author | haftmann |
Thu, 14 Jul 2011 00:16:41 +0200 | |
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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, with special focus on sets *} |
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theory Complete_Lattice |
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imports Set |
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begin |
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lemma ball_conj_distrib: |
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"(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))" |
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by blast |
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lemma bex_disj_distrib: |
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"(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x))" |
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by blast |
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notation |
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less_eq (infix "\<sqsubseteq>" 50) and |
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less (infix "\<sqsubset>" 50) and |
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inf (infixl "\<sqinter>" 70) and |
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sup (infixl "\<squnion>" 65) and |
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top ("\<top>") and |
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bot ("\<bottom>") |
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subsection {* Syntactic infimum and supremum operations *} |
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class Inf = |
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fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) |
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class Sup = |
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fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) |
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subsection {* Abstract complete lattices *} |
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class complete_lattice = bounded_lattice + Inf + Sup + |
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assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" |
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and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" |
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assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" |
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and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" |
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begin |
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lemma dual_complete_lattice: |
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"class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>" |
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by (auto intro!: class.complete_lattice.intro dual_bounded_lattice) |
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(unfold_locales, (fact bot_least top_greatest |
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Sup_upper Sup_least Inf_lower Inf_greatest)+) |
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}" |
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
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lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}" |
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
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lemma Inf_empty [simp]: |
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"\<Sqinter>{} = \<top>" |
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by (auto intro: antisym Inf_greatest) |
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lemma Sup_empty [simp]: |
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"\<Squnion>{} = \<bottom>" |
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by (auto intro: antisym Sup_least) |
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lemma Inf_UNIV [simp]: |
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"\<Sqinter>UNIV = \<bottom>" |
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by (simp add: Sup_Inf Sup_empty [symmetric]) |
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lemma Sup_UNIV [simp]: |
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"\<Squnion>UNIV = \<top>" |
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by (simp add: Inf_Sup Inf_empty [symmetric]) |
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lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" |
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by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) |
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lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" |
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by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) |
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lemma Inf_singleton [simp]: |
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"\<Sqinter>{a} = a" |
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by (auto intro: antisym Inf_lower Inf_greatest) |
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lemma Sup_singleton [simp]: |
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"\<Squnion>{a} = a" |
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by (auto intro: antisym Sup_upper Sup_least) |
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lemma Inf_binary: |
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"\<Sqinter>{a, b} = a \<sqinter> b" |
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by (simp add: Inf_empty Inf_insert) |
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lemma Sup_binary: |
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"\<Squnion>{a, b} = a \<squnion> b" |
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by (simp add: Sup_empty Sup_insert) |
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lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)" |
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by (auto intro: Inf_greatest dest: Inf_lower) |
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lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)" |
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by (auto intro: Sup_least dest: Sup_upper) |
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lemma Inf_mono: |
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assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b" |
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shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B" |
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proof (rule Inf_greatest) |
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fix b assume "b \<in> B" |
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with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast |
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from `a \<in> A` have "\<Sqinter>A \<sqsubseteq> a" by (rule Inf_lower) |
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with `a \<sqsubseteq> b` show "\<Sqinter>A \<sqsubseteq> b" by auto |
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qed |
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lemma Sup_mono: |
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assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b" |
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shows "\<Squnion>A \<sqsubseteq> \<Squnion>B" |
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proof (rule Sup_least) |
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fix a assume "a \<in> A" |
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with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast |
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from `b \<in> B` have "b \<sqsubseteq> \<Squnion>B" by (rule Sup_upper) |
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with `a \<sqsubseteq> b` show "a \<sqsubseteq> \<Squnion>B" by auto |
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qed |
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lemma top_le: |
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"\<top> \<sqsubseteq> x \<Longrightarrow> x = \<top>" |
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by (rule antisym) auto |
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lemma le_bot: |
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"x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>" |
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by (rule antisym) auto |
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lemma not_less_bot[simp]: "\<not> (x \<sqsubset> \<bottom>)" |
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using bot_least[of x] by (auto simp: le_less) |
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lemma not_top_less[simp]: "\<not> (\<top> \<sqsubset> x)" |
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using top_greatest[of x] by (auto simp: le_less) |
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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A" |
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using Sup_upper[of u A] by auto |
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v" |
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using Inf_lower[of u A] by auto |
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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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"INFI A f = \<Sqinter> (f ` A)" |
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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"SUPR A f = \<Squnion> (f ` A)" |
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end |
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syntax |
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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10) |
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) |
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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) |
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) |
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syntax (xsymbols) |
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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) |
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) |
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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) |
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) |
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translations |
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"INF x y. B" == "INF x. INF y. B" |
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"INF x. B" == "CONST INFI CONST UNIV (%x. B)" |
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"INF x. B" == "INF x:CONST UNIV. B" |
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"INF x:A. B" == "CONST INFI A (%x. B)" |
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"SUP x y. B" == "SUP x. SUP y. B" |
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"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" |
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"SUP x. B" == "SUP x:CONST UNIV. B" |
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"SUP x:A. B" == "CONST SUPR A (%x. B)" |
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print_translation {* |
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[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, |
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Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] |
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*} -- {* to avoid eta-contraction of body *} |
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context complete_lattice |
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begin |
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lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g" |
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by (simp add: SUPR_def cong: image_cong) |
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lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g" |
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by (simp add: INFI_def cong: image_cong) |
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lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)" |
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by (auto simp add: SUPR_def intro: Sup_upper) |
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lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)" |
41971 | 187 |
using le_SUPI[of i A M] by auto |
188 |
||
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lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u" |
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by (auto simp add: SUPR_def intro: Sup_least) |
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lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i" |
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by (auto simp add: INFI_def intro: Inf_lower) |
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194 |
|
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lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u" |
41971 | 196 |
using INF_leI[of i A M] by auto |
197 |
||
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lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)" |
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by (auto simp add: INFI_def intro: Inf_greatest) |
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lemma SUP_le_iff: "(\<Squnion>i\<in>A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)" |
35629 | 202 |
unfolding SUPR_def by (auto simp add: Sup_le_iff) |
203 |
||
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lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)" |
35629 | 205 |
unfolding INFI_def by (auto simp add: le_Inf_iff) |
206 |
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lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. M) = M" |
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by (auto intro: antisym INF_leI le_INFI) |
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lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. M) = M" |
41082 | 211 |
by (auto intro: antisym SUP_leI le_SUPI) |
38705 | 212 |
|
213 |
lemma INF_mono: |
|
43753 | 214 |
"(\<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)" |
38705 | 215 |
by (force intro!: Inf_mono simp: INFI_def) |
216 |
||
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lemma SUP_mono: |
43753 | 218 |
"(\<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)" |
41082 | 219 |
by (force intro!: Sup_mono simp: SUPR_def) |
40872 | 220 |
|
43753 | 221 |
lemma INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<sqsubseteq> INFI A f" |
40872 | 222 |
by (intro INF_mono) auto |
223 |
||
43753 | 224 |
lemma SUP_subset: "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f" |
41082 | 225 |
by (intro SUP_mono) auto |
40872 | 226 |
|
43753 | 227 |
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)" |
40872 | 228 |
by (iprover intro: INF_leI le_INFI order_trans antisym) |
229 |
||
43753 | 230 |
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)" |
41082 | 231 |
by (iprover intro: SUP_leI le_SUPI order_trans antisym) |
232 |
||
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end |
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|
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lemma Inf_less_iff: |
236 |
fixes a :: "'a\<Colon>{complete_lattice,linorder}" |
|
43753 | 237 |
shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)" |
43754 | 238 |
unfolding not_le [symmetric] le_Inf_iff by auto |
41082 | 239 |
|
38705 | 240 |
lemma less_Sup_iff: |
241 |
fixes a :: "'a\<Colon>{complete_lattice,linorder}" |
|
43753 | 242 |
shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)" |
43754 | 243 |
unfolding not_le [symmetric] Sup_le_iff by auto |
38705 | 244 |
|
41082 | 245 |
lemma INF_less_iff: |
246 |
fixes a :: "'a::{complete_lattice,linorder}" |
|
43753 | 247 |
shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)" |
41082 | 248 |
unfolding INFI_def Inf_less_iff by auto |
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249 |
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lemma less_SUP_iff: |
251 |
fixes a :: "'a::{complete_lattice,linorder}" |
|
43753 | 252 |
shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)" |
40872 | 253 |
unfolding SUPR_def less_Sup_iff by auto |
254 |
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32139 | 255 |
subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *} |
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instantiation bool :: complete_lattice |
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begin |
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definition |
41080 | 261 |
"\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" |
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definition |
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"\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" |
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instance proof |
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qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) |
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end |
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lemma INFI_bool_eq [simp]: |
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"INFI = Ball" |
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proof (rule ext)+ |
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fix A :: "'a set" |
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fix P :: "'a \<Rightarrow> bool" |
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show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)" |
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by (auto simp add: Ball_def INFI_def Inf_bool_def) |
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qed |
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lemma SUPR_bool_eq [simp]: |
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"SUPR = Bex" |
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proof (rule ext)+ |
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fix A :: "'a set" |
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fix P :: "'a \<Rightarrow> bool" |
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show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)" |
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by (auto simp add: Bex_def SUPR_def Sup_bool_def) |
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qed |
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instantiation "fun" :: (type, complete_lattice) complete_lattice |
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begin |
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definition |
41080 | 293 |
"\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" |
294 |
||
295 |
lemma Inf_apply: |
|
296 |
"(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}" |
|
297 |
by (simp add: Inf_fun_def) |
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definition |
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"\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" |
301 |
||
302 |
lemma Sup_apply: |
|
303 |
"(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}" |
|
304 |
by (simp add: Sup_fun_def) |
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instance proof |
41080 | 307 |
qed (auto simp add: le_fun_def Inf_apply Sup_apply |
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intro: Inf_lower Sup_upper Inf_greatest Sup_least) |
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end |
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311 |
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lemma INFI_apply: |
313 |
"(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" |
|
314 |
by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply) |
|
38705 | 315 |
|
41080 | 316 |
lemma SUPR_apply: |
317 |
"(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" |
|
318 |
by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply) |
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41082 | 321 |
subsection {* Inter *} |
322 |
||
323 |
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where |
|
324 |
"Inter S \<equiv> \<Sqinter>S" |
|
325 |
||
326 |
notation (xsymbols) |
|
327 |
Inter ("\<Inter>_" [90] 90) |
|
328 |
||
329 |
lemma Inter_eq: |
|
330 |
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" |
|
331 |
proof (rule set_eqI) |
|
332 |
fix x |
|
333 |
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" |
|
334 |
by auto |
|
335 |
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" |
|
336 |
by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def) |
|
337 |
qed |
|
338 |
||
43741 | 339 |
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" |
41082 | 340 |
by (unfold Inter_eq) blast |
341 |
||
43741 | 342 |
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" |
41082 | 343 |
by (simp add: Inter_eq) |
344 |
||
345 |
text {* |
|
346 |
\medskip A ``destruct'' rule -- every @{term X} in @{term C} |
|
43741 | 347 |
contains @{term A} as an element, but @{prop "A \<in> X"} can hold when |
348 |
@{prop "X \<in> C"} does not! This rule is analogous to @{text spec}. |
|
41082 | 349 |
*} |
350 |
||
43741 | 351 |
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" |
41082 | 352 |
by auto |
353 |
||
43741 | 354 |
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" |
41082 | 355 |
-- {* ``Classical'' elimination rule -- does not require proving |
43741 | 356 |
@{prop "X \<in> C"}. *} |
41082 | 357 |
by (unfold Inter_eq) blast |
358 |
||
43741 | 359 |
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 360 |
by (fact Inf_lower) |
361 |
||
362 |
lemma (in complete_lattice) Inf_less_eq: |
|
363 |
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<sqsubseteq> u" |
|
364 |
and "A \<noteq> {}" |
|
43753 | 365 |
shows "\<Sqinter>A \<sqsubseteq> u" |
43740 | 366 |
proof - |
367 |
from `A \<noteq> {}` obtain v where "v \<in> A" by blast |
|
368 |
moreover with assms have "v \<sqsubseteq> u" by blast |
|
369 |
ultimately show ?thesis by (rule Inf_lower2) |
|
370 |
qed |
|
41082 | 371 |
|
372 |
lemma Inter_subset: |
|
43755 | 373 |
"(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 374 |
by (fact Inf_less_eq) |
41082 | 375 |
|
43755 | 376 |
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> Inter A" |
43740 | 377 |
by (fact Inf_greatest) |
41082 | 378 |
|
379 |
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}" |
|
43739 | 380 |
by (fact Inf_binary [symmetric]) |
41082 | 381 |
|
382 |
lemma Inter_empty [simp]: "\<Inter>{} = UNIV" |
|
383 |
by (fact Inf_empty) |
|
384 |
||
385 |
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}" |
|
43739 | 386 |
by (fact Inf_UNIV) |
41082 | 387 |
|
388 |
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B" |
|
43739 | 389 |
by (fact Inf_insert) |
41082 | 390 |
|
43741 | 391 |
lemma (in complete_lattice) Inf_inter_less: "\<Sqinter>A \<squnion> \<Sqinter>B \<sqsubseteq> \<Sqinter>(A \<inter> B)" |
392 |
by (auto intro: Inf_greatest Inf_lower) |
|
393 |
||
41082 | 394 |
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" |
43741 | 395 |
by (fact Inf_inter_less) |
396 |
||
43756 | 397 |
lemma (in complete_lattice) Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" |
398 |
by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) |
|
41082 | 399 |
|
400 |
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" |
|
43756 | 401 |
by (fact Inf_union_distrib) |
402 |
||
43801 | 403 |
lemma (in complete_lattice) Inf_top_conv [no_atp]: |
404 |
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
405 |
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
406 |
proof - |
|
407 |
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
408 |
proof |
|
409 |
assume "\<forall>x\<in>A. x = \<top>" |
|
410 |
then have "A = {} \<or> A = {\<top>}" by auto |
|
411 |
then show "\<Sqinter>A = \<top>" by auto |
|
412 |
next |
|
413 |
assume "\<Sqinter>A = \<top>" |
|
414 |
show "\<forall>x\<in>A. x = \<top>" |
|
415 |
proof (rule ccontr) |
|
416 |
assume "\<not> (\<forall>x\<in>A. x = \<top>)" |
|
417 |
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast |
|
418 |
then obtain B where "A = insert x B" by blast |
|
419 |
with `\<Sqinter>A = \<top>` `x \<noteq> \<top>` show False by (simp add: Inf_insert) |
|
420 |
qed |
|
421 |
qed |
|
422 |
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto |
|
423 |
qed |
|
41082 | 424 |
|
425 |
lemma Inter_UNIV_conv [simp,no_atp]: |
|
43741 | 426 |
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
427 |
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
|
43801 | 428 |
by (fact Inf_top_conv)+ |
41082 | 429 |
|
43756 | 430 |
lemma (in complete_lattice) Inf_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> \<Sqinter>B" |
431 |
by (auto intro: Inf_greatest Inf_lower) |
|
432 |
||
43741 | 433 |
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" |
43756 | 434 |
by (fact Inf_anti_mono) |
41082 | 435 |
|
436 |
||
437 |
subsection {* Intersections of families *} |
|
438 |
||
439 |
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where |
|
440 |
"INTER \<equiv> INFI" |
|
441 |
||
442 |
syntax |
|
443 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) |
|
444 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) |
|
445 |
||
446 |
syntax (xsymbols) |
|
447 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) |
|
448 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) |
|
449 |
||
450 |
syntax (latex output) |
|
451 |
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
|
452 |
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) |
|
453 |
||
454 |
translations |
|
455 |
"INT x y. B" == "INT x. INT y. B" |
|
456 |
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" |
|
457 |
"INT x. B" == "INT x:CONST UNIV. B" |
|
458 |
"INT x:A. B" == "CONST INTER A (%x. B)" |
|
459 |
||
460 |
print_translation {* |
|
42284 | 461 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] |
41082 | 462 |
*} -- {* to avoid eta-contraction of body *} |
463 |
||
464 |
lemma INTER_eq_Inter_image: |
|
465 |
"(\<Inter>x\<in>A. B x) = \<Inter>(B`A)" |
|
466 |
by (fact INFI_def) |
|
467 |
||
468 |
lemma Inter_def: |
|
469 |
"\<Inter>S = (\<Inter>x\<in>S. x)" |
|
470 |
by (simp add: INTER_eq_Inter_image image_def) |
|
471 |
||
472 |
lemma INTER_def: |
|
473 |
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" |
|
474 |
by (auto simp add: INTER_eq_Inter_image Inter_eq) |
|
475 |
||
476 |
lemma Inter_image_eq [simp]: |
|
477 |
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" |
|
43801 | 478 |
by (rule sym) (fact INFI_def) |
41082 | 479 |
|
43817 | 480 |
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)" |
41082 | 481 |
by (unfold INTER_def) blast |
482 |
||
43817 | 483 |
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)" |
41082 | 484 |
by (unfold INTER_def) blast |
485 |
||
43817 | 486 |
lemma INT_D [elim, Pure.elim]: "b : (\<Inter>x\<in>A. B x) \<Longrightarrow> a:A \<Longrightarrow> b: B a" |
41082 | 487 |
by auto |
488 |
||
43817 | 489 |
lemma INT_E [elim]: "b : (\<Inter>x\<in>A. B x) \<Longrightarrow> (b: B a \<Longrightarrow> R) \<Longrightarrow> (a~:A \<Longrightarrow> R) \<Longrightarrow> R" |
41082 | 490 |
-- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *} |
491 |
by (unfold INTER_def) blast |
|
492 |
||
493 |
lemma INT_cong [cong]: |
|
43817 | 494 |
"A = B \<Longrightarrow> (\<And>x. x:B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Inter>x\<in>A. C x) = (\<Inter>x\<in>B. D x)" |
41082 | 495 |
by (simp add: INTER_def) |
496 |
||
497 |
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" |
|
498 |
by blast |
|
499 |
||
500 |
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" |
|
501 |
by blast |
|
502 |
||
43817 | 503 |
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a" |
41082 | 504 |
by (fact INF_leI) |
505 |
||
43817 | 506 |
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)" |
41082 | 507 |
by (fact le_INFI) |
508 |
||
509 |
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV" |
|
510 |
by blast |
|
511 |
||
43817 | 512 |
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" |
41082 | 513 |
by blast |
514 |
||
515 |
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)" |
|
516 |
by (fact le_INF_iff) |
|
517 |
||
518 |
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" |
|
519 |
by blast |
|
520 |
||
521 |
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" |
|
522 |
by blast |
|
523 |
||
524 |
lemma INT_insert_distrib: |
|
43817 | 525 |
"u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" |
41082 | 526 |
by blast |
527 |
||
528 |
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" |
|
529 |
by auto |
|
530 |
||
531 |
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})" |
|
532 |
-- {* Look: it has an \emph{existential} quantifier *} |
|
533 |
by blast |
|
534 |
||
535 |
lemma INTER_UNIV_conv[simp]: |
|
43817 | 536 |
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" |
537 |
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" |
|
41082 | 538 |
by blast+ |
539 |
||
43817 | 540 |
lemma INT_bool_eq: "(\<Inter>b. A b) = (A True \<inter> A False)" |
41082 | 541 |
by (auto intro: bool_induct) |
542 |
||
543 |
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" |
|
544 |
by blast |
|
545 |
||
546 |
lemma INT_anti_mono: |
|
43817 | 547 |
"B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> |
41082 | 548 |
(\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)" |
549 |
-- {* The last inclusion is POSITIVE! *} |
|
550 |
by (blast dest: subsetD) |
|
551 |
||
43817 | 552 |
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)" |
41082 | 553 |
by blast |
554 |
||
555 |
||
32139 | 556 |
subsection {* Union *} |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
557 |
|
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset
|
558 |
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
|
559 |
"Union S \<equiv> \<Squnion>S" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
560 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
561 |
notation (xsymbols) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
562 |
Union ("\<Union>_" [90] 90) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
563 |
|
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
|
564 |
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
|
565 |
"\<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
|
566 |
proof (rule set_eqI) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
567 |
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
|
568 |
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
|
569 |
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
|
570 |
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" |
32587
caa5ada96a00
Inter and Union are mere abbreviations for Inf and Sup
haftmann
parents:
32436
diff
changeset
|
571 |
by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
572 |
qed |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
573 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
574 |
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
|
575 |
"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
|
576 |
by (unfold Union_eq) blast |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
577 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
578 |
lemma UnionI [intro]: |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
579 |
"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
|
580 |
-- {* 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
|
581 |
@{term A} may be flexible. *} |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
582 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
583 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
584 |
lemma UnionE [elim!]: |
43817 | 585 |
"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
|
586 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
587 |
|
43817 | 588 |
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
589 |
by (iprover intro: subsetI UnionI) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
590 |
|
43817 | 591 |
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
592 |
by (iprover intro: subsetI elim: UnionE dest: subsetD) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
593 |
|
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
|
594 |
lemma Un_eq_Union: "A \<union> B = \<Union>{A, B}" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
595 |
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
|
596 |
|
43817 | 597 |
lemma Union_empty [simp]: "\<Union>{} = {}" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
598 |
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
|
599 |
|
43817 | 600 |
lemma Union_UNIV [simp]: "\<Union>UNIV = UNIV" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
601 |
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
|
602 |
|
43817 | 603 |
lemma Union_insert [simp]: "\<Union>insert a B = a \<union> \<Union>B" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
604 |
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
|
605 |
|
43817 | 606 |
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
607 |
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
|
608 |
|
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
|
609 |
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
610 |
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
|
611 |
|
43817 | 612 |
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
613 |
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
|
614 |
|
43817 | 615 |
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
616 |
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
|
617 |
|
43817 | 618 |
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
619 |
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
|
620 |
|
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
|
621 |
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
|
622 |
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
|
623 |
|
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
|
624 |
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
|
625 |
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
|
626 |
|
43817 | 627 |
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
628 |
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
|
629 |
|
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
630 |
|
32139 | 631 |
subsection {* Unions of families *} |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
632 |
|
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset
|
633 |
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
|
634 |
"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
|
635 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
636 |
syntax |
35115 | 637 |
"_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
|
638 |
"_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
|
639 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
640 |
syntax (xsymbols) |
35115 | 641 |
"_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
|
642 |
"_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
|
643 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
644 |
syntax (latex output) |
35115 | 645 |
"_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
|
646 |
"_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
|
647 |
|
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 |
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
|
649 |
"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
|
650 |
"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
|
651 |
"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
|
652 |
"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
|
653 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
654 |
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
|
655 |
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
|
656 |
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
|
657 |
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
|
658 |
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
|
659 |
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
|
660 |
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
|
661 |
*} |
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 |
|
35115 | 663 |
print_translation {* |
42284 | 664 |
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] |
35115 | 665 |
*} -- {* 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
|
666 |
|
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
|
667 |
lemma UNION_eq_Union_image: |
43817 | 668 |
"(\<Union>x\<in>A. B x) = \<Union>(B ` A)" |
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset
|
669 |
by (fact SUPR_def) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
670 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
671 |
lemma Union_def: |
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
672 |
"\<Union>S = (\<Union>x\<in>S. x)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
673 |
by (simp add: UNION_eq_Union_image image_def) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
674 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
675 |
lemma UNION_def [no_atp]: |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
676 |
"(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" |
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset
|
677 |
by (auto simp add: UNION_eq_Union_image Union_eq) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
678 |
|
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
679 |
lemma Union_image_eq [simp]: |
43817 | 680 |
"\<Union>(B ` A) = (\<Union>x\<in>A. B x)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
681 |
by (rule sym) (fact UNION_eq_Union_image) |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
682 |
|
43817 | 683 |
lemma UN_iff [simp]: "(b: (\<Union>x\<in>A. B x)) = (\<exists>x\<in>A. b: B x)" |
11979 | 684 |
by (unfold UNION_def) blast |
685 |
||
43817 | 686 |
lemma UN_I [intro]: "a:A \<Longrightarrow> b: B a \<Longrightarrow> b: (\<Union>x\<in>A. B x)" |
11979 | 687 |
-- {* The order of the premises presupposes that @{term A} is rigid; |
688 |
@{term b} may be flexible. *} |
|
689 |
by auto |
|
690 |
||
43817 | 691 |
lemma UN_E [elim!]: "b : (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x:A \<Longrightarrow> b: B x \<Longrightarrow> R) \<Longrightarrow> R" |
11979 | 692 |
by (unfold UNION_def) blast |
923 | 693 |
|
11979 | 694 |
lemma UN_cong [cong]: |
43817 | 695 |
"A = B \<Longrightarrow> (\<And>x. x:B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)" |
11979 | 696 |
by (simp add: UNION_def) |
697 |
||
29691 | 698 |
lemma strong_UN_cong: |
43817 | 699 |
"A = B \<Longrightarrow> (\<And>x. x:B =simp=> C x = D x) \<Longrightarrow> (\<Union>x\<in>A. C x) = (\<Union>x\<in>B. D x)" |
29691 | 700 |
by (simp add: UNION_def simp_implies_def) |
701 |
||
43817 | 702 |
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
|
703 |
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
|
704 |
|
43817 | 705 |
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)" |
32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset
|
706 |
by (fact le_SUPI) |
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
|
707 |
|
43817 | 708 |
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
709 |
by (iprover intro: subsetI elim: UN_E dest: subsetD) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
710 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
711 |
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
|
712 |
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
|
713 |
|
43817 | 714 |
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" |
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
|
715 |
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
|
716 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
717 |
lemma UN_empty [simp,no_atp]: "(\<Union>x\<in>{}. B x) = {}" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
718 |
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
|
719 |
|
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
|
720 |
lemma UN_empty2 [simp]: "(\<Union>x\<in>A. {}) = {}" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
721 |
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
|
722 |
|
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
|
723 |
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
724 |
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
|
725 |
|
43817 | 726 |
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
727 |
by auto |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
728 |
|
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
|
729 |
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
730 |
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
|
731 |
|
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
|
732 |
lemma UN_Un[simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)" |
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
|
733 |
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
|
734 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
735 |
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
736 |
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
|
737 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
738 |
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" |
35629 | 739 |
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
|
740 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
741 |
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
742 |
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
|
743 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
744 |
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
745 |
by auto |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
746 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
747 |
lemma UN_eq: "(\<Union>x\<in>A. B x) = \<Union>({Y. \<exists>x\<in>A. Y = B x})" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
748 |
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
|
749 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
750 |
lemma UNION_empty_conv[simp]: |
43817 | 751 |
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
752 |
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
753 |
by blast+ |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
754 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
755 |
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
|
756 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
757 |
|
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
|
758 |
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) = (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
759 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
760 |
|
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
|
761 |
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
762 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
763 |
|
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
|
764 |
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
|
765 |
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
|
766 |
|
43817 | 767 |
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
768 |
by (auto intro: bool_contrapos) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
769 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
770 |
lemma 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
|
771 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
772 |
|
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
|
773 |
lemma UN_mono: |
43817 | 774 |
"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
|
775 |
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
776 |
by (blast dest: subsetD) |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
777 |
|
43817 | 778 |
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
|
779 |
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
|
780 |
|
43817 | 781 |
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
|
782 |
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
|
783 |
|
43817 | 784 |
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
|
785 |
-- {* 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
|
786 |
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
|
787 |
|
43817 | 788 |
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)" |
789 |
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
|
790 |
|
11979 | 791 |
|
32139 | 792 |
subsection {* Distributive laws *} |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
793 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
794 |
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
795 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
796 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
797 |
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
798 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
799 |
|
43817 | 800 |
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
|
801 |
-- {* 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
|
802 |
-- {* Union of a family of unions *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
803 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
804 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
805 |
lemma UN_Un_distrib: "(\<Union>i\<in>I. A i \<union> B i) = (\<Union>i\<in>I. A i) \<union> (\<Union>i\<in>I. B i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
806 |
-- {* Equivalent version *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
807 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
808 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
809 |
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
810 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
811 |
|
43817 | 812 |
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
813 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
814 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
815 |
lemma INT_Int_distrib: "(\<Inter>i\<in>I. A i \<inter> B i) = (\<Inter>i\<in>I. A i) \<inter> (\<Inter>i\<in>I. B i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
816 |
-- {* Equivalent version *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
817 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
818 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
819 |
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
|
820 |
-- {* Halmos, Naive Set Theory, page 35. *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
821 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
822 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
823 |
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
824 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
825 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
826 |
lemma Int_UN_distrib2: "(\<Union>i\<in>I. A i) \<inter> (\<Union>j\<in>J. B j) = (\<Union>i\<in>I. \<Union>j\<in>J. A i \<inter> B j)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
827 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
828 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
829 |
lemma Un_INT_distrib2: "(\<Inter>i\<in>I. A i) \<union> (\<Inter>j\<in>J. B j) = (\<Inter>i\<in>I. \<Inter>j\<in>J. A i \<union> B j)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
830 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
831 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
832 |
|
32139 | 833 |
subsection {* Complement *} |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
834 |
|
43817 | 835 |
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
836 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
837 |
|
43817 | 838 |
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
839 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
840 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
841 |
|
32139 | 842 |
subsection {* Miniscoping and maxiscoping *} |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
843 |
|
13860 | 844 |
text {* \medskip Miniscoping: pushing in quantifiers and big Unions |
845 |
and Intersections. *} |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
846 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
847 |
lemma UN_simps [simp]: |
43817 | 848 |
"\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))" |
849 |
"\<And>A B C. (\<Union>x\<in>C. A x Un B) = ((if C={} then {} else (\<Union>x\<in>C. A x) Un B))" |
|
850 |
"\<And>A B C. (\<Union>x\<in>C. A Un B x) = ((if C={} then {} else A Un (\<Union>x\<in>C. B x)))" |
|
851 |
"\<And>A B C. (\<Union>x\<in>C. A x Int B) = ((\<Union>x\<in>C. A x) Int B)" |
|
852 |
"\<And>A B C. (\<Union>x\<in>C. A Int B x) = (A Int (\<Union>x\<in>C. B x))" |
|
853 |
"\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)" |
|
854 |
"\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))" |
|
855 |
"\<And>A B. (UN x: \<Union>A. B x) = (UN y:A. UN x:y. B x)" |
|
856 |
"\<And>A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)" |
|
857 |
"\<And>A B f. (UN x:f`A. B x) = (UN a:A. B (f a))" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
858 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
859 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
860 |
lemma INT_simps [simp]: |
43817 | 861 |
"\<And>A B C. (\<Inter>x\<in>C. A x Int B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) Int B)" |
862 |
"\<And>A B C. (\<Inter>x\<in>C. A Int B x) = (if C={} then UNIV else A Int (\<Inter>x\<in>C. B x))" |
|
863 |
"\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)" |
|
864 |
"\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))" |
|
865 |
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)" |
|
866 |
"\<And>A B C. (\<Inter>x\<in>C. A x Un B) = ((\<Inter>x\<in>C. A x) Un B)" |
|
867 |
"\<And>A B C. (\<Inter>x\<in>C. A Un B x) = (A Un (\<Inter>x\<in>C. B x))" |
|
868 |
"\<And>A B. (INT x: \<Union>A. B x) = (\<Inter>y\<in>A. INT x:y. B x)" |
|
869 |
"\<And>A B C. (INT z: UNION A B. C z) = (\<Inter>x\<in>A. INT z: B(x). C z)" |
|
870 |
"\<And>A B f. (INT x:f`A. B x) = (INT a:A. B (f a))" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
871 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
872 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
873 |
lemma ball_simps [simp,no_atp]: |
43817 | 874 |
"\<And>A P Q. (\<forall>x\<in>A. P x | Q) = ((\<forall>x\<in>A. P x) | Q)" |
875 |
"\<And>A P Q. (\<forall>x\<in>A. P | Q x) = (P | (\<forall>x\<in>A. Q x))" |
|
876 |
"\<And>A P Q. (\<forall>x\<in>A. P --> Q x) = (P --> (\<forall>x\<in>A. Q x))" |
|
877 |
"\<And>A P Q. (\<forall>x\<in>A. P x --> Q) = ((\<exists>x\<in>A. P x) --> Q)" |
|
878 |
"\<And>P. (ALL x:{}. P x) = True" |
|
879 |
"\<And>P. (ALL x:UNIV. P x) = (ALL x. P x)" |
|
880 |
"\<And>a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))" |
|
881 |
"\<And>A P. (ALL x:\<Union>A. P x) = (ALL y:A. ALL x:y. P x)" |
|
882 |
"\<And>A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)" |
|
883 |
"\<And>P Q. (ALL x:Collect Q. P x) = (ALL x. Q x --> P x)" |
|
884 |
"\<And>A P f. (ALL x:f`A. P x) = (\<forall>x\<in>A. P (f x))" |
|
885 |
"\<And>A P. (~(\<forall>x\<in>A. P x)) = (\<exists>x\<in>A. ~P x)" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
886 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
887 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset
|
888 |
lemma bex_simps [simp,no_atp]: |
43817 | 889 |
"\<And>A P Q. (\<exists>x\<in>A. P x & Q) = ((\<exists>x\<in>A. P x) & Q)" |
890 |
"\<And>A P Q. (\<exists>x\<in>A. P & Q x) = (P & (\<exists>x\<in>A. Q x))" |
|
891 |
"\<And>P. (EX x:{}. P x) = False" |
|
892 |
"\<And>P. (EX x:UNIV. P x) = (EX x. P x)" |
|
893 |
"\<And>a B P. (EX x:insert a B. P x) = (P(a) | (EX x:B. P x))" |
|
894 |
"\<And>A P. (EX x:\<Union>A. P x) = (EX y:A. EX x:y. P x)" |
|
895 |
"\<And>A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)" |
|
896 |
"\<And>P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)" |
|
897 |
"\<And>A P f. (EX x:f`A. P x) = (\<exists>x\<in>A. P (f x))" |
|
898 |
"\<And>A P. (~(\<exists>x\<in>A. P x)) = (\<forall>x\<in>A. ~P x)" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
899 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
900 |
|
13860 | 901 |
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} |
902 |
||
903 |
lemma UN_extend_simps: |
|
43817 | 904 |
"\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))" |
905 |
"\<And>A B C. (\<Union>x\<in>C. A x) Un B = (if C={} then B else (\<Union>x\<in>C. A x Un B))" |
|
906 |
"\<And>A B C. A Un (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A Un B x))" |
|
907 |
"\<And>A B C. ((\<Union>x\<in>C. A x) Int B) = (\<Union>x\<in>C. A x Int B)" |
|
908 |
"\<And>A B C. (A Int (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A Int B x)" |
|
909 |
"\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)" |
|
910 |
"\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)" |
|
911 |
"\<And>A B. (UN y:A. UN x:y. B x) = (UN x: \<Union>A. B x)" |
|
912 |
"\<And>A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)" |
|
913 |
"\<And>A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)" |
|
13860 | 914 |
by auto |
915 |
||
916 |
lemma INT_extend_simps: |
|
43817 | 917 |
"\<And>A B C. (\<Inter>x\<in>C. A x) Int B = (if C={} then B else (\<Inter>x\<in>C. A x Int B))" |
918 |
"\<And>A B C. A Int (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A Int B x))" |
|
919 |
"\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV-B else (\<Inter>x\<in>C. A x - B))" |
|
920 |
"\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))" |
|
921 |
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))" |
|
922 |
"\<And>A B C. ((\<Inter>x\<in>C. A x) Un B) = (\<Inter>x\<in>C. A x Un B)" |
|
923 |
"\<And>A B C. A Un (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A Un B x)" |
|
924 |
"\<And>A B. (\<Inter>y\<in>A. INT x:y. B x) = (INT x: \<Union>A. B x)" |
|
925 |
"\<And>A B C. (\<Inter>x\<in>A. INT z: B(x). C z) = (INT z: UNION A B. C z)" |
|
926 |
"\<And>A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)" |
|
13860 | 927 |
by auto |
928 |
||
929 |
||
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
930 |
no_notation |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
931 |
less_eq (infix "\<sqsubseteq>" 50) and |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
932 |
less (infix "\<sqsubset>" 50) and |
41082 | 933 |
bot ("\<bottom>") and |
934 |
top ("\<top>") and |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
935 |
inf (infixl "\<sqinter>" 70) and |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
936 |
sup (infixl "\<squnion>" 65) and |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
937 |
Inf ("\<Sqinter>_" [900] 900) and |
41082 | 938 |
Sup ("\<Squnion>_" [900] 900) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
939 |
|
41080 | 940 |
no_syntax (xsymbols) |
41082 | 941 |
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) |
942 |
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) |
|
41080 | 943 |
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) |
944 |
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) |
|
945 |
||
30596 | 946 |
lemmas mem_simps = |
947 |
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff |
|
948 |
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff |
|
949 |
-- {* Each of these has ALREADY been added @{text "[simp]"} above. *} |
|
21669 | 950 |
|
11979 | 951 |
end |