(* Title: HOL/Conditionally_Complete_Lattices.thy
Author: Amine Chaieb and L C Paulson, University of Cambridge
Author: Johannes Hölzl, TU München
Author: Luke S. Serafin, Carnegie Mellon University
*)
section \<open>Conditionally-complete Lattices\<close>
theory Conditionally_Complete_Lattices
imports Finite_Set Lattices_Big Set_Interval
begin
context preorder
begin
definition "bdd_above A \<longleftrightarrow> (\<exists>M. \<forall>x \<in> A. x \<le> M)"
definition "bdd_below A \<longleftrightarrow> (\<exists>m. \<forall>x \<in> A. m \<le> x)"
lemma bdd_aboveI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> M) \<Longrightarrow> bdd_above A"
by (auto simp: bdd_above_def)
lemma bdd_belowI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> x) \<Longrightarrow> bdd_below A"
by (auto simp: bdd_below_def)
lemma bdd_aboveI2: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> bdd_above (f`A)"
by force
lemma bdd_belowI2: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> bdd_below (f`A)"
by force
lemma bdd_above_empty [simp, intro]: "bdd_above {}"
unfolding bdd_above_def by auto
lemma bdd_below_empty [simp, intro]: "bdd_below {}"
unfolding bdd_below_def by auto
lemma bdd_above_mono: "bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_above A"
by (metis (full_types) bdd_above_def order_class.le_neq_trans psubsetD)
lemma bdd_below_mono: "bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_below A"
by (metis bdd_below_def order_class.le_neq_trans psubsetD)
lemma bdd_above_Int1 [simp]: "bdd_above A \<Longrightarrow> bdd_above (A \<inter> B)"
using bdd_above_mono by auto
lemma bdd_above_Int2 [simp]: "bdd_above B \<Longrightarrow> bdd_above (A \<inter> B)"
using bdd_above_mono by auto
lemma bdd_below_Int1 [simp]: "bdd_below A \<Longrightarrow> bdd_below (A \<inter> B)"
using bdd_below_mono by auto
lemma bdd_below_Int2 [simp]: "bdd_below B \<Longrightarrow> bdd_below (A \<inter> B)"
using bdd_below_mono by auto
lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}"
by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}"
by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}"
by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}"
by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}"
by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}"
by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}"
by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}"
by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}"
by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}"
by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}"
by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}"
by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
end
lemma (in order_top) bdd_above_top[simp, intro!]: "bdd_above A"
by (rule bdd_aboveI[of _ top]) simp
lemma (in order_bot) bdd_above_bot[simp, intro!]: "bdd_below A"
by (rule bdd_belowI[of _ bot]) simp
lemma bdd_above_image_mono: "mono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_above (f`A)"
by (auto simp: bdd_above_def mono_def)
lemma bdd_below_image_mono: "mono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_below (f`A)"
by (auto simp: bdd_below_def mono_def)
lemma bdd_above_image_antimono: "antimono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_above (f`A)"
by (auto simp: bdd_above_def bdd_below_def antimono_def)
lemma bdd_below_image_antimono: "antimono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below (f`A)"
by (auto simp: bdd_above_def bdd_below_def antimono_def)
lemma
fixes X :: "'a::ordered_ab_group_add set"
shows bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X"
and bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X"
using bdd_above_image_antimono[of uminus X] bdd_below_image_antimono[of uminus "uminus`X"]
using bdd_below_image_antimono[of uminus X] bdd_above_image_antimono[of uminus "uminus`X"]
by (auto simp: antimono_def image_image)
context lattice
begin
lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)
lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
by (auto simp: bdd_below_def intro: le_infI2 inf_le1)
lemma bdd_finite [simp]:
assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
using assms by (induct rule: finite_induct, auto)
lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)"
proof
assume "bdd_above (A \<union> B)"
thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto
next
assume "bdd_above A \<and> bdd_above B"
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
hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
thus "bdd_above (A \<union> B)" unfolding bdd_above_def ..
qed
lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)"
proof
assume "bdd_below (A \<union> B)"
thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto
next
assume "bdd_below A \<and> bdd_below B"
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
hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2)
thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
qed
lemma bdd_above_image_sup[simp]:
"bdd_above ((\<lambda>x. sup (f x) (g x)) ` A) \<longleftrightarrow> bdd_above (f`A) \<and> bdd_above (g`A)"
by (auto simp: bdd_above_def intro: le_supI1 le_supI2)
lemma bdd_below_image_inf[simp]:
"bdd_below ((\<lambda>x. inf (f x) (g x)) ` A) \<longleftrightarrow> bdd_below (f`A) \<and> bdd_below (g`A)"
by (auto simp: bdd_below_def intro: le_infI1 le_infI2)
lemma bdd_below_UN[simp]: "finite I \<Longrightarrow> bdd_below (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_below (A i))"
by (induction I rule: finite.induct) auto
lemma bdd_above_UN[simp]: "finite I \<Longrightarrow> bdd_above (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_above (A i))"
by (induction I rule: finite.induct) auto
end
text \<open>
To avoid name classes with the \<^class>\<open>complete_lattice\<close>-class we prefix \<^const>\<open>Sup\<close> and
\<^const>\<open>Inf\<close> in theorem names with c.
\<close>
class conditionally_complete_lattice = lattice + Sup + Inf +
assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x"
and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X"
assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X"
and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
begin
lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
by (metis cSup_upper order_trans)
lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
by (metis cInf_lower order_trans)
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"
by (metis cSup_least cSup_upper2)
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"
by (metis cInf_greatest cInf_lower2)
lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
by (metis cSup_least cSup_upper subsetD)
lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A"
by (metis cInf_greatest cInf_lower subsetD)
lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto
lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
by (metis order_trans cSup_upper cSup_least)
lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
by (metis order_trans cInf_lower cInf_greatest)
lemma cSup_eq_non_empty:
assumes 1: "X \<noteq> {}"
assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
shows "Sup X = a"
by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper)
lemma cInf_eq_non_empty:
assumes 1: "X \<noteq> {}"
assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
shows "Inf X = a"
by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower)
lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)
lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
lemma cSup_singleton [simp]: "Sup {x} = x"
by (intro cSup_eq_maximum) auto
lemma cInf_singleton [simp]: "Inf {x} = x"
by (intro cInf_eq_minimum) auto
lemma cSup_insert_If: "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
using cSup_insert[of X] by simp
lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
using cInf_insert[of X] by simp
lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
proof (induct X arbitrary: x rule: finite_induct)
case (insert x X y) then show ?case
by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
qed simp
lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
proof (induct X arbitrary: x rule: finite_induct)
case (insert x X y) then show ?case
by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
qed simp
lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
lemma cSup_atMost[simp]: "Sup {..x} = x"
by (auto intro!: cSup_eq_maximum)
lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x"
by (auto intro!: cSup_eq_maximum)
lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x"
by (auto intro!: cSup_eq_maximum)
lemma cInf_atLeast[simp]: "Inf {x..} = x"
by (auto intro!: cInf_eq_minimum)
lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y"
by (auto intro!: cInf_eq_minimum)
lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
by (auto intro!: cInf_eq_minimum)
lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> \<Sqinter>(f ` A) \<le> f x"
using cInf_lower [of _ "f ` A"] by simp
lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> \<Sqinter>(f ` A)"
using cInf_greatest [of "f ` A"] by auto
lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> \<Squnion>(f ` A)"
using cSup_upper [of _ "f ` A"] by simp
lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> \<Squnion>(f ` A) \<le> M"
using cSup_least [of "f ` A"] by auto
lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> \<Sqinter>(f ` A) \<le> u"
by (auto intro: cINF_lower order_trans)
lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> \<Squnion>(f ` A)"
by (auto intro: cSUP_upper order_trans)
lemma cSUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>x\<in>A. c) = c"
by (intro antisym cSUP_least) (auto intro: cSUP_upper)
lemma cINF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>x\<in>A. c) = c"
by (intro antisym cINF_greatest) (auto intro: cINF_lower)
lemma le_cINF_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> u \<le> \<Sqinter>(f ` A) \<longleftrightarrow> (\<forall>x\<in>A. u \<le> f x)"
by (metis cINF_greatest cINF_lower order_trans)
lemma cSUP_le_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> \<Squnion>(f ` A) \<le> u \<longleftrightarrow> (\<forall>x\<in>A. f x \<le> u)"
by (metis cSUP_least cSUP_upper order_trans)
lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (\<Sqinter>i\<in>A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i"
by (metis cINF_lower less_le_trans)
lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (\<Squnion>i\<in>A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y"
by (metis cSUP_upper le_less_trans)
lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> \<Sqinter>(f ` insert a A) = inf (f a) (\<Sqinter>(f ` A))"
by (simp add: cInf_insert cong del: INF_cong)
lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> \<Squnion>(f ` insert a A) = sup (f a) (\<Squnion>(f ` A))"
by (simp add: cSup_insert cong del: SUP_cong)
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> \<Sqinter>(f ` A) \<le> \<Sqinter>(g ` B)"
using cInf_mono [of "g ` B" "f ` A"] by auto
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> \<Squnion>(f ` A) \<le> \<Squnion>(g ` B)"
using cSup_mono [of "f ` A" "g ` B"] by auto
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> \<Sqinter>(g ` B) \<le> \<Sqinter>(f ` A)"
by (rule cINF_mono) auto
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> \<Squnion>(f ` A) \<le> \<Squnion>(g ` B)"
by (rule cSUP_mono) auto
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)"
by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
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) "
by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
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)"
by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
lemma cINF_union: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_below (f ` B) \<Longrightarrow> \<Sqinter> (f ` (A \<union> B)) = \<Sqinter> (f ` A) \<sqinter> \<Sqinter> (f ` B)"
using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un cong del: INF_cong)
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)"
by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
lemma cSUP_union: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_above (f ` B) \<Longrightarrow> \<Squnion> (f ` (A \<union> B)) = \<Squnion> (f ` A) \<squnion> \<Squnion> (f ` B)"
using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un cong del: SUP_cong)
lemma cINF_inf_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> bdd_below (g`A) \<Longrightarrow> \<Sqinter> (f ` A) \<sqinter> \<Sqinter> (g ` A) = (\<Sqinter>a\<in>A. inf (f a) (g a))"
by (intro antisym le_infI cINF_greatest cINF_lower2)
(auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
lemma SUP_sup_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> bdd_above (g`A) \<Longrightarrow> \<Squnion> (f ` A) \<squnion> \<Squnion> (g ` A) = (\<Squnion>a\<in>A. sup (f a) (g a))"
by (intro antisym le_supI cSUP_least cSUP_upper2)
(auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
lemma cInf_le_cSup:
"A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<le> Sup A"
by (auto intro!: cSup_upper2[of "SOME a. a \<in> A"] intro: someI cInf_lower)
end
instance complete_lattice \<subseteq> conditionally_complete_lattice
by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
lemma cSup_eq:
fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
shows "Sup X = a"
proof cases
assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
qed (intro cSup_eq_non_empty assms)
lemma cInf_eq:
fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
shows "Inf X = a"
proof cases
assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
qed (intro cInf_eq_non_empty assms)
class conditionally_complete_linorder = conditionally_complete_lattice + linorder
begin
lemma less_cSup_iff:
"X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
lemma cINF_less_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> (\<Sqinter>i\<in>A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
using cInf_less_iff[of "f`A"] by auto
lemma less_cSUP_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> a < (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
using less_cSup_iff[of "f`A"] by auto
lemma less_cSupE:
assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x"
by (metis cSup_least assms not_le that)
lemma less_cSupD:
"X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
by (metis less_cSup_iff not_le_imp_less bdd_above_def)
lemma cInf_lessD:
"X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
by (metis cInf_less_iff not_le_imp_less bdd_below_def)
lemma complete_interval:
assumes "a < b" and "P a" and "\<not> P b"
shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and>
(\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x}"], auto)
show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
by (rule cSup_upper, auto simp: bdd_above_def)
(metis \<open>a < b\<close> \<open>\<not> P b\<close> linear less_le)
next
show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
apply (rule cSup_least)
apply auto
apply (metis less_le_not_le)
apply (metis \<open>a<b\<close> \<open>\<not> P b\<close> linear less_le)
done
next
fix x
assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
show "P x"
apply (rule less_cSupE [OF lt], auto)
apply (metis less_le_not_le)
apply (metis x)
done
next
fix d
assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
by (rule_tac cSup_upper, auto simp: bdd_above_def)
(metis \<open>a<b\<close> \<open>\<not> P b\<close> linear less_le)
qed
end
instance complete_linorder < conditionally_complete_linorder
..
lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
using cSup_eq_Sup_fin[of X] by (simp add: Sup_fin_Max)
lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
using cInf_eq_Inf_fin[of X] by (simp add: Inf_fin_Min)
lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
by (auto intro!: cSup_eq_non_empty intro: dense_le)
lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
by (auto intro!: cInf_eq_non_empty intro: dense_ge)
lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
lemma Inf_insert_finite:
fixes S :: "'a::conditionally_complete_linorder set"
shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
by (simp add: cInf_eq_Min)
lemma Sup_insert_finite:
fixes S :: "'a::conditionally_complete_linorder set"
shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
by (simp add: cSup_insert sup_max)
lemma finite_imp_less_Inf:
fixes a :: "'a::conditionally_complete_linorder"
shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a < x\<rbrakk> \<Longrightarrow> a < Inf X"
by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)
lemma finite_less_Inf_iff:
fixes a :: "'a :: conditionally_complete_linorder"
shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a < Inf X \<longleftrightarrow> (\<forall>x \<in> X. a < x)"
by (auto simp: cInf_eq_Min)
lemma finite_imp_Sup_less:
fixes a :: "'a::conditionally_complete_linorder"
shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a > x\<rbrakk> \<Longrightarrow> a > Sup X"
by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)
lemma finite_Sup_less_iff:
fixes a :: "'a :: conditionally_complete_linorder"
shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a > Sup X \<longleftrightarrow> (\<forall>x \<in> X. a > x)"
by (auto simp: cSup_eq_Max)
class linear_continuum = conditionally_complete_linorder + dense_linorder +
assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
begin
lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
by (metis UNIV_not_singleton neq_iff)
end
instantiation nat :: conditionally_complete_linorder
begin
definition "Sup (X::nat set) = Max X"
definition "Inf (X::nat set) = (LEAST n. n \<in> X)"
lemma bdd_above_nat: "bdd_above X \<longleftrightarrow> finite (X::nat set)"
proof
assume "bdd_above X"
then obtain z where "X \<subseteq> {.. z}"
by (auto simp: bdd_above_def)
then show "finite X"
by (rule finite_subset) simp
qed simp
instance
proof
fix x :: nat
fix X :: "nat set"
show "Inf X \<le> x" if "x \<in> X" "bdd_below X"
using that by (simp add: Inf_nat_def Least_le)
show "x \<le> Inf X" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y"
using that unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex)
show "x \<le> Sup X" if "x \<in> X" "bdd_above X"
using that by (simp add: Sup_nat_def bdd_above_nat)
show "Sup X \<le> x" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x"
proof -
from that have "bdd_above X"
by (auto simp: bdd_above_def)
with that show ?thesis
by (simp add: Sup_nat_def bdd_above_nat)
qed
qed
end
lemma Inf_nat_def1:
fixes K::"nat set"
assumes "K \<noteq> {}"
shows "Inf K \<in> K"
by (auto simp add: Min_def Inf_nat_def) (meson LeastI assms bot.extremum_unique subsetI)
instantiation int :: conditionally_complete_linorder
begin
definition "Sup (X::int set) = (THE x. x \<in> X \<and> (\<forall>y\<in>X. y \<le> x))"
definition "Inf (X::int set) = - (Sup (uminus ` X))"
instance
proof
{ fix x :: int and X :: "int set" assume "X \<noteq> {}" "bdd_above X"
then obtain x y where "X \<subseteq> {..y}" "x \<in> X"
by (auto simp: bdd_above_def)
then have *: "finite (X \<inter> {x..y})" "X \<inter> {x..y} \<noteq> {}" and "x \<le> y"
by (auto simp: subset_eq)
have "\<exists>!x\<in>X. (\<forall>y\<in>X. y \<le> x)"
proof
{ fix z assume "z \<in> X"
have "z \<le> Max (X \<inter> {x..y})"
proof cases
assume "x \<le> z" with \<open>z \<in> X\<close> \<open>X \<subseteq> {..y}\<close> *(1) show ?thesis
by (auto intro!: Max_ge)
next
assume "\<not> x \<le> z"
then have "z < x" by simp
also have "x \<le> Max (X \<inter> {x..y})"
using \<open>x \<in> X\<close> *(1) \<open>x \<le> y\<close> by (intro Max_ge) auto
finally show ?thesis by simp
qed }
note le = this
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
fix z assume *: "z \<in> X \<and> (\<forall>y\<in>X. y \<le> z)"
with le have "z \<le> Max (X \<inter> {x..y})"
by auto
moreover have "Max (X \<inter> {x..y}) \<le> z"
using * ex by auto
ultimately show "z = Max (X \<inter> {x..y})"
by auto
qed
then have "Sup X \<in> X \<and> (\<forall>y\<in>X. y \<le> Sup X)"
unfolding Sup_int_def by (rule theI') }
note Sup_int = this
{ fix x :: int and X :: "int set" assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
using Sup_int[of X] by auto }
note le_Sup = this
{ 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"
using Sup_int[of X] by (auto simp: bdd_above_def) }
note Sup_le = this
{ fix x :: int and X :: "int set" assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) }
{ 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"
using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) }
qed
end
lemma interval_cases:
fixes S :: "'a :: conditionally_complete_linorder set"
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"
shows "\<exists>a b. S = {} \<or>
S = UNIV \<or>
S = {..<b} \<or>
S = {..b} \<or>
S = {a<..} \<or>
S = {a..} \<or>
S = {a<..<b} \<or>
S = {a<..b} \<or>
S = {a..<b} \<or>
S = {a..b}"
proof -
define lower upper where "lower = {x. \<exists>s\<in>S. s \<le> x}" and "upper = {x. \<exists>s\<in>S. x \<le> s}"
with ivl have "S = lower \<inter> upper"
by auto
moreover
have "\<exists>a. upper = UNIV \<or> upper = {} \<or> upper = {.. a} \<or> upper = {..< a}"
proof cases
assume *: "bdd_above S \<and> S \<noteq> {}"
from * have "upper \<subseteq> {.. Sup S}"
by (auto simp: upper_def intro: cSup_upper2)
moreover from * have "{..< Sup S} \<subseteq> upper"
by (force simp add: less_cSup_iff upper_def subset_eq Ball_def)
ultimately have "upper = {.. Sup S} \<or> upper = {..< Sup S}"
unfolding ivl_disj_un(2)[symmetric] by auto
then show ?thesis by auto
next
assume "\<not> (bdd_above S \<and> S \<noteq> {})"
then have "upper = UNIV \<or> upper = {}"
by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le)
then show ?thesis
by auto
qed
moreover
have "\<exists>b. lower = UNIV \<or> lower = {} \<or> lower = {b ..} \<or> lower = {b <..}"
proof cases
assume *: "bdd_below S \<and> S \<noteq> {}"
from * have "lower \<subseteq> {Inf S ..}"
by (auto simp: lower_def intro: cInf_lower2)
moreover from * have "{Inf S <..} \<subseteq> lower"
by (force simp add: cInf_less_iff lower_def subset_eq Ball_def)
ultimately have "lower = {Inf S ..} \<or> lower = {Inf S <..}"
unfolding ivl_disj_un(1)[symmetric] by auto
then show ?thesis by auto
next
assume "\<not> (bdd_below S \<and> S \<noteq> {})"
then have "lower = UNIV \<or> lower = {}"
by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le)
then show ?thesis
by auto
qed
ultimately show ?thesis
unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def
by (metis inf_bot_left inf_bot_right inf_top.left_neutral inf_top.right_neutral)
qed
lemma cSUP_eq_cINF_D:
fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
assumes eq: "(\<Squnion>x\<in>A. f x) = (\<Sqinter>x\<in>A. f x)"
and bdd: "bdd_above (f ` A)" "bdd_below (f ` A)"
and a: "a \<in> A"
shows "f a = (\<Sqinter>x\<in>A. f x)"
apply (rule antisym)
using a bdd
apply (auto simp: cINF_lower)
apply (metis eq cSUP_upper)
done
lemma cSUP_UNION:
fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
and bdd_UN: "bdd_above (\<Union>x\<in>A. f ` B x)"
shows "(\<Squnion>z \<in> \<Union>x\<in>A. B x. f z) = (\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z)"
proof -
have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_above (f ` B x)"
using bdd_UN by (meson UN_upper bdd_above_mono)
obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<le> M"
using bdd_UN by (auto simp: bdd_above_def)
then have bdd2: "bdd_above ((\<lambda>x. \<Squnion>z\<in>B x. f z) ` A)"
unfolding bdd_above_def by (force simp: bdd cSUP_le_iff ne(2))
have "(\<Squnion>z \<in> \<Union>x\<in>A. B x. f z) \<le> (\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z)"
using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper2 simp: bdd2 bdd)
moreover have "(\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z) \<le> (\<Squnion> z \<in> \<Union>x\<in>A. B x. f z)"
using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper simp: image_UN bdd_UN)
ultimately show ?thesis
by (rule order_antisym)
qed
lemma cINF_UNION:
fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
and bdd_UN: "bdd_below (\<Union>x\<in>A. f ` B x)"
shows "(\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z) = (\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z)"
proof -
have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_below (f ` B x)"
using bdd_UN by (meson UN_upper bdd_below_mono)
obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<ge> M"
using bdd_UN by (auto simp: bdd_below_def)
then have bdd2: "bdd_below ((\<lambda>x. \<Sqinter>z\<in>B x. f z) ` A)"
unfolding bdd_below_def by (force simp: bdd le_cINF_iff ne(2))
have "(\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z) \<le> (\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z)"
using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower simp: bdd2 bdd)
moreover have "(\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z) \<le> (\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z)"
using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower2 simp: bdd bdd_UN bdd2)
ultimately show ?thesis
by (rule order_antisym)
qed
lemma cSup_abs_le:
fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
shows "S \<noteq> {} \<Longrightarrow> (\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
apply (auto simp add: abs_le_iff intro: cSup_least)
by (metis bdd_aboveI cSup_upper neg_le_iff_le order_trans)
end