(*  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

 *)

 header {* Conditionally-complete Lattices *}

 theory Conditionally_Complete_Lattices

 imports Main Lubs

 begin

 lemma (in linorder) Sup_fin_eq_Max: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X"

   by (induct X rule: finite_ne_induct) (simp_all add: sup_max)

 lemma (in linorder) Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"

   by (induct X rule: finite_ne_induct) (simp_all add: inf_min)

 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_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

 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_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_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)

 end

 text {*

 To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and

 @{const Inf} in theorem names with c.

 *}

 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> INFI A f \<le> f x"

   unfolding INF_def by (rule cInf_lower) auto

 lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFI A f"

   unfolding INF_def by (rule cInf_greatest) auto

 lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPR A f"

   unfolding SUP_def by (rule cSup_upper) auto

 lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPR A f \<le> M"

   unfolding SUP_def by (rule cSup_least) auto

 lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFI A f \<le> u"

   by (auto intro: cINF_lower assms order_trans)

 lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPR A f"

   by (auto intro: cSUP_upper assms order_trans)

 lemma cSUP_const: "A \<noteq> {} \<Longrightarrow> (SUP x:A. c) = c"

   by (intro antisym cSUP_least) (auto intro: cSUP_upper)

 lemma cINF_const: "A \<noteq> {} \<Longrightarrow> (INF x: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> INFI A f \<longleftrightarrow> (\<forall>x\<in>A. u \<le> f x)"

   by (metis cINF_greatest cINF_lower assms order_trans)

 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)"

   by (metis cSUP_least cSUP_upper assms order_trans)

 lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFI (insert a A) f = inf (f a) (INFI A f)"

   by (metis INF_def cInf_insert assms empty_is_image image_insert)

 lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPR (insert a A) f = sup (f a) (SUPR A f)"

   by (metis SUP_def cSup_insert assms empty_is_image image_insert)

 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"

   unfolding INF_def by (auto intro: cInf_mono)

 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"

   unfolding SUP_def by (auto intro: cSup_mono)

 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"

   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> SUPR A f \<le> SUPR B g"

   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> INFI (A \<union> B) f = inf (INFI A f) (INFI B f)"

   unfolding INF_def using assms by (auto simp add: image_Un intro: cInf_union_distrib)

 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> SUPR (A \<union> B) f = sup (SUPR A f) (SUPR B f)"

   unfolding SUP_def by (auto simp add: image_Un intro: cSup_union_distrib)

 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))"

   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> sup (SUPR A f) (SUPR A g) = (SUP a: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)

 end

 instance complete_lattice \<subseteq> conditionally_complete_lattice

   by default (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)

 lemma isLub_cSup: 

   "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> (\<exists>b. S *<= b) \<Longrightarrow> isLub UNIV S (Sup S)"

   by  (auto simp add: isLub_def setle_def leastP_def isUb_def

             intro!: setgeI cSup_upper cSup_least)

 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)

 lemma cSup_le: "(S::'a::conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> S *<= b \<Longrightarrow> Sup S \<le> b"

   by (metis cSup_least setle_def)

 lemma cInf_ge: "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> b <=* S \<Longrightarrow> Inf S \<ge> b"

   by (metis cInf_greatest setge_def)

 class conditionally_complete_linorder = conditionally_complete_lattice + linorder

 begin

 lemma less_cSup_iff : (*REAL_SUP_LE in HOL4*)

   "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 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_leE 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_leE 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 & x < d --> 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 `a < b` `\<not> P b` 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 `a<b` `~ P b` 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 `a<b` `~ P b` linear less_le)

 qed

 end

 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] Sup_fin_eq_Max[of X] by simp

 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] Inf_fin_eq_Min[of X] by simp

 lemma cSup_bounds:

   fixes S :: "'a :: conditionally_complete_lattice set"

   assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"

   shows "a \<le> Sup S \<and> Sup S \<le> b"

 proof-

   from isLub_cSup[OF Se] u have lub: "isLub UNIV S (Sup S)" by blast

   hence b: "Sup S \<le> b" using u 

     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) 

   from Se obtain y where y: "y \<in> S" by blast

   from lub l have "a \<le> Sup S"

     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)

        (metis le_iff_sup le_sup_iff y)

   with b show ?thesis by blast

 qed

 lemma cSup_unique: "(S::'a :: {conditionally_complete_linorder, no_bot} set) *<= b \<Longrightarrow> (\<forall>b'<b. \<exists>x\<in>S. b' < x) \<Longrightarrow> Sup S = b"

   by (rule cSup_eq) (auto simp: not_le[symmetric] setle_def)

 lemma cInf_unique: "b <=* (S::'a :: {conditionally_complete_linorder, no_top} set) \<Longrightarrow> (\<forall>b'>b. \<exists>x\<in>S. b' > x) \<Longrightarrow> Inf S = b"

   by (rule cInf_eq) (auto simp: not_le[symmetric] setge_def)

 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, no_bot, dense_linorder}} = x"

   by (auto intro!: cSup_eq intro: dense_le_bounded)

 lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"

   by (auto intro!: cSup_eq intro: dense_le_bounded)

 lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"

   by (auto intro!: cInf_eq intro: dense_ge)

 lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, no_top, dense_linorder}} = y"

   by (auto intro!: cInf_eq intro: dense_ge_bounded)

 lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, no_top, dense_linorder}} = y"

   by (auto intro!: cInf_eq intro: dense_ge_bounded)

 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

 end