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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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header {* Conditionally-complete Lattices *}
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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_uminus[simp]:
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fixes X :: "'a::ordered_ab_group_add set"
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shows "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X"
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by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
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lemma bdd_below_uminus[simp]:
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fixes X :: "'a::ordered_ab_group_add set"
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shows"bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X"
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by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
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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 {*
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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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*}
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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}"
|
hoelzl@55710
|
221 |
by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
|
hoelzl@52655
|
222 |
|
hoelzl@55710
|
223 |
lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
|
hoelzl@55710
|
224 |
by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
|
hoelzl@52612
|
225 |
|
hoelzl@55710
|
226 |
lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
|
hoelzl@55710
|
227 |
by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
|
hoelzl@52612
|
228 |
|
hoelzl@55711
|
229 |
lemma cSup_singleton [simp]: "Sup {x} = x"
|
hoelzl@55711
|
230 |
by (intro cSup_eq_maximum) auto
|
hoelzl@55711
|
231 |
|
hoelzl@55711
|
232 |
lemma cInf_singleton [simp]: "Inf {x} = x"
|
hoelzl@55711
|
233 |
by (intro cInf_eq_minimum) auto
|
hoelzl@55711
|
234 |
|
hoelzl@55710
|
235 |
lemma cSup_insert_If: "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
|
hoelzl@55710
|
236 |
using cSup_insert[of X] by simp
|
hoelzl@52612
|
237 |
|
hoelzl@55711
|
238 |
lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
|
hoelzl@55710
|
239 |
using cInf_insert[of X] by simp
|
hoelzl@52612
|
240 |
|
hoelzl@52612
|
241 |
lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
|
hoelzl@52612
|
242 |
proof (induct X arbitrary: x rule: finite_induct)
|
hoelzl@52612
|
243 |
case (insert x X y) then show ?case
|
hoelzl@55710
|
244 |
by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
|
hoelzl@52612
|
245 |
qed simp
|
hoelzl@52612
|
246 |
|
hoelzl@52612
|
247 |
lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
|
hoelzl@52612
|
248 |
proof (induct X arbitrary: x rule: finite_induct)
|
hoelzl@52612
|
249 |
case (insert x X y) then show ?case
|
hoelzl@55710
|
250 |
by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
|
hoelzl@52612
|
251 |
qed simp
|
hoelzl@52612
|
252 |
|
hoelzl@52612
|
253 |
lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
|
hoelzl@55710
|
254 |
by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
|
hoelzl@52612
|
255 |
|
hoelzl@52612
|
256 |
lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
|
hoelzl@55710
|
257 |
by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
|
hoelzl@52612
|
258 |
|
hoelzl@52612
|
259 |
lemma cSup_atMost[simp]: "Sup {..x} = x"
|
hoelzl@52612
|
260 |
by (auto intro!: cSup_eq_maximum)
|
hoelzl@52612
|
261 |
|
hoelzl@52612
|
262 |
lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x"
|
hoelzl@52612
|
263 |
by (auto intro!: cSup_eq_maximum)
|
hoelzl@52612
|
264 |
|
hoelzl@52612
|
265 |
lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x"
|
hoelzl@52612
|
266 |
by (auto intro!: cSup_eq_maximum)
|
hoelzl@52612
|
267 |
|
hoelzl@52612
|
268 |
lemma cInf_atLeast[simp]: "Inf {x..} = x"
|
hoelzl@52612
|
269 |
by (auto intro!: cInf_eq_minimum)
|
hoelzl@52612
|
270 |
|
hoelzl@52612
|
271 |
lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y"
|
hoelzl@52612
|
272 |
by (auto intro!: cInf_eq_minimum)
|
hoelzl@52612
|
273 |
|
hoelzl@52612
|
274 |
lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
|
hoelzl@52612
|
275 |
by (auto intro!: cInf_eq_minimum)
|
hoelzl@52612
|
276 |
|
hoelzl@55711
|
277 |
lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> INFI A f \<le> f x"
|
hoelzl@55711
|
278 |
unfolding INF_def by (rule cInf_lower) auto
|
hoelzl@55711
|
279 |
|
hoelzl@55711
|
280 |
lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFI A f"
|
hoelzl@55711
|
281 |
unfolding INF_def by (rule cInf_greatest) auto
|
hoelzl@55711
|
282 |
|
hoelzl@55711
|
283 |
lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPR A f"
|
hoelzl@55711
|
284 |
unfolding SUP_def by (rule cSup_upper) auto
|
hoelzl@55711
|
285 |
|
hoelzl@55711
|
286 |
lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPR A f \<le> M"
|
hoelzl@55711
|
287 |
unfolding SUP_def by (rule cSup_least) auto
|
hoelzl@55711
|
288 |
|
hoelzl@55711
|
289 |
lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFI A f \<le> u"
|
hoelzl@55711
|
290 |
by (auto intro: cINF_lower assms order_trans)
|
hoelzl@55711
|
291 |
|
hoelzl@55711
|
292 |
lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPR A f"
|
hoelzl@55711
|
293 |
by (auto intro: cSUP_upper assms order_trans)
|
hoelzl@55711
|
294 |
|
hoelzl@55713
|
295 |
lemma cSUP_const: "A \<noteq> {} \<Longrightarrow> (SUP x:A. c) = c"
|
hoelzl@55713
|
296 |
by (intro antisym cSUP_least) (auto intro: cSUP_upper)
|
hoelzl@55713
|
297 |
|
hoelzl@55713
|
298 |
lemma cINF_const: "A \<noteq> {} \<Longrightarrow> (INF x:A. c) = c"
|
hoelzl@55713
|
299 |
by (intro antisym cINF_greatest) (auto intro: cINF_lower)
|
hoelzl@55713
|
300 |
|
hoelzl@55711
|
301 |
lemma le_cINF_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> u \<le> INFI A f \<longleftrightarrow> (\<forall>x\<in>A. u \<le> f x)"
|
hoelzl@55711
|
302 |
by (metis cINF_greatest cINF_lower assms order_trans)
|
hoelzl@55711
|
303 |
|
hoelzl@55711
|
304 |
lemma cSUP_le_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPR A f \<le> u \<longleftrightarrow> (\<forall>x\<in>A. f x \<le> u)"
|
hoelzl@55711
|
305 |
by (metis cSUP_least cSUP_upper assms order_trans)
|
hoelzl@55711
|
306 |
|
hoelzl@55715
|
307 |
lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (INF i:A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i"
|
hoelzl@55715
|
308 |
by (metis cINF_lower less_le_trans)
|
hoelzl@55715
|
309 |
|
hoelzl@55715
|
310 |
lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (SUP i:A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y"
|
hoelzl@55715
|
311 |
by (metis cSUP_upper le_less_trans)
|
hoelzl@55715
|
312 |
|
hoelzl@55711
|
313 |
lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFI (insert a A) f = inf (f a) (INFI A f)"
|
hoelzl@55711
|
314 |
by (metis INF_def cInf_insert assms empty_is_image image_insert)
|
hoelzl@55711
|
315 |
|
hoelzl@55711
|
316 |
lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPR (insert a A) f = sup (f a) (SUPR A f)"
|
hoelzl@55711
|
317 |
by (metis SUP_def cSup_insert assms empty_is_image image_insert)
|
hoelzl@55711
|
318 |
|
hoelzl@55711
|
319 |
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> INFI A f \<le> INFI B g"
|
hoelzl@55711
|
320 |
unfolding INF_def by (auto intro: cInf_mono)
|
hoelzl@55711
|
321 |
|
hoelzl@55711
|
322 |
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> SUPR A f \<le> SUPR B g"
|
hoelzl@55711
|
323 |
unfolding SUP_def by (auto intro: cSup_mono)
|
hoelzl@55711
|
324 |
|
hoelzl@55711
|
325 |
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> INFI B g \<le> INFI A f"
|
hoelzl@55711
|
326 |
by (rule cINF_mono) auto
|
hoelzl@55711
|
327 |
|
hoelzl@55711
|
328 |
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> SUPR A f \<le> SUPR B g"
|
hoelzl@55711
|
329 |
by (rule cSUP_mono) auto
|
hoelzl@55711
|
330 |
|
hoelzl@55711
|
331 |
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@55711
|
332 |
by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
|
hoelzl@55711
|
333 |
|
hoelzl@55711
|
334 |
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@55711
|
335 |
by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
|
hoelzl@55711
|
336 |
|
hoelzl@55711
|
337 |
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@55711
|
338 |
by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
|
hoelzl@55711
|
339 |
|
hoelzl@55711
|
340 |
lemma cINF_union: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_below (f`B) \<Longrightarrow> INFI (A \<union> B) f = inf (INFI A f) (INFI B f)"
|
hoelzl@55711
|
341 |
unfolding INF_def using assms by (auto simp add: image_Un intro: cInf_union_distrib)
|
hoelzl@55711
|
342 |
|
hoelzl@55711
|
343 |
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@55711
|
344 |
by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
|
hoelzl@55711
|
345 |
|
hoelzl@55711
|
346 |
lemma cSUP_union: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_above (f`B) \<Longrightarrow> SUPR (A \<union> B) f = sup (SUPR A f) (SUPR B f)"
|
hoelzl@55711
|
347 |
unfolding SUP_def by (auto simp add: image_Un intro: cSup_union_distrib)
|
hoelzl@55711
|
348 |
|
hoelzl@55711
|
349 |
lemma cINF_inf_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> bdd_below (g`A) \<Longrightarrow> inf (INFI A f) (INFI A g) = (INF a:A. inf (f a) (g a))"
|
hoelzl@55711
|
350 |
by (intro antisym le_infI cINF_greatest cINF_lower2)
|
hoelzl@55711
|
351 |
(auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
|
hoelzl@55711
|
352 |
|
hoelzl@55711
|
353 |
lemma SUP_sup_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> bdd_above (g`A) \<Longrightarrow> sup (SUPR A f) (SUPR A g) = (SUP a:A. sup (f a) (g a))"
|
hoelzl@55711
|
354 |
by (intro antisym le_supI cSUP_least cSUP_upper2)
|
hoelzl@55711
|
355 |
(auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
|
hoelzl@55711
|
356 |
|
paulson@33269
|
357 |
end
|
paulson@33269
|
358 |
|
hoelzl@52910
|
359 |
instance complete_lattice \<subseteq> conditionally_complete_lattice
|
hoelzl@52612
|
360 |
by default (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
|
hoelzl@52612
|
361 |
|
hoelzl@52612
|
362 |
lemma cSup_eq:
|
hoelzl@52910
|
363 |
fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
|
hoelzl@52612
|
364 |
assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
|
hoelzl@52612
|
365 |
assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
|
hoelzl@52612
|
366 |
shows "Sup X = a"
|
hoelzl@52612
|
367 |
proof cases
|
hoelzl@52612
|
368 |
assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
|
hoelzl@52612
|
369 |
qed (intro cSup_eq_non_empty assms)
|
hoelzl@52612
|
370 |
|
hoelzl@52612
|
371 |
lemma cInf_eq:
|
hoelzl@52910
|
372 |
fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
|
hoelzl@52612
|
373 |
assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
|
hoelzl@52612
|
374 |
assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
|
hoelzl@52612
|
375 |
shows "Inf X = a"
|
hoelzl@52612
|
376 |
proof cases
|
hoelzl@52612
|
377 |
assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
|
hoelzl@52612
|
378 |
qed (intro cInf_eq_non_empty assms)
|
hoelzl@52612
|
379 |
|
hoelzl@52910
|
380 |
class conditionally_complete_linorder = conditionally_complete_lattice + linorder
|
paulson@33269
|
381 |
begin
|
hoelzl@52612
|
382 |
|
hoelzl@52612
|
383 |
lemma less_cSup_iff : (*REAL_SUP_LE in HOL4*)
|
hoelzl@55710
|
384 |
"X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
|
hoelzl@52612
|
385 |
by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
|
hoelzl@52612
|
386 |
|
hoelzl@55710
|
387 |
lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
|
hoelzl@52612
|
388 |
by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
|
hoelzl@52612
|
389 |
|
hoelzl@55715
|
390 |
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)"
|
hoelzl@55715
|
391 |
unfolding INF_def using cInf_less_iff[of "f`A"] by auto
|
hoelzl@55715
|
392 |
|
hoelzl@55715
|
393 |
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)"
|
hoelzl@55715
|
394 |
unfolding SUP_def using less_cSup_iff[of "f`A"] by auto
|
hoelzl@55715
|
395 |
|
hoelzl@52612
|
396 |
lemma less_cSupE:
|
hoelzl@52612
|
397 |
assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x"
|
hoelzl@52612
|
398 |
by (metis cSup_least assms not_le that)
|
hoelzl@52612
|
399 |
|
hoelzl@52655
|
400 |
lemma less_cSupD:
|
hoelzl@52655
|
401 |
"X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
|
hoelzl@55710
|
402 |
by (metis less_cSup_iff not_leE bdd_above_def)
|
hoelzl@52655
|
403 |
|
hoelzl@52655
|
404 |
lemma cInf_lessD:
|
hoelzl@52655
|
405 |
"X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
|
hoelzl@55710
|
406 |
by (metis cInf_less_iff not_leE bdd_below_def)
|
hoelzl@52655
|
407 |
|
hoelzl@52612
|
408 |
lemma complete_interval:
|
hoelzl@52612
|
409 |
assumes "a < b" and "P a" and "\<not> P b"
|
hoelzl@52612
|
410 |
shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and>
|
hoelzl@52612
|
411 |
(\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
|
hoelzl@52612
|
412 |
proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
|
hoelzl@52612
|
413 |
show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
|
hoelzl@55710
|
414 |
by (rule cSup_upper, auto simp: bdd_above_def)
|
hoelzl@52612
|
415 |
(metis `a < b` `\<not> P b` linear less_le)
|
hoelzl@52612
|
416 |
next
|
hoelzl@52612
|
417 |
show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
|
hoelzl@52612
|
418 |
apply (rule cSup_least)
|
hoelzl@52612
|
419 |
apply auto
|
hoelzl@52612
|
420 |
apply (metis less_le_not_le)
|
hoelzl@52612
|
421 |
apply (metis `a<b` `~ P b` linear less_le)
|
hoelzl@52612
|
422 |
done
|
hoelzl@52612
|
423 |
next
|
hoelzl@52612
|
424 |
fix x
|
hoelzl@52612
|
425 |
assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
|
hoelzl@52612
|
426 |
show "P x"
|
hoelzl@52612
|
427 |
apply (rule less_cSupE [OF lt], auto)
|
hoelzl@52612
|
428 |
apply (metis less_le_not_le)
|
hoelzl@52612
|
429 |
apply (metis x)
|
hoelzl@52612
|
430 |
done
|
hoelzl@52612
|
431 |
next
|
hoelzl@52612
|
432 |
fix d
|
hoelzl@52612
|
433 |
assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
|
hoelzl@52612
|
434 |
thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
|
hoelzl@55710
|
435 |
by (rule_tac cSup_upper, auto simp: bdd_above_def)
|
hoelzl@52612
|
436 |
(metis `a<b` `~ P b` linear less_le)
|
hoelzl@52612
|
437 |
qed
|
hoelzl@52612
|
438 |
|
hoelzl@52612
|
439 |
end
|
hoelzl@52612
|
440 |
|
hoelzl@55711
|
441 |
lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
|
hoelzl@55711
|
442 |
using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp
|
hoelzl@52912
|
443 |
|
hoelzl@55711
|
444 |
lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
|
hoelzl@55711
|
445 |
using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp
|
hoelzl@52912
|
446 |
|
hoelzl@55709
|
447 |
lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
|
hoelzl@52612
|
448 |
by (auto intro!: cSup_eq_non_empty intro: dense_le)
|
hoelzl@52612
|
449 |
|
hoelzl@55709
|
450 |
lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
|
hoelzl@52612
|
451 |
by (auto intro!: cSup_eq intro: dense_le_bounded)
|
hoelzl@52612
|
452 |
|
hoelzl@55709
|
453 |
lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
|
hoelzl@52612
|
454 |
by (auto intro!: cSup_eq intro: dense_le_bounded)
|
hoelzl@52612
|
455 |
|
hoelzl@55709
|
456 |
lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
|
hoelzl@52612
|
457 |
by (auto intro!: cInf_eq intro: dense_ge)
|
hoelzl@52612
|
458 |
|
hoelzl@55709
|
459 |
lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, no_top, dense_linorder}} = y"
|
hoelzl@52612
|
460 |
by (auto intro!: cInf_eq intro: dense_ge_bounded)
|
hoelzl@52612
|
461 |
|
hoelzl@55709
|
462 |
lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, no_top, dense_linorder}} = y"
|
hoelzl@52612
|
463 |
by (auto intro!: cInf_eq intro: dense_ge_bounded)
|
hoelzl@52612
|
464 |
|
hoelzl@55711
|
465 |
class linear_continuum = conditionally_complete_linorder + dense_linorder +
|
hoelzl@55711
|
466 |
assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
|
hoelzl@55711
|
467 |
begin
|
hoelzl@55711
|
468 |
|
hoelzl@55711
|
469 |
lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
|
hoelzl@55711
|
470 |
by (metis UNIV_not_singleton neq_iff)
|
hoelzl@55711
|
471 |
|
paulson@33269
|
472 |
end
|
hoelzl@55711
|
473 |
|
hoelzl@55711
|
474 |
end
|