src/HOL/Conditionally_Complete_Lattices.thy
author paulson <lp15@cam.ac.uk>
Thu Dec 10 13:38:40 2015 +0000 (2015-12-10)
changeset 61824 dcbe9f756ae0
parent 61169 4de9ff3ea29a
child 62343 24106dc44def
permissions -rw-r--r--
not_leE -> not_le_imp_less and other tidying
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(*  Title:      HOL/Conditionally_Complete_Lattices.thy
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    Author:     Amine Chaieb and L C Paulson, University of Cambridge
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    Author:     Johannes Hölzl, TU München
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    Author:     Luke S. Serafin, Carnegie Mellon University
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*)
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section \<open>Conditionally-complete Lattices\<close>
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theory Conditionally_Complete_Lattices
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imports Main
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begin
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lemma (in linorder) Sup_fin_eq_Max: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X"
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  by (induct X rule: finite_ne_induct) (simp_all add: sup_max)
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lemma (in linorder) Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"
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  by (induct X rule: finite_ne_induct) (simp_all add: inf_min)
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context preorder
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begin
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definition "bdd_above A \<longleftrightarrow> (\<exists>M. \<forall>x \<in> A. x \<le> M)"
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definition "bdd_below A \<longleftrightarrow> (\<exists>m. \<forall>x \<in> A. m \<le> x)"
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lemma bdd_aboveI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> M) \<Longrightarrow> bdd_above A"
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  by (auto simp: bdd_above_def)
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lemma bdd_belowI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> x) \<Longrightarrow> bdd_below A"
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  by (auto simp: bdd_below_def)
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lemma bdd_aboveI2: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> bdd_above (f`A)"
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  by force
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lemma bdd_belowI2: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> bdd_below (f`A)"
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  by force
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lemma bdd_above_empty [simp, intro]: "bdd_above {}"
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  unfolding bdd_above_def by auto
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lemma bdd_below_empty [simp, intro]: "bdd_below {}"
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  unfolding bdd_below_def by auto
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lemma bdd_above_mono: "bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_above A"
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  by (metis (full_types) bdd_above_def order_class.le_neq_trans psubsetD)
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lemma bdd_below_mono: "bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_below A"
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  by (metis bdd_below_def order_class.le_neq_trans psubsetD)
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lemma bdd_above_Int1 [simp]: "bdd_above A \<Longrightarrow> bdd_above (A \<inter> B)"
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  using bdd_above_mono by auto
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lemma bdd_above_Int2 [simp]: "bdd_above B \<Longrightarrow> bdd_above (A \<inter> B)"
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  using bdd_above_mono by auto
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lemma bdd_below_Int1 [simp]: "bdd_below A \<Longrightarrow> bdd_below (A \<inter> B)"
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  using bdd_below_mono by auto
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lemma bdd_below_Int2 [simp]: "bdd_below B \<Longrightarrow> bdd_below (A \<inter> B)"
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  using bdd_below_mono by auto
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lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}"
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  by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
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lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}"
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  by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
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lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}"
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  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
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lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}"
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  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
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lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}"
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  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
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lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}"
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  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
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lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}"
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  by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
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lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}"
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  by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
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lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}"
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  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
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lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}"
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  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
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lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}"
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  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
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lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}"
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  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
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end
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lemma (in order_top) bdd_above_top[simp, intro!]: "bdd_above A"
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  by (rule bdd_aboveI[of _ top]) simp
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lemma (in order_bot) bdd_above_bot[simp, intro!]: "bdd_below A"
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  by (rule bdd_belowI[of _ bot]) simp
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lemma bdd_above_image_mono: "mono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_above (f`A)"
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  by (auto simp: bdd_above_def mono_def)
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lemma bdd_below_image_mono: "mono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_below (f`A)"
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  by (auto simp: bdd_below_def mono_def)
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lemma bdd_above_image_antimono: "antimono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_above (f`A)"
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  by (auto simp: bdd_above_def bdd_below_def antimono_def)
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lemma bdd_below_image_antimono: "antimono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below (f`A)"
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  by (auto simp: bdd_above_def bdd_below_def antimono_def)
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lemma
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  fixes X :: "'a::ordered_ab_group_add set"
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  shows bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X"
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    and bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X"
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  using bdd_above_image_antimono[of uminus X] bdd_below_image_antimono[of uminus "uminus`X"]
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  using bdd_below_image_antimono[of uminus X] bdd_above_image_antimono[of uminus "uminus`X"]
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  by (auto simp: antimono_def image_image)
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context lattice
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begin
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lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
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  by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)
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lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
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  by (auto simp: bdd_below_def intro: le_infI2 inf_le1)
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lemma bdd_finite [simp]:
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  assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
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  using assms by (induct rule: finite_induct, auto)
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lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)"
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proof
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  assume "bdd_above (A \<union> B)"
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  thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto
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next
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  assume "bdd_above A \<and> bdd_above B"
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  then obtain a b where "\<forall>x\<in>A. x \<le> a" "\<forall>x\<in>B. x \<le> b" unfolding bdd_above_def by auto
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  hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
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  thus "bdd_above (A \<union> B)" unfolding bdd_above_def ..
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qed
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lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)"
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proof
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  assume "bdd_below (A \<union> B)"
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  thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto
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next
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  assume "bdd_below A \<and> bdd_below B"
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  then obtain a b where "\<forall>x\<in>A. a \<le> x" "\<forall>x\<in>B. b \<le> x" unfolding bdd_below_def by auto
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  hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2)
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  thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
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qed
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lemma bdd_above_sup[simp]: "bdd_above ((\<lambda>x. sup (f x) (g x)) ` A) \<longleftrightarrow> bdd_above (f`A) \<and> bdd_above (g`A)"
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  by (auto simp: bdd_above_def intro: le_supI1 le_supI2)
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lemma bdd_below_inf[simp]: "bdd_below ((\<lambda>x. inf (f x) (g x)) ` A) \<longleftrightarrow> bdd_below (f`A) \<and> bdd_below (g`A)"
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  by (auto simp: bdd_below_def intro: le_infI1 le_infI2)
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end
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text \<open>
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To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and
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@{const Inf} in theorem names with c.
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\<close>
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class conditionally_complete_lattice = lattice + Sup + Inf +
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  assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x"
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    and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X"
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  assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X"
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    and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
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begin
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lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
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  by (metis cSup_upper order_trans)
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lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
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  by (metis cInf_lower order_trans)
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lemma cSup_mono: "B \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b \<le> a) \<Longrightarrow> Sup B \<le> Sup A"
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  by (metis cSup_least cSup_upper2)
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lemma cInf_mono: "B \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> (\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b) \<Longrightarrow> Inf A \<le> Inf B"
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  by (metis cInf_greatest cInf_lower2)
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lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
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  by (metis cSup_least cSup_upper subsetD)
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lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A"
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  by (metis cInf_greatest cInf_lower subsetD)
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lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
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  by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
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lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
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  by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto
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lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
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  by (metis order_trans cSup_upper cSup_least)
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lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
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  by (metis order_trans cInf_lower cInf_greatest)
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lemma cSup_eq_non_empty:
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  assumes 1: "X \<noteq> {}"
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  assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
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  assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
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  shows "Sup X = a"
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  by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper)
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lemma cInf_eq_non_empty:
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  assumes 1: "X \<noteq> {}"
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  assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
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  assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
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  shows "Inf X = a"
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  by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower)
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lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
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  by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)
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lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
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  by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
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lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
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  by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
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lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
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  by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
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lemma cSup_singleton [simp]: "Sup {x} = x"
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  by (intro cSup_eq_maximum) auto
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lemma cInf_singleton [simp]: "Inf {x} = x"
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  by (intro cInf_eq_minimum) auto
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lemma cSup_insert_If:  "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
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  using cSup_insert[of X] by simp
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lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
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  using cInf_insert[of X] by simp
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lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
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proof (induct X arbitrary: x rule: finite_induct)
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  case (insert x X y) then show ?case
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   254
    by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
hoelzl@51475
   255
qed simp
hoelzl@51475
   256
hoelzl@51475
   257
lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
hoelzl@51475
   258
proof (induct X arbitrary: x rule: finite_induct)
hoelzl@51475
   259
  case (insert x X y) then show ?case
hoelzl@54258
   260
    by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
hoelzl@51475
   261
qed simp
hoelzl@51475
   262
hoelzl@51475
   263
lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
hoelzl@54258
   264
  by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
hoelzl@51475
   265
hoelzl@51475
   266
lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
hoelzl@54258
   267
  by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
hoelzl@51475
   268
hoelzl@51475
   269
lemma cSup_atMost[simp]: "Sup {..x} = x"
hoelzl@51475
   270
  by (auto intro!: cSup_eq_maximum)
hoelzl@51475
   271
hoelzl@51475
   272
lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x"
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   273
  by (auto intro!: cSup_eq_maximum)
hoelzl@51475
   274
hoelzl@51475
   275
lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x"
hoelzl@51475
   276
  by (auto intro!: cSup_eq_maximum)
hoelzl@51475
   277
hoelzl@51475
   278
lemma cInf_atLeast[simp]: "Inf {x..} = x"
hoelzl@51475
   279
  by (auto intro!: cInf_eq_minimum)
hoelzl@51475
   280
hoelzl@51475
   281
lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y"
hoelzl@51475
   282
  by (auto intro!: cInf_eq_minimum)
hoelzl@51475
   283
hoelzl@51475
   284
lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
hoelzl@51475
   285
  by (auto intro!: cInf_eq_minimum)
hoelzl@51475
   286
haftmann@56218
   287
lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> INFIMUM A f \<le> f x"
haftmann@56166
   288
  using cInf_lower [of _ "f ` A"] by simp
hoelzl@54259
   289
haftmann@56218
   290
lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFIMUM A f"
haftmann@56166
   291
  using cInf_greatest [of "f ` A"] by auto
hoelzl@54259
   292
haftmann@56218
   293
lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPREMUM A f"
haftmann@56166
   294
  using cSup_upper [of _ "f ` A"] by simp
hoelzl@54259
   295
haftmann@56218
   296
lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPREMUM A f \<le> M"
haftmann@56166
   297
  using cSup_least [of "f ` A"] by auto
hoelzl@54259
   298
haftmann@56218
   299
lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFIMUM A f \<le> u"
hoelzl@54259
   300
  by (auto intro: cINF_lower assms order_trans)
hoelzl@54259
   301
haftmann@56218
   302
lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPREMUM A f"
hoelzl@54259
   303
  by (auto intro: cSUP_upper assms order_trans)
hoelzl@54259
   304
lp15@60615
   305
lemma cSUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (SUP x:A. c) = c"
hoelzl@54261
   306
  by (intro antisym cSUP_least) (auto intro: cSUP_upper)
hoelzl@54261
   307
lp15@60615
   308
lemma cINF_const [simp]: "A \<noteq> {} \<Longrightarrow> (INF x:A. c) = c"
hoelzl@54261
   309
  by (intro antisym cINF_greatest) (auto intro: cINF_lower)
hoelzl@54261
   310
haftmann@56218
   311
lemma le_cINF_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> u \<le> INFIMUM A f \<longleftrightarrow> (\<forall>x\<in>A. u \<le> f x)"
hoelzl@54259
   312
  by (metis cINF_greatest cINF_lower assms order_trans)
hoelzl@54259
   313
haftmann@56218
   314
lemma cSUP_le_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPREMUM A f \<le> u \<longleftrightarrow> (\<forall>x\<in>A. f x \<le> u)"
hoelzl@54259
   315
  by (metis cSUP_least cSUP_upper assms order_trans)
hoelzl@54259
   316
hoelzl@54263
   317
lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (INF i:A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i"
hoelzl@54263
   318
  by (metis cINF_lower less_le_trans)
hoelzl@54263
   319
hoelzl@54263
   320
lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (SUP i:A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y"
hoelzl@54263
   321
  by (metis cSUP_upper le_less_trans)
hoelzl@54263
   322
haftmann@56218
   323
lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFIMUM (insert a A) f = inf (f a) (INFIMUM A f)"
haftmann@56166
   324
  by (metis cInf_insert Inf_image_eq image_insert image_is_empty)
hoelzl@54259
   325
haftmann@56218
   326
lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPREMUM (insert a A) f = sup (f a) (SUPREMUM A f)"
haftmann@56166
   327
  by (metis cSup_insert Sup_image_eq image_insert image_is_empty)
hoelzl@54259
   328
haftmann@56218
   329
lemma cINF_mono: "B \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> (\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m) \<Longrightarrow> INFIMUM A f \<le> INFIMUM B g"
haftmann@56166
   330
  using cInf_mono [of "g ` B" "f ` A"] by auto
hoelzl@54259
   331
haftmann@56218
   332
lemma cSUP_mono: "A \<noteq> {} \<Longrightarrow> bdd_above (g ` B) \<Longrightarrow> (\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m) \<Longrightarrow> SUPREMUM A f \<le> SUPREMUM B g"
haftmann@56166
   333
  using cSup_mono [of "f ` A" "g ` B"] by auto
hoelzl@54259
   334
haftmann@56218
   335
lemma cINF_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below (g ` B) \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> g x \<le> f x) \<Longrightarrow> INFIMUM B g \<le> INFIMUM A f"
hoelzl@54259
   336
  by (rule cINF_mono) auto
hoelzl@54259
   337
haftmann@56218
   338
lemma cSUP_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above (g ` B) \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<le> g x) \<Longrightarrow> SUPREMUM A f \<le> SUPREMUM B g"
hoelzl@54259
   339
  by (rule cSUP_mono) auto
hoelzl@54259
   340
hoelzl@54259
   341
lemma less_eq_cInf_inter: "bdd_below A \<Longrightarrow> bdd_below B \<Longrightarrow> A \<inter> B \<noteq> {} \<Longrightarrow> inf (Inf A) (Inf B) \<le> Inf (A \<inter> B)"
hoelzl@54259
   342
  by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
hoelzl@54259
   343
hoelzl@54259
   344
lemma cSup_inter_less_eq: "bdd_above A \<Longrightarrow> bdd_above B \<Longrightarrow> A \<inter> B \<noteq> {} \<Longrightarrow> Sup (A \<inter> B) \<le> sup (Sup A) (Sup B) "
hoelzl@54259
   345
  by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
hoelzl@54259
   346
hoelzl@54259
   347
lemma cInf_union_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below A \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> Inf (A \<union> B) = inf (Inf A) (Inf B)"
hoelzl@54259
   348
  by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
hoelzl@54259
   349
haftmann@56218
   350
lemma cINF_union: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_below (f`B) \<Longrightarrow> INFIMUM (A \<union> B) f = inf (INFIMUM A f) (INFIMUM B f)"
haftmann@56166
   351
  using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
hoelzl@54259
   352
hoelzl@54259
   353
lemma cSup_union_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> Sup (A \<union> B) = sup (Sup A) (Sup B)"
hoelzl@54259
   354
  by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
hoelzl@54259
   355
haftmann@56218
   356
lemma cSUP_union: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_above (f`B) \<Longrightarrow> SUPREMUM (A \<union> B) f = sup (SUPREMUM A f) (SUPREMUM B f)"
haftmann@56166
   357
  using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
hoelzl@54259
   358
haftmann@56218
   359
lemma cINF_inf_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> bdd_below (g`A) \<Longrightarrow> inf (INFIMUM A f) (INFIMUM A g) = (INF a:A. inf (f a) (g a))"
hoelzl@54259
   360
  by (intro antisym le_infI cINF_greatest cINF_lower2)
hoelzl@54259
   361
     (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
hoelzl@54259
   362
haftmann@56218
   363
lemma SUP_sup_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> bdd_above (g`A) \<Longrightarrow> sup (SUPREMUM A f) (SUPREMUM A g) = (SUP a:A. sup (f a) (g a))"
hoelzl@54259
   364
  by (intro antisym le_supI cSUP_least cSUP_upper2)
hoelzl@54259
   365
     (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
hoelzl@54259
   366
hoelzl@57447
   367
lemma cInf_le_cSup:
hoelzl@57447
   368
  "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<le> Sup A"
hoelzl@57447
   369
  by (auto intro!: cSup_upper2[of "SOME a. a \<in> A"] intro: someI cInf_lower)
hoelzl@57447
   370
paulson@33269
   371
end
paulson@33269
   372
hoelzl@51773
   373
instance complete_lattice \<subseteq> conditionally_complete_lattice
wenzelm@61169
   374
  by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
hoelzl@51475
   375
hoelzl@51475
   376
lemma cSup_eq:
hoelzl@51773
   377
  fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
hoelzl@51475
   378
  assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
hoelzl@51475
   379
  assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
hoelzl@51475
   380
  shows "Sup X = a"
hoelzl@51475
   381
proof cases
hoelzl@51475
   382
  assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
hoelzl@51475
   383
qed (intro cSup_eq_non_empty assms)
hoelzl@51475
   384
hoelzl@51475
   385
lemma cInf_eq:
hoelzl@51773
   386
  fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
hoelzl@51475
   387
  assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
hoelzl@51475
   388
  assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
hoelzl@51475
   389
  shows "Inf X = a"
hoelzl@51475
   390
proof cases
hoelzl@51475
   391
  assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
hoelzl@51475
   392
qed (intro cInf_eq_non_empty assms)
hoelzl@51475
   393
hoelzl@51773
   394
class conditionally_complete_linorder = conditionally_complete_lattice + linorder
paulson@33269
   395
begin
hoelzl@51475
   396
hoelzl@51475
   397
lemma less_cSup_iff : (*REAL_SUP_LE in HOL4*)
hoelzl@54258
   398
  "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
hoelzl@51475
   399
  by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
hoelzl@51475
   400
hoelzl@54258
   401
lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
hoelzl@51475
   402
  by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
hoelzl@51475
   403
hoelzl@54263
   404
lemma cINF_less_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> (INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
haftmann@56166
   405
  using cInf_less_iff[of "f`A"] by auto
hoelzl@54263
   406
hoelzl@54263
   407
lemma less_cSUP_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
haftmann@56166
   408
  using less_cSup_iff[of "f`A"] by auto
hoelzl@54263
   409
hoelzl@51475
   410
lemma less_cSupE:
hoelzl@51475
   411
  assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x"
hoelzl@51475
   412
  by (metis cSup_least assms not_le that)
hoelzl@51475
   413
hoelzl@51518
   414
lemma less_cSupD:
hoelzl@51518
   415
  "X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
lp15@61824
   416
  by (metis less_cSup_iff not_le_imp_less bdd_above_def)
hoelzl@51518
   417
hoelzl@51518
   418
lemma cInf_lessD:
hoelzl@51518
   419
  "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
lp15@61824
   420
  by (metis cInf_less_iff not_le_imp_less bdd_below_def)
hoelzl@51518
   421
hoelzl@51475
   422
lemma complete_interval:
hoelzl@51475
   423
  assumes "a < b" and "P a" and "\<not> P b"
hoelzl@51475
   424
  shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and>
hoelzl@51475
   425
             (\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
hoelzl@51475
   426
proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
hoelzl@51475
   427
  show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
hoelzl@54258
   428
    by (rule cSup_upper, auto simp: bdd_above_def)
wenzelm@60758
   429
       (metis \<open>a < b\<close> \<open>\<not> P b\<close> linear less_le)
hoelzl@51475
   430
next
hoelzl@51475
   431
  show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
hoelzl@51475
   432
    apply (rule cSup_least) 
hoelzl@51475
   433
    apply auto
hoelzl@51475
   434
    apply (metis less_le_not_le)
wenzelm@60758
   435
    apply (metis \<open>a<b\<close> \<open>~ P b\<close> linear less_le)
hoelzl@51475
   436
    done
hoelzl@51475
   437
next
hoelzl@51475
   438
  fix x
hoelzl@51475
   439
  assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
hoelzl@51475
   440
  show "P x"
hoelzl@51475
   441
    apply (rule less_cSupE [OF lt], auto)
hoelzl@51475
   442
    apply (metis less_le_not_le)
hoelzl@51475
   443
    apply (metis x) 
hoelzl@51475
   444
    done
hoelzl@51475
   445
next
hoelzl@51475
   446
  fix d
hoelzl@51475
   447
    assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
hoelzl@51475
   448
    thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
hoelzl@54258
   449
      by (rule_tac cSup_upper, auto simp: bdd_above_def)
wenzelm@60758
   450
         (metis \<open>a<b\<close> \<open>~ P b\<close> linear less_le)
hoelzl@51475
   451
qed
hoelzl@51475
   452
hoelzl@51475
   453
end
hoelzl@51475
   454
hoelzl@60172
   455
instance complete_linorder < conditionally_complete_linorder
hoelzl@60172
   456
  ..
hoelzl@60172
   457
hoelzl@54259
   458
lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
hoelzl@54259
   459
  using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp
hoelzl@51775
   460
hoelzl@54259
   461
lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
hoelzl@54259
   462
  using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp
hoelzl@51775
   463
hoelzl@54257
   464
lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
hoelzl@51475
   465
  by (auto intro!: cSup_eq_non_empty intro: dense_le)
hoelzl@51475
   466
hoelzl@57447
   467
lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
hoelzl@57447
   468
  by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
hoelzl@51475
   469
hoelzl@57447
   470
lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
hoelzl@57447
   471
  by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
hoelzl@51475
   472
hoelzl@54257
   473
lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
hoelzl@57447
   474
  by (auto intro!: cInf_eq_non_empty intro: dense_ge)
hoelzl@51475
   475
hoelzl@57447
   476
lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
hoelzl@57447
   477
  by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
hoelzl@51475
   478
hoelzl@57447
   479
lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
hoelzl@57447
   480
  by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
hoelzl@51475
   481
hoelzl@54259
   482
class linear_continuum = conditionally_complete_linorder + dense_linorder +
hoelzl@54259
   483
  assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
hoelzl@54259
   484
begin
hoelzl@54259
   485
hoelzl@54259
   486
lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
hoelzl@54259
   487
  by (metis UNIV_not_singleton neq_iff)
hoelzl@54259
   488
paulson@33269
   489
end
hoelzl@54259
   490
hoelzl@54281
   491
instantiation nat :: conditionally_complete_linorder
hoelzl@54281
   492
begin
hoelzl@54281
   493
hoelzl@54281
   494
definition "Sup (X::nat set) = Max X"
hoelzl@54281
   495
definition "Inf (X::nat set) = (LEAST n. n \<in> X)"
hoelzl@54281
   496
hoelzl@54281
   497
lemma bdd_above_nat: "bdd_above X \<longleftrightarrow> finite (X::nat set)"
hoelzl@54281
   498
proof
hoelzl@54281
   499
  assume "bdd_above X"
hoelzl@54281
   500
  then obtain z where "X \<subseteq> {.. z}"
hoelzl@54281
   501
    by (auto simp: bdd_above_def)
hoelzl@54281
   502
  then show "finite X"
hoelzl@54281
   503
    by (rule finite_subset) simp
hoelzl@54281
   504
qed simp
hoelzl@54281
   505
hoelzl@54281
   506
instance
hoelzl@54281
   507
proof
hoelzl@54281
   508
  fix x :: nat and X :: "nat set"
hoelzl@54281
   509
  { assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
hoelzl@54281
   510
      by (simp add: Inf_nat_def Least_le) }
hoelzl@54281
   511
  { assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y" then show "x \<le> Inf X"
hoelzl@54281
   512
      unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex) }
hoelzl@54281
   513
  { assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
hoelzl@54281
   514
      by (simp add: Sup_nat_def bdd_above_nat) }
hoelzl@54281
   515
  { assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x" 
hoelzl@54281
   516
    moreover then have "bdd_above X"
hoelzl@54281
   517
      by (auto simp: bdd_above_def)
hoelzl@54281
   518
    ultimately show "Sup X \<le> x"
hoelzl@54281
   519
      by (simp add: Sup_nat_def bdd_above_nat) }
hoelzl@54281
   520
qed
hoelzl@54259
   521
end
hoelzl@54281
   522
hoelzl@54281
   523
instantiation int :: conditionally_complete_linorder
hoelzl@54281
   524
begin
hoelzl@54281
   525
hoelzl@54281
   526
definition "Sup (X::int set) = (THE x. x \<in> X \<and> (\<forall>y\<in>X. y \<le> x))"
hoelzl@54281
   527
definition "Inf (X::int set) = - (Sup (uminus ` X))"
hoelzl@54281
   528
hoelzl@54281
   529
instance
hoelzl@54281
   530
proof
hoelzl@54281
   531
  { fix x :: int and X :: "int set" assume "X \<noteq> {}" "bdd_above X"
hoelzl@54281
   532
    then obtain x y where "X \<subseteq> {..y}" "x \<in> X"
hoelzl@54281
   533
      by (auto simp: bdd_above_def)
hoelzl@54281
   534
    then have *: "finite (X \<inter> {x..y})" "X \<inter> {x..y} \<noteq> {}" and "x \<le> y"
hoelzl@54281
   535
      by (auto simp: subset_eq)
hoelzl@54281
   536
    have "\<exists>!x\<in>X. (\<forall>y\<in>X. y \<le> x)"
hoelzl@54281
   537
    proof
hoelzl@54281
   538
      { fix z assume "z \<in> X"
hoelzl@54281
   539
        have "z \<le> Max (X \<inter> {x..y})"
hoelzl@54281
   540
        proof cases
wenzelm@60758
   541
          assume "x \<le> z" with \<open>z \<in> X\<close> \<open>X \<subseteq> {..y}\<close> *(1) show ?thesis
hoelzl@54281
   542
            by (auto intro!: Max_ge)
hoelzl@54281
   543
        next
hoelzl@54281
   544
          assume "\<not> x \<le> z"
hoelzl@54281
   545
          then have "z < x" by simp
hoelzl@54281
   546
          also have "x \<le> Max (X \<inter> {x..y})"
wenzelm@60758
   547
            using \<open>x \<in> X\<close> *(1) \<open>x \<le> y\<close> by (intro Max_ge) auto
hoelzl@54281
   548
          finally show ?thesis by simp
hoelzl@54281
   549
        qed }
hoelzl@54281
   550
      note le = this
hoelzl@54281
   551
      with Max_in[OF *] show ex: "Max (X \<inter> {x..y}) \<in> X \<and> (\<forall>z\<in>X. z \<le> Max (X \<inter> {x..y}))" by auto
hoelzl@54281
   552
hoelzl@54281
   553
      fix z assume *: "z \<in> X \<and> (\<forall>y\<in>X. y \<le> z)"
hoelzl@54281
   554
      with le have "z \<le> Max (X \<inter> {x..y})"
hoelzl@54281
   555
        by auto
hoelzl@54281
   556
      moreover have "Max (X \<inter> {x..y}) \<le> z"
hoelzl@54281
   557
        using * ex by auto
hoelzl@54281
   558
      ultimately show "z = Max (X \<inter> {x..y})"
hoelzl@54281
   559
        by auto
hoelzl@54281
   560
    qed
hoelzl@54281
   561
    then have "Sup X \<in> X \<and> (\<forall>y\<in>X. y \<le> Sup X)"
hoelzl@54281
   562
      unfolding Sup_int_def by (rule theI') }
hoelzl@54281
   563
  note Sup_int = this
hoelzl@54281
   564
hoelzl@54281
   565
  { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
hoelzl@54281
   566
      using Sup_int[of X] by auto }
hoelzl@54281
   567
  note le_Sup = this
hoelzl@54281
   568
  { fix x :: int and X :: "int set" assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x" then show "Sup X \<le> x"
hoelzl@54281
   569
      using Sup_int[of X] by (auto simp: bdd_above_def) }
hoelzl@54281
   570
  note Sup_le = this
hoelzl@54281
   571
hoelzl@54281
   572
  { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
hoelzl@54281
   573
      using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) }
hoelzl@54281
   574
  { fix x :: int and X :: "int set" assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y" then show "x \<le> Inf X"
hoelzl@54281
   575
      using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) }
hoelzl@54281
   576
qed
hoelzl@54281
   577
end
hoelzl@54281
   578
hoelzl@57275
   579
lemma interval_cases:
hoelzl@57275
   580
  fixes S :: "'a :: conditionally_complete_linorder set"
hoelzl@57275
   581
  assumes ivl: "\<And>a b x. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> x \<in> S"
hoelzl@57275
   582
  shows "\<exists>a b. S = {} \<or>
hoelzl@57275
   583
    S = UNIV \<or>
hoelzl@57275
   584
    S = {..<b} \<or>
hoelzl@57275
   585
    S = {..b} \<or>
hoelzl@57275
   586
    S = {a<..} \<or>
hoelzl@57275
   587
    S = {a..} \<or>
hoelzl@57275
   588
    S = {a<..<b} \<or>
hoelzl@57275
   589
    S = {a<..b} \<or>
hoelzl@57275
   590
    S = {a..<b} \<or>
hoelzl@57275
   591
    S = {a..b}"
hoelzl@57275
   592
proof -
hoelzl@57275
   593
  def lower \<equiv> "{x. \<exists>s\<in>S. s \<le> x}" and upper \<equiv> "{x. \<exists>s\<in>S. x \<le> s}"
hoelzl@57275
   594
  with ivl have "S = lower \<inter> upper"
hoelzl@57275
   595
    by auto
hoelzl@57275
   596
  moreover 
hoelzl@57275
   597
  have "\<exists>a. upper = UNIV \<or> upper = {} \<or> upper = {.. a} \<or> upper = {..< a}"
hoelzl@57275
   598
  proof cases
hoelzl@57275
   599
    assume *: "bdd_above S \<and> S \<noteq> {}"
hoelzl@57275
   600
    from * have "upper \<subseteq> {.. Sup S}"
hoelzl@57275
   601
      by (auto simp: upper_def intro: cSup_upper2)
hoelzl@57275
   602
    moreover from * have "{..< Sup S} \<subseteq> upper"
hoelzl@57275
   603
      by (force simp add: less_cSup_iff upper_def subset_eq Ball_def)
hoelzl@57275
   604
    ultimately have "upper = {.. Sup S} \<or> upper = {..< Sup S}"
hoelzl@57275
   605
      unfolding ivl_disj_un(2)[symmetric] by auto
hoelzl@57275
   606
    then show ?thesis by auto
hoelzl@57275
   607
  next
hoelzl@57275
   608
    assume "\<not> (bdd_above S \<and> S \<noteq> {})"
hoelzl@57275
   609
    then have "upper = UNIV \<or> upper = {}"
hoelzl@57275
   610
      by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le)
hoelzl@57275
   611
    then show ?thesis
hoelzl@57275
   612
      by auto
hoelzl@57275
   613
  qed
hoelzl@57275
   614
  moreover
hoelzl@57275
   615
  have "\<exists>b. lower = UNIV \<or> lower = {} \<or> lower = {b ..} \<or> lower = {b <..}"
hoelzl@57275
   616
  proof cases
hoelzl@57275
   617
    assume *: "bdd_below S \<and> S \<noteq> {}"
hoelzl@57275
   618
    from * have "lower \<subseteq> {Inf S ..}"
hoelzl@57275
   619
      by (auto simp: lower_def intro: cInf_lower2)
hoelzl@57275
   620
    moreover from * have "{Inf S <..} \<subseteq> lower"
hoelzl@57275
   621
      by (force simp add: cInf_less_iff lower_def subset_eq Ball_def)
hoelzl@57275
   622
    ultimately have "lower = {Inf S ..} \<or> lower = {Inf S <..}"
hoelzl@57275
   623
      unfolding ivl_disj_un(1)[symmetric] by auto
hoelzl@57275
   624
    then show ?thesis by auto
hoelzl@57275
   625
  next
hoelzl@57275
   626
    assume "\<not> (bdd_below S \<and> S \<noteq> {})"
hoelzl@57275
   627
    then have "lower = UNIV \<or> lower = {}"
hoelzl@57275
   628
      by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le)
hoelzl@57275
   629
    then show ?thesis
hoelzl@57275
   630
      by auto
hoelzl@57275
   631
  qed
hoelzl@57275
   632
  ultimately show ?thesis
hoelzl@57275
   633
    unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def
hoelzl@57275
   634
    by (elim exE disjE) auto
hoelzl@57275
   635
qed
hoelzl@57275
   636
lp15@60615
   637
lemma cSUP_eq_cINF_D:
lp15@60615
   638
  fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
lp15@60615
   639
  assumes eq: "(SUP x:A. f x) = (INF x:A. f x)"
lp15@60615
   640
     and bdd: "bdd_above (f ` A)" "bdd_below (f ` A)"
lp15@60615
   641
     and a: "a \<in> A"
lp15@60615
   642
  shows "f a = (INF x:A. f x)"
lp15@60615
   643
apply (rule antisym)
lp15@60615
   644
using a bdd
lp15@60615
   645
apply (auto simp: cINF_lower)
lp15@60615
   646
apply (metis eq cSUP_upper)
lp15@60615
   647
done 
lp15@60615
   648
hoelzl@54281
   649
end