src/HOL/Library/Countable_Complete_Lattices.thy
author hoelzl
Fri Feb 19 12:25:57 2016 +0100 (2016-02-19)
changeset 62374 cb27a55d868a
parent 62373 ea7a442e9a56
child 62390 842917225d56
permissions -rw-r--r--
remove lattice syntax from countable complete lattice
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(*  Title:      HOL/Library/Countable_Complete_Lattices.thy
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    Author:     Johannes Hölzl
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*)
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section \<open>Countable Complete Lattices\<close>
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theory Countable_Complete_Lattices
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  imports Main Countable_Set
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begin
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lemma UNIV_nat_eq: "UNIV = insert 0 (range Suc)"
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  by (metis UNIV_eq_I nat.nchotomy insertCI rangeI)
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class countable_complete_lattice = lattice + Inf + Sup + bot + top +
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  assumes ccInf_lower: "countable A \<Longrightarrow> x \<in> A \<Longrightarrow> Inf A \<le> x"
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  assumes ccInf_greatest: "countable A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf A"
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  assumes ccSup_upper: "countable A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> Sup A"
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  assumes ccSup_least: "countable A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup A \<le> z"
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  assumes ccInf_empty [simp]: "Inf {} = top"
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  assumes ccSup_empty [simp]: "Sup {} = bot"
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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 "bot \<le> a" by (auto intro: ccSup_least simp only: ccSup_empty [symmetric])
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  show "a \<le> top" by (auto intro: ccInf_greatest simp only: ccInf_empty [symmetric])
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qed
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lemma ccINF_lower: "countable A \<Longrightarrow> i \<in> A \<Longrightarrow> (INF i :A. f i) \<le> f i"
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  using ccInf_lower [of "f ` A"] by simp
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lemma ccINF_greatest: "countable A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> u \<le> f i) \<Longrightarrow> u \<le> (INF i :A. f i)"
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  using ccInf_greatest [of "f ` A"] by auto
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lemma ccSUP_upper: "countable A \<Longrightarrow> i \<in> A \<Longrightarrow> f i \<le> (SUP i :A. f i)"
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  using ccSup_upper [of "f ` A"] by simp
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lemma ccSUP_least: "countable A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> f i \<le> u) \<Longrightarrow> (SUP i :A. f i) \<le> u"
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  using ccSup_least [of "f ` A"] by auto
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lemma ccInf_lower2: "countable A \<Longrightarrow> u \<in> A \<Longrightarrow> u \<le> v \<Longrightarrow> Inf A \<le> v"
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  using ccInf_lower [of A u] by auto
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lemma ccINF_lower2: "countable A \<Longrightarrow> i \<in> A \<Longrightarrow> f i \<le> u \<Longrightarrow> (INF i :A. f i) \<le> u"
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  using ccINF_lower [of A i f] by auto
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lemma ccSup_upper2: "countable A \<Longrightarrow> u \<in> A \<Longrightarrow> v \<le> u \<Longrightarrow> v \<le> Sup A"
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  using ccSup_upper [of A u] by auto
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lemma ccSUP_upper2: "countable A \<Longrightarrow> i \<in> A \<Longrightarrow> u \<le> f i \<Longrightarrow> u \<le> (SUP i :A. f i)"
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  using ccSUP_upper [of A i f] by auto
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lemma le_ccInf_iff: "countable A \<Longrightarrow> b \<le> Inf A \<longleftrightarrow> (\<forall>a\<in>A. b \<le> a)"
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  by (auto intro: ccInf_greatest dest: ccInf_lower)
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lemma le_ccINF_iff: "countable A \<Longrightarrow> u \<le> (INF i :A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<le> f i)"
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  using le_ccInf_iff [of "f ` A"] by simp
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lemma ccSup_le_iff: "countable A \<Longrightarrow> Sup A \<le> b \<longleftrightarrow> (\<forall>a\<in>A. a \<le> b)"
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  by (auto intro: ccSup_least dest: ccSup_upper)
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lemma ccSUP_le_iff: "countable A \<Longrightarrow> (SUP i :A. f i) \<le> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<le> u)"
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  using ccSup_le_iff [of "f ` A"] by simp
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lemma ccInf_insert [simp]: "countable A \<Longrightarrow> Inf (insert a A) = inf a (Inf A)"
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  by (force intro: le_infI le_infI1 le_infI2 antisym ccInf_greatest ccInf_lower)
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lemma ccINF_insert [simp]: "countable A \<Longrightarrow> (INF x:insert a A. f x) = inf (f a) (INFIMUM A f)"
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  unfolding image_insert by simp
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lemma ccSup_insert [simp]: "countable A \<Longrightarrow> Sup (insert a A) = sup a (Sup A)"
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  by (force intro: le_supI le_supI1 le_supI2 antisym ccSup_least ccSup_upper)
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lemma ccSUP_insert [simp]: "countable A \<Longrightarrow> (SUP x:insert a A. f x) = sup (f a) (SUPREMUM A f)"
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  unfolding image_insert by simp
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lemma ccINF_empty [simp]: "(INF x:{}. f x) = top"
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  unfolding image_empty by simp
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lemma ccSUP_empty [simp]: "(SUP x:{}. f x) = bot"
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  unfolding image_empty by simp
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lemma ccInf_superset_mono: "countable A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> Inf A \<le> Inf B"
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  by (auto intro: ccInf_greatest ccInf_lower countable_subset)
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lemma ccSup_subset_mono: "countable B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
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  by (auto intro: ccSup_least ccSup_upper countable_subset)
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lemma ccInf_mono:
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  assumes [intro]: "countable B" "countable A"
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  assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"
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  shows "Inf A \<le> Inf B"
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proof (rule ccInf_greatest)
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  fix b assume "b \<in> B"
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  with assms obtain a where "a \<in> A" and "a \<le> b" by blast
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  from \<open>a \<in> A\<close> have "Inf A \<le> a" by (rule ccInf_lower[rotated]) auto
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  with \<open>a \<le> b\<close> show "Inf A \<le> b" by auto
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qed auto
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lemma ccINF_mono:
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  "countable A \<Longrightarrow> countable B \<Longrightarrow> (\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> (INF n:A. f n) \<le> (INF n:B. g n)"
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  using ccInf_mono [of "g ` B" "f ` A"] by auto
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lemma ccSup_mono:
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  assumes [intro]: "countable B" "countable A"
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  assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"
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  shows "Sup A \<le> Sup B"
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proof (rule ccSup_least)
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  fix a assume "a \<in> A"
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  with assms obtain b where "b \<in> B" and "a \<le> b" by blast
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  from \<open>b \<in> B\<close> have "b \<le> Sup B" by (rule ccSup_upper[rotated]) auto
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  with \<open>a \<le> b\<close> show "a \<le> Sup B" by auto
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qed auto
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lemma ccSUP_mono:
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  "countable A \<Longrightarrow> countable B \<Longrightarrow> (\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> (SUP n:A. f n) \<le> (SUP n:B. g n)"
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  using ccSup_mono [of "g ` B" "f ` A"] by auto
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lemma ccINF_superset_mono:
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  "countable A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (INF x:A. f x) \<le> (INF x:B. g x)"
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  by (blast intro: ccINF_mono countable_subset dest: subsetD)
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lemma ccSUP_subset_mono:
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  "countable B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (SUP x:A. f x) \<le> (SUP x:B. g x)"
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  by (blast intro: ccSUP_mono countable_subset dest: subsetD)
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lemma less_eq_ccInf_inter: "countable A \<Longrightarrow> countable B \<Longrightarrow> sup (Inf A) (Inf B) \<le> Inf (A \<inter> B)"
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  by (auto intro: ccInf_greatest ccInf_lower)
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lemma ccSup_inter_less_eq: "countable A \<Longrightarrow> countable B \<Longrightarrow> Sup (A \<inter> B) \<le> inf (Sup A) (Sup B)"
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  by (auto intro: ccSup_least ccSup_upper)
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lemma ccInf_union_distrib: "countable A \<Longrightarrow> countable B \<Longrightarrow> Inf (A \<union> B) = inf (Inf A) (Inf B)"
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  by (rule antisym) (auto intro: ccInf_greatest ccInf_lower le_infI1 le_infI2)
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lemma ccINF_union:
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  "countable A \<Longrightarrow> countable B \<Longrightarrow> (INF i:A \<union> B. M i) = inf (INF i:A. M i) (INF i:B. M i)"
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  by (auto intro!: antisym ccINF_mono intro: le_infI1 le_infI2 ccINF_greatest ccINF_lower)
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lemma ccSup_union_distrib: "countable A \<Longrightarrow> countable B \<Longrightarrow> Sup (A \<union> B) = sup (Sup A) (Sup B)"
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  by (rule antisym) (auto intro: ccSup_least ccSup_upper le_supI1 le_supI2)
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lemma ccSUP_union:
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  "countable A \<Longrightarrow> countable B \<Longrightarrow> (SUP i:A \<union> B. M i) = sup (SUP i:A. M i) (SUP i:B. M i)"
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  by (auto intro!: antisym ccSUP_mono intro: le_supI1 le_supI2 ccSUP_least ccSUP_upper)
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lemma ccINF_inf_distrib: "countable A \<Longrightarrow> inf (INF a:A. f a) (INF a:A. g a) = (INF a:A. inf (f a) (g a))"
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  by (rule antisym) (rule ccINF_greatest, auto intro: le_infI1 le_infI2 ccINF_lower ccINF_mono)
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lemma ccSUP_sup_distrib: "countable A \<Longrightarrow> sup (SUP a:A. f a) (SUP a:A. g a) = (SUP a:A. sup (f a) (g a))"
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  by (rule antisym[rotated]) (rule ccSUP_least, auto intro: le_supI1 le_supI2 ccSUP_upper ccSUP_mono)
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lemma ccINF_const [simp]: "A \<noteq> {} \<Longrightarrow> (INF i :A. f) = f"
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  unfolding image_constant_conv by auto
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lemma ccSUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (SUP i :A. f) = f"
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  unfolding image_constant_conv by auto
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lemma ccINF_top [simp]: "(INF x:A. top) = top"
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  by (cases "A = {}") simp_all
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lemma ccSUP_bot [simp]: "(SUP x:A. bot) = bot"
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  by (cases "A = {}") simp_all
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lemma ccINF_commute: "countable A \<Longrightarrow> countable B \<Longrightarrow> (INF i:A. INF j:B. f i j) = (INF j:B. INF i:A. f i j)"
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  by (iprover intro: ccINF_lower ccINF_greatest order_trans antisym)
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lemma ccSUP_commute: "countable A \<Longrightarrow> countable B \<Longrightarrow> (SUP i:A. SUP j:B. f i j) = (SUP j:B. SUP i:A. f i j)"
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  by (iprover intro: ccSUP_upper ccSUP_least order_trans antisym)
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end
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context
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  fixes a :: "'a::{countable_complete_lattice, linorder}"
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begin
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lemma less_ccSup_iff: "countable S \<Longrightarrow> a < Sup S \<longleftrightarrow> (\<exists>x\<in>S. a < x)"
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  unfolding not_le [symmetric] by (subst ccSup_le_iff) auto
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lemma less_ccSUP_iff: "countable A \<Longrightarrow> a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
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  using less_ccSup_iff [of "f ` A"] by simp
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lemma ccInf_less_iff: "countable S \<Longrightarrow> Inf S < a \<longleftrightarrow> (\<exists>x\<in>S. x < a)"
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  unfolding not_le [symmetric] by (subst le_ccInf_iff) auto
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lemma ccINF_less_iff: "countable A \<Longrightarrow> (INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
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  using ccInf_less_iff [of "f ` A"] by simp
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end
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class countable_complete_distrib_lattice = countable_complete_lattice +
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  assumes sup_ccInf: "countable B \<Longrightarrow> sup a (Inf B) = (INF b:B. sup a b)"
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  assumes inf_ccSup: "countable B \<Longrightarrow> inf a (Sup B) = (SUP b:B. inf a b)"
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begin
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lemma sup_ccINF:
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  "countable B \<Longrightarrow> sup a (INF b:B. f b) = (INF b:B. sup a (f b))"
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  by (simp only: sup_ccInf image_image countable_image)
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lemma inf_ccSUP:
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  "countable B \<Longrightarrow> inf a (SUP b:B. f b) = (SUP b:B. inf a (f b))"
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  by (simp only: inf_ccSup image_image countable_image)
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subclass distrib_lattice
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proof
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  fix a b c
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  from sup_ccInf[of "{b, c}" a] have "sup a (Inf {b, c}) = (INF d:{b, c}. sup a d)"
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    by simp
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  then show "sup a (inf b c) = inf (sup a b) (sup a c)"
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    by simp
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qed
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lemma ccInf_sup:
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  "countable B \<Longrightarrow> sup (Inf B) a = (INF b:B. sup b a)"
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  by (simp add: sup_ccInf sup_commute)
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lemma ccSup_inf:
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  "countable B \<Longrightarrow> inf (Sup B) a = (SUP b:B. inf b a)"
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  by (simp add: inf_ccSup inf_commute)
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lemma ccINF_sup:
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  "countable B \<Longrightarrow> sup (INF b:B. f b) a = (INF b:B. sup (f b) a)"
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  by (simp add: sup_ccINF sup_commute)
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lemma ccSUP_inf:
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  "countable B \<Longrightarrow> inf (SUP b:B. f b) a = (SUP b:B. inf (f b) a)"
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  by (simp add: inf_ccSUP inf_commute)
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lemma ccINF_sup_distrib2:
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  "countable A \<Longrightarrow> countable B \<Longrightarrow> sup (INF a:A. f a) (INF b:B. g b) = (INF a:A. INF b:B. sup (f a) (g b))"
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  by (subst ccINF_commute) (simp_all add: sup_ccINF ccINF_sup)
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lemma ccSUP_inf_distrib2:
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  "countable A \<Longrightarrow> countable B \<Longrightarrow> inf (SUP a:A. f a) (SUP b:B. g b) = (SUP a:A. SUP b:B. inf (f a) (g b))"
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  by (subst ccSUP_commute) (simp_all add: inf_ccSUP ccSUP_inf)
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context
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  fixes f :: "'a \<Rightarrow> 'b::countable_complete_lattice"
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  assumes "mono f"
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begin
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lemma mono_ccInf:
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  "countable A \<Longrightarrow> f (Inf A) \<le> (INF x:A. f x)"
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  using \<open>mono f\<close>
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  by (auto intro!: countable_complete_lattice_class.ccINF_greatest intro: ccInf_lower dest: monoD)
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lemma mono_ccSup:
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  "countable A \<Longrightarrow> (SUP x:A. f x) \<le> f (Sup A)"
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  using \<open>mono f\<close> by (auto intro: countable_complete_lattice_class.ccSUP_least ccSup_upper dest: monoD)
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lemma mono_ccINF:
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  "countable I \<Longrightarrow> f (INF i : I. A i) \<le> (INF x : I. f (A x))"
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  by (intro countable_complete_lattice_class.ccINF_greatest monoD[OF \<open>mono f\<close>] ccINF_lower)
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lemma mono_ccSUP:
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  "countable I \<Longrightarrow> (SUP x : I. f (A x)) \<le> f (SUP i : I. A i)"
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  by (intro countable_complete_lattice_class.ccSUP_least monoD[OF \<open>mono f\<close>] ccSUP_upper)
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end
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end
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subsubsection \<open>Instances of countable complete lattices\<close>
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instance "fun" :: (type, countable_complete_lattice) countable_complete_lattice
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  by standard
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     (auto simp: le_fun_def intro!: ccSUP_upper ccSUP_least ccINF_lower ccINF_greatest)
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subclass (in complete_lattice) countable_complete_lattice
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  by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
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subclass (in complete_distrib_lattice) countable_complete_distrib_lattice
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  by standard (auto intro: sup_Inf inf_Sup)
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end