author  wenzelm 
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parent 41971  a54e8e95fe96 
child 43739  4529a3c56609 
permissions  rwrr 
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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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32139  5 
theory Complete_Lattice 
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imports Set 

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begin 

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notation 
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less_eq (infix "\<sqsubseteq>" 50) and 
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less (infix "\<sqsubset>" 50) and 
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inf (infixl "\<sqinter>" 70) and 
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sup (infixl "\<squnion>" 65) and 
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top ("\<top>") and 
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bot ("\<bottom>") 

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subsection {* Syntactic infimum and supremum operations *} 
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class Inf = 

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fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) 

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class Sup = 

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fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) 

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subsection {* Abstract complete lattices *} 
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class complete_lattice = bounded_lattice + Inf + Sup + 
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assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" 
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and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" 
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assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" 
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and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" 
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begin 
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lemma dual_complete_lattice: 
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"class.complete_lattice Sup Inf (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>" 
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by (auto intro!: class.complete_lattice.intro dual_bounded_lattice) 
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(unfold_locales, (fact bot_least top_greatest 
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Sup_upper Sup_least Inf_lower Inf_greatest)+) 
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}" 
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
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lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<sqsubseteq> b}" 
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by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) 
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lemma Inf_empty [simp]: 
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"\<Sqinter>{} = \<top>" 
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by (auto intro: antisym Inf_greatest) 
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lemma Sup_empty [simp]: 
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"\<Squnion>{} = \<bottom>" 
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by (auto intro: antisym Sup_least) 
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lemma Inf_UNIV [simp]: 
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"\<Sqinter>UNIV = \<bottom>" 

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by (simp add: Sup_Inf Sup_empty [symmetric]) 

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lemma Sup_UNIV [simp]: 

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"\<Squnion>UNIV = \<top>" 

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by (simp add: Inf_Sup Inf_empty [symmetric]) 

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lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" 
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by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) 
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lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" 
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by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) 
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lemma Inf_singleton [simp]: 
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"\<Sqinter>{a} = a" 
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by (auto intro: antisym Inf_lower Inf_greatest) 
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lemma Sup_singleton [simp]: 
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"\<Squnion>{a} = a" 
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by (auto intro: antisym Sup_upper Sup_least) 
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lemma Inf_binary: 
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"\<Sqinter>{a, b} = a \<sqinter> b" 
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by (simp add: Inf_empty Inf_insert) 
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lemma Sup_binary: 
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"\<Squnion>{a, b} = a \<squnion> b" 
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by (simp add: Sup_empty Sup_insert) 
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lemma le_Inf_iff: "b \<sqsubseteq> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)" 
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by (auto intro: Inf_greatest dest: Inf_lower) 

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lemma Sup_le_iff: "Sup A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)" 
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by (auto intro: Sup_least dest: Sup_upper) 

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lemma Inf_mono: 

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assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b" 
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shows "Inf A \<sqsubseteq> Inf B" 

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proof (rule Inf_greatest) 
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fix b assume "b \<in> B" 

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with assms obtain a where "a \<in> A" and "a \<sqsubseteq> b" by blast 
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from `a \<in> A` have "Inf A \<sqsubseteq> a" by (rule Inf_lower) 

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with `a \<sqsubseteq> b` show "Inf A \<sqsubseteq> b" by auto 

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qed 
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lemma Sup_mono: 
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assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b" 
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shows "Sup A \<sqsubseteq> Sup B" 

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proof (rule Sup_least) 
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fix a assume "a \<in> A" 

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with assms obtain b where "b \<in> B" and "a \<sqsubseteq> b" by blast 
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from `b \<in> B` have "b \<sqsubseteq> Sup B" by (rule Sup_upper) 

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with `a \<sqsubseteq> b` show "a \<sqsubseteq> Sup B" by auto 

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qed 
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lemma top_le: 
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"top \<sqsubseteq> x \<Longrightarrow> x = top" 

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by (rule antisym) auto 

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lemma le_bot: 

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"x \<sqsubseteq> bot \<Longrightarrow> x = bot" 

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by (rule antisym) auto 

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lemma not_less_bot[simp]: "\<not> (x \<sqsubset> bot)" 

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using bot_least[of x] by (auto simp: le_less) 

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lemma not_top_less[simp]: "\<not> (top \<sqsubset> x)" 

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using top_greatest[of x] by (auto simp: le_less) 

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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> Sup A" 

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using Sup_upper[of u A] by auto 

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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> Inf A \<sqsubseteq> v" 

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using Inf_lower[of u A] by auto 

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definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
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"INFI A f = \<Sqinter> (f ` A)" 
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definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where 
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"SUPR A f = \<Squnion> (f ` A)" 

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end 
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syntax 
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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _./ _)" [0, 10] 10) 
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3INF _:_./ _)" [0, 0, 10] 10) 

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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _./ _)" [0, 10] 10) 
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3SUP _:_./ _)" [0, 0, 10] 10) 

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syntax (xsymbols) 

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"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

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"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

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translations 
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"INF x y. B" == "INF x. INF y. B" 
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"INF x. B" == "CONST INFI CONST UNIV (%x. B)" 

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"INF x. B" == "INF x:CONST UNIV. B" 

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"INF x:A. B" == "CONST INFI A (%x. B)" 

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"SUP x y. B" == "SUP x. SUP y. B" 
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"SUP x. B" == "CONST SUPR CONST UNIV (%x. B)" 
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"SUP x. B" == "SUP x:CONST UNIV. B" 
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"SUP x:A. B" == "CONST SUPR A (%x. B)" 
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print_translation {* 
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[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"}, 
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Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}] 

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*}  {* to avoid etacontraction of body *} 
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context complete_lattice 
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begin 
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lemma SUP_cong: "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> SUPR A f = SUPR A g" 
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by (simp add: SUPR_def cong: image_cong) 

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

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by (simp add: INFI_def cong: image_cong) 

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lemma le_SUPI: "i : A \<Longrightarrow> M i \<sqsubseteq> (SUP i:A. M i)" 
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by (auto simp add: SUPR_def intro: Sup_upper) 
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lemma le_SUPI2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> M i \<Longrightarrow> u \<sqsubseteq> (SUP i:A. M i)" 
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using le_SUPI[of i A M] by auto 

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lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<sqsubseteq> u) \<Longrightarrow> (SUP i:A. M i) \<sqsubseteq> u" 
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by (auto simp add: SUPR_def intro: Sup_least) 
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lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> M i" 
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by (auto simp add: INFI_def intro: Inf_lower) 
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lemma INF_leI2: "i \<in> A \<Longrightarrow> M i \<sqsubseteq> u \<Longrightarrow> (INF i:A. M i) \<sqsubseteq> u" 
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using INF_leI[of i A M] by auto 

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lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<sqsubseteq> M i) \<Longrightarrow> u \<sqsubseteq> (INF i:A. M i)" 
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by (auto simp add: INFI_def intro: Inf_greatest) 
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lemma SUP_le_iff: "(SUP i:A. M i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i \<in> A. M i \<sqsubseteq> u)" 
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unfolding SUPR_def by (auto simp add: Sup_le_iff) 

195 

196 
lemma le_INF_iff: "u \<sqsubseteq> (INF i:A. M i) \<longleftrightarrow> (\<forall>i \<in> A. u \<sqsubseteq> M i)" 

197 
unfolding INFI_def by (auto simp add: le_Inf_iff) 

198 

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lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" 
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by (auto intro: antisym INF_leI le_INFI) 
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41082  202 
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" 
203 
by (auto intro: antisym SUP_leI le_SUPI) 

38705  204 

205 
lemma INF_mono: 

206 
"(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)" 

207 
by (force intro!: Inf_mono simp: INFI_def) 

208 

41082  209 
lemma SUP_mono: 
210 
"(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (SUP n:A. f n) \<le> (SUP n:B. g n)" 

211 
by (force intro!: Sup_mono simp: SUPR_def) 

40872  212 

213 
lemma INF_subset: "A \<subseteq> B \<Longrightarrow> INFI B f \<le> INFI A f" 

214 
by (intro INF_mono) auto 

215 

41082  216 
lemma SUP_subset: "A \<subseteq> B \<Longrightarrow> SUPR A f \<le> SUPR B f" 
217 
by (intro SUP_mono) auto 

40872  218 

219 
lemma INF_commute: "(INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)" 

220 
by (iprover intro: INF_leI le_INFI order_trans antisym) 

221 

41082  222 
lemma SUP_commute: "(SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)" 
223 
by (iprover intro: SUP_leI le_SUPI order_trans antisym) 

224 

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end 
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lemma Inf_less_iff: 
228 
fixes a :: "'a\<Colon>{complete_lattice,linorder}" 

229 
shows "Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)" 

230 
unfolding not_le[symmetric] le_Inf_iff by auto 

231 

38705  232 
lemma less_Sup_iff: 
233 
fixes a :: "'a\<Colon>{complete_lattice,linorder}" 

234 
shows "a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)" 

235 
unfolding not_le[symmetric] Sup_le_iff by auto 

236 

41082  237 
lemma INF_less_iff: 
238 
fixes a :: "'a::{complete_lattice,linorder}" 

239 
shows "(INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)" 

240 
unfolding INFI_def Inf_less_iff by auto 

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40872  242 
lemma less_SUP_iff: 
243 
fixes a :: "'a::{complete_lattice,linorder}" 

244 
shows "a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)" 

245 
unfolding SUPR_def less_Sup_iff by auto 

246 

32139  247 
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  253 
"\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" 
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definition 
41080  256 
"\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" 
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instance proof 
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qed (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) 
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end 
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lemma INFI_bool_eq [simp]: 
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"INFI = Ball" 
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proof (rule ext)+ 
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fix A :: "'a set" 
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fix P :: "'a \<Rightarrow> bool" 
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show "(INF x:A. P x) \<longleftrightarrow> (\<forall>x \<in> A. P x)" 
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by (auto simp add: Ball_def INFI_def Inf_bool_def) 
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qed 
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lemma SUPR_bool_eq [simp]: 
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"SUPR = Bex" 
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proof (rule ext)+ 
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fix A :: "'a set" 
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fix P :: "'a \<Rightarrow> bool" 
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show "(SUP x:A. P x) \<longleftrightarrow> (\<exists>x \<in> A. P x)" 
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by (auto simp add: Bex_def SUPR_def Sup_bool_def) 
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qed 
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instantiation "fun" :: (type, complete_lattice) complete_lattice 
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begin 
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definition 
41080  285 
"\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" 
286 

287 
lemma Inf_apply: 

288 
"(\<Sqinter>A) x = \<Sqinter>{y. \<exists>f\<in>A. y = f x}" 

289 
by (simp add: Inf_fun_def) 

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definition 
41080  292 
"\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" 
293 

294 
lemma Sup_apply: 

295 
"(\<Squnion>A) x = \<Squnion>{y. \<exists>f\<in>A. y = f x}" 

296 
by (simp add: Sup_fun_def) 

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instance proof 
41080  299 
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  304 
lemma INFI_apply: 
305 
"(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" 

306 
by (auto intro: arg_cong [of _ _ Inf] simp add: INFI_def Inf_apply) 

38705  307 

41080  308 
lemma SUPR_apply: 
309 
"(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" 

310 
by (auto intro: arg_cong [of _ _ Sup] simp add: SUPR_def Sup_apply) 

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41082  313 
subsection {* Inter *} 
314 

315 
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" where 

316 
"Inter S \<equiv> \<Sqinter>S" 

317 

318 
notation (xsymbols) 

319 
Inter ("\<Inter>_" [90] 90) 

320 

321 
lemma Inter_eq: 

322 
"\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" 

323 
proof (rule set_eqI) 

324 
fix x 

325 
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" 

326 
by auto 

327 
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" 

328 
by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def) 

329 
qed 

330 

331 
lemma Inter_iff [simp,no_atp]: "(A : Inter C) = (ALL X:C. A:X)" 

332 
by (unfold Inter_eq) blast 

333 

334 
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" 

335 
by (simp add: Inter_eq) 

336 

337 
text {* 

338 
\medskip A ``destruct'' rule  every @{term X} in @{term C} 

339 
contains @{term A} as an element, but @{prop "A:X"} can hold when 

340 
@{prop "X:C"} does not! This rule is analogous to @{text spec}. 

341 
*} 

342 

343 
lemma InterD [elim, Pure.elim]: "A : Inter C ==> X:C ==> A:X" 

344 
by auto 

345 

346 
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" 

347 
 {* ``Classical'' elimination rule  does not require proving 

348 
@{prop "X:C"}. *} 

349 
by (unfold Inter_eq) blast 

350 

351 
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B" 

352 
by blast 

353 

354 
lemma Inter_subset: 

355 
"[ !!X. X \<in> A ==> X \<subseteq> B; A ~= {} ] ==> \<Inter>A \<subseteq> B" 

356 
by blast 

357 

358 
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" 

359 
by (iprover intro: InterI subsetI dest: subsetD) 

360 

361 
lemma Int_eq_Inter: "A \<inter> B = \<Inter>{A, B}" 

362 
by blast 

363 

364 
lemma Inter_empty [simp]: "\<Inter>{} = UNIV" 

365 
by (fact Inf_empty) 

366 

367 
lemma Inter_UNIV [simp]: "\<Inter>UNIV = {}" 

368 
by blast 

369 

370 
lemma Inter_insert [simp]: "\<Inter>(insert a B) = a \<inter> \<Inter>B" 

371 
by blast 

372 

373 
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" 

374 
by blast 

375 

376 
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" 

377 
by blast 

378 

379 
lemma Inter_UNIV_conv [simp,no_atp]: 

380 
"(\<Inter>A = UNIV) = (\<forall>x\<in>A. x = UNIV)" 

381 
"(UNIV = \<Inter>A) = (\<forall>x\<in>A. x = UNIV)" 

382 
by blast+ 

383 

384 
lemma Inter_anti_mono: "B \<subseteq> A ==> \<Inter>A \<subseteq> \<Inter>B" 

385 
by blast 

386 

387 

388 
subsection {* Intersections of families *} 

389 

390 
abbreviation INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where 

391 
"INTER \<equiv> INFI" 

392 

393 
syntax 

394 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) 

395 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) 

396 

397 
syntax (xsymbols) 

398 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) 

399 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) 

400 

401 
syntax (latex output) 

402 
"_INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) 

403 
"_INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) 

404 

405 
translations 

406 
"INT x y. B" == "INT x. INT y. B" 

407 
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" 

408 
"INT x. B" == "INT x:CONST UNIV. B" 

409 
"INT x:A. B" == "CONST INTER A (%x. B)" 

410 

411 
print_translation {* 

42284  412 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INTER} @{syntax_const "_INTER"}] 
41082  413 
*}  {* to avoid etacontraction of body *} 
414 

415 
lemma INTER_eq_Inter_image: 

416 
"(\<Inter>x\<in>A. B x) = \<Inter>(B`A)" 

417 
by (fact INFI_def) 

418 

419 
lemma Inter_def: 

420 
"\<Inter>S = (\<Inter>x\<in>S. x)" 

421 
by (simp add: INTER_eq_Inter_image image_def) 

422 

423 
lemma INTER_def: 

424 
"(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" 

425 
by (auto simp add: INTER_eq_Inter_image Inter_eq) 

426 

427 
lemma Inter_image_eq [simp]: 

428 
"\<Inter>(B`A) = (\<Inter>x\<in>A. B x)" 

429 
by (rule sym) (fact INTER_eq_Inter_image) 

430 

431 
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" 

432 
by (unfold INTER_def) blast 

433 

434 
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" 

435 
by (unfold INTER_def) blast 

436 

437 
lemma INT_D [elim, Pure.elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" 

438 
by auto 

439 

440 
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" 

441 
 {* "Classical" elimination  by the Excluded Middle on @{prop "a:A"}. *} 

442 
by (unfold INTER_def) blast 

443 

444 
lemma INT_cong [cong]: 

445 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" 

446 
by (simp add: INTER_def) 

447 

448 
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" 

449 
by blast 

450 

451 
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" 

452 
by blast 

453 

454 
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a" 

455 
by (fact INF_leI) 

456 

457 
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" 

458 
by (fact le_INFI) 

459 

460 
lemma INT_empty [simp]: "(\<Inter>x\<in>{}. B x) = UNIV" 

461 
by blast 

462 

463 
lemma INT_absorb: "k \<in> I ==> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" 

464 
by blast 

465 

466 
lemma INT_subset_iff: "(B \<subseteq> (\<Inter>i\<in>I. A i)) = (\<forall>i\<in>I. B \<subseteq> A i)" 

467 
by (fact le_INF_iff) 

468 

469 
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" 

470 
by blast 

471 

472 
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" 

473 
by blast 

474 

475 
lemma INT_insert_distrib: 

476 
"u \<in> A ==> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" 

477 
by blast 

478 

479 
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" 

480 
by auto 

481 

482 
lemma INT_eq: "(\<Inter>x\<in>A. B x) = \<Inter>({Y. \<exists>x\<in>A. Y = B x})" 

483 
 {* Look: it has an \emph{existential} quantifier *} 

484 
by blast 

485 

486 
lemma INTER_UNIV_conv[simp]: 

487 
"(UNIV = (INT x:A. B x)) = (\<forall>x\<in>A. B x = UNIV)" 

488 
"((INT x:A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" 

489 
by blast+ 

490 

491 
lemma INT_bool_eq: "(\<Inter>b::bool. A b) = (A True \<inter> A False)" 

492 
by (auto intro: bool_induct) 

493 

494 
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" 

495 
by blast 

496 

497 
lemma INT_anti_mono: 

498 
"B \<subseteq> A ==> (!!x. x \<in> A ==> f x \<subseteq> g x) ==> 

499 
(\<Inter>x\<in>A. f x) \<subseteq> (\<Inter>x\<in>A. g x)" 

500 
 {* The last inclusion is POSITIVE! *} 

501 
by (blast dest: subsetD) 

502 

503 
lemma vimage_INT: "f`(INT x:A. B x) = (INT x:A. f ` B x)" 

504 
by blast 

505 

506 

32139  507 
subsection {* Union *} 
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abbreviation Union :: "'a set set \<Rightarrow> 'a set" where 
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"Union S \<equiv> \<Squnion>S" 
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swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

511 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

512 
notation (xsymbols) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

513 
Union ("\<Union>_" [90] 90) 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

514 

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

515 
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

516 
"\<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

517 
proof (rule set_eqI) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

518 
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

519 
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

520 
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

521 
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

522 
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

523 
qed 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

524 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

525 
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

526 
"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

527 
by (unfold Union_eq) blast 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

528 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

529 
lemma UnionI [intro]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

530 
"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

531 
 {* 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

532 
@{term A} may be flexible. *} 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

533 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

534 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

535 
lemma UnionE [elim!]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

536 
"A \<in> \<Union>C \<Longrightarrow> (\<And>X. A\<in>X \<Longrightarrow> X\<in>C \<Longrightarrow> R) \<Longrightarrow> R" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

537 
by auto 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

538 

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

539 
lemma Union_upper: "B \<in> A ==> B \<subseteq> 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

540 
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

541 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

542 
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> 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

543 
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

544 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

545 
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

546 
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

547 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

548 
lemma Union_empty [simp]: "Union({}) = {}" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

549 
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

550 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

551 
lemma Union_UNIV [simp]: "Union UNIV = UNIV" 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

552 
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

553 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

554 
lemma Union_insert [simp]: "Union (insert a B) = a \<union> \<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

555 
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

556 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

557 
lemma Union_Un_distrib [simp]: "\<Union>(A Un B) = \<Union>A \<union> \<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

558 
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

559 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

560 
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

561 
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

562 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

563 
lemma Union_empty_conv [simp,no_atp]: "(\<Union>A = {}) = (\<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

564 
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

565 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

566 
lemma empty_Union_conv [simp,no_atp]: "({} = \<Union>A) = (\<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

567 
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

568 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

569 
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> 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

570 
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

571 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

572 
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

573 
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

574 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

575 
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

576 
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

577 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

578 
lemma Union_mono: "A \<subseteq> B ==> \<Union>A \<subseteq> \<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

579 
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

580 

32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

581 

32139  582 
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

583 

32606
b5c3a8a75772
INTER and UNION are mere abbreviations for INFI and SUPR
haftmann
parents:
32587
diff
changeset

584 
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

585 
"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

586 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

587 
syntax 
35115  588 
"_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

589 
"_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

590 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

591 
syntax (xsymbols) 
35115  592 
"_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

593 
"_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

594 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

595 
syntax (latex output) 
35115  596 
"_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

597 
"_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

598 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

599 
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

600 
"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

601 
"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

602 
"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

603 
"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

604 

3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

605 
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

606 
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

607 
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

608 
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

609 
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

610 
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

611 
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

612 
*} 
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset

613 

35115  614 
print_translation {* 
42284  615 
[Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax UNION} @{syntax_const "_UNION"}] 
35115  616 
*}  {* 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

617 

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

618 
lemma UNION_eq_Union_image: 
f645b51e8e54
set 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 
"(\<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

620 
by (fact SUPR_def) 
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

621 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

622 
lemma Union_def: 
32117
0762b9ad83df
Set.thy: prefer = over == where possible; tuned ML setup; dropped (moved) ML legacy
haftmann
parents:
32115
diff
changeset

623 
"\<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

624 
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

625 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

626 
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

627 
"(\<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

628 
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

629 

8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

630 
lemma Union_image_eq [simp]: 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

631 
"\<Union>(B`A) = (\<Union>x\<in>A. B x)" 
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset

632 
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

633 

11979  634 
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" 
635 
by (unfold UNION_def) blast 

636 

637 
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" 

638 
 {* The order of the premises presupposes that @{term A} is rigid; 

639 
@{term b} may be flexible. *} 

640 
by auto 

641 

642 
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" 

643 
by (unfold UNION_def) blast 

923  644 

11979  645 
lemma UN_cong [cong]: 
646 
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

647 
by (simp add: UNION_def) 

648 

29691  649 
lemma strong_UN_cong: 
650 
"A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" 

651 
by (simp add: UNION_def simp_implies_def) 

652 

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

653 
lemma image_eq_UN: "f`A = (UN x:A. {f x})" 
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 
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

655 

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

656 
lemma UN_upper: "a \<in> A ==> 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

657 
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

658 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

659 
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> 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

660 
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

661 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

662 
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

663 
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

664 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

665 
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<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

666 
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

667 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

668 
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

669 
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

670 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

671 
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

672 
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

673 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

674 
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

675 
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

676 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

677 
lemma UN_absorb: "k \<in> I ==> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A 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

678 
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

679 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

680 
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

681 
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

682 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

683 
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

684 
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

685 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

686 
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

687 
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

688 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

689 
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" 
35629  690 
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

691 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

692 
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

693 
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

694 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

695 
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

696 
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

697 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

698 
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

699 
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

700 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

701 
lemma UNION_empty_conv[simp]: 
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

702 
"({} = (UN x:A. B x)) = (\<forall>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

703 
"((UN x:A. B x) = {}) = (\<forall>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

704 
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

705 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

706 
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

707 
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

708 

f645b51e8e54
set 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 
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

710 
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

711 

f645b51e8e54
set 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 
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

713 
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

714 

f645b51e8e54
set 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 
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

716 
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

717 

f645b51e8e54
set 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 
lemma UN_bool_eq: "(\<Union>b::bool. A b) = (A True \<union> A False)" 
f645b51e8e54
set 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 
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

720 

f645b51e8e54
set 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 
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

722 
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

723 

f645b51e8e54
set 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 
lemma UN_mono: 
f645b51e8e54
set 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 
"A \<subseteq> B ==> (!!x. x \<in> A ==> f x \<subseteq> 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

726 
(\<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

727 
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

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 vimage_Union: "f ` (Union A) = (UN X:A. 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

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 vimage_UN: "f`(UN x:A. B x) = (UN x:A. f ` 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

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 vimage_eq_UN: "f`B = (UN y: B. f`{y})" 
f645b51e8e54
set 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 
 {* 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

737 
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

738 

f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset

739 
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (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

740 
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

741 

11979  742 

32139  743 
subsection {* Distributive laws *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

744 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

745 
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

746 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

747 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

748 
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

749 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

750 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

751 
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A`C) \<union> \<Union>(B`C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

752 
 {* 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

753 
 {* Union of a family of unions *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

754 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

755 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

756 
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

757 
 {* Equivalent version *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

758 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

759 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

760 
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

761 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

762 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

763 
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A`C) \<inter> \<Inter>(B`C)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

764 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

765 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

766 
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

767 
 {* Equivalent version *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

768 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

769 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

770 
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

771 
 {* Halmos, Naive Set Theory, page 35. *} 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

772 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

773 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

774 
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

775 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

776 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

777 
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

778 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

779 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

780 
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

781 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

782 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

783 

32139  784 
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

785 

f645b51e8e54
set 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 
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

787 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

788 

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

789 
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

790 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

791 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

792 

32139  793 
subsection {* Miniscoping and maxiscoping *} 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

794 

13860  795 
text {* \medskip Miniscoping: pushing in quantifiers and big Unions 
796 
and Intersections. *} 

12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

797 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

798 
lemma UN_simps [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

799 
"!!a B C. (UN x:C. insert a (B x)) = (if C={} then {} else insert a (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

800 
"!!A B C. (UN x:C. A x Un B) = ((if C={} then {} else (UN x:C. A x) Un B))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

801 
"!!A B C. (UN x:C. A Un B x) = ((if C={} then {} else A Un (UN x:C. B x)))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

802 
"!!A B C. (UN x:C. A x Int B) = ((UN x:C. A x) Int B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

803 
"!!A B C. (UN x:C. A Int B x) = (A Int (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

804 
"!!A B C. (UN x:C. A x  B) = ((UN x:C. A x)  B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

805 
"!!A B C. (UN x:C. A  B x) = (A  (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

806 
"!!A B. (UN x: Union A. B x) = (UN y:A. UN x:y. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

807 
"!!A B C. (UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

808 
"!!A B f. (UN x:f`A. B x) = (UN a:A. B (f a))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

809 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

810 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

811 
lemma INT_simps [simp]: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

812 
"!!A B C. (INT x:C. A x Int B) = (if C={} then UNIV else (INT x:C. A x) Int B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

813 
"!!A B C. (INT x:C. A Int B x) = (if C={} then UNIV else A Int (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

814 
"!!A B C. (INT x:C. A x  B) = (if C={} then UNIV else (INT x:C. A x)  B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

815 
"!!A B C. (INT x:C. A  B x) = (if C={} then UNIV else A  (UN x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

816 
"!!a B C. (INT x:C. insert a (B x)) = insert a (INT x:C. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

817 
"!!A B C. (INT x:C. A x Un B) = ((INT x:C. A x) Un B)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

818 
"!!A B C. (INT x:C. A Un B x) = (A Un (INT x:C. B x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

819 
"!!A B. (INT x: Union A. B x) = (INT y:A. INT x:y. B x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

820 
"!!A B C. (INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

821 
"!!A B f. (INT x:f`A. B x) = (INT a:A. B (f a))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

822 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

823 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

824 
lemma ball_simps [simp,no_atp]: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

825 
"!!A P Q. (ALL x:A. P x  Q) = ((ALL x:A. P x)  Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

826 
"!!A P Q. (ALL x:A. P  Q x) = (P  (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

827 
"!!A P Q. (ALL x:A. P > Q x) = (P > (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

828 
"!!A P Q. (ALL x:A. P x > Q) = ((EX x:A. P x) > Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

829 
"!!P. (ALL x:{}. P x) = True" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

830 
"!!P. (ALL x:UNIV. P x) = (ALL x. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

831 
"!!a B P. (ALL x:insert a B. P x) = (P a & (ALL x:B. P x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

832 
"!!A P. (ALL x:Union A. P x) = (ALL y:A. ALL x:y. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

833 
"!!A B P. (ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

834 
"!!P Q. (ALL x:Collect Q. P x) = (ALL x. Q x > P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

835 
"!!A P f. (ALL x:f`A. P x) = (ALL x:A. P (f x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

836 
"!!A P. (~(ALL x:A. P x)) = (EX x:A. ~P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

837 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

838 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35629
diff
changeset

839 
lemma bex_simps [simp,no_atp]: 
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

840 
"!!A P Q. (EX x:A. P x & Q) = ((EX x:A. P x) & Q)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

841 
"!!A P Q. (EX x:A. P & Q x) = (P & (EX x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

842 
"!!P. (EX x:{}. P x) = False" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

843 
"!!P. (EX x:UNIV. P x) = (EX x. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

844 
"!!a B P. (EX x:insert a B. P x) = (P(a)  (EX x:B. P x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

845 
"!!A P. (EX x:Union A. P x) = (EX y:A. EX x:y. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

846 
"!!A B P. (EX x: UNION A B. P x) = (EX a:A. EX x:B a. P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

847 
"!!P Q. (EX x:Collect Q. P x) = (EX x. Q x & P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

848 
"!!A P f. (EX x:f`A. P x) = (EX x:A. P (f x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

849 
"!!A P. (~(EX x:A. P x)) = (ALL x:A. ~P x)" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

850 
by auto 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

851 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

852 
lemma ball_conj_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

853 
"(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

854 
by blast 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

855 

f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

856 
lemma bex_disj_distrib: 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

857 
"(EX x:A. P x  Q x) = ((EX x:A. P x)  (EX x:A. Q x))" 
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset

858 
by blast 
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 

13860  861 
text {* \medskip Maxiscoping: pulling out big Unions and Intersections. *} 
862 

863 
lemma UN_extend_simps: 

864 
"!!a B C. insert a (UN x:C. B x) = (if C={} then {a} else (UN x:C. insert a (B x)))" 

865 
"!!A B C. (UN x:C. A x) Un B = (if C={} then B else (UN x:C. A x Un B))" 

866 
"!!A B C. A Un (UN x:C. B x) = (if C={} then A else (UN x:C. A Un B x))" 

867 
"!!A B C. ((UN x:C. A x) Int B) = (UN x:C. A x Int B)" 

868 
"!!A B C. (A Int (UN x:C. B x)) = (UN x:C. A Int B x)" 

869 
"!!A B C. ((UN x:C. A x)  B) = (UN x:C. A x  B)" 

870 
"!!A B C. (A  (INT x:C. B x)) = (UN x:C. A  B x)" 

871 
"!!A B. (UN y:A. UN x:y. B x) = (UN x: Union A. B x)" 

872 
"!!A B C. (UN x:A. UN z: B(x). C z) = (UN z: UNION A B. C z)" 

873 
"!!A B f. (UN a:A. B (f a)) = (UN x:f`A. B x)" 

874 
by auto 

875 

876 
lemma INT_extend_simps: 

877 
"!!A B C. (INT x:C. A x) Int B = (if C={} then B else (INT x:C. A x Int B))" 

878 
"!!A B C. A Int (INT x:C. B x) = (if C={} then A else (INT x:C. A Int B x))" 

879 
"!!A B C. (INT x:C. A x)  B = (if C={} then UNIVB else (INT x:C. A x  B))" 

880 
"!!A B C. A  (UN x:C. B x) = (if C={} then A else (INT x:C. A  B x))" 

881 
"!!a B C. insert a (INT x:C. B x) = (INT x:C. insert a (B x))" 

882 
"!!A B C. ((INT x:C. A x) Un B) = (INT x:C. A x Un B)" 

883 
"!!A B C. A Un (INT x:C. B x) = (INT x:C. A Un B x)" 

884 
"!!A B. (INT y:A. INT x:y. B x) = (INT x: Union A. B x)" 

885 
"!!A B C. (INT x:A. INT z: B(x). C z) = (INT z: UNION A B. C z)" 

886 
"!!A B f. (INT a:A. B (f a)) = (INT x:f`A. B x)" 

887 
by auto 

888 

889 

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

890 
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

891 
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

892 
less (infix "\<sqsubset>" 50) and 
41082  893 
bot ("\<bottom>") and 
894 
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

895 
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

896 
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

897 
Inf ("\<Sqinter>_" [900] 900) and 
41082  898 
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

899 

41080  900 
no_syntax (xsymbols) 
41082  901 
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 
902 
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

41080  903 
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 
904 
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

905 

30596  906 
lemmas mem_simps = 
907 
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff 

908 
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff 

909 
 {* Each of these has ALREADY been added @{text "[simp]"} above. *} 

21669  910 

11979  911 
end 