author | haftmann |
Sun, 10 Mar 2013 11:21:16 +0100 | |
changeset 51386 | 616f68ddcb7f |
parent 51341 | 8c10293e7ea7 |
child 51489 | f738e6dbd844 |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *) |
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header {* Complete lattices *} |
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theory Complete_Lattices |
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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) |
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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 sup (op \<ge>) (op >) inf \<top> \<bottom>" |
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by (auto intro!: class.complete_lattice.intro dual_bounded_lattice) |
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(unfold_locales, (fact bot_least top_greatest |
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Sup_upper Sup_least Inf_lower Inf_greatest)+) |
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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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INF_def: "INFI A f = \<Sqinter>(f ` A)" |
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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SUP_def: "SUPR A f = \<Squnion>(f ` A)" |
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text {* |
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Note: must use names @{const INFI} and @{const SUPR} here instead of |
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@{text INF} and @{text SUP} to allow the following syntax coexist |
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with the plain constant names. |
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*} |
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end |
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syntax |
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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10) |
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) |
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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) |
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) |
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syntax (xsymbols) |
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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) |
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) |
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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) |
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) |
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translations |
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"INF x y. B" == "INF x. INF y. B" |
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"INF x. B" == "CONST INFI CONST UNIV (%x. B)" |
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"INF x. B" == "INF x:CONST UNIV. B" |
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"INF x:A. B" == "CONST INFI A (%x. B)" |
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"SUP x y. B" == "SUP x. SUP y. B" |
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"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" |
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"SUP x. B" == "SUP x:CONST UNIV. B" |
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"SUP x:A. B" == "CONST SUPR A (%x. B)" |
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print_translation {* |
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[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, |
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Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] |
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*} -- {* to avoid eta-contraction of body *} |
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context complete_lattice |
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begin |
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lemma INF_foundation_dual [no_atp]: |
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"complete_lattice.SUPR Inf = INFI" |
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by (simp add: fun_eq_iff INF_def |
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complete_lattice.SUP_def [OF dual_complete_lattice]) |
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lemma SUP_foundation_dual [no_atp]: |
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"complete_lattice.INFI Sup = SUPR" |
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by (simp add: fun_eq_iff SUP_def |
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complete_lattice.INF_def [OF dual_complete_lattice]) |
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lemma Sup_eqI: |
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"(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x" |
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by (blast intro: antisym Sup_least Sup_upper) |
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lemma Inf_eqI: |
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"(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x" |
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by (blast intro: antisym Inf_greatest Inf_lower) |
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lemma SUP_eqI: |
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"(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x" |
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unfolding SUP_def by (rule Sup_eqI) auto |
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lemma INF_eqI: |
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"(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x" |
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unfolding INF_def by (rule Inf_eqI) auto |
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lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i" |
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by (auto simp add: INF_def intro: Inf_lower) |
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lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)" |
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by (auto simp add: INF_def intro: Inf_greatest) |
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lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)" |
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by (auto simp add: SUP_def intro: Sup_upper) |
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lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u" |
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by (auto simp add: SUP_def intro: Sup_least) |
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v" |
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using Inf_lower [of u A] by auto |
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lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u" |
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using INF_lower [of i A f] by auto |
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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 SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)" |
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using SUP_upper [of i A f] by auto |
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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 le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)" |
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by (auto simp add: INF_def le_Inf_iff) |
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lemma Sup_le_iff: "\<Squnion>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 SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)" |
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by (auto simp add: SUP_def Sup_le_iff) |
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lemma Inf_empty [simp]: |
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"\<Sqinter>{} = \<top>" |
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by (auto intro: antisym Inf_greatest) |
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lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>" |
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by (simp add: INF_def) |
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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 SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>" |
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by (simp add: SUP_def) |
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lemma Inf_UNIV [simp]: |
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"\<Sqinter>UNIV = \<bottom>" |
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by (auto intro!: antisym Inf_lower) |
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lemma Sup_UNIV [simp]: |
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"\<Squnion>UNIV = \<top>" |
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by (auto intro!: antisym Sup_upper) |
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lemma Inf_insert [simp]: "\<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 INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f" |
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by (simp add: INF_def) |
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lemma Sup_insert [simp]: "\<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 SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f" |
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by (simp add: SUP_def) |
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lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))" |
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by (simp add: INF_def image_image) |
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lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))" |
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by (simp add: SUP_def image_image) |
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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_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B" |
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by (auto intro: Inf_greatest Inf_lower) |
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lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B" |
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by (auto intro: Sup_least Sup_upper) |
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lemma INF_cong: |
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"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)" |
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by (simp add: INF_def image_def) |
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lemma SUP_cong: |
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"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)" |
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by (simp add: SUP_def image_def) |
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lemma Inf_mono: |
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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" |
|
41971 | 209 |
with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast |
43741 | 210 |
from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower) |
211 |
with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto |
|
38705 | 212 |
qed |
213 |
||
44041 | 214 |
lemma INF_mono: |
215 |
"(\<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)" |
|
44918 | 216 |
unfolding INF_def by (rule Inf_mono) fast |
44041 | 217 |
|
41082 | 218 |
lemma Sup_mono: |
41971 | 219 |
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b" |
43741 | 220 |
shows "\<Squnion>A \<sqsubseteq> \<Squnion>B" |
41082 | 221 |
proof (rule Sup_least) |
222 |
fix a assume "a \<in> A" |
|
41971 | 223 |
with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast |
43741 | 224 |
from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper) |
225 |
with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto |
|
41082 | 226 |
qed |
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227 |
|
44041 | 228 |
lemma SUP_mono: |
229 |
"(\<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)" |
|
44918 | 230 |
unfolding SUP_def by (rule Sup_mono) fast |
44041 | 231 |
|
232 |
lemma INF_superset_mono: |
|
233 |
"B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)" |
|
234 |
-- {* The last inclusion is POSITIVE! *} |
|
235 |
by (blast intro: INF_mono dest: subsetD) |
|
236 |
||
237 |
lemma SUP_subset_mono: |
|
238 |
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)" |
|
239 |
by (blast intro: SUP_mono dest: subsetD) |
|
240 |
||
43868 | 241 |
lemma Inf_less_eq: |
242 |
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u" |
|
243 |
and "A \<noteq> {}" |
|
244 |
shows "\<Sqinter>A \<sqsubseteq> u" |
|
245 |
proof - |
|
246 |
from `A \<noteq> {}` obtain v where "v \<in> A" by blast |
|
247 |
moreover with assms have "v \<sqsubseteq> u" by blast |
|
248 |
ultimately show ?thesis by (rule Inf_lower2) |
|
249 |
qed |
|
250 |
||
251 |
lemma less_eq_Sup: |
|
252 |
assumes "\<And>v. v \<in> A \<Longrightarrow> u \<sqsubseteq> v" |
|
253 |
and "A \<noteq> {}" |
|
254 |
shows "u \<sqsubseteq> \<Squnion>A" |
|
255 |
proof - |
|
256 |
from `A \<noteq> {}` obtain v where "v \<in> A" by blast |
|
257 |
moreover with assms have "u \<sqsubseteq> v" by blast |
|
258 |
ultimately show ?thesis by (rule Sup_upper2) |
|
259 |
qed |
|
260 |
||
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261 |
lemma SUPR_eq: |
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|
262 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j" |
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263 |
assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i" |
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264 |
shows "(SUP i:A. f i) = (SUP j:B. g j)" |
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|
265 |
by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+ |
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|
266 |
|
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267 |
lemma INFI_eq: |
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|
268 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j" |
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|
269 |
assumes "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i" |
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|
270 |
shows "(INF i:A. f i) = (INF j:B. g j)" |
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271 |
by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+ |
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272 |
|
43899 | 273 |
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)" |
43868 | 274 |
by (auto intro: Inf_greatest Inf_lower) |
275 |
||
43899 | 276 |
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<sqsubseteq> \<Squnion>A \<sqinter> \<Squnion>B " |
43868 | 277 |
by (auto intro: Sup_least Sup_upper) |
278 |
||
279 |
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" |
|
280 |
by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) |
|
281 |
||
44041 | 282 |
lemma INF_union: |
283 |
"(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)" |
|
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284 |
by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower) |
44041 | 285 |
|
43868 | 286 |
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B" |
287 |
by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2) |
|
288 |
||
44041 | 289 |
lemma SUP_union: |
290 |
"(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)" |
|
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|
291 |
by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper) |
44041 | 292 |
|
293 |
lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)" |
|
44103
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|
294 |
by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono) |
44041 | 295 |
|
44918 | 296 |
lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" (is "?L = ?R") |
297 |
proof (rule antisym) |
|
298 |
show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono) |
|
299 |
next |
|
300 |
show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper) |
|
301 |
qed |
|
44041 | 302 |
|
44918 | 303 |
lemma Inf_top_conv [simp, no_atp]: |
43868 | 304 |
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
305 |
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
306 |
proof - |
|
307 |
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
308 |
proof |
|
309 |
assume "\<forall>x\<in>A. x = \<top>" |
|
310 |
then have "A = {} \<or> A = {\<top>}" by auto |
|
44919 | 311 |
then show "\<Sqinter>A = \<top>" by auto |
43868 | 312 |
next |
313 |
assume "\<Sqinter>A = \<top>" |
|
314 |
show "\<forall>x\<in>A. x = \<top>" |
|
315 |
proof (rule ccontr) |
|
316 |
assume "\<not> (\<forall>x\<in>A. x = \<top>)" |
|
317 |
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast |
|
318 |
then obtain B where "A = insert x B" by blast |
|
44919 | 319 |
with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by simp |
43868 | 320 |
qed |
321 |
qed |
|
322 |
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto |
|
323 |
qed |
|
324 |
||
44918 | 325 |
lemma INF_top_conv [simp]: |
44041 | 326 |
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
327 |
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
|
44919 | 328 |
by (auto simp add: INF_def) |
44041 | 329 |
|
44918 | 330 |
lemma Sup_bot_conv [simp, no_atp]: |
43868 | 331 |
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P) |
332 |
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q) |
|
44920 | 333 |
using dual_complete_lattice |
334 |
by (rule complete_lattice.Inf_top_conv)+ |
|
43868 | 335 |
|
44918 | 336 |
lemma SUP_bot_conv [simp]: |
44041 | 337 |
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
338 |
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
|
44919 | 339 |
by (auto simp add: SUP_def) |
44041 | 340 |
|
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|
341 |
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f" |
44103
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44085
diff
changeset
|
342 |
by (auto intro: antisym INF_lower INF_greatest) |
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|
343 |
|
43870 | 344 |
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f" |
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changeset
|
345 |
by (auto intro: antisym SUP_upper SUP_least) |
43870 | 346 |
|
44918 | 347 |
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>" |
44921 | 348 |
by (cases "A = {}") simp_all |
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|
349 |
|
44918 | 350 |
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>" |
44921 | 351 |
by (cases "A = {}") simp_all |
43900
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|
352 |
|
43865
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|
353 |
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)" |
44103
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diff
changeset
|
354 |
by (iprover intro: INF_lower INF_greatest order_trans antisym) |
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changeset
|
355 |
|
43870 | 356 |
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)" |
44103
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haftmann
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44085
diff
changeset
|
357 |
by (iprover intro: SUP_upper SUP_least order_trans antisym) |
43870 | 358 |
|
43871 | 359 |
lemma INF_absorb: |
43868 | 360 |
assumes "k \<in> I" |
361 |
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)" |
|
362 |
proof - |
|
363 |
from assms obtain J where "I = insert k J" by blast |
|
364 |
then show ?thesis by (simp add: INF_insert) |
|
365 |
qed |
|
366 |
||
43871 | 367 |
lemma SUP_absorb: |
368 |
assumes "k \<in> I" |
|
369 |
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)" |
|
370 |
proof - |
|
371 |
from assms obtain J where "I = insert k J" by blast |
|
372 |
then show ?thesis by (simp add: SUP_insert) |
|
373 |
qed |
|
374 |
||
375 |
lemma INF_constant: |
|
43868 | 376 |
"(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)" |
44921 | 377 |
by simp |
43868 | 378 |
|
43871 | 379 |
lemma SUP_constant: |
380 |
"(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)" |
|
44921 | 381 |
by simp |
43871 | 382 |
|
43943 | 383 |
lemma less_INF_D: |
384 |
assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" shows "y < f i" |
|
385 |
proof - |
|
386 |
note `y < (\<Sqinter>i\<in>A. f i)` |
|
387 |
also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A` |
|
44103
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diff
changeset
|
388 |
by (rule INF_lower) |
43943 | 389 |
finally show "y < f i" . |
390 |
qed |
|
391 |
||
392 |
lemma SUP_lessD: |
|
393 |
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y" |
|
394 |
proof - |
|
395 |
have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A` |
|
44103
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haftmann
parents:
44085
diff
changeset
|
396 |
by (rule SUP_upper) |
43943 | 397 |
also note `(\<Squnion>i\<in>A. f i) < y` |
398 |
finally show "f i < y" . |
|
399 |
qed |
|
400 |
||
43873 | 401 |
lemma INF_UNIV_bool_expand: |
43868 | 402 |
"(\<Sqinter>b. A b) = A True \<sqinter> A False" |
44921 | 403 |
by (simp add: UNIV_bool INF_insert inf_commute) |
43868 | 404 |
|
43873 | 405 |
lemma SUP_UNIV_bool_expand: |
43871 | 406 |
"(\<Squnion>b. A b) = A True \<squnion> A False" |
44921 | 407 |
by (simp add: UNIV_bool SUP_insert sup_commute) |
43871 | 408 |
|
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|
409 |
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A" |
d63ec23c9125
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diff
changeset
|
410 |
by (blast intro: Sup_upper2 Inf_lower ex_in_conv) |
d63ec23c9125
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changeset
|
411 |
|
d63ec23c9125
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changeset
|
412 |
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> INFI A f \<le> SUPR A f" |
d63ec23c9125
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diff
changeset
|
413 |
unfolding INF_def SUP_def by (rule Inf_le_Sup) auto |
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|
414 |
|
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haftmann
parents:
32064
diff
changeset
|
415 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
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32064
diff
changeset
|
416 |
|
44024 | 417 |
class complete_distrib_lattice = complete_lattice + |
44039 | 418 |
assumes sup_Inf: "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" |
44024 | 419 |
assumes inf_Sup: "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)" |
420 |
begin |
|
421 |
||
44039 | 422 |
lemma sup_INF: |
423 |
"a \<squnion> (\<Sqinter>b\<in>B. f b) = (\<Sqinter>b\<in>B. a \<squnion> f b)" |
|
424 |
by (simp add: INF_def sup_Inf image_image) |
|
425 |
||
426 |
lemma inf_SUP: |
|
427 |
"a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)" |
|
428 |
by (simp add: SUP_def inf_Sup image_image) |
|
429 |
||
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
430 |
lemma dual_complete_distrib_lattice: |
44845 | 431 |
"class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" |
44024 | 432 |
apply (rule class.complete_distrib_lattice.intro) |
433 |
apply (fact dual_complete_lattice) |
|
434 |
apply (rule class.complete_distrib_lattice_axioms.intro) |
|
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
435 |
apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf) |
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
436 |
done |
44024 | 437 |
|
44322 | 438 |
subclass distrib_lattice proof |
44024 | 439 |
fix a b c |
440 |
from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" . |
|
44919 | 441 |
then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def) |
44024 | 442 |
qed |
443 |
||
44039 | 444 |
lemma Inf_sup: |
445 |
"\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)" |
|
446 |
by (simp add: sup_Inf sup_commute) |
|
447 |
||
448 |
lemma Sup_inf: |
|
449 |
"\<Squnion>B \<sqinter> a = (\<Squnion>b\<in>B. b \<sqinter> a)" |
|
450 |
by (simp add: inf_Sup inf_commute) |
|
451 |
||
452 |
lemma INF_sup: |
|
453 |
"(\<Sqinter>b\<in>B. f b) \<squnion> a = (\<Sqinter>b\<in>B. f b \<squnion> a)" |
|
454 |
by (simp add: sup_INF sup_commute) |
|
455 |
||
456 |
lemma SUP_inf: |
|
457 |
"(\<Squnion>b\<in>B. f b) \<sqinter> a = (\<Squnion>b\<in>B. f b \<sqinter> a)" |
|
458 |
by (simp add: inf_SUP inf_commute) |
|
459 |
||
460 |
lemma Inf_sup_eq_top_iff: |
|
461 |
"(\<Sqinter>B \<squnion> a = \<top>) \<longleftrightarrow> (\<forall>b\<in>B. b \<squnion> a = \<top>)" |
|
462 |
by (simp only: Inf_sup INF_top_conv) |
|
463 |
||
464 |
lemma Sup_inf_eq_bot_iff: |
|
465 |
"(\<Squnion>B \<sqinter> a = \<bottom>) \<longleftrightarrow> (\<forall>b\<in>B. b \<sqinter> a = \<bottom>)" |
|
466 |
by (simp only: Sup_inf SUP_bot_conv) |
|
467 |
||
468 |
lemma INF_sup_distrib2: |
|
469 |
"(\<Sqinter>a\<in>A. f a) \<squnion> (\<Sqinter>b\<in>B. g b) = (\<Sqinter>a\<in>A. \<Sqinter>b\<in>B. f a \<squnion> g b)" |
|
470 |
by (subst INF_commute) (simp add: sup_INF INF_sup) |
|
471 |
||
472 |
lemma SUP_inf_distrib2: |
|
473 |
"(\<Squnion>a\<in>A. f a) \<sqinter> (\<Squnion>b\<in>B. g b) = (\<Squnion>a\<in>A. \<Squnion>b\<in>B. f a \<sqinter> g b)" |
|
474 |
by (subst SUP_commute) (simp add: inf_SUP SUP_inf) |
|
475 |
||
44024 | 476 |
end |
477 |
||
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
478 |
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice |
43873 | 479 |
begin |
480 |
||
43943 | 481 |
lemma dual_complete_boolean_algebra: |
44845 | 482 |
"class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
483 |
by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra) |
43943 | 484 |
|
43873 | 485 |
lemma uminus_Inf: |
486 |
"- (\<Sqinter>A) = \<Squnion>(uminus ` A)" |
|
487 |
proof (rule antisym) |
|
488 |
show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)" |
|
489 |
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp |
|
490 |
show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A" |
|
491 |
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto |
|
492 |
qed |
|
493 |
||
44041 | 494 |
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)" |
495 |
by (simp add: INF_def SUP_def uminus_Inf image_image) |
|
496 |
||
43873 | 497 |
lemma uminus_Sup: |
498 |
"- (\<Squnion>A) = \<Sqinter>(uminus ` A)" |
|
499 |
proof - |
|
500 |
have "\<Squnion>A = - \<Sqinter>(uminus ` A)" by (simp add: image_image uminus_Inf) |
|
501 |
then show ?thesis by simp |
|
502 |
qed |
|
503 |
||
504 |
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)" |
|
505 |
by (simp add: INF_def SUP_def uminus_Sup image_image) |
|
506 |
||
507 |
end |
|
508 |
||
43940 | 509 |
class complete_linorder = linorder + complete_lattice |
510 |
begin |
|
511 |
||
43943 | 512 |
lemma dual_complete_linorder: |
44845 | 513 |
"class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>" |
43943 | 514 |
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder) |
515 |
||
51386 | 516 |
lemma complete_linorder_inf_min: "inf = min" |
517 |
by (rule ext)+ (auto intro: antisym) |
|
518 |
||
519 |
lemma complete_linorder_sup_max: "sup = max" |
|
520 |
by (rule ext)+ (auto intro: antisym) |
|
521 |
||
44918 | 522 |
lemma Inf_less_iff: |
43940 | 523 |
"\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)" |
524 |
unfolding not_le [symmetric] le_Inf_iff by auto |
|
525 |
||
44918 | 526 |
lemma INF_less_iff: |
44041 | 527 |
"(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)" |
528 |
unfolding INF_def Inf_less_iff by auto |
|
529 |
||
44918 | 530 |
lemma less_Sup_iff: |
43940 | 531 |
"a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)" |
532 |
unfolding not_le [symmetric] Sup_le_iff by auto |
|
533 |
||
44918 | 534 |
lemma less_SUP_iff: |
43940 | 535 |
"a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)" |
536 |
unfolding SUP_def less_Sup_iff by auto |
|
537 |
||
44918 | 538 |
lemma Sup_eq_top_iff [simp]: |
43943 | 539 |
"\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)" |
540 |
proof |
|
541 |
assume *: "\<Squnion>A = \<top>" |
|
542 |
show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" unfolding * [symmetric] |
|
543 |
proof (intro allI impI) |
|
544 |
fix x assume "x < \<Squnion>A" then show "\<exists>i\<in>A. x < i" |
|
545 |
unfolding less_Sup_iff by auto |
|
546 |
qed |
|
547 |
next |
|
548 |
assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i" |
|
549 |
show "\<Squnion>A = \<top>" |
|
550 |
proof (rule ccontr) |
|
551 |
assume "\<Squnion>A \<noteq> \<top>" |
|
552 |
with top_greatest [of "\<Squnion>A"] |
|
553 |
have "\<Squnion>A < \<top>" unfolding le_less by auto |
|
554 |
then have "\<Squnion>A < \<Squnion>A" |
|
555 |
using * unfolding less_Sup_iff by auto |
|
556 |
then show False by auto |
|
557 |
qed |
|
558 |
qed |
|
559 |
||
44918 | 560 |
lemma SUP_eq_top_iff [simp]: |
44041 | 561 |
"(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)" |
44919 | 562 |
unfolding SUP_def by auto |
44041 | 563 |
|
44918 | 564 |
lemma Inf_eq_bot_iff [simp]: |
43943 | 565 |
"\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)" |
44920 | 566 |
using dual_complete_linorder |
567 |
by (rule complete_linorder.Sup_eq_top_iff) |
|
43943 | 568 |
|
44918 | 569 |
lemma INF_eq_bot_iff [simp]: |
43967 | 570 |
"(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)" |
44919 | 571 |
unfolding INF_def by auto |
43967 | 572 |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
573 |
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
574 |
proof safe |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
575 |
fix y assume "x \<le> \<Squnion>A" "y < x" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
576 |
then have "y < \<Squnion>A" by auto |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
577 |
then show "\<exists>a\<in>A. y < a" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
578 |
unfolding less_Sup_iff . |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
579 |
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
580 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
581 |
lemma le_SUP_iff: "x \<le> SUPR A f \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
582 |
unfolding le_Sup_iff SUP_def by simp |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
583 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
584 |
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
585 |
proof safe |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
586 |
fix y assume "x \<ge> \<Sqinter>A" "y > x" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
587 |
then have "y > \<Sqinter>A" by auto |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
588 |
then show "\<exists>a\<in>A. y > a" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
589 |
unfolding Inf_less_iff . |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
590 |
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
591 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
592 |
lemma INF_le_iff: |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
593 |
"INFI A f \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
594 |
unfolding Inf_le_iff INF_def by simp |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
595 |
|
51386 | 596 |
subclass complete_distrib_lattice |
597 |
proof |
|
598 |
fix a and B |
|
599 |
show "a \<squnion> \<Sqinter>B = (\<Sqinter>b\<in>B. a \<squnion> b)" and "a \<sqinter> \<Squnion>B = (\<Squnion>b\<in>B. a \<sqinter> b)" |
|
600 |
by (safe intro!: INF_eqI [symmetric] sup_mono Inf_lower SUP_eqI [symmetric] inf_mono Sup_upper) |
|
601 |
(auto simp: not_less [symmetric] Inf_less_iff less_Sup_iff |
|
602 |
le_max_iff_disj complete_linorder_sup_max min_le_iff_disj complete_linorder_inf_min) |
|
603 |
qed |
|
604 |
||
43940 | 605 |
end |
606 |
||
51341
8c10293e7ea7
complete_linorder is also a complete_distrib_lattice
hoelzl
parents:
51328
diff
changeset
|
607 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
608 |
subsection {* Complete lattice on @{typ bool} *} |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
609 |
|
44024 | 610 |
instantiation bool :: complete_lattice |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
611 |
begin |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
612 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
613 |
definition |
46154 | 614 |
[simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
615 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
616 |
definition |
46154 | 617 |
[simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
618 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
619 |
instance proof |
44322 | 620 |
qed (auto intro: bool_induct) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
621 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
622 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
623 |
|
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
624 |
lemma not_False_in_image_Ball [simp]: |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
625 |
"False \<notin> P ` A \<longleftrightarrow> Ball A P" |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
626 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
627 |
|
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
628 |
lemma True_in_image_Bex [simp]: |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
629 |
"True \<in> P ` A \<longleftrightarrow> Bex A P" |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
630 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
631 |
|
43873 | 632 |
lemma INF_bool_eq [simp]: |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
633 |
"INFI = Ball" |
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
634 |
by (simp add: fun_eq_iff INF_def) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
635 |
|
43873 | 636 |
lemma SUP_bool_eq [simp]: |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
637 |
"SUPR = Bex" |
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
638 |
by (simp add: fun_eq_iff SUP_def) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
639 |
|
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
640 |
instance bool :: complete_boolean_algebra proof |
44322 | 641 |
qed (auto intro: bool_induct) |
44024 | 642 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
643 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
644 |
subsection {* Complete lattice on @{typ "_ \<Rightarrow> _"} *} |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
645 |
|
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
646 |
instantiation "fun" :: (type, complete_lattice) complete_lattice |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
647 |
begin |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
648 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
649 |
definition |
44024 | 650 |
"\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)" |
41080 | 651 |
|
46882 | 652 |
lemma Inf_apply [simp, code]: |
44024 | 653 |
"(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)" |
41080 | 654 |
by (simp add: Inf_fun_def) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
655 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
656 |
definition |
44024 | 657 |
"\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)" |
41080 | 658 |
|
46882 | 659 |
lemma Sup_apply [simp, code]: |
44024 | 660 |
"(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)" |
41080 | 661 |
by (simp add: Sup_fun_def) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
662 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
663 |
instance proof |
46884 | 664 |
qed (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
665 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
666 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
667 |
|
46882 | 668 |
lemma INF_apply [simp]: |
41080 | 669 |
"(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" |
46884 | 670 |
by (auto intro: arg_cong [of _ _ Inf] simp add: INF_def) |
38705 | 671 |
|
46882 | 672 |
lemma SUP_apply [simp]: |
41080 | 673 |
"(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" |
46884 | 674 |
by (auto intro: arg_cong [of _ _ Sup] simp add: SUP_def) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
675 |
|
44024 | 676 |
instance "fun" :: (type, complete_distrib_lattice) complete_distrib_lattice proof |
46884 | 677 |
qed (auto simp add: INF_def SUP_def inf_Sup sup_Inf image_image) |
44024 | 678 |
|
43873 | 679 |
instance "fun" :: (type, complete_boolean_algebra) complete_boolean_algebra .. |
680 |
||
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
681 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
682 |
subsection {* Complete lattice on unary and binary predicates *} |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
683 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
684 |
lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)" |
46884 | 685 |
by simp |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
686 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
687 |
lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)" |
46884 | 688 |
by simp |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
689 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
690 |
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b" |
46884 | 691 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
692 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
693 |
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c" |
46884 | 694 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
695 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
696 |
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b" |
46884 | 697 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
698 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
699 |
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c" |
46884 | 700 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
701 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
702 |
lemma INF1_E: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" |
46884 | 703 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
704 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
705 |
lemma INF2_E: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" |
46884 | 706 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
707 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
708 |
lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)" |
46884 | 709 |
by simp |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
710 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
711 |
lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)" |
46884 | 712 |
by simp |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
713 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
714 |
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b" |
46884 | 715 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
716 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
717 |
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" |
46884 | 718 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
719 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
720 |
lemma SUP1_E: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R" |
46884 | 721 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
722 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
723 |
lemma SUP2_E: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R" |
46884 | 724 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
725 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
726 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
727 |
subsection {* Complete lattice on @{typ "_ set"} *} |
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
728 |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
729 |
instantiation "set" :: (type) complete_lattice |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
730 |
begin |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
731 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
732 |
definition |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
733 |
"\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}" |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
734 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
735 |
definition |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
736 |
"\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}" |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
737 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
738 |
instance proof |
51386 | 739 |
qed (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def) |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
740 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
741 |
end |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
742 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
743 |
instance "set" :: (type) complete_boolean_algebra |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
744 |
proof |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
745 |
qed (auto simp add: INF_def SUP_def Inf_set_def Sup_set_def image_def) |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
746 |
|
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
747 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
748 |
subsubsection {* Inter *} |
41082 | 749 |
|
750 |
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where |
|
751 |
"Inter S \<equiv> \<Sqinter>S" |
|
752 |
||
753 |
notation (xsymbols) |
|
754 |
Inter ("\<Inter>_" [90] 90) |
|
755 |
||
756 |
lemma Inter_eq: |
|
757 |
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" |
|
758 |
proof (rule set_eqI) |
|
759 |
fix x |
|
760 |
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" |
|
761 |
by auto |
|
762 |
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
763 |
by (simp add: Inf_set_def image_def) |
41082 | 764 |
qed |
765 |
||
43741 | 766 |
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" |
41082 | 767 |
by (unfold Inter_eq) blast |
768 |
||
43741 | 769 |
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" |
41082 | 770 |
by (simp add: Inter_eq) |
771 |
||
772 |
text {* |
|
773 |
\medskip A ``destruct'' rule -- every @{term X} in @{term C} |
|
43741 | 774 |
contains @{term A} as an element, but @{prop "A \<in> X"} can hold when |
775 |
@{prop "X \<in> C"} does not! This rule is analogous to @{text spec}. |
|
41082 | 776 |
*} |
777 |
||
43741 | 778 |
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" |
41082 | 779 |
by auto |
780 |
||
43741 | 781 |
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" |
41082 | 782 |
-- {* ``Classical'' elimination rule -- does not require proving |
43741 | 783 |
@{prop "X \<in> C"}. *} |
41082 | 784 |
by (unfold Inter_eq) blast |
785 |
||
43741 | 786 |
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 787 |
by (fact Inf_lower) |
788 |
||
41082 | 789 |
lemma Inter_subset: |
43755 | 790 |
"(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 791 |
by (fact Inf_less_eq) |
41082 | 792 |
|
43755 | 793 |
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A" |
43740 | 794 |
by (fact Inf_greatest) |
41082 | 795 |
|
44067 | 796 |
lemma Inter_empty: "\<Inter>{} = UNIV" |
797 |
by (fact Inf_empty) (* already simp *) |
|
41082 | 798 |
|
44067 | 799 |
lemma Inter_UNIV: "\<Inter>UNIV = {}" |
800 |
by (fact Inf_UNIV) (* already simp *) |
|
41082 | 801 |
|
44920 | 802 |
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B" |
803 |
by (fact Inf_insert) (* already simp *) |
|
41082 | 804 |
|
805 |
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" |
|
43899 | 806 |
by (fact less_eq_Inf_inter) |
41082 | 807 |
|
808 |
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" |
|
43756 | 809 |
by (fact Inf_union_distrib) |
810 |
||
43868 | 811 |
lemma Inter_UNIV_conv [simp, no_atp]: |
43741 | 812 |
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
813 |
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
|
43801 | 814 |
by (fact Inf_top_conv)+ |
41082 | 815 |
|
43741 | 816 |
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" |
43899 | 817 |
by (fact Inf_superset_mono) |
41082 | 818 |
|
819 |
||
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
820 |
subsubsection {* Intersections of families *} |
41082 | 821 |
|
822 |
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where |
|
823 |
"INTER \<equiv> INFI" |
|
824 |
||
43872 | 825 |
text {* |
826 |
Note: must use name @{const INTER} here instead of @{text INT} |
|
827 |
to allow the following syntax coexist with the plain constant name. |
|
828 |
*} |
|
829 |
||
41082 | 830 |
syntax |
831 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) |
|
832 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) |
|
833 |
||
834 |
syntax (xsymbols) |
|
835 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) |
|
836 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) |
|
837 |
||
838 |
syntax (latex output) |
|
839 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
|
840 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) |
|
841 |
||
842 |
translations |
|
843 |
"INT x y. B" == "INT x. INT y. B" |
|
844 |
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" |
|
845 |
"INT x. B" == "INT x:CONST UNIV. B" |
|
846 |
"INT x:A. B" == "CONST INTER A (%x. B)" |
|
847 |
||
848 |
print_translation {* |
|
42284 | 849 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] |
41082 | 850 |
*} -- {* to avoid eta-contraction of body *} |
851 |
||
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
852 |
lemma INTER_eq: |
41082 | 853 |
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
854 |
by (auto simp add: INF_def) |
41082 | 855 |
|
856 |
lemma Inter_image_eq [simp]: |
|
857 |
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" |
|
43872 | 858 |
by (rule sym) (fact INF_def) |
41082 | 859 |
|
43817 | 860 |
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
861 |
by (auto simp add: INF_def image_def) |
41082 | 862 |
|
43817 | 863 |
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
864 |
by (auto simp add: INF_def image_def) |
41082 | 865 |
|
43852 | 866 |
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a" |
41082 | 867 |
by auto |
868 |
||
43852 | 869 |
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" |
870 |
-- {* "Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}. *} |
|
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
871 |
by (auto simp add: INF_def image_def) |
41082 | 872 |
|
873 |
lemma INT_cong [cong]: |
|
43854 | 874 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
875 |
by (fact INF_cong) |
41082 | 876 |
|
877 |
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" |
|
878 |
by blast |
|
879 |
||
880 |
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" |
|
881 |
by blast |
|
882 |
||
43817 | 883 |
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
884 |
by (fact INF_lower) |
41082 | 885 |
|
43817 | 886 |
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
887 |
by (fact INF_greatest) |
41082 | 888 |
|
44067 | 889 |
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
890 |
by (fact INF_empty) |
43854 | 891 |
|
43817 | 892 |
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" |
43872 | 893 |
by (fact INF_absorb) |
41082 | 894 |
|
43854 | 895 |
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)" |
41082 | 896 |
by (fact le_INF_iff) |
897 |
||
898 |
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
899 |
by (fact INF_insert) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
900 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
901 |
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
902 |
by (fact INF_union) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
903 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
904 |
lemma INT_insert_distrib: |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
905 |
"u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
906 |
by blast |
43854 | 907 |
|
41082 | 908 |
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
909 |
by (fact INF_constant) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
910 |
|
44920 | 911 |
lemma INTER_UNIV_conv: |
43817 | 912 |
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" |
913 |
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" |
|
44920 | 914 |
by (fact INF_top_conv)+ (* already simp *) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
915 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
916 |
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False" |
43873 | 917 |
by (fact INF_UNIV_bool_expand) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
918 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
919 |
lemma INT_anti_mono: |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
920 |
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
921 |
-- {* The last inclusion is POSITIVE! *} |
43940 | 922 |
by (fact INF_superset_mono) |
41082 | 923 |
|
924 |
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" |
|
925 |
by blast |
|
926 |
||
43817 | 927 |
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)" |
41082 | 928 |
by blast |
929 |
||
930 |
||
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
931 |
subsubsection {* Union *} |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
932 |
|
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset
|
933 |
abbreviation Union :: "'a set set \<Rightarrow> 'a set" where |
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset
|
934 |
"Union S \<equiv> \<Squnion>S" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
935 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
936 |
notation (xsymbols) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
937 |
Union ("\<Union>_" [90] 90) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
938 |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
939 |
lemma Union_eq: |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
940 |
"\<Union>A = {x. \<exists>B \<in> A. x \<in> B}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
38705
diff
changeset
|
941 |
proof (rule set_eqI) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
942 |
fix x |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
943 |
have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
944 |
by auto |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
945 |
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
946 |
by (simp add: Sup_set_def image_def) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
947 |
qed |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
948 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
949 |
lemma Union_iff [simp, no_atp]: |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
950 |
"A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)" |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
951 |
by (unfold Union_eq) blast |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
952 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
953 |
lemma UnionI [intro]: |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
954 |
"X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C" |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
955 |
-- {* The order of the premises presupposes that @{term C} is rigid; |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
956 |
@{term A} may be flexible. *} |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
957 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
958 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
959 |
lemma UnionE [elim!]: |
43817 | 960 |
"A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
961 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
962 |
|
43817 | 963 |
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" |
43901 | 964 |
by (fact Sup_upper) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
965 |
|
43817 | 966 |
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" |
43901 | 967 |
by (fact Sup_least) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
968 |
|
44920 | 969 |
lemma Union_empty: "\<Union>{} = {}" |
970 |
by (fact Sup_empty) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
971 |
|
44920 | 972 |
lemma Union_UNIV: "\<Union>UNIV = UNIV" |
973 |
by (fact Sup_UNIV) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
974 |
|
44920 | 975 |
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B" |
976 |
by (fact Sup_insert) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
977 |
|
43817 | 978 |
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" |
43901 | 979 |
by (fact Sup_union_distrib) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
980 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
981 |
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" |
43901 | 982 |
by (fact Sup_inter_less_eq) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
983 |
|
44920 | 984 |
lemma Union_empty_conv [no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
985 |
by (fact Sup_bot_conv) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
986 |
|
44920 | 987 |
lemma empty_Union_conv [no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
988 |
by (fact Sup_bot_conv) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
989 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
990 |
lemma subset_Pow_Union: "A \<subseteq> Pow (\<Union>A)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
991 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
992 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
993 |
lemma Union_Pow_eq [simp]: "\<Union>(Pow A) = A" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
994 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
995 |
|
43817 | 996 |
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
43901 | 997 |
by (fact Sup_subset_mono) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
998 |
|
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
999 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1000 |
subsubsection {* Unions of families *} |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1001 |
|
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset
|
1002 |
abbreviation UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where |
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset
|
1003 |
"UNION \<equiv> SUPR" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1004 |
|
43872 | 1005 |
text {* |
1006 |
Note: must use name @{const UNION} here instead of @{text UN} |
|
1007 |
to allow the following syntax coexist with the plain constant name. |
|
1008 |
*} |
|
1009 |
||
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1010 |
syntax |
35115 | 1011 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) |
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset
|
1012 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 0, 10] 10) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1013 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1014 |
syntax (xsymbols) |
35115 | 1015 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) |
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset
|
1016 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1017 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1018 |
syntax (latex output) |
35115 | 1019 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
36364
0e2679025aeb
fix syntax precedence declarations for UNION, INTER, SUP, INF
huffman
parents:
35828
diff
changeset
|
1020 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1021 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1022 |
translations |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1023 |
"UN x y. B" == "UN x. UN y. B" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1024 |
"UN x. B" == "CONST UNION CONST UNIV (%x. B)" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1025 |
"UN x. B" == "UN x:CONST UNIV. B" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1026 |
"UN x:A. B" == "CONST UNION A (%x. B)" |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1027 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1028 |
text {* |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1029 |
Note the difference between ordinary xsymbol syntax of indexed |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1030 |
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1031 |
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1032 |
former does not make the index expression a subscript of the |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1033 |
union/intersection symbol because this leads to problems with nested |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1034 |
subscripts in Proof General. |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1035 |
*} |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1036 |
|
35115 | 1037 |
print_translation {* |
42284 | 1038 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] |
35115 | 1039 |
*} -- {* to avoid eta-contraction of body *} |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1040 |
|
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1041 |
lemma UNION_eq [no_atp]: |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1042 |
"(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1043 |
by (auto simp add: SUP_def) |
44920 | 1044 |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1045 |
lemma bind_UNION [code]: |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1046 |
"Set.bind A f = UNION A f" |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1047 |
by (simp add: bind_def UNION_eq) |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1048 |
|
46036 | 1049 |
lemma member_bind [simp]: |
1050 |
"x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f " |
|
1051 |
by (simp add: bind_UNION) |
|
1052 |
||
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1053 |
lemma Union_image_eq [simp]: |
43817 | 1054 |
"\<Union>(B ` A) = (\<Union>x\<in>A. B x)" |
44920 | 1055 |
by (rule sym) (fact SUP_def) |
1056 |
||
46036 | 1057 |
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1058 |
by (auto simp add: SUP_def image_def) |
11979 | 1059 |
|
43852 | 1060 |
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)" |
11979 | 1061 |
-- {* The order of the premises presupposes that @{term A} is rigid; |
1062 |
@{term b} may be flexible. *} |
|
1063 |
by auto |
|
1064 |
||
43852 | 1065 |
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1066 |
by (auto simp add: SUP_def image_def) |
923 | 1067 |
|
11979 | 1068 |
lemma UN_cong [cong]: |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1069 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)" |
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1070 |
by (fact SUP_cong) |
11979 | 1071 |
|
29691 | 1072 |
lemma strong_UN_cong: |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1073 |
"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)" |
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1074 |
by (unfold simp_implies_def) (fact UN_cong) |
29691 | 1075 |
|
43817 | 1076 |
lemma image_eq_UN: "f ` A = (\<Union>x\<in>A. {f x})" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1077 |
by blast |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1078 |
|
43817 | 1079 |
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
1080 |
by (fact SUP_upper) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1081 |
|
43817 | 1082 |
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
1083 |
by (fact SUP_least) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1084 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
1085 |
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1086 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1087 |
|
43817 | 1088 |
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1089 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1090 |
|
44067 | 1091 |
lemma UN_empty [no_atp]: "(\<Union>x\<in>{}. B x) = {}" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1092 |
by (fact SUP_empty) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1093 |
|
44920 | 1094 |
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}" |
1095 |
by (fact SUP_bot) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1096 |
|
43817 | 1097 |
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1098 |
by (fact SUP_absorb) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1099 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1100 |
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1101 |
by (fact SUP_insert) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1102 |
|
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1103 |
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)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1104 |
by (fact SUP_union) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1105 |
|
43967 | 1106 |
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)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1107 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1108 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1109 |
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" |
35629 | 1110 |
by (fact SUP_le_iff) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1111 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1112 |
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1113 |
by (fact SUP_constant) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1114 |
|
43944 | 1115 |
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1116 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1117 |
|
44920 | 1118 |
lemma UNION_empty_conv: |
43817 | 1119 |
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
1120 |
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
|
44920 | 1121 |
by (fact SUP_bot_conv)+ (* already simp *) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1122 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
1123 |
lemma Collect_ex_eq [no_atp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1124 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1125 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1126 |
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1127 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1128 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1129 |
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1130 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1131 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1132 |
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1133 |
by (auto simp add: split_if_mem2) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1134 |
|
43817 | 1135 |
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1136 |
by (fact SUP_UNIV_bool_expand) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1137 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1138 |
lemma UN_Pow_subset: "(\<Union>x\<in>A. Pow (B x)) \<subseteq> Pow (\<Union>x\<in>A. B x)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1139 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1140 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1141 |
lemma UN_mono: |
43817 | 1142 |
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1143 |
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" |
43940 | 1144 |
by (fact SUP_subset_mono) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1145 |
|
43817 | 1146 |
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1147 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1148 |
|
43817 | 1149 |
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1150 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1151 |
|
43817 | 1152 |
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1153 |
-- {* NOT suitable for rewriting *} |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1154 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1155 |
|
43817 | 1156 |
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)" |
1157 |
by blast |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1158 |
|
45013 | 1159 |
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" |
1160 |
by blast |
|
1161 |
||
11979 | 1162 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1163 |
subsubsection {* Distributive laws *} |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1164 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1165 |
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1166 |
by (fact inf_Sup) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1167 |
|
44039 | 1168 |
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" |
1169 |
by (fact sup_Inf) |
|
1170 |
||
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1171 |
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" |
44039 | 1172 |
by (fact Sup_inf) |
1173 |
||
1174 |
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)" |
|
1175 |
by (rule sym) (rule INF_inf_distrib) |
|
1176 |
||
1177 |
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)" |
|
1178 |
by (rule sym) (rule SUP_sup_distrib) |
|
1179 |
||
1180 |
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" |
|
1181 |
by (simp only: INT_Int_distrib INF_def) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1182 |
|
43817 | 1183 |
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1184 |
-- {* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1185 |
-- {* Union of a family of unions *} |
44039 | 1186 |
by (simp only: UN_Un_distrib SUP_def) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1187 |
|
44039 | 1188 |
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" |
1189 |
by (fact sup_INF) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1190 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1191 |
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1192 |
-- {* Halmos, Naive Set Theory, page 35. *} |
44039 | 1193 |
by (fact inf_SUP) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1194 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1195 |
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)" |
44039 | 1196 |
by (fact SUP_inf_distrib2) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1197 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1198 |
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)" |
44039 | 1199 |
by (fact INF_sup_distrib2) |
1200 |
||
1201 |
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" |
|
1202 |
by (fact Sup_inf_eq_bot_iff) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1203 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1204 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1205 |
subsubsection {* Complement *} |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1206 |
|
43873 | 1207 |
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)" |
1208 |
by (fact uminus_INF) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1209 |
|
43873 | 1210 |
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)" |
1211 |
by (fact uminus_SUP) |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1212 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1213 |
|
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
1214 |
subsubsection {* Miniscoping and maxiscoping *} |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1215 |
|
13860 | 1216 |
text {* \medskip Miniscoping: pushing in quantifiers and big Unions |
1217 |
and Intersections. *} |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1218 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1219 |
lemma UN_simps [simp]: |
43817 | 1220 |
"\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1221 |
"\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))" |
43852 | 1222 |
"\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1223 |
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)" |
43852 | 1224 |
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))" |
1225 |
"\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)" |
|
1226 |
"\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))" |
|
1227 |
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)" |
|
1228 |
"\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)" |
|
43831 | 1229 |
"\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1230 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1231 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1232 |
lemma INT_simps [simp]: |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1233 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter> B)" |
43831 | 1234 |
"\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))" |
43852 | 1235 |
"\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)" |
1236 |
"\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))" |
|
43817 | 1237 |
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)" |
43852 | 1238 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)" |
1239 |
"\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))" |
|
1240 |
"\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)" |
|
1241 |
"\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)" |
|
1242 |
"\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1243 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1244 |
|
43967 | 1245 |
lemma UN_ball_bex_simps [simp, no_atp]: |
43852 | 1246 |
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)" |
43967 | 1247 |
"\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)" |
43852 | 1248 |
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)" |
1249 |
"\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1250 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1251 |
|
43943 | 1252 |
|
13860 | 1253 |
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} |
1254 |
||
1255 |
lemma UN_extend_simps: |
|
43817 | 1256 |
"\<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)))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1257 |
"\<And>A B C. (\<Union>x\<in>C. A x) \<union> B = (if C={} then B else (\<Union>x\<in>C. A x \<union> B))" |
43852 | 1258 |
"\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))" |
1259 |
"\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)" |
|
1260 |
"\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)" |
|
43817 | 1261 |
"\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)" |
1262 |
"\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)" |
|
43852 | 1263 |
"\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)" |
1264 |
"\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)" |
|
43831 | 1265 |
"\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)" |
13860 | 1266 |
by auto |
1267 |
||
1268 |
lemma INT_extend_simps: |
|
43852 | 1269 |
"\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))" |
1270 |
"\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))" |
|
1271 |
"\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))" |
|
1272 |
"\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))" |
|
43817 | 1273 |
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))" |
43852 | 1274 |
"\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)" |
1275 |
"\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)" |
|
1276 |
"\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)" |
|
1277 |
"\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)" |
|
1278 |
"\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)" |
|
13860 | 1279 |
by auto |
1280 |
||
43872 | 1281 |
text {* Finally *} |
1282 |
||
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1283 |
no_notation |
46691 | 1284 |
less_eq (infix "\<sqsubseteq>" 50) and |
1285 |
less (infix "\<sqsubset>" 50) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1286 |
|
30596 | 1287 |
lemmas mem_simps = |
1288 |
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff |
|
1289 |
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff |
|
1290 |
-- {* Each of these has ALREADY been added @{text "[simp]"} above. *} |
|
21669 | 1291 |
|
11979 | 1292 |
end |
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
1293 |