src/HOL/Complete_Lattice.thy
author haftmann
Mon Jul 18 21:34:01 2011 +0200 (2011-07-18)
changeset 43899 60ef6abb2f92
parent 43898 935359fd8210
child 43900 7162691e740b
permissions -rw-r--r--
avoid misunderstandable names
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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_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)"
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  by (simp add: INF_def image_def)
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lemma SUP_cong:
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  "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)"
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  by (simp add: SUP_def image_def)
haftmann@43870
   279
hoelzl@38705
   280
lemma INF_mono:
haftmann@43753
   281
  "(\<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
   282
  by (force intro!: Inf_mono simp: INF_def)
hoelzl@38705
   283
haftmann@43870
   284
lemma SUP_mono:
haftmann@43870
   285
  "(\<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
   286
  by (force intro!: Sup_mono simp: SUP_def)
haftmann@43870
   287
haftmann@43899
   288
lemma INF_superset_mono:
haftmann@43899
   289
  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
haftmann@43899
   290
  by (rule INF_mono) auto
haftmann@43865
   291
haftmann@43899
   292
lemma SUPO_subset_mono:
haftmann@43870
   293
  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
haftmann@43899
   294
  by (rule SUP_mono) auto
haftmann@43870
   295
haftmann@43865
   296
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
   297
  by (iprover intro: INF_leI le_INF_I order_trans antisym)
haftmann@43865
   298
haftmann@43870
   299
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
   300
  by (iprover intro: SUP_leI le_SUP_I order_trans antisym)
haftmann@43870
   301
haftmann@43871
   302
lemma INF_absorb:
haftmann@43868
   303
  assumes "k \<in> I"
haftmann@43868
   304
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   305
proof -
haftmann@43868
   306
  from assms obtain J where "I = insert k J" by blast
haftmann@43868
   307
  then show ?thesis by (simp add: INF_insert)
haftmann@43868
   308
qed
haftmann@43868
   309
haftmann@43871
   310
lemma SUP_absorb:
haftmann@43871
   311
  assumes "k \<in> I"
haftmann@43871
   312
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   313
proof -
haftmann@43871
   314
  from assms obtain J where "I = insert k J" by blast
haftmann@43871
   315
  then show ?thesis by (simp add: SUP_insert)
haftmann@43871
   316
qed
haftmann@43871
   317
haftmann@43871
   318
lemma INF_union:
haftmann@43868
   319
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@43872
   320
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INF_I INF_leI)
haftmann@43868
   321
haftmann@43871
   322
lemma SUP_union:
haftmann@43871
   323
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@43872
   324
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_leI le_SUP_I)
haftmann@43871
   325
haftmann@43871
   326
lemma INF_constant:
haftmann@43868
   327
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
haftmann@43868
   328
  by (simp add: INF_empty)
haftmann@43868
   329
haftmann@43871
   330
lemma SUP_constant:
haftmann@43871
   331
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
haftmann@43871
   332
  by (simp add: SUP_empty)
haftmann@43871
   333
haftmann@43871
   334
lemma INF_eq:
haftmann@43868
   335
  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43872
   336
  by (simp add: INF_def image_def)
haftmann@43868
   337
haftmann@43871
   338
lemma SUP_eq:
haftmann@43871
   339
  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43872
   340
  by (simp add: SUP_def image_def)
haftmann@43871
   341
haftmann@43871
   342
lemma INF_top_conv:
haftmann@43868
   343
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@43868
   344
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@43872
   345
  by (auto simp add: INF_def Inf_top_conv)
haftmann@43868
   346
haftmann@43871
   347
lemma SUP_bot_conv:
haftmann@43871
   348
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@43871
   349
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@43872
   350
  by (auto simp add: SUP_def Sup_bot_conv)
haftmann@43868
   351
haftmann@43873
   352
lemma INF_UNIV_range:
haftmann@43871
   353
  "(\<Sqinter>x. f x) = \<Sqinter>range f"
haftmann@43872
   354
  by (fact INF_def)
haftmann@43871
   355
haftmann@43873
   356
lemma SUP_UNIV_range:
haftmann@43871
   357
  "(\<Squnion>x. f x) = \<Squnion>range f"
haftmann@43872
   358
  by (fact SUP_def)
haftmann@43871
   359
haftmann@43873
   360
lemma INF_UNIV_bool_expand:
haftmann@43868
   361
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@43868
   362
  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
haftmann@43868
   363
haftmann@43873
   364
lemma SUP_UNIV_bool_expand:
haftmann@43871
   365
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@43871
   366
  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
haftmann@43871
   367
haftmann@43899
   368
lemma INF_mono':
haftmann@43868
   369
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>B. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
haftmann@43868
   370
  -- {* The last inclusion is POSITIVE! *}
haftmann@43899
   371
  by (rule INF_mono) auto
haftmann@43868
   372
haftmann@43899
   373
lemma SUP_mono':
haftmann@43899
   374
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>B. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
haftmann@43871
   375
  -- {* The last inclusion is POSITIVE! *}
haftmann@43871
   376
  by (blast intro: SUP_mono dest: subsetD)
haftmann@43871
   377
haftmann@32077
   378
end
haftmann@32077
   379
haftmann@41082
   380
lemma Inf_less_iff:
haftmann@41082
   381
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43753
   382
  shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43754
   383
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@41082
   384
haftmann@43871
   385
lemma less_Sup_iff:
haftmann@43871
   386
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43871
   387
  shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43871
   388
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43871
   389
haftmann@43865
   390
lemma INF_less_iff:
haftmann@43865
   391
  fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43865
   392
  shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@43872
   393
  unfolding INF_def Inf_less_iff by auto
haftmann@43865
   394
hoelzl@40872
   395
lemma less_SUP_iff:
hoelzl@40872
   396
  fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43753
   397
  shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@43872
   398
  unfolding SUP_def less_Sup_iff by auto
hoelzl@40872
   399
haftmann@43873
   400
class complete_boolean_algebra = boolean_algebra + complete_lattice
haftmann@43873
   401
begin
haftmann@43873
   402
haftmann@43873
   403
lemma uminus_Inf:
haftmann@43873
   404
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   405
proof (rule antisym)
haftmann@43873
   406
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   407
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   408
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   409
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   410
qed
haftmann@43873
   411
haftmann@43873
   412
lemma uminus_Sup:
haftmann@43873
   413
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   414
proof -
haftmann@43873
   415
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
haftmann@43873
   416
  then show ?thesis by simp
haftmann@43873
   417
qed
haftmann@43873
   418
  
haftmann@43873
   419
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@43873
   420
  by (simp add: INF_def SUP_def uminus_Inf image_image)
haftmann@43873
   421
haftmann@43873
   422
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@43873
   423
  by (simp add: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   424
haftmann@43873
   425
end
haftmann@43873
   426
haftmann@43873
   427
haftmann@32139
   428
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077
   429
haftmann@43873
   430
instantiation bool :: complete_boolean_algebra
haftmann@32077
   431
begin
haftmann@32077
   432
haftmann@32077
   433
definition
haftmann@41080
   434
  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@32077
   435
haftmann@32077
   436
definition
haftmann@41080
   437
  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@32077
   438
haftmann@32077
   439
instance proof
haftmann@43852
   440
qed (auto simp add: Inf_bool_def Sup_bool_def)
haftmann@32077
   441
haftmann@32077
   442
end
haftmann@32077
   443
haftmann@43873
   444
lemma INF_bool_eq [simp]:
haftmann@32120
   445
  "INFI = Ball"
haftmann@32120
   446
proof (rule ext)+
haftmann@32120
   447
  fix A :: "'a set"
haftmann@32120
   448
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   449
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
haftmann@43872
   450
    by (auto simp add: Ball_def INF_def Inf_bool_def)
haftmann@32120
   451
qed
haftmann@32120
   452
haftmann@43873
   453
lemma SUP_bool_eq [simp]:
haftmann@32120
   454
  "SUPR = Bex"
haftmann@32120
   455
proof (rule ext)+
haftmann@32120
   456
  fix A :: "'a set"
haftmann@32120
   457
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   458
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@43872
   459
    by (auto simp add: Bex_def SUP_def Sup_bool_def)
haftmann@32120
   460
qed
haftmann@32120
   461
haftmann@32077
   462
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   463
begin
haftmann@32077
   464
haftmann@32077
   465
definition
haftmann@41080
   466
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   467
haftmann@41080
   468
lemma Inf_apply:
haftmann@41080
   469
  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   470
  by (simp add: Inf_fun_def)
haftmann@32077
   471
haftmann@32077
   472
definition
haftmann@41080
   473
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   474
haftmann@41080
   475
lemma Sup_apply:
haftmann@41080
   476
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   477
  by (simp add: Sup_fun_def)
haftmann@32077
   478
haftmann@32077
   479
instance proof
haftmann@41080
   480
qed (auto simp add: le_fun_def Inf_apply Sup_apply
haftmann@32077
   481
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@32077
   482
haftmann@32077
   483
end
haftmann@32077
   484
haftmann@43873
   485
lemma INF_apply:
haftmann@41080
   486
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@43872
   487
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
hoelzl@38705
   488
haftmann@43873
   489
lemma SUP_apply:
haftmann@41080
   490
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@43872
   491
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
haftmann@32077
   492
haftmann@43873
   493
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   494
haftmann@32077
   495
haftmann@41082
   496
subsection {* Inter *}
haftmann@41082
   497
haftmann@41082
   498
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   499
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   500
  
haftmann@41082
   501
notation (xsymbols)
haftmann@41082
   502
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   503
haftmann@41082
   504
lemma Inter_eq:
haftmann@41082
   505
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   506
proof (rule set_eqI)
haftmann@41082
   507
  fix x
haftmann@41082
   508
  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
   509
    by auto
haftmann@41082
   510
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   511
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082
   512
qed
haftmann@41082
   513
haftmann@43741
   514
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   515
  by (unfold Inter_eq) blast
haftmann@41082
   516
haftmann@43741
   517
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   518
  by (simp add: Inter_eq)
haftmann@41082
   519
haftmann@41082
   520
text {*
haftmann@41082
   521
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   522
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   523
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   524
*}
haftmann@41082
   525
haftmann@43741
   526
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   527
  by auto
haftmann@41082
   528
haftmann@43741
   529
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   530
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   531
    @{prop "X \<in> C"}. *}
haftmann@41082
   532
  by (unfold Inter_eq) blast
haftmann@41082
   533
haftmann@43741
   534
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   535
  by (fact Inf_lower)
haftmann@43740
   536
haftmann@41082
   537
lemma Inter_subset:
haftmann@43755
   538
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   539
  by (fact Inf_less_eq)
haftmann@41082
   540
haftmann@43755
   541
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   542
  by (fact Inf_greatest)
haftmann@41082
   543
haftmann@41082
   544
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@43739
   545
  by (fact Inf_binary [symmetric])
haftmann@41082
   546
haftmann@41082
   547
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082
   548
  by (fact Inf_empty)
haftmann@41082
   549
haftmann@41082
   550
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@43739
   551
  by (fact Inf_UNIV)
haftmann@41082
   552
haftmann@41082
   553
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739
   554
  by (fact Inf_insert)
haftmann@41082
   555
haftmann@41082
   556
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   557
  by (fact less_eq_Inf_inter)
haftmann@41082
   558
haftmann@41082
   559
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   560
  by (fact Inf_union_distrib)
haftmann@43756
   561
haftmann@43868
   562
lemma Inter_UNIV_conv [simp, no_atp]:
haftmann@43741
   563
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   564
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   565
  by (fact Inf_top_conv)+
haftmann@41082
   566
haftmann@43741
   567
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   568
  by (fact Inf_superset_mono)
haftmann@41082
   569
haftmann@41082
   570
haftmann@41082
   571
subsection {* Intersections of families *}
haftmann@41082
   572
haftmann@41082
   573
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   574
  "INTER \<equiv> INFI"
haftmann@41082
   575
haftmann@43872
   576
text {*
haftmann@43872
   577
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   578
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   579
*}
haftmann@43872
   580
haftmann@41082
   581
syntax
haftmann@41082
   582
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   583
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   584
haftmann@41082
   585
syntax (xsymbols)
haftmann@41082
   586
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   587
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   588
haftmann@41082
   589
syntax (latex output)
haftmann@41082
   590
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   591
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   592
haftmann@41082
   593
translations
haftmann@41082
   594
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   595
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   596
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   597
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   598
haftmann@41082
   599
print_translation {*
wenzelm@42284
   600
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   601
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   602
haftmann@41082
   603
lemma INTER_eq_Inter_image:
haftmann@41082
   604
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@43872
   605
  by (fact INF_def)
haftmann@41082
   606
  
haftmann@41082
   607
lemma Inter_def:
haftmann@41082
   608
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082
   609
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082
   610
haftmann@41082
   611
lemma INTER_def:
haftmann@41082
   612
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082
   613
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082
   614
haftmann@41082
   615
lemma Inter_image_eq [simp]:
haftmann@41082
   616
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43872
   617
  by (rule sym) (fact INF_def)
haftmann@41082
   618
haftmann@43817
   619
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@41082
   620
  by (unfold INTER_def) blast
haftmann@41082
   621
haftmann@43817
   622
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
   623
  by (unfold INTER_def) blast
haftmann@41082
   624
haftmann@43852
   625
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
   626
  by auto
haftmann@41082
   627
haftmann@43852
   628
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
   629
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@41082
   630
  by (unfold INTER_def) blast
haftmann@41082
   631
haftmann@41082
   632
lemma INT_cong [cong]:
haftmann@43854
   633
  "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
   634
  by (fact INF_cong)
haftmann@41082
   635
haftmann@41082
   636
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   637
  by blast
haftmann@41082
   638
haftmann@41082
   639
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   640
  by blast
haftmann@41082
   641
haftmann@43817
   642
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082
   643
  by (fact INF_leI)
haftmann@41082
   644
haftmann@43817
   645
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@43872
   646
  by (fact le_INF_I)
haftmann@41082
   647
haftmann@41082
   648
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@43872
   649
  by (fact INF_empty)
haftmann@43854
   650
haftmann@43817
   651
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   652
  by (fact INF_absorb)
haftmann@41082
   653
haftmann@43854
   654
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   655
  by (fact le_INF_iff)
haftmann@41082
   656
haftmann@41082
   657
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   658
  by (fact INF_insert)
haftmann@43865
   659
haftmann@43865
   660
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
   661
  by (fact INF_union)
haftmann@43865
   662
haftmann@43865
   663
lemma INT_insert_distrib:
haftmann@43865
   664
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   665
  by blast
haftmann@43854
   666
haftmann@41082
   667
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   668
  by (fact INF_constant)
haftmann@43865
   669
haftmann@41082
   670
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082
   671
  -- {* Look: it has an \emph{existential} quantifier *}
haftmann@43865
   672
  by (fact INF_eq)
haftmann@43865
   673
haftmann@43854
   674
lemma INTER_UNIV_conv [simp]:
haftmann@43817
   675
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   676
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43865
   677
  by (fact INF_top_conv)+
haftmann@43865
   678
haftmann@43865
   679
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   680
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   681
haftmann@43865
   682
lemma INT_anti_mono:
haftmann@43867
   683
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>B. g x)"
haftmann@43865
   684
  -- {* The last inclusion is POSITIVE! *}
haftmann@43899
   685
  by (fact INF_mono')
haftmann@41082
   686
haftmann@41082
   687
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   688
  by blast
haftmann@41082
   689
haftmann@43817
   690
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   691
  by blast
haftmann@41082
   692
haftmann@41082
   693
haftmann@32139
   694
subsection {* Union *}
haftmann@32115
   695
haftmann@32587
   696
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   697
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   698
haftmann@32115
   699
notation (xsymbols)
haftmann@32115
   700
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   701
haftmann@32135
   702
lemma Union_eq:
haftmann@32135
   703
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   704
proof (rule set_eqI)
haftmann@32115
   705
  fix x
haftmann@32135
   706
  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
   707
    by auto
haftmann@32135
   708
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@32587
   709
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115
   710
qed
haftmann@32115
   711
blanchet@35828
   712
lemma Union_iff [simp, no_atp]:
haftmann@32115
   713
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   714
  by (unfold Union_eq) blast
haftmann@32115
   715
haftmann@32115
   716
lemma UnionI [intro]:
haftmann@32115
   717
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   718
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   719
    @{term A} may be flexible. *}
haftmann@32115
   720
  by auto
haftmann@32115
   721
haftmann@32115
   722
lemma UnionE [elim!]:
haftmann@43817
   723
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   724
  by auto
haftmann@32115
   725
haftmann@43817
   726
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@32135
   727
  by (iprover intro: subsetI UnionI)
haftmann@32135
   728
haftmann@43817
   729
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@32135
   730
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
haftmann@32135
   731
haftmann@32135
   732
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135
   733
  by blast
haftmann@32135
   734
haftmann@43817
   735
lemma Union_empty [simp]: "\<Union>{} = {}"
haftmann@32135
   736
  by blast
haftmann@32135
   737
haftmann@43817
   738
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
haftmann@32135
   739
  by blast
haftmann@32135
   740
haftmann@43817
   741
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
haftmann@32135
   742
  by blast
haftmann@32135
   743
haftmann@43817
   744
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@32135
   745
  by blast
haftmann@32135
   746
haftmann@32135
   747
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@32135
   748
  by blast
haftmann@32135
   749
haftmann@43817
   750
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@32135
   751
  by blast
haftmann@32135
   752
haftmann@43817
   753
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@32135
   754
  by blast
haftmann@32135
   755
haftmann@43817
   756
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@32135
   757
  by blast
haftmann@32135
   758
haftmann@32135
   759
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   760
  by blast
haftmann@32135
   761
haftmann@32135
   762
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   763
  by blast
haftmann@32135
   764
haftmann@43817
   765
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@32135
   766
  by blast
haftmann@32135
   767
haftmann@32115
   768
haftmann@32139
   769
subsection {* Unions of families *}
haftmann@32077
   770
haftmann@32606
   771
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   772
  "UNION \<equiv> SUPR"
haftmann@32077
   773
haftmann@43872
   774
text {*
haftmann@43872
   775
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
   776
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   777
*}
haftmann@43872
   778
haftmann@32077
   779
syntax
wenzelm@35115
   780
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   781
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   782
haftmann@32077
   783
syntax (xsymbols)
wenzelm@35115
   784
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   785
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   786
haftmann@32077
   787
syntax (latex output)
wenzelm@35115
   788
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   789
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   790
haftmann@32077
   791
translations
haftmann@32077
   792
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   793
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   794
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   795
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   796
haftmann@32077
   797
text {*
haftmann@32077
   798
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   799
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   800
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   801
  former does not make the index expression a subscript of the
haftmann@32077
   802
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   803
  subscripts in Proof General.
haftmann@32077
   804
*}
haftmann@32077
   805
wenzelm@35115
   806
print_translation {*
wenzelm@42284
   807
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   808
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   809
haftmann@32135
   810
lemma UNION_eq_Union_image:
haftmann@43817
   811
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
haftmann@43872
   812
  by (fact SUP_def)
haftmann@32115
   813
haftmann@32115
   814
lemma Union_def:
haftmann@32117
   815
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   816
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   817
blanchet@35828
   818
lemma UNION_def [no_atp]:
haftmann@32135
   819
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   820
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   821
  
haftmann@32115
   822
lemma Union_image_eq [simp]:
haftmann@43817
   823
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   824
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   825
  
haftmann@43852
   826
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
wenzelm@11979
   827
  by (unfold UNION_def) blast
wenzelm@11979
   828
haftmann@43852
   829
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
   830
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   831
    @{term b} may be flexible. *}
wenzelm@11979
   832
  by auto
wenzelm@11979
   833
haftmann@43852
   834
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
   835
  by (unfold UNION_def) blast
clasohm@923
   836
wenzelm@11979
   837
lemma UN_cong [cong]:
haftmann@43852
   838
    "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)"
wenzelm@11979
   839
  by (simp add: UNION_def)
wenzelm@11979
   840
berghofe@29691
   841
lemma strong_UN_cong:
haftmann@43852
   842
    "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)"
berghofe@29691
   843
  by (simp add: UNION_def simp_implies_def)
berghofe@29691
   844
haftmann@43817
   845
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
   846
  by blast
haftmann@32077
   847
haftmann@43817
   848
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@43872
   849
  by (fact le_SUP_I)
haftmann@32135
   850
haftmann@43817
   851
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135
   852
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135
   853
blanchet@35828
   854
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   855
  by blast
haftmann@32135
   856
haftmann@43817
   857
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
   858
  by blast
haftmann@32135
   859
blanchet@35828
   860
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135
   861
  by blast
haftmann@32135
   862
haftmann@32135
   863
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135
   864
  by blast
haftmann@32135
   865
haftmann@32135
   866
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   867
  by blast
haftmann@32135
   868
haftmann@43817
   869
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135
   870
  by auto
haftmann@32135
   871
haftmann@32135
   872
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135
   873
  by blast
haftmann@32135
   874
haftmann@32135
   875
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@32135
   876
  by blast
haftmann@32135
   877
haftmann@32135
   878
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
   879
  by blast
haftmann@32135
   880
haftmann@32135
   881
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   882
  by (fact SUP_le_iff)
haftmann@32135
   883
haftmann@32135
   884
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   885
  by blast
haftmann@32135
   886
haftmann@32135
   887
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135
   888
  by auto
haftmann@32135
   889
haftmann@32135
   890
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135
   891
  by blast
haftmann@32135
   892
haftmann@32135
   893
lemma UNION_empty_conv[simp]:
haftmann@43817
   894
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
   895
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@32135
   896
by blast+
haftmann@32135
   897
blanchet@35828
   898
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   899
  by blast
haftmann@32135
   900
haftmann@32135
   901
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
   902
  by blast
haftmann@32135
   903
haftmann@32135
   904
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
   905
  by blast
haftmann@32135
   906
haftmann@32135
   907
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   908
  by (auto simp add: split_if_mem2)
haftmann@32135
   909
haftmann@43817
   910
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@32135
   911
  by (auto intro: bool_contrapos)
haftmann@32135
   912
haftmann@32135
   913
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   914
  by blast
haftmann@32135
   915
haftmann@32135
   916
lemma UN_mono:
haftmann@43817
   917
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
   918
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135
   919
  by (blast dest: subsetD)
haftmann@32135
   920
haftmann@43817
   921
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
   922
  by blast
haftmann@32135
   923
haftmann@43817
   924
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
   925
  by blast
haftmann@32135
   926
haftmann@43817
   927
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
   928
  -- {* NOT suitable for rewriting *}
haftmann@32135
   929
  by blast
haftmann@32135
   930
haftmann@43817
   931
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
   932
  by blast
haftmann@32135
   933
wenzelm@11979
   934
haftmann@32139
   935
subsection {* Distributive laws *}
wenzelm@12897
   936
wenzelm@12897
   937
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   938
  by blast
wenzelm@12897
   939
wenzelm@12897
   940
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   941
  by blast
wenzelm@12897
   942
haftmann@43817
   943
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897
   944
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   945
  -- {* Union of a family of unions *}
wenzelm@12897
   946
  by blast
wenzelm@12897
   947
wenzelm@12897
   948
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
   949
  -- {* Equivalent version *}
wenzelm@12897
   950
  by blast
wenzelm@12897
   951
wenzelm@12897
   952
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   953
  by blast
wenzelm@12897
   954
haftmann@43817
   955
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
wenzelm@12897
   956
  by blast
wenzelm@12897
   957
wenzelm@12897
   958
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
   959
  -- {* Equivalent version *}
wenzelm@12897
   960
  by blast
wenzelm@12897
   961
wenzelm@12897
   962
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   963
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   964
  by blast
wenzelm@12897
   965
wenzelm@12897
   966
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   967
  by blast
wenzelm@12897
   968
wenzelm@12897
   969
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
   970
  by blast
wenzelm@12897
   971
wenzelm@12897
   972
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
   973
  by blast
wenzelm@12897
   974
wenzelm@12897
   975
haftmann@32139
   976
subsection {* Complement *}
haftmann@32135
   977
haftmann@43873
   978
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
   979
  by (fact uminus_INF)
wenzelm@12897
   980
haftmann@43873
   981
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
   982
  by (fact uminus_SUP)
wenzelm@12897
   983
wenzelm@12897
   984
haftmann@32139
   985
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   986
paulson@13860
   987
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   988
           and Intersections. *}
wenzelm@12897
   989
wenzelm@12897
   990
lemma UN_simps [simp]:
haftmann@43817
   991
  "\<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@43852
   992
  "\<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
   993
  "\<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@43852
   994
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter>B)"
haftmann@43852
   995
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
   996
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
   997
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
   998
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
   999
  "\<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
  1000
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1001
  by auto
wenzelm@12897
  1002
wenzelm@12897
  1003
lemma INT_simps [simp]:
haftmann@43831
  1004
  "\<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
  1005
  "\<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
  1006
  "\<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
  1007
  "\<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
  1008
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1009
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1010
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1011
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1012
  "\<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
  1013
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1014
  by auto
wenzelm@12897
  1015
blanchet@35828
  1016
lemma ball_simps [simp,no_atp]:
haftmann@43852
  1017
  "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
haftmann@43852
  1018
  "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
haftmann@43852
  1019
  "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
haftmann@43852
  1020
  "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
haftmann@43852
  1021
  "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
haftmann@43852
  1022
  "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
haftmann@43852
  1023
  "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
haftmann@43852
  1024
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43852
  1025
  "\<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
  1026
  "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
haftmann@43852
  1027
  "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
haftmann@43852
  1028
  "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
wenzelm@12897
  1029
  by auto
wenzelm@12897
  1030
blanchet@35828
  1031
lemma bex_simps [simp,no_atp]:
haftmann@43852
  1032
  "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
haftmann@43852
  1033
  "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
haftmann@43852
  1034
  "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
haftmann@43852
  1035
  "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
haftmann@43852
  1036
  "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
haftmann@43852
  1037
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1038
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
haftmann@43852
  1039
  "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
haftmann@43852
  1040
  "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
haftmann@43852
  1041
  "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
wenzelm@12897
  1042
  by auto
wenzelm@12897
  1043
paulson@13860
  1044
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1045
paulson@13860
  1046
lemma UN_extend_simps:
haftmann@43817
  1047
  "\<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@43852
  1048
  "\<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
  1049
  "\<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
  1050
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1051
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1052
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1053
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1054
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1055
  "\<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
  1056
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1057
  by auto
paulson@13860
  1058
paulson@13860
  1059
lemma INT_extend_simps:
haftmann@43852
  1060
  "\<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
  1061
  "\<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
  1062
  "\<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
  1063
  "\<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
  1064
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1065
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1066
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1067
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1068
  "\<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
  1069
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1070
  by auto
paulson@13860
  1071
paulson@13860
  1072
haftmann@43872
  1073
text {* Legacy names *}
haftmann@43872
  1074
haftmann@43872
  1075
lemmas (in complete_lattice) INFI_def = INF_def
haftmann@43872
  1076
lemmas (in complete_lattice) SUPR_def = SUP_def
haftmann@43872
  1077
lemmas (in complete_lattice) le_SUPI = le_SUP_I
haftmann@43872
  1078
lemmas (in complete_lattice) le_SUPI2 = le_SUP_I2
haftmann@43872
  1079
lemmas (in complete_lattice) le_INFI = le_INF_I
haftmann@43899
  1080
lemmas (in complete_lattice) INF_subset = INF_superset_mono 
haftmann@43873
  1081
lemmas INFI_apply = INF_apply
haftmann@43873
  1082
lemmas SUPR_apply = SUP_apply
haftmann@43872
  1083
haftmann@43872
  1084
text {* Finally *}
haftmann@43872
  1085
haftmann@32135
  1086
no_notation
haftmann@32135
  1087
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
  1088
  less (infix "\<sqsubset>" 50) and
haftmann@41082
  1089
  bot ("\<bottom>") and
haftmann@41082
  1090
  top ("\<top>") and
haftmann@32135
  1091
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
  1092
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
  1093
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
  1094
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
  1095
haftmann@41080
  1096
no_syntax (xsymbols)
haftmann@41082
  1097
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
  1098
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1099
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
  1100
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1101
haftmann@30596
  1102
lemmas mem_simps =
haftmann@30596
  1103
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1104
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1105
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1106
wenzelm@11979
  1107
end