author | nipkow |
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(* Title: HOL/Complete_Lattices.thy |
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Author: Tobias Nipkow |
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Author: Lawrence C Paulson |
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Author: Markus Wenzel |
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Author: Florian Haftmann |
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Changes to complete distributive lattices due to Viorel Preoteasa
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Author: Viorel Preoteasa (Complete Distributive Lattices) |
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*) |
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section \<open>Complete lattices\<close> |
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theory Complete_Lattices |
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imports Fun |
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begin |
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subsection \<open>Syntactic infimum and supremum operations\<close> |
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class Inf = |
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fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>") |
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class Sup = |
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fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>") |
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syntax (ASCII) |
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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 |
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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 _\<in>_./ _)" [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 _\<in>_./ _)" [0, 0, 10] 10) |
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syntax |
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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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"\<Sqinter>x y. f" \<rightleftharpoons> "\<Sqinter>x. \<Sqinter>y. f" |
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"\<Sqinter>x. f" \<rightleftharpoons> "\<Sqinter>(CONST range (\<lambda>x. f))" |
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"\<Sqinter>x\<in>A. f" \<rightleftharpoons> "CONST Inf ((\<lambda>x. f) ` A)" |
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"\<Squnion>x y. f" \<rightleftharpoons> "\<Squnion>x. \<Squnion>y. f" |
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"\<Squnion>x. f" \<rightleftharpoons> "\<Squnion>(CONST range (\<lambda>x. f))" |
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"\<Squnion>x\<in>A. f" \<rightleftharpoons> "CONST Sup ((\<lambda>x. f) ` A)" |
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context Inf |
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begin |
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lemma INF_image [simp]: " \<Sqinter>(g ` f ` A) = \<Sqinter>((g \<circ> f) ` A)" |
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by (simp add: image_comp) |
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lemma INF_identity_eq [simp]: "(\<Sqinter>x\<in>A. x) = \<Sqinter>A" |
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by simp |
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lemma INF_id_eq [simp]: "\<Sqinter>(id ` A) = \<Sqinter>A" |
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by simp |
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lemma INF_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> \<Sqinter>(C ` A) = \<Sqinter>(D ` B)" |
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by (simp add: image_def) |
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lemma INF_cong_simp [cong]: |
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"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> \<Sqinter>(C ` A) = \<Sqinter>(D ` B)" |
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unfolding simp_implies_def by (fact INF_cong) |
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end |
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context Sup |
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begin |
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lemma SUP_image [simp]: "\<Squnion>(g ` f ` A) = \<Squnion>((g \<circ> f) ` A)" |
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by(fact Inf.INF_image) |
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lemma SUP_identity_eq [simp]: "(\<Squnion>x\<in>A. x) = \<Squnion>A" |
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by(fact Inf.INF_identity_eq) |
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lemma SUP_id_eq [simp]: "\<Squnion>(id ` A) = \<Squnion>A" |
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by(fact Inf.INF_id_eq) |
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lemma SUP_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> \<Squnion>(C ` A) = \<Squnion>(D ` B)" |
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by (fact Inf.INF_cong) |
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lemma SUP_cong_simp [cong]: |
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"A = B \<Longrightarrow> (\<And>x. x \<in> B =simp=> C x = D x) \<Longrightarrow> \<Squnion>(C ` A) = \<Squnion>(D ` B)" |
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by (fact Inf.INF_cong_simp) |
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end |
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subsection \<open>Abstract complete lattices\<close> |
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text \<open>A complete lattice always has a bottom and a top, |
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so we include them into the following type class, |
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along with assumptions that define bottom and top |
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in terms of infimum and supremum.\<close> |
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class complete_lattice = lattice + Inf + Sup + bot + top + |
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assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<le> x" |
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and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> \<Sqinter>A" |
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and Sup_upper: "x \<in> A \<Longrightarrow> x \<le> \<Squnion>A" |
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and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> \<Squnion>A \<le> z" |
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and Inf_empty [simp]: "\<Sqinter>{} = \<top>" |
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and Sup_empty [simp]: "\<Squnion>{} = \<bottom>" |
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begin |
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subclass bounded_lattice |
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proof |
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fix a |
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show "\<bottom> \<le> a" |
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by (auto intro: Sup_least simp only: Sup_empty [symmetric]) |
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show "a \<le> \<top>" |
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by (auto intro: Inf_greatest simp only: Inf_empty [symmetric]) |
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qed |
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lemma dual_complete_lattice: "class.complete_lattice Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>" |
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by (auto intro!: class.complete_lattice.intro dual_lattice) |
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(unfold_locales, (fact Inf_empty Sup_empty Sup_upper Sup_least Inf_lower Inf_greatest)+) |
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end |
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context complete_lattice |
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begin |
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lemma Sup_eqI: |
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"(\<And>y. y \<in> A \<Longrightarrow> y \<le> x) \<Longrightarrow> (\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> \<Squnion>A = x" |
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by (blast intro: antisym Sup_least Sup_upper) |
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lemma Inf_eqI: |
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"(\<And>i. i \<in> A \<Longrightarrow> x \<le> i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x) \<Longrightarrow> \<Sqinter>A = x" |
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by (blast intro: antisym Inf_greatest Inf_lower) |
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lemma SUP_eqI: |
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"(\<And>i. i \<in> A \<Longrightarrow> f i \<le> x) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> (\<Squnion>i\<in>A. f i) = x" |
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using Sup_eqI [of "f ` A" x] by auto |
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lemma INF_eqI: |
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"(\<And>i. i \<in> A \<Longrightarrow> x \<le> f i) \<Longrightarrow> (\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<ge> y) \<Longrightarrow> x \<ge> y) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) = x" |
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using Inf_eqI [of "f ` A" x] by auto |
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lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<le> f i" |
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using Inf_lower [of _ "f ` A"] by simp |
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lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<le> f i) \<Longrightarrow> u \<le> (\<Sqinter>i\<in>A. f i)" |
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using Inf_greatest [of "f ` A"] by auto |
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lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<le> (\<Squnion>i\<in>A. f i)" |
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using Sup_upper [of _ "f ` A"] by simp |
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lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<le> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<le> u" |
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using Sup_least [of "f ` A"] by auto |
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lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<le> v \<Longrightarrow> \<Sqinter>A \<le> v" |
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using Inf_lower [of u A] by auto |
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lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<le> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<le> u" |
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using INF_lower [of i A f] by auto |
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lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<le> u \<Longrightarrow> v \<le> \<Squnion>A" |
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using Sup_upper [of u A] by auto |
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lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<le> f i \<Longrightarrow> u \<le> (\<Squnion>i\<in>A. f i)" |
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using SUP_upper [of i A f] by auto |
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lemma le_Inf_iff: "b \<le> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<le> a)" |
44040 | 167 |
by (auto intro: Inf_greatest dest: Inf_lower) |
168 |
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169 |
lemma le_INF_iff: "u \<le> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<le> f i)" |
56166 | 170 |
using le_Inf_iff [of _ "f ` A"] by simp |
44040 | 171 |
|
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172 |
lemma Sup_le_iff: "\<Squnion>A \<le> b \<longleftrightarrow> (\<forall>a\<in>A. a \<le> b)" |
44040 | 173 |
by (auto intro: Sup_least dest: Sup_upper) |
174 |
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|
175 |
lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<le> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<le> u)" |
56166 | 176 |
using Sup_le_iff [of "f ` A"] by simp |
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177 |
|
69745 | 178 |
lemma Inf_insert [simp]: "\<Sqinter>(insert a A) = a \<sqinter> \<Sqinter>A" |
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179 |
by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) |
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180 |
|
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lemma INF_insert [simp]: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> \<Sqinter>(f ` A)" |
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182 |
by (simp cong del: INF_cong_simp) |
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183 |
|
69745 | 184 |
lemma Sup_insert [simp]: "\<Squnion>(insert a A) = a \<squnion> \<Squnion>A" |
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185 |
by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) |
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186 |
|
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lemma SUP_insert [simp]: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> \<Squnion>(f ` A)" |
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188 |
by (simp cong del: SUP_cong_simp) |
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189 |
|
44067 | 190 |
lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>" |
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191 |
by (simp cong del: INF_cong_simp) |
44040 | 192 |
|
44067 | 193 |
lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>" |
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194 |
by (simp cong del: SUP_cong_simp) |
44040 | 195 |
|
63575 | 196 |
lemma Inf_UNIV [simp]: "\<Sqinter>UNIV = \<bottom>" |
44040 | 197 |
by (auto intro!: antisym Inf_lower) |
41080 | 198 |
|
63575 | 199 |
lemma Sup_UNIV [simp]: "\<Squnion>UNIV = \<top>" |
44040 | 200 |
by (auto intro!: antisym Sup_upper) |
41080 | 201 |
|
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202 |
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}" |
44040 | 203 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
204 |
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205 |
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}" |
44040 | 206 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
207 |
||
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208 |
lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<le> \<Sqinter>B" |
43899 | 209 |
by (auto intro: Inf_greatest Inf_lower) |
210 |
||
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|
211 |
lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<le> \<Squnion>B" |
43899 | 212 |
by (auto intro: Sup_least Sup_upper) |
213 |
||
38705 | 214 |
lemma Inf_mono: |
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215 |
assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b" |
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|
216 |
shows "\<Sqinter>A \<le> \<Sqinter>B" |
38705 | 217 |
proof (rule Inf_greatest) |
218 |
fix b assume "b \<in> B" |
|
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219 |
with assms obtain a where "a \<in> A" and "a \<le> b" by blast |
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|
220 |
from \<open>a \<in> A\<close> have "\<Sqinter>A \<le> a" by (rule Inf_lower) |
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221 |
with \<open>a \<le> b\<close> show "\<Sqinter>A \<le> b" by auto |
38705 | 222 |
qed |
223 |
||
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224 |
lemma INF_mono: "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<le> (\<Sqinter>n\<in>B. g n)" |
56166 | 225 |
using Inf_mono [of "g ` B" "f ` A"] by auto |
44041 | 226 |
|
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227 |
lemma INF_mono': "(\<And>x. f x \<le> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<le> (\<Sqinter>x\<in>A. g x)" |
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228 |
by (rule INF_mono) auto |
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229 |
|
41082 | 230 |
lemma Sup_mono: |
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231 |
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b" |
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232 |
shows "\<Squnion>A \<le> \<Squnion>B" |
41082 | 233 |
proof (rule Sup_least) |
234 |
fix a assume "a \<in> A" |
|
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|
235 |
with assms obtain b where "b \<in> B" and "a \<le> b" by blast |
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changeset
|
236 |
from \<open>b \<in> B\<close> have "b \<le> \<Squnion>B" by (rule Sup_upper) |
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|
237 |
with \<open>a \<le> b\<close> show "a \<le> \<Squnion>B" by auto |
41082 | 238 |
qed |
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239 |
|
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|
240 |
lemma SUP_mono: "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<le> (\<Squnion>n\<in>B. g n)" |
56166 | 241 |
using Sup_mono [of "f ` A" "g ` B"] by auto |
44041 | 242 |
|
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|
243 |
lemma SUP_mono': "(\<And>x. f x \<le> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<le> (\<Squnion>x\<in>A. g x)" |
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Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
68802
diff
changeset
|
244 |
by (rule SUP_mono) auto |
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Some basic materials on filters and topology
Manuel Eberl <eberlm@in.tum.de>
parents:
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diff
changeset
|
245 |
|
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|
246 |
lemma INF_superset_mono: "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<le> (\<Sqinter>x\<in>B. g x)" |
61799 | 247 |
\<comment> \<open>The last inclusion is POSITIVE!\<close> |
44041 | 248 |
by (blast intro: INF_mono dest: subsetD) |
249 |
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|
250 |
lemma SUP_subset_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<le> (\<Squnion>x\<in>B. g x)" |
44041 | 251 |
by (blast intro: SUP_mono dest: subsetD) |
252 |
||
43868 | 253 |
lemma Inf_less_eq: |
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|
254 |
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<le> u" |
43868 | 255 |
and "A \<noteq> {}" |
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changeset
|
256 |
shows "\<Sqinter>A \<le> u" |
43868 | 257 |
proof - |
60758 | 258 |
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast |
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|
259 |
moreover from \<open>v \<in> A\<close> assms(1) have "v \<le> u" by blast |
43868 | 260 |
ultimately show ?thesis by (rule Inf_lower2) |
261 |
qed |
|
262 |
||
263 |
lemma less_eq_Sup: |
|
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|
264 |
assumes "\<And>v. v \<in> A \<Longrightarrow> u \<le> v" |
43868 | 265 |
and "A \<noteq> {}" |
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changeset
|
266 |
shows "u \<le> \<Squnion>A" |
43868 | 267 |
proof - |
60758 | 268 |
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast |
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changeset
|
269 |
moreover from \<open>v \<in> A\<close> assms(1) have "u \<le> v" by blast |
43868 | 270 |
ultimately show ?thesis by (rule Sup_upper2) |
271 |
qed |
|
272 |
||
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|
273 |
lemma INF_eq: |
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|
274 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j" |
63575 | 275 |
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i" |
68797 | 276 |
shows "\<Sqinter>(f ` A) = \<Sqinter>(g ` B)" |
62343
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prefer abbreviations for compound operators INFIMUM and SUPREMUM
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changeset
|
277 |
by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+ |
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prefer abbreviations for compound operators INFIMUM and SUPREMUM
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|
278 |
|
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|
279 |
lemma SUP_eq: |
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|
280 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j" |
63575 | 281 |
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i" |
68797 | 282 |
shows "\<Squnion>(f ` A) = \<Squnion>(g ` B)" |
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changeset
|
283 |
by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+ |
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changeset
|
284 |
|
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changeset
|
285 |
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<le> \<Sqinter>(A \<inter> B)" |
43868 | 286 |
by (auto intro: Inf_greatest Inf_lower) |
287 |
||
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changeset
|
288 |
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<le> \<Squnion>A \<sqinter> \<Squnion>B " |
43868 | 289 |
by (auto intro: Sup_least Sup_upper) |
290 |
||
291 |
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" |
|
292 |
by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) |
|
293 |
||
63575 | 294 |
lemma INF_union: "(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)" |
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44085
diff
changeset
|
295 |
by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower) |
44041 | 296 |
|
43868 | 297 |
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B" |
298 |
by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2) |
|
299 |
||
63575 | 300 |
lemma SUP_union: "(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)" |
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haftmann
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44085
diff
changeset
|
301 |
by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper) |
44041 | 302 |
|
303 |
lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)" |
|
44103
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haftmann
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44085
diff
changeset
|
304 |
by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono) |
44041 | 305 |
|
63575 | 306 |
lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" |
307 |
(is "?L = ?R") |
|
44918 | 308 |
proof (rule antisym) |
63575 | 309 |
show "?L \<le> ?R" |
310 |
by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono) |
|
311 |
show "?R \<le> ?L" |
|
312 |
by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper) |
|
44918 | 313 |
qed |
44041 | 314 |
|
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diff
changeset
|
315 |
lemma Inf_top_conv [simp]: |
43868 | 316 |
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
317 |
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
318 |
proof - |
|
319 |
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
320 |
proof |
|
321 |
assume "\<forall>x\<in>A. x = \<top>" |
|
322 |
then have "A = {} \<or> A = {\<top>}" by auto |
|
44919 | 323 |
then show "\<Sqinter>A = \<top>" by auto |
43868 | 324 |
next |
325 |
assume "\<Sqinter>A = \<top>" |
|
326 |
show "\<forall>x\<in>A. x = \<top>" |
|
327 |
proof (rule ccontr) |
|
328 |
assume "\<not> (\<forall>x\<in>A. x = \<top>)" |
|
329 |
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast |
|
330 |
then obtain B where "A = insert x B" by blast |
|
60758 | 331 |
with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp |
43868 | 332 |
qed |
333 |
qed |
|
334 |
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto |
|
335 |
qed |
|
336 |
||
44918 | 337 |
lemma INF_top_conv [simp]: |
56166 | 338 |
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
339 |
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
|
340 |
using Inf_top_conv [of "B ` A"] by simp_all |
|
44041 | 341 |
|
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changeset
|
342 |
lemma Sup_bot_conv [simp]: |
63575 | 343 |
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" |
344 |
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" |
|
44920 | 345 |
using dual_complete_lattice |
346 |
by (rule complete_lattice.Inf_top_conv)+ |
|
43868 | 347 |
|
44918 | 348 |
lemma SUP_bot_conv [simp]: |
63575 | 349 |
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
350 |
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
|
56166 | 351 |
using Sup_bot_conv [of "B ` A"] by simp_all |
44041 | 352 |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
353 |
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
354 |
by (auto intro: antisym INF_lower INF_greatest) |
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
|
355 |
|
43870 | 356 |
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
357 |
by (auto intro: antisym SUP_upper SUP_least) |
43870 | 358 |
|
44918 | 359 |
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>" |
44921 | 360 |
by (cases "A = {}") simp_all |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
361 |
|
44918 | 362 |
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>" |
44921 | 363 |
by (cases "A = {}") simp_all |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
364 |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
365 |
lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
366 |
by (iprover intro: INF_lower INF_greatest order_trans antisym) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
367 |
|
43870 | 368 |
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
369 |
by (iprover intro: SUP_upper SUP_least order_trans antisym) |
43870 | 370 |
|
43871 | 371 |
lemma INF_absorb: |
43868 | 372 |
assumes "k \<in> I" |
373 |
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)" |
|
374 |
proof - |
|
375 |
from assms obtain J where "I = insert k J" by blast |
|
56166 | 376 |
then show ?thesis by simp |
43868 | 377 |
qed |
378 |
||
43871 | 379 |
lemma SUP_absorb: |
380 |
assumes "k \<in> I" |
|
381 |
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)" |
|
382 |
proof - |
|
383 |
from assms obtain J where "I = insert k J" by blast |
|
56166 | 384 |
then show ?thesis by simp |
43871 | 385 |
qed |
386 |
||
67613 | 387 |
lemma INF_inf_const1: "I \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>I. inf x (f i)) = inf x (\<Sqinter>i\<in>I. f i)" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
388 |
by (intro antisym INF_greatest inf_mono order_refl INF_lower) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
389 |
(auto intro: INF_lower2 le_infI2 intro!: INF_mono) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
390 |
|
67613 | 391 |
lemma INF_inf_const2: "I \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>I. inf (f i) x) = inf (\<Sqinter>i\<in>I. f i) x" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
392 |
using INF_inf_const1[of I x f] by (simp add: inf_commute) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57197
diff
changeset
|
393 |
|
63575 | 394 |
lemma INF_constant: "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)" |
44921 | 395 |
by simp |
43868 | 396 |
|
63575 | 397 |
lemma SUP_constant: "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)" |
44921 | 398 |
by simp |
43871 | 399 |
|
43943 | 400 |
lemma less_INF_D: |
63575 | 401 |
assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" |
402 |
shows "y < f i" |
|
43943 | 403 |
proof - |
60758 | 404 |
note \<open>y < (\<Sqinter>i\<in>A. f i)\<close> |
405 |
also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using \<open>i \<in> A\<close> |
|
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
406 |
by (rule INF_lower) |
43943 | 407 |
finally show "y < f i" . |
408 |
qed |
|
409 |
||
410 |
lemma SUP_lessD: |
|
63575 | 411 |
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" |
412 |
shows "f i < y" |
|
43943 | 413 |
proof - |
63575 | 414 |
have "f i \<le> (\<Squnion>i\<in>A. f i)" |
415 |
using \<open>i \<in> A\<close> by (rule SUP_upper) |
|
60758 | 416 |
also note \<open>(\<Squnion>i\<in>A. f i) < y\<close> |
43943 | 417 |
finally show "f i < y" . |
418 |
qed |
|
419 |
||
63575 | 420 |
lemma INF_UNIV_bool_expand: "(\<Sqinter>b. A b) = A True \<sqinter> A False" |
56166 | 421 |
by (simp add: UNIV_bool inf_commute) |
43868 | 422 |
|
63575 | 423 |
lemma SUP_UNIV_bool_expand: "(\<Squnion>b. A b) = A True \<squnion> A False" |
56166 | 424 |
by (simp add: UNIV_bool sup_commute) |
43871 | 425 |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
426 |
lemma Inf_le_Sup: "A \<noteq> {} \<Longrightarrow> Inf A \<le> Sup A" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
427 |
by (blast intro: Sup_upper2 Inf_lower ex_in_conv) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
428 |
|
68797 | 429 |
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> \<Sqinter>(f ` A) \<le> \<Squnion>(f ` A)" |
56166 | 430 |
using Inf_le_Sup [of "f ` A"] by simp |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
431 |
|
68797 | 432 |
lemma INF_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> \<Sqinter>(f ` I) = x" |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
433 |
by (auto intro: INF_eqI) |
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
434 |
|
68797 | 435 |
lemma SUP_eq_const: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i = x) \<Longrightarrow> \<Squnion>(f ` I) = x" |
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
436 |
by (auto intro: SUP_eqI) |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
437 |
|
68797 | 438 |
lemma INF_eq_iff: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i \<le> c) \<Longrightarrow> \<Sqinter>(f ` I) = c \<longleftrightarrow> (\<forall>i\<in>I. f i = c)" |
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
439 |
using INF_eq_const [of I f c] INF_lower [of _ I f] |
69546
27dae626822b
prefer naming convention from datatype package for strong congruence rules
haftmann
parents:
69275
diff
changeset
|
440 |
by (auto intro: antisym cong del: INF_cong_simp) |
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
441 |
|
68797 | 442 |
lemma SUP_eq_iff: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> c \<le> f i) \<Longrightarrow> \<Squnion>(f ` I) = c \<longleftrightarrow> (\<forall>i\<in>I. f i = c)" |
56248
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
443 |
using SUP_eq_const [of I f c] SUP_upper [of _ I f] |
69546
27dae626822b
prefer naming convention from datatype package for strong congruence rules
haftmann
parents:
69275
diff
changeset
|
444 |
by (auto intro: antisym cong del: SUP_cong_simp) |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
445 |
|
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
|
446 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
447 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
448 |
context complete_lattice |
44024 | 449 |
begin |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
450 |
lemma Sup_Inf_le: "Sup (Inf ` {f ` A | f . (\<forall> Y \<in> A . f Y \<in> Y)}) \<le> Inf (Sup ` A)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
451 |
by (rule SUP_least, clarify, rule INF_greatest, simp add: INF_lower2 Sup_upper) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
452 |
end |
44039 | 453 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
454 |
class complete_distrib_lattice = complete_lattice + |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
455 |
assumes Inf_Sup_le: "Inf (Sup ` A) \<le> Sup (Inf ` {f ` A | f . (\<forall> Y \<in> A . f Y \<in> Y)})" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
456 |
begin |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
457 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
458 |
lemma Inf_Sup: "Inf (Sup ` A) = Sup (Inf ` {f ` A | f . (\<forall> Y \<in> A . f Y \<in> Y)})" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
459 |
by (rule antisym, rule Inf_Sup_le, rule Sup_Inf_le) |
44024 | 460 |
|
63575 | 461 |
subclass distrib_lattice |
462 |
proof |
|
44024 | 463 |
fix a b c |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
464 |
show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
465 |
proof (rule antisym, simp_all, safe) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
466 |
show "b \<sqinter> c \<le> a \<squnion> b" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
467 |
by (rule le_infI1, simp) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
468 |
show "b \<sqinter> c \<le> a \<squnion> c" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
469 |
by (rule le_infI2, simp) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
470 |
have [simp]: "a \<sqinter> c \<le> a \<squnion> b \<sqinter> c" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
471 |
by (rule le_infI1, simp) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
472 |
have [simp]: "b \<sqinter> a \<le> a \<squnion> b \<sqinter> c" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
473 |
by (rule le_infI2, simp) |
68797 | 474 |
have "\<Sqinter>(Sup ` {{a, b}, {a, c}}) = |
475 |
\<Squnion>(Inf ` {f ` {{a, b}, {a, c}} | f. \<forall>Y\<in>{{a, b}, {a, c}}. f Y \<in> Y})" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
476 |
by (rule Inf_Sup) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
477 |
from this show "(a \<squnion> b) \<sqinter> (a \<squnion> c) \<le> a \<squnion> b \<sqinter> c" |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
478 |
apply simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
479 |
by (rule SUP_least, safe, simp_all) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
480 |
qed |
44024 | 481 |
qed |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
482 |
end |
44039 | 483 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
484 |
context complete_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
485 |
begin |
56074 | 486 |
context |
487 |
fixes f :: "'a \<Rightarrow> 'b::complete_lattice" |
|
488 |
assumes "mono f" |
|
489 |
begin |
|
490 |
||
63575 | 491 |
lemma mono_Inf: "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)" |
60758 | 492 |
using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD) |
56074 | 493 |
|
63575 | 494 |
lemma mono_Sup: "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)" |
60758 | 495 |
using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD) |
56074 | 496 |
|
67613 | 497 |
lemma mono_INF: "f (\<Sqinter>i\<in>I. A i) \<le> (\<Sqinter>x\<in>I. f (A x))" |
60758 | 498 |
by (intro complete_lattice_class.INF_greatest monoD[OF \<open>mono f\<close>] INF_lower) |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
58889
diff
changeset
|
499 |
|
67613 | 500 |
lemma mono_SUP: "(\<Squnion>x\<in>I. f (A x)) \<le> f (\<Squnion>i\<in>I. A i)" |
60758 | 501 |
by (intro complete_lattice_class.SUP_least monoD[OF \<open>mono f\<close>] SUP_upper) |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
58889
diff
changeset
|
502 |
|
56074 | 503 |
end |
504 |
||
44024 | 505 |
end |
506 |
||
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
507 |
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice |
43873 | 508 |
begin |
509 |
||
63575 | 510 |
lemma uminus_Inf: "- (\<Sqinter>A) = \<Squnion>(uminus ` A)" |
43873 | 511 |
proof (rule antisym) |
512 |
show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)" |
|
513 |
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp |
|
514 |
show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A" |
|
515 |
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto |
|
516 |
qed |
|
517 |
||
44041 | 518 |
lemma uminus_INF: "- (\<Sqinter>x\<in>A. B x) = (\<Squnion>x\<in>A. - B x)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
519 |
by (simp add: uminus_Inf image_image) |
44041 | 520 |
|
63575 | 521 |
lemma uminus_Sup: "- (\<Squnion>A) = \<Sqinter>(uminus ` A)" |
43873 | 522 |
proof - |
63575 | 523 |
have "\<Squnion>A = - \<Sqinter>(uminus ` A)" |
524 |
by (simp add: image_image uminus_INF) |
|
43873 | 525 |
then show ?thesis by simp |
526 |
qed |
|
63575 | 527 |
|
43873 | 528 |
lemma uminus_SUP: "- (\<Squnion>x\<in>A. B x) = (\<Sqinter>x\<in>A. - B x)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
529 |
by (simp add: uminus_Sup image_image) |
43873 | 530 |
|
531 |
end |
|
532 |
||
43940 | 533 |
class complete_linorder = linorder + complete_lattice |
534 |
begin |
|
535 |
||
43943 | 536 |
lemma dual_complete_linorder: |
67399 | 537 |
"class.complete_linorder Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>" |
43943 | 538 |
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder) |
539 |
||
51386 | 540 |
lemma complete_linorder_inf_min: "inf = min" |
51540
eea5c4ca4a0e
explicit sublocale dependency for Min/Max yields more appropriate Min/Max prefix for a couple of facts
haftmann
parents:
51489
diff
changeset
|
541 |
by (auto intro: antisym simp add: min_def fun_eq_iff) |
51386 | 542 |
|
543 |
lemma complete_linorder_sup_max: "sup = max" |
|
51540
eea5c4ca4a0e
explicit sublocale dependency for Min/Max yields more appropriate Min/Max prefix for a couple of facts
haftmann
parents:
51489
diff
changeset
|
544 |
by (auto intro: antisym simp add: max_def fun_eq_iff) |
51386 | 545 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
546 |
lemma Inf_less_iff: "\<Sqinter>S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)" |
63172 | 547 |
by (simp add: not_le [symmetric] le_Inf_iff) |
43940 | 548 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
549 |
lemma INF_less_iff: "(\<Sqinter>i\<in>A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)" |
63172 | 550 |
by (simp add: Inf_less_iff [of "f ` A"]) |
44041 | 551 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
552 |
lemma less_Sup_iff: "a < \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a < x)" |
63172 | 553 |
by (simp add: not_le [symmetric] Sup_le_iff) |
43940 | 554 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
555 |
lemma less_SUP_iff: "a < (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)" |
63172 | 556 |
by (simp add: less_Sup_iff [of _ "f ` A"]) |
43940 | 557 |
|
63575 | 558 |
lemma Sup_eq_top_iff [simp]: "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)" |
43943 | 559 |
proof |
560 |
assume *: "\<Squnion>A = \<top>" |
|
63575 | 561 |
show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" |
562 |
unfolding * [symmetric] |
|
43943 | 563 |
proof (intro allI impI) |
63575 | 564 |
fix x |
565 |
assume "x < \<Squnion>A" |
|
566 |
then show "\<exists>i\<in>A. x < i" |
|
63172 | 567 |
by (simp add: less_Sup_iff) |
43943 | 568 |
qed |
569 |
next |
|
570 |
assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i" |
|
571 |
show "\<Squnion>A = \<top>" |
|
572 |
proof (rule ccontr) |
|
573 |
assume "\<Squnion>A \<noteq> \<top>" |
|
63575 | 574 |
with top_greatest [of "\<Squnion>A"] have "\<Squnion>A < \<top>" |
575 |
unfolding le_less by auto |
|
576 |
with * have "\<Squnion>A < \<Squnion>A" |
|
577 |
unfolding less_Sup_iff by auto |
|
43943 | 578 |
then show False by auto |
579 |
qed |
|
580 |
qed |
|
581 |
||
63575 | 582 |
lemma SUP_eq_top_iff [simp]: "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)" |
56166 | 583 |
using Sup_eq_top_iff [of "f ` A"] by simp |
44041 | 584 |
|
63575 | 585 |
lemma Inf_eq_bot_iff [simp]: "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)" |
44920 | 586 |
using dual_complete_linorder |
587 |
by (rule complete_linorder.Sup_eq_top_iff) |
|
43943 | 588 |
|
63575 | 589 |
lemma INF_eq_bot_iff [simp]: "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)" |
56166 | 590 |
using Inf_eq_bot_iff [of "f ` A"] by simp |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
591 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
592 |
lemma Inf_le_iff: "\<Sqinter>A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
593 |
proof safe |
63575 | 594 |
fix y |
595 |
assume "x \<ge> \<Sqinter>A" "y > x" |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
596 |
then have "y > \<Sqinter>A" by auto |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
597 |
then show "\<exists>a\<in>A. y > a" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
598 |
unfolding Inf_less_iff . |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
599 |
qed (auto elim!: allE[of _ "\<Sqinter>A"] simp add: not_le[symmetric] Inf_lower) |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
600 |
|
68802 | 601 |
lemma INF_le_iff: "\<Sqinter>(f ` A) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)" |
56166 | 602 |
using Inf_le_iff [of "f ` A"] by simp |
603 |
||
604 |
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)" |
|
605 |
proof safe |
|
63575 | 606 |
fix y |
607 |
assume "x \<le> \<Squnion>A" "y < x" |
|
56166 | 608 |
then have "y < \<Squnion>A" by auto |
609 |
then show "\<exists>a\<in>A. y < a" |
|
610 |
unfolding less_Sup_iff . |
|
611 |
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper) |
|
612 |
||
68802 | 613 |
lemma le_SUP_iff: "x \<le> \<Squnion>(f ` A) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)" |
56166 | 614 |
using le_Sup_iff [of _ "f ` A"] by simp |
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
615 |
|
43940 | 616 |
end |
617 |
||
69593 | 618 |
subsection \<open>Complete lattice on \<^typ>\<open>bool\<close>\<close> |
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
|
619 |
|
44024 | 620 |
instantiation bool :: complete_lattice |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
621 |
begin |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
622 |
|
63575 | 623 |
definition [simp, code]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
624 |
|
63575 | 625 |
definition [simp, code]: "\<Squnion>A \<longleftrightarrow> True \<in> A" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
626 |
|
63575 | 627 |
instance |
628 |
by standard (auto intro: bool_induct) |
|
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
629 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
630 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
631 |
|
63575 | 632 |
lemma not_False_in_image_Ball [simp]: "False \<notin> P ` A \<longleftrightarrow> Ball A P" |
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
633 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
634 |
|
63575 | 635 |
lemma True_in_image_Bex [simp]: "True \<in> P ` A \<longleftrightarrow> Bex A P" |
49905
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
636 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
637 |
|
68802 | 638 |
lemma INF_bool_eq [simp]: "(\<lambda>A f. \<Sqinter>(f ` A)) = Ball" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
639 |
by (simp add: fun_eq_iff) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
640 |
|
68802 | 641 |
lemma SUP_bool_eq [simp]: "(\<lambda>A f. \<Squnion>(f ` A)) = Bex" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
642 |
by (simp add: fun_eq_iff) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
643 |
|
63575 | 644 |
instance bool :: complete_boolean_algebra |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
645 |
by (standard, fastforce) |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
646 |
|
69593 | 647 |
subsection \<open>Complete lattice on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
648 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
649 |
instantiation "fun" :: (type, Inf) Inf |
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
|
650 |
begin |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
651 |
|
63575 | 652 |
definition "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)" |
41080 | 653 |
|
63575 | 654 |
lemma Inf_apply [simp, code]: "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)" |
41080 | 655 |
by (simp add: Inf_fun_def) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
656 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
657 |
instance .. |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
658 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
659 |
end |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
660 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
661 |
instantiation "fun" :: (type, Sup) Sup |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
662 |
begin |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
663 |
|
63575 | 664 |
definition "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)" |
41080 | 665 |
|
63575 | 666 |
lemma Sup_apply [simp, code]: "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)" |
41080 | 667 |
by (simp add: Sup_fun_def) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
668 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
669 |
instance .. |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
670 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
671 |
end |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
672 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
673 |
instantiation "fun" :: (type, complete_lattice) complete_lattice |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
674 |
begin |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
675 |
|
63575 | 676 |
instance |
677 |
by standard (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least) |
|
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
678 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
679 |
end |
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
680 |
|
63575 | 681 |
lemma INF_apply [simp]: "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" |
56166 | 682 |
using Inf_apply [of "f ` A"] by (simp add: comp_def) |
38705 | 683 |
|
63575 | 684 |
lemma SUP_apply [simp]: "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" |
56166 | 685 |
using Sup_apply [of "f ` A"] by (simp add: comp_def) |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
686 |
|
60758 | 687 |
subsection \<open>Complete lattice on unary and binary predicates\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
688 |
|
63575 | 689 |
lemma Inf1_I: "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a" |
46884 | 690 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
691 |
|
63575 | 692 |
lemma INF1_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
693 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
694 |
|
63575 | 695 |
lemma INF2_I: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
696 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
697 |
|
63575 | 698 |
lemma Inf2_I: "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b" |
46884 | 699 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
700 |
|
63575 | 701 |
lemma Inf1_D: "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a" |
46884 | 702 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
703 |
|
63575 | 704 |
lemma INF1_D: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
705 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
706 |
|
63575 | 707 |
lemma Inf2_D: "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b" |
46884 | 708 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
709 |
|
63575 | 710 |
lemma INF2_D: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
711 |
by simp |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
712 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
713 |
lemma Inf1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
714 |
assumes "(\<Sqinter>A) a" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
715 |
obtains "P a" | "P \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
716 |
using assms by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
717 |
|
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
718 |
lemma INF1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
719 |
assumes "(\<Sqinter>x\<in>A. B x) b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
720 |
obtains "B a b" | "a \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
721 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
722 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
723 |
lemma Inf2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
724 |
assumes "(\<Sqinter>A) a b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
725 |
obtains "r a b" | "r \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
726 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
727 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
728 |
lemma INF2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
729 |
assumes "(\<Sqinter>x\<in>A. B x) b c" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
730 |
obtains "B a b c" | "a \<notin> A" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
731 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
732 |
|
63575 | 733 |
lemma Sup1_I: "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a" |
46884 | 734 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
735 |
|
63575 | 736 |
lemma SUP1_I: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
737 |
by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
738 |
|
63575 | 739 |
lemma Sup2_I: "r \<in> A \<Longrightarrow> r a b \<Longrightarrow> (\<Squnion>A) a b" |
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
740 |
by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
741 |
|
63575 | 742 |
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" |
46884 | 743 |
by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
744 |
|
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
745 |
lemma Sup1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
746 |
assumes "(\<Squnion>A) a" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
747 |
obtains P where "P \<in> A" and "P a" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
748 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
749 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
750 |
lemma SUP1_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
751 |
assumes "(\<Squnion>x\<in>A. B x) b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
752 |
obtains x where "x \<in> A" and "B x b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
753 |
using assms by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
754 |
|
56742
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
755 |
lemma Sup2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
756 |
assumes "(\<Squnion>A) a b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
757 |
obtains r where "r \<in> A" "r a b" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
758 |
using assms by auto |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
759 |
|
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
760 |
lemma SUP2_E: |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
761 |
assumes "(\<Squnion>x\<in>A. B x) b c" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
762 |
obtains x where "x \<in> A" "B x b c" |
678a52e676b6
more complete classical rules for Inf and Sup, modelled after theiry counterparts on Inter and Union (and INF and SUP)
haftmann
parents:
56741
diff
changeset
|
763 |
using assms by auto |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
764 |
|
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
765 |
|
69593 | 766 |
subsection \<open>Complete lattice on \<^typ>\<open>_ set\<close>\<close> |
46631
2c5c003cee35
moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents:
46557
diff
changeset
|
767 |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
768 |
instantiation "set" :: (type) complete_lattice |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
769 |
begin |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
770 |
|
63575 | 771 |
definition "\<Sqinter>A = {x. \<Sqinter>((\<lambda>B. x \<in> B) ` A)}" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
772 |
|
63575 | 773 |
definition "\<Squnion>A = {x. \<Squnion>((\<lambda>B. x \<in> B) ` A)}" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
774 |
|
63575 | 775 |
instance |
776 |
by standard (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def) |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
777 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
778 |
end |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
779 |
|
60758 | 780 |
subsubsection \<open>Inter\<close> |
41082 | 781 |
|
69745 | 782 |
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" ("\<Inter>") |
61952 | 783 |
where "\<Inter>S \<equiv> \<Sqinter>S" |
63575 | 784 |
|
785 |
lemma Inter_eq: "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" |
|
41082 | 786 |
proof (rule set_eqI) |
787 |
fix x |
|
788 |
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" |
|
789 |
by auto |
|
790 |
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}" |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
791 |
by (simp add: Inf_set_def image_def) |
41082 | 792 |
qed |
793 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
794 |
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" |
41082 | 795 |
by (unfold Inter_eq) blast |
796 |
||
43741 | 797 |
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" |
41082 | 798 |
by (simp add: Inter_eq) |
799 |
||
60758 | 800 |
text \<open> |
69593 | 801 |
\<^medskip> A ``destruct'' rule -- every \<^term>\<open>X\<close> in \<^term>\<open>C\<close> |
802 |
contains \<^term>\<open>A\<close> as an element, but \<^prop>\<open>A \<in> X\<close> can hold when |
|
803 |
\<^prop>\<open>X \<in> C\<close> does not! This rule is analogous to \<open>spec\<close>. |
|
60758 | 804 |
\<close> |
41082 | 805 |
|
43741 | 806 |
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" |
41082 | 807 |
by auto |
808 |
||
43741 | 809 |
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" |
61799 | 810 |
\<comment> \<open>``Classical'' elimination rule -- does not require proving |
69593 | 811 |
\<^prop>\<open>X \<in> C\<close>.\<close> |
63575 | 812 |
unfolding Inter_eq by blast |
41082 | 813 |
|
43741 | 814 |
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 815 |
by (fact Inf_lower) |
816 |
||
63575 | 817 |
lemma Inter_subset: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 818 |
by (fact Inf_less_eq) |
41082 | 819 |
|
61952 | 820 |
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A" |
43740 | 821 |
by (fact Inf_greatest) |
41082 | 822 |
|
44067 | 823 |
lemma Inter_empty: "\<Inter>{} = UNIV" |
824 |
by (fact Inf_empty) (* already simp *) |
|
41082 | 825 |
|
44067 | 826 |
lemma Inter_UNIV: "\<Inter>UNIV = {}" |
827 |
by (fact Inf_UNIV) (* already simp *) |
|
41082 | 828 |
|
44920 | 829 |
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B" |
830 |
by (fact Inf_insert) (* already simp *) |
|
41082 | 831 |
|
832 |
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" |
|
43899 | 833 |
by (fact less_eq_Inf_inter) |
41082 | 834 |
|
835 |
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" |
|
43756 | 836 |
by (fact Inf_union_distrib) |
837 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
838 |
lemma Inter_UNIV_conv [simp]: |
43741 | 839 |
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
840 |
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
|
43801 | 841 |
by (fact Inf_top_conv)+ |
41082 | 842 |
|
43741 | 843 |
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" |
43899 | 844 |
by (fact Inf_superset_mono) |
41082 | 845 |
|
846 |
||
60758 | 847 |
subsubsection \<open>Intersections of families\<close> |
41082 | 848 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
849 |
syntax (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
850 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3INT _./ _)" [0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
851 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) |
41082 | 852 |
|
69274
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
853 |
syntax |
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
854 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) |
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
855 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) |
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
856 |
|
41082 | 857 |
syntax (latex output) |
62789 | 858 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
859 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<in>_\<^esub>)/ _)" [0, 0, 10] 10) |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
860 |
|
41082 | 861 |
translations |
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
862 |
"\<Inter>x y. f" \<rightleftharpoons> "\<Inter>x. \<Inter>y. f" |
69745 | 863 |
"\<Inter>x. f" \<rightleftharpoons> "\<Inter>(CONST range (\<lambda>x. f))" |
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
864 |
"\<Inter>x\<in>A. f" \<rightleftharpoons> "CONST Inter ((\<lambda>x. f) ` A)" |
41082 | 865 |
|
63575 | 866 |
lemma INTER_eq: "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" |
56166 | 867 |
by (auto intro!: INF_eqI) |
41082 | 868 |
|
43817 | 869 |
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)" |
56166 | 870 |
using Inter_iff [of _ "B ` A"] by simp |
41082 | 871 |
|
43817 | 872 |
lemma INT_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> b \<in> B x) \<Longrightarrow> b \<in> (\<Inter>x\<in>A. B x)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
873 |
by auto |
41082 | 874 |
|
43852 | 875 |
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a" |
41082 | 876 |
by auto |
877 |
||
43852 | 878 |
lemma INT_E [elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> (b \<in> B a \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" |
69593 | 879 |
\<comment> \<open>"Classical" elimination -- by the Excluded Middle on \<^prop>\<open>a\<in>A\<close>.\<close> |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
880 |
by auto |
41082 | 881 |
|
882 |
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" |
|
883 |
by blast |
|
884 |
||
885 |
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" |
|
886 |
by blast |
|
887 |
||
43817 | 888 |
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
889 |
by (fact INF_lower) |
41082 | 890 |
|
43817 | 891 |
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
892 |
by (fact INF_greatest) |
41082 | 893 |
|
44067 | 894 |
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
895 |
by (fact INF_empty) |
43854 | 896 |
|
43817 | 897 |
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" |
43872 | 898 |
by (fact INF_absorb) |
41082 | 899 |
|
43854 | 900 |
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)" |
41082 | 901 |
by (fact le_INF_iff) |
902 |
||
69275 | 903 |
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> \<Inter> (B ` A)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
904 |
by (fact INF_insert) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
905 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
906 |
lemma INT_Un: "(\<Inter>i \<in> A \<union> B. M i) = (\<Inter>i \<in> A. M i) \<inter> (\<Inter>i\<in>B. M i)" |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
907 |
by (fact INF_union) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
908 |
|
63575 | 909 |
lemma INT_insert_distrib: "u \<in> A \<Longrightarrow> (\<Inter>x\<in>A. insert a (B x)) = insert a (\<Inter>x\<in>A. B x)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
910 |
by blast |
43854 | 911 |
|
41082 | 912 |
lemma INT_constant [simp]: "(\<Inter>y\<in>A. c) = (if A = {} then UNIV else c)" |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
913 |
by (fact INF_constant) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
914 |
|
44920 | 915 |
lemma INTER_UNIV_conv: |
63575 | 916 |
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" |
917 |
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" |
|
44920 | 918 |
by (fact INF_top_conv)+ (* already simp *) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
919 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
920 |
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False" |
43873 | 921 |
by (fact INF_UNIV_bool_expand) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
922 |
|
63575 | 923 |
lemma INT_anti_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)" |
61799 | 924 |
\<comment> \<open>The last inclusion is POSITIVE!\<close> |
43940 | 925 |
by (fact INF_superset_mono) |
41082 | 926 |
|
927 |
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" |
|
928 |
by blast |
|
929 |
||
43817 | 930 |
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)" |
41082 | 931 |
by blast |
932 |
||
933 |
||
60758 | 934 |
subsubsection \<open>Union\<close> |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
935 |
|
69745 | 936 |
abbreviation Union :: "'a set set \<Rightarrow> 'a set" ("\<Union>") |
61952 | 937 |
where "\<Union>S \<equiv> \<Squnion>S" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
938 |
|
63575 | 939 |
lemma Union_eq: "\<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
|
940 |
proof (rule set_eqI) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
941 |
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
|
942 |
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
|
943 |
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
|
944 |
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
945 |
by (simp add: Sup_set_def image_def) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
946 |
qed |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
947 |
|
63575 | 948 |
lemma Union_iff [simp]: "A \<in> \<Union>C \<longleftrightarrow> (\<exists>X\<in>C. A\<in>X)" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
949 |
by (unfold Union_eq) blast |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
950 |
|
63575 | 951 |
lemma UnionI [intro]: "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C" |
69593 | 952 |
\<comment> \<open>The order of the premises presupposes that \<^term>\<open>C\<close> is rigid; |
953 |
\<^term>\<open>A\<close> may be flexible.\<close> |
|
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
954 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
955 |
|
63575 | 956 |
lemma UnionE [elim!]: "A \<in> \<Union>C \<Longrightarrow> (\<And>X. A \<in> X \<Longrightarrow> X \<in> C \<Longrightarrow> R) \<Longrightarrow> R" |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
957 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
958 |
|
43817 | 959 |
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" |
43901 | 960 |
by (fact Sup_upper) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
961 |
|
43817 | 962 |
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" |
43901 | 963 |
by (fact Sup_least) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
964 |
|
44920 | 965 |
lemma Union_empty: "\<Union>{} = {}" |
966 |
by (fact Sup_empty) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
967 |
|
44920 | 968 |
lemma Union_UNIV: "\<Union>UNIV = UNIV" |
969 |
by (fact Sup_UNIV) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
970 |
|
69745 | 971 |
lemma Union_insert: "\<Union>(insert a B) = a \<union> \<Union>B" |
44920 | 972 |
by (fact Sup_insert) (* already simp *) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
973 |
|
43817 | 974 |
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" |
43901 | 975 |
by (fact Sup_union_distrib) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
976 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
977 |
lemma Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" |
43901 | 978 |
by (fact Sup_inter_less_eq) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
979 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
980 |
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
44920 | 981 |
by (fact Sup_bot_conv) (* already simp *) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
982 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
983 |
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
44920 | 984 |
by (fact Sup_bot_conv) (* already simp *) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
985 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
986 |
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
|
987 |
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
|
988 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
989 |
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
|
990 |
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
|
991 |
|
43817 | 992 |
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
43901 | 993 |
by (fact Sup_subset_mono) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
994 |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63365
diff
changeset
|
995 |
lemma Union_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> B \<and> x \<subseteq> y) \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63365
diff
changeset
|
996 |
by blast |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
997 |
|
63879
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
998 |
lemma disjnt_inj_on_iff: |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
999 |
"\<lbrakk>inj_on f (\<Union>\<A>); X \<in> \<A>; Y \<in> \<A>\<rbrakk> \<Longrightarrow> disjnt (f ` X) (f ` Y) \<longleftrightarrow> disjnt X Y" |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
1000 |
apply (auto simp: disjnt_def) |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
1001 |
using inj_on_eq_iff by fastforce |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
1002 |
|
63575 | 1003 |
|
60758 | 1004 |
subsubsection \<open>Unions of families\<close> |
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
|
1005 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1006 |
syntax (ASCII) |
35115 | 1007 |
"_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
|
1008 |
"_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
|
1009 |
|
69274
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
1010 |
syntax |
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
1011 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) |
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
1012 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10) |
ff7e6751a1a7
clarified status of ancient ASCII syntax for big union and inter
haftmann
parents:
69260
diff
changeset
|
1013 |
|
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
|
1014 |
syntax (latex output) |
62789 | 1015 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
1016 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(\<open>unbreakable\<close>\<^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
|
1017 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1018 |
translations |
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
1019 |
"\<Union>x y. f" \<rightleftharpoons> "\<Union>x. \<Union>y. f" |
69745 | 1020 |
"\<Union>x. f" \<rightleftharpoons> "\<Union>(CONST range (\<lambda>x. f))" |
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
1021 |
"\<Union>x\<in>A. f" \<rightleftharpoons> "CONST Union ((\<lambda>x. f) ` A)" |
32077
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1022 |
|
60758 | 1023 |
text \<open> |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1024 |
Note the difference between ordinary syntax of indexed |
61799 | 1025 |
unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>) |
69593 | 1026 |
and their \LaTeX\ rendition: \<^term>\<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>. |
60758 | 1027 |
\<close> |
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
|
1028 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1029 |
lemma disjoint_UN_iff: "disjnt A (\<Union>i\<in>I. B i) \<longleftrightarrow> (\<forall>i\<in>I. disjnt A (B i))" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1030 |
by (auto simp: disjnt_def) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1031 |
|
63575 | 1032 |
lemma UNION_eq: "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" |
56166 | 1033 |
by (auto intro!: SUP_eqI) |
44920 | 1034 |
|
69275 | 1035 |
lemma bind_UNION [code]: "Set.bind A f = \<Union>(f ` A)" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1036 |
by (simp add: bind_def UNION_eq) |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1037 |
|
69275 | 1038 |
lemma member_bind [simp]: "x \<in> Set.bind A f \<longleftrightarrow> x \<in> \<Union>(f ` A)" |
46036 | 1039 |
by (simp add: bind_UNION) |
1040 |
||
60585 | 1041 |
lemma Union_SetCompr_eq: "\<Union>{f x| x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}" |
60307
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60172
diff
changeset
|
1042 |
by blast |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60172
diff
changeset
|
1043 |
|
46036 | 1044 |
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)" |
56166 | 1045 |
using Union_iff [of _ "B ` A"] by simp |
11979 | 1046 |
|
43852 | 1047 |
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)" |
69593 | 1048 |
\<comment> \<open>The order of the premises presupposes that \<^term>\<open>A\<close> is rigid; |
1049 |
\<^term>\<open>b\<close> may be flexible.\<close> |
|
11979 | 1050 |
by auto |
1051 |
||
43852 | 1052 |
lemma UN_E [elim!]: "b \<in> (\<Union>x\<in>A. B x) \<Longrightarrow> (\<And>x. x\<in>A \<Longrightarrow> b \<in> B x \<Longrightarrow> R) \<Longrightarrow> R" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1053 |
by auto |
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
|
1054 |
|
43817 | 1055 |
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
1056 |
by (fact SUP_upper) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1057 |
|
43817 | 1058 |
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C" |
44103
cedaca00789f
more uniform naming scheme for Inf/INF and Sup/SUP lemmas
haftmann
parents:
44085
diff
changeset
|
1059 |
by (fact SUP_least) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1060 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1061 |
lemma Collect_bex_eq: "{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
|
1062 |
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
|
1063 |
|
43817 | 1064 |
lemma UN_insert_distrib: "u \<in> A \<Longrightarrow> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1065 |
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
|
1066 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1067 |
lemma UN_empty: "(\<Union>x\<in>{}. B x) = {}" |
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1068 |
by (fact SUP_empty) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1069 |
|
44920 | 1070 |
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}" |
1071 |
by (fact SUP_bot) (* already simp *) |
|
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1072 |
|
43817 | 1073 |
lemma UN_absorb: "k \<in> I \<Longrightarrow> A k \<union> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. A i)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1074 |
by (fact SUP_absorb) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1075 |
|
69275 | 1076 |
lemma UN_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> \<Union>(B ` A)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1077 |
by (fact SUP_insert) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1078 |
|
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1079 |
lemma UN_Un [simp]: "(\<Union>i \<in> A \<union> B. M i) = (\<Union>i\<in>A. M i) \<union> (\<Union>i\<in>B. M i)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1080 |
by (fact SUP_union) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1081 |
|
43967 | 1082 |
lemma UN_UN_flatten: "(\<Union>x \<in> (\<Union>y\<in>A. B y). C x) = (\<Union>y\<in>A. \<Union>x\<in>B y. C x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1083 |
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
|
1084 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1085 |
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" |
35629 | 1086 |
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
|
1087 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1088 |
lemma UN_constant [simp]: "(\<Union>y\<in>A. c) = (if A = {} then {} else c)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1089 |
by (fact SUP_constant) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1090 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1091 |
lemma UNION_singleton_eq_range: "(\<Union>x\<in>A. {f x}) = f ` A" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1092 |
by blast |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1093 |
|
43944 | 1094 |
lemma image_Union: "f ` \<Union>S = (\<Union>x\<in>S. f ` x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1095 |
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
|
1096 |
|
44920 | 1097 |
lemma UNION_empty_conv: |
43817 | 1098 |
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
1099 |
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
|
44920 | 1100 |
by (fact SUP_bot_conv)+ (* already simp *) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1101 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1102 |
lemma Collect_ex_eq: "{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
|
1103 |
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
|
1104 |
|
69275 | 1105 |
lemma ball_UN: "(\<forall>z \<in> \<Union>(B ` A). P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1106 |
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
|
1107 |
|
69275 | 1108 |
lemma bex_UN: "(\<exists>z \<in> \<Union>(B ` A). P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1109 |
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
|
1110 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1111 |
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)" |
62390 | 1112 |
by safe (auto simp add: if_split_mem2) |
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
|
1113 |
|
43817 | 1114 |
lemma UN_bool_eq: "(\<Union>b. A b) = (A True \<union> A False)" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1115 |
by (fact SUP_UNIV_bool_expand) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1116 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1117 |
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
|
1118 |
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
|
1119 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1120 |
lemma UN_mono: |
43817 | 1121 |
"A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1122 |
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" |
43940 | 1123 |
by (fact SUP_subset_mono) |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1124 |
|
43817 | 1125 |
lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1126 |
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
|
1127 |
|
43817 | 1128 |
lemma vimage_UN: "f -` (\<Union>x\<in>A. B x) = (\<Union>x\<in>A. f -` B x)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1129 |
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
|
1130 |
|
43817 | 1131 |
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})" |
61799 | 1132 |
\<comment> \<open>NOT suitable for rewriting\<close> |
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
|
1133 |
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
|
1134 |
|
69275 | 1135 |
lemma image_UN: "f ` \<Union>(B ` A) = (\<Union>x\<in>A. f ` B x)" |
43817 | 1136 |
by blast |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1137 |
|
45013 | 1138 |
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" |
1139 |
by blast |
|
1140 |
||
67399 | 1141 |
lemma inj_on_image: "inj_on f (\<Union>A) \<Longrightarrow> inj_on ((`) f) A" |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62789
diff
changeset
|
1142 |
unfolding inj_on_def by blast |
11979 | 1143 |
|
63575 | 1144 |
|
60758 | 1145 |
subsubsection \<open>Distributive laws\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1146 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1147 |
lemma Int_Union: "A \<inter> \<Union>B = (\<Union>C\<in>B. A \<inter> C)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1148 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1149 |
|
44039 | 1150 |
lemma Un_Inter: "A \<union> \<Inter>B = (\<Inter>C\<in>B. A \<union> C)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1151 |
by blast |
44039 | 1152 |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1153 |
lemma Int_Union2: "\<Union>B \<inter> A = (\<Union>C\<in>B. C \<inter> A)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1154 |
by blast |
44039 | 1155 |
|
1156 |
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)" |
|
1157 |
by (rule sym) (rule INF_inf_distrib) |
|
1158 |
||
1159 |
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)" |
|
1160 |
by (rule sym) (rule SUP_sup_distrib) |
|
1161 |
||
63575 | 1162 |
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" (* FIXME drop *) |
56166 | 1163 |
by (simp add: INT_Int_distrib) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1164 |
|
69020
4f94e262976d
elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents:
68980
diff
changeset
|
1165 |
lemma Int_Inter_eq: "A \<inter> \<Inter>\<B> = (if \<B>={} then A else (\<Inter>B\<in>\<B>. A \<inter> B))" |
4f94e262976d
elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents:
68980
diff
changeset
|
1166 |
"\<Inter>\<B> \<inter> A = (if \<B>={} then A else (\<Inter>B\<in>\<B>. B \<inter> A))" |
4f94e262976d
elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents:
68980
diff
changeset
|
1167 |
by auto |
4f94e262976d
elimination of near duplication involving Rolle's theorem and the MVT
paulson <lp15@cam.ac.uk>
parents:
68980
diff
changeset
|
1168 |
|
63575 | 1169 |
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" (* FIXME drop *) |
61799 | 1170 |
\<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close> |
1171 |
\<comment> \<open>Union of a family of unions\<close> |
|
56166 | 1172 |
by (simp add: UN_Un_distrib) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1173 |
|
44039 | 1174 |
lemma Un_INT_distrib: "B \<union> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. B \<union> A i)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1175 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1176 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1177 |
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)" |
61799 | 1178 |
\<comment> \<open>Halmos, Naive Set Theory, page 35.\<close> |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1179 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1180 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1181 |
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)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1182 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1183 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1184 |
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)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1185 |
by blast |
44039 | 1186 |
|
1187 |
lemma Union_disjoint: "(\<Union>C \<inter> A = {}) \<longleftrightarrow> (\<forall>B\<in>C. B \<inter> A = {})" |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1188 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1189 |
|
67613 | 1190 |
lemma SUP_UNION: "(\<Squnion>x\<in>(\<Union>y\<in>A. g y). f x) = (\<Squnion>y\<in>A. \<Squnion>x\<in>g y. f x :: _ :: complete_lattice)" |
63575 | 1191 |
by (rule order_antisym) (blast intro: SUP_least SUP_upper2)+ |
1192 |
||
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1193 |
|
60758 | 1194 |
subsection \<open>Injections and bijections\<close> |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1195 |
|
63575 | 1196 |
lemma inj_on_Inter: "S \<noteq> {} \<Longrightarrow> (\<And>A. A \<in> S \<Longrightarrow> inj_on f A) \<Longrightarrow> inj_on f (\<Inter>S)" |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1197 |
unfolding inj_on_def by blast |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1198 |
|
63575 | 1199 |
lemma inj_on_INTER: "I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)) \<Longrightarrow> inj_on f (\<Inter>i \<in> I. A i)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1200 |
unfolding inj_on_def by safe simp |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1201 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1202 |
lemma inj_on_UNION_chain: |
63575 | 1203 |
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" |
1204 |
and inj: "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
|
60585 | 1205 |
shows "inj_on f (\<Union>i \<in> I. A i)" |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1206 |
proof - |
63575 | 1207 |
have "x = y" |
1208 |
if *: "i \<in> I" "j \<in> I" |
|
1209 |
and **: "x \<in> A i" "y \<in> A j" |
|
1210 |
and ***: "f x = f y" |
|
1211 |
for i j x y |
|
1212 |
using chain [OF *] |
|
1213 |
proof |
|
1214 |
assume "A i \<le> A j" |
|
1215 |
with ** have "x \<in> A j" by auto |
|
1216 |
with inj * ** *** show ?thesis |
|
1217 |
by (auto simp add: inj_on_def) |
|
1218 |
next |
|
1219 |
assume "A j \<le> A i" |
|
1220 |
with ** have "y \<in> A i" by auto |
|
1221 |
with inj * ** *** show ?thesis |
|
1222 |
by (auto simp add: inj_on_def) |
|
1223 |
qed |
|
1224 |
then show ?thesis |
|
1225 |
by (unfold inj_on_def UNION_eq) auto |
|
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1226 |
qed |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1227 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1228 |
lemma bij_betw_UNION_chain: |
63575 | 1229 |
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" |
1230 |
and bij: "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" |
|
60585 | 1231 |
shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)" |
63575 | 1232 |
unfolding bij_betw_def |
63576 | 1233 |
proof safe |
63575 | 1234 |
have "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
1235 |
using bij bij_betw_def[of f] by auto |
|
69275 | 1236 |
then show "inj_on f (\<Union>(A ` I))" |
63575 | 1237 |
using chain inj_on_UNION_chain[of I A f] by auto |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1238 |
next |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1239 |
fix i x |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1240 |
assume *: "i \<in> I" "x \<in> A i" |
63576 | 1241 |
with bij have "f x \<in> A' i" |
1242 |
by (auto simp: bij_betw_def) |
|
69275 | 1243 |
with * show "f x \<in> \<Union>(A' ` I)" by blast |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1244 |
next |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1245 |
fix i x' |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1246 |
assume *: "i \<in> I" "x' \<in> A' i" |
63576 | 1247 |
with bij have "\<exists>x \<in> A i. x' = f x" |
1248 |
unfolding bij_betw_def by blast |
|
63575 | 1249 |
with * have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x" |
1250 |
by blast |
|
69275 | 1251 |
then show "x' \<in> f ` \<Union>(A ` I)" |
63575 | 1252 |
by blast |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1253 |
qed |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1254 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1255 |
(*injectivity's required. Left-to-right inclusion holds even if A is empty*) |
69275 | 1256 |
lemma image_INT: "inj_on f C \<Longrightarrow> \<forall>x\<in>A. B x \<subseteq> C \<Longrightarrow> j \<in> A \<Longrightarrow> f ` (\<Inter>(B ` A)) = (\<Inter>x\<in>A. f ` B x)" |
63575 | 1257 |
by (auto simp add: inj_on_def) blast |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1258 |
|
69275 | 1259 |
lemma bij_image_INT: "bij f \<Longrightarrow> f ` (\<Inter>(B ` A)) = (\<Inter>x\<in>A. f ` B x)" |
64966
d53d7ca3303e
added inj_def (redundant, analogous to surj_def, bij_def);
wenzelm
parents:
63879
diff
changeset
|
1260 |
by (auto simp: bij_def inj_def surj_def) blast |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1261 |
|
69275 | 1262 |
lemma UNION_fun_upd: "\<Union>(A(i := B) ` J) = \<Union>(A ` (J - {i})) \<union> (if i \<in> J then B else {})" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1263 |
by (auto simp add: set_eq_iff) |
63365 | 1264 |
|
1265 |
lemma bij_betw_Pow: |
|
1266 |
assumes "bij_betw f A B" |
|
1267 |
shows "bij_betw (image f) (Pow A) (Pow B)" |
|
1268 |
proof - |
|
1269 |
from assms have "inj_on f A" |
|
1270 |
by (rule bij_betw_imp_inj_on) |
|
69745 | 1271 |
then have "inj_on f (\<Union>(Pow A))" |
63365 | 1272 |
by simp |
1273 |
then have "inj_on (image f) (Pow A)" |
|
1274 |
by (rule inj_on_image) |
|
1275 |
then have "bij_betw (image f) (Pow A) (image f ` Pow A)" |
|
1276 |
by (rule inj_on_imp_bij_betw) |
|
1277 |
moreover from assms have "f ` A = B" |
|
1278 |
by (rule bij_betw_imp_surj_on) |
|
1279 |
then have "image f ` Pow A = Pow B" |
|
1280 |
by (rule image_Pow_surj) |
|
1281 |
ultimately show ?thesis by simp |
|
1282 |
qed |
|
1283 |
||
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1284 |
|
60758 | 1285 |
subsubsection \<open>Complement\<close> |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1286 |
|
43873 | 1287 |
lemma Compl_INT [simp]: "- (\<Inter>x\<in>A. B x) = (\<Union>x\<in>A. -B x)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1288 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1289 |
|
43873 | 1290 |
lemma Compl_UN [simp]: "- (\<Union>x\<in>A. B x) = (\<Inter>x\<in>A. -B x)" |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
1291 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1292 |
|
60758 | 1293 |
subsubsection \<open>Miniscoping and maxiscoping\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1294 |
|
63575 | 1295 |
text \<open>\<^medskip> Miniscoping: pushing in quantifiers and big Unions and Intersections.\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1296 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1297 |
lemma UN_simps [simp]: |
43817 | 1298 |
"\<And>a B C. (\<Union>x\<in>C. insert a (B x)) = (if C={} then {} else insert a (\<Union>x\<in>C. B x))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1299 |
"\<And>A B C. (\<Union>x\<in>C. A x \<union> B) = ((if C={} then {} else (\<Union>x\<in>C. A x) \<union> B))" |
43852 | 1300 |
"\<And>A B C. (\<Union>x\<in>C. A \<union> B x) = ((if C={} then {} else A \<union> (\<Union>x\<in>C. B x)))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1301 |
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)" |
43852 | 1302 |
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))" |
1303 |
"\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)" |
|
1304 |
"\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))" |
|
1305 |
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)" |
|
69275 | 1306 |
"\<And>A B C. (\<Union>z\<in>(\<Union>(B ` A)). C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)" |
43831 | 1307 |
"\<And>A B f. (\<Union>x\<in>f`A. B x) = (\<Union>a\<in>A. B (f a))" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1308 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1309 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1310 |
lemma INT_simps [simp]: |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1311 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<inter> B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) \<inter> B)" |
43831 | 1312 |
"\<And>A B C. (\<Inter>x\<in>C. A \<inter> B x) = (if C={} then UNIV else A \<inter>(\<Inter>x\<in>C. B x))" |
43852 | 1313 |
"\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)" |
1314 |
"\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))" |
|
43817 | 1315 |
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)" |
43852 | 1316 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)" |
1317 |
"\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))" |
|
1318 |
"\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)" |
|
69275 | 1319 |
"\<And>A B C. (\<Inter>z\<in>(\<Union>(B ` A)). C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)" |
43852 | 1320 |
"\<And>A B f. (\<Inter>x\<in>f`A. B x) = (\<Inter>a\<in>A. B (f a))" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1321 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1322 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1323 |
lemma UN_ball_bex_simps [simp]: |
43852 | 1324 |
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)" |
69275 | 1325 |
"\<And>A B P. (\<forall>x\<in>(\<Union>(B ` A)). P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)" |
43852 | 1326 |
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)" |
69275 | 1327 |
"\<And>A B P. (\<exists>x\<in>(\<Union>(B ` A)). P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1328 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1329 |
|
43943 | 1330 |
|
63575 | 1331 |
text \<open>\<^medskip> Maxiscoping: pulling out big Unions and Intersections.\<close> |
13860 | 1332 |
|
1333 |
lemma UN_extend_simps: |
|
43817 | 1334 |
"\<And>a B C. insert a (\<Union>x\<in>C. B x) = (if C={} then {a} else (\<Union>x\<in>C. insert a (B x)))" |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1335 |
"\<And>A B C. (\<Union>x\<in>C. A x) \<union> B = (if C={} then B else (\<Union>x\<in>C. A x \<union> B))" |
43852 | 1336 |
"\<And>A B C. A \<union> (\<Union>x\<in>C. B x) = (if C={} then A else (\<Union>x\<in>C. A \<union> B x))" |
1337 |
"\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)" |
|
1338 |
"\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)" |
|
43817 | 1339 |
"\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)" |
1340 |
"\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)" |
|
43852 | 1341 |
"\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)" |
69275 | 1342 |
"\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>(\<Union>(B ` A)). C z)" |
43831 | 1343 |
"\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)" |
13860 | 1344 |
by auto |
1345 |
||
1346 |
lemma INT_extend_simps: |
|
43852 | 1347 |
"\<And>A B C. (\<Inter>x\<in>C. A x) \<inter> B = (if C={} then B else (\<Inter>x\<in>C. A x \<inter> B))" |
1348 |
"\<And>A B C. A \<inter> (\<Inter>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A \<inter> B x))" |
|
1349 |
"\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))" |
|
1350 |
"\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))" |
|
43817 | 1351 |
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))" |
43852 | 1352 |
"\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)" |
1353 |
"\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)" |
|
1354 |
"\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)" |
|
69275 | 1355 |
"\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>(\<Union>(B ` A)). C z)" |
43852 | 1356 |
"\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)" |
13860 | 1357 |
by auto |
1358 |
||
60758 | 1359 |
text \<open>Finally\<close> |
43872 | 1360 |
|
30596 | 1361 |
lemmas mem_simps = |
1362 |
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff |
|
1363 |
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff |
|
61799 | 1364 |
\<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close> |
21669 | 1365 |
|
11979 | 1366 |
end |