest)
haftmann@41082
   652
haftmann@41082
   653
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@43739
   654
  by (fact Inf_binary [symmetric])
haftmann@41082
   655
haftmann@41082
   656
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082
   657
  by (fact Inf_empty)
haftmann@41082
   658
haftmann@41082
   659
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@43739
   660
  by (fact Inf_UNIV)
haftmann@41082
   661
haftmann@41082
   662
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739
   663
  by (fact Inf_insert)
haftmann@41082
   664
haftmann@41082
   665
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   666
  by (fact less_eq_Inf_inter)
haftmann@41082
   667
haftmann@41082
   668
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   669
  by (fact Inf_union_distrib)
haftmann@43756
   670
haftmann@43868
   671
lemma Inter_UNIV_conv [simp, no_atp]:
haftmann@43741
   672
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   673
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   674
  by (fact Inf_top_conv)+
haftmann@41082
   675
haftmann@43741
   676
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   677
  by (fact Inf_superset_mono)
haftmann@41082
   678
haftmann@41082
   679
haftmann@41082
   680
subsection {* Intersections of families *}
haftmann@41082
   681
haftmann@41082
   682
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   683
  "INTER \<equiv> INFI"
haftmann@41082
   684
haftmann@43872
   685
text {*
haftmann@43872
   686
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   687
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   688
*}
haftmann@43872
   689
haftmann@41082
   690
syntax
haftmann@41082
   691
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   692
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   693
haftmann@41082
   694
syntax (xsymbols)
haftmann@41082
   695
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   696
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   697
haftmann@41082
   698
syntax (latex output)
haftmann@41082
   699
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   700
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   701
haftmann@41082
   702
translations
haftmann@41082
   703
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   704
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   705
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   706
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   707
haftmann@41082
   708
print_translation {*
wenzelm@42284
   709
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   710
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   711
haftmann@41082
   712
lemma INTER_eq_Inter_image:
haftmann@41082
   713
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@43872
   714
  by (fact INF_def)
haftmann@41082
   715
  
haftmann@41082
   716
lemma Inter_def:
haftmann@41082
   717
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082
   718
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082
   719
haftmann@41082
   720
lemma INTER_def:
haftmann@41082
   721
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082
   722
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082
   723
haftmann@41082
   724
lemma Inter_image_eq [simp]:
haftmann@41082
   725
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43872
   726
  by (rule sym) (fact INF_def)
haftmann@41082
   727
haftmann@43817
   728
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@41082
   729
  by (unfold INTER_def) blast
haftmann@41082
   730
haftmann@43817
   731
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
haftmann@41082
   732
  by (unfold INTER_def) blast
haftmann@41082
   733
haftmann@43852
   734
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
haftmann@41082
   735
  by auto
haftmann@41082
   736
haftmann@43852
   737
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"
haftmann@43852
   738
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@41082
   739
  by (unfold INTER_def) blast
haftmann@41082
   740
haftmann@41082
   741
lemma INT_cong [cong]:
haftmann@43854
   742
  "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)"
haftmann@43865
   743
  by (fact INF_cong)
haftmann@41082
   744
haftmann@41082
   745
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   746
  by blast
haftmann@41082
   747
haftmann@41082
   748
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   749
  by blast
haftmann@41082
   750
haftmann@43817
   751
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082
   752
  by (fact INF_leI)
haftmann@41082
   753
haftmann@43817
   754
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@43872
   755
  by (fact le_INF_I)
haftmann@41082
   756
haftmann@41082
   757
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@43872
   758
  by (fact INF_empty)
haftmann@43854
   759
haftmann@43817
   760
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   761
  by (fact INF_absorb)
haftmann@41082
   762
haftmann@43854
   763
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   764
  by (fact le_INF_iff)
haftmann@41082
   765
haftmann@41082
   766
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   767
  by (fact INF_insert)
haftmann@43865
   768
haftmann@43865
   769
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
haftmann@43865
   770
  by (fact INF_union)
haftmann@43865
   771
haftmann@43865
   772
lemma INT_insert_distrib:
haftmann@43865
   773
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   774
  by blast
haftmann@43854
   775
haftmann@41082
   776
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   777
  by (fact INF_constant)
haftmann@43865
   778
haftmann@41082
   779
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082
   780
  -- {* Look: it has an \emph{existential} quantifier *}
haftmann@43865
   781
  by (fact INF_eq)
haftmann@43865
   782
haftmann@43854
   783
lemma INTER_UNIV_conv [simp]:
haftmann@43817
   784
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   785
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43865
   786
  by (fact INF_top_conv)+
haftmann@43865
   787
haftmann@43865
   788
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   789
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   790
haftmann@43865
   791
lemma INT_anti_mono:
haftmann@43900
   792
  "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)"
haftmann@43865
   793
  -- {* The last inclusion is POSITIVE! *}
haftmann@43940
   794
  by (fact INF_superset_mono)
haftmann@41082
   795
haftmann@41082
   796
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   797
  by blast
haftmann@41082
   798
haftmann@43817
   799
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   800
  by blast
haftmann@41082
   801
haftmann@41082
   802
haftmann@32139
   803
subsection {* Union *}
haftmann@32115
   804
haftmann@32587
   805
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   806
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   807
haftmann@32115
   808
notation (xsymbols)
haftmann@32115
   809
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   810
haftmann@32135
   811
lemma Union_eq:
haftmann@32135
   812
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   813
proof (rule set_eqI)
haftmann@32115
   814
  fix x
haftmann@32135
   815
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
haftmann@32115
   816
    by auto
haftmann@32135
   817
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@32587
   818
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115
   819
qed
haftmann@32115
   820
blanchet@35828
   821
lemma Union_iff [simp, no_atp]:
haftmann@32115
   822
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   823
  by (unfold Union_eq) blast
haftmann@32115
   824
haftmann@32115
   825
lemma UnionI [intro]:
haftmann@32115
   826
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   827
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   828
    @{term A} may be flexible. *}
haftmann@32115
   829
  by auto
haftmann@32115
   830
haftmann@32115
   831
lemma UnionE [elim!]:
haftmann@43817
   832
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   833
  by auto
haftmann@32115
   834
haftmann@43817
   835
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
   836
  by (fact Sup_upper)
haftmann@32135
   837
haftmann@43817
   838
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
   839
  by (fact Sup_least)
haftmann@32135
   840
haftmann@32135
   841
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135
   842
  by blast
haftmann@32135
   843
haftmann@43817
   844
lemma Union_empty [simp]: "\<Union>{} = {}"
haftmann@43901
   845
  by (fact Sup_empty)
haftmann@32135
   846
haftmann@43817
   847
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
haftmann@43901
   848
  by (fact Sup_UNIV)
haftmann@32135
   849
haftmann@43817
   850
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
haftmann@43901
   851
  by (fact Sup_insert)
haftmann@32135
   852
haftmann@43817
   853
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
   854
  by (fact Sup_union_distrib)
haftmann@32135
   855
haftmann@32135
   856
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
   857
  by (fact Sup_inter_less_eq)
haftmann@32135
   858
haftmann@43817
   859
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@43901
   860
  by (fact Sup_bot_conv)
haftmann@32135
   861
haftmann@43817
   862
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@43901
   863
  by (fact Sup_bot_conv)
haftmann@32135
   864
haftmann@43901
   865
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" -- "FIXME generalize"
haftmann@32135
   866
  by blast
haftmann@32135
   867
haftmann@32135
   868
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   869
  by blast
haftmann@32135
   870
haftmann@32135
   871
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   872
  by blast
haftmann@32135
   873
haftmann@43817
   874
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
   875
  by (fact Sup_subset_mono)
haftmann@32135
   876
haftmann@32115
   877
haftmann@32139
   878
subsection {* Unions of families *}
haftmann@32077
   879
haftmann@32606
   880
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   881
  "UNION \<equiv> SUPR"
haftmann@32077
   882
haftmann@43872
   883
text {*
haftmann@43872
   884
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
   885
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   886
*}
haftmann@43872
   887
haftmann@32077
   888
syntax
wenzelm@35115
   889
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   890
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   891
haftmann@32077
   892
syntax (xsymbols)
wenzelm@35115
   893
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   894
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   895
haftmann@32077
   896
syntax (latex output)
wenzelm@35115
   897
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   898
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   899
haftmann@32077
   900
translations
haftmann@32077
   901
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   902
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   903
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   904
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   905
haftmann@32077
   906
text {*
haftmann@32077
   907
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   908
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   909
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   910
  former does not make the index expression a subscript of the
haftmann@32077
   911
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   912
  subscripts in Proof General.
haftmann@32077
   913
*}
haftmann@32077
   914
wenzelm@35115
   915
print_translation {*
wenzelm@42284
   916
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   917
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   918
haftmann@32135
   919
lemma UNION_eq_Union_image:
haftmann@43817
   920
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
haftmann@43872
   921
  by (fact SUP_def)
haftmann@32115
   922
haftmann@32115
   923
lemma Union_def:
haftmann@32117
   924
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   925
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   926
blanchet@35828
   927
lemma UNION_def [no_atp]:
haftmann@32135
   928
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   929
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   930
  
haftmann@32115
   931
lemma Union_image_eq [simp]:
haftmann@43817
   932
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   933
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   934
  
haftmann@43852
   935
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
wenzelm@11979
   936
  by (unfold UNION_def) blast
wenzelm@11979
   937
haftmann@43852
   938
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
   939
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   940
    @{term b} may be flexible. *}
wenzelm@11979
   941
  by auto
wenzelm@11979
   942
haftmann@43852
   943
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"
wenzelm@11979
   944
  by (unfold UNION_def) blast
clasohm@923
   945
wenzelm@11979
   946
lemma UN_cong [cong]:
haftmann@43900
   947
  "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)"
haftmann@43900
   948
  by (fact SUP_cong)
wenzelm@11979
   949
berghofe@29691
   950
lemma strong_UN_cong:
haftmann@43900
   951
  "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)"
haftmann@43900
   952
  by (unfold simp_implies_def) (fact UN_cong)
berghofe@29691
   953
haftmann@43817
   954
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
   955
  by blast
haftmann@32077
   956
haftmann@43817
   957
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@43872
   958
  by (fact le_SUP_I)
haftmann@32135
   959
haftmann@43817
   960
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@43900
   961
  by (fact SUP_leI)
haftmann@32135
   962
blanchet@35828
   963
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   964
  by blast
haftmann@32135
   965
haftmann@43817
   966
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
   967
  by blast
haftmann@32135
   968
haftmann@43900
   969
lemma UN_empty [simp, no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@43900
   970
  by (fact SUP_empty)
haftmann@32135
   971
haftmann@32135
   972
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@43900
   973
  by (fact SUP_bot)
haftmann@32135
   974
haftmann@32135
   975
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   976
  by blast
haftmann@32135
   977
haftmann@43817
   978
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
   979
  by (fact SUP_absorb)
haftmann@32135
   980
haftmann@32135
   981
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
   982
  by (fact SUP_insert)
haftmann@32135
   983
haftmann@32135
   984
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)"
haftmann@43900
   985
  by (fact SUP_union)
haftmann@32135
   986
haftmann@43967
   987
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)"
haftmann@32135
   988
  by blast
haftmann@32135
   989
haftmann@32135
   990
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   991
  by (fact SUP_le_iff)
haftmann@32135
   992
haftmann@32135
   993
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
   994
  by (fact SUP_constant)
haftmann@32135
   995
haftmann@32135
   996
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43900
   997
  by (fact SUP_eq)
haftmann@43900
   998
haftmann@43944
   999
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
  1000
  by blast
haftmann@32135
  1001
haftmann@32135
  1002
lemma UNION_empty_conv[simp]:
haftmann@43817
  1003
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
  1004
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43900
  1005
  by (fact SUP_bot_conv)+
haftmann@32135
  1006
blanchet@35828
  1007
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
  1008
  by blast
haftmann@32135
  1009
haftmann@43900
  1010
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
  1011
  by blast
haftmann@32135
  1012
haftmann@43900
  1013
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
  1014
  by blast
haftmann@32135
  1015
haftmann@32135
  1016
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
  1017
  by (auto simp add: split_if_mem2)
haftmann@32135
  1018
haftmann@43817
  1019
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
  1020
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
  1021
haftmann@32135
  1022
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
  1023
  by blast
haftmann@32135
  1024
haftmann@32135
  1025
lemma UN_mono:
haftmann@43817
  1026
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
  1027
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43940
  1028
  by (fact SUP_subset_mono)
haftmann@32135
  1029
haftmann@43817
  1030
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
  1031
  by blast
haftmann@32135
  1032
haftmann@43817
  1033
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
  1034
  by blast
haftmann@32135
  1035
haftmann@43817
  1036
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
  1037
  -- {* NOT suitable for rewriting *}
haftmann@32135
  1038
  by blast
haftmann@32135
  1039
haftmann@43817
  1040
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
  1041
  by blast
haftmann@32135
  1042
wenzelm@11979
  1043
haftmann@32139
  1044
subsection {* Distributive laws *}
wenzelm@12897
  1045
wenzelm@12897
  1046
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
haftmann@44032
  1047
  by (fact inf_Sup)
wenzelm@12897
  1048
wenzelm@12897
  1049
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
  1050
  by blast
wenzelm@12897
  1051
haftmann@43817
  1052
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897
  1053
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
  1054
  -- {* Union of a family of unions *}
wenzelm@12897
  1055
  by blast
wenzelm@12897
  1056
wenzelm@12897
  1057
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)"
wenzelm@12897
  1058
  -- {* Equivalent version *}
wenzelm@12897
  1059
  by blast
wenzelm@12897
  1060
wenzelm@12897
  1061
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
haftmann@44032
  1062
  by (fact sup_Inf)
wenzelm@12897
  1063
haftmann@43817
  1064
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
wenzelm@12897
  1065
  by blast
wenzelm@12897
  1066
wenzelm@12897
  1067
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)"
wenzelm@12897
  1068
  -- {* Equivalent version *}
wenzelm@12897
  1069
  by blast
wenzelm@12897
  1070
wenzelm@12897
  1071
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
  1072
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
  1073
  by blast
wenzelm@12897
  1074
wenzelm@12897
  1075
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
  1076
  by blast
wenzelm@12897
  1077
wenzelm@12897
  1078
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)"
wenzelm@12897
  1079
  by blast
wenzelm@12897
  1080
wenzelm@12897
  1081
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)"
wenzelm@12897
  1082
  by blast
wenzelm@12897
  1083
wenzelm@12897
  1084
haftmann@32139
  1085
subsection {* Complement *}
haftmann@32135
  1086
haftmann@43873
  1087
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
  1088
  by (fact uminus_INF)
wenzelm@12897
  1089
haftmann@43873
  1090
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
  1091
  by (fact uminus_SUP)
wenzelm@12897
  1092
wenzelm@12897
  1093
haftmann@32139
  1094
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
  1095
paulson@13860
  1096
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
  1097
           and Intersections. *}
wenzelm@12897
  1098
wenzelm@12897
  1099
lemma UN_simps [simp]:
haftmann@43817
  1100
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
haftmann@44032
  1101
  "\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
haftmann@43852
  1102
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
haftmann@44032
  1103
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)"
haftmann@43852
  1104
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1105
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1106
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1107
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1108
  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
haftmann@43831
  1109
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1110
  by auto
wenzelm@12897
  1111
wenzelm@12897
  1112
lemma INT_simps [simp]:
haftmann@44032
  1113
  "\<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)"
haftmann@43831
  1114
  "\<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))"
haftmann@43852
  1115
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
haftmann@43852
  1116
  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
haftmann@43817
  1117
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1118
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1119
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1120
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1121
  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
haftmann@43852
  1122
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1123
  by auto
wenzelm@12897
  1124
haftmann@43967
  1125
lemma UN_ball_bex_simps [simp, no_atp]:
haftmann@43852
  1126
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43967
  1127
  "\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
haftmann@43852
  1128
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1129
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
wenzelm@12897
  1130
  by auto
wenzelm@12897
  1131
haftmann@43943
  1132
paulson@13860
  1133
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1134
paulson@13860
  1135
lemma UN_extend_simps:
haftmann@43817
  1136
  "\<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)))"
haftmann@44032
  1137
  "\<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))"
haftmann@43852
  1138
  "\<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))"
haftmann@43852
  1139
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1140
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1141
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1142
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1143
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1144
  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
haftmann@43831
  1145
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1146
  by auto
paulson@13860
  1147
paulson@13860
  1148
lemma INT_extend_simps:
haftmann@43852
  1149
  "\<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))"
haftmann@43852
  1150
  "\<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))"
haftmann@43852
  1151
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
haftmann@43852
  1152
  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
haftmann@43817
  1153
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1154
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1155
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1156
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1157
  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
haftmann@43852
  1158
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1159
  by auto
paulson@13860
  1160
paulson@13860
  1161
haftmann@43872
  1162
text {* Legacy names *}
haftmann@43872
  1163
haftmann@43872
  1164
lemmas (in complete_lattice) INFI_def = INF_def
haftmann@43872
  1165
lemmas (in complete_lattice) SUPR_def = SUP_def
haftmann@43872
  1166
lemmas (in complete_lattice) le_SUPI = le_SUP_I
haftmann@43872
  1167
lemmas (in complete_lattice) le_SUPI2 = le_SUP_I2
haftmann@43872
  1168
lemmas (in complete_lattice) le_INFI = le_INF_I
haftmann@43943
  1169
lemmas (in complete_lattice) less_INFD = less_INF_D
haftmann@43940
  1170
haftmann@43940
  1171
lemma (in complete_lattice) INF_subset:
haftmann@43940
  1172
  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
haftmann@43940
  1173
  by (rule INF_superset_mono) auto
haftmann@43940
  1174
haftmann@43873
  1175
lemmas INFI_apply = INF_apply
haftmann@43873
  1176
lemmas SUPR_apply = SUP_apply
haftmann@43872
  1177
haftmann@43872
  1178
text {* Finally *}
haftmann@43872
  1179
haftmann@32135
  1180
no_notation
haftmann@32135
  1181
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
  1182
  less (infix "\<sqsubset>" 50) and
haftmann@41082
  1183
  bot ("\<bottom>") and
haftmann@41082
  1184
  top ("\<top>") and
haftmann@32135
  1185
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
  1186
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
  1187
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
  1188
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
  1189
haftmann@41080
  1190
no_syntax (xsymbols)
haftmann@41082
  1191
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
  1192
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1193
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
  1194
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1195
haftmann@30596
  1196
lemmas mem_simps =
haftmann@30596
  1197
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1198
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1199
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1200
wenzelm@11979
  1201
end