src/HOL/Complete_Lattice.thy
author haftmann
Thu Jul 14 00:16:41 2011 +0200 (2011-07-14)
changeset 43817 d53350bc65a4
parent 43814 58791b75cf1f
child 43818 fcc5d3ffb6f5
permissions -rw-r--r--
tuned notation
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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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lemma ball_conj_distrib:
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  "(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"
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  by blast
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lemma bex_disj_distrib:
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  "(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x))"
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  by blast
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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 >) (op \<squnion>) (op \<sqinter>) \<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_empty 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_empty 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 top_le:
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  "\<top> \<sqsubseteq> x \<Longrightarrow> x = \<top>"
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  by (rule antisym) auto
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lemma le_bot:
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  "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
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  by (rule antisym) auto
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lemma not_less_bot[simp]: "\<not> (x \<sqsubset> \<bottom>)"
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  using bot_least[of x] by (auto simp: le_less)
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lemma not_top_less[simp]: "\<not> (\<top> \<sqsubset> x)"
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  using top_greatest[of x] by (auto simp: le_less)
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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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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
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  "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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  "SUPR A f = \<Squnion> (f ` A)"
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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 SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g"
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  by (simp add: SUPR_def cong: image_cong)
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lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g"
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  by (simp add: INFI_def cong: image_cong)
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lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
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  by (auto simp add: SUPR_def intro: Sup_upper)
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lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)"
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  using le_SUPI[of i A M] by auto
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lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u"
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  by (auto simp add: SUPR_def intro: Sup_least)
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lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i"
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  by (auto simp add: INFI_def intro: Inf_lower)
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lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u"
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  using INF_leI[of i A M] by auto
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lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)"
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  by (auto simp add: INFI_def intro: Inf_greatest)
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lemma SUP_le_iff: "(\<Squnion>i\<in>A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)"
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  unfolding SUPR_def by (auto simp add: Sup_le_iff)
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lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)"
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  unfolding INFI_def by (auto simp add: le_Inf_iff)
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lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. M) = M"
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  by (auto intro: antisym INF_leI le_INFI)
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lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. M) = M"
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  by (auto intro: antisym SUP_leI le_SUPI)
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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)"
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  by (force intro!: Inf_mono simp: INFI_def)
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lemma SUP_mono:
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  "(\<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)"
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  by (force intro!: Sup_mono simp: SUPR_def)
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lemma INF_subset:  "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f"
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  by (intro INF_mono) auto
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lemma SUP_subset:  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
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  by (intro SUP_mono) auto
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lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
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  by (iprover intro: INF_leI le_INFI order_trans antisym)
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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)"
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  by (iprover intro: SUP_leI le_SUPI order_trans antisym)
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end
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lemma Inf_less_iff:
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  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
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  shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
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  unfolding not_le [symmetric] le_Inf_iff by auto
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lemma less_Sup_iff:
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  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
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  shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
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  unfolding not_le [symmetric] Sup_le_iff by auto
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lemma INF_less_iff:
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  fixes a :: "'a::{complete_lattice,linorder}"
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  shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
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  unfolding INFI_def Inf_less_iff by auto
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lemma less_SUP_iff:
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  fixes a :: "'a::{complete_lattice,linorder}"
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  shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
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  unfolding SUPR_def less_Sup_iff by auto
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subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
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instantiation bool :: complete_lattice
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begin
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definition
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  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
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definition
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  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
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instance proof
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qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
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end
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lemma INFI_bool_eq [simp]:
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  "INFI = Ball"
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proof (rule ext)+
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  fix A :: "'a set"
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  fix P :: "'a \<Rightarrow> bool"
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  show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
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    by (auto simp add: Ball_def INFI_def Inf_bool_def)
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qed
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lemma SUPR_bool_eq [simp]:
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   281
  "SUPR = Bex"
haftmann@32120
   282
proof (rule ext)+
haftmann@32120
   283
  fix A :: "'a set"
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   284
  fix P :: "'a \<Rightarrow> bool"
haftmann@43753
   285
  show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
haftmann@32120
   286
    by (auto simp add: Bex_def SUPR_def Sup_bool_def)
haftmann@32120
   287
qed
haftmann@32120
   288
haftmann@32077
   289
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@32077
   290
begin
haftmann@32077
   291
haftmann@32077
   292
definition
haftmann@41080
   293
  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   294
haftmann@41080
   295
lemma Inf_apply:
haftmann@41080
   296
  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   297
  by (simp add: Inf_fun_def)
haftmann@32077
   298
haftmann@32077
   299
definition
haftmann@41080
   300
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   301
haftmann@41080
   302
lemma Sup_apply:
haftmann@41080
   303
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   304
  by (simp add: Sup_fun_def)
haftmann@32077
   305
haftmann@32077
   306
instance proof
haftmann@41080
   307
qed (auto simp add: le_fun_def Inf_apply Sup_apply
haftmann@32077
   308
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@32077
   309
haftmann@32077
   310
end
haftmann@32077
   311
haftmann@41080
   312
lemma INFI_apply:
haftmann@41080
   313
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@41080
   314
  by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
hoelzl@38705
   315
haftmann@41080
   316
lemma SUPR_apply:
haftmann@41080
   317
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@41080
   318
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
haftmann@32077
   319
haftmann@32077
   320
haftmann@41082
   321
subsection {* Inter *}
haftmann@41082
   322
haftmann@41082
   323
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   324
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   325
  
haftmann@41082
   326
notation (xsymbols)
haftmann@41082
   327
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   328
haftmann@41082
   329
lemma Inter_eq:
haftmann@41082
   330
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   331
proof (rule set_eqI)
haftmann@41082
   332
  fix x
haftmann@41082
   333
  have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
haftmann@41082
   334
    by auto
haftmann@41082
   335
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   336
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082
   337
qed
haftmann@41082
   338
haftmann@43741
   339
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
haftmann@41082
   340
  by (unfold Inter_eq) blast
haftmann@41082
   341
haftmann@43741
   342
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C"
haftmann@41082
   343
  by (simp add: Inter_eq)
haftmann@41082
   344
haftmann@41082
   345
text {*
haftmann@41082
   346
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@43741
   347
  contains @{term A} as an element, but @{prop "A \<in> X"} can hold when
haftmann@43741
   348
  @{prop "X \<in> C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   349
*}
haftmann@41082
   350
haftmann@43741
   351
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X"
haftmann@41082
   352
  by auto
haftmann@41082
   353
haftmann@43741
   354
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   355
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@43741
   356
    @{prop "X \<in> C"}. *}
haftmann@41082
   357
  by (unfold Inter_eq) blast
haftmann@41082
   358
haftmann@43741
   359
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   360
  by (fact Inf_lower)
haftmann@43740
   361
haftmann@43740
   362
lemma (in complete_lattice) Inf_less_eq:
haftmann@43740
   363
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43740
   364
    and "A \<noteq> {}"
haftmann@43753
   365
  shows "\<Sqinter>A \<sqsubseteq> u"
haftmann@43740
   366
proof -
haftmann@43740
   367
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
haftmann@43740
   368
  moreover with assms have "v \<sqsubseteq> u" by blast
haftmann@43740
   369
  ultimately show ?thesis by (rule Inf_lower2)
haftmann@43740
   370
qed
haftmann@41082
   371
haftmann@41082
   372
lemma Inter_subset:
haftmann@43755
   373
  "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B"
haftmann@43740
   374
  by (fact Inf_less_eq)
haftmann@41082
   375
haftmann@43755
   376
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A"
haftmann@43740
   377
  by (fact Inf_greatest)
haftmann@41082
   378
haftmann@41082
   379
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@43739
   380
  by (fact Inf_binary [symmetric])
haftmann@41082
   381
haftmann@41082
   382
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082
   383
  by (fact Inf_empty)
haftmann@41082
   384
haftmann@41082
   385
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@43739
   386
  by (fact Inf_UNIV)
haftmann@41082
   387
haftmann@41082
   388
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739
   389
  by (fact Inf_insert)
haftmann@41082
   390
haftmann@43741
   391
lemma (in complete_lattice) Inf_inter_less: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)"
haftmann@43741
   392
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43741
   393
haftmann@41082
   394
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@43741
   395
  by (fact Inf_inter_less)
haftmann@43741
   396
haftmann@43756
   397
lemma (in complete_lattice) Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
haftmann@43756
   398
  by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)
haftmann@41082
   399
haftmann@41082
   400
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@43756
   401
  by (fact Inf_union_distrib)
haftmann@43756
   402
haftmann@43801
   403
lemma (in complete_lattice) Inf_top_conv [no_atp]:
haftmann@43801
   404
  "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801
   405
  "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801
   406
proof -
haftmann@43801
   407
  show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801
   408
  proof
haftmann@43801
   409
    assume "\<forall>x\<in>A. x = \<top>"
haftmann@43801
   410
    then have "A = {} \<or> A = {\<top>}" by auto
haftmann@43801
   411
    then show "\<Sqinter>A = \<top>" by auto
haftmann@43801
   412
  next
haftmann@43801
   413
    assume "\<Sqinter>A = \<top>"
haftmann@43801
   414
    show "\<forall>x\<in>A. x = \<top>"
haftmann@43801
   415
    proof (rule ccontr)
haftmann@43801
   416
      assume "\<not> (\<forall>x\<in>A. x = \<top>)"
haftmann@43801
   417
      then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast
haftmann@43801
   418
      then obtain B where "A = insert x B" by blast
haftmann@43801
   419
      with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert)
haftmann@43801
   420
    qed
haftmann@43801
   421
  qed
haftmann@43801
   422
  then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
haftmann@43801
   423
qed
haftmann@41082
   424
haftmann@41082
   425
lemma Inter_UNIV_conv [simp,no_atp]:
haftmann@43741
   426
  "\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43741
   427
  "UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)"
haftmann@43801
   428
  by (fact Inf_top_conv)+
haftmann@41082
   429
haftmann@43756
   430
lemma (in complete_lattice) Inf_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B"
haftmann@43756
   431
  by (auto intro: Inf_greatest Inf_lower)
haftmann@43756
   432
haftmann@43741
   433
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B"
haftmann@43756
   434
  by (fact Inf_anti_mono)
haftmann@41082
   435
haftmann@41082
   436
haftmann@41082
   437
subsection {* Intersections of families *}
haftmann@41082
   438
haftmann@41082
   439
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   440
  "INTER \<equiv> INFI"
haftmann@41082
   441
haftmann@41082
   442
syntax
haftmann@41082
   443
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   444
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   445
haftmann@41082
   446
syntax (xsymbols)
haftmann@41082
   447
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   448
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   449
haftmann@41082
   450
syntax (latex output)
haftmann@41082
   451
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   452
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   453
haftmann@41082
   454
translations
haftmann@41082
   455
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   456
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   457
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   458
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   459
haftmann@41082
   460
print_translation {*
wenzelm@42284
   461
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   462
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   463
haftmann@41082
   464
lemma INTER_eq_Inter_image:
haftmann@41082
   465
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@41082
   466
  by (fact INFI_def)
haftmann@41082
   467
  
haftmann@41082
   468
lemma Inter_def:
haftmann@41082
   469
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082
   470
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082
   471
haftmann@41082
   472
lemma INTER_def:
haftmann@41082
   473
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082
   474
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082
   475
haftmann@41082
   476
lemma Inter_image_eq [simp]:
haftmann@41082
   477
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@43801
   478
  by (rule sym) (fact INFI_def)
haftmann@41082
   479
haftmann@43817
   480
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)"
haftmann@41082
   481
  by (unfold INTER_def) blast
haftmann@41082
   482
haftmann@43817
   483
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)"
haftmann@41082
   484
  by (unfold INTER_def) blast
haftmann@41082
   485
haftmann@43817
   486
lemma INT_D [elim, Pure.elim]: "b : (\<Inter>x\<in>A. B x) \<Longrightarrow> a:A \<Longrightarrow> b: B a"
haftmann@41082
   487
  by auto
haftmann@41082
   488
haftmann@43817
   489
lemma INT_E [elim]: "b : (\<Inter>x\<in>A. B x) \<Longrightarrow> (b: B a \<Longrightarrow> R) \<Longrightarrow> (a~:A \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@41082
   490
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
haftmann@41082
   491
  by (unfold INTER_def) blast
haftmann@41082
   492
haftmann@41082
   493
lemma INT_cong [cong]:
haftmann@43817
   494
    "A = B \<Longrightarrow> (\<And>x. x:B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)"
haftmann@41082
   495
  by (simp add: INTER_def)
haftmann@41082
   496
haftmann@41082
   497
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   498
  by blast
haftmann@41082
   499
haftmann@41082
   500
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   501
  by blast
haftmann@41082
   502
haftmann@43817
   503
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082
   504
  by (fact INF_leI)
haftmann@41082
   505
haftmann@43817
   506
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@41082
   507
  by (fact le_INFI)
haftmann@41082
   508
haftmann@41082
   509
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@41082
   510
  by blast
haftmann@41082
   511
haftmann@43817
   512
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@41082
   513
  by blast
haftmann@41082
   514
haftmann@41082
   515
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   516
  by (fact le_INF_iff)
haftmann@41082
   517
haftmann@41082
   518
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@41082
   519
  by blast
haftmann@41082
   520
haftmann@41082
   521
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)"
haftmann@41082
   522
  by blast
haftmann@41082
   523
haftmann@41082
   524
lemma INT_insert_distrib:
haftmann@43817
   525
    "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@41082
   526
  by blast
haftmann@41082
   527
haftmann@41082
   528
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@41082
   529
  by auto
haftmann@41082
   530
haftmann@41082
   531
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082
   532
  -- {* Look: it has an \emph{existential} quantifier *}
haftmann@41082
   533
  by blast
haftmann@41082
   534
haftmann@41082
   535
lemma INTER_UNIV_conv[simp]:
haftmann@43817
   536
 "(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@43817
   537
 "((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082
   538
by blast+
haftmann@41082
   539
haftmann@43817
   540
lemma INT_bool_eq: "(\<Inter>b. A b) = (A True \<inter> A False)"
haftmann@41082
   541
  by (auto intro: bool_induct)
haftmann@41082
   542
haftmann@41082
   543
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   544
  by blast
haftmann@41082
   545
haftmann@41082
   546
lemma INT_anti_mono:
haftmann@43817
   547
  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@41082
   548
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@41082
   549
  -- {* The last inclusion is POSITIVE! *}
haftmann@41082
   550
  by (blast dest: subsetD)
haftmann@41082
   551
haftmann@43817
   552
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)"
haftmann@41082
   553
  by blast
haftmann@41082
   554
haftmann@41082
   555
haftmann@32139
   556
subsection {* Union *}
haftmann@32115
   557
haftmann@32587
   558
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   559
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   560
haftmann@32115
   561
notation (xsymbols)
haftmann@32115
   562
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   563
haftmann@32135
   564
lemma Union_eq:
haftmann@32135
   565
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   566
proof (rule set_eqI)
haftmann@32115
   567
  fix x
haftmann@32135
   568
  have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
haftmann@32115
   569
    by auto
haftmann@32135
   570
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@32587
   571
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115
   572
qed
haftmann@32115
   573
blanchet@35828
   574
lemma Union_iff [simp, no_atp]:
haftmann@32115
   575
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   576
  by (unfold Union_eq) blast
haftmann@32115
   577
haftmann@32115
   578
lemma UnionI [intro]:
haftmann@32115
   579
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   580
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   581
    @{term A} may be flexible. *}
haftmann@32115
   582
  by auto
haftmann@32115
   583
haftmann@32115
   584
lemma UnionE [elim!]:
haftmann@43817
   585
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   586
  by auto
haftmann@32115
   587
haftmann@43817
   588
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A"
haftmann@32135
   589
  by (iprover intro: subsetI UnionI)
haftmann@32135
   590
haftmann@43817
   591
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C"
haftmann@32135
   592
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
haftmann@32135
   593
haftmann@32135
   594
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135
   595
  by blast
haftmann@32135
   596
haftmann@43817
   597
lemma Union_empty [simp]: "\<Union>{} = {}"
haftmann@32135
   598
  by blast
haftmann@32135
   599
haftmann@43817
   600
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV"
haftmann@32135
   601
  by blast
haftmann@32135
   602
haftmann@43817
   603
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B"
haftmann@32135
   604
  by blast
haftmann@32135
   605
haftmann@43817
   606
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B"
haftmann@32135
   607
  by blast
haftmann@32135
   608
haftmann@32135
   609
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@32135
   610
  by blast
haftmann@32135
   611
haftmann@43817
   612
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@32135
   613
  by blast
haftmann@32135
   614
haftmann@43817
   615
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})"
haftmann@32135
   616
  by blast
haftmann@32135
   617
haftmann@43817
   618
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@32135
   619
  by blast
haftmann@32135
   620
haftmann@32135
   621
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   622
  by blast
haftmann@32135
   623
haftmann@32135
   624
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   625
  by blast
haftmann@32135
   626
haftmann@43817
   627
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B"
haftmann@32135
   628
  by blast
haftmann@32135
   629
haftmann@32115
   630
haftmann@32139
   631
subsection {* Unions of families *}
haftmann@32077
   632
haftmann@32606
   633
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   634
  "UNION \<equiv> SUPR"
haftmann@32077
   635
haftmann@32077
   636
syntax
wenzelm@35115
   637
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   638
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   639
haftmann@32077
   640
syntax (xsymbols)
wenzelm@35115
   641
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   642
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   643
haftmann@32077
   644
syntax (latex output)
wenzelm@35115
   645
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   646
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   647
haftmann@32077
   648
translations
haftmann@32077
   649
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   650
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   651
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   652
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   653
haftmann@32077
   654
text {*
haftmann@32077
   655
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   656
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   657
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   658
  former does not make the index expression a subscript of the
haftmann@32077
   659
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   660
  subscripts in Proof General.
haftmann@32077
   661
*}
haftmann@32077
   662
wenzelm@35115
   663
print_translation {*
wenzelm@42284
   664
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   665
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   666
haftmann@32135
   667
lemma UNION_eq_Union_image:
haftmann@43817
   668
  "(\<Union>x\<in>A. B x) = \<Union>(B ` A)"
haftmann@32606
   669
  by (fact SUPR_def)
haftmann@32115
   670
haftmann@32115
   671
lemma Union_def:
haftmann@32117
   672
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   673
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   674
blanchet@35828
   675
lemma UNION_def [no_atp]:
haftmann@32135
   676
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   677
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   678
  
haftmann@32115
   679
lemma Union_image_eq [simp]:
haftmann@43817
   680
  "\<Union>(B ` A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   681
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   682
  
haftmann@43817
   683
lemma UN_iff [simp]: "(b: (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b: B x)"
wenzelm@11979
   684
  by (unfold UNION_def) blast
wenzelm@11979
   685
haftmann@43817
   686
lemma UN_I [intro]: "a:A \<Longrightarrow> b: B a \<Longrightarrow> b: (\<Union>x\<in>A. B x)"
wenzelm@11979
   687
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   688
    @{term b} may be flexible. *}
wenzelm@11979
   689
  by auto
wenzelm@11979
   690
haftmann@43817
   691
lemma UN_E [elim!]: "b : (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x:A \<Longrightarrow> b: B x \<Longrightarrow> R) \<Longrightarrow> R"
wenzelm@11979
   692
  by (unfold UNION_def) blast
clasohm@923
   693
wenzelm@11979
   694
lemma UN_cong [cong]:
haftmann@43817
   695
    "A = B \<Longrightarrow> (\<And>x. x:B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
wenzelm@11979
   696
  by (simp add: UNION_def)
wenzelm@11979
   697
berghofe@29691
   698
lemma strong_UN_cong:
haftmann@43817
   699
    "A = B \<Longrightarrow> (\<And>x. x:B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)"
berghofe@29691
   700
  by (simp add: UNION_def simp_implies_def)
berghofe@29691
   701
haftmann@43817
   702
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})"
haftmann@32077
   703
  by blast
haftmann@32077
   704
haftmann@43817
   705
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@32606
   706
  by (fact le_SUPI)
haftmann@32135
   707
haftmann@43817
   708
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135
   709
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135
   710
blanchet@35828
   711
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   712
  by blast
haftmann@32135
   713
haftmann@43817
   714
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
   715
  by blast
haftmann@32135
   716
blanchet@35828
   717
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135
   718
  by blast
haftmann@32135
   719
haftmann@32135
   720
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135
   721
  by blast
haftmann@32135
   722
haftmann@32135
   723
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   724
  by blast
haftmann@32135
   725
haftmann@43817
   726
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135
   727
  by auto
haftmann@32135
   728
haftmann@32135
   729
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135
   730
  by blast
haftmann@32135
   731
haftmann@32135
   732
lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)"
haftmann@32135
   733
  by blast
haftmann@32135
   734
haftmann@32135
   735
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)"
haftmann@32135
   736
  by blast
haftmann@32135
   737
haftmann@32135
   738
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   739
  by (fact SUP_le_iff)
haftmann@32135
   740
haftmann@32135
   741
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   742
  by blast
haftmann@32135
   743
haftmann@32135
   744
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135
   745
  by auto
haftmann@32135
   746
haftmann@32135
   747
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135
   748
  by blast
haftmann@32135
   749
haftmann@32135
   750
lemma UNION_empty_conv[simp]:
haftmann@43817
   751
  "{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@43817
   752
  "(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})"
haftmann@32135
   753
by blast+
haftmann@32135
   754
blanchet@35828
   755
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   756
  by blast
haftmann@32135
   757
haftmann@32135
   758
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
   759
  by blast
haftmann@32135
   760
haftmann@32135
   761
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
   762
  by blast
haftmann@32135
   763
haftmann@32135
   764
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   765
  by (auto simp add: split_if_mem2)
haftmann@32135
   766
haftmann@43817
   767
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)"
haftmann@32135
   768
  by (auto intro: bool_contrapos)
haftmann@32135
   769
haftmann@32135
   770
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   771
  by blast
haftmann@32135
   772
haftmann@32135
   773
lemma UN_mono:
haftmann@43817
   774
  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
haftmann@32135
   775
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135
   776
  by (blast dest: subsetD)
haftmann@32135
   777
haftmann@43817
   778
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
haftmann@32135
   779
  by blast
haftmann@32135
   780
haftmann@43817
   781
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)"
haftmann@32135
   782
  by blast
haftmann@32135
   783
haftmann@43817
   784
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})"
haftmann@32135
   785
  -- {* NOT suitable for rewriting *}
haftmann@32135
   786
  by blast
haftmann@32135
   787
haftmann@43817
   788
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)"
haftmann@43817
   789
  by blast
haftmann@32135
   790
wenzelm@11979
   791
haftmann@32139
   792
subsection {* Distributive laws *}
wenzelm@12897
   793
wenzelm@12897
   794
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   795
  by blast
wenzelm@12897
   796
wenzelm@12897
   797
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   798
  by blast
wenzelm@12897
   799
haftmann@43817
   800
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)"
wenzelm@12897
   801
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   802
  -- {* Union of a family of unions *}
wenzelm@12897
   803
  by blast
wenzelm@12897
   804
wenzelm@12897
   805
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)"
wenzelm@12897
   806
  -- {* Equivalent version *}
wenzelm@12897
   807
  by blast
wenzelm@12897
   808
wenzelm@12897
   809
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   810
  by blast
wenzelm@12897
   811
haftmann@43817
   812
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)"
wenzelm@12897
   813
  by blast
wenzelm@12897
   814
wenzelm@12897
   815
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)"
wenzelm@12897
   816
  -- {* Equivalent version *}
wenzelm@12897
   817
  by blast
wenzelm@12897
   818
wenzelm@12897
   819
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   820
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   821
  by blast
wenzelm@12897
   822
wenzelm@12897
   823
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   824
  by blast
wenzelm@12897
   825
wenzelm@12897
   826
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)"
wenzelm@12897
   827
  by blast
wenzelm@12897
   828
wenzelm@12897
   829
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)"
wenzelm@12897
   830
  by blast
wenzelm@12897
   831
wenzelm@12897
   832
haftmann@32139
   833
subsection {* Complement *}
haftmann@32135
   834
haftmann@43817
   835
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
   836
  by blast
wenzelm@12897
   837
haftmann@43817
   838
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
   839
  by blast
wenzelm@12897
   840
wenzelm@12897
   841
haftmann@32139
   842
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   843
paulson@13860
   844
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   845
           and Intersections. *}
wenzelm@12897
   846
wenzelm@12897
   847
lemma UN_simps [simp]:
haftmann@43817
   848
  "\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))"
haftmann@43817
   849
  "\<And>A B C. (\<Union>x\<in>C. A x Un B)   = ((if C={} then {} else (\<Union>x\<in>C. A x) Un B))"
haftmann@43817
   850
  "\<And>A B C. (\<Union>x\<in>C. A Un B x)   = ((if C={} then {} else A Un (\<Union>x\<in>C. B x)))"
haftmann@43817
   851
  "\<And>A B C. (\<Union>x\<in>C. A x Int B)  = ((\<Union>x\<in>C. A x) Int B)"
haftmann@43817
   852
  "\<And>A B C. (\<Union>x\<in>C. A Int B x)  = (A Int (\<Union>x\<in>C. B x))"
haftmann@43817
   853
  "\<And>A B C. (\<Union>x\<in>C. A x - B)    = ((\<Union>x\<in>C. A x) - B)"
haftmann@43817
   854
  "\<And>A B C. (\<Union>x\<in>C. A - B x)    = (A - (\<Inter>x\<in>C. B x))"
haftmann@43817
   855
  "\<And>A B. (UN x: \<Union>A. B x) = (UN y:A. UN x:y. B x)"
haftmann@43817
   856
  "\<And>A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
haftmann@43817
   857
  "\<And>A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
   858
  by auto
wenzelm@12897
   859
wenzelm@12897
   860
lemma INT_simps [simp]:
haftmann@43817
   861
  "\<And>A B C. (\<Inter>x\<in>C. A x Int B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) Int B)"
haftmann@43817
   862
  "\<And>A B C. (\<Inter>x\<in>C. A Int B x) = (if C={} then UNIV else A Int (\<Inter>x\<in>C. B x))"
haftmann@43817
   863
  "\<And>A B C. (\<Inter>x\<in>C. A x - B)   = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)"
haftmann@43817
   864
  "\<And>A B C. (\<Inter>x\<in>C. A - B x)   = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))"
haftmann@43817
   865
  "\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)"
haftmann@43817
   866
  "\<And>A B C. (\<Inter>x\<in>C. A x Un B)  = ((\<Inter>x\<in>C. A x) Un B)"
haftmann@43817
   867
  "\<And>A B C. (\<Inter>x\<in>C. A Un B x)  = (A Un (\<Inter>x\<in>C. B x))"
haftmann@43817
   868
  "\<And>A B. (INT x: \<Union>A. B x) = (\<Inter>y\<in>A. INT x:y. B x)"
haftmann@43817
   869
  "\<And>A B C. (INT z: UNION A B. C z) = (\<Inter>x\<in>A. INT z: B(x). C z)"
haftmann@43817
   870
  "\<And>A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
   871
  by auto
wenzelm@12897
   872
blanchet@35828
   873
lemma ball_simps [simp,no_atp]:
haftmann@43817
   874
  "\<And>A P Q. (\<forall>x\<in>A. P x | Q) = ((\<forall>x\<in>A. P x) | Q)"
haftmann@43817
   875
  "\<And>A P Q. (\<forall>x\<in>A. P | Q x) = (P | (\<forall>x\<in>A. Q x))"
haftmann@43817
   876
  "\<And>A P Q. (\<forall>x\<in>A. P --> Q x) = (P --> (\<forall>x\<in>A. Q x))"
haftmann@43817
   877
  "\<And>A P Q. (\<forall>x\<in>A. P x --> Q) = ((\<exists>x\<in>A. P x) --> Q)"
haftmann@43817
   878
  "\<And>P. (ALL x:{}. P x) = True"
haftmann@43817
   879
  "\<And>P. (ALL x:UNIV. P x) = (ALL x. P x)"
haftmann@43817
   880
  "\<And>a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
haftmann@43817
   881
  "\<And>A P. (ALL x:\<Union>A. P x) = (ALL y:A. ALL x:y. P x)"
haftmann@43817
   882
  "\<And>A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
haftmann@43817
   883
  "\<And>P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
haftmann@43817
   884
  "\<And>A P f. (ALL x:f`A. P x) = (\<forall>x\<in>A. P (f x))"
haftmann@43817
   885
  "\<And>A P. (~(\<forall>x\<in>A. P x)) = (\<exists>x\<in>A. ~P x)"
wenzelm@12897
   886
  by auto
wenzelm@12897
   887
blanchet@35828
   888
lemma bex_simps [simp,no_atp]:
haftmann@43817
   889
  "\<And>A P Q. (\<exists>x\<in>A. P x & Q) = ((\<exists>x\<in>A. P x) & Q)"
haftmann@43817
   890
  "\<And>A P Q. (\<exists>x\<in>A. P & Q x) = (P & (\<exists>x\<in>A. Q x))"
haftmann@43817
   891
  "\<And>P. (EX x:{}. P x) = False"
haftmann@43817
   892
  "\<And>P. (EX x:UNIV. P x) = (EX x. P x)"
haftmann@43817
   893
  "\<And>a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
haftmann@43817
   894
  "\<And>A P. (EX x:\<Union>A. P x) = (EX y:A. EX x:y. P x)"
haftmann@43817
   895
  "\<And>A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
haftmann@43817
   896
  "\<And>P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
haftmann@43817
   897
  "\<And>A P f. (EX x:f`A. P x) = (\<exists>x\<in>A. P (f x))"
haftmann@43817
   898
  "\<And>A P. (~(\<exists>x\<in>A. P x)) = (\<forall>x\<in>A. ~P x)"
wenzelm@12897
   899
  by auto
wenzelm@12897
   900
paulson@13860
   901
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
   902
paulson@13860
   903
lemma UN_extend_simps:
haftmann@43817
   904
  "\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))"
haftmann@43817
   905
  "\<And>A B C. (\<Union>x\<in>C. A x) Un B    = (if C={} then B else (\<Union>x\<in>C. A x Un B))"
haftmann@43817
   906
  "\<And>A B C. A Un (\<Union>x\<in>C. B x)   = (if C={} then A else (\<Union>x\<in>C. A Un B x))"
haftmann@43817
   907
  "\<And>A B C. ((\<Union>x\<in>C. A x) Int B) = (\<Union>x\<in>C. A x Int B)"
haftmann@43817
   908
  "\<And>A B C. (A Int (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A Int B x)"
haftmann@43817
   909
  "\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)"
haftmann@43817
   910
  "\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)"
haftmann@43817
   911
  "\<And>A B. (UN y:A. UN x:y. B x) = (UN x: \<Union>A. B x)"
haftmann@43817
   912
  "\<And>A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
haftmann@43817
   913
  "\<And>A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
   914
  by auto
paulson@13860
   915
paulson@13860
   916
lemma INT_extend_simps:
haftmann@43817
   917
  "\<And>A B C. (\<Inter>x\<in>C. A x) Int B = (if C={} then B else (\<Inter>x\<in>C. A x Int B))"
haftmann@43817
   918
  "\<And>A B C. A Int (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A Int B x))"
haftmann@43817
   919
  "\<And>A B C. (\<Inter>x\<in>C. A x) - B   = (if C={} then UNIV-B else (\<Inter>x\<in>C. A x - B))"
haftmann@43817
   920
  "\<And>A B C. A - (\<Union>x\<in>C. B x)   = (if C={} then A else (\<Inter>x\<in>C. A - B x))"
haftmann@43817
   921
  "\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))"
haftmann@43817
   922
  "\<And>A B C. ((\<Inter>x\<in>C. A x) Un B)  = (\<Inter>x\<in>C. A x Un B)"
haftmann@43817
   923
  "\<And>A B C. A Un (\<Inter>x\<in>C. B x)  = (\<Inter>x\<in>C. A Un B x)"
haftmann@43817
   924
  "\<And>A B. (\<Inter>y\<in>A. INT x:y. B x) = (INT x: \<Union>A. B x)"
haftmann@43817
   925
  "\<And>A B C. (\<Inter>x\<in>A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
haftmann@43817
   926
  "\<And>A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
   927
  by auto
paulson@13860
   928
paulson@13860
   929
haftmann@32135
   930
no_notation
haftmann@32135
   931
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
   932
  less (infix "\<sqsubset>" 50) and
haftmann@41082
   933
  bot ("\<bottom>") and
haftmann@41082
   934
  top ("\<top>") and
haftmann@32135
   935
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
   936
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
   937
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
   938
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
   939
haftmann@41080
   940
no_syntax (xsymbols)
haftmann@41082
   941
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
   942
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   943
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
   944
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   945
haftmann@30596
   946
lemmas mem_simps =
haftmann@30596
   947
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
   948
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
   949
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
   950
wenzelm@11979
   951
end