src/HOL/Complete_Lattice.thy
author haftmann
Sun Jul 10 14:26:07 2011 +0200 (2011-07-10)
changeset 43740 3316e6831801
parent 43739 4529a3c56609
child 43741 fac11b64713c
permissions -rw-r--r--
more succinct proofs
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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 >) (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> Inf 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: "Sup 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 "Inf A \<sqsubseteq> Inf 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 "Inf A \<sqsubseteq> a" by (rule Inf_lower)
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  with `a \<sqsubseteq> b` show "Inf 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 "Sup A \<sqsubseteq> Sup 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> Sup B" by (rule Sup_upper)
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  with `a \<sqsubseteq> b` show "a \<sqsubseteq> Sup 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> bot \<Longrightarrow> x = bot"
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  by (rule antisym) auto
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lemma not_less_bot[simp]: "\<not> (x \<sqsubset> bot)"
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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> Sup 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> Inf 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 : A \<Longrightarrow> M i \<sqsubseteq> (SUP i: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> (SUP i: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 : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i: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 : A \<Longrightarrow> (INF i: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> (INF i: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 : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)"
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  by (auto simp add: INFI_def intro: Inf_greatest)
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lemma SUP_le_iff: "(SUP i: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> (INF i: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> (INF i: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> (SUP i: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 \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n: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 \<le> g m) \<Longrightarrow> (SUP n:A. f n) \<le> (SUP n: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 \<le> 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 \<le> SUPR B f"
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  by (intro SUP_mono) auto
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lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i: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: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i: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 "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < 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 < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < 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 "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < 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 < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < 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 "(INF x: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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  "SUPR = Bex"
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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 "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)"
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    by (auto simp add: Bex_def SUPR_def Sup_bool_def)
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qed
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instantiation "fun" :: (type, complete_lattice) complete_lattice
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begin
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definition
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  "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
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lemma Inf_apply:
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  "(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}"
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  by (simp add: Inf_fun_def)
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definition
haftmann@41080
   292
  "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@41080
   293
haftmann@41080
   294
lemma Sup_apply:
haftmann@41080
   295
  "(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}"
haftmann@41080
   296
  by (simp add: Sup_fun_def)
haftmann@32077
   297
haftmann@32077
   298
instance proof
haftmann@41080
   299
qed (auto simp add: le_fun_def Inf_apply Sup_apply
haftmann@32077
   300
  intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@32077
   301
haftmann@32077
   302
end
haftmann@32077
   303
haftmann@41080
   304
lemma INFI_apply:
haftmann@41080
   305
  "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)"
haftmann@41080
   306
  by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply)
hoelzl@38705
   307
haftmann@41080
   308
lemma SUPR_apply:
haftmann@41080
   309
  "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)"
haftmann@41080
   310
  by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply)
haftmann@32077
   311
haftmann@32077
   312
haftmann@41082
   313
subsection {* Inter *}
haftmann@41082
   314
haftmann@41082
   315
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where
haftmann@41082
   316
  "Inter S \<equiv> \<Sqinter>S"
haftmann@41082
   317
  
haftmann@41082
   318
notation (xsymbols)
haftmann@41082
   319
  Inter  ("\<Inter>_" [90] 90)
haftmann@41082
   320
haftmann@41082
   321
lemma Inter_eq:
haftmann@41082
   322
  "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   323
proof (rule set_eqI)
haftmann@41082
   324
  fix x
haftmann@41082
   325
  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
   326
    by auto
haftmann@41082
   327
  then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
haftmann@41082
   328
    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
haftmann@41082
   329
qed
haftmann@41082
   330
haftmann@41082
   331
lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)"
haftmann@41082
   332
  by (unfold Inter_eq) blast
haftmann@41082
   333
haftmann@41082
   334
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
haftmann@41082
   335
  by (simp add: Inter_eq)
haftmann@41082
   336
haftmann@41082
   337
text {*
haftmann@41082
   338
  \medskip A ``destruct'' rule -- every @{term X} in @{term C}
haftmann@41082
   339
  contains @{term A} as an element, but @{prop "A:X"} can hold when
haftmann@41082
   340
  @{prop "X:C"} does not!  This rule is analogous to @{text spec}.
haftmann@41082
   341
*}
haftmann@41082
   342
haftmann@41082
   343
lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X"
haftmann@41082
   344
  by auto
haftmann@41082
   345
haftmann@41082
   346
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
haftmann@41082
   347
  -- {* ``Classical'' elimination rule -- does not require proving
haftmann@41082
   348
    @{prop "X:C"}. *}
haftmann@41082
   349
  by (unfold Inter_eq) blast
haftmann@41082
   350
haftmann@41082
   351
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B"
haftmann@43740
   352
  by (fact Inf_lower)
haftmann@43740
   353
haftmann@43740
   354
lemma (in complete_lattice) Inf_less_eq:
haftmann@43740
   355
  assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u"
haftmann@43740
   356
    and "A \<noteq> {}"
haftmann@43740
   357
  shows "\<Sqinter>A \<le> u"
haftmann@43740
   358
proof -
haftmann@43740
   359
  from `A \<noteq> {}` obtain v where "v \<in> A" by blast
haftmann@43740
   360
  moreover with assms have "v \<sqsubseteq> u" by blast
haftmann@43740
   361
  ultimately show ?thesis by (rule Inf_lower2)
haftmann@43740
   362
qed
haftmann@41082
   363
haftmann@41082
   364
lemma Inter_subset:
haftmann@41082
   365
  "[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B"
haftmann@43740
   366
  by (fact Inf_less_eq)
haftmann@41082
   367
haftmann@41082
   368
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A"
haftmann@43740
   369
  by (fact Inf_greatest)
haftmann@41082
   370
haftmann@41082
   371
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}"
haftmann@43739
   372
  by (fact Inf_binary [symmetric])
haftmann@41082
   373
haftmann@41082
   374
lemma Inter_empty [simp]: "\<Inter>{} = UNIV"
haftmann@41082
   375
  by (fact Inf_empty)
haftmann@41082
   376
haftmann@41082
   377
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}"
haftmann@43739
   378
  by (fact Inf_UNIV)
haftmann@41082
   379
haftmann@41082
   380
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B"
haftmann@43739
   381
  by (fact Inf_insert)
haftmann@41082
   382
haftmann@41082
   383
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)"
haftmann@41082
   384
  by blast
haftmann@41082
   385
haftmann@41082
   386
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B"
haftmann@41082
   387
  by blast
haftmann@41082
   388
haftmann@41082
   389
lemma Inter_UNIV_conv [simp,no_atp]:
haftmann@41082
   390
  "(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
haftmann@41082
   391
  "(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)"
haftmann@41082
   392
  by blast+
haftmann@41082
   393
haftmann@41082
   394
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B"
haftmann@41082
   395
  by blast
haftmann@41082
   396
haftmann@41082
   397
haftmann@41082
   398
subsection {* Intersections of families *}
haftmann@41082
   399
haftmann@41082
   400
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@41082
   401
  "INTER \<equiv> INFI"
haftmann@41082
   402
haftmann@41082
   403
syntax
haftmann@41082
   404
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3INT _./ _)" [0, 10] 10)
haftmann@41082
   405
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)
haftmann@41082
   406
haftmann@41082
   407
syntax (xsymbols)
haftmann@41082
   408
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>_./ _)" [0, 10] 10)
haftmann@41082
   409
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41082
   410
haftmann@41082
   411
syntax (latex output)
haftmann@41082
   412
  "_INTER1"     :: "pttrns => 'b set => 'b set"           ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
haftmann@41082
   413
  "_INTER"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@41082
   414
haftmann@41082
   415
translations
haftmann@41082
   416
  "INT x y. B"  == "INT x. INT y. B"
haftmann@41082
   417
  "INT x. B"    == "CONST INTER CONST UNIV (%x. B)"
haftmann@41082
   418
  "INT x. B"    == "INT x:CONST UNIV. B"
haftmann@41082
   419
  "INT x:A. B"  == "CONST INTER A (%x. B)"
haftmann@41082
   420
haftmann@41082
   421
print_translation {*
wenzelm@42284
   422
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}]
haftmann@41082
   423
*} -- {* to avoid eta-contraction of body *}
haftmann@41082
   424
haftmann@41082
   425
lemma INTER_eq_Inter_image:
haftmann@41082
   426
  "(\<Inter>x\<in>A. B x) = \<Inter>(B`A)"
haftmann@41082
   427
  by (fact INFI_def)
haftmann@41082
   428
  
haftmann@41082
   429
lemma Inter_def:
haftmann@41082
   430
  "\<Inter>S = (\<Inter>x\<in>S. x)"
haftmann@41082
   431
  by (simp add: INTER_eq_Inter_image image_def)
haftmann@41082
   432
haftmann@41082
   433
lemma INTER_def:
haftmann@41082
   434
  "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}"
haftmann@41082
   435
  by (auto simp add: INTER_eq_Inter_image Inter_eq)
haftmann@41082
   436
haftmann@41082
   437
lemma Inter_image_eq [simp]:
haftmann@41082
   438
  "\<Inter>(B`A) = (\<Inter>x\<in>A. B x)"
haftmann@41082
   439
  by (rule sym) (fact INTER_eq_Inter_image)
haftmann@41082
   440
haftmann@41082
   441
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
haftmann@41082
   442
  by (unfold INTER_def) blast
haftmann@41082
   443
haftmann@41082
   444
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
haftmann@41082
   445
  by (unfold INTER_def) blast
haftmann@41082
   446
haftmann@41082
   447
lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
haftmann@41082
   448
  by auto
haftmann@41082
   449
haftmann@41082
   450
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
haftmann@41082
   451
  -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
haftmann@41082
   452
  by (unfold INTER_def) blast
haftmann@41082
   453
haftmann@41082
   454
lemma INT_cong [cong]:
haftmann@41082
   455
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
haftmann@41082
   456
  by (simp add: INTER_def)
haftmann@41082
   457
haftmann@41082
   458
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
haftmann@41082
   459
  by blast
haftmann@41082
   460
haftmann@41082
   461
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
haftmann@41082
   462
  by blast
haftmann@41082
   463
haftmann@41082
   464
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a"
haftmann@41082
   465
  by (fact INF_leI)
haftmann@41082
   466
haftmann@41082
   467
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)"
haftmann@41082
   468
  by (fact le_INFI)
haftmann@41082
   469
haftmann@41082
   470
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV"
haftmann@41082
   471
  by blast
haftmann@41082
   472
haftmann@41082
   473
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)"
haftmann@41082
   474
  by blast
haftmann@41082
   475
haftmann@41082
   476
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)"
haftmann@41082
   477
  by (fact le_INF_iff)
haftmann@41082
   478
haftmann@41082
   479
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B"
haftmann@41082
   480
  by blast
haftmann@41082
   481
haftmann@41082
   482
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
   483
  by blast
haftmann@41082
   484
haftmann@41082
   485
lemma INT_insert_distrib:
haftmann@41082
   486
    "u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)"
haftmann@41082
   487
  by blast
haftmann@41082
   488
haftmann@41082
   489
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)"
haftmann@41082
   490
  by auto
haftmann@41082
   491
haftmann@41082
   492
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@41082
   493
  -- {* Look: it has an \emph{existential} quantifier *}
haftmann@41082
   494
  by blast
haftmann@41082
   495
haftmann@41082
   496
lemma INTER_UNIV_conv[simp]:
haftmann@41082
   497
 "(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082
   498
 "((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)"
haftmann@41082
   499
by blast+
haftmann@41082
   500
haftmann@41082
   501
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)"
haftmann@41082
   502
  by (auto intro: bool_induct)
haftmann@41082
   503
haftmann@41082
   504
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
haftmann@41082
   505
  by blast
haftmann@41082
   506
haftmann@41082
   507
lemma INT_anti_mono:
haftmann@41082
   508
  "B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@41082
   509
    (\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
haftmann@41082
   510
  -- {* The last inclusion is POSITIVE! *}
haftmann@41082
   511
  by (blast dest: subsetD)
haftmann@41082
   512
haftmann@41082
   513
lemma vimage_INT: "f-`(INT x:A. B x) = (INT x:A. f -` B x)"
haftmann@41082
   514
  by blast
haftmann@41082
   515
haftmann@41082
   516
haftmann@32139
   517
subsection {* Union *}
haftmann@32115
   518
haftmann@32587
   519
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where
haftmann@32587
   520
  "Union S \<equiv> \<Squnion>S"
haftmann@32115
   521
haftmann@32115
   522
notation (xsymbols)
haftmann@32115
   523
  Union  ("\<Union>_" [90] 90)
haftmann@32115
   524
haftmann@32135
   525
lemma Union_eq:
haftmann@32135
   526
  "\<Union>A = {x. \<exists>B \<in> A. x \<in> B}"
nipkow@39302
   527
proof (rule set_eqI)
haftmann@32115
   528
  fix x
haftmann@32135
   529
  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
   530
    by auto
haftmann@32135
   531
  then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
haftmann@32587
   532
    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
haftmann@32115
   533
qed
haftmann@32115
   534
blanchet@35828
   535
lemma Union_iff [simp, no_atp]:
haftmann@32115
   536
  "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)"
haftmann@32115
   537
  by (unfold Union_eq) blast
haftmann@32115
   538
haftmann@32115
   539
lemma UnionI [intro]:
haftmann@32115
   540
  "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C"
haftmann@32115
   541
  -- {* The order of the premises presupposes that @{term C} is rigid;
haftmann@32115
   542
    @{term A} may be flexible. *}
haftmann@32115
   543
  by auto
haftmann@32115
   544
haftmann@32115
   545
lemma UnionE [elim!]:
haftmann@32115
   546
  "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@32115
   547
  by auto
haftmann@32115
   548
haftmann@32135
   549
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A"
haftmann@32135
   550
  by (iprover intro: subsetI UnionI)
haftmann@32135
   551
haftmann@32135
   552
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C"
haftmann@32135
   553
  by (iprover intro: subsetI elim: UnionE dest: subsetD)
haftmann@32135
   554
haftmann@32135
   555
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}"
haftmann@32135
   556
  by blast
haftmann@32135
   557
haftmann@32135
   558
lemma Union_empty [simp]: "Union({}) = {}"
haftmann@32135
   559
  by blast
haftmann@32135
   560
haftmann@32135
   561
lemma Union_UNIV [simp]: "Union UNIV = UNIV"
haftmann@32135
   562
  by blast
haftmann@32135
   563
haftmann@32135
   564
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<Union>B"
haftmann@32135
   565
  by blast
haftmann@32135
   566
haftmann@32135
   567
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<Union>B"
haftmann@32135
   568
  by blast
haftmann@32135
   569
haftmann@32135
   570
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B"
haftmann@32135
   571
  by blast
haftmann@32135
   572
blanchet@35828
   573
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<forall>x\<in>A. x = {})"
haftmann@32135
   574
  by blast
haftmann@32135
   575
blanchet@35828
   576
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<forall>x\<in>A. x = {})"
haftmann@32135
   577
  by blast
haftmann@32135
   578
haftmann@32135
   579
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
haftmann@32135
   580
  by blast
haftmann@32135
   581
haftmann@32135
   582
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)"
haftmann@32135
   583
  by blast
haftmann@32135
   584
haftmann@32135
   585
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A"
haftmann@32135
   586
  by blast
haftmann@32135
   587
haftmann@32135
   588
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<Union>B"
haftmann@32135
   589
  by blast
haftmann@32135
   590
haftmann@32115
   591
haftmann@32139
   592
subsection {* Unions of families *}
haftmann@32077
   593
haftmann@32606
   594
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where
haftmann@32606
   595
  "UNION \<equiv> SUPR"
haftmann@32077
   596
haftmann@32077
   597
syntax
wenzelm@35115
   598
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
huffman@36364
   599
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)
haftmann@32077
   600
haftmann@32077
   601
syntax (xsymbols)
wenzelm@35115
   602
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>_./ _)" [0, 10] 10)
huffman@36364
   603
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@32077
   604
haftmann@32077
   605
syntax (latex output)
wenzelm@35115
   606
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10)
huffman@36364
   607
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10)
haftmann@32077
   608
haftmann@32077
   609
translations
haftmann@32077
   610
  "UN x y. B"   == "UN x. UN y. B"
haftmann@32077
   611
  "UN x. B"     == "CONST UNION CONST UNIV (%x. B)"
haftmann@32077
   612
  "UN x. B"     == "UN x:CONST UNIV. B"
haftmann@32077
   613
  "UN x:A. B"   == "CONST UNION A (%x. B)"
haftmann@32077
   614
haftmann@32077
   615
text {*
haftmann@32077
   616
  Note the difference between ordinary xsymbol syntax of indexed
haftmann@32077
   617
  unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"})
haftmann@32077
   618
  and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The
haftmann@32077
   619
  former does not make the index expression a subscript of the
haftmann@32077
   620
  union/intersection symbol because this leads to problems with nested
haftmann@32077
   621
  subscripts in Proof General.
haftmann@32077
   622
*}
haftmann@32077
   623
wenzelm@35115
   624
print_translation {*
wenzelm@42284
   625
  [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}]
wenzelm@35115
   626
*} -- {* to avoid eta-contraction of body *}
haftmann@32077
   627
haftmann@32135
   628
lemma UNION_eq_Union_image:
haftmann@32135
   629
  "(\<Union>x\<in>A. B x) = \<Union>(B`A)"
haftmann@32606
   630
  by (fact SUPR_def)
haftmann@32115
   631
haftmann@32115
   632
lemma Union_def:
haftmann@32117
   633
  "\<Union>S = (\<Union>x\<in>S. x)"
haftmann@32115
   634
  by (simp add: UNION_eq_Union_image image_def)
haftmann@32115
   635
blanchet@35828
   636
lemma UNION_def [no_atp]:
haftmann@32135
   637
  "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
haftmann@32117
   638
  by (auto simp add: UNION_eq_Union_image Union_eq)
haftmann@32115
   639
  
haftmann@32115
   640
lemma Union_image_eq [simp]:
haftmann@32115
   641
  "\<Union>(B`A) = (\<Union>x\<in>A. B x)"
haftmann@32115
   642
  by (rule sym) (fact UNION_eq_Union_image)
haftmann@32115
   643
  
wenzelm@11979
   644
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
wenzelm@11979
   645
  by (unfold UNION_def) blast
wenzelm@11979
   646
wenzelm@11979
   647
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
wenzelm@11979
   648
  -- {* The order of the premises presupposes that @{term A} is rigid;
wenzelm@11979
   649
    @{term b} may be flexible. *}
wenzelm@11979
   650
  by auto
wenzelm@11979
   651
wenzelm@11979
   652
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
wenzelm@11979
   653
  by (unfold UNION_def) blast
clasohm@923
   654
wenzelm@11979
   655
lemma UN_cong [cong]:
wenzelm@11979
   656
    "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
wenzelm@11979
   657
  by (simp add: UNION_def)
wenzelm@11979
   658
berghofe@29691
   659
lemma strong_UN_cong:
berghofe@29691
   660
    "A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
berghofe@29691
   661
  by (simp add: UNION_def simp_implies_def)
berghofe@29691
   662
haftmann@32077
   663
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
haftmann@32077
   664
  by blast
haftmann@32077
   665
haftmann@32135
   666
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)"
haftmann@32606
   667
  by (fact le_SUPI)
haftmann@32135
   668
haftmann@32135
   669
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C"
haftmann@32135
   670
  by (iprover intro: subsetI elim: UN_E dest: subsetD)
haftmann@32135
   671
blanchet@35828
   672
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
haftmann@32135
   673
  by blast
haftmann@32135
   674
haftmann@32135
   675
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)"
haftmann@32135
   676
  by blast
haftmann@32135
   677
blanchet@35828
   678
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}"
haftmann@32135
   679
  by blast
haftmann@32135
   680
haftmann@32135
   681
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}"
haftmann@32135
   682
  by blast
haftmann@32135
   683
haftmann@32135
   684
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A"
haftmann@32135
   685
  by blast
haftmann@32135
   686
haftmann@32135
   687
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)"
haftmann@32135
   688
  by auto
haftmann@32135
   689
haftmann@32135
   690
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B"
haftmann@32135
   691
  by blast
haftmann@32135
   692
haftmann@32135
   693
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
   694
  by blast
haftmann@32135
   695
haftmann@32135
   696
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
   697
  by blast
haftmann@32135
   698
haftmann@32135
   699
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
huffman@35629
   700
  by (fact SUP_le_iff)
haftmann@32135
   701
haftmann@32135
   702
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)"
haftmann@32135
   703
  by blast
haftmann@32135
   704
haftmann@32135
   705
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
haftmann@32135
   706
  by auto
haftmann@32135
   707
haftmann@32135
   708
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})"
haftmann@32135
   709
  by blast
haftmann@32135
   710
haftmann@32135
   711
lemma UNION_empty_conv[simp]:
haftmann@32135
   712
  "({} = (UN x:A. B x)) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   713
  "((UN x:A. B x) = {}) = (\<forall>x\<in>A. B x = {})"
haftmann@32135
   714
by blast+
haftmann@32135
   715
blanchet@35828
   716
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})"
haftmann@32135
   717
  by blast
haftmann@32135
   718
haftmann@32135
   719
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)"
haftmann@32135
   720
  by blast
haftmann@32135
   721
haftmann@32135
   722
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
haftmann@32135
   723
  by blast
haftmann@32135
   724
haftmann@32135
   725
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)"
haftmann@32135
   726
  by (auto simp add: split_if_mem2)
haftmann@32135
   727
haftmann@32135
   728
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)"
haftmann@32135
   729
  by (auto intro: bool_contrapos)
haftmann@32135
   730
haftmann@32135
   731
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)"
haftmann@32135
   732
  by blast
haftmann@32135
   733
haftmann@32135
   734
lemma UN_mono:
haftmann@32135
   735
  "A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==>
haftmann@32135
   736
    (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
haftmann@32135
   737
  by (blast dest: subsetD)
haftmann@32135
   738
haftmann@32135
   739
lemma vimage_Union: "f -` (Union A) = (UN X:A. f -` X)"
haftmann@32135
   740
  by blast
haftmann@32135
   741
haftmann@32135
   742
lemma vimage_UN: "f-`(UN x:A. B x) = (UN x:A. f -` B x)"
haftmann@32135
   743
  by blast
haftmann@32135
   744
haftmann@32135
   745
lemma vimage_eq_UN: "f-`B = (UN y: B. f-`{y})"
haftmann@32135
   746
  -- {* NOT suitable for rewriting *}
haftmann@32135
   747
  by blast
haftmann@32135
   748
haftmann@32135
   749
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
haftmann@32135
   750
by blast
haftmann@32135
   751
wenzelm@11979
   752
haftmann@32139
   753
subsection {* Distributive laws *}
wenzelm@12897
   754
wenzelm@12897
   755
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)"
wenzelm@12897
   756
  by blast
wenzelm@12897
   757
wenzelm@12897
   758
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
wenzelm@12897
   759
  by blast
wenzelm@12897
   760
wenzelm@12897
   761
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)"
wenzelm@12897
   762
  -- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *}
wenzelm@12897
   763
  -- {* Union of a family of unions *}
wenzelm@12897
   764
  by blast
wenzelm@12897
   765
wenzelm@12897
   766
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
   767
  -- {* Equivalent version *}
wenzelm@12897
   768
  by blast
wenzelm@12897
   769
wenzelm@12897
   770
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)"
wenzelm@12897
   771
  by blast
wenzelm@12897
   772
wenzelm@12897
   773
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)"
wenzelm@12897
   774
  by blast
wenzelm@12897
   775
wenzelm@12897
   776
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
   777
  -- {* Equivalent version *}
wenzelm@12897
   778
  by blast
wenzelm@12897
   779
wenzelm@12897
   780
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)"
wenzelm@12897
   781
  -- {* Halmos, Naive Set Theory, page 35. *}
wenzelm@12897
   782
  by blast
wenzelm@12897
   783
wenzelm@12897
   784
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)"
wenzelm@12897
   785
  by blast
wenzelm@12897
   786
wenzelm@12897
   787
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
   788
  by blast
wenzelm@12897
   789
wenzelm@12897
   790
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
   791
  by blast
wenzelm@12897
   792
wenzelm@12897
   793
haftmann@32139
   794
subsection {* Complement *}
haftmann@32135
   795
haftmann@32135
   796
lemma Compl_UN [simp]: "-(\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)"
wenzelm@12897
   797
  by blast
wenzelm@12897
   798
haftmann@32135
   799
lemma Compl_INT [simp]: "-(\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)"
wenzelm@12897
   800
  by blast
wenzelm@12897
   801
wenzelm@12897
   802
haftmann@32139
   803
subsection {* Miniscoping and maxiscoping *}
wenzelm@12897
   804
paulson@13860
   805
text {* \medskip Miniscoping: pushing in quantifiers and big Unions
paulson@13860
   806
           and Intersections. *}
wenzelm@12897
   807
wenzelm@12897
   808
lemma UN_simps [simp]:
wenzelm@12897
   809
  "!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))"
wenzelm@12897
   810
  "!!A B C. (UN x:C. A x Un B)   = ((if C={} then {} else (UN x:C. A x) Un B))"
wenzelm@12897
   811
  "!!A B C. (UN x:C. A Un B x)   = ((if C={} then {} else A Un (UN x:C. B x)))"
wenzelm@12897
   812
  "!!A B C. (UN x:C. A x Int B)  = ((UN x:C. A x) Int B)"
wenzelm@12897
   813
  "!!A B C. (UN x:C. A Int B x)  = (A Int (UN x:C. B x))"
wenzelm@12897
   814
  "!!A B C. (UN x:C. A x - B)    = ((UN x:C. A x) - B)"
wenzelm@12897
   815
  "!!A B C. (UN x:C. A - B x)    = (A - (INT x:C. B x))"
wenzelm@12897
   816
  "!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)"
wenzelm@12897
   817
  "!!A B C. (UN z: UNION A B. C z) = (UN  x:A. UN z: B(x). C z)"
wenzelm@12897
   818
  "!!A B f. (UN x:f`A. B x)     = (UN a:A. B (f a))"
wenzelm@12897
   819
  by auto
wenzelm@12897
   820
wenzelm@12897
   821
lemma INT_simps [simp]:
wenzelm@12897
   822
  "!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)"
wenzelm@12897
   823
  "!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))"
wenzelm@12897
   824
  "!!A B C. (INT x:C. A x - B)   = (if C={} then UNIV else (INT x:C. A x) - B)"
wenzelm@12897
   825
  "!!A B C. (INT x:C. A - B x)   = (if C={} then UNIV else A - (UN x:C. B x))"
wenzelm@12897
   826
  "!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)"
wenzelm@12897
   827
  "!!A B C. (INT x:C. A x Un B)  = ((INT x:C. A x) Un B)"
wenzelm@12897
   828
  "!!A B C. (INT x:C. A Un B x)  = (A Un (INT x:C. B x))"
wenzelm@12897
   829
  "!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)"
wenzelm@12897
   830
  "!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)"
wenzelm@12897
   831
  "!!A B f. (INT x:f`A. B x)    = (INT a:A. B (f a))"
wenzelm@12897
   832
  by auto
wenzelm@12897
   833
blanchet@35828
   834
lemma ball_simps [simp,no_atp]:
wenzelm@12897
   835
  "!!A P Q. (ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)"
wenzelm@12897
   836
  "!!A P Q. (ALL x:A. P | Q x) = (P | (ALL x:A. Q x))"
wenzelm@12897
   837
  "!!A P Q. (ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))"
wenzelm@12897
   838
  "!!A P Q. (ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)"
wenzelm@12897
   839
  "!!P. (ALL x:{}. P x) = True"
wenzelm@12897
   840
  "!!P. (ALL x:UNIV. P x) = (ALL x. P x)"
wenzelm@12897
   841
  "!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))"
wenzelm@12897
   842
  "!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)"
wenzelm@12897
   843
  "!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)"
wenzelm@12897
   844
  "!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"
wenzelm@12897
   845
  "!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))"
wenzelm@12897
   846
  "!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)"
wenzelm@12897
   847
  by auto
wenzelm@12897
   848
blanchet@35828
   849
lemma bex_simps [simp,no_atp]:
wenzelm@12897
   850
  "!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)"
wenzelm@12897
   851
  "!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))"
wenzelm@12897
   852
  "!!P. (EX x:{}. P x) = False"
wenzelm@12897
   853
  "!!P. (EX x:UNIV. P x) = (EX x. P x)"
wenzelm@12897
   854
  "!!a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))"
wenzelm@12897
   855
  "!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)"
wenzelm@12897
   856
  "!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)"
wenzelm@12897
   857
  "!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)"
wenzelm@12897
   858
  "!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))"
wenzelm@12897
   859
  "!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)"
wenzelm@12897
   860
  by auto
wenzelm@12897
   861
wenzelm@12897
   862
lemma ball_conj_distrib:
wenzelm@12897
   863
  "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"
wenzelm@12897
   864
  by blast
wenzelm@12897
   865
wenzelm@12897
   866
lemma bex_disj_distrib:
wenzelm@12897
   867
  "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"
wenzelm@12897
   868
  by blast
wenzelm@12897
   869
wenzelm@12897
   870
paulson@13860
   871
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *}
paulson@13860
   872
paulson@13860
   873
lemma UN_extend_simps:
paulson@13860
   874
  "!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))"
paulson@13860
   875
  "!!A B C. (UN x:C. A x) Un B    = (if C={} then B else (UN x:C. A x Un B))"
paulson@13860
   876
  "!!A B C. A Un (UN x:C. B x)   = (if C={} then A else (UN x:C. A Un B x))"
paulson@13860
   877
  "!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)"
paulson@13860
   878
  "!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)"
paulson@13860
   879
  "!!A B C. ((UN x:C. A x) - B) = (UN x:C. A x - B)"
paulson@13860
   880
  "!!A B C. (A - (INT x:C. B x)) = (UN x:C. A - B x)"
paulson@13860
   881
  "!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)"
paulson@13860
   882
  "!!A B C. (UN  x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)"
paulson@13860
   883
  "!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)"
paulson@13860
   884
  by auto
paulson@13860
   885
paulson@13860
   886
lemma INT_extend_simps:
paulson@13860
   887
  "!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))"
paulson@13860
   888
  "!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))"
paulson@13860
   889
  "!!A B C. (INT x:C. A x) - B   = (if C={} then UNIV-B else (INT x:C. A x - B))"
paulson@13860
   890
  "!!A B C. A - (UN x:C. B x)   = (if C={} then A else (INT x:C. A - B x))"
paulson@13860
   891
  "!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))"
paulson@13860
   892
  "!!A B C. ((INT x:C. A x) Un B)  = (INT x:C. A x Un B)"
paulson@13860
   893
  "!!A B C. A Un (INT x:C. B x)  = (INT x:C. A Un B x)"
paulson@13860
   894
  "!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)"
paulson@13860
   895
  "!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)"
paulson@13860
   896
  "!!A B f. (INT a:A. B (f a))    = (INT x:f`A. B x)"
paulson@13860
   897
  by auto
paulson@13860
   898
paulson@13860
   899
haftmann@32135
   900
no_notation
haftmann@32135
   901
  less_eq  (infix "\<sqsubseteq>" 50) and
haftmann@32135
   902
  less (infix "\<sqsubset>" 50) and
haftmann@41082
   903
  bot ("\<bottom>") and
haftmann@41082
   904
  top ("\<top>") and
haftmann@32135
   905
  inf  (infixl "\<sqinter>" 70) and
haftmann@32135
   906
  sup  (infixl "\<squnion>" 65) and
haftmann@32135
   907
  Inf  ("\<Sqinter>_" [900] 900) and
haftmann@41082
   908
  Sup  ("\<Squnion>_" [900] 900)
haftmann@32135
   909
haftmann@41080
   910
no_syntax (xsymbols)
haftmann@41082
   911
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
   912
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   913
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
   914
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
   915
haftmann@30596
   916
lemmas mem_simps =
haftmann@30596
   917
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
haftmann@30596
   918
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
haftmann@30596
   919
  -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
wenzelm@21669
   920
wenzelm@11979
   921
end