src/HOL/Conditionally_Complete_Lattices.thy
 author haftmann Fri Oct 10 19:55:32 2014 +0200 (2014-10-10) changeset 58646 cd63a4b12a33 parent 57447 87429bdecad5 child 58889 5b7a9633cfa8 permissions -rw-r--r--
specialized specification: avoid trivial instances
```     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 header {* Conditionally-complete Lattices *}
```
```     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_uminus[simp]:
```
```   106   fixes X :: "'a::ordered_ab_group_add set"
```
```   107   shows "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below X"
```
```   108   by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
```
```   109
```
```   110 lemma bdd_below_uminus[simp]:
```
```   111   fixes X :: "'a::ordered_ab_group_add set"
```
```   112   shows"bdd_below (uminus ` X) \<longleftrightarrow> bdd_above X"
```
```   113   by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
```
```   114
```
```   115 context lattice
```
```   116 begin
```
```   117
```
```   118 lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
```
```   119   by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)
```
```   120
```
```   121 lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
```
```   122   by (auto simp: bdd_below_def intro: le_infI2 inf_le1)
```
```   123
```
```   124 lemma bdd_finite [simp]:
```
```   125   assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
```
```   126   using assms by (induct rule: finite_induct, auto)
```
```   127
```
```   128 lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)"
```
```   129 proof
```
```   130   assume "bdd_above (A \<union> B)"
```
```   131   thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto
```
```   132 next
```
```   133   assume "bdd_above A \<and> bdd_above B"
```
```   134   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
```
```   135   hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
```
```   136   thus "bdd_above (A \<union> B)" unfolding bdd_above_def ..
```
```   137 qed
```
```   138
```
```   139 lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)"
```
```   140 proof
```
```   141   assume "bdd_below (A \<union> B)"
```
```   142   thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto
```
```   143 next
```
```   144   assume "bdd_below A \<and> bdd_below B"
```
```   145   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
```
```   146   hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2)
```
```   147   thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
```
```   148 qed
```
```   149
```
```   150 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)"
```
```   151   by (auto simp: bdd_above_def intro: le_supI1 le_supI2)
```
```   152
```
```   153 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)"
```
```   154   by (auto simp: bdd_below_def intro: le_infI1 le_infI2)
```
```   155
```
```   156 end
```
```   157
```
```   158
```
```   159 text {*
```
```   160
```
```   161 To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and
```
```   162 @{const Inf} in theorem names with c.
```
```   163
```
```   164 *}
```
```   165
```
```   166 class conditionally_complete_lattice = lattice + Sup + Inf +
```
```   167   assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x"
```
```   168     and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X"
```
```   169   assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X"
```
```   170     and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
```
```   171 begin
```
```   172
```
```   173 lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
```
```   174   by (metis cSup_upper order_trans)
```
```   175
```
```   176 lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
```
```   177   by (metis cInf_lower order_trans)
```
```   178
```
```   179 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"
```
```   180   by (metis cSup_least cSup_upper2)
```
```   181
```
```   182 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"
```
```   183   by (metis cInf_greatest cInf_lower2)
```
```   184
```
```   185 lemma cSup_subset_mono: "A \<noteq> {} \<Longrightarrow> bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Sup A \<le> Sup B"
```
```   186   by (metis cSup_least cSup_upper subsetD)
```
```   187
```
```   188 lemma cInf_superset_mono: "A \<noteq> {} \<Longrightarrow> bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> Inf B \<le> Inf A"
```
```   189   by (metis cInf_greatest cInf_lower subsetD)
```
```   190
```
```   191 lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
```
```   192   by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
```
```   193
```
```   194 lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
```
```   195   by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto
```
```   196
```
```   197 lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
```
```   198   by (metis order_trans cSup_upper cSup_least)
```
```   199
```
```   200 lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
```
```   201   by (metis order_trans cInf_lower cInf_greatest)
```
```   202
```
```   203 lemma cSup_eq_non_empty:
```
```   204   assumes 1: "X \<noteq> {}"
```
```   205   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
```
```   206   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
```
```   207   shows "Sup X = a"
```
```   208   by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper)
```
```   209
```
```   210 lemma cInf_eq_non_empty:
```
```   211   assumes 1: "X \<noteq> {}"
```
```   212   assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
```
```   213   assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
```
```   214   shows "Inf X = a"
```
```   215   by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower)
```
```   216
```
```   217 lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
```
```   218   by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)
```
```   219
```
```   220 lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
```
```   221   by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
```
```   222
```
```   223 lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
```
```   224   by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
```
```   225
```
```   226 lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
```
```   227   by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
```
```   228
```
```   229 lemma cSup_singleton [simp]: "Sup {x} = x"
```
```   230   by (intro cSup_eq_maximum) auto
```
```   231
```
```   232 lemma cInf_singleton [simp]: "Inf {x} = x"
```
```   233   by (intro cInf_eq_minimum) auto
```
```   234
```
```   235 lemma cSup_insert_If:  "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
```
```   236   using cSup_insert[of X] by simp
```
```   237
```
```   238 lemma cInf_insert_If: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
```
```   239   using cInf_insert[of X] by simp
```
```   240
```
```   241 lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
```
```   242 proof (induct X arbitrary: x rule: finite_induct)
```
```   243   case (insert x X y) then show ?case
```
```   244     by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
```
```   245 qed simp
```
```   246
```
```   247 lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
```
```   248 proof (induct X arbitrary: x rule: finite_induct)
```
```   249   case (insert x X y) then show ?case
```
```   250     by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
```
```   251 qed simp
```
```   252
```
```   253 lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
```
```   254   by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
```
```   255
```
```   256 lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
```
```   257   by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
```
```   258
```
```   259 lemma cSup_atMost[simp]: "Sup {..x} = x"
```
```   260   by (auto intro!: cSup_eq_maximum)
```
```   261
```
```   262 lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x"
```
```   263   by (auto intro!: cSup_eq_maximum)
```
```   264
```
```   265 lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x"
```
```   266   by (auto intro!: cSup_eq_maximum)
```
```   267
```
```   268 lemma cInf_atLeast[simp]: "Inf {x..} = x"
```
```   269   by (auto intro!: cInf_eq_minimum)
```
```   270
```
```   271 lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y"
```
```   272   by (auto intro!: cInf_eq_minimum)
```
```   273
```
```   274 lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y"
```
```   275   by (auto intro!: cInf_eq_minimum)
```
```   276
```
```   277 lemma cINF_lower: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> INFIMUM A f \<le> f x"
```
```   278   using cInf_lower [of _ "f ` A"] by simp
```
```   279
```
```   280 lemma cINF_greatest: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> m \<le> f x) \<Longrightarrow> m \<le> INFIMUM A f"
```
```   281   using cInf_greatest [of "f ` A"] by auto
```
```   282
```
```   283 lemma cSUP_upper: "x \<in> A \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> f x \<le> SUPREMUM A f"
```
```   284   using cSup_upper [of _ "f ` A"] by simp
```
```   285
```
```   286 lemma cSUP_least: "A \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<le> M) \<Longrightarrow> SUPREMUM A f \<le> M"
```
```   287   using cSup_least [of "f ` A"] by auto
```
```   288
```
```   289 lemma cINF_lower2: "bdd_below (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> f x \<le> u \<Longrightarrow> INFIMUM A f \<le> u"
```
```   290   by (auto intro: cINF_lower assms order_trans)
```
```   291
```
```   292 lemma cSUP_upper2: "bdd_above (f ` A) \<Longrightarrow> x \<in> A \<Longrightarrow> u \<le> f x \<Longrightarrow> u \<le> SUPREMUM A f"
```
```   293   by (auto intro: cSUP_upper assms order_trans)
```
```   294
```
```   295 lemma cSUP_const: "A \<noteq> {} \<Longrightarrow> (SUP x:A. c) = c"
```
```   296   by (intro antisym cSUP_least) (auto intro: cSUP_upper)
```
```   297
```
```   298 lemma cINF_const: "A \<noteq> {} \<Longrightarrow> (INF x:A. c) = c"
```
```   299   by (intro antisym cINF_greatest) (auto intro: cINF_lower)
```
```   300
```
```   301 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)"
```
```   302   by (metis cINF_greatest cINF_lower assms order_trans)
```
```   303
```
```   304 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)"
```
```   305   by (metis cSUP_least cSUP_upper assms order_trans)
```
```   306
```
```   307 lemma less_cINF_D: "bdd_below (f`A) \<Longrightarrow> y < (INF i:A. f i) \<Longrightarrow> i \<in> A \<Longrightarrow> y < f i"
```
```   308   by (metis cINF_lower less_le_trans)
```
```   309
```
```   310 lemma cSUP_lessD: "bdd_above (f`A) \<Longrightarrow> (SUP i:A. f i) < y \<Longrightarrow> i \<in> A \<Longrightarrow> f i < y"
```
```   311   by (metis cSUP_upper le_less_trans)
```
```   312
```
```   313 lemma cINF_insert: "A \<noteq> {} \<Longrightarrow> bdd_below (f ` A) \<Longrightarrow> INFIMUM (insert a A) f = inf (f a) (INFIMUM A f)"
```
```   314   by (metis cInf_insert Inf_image_eq image_insert image_is_empty)
```
```   315
```
```   316 lemma cSUP_insert: "A \<noteq> {} \<Longrightarrow> bdd_above (f ` A) \<Longrightarrow> SUPREMUM (insert a A) f = sup (f a) (SUPREMUM A f)"
```
```   317   by (metis cSup_insert Sup_image_eq image_insert image_is_empty)
```
```   318
```
```   319 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"
```
```   320   using cInf_mono [of "g ` B" "f ` A"] by auto
```
```   321
```
```   322 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"
```
```   323   using cSup_mono [of "f ` A" "g ` B"] by auto
```
```   324
```
```   325 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"
```
```   326   by (rule cINF_mono) auto
```
```   327
```
```   328 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"
```
```   329   by (rule cSUP_mono) auto
```
```   330
```
```   331 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)"
```
```   332   by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)
```
```   333
```
```   334 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) "
```
```   335   by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)
```
```   336
```
```   337 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)"
```
```   338   by (intro antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)
```
```   339
```
```   340 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)"
```
```   341   using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
```
```   342
```
```   343 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)"
```
```   344   by (intro antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)
```
```   345
```
```   346 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)"
```
```   347   using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un [symmetric])
```
```   348
```
```   349 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))"
```
```   350   by (intro antisym le_infI cINF_greatest cINF_lower2)
```
```   351      (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)
```
```   352
```
```   353 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))"
```
```   354   by (intro antisym le_supI cSUP_least cSUP_upper2)
```
```   355      (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)
```
```   356
```
```   357 lemma cInf_le_cSup:
```
```   358   "A \<noteq> {} \<Longrightarrow> bdd_above A \<Longrightarrow> bdd_below A \<Longrightarrow> Inf A \<le> Sup A"
```
```   359   by (auto intro!: cSup_upper2[of "SOME a. a \<in> A"] intro: someI cInf_lower)
```
```   360
```
```   361 end
```
```   362
```
```   363 instance complete_lattice \<subseteq> conditionally_complete_lattice
```
```   364   by default (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)
```
```   365
```
```   366 lemma cSup_eq:
```
```   367   fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
```
```   368   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
```
```   369   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y"
```
```   370   shows "Sup X = a"
```
```   371 proof cases
```
```   372   assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
```
```   373 qed (intro cSup_eq_non_empty assms)
```
```   374
```
```   375 lemma cInf_eq:
```
```   376   fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
```
```   377   assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"
```
```   378   assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a"
```
```   379   shows "Inf X = a"
```
```   380 proof cases
```
```   381   assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
```
```   382 qed (intro cInf_eq_non_empty assms)
```
```   383
```
```   384 class conditionally_complete_linorder = conditionally_complete_lattice + linorder
```
```   385 begin
```
```   386
```
```   387 lemma less_cSup_iff : (*REAL_SUP_LE in HOL4*)
```
```   388   "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
```
```   389   by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
```
```   390
```
```   391 lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
```
```   392   by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
```
```   393
```
```   394 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)"
```
```   395   using cInf_less_iff[of "f`A"] by auto
```
```   396
```
```   397 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)"
```
```   398   using less_cSup_iff[of "f`A"] by auto
```
```   399
```
```   400 lemma less_cSupE:
```
```   401   assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x"
```
```   402   by (metis cSup_least assms not_le that)
```
```   403
```
```   404 lemma less_cSupD:
```
```   405   "X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
```
```   406   by (metis less_cSup_iff not_leE bdd_above_def)
```
```   407
```
```   408 lemma cInf_lessD:
```
```   409   "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
```
```   410   by (metis cInf_less_iff not_leE bdd_below_def)
```
```   411
```
```   412 lemma complete_interval:
```
```   413   assumes "a < b" and "P a" and "\<not> P b"
```
```   414   shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and>
```
```   415              (\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
```
```   416 proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
```
```   417   show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   418     by (rule cSup_upper, auto simp: bdd_above_def)
```
```   419        (metis `a < b` `\<not> P b` linear less_le)
```
```   420 next
```
```   421   show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
```
```   422     apply (rule cSup_least)
```
```   423     apply auto
```
```   424     apply (metis less_le_not_le)
```
```   425     apply (metis `a<b` `~ P b` linear less_le)
```
```   426     done
```
```   427 next
```
```   428   fix x
```
```   429   assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   430   show "P x"
```
```   431     apply (rule less_cSupE [OF lt], auto)
```
```   432     apply (metis less_le_not_le)
```
```   433     apply (metis x)
```
```   434     done
```
```   435 next
```
```   436   fix d
```
```   437     assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
```
```   438     thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
```
```   439       by (rule_tac cSup_upper, auto simp: bdd_above_def)
```
```   440          (metis `a<b` `~ P b` linear less_le)
```
```   441 qed
```
```   442
```
```   443 end
```
```   444
```
```   445 lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X"
```
```   446   using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp
```
```   447
```
```   448 lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X"
```
```   449   using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp
```
```   450
```
```   451 lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
```
```   452   by (auto intro!: cSup_eq_non_empty intro: dense_le)
```
```   453
```
```   454 lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
```
```   455   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
```
```   456
```
```   457 lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
```
```   458   by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)
```
```   459
```
```   460 lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
```
```   461   by (auto intro!: cInf_eq_non_empty intro: dense_ge)
```
```   462
```
```   463 lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
```
```   464   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
```
```   465
```
```   466 lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
```
```   467   by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)
```
```   468
```
```   469 class linear_continuum = conditionally_complete_linorder + dense_linorder +
```
```   470   assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b"
```
```   471 begin
```
```   472
```
```   473 lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a"
```
```   474   by (metis UNIV_not_singleton neq_iff)
```
```   475
```
```   476 end
```
```   477
```
```   478 instantiation nat :: conditionally_complete_linorder
```
```   479 begin
```
```   480
```
```   481 definition "Sup (X::nat set) = Max X"
```
```   482 definition "Inf (X::nat set) = (LEAST n. n \<in> X)"
```
```   483
```
```   484 lemma bdd_above_nat: "bdd_above X \<longleftrightarrow> finite (X::nat set)"
```
```   485 proof
```
```   486   assume "bdd_above X"
```
```   487   then obtain z where "X \<subseteq> {.. z}"
```
```   488     by (auto simp: bdd_above_def)
```
```   489   then show "finite X"
```
```   490     by (rule finite_subset) simp
```
```   491 qed simp
```
```   492
```
```   493 instance
```
```   494 proof
```
```   495   fix x :: nat and X :: "nat set"
```
```   496   { assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
```
```   497       by (simp add: Inf_nat_def Least_le) }
```
```   498   { assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> x \<le> y" then show "x \<le> Inf X"
```
```   499       unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex) }
```
```   500   { assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
```
```   501       by (simp add: Sup_nat_def bdd_above_nat) }
```
```   502   { assume "X \<noteq> {}" "\<And>y. y \<in> X \<Longrightarrow> y \<le> x"
```
```   503     moreover then have "bdd_above X"
```
```   504       by (auto simp: bdd_above_def)
```
```   505     ultimately show "Sup X \<le> x"
```
```   506       by (simp add: Sup_nat_def bdd_above_nat) }
```
```   507 qed
```
```   508 end
```
```   509
```
```   510 instantiation int :: conditionally_complete_linorder
```
```   511 begin
```
```   512
```
```   513 definition "Sup (X::int set) = (THE x. x \<in> X \<and> (\<forall>y\<in>X. y \<le> x))"
```
```   514 definition "Inf (X::int set) = - (Sup (uminus ` X))"
```
```   515
```
```   516 instance
```
```   517 proof
```
```   518   { fix x :: int and X :: "int set" assume "X \<noteq> {}" "bdd_above X"
```
```   519     then obtain x y where "X \<subseteq> {..y}" "x \<in> X"
```
```   520       by (auto simp: bdd_above_def)
```
```   521     then have *: "finite (X \<inter> {x..y})" "X \<inter> {x..y} \<noteq> {}" and "x \<le> y"
```
```   522       by (auto simp: subset_eq)
```
```   523     have "\<exists>!x\<in>X. (\<forall>y\<in>X. y \<le> x)"
```
```   524     proof
```
```   525       { fix z assume "z \<in> X"
```
```   526         have "z \<le> Max (X \<inter> {x..y})"
```
```   527         proof cases
```
```   528           assume "x \<le> z" with `z \<in> X` `X \<subseteq> {..y}` *(1) show ?thesis
```
```   529             by (auto intro!: Max_ge)
```
```   530         next
```
```   531           assume "\<not> x \<le> z"
```
```   532           then have "z < x" by simp
```
```   533           also have "x \<le> Max (X \<inter> {x..y})"
```
```   534             using `x \<in> X` *(1) `x \<le> y` by (intro Max_ge) auto
```
```   535           finally show ?thesis by simp
```
```   536         qed }
```
```   537       note le = this
```
```   538       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
```
```   539
```
```   540       fix z assume *: "z \<in> X \<and> (\<forall>y\<in>X. y \<le> z)"
```
```   541       with le have "z \<le> Max (X \<inter> {x..y})"
```
```   542         by auto
```
```   543       moreover have "Max (X \<inter> {x..y}) \<le> z"
```
```   544         using * ex by auto
```
```   545       ultimately show "z = Max (X \<inter> {x..y})"
```
```   546         by auto
```
```   547     qed
```
```   548     then have "Sup X \<in> X \<and> (\<forall>y\<in>X. y \<le> Sup X)"
```
```   549       unfolding Sup_int_def by (rule theI') }
```
```   550   note Sup_int = this
```
```   551
```
```   552   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_above X" then show "x \<le> Sup X"
```
```   553       using Sup_int[of X] by auto }
```
```   554   note le_Sup = this
```
```   555   { 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"
```
```   556       using Sup_int[of X] by (auto simp: bdd_above_def) }
```
```   557   note Sup_le = this
```
```   558
```
```   559   { fix x :: int and X :: "int set" assume "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
```
```   560       using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) }
```
```   561   { 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"
```
```   562       using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) }
```
```   563 qed
```
```   564 end
```
```   565
```
```   566 lemma interval_cases:
```
```   567   fixes S :: "'a :: conditionally_complete_linorder set"
```
```   568   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"
```
```   569   shows "\<exists>a b. S = {} \<or>
```
```   570     S = UNIV \<or>
```
```   571     S = {..<b} \<or>
```
```   572     S = {..b} \<or>
```
```   573     S = {a<..} \<or>
```
```   574     S = {a..} \<or>
```
```   575     S = {a<..<b} \<or>
```
```   576     S = {a<..b} \<or>
```
```   577     S = {a..<b} \<or>
```
```   578     S = {a..b}"
```
```   579 proof -
```
```   580   def lower \<equiv> "{x. \<exists>s\<in>S. s \<le> x}" and upper \<equiv> "{x. \<exists>s\<in>S. x \<le> s}"
```
```   581   with ivl have "S = lower \<inter> upper"
```
```   582     by auto
```
```   583   moreover
```
```   584   have "\<exists>a. upper = UNIV \<or> upper = {} \<or> upper = {.. a} \<or> upper = {..< a}"
```
```   585   proof cases
```
```   586     assume *: "bdd_above S \<and> S \<noteq> {}"
```
```   587     from * have "upper \<subseteq> {.. Sup S}"
```
```   588       by (auto simp: upper_def intro: cSup_upper2)
```
```   589     moreover from * have "{..< Sup S} \<subseteq> upper"
```
```   590       by (force simp add: less_cSup_iff upper_def subset_eq Ball_def)
```
```   591     ultimately have "upper = {.. Sup S} \<or> upper = {..< Sup S}"
```
```   592       unfolding ivl_disj_un(2)[symmetric] by auto
```
```   593     then show ?thesis by auto
```
```   594   next
```
```   595     assume "\<not> (bdd_above S \<and> S \<noteq> {})"
```
```   596     then have "upper = UNIV \<or> upper = {}"
```
```   597       by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le)
```
```   598     then show ?thesis
```
```   599       by auto
```
```   600   qed
```
```   601   moreover
```
```   602   have "\<exists>b. lower = UNIV \<or> lower = {} \<or> lower = {b ..} \<or> lower = {b <..}"
```
```   603   proof cases
```
```   604     assume *: "bdd_below S \<and> S \<noteq> {}"
```
```   605     from * have "lower \<subseteq> {Inf S ..}"
```
```   606       by (auto simp: lower_def intro: cInf_lower2)
```
```   607     moreover from * have "{Inf S <..} \<subseteq> lower"
```
```   608       by (force simp add: cInf_less_iff lower_def subset_eq Ball_def)
```
```   609     ultimately have "lower = {Inf S ..} \<or> lower = {Inf S <..}"
```
```   610       unfolding ivl_disj_un(1)[symmetric] by auto
```
```   611     then show ?thesis by auto
```
```   612   next
```
```   613     assume "\<not> (bdd_below S \<and> S \<noteq> {})"
```
```   614     then have "lower = UNIV \<or> lower = {}"
```
```   615       by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le)
```
```   616     then show ?thesis
```
```   617       by auto
```
```   618   qed
```
```   619   ultimately show ?thesis
```
```   620     unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def
```
```   621     by (elim exE disjE) auto
```
```   622 qed
```
```   623
```
```   624 end
```