1(* Title: HOL/Conditionally_Complete_Lattices.thy 2 Author: Amine Chaieb and L C Paulson, University of Cambridge 3 Author: Johannes H��lzl, TU M��nchen 4 Author: Luke S. Serafin, Carnegie Mellon University 5*) 6 7section \<open>Conditionally-complete Lattices\<close> 8 9theory Conditionally_Complete_Lattices 10imports Finite_Set Lattices_Big Set_Interval 11begin 12 13context preorder 14begin 15 16definition "bdd_above A \<longleftrightarrow> (\<exists>M. \<forall>x \<in> A. x \<le> M)" 17definition "bdd_below A \<longleftrightarrow> (\<exists>m. \<forall>x \<in> A. m \<le> x)" 18 19lemma bdd_aboveI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> M) \<Longrightarrow> bdd_above A" 20 by (auto simp: bdd_above_def) 21 22lemma bdd_belowI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> x) \<Longrightarrow> bdd_below A" 23 by (auto simp: bdd_below_def) 24 25lemma bdd_aboveI2: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> bdd_above (f`A)" 26 by force 27 28lemma bdd_belowI2: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> bdd_below (f`A)" 29 by force 30 31lemma bdd_above_empty [simp, intro]: "bdd_above {}" 32 unfolding bdd_above_def by auto 33 34lemma bdd_below_empty [simp, intro]: "bdd_below {}" 35 unfolding bdd_below_def by auto 36 37lemma bdd_above_mono: "bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_above A" 38 by (metis (full_types) bdd_above_def order_class.le_neq_trans psubsetD) 39 40lemma bdd_below_mono: "bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_below A" 41 by (metis bdd_below_def order_class.le_neq_trans psubsetD) 42 43lemma bdd_above_Int1 [simp]: "bdd_above A \<Longrightarrow> bdd_above (A \<inter> B)" 44 using bdd_above_mono by auto 45 46lemma bdd_above_Int2 [simp]: "bdd_above B \<Longrightarrow> bdd_above (A \<inter> B)" 47 using bdd_above_mono by auto 48 49lemma bdd_below_Int1 [simp]: "bdd_below A \<Longrightarrow> bdd_below (A \<inter> B)" 50 using bdd_below_mono by auto 51 52lemma bdd_below_Int2 [simp]: "bdd_below B \<Longrightarrow> bdd_below (A \<inter> B)" 53 using bdd_below_mono by auto 54 55lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}" 56 by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le) 57 58lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}" 59 by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le) 60 61lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}" 62 by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le) 63 64lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}" 65 by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le) 66 67lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}" 68 by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le) 69 70lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}" 71 by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le) 72 73lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}" 74 by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le) 75 76lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}" 77 by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le) 78 79lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}" 80 by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le) 81 82lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}" 83 by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le) 84 85lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}" 86 by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le) 87 88lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}" 89 by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le) 90 91end 92 93lemma (in order_top) bdd_above_top[simp, intro!]: "bdd_above A" 94 by (rule bdd_aboveI[of _ top]) simp 95 96lemma (in order_bot) bdd_above_bot[simp, intro!]: "bdd_below A" 97 by (rule bdd_belowI[of _ bot]) simp 98 99lemma bdd_above_image_mono: "mono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_above (f`A)" 100 by (auto simp: bdd_above_def mono_def) 101 102lemma bdd_below_image_mono: "mono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_below (f`A)" 103 by (auto simp: bdd_below_def mono_def) 104 105lemma bdd_above_image_antimono: "antimono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_above (f`A)" 106 by (auto simp: bdd_above_def bdd_below_def antimono_def) 107 108lemma bdd_below_image_antimono: "antimono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below (f`A)" 109 by (auto simp: bdd_above_def bdd_below_def antimono_def) 110 111lemma 112 fixes X :: "'a::ordered_ab_group_add set" 113 shows bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X" 114 and bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X" 115 using bdd_above_image_antimono[of uminus X] bdd_below_image_antimono[of uminus "uminus`X"] 116 using bdd_below_image_antimono[of uminus X] bdd_above_image_antimono[of uminus "uminus`X"] 117 by (auto simp: antimono_def image_image) 118 119context lattice 120begin 121 122lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A" 123 by (auto simp: bdd_above_def intro: le_supI2 sup_ge1) 124 125lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A" 126 by (auto simp: bdd_below_def intro: le_infI2 inf_le1) 127 128lemma bdd_finite [simp]: 129 assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A" 130 using assms by (induct rule: finite_induct, auto) 131 132lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)" 133proof 134 assume "bdd_above (A \<union> B)" 135 thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto 136next 137 assume "bdd_above A \<and> bdd_above B" 138 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 139 hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2) 140 thus "bdd_above (A \<union> B)" unfolding bdd_above_def .. 141qed 142 143lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)" 144proof 145 assume "bdd_below (A \<union> B)" 146 thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto 147next 148 assume "bdd_below A \<and> bdd_below B" 149 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 150 hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2) 151 thus "bdd_below (A \<union> B)" unfolding bdd_below_def .. 152qed 153 154lemma bdd_above_image_sup[simp]: 155 "bdd_above ((\<lambda>x. sup (f x) (g x)) ` A) \<longleftrightarrow> bdd_above (f`A) \<and> bdd_above (g`A)" 156by (auto simp: bdd_above_def intro: le_supI1 le_supI2) 157 158lemma bdd_below_image_inf[simp]: 159 "bdd_below ((\<lambda>x. inf (f x) (g x)) ` A) \<longleftrightarrow> bdd_below (f`A) \<and> bdd_below (g`A)" 160by (auto simp: bdd_below_def intro: le_infI1 le_infI2) 161 162lemma bdd_below_UN[simp]: "finite I \<Longrightarrow> bdd_below (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_below (A i))" 163by (induction I rule: finite.induct) auto 164 165lemma bdd_above_UN[simp]: "finite I \<Longrightarrow> bdd_above (\<Union>i\<in>I. A i) = (\<forall>i \<in> I. bdd_above (A i))" 166by (induction I rule: finite.induct) auto 167 168end 169 170 171text \<open> 172 173To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and 174@{const Inf} in theorem names with c. 175 176\<close> 177 178class conditionally_complete_lattice = lattice + Sup + Inf + 179 assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x" 180 and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X" 181 assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X" 182 and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z" 183begin 184 185lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X" 186 by (metis cSup_upper order_trans) 187 188lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y" 189 by (metis cInf_lower order_trans) 190 191lemma 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" 192 by (metis cSup_least cSup_upper2) 193 194lemma 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" 195 by (metis cInf_greatest cInf_lower2) 196 197lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B" 198 by (metis cSup_least cSup_upper subsetD) 199 200lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A" 201 by (metis cInf_greatest cInf_lower subsetD) 202 203lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z" 204 by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto 205 206lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z" 207 by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto 208 209lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)" 210 by (metis order_trans cSup_upper cSup_least) 211 212lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)" 213 by (metis order_trans cInf_lower cInf_greatest) 214 215lemma cSup_eq_non_empty: 216 assumes 1: "X \<noteq> {}" 217 assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a" 218 assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y" 219 shows "Sup X = a" 220 by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper) 221 222lemma cInf_eq_non_empty: 223 assumes 1: "X \<noteq> {}" 224 assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x" 225 assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a" 226 shows "Inf X = a" 227 by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower) 228 229lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}" 230 by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def) 231 232lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}" 233 by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def) 234 235lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)" 236 by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least) 237 238lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)" 239 by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest) 240 241lemma cSup_singleton [simp]: "Sup {x} = x" 242 by (intro cSup_eq_maximum) auto 243 244lemma cInf_singleton [simp]: "Inf {x} = x" 245 by (intro cInf_eq_minimum) auto 246 247lemma cSup_insert_If: "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))" 248 using cSup_insert[of X] by simp 249 250lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))" 251 using cInf_insert[of X] by simp 252 253lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X" 254proof (induct X arbitrary: x rule: finite_induct) 255 case (insert x X y) then show ?case 256 by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2) 257qed simp 258 259lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x" 260proof (induct X arbitrary: x rule: finite_induct) 261 case (insert x X y) then show ?case 262 by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2) 263qed simp 264 265lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X" 266 by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert) 267 268lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X" 269 by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert) 270 271lemma cSup_atMost[simp]: "Sup {..x} = x" 272 by (auto intro!: cSup_eq_maximum) 273 274lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x" 275 by (auto intro!: cSup_eq_maximum) 276 277lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x" 278 by (auto intro!: cSup_eq_maximum) 279 280lemma cInf_atLeast[simp]: "Inf {x..} = x" 281 by (auto intro!: cInf_eq_minimum) 282 283lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y" 284 by (auto intro!: cInf_eq_minimum) 285 286lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y" 287 by (auto intro!: cInf_eq_minimum) 288 289lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> INFIMUM A f \<le> f x" 290 using cInf_lower [of _ "f ` A"] by simp 291 292lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFIMUM A f" 293 using cInf_greatest [of "f ` A"] by auto 294 295lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPREMUM A f" 296 using cSup_upper [of _ "f ` A"] by simp 297 298lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPREMUM A f \<le> M" 299 using cSup_least [of "f ` A"] by auto 300 301lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFIMUM A f \<le> u" 302 by (auto intro: cINF_lower order_trans) 303 304lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPREMUM A f" 305 by (auto intro: cSUP_upper order_trans) 306 307lemma cSUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>x\<in>A. c) = c" 308 by (intro antisym cSUP_least) (auto intro: cSUP_upper) 309 310lemma cINF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>x\<in>A. c) = c" 311 by (intro antisym cINF_greatest) (auto intro: cINF_lower) 312 313lemma le_cINF_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> u \<le> INFIMUM A f \<longleftrightarrow> (\<forall>x\<in>A. u \<le> f x)" 314 by (metis cINF_greatest cINF_lower order_trans) 315 316lemma cSUP_le_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPREMUM A f \<le> u \<longleftrightarrow> (\<forall>x\<in>A. f x \<le> u)" 317 by (metis cSUP_least cSUP_upper order_trans) 318 319lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (\<Sqinter>i\<in>A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i" 320 by (metis cINF_lower less_le_trans) 321 322lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (\<Squnion>i\<in>A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y" 323 by (metis cSUP_upper le_less_trans) 324 325lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFIMUM (insert a A) f = inf (f a) (INFIMUM A f)" 326 by (metis cInf_insert image_insert image_is_empty) 327 328lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPREMUM (insert a A) f = sup (f a) (SUPREMUM A f)" 329 by (metis cSup_insert image_insert image_is_empty) 330 331lemma 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> INFIMUM A f \<le> INFIMUM B g" 332 using cInf_mono [of "g ` B" "f ` A"] by auto 333 334lemma 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> SUPREMUM A f \<le> SUPREMUM B g" 335 using cSup_mono [of "f ` A" "g ` B"] by auto 336 337lemma 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> INFIMUM B g \<le> INFIMUM A f" 338 by (rule cINF_mono) auto 339 340lemma 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> SUPREMUM A f \<le> SUPREMUM B g" 341 by (rule cSUP_mono) auto 342 343lemma 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)" 344 by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1) 345 346lemma 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) " 347 by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1) 348 349lemma 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)" 350 by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower) 351 352lemma cINF_union: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_below (f`B) \<Longrightarrow> INFIMUM (A \<union> B) f = inf (INFIMUM A f) (INFIMUM B f)" 353 using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric]) 354 355lemma 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)" 356 by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper) 357 358lemma cSUP_union: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> B \<noteq> {} \<Longrightarrow> bdd_above (f`B) \<Longrightarrow> SUPREMUM (A \<union> B) f = sup (SUPREMUM A f) (SUPREMUM B f)" 359 using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric]) 360 361lemma cINF_inf_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> bdd_below (g`A) \<Longrightarrow> inf (INFIMUM A f) (INFIMUM A g) = (\<Sqinter>a\<in>A. inf (f a) (g a))" 362 by (intro antisym le_infI cINF_greatest cINF_lower2) 363 (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI) 364 365lemma SUP_sup_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> bdd_above (g`A) \<Longrightarrow> sup (SUPREMUM A f) (SUPREMUM A g) = (\<Squnion>a\<in>A. sup (f a) (g a))" 366 by (intro antisym le_supI cSUP_least cSUP_upper2) 367 (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI) 368 369lemma cInf_le_cSup: 370 "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<le> Sup A" 371 by (auto intro!: cSup_upper2[of "SOME a. a \<in> A"] intro: someI cInf_lower) 372 373end 374 375instance complete_lattice \<subseteq> conditionally_complete_lattice 376 by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest) 377 378lemma cSup_eq: 379 fixes a :: "'a :: {conditionally_complete_lattice, no_bot}" 380 assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a" 381 assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y" 382 shows "Sup X = a" 383proof cases 384 assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le) 385qed (intro cSup_eq_non_empty assms) 386 387lemma cInf_eq: 388 fixes a :: "'a :: {conditionally_complete_lattice, no_top}" 389 assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x" 390 assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a" 391 shows "Inf X = a" 392proof cases 393 assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le) 394qed (intro cInf_eq_non_empty assms) 395 396class conditionally_complete_linorder = conditionally_complete_lattice + linorder 397begin 398 399lemma less_cSup_iff: 400 "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)" 401 by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans) 402 403lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)" 404 by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans) 405 406lemma 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)" 407 using cInf_less_iff[of "f`A"] by auto 408 409lemma 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)" 410 using less_cSup_iff[of "f`A"] by auto 411 412lemma less_cSupE: 413 assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x" 414 by (metis cSup_least assms not_le that) 415 416lemma less_cSupD: 417 "X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x" 418 by (metis less_cSup_iff not_le_imp_less bdd_above_def) 419 420lemma cInf_lessD: 421 "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z" 422 by (metis cInf_less_iff not_le_imp_less bdd_below_def) 423 424lemma complete_interval: 425 assumes "a < b" and "P a" and "\<not> P b" 426 shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and> 427 (\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)" 428proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x}"], auto) 429 show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}" 430 by (rule cSup_upper, auto simp: bdd_above_def) 431 (metis \<open>a < b\<close> \<open>\<not> P b\<close> linear less_le) 432next 433 show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b" 434 apply (rule cSup_least) 435 apply auto 436 apply (metis less_le_not_le) 437 apply (metis \<open>a<b\<close> \<open>\<not> P b\<close> linear less_le) 438 done 439next 440 fix x 441 assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}" 442 show "P x" 443 apply (rule less_cSupE [OF lt], auto) 444 apply (metis less_le_not_le) 445 apply (metis x) 446 done 447next 448 fix d 449 assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x" 450 thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}" 451 by (rule_tac cSup_upper, auto simp: bdd_above_def) 452 (metis \<open>a<b\<close> \<open>\<not> P b\<close> linear less_le) 453qed 454 455end 456 457instance complete_linorder < conditionally_complete_linorder 458 .. 459 460lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X" 461 using cSup_eq_Sup_fin[of X] by (simp add: Sup_fin_Max) 462 463lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X" 464 using cInf_eq_Inf_fin[of X] by (simp add: Inf_fin_Min) 465 466lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x" 467 by (auto intro!: cSup_eq_non_empty intro: dense_le) 468 469lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x" 470 by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded) 471 472lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x" 473 by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded) 474 475lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x" 476 by (auto intro!: cInf_eq_non_empty intro: dense_ge) 477 478lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y" 479 by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded) 480 481lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y" 482 by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded) 483 484class linear_continuum = conditionally_complete_linorder + dense_linorder + 485 assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b" 486begin 487 488lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a" 489 by (metis UNIV_not_singleton neq_iff) 490 491end 492 493instantiation nat :: conditionally_complete_linorder 494begin 495 496definition "Sup (X::nat set) = Max X" 497definition "Inf (X::nat set) = (LEAST n. n \<in> X)" 498 499lemma bdd_above_nat: "bdd_above X \<longleftrightarrow> finite (X::nat set)" 500proof 501 assume "bdd_above X" 502 then obtain z where "X \<subseteq> {.. z}" 503 by (auto simp: bdd_above_def) 504 then show "finite X" 505 by (rule finite_subset) simp 506qed simp 507 508instance 509proof 510 fix x :: nat 511 fix X :: "nat set" 512 show "Inf X \<le> x" if "x \<in> X" "bdd_below X" 513 using that by (simp add: Inf_nat_def Least_le) 514 show "x \<le> Inf X" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y" 515 using that unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex) 516 show "x \<le> Sup X" if "x \<in> X" "bdd_above X" 517 using that by (simp add: Sup_nat_def bdd_above_nat) 518 show "Sup X \<le> x" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x" 519 proof - 520 from that have "bdd_above X" 521 by (auto simp: bdd_above_def) 522 with that show ?thesis 523 by (simp add: Sup_nat_def bdd_above_nat) 524 qed 525qed 526 527end 528 529lemma Inf_nat_def1: 530 fixes K::"nat set" 531 assumes "K \<noteq> {}" 532 shows "Inf K \<in> K" 533by (auto simp add: Min_def Inf_nat_def) (meson LeastI assms bot.extremum_unique subsetI) 534 535 536instantiation int :: conditionally_complete_linorder 537begin 538 539definition "Sup (X::int set) = (THE x. x \<in> X \<and> (\<forall>y\<in>X. y \<le> x))" 540definition "Inf (X::int set) = - (Sup (uminus ` X))" 541 542instance 543proof 544 { fix x :: int and X :: "int set" assume "X \<noteq> {}" "bdd_above X" 545 then obtain x y where "X \<subseteq> {..y}" "x \<in> X" 546 by (auto simp: bdd_above_def) 547 then have *: "finite (X \<inter> {x..y})" "X \<inter> {x..y} \<noteq> {}" and "x \<le> y" 548 by (auto simp: subset_eq) 549 have "\<exists>!x\<in>X. (\<forall>y\<in>X. y \<le> x)" 550 proof 551 { fix z assume "z \<in> X" 552 have "z \<le> Max (X \<inter> {x..y})" 553 proof cases 554 assume "x \<le> z" with \<open>z \<in> X\<close> \<open>X \<subseteq> {..y}\<close> *(1) show ?thesis 555 by (auto intro!: Max_ge) 556 next 557 assume "\<not> x \<le> z" 558 then have "z < x" by simp 559 also have "x \<le> Max (X \<inter> {x..y})" 560 using \<open>x \<in> X\<close> *(1) \<open>x \<le> y\<close> by (intro Max_ge) auto 561 finally show ?thesis by simp 562 qed } 563 note le = this 564 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 565 566 fix z assume *: "z \<in> X \<and> (\<forall>y\<in>X. y \<le> z)" 567 with le have "z \<le> Max (X \<inter> {x..y})" 568 by auto 569 moreover have "Max (X \<inter> {x..y}) \<le> z" 570 using * ex by auto 571 ultimately show "z = Max (X \<inter> {x..y})" 572 by auto 573 qed 574 then have "Sup X \<in> X \<and> (\<forall>y\<in>X. y \<le> Sup X)" 575 unfolding Sup_int_def by (rule theI') } 576 note Sup_int = this 577 578 { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X" 579 using Sup_int[of X] by auto } 580 note le_Sup = this 581 { 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" 582 using Sup_int[of X] by (auto simp: bdd_above_def) } 583 note Sup_le = this 584 585 { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x" 586 using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) } 587 { 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" 588 using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) } 589qed 590end 591 592lemma interval_cases: 593 fixes S :: "'a :: conditionally_complete_linorder set" 594 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" 595 shows "\<exists>a b. S = {} \<or> 596 S = UNIV \<or> 597 S = {..<b} \<or> 598 S = {..b} \<or> 599 S = {a<..} \<or> 600 S = {a..} \<or> 601 S = {a<..<b} \<or> 602 S = {a<..b} \<or> 603 S = {a..<b} \<or> 604 S = {a..b}" 605proof - 606 define lower upper where "lower = {x. \<exists>s\<in>S. s \<le> x}" and "upper = {x. \<exists>s\<in>S. x \<le> s}" 607 with ivl have "S = lower \<inter> upper" 608 by auto 609 moreover 610 have "\<exists>a. upper = UNIV \<or> upper = {} \<or> upper = {.. a} \<or> upper = {..< a}" 611 proof cases 612 assume *: "bdd_above S \<and> S \<noteq> {}" 613 from * have "upper \<subseteq> {.. Sup S}" 614 by (auto simp: upper_def intro: cSup_upper2) 615 moreover from * have "{..< Sup S} \<subseteq> upper" 616 by (force simp add: less_cSup_iff upper_def subset_eq Ball_def) 617 ultimately have "upper = {.. Sup S} \<or> upper = {..< Sup S}" 618 unfolding ivl_disj_un(2)[symmetric] by auto 619 then show ?thesis by auto 620 next 621 assume "\<not> (bdd_above S \<and> S \<noteq> {})" 622 then have "upper = UNIV \<or> upper = {}" 623 by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le) 624 then show ?thesis 625 by auto 626 qed 627 moreover 628 have "\<exists>b. lower = UNIV \<or> lower = {} \<or> lower = {b ..} \<or> lower = {b <..}" 629 proof cases 630 assume *: "bdd_below S \<and> S \<noteq> {}" 631 from * have "lower \<subseteq> {Inf S ..}" 632 by (auto simp: lower_def intro: cInf_lower2) 633 moreover from * have "{Inf S <..} \<subseteq> lower" 634 by (force simp add: cInf_less_iff lower_def subset_eq Ball_def) 635 ultimately have "lower = {Inf S ..} \<or> lower = {Inf S <..}" 636 unfolding ivl_disj_un(1)[symmetric] by auto 637 then show ?thesis by auto 638 next 639 assume "\<not> (bdd_below S \<and> S \<noteq> {})" 640 then have "lower = UNIV \<or> lower = {}" 641 by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le) 642 then show ?thesis 643 by auto 644 qed 645 ultimately show ?thesis 646 unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def 647 by (metis inf_bot_left inf_bot_right inf_top.left_neutral inf_top.right_neutral) 648qed 649 650lemma cSUP_eq_cINF_D: 651 fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice" 652 assumes eq: "(\<Squnion>x\<in>A. f x) = (\<Sqinter>x\<in>A. f x)" 653 and bdd: "bdd_above (f ` A)" "bdd_below (f ` A)" 654 and a: "a \<in> A" 655 shows "f a = (\<Sqinter>x\<in>A. f x)" 656apply (rule antisym) 657using a bdd 658apply (auto simp: cINF_lower) 659apply (metis eq cSUP_upper) 660done 661 662lemma cSUP_UNION: 663 fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice" 664 assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}" 665 and bdd_UN: "bdd_above (\<Union>x\<in>A. f ` B x)" 666 shows "(\<Squnion>z \<in> \<Union>x\<in>A. B x. f z) = (\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z)" 667proof - 668 have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_above (f ` B x)" 669 using bdd_UN by (meson UN_upper bdd_above_mono) 670 obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<le> M" 671 using bdd_UN by (auto simp: bdd_above_def) 672 then have bdd2: "bdd_above ((\<lambda>x. \<Squnion>z\<in>B x. f z) ` A)" 673 unfolding bdd_above_def by (force simp: bdd cSUP_le_iff ne(2)) 674 have "(\<Squnion>z \<in> \<Union>x\<in>A. B x. f z) \<le> (\<Squnion>x\<in>A. \<Squnion>z\<in>B x. f z)" 675 using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper2 simp: bdd2 bdd) 676 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)" 677 using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper simp: image_UN bdd_UN) 678 ultimately show ?thesis 679 by (rule order_antisym) 680qed 681 682lemma cINF_UNION: 683 fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice" 684 assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}" 685 and bdd_UN: "bdd_below (\<Union>x\<in>A. f ` B x)" 686 shows "(\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z) = (\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z)" 687proof - 688 have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_below (f ` B x)" 689 using bdd_UN by (meson UN_upper bdd_below_mono) 690 obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<ge> M" 691 using bdd_UN by (auto simp: bdd_below_def) 692 then have bdd2: "bdd_below ((\<lambda>x. \<Sqinter>z\<in>B x. f z) ` A)" 693 unfolding bdd_below_def by (force simp: bdd le_cINF_iff ne(2)) 694 have "(\<Sqinter>z \<in> \<Union>x\<in>A. B x. f z) \<le> (\<Sqinter>x\<in>A. \<Sqinter>z\<in>B x. f z)" 695 using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower simp: bdd2 bdd) 696 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)" 697 using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower2 simp: bdd bdd_UN bdd2) 698 ultimately show ?thesis 699 by (rule order_antisym) 700qed 701 702lemma cSup_abs_le: 703 fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set" 704 shows "S \<noteq> {} \<Longrightarrow> (\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a" 705 apply (auto simp add: abs_le_iff intro: cSup_least) 706 by (metis bdd_aboveI cSup_upper neg_le_iff_le order_trans) 707 708end 709