author | hoelzl |
Thu, 25 Apr 2013 11:59:21 +0200 | |
changeset 51775 | 408d937c9486 |
parent 51773 | 9328c6681f3c |
child 52265 | bb907eba5902 |
permissions | -rw-r--r-- |
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(* Title: HOL/Conditional_Complete_Lattices.thy |
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Author: Amine Chaieb and L C Paulson, University of Cambridge |
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Author: Johannes Hölzl, TU München |
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*) |
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header {* Conditionally-complete Lattices *} |
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theory Conditionally_Complete_Lattices |
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imports Main Lubs |
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begin |
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|
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lemma Sup_fin_eq_Max: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup_fin X = Max X" |
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by (induct X rule: finite_ne_induct) (simp_all add: sup_max) |
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|
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lemma Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X" |
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by (induct X rule: finite_ne_induct) (simp_all add: inf_min) |
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|
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text {* |
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|
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To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and |
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@{const Inf} in theorem names with c. |
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|
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*} |
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class conditionally_complete_lattice = lattice + Sup + Inf + |
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assumes cInf_lower: "x \<in> X \<Longrightarrow> (\<And>a. a \<in> X \<Longrightarrow> z \<le> a) \<Longrightarrow> Inf X \<le> x" |
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and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X" |
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assumes cSup_upper: "x \<in> X \<Longrightarrow> (\<And>a. a \<in> X \<Longrightarrow> a \<le> z) \<Longrightarrow> x \<le> Sup X" |
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and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z" |
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begin |
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|
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lemma cSup_eq_maximum: (*REAL_SUP_MAX in HOL4*) |
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"z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z" |
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by (blast intro: antisym cSup_upper cSup_least) |
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|
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lemma cInf_eq_minimum: (*REAL_INF_MIN in HOL4*) |
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"z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z" |
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by (intro antisym cInf_lower[of z X z] cInf_greatest[of X z]) auto |
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lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> (\<And>a. a \<in> S \<Longrightarrow> a \<le> z) \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)" |
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by (metis order_trans cSup_upper cSup_least) |
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lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> (\<And>a. a \<in> S \<Longrightarrow> z \<le> a) \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)" |
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by (metis order_trans cInf_lower cInf_greatest) |
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lemma cSup_singleton [simp]: "Sup {x} = x" |
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by (intro cSup_eq_maximum) auto |
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lemma cInf_singleton [simp]: "Inf {x} = x" |
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by (intro cInf_eq_minimum) auto |
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lemma cSup_upper2: (*REAL_IMP_LE_SUP in HOL4*) |
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"x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> y \<le> Sup X" |
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by (metis cSup_upper order_trans) |
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lemma cInf_lower2: |
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"x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X \<le> y" |
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by (metis cInf_lower order_trans) |
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lemma cSup_upper_EX: "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> x \<le> z \<Longrightarrow> x \<le> Sup X" |
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by (blast intro: cSup_upper) |
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lemma cInf_lower_EX: "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> z \<le> x \<Longrightarrow> Inf X \<le> x" |
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by (blast intro: cInf_lower) |
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lemma cSup_eq_non_empty: |
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assumes 1: "X \<noteq> {}" |
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assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a" |
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assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y" |
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shows "Sup X = a" |
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by (intro 3 1 antisym cSup_least) (auto intro: 2 1 cSup_upper) |
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lemma cInf_eq_non_empty: |
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assumes 1: "X \<noteq> {}" |
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assumes 2: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x" |
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assumes 3: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a" |
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shows "Inf X = a" |
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by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower) |
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lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}" |
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by (rule cInf_eq_non_empty) (auto intro: cSup_upper cSup_least) |
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lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}" |
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by (rule cSup_eq_non_empty) (auto intro: cInf_lower cInf_greatest) |
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lemma cSup_insert: |
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assumes x: "X \<noteq> {}" |
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and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z" |
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shows "Sup (insert a X) = sup a (Sup X)" |
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proof (intro cSup_eq_non_empty) |
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fix y assume "\<And>x. x \<in> insert a X \<Longrightarrow> x \<le> y" with x show "sup a (Sup X) \<le> y" by (auto intro: cSup_least) |
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qed (auto intro: le_supI2 z cSup_upper) |
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lemma cInf_insert: |
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assumes x: "X \<noteq> {}" |
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and z: "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" |
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shows "Inf (insert a X) = inf a (Inf X)" |
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proof (intro cInf_eq_non_empty) |
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fix y assume "\<And>x. x \<in> insert a X \<Longrightarrow> y \<le> x" with x show "y \<le> inf a (Inf X)" by (auto intro: cInf_greatest) |
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qed (auto intro: le_infI2 z cInf_lower) |
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lemma cSup_insert_If: |
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"(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))" |
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104 |
using cSup_insert[of X z] by simp |
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105 |
|
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106 |
lemma cInf_insert_if: |
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107 |
"(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))" |
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108 |
using cInf_insert[of X z] by simp |
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109 |
|
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110 |
lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X" |
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proof (induct X arbitrary: x rule: finite_induct) |
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case (insert x X y) then show ?case |
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apply (cases "X = {}") |
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apply simp |
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apply (subst cSup_insert[of _ "Sup X"]) |
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apply (auto intro: le_supI2) |
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117 |
done |
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118 |
qed simp |
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119 |
|
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lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x" |
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proof (induct X arbitrary: x rule: finite_induct) |
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case (insert x X y) then show ?case |
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123 |
apply (cases "X = {}") |
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124 |
apply simp |
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apply (subst cInf_insert[of _ "Inf X"]) |
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126 |
apply (auto intro: le_infI2) |
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127 |
done |
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128 |
qed simp |
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129 |
|
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130 |
lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X" |
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proof (induct X rule: finite_ne_induct) |
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132 |
case (insert x X) then show ?case |
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133 |
using cSup_insert[of X "Sup_fin X" x] le_cSup_finite[of X] by simp |
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134 |
qed simp |
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135 |
|
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136 |
lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X" |
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proof (induct X rule: finite_ne_induct) |
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138 |
case (insert x X) then show ?case |
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139 |
using cInf_insert[of X "Inf_fin X" x] cInf_le_finite[of X] by simp |
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140 |
qed simp |
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141 |
|
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142 |
lemma cSup_atMost[simp]: "Sup {..x} = x" |
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by (auto intro!: cSup_eq_maximum) |
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144 |
|
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lemma cSup_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Sup {y<..x} = x" |
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by (auto intro!: cSup_eq_maximum) |
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147 |
|
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lemma cSup_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Sup {y..x} = x" |
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by (auto intro!: cSup_eq_maximum) |
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150 |
|
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151 |
lemma cInf_atLeast[simp]: "Inf {x..} = x" |
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by (auto intro!: cInf_eq_minimum) |
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153 |
|
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lemma cInf_atLeastLessThan[simp]: "y < x \<Longrightarrow> Inf {y..<x} = y" |
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by (auto intro!: cInf_eq_minimum) |
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156 |
|
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157 |
lemma cInf_atLeastAtMost[simp]: "y \<le> x \<Longrightarrow> Inf {y..x} = y" |
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by (auto intro!: cInf_eq_minimum) |
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159 |
|
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160 |
end |
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|
51773 | 162 |
instance complete_lattice \<subseteq> conditionally_complete_lattice |
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by default (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest) |
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164 |
|
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165 |
lemma isLub_cSup: |
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"(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> (\<exists>b. S *<= b) \<Longrightarrow> isLub UNIV S (Sup S)" |
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by (auto simp add: isLub_def setle_def leastP_def isUb_def |
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intro!: setgeI intro: cSup_upper cSup_least) |
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169 |
|
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170 |
lemma cSup_eq: |
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fixes a :: "'a :: {conditionally_complete_lattice, no_bot}" |
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172 |
assumes upper: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a" |
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173 |
assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y" |
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174 |
shows "Sup X = a" |
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175 |
proof cases |
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assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le) |
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177 |
qed (intro cSup_eq_non_empty assms) |
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178 |
|
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179 |
lemma cInf_eq: |
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fixes a :: "'a :: {conditionally_complete_lattice, no_top}" |
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181 |
assumes upper: "\<And>x. x \<in> X \<Longrightarrow> a \<le> x" |
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182 |
assumes least: "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a" |
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183 |
shows "Inf X = a" |
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184 |
proof cases |
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assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le) |
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186 |
qed (intro cInf_eq_non_empty assms) |
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187 |
|
51773 | 188 |
lemma cSup_le: "(S::'a::conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> S *<= b \<Longrightarrow> Sup S \<le> b" |
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by (metis cSup_least setle_def) |
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190 |
|
51773 | 191 |
lemma cInf_ge: "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> b <=* S \<Longrightarrow> Inf S \<ge> b" |
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192 |
by (metis cInf_greatest setge_def) |
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193 |
|
51773 | 194 |
class conditionally_complete_linorder = conditionally_complete_lattice + linorder |
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195 |
begin |
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196 |
|
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197 |
lemma less_cSup_iff : (*REAL_SUP_LE in HOL4*) |
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"X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)" |
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by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans) |
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200 |
|
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201 |
lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)" |
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by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans) |
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203 |
|
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204 |
lemma less_cSupE: |
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assumes "y < Sup X" "X \<noteq> {}" obtains x where "x \<in> X" "y < x" |
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by (metis cSup_least assms not_le that) |
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207 |
|
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208 |
lemma less_cSupD: |
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"X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x" |
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by (metis less_cSup_iff not_leE) |
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|
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lemma cInf_lessD: |
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"X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z" |
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by (metis cInf_less_iff not_leE) |
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|
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lemma complete_interval: |
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assumes "a < b" and "P a" and "\<not> P b" |
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shows "\<exists>c. a \<le> c \<and> c \<le> b \<and> (\<forall>x. a \<le> x \<and> x < c \<longrightarrow> P x) \<and> |
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(\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)" |
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proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto) |
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show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}" |
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by (rule cSup_upper [where z=b], auto) |
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(metis `a < b` `\<not> P b` linear less_le) |
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next |
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show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b" |
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apply (rule cSup_least) |
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apply auto |
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apply (metis less_le_not_le) |
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apply (metis `a<b` `~ P b` linear less_le) |
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done |
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next |
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fix x |
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assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}" |
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show "P x" |
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apply (rule less_cSupE [OF lt], auto) |
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apply (metis less_le_not_le) |
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apply (metis x) |
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done |
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239 |
next |
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fix d |
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assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x" |
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thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}" |
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by (rule_tac z="b" in cSup_upper, auto) |
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(metis `a<b` `~ P b` linear less_le) |
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qed |
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|
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end |
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|
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class linear_continuum = conditionally_complete_linorder + inner_dense_linorder + |
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assumes UNIV_not_singleton: "\<exists>a b::'a. a \<noteq> b" |
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begin |
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|
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lemma ex_gt_or_lt: "\<exists>b. a < b \<or> b < a" |
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by (metis UNIV_not_singleton neq_iff) |
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|
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256 |
end |
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257 |
|
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lemma cSup_bounds: |
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fixes S :: "'a :: conditionally_complete_lattice set" |
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assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b" |
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shows "a \<le> Sup S \<and> Sup S \<le> b" |
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proof- |
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from isLub_cSup[OF Se] u have lub: "isLub UNIV S (Sup S)" by blast |
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hence b: "Sup S \<le> b" using u |
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by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) |
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from Se obtain y where y: "y \<in> S" by blast |
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from lub l have "a \<le> Sup S" |
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by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) |
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(metis le_iff_sup le_sup_iff y) |
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with b show ?thesis by blast |
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qed |
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272 |
|
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|
51773 | 274 |
lemma cSup_unique: "(S::'a :: {conditionally_complete_linorder, no_bot} set) *<= b \<Longrightarrow> (\<forall>b'<b. \<exists>x\<in>S. b' < x) \<Longrightarrow> Sup S = b" |
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by (rule cSup_eq) (auto simp: not_le[symmetric] setle_def) |
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|
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lemma cInf_unique: "b <=* (S::'a :: {conditionally_complete_linorder, no_top} set) \<Longrightarrow> (\<forall>b'>b. \<exists>x\<in>S. b' > x) \<Longrightarrow> Inf S = b" |
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by (rule cInf_eq) (auto simp: not_le[symmetric] setge_def) |
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|
51773 | 280 |
lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Max X" |
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using cSup_eq_Sup_fin[of X] Sup_fin_eq_Max[of X] by simp |
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282 |
|
51773 | 283 |
lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set) \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Min X" |
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using cInf_eq_Inf_fin[of X] Inf_fin_eq_Min[of X] by simp |
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285 |
|
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lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x" |
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by (auto intro!: cSup_eq_non_empty intro: dense_le) |
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288 |
|
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lemma cSup_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x" |
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by (auto intro!: cSup_eq intro: dense_le_bounded) |
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291 |
|
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lemma cSup_atLeastLessThan[simp]: "y < x \<Longrightarrow> Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x" |
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by (auto intro!: cSup_eq intro: dense_le_bounded) |
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|
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lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, dense_linorder} <..} = x" |
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by (auto intro!: cInf_eq intro: dense_ge) |
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297 |
|
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lemma cInf_greaterThanAtMost[simp]: "y < x \<Longrightarrow> Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y" |
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by (auto intro!: cInf_eq intro: dense_ge_bounded) |
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300 |
|
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lemma cInf_greaterThanLessThan[simp]: "y < x \<Longrightarrow> Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y" |
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by (auto intro!: cInf_eq intro: dense_ge_bounded) |
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303 |
|
33269
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New theory SupInf of the supremum and infimum operators for sets of reals.
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|
304 |
end |