author | haftmann |
Fri, 24 Aug 2018 20:22:14 +0000 | |
changeset 68802 | 3974935e0252 |
parent 68801 | c898c2b1fd58 |
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permissions | -rw-r--r-- |
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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>_" [900] 900) |
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begin |
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abbreviation (input) INFIMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" \<comment> \<open>legacy\<close> |
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where "INFIMUM A f \<equiv> \<Sqinter>(f ` A)" |
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end |
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class Sup = |
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fixes Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) |
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begin |
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abbreviation (input) SUPREMUM :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" \<comment> \<open>legacy\<close> |
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where "SUPREMUM A f \<equiv> \<Squnion>(f ` A)" |
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end |
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syntax (input) |
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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 strong_INF_cong [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 (simp add: image_comp) |
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lemma SUP_identity_eq [simp]: "(\<Squnion>x\<in>A. x) = \<Squnion>A" |
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by simp |
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lemma SUP_id_eq [simp]: "\<Squnion>(id ` A) = \<Squnion>A" |
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by (simp add: id_def) |
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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 (simp add: image_def) |
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lemma strong_SUP_cong [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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unfolding simp_implies_def by (fact SUP_cong) |
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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" |
44040 | 167 |
using Inf_lower [of u A] by auto |
168 |
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169 |
lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<le> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<le> u" |
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170 |
using INF_lower [of i A f] by auto |
44040 | 171 |
|
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172 |
lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<le> u \<Longrightarrow> v \<le> \<Squnion>A" |
44040 | 173 |
using Sup_upper [of u A] by auto |
174 |
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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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176 |
using SUP_upper [of i A f] by auto |
44040 | 177 |
|
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178 |
lemma le_Inf_iff: "b \<le> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<le> a)" |
44040 | 179 |
by (auto intro: Inf_greatest dest: Inf_lower) |
180 |
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181 |
lemma le_INF_iff: "u \<le> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<le> f i)" |
56166 | 182 |
using le_Inf_iff [of _ "f ` A"] by simp |
44040 | 183 |
|
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184 |
lemma Sup_le_iff: "\<Squnion>A \<le> b \<longleftrightarrow> (\<forall>a\<in>A. a \<le> b)" |
44040 | 185 |
by (auto intro: Sup_least dest: Sup_upper) |
186 |
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187 |
lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<le> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<le> u)" |
56166 | 188 |
using Sup_le_iff [of "f ` A"] by simp |
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189 |
|
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190 |
lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" |
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191 |
by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower) |
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192 |
|
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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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194 |
by (simp cong del: strong_INF_cong) |
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195 |
|
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lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" |
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by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper) |
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198 |
|
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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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200 |
by (simp cong del: strong_SUP_cong) |
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201 |
|
44067 | 202 |
lemma INF_empty [simp]: "(\<Sqinter>x\<in>{}. f x) = \<top>" |
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203 |
by (simp cong del: strong_INF_cong) |
44040 | 204 |
|
44067 | 205 |
lemma SUP_empty [simp]: "(\<Squnion>x\<in>{}. f x) = \<bottom>" |
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206 |
by (simp cong del: strong_SUP_cong) |
44040 | 207 |
|
63575 | 208 |
lemma Inf_UNIV [simp]: "\<Sqinter>UNIV = \<bottom>" |
44040 | 209 |
by (auto intro!: antisym Inf_lower) |
41080 | 210 |
|
63575 | 211 |
lemma Sup_UNIV [simp]: "\<Squnion>UNIV = \<top>" |
44040 | 212 |
by (auto intro!: antisym Sup_upper) |
41080 | 213 |
|
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214 |
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}" |
44040 | 215 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
216 |
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217 |
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}" |
44040 | 218 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
219 |
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220 |
lemma Inf_superset_mono: "B \<subseteq> A \<Longrightarrow> \<Sqinter>A \<le> \<Sqinter>B" |
43899 | 221 |
by (auto intro: Inf_greatest Inf_lower) |
222 |
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223 |
lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<le> \<Squnion>B" |
43899 | 224 |
by (auto intro: Sup_least Sup_upper) |
225 |
||
38705 | 226 |
lemma Inf_mono: |
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assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b" |
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228 |
shows "\<Sqinter>A \<le> \<Sqinter>B" |
38705 | 229 |
proof (rule Inf_greatest) |
230 |
fix b assume "b \<in> B" |
|
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231 |
with assms obtain a where "a \<in> A" and "a \<le> b" by blast |
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232 |
from \<open>a \<in> A\<close> have "\<Sqinter>A \<le> a" by (rule Inf_lower) |
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233 |
with \<open>a \<le> b\<close> show "\<Sqinter>A \<le> b" by auto |
38705 | 234 |
qed |
235 |
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236 |
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 | 237 |
using Inf_mono [of "g ` B" "f ` A"] by auto |
44041 | 238 |
|
41082 | 239 |
lemma Sup_mono: |
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240 |
assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b" |
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241 |
shows "\<Squnion>A \<le> \<Squnion>B" |
41082 | 242 |
proof (rule Sup_least) |
243 |
fix a assume "a \<in> A" |
|
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244 |
with assms obtain b where "b \<in> B" and "a \<le> b" by blast |
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|
245 |
from \<open>b \<in> B\<close> have "b \<le> \<Squnion>B" by (rule Sup_upper) |
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246 |
with \<open>a \<le> b\<close> show "a \<le> \<Squnion>B" by auto |
41082 | 247 |
qed |
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248 |
|
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249 |
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 | 250 |
using Sup_mono [of "f ` A" "g ` B"] by auto |
44041 | 251 |
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252 |
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 | 253 |
\<comment> \<open>The last inclusion is POSITIVE!\<close> |
44041 | 254 |
by (blast intro: INF_mono dest: subsetD) |
255 |
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256 |
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 | 257 |
by (blast intro: SUP_mono dest: subsetD) |
258 |
||
43868 | 259 |
lemma Inf_less_eq: |
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260 |
assumes "\<And>v. v \<in> A \<Longrightarrow> v \<le> u" |
43868 | 261 |
and "A \<noteq> {}" |
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262 |
shows "\<Sqinter>A \<le> u" |
43868 | 263 |
proof - |
60758 | 264 |
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast |
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265 |
moreover from \<open>v \<in> A\<close> assms(1) have "v \<le> u" by blast |
43868 | 266 |
ultimately show ?thesis by (rule Inf_lower2) |
267 |
qed |
|
268 |
||
269 |
lemma less_eq_Sup: |
|
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270 |
assumes "\<And>v. v \<in> A \<Longrightarrow> u \<le> v" |
43868 | 271 |
and "A \<noteq> {}" |
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272 |
shows "u \<le> \<Squnion>A" |
43868 | 273 |
proof - |
60758 | 274 |
from \<open>A \<noteq> {}\<close> obtain v where "v \<in> A" by blast |
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275 |
moreover from \<open>v \<in> A\<close> assms(1) have "u \<le> v" by blast |
43868 | 276 |
ultimately show ?thesis by (rule Sup_upper2) |
277 |
qed |
|
278 |
||
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279 |
lemma INF_eq: |
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280 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<ge> g j" |
63575 | 281 |
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<ge> f i" |
68797 | 282 |
shows "\<Sqinter>(f ` A) = \<Sqinter>(g ` B)" |
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283 |
by (intro antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+ |
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|
284 |
|
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|
285 |
lemma SUP_eq: |
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|
286 |
assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i \<le> g j" |
63575 | 287 |
and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j \<le> f i" |
68797 | 288 |
shows "\<Squnion>(f ` A) = \<Squnion>(g ` B)" |
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289 |
by (intro antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+ |
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|
290 |
|
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|
291 |
lemma less_eq_Inf_inter: "\<Sqinter>A \<squnion> \<Sqinter>B \<le> \<Sqinter>(A \<inter> B)" |
43868 | 292 |
by (auto intro: Inf_greatest Inf_lower) |
293 |
||
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|
294 |
lemma Sup_inter_less_eq: "\<Squnion>(A \<inter> B) \<le> \<Squnion>A \<sqinter> \<Squnion>B " |
43868 | 295 |
by (auto intro: Sup_least Sup_upper) |
296 |
||
297 |
lemma Inf_union_distrib: "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B" |
|
298 |
by (rule antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2) |
|
299 |
||
63575 | 300 |
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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|
301 |
by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower) |
44041 | 302 |
|
43868 | 303 |
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B" |
304 |
by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2) |
|
305 |
||
63575 | 306 |
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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changeset
|
307 |
by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper) |
44041 | 308 |
|
309 |
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)" |
|
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|
310 |
by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono) |
44041 | 311 |
|
63575 | 312 |
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)" |
313 |
(is "?L = ?R") |
|
44918 | 314 |
proof (rule antisym) |
63575 | 315 |
show "?L \<le> ?R" |
316 |
by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono) |
|
317 |
show "?R \<le> ?L" |
|
318 |
by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper) |
|
44918 | 319 |
qed |
44041 | 320 |
|
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|
321 |
lemma Inf_top_conv [simp]: |
43868 | 322 |
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
323 |
"\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
324 |
proof - |
|
325 |
show "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" |
|
326 |
proof |
|
327 |
assume "\<forall>x\<in>A. x = \<top>" |
|
328 |
then have "A = {} \<or> A = {\<top>}" by auto |
|
44919 | 329 |
then show "\<Sqinter>A = \<top>" by auto |
43868 | 330 |
next |
331 |
assume "\<Sqinter>A = \<top>" |
|
332 |
show "\<forall>x\<in>A. x = \<top>" |
|
333 |
proof (rule ccontr) |
|
334 |
assume "\<not> (\<forall>x\<in>A. x = \<top>)" |
|
335 |
then obtain x where "x \<in> A" and "x \<noteq> \<top>" by blast |
|
336 |
then obtain B where "A = insert x B" by blast |
|
60758 | 337 |
with \<open>\<Sqinter>A = \<top>\<close> \<open>x \<noteq> \<top>\<close> show False by simp |
43868 | 338 |
qed |
339 |
qed |
|
340 |
then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto |
|
341 |
qed |
|
342 |
||
44918 | 343 |
lemma INF_top_conv [simp]: |
56166 | 344 |
"(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
345 |
"\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)" |
|
346 |
using Inf_top_conv [of "B ` A"] by simp_all |
|
44041 | 347 |
|
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348 |
lemma Sup_bot_conv [simp]: |
63575 | 349 |
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" |
350 |
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" |
|
44920 | 351 |
using dual_complete_lattice |
352 |
by (rule complete_lattice.Inf_top_conv)+ |
|
43868 | 353 |
|
44918 | 354 |
lemma SUP_bot_conv [simp]: |
63575 | 355 |
"(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
356 |
"\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)" |
|
56166 | 357 |
using Sup_bot_conv [of "B ` A"] by simp_all |
44041 | 358 |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
359 |
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
|
360 |
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
|
361 |
|
43870 | 362 |
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
|
363 |
by (auto intro: antisym SUP_upper SUP_least) |
43870 | 364 |
|
44918 | 365 |
lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>" |
44921 | 366 |
by (cases "A = {}") simp_all |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
367 |
|
44918 | 368 |
lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>" |
44921 | 369 |
by (cases "A = {}") simp_all |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
370 |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
371 |
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
|
372 |
by (iprover intro: INF_lower INF_greatest order_trans antisym) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
373 |
|
43870 | 374 |
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
|
375 |
by (iprover intro: SUP_upper SUP_least order_trans antisym) |
43870 | 376 |
|
43871 | 377 |
lemma INF_absorb: |
43868 | 378 |
assumes "k \<in> I" |
379 |
shows "A k \<sqinter> (\<Sqinter>i\<in>I. A i) = (\<Sqinter>i\<in>I. A i)" |
|
380 |
proof - |
|
381 |
from assms obtain J where "I = insert k J" by blast |
|
56166 | 382 |
then show ?thesis by simp |
43868 | 383 |
qed |
384 |
||
43871 | 385 |
lemma SUP_absorb: |
386 |
assumes "k \<in> I" |
|
387 |
shows "A k \<squnion> (\<Squnion>i\<in>I. A i) = (\<Squnion>i\<in>I. A i)" |
|
388 |
proof - |
|
389 |
from assms obtain J where "I = insert k J" by blast |
|
56166 | 390 |
then show ?thesis by simp |
43871 | 391 |
qed |
392 |
||
67613 | 393 |
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
|
394 |
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
|
395 |
(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
|
396 |
|
67613 | 397 |
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
|
398 |
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
|
399 |
|
63575 | 400 |
lemma INF_constant: "(\<Sqinter>y\<in>A. c) = (if A = {} then \<top> else c)" |
44921 | 401 |
by simp |
43868 | 402 |
|
63575 | 403 |
lemma SUP_constant: "(\<Squnion>y\<in>A. c) = (if A = {} then \<bottom> else c)" |
44921 | 404 |
by simp |
43871 | 405 |
|
43943 | 406 |
lemma less_INF_D: |
63575 | 407 |
assumes "y < (\<Sqinter>i\<in>A. f i)" "i \<in> A" |
408 |
shows "y < f i" |
|
43943 | 409 |
proof - |
60758 | 410 |
note \<open>y < (\<Sqinter>i\<in>A. f i)\<close> |
411 |
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
|
412 |
by (rule INF_lower) |
43943 | 413 |
finally show "y < f i" . |
414 |
qed |
|
415 |
||
416 |
lemma SUP_lessD: |
|
63575 | 417 |
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" |
418 |
shows "f i < y" |
|
43943 | 419 |
proof - |
63575 | 420 |
have "f i \<le> (\<Squnion>i\<in>A. f i)" |
421 |
using \<open>i \<in> A\<close> by (rule SUP_upper) |
|
60758 | 422 |
also note \<open>(\<Squnion>i\<in>A. f i) < y\<close> |
43943 | 423 |
finally show "f i < y" . |
424 |
qed |
|
425 |
||
63575 | 426 |
lemma INF_UNIV_bool_expand: "(\<Sqinter>b. A b) = A True \<sqinter> A False" |
56166 | 427 |
by (simp add: UNIV_bool inf_commute) |
43868 | 428 |
|
63575 | 429 |
lemma SUP_UNIV_bool_expand: "(\<Squnion>b. A b) = A True \<squnion> A False" |
56166 | 430 |
by (simp add: UNIV_bool sup_commute) |
43871 | 431 |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
432 |
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
|
433 |
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
|
434 |
|
68797 | 435 |
lemma INF_le_SUP: "A \<noteq> {} \<Longrightarrow> \<Sqinter>(f ` A) \<le> \<Squnion>(f ` A)" |
56166 | 436 |
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
|
437 |
|
68797 | 438 |
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
|
439 |
by (auto intro: INF_eqI) |
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
440 |
|
68797 | 441 |
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
|
442 |
by (auto intro: SUP_eqI) |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
443 |
|
68797 | 444 |
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
|
445 |
using INF_eq_const [of I f c] INF_lower [of _ I f] |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
446 |
by (auto intro: antisym cong del: strong_INF_cong) |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
447 |
|
68797 | 448 |
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
|
449 |
using SUP_eq_const [of I f c] SUP_upper [of _ I f] |
67dc9549fa15
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
haftmann
parents:
56218
diff
changeset
|
450 |
by (auto intro: antisym cong del: strong_SUP_cong) |
54414
72949fae4f81
add equalities for SUP and INF over constant functions
hoelzl
parents:
54259
diff
changeset
|
451 |
|
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
|
452 |
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
|
453 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
454 |
context complete_lattice |
44024 | 455 |
begin |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
456 |
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
|
457 |
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
|
458 |
end |
44039 | 459 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
460 |
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
|
461 |
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
|
462 |
begin |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
463 |
|
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
464 |
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
|
465 |
by (rule antisym, rule Inf_Sup_le, rule Sup_Inf_le) |
44024 | 466 |
|
63575 | 467 |
subclass distrib_lattice |
468 |
proof |
|
44024 | 469 |
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
|
470 |
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
|
471 |
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
|
472 |
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
|
473 |
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
|
474 |
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
|
475 |
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
|
476 |
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
|
477 |
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
|
478 |
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
|
479 |
by (rule le_infI2, simp) |
68797 | 480 |
have "\<Sqinter>(Sup ` {{a, b}, {a, c}}) = |
481 |
\<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
|
482 |
by (rule Inf_Sup) |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
483 |
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
|
484 |
apply simp |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
485 |
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
|
486 |
qed |
44024 | 487 |
qed |
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
488 |
end |
44039 | 489 |
|
67829
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
490 |
context complete_lattice |
2a6ef5ba4822
Changes to complete distributive lattices due to Viorel Preoteasa
Manuel Eberl <eberlm@in.tum.de>
parents:
67673
diff
changeset
|
491 |
begin |
56074 | 492 |
context |
493 |
fixes f :: "'a \<Rightarrow> 'b::complete_lattice" |
|
494 |
assumes "mono f" |
|
495 |
begin |
|
496 |
||
63575 | 497 |
lemma mono_Inf: "f (\<Sqinter>A) \<le> (\<Sqinter>x\<in>A. f x)" |
60758 | 498 |
using \<open>mono f\<close> by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD) |
56074 | 499 |
|
63575 | 500 |
lemma mono_Sup: "(\<Squnion>x\<in>A. f x) \<le> f (\<Squnion>A)" |
60758 | 501 |
using \<open>mono f\<close> by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD) |
56074 | 502 |
|
67613 | 503 |
lemma mono_INF: "f (\<Sqinter>i\<in>I. A i) \<le> (\<Sqinter>x\<in>I. f (A x))" |
60758 | 504 |
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
|
505 |
|
67613 | 506 |
lemma mono_SUP: "(\<Squnion>x\<in>I. f (A x)) \<le> f (\<Squnion>i\<in>I. A i)" |
60758 | 507 |
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
|
508 |
|
56074 | 509 |
end |
510 |
||
44024 | 511 |
end |
512 |
||
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
513 |
class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice |
43873 | 514 |
begin |
515 |
||
63575 | 516 |
lemma uminus_Inf: "- (\<Sqinter>A) = \<Squnion>(uminus ` A)" |
43873 | 517 |
proof (rule antisym) |
518 |
show "- \<Sqinter>A \<le> \<Squnion>(uminus ` A)" |
|
519 |
by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp |
|
520 |
show "\<Squnion>(uminus ` A) \<le> - \<Sqinter>A" |
|
521 |
by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto |
|
522 |
qed |
|
523 |
||
44041 | 524 |
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
|
525 |
by (simp add: uminus_Inf image_image) |
44041 | 526 |
|
63575 | 527 |
lemma uminus_Sup: "- (\<Squnion>A) = \<Sqinter>(uminus ` A)" |
43873 | 528 |
proof - |
63575 | 529 |
have "\<Squnion>A = - \<Sqinter>(uminus ` A)" |
530 |
by (simp add: image_image uminus_INF) |
|
43873 | 531 |
then show ?thesis by simp |
532 |
qed |
|
63575 | 533 |
|
43873 | 534 |
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
|
535 |
by (simp add: uminus_Sup image_image) |
43873 | 536 |
|
537 |
end |
|
538 |
||
43940 | 539 |
class complete_linorder = linorder + complete_lattice |
540 |
begin |
|
541 |
||
43943 | 542 |
lemma dual_complete_linorder: |
67399 | 543 |
"class.complete_linorder Sup Inf sup (\<ge>) (>) inf \<top> \<bottom>" |
43943 | 544 |
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder) |
545 |
||
51386 | 546 |
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
|
547 |
by (auto intro: antisym simp add: min_def fun_eq_iff) |
51386 | 548 |
|
549 |
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
|
550 |
by (auto intro: antisym simp add: max_def fun_eq_iff) |
51386 | 551 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
552 |
lemma Inf_less_iff: "\<Sqinter>S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)" |
63172 | 553 |
by (simp add: not_le [symmetric] le_Inf_iff) |
43940 | 554 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
555 |
lemma INF_less_iff: "(\<Sqinter>i\<in>A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)" |
63172 | 556 |
by (simp add: Inf_less_iff [of "f ` A"]) |
44041 | 557 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
558 |
lemma less_Sup_iff: "a < \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a < x)" |
63172 | 559 |
by (simp add: not_le [symmetric] Sup_le_iff) |
43940 | 560 |
|
63820
9f004fbf9d5c
discontinued theory-local special syntax for lattice orderings
haftmann
parents:
63576
diff
changeset
|
561 |
lemma less_SUP_iff: "a < (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)" |
63172 | 562 |
by (simp add: less_Sup_iff [of _ "f ` A"]) |
43940 | 563 |
|
63575 | 564 |
lemma Sup_eq_top_iff [simp]: "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)" |
43943 | 565 |
proof |
566 |
assume *: "\<Squnion>A = \<top>" |
|
63575 | 567 |
show "(\<forall>x<\<top>. \<exists>i\<in>A. x < i)" |
568 |
unfolding * [symmetric] |
|
43943 | 569 |
proof (intro allI impI) |
63575 | 570 |
fix x |
571 |
assume "x < \<Squnion>A" |
|
572 |
then show "\<exists>i\<in>A. x < i" |
|
63172 | 573 |
by (simp add: less_Sup_iff) |
43943 | 574 |
qed |
575 |
next |
|
576 |
assume *: "\<forall>x<\<top>. \<exists>i\<in>A. x < i" |
|
577 |
show "\<Squnion>A = \<top>" |
|
578 |
proof (rule ccontr) |
|
579 |
assume "\<Squnion>A \<noteq> \<top>" |
|
63575 | 580 |
with top_greatest [of "\<Squnion>A"] have "\<Squnion>A < \<top>" |
581 |
unfolding le_less by auto |
|
582 |
with * have "\<Squnion>A < \<Squnion>A" |
|
583 |
unfolding less_Sup_iff by auto |
|
43943 | 584 |
then show False by auto |
585 |
qed |
|
586 |
qed |
|
587 |
||
63575 | 588 |
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 | 589 |
using Sup_eq_top_iff [of "f ` A"] by simp |
44041 | 590 |
|
63575 | 591 |
lemma Inf_eq_bot_iff [simp]: "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)" |
44920 | 592 |
using dual_complete_linorder |
593 |
by (rule complete_linorder.Sup_eq_top_iff) |
|
43943 | 594 |
|
63575 | 595 |
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 | 596 |
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
|
597 |
|
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
598 |
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
|
599 |
proof safe |
63575 | 600 |
fix y |
601 |
assume "x \<ge> \<Sqinter>A" "y > x" |
|
51328
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
602 |
then have "y > \<Sqinter>A" by auto |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
603 |
then show "\<exists>a\<in>A. y > a" |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
604 |
unfolding Inf_less_iff . |
d63ec23c9125
move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents:
49905
diff
changeset
|
605 |
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
|
606 |
|
68802 | 607 |
lemma INF_le_iff: "\<Sqinter>(f ` A) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)" |
56166 | 608 |
using Inf_le_iff [of "f ` A"] by simp |
609 |
||
610 |
lemma le_Sup_iff: "x \<le> \<Squnion>A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)" |
|
611 |
proof safe |
|
63575 | 612 |
fix y |
613 |
assume "x \<le> \<Squnion>A" "y < x" |
|
56166 | 614 |
then have "y < \<Squnion>A" by auto |
615 |
then show "\<exists>a\<in>A. y < a" |
|
616 |
unfolding less_Sup_iff . |
|
617 |
qed (auto elim!: allE[of _ "\<Squnion>A"] simp add: not_le[symmetric] Sup_upper) |
|
618 |
||
68802 | 619 |
lemma le_SUP_iff: "x \<le> \<Squnion>(f ` A) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)" |
56166 | 620 |
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
|
621 |
|
43940 | 622 |
end |
623 |
||
60758 | 624 |
subsection \<open>Complete lattice on @{typ bool}\<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
|
625 |
|
44024 | 626 |
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
|
627 |
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
|
628 |
|
63575 | 629 |
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
|
630 |
|
63575 | 631 |
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
|
632 |
|
63575 | 633 |
instance |
634 |
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
|
635 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
636 |
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
|
637 |
|
63575 | 638 |
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
|
639 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
640 |
|
63575 | 641 |
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
|
642 |
by auto |
a81f95693c68
simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents:
46884
diff
changeset
|
643 |
|
68802 | 644 |
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
|
645 |
by (simp add: fun_eq_iff) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
646 |
|
68802 | 647 |
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
|
648 |
by (simp add: fun_eq_iff) |
32120
53a21a5e6889
attempt for more concise setup of non-etacontracting binders
haftmann
parents:
32117
diff
changeset
|
649 |
|
63575 | 650 |
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
|
651 |
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
|
652 |
|
60758 | 653 |
subsection \<open>Complete lattice on @{typ "_ \<Rightarrow> _"}\<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
|
654 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
655 |
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
|
656 |
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
|
657 |
|
63575 | 658 |
definition "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)" |
41080 | 659 |
|
63575 | 660 |
lemma Inf_apply [simp, code]: "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)" |
41080 | 661 |
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
|
662 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
663 |
instance .. |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
664 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
665 |
end |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
666 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
667 |
instantiation "fun" :: (type, Sup) Sup |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
668 |
begin |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
669 |
|
63575 | 670 |
definition "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)" |
41080 | 671 |
|
63575 | 672 |
lemma Sup_apply [simp, code]: "(\<Squnion>A) x = (\<Squnion>f\<in>A. f x)" |
41080 | 673 |
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
|
674 |
|
57197
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
675 |
instance .. |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
676 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
677 |
end |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
678 |
|
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
679 |
instantiation "fun" :: (type, complete_lattice) complete_lattice |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
680 |
begin |
4cf607675df8
Sup/Inf on functions decoupled from complete_lattice.
nipkow
parents:
56742
diff
changeset
|
681 |
|
63575 | 682 |
instance |
683 |
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
|
684 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
685 |
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
|
686 |
|
63575 | 687 |
lemma INF_apply [simp]: "(\<Sqinter>y\<in>A. f y) x = (\<Sqinter>y\<in>A. f y x)" |
56166 | 688 |
using Inf_apply [of "f ` A"] by (simp add: comp_def) |
38705 | 689 |
|
63575 | 690 |
lemma SUP_apply [simp]: "(\<Squnion>y\<in>A. f y) x = (\<Squnion>y\<in>A. f y x)" |
56166 | 691 |
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
|
692 |
|
60758 | 693 |
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
|
694 |
|
63575 | 695 |
lemma Inf1_I: "(\<And>P. P \<in> A \<Longrightarrow> P a) \<Longrightarrow> (\<Sqinter>A) a" |
46884 | 696 |
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
|
697 |
|
63575 | 698 |
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
|
699 |
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
|
700 |
|
63575 | 701 |
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
|
702 |
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
|
703 |
|
63575 | 704 |
lemma Inf2_I: "(\<And>r. r \<in> A \<Longrightarrow> r a b) \<Longrightarrow> (\<Sqinter>A) a b" |
46884 | 705 |
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
|
706 |
|
63575 | 707 |
lemma Inf1_D: "(\<Sqinter>A) a \<Longrightarrow> P \<in> A \<Longrightarrow> P a" |
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 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
|
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 |
|
63575 | 713 |
lemma Inf2_D: "(\<Sqinter>A) a b \<Longrightarrow> r \<in> A \<Longrightarrow> r a b" |
46884 | 714 |
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
|
715 |
|
63575 | 716 |
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
|
717 |
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
|
718 |
|
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 |
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
|
720 |
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
|
721 |
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
|
722 |
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
|
723 |
|
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
|
724 |
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
|
725 |
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
|
726 |
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
|
727 |
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
|
728 |
|
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 |
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
|
730 |
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
|
731 |
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
|
732 |
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
|
733 |
|
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
|
734 |
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
|
735 |
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
|
736 |
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
|
737 |
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
|
738 |
|
63575 | 739 |
lemma Sup1_I: "P \<in> A \<Longrightarrow> P a \<Longrightarrow> (\<Squnion>A) a" |
46884 | 740 |
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
|
741 |
|
63575 | 742 |
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
|
743 |
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
|
744 |
|
63575 | 745 |
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
|
746 |
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
|
747 |
|
63575 | 748 |
lemma SUP2_I: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" |
46884 | 749 |
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
|
750 |
|
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
|
751 |
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
|
752 |
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
|
753 |
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
|
754 |
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
|
755 |
|
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 |
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
|
757 |
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
|
758 |
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
|
759 |
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
|
760 |
|
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
|
761 |
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
|
762 |
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
|
763 |
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
|
764 |
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
|
765 |
|
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
|
766 |
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
|
767 |
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
|
768 |
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
|
769 |
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
|
770 |
|
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
|
771 |
|
60758 | 772 |
subsection \<open>Complete lattice on @{typ "_ set"}\<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
|
773 |
|
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
774 |
instantiation "set" :: (type) complete_lattice |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
775 |
begin |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
776 |
|
63575 | 777 |
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
|
778 |
|
63575 | 779 |
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
|
780 |
|
63575 | 781 |
instance |
782 |
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
|
783 |
|
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
784 |
end |
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
785 |
|
60758 | 786 |
subsubsection \<open>Inter\<close> |
41082 | 787 |
|
61952 | 788 |
abbreviation Inter :: "'a set set \<Rightarrow> 'a set" ("\<Inter>_" [900] 900) |
789 |
where "\<Inter>S \<equiv> \<Sqinter>S" |
|
63575 | 790 |
|
791 |
lemma Inter_eq: "\<Inter>A = {x. \<forall>B \<in> A. x \<in> B}" |
|
41082 | 792 |
proof (rule set_eqI) |
793 |
fix x |
|
794 |
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)" |
|
795 |
by auto |
|
796 |
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
|
797 |
by (simp add: Inf_set_def image_def) |
41082 | 798 |
qed |
799 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
800 |
lemma Inter_iff [simp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)" |
41082 | 801 |
by (unfold Inter_eq) blast |
802 |
||
43741 | 803 |
lemma InterI [intro!]: "(\<And>X. X \<in> C \<Longrightarrow> A \<in> X) \<Longrightarrow> A \<in> \<Inter>C" |
41082 | 804 |
by (simp add: Inter_eq) |
805 |
||
60758 | 806 |
text \<open> |
63575 | 807 |
\<^medskip> A ``destruct'' rule -- every @{term X} in @{term C} |
43741 | 808 |
contains @{term A} as an element, but @{prop "A \<in> X"} can hold when |
61799 | 809 |
@{prop "X \<in> C"} does not! This rule is analogous to \<open>spec\<close>. |
60758 | 810 |
\<close> |
41082 | 811 |
|
43741 | 812 |
lemma InterD [elim, Pure.elim]: "A \<in> \<Inter>C \<Longrightarrow> X \<in> C \<Longrightarrow> A \<in> X" |
41082 | 813 |
by auto |
814 |
||
43741 | 815 |
lemma InterE [elim]: "A \<in> \<Inter>C \<Longrightarrow> (X \<notin> C \<Longrightarrow> R) \<Longrightarrow> (A \<in> X \<Longrightarrow> R) \<Longrightarrow> R" |
61799 | 816 |
\<comment> \<open>``Classical'' elimination rule -- does not require proving |
60758 | 817 |
@{prop "X \<in> C"}.\<close> |
63575 | 818 |
unfolding Inter_eq by blast |
41082 | 819 |
|
43741 | 820 |
lemma Inter_lower: "B \<in> A \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 821 |
by (fact Inf_lower) |
822 |
||
63575 | 823 |
lemma Inter_subset: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<Inter>A \<subseteq> B" |
43740 | 824 |
by (fact Inf_less_eq) |
41082 | 825 |
|
61952 | 826 |
lemma Inter_greatest: "(\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X) \<Longrightarrow> C \<subseteq> \<Inter>A" |
43740 | 827 |
by (fact Inf_greatest) |
41082 | 828 |
|
44067 | 829 |
lemma Inter_empty: "\<Inter>{} = UNIV" |
830 |
by (fact Inf_empty) (* already simp *) |
|
41082 | 831 |
|
44067 | 832 |
lemma Inter_UNIV: "\<Inter>UNIV = {}" |
833 |
by (fact Inf_UNIV) (* already simp *) |
|
41082 | 834 |
|
44920 | 835 |
lemma Inter_insert: "\<Inter>(insert a B) = a \<inter> \<Inter>B" |
836 |
by (fact Inf_insert) (* already simp *) |
|
41082 | 837 |
|
838 |
lemma Inter_Un_subset: "\<Inter>A \<union> \<Inter>B \<subseteq> \<Inter>(A \<inter> B)" |
|
43899 | 839 |
by (fact less_eq_Inf_inter) |
41082 | 840 |
|
841 |
lemma Inter_Un_distrib: "\<Inter>(A \<union> B) = \<Inter>A \<inter> \<Inter>B" |
|
43756 | 842 |
by (fact Inf_union_distrib) |
843 |
||
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
844 |
lemma Inter_UNIV_conv [simp]: |
43741 | 845 |
"\<Inter>A = UNIV \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
846 |
"UNIV = \<Inter>A \<longleftrightarrow> (\<forall>x\<in>A. x = UNIV)" |
|
43801 | 847 |
by (fact Inf_top_conv)+ |
41082 | 848 |
|
43741 | 849 |
lemma Inter_anti_mono: "B \<subseteq> A \<Longrightarrow> \<Inter>A \<subseteq> \<Inter>B" |
43899 | 850 |
by (fact Inf_superset_mono) |
41082 | 851 |
|
852 |
||
60758 | 853 |
subsubsection \<open>Intersections of families\<close> |
41082 | 854 |
|
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
855 |
abbreviation (input) INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" \<comment> \<open>legacy\<close> |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
856 |
where "INTER \<equiv> INFIMUM" |
41082 | 857 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
858 |
syntax (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
859 |
"_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
|
860 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3INT _:_./ _)" [0, 0, 10] 10) |
41082 | 861 |
|
862 |
syntax (latex output) |
|
62789 | 863 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
864 |
"_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
|
865 |
|
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
866 |
syntax |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
867 |
"_INTER1" :: "pttrns \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
868 |
"_INTER" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 0, 10] 10) |
41082 | 869 |
|
870 |
translations |
|
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
871 |
"\<Inter>x y. f" \<rightleftharpoons> "\<Inter>x. \<Inter>y. f" |
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
872 |
"\<Inter>x. f" \<rightleftharpoons> "\<Inter>CONST range (\<lambda>x. f)" |
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
873 |
"\<Inter>x\<in>A. f" \<rightleftharpoons> "CONST Inter ((\<lambda>x. f) ` A)" |
41082 | 874 |
|
63575 | 875 |
lemma INTER_eq: "(\<Inter>x\<in>A. B x) = {y. \<forall>x\<in>A. y \<in> B x}" |
56166 | 876 |
by (auto intro!: INF_eqI) |
41082 | 877 |
|
43817 | 878 |
lemma INT_iff [simp]: "b \<in> (\<Inter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. b \<in> B x)" |
56166 | 879 |
using Inter_iff [of _ "B ` A"] by simp |
41082 | 880 |
|
43817 | 881 |
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
|
882 |
by auto |
41082 | 883 |
|
43852 | 884 |
lemma INT_D [elim, Pure.elim]: "b \<in> (\<Inter>x\<in>A. B x) \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> B a" |
41082 | 885 |
by auto |
886 |
||
43852 | 887 |
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" |
61799 | 888 |
\<comment> \<open>"Classical" elimination -- by the Excluded Middle on @{prop "a\<in>A"}.\<close> |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
889 |
by auto |
41082 | 890 |
|
891 |
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" |
|
892 |
by blast |
|
893 |
||
894 |
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" |
|
895 |
by blast |
|
896 |
||
43817 | 897 |
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
|
898 |
by (fact INF_lower) |
41082 | 899 |
|
43817 | 900 |
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
|
901 |
by (fact INF_greatest) |
41082 | 902 |
|
44067 | 903 |
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
|
904 |
by (fact INF_empty) |
43854 | 905 |
|
43817 | 906 |
lemma INT_absorb: "k \<in> I \<Longrightarrow> A k \<inter> (\<Inter>i\<in>I. A i) = (\<Inter>i\<in>I. A i)" |
43872 | 907 |
by (fact INF_absorb) |
41082 | 908 |
|
43854 | 909 |
lemma INT_subset_iff: "B \<subseteq> (\<Inter>i\<in>I. A i) \<longleftrightarrow> (\<forall>i\<in>I. B \<subseteq> A i)" |
41082 | 910 |
by (fact le_INF_iff) |
911 |
||
912 |
lemma INT_insert [simp]: "(\<Inter>x \<in> insert a A. B x) = B a \<inter> INTER A B" |
|
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
913 |
by (fact INF_insert) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
914 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
915 |
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
|
916 |
by (fact INF_union) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
917 |
|
63575 | 918 |
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
|
919 |
by blast |
43854 | 920 |
|
41082 | 921 |
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
|
922 |
by (fact INF_constant) |
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
923 |
|
44920 | 924 |
lemma INTER_UNIV_conv: |
63575 | 925 |
"(UNIV = (\<Inter>x\<in>A. B x)) = (\<forall>x\<in>A. B x = UNIV)" |
926 |
"((\<Inter>x\<in>A. B x) = UNIV) = (\<forall>x\<in>A. B x = UNIV)" |
|
44920 | 927 |
by (fact INF_top_conv)+ (* already simp *) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
928 |
|
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
929 |
lemma INT_bool_eq: "(\<Inter>b. A b) = A True \<inter> A False" |
43873 | 930 |
by (fact INF_UNIV_bool_expand) |
43865
db18f4d0cc7d
further generalization from sets to complete lattices
haftmann
parents:
43854
diff
changeset
|
931 |
|
63575 | 932 |
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 | 933 |
\<comment> \<open>The last inclusion is POSITIVE!\<close> |
43940 | 934 |
by (fact INF_superset_mono) |
41082 | 935 |
|
936 |
lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))" |
|
937 |
by blast |
|
938 |
||
43817 | 939 |
lemma vimage_INT: "f -` (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. f -` B x)" |
41082 | 940 |
by blast |
941 |
||
942 |
||
60758 | 943 |
subsubsection \<open>Union\<close> |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
944 |
|
61952 | 945 |
abbreviation Union :: "'a set set \<Rightarrow> 'a set" ("\<Union>_" [900] 900) |
946 |
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
|
947 |
|
63575 | 948 |
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
|
949 |
proof (rule set_eqI) |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
950 |
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
|
951 |
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
|
952 |
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
|
953 |
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
|
954 |
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
|
955 |
qed |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
956 |
|
63575 | 957 |
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
|
958 |
by (unfold Union_eq) blast |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
959 |
|
63575 | 960 |
lemma UnionI [intro]: "X \<in> C \<Longrightarrow> A \<in> X \<Longrightarrow> A \<in> \<Union>C" |
61799 | 961 |
\<comment> \<open>The order of the premises presupposes that @{term C} is rigid; |
60758 | 962 |
@{term A} may be flexible.\<close> |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
963 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
964 |
|
63575 | 965 |
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
|
966 |
by auto |
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
967 |
|
43817 | 968 |
lemma Union_upper: "B \<in> A \<Longrightarrow> B \<subseteq> \<Union>A" |
43901 | 969 |
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
|
970 |
|
43817 | 971 |
lemma Union_least: "(\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C) \<Longrightarrow> \<Union>A \<subseteq> C" |
43901 | 972 |
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
|
973 |
|
44920 | 974 |
lemma Union_empty: "\<Union>{} = {}" |
975 |
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
|
976 |
|
44920 | 977 |
lemma Union_UNIV: "\<Union>UNIV = UNIV" |
978 |
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
|
979 |
|
44920 | 980 |
lemma Union_insert: "\<Union>insert a B = a \<union> \<Union>B" |
981 |
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
|
982 |
|
43817 | 983 |
lemma Union_Un_distrib [simp]: "\<Union>(A \<union> B) = \<Union>A \<union> \<Union>B" |
43901 | 984 |
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
|
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 Union_Int_subset: "\<Union>(A \<inter> B) \<subseteq> \<Union>A \<inter> \<Union>B" |
43901 | 987 |
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
|
988 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
989 |
lemma Union_empty_conv: "(\<Union>A = {}) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
44920 | 990 |
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
|
991 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
992 |
lemma empty_Union_conv: "({} = \<Union>A) \<longleftrightarrow> (\<forall>x\<in>A. x = {})" |
44920 | 993 |
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
|
994 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
995 |
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
|
996 |
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
|
997 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
998 |
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
|
999 |
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
|
1000 |
|
43817 | 1001 |
lemma Union_mono: "A \<subseteq> B \<Longrightarrow> \<Union>A \<subseteq> \<Union>B" |
43901 | 1002 |
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
|
1003 |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63365
diff
changeset
|
1004 |
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
|
1005 |
by blast |
32115
8f10fb3bb46e
swapped bootstrap order of UNION/Union and INTER/Inter in theory Set
haftmann
parents:
32082
diff
changeset
|
1006 |
|
63879
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
1007 |
lemma disjnt_inj_on_iff: |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
1008 |
"\<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
|
1009 |
apply (auto simp: disjnt_def) |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
1010 |
using inj_on_eq_iff by fastforce |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63820
diff
changeset
|
1011 |
|
63575 | 1012 |
|
60758 | 1013 |
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
|
1014 |
|
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
1015 |
abbreviation (input) UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" \<comment> \<open>legacy\<close> |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1016 |
where "UNION \<equiv> SUPREMUM" |
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 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1018 |
syntax (ASCII) |
35115 | 1019 |
"_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
|
1020 |
"_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
|
1021 |
|
3698947146b2
closer relation of sets and complete lattices; corresponding consts, defs and syntax at similar places in theory text
haftmann
parents:
32064
diff
changeset
|
1022 |
syntax (latex output) |
62789 | 1023 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(\<open>unbreakable\<close>\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
1024 |
"_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
|
1025 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1026 |
syntax |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1027 |
"_UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1028 |
"_UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 0, 10] 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1029 |
|
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
|
1030 |
translations |
68796
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
1031 |
"\<Union>x y. f" \<rightleftharpoons> "\<Union>x. \<Union>y. f" |
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
1032 |
"\<Union>x. f" \<rightleftharpoons> "\<Union>CONST range (\<lambda>x. f)" |
9ca183045102
simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents:
68795
diff
changeset
|
1033 |
"\<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
|
1034 |
|
60758 | 1035 |
text \<open> |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1036 |
Note the difference between ordinary syntax of indexed |
61799 | 1037 |
unions and intersections (e.g.\ \<open>\<Union>a\<^sub>1\<in>A\<^sub>1. B\<close>) |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61952
diff
changeset
|
1038 |
and their \LaTeX\ rendition: @{term"\<Union>a\<^sub>1\<in>A\<^sub>1. B"}. |
60758 | 1039 |
\<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
|
1040 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1041 |
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
|
1042 |
by (auto simp: disjnt_def) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1043 |
|
63575 | 1044 |
lemma UNION_eq: "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}" |
56166 | 1045 |
by (auto intro!: SUP_eqI) |
44920 | 1046 |
|
63575 | 1047 |
lemma bind_UNION [code]: "Set.bind A f = UNION A f" |
45960
e1b09bfb52f1
lattice type class instances for `set`; added code lemma for Set.bind
haftmann
parents:
45013
diff
changeset
|
1048 |
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
|
1049 |
|
63575 | 1050 |
lemma member_bind [simp]: "x \<in> Set.bind P f \<longleftrightarrow> x \<in> UNION P f " |
46036 | 1051 |
by (simp add: bind_UNION) |
1052 |
||
60585 | 1053 |
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
|
1054 |
by blast |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60172
diff
changeset
|
1055 |
|
46036 | 1056 |
lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<exists>x\<in>A. b \<in> B x)" |
56166 | 1057 |
using Union_iff [of _ "B ` A"] by simp |
11979 | 1058 |
|
43852 | 1059 |
lemma UN_I [intro]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> b \<in> (\<Union>x\<in>A. B x)" |
61799 | 1060 |
\<comment> \<open>The order of the premises presupposes that @{term A} is rigid; |
60758 | 1061 |
@{term b} may be flexible.\<close> |
11979 | 1062 |
by auto |
1063 |
||
43852 | 1064 |
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
|
1065 |
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
|
1066 |
|
43817 | 1067 |
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
|
1068 |
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
|
1069 |
|
43817 | 1070 |
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
|
1071 |
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
|
1072 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1073 |
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
|
1074 |
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
|
1075 |
|
43817 | 1076 |
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
|
1077 |
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
|
1078 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1079 |
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
|
1080 |
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
|
1081 |
|
44920 | 1082 |
lemma UN_empty2: "(\<Union>x\<in>A. {}) = {}" |
1083 |
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
|
1084 |
|
43817 | 1085 |
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
|
1086 |
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
|
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_insert [simp]: "(\<Union>x\<in>insert a A. B x) = B a \<union> UNION A B" |
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1089 |
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
|
1090 |
|
44085
a65e26f1427b
move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents:
44084
diff
changeset
|
1091 |
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
|
1092 |
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
|
1093 |
|
43967 | 1094 |
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
|
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 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1097 |
lemma UN_subset_iff: "((\<Union>i\<in>I. A i) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)" |
35629 | 1098 |
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
|
1099 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1100 |
lemma UN_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
|
1101 |
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
|
1102 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1103 |
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
|
1104 |
by blast |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67613
diff
changeset
|
1105 |
|
43944 | 1106 |
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
|
1107 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1108 |
|
44920 | 1109 |
lemma UNION_empty_conv: |
43817 | 1110 |
"{} = (\<Union>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
1111 |
"(\<Union>x\<in>A. B x) = {} \<longleftrightarrow> (\<forall>x\<in>A. B x = {})" |
|
44920 | 1112 |
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
|
1113 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1114 |
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
|
1115 |
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
|
1116 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1117 |
lemma ball_UN: "(\<forall>z \<in> UNION A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>z \<in> B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
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 |
|
43900
7162691e740b
generalization; various notation and proof tuning
haftmann
parents:
43899
diff
changeset
|
1120 |
lemma bex_UN: "(\<exists>z \<in> UNION A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>z\<in>B x. P z)" |
32135
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1121 |
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
|
1122 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1123 |
lemma Un_eq_UN: "A \<union> B = (\<Union>b. if b then A else B)" |
62390 | 1124 |
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
|
1125 |
|
43817 | 1126 |
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
|
1127 |
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
|
1128 |
|
f645b51e8e54
set 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 |
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
|
1130 |
by blast |
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1131 |
|
f645b51e8e54
set intersection and union now named inter and union; closer connection between set and lattice operations; factored out complete lattice
haftmann
parents:
32120
diff
changeset
|
1132 |
lemma UN_mono: |
43817 | 1133 |
"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
|
1134 |
(\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)" |
43940 | 1135 |
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
|
1136 |
|
43817 | 1137 |
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
|
1138 |
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
|
1139 |
|
43817 | 1140 |
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
|
1141 |
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
|
1142 |
|
43817 | 1143 |
lemma vimage_eq_UN: "f -` B = (\<Union>y\<in>B. f -` {y})" |
61799 | 1144 |
\<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
|
1145 |
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
|
1146 |
|
43817 | 1147 |
lemma image_UN: "f ` UNION A B = (\<Union>x\<in>A. f ` B x)" |
1148 |
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
|
1149 |
|
45013 | 1150 |
lemma UN_singleton [simp]: "(\<Union>x\<in>A. {x}) = A" |
1151 |
by blast |
|
1152 |
||
67399 | 1153 |
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
|
1154 |
unfolding inj_on_def by blast |
11979 | 1155 |
|
63575 | 1156 |
|
60758 | 1157 |
subsubsection \<open>Distributive laws\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1158 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1159 |
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
|
1160 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1161 |
|
44039 | 1162 |
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
|
1163 |
by blast |
44039 | 1164 |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1165 |
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
|
1166 |
by blast |
44039 | 1167 |
|
1168 |
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)" |
|
1169 |
by (rule sym) (rule INF_inf_distrib) |
|
1170 |
||
1171 |
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)" |
|
1172 |
by (rule sym) (rule SUP_sup_distrib) |
|
1173 |
||
63575 | 1174 |
lemma Int_Inter_image: "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter>(A ` C) \<inter> \<Inter>(B ` C)" (* FIXME drop *) |
56166 | 1175 |
by (simp add: INT_Int_distrib) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1176 |
|
63575 | 1177 |
lemma Un_Union_image: "(\<Union>x\<in>C. A x \<union> B x) = \<Union>(A ` C) \<union> \<Union>(B ` C)" (* FIXME drop *) |
61799 | 1178 |
\<comment> \<open>Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:\<close> |
1179 |
\<comment> \<open>Union of a family of unions\<close> |
|
56166 | 1180 |
by (simp add: UN_Un_distrib) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1181 |
|
44039 | 1182 |
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
|
1183 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1184 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1185 |
lemma Int_UN_distrib: "B \<inter> (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. B \<inter> A i)" |
61799 | 1186 |
\<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
|
1187 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1188 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1189 |
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
|
1190 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1191 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1192 |
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
|
1193 |
by blast |
44039 | 1194 |
|
1195 |
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
|
1196 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1197 |
|
67613 | 1198 |
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 | 1199 |
by (rule order_antisym) (blast intro: SUP_least SUP_upper2)+ |
1200 |
||
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1201 |
|
60758 | 1202 |
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
|
1203 |
|
63575 | 1204 |
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
|
1205 |
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
|
1206 |
|
63575 | 1207 |
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
|
1208 |
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
|
1209 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1210 |
lemma inj_on_UNION_chain: |
63575 | 1211 |
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" |
1212 |
and inj: "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
|
60585 | 1213 |
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
|
1214 |
proof - |
63575 | 1215 |
have "x = y" |
1216 |
if *: "i \<in> I" "j \<in> I" |
|
1217 |
and **: "x \<in> A i" "y \<in> A j" |
|
1218 |
and ***: "f x = f y" |
|
1219 |
for i j x y |
|
1220 |
using chain [OF *] |
|
1221 |
proof |
|
1222 |
assume "A i \<le> A j" |
|
1223 |
with ** have "x \<in> A j" by auto |
|
1224 |
with inj * ** *** show ?thesis |
|
1225 |
by (auto simp add: inj_on_def) |
|
1226 |
next |
|
1227 |
assume "A j \<le> A i" |
|
1228 |
with ** have "y \<in> A i" by auto |
|
1229 |
with inj * ** *** show ?thesis |
|
1230 |
by (auto simp add: inj_on_def) |
|
1231 |
qed |
|
1232 |
then show ?thesis |
|
1233 |
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
|
1234 |
qed |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1235 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1236 |
lemma bij_betw_UNION_chain: |
63575 | 1237 |
assumes chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" |
1238 |
and bij: "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" |
|
60585 | 1239 |
shows "bij_betw f (\<Union>i \<in> I. A i) (\<Union>i \<in> I. A' i)" |
63575 | 1240 |
unfolding bij_betw_def |
63576 | 1241 |
proof safe |
63575 | 1242 |
have "\<And>i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
1243 |
using bij bij_betw_def[of f] by auto |
|
63576 | 1244 |
then show "inj_on f (UNION I A)" |
63575 | 1245 |
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
|
1246 |
next |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1247 |
fix i x |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1248 |
assume *: "i \<in> I" "x \<in> A i" |
63576 | 1249 |
with bij have "f x \<in> A' i" |
1250 |
by (auto simp: bij_betw_def) |
|
1251 |
with * show "f x \<in> UNION I A'" by blast |
|
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1252 |
next |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1253 |
fix i x' |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1254 |
assume *: "i \<in> I" "x' \<in> A' i" |
63576 | 1255 |
with bij have "\<exists>x \<in> A i. x' = f x" |
1256 |
unfolding bij_betw_def by blast |
|
63575 | 1257 |
with * have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x" |
1258 |
by blast |
|
63576 | 1259 |
then show "x' \<in> f ` UNION I A" |
63575 | 1260 |
by blast |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1261 |
qed |
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1262 |
|
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1263 |
(*injectivity's required. Left-to-right inclusion holds even if A is empty*) |
63575 | 1264 |
lemma image_INT: "inj_on f C \<Longrightarrow> \<forall>x\<in>A. B x \<subseteq> C \<Longrightarrow> j \<in> A \<Longrightarrow> f ` (INTER A B) = (INT x:A. f ` B x)" |
1265 |
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
|
1266 |
|
63575 | 1267 |
lemma bij_image_INT: "bij f \<Longrightarrow> f ` (INTER A B) = (INT x:A. f ` B x)" |
64966
d53d7ca3303e
added inj_def (redundant, analogous to surj_def, bij_def);
wenzelm
parents:
63879
diff
changeset
|
1268 |
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
|
1269 |
|
63575 | 1270 |
lemma UNION_fun_upd: "UNION J (A(i := B)) = UNION (J - {i}) A \<union> (if i \<in> J then B else {})" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
62048
diff
changeset
|
1271 |
by (auto simp add: set_eq_iff) |
63365 | 1272 |
|
1273 |
lemma bij_betw_Pow: |
|
1274 |
assumes "bij_betw f A B" |
|
1275 |
shows "bij_betw (image f) (Pow A) (Pow B)" |
|
1276 |
proof - |
|
1277 |
from assms have "inj_on f A" |
|
1278 |
by (rule bij_betw_imp_inj_on) |
|
1279 |
then have "inj_on f (\<Union>Pow A)" |
|
1280 |
by simp |
|
1281 |
then have "inj_on (image f) (Pow A)" |
|
1282 |
by (rule inj_on_image) |
|
1283 |
then have "bij_betw (image f) (Pow A) (image f ` Pow A)" |
|
1284 |
by (rule inj_on_imp_bij_betw) |
|
1285 |
moreover from assms have "f ` A = B" |
|
1286 |
by (rule bij_betw_imp_surj_on) |
|
1287 |
then have "image f ` Pow A = Pow B" |
|
1288 |
by (rule image_Pow_surj) |
|
1289 |
ultimately show ?thesis by simp |
|
1290 |
qed |
|
1291 |
||
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
54414
diff
changeset
|
1292 |
|
60758 | 1293 |
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
|
1294 |
|
43873 | 1295 |
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
|
1296 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1297 |
|
43873 | 1298 |
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
|
1299 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1300 |
|
60758 | 1301 |
subsubsection \<open>Miniscoping and maxiscoping\<close> |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1302 |
|
63575 | 1303 |
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
|
1304 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1305 |
lemma UN_simps [simp]: |
43817 | 1306 |
"\<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
|
1307 |
"\<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 | 1308 |
"\<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
|
1309 |
"\<And>A B C. (\<Union>x\<in>C. A x \<inter> B) = ((\<Union>x\<in>C. A x) \<inter> B)" |
43852 | 1310 |
"\<And>A B C. (\<Union>x\<in>C. A \<inter> B x) = (A \<inter>(\<Union>x\<in>C. B x))" |
1311 |
"\<And>A B C. (\<Union>x\<in>C. A x - B) = ((\<Union>x\<in>C. A x) - B)" |
|
1312 |
"\<And>A B C. (\<Union>x\<in>C. A - B x) = (A - (\<Inter>x\<in>C. B x))" |
|
1313 |
"\<And>A B. (\<Union>x\<in>\<Union>A. B x) = (\<Union>y\<in>A. \<Union>x\<in>y. B x)" |
|
1314 |
"\<And>A B C. (\<Union>z\<in>UNION A B. C z) = (\<Union>x\<in>A. \<Union>z\<in>B x. C z)" |
|
43831 | 1315 |
"\<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
|
1316 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1317 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1318 |
lemma INT_simps [simp]: |
44032
cb768f4ceaf9
solving duality problem for complete_distrib_lattice; tuned
haftmann
parents:
44029
diff
changeset
|
1319 |
"\<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 | 1320 |
"\<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 | 1321 |
"\<And>A B C. (\<Inter>x\<in>C. A x - B) = (if C={} then UNIV else (\<Inter>x\<in>C. A x) - B)" |
1322 |
"\<And>A B C. (\<Inter>x\<in>C. A - B x) = (if C={} then UNIV else A - (\<Union>x\<in>C. B x))" |
|
43817 | 1323 |
"\<And>a B C. (\<Inter>x\<in>C. insert a (B x)) = insert a (\<Inter>x\<in>C. B x)" |
43852 | 1324 |
"\<And>A B C. (\<Inter>x\<in>C. A x \<union> B) = ((\<Inter>x\<in>C. A x) \<union> B)" |
1325 |
"\<And>A B C. (\<Inter>x\<in>C. A \<union> B x) = (A \<union> (\<Inter>x\<in>C. B x))" |
|
1326 |
"\<And>A B. (\<Inter>x\<in>\<Union>A. B x) = (\<Inter>y\<in>A. \<Inter>x\<in>y. B x)" |
|
1327 |
"\<And>A B C. (\<Inter>z\<in>UNION A B. C z) = (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z)" |
|
1328 |
"\<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
|
1329 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1330 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53374
diff
changeset
|
1331 |
lemma UN_ball_bex_simps [simp]: |
43852 | 1332 |
"\<And>A P. (\<forall>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<forall>y\<in>A. \<forall>x\<in>y. P x)" |
43967 | 1333 |
"\<And>A B P. (\<forall>x\<in>UNION A B. P x) = (\<forall>a\<in>A. \<forall>x\<in> B a. P x)" |
43852 | 1334 |
"\<And>A P. (\<exists>x\<in>\<Union>A. P x) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>x\<in>y. P x)" |
1335 |
"\<And>A B P. (\<exists>x\<in>UNION A B. P x) \<longleftrightarrow> (\<exists>a\<in>A. \<exists>x\<in>B a. P x)" |
|
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1336 |
by auto |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1337 |
|
43943 | 1338 |
|
63575 | 1339 |
text \<open>\<^medskip> Maxiscoping: pulling out big Unions and Intersections.\<close> |
13860 | 1340 |
|
1341 |
lemma UN_extend_simps: |
|
43817 | 1342 |
"\<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
|
1343 |
"\<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 | 1344 |
"\<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))" |
1345 |
"\<And>A B C. ((\<Union>x\<in>C. A x) \<inter> B) = (\<Union>x\<in>C. A x \<inter> B)" |
|
1346 |
"\<And>A B C. (A \<inter> (\<Union>x\<in>C. B x)) = (\<Union>x\<in>C. A \<inter> B x)" |
|
43817 | 1347 |
"\<And>A B C. ((\<Union>x\<in>C. A x) - B) = (\<Union>x\<in>C. A x - B)" |
1348 |
"\<And>A B C. (A - (\<Inter>x\<in>C. B x)) = (\<Union>x\<in>C. A - B x)" |
|
43852 | 1349 |
"\<And>A B. (\<Union>y\<in>A. \<Union>x\<in>y. B x) = (\<Union>x\<in>\<Union>A. B x)" |
1350 |
"\<And>A B C. (\<Union>x\<in>A. \<Union>z\<in>B x. C z) = (\<Union>z\<in>UNION A B. C z)" |
|
43831 | 1351 |
"\<And>A B f. (\<Union>a\<in>A. B (f a)) = (\<Union>x\<in>f`A. B x)" |
13860 | 1352 |
by auto |
1353 |
||
1354 |
lemma INT_extend_simps: |
|
43852 | 1355 |
"\<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))" |
1356 |
"\<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))" |
|
1357 |
"\<And>A B C. (\<Inter>x\<in>C. A x) - B = (if C={} then UNIV - B else (\<Inter>x\<in>C. A x - B))" |
|
1358 |
"\<And>A B C. A - (\<Union>x\<in>C. B x) = (if C={} then A else (\<Inter>x\<in>C. A - B x))" |
|
43817 | 1359 |
"\<And>a B C. insert a (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. insert a (B x))" |
43852 | 1360 |
"\<And>A B C. ((\<Inter>x\<in>C. A x) \<union> B) = (\<Inter>x\<in>C. A x \<union> B)" |
1361 |
"\<And>A B C. A \<union> (\<Inter>x\<in>C. B x) = (\<Inter>x\<in>C. A \<union> B x)" |
|
1362 |
"\<And>A B. (\<Inter>y\<in>A. \<Inter>x\<in>y. B x) = (\<Inter>x\<in>\<Union>A. B x)" |
|
1363 |
"\<And>A B C. (\<Inter>x\<in>A. \<Inter>z\<in>B x. C z) = (\<Inter>z\<in>UNION A B. C z)" |
|
1364 |
"\<And>A B f. (\<Inter>a\<in>A. B (f a)) = (\<Inter>x\<in>f`A. B x)" |
|
13860 | 1365 |
by auto |
1366 |
||
60758 | 1367 |
text \<open>Finally\<close> |
43872 | 1368 |
|
30596 | 1369 |
lemmas mem_simps = |
1370 |
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff |
|
1371 |
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff |
|
61799 | 1372 |
\<comment> \<open>Each of these has ALREADY been added \<open>[simp]\<close> above.\<close> |
21669 | 1373 |
|
11979 | 1374 |
end |