src/HOL/Complete_Lattice.thy
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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_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 Inf_inter_less_eq: "\<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_greater_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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lemma Inf_anti_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_anti_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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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)
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lemma INF_mono:
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  "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   282
  by (force intro!: Inf_mono simp: INF_def)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   283
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   284
lemma SUP_mono:
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   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)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   286
  by (force intro!: Sup_mono simp: SUP_def)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   287
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   288
lemma INF_subset:
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   289
  "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   290
  by (intro INF_mono) auto
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   291
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   292
lemma SUP_subset:
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   293
  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   294
  by (intro SUP_mono) auto
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   295
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   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)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   297
  by (iprover intro: INF_leI le_INF_I order_trans antisym)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   298
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   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)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   300
  by (iprover intro: SUP_leI le_SUP_I order_trans antisym)
43870
92129f505125 structuring duals together
haftmann
parents: 43868
diff changeset
   301
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   302
lemma INF_absorb:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   303
  assumes "k \<in> I"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   304
  shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   305
proof -
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   306
  from assms obtain J where "I = insert k J" by blast
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   307
  then show ?thesis by (simp add: INF_insert)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   308
qed
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   309
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   310
lemma SUP_absorb:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   311
  assumes "k \<in> I"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   312
  shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   313
proof -
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   314
  from assms obtain J where "I = insert k J" by blast
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   315
  then show ?thesis by (simp add: SUP_insert)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   316
qed
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   317
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   318
lemma INF_union:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   319
  "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   320
  by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INF_I INF_leI)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   321
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   322
lemma SUP_union:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   323
  "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   324
  by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_leI le_SUP_I)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   325
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   326
lemma INF_constant:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   327
  "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   328
  by (simp add: INF_empty)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   329
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   330
lemma SUP_constant:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   331
  "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   332
  by (simp add: SUP_empty)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   333
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   334
lemma INF_eq:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   335
  "(\<Sqinter>x\<in>A. B x) = \<Sqinter>({Y. \<exists>x\<in>A. Y = B x})"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   336
  by (simp add: INF_def image_def)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   337
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   338
lemma SUP_eq:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   339
  "(\<Squnion>x\<in>A. B x) = \<Squnion>({Y. \<exists>x\<in>A. Y = B x})"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   340
  by (simp add: SUP_def image_def)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   341
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   342
lemma INF_top_conv:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   343
 "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   344
 "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   345
  by (auto simp add: INF_def Inf_top_conv)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   346
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   347
lemma SUP_bot_conv:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   348
 "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   349
 "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   350
  by (auto simp add: SUP_def Sup_bot_conv)
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   351
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   352
lemma INF_UNIV_range:
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   353
  "(\<Sqinter>x. f x) = \<Sqinter>range f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   354
  by (fact INF_def)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   355
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   356
lemma SUP_UNIV_range:
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   357
  "(\<Squnion>x. f x) = \<Squnion>range f"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   358
  by (fact SUP_def)
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   359
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   360
lemma INF_UNIV_bool_expand:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   361
  "(\<Sqinter>b. A b) = A True \<sqinter> A False"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   362
  by (simp add: UNIV_bool INF_empty INF_insert inf_commute)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   363
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   364
lemma SUP_UNIV_bool_expand:
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   365
  "(\<Squnion>b. A b) = A True \<squnion> A False"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   366
  by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   367
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   368
lemma INF_anti_mono:
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   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)"
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   370
  -- {* The last inclusion is POSITIVE! *}
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   371
  by (blast intro: INF_mono dest: subsetD)
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   372
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   373
lemma SUP_anti_mono:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   374
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> g x \<sqsubseteq> f x) \<Longrightarrow> (\<Squnion>x\<in>B. g x) \<sqsubseteq> (\<Squnion>x\<in>B. f x)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   375
  -- {* The last inclusion is POSITIVE! *}
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   376
  by (blast intro: SUP_mono dest: subsetD)
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   377
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   378
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   379
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   380
lemma Inf_less_iff:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   381
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
43753
fe5e846c0839 tuned notation
haftmann
parents: 43741
diff changeset
   382
  shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
43754
09d455c93bf8 tuned notation
haftmann
parents: 43753
diff changeset
   383
  unfolding not_le [symmetric] le_Inf_iff by auto
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   384
43871
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   385
lemma less_Sup_iff:
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   386
  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   387
  shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   388
  unfolding not_le [symmetric] Sup_le_iff by auto
79c3231e0593 more lemmas about SUP
haftmann
parents: 43870
diff changeset
   389
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   390
lemma INF_less_iff:
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   391
  fixes a :: "'a::{complete_lattice,linorder}"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   392
  shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   393
  unfolding INF_def Inf_less_iff by auto
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   394
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40714
diff changeset
   395
lemma less_SUP_iff:
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40714
diff changeset
   396
  fixes a :: "'a::{complete_lattice,linorder}"
43753
fe5e846c0839 tuned notation
haftmann
parents: 43741
diff changeset
   397
  shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   398
  unfolding SUP_def less_Sup_iff by auto
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40714
diff changeset
   399
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   400
-- "FIXME move"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   401
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   402
lemma image_ident [simp]: "(%x. x) ` Y = Y"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   403
  by blast
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   404
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   405
lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   406
  by blast
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   407
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   408
class complete_boolean_algebra = boolean_algebra + complete_lattice
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   409
begin
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   410
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   411
lemma uminus_Inf:
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   412
  "- (\<Sqinter>A) = \<Squnion>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   413
proof (rule antisym)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   414
  show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   415
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   416
  show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   417
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   418
qed
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   419
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   420
lemma uminus_Sup:
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   421
  "- (\<Squnion>A) = \<Sqinter>(uminus ` A)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   422
proof -
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   423
  have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   424
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   425
qed
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   426
  
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   427
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   428
  by (simp add: INF_def SUP_def uminus_Inf image_image)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   429
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   430
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   431
  by (simp add: INF_def SUP_def uminus_Sup image_image)
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   432
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   433
end
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   434
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   435
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   436
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   437
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   438
instantiation bool :: complete_boolean_algebra
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   439
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   440
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   441
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   442
  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   443
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   444
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   445
  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   446
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   447
instance proof
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   448
qed (auto simp add: Inf_bool_def Sup_bool_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   449
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   450
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   451
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   452
lemma INF_bool_eq [simp]:
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   453
  "INFI = Ball"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   454
proof (rule ext)+
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   455
  fix A :: "'a set"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   456
  fix P :: "'a \<Rightarrow> bool"
43753
fe5e846c0839 tuned notation
haftmann
parents: 43741
diff changeset
   457
  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   458
    by (auto simp add: Ball_def INF_def Inf_bool_def)
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   459
qed
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   460
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   461
lemma SUP_bool_eq [simp]:
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   462
  "SUPR = Bex"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   463
proof (rule ext)+
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   464
  fix A :: "'a set"
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   465
  fix P :: "'a \<Rightarrow> bool"
43753
fe5e846c0839 tuned notation
haftmann
parents: 43741
diff changeset
   466
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   467
    by (auto simp add: Bex_def SUP_def Sup_bool_def)
32120
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   468
qed
53a21a5e6889 attempt for more concise setup of non-etacontracting binders
haftmann
parents: 32117
diff changeset
   469
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   470
instantiation "fun" :: (type, complete_lattice) complete_lattice
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   471
begin
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   472
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   473
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   474
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   475
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   476
lemma Inf_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   477
  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   478
  by (simp add: Inf_fun_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   479
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   480
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   481
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   482
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   483
lemma Sup_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   484
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   485
  by (simp add: Sup_fun_def)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   486
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   487
instance proof
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   488
qed (auto simp add: le_fun_def Inf_apply Sup_apply
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   489
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   490
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   491
end
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   492
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   493
lemma INF_apply:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   494
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   495
  by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def Inf_apply)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 37767
diff changeset
   496
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   497
lemma SUP_apply:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
   498
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   499
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def Sup_apply)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   500
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   501
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra ..
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   502
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   503
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   504
subsection {* Inter *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   505
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   506
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   507
  "Inter S \<equiv> \<Sqinter>S"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   508
  
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   509
notation (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   510
  Inter  ("\<Inter>_" [90] 90)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   511
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   512
lemma Inter_eq:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   513
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   514
proof (rule set_eqI)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   515
  fix x
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   516
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   517
    by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   518
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   519
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   520
qed
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   521
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   522
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   523
  by (unfold Inter_eq) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   524
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   525
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   526
  by (simp add: Inter_eq)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   527
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   528
text {*
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   529
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   530
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   531
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   532
*}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   533
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   534
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   535
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   536
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   537
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   538
  -- {* ``Classical'' elimination rule -- does not require proving
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   539
    @{prop "X \<in> C"}. *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   540
  by (unfold Inter_eq) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   541
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   542
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   543
  by (fact Inf_lower)
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   544
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   545
lemma Inter_subset:
43755
5e4a595e63fc tuned notation
haftmann
parents: 43754
diff changeset
   546
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   547
  by (fact Inf_less_eq)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   548
43755
5e4a595e63fc tuned notation
haftmann
parents: 43754
diff changeset
   549
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
43740
3316e6831801 more succinct proofs
haftmann
parents: 43739
diff changeset
   550
  by (fact Inf_greatest)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   551
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   552
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
43739
4529a3c56609 more succinct proofs
haftmann
parents: 42284
diff changeset
   553
  by (fact Inf_binary [symmetric])
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   554
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   555
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   556
  by (fact Inf_empty)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   557
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   558
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
43739
4529a3c56609 more succinct proofs
haftmann
parents: 42284
diff changeset
   559
  by (fact Inf_UNIV)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   560
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   561
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
43739
4529a3c56609 more succinct proofs
haftmann
parents: 42284
diff changeset
   562
  by (fact Inf_insert)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   563
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   564
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   565
  by (fact Inf_inter_less_eq)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   566
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   567
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
43756
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   568
  by (fact Inf_union_distrib)
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   569
43868
9684251c7ec1 more lemmas about Sup
haftmann
parents: 43867
diff changeset
   570
lemma Inter_UNIV_conv [simp, no_atp]:
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   571
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   572
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
43801
097732301fc0 more generalization towards complete lattices
haftmann
parents: 43756
diff changeset
   573
  by (fact Inf_top_conv)+
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   574
43741
fac11b64713c tuned proofs and notation
haftmann
parents: 43740
diff changeset
   575
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
43756
15ea1a07a8cf tuned proofs
haftmann
parents: 43755
diff changeset
   576
  by (fact Inf_anti_mono)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   577
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   578
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   579
subsection {* Intersections of families *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   580
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   581
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   582
  "INTER \<equiv> INFI"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   583
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   584
text {*
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   585
  Note: must use name @{const INTER} here instead of @{text INT}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   586
  to allow the following syntax coexist with the plain constant name.
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   587
*}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   588
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   589
syntax
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   590
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   591
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   592
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   593
syntax (xsymbols)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   594
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   595
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   596
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   597
syntax (latex output)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   598
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   599
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   600
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   601
translations
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   602
  "INT x y. B"  == "INT x. INT y. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   603
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   604
  "INT x. B"    == "INT x:CONST UNIV. B"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   605
  "INT x:A. B"  == "CONST INTER A (%x. B)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   606
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   607
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   608
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   609
*} -- {* to avoid eta-contraction of body *}
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   610
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   611
lemma INTER_eq_Inter_image:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   612
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   613
  by (fact INF_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   614
  
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   615
lemma Inter_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   616
  "\<Inter>S = (\<Inter>x\<in>S. x)"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   617
  by (simp add: INTER_eq_Inter_image image_def)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   618
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   619
lemma INTER_def:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   620
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   621
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   622
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   623
lemma Inter_image_eq [simp]:
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   624
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   625
  by (rule sym) (fact INF_def)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   626
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   627
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   628
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   629
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   630
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   631
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   632
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   633
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   634
  by auto
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   635
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   636
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"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   637
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *}
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   638
  by (unfold INTER_def) blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   639
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   640
lemma INT_cong [cong]:
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   641
  "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   642
  by (fact INF_cong)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   643
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   644
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   645
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   646
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   647
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   648
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   649
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   650
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   651
  by (fact INF_leI)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   652
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   653
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   654
  by (fact le_INF_I)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   655
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   656
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   657
  by (fact INF_empty)
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   658
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   659
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   660
  by (fact INF_absorb)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   661
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   662
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   663
  by (fact le_INF_iff)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   664
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   665
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   666
  by (fact INF_insert)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   667
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   668
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   669
  by (fact INF_union)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   670
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   671
lemma INT_insert_distrib:
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   672
  "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   673
  by blast
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   674
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   675
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   676
  by (fact INF_constant)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   677
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   678
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   679
  -- {* Look: it has an \emph{existential} quantifier *}
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   680
  by (fact INF_eq)
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   681
43854
f1d23df1adde generalized some lemmas
haftmann
parents: 43853
diff changeset
   682
lemma INTER_UNIV_conv [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   683
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   684
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   685
  by (fact INF_top_conv)+
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   686
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   687
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False"
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   688
  by (fact INF_UNIV_bool_expand)
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   689
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   690
lemma INT_anti_mono:
43867
771014555553 generalized INT_anti_mono
haftmann
parents: 43866
diff changeset
   691
  "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)"
43865
db18f4d0cc7d further generalization from sets to complete lattices
haftmann
parents: 43854
diff changeset
   692
  -- {* The last inclusion is POSITIVE! *}
43867
771014555553 generalized INT_anti_mono
haftmann
parents: 43866
diff changeset
   693
  by (fact INF_anti_mono)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   694
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   695
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   696
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   697
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   698
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   699
  by blast
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   700
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
   701
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   702
subsection {* Union *}
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   703
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   704
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   705
  "Union S \<equiv> \<Squnion>S"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   706
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   707
notation (xsymbols)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   708
  Union  ("\<Union>_" [90] 90)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   709
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   710
lemma Union_eq:
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   711
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 38705
diff changeset
   712
proof (rule set_eqI)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   713
  fix x
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   714
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   715
    by auto
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   716
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
32587
caa5ada96a00 Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents: 32436
diff changeset
   717
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   718
qed
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   719
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   720
lemma Union_iff [simp, no_atp]:
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   721
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   722
  by (unfold Union_eq) blast
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   723
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   724
lemma UnionI [intro]:
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   725
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   726
  -- {* The order of the premises presupposes that @{term C} is rigid;
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   727
    @{term A} may be flexible. *}
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   728
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   729
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   730
lemma UnionE [elim!]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   731
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   732
  by auto
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   733
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   734
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   735
  by (iprover intro: subsetI UnionI)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   736
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   737
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   738
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   739
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   740
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   741
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   742
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   743
lemma Union_empty [simp]: "\<Union>{} = {}"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   744
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   745
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   746
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   747
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   748
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   749
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   750
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   751
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   752
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   753
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   754
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   755
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   756
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   757
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   758
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   759
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   760
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   761
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   762
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   763
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   764
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   765
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   766
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   767
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   768
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   769
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   770
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   771
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   772
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   773
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   774
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   775
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   776
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   777
subsection {* Unions of families *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   778
32606
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   779
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
b5c3a8a75772 INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents: 32587
diff changeset
   780
  "UNION \<equiv> SUPR"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   781
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   782
text {*
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   783
  Note: must use name @{const UNION} here instead of @{text UN}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   784
  to allow the following syntax coexist with the plain constant name.
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   785
*}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   786
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   787
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   788
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   789
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   790
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   791
syntax (xsymbols)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   792
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   793
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   794
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   795
syntax (latex output)
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   796
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
36364
0e2679025aeb fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents: 35828
diff changeset
   797
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   798
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   799
translations
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   800
  "UN x y. B"   == "UN x. UN y. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   801
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   802
  "UN x. B"     == "UN x:CONST UNIV. B"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   803
  "UN x:A. B"   == "CONST UNION A (%x. B)"
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   804
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   805
text {*
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   806
  Note the difference between ordinary xsymbol syntax of indexed
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   807
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   808
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   809
  former does not make the index expression a subscript of the
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   810
  union/intersection symbol because this leads to problems with nested
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   811
  subscripts in Proof General.
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   812
*}
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   813
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   814
print_translation {*
42284
326f57825e1a explicit structure Syntax_Trans;
wenzelm
parents: 41971
diff changeset
   815
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34007
diff changeset
   816
*} -- {* to avoid eta-contraction of body *}
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   817
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   818
lemma UNION_eq_Union_image:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   819
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   820
  by (fact SUP_def)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   821
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   822
lemma Union_def:
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   823
  "\<Union>S = (\<Union>x\<in>S. x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   824
  by (simp add: UNION_eq_Union_image image_def)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   825
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   826
lemma UNION_def [no_atp]:
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   827
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
32117
0762b9ad83df Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents: 32115
diff changeset
   828
  by (auto simp add: UNION_eq_Union_image Union_eq)
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   829
  
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   830
lemma Union_image_eq [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   831
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
32115
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   832
  by (rule sym) (fact UNION_eq_Union_image)
8f10fb3bb46e swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents: 32082
diff changeset
   833
  
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   834
lemma UN_iff [simp]: "(b \<in> (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b \<in> B x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   835
  by (unfold UNION_def) blast
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   836
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   837
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   838
  -- {* The order of the premises presupposes that @{term A} is rigid;
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   839
    @{term b} may be flexible. *}
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   840
  by auto
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   841
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   842
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   843
  by (unfold UNION_def) blast
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   844
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   845
lemma UN_cong [cong]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   846
    "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)"
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   847
  by (simp add: UNION_def)
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   848
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   849
lemma strong_UN_cong:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
   850
    "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)"
29691
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   851
  by (simp add: UNION_def simp_implies_def)
9f03b5f847cd Added strong congruence rule for UN.
berghofe
parents: 28562
diff changeset
   852
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   853
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
32077
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   854
  by blast
3698947146b2 closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents: 32064
diff changeset
   855
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   856
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
   857
  by (fact le_SUP_I)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   858
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   859
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   860
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   861
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   862
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   863
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   864
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   865
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   866
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   867
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   868
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   869
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   870
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   871
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   872
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   873
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   874
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   875
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   876
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   877
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   878
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   879
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   880
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   881
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   882
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   883
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)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   884
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   885
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   886
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)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   887
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   888
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   889
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
35629
57f1a5e93b6b add some lemmas about complete lattices
huffman
parents: 35115
diff changeset
   890
  by (fact SUP_le_iff)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   891
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   892
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   893
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   894
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   895
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   896
  by auto
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   897
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   898
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   899
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   900
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   901
lemma UNION_empty_conv[simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   902
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   903
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   904
by blast+
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   905
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
   906
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   907
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   908
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   909
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   910
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   911
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   912
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   913
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   914
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   915
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   916
  by (auto simp add: split_if_mem2)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   917
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   918
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   919
  by (auto intro: bool_contrapos)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   920
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   921
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   922
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   923
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   924
lemma UN_mono:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   925
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   926
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   927
  by (blast dest: subsetD)
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   928
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   929
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   930
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   931
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   932
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   933
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   934
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   935
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   936
  -- {* NOT suitable for rewriting *}
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   937
  by blast
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   938
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   939
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   940
  by blast
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   941
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
   942
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   943
subsection {* Distributive laws *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   944
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   945
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   946
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   947
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   948
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   949
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   950
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   951
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   952
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   953
  -- {* Union of a family of unions *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   954
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   955
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   956
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   957
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   958
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   959
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   960
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   961
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   962
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   963
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   964
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   965
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   966
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   967
  -- {* Equivalent version *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   968
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   969
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   970
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   971
  -- {* Halmos, Naive Set Theory, page 35. *}
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   972
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   973
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   974
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   975
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   976
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   977
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   978
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   979
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   980
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   981
  by blast
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   982
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   983
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   984
subsection {* Complement *}
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
   985
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   986
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   987
  by (fact uminus_INF)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   988
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   989
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
   990
  by (fact uminus_SUP)
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   991
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   992
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 32135
diff changeset
   993
subsection {* Miniscoping and maxiscoping *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   994
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   995
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
   996
           and Intersections. *}
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   997
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
   998
lemma UN_simps [simp]:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
   999
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1000
  "\<And>A B C. (\<Union>x\<in>C. A x \<union>  B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1001
  "\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1002
  "\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter>B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1003
  "\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1004
  "\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1005
  "\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1006
  "\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1007
  "\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1008
  "\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1009
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1010
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1011
lemma INT_simps [simp]:
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1012
  "\<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)"
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1013
  "\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1014
  "\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1015
  "\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1016
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1017
  "\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1018
  "\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1019
  "\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1020
  "\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1021
  "\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1022
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1023
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
  1024
lemma ball_simps [simp,no_atp]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1025
  "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1026
  "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1027
  "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1028
  "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1029
  "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1030
  "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1031
  "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1032
  "\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1033
  "\<And>A B P. (\<forall>x\<in> UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1034
  "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1035
  "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1036
  "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1037
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1038
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35629
diff changeset
  1039
lemma bex_simps [simp,no_atp]:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1040
  "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1041
  "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1042
  "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1043
  "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1044
  "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a | (\<exists>x\<in>B. P x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1045
  "\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1046
  "\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1047
  "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1048
  "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1049
  "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
12897
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1050
  by auto
f4d10ad0ea7b converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents: 12633
diff changeset
  1051
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1052
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1053
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1054
lemma UN_extend_simps:
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1055
  "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1056
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1057
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1058
  "\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1059
  "\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1060
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1061
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1062
  "\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1063
  "\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)"
43831
e323be6b02a5 tuned notation and proofs
haftmann
parents: 43818
diff changeset
  1064
  "\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1065
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1066
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1067
lemma INT_extend_simps:
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1068
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1069
  "\<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))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1070
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1071
  "\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
43817
d53350bc65a4 tuned notation
haftmann
parents: 43814
diff changeset
  1072
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
43852
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1073
  "\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1074
  "\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1075
  "\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1076
  "\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)"
7411fbf0a325 tuned notation
haftmann
parents: 43831
diff changeset
  1077
  "\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)"
13860
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1078
  by auto
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1079
b681a3cb0beb new UN/INT simprules
paulson
parents: 13858
diff changeset
  1080
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1081
text {* Legacy names *}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1082
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1083
lemmas (in complete_lattice) INFI_def = INF_def
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1084
lemmas (in complete_lattice) SUPR_def = SUP_def
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1085
lemmas (in complete_lattice) le_SUPI = le_SUP_I
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1086
lemmas (in complete_lattice) le_SUPI2 = le_SUP_I2
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1087
lemmas (in complete_lattice) le_INFI = le_INF_I
43873
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1088
lemmas INFI_apply = INF_apply
8a2f339641c1 more on complement
haftmann
parents: 43872
diff changeset
  1089
lemmas SUPR_apply = SUP_apply
43872
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1090
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1091
text {* Finally *}
6b917e5877d2 more consistent theorem names
haftmann
parents: 43871
diff changeset
  1092
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1093
no_notation
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1094
  less_eq  (infix "\<sqsubseteq>" 50) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1095
  less (infix "\<sqsubset>" 50) and
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1096
  bot ("\<bottom>") and
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1097
  top ("\<top>") and
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1098
  inf  (infixl "\<sqinter>" 70) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1099
  sup  (infixl "\<squnion>" 65) and
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1100
  Inf  ("\<Sqinter>_" [900] 900) and
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1101
  Sup  ("\<Squnion>_" [900] 900)
32135
f645b51e8e54 set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents: 32120
diff changeset
  1102
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1103
no_syntax (xsymbols)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1104
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1105
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1106
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1107
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 40872
diff changeset
  1108
30596
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1109
lemmas mem_simps =
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1110
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1111
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
140b22f22071 tuned some theorem and attribute bindings
haftmann
parents: 30531
diff changeset
  1112
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
21669
c68717c16013 removed legacy ML bindings;
wenzelm
parents: 21549
diff changeset
  1113
11979
0a3dace545c5 converted theory "Set";
wenzelm
parents: 11752
diff changeset
  1114
end