src/HOL/Conditionally_Complete_Lattices.thy
 author haftmann Wed Apr 12 09:27:47 2017 +0200 (2017-04-12) changeset 65466 b0f89998c2a1 parent 63540 f8652d0534fa child 67091 1393c2340eec permissions -rw-r--r--
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```     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 Finite_Set Lattices_Big Set_Interval
```
```    11 begin
```
```    12
```
```    13 context linorder
```
```    14 begin
```
```    15
```
```    16 lemma Sup_fin_eq_Max:
```
```    17   "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X"
```
```    18   by (induct X rule: finite_ne_induct) (simp_all add: sup_max)
```
```    19
```
```    20 lemma Inf_fin_eq_Min:
```
```    21   "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"
```
```    22   by (induct X rule: finite_ne_induct) (simp_all add: inf_min)
```
```    23
```
```    24 end
```
```    25
```
```    26 context preorder
```
```    27 begin
```
```    28
```
```    29 definition "bdd_above A \<longleftrightarrow> (\<exists>M. \<forall>x \<in> A. x \<le> M)"
```
```    30 definition "bdd_below A \<longleftrightarrow> (\<exists>m. \<forall>x \<in> A. m \<le> x)"
```
```    31
```
```    32 lemma bdd_aboveI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> M) \<Longrightarrow> bdd_above A"
```
```    33   by (auto simp: bdd_above_def)
```
```    34
```
```    35 lemma bdd_belowI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> x) \<Longrightarrow> bdd_below A"
```
```    36   by (auto simp: bdd_below_def)
```
```    37
```
```    38 lemma bdd_aboveI2: "(\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> bdd_above (f`A)"
```
```    39   by force
```
```    40
```
```    41 lemma bdd_belowI2: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> bdd_below (f`A)"
```
```    42   by force
```
```    43
```
```    44 lemma bdd_above_empty [simp, intro]: "bdd_above {}"
```
```    45   unfolding bdd_above_def by auto
```
```    46
```
```    47 lemma bdd_below_empty [simp, intro]: "bdd_below {}"
```
```    48   unfolding bdd_below_def by auto
```
```    49
```
```    50 lemma bdd_above_mono: "bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_above A"
```
```    51   by (metis (full_types) bdd_above_def order_class.le_neq_trans psubsetD)
```
```    52
```
```    53 lemma bdd_below_mono: "bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_below A"
```
```    54   by (metis bdd_below_def order_class.le_neq_trans psubsetD)
```
```    55
```
```    56 lemma bdd_above_Int1 [simp]: "bdd_above A \<Longrightarrow> bdd_above (A \<inter> B)"
```
```    57   using bdd_above_mono by auto
```
```    58
```
```    59 lemma bdd_above_Int2 [simp]: "bdd_above B \<Longrightarrow> bdd_above (A \<inter> B)"
```
```    60   using bdd_above_mono by auto
```
```    61
```
```    62 lemma bdd_below_Int1 [simp]: "bdd_below A \<Longrightarrow> bdd_below (A \<inter> B)"
```
```    63   using bdd_below_mono by auto
```
```    64
```
```    65 lemma bdd_below_Int2 [simp]: "bdd_below B \<Longrightarrow> bdd_below (A \<inter> B)"
```
```    66   using bdd_below_mono by auto
```
```    67
```
```    68 lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}"
```
```    69   by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
```
```    70
```
```    71 lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}"
```
```    72   by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
```
```    73
```
```    74 lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}"
```
```    75   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    76
```
```    77 lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}"
```
```    78   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    79
```
```    80 lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}"
```
```    81   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    82
```
```    83 lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}"
```
```    84   by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
```
```    85
```
```    86 lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}"
```
```    87   by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
```
```    88
```
```    89 lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}"
```
```    90   by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
```
```    91
```
```    92 lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}"
```
```    93   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```    94
```
```    95 lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}"
```
```    96   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```    97
```
```    98 lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}"
```
```    99   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```   100
```
```   101 lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}"
```
```   102   by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
```
```   103
```
```   104 end
```
```   105
```
```   106 lemma (in order_top) bdd_above_top[simp, intro!]: "bdd_above A"
```
```   107   by (rule bdd_aboveI[of _ top]) simp
```
```   108
```
```   109 lemma (in order_bot) bdd_above_bot[simp, intro!]: "bdd_below A"
```
```   110   by (rule bdd_belowI[of _ bot]) simp
```
```   111
```
```   112 lemma bdd_above_image_mono: "mono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_above (f`A)"
```
```   113   by (auto simp: bdd_above_def mono_def)
```
```   114
```
```   115 lemma bdd_below_image_mono: "mono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_below (f`A)"
```
```   116   by (auto simp: bdd_below_def mono_def)
```
```   117
```
```   118 lemma bdd_above_image_antimono: "antimono f \<Longrightarrow> bdd_below A \<Longrightarrow> bdd_above (f`A)"
```
```   119   by (auto simp: bdd_above_def bdd_below_def antimono_def)
```
```   120
```
```   121 lemma bdd_below_image_antimono: "antimono f \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below (f`A)"
```
```   122   by (auto simp: bdd_above_def bdd_below_def antimono_def)
```
```   123
```
```   124 lemma
```
```   125   fixes X :: "'a::ordered_ab_group_add set"
```
```   126   shows bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X"
```
```   127     and bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X"
```
```   128   using bdd_above_image_antimono[of uminus X] bdd_below_image_antimono[of uminus "uminus`X"]
```
```   129   using bdd_below_image_antimono[of uminus X] bdd_above_image_antimono[of uminus "uminus`X"]
```
```   130   by (auto simp: antimono_def image_image)
```
```   131
```
```   132 context lattice
```
```   133 begin
```
```   134
```
```   135 lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
```
```   136   by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)
```
```   137
```
```   138 lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
```
```   139   by (auto simp: bdd_below_def intro: le_infI2 inf_le1)
```
```   140
```
```   141 lemma bdd_finite [simp]:
```
```   142   assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
```
```   143   using assms by (induct rule: finite_induct, auto)
```
```   144
```
```   145 lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)"
```
```   146 proof
```
```   147   assume "bdd_above (A \<union> B)"
```
```   148   thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto
```
```   149 next
```
```   150   assume "bdd_above A \<and> bdd_above B"
```
```   151   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
```
```   152   hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
```
```   153   thus "bdd_above (A \<union> B)" unfolding bdd_above_def ..
```
```   154 qed
```
```   155
```
```   156 lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)"
```
```   157 proof
```
```   158   assume "bdd_below (A \<union> B)"
```
```   159   thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto
```
```   160 next
```
```   161   assume "bdd_below A \<and> bdd_below B"
```
```   162   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
```
```   163   hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2)
```
```   164   thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
```
```   165 qed
```
```   166
```
```   167 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)"
```
```   168   by (auto simp: bdd_above_def intro: le_supI1 le_supI2)
```
```   169
```
```   170 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)"
```
```   171   by (auto simp: bdd_below_def intro: le_infI1 le_infI2)
```
```   172
```
```   173 end
```
```   174
```
```   175
```
```   176 text \<open>
```
```   177
```
```   178 To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and
```
```   179 @{const Inf} in theorem names with c.
```
```   180
```
```   181 \<close>
```
```   182
```
```   183 class conditionally_complete_lattice = lattice + Sup + Inf +
```
```   184   assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x"
```
```   185     and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X"
```
```   186   assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X"
```
```   187     and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
```
```   188 begin
```
```   189
```
```   190 lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
```
```   191   by (metis cSup_upper order_trans)
```
```   192
```
```   193 lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
```
```   194   by (metis cInf_lower order_trans)
```
```   195
```
```   196 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"
```
```   197   by (metis cSup_least cSup_upper2)
```
```   198
```
```   199 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"
```
```   200   by (metis cInf_greatest cInf_lower2)
```
```   201
```
```   202 lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
```
```   203   by (metis cSup_least cSup_upper subsetD)
```
```   204
```
```   205 lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A"
```
```   206   by (metis cInf_greatest cInf_lower subsetD)
```
```   207
```
```   208 lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
```
```   209   by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
```
```   210
```
```   211 lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
```
```   212   by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto
```
```   213
```
```   214 lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
```
```   215   by (metis order_trans cSup_upper cSup_least)
```
```   216
```
```   217 lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
```
```   218   by (metis order_trans cInf_lower cInf_greatest)
```
```   219
```
```   220 lemma cSup_eq_non_empty:
```
```   221   assumes 1: "X \<noteq> {}"
```
```   222   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
```
```   223   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
```
```   224   shows "Sup X = a"
```
```   225   by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper)
```
```   226
```
```   227 lemma cInf_eq_non_empty:
```
```   228   assumes 1: "X \<noteq> {}"
```
```   229   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
```
```   230   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
```
```   231   shows "Inf X = a"
```
```   232   by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower)
```
```   233
```
```   234 lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
```
```   235   by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)
```
```   236
```
```   237 lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
```
```   238   by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
```
```   239
```
```   240 lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
```
```   241   by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
```
```   242
```
```   243 lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
```
```   244   by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
```
```   245
```
```   246 lemma cSup_singleton [simp]: "Sup {x} = x"
```
```   247   by (intro cSup_eq_maximum) auto
```
```   248
```
```   249 lemma cInf_singleton [simp]: "Inf {x} = x"
```
```   250   by (intro cInf_eq_minimum) auto
```
```   251
```
```   252 lemma cSup_insert_If:  "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
```
```   253   using cSup_insert[of X] by simp
```
```   254
```
```   255 lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
```
```   256   using cInf_insert[of X] by simp
```
```   257
```
```   258 lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
```
```   259 proof (induct X arbitrary: x rule: finite_induct)
```
```   260   case (insert x X y) then show ?case
```
```   261     by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
```
```   262 qed simp
```
```   263
```
```   264 lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
```
```   265 proof (induct X arbitrary: x rule: finite_induct)
```
```   266   case (insert x X y) then show ?case
```
```   267     by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
```
```   268 qed simp
```
```   269
```
```   270 lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
```
```   271   by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
```
```   272
```
```   273 lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
```
```   274   by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
```
```   275
```
```   276 lemma cSup_atMost[simp]: "Sup {..x} = x"
```
```   277   by (auto intro!: cSup_eq_maximum)
```
```   278
```
```   279 lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x"
```
```   280   by (auto intro!: cSup_eq_maximum)
```
```   281
```
```   282 lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x"
```
```   283   by (auto intro!: cSup_eq_maximum)
```
```   284
```
```   285 lemma cInf_atLeast[simp]: "Inf {x..} = x"
```
```   286   by (auto intro!: cInf_eq_minimum)
```
```   287
```
```   288 lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y"
```
```   289   by (auto intro!: cInf_eq_minimum)
```
```   290
```
```   291 lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
```
```   292   by (auto intro!: cInf_eq_minimum)
```
```   293
```
```   294 lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> INFIMUM A f \<le> f x"
```
```   295   using cInf_lower [of _ "f ` A"] by simp
```
```   296
```
```   297 lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFIMUM A f"
```
```   298   using cInf_greatest [of "f ` A"] by auto
```
```   299
```
```   300 lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPREMUM A f"
```
```   301   using cSup_upper [of _ "f ` A"] by simp
```
```   302
```
```   303 lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPREMUM A f \<le> M"
```
```   304   using cSup_least [of "f ` A"] by auto
```
```   305
```
```   306 lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFIMUM A f \<le> u"
```
```   307   by (auto intro: cINF_lower order_trans)
```
```   308
```
```   309 lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPREMUM A f"
```
```   310   by (auto intro: cSUP_upper order_trans)
```
```   311
```
```   312 lemma cSUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (SUP x:A. c) = c"
```
```   313   by (intro antisym cSUP_least) (auto intro: cSUP_upper)
```
```   314
```
```   315 lemma cINF_const [simp]: "A \<noteq> {} \<Longrightarrow> (INF x:A. c) = c"
```
```   316   by (intro antisym cINF_greatest) (auto intro: cINF_lower)
```
```   317
```
```   318 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)"
```
```   319   by (metis cINF_greatest cINF_lower order_trans)
```
```   320
```
```   321 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)"
```
```   322   by (metis cSUP_least cSUP_upper order_trans)
```
```   323
```
```   324 lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (INF i:A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i"
```
```   325   by (metis cINF_lower less_le_trans)
```
```   326
```
```   327 lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (SUP i:A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y"
```
```   328   by (metis cSUP_upper le_less_trans)
```
```   329
```
```   330 lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFIMUM (insert a A) f = inf (f a) (INFIMUM A f)"
```
```   331   by (metis cInf_insert image_insert image_is_empty)
```
```   332
```
```   333 lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPREMUM (insert a A) f = sup (f a) (SUPREMUM A f)"
```
```   334   by (metis cSup_insert image_insert image_is_empty)
```
```   335
```
```   336 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"
```
```   337   using cInf_mono [of "g ` B" "f ` A"] by auto
```
```   338
```
```   339 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"
```
```   340   using cSup_mono [of "f ` A" "g ` B"] by auto
```
```   341
```
```   342 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"
```
```   343   by (rule cINF_mono) auto
```
```   344
```
```   345 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"
```
```   346   by (rule cSUP_mono) auto
```
```   347
```
```   348 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)"
```
```   349   by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
```
```   350
```
```   351 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) "
```
```   352   by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
```
```   353
```
```   354 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)"
```
```   355   by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
```
```   356
```
```   357 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)"
```
```   358   using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
```
```   359
```
```   360 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)"
```
```   361   by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
```
```   362
```
```   363 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)"
```
```   364   using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
```
```   365
```
```   366 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))"
```
```   367   by (intro antisym le_infI cINF_greatest cINF_lower2)
```
```   368      (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
```
```   369
```
```   370 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))"
```
```   371   by (intro antisym le_supI cSUP_least cSUP_upper2)
```
```   372      (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
```
```   373
```
```   374 lemma cInf_le_cSup:
```
```   375   "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<le> Sup A"
```
```   376   by (auto intro!: cSup_upper2[of "SOME a. a \<in> A"] intro: someI cInf_lower)
```
```   377
```
```   378 end
```
```   379
```
```   380 instance complete_lattice \<subseteq> conditionally_complete_lattice
```
```   381   by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
```
```   382
```
```   383 lemma cSup_eq:
```
```   384   fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
```
```   385   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
```
```   386   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
```
```   387   shows "Sup X = a"
```
```   388 proof cases
```
```   389   assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
```
```   390 qed (intro cSup_eq_non_empty assms)
```
```   391
```
```   392 lemma cInf_eq:
```
```   393   fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
```
```   394   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
```
```   395   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
```
```   396   shows "Inf X = a"
```
```   397 proof cases
```
```   398   assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
```
```   399 qed (intro cInf_eq_non_empty assms)
```
```   400
```
```   401 class conditionally_complete_linorder = conditionally_complete_lattice + linorder
```
```   402 begin
```
```   403
```
```   404 lemma less_cSup_iff:
```
```   405   "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
```
```   406   by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
```
```   407
```
```   408 lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
```
```   409   by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
```
```   410
```
```   411 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)"
```
```   412   using cInf_less_iff[of "f`A"] by auto
```
```   413
```
```   414 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)"
```
```   415   using less_cSup_iff[of "f`A"] by auto
```
```   416
```
```   417 lemma less_cSupE:
```
```   418   assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x"
```
```   419   by (metis cSup_least assms not_le that)
```
```   420
```
```   421 lemma less_cSupD:
```
```   422   "X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
```
```   423   by (metis less_cSup_iff not_le_imp_less bdd_above_def)
```
```   424
```
```   425 lemma cInf_lessD:
```
```   426   "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
```
```   427   by (metis cInf_less_iff not_le_imp_less bdd_below_def)
```
```   428
```
```   429 lemma complete_interval:
```
```   430   assumes "a < b" and "P a" and "\<not> P b"
```
```   431   shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and>
```
```   432              (\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
```
```   433 proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
```
```   434   show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   435     by (rule cSup_upper, auto simp: bdd_above_def)
```
```   436        (metis \<open>a < b\<close> \<open>\<not> P b\<close> linear less_le)
```
```   437 next
```
```   438   show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
```
```   439     apply (rule cSup_least)
```
```   440     apply auto
```
```   441     apply (metis less_le_not_le)
```
```   442     apply (metis \<open>a<b\<close> \<open>~ P b\<close> linear less_le)
```
```   443     done
```
```   444 next
```
```   445   fix x
```
```   446   assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   447   show "P x"
```
```   448     apply (rule less_cSupE [OF lt], auto)
```
```   449     apply (metis less_le_not_le)
```
```   450     apply (metis x)
```
```   451     done
```
```   452 next
```
```   453   fix d
```
```   454     assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
```
```   455     thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   456       by (rule_tac cSup_upper, auto simp: bdd_above_def)
```
```   457          (metis \<open>a<b\<close> \<open>~ P b\<close> linear less_le)
```
```   458 qed
```
```   459
```
```   460 end
```
```   461
```
```   462 instance complete_linorder < conditionally_complete_linorder
```
```   463   ..
```
```   464
```
```   465 lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
```
```   466   using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp
```
```   467
```
```   468 lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
```
```   469   using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp
```
```   470
```
```   471 lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
```
```   472   by (auto intro!: cSup_eq_non_empty intro: dense_le)
```
```   473
```
```   474 lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
```
```   475   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
```
```   476
```
```   477 lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
```
```   478   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
```
```   479
```
```   480 lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
```
```   481   by (auto intro!: cInf_eq_non_empty intro: dense_ge)
```
```   482
```
```   483 lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
```
```   484   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
```
```   485
```
```   486 lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
```
```   487   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
```
```   488
```
```   489 class linear_continuum = conditionally_complete_linorder + dense_linorder +
```
```   490   assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
```
```   491 begin
```
```   492
```
```   493 lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
```
```   494   by (metis UNIV_not_singleton neq_iff)
```
```   495
```
```   496 end
```
```   497
```
```   498 instantiation nat :: conditionally_complete_linorder
```
```   499 begin
```
```   500
```
```   501 definition "Sup (X::nat set) = Max X"
```
```   502 definition "Inf (X::nat set) = (LEAST n. n \<in> X)"
```
```   503
```
```   504 lemma bdd_above_nat: "bdd_above X \<longleftrightarrow> finite (X::nat set)"
```
```   505 proof
```
```   506   assume "bdd_above X"
```
```   507   then obtain z where "X \<subseteq> {.. z}"
```
```   508     by (auto simp: bdd_above_def)
```
```   509   then show "finite X"
```
```   510     by (rule finite_subset) simp
```
```   511 qed simp
```
```   512
```
```   513 instance
```
```   514 proof
```
```   515   fix x :: nat
```
```   516   fix X :: "nat set"
```
```   517   show "Inf X \<le> x" if "x \<in> X" "bdd_below X"
```
```   518     using that by (simp add: Inf_nat_def Least_le)
```
```   519   show "x \<le> Inf X" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y"
```
```   520     using that unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex)
```
```   521   show "x \<le> Sup X" if "x \<in> X" "bdd_above X"
```
```   522     using that by (simp add: Sup_nat_def bdd_above_nat)
```
```   523   show "Sup X \<le> x" if "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x"
```
```   524   proof -
```
```   525     from that have "bdd_above X"
```
```   526       by (auto simp: bdd_above_def)
```
```   527     with that show ?thesis
```
```   528       by (simp add: Sup_nat_def bdd_above_nat)
```
```   529   qed
```
```   530 qed
```
```   531
```
```   532 end
```
```   533
```
```   534 instantiation int :: conditionally_complete_linorder
```
```   535 begin
```
```   536
```
```   537 definition "Sup (X::int set) = (THE x. x \<in> X \<and> (\<forall>y\<in>X. y \<le> x))"
```
```   538 definition "Inf (X::int set) = - (Sup (uminus ` X))"
```
```   539
```
```   540 instance
```
```   541 proof
```
```   542   { fix x :: int and X :: "int set" assume "X \<noteq> {}" "bdd_above X"
```
```   543     then obtain x y where "X \<subseteq> {..y}" "x \<in> X"
```
```   544       by (auto simp: bdd_above_def)
```
```   545     then have *: "finite (X \<inter> {x..y})" "X \<inter> {x..y} \<noteq> {}" and "x \<le> y"
```
```   546       by (auto simp: subset_eq)
```
```   547     have "\<exists>!x\<in>X. (\<forall>y\<in>X. y \<le> x)"
```
```   548     proof
```
```   549       { fix z assume "z \<in> X"
```
```   550         have "z \<le> Max (X \<inter> {x..y})"
```
```   551         proof cases
```
```   552           assume "x \<le> z" with \<open>z \<in> X\<close> \<open>X \<subseteq> {..y}\<close> *(1) show ?thesis
```
```   553             by (auto intro!: Max_ge)
```
```   554         next
```
```   555           assume "\<not> x \<le> z"
```
```   556           then have "z < x" by simp
```
```   557           also have "x \<le> Max (X \<inter> {x..y})"
```
```   558             using \<open>x \<in> X\<close> *(1) \<open>x \<le> y\<close> by (intro Max_ge) auto
```
```   559           finally show ?thesis by simp
```
```   560         qed }
```
```   561       note le = this
```
```   562       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
```
```   563
```
```   564       fix z assume *: "z \<in> X \<and> (\<forall>y\<in>X. y \<le> z)"
```
```   565       with le have "z \<le> Max (X \<inter> {x..y})"
```
```   566         by auto
```
```   567       moreover have "Max (X \<inter> {x..y}) \<le> z"
```
```   568         using * ex by auto
```
```   569       ultimately show "z = Max (X \<inter> {x..y})"
```
```   570         by auto
```
```   571     qed
```
```   572     then have "Sup X \<in> X \<and> (\<forall>y\<in>X. y \<le> Sup X)"
```
```   573       unfolding Sup_int_def by (rule theI') }
```
```   574   note Sup_int = this
```
```   575
```
```   576   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
```
```   577       using Sup_int[of X] by auto }
```
```   578   note le_Sup = this
```
```   579   { 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"
```
```   580       using Sup_int[of X] by (auto simp: bdd_above_def) }
```
```   581   note Sup_le = this
```
```   582
```
```   583   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
```
```   584       using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) }
```
```   585   { 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"
```
```   586       using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) }
```
```   587 qed
```
```   588 end
```
```   589
```
```   590 lemma interval_cases:
```
```   591   fixes S :: "'a :: conditionally_complete_linorder set"
```
```   592   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"
```
```   593   shows "\<exists>a b. S = {} \<or>
```
```   594     S = UNIV \<or>
```
```   595     S = {..<b} \<or>
```
```   596     S = {..b} \<or>
```
```   597     S = {a<..} \<or>
```
```   598     S = {a..} \<or>
```
```   599     S = {a<..<b} \<or>
```
```   600     S = {a<..b} \<or>
```
```   601     S = {a..<b} \<or>
```
```   602     S = {a..b}"
```
```   603 proof -
```
```   604   define lower upper where "lower = {x. \<exists>s\<in>S. s \<le> x}" and "upper = {x. \<exists>s\<in>S. x \<le> s}"
```
```   605   with ivl have "S = lower \<inter> upper"
```
```   606     by auto
```
```   607   moreover
```
```   608   have "\<exists>a. upper = UNIV \<or> upper = {} \<or> upper = {.. a} \<or> upper = {..< a}"
```
```   609   proof cases
```
```   610     assume *: "bdd_above S \<and> S \<noteq> {}"
```
```   611     from * have "upper \<subseteq> {.. Sup S}"
```
```   612       by (auto simp: upper_def intro: cSup_upper2)
```
```   613     moreover from * have "{..< Sup S} \<subseteq> upper"
```
```   614       by (force simp add: less_cSup_iff upper_def subset_eq Ball_def)
```
```   615     ultimately have "upper = {.. Sup S} \<or> upper = {..< Sup S}"
```
```   616       unfolding ivl_disj_un(2)[symmetric] by auto
```
```   617     then show ?thesis by auto
```
```   618   next
```
```   619     assume "\<not> (bdd_above S \<and> S \<noteq> {})"
```
```   620     then have "upper = UNIV \<or> upper = {}"
```
```   621       by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le)
```
```   622     then show ?thesis
```
```   623       by auto
```
```   624   qed
```
```   625   moreover
```
```   626   have "\<exists>b. lower = UNIV \<or> lower = {} \<or> lower = {b ..} \<or> lower = {b <..}"
```
```   627   proof cases
```
```   628     assume *: "bdd_below S \<and> S \<noteq> {}"
```
```   629     from * have "lower \<subseteq> {Inf S ..}"
```
```   630       by (auto simp: lower_def intro: cInf_lower2)
```
```   631     moreover from * have "{Inf S <..} \<subseteq> lower"
```
```   632       by (force simp add: cInf_less_iff lower_def subset_eq Ball_def)
```
```   633     ultimately have "lower = {Inf S ..} \<or> lower = {Inf S <..}"
```
```   634       unfolding ivl_disj_un(1)[symmetric] by auto
```
```   635     then show ?thesis by auto
```
```   636   next
```
```   637     assume "\<not> (bdd_below S \<and> S \<noteq> {})"
```
```   638     then have "lower = UNIV \<or> lower = {}"
```
```   639       by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le)
```
```   640     then show ?thesis
```
```   641       by auto
```
```   642   qed
```
```   643   ultimately show ?thesis
```
```   644     unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def
```
```   645     by (metis inf_bot_left inf_bot_right inf_top.left_neutral inf_top.right_neutral)
```
```   646 qed
```
```   647
```
```   648 lemma cSUP_eq_cINF_D:
```
```   649   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
```
```   650   assumes eq: "(SUP x:A. f x) = (INF x:A. f x)"
```
```   651      and bdd: "bdd_above (f ` A)" "bdd_below (f ` A)"
```
```   652      and a: "a \<in> A"
```
```   653   shows "f a = (INF x:A. f x)"
```
```   654 apply (rule antisym)
```
```   655 using a bdd
```
```   656 apply (auto simp: cINF_lower)
```
```   657 apply (metis eq cSUP_upper)
```
```   658 done
```
```   659
```
```   660 lemma cSUP_UNION:
```
```   661   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
```
```   662   assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
```
```   663       and bdd_UN: "bdd_above (\<Union>x\<in>A. f ` B x)"
```
```   664   shows "(SUP z : \<Union>x\<in>A. B x. f z) = (SUP x:A. SUP z:B x. f z)"
```
```   665 proof -
```
```   666   have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_above (f ` B x)"
```
```   667     using bdd_UN by (meson UN_upper bdd_above_mono)
```
```   668   obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<le> M"
```
```   669     using bdd_UN by (auto simp: bdd_above_def)
```
```   670   then have bdd2: "bdd_above ((\<lambda>x. SUP z:B x. f z) ` A)"
```
```   671     unfolding bdd_above_def by (force simp: bdd cSUP_le_iff ne(2))
```
```   672   have "(SUP z:\<Union>x\<in>A. B x. f z) \<le> (SUP x:A. SUP z:B x. f z)"
```
```   673     using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper2 simp: bdd2 bdd)
```
```   674   moreover have "(SUP x:A. SUP z:B x. f z) \<le> (SUP z:\<Union>x\<in>A. B x. f z)"
```
```   675     using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper simp: image_UN bdd_UN)
```
```   676   ultimately show ?thesis
```
```   677     by (rule order_antisym)
```
```   678 qed
```
```   679
```
```   680 lemma cINF_UNION:
```
```   681   fixes f :: "_ \<Rightarrow> 'b::conditionally_complete_lattice"
```
```   682   assumes ne: "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> B(x) \<noteq> {}"
```
```   683       and bdd_UN: "bdd_below (\<Union>x\<in>A. f ` B x)"
```
```   684   shows "(INF z : \<Union>x\<in>A. B x. f z) = (INF x:A. INF z:B x. f z)"
```
```   685 proof -
```
```   686   have bdd: "\<And>x. x \<in> A \<Longrightarrow> bdd_below (f ` B x)"
```
```   687     using bdd_UN by (meson UN_upper bdd_below_mono)
```
```   688   obtain M where "\<And>x y. x \<in> A \<Longrightarrow> y \<in> B(x) \<Longrightarrow> f y \<ge> M"
```
```   689     using bdd_UN by (auto simp: bdd_below_def)
```
```   690   then have bdd2: "bdd_below ((\<lambda>x. INF z:B x. f z) ` A)"
```
```   691     unfolding bdd_below_def by (force simp: bdd le_cINF_iff ne(2))
```
```   692   have "(INF z:\<Union>x\<in>A. B x. f z) \<le> (INF x:A. INF z:B x. f z)"
```
```   693     using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower simp: bdd2 bdd)
```
```   694   moreover have "(INF x:A. INF z:B x. f z) \<le> (INF z:\<Union>x\<in>A. B x. f z)"
```
```   695     using assms  by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower2  simp: bdd bdd_UN bdd2)
```
```   696   ultimately show ?thesis
```
```   697     by (rule order_antisym)
```
```   698 qed
```
```   699
```
```   700 lemma cSup_abs_le:
```
```   701   fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
```
```   702   shows "S \<noteq> {} \<Longrightarrow> (\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
```
```   703   apply (auto simp add: abs_le_iff intro: cSup_least)
```
```   704   by (metis bdd_aboveI cSup_upper neg_le_iff_le order_trans)
```
```   705
```
```   706 end
```