src/HOL/Conditionally_Complete_Lattices.thy
 author wenzelm Mon Apr 25 16:09:26 2016 +0200 (2016-04-25) changeset 63040 eb4ddd18d635 parent 62626 de25474ce728 child 63092 a949b2a5f51d permissions -rw-r--r--
eliminated old 'def';
```     1 (*  Title:      HOL/Conditionally_Complete_Lattices.thy
```
```     2     Author:     Amine Chaieb and L C Paulson, University of Cambridge
```
```     3     Author:     Johannes Hölzl, TU München
```
```     4     Author:     Luke S. Serafin, Carnegie Mellon University
```
```     5 *)
```
```     6
```
```     7 section \<open>Conditionally-complete Lattices\<close>
```
```     8
```
```     9 theory Conditionally_Complete_Lattices
```
```    10 imports Main
```
```    11 begin
```
```    12
```
```    13 lemma (in linorder) Sup_fin_eq_Max: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X"
```
```    14   by (induct X rule: finite_ne_induct) (simp_all add: sup_max)
```
```    15
```
```    16 lemma (in linorder) Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"
```
```    17   by (induct X rule: finite_ne_induct) (simp_all add: inf_min)
```
```    18
```
```    19 context preorder
```
```    20 begin
```
```    21
```
```    22 definition "bdd_above A \<longleftrightarrow> (\<exists>M. \<forall>x \<in> A. x \<le> M)"
```
```    23 definition "bdd_below A \<longleftrightarrow> (\<exists>m. \<forall>x \<in> A. m \<le> x)"
```
```    24
```
```    25 lemma bdd_aboveI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> M) \<Longrightarrow> bdd_above A"
```
```    26   by (auto simp: bdd_above_def)
```
```    27
```
```    28 lemma bdd_belowI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> x) \<Longrightarrow> bdd_below A"
```
```    29   by (auto simp: bdd_below_def)
```
```    30
```
```    31 lemma bdd_aboveI2: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> bdd_above (f`A)"
```
```    32   by force
```
```    33
```
```    34 lemma bdd_belowI2: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> bdd_below (f`A)"
```
```    35   by force
```
```    36
```
```    37 lemma bdd_above_empty [simp, intro]: "bdd_above {}"
```
```    38   unfolding bdd_above_def by auto
```
```    39
```
```    40 lemma bdd_below_empty [simp, intro]: "bdd_below {}"
```
```    41   unfolding bdd_below_def by auto
```
```    42
```
```    43 lemma bdd_above_mono: "bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_above A"
```
```    44   by (metis (full_types) bdd_above_def order_class.le_neq_trans psubsetD)
```
```    45
```
```    46 lemma bdd_below_mono: "bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_below A"
```
```    47   by (metis bdd_below_def order_class.le_neq_trans psubsetD)
```
```    48
```
```    49 lemma bdd_above_Int1 [simp]: "bdd_above A \<Longrightarrow> bdd_above (A \<inter> B)"
```
```    50   using bdd_above_mono by auto
```
```    51
```
```    52 lemma bdd_above_Int2 [simp]: "bdd_above B \<Longrightarrow> bdd_above (A \<inter> B)"
```
```    53   using bdd_above_mono by auto
```
```    54
```
```    55 lemma bdd_below_Int1 [simp]: "bdd_below A \<Longrightarrow> bdd_below (A \<inter> B)"
```
```    56   using bdd_below_mono by auto
```
```    57
```
```    58 lemma bdd_below_Int2 [simp]: "bdd_below B \<Longrightarrow> bdd_below (A \<inter> B)"
```
```    59   using bdd_below_mono by auto
```
```    60
```
```    61 lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}"
```
```    62   by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
```
```    63
```
```    64 lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}"
```
```    65   by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
```
```    66
```
```    67 lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}"
```
```    68   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    69
```
```    70 lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}"
```
```    71   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    72
```
```    73 lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}"
```
```    74   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    75
```
```    76 lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}"
```
```    77   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    78
```
```    79 lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}"
```
```    80   by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
```
```    81
```
```    82 lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}"
```
```    83   by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
```
```    84
```
```    85 lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}"
```
```    86   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```    87
```
```    88 lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}"
```
```    89   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```    90
```
```    91 lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}"
```
```    92   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```    93
```
```    94 lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}"
```
```    95   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```    96
```
```    97 end
```
```    98
```
```    99 lemma (in order_top) bdd_above_top[simp, intro!]: "bdd_above A"
```
```   100   by (rule bdd_aboveI[of _ top]) simp
```
```   101
```
```   102 lemma (in order_bot) bdd_above_bot[simp, intro!]: "bdd_below A"
```
```   103   by (rule bdd_belowI[of _ bot]) simp
```
```   104
```
```   105 lemma bdd_above_image_mono: "mono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_above (f`A)"
```
```   106   by (auto simp: bdd_above_def mono_def)
```
```   107
```
```   108 lemma bdd_below_image_mono: "mono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_below (f`A)"
```
```   109   by (auto simp: bdd_below_def mono_def)
```
```   110
```
```   111 lemma bdd_above_image_antimono: "antimono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_above (f`A)"
```
```   112   by (auto simp: bdd_above_def bdd_below_def antimono_def)
```
```   113
```
```   114 lemma bdd_below_image_antimono: "antimono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below (f`A)"
```
```   115   by (auto simp: bdd_above_def bdd_below_def antimono_def)
```
```   116
```
```   117 lemma
```
```   118   fixes X :: "'a::ordered_ab_group_add set"
```
```   119   shows bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X"
```
```   120     and bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X"
```
```   121   using bdd_above_image_antimono[of uminus X] bdd_below_image_antimono[of uminus "uminus`X"]
```
```   122   using bdd_below_image_antimono[of uminus X] bdd_above_image_antimono[of uminus "uminus`X"]
```
```   123   by (auto simp: antimono_def image_image)
```
```   124
```
```   125 context lattice
```
```   126 begin
```
```   127
```
```   128 lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
```
```   129   by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)
```
```   130
```
```   131 lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
```
```   132   by (auto simp: bdd_below_def intro: le_infI2 inf_le1)
```
```   133
```
```   134 lemma bdd_finite [simp]:
```
```   135   assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
```
```   136   using assms by (induct rule: finite_induct, auto)
```
```   137
```
```   138 lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)"
```
```   139 proof
```
```   140   assume "bdd_above (A \<union> B)"
```
```   141   thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto
```
```   142 next
```
```   143   assume "bdd_above A \<and> bdd_above B"
```
```   144   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
```
```   145   hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
```
```   146   thus "bdd_above (A \<union> B)" unfolding bdd_above_def ..
```
```   147 qed
```
```   148
```
```   149 lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)"
```
```   150 proof
```
```   151   assume "bdd_below (A \<union> B)"
```
```   152   thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto
```
```   153 next
```
```   154   assume "bdd_below A \<and> bdd_below B"
```
```   155   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
```
```   156   hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2)
```
```   157   thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
```
```   158 qed
```
```   159
```
```   160 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)"
```
```   161   by (auto simp: bdd_above_def intro: le_supI1 le_supI2)
```
```   162
```
```   163 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)"
```
```   164   by (auto simp: bdd_below_def intro: le_infI1 le_infI2)
```
```   165
```
```   166 end
```
```   167
```
```   168
```
```   169 text \<open>
```
```   170
```
```   171 To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and
```
```   172 @{const Inf} in theorem names with c.
```
```   173
```
```   174 \<close>
```
```   175
```
```   176 class conditionally_complete_lattice = lattice + Sup + Inf +
```
```   177   assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x"
```
```   178     and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X"
```
```   179   assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X"
```
```   180     and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
```
```   181 begin
```
```   182
```
```   183 lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
```
```   184   by (metis cSup_upper order_trans)
```
```   185
```
```   186 lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
```
```   187   by (metis cInf_lower order_trans)
```
```   188
```
```   189 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"
```
```   190   by (metis cSup_least cSup_upper2)
```
```   191
```
```   192 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"
```
```   193   by (metis cInf_greatest cInf_lower2)
```
```   194
```
```   195 lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
```
```   196   by (metis cSup_least cSup_upper subsetD)
```
```   197
```
```   198 lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A"
```
```   199   by (metis cInf_greatest cInf_lower subsetD)
```
```   200
```
```   201 lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
```
```   202   by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
```
```   203
```
```   204 lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
```
```   205   by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto
```
```   206
```
```   207 lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
```
```   208   by (metis order_trans cSup_upper cSup_least)
```
```   209
```
```   210 lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
```
```   211   by (metis order_trans cInf_lower cInf_greatest)
```
```   212
```
```   213 lemma cSup_eq_non_empty:
```
```   214   assumes 1: "X \<noteq> {}"
```
```   215   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
```
```   216   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
```
```   217   shows "Sup X = a"
```
```   218   by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper)
```
```   219
```
```   220 lemma cInf_eq_non_empty:
```
```   221   assumes 1: "X \<noteq> {}"
```
```   222   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
```
```   223   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
```
```   224   shows "Inf X = a"
```
```   225   by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower)
```
```   226
```
```   227 lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
```
```   228   by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)
```
```   229
```
```   230 lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
```
```   231   by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
```
```   232
```
```   233 lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
```
```   234   by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
```
```   235
```
```   236 lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
```
```   237   by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
```
```   238
```
```   239 lemma cSup_singleton [simp]: "Sup {x} = x"
```
```   240   by (intro cSup_eq_maximum) auto
```
```   241
```
```   242 lemma cInf_singleton [simp]: "Inf {x} = x"
```
```   243   by (intro cInf_eq_minimum) auto
```
```   244
```
```   245 lemma cSup_insert_If:  "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
```
```   246   using cSup_insert[of X] by simp
```
```   247
```
```   248 lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
```
```   249   using cInf_insert[of X] by simp
```
```   250
```
```   251 lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
```
```   252 proof (induct X arbitrary: x rule: finite_induct)
```
```   253   case (insert x X y) then show ?case
```
```   254     by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
```
```   255 qed simp
```
```   256
```
```   257 lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
```
```   258 proof (induct X arbitrary: x rule: finite_induct)
```
```   259   case (insert x X y) then show ?case
```
```   260     by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
```
```   261 qed simp
```
```   262
```
```   263 lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
```
```   264   by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
```
```   265
```
```   266 lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
```
```   267   by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
```
```   268
```
```   269 lemma cSup_atMost[simp]: "Sup {..x} = x"
```
```   270   by (auto intro!: cSup_eq_maximum)
```
```   271
```
```   272 lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x"
```
```   273   by (auto intro!: cSup_eq_maximum)
```
```   274
```
```   275 lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x"
```
```   276   by (auto intro!: cSup_eq_maximum)
```
```   277
```
```   278 lemma cInf_atLeast[simp]: "Inf {x..} = x"
```
```   279   by (auto intro!: cInf_eq_minimum)
```
```   280
```
```   281 lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y"
```
```   282   by (auto intro!: cInf_eq_minimum)
```
```   283
```
```   284 lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
```
```   285   by (auto intro!: cInf_eq_minimum)
```
```   286
```
```   287 lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> INFIMUM A f \<le> f x"
```
```   288   using cInf_lower [of _ "f ` A"] by simp
```
```   289
```
```   290 lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFIMUM A f"
```
```   291   using cInf_greatest [of "f ` A"] by auto
```
```   292
```
```   293 lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPREMUM A f"
```
```   294   using cSup_upper [of _ "f ` A"] by simp
```
```   295
```
```   296 lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPREMUM A f \<le> M"
```
```   297   using cSup_least [of "f ` A"] by auto
```
```   298
```
```   299 lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFIMUM A f \<le> u"
```
```   300   by (auto intro: cINF_lower assms order_trans)
```
```   301
```
```   302 lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPREMUM A f"
```
```   303   by (auto intro: cSUP_upper assms order_trans)
```
```   304
```
```   305 lemma cSUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (SUP x:A. c) = c"
```
```   306   by (intro antisym cSUP_least) (auto intro: cSUP_upper)
```
```   307
```
```   308 lemma cINF_const [simp]: "A \<noteq> {} \<Longrightarrow> (INF x:A. c) = c"
```
```   309   by (intro antisym cINF_greatest) (auto intro: cINF_lower)
```
```   310
```
```   311 lemma 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)"
```
```   312   by (metis cINF_greatest cINF_lower assms order_trans)
```
```   313
```
```   314 lemma 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)"
```
```   315   by (metis cSUP_least cSUP_upper assms order_trans)
```
```   316
```
```   317 lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (INF i:A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i"
```
```   318   by (metis cINF_lower less_le_trans)
```
```   319
```
```   320 lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (SUP i:A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y"
```
```   321   by (metis cSUP_upper le_less_trans)
```
```   322
```
```   323 lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFIMUM (insert a A) f = inf (f a) (INFIMUM A f)"
```
```   324   by (metis cInf_insert image_insert image_is_empty)
```
```   325
```
```   326 lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPREMUM (insert a A) f = sup (f a) (SUPREMUM A f)"
```
```   327   by (metis cSup_insert image_insert image_is_empty)
```
```   328
```
```   329 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> INFIMUM A f \<le> INFIMUM B g"
```
```   330   using cInf_mono [of "g ` B" "f ` A"] by auto
```
```   331
```
```   332 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> SUPREMUM A f \<le> SUPREMUM B g"
```
```   333   using cSup_mono [of "f ` A" "g ` B"] by auto
```
```   334
```
```   335 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> INFIMUM B g \<le> INFIMUM A f"
```
```   336   by (rule cINF_mono) auto
```
```   337
```
```   338 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> SUPREMUM A f \<le> SUPREMUM B g"
```
```   339   by (rule cSUP_mono) auto
```
```   340
```
```   341 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)"
```
```   342   by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
```
```   343
```
```   344 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) "
```
```   345   by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
```
```   346
```
```   347 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)"
```
```   348   by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
```
```   349
```
```   350 lemma 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)"
```
```   351   using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
```
```   352
```
```   353 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)"
```
```   354   by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
```
```   355
```
```   356 lemma 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)"
```
```   357   using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
```
```   358
```
```   359 lemma cINF_inf_distrib: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> bdd_below (g`A) \<Longrightarrow> inf (INFIMUM A f) (INFIMUM A g) = (INF a:A. inf (f a) (g a))"
```
```   360   by (intro antisym le_infI cINF_greatest cINF_lower2)
```
```   361      (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
```
```   362
```
```   363 lemma SUP_sup_distrib: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> bdd_above (g`A) \<Longrightarrow> sup (SUPREMUM A f) (SUPREMUM A g) = (SUP a:A. sup (f a) (g a))"
```
```   364   by (intro antisym le_supI cSUP_least cSUP_upper2)
```
```   365      (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
```
```   366
```
```   367 lemma cInf_le_cSup:
```
```   368   "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<le> Sup A"
```
```   369   by (auto intro!: cSup_upper2[of "SOME a. a \<in> A"] intro: someI cInf_lower)
```
```   370
```
```   371 end
```
```   372
```
```   373 instance complete_lattice \<subseteq> conditionally_complete_lattice
```
```   374   by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
```
```   375
```
```   376 lemma cSup_eq:
```
```   377   fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
```
```   378   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
```
```   379   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
```
```   380   shows "Sup X = a"
```
```   381 proof cases
```
```   382   assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
```
```   383 qed (intro cSup_eq_non_empty assms)
```
```   384
```
```   385 lemma cInf_eq:
```
```   386   fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
```
```   387   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
```
```   388   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
```
```   389   shows "Inf X = a"
```
```   390 proof cases
```
```   391   assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
```
```   392 qed (intro cInf_eq_non_empty assms)
```
```   393
```
```   394 class conditionally_complete_linorder = conditionally_complete_lattice + linorder
```
```   395 begin
```
```   396
```
```   397 lemma less_cSup_iff : (*REAL_SUP_LE in HOL4*)
```
```   398   "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
```
```   399   by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
```
```   400
```
```   401 lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
```
```   402   by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
```
```   403
```
```   404 lemma cINF_less_iff: "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> (INF i:A. f i) < a \<longleftrightarrow> (\<exists>x\<in>A. f x < a)"
```
```   405   using cInf_less_iff[of "f`A"] by auto
```
```   406
```
```   407 lemma less_cSUP_iff: "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> a < (SUP i:A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a < f x)"
```
```   408   using less_cSup_iff[of "f`A"] by auto
```
```   409
```
```   410 lemma less_cSupE:
```
```   411   assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x"
```
```   412   by (metis cSup_least assms not_le that)
```
```   413
```
```   414 lemma less_cSupD:
```
```   415   "X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
```
```   416   by (metis less_cSup_iff not_le_imp_less bdd_above_def)
```
```   417
```
```   418 lemma cInf_lessD:
```
```   419   "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
```
```   420   by (metis cInf_less_iff not_le_imp_less bdd_below_def)
```
```   421
```
```   422 lemma complete_interval:
```
```   423   assumes "a < b" and "P a" and "\<not> P b"
```
```   424   shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and>
```
```   425              (\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
```
```   426 proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
```
```   427   show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   428     by (rule cSup_upper, auto simp: bdd_above_def)
```
```   429        (metis \<open>a < b\<close> \<open>\<not> P b\<close> linear less_le)
```
```   430 next
```
```   431   show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
```
```   432     apply (rule cSup_least)
```
```   433     apply auto
```
```   434     apply (metis less_le_not_le)
```
```   435     apply (metis \<open>a<b\<close> \<open>~ P b\<close> linear less_le)
```
```   436     done
```
```   437 next
```
```   438   fix x
```
```   439   assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   440   show "P x"
```
```   441     apply (rule less_cSupE [OF lt], auto)
```
```   442     apply (metis less_le_not_le)
```
```   443     apply (metis x)
```
```   444     done
```
```   445 next
```
```   446   fix d
```
```   447     assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
```
```   448     thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   449       by (rule_tac cSup_upper, auto simp: bdd_above_def)
```
```   450          (metis \<open>a<b\<close> \<open>~ P b\<close> linear less_le)
```
```   451 qed
```
```   452
```
```   453 end
```
```   454
```
```   455 instance complete_linorder < conditionally_complete_linorder
```
```   456   ..
```
```   457
```
```   458 lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
```
```   459   using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp
```
```   460
```
```   461 lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
```
```   462   using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp
```
```   463
```
```   464 lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
```
```   465   by (auto intro!: cSup_eq_non_empty intro: dense_le)
```
```   466
```
```   467 lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
```
```   468   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
```
```   469
```
```   470 lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
```
```   471   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
```
```   472
```
```   473 lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
```
```   474   by (auto intro!: cInf_eq_non_empty intro: dense_ge)
```
```   475
```
```   476 lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
```
```   477   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
```
```   478
```
```   479 lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
```
```   480   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
```
```   481
```
```   482 class linear_continuum = conditionally_complete_linorder + dense_linorder +
```
```   483   assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
```
```   484 begin
```
```   485
```
```   486 lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
```
```   487   by (metis UNIV_not_singleton neq_iff)
```
```   488
```
```   489 end
```
```   490
```
```   491 instantiation nat :: conditionally_complete_linorder
```
```   492 begin
```
```   493
```
```   494 definition "Sup (X::nat set) = Max X"
```
```   495 definition "Inf (X::nat set) = (LEAST n. n \<in> X)"
```
```   496
```
```   497 lemma bdd_above_nat: "bdd_above X \<longleftrightarrow> finite (X::nat set)"
```
```   498 proof
```
```   499   assume "bdd_above X"
```
```   500   then obtain z where "X \<subseteq> {.. z}"
```
```   501     by (auto simp: bdd_above_def)
```
```   502   then show "finite X"
```
```   503     by (rule finite_subset) simp
```
```   504 qed simp
```
```   505
```
```   506 instance
```
```   507 proof
```
```   508   fix x :: nat and X :: "nat set"
```
```   509   { assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
```
```   510       by (simp add: Inf_nat_def Least_le) }
```
```   511   { assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y" then show "x \<le> Inf X"
```
```   512       unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex) }
```
```   513   { assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
```
```   514       by (simp add: Sup_nat_def bdd_above_nat) }
```
```   515   { assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x"
```
```   516     moreover then have "bdd_above X"
```
```   517       by (auto simp: bdd_above_def)
```
```   518     ultimately show "Sup X \<le> x"
```
```   519       by (simp add: Sup_nat_def bdd_above_nat) }
```
```   520 qed
```
```   521 end
```
```   522
```
```   523 instantiation int :: conditionally_complete_linorder
```
```   524 begin
```
```   525
```
```   526 definition "Sup (X::int set) = (THE x. x \<in> X \<and> (\<forall>y\<in>X. y \<le> x))"
```
```   527 definition "Inf (X::int set) = - (Sup (uminus ` X))"
```
```   528
```
```   529 instance
```
```   530 proof
```
```   531   { fix x :: int and X :: "int set" assume "X \<noteq> {}" "bdd_above X"
```
```   532     then obtain x y where "X \<subseteq> {..y}" "x \<in> X"
```
```   533       by (auto simp: bdd_above_def)
```
```   534     then have *: "finite (X \<inter> {x..y})" "X \<inter> {x..y} \<noteq> {}" and "x \<le> y"
```
```   535       by (auto simp: subset_eq)
```
```   536     have "\<exists>!x\<in>X. (\<forall>y\<in>X. y \<le> x)"
```
```   537     proof
```
```   538       { fix z assume "z \<in> X"
```
```   539         have "z \<le> Max (X \<inter> {x..y})"
```
```   540         proof cases
```
```   541           assume "x \<le> z" with \<open>z \<in> X\<close> \<open>X \<subseteq> {..y}\<close> *(1) show ?thesis
```
```   542             by (auto intro!: Max_ge)
```
```   543         next
```
```   544           assume "\<not> x \<le> z"
```
```   545           then have "z < x" by simp
```
```   546           also have "x \<le> Max (X \<inter> {x..y})"
```
```   547             using \<open>x \<in> X\<close> *(1) \<open>x \<le> y\<close> by (intro Max_ge) auto
```
```   548           finally show ?thesis by simp
```
```   549         qed }
```
```   550       note le = this
```
```   551       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
```
```   552
```
```   553       fix z assume *: "z \<in> X \<and> (\<forall>y\<in>X. y \<le> z)"
```
```   554       with le have "z \<le> Max (X \<inter> {x..y})"
```
```   555         by auto
```
```   556       moreover have "Max (X \<inter> {x..y}) \<le> z"
```
```   557         using * ex by auto
```
```   558       ultimately show "z = Max (X \<inter> {x..y})"
```
```   559         by auto
```
```   560     qed
```
```   561     then have "Sup X \<in> X \<and> (\<forall>y\<in>X. y \<le> Sup X)"
```
```   562       unfolding Sup_int_def by (rule theI') }
```
```   563   note Sup_int = this
```
```   564
```
```   565   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
```
```   566       using Sup_int[of X] by auto }
```
```   567   note le_Sup = this
```
```   568   { 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"
```
```   569       using Sup_int[of X] by (auto simp: bdd_above_def) }
```
```   570   note Sup_le = this
```
```   571
```
```   572   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
```
```   573       using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) }
```
```   574   { 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"
```
```   575       using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) }
```
```   576 qed
```
```   577 end
```
```   578
```
```   579 lemma interval_cases:
```
```   580   fixes S :: "'a :: conditionally_complete_linorder set"
```
```   581   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"
```
```   582   shows "\<exists>a b. S = {} \<or>
```
```   583     S = UNIV \<or>
```
```   584     S = {..<b} \<or>
```
```   585     S = {..b} \<or>
```
```   586     S = {a<..} \<or>
```
```   587     S = {a..} \<or>
```
```   588     S = {a<..<b} \<or>
```
```   589     S = {a<..b} \<or>
```
```   590     S = {a..<b} \<or>
```
```   591     S = {a..b}"
```
```   592 proof -
```
```   593   define lower upper where "lower = {x. \<exists>s\<in>S. s \<le> x}" and "upper = {x. \<exists>s\<in>S. x \<le> s}"
```
```   594   with ivl have "S = lower \<inter> upper"
```
```   595     by auto
```
```   596   moreover
```
```   597   have "\<exists>a. upper = UNIV \<or> upper = {} \<or> upper = {.. a} \<or> upper = {..< a}"
```
```   598   proof cases
```
```   599     assume *: "bdd_above S \<and> S \<noteq> {}"
```
```   600     from * have "upper \<subseteq> {.. Sup S}"
```
```   601       by (auto simp: upper_def intro: cSup_upper2)
```
```   602     moreover from * have "{..< Sup S} \<subseteq> upper"
```
```   603       by (force simp add: less_cSup_iff upper_def subset_eq Ball_def)
```
```   604     ultimately have "upper = {.. Sup S} \<or> upper = {..< Sup S}"
```
```   605       unfolding ivl_disj_un(2)[symmetric] by auto
```
```   606     then show ?thesis by auto
```
```   607   next
```
```   608     assume "\<not> (bdd_above S \<and> S \<noteq> {})"
```
```   609     then have "upper = UNIV \<or> upper = {}"
```
```   610       by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le)
```
```   611     then show ?thesis
```
```   612       by auto
```
```   613   qed
```
```   614   moreover
```
```   615   have "\<exists>b. lower = UNIV \<or> lower = {} \<or> lower = {b ..} \<or> lower = {b <..}"
```
```   616   proof cases
```
```   617     assume *: "bdd_below S \<and> S \<noteq> {}"
```
```   618     from * have "lower \<subseteq> {Inf S ..}"
```
```   619       by (auto simp: lower_def intro: cInf_lower2)
```
```   620     moreover from * have "{Inf S <..} \<subseteq> lower"
```
```   621       by (force simp add: cInf_less_iff lower_def subset_eq Ball_def)
```
```   622     ultimately have "lower = {Inf S ..} \<or> lower = {Inf S <..}"
```
```   623       unfolding ivl_disj_un(1)[symmetric] by auto
```
```   624     then show ?thesis by auto
```
```   625   next
```
```   626     assume "\<not> (bdd_below S \<and> S \<noteq> {})"
```
```   627     then have "lower = UNIV \<or> lower = {}"
```
```   628       by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le)
```
```   629     then show ?thesis
```
```   630       by auto
```
```   631   qed
```
```   632   ultimately show ?thesis
```
```   633     unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def
```
```   634     by (elim exE disjE) auto
```
```   635 qed
```
```   636
```
```   637 lemma cSUP_eq_cINF_D:
```
```   638   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
```
```   639   assumes eq: "(SUP x:A. f x) = (INF x:A. f x)"
```
```   640      and bdd: "bdd_above (f ` A)" "bdd_below (f ` A)"
```
```   641      and a: "a \<in> A"
```
```   642   shows "f a = (INF x:A. f x)"
```
```   643 apply (rule antisym)
```
```   644 using a bdd
```
```   645 apply (auto simp: cINF_lower)
```
```   646 apply (metis eq cSUP_upper)
```
```   647 done
```
```   648
```
```   649 lemma cSUP_UNION:
```
```   650   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
```
```   651   assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
```
```   652       and bdd_UN: "bdd_above (\<Union>x\<in>A. f ` B x)"
```
```   653   shows "(SUP z : \<Union>x\<in>A. B x. f z) = (SUP x:A. SUP z:B x. f z)"
```
```   654 proof -
```
```   655   have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_above (f ` B x)"
```
```   656     using bdd_UN by (meson UN_upper bdd_above_mono)
```
```   657   obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<le> M"
```
```   658     using bdd_UN by (auto simp: bdd_above_def)
```
```   659   then have bdd2: "bdd_above ((\<lambda>x. SUP z:B x. f z) ` A)"
```
```   660     unfolding bdd_above_def by (force simp: bdd cSUP_le_iff ne(2))
```
```   661   have "(SUP z:\<Union>x\<in>A. B x. f z) \<le> (SUP x:A. SUP z:B x. f z)"
```
```   662     using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper2 simp: bdd2 bdd)
```
```   663   moreover have "(SUP x:A. SUP z:B x. f z) \<le> (SUP z:\<Union>x\<in>A. B x. f z)"
```
```   664     using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper simp: image_UN bdd_UN)
```
```   665   ultimately show ?thesis
```
```   666     by (rule order_antisym)
```
```   667 qed
```
```   668
```
```   669 lemma cINF_UNION:
```
```   670   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
```
```   671   assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
```
```   672       and bdd_UN: "bdd_below (\<Union>x\<in>A. f ` B x)"
```
```   673   shows "(INF z : \<Union>x\<in>A. B x. f z) = (INF x:A. INF z:B x. f z)"
```
```   674 proof -
```
```   675   have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_below (f ` B x)"
```
```   676     using bdd_UN by (meson UN_upper bdd_below_mono)
```
```   677   obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<ge> M"
```
```   678     using bdd_UN by (auto simp: bdd_below_def)
```
```   679   then have bdd2: "bdd_below ((\<lambda>x. INF z:B x. f z) ` A)"
```
```   680     unfolding bdd_below_def by (force simp: bdd le_cINF_iff ne(2))
```
```   681   have "(INF z:\<Union>x\<in>A. B x. f z) \<le> (INF x:A. INF z:B x. f z)"
```
```   682     using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower simp: bdd2 bdd)
```
```   683   moreover have "(INF x:A. INF z:B x. f z) \<le> (INF z:\<Union>x\<in>A. B x. f z)"
```
```   684     using assms  by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower2  simp: bdd bdd_UN bdd2)
```
```   685   ultimately show ?thesis
```
```   686     by (rule order_antisym)
```
```   687 qed
```
```   688
```
```   689 lemma cSup_abs_le:
```
```   690   fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
```
```   691   shows "S \<noteq> {} \<Longrightarrow> (\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
```
```   692   apply (auto simp add: abs_le_iff intro: cSup_least)
```
```   693   by (metis bdd_aboveI cSup_upper neg_le_iff_le order_trans)
```
```   694
```
```   695 lemma cInf_abs_ge:
```
```   696   fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
```
```   697   assumes "S \<noteq> {}" and bdd: "\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a"
```
```   698   shows "\<bar>Inf S\<bar> \<le> a"
```
```   699 proof -
```
```   700   have "Sup (uminus ` S) = - (Inf S)"
```
```   701   proof (rule antisym)
```
```   702     show "- (Inf S) \<le> Sup(uminus ` S)"
```
```   703       apply (subst minus_le_iff)
```
```   704       apply (rule cInf_greatest [OF \<open>S \<noteq> {}\<close>])
```
```   705       apply (subst minus_le_iff)
```
```   706       apply (rule cSup_upper, force)
```
```   707       using bdd apply (force simp add: abs_le_iff bdd_above_def)
```
```   708       done
```
```   709   next
```
```   710     show "Sup (uminus ` S) \<le> - Inf S"
```
```   711       apply (rule cSup_least)
```
```   712        using \<open>S \<noteq> {}\<close> apply (force simp add: )
```
```   713       apply clarsimp
```
```   714       apply (rule cInf_lower)
```
```   715       apply assumption
```
```   716       using bdd apply (simp add: bdd_below_def)
```
```   717       apply (rule_tac x="-a" in exI)
```
```   718       apply force
```
```   719       done
```
```   720   qed
```
```   721   with cSup_abs_le [of "uminus ` S"] assms show ?thesis by fastforce
```
```   722 qed
```
```   723
```
```   724 lemma cSup_asclose:
```
```   725   fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
```
```   726   assumes S: "S \<noteq> {}"
```
```   727     and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
```
```   728   shows "\<bar>Sup S - l\<bar> \<le> e"
```
```   729 proof -
```
```   730   have th: "\<And>(x::'a) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e"
```
```   731     by arith
```
```   732   have "bdd_above S"
```
```   733     using b by (auto intro!: bdd_aboveI[of _ "l + e"])
```
```   734   with S b show ?thesis
```
```   735     unfolding th by (auto intro!: cSup_upper2 cSup_least)
```
```   736 qed
```
```   737
```
```   738 lemma cInf_asclose:
```
```   739   fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
```
```   740   assumes S: "S \<noteq> {}"
```
```   741     and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
```
```   742   shows "\<bar>Inf S - l\<bar> \<le> e"
```
```   743 proof -
```
```   744   have th: "\<And>(x::'a) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e"
```
```   745     by arith
```
```   746   have "bdd_below S"
```
```   747     using b by (auto intro!: bdd_belowI[of _ "l - e"])
```
```   748   with S b show ?thesis
```
```   749     unfolding th by (auto intro!: cInf_lower2 cInf_greatest)
```
```   750 qed
```
```   751
```
```   752 end
```