src/HOL/Complete_Lattice.thy
author haftmann
Mon Jul 18 21:52:34 2011 +0200 (2011-07-18)
changeset 43901 3ab6c30d256d
parent 43900 7162691e740b
child 43940 26ca0bad226a
permissions -rw-r--r--
proof tuning
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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@43899
   295
  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
haftmann@43899
   296
  by (rule INF_mono) auto
haftmann@43865
   297
haftmann@43900
   298
lemma SUP_subset_mono:
haftmann@43870
   299
  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
haftmann@43899
   300
  by (rule SUP_mono) auto
haftmann@43870
   301
haftmann@43865
   302
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
   303
  by (iprover intro: INF_leI le_INF_I order_trans antisym)
haftmann@43865
   304
haftmann@43870
   305
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)"
haftmann@43872
   306
  by (iprover intro: SUP_leI le_SUP_I order_trans antisym)
haftmann@43870
   307
haftmann@43871
   308
lemma INF_absorb:
haftmann@43868
   309
  assumes "k \<in> I"
haftmann@43868
   310
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
haftmann@43868
   311
proof -
haftmann@43868
   312
  from assms obtain J where "I = insert k J" by blast
haftmann@43868
   313
  then show ?thesis by (simp add: INF_insert)
haftmann@43868
   314
qed
haftmann@43868
   315
haftmann@43871
   316
lemma SUP_absorb:
haftmann@43871
   317
  assumes "k \<in> I"
haftmann@43871
   318
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
haftmann@43871
   319
proof -
haftmann@43871
   320
  from assms obtain J where "I = insert k J" by blast
haftmann@43871
   321
  then show ?thesis by (simp add: SUP_insert)
haftmann@43871
   322
qed
haftmann@43871
   323
haftmann@43871
   324
lemma INF_union:
haftmann@43868
   325
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
haftmann@43872
   326
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INF_I INF_leI)
haftmann@43868
   327
haftmann@43871
   328
lemma SUP_union:
haftmann@43871
   329
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
haftmann@43872
   330
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_leI le_SUP_I)
haftmann@43871
   331
haftmann@43871
   332
lemma INF_constant:
haftmann@43868
   333
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
haftmann@43868
   334
  by (simp add: INF_empty)
haftmann@43868
   335
haftmann@43871
   336
lemma SUP_constant:
haftmann@43871
   337
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
haftmann@43871
   338
  by (simp add: SUP_empty)
haftmann@43871
   339
haftmann@43871
   340
lemma INF_eq:
haftmann@43868
   341
  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43872
   342
  by (simp add: INF_def image_def)
haftmann@43868
   343
haftmann@43871
   344
lemma SUP_eq:
haftmann@43871
   345
  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43872
   346
  by (simp add: SUP_def image_def)
haftmann@43871
   347
haftmann@43871
   348
lemma INF_top_conv:
haftmann@43868
   349
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@43868
   350
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
haftmann@43872
   351
  by (auto simp add: INF_def Inf_top_conv)
haftmann@43868
   352
haftmann@43871
   353
lemma SUP_bot_conv:
haftmann@43871
   354
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@43871
   355
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
haftmann@43872
   356
  by (auto simp add: SUP_def Sup_bot_conv)
haftmann@43868
   357
haftmann@43873
   358
lemma INF_UNIV_range:
haftmann@43871
   359
  "(\<Sqinter>x. f x) = \<Sqinter>range f"
haftmann@43872
   360
  by (fact INF_def)
haftmann@43871
   361
haftmann@43873
   362
lemma SUP_UNIV_range:
haftmann@43871
   363
  "(\<Squnion>x. f x) = \<Squnion>range f"
haftmann@43872
   364
  by (fact SUP_def)
haftmann@43871
   365
haftmann@43873
   366
lemma INF_UNIV_bool_expand:
haftmann@43868
   367
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
haftmann@43868
   368
  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
haftmann@43868
   369
haftmann@43873
   370
lemma SUP_UNIV_bool_expand:
haftmann@43871
   371
  "(\<Squnion>b. A b) = A True \<squnion> A False"
haftmann@43871
   372
  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
haftmann@43871
   373
haftmann@43899
   374
lemma INF_mono':
haftmann@43900
   375
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
haftmann@43868
   376
  -- {* The last inclusion is POSITIVE! *}
haftmann@43899
   377
  by (rule INF_mono) auto
haftmann@43868
   378
haftmann@43899
   379
lemma SUP_mono':
haftmann@43900
   380
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
haftmann@43871
   381
  -- {* The last inclusion is POSITIVE! *}
haftmann@43871
   382
  by (blast intro: SUP_mono dest: subsetD)
haftmann@43871
   383
haftmann@32077
   384
end
haftmann@32077
   385
haftmann@41082
   386
lemma Inf_less_iff:
haftmann@41082
   387
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43753
   388
  shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
haftmann@43754
   389
  unfolding not_le [symmetric] le_Inf_iff by auto
haftmann@41082
   390
haftmann@43871
   391
lemma less_Sup_iff:
haftmann@43871
   392
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
haftmann@43871
   393
  shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
haftmann@43871
   394
  unfolding not_le [symmetric] Sup_le_iff by auto
haftmann@43871
   395
haftmann@43865
   396
lemma INF_less_iff:
haftmann@43865
   397
  fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43865
   398
  shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
haftmann@43872
   399
  unfolding INF_def Inf_less_iff by auto
haftmann@43865
   400
hoelzl@40872
   401
lemma less_SUP_iff:
hoelzl@40872
   402
  fixes a :: "'a::{complete_lattice,linorder}"
haftmann@43753
   403
  shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
haftmann@43872
   404
  unfolding SUP_def less_Sup_iff by auto
hoelzl@40872
   405
haftmann@43873
   406
class complete_boolean_algebra = boolean_algebra + complete_lattice
haftmann@43873
   407
begin
haftmann@43873
   408
haftmann@43873
   409
lemma uminus_Inf:
haftmann@43873
   410
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
haftmann@43873
   411
proof (rule antisym)
haftmann@43873
   412
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
haftmann@43873
   413
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
haftmann@43873
   414
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
haftmann@43873
   415
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
haftmann@43873
   416
qed
haftmann@43873
   417
haftmann@43873
   418
lemma uminus_Sup:
haftmann@43873
   419
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
haftmann@43873
   420
proof -
haftmann@43873
   421
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
haftmann@43873
   422
  then show ?thesis by simp
haftmann@43873
   423
qed
haftmann@43873
   424
  
haftmann@43873
   425
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
haftmann@43873
   426
  by (simp add: INF_def SUP_def uminus_Inf image_image)
haftmann@43873
   427
haftmann@43873
   428
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
haftmann@43873
   429
  by (simp add: INF_def SUP_def uminus_Sup image_image)
haftmann@43873
   430
haftmann@43873
   431
end
haftmann@43873
   432
haftmann@43873
   433
haftmann@32139
   434
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
haftmann@32077
   435
haftmann@43873
   436
instantiation bool :: complete_boolean_algebra
haftmann@32077
   437
begin
haftmann@32077
   438
haftmann@32077
   439
definition
haftmann@41080
   440
  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@32077
   441
haftmann@32077
   442
definition
haftmann@41080
   443
  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@32077
   444
haftmann@32077
   445
instance proof
haftmann@43852
   446
qed (auto simp add: Inf_bool_def Sup_bool_def)
haftmann@32077
   447
haftmann@32077
   448
end
haftmann@32077
   449
haftmann@43873
   450
lemma INF_bool_eq [simp]:
haftmann@32120
   451
  "INFI = Ball"
haftmann@32120
   452
proof (rule ext)+
haftmann@32120
   453
  fix A :: "'a set"
haftmann@32120
   454
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   455
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
haftmann@43872
   456
    by (auto simp add: Ball_def INF_def Inf_bool_def)
haftmann@32120
   457
qed
haftmann@32120
   458
haftmann@43873
   459
lemma SUP_bool_eq [simp]:
haftmann@32120
   460
  "SUPR = Bex"
haftmann@32120
   461
proof (rule ext)+
haftmann@32120
   462
  fix A :: "'a set"
haftmann@32120
   463
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   464
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@43872
   465
    by (auto simp add: Bex_def SUP_def Sup_bool_def)
haftmann@32120
   466
qed
haftmann@32120
   467
haftmann@32077
   468
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   469
begin
haftmann@32077
   470
haftmann@32077
   471
definition
haftmann@41080
   472
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   473
haftmann@41080
   474
lemma Inf_apply:
haftmann@41080
   475
  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   476
  by (simp add: Inf_fun_def)
haftmann@32077
   477
haftmann@32077
   478
definition
haftmann@41080
   479
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   480
haftmann@41080
   481
lemma Sup_apply:
haftmann@41080
   482
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   483
  by (simp add: Sup_fun_def)
haftmann@32077
   484
haftmann@32077
   485
instance proof
haftmann@41080
   486
qed (auto simp add: le_fun_def Inf_apply Sup_apply
haftmann@32077
   487
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@32077
   488
haftmann@32077
   489
end
haftmann@32077
   490
haftmann@43873
   491
lemma INF_apply:
haftmann@41080
   492
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@43872
   493
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
hoelzl@38705
   494
haftmann@43873
   495
lemma SUP_apply:
haftmann@41080
   496
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@43872
   497
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
haftmann@32077
   498
haftmann@43873
   499
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
haftmann@43873
   500
haftmann@32077
   501
haftmann@41082
   502
subsection {* Inter *}
haftmann@41082
   503
haftmann@41082
   504
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   505
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   506
  
haftmann@41082
   507
notation (xsymbols)
haftmann@41082
   508
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   509
haftmann@41082
   510
lemma Inter_eq:
haftmann@41082
   511
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   512
proof (rule set_eqI)
haftmann@41082
   513
  fix x
haftmann@41082
   514
  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
   515
    by auto
haftmann@41082
   516
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   517
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082
   518
qed
haftmann@41082
   519
haftmann@43741
   520
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   521
  by (unfold Inter_eq) blast
haftmann@41082
   522
haftmann@43741
   523
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   524
  by (simp add: Inter_eq)
haftmann@41082
   525
haftmann@41082
   526
text {*
haftmann@41082
   527
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   528
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   529
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   530
*}
haftmann@41082
   531
haftmann@43741
   532
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   533
  by auto
haftmann@41082
   534
haftmann@43741
   535
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   536
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   537
    @{prop "X \<in> C"}. *}
haftmann@41082
   538
  by (unfold Inter_eq) blast
haftmann@41082
   539
haftmann@43741
   540
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   541
  by (fact Inf_lower)
haftmann@43740
   542
haftmann@41082
   543
lemma Inter_subset:
haftmann@43755
   544
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   545
  by (fact Inf_less_eq)
haftmann@41082
   546
haftmann@43755
   547
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   548
  by (fact Inf_greatest)
haftmann@41082
   549
haftmann@41082
   550
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@43739
   551
  by (fact Inf_binary [symmetric])
haftmann@41082
   552
haftmann@41082
   553
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082
   554
  by (fact Inf_empty)
haftmann@41082
   555
haftmann@41082
   556
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@43739
   557
  by (fact Inf_UNIV)
haftmann@41082
   558
haftmann@41082
   559
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739
   560
  by (fact Inf_insert)
haftmann@41082
   561
haftmann@41082
   562
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43899
   563
  by (fact less_eq_Inf_inter)
haftmann@41082
   564
haftmann@41082
   565
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   566
  by (fact Inf_union_distrib)
haftmann@43756
   567
haftmann@43868
   568
lemma Inter_UNIV_conv [simp, no_atp]:
haftmann@43741
   569
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   570
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   571
  by (fact Inf_top_conv)+
haftmann@41082
   572
haftmann@43741
   573
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43899
   574
  by (fact Inf_superset_mono)
haftmann@41082
   575
haftmann@41082
   576
haftmann@41082
   577
subsection {* Intersections of families *}
haftmann@41082
   578
haftmann@41082
   579
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   580
  "INTER \<equiv> INFI"
haftmann@41082
   581
haftmann@43872
   582
text {*
haftmann@43872
   583
  Note: must use name @{const INTER} here instead of @{text INT}
haftmann@43872
   584
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   585
*}
haftmann@43872
   586
haftmann@41082
   587
syntax
haftmann@41082
   588
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   589
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   590
haftmann@41082
   591
syntax (xsymbols)
haftmann@41082
   592
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   593
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   594
haftmann@41082
   595
syntax (latex output)
haftmann@41082
   596
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   597
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   598
haftmann@41082
   599
translations
haftmann@41082
   600
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   601
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   602
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   603
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   604
haftmann@41082
   605
print_translation {*
wenzelm@42284
   606
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   607
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   608
haftmann@41082
   609
lemma INTER_eq_Inter_image:
haftmann@41082
   610
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@43872
   611
  by (fact INF_def)
haftmann@41082
   612
  
haftmann@41082
   613
lemma Inter_def:
haftmann@41082
   614
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082
   615
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082
   616
haftmann@41082
   617
lemma INTER_def:
haftmann@41082
   618
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082
   619
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082
   620
haftmann@41082
   621
lemma Inter_image_eq [simp]:
haftmann@41082
   622
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43872
   623
  by (rule sym) (fact INF_def)
haftmann@41082
   624
haftmann@43817
   625
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@41082
   626
  by (unfold INTER_def) blast
haftmann@41082
   627
haftmann@43817
   628
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
haftmann@41082
   629
  by (unfold INTER_def) blast
haftmann@41082
   630
haftmann@43852
   631
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
haftmann@41082
   632
  by auto
haftmann@41082
   633
haftmann@43852
   634
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@43852
   635
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
haftmann@41082
   636
  by (unfold INTER_def) blast
haftmann@41082
   637
haftmann@41082
   638
lemma INT_cong [cong]:
haftmann@43854
   639
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
haftmann@43865
   640
  by (fact INF_cong)
haftmann@41082
   641
haftmann@41082
   642
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   643
  by blast
haftmann@41082
   644
haftmann@41082
   645
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   646
  by blast
haftmann@41082
   647
haftmann@43817
   648
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082
   649
  by (fact INF_leI)
haftmann@41082
   650
haftmann@43817
   651
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@43872
   652
  by (fact le_INF_I)
haftmann@41082
   653
haftmann@41082
   654
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@43872
   655
  by (fact INF_empty)
haftmann@43854
   656
haftmann@43817
   657
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@43872
   658
  by (fact INF_absorb)
haftmann@41082
   659
haftmann@43854
   660
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   661
  by (fact le_INF_iff)
haftmann@41082
   662
haftmann@41082
   663
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@43865
   664
  by (fact INF_insert)
haftmann@43865
   665
haftmann@43865
   666
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
haftmann@43865
   667
  by (fact INF_union)
haftmann@43865
   668
haftmann@43865
   669
lemma INT_insert_distrib:
haftmann@43865
   670
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@43865
   671
  by blast
haftmann@43854
   672
haftmann@41082
   673
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@43865
   674
  by (fact INF_constant)
haftmann@43865
   675
haftmann@41082
   676
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082
   677
  -- {* Look: it has an \emph{existential} quantifier *}
haftmann@43865
   678
  by (fact INF_eq)
haftmann@43865
   679
haftmann@43854
   680
lemma INTER_UNIV_conv [simp]:
haftmann@43817
   681
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   682
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43865
   683
  by (fact INF_top_conv)+
haftmann@43865
   684
haftmann@43865
   685
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
haftmann@43873
   686
  by (fact INF_UNIV_bool_expand)
haftmann@43865
   687
haftmann@43865
   688
lemma INT_anti_mono:
haftmann@43900
   689
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@43865
   690
  -- {* The last inclusion is POSITIVE! *}
haftmann@43899
   691
  by (fact INF_mono')
haftmann@41082
   692
haftmann@41082
   693
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   694
  by blast
haftmann@41082
   695
haftmann@43817
   696
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   697
  by blast
haftmann@41082
   698
haftmann@41082
   699
haftmann@32139
   700
subsection {* Union *}
haftmann@32115
   701
haftmann@32587
   702
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   703
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   704
haftmann@32115
   705
notation (xsymbols)
haftmann@32115
   706
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   707
haftmann@32135
   708
lemma Union_eq:
haftmann@32135
   709
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   710
proof (rule set_eqI)
haftmann@32115
   711
  fix x
haftmann@32135
   712
  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
   713
    by auto
haftmann@32135
   714
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@32587
   715
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115
   716
qed
haftmann@32115
   717
blanchet@35828
   718
lemma Union_iff [simp, no_atp]:
haftmann@32115
   719
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   720
  by (unfold Union_eq) blast
haftmann@32115
   721
haftmann@32115
   722
lemma UnionI [intro]:
haftmann@32115
   723
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   724
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   725
    @{term A} may be flexible. *}
haftmann@32115
   726
  by auto
haftmann@32115
   727
haftmann@32115
   728
lemma UnionE [elim!]:
haftmann@43817
   729
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   730
  by auto
haftmann@32115
   731
haftmann@43817
   732
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@43901
   733
  by (fact Sup_upper)
haftmann@32135
   734
haftmann@43817
   735
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@43901
   736
  by (fact Sup_least)
haftmann@32135
   737
haftmann@32135
   738
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135
   739
  by blast
haftmann@32135
   740
haftmann@43817
   741
lemma Union_empty [simp]: "\<Union>{} = {}"
haftmann@43901
   742
  by (fact Sup_empty)
haftmann@32135
   743
haftmann@43817
   744
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
haftmann@43901
   745
  by (fact Sup_UNIV)
haftmann@32135
   746
haftmann@43817
   747
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
haftmann@43901
   748
  by (fact Sup_insert)
haftmann@32135
   749
haftmann@43817
   750
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@43901
   751
  by (fact Sup_union_distrib)
haftmann@32135
   752
haftmann@32135
   753
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@43901
   754
  by (fact Sup_inter_less_eq)
haftmann@32135
   755
haftmann@43817
   756
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@43901
   757
  by (fact Sup_bot_conv)
haftmann@32135
   758
haftmann@43817
   759
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@43901
   760
  by (fact Sup_bot_conv)
haftmann@32135
   761
haftmann@43901
   762
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" -- "FIXME generalize"
haftmann@32135
   763
  by blast
haftmann@32135
   764
haftmann@32135
   765
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   766
  by blast
haftmann@32135
   767
haftmann@32135
   768
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   769
  by blast
haftmann@32135
   770
haftmann@43817
   771
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@43901
   772
  by (fact Sup_subset_mono)
haftmann@32135
   773
haftmann@32115
   774
haftmann@32139
   775
subsection {* Unions of families *}
haftmann@32077
   776
haftmann@32606
   777
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   778
  "UNION \<equiv> SUPR"
haftmann@32077
   779
haftmann@43872
   780
text {*
haftmann@43872
   781
  Note: must use name @{const UNION} here instead of @{text UN}
haftmann@43872
   782
  to allow the following syntax coexist with the plain constant name.
haftmann@43872
   783
*}
haftmann@43872
   784
haftmann@32077
   785
syntax
wenzelm@35115
   786
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   787
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   788
haftmann@32077
   789
syntax (xsymbols)
wenzelm@35115
   790
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   791
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   792
haftmann@32077
   793
syntax (latex output)
wenzelm@35115
   794
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   795
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   796
haftmann@32077
   797
translations
haftmann@32077
   798
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   799
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   800
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   801
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   802
haftmann@32077
   803
text {*
haftmann@32077
   804
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   805
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   806
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   807
  former does not make the index expression a subscript of the
haftmann@32077
   808
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   809
  subscripts in Proof General.
haftmann@32077
   810
*}
haftmann@32077
   811
wenzelm@35115
   812
print_translation {*
wenzelm@42284
   813
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   814
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   815
haftmann@32135
   816
lemma UNION_eq_Union_image:
haftmann@43817
   817
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
haftmann@43872
   818
  by (fact SUP_def)
haftmann@32115
   819
haftmann@32115
   820
lemma Union_def:
haftmann@32117
   821
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   822
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   823
blanchet@35828
   824
lemma UNION_def [no_atp]:
haftmann@32135
   825
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   826
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   827
  
haftmann@32115
   828
lemma Union_image_eq [simp]:
haftmann@43817
   829
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   830
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   831
  
haftmann@43852
   832
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
wenzelm@11979
   833
  by (unfold UNION_def) blast
wenzelm@11979
   834
haftmann@43852
   835
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
wenzelm@11979
   836
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   837
    @{term b} may be flexible. *}
wenzelm@11979
   838
  by auto
wenzelm@11979
   839
haftmann@43852
   840
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@11979
   841
  by (unfold UNION_def) blast
clasohm@923
   842
wenzelm@11979
   843
lemma UN_cong [cong]:
haftmann@43900
   844
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
haftmann@43900
   845
  by (fact SUP_cong)
wenzelm@11979
   846
berghofe@29691
   847
lemma strong_UN_cong:
haftmann@43900
   848
  "A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
haftmann@43900
   849
  by (unfold simp_implies_def) (fact UN_cong)
berghofe@29691
   850
haftmann@43817
   851
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
   852
  by blast
haftmann@32077
   853
haftmann@43817
   854
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@43872
   855
  by (fact le_SUP_I)
haftmann@32135
   856
haftmann@43817
   857
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@43900
   858
  by (fact SUP_leI)
haftmann@32135
   859
blanchet@35828
   860
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   861
  by blast
haftmann@32135
   862
haftmann@43817
   863
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
   864
  by blast
haftmann@32135
   865
haftmann@43900
   866
lemma UN_empty [simp, no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@43900
   867
  by (fact SUP_empty)
haftmann@32135
   868
haftmann@32135
   869
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@43900
   870
  by (fact SUP_bot)
haftmann@32135
   871
haftmann@32135
   872
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   873
  by blast
haftmann@32135
   874
haftmann@43817
   875
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@43900
   876
  by (fact SUP_absorb)
haftmann@32135
   877
haftmann@32135
   878
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@43900
   879
  by (fact SUP_insert)
haftmann@32135
   880
haftmann@32135
   881
lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
haftmann@43900
   882
  by (fact SUP_union)
haftmann@32135
   883
haftmann@43900
   884
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)" -- "FIXME generalize"
haftmann@32135
   885
  by blast
haftmann@32135
   886
haftmann@32135
   887
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   888
  by (fact SUP_le_iff)
haftmann@32135
   889
haftmann@32135
   890
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@43900
   891
  by (fact SUP_constant)
haftmann@32135
   892
haftmann@32135
   893
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@43900
   894
  by (fact SUP_eq)
haftmann@43900
   895
haftmann@43900
   896
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" -- "FIXME generalize"
haftmann@32135
   897
  by blast
haftmann@32135
   898
haftmann@32135
   899
lemma UNION_empty_conv[simp]:
haftmann@43817
   900
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
   901
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43900
   902
  by (fact SUP_bot_conv)+
haftmann@32135
   903
blanchet@35828
   904
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   905
  by blast
haftmann@32135
   906
haftmann@43900
   907
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
   908
  by blast
haftmann@32135
   909
haftmann@43900
   910
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
   911
  by blast
haftmann@32135
   912
haftmann@32135
   913
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   914
  by (auto simp add: split_if_mem2)
haftmann@32135
   915
haftmann@43817
   916
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@43900
   917
  by (fact SUP_UNIV_bool_expand)
haftmann@32135
   918
haftmann@32135
   919
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   920
  by blast
haftmann@32135
   921
haftmann@32135
   922
lemma UN_mono:
haftmann@43817
   923
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
   924
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@43900
   925
  by (fact SUP_mono')
haftmann@32135
   926
haftmann@43817
   927
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
   928
  by blast
haftmann@32135
   929
haftmann@43817
   930
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
   931
  by blast
haftmann@32135
   932
haftmann@43817
   933
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
   934
  -- {* NOT suitable for rewriting *}
haftmann@32135
   935
  by blast
haftmann@32135
   936
haftmann@43817
   937
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
   938
  by blast
haftmann@32135
   939
wenzelm@11979
   940
haftmann@32139
   941
subsection {* Distributive laws *}
wenzelm@12897
   942
wenzelm@12897
   943
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   944
  by blast
wenzelm@12897
   945
wenzelm@12897
   946
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   947
  by blast
wenzelm@12897
   948
haftmann@43817
   949
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897
   950
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   951
  -- {* Union of a family of unions *}
wenzelm@12897
   952
  by blast
wenzelm@12897
   953
wenzelm@12897
   954
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
wenzelm@12897
   955
  -- {* Equivalent version *}
wenzelm@12897
   956
  by blast
wenzelm@12897
   957
wenzelm@12897
   958
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   959
  by blast
wenzelm@12897
   960
haftmann@43817
   961
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
wenzelm@12897
   962
  by blast
wenzelm@12897
   963
wenzelm@12897
   964
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
wenzelm@12897
   965
  -- {* Equivalent version *}
wenzelm@12897
   966
  by blast
wenzelm@12897
   967
wenzelm@12897
   968
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   969
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   970
  by blast
wenzelm@12897
   971
wenzelm@12897
   972
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   973
  by blast
wenzelm@12897
   974
wenzelm@12897
   975
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
wenzelm@12897
   976
  by blast
wenzelm@12897
   977
wenzelm@12897
   978
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
wenzelm@12897
   979
  by blast
wenzelm@12897
   980
wenzelm@12897
   981
haftmann@32139
   982
subsection {* Complement *}
haftmann@32135
   983
haftmann@43873
   984
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
haftmann@43873
   985
  by (fact uminus_INF)
wenzelm@12897
   986
haftmann@43873
   987
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
haftmann@43873
   988
  by (fact uminus_SUP)
wenzelm@12897
   989
wenzelm@12897
   990
haftmann@32139
   991
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   992
paulson@13860
   993
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   994
           and Intersections. *}
wenzelm@12897
   995
wenzelm@12897
   996
lemma UN_simps [simp]:
haftmann@43817
   997
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
haftmann@43852
   998
  "\<And>A B C. (\<Union>x\<in>C. A x \<union>  B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
haftmann@43852
   999
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
haftmann@43852
  1000
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter>B)"
haftmann@43852
  1001
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
haftmann@43852
  1002
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
haftmann@43852
  1003
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
haftmann@43852
  1004
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
haftmann@43852
  1005
  "\<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
  1006
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
wenzelm@12897
  1007
  by auto
wenzelm@12897
  1008
wenzelm@12897
  1009
lemma INT_simps [simp]:
haftmann@43831
  1010
  "\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter>B)"
haftmann@43831
  1011
  "\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))"
haftmann@43852
  1012
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
haftmann@43852
  1013
  "\<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
  1014
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43852
  1015
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
haftmann@43852
  1016
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
haftmann@43852
  1017
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
haftmann@43852
  1018
  "\<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
  1019
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
wenzelm@12897
  1020
  by auto
wenzelm@12897
  1021
blanchet@35828
  1022
lemma ball_simps [simp,no_atp]:
haftmann@43852
  1023
  "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
haftmann@43852
  1024
  "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
haftmann@43852
  1025
  "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
haftmann@43852
  1026
  "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
haftmann@43852
  1027
  "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
haftmann@43852
  1028
  "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
haftmann@43852
  1029
  "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
haftmann@43852
  1030
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
haftmann@43852
  1031
  "\<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
  1032
  "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
haftmann@43852
  1033
  "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
haftmann@43852
  1034
  "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
wenzelm@12897
  1035
  by auto
wenzelm@12897
  1036
blanchet@35828
  1037
lemma bex_simps [simp,no_atp]:
haftmann@43852
  1038
  "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
haftmann@43852
  1039
  "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
haftmann@43852
  1040
  "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
haftmann@43852
  1041
  "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
haftmann@43852
  1042
  "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
haftmann@43852
  1043
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
haftmann@43852
  1044
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
haftmann@43852
  1045
  "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
haftmann@43852
  1046
  "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
haftmann@43852
  1047
  "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
wenzelm@12897
  1048
  by auto
wenzelm@12897
  1049
paulson@13860
  1050
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
  1051
paulson@13860
  1052
lemma UN_extend_simps:
haftmann@43817
  1053
  "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"
haftmann@43852
  1054
  "\<And>A B C. (\<Union>x\<in>C. A x) \<union>  B  = (if C={} then B else (\<Union>x\<in>C. A x \<union>  B))"
haftmann@43852
  1055
  "\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))"
haftmann@43852
  1056
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
haftmann@43852
  1057
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
haftmann@43817
  1058
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
  1059
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43852
  1060
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
haftmann@43852
  1061
  "\<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
  1062
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
paulson@13860
  1063
  by auto
paulson@13860
  1064
paulson@13860
  1065
lemma INT_extend_simps:
haftmann@43852
  1066
  "\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))"
haftmann@43852
  1067
  "\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))"
haftmann@43852
  1068
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
haftmann@43852
  1069
  "\<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
  1070
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43852
  1071
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
haftmann@43852
  1072
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
haftmann@43852
  1073
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
haftmann@43852
  1074
  "\<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
  1075
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
paulson@13860
  1076
  by auto
paulson@13860
  1077
paulson@13860
  1078
haftmann@43872
  1079
text {* Legacy names *}
haftmann@43872
  1080
haftmann@43872
  1081
lemmas (in complete_lattice) INFI_def = INF_def
haftmann@43872
  1082
lemmas (in complete_lattice) SUPR_def = SUP_def
haftmann@43872
  1083
lemmas (in complete_lattice) le_SUPI = le_SUP_I
haftmann@43872
  1084
lemmas (in complete_lattice) le_SUPI2 = le_SUP_I2
haftmann@43872
  1085
lemmas (in complete_lattice) le_INFI = le_INF_I
haftmann@43899
  1086
lemmas (in complete_lattice) INF_subset = INF_superset_mono 
haftmann@43873
  1087
lemmas INFI_apply = INF_apply
haftmann@43873
  1088
lemmas SUPR_apply = SUP_apply
haftmann@43872
  1089
haftmann@43872
  1090
text {* Finally *}
haftmann@43872
  1091
haftmann@32135
  1092
no_notation
haftmann@32135
  1093
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
  1094
  less (infix "\<sqsubset>" 50) and
haftmann@41082
  1095
  bot ("\<bottom>") and
haftmann@41082
  1096
  top ("\<top>") and
haftmann@32135
  1097
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
  1098
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
  1099
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
  1100
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
  1101
haftmann@41080
  1102
no_syntax (xsymbols)
haftmann@41082
  1103
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
  1104
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1105
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
  1106
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1107
haftmann@30596
  1108
lemmas mem_simps =
haftmann@30596
  1109
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
  1110
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
  1111
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
  1112
wenzelm@11979
  1113
end