1.1 --- a/src/HOL/Complete_Lattices.thy Tue Nov 05 09:44:58 2013 +0100
1.2 +++ b/src/HOL/Complete_Lattices.thy Tue Nov 05 09:44:59 2013 +0100
1.3 @@ -20,6 +20,12 @@
1.4 definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
1.5 INF_def: "INFI A f = \<Sqinter>(f ` A)"
1.6
1.7 +lemma INF_image [simp]: "INFI (f`A) g = INFI A (\<lambda>x. g (f x))"
1.8 + by (simp add: INF_def image_image)
1.9 +
1.10 +lemma INF_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> INFI A C = INFI B D"
1.11 + by (simp add: INF_def image_def)
1.12 +
1.13 end
1.14
1.15 class Sup =
1.16 @@ -29,6 +35,12 @@
1.17 definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
1.18 SUP_def: "SUPR A f = \<Squnion>(f ` A)"
1.19
1.20 +lemma SUP_image [simp]: "SUPR (f`A) g = SUPR A (%x. g (f x))"
1.21 + by (simp add: SUP_def image_image)
1.22 +
1.23 +lemma SUP_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> SUPR A C = SUPR B D"
1.24 + by (simp add: SUP_def image_def)
1.25 +
1.26 end
1.27
1.28 text {*
1.29 @@ -183,12 +195,6 @@
1.30 "\<Squnion>UNIV = \<top>"
1.31 by (auto intro!: antisym Sup_upper)
1.32
1.33 -lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
1.34 - by (simp add: INF_def image_image)
1.35 -
1.36 -lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
1.37 - by (simp add: SUP_def image_image)
1.38 -
1.39 lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
1.40 by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
1.41
1.42 @@ -201,14 +207,6 @@
1.43 lemma Sup_subset_mono: "A \<subseteq> B \<Longrightarrow> \<Squnion>A \<sqsubseteq> \<Squnion>B"
1.44 by (auto intro: Sup_least Sup_upper)
1.45
1.46 -lemma INF_cong:
1.47 - "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Sqinter>x\<in>A. C x) = (\<Sqinter>x\<in>B. D x)"
1.48 - by (simp add: INF_def image_def)
1.49 -
1.50 -lemma SUP_cong:
1.51 - "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (\<Squnion>x\<in>A. C x) = (\<Squnion>x\<in>B. D x)"
1.52 - by (simp add: SUP_def image_def)
1.53 -
1.54 lemma Inf_mono:
1.55 assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<sqsubseteq> b"
1.56 shows "\<Sqinter>A \<sqsubseteq> \<Sqinter>B"
2.1 --- a/src/HOL/Conditionally_Complete_Lattices.thy Tue Nov 05 09:44:58 2013 +0100
2.2 +++ b/src/HOL/Conditionally_Complete_Lattices.thy Tue Nov 05 09:44:59 2013 +0100
2.3 @@ -10,10 +10,10 @@
2.4 imports Main Lubs
2.5 begin
2.6
2.7 -lemma Sup_fin_eq_Max: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X"
2.8 +lemma (in linorder) Sup_fin_eq_Max: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X"
2.9 by (induct X rule: finite_ne_induct) (simp_all add: sup_max)
2.10
2.11 -lemma Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"
2.12 +lemma (in linorder) Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"
2.13 by (induct X rule: finite_ne_induct) (simp_all add: inf_min)
2.14
2.15 context preorder
2.16 @@ -125,6 +125,12 @@
2.17 thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
2.18 qed
2.19
2.20 +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)"
2.21 + by (auto simp: bdd_above_def intro: le_supI1 le_supI2)
2.22 +
2.23 +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)"
2.24 + by (auto simp: bdd_below_def intro: le_infI1 le_infI2)
2.25 +
2.26 end
2.27
2.28
2.29 @@ -142,6 +148,24 @@
2.30 and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
2.31 begin
2.32
2.33 +lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
2.34 + by (metis cSup_upper order_trans)
2.35 +
2.36 +lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
2.37 + by (metis cInf_lower order_trans)
2.38 +
2.39 +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"
2.40 + by (metis cSup_least cSup_upper2)
2.41 +
2.42 +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"
2.43 + by (metis cInf_greatest cInf_lower2)
2.44 +
2.45 +lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
2.46 + by (metis cSup_least cSup_upper subsetD)
2.47 +
2.48 +lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A"
2.49 + by (metis cInf_greatest cInf_lower subsetD)
2.50 +
2.51 lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
2.52 by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
2.53
2.54 @@ -154,18 +178,6 @@
2.55 lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
2.56 by (metis order_trans cInf_lower cInf_greatest)
2.57
2.58 -lemma cSup_singleton [simp]: "Sup {x} = x"
2.59 - by (intro cSup_eq_maximum) auto
2.60 -
2.61 -lemma cInf_singleton [simp]: "Inf {x} = x"
2.62 - by (intro cInf_eq_minimum) auto
2.63 -
2.64 -lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
2.65 - by (metis cSup_upper order_trans)
2.66 -
2.67 -lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
2.68 - by (metis cInf_lower order_trans)
2.69 -
2.70 lemma cSup_eq_non_empty:
2.71 assumes 1: "X \<noteq> {}"
2.72 assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
2.73 @@ -192,10 +204,16 @@
2.74 lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
2.75 by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
2.76
2.77 +lemma cSup_singleton [simp]: "Sup {x} = x"
2.78 + by (intro cSup_eq_maximum) auto
2.79 +
2.80 +lemma cInf_singleton [simp]: "Inf {x} = x"
2.81 + by (intro cInf_eq_minimum) auto
2.82 +
2.83 lemma cSup_insert_If: "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
2.84 using cSup_insert[of X] by simp
2.85
2.86 -lemma cInf_insert_if: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
2.87 +lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
2.88 using cInf_insert[of X] by simp
2.89
2.90 lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
2.91 @@ -234,6 +252,74 @@
2.92 lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
2.93 by (auto intro!: cInf_eq_minimum)
2.94
2.95 +lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> INFI A f \<le> f x"
2.96 + unfolding INF_def by (rule cInf_lower) auto
2.97 +
2.98 +lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFI A f"
2.99 + unfolding INF_def by (rule cInf_greatest) auto
2.100 +
2.101 +lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPR A f"
2.102 + unfolding SUP_def by (rule cSup_upper) auto
2.103 +
2.104 +lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPR A f \<le> M"
2.105 + unfolding SUP_def by (rule cSup_least) auto
2.106 +
2.107 +lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFI A f \<le> u"
2.108 + by (auto intro: cINF_lower assms order_trans)
2.109 +
2.110 +lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPR A f"
2.111 + by (auto intro: cSUP_upper assms order_trans)
2.112 +
2.113 +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)"
2.114 + by (metis cINF_greatest cINF_lower assms order_trans)
2.115 +
2.116 +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)"
2.117 + by (metis cSUP_least cSUP_upper assms order_trans)
2.118 +
2.119 +lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFI (insert a A) f = inf (f a) (INFI A f)"
2.120 + by (metis INF_def cInf_insert assms empty_is_image image_insert)
2.121 +
2.122 +lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPR (insert a A) f = sup (f a) (SUPR A f)"
2.123 + by (metis SUP_def cSup_insert assms empty_is_image image_insert)
2.124 +
2.125 +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"
2.126 + unfolding INF_def by (auto intro: cInf_mono)
2.127 +
2.128 +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"
2.129 + unfolding SUP_def by (auto intro: cSup_mono)
2.130 +
2.131 +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"
2.132 + by (rule cINF_mono) auto
2.133 +
2.134 +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"
2.135 + by (rule cSUP_mono) auto
2.136 +
2.137 +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)"
2.138 + by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
2.139 +
2.140 +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) "
2.141 + by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
2.142 +
2.143 +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)"
2.144 + by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
2.145 +
2.146 +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)"
2.147 + unfolding INF_def using assms by (auto simp add: image_Un intro: cInf_union_distrib)
2.148 +
2.149 +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)"
2.150 + by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
2.151 +
2.152 +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)"
2.153 + unfolding SUP_def by (auto simp add: image_Un intro: cSup_union_distrib)
2.154 +
2.155 +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))"
2.156 + by (intro antisym le_infI cINF_greatest cINF_lower2)
2.157 + (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
2.158 +
2.159 +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))"
2.160 + by (intro antisym le_supI cSUP_least cSUP_upper2)
2.161 + (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
2.162 +
2.163 end
2.164
2.165 instance complete_lattice \<subseteq> conditionally_complete_lattice
2.166 @@ -323,14 +409,11 @@
2.167
2.168 end
2.169
2.170 -class linear_continuum = conditionally_complete_linorder + dense_linorder +
2.171 - assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
2.172 -begin
2.173 +lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
2.174 + using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp
2.175
2.176 -lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
2.177 - by (metis UNIV_not_singleton neq_iff)
2.178 -
2.179 -end
2.180 +lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
2.181 + using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp
2.182
2.183 lemma cSup_bounds:
2.184 fixes S :: "'a :: conditionally_complete_lattice set"
2.185 @@ -347,19 +430,12 @@
2.186 with b show ?thesis by blast
2.187 qed
2.188
2.189 -
2.190 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"
2.191 by (rule cSup_eq) (auto simp: not_le[symmetric] setle_def)
2.192
2.193 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"
2.194 by (rule cInf_eq) (auto simp: not_le[symmetric] setge_def)
2.195
2.196 -lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
2.197 - using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp
2.198 -
2.199 -lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
2.200 - using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp
2.201 -
2.202 lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
2.203 by (auto intro!: cSup_eq_non_empty intro: dense_le)
2.204
2.205 @@ -378,4 +454,13 @@
2.206 lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, no_top, dense_linorder}} = y"
2.207 by (auto intro!: cInf_eq intro: dense_ge_bounded)
2.208
2.209 +class linear_continuum = conditionally_complete_linorder + dense_linorder +
2.210 + assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
2.211 +begin
2.212 +
2.213 +lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
2.214 + by (metis UNIV_not_singleton neq_iff)
2.215 +
2.216 end
2.217 +
2.218 +end
3.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Nov 05 09:44:58 2013 +0100
3.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Nov 05 09:44:59 2013 +0100
3.3 @@ -2686,7 +2686,7 @@
3.4 lemma Inf_insert:
3.5 fixes S :: "real set"
3.6 shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
3.7 - by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_if)
3.8 + by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)
3.9
3.10 lemma Inf_insert_finite:
3.11 fixes S :: "real set"