src/HOL/Conditionally_Complete_Lattices.thy

 
 theory Conditionally_Complete_Lattices
 imports Main Lubs
 begin
 
-lemma Sup_fin_eq_Max: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X"
+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 Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"
+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
 

   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 {*
 

     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_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_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_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"

   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))"
+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

 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 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)

          (metis `a<b` `~ P b` linear less_le)
 qed
 
 end
 
-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
+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"

     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_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_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)

   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)
 
-end
+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