author  haftmann 
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changeset 43817  d53350bc65a4 
parent 43814  58791b75cf1f 
child 43818  fcc5d3ffb6f5 
permissions  rwrr 
32139  1 
(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *) 
11979  2 

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header {* Complete lattices, with special focus on sets *} 
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32139  5 
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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32139  25 

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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 etacontraction of body *} 
11979  173 

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context complete_lattice 
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begin 
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lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g" 
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by (simp add: SUPR_def cong: image_cong) 

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lemma INF_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> INFI A f = INFI A g" 

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by (simp add: INFI_def cong: image_cong) 

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lemma le_SUPI: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> (\<Squnion>i\<in>A. M i)" 
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by (auto simp add: SUPR_def intro: Sup_upper) 
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lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. M i)" 
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using le_SUPI[of i A M] by auto 
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43741  189 
lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. M i) \<sqsubseteq> u" 
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by (auto simp add: SUPR_def intro: Sup_least) 
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lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> M i" 
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by (auto simp add: INFI_def intro: Inf_lower) 
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lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. M i) \<sqsubseteq> u" 
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using INF_leI[of i A M] by auto 
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lemma le_INFI: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. M i)" 
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by (auto simp add: INFI_def intro: Inf_greatest) 
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43753  201 
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 

43753  204 
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 

43753  207 
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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43753  210 
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 

41082  217 
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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41082  235 
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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40872  250 
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 

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 
41080  264 
"\<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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41080  271 
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" 
43753  276 
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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41080  280 
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" 
43753  285 
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 
41080  300 
"\<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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41080  312 
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 etacontraction 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 etacontraction 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 UNIVB 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 