src/HOL/Library/Lub_Glb.thy
 author wenzelm Mon Dec 28 01:28:28 2015 +0100 (2015-12-28) changeset 61945 1135b8de26c3 parent 61585 a9599d3d7610 child 61969 e01015e49041 permissions -rw-r--r--
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```     1 (*  Title:      HOL/Library/Lub_Glb.thy
```
```     2     Author:     Jacques D. Fleuriot, University of Cambridge
```
```     3     Author:     Amine Chaieb, University of Cambridge *)
```
```     4
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```     5 section \<open>Definitions of Least Upper Bounds and Greatest Lower Bounds\<close>
```
```     6
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```     7 theory Lub_Glb
```
```     8 imports Complex_Main
```
```     9 begin
```
```    10
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```    11 text \<open>Thanks to suggestions by James Margetson\<close>
```
```    12
```
```    13 definition setle :: "'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"  (infixl "*<=" 70)
```
```    14   where "S *<= x = (ALL y: S. y \<le> x)"
```
```    15
```
```    16 definition setge :: "'a::ord \<Rightarrow> 'a set \<Rightarrow> bool"  (infixl "<=*" 70)
```
```    17   where "x <=* S = (ALL y: S. x \<le> y)"
```
```    18
```
```    19
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```    20 subsection \<open>Rules for the Relations \<open>*<=\<close> and \<open><=*\<close>\<close>
```
```    21
```
```    22 lemma setleI: "ALL y: S. y \<le> x \<Longrightarrow> S *<= x"
```
```    23   by (simp add: setle_def)
```
```    24
```
```    25 lemma setleD: "S *<= x \<Longrightarrow> y: S \<Longrightarrow> y \<le> x"
```
```    26   by (simp add: setle_def)
```
```    27
```
```    28 lemma setgeI: "ALL y: S. x \<le> y \<Longrightarrow> x <=* S"
```
```    29   by (simp add: setge_def)
```
```    30
```
```    31 lemma setgeD: "x <=* S \<Longrightarrow> y: S \<Longrightarrow> x \<le> y"
```
```    32   by (simp add: setge_def)
```
```    33
```
```    34
```
```    35 definition leastP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
```
```    36   where "leastP P x = (P x \<and> x <=* Collect P)"
```
```    37
```
```    38 definition isUb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
```
```    39   where "isUb R S x = (S *<= x \<and> x: R)"
```
```    40
```
```    41 definition isLub :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
```
```    42   where "isLub R S x = leastP (isUb R S) x"
```
```    43
```
```    44 definition ubs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
```
```    45   where "ubs R S = Collect (isUb R S)"
```
```    46
```
```    47
```
```    48 subsection \<open>Rules about the Operators @{term leastP}, @{term ub} and @{term lub}\<close>
```
```    49
```
```    50 lemma leastPD1: "leastP P x \<Longrightarrow> P x"
```
```    51   by (simp add: leastP_def)
```
```    52
```
```    53 lemma leastPD2: "leastP P x \<Longrightarrow> x <=* Collect P"
```
```    54   by (simp add: leastP_def)
```
```    55
```
```    56 lemma leastPD3: "leastP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<le> y"
```
```    57   by (blast dest!: leastPD2 setgeD)
```
```    58
```
```    59 lemma isLubD1: "isLub R S x \<Longrightarrow> S *<= x"
```
```    60   by (simp add: isLub_def isUb_def leastP_def)
```
```    61
```
```    62 lemma isLubD1a: "isLub R S x \<Longrightarrow> x: R"
```
```    63   by (simp add: isLub_def isUb_def leastP_def)
```
```    64
```
```    65 lemma isLub_isUb: "isLub R S x \<Longrightarrow> isUb R S x"
```
```    66   unfolding isUb_def by (blast dest: isLubD1 isLubD1a)
```
```    67
```
```    68 lemma isLubD2: "isLub R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
```
```    69   by (blast dest!: isLubD1 setleD)
```
```    70
```
```    71 lemma isLubD3: "isLub R S x \<Longrightarrow> leastP (isUb R S) x"
```
```    72   by (simp add: isLub_def)
```
```    73
```
```    74 lemma isLubI1: "leastP(isUb R S) x \<Longrightarrow> isLub R S x"
```
```    75   by (simp add: isLub_def)
```
```    76
```
```    77 lemma isLubI2: "isUb R S x \<Longrightarrow> x <=* Collect (isUb R S) \<Longrightarrow> isLub R S x"
```
```    78   by (simp add: isLub_def leastP_def)
```
```    79
```
```    80 lemma isUbD: "isUb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
```
```    81   by (simp add: isUb_def setle_def)
```
```    82
```
```    83 lemma isUbD2: "isUb R S x \<Longrightarrow> S *<= x"
```
```    84   by (simp add: isUb_def)
```
```    85
```
```    86 lemma isUbD2a: "isUb R S x \<Longrightarrow> x: R"
```
```    87   by (simp add: isUb_def)
```
```    88
```
```    89 lemma isUbI: "S *<= x \<Longrightarrow> x: R \<Longrightarrow> isUb R S x"
```
```    90   by (simp add: isUb_def)
```
```    91
```
```    92 lemma isLub_le_isUb: "isLub R S x \<Longrightarrow> isUb R S y \<Longrightarrow> x \<le> y"
```
```    93   unfolding isLub_def by (blast intro!: leastPD3)
```
```    94
```
```    95 lemma isLub_ubs: "isLub R S x \<Longrightarrow> x <=* ubs R S"
```
```    96   unfolding ubs_def isLub_def by (rule leastPD2)
```
```    97
```
```    98 lemma isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::'a::linorder)"
```
```    99   apply (frule isLub_isUb)
```
```   100   apply (frule_tac x = y in isLub_isUb)
```
```   101   apply (blast intro!: order_antisym dest!: isLub_le_isUb)
```
```   102   done
```
```   103
```
```   104 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```   105   by (simp add: isUbI setleI)
```
```   106
```
```   107
```
```   108 definition greatestP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
```
```   109   where "greatestP P x = (P x \<and> Collect P *<=  x)"
```
```   110
```
```   111 definition isLb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
```
```   112   where "isLb R S x = (x <=* S \<and> x: R)"
```
```   113
```
```   114 definition isGlb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
```
```   115   where "isGlb R S x = greatestP (isLb R S) x"
```
```   116
```
```   117 definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
```
```   118   where "lbs R S = Collect (isLb R S)"
```
```   119
```
```   120
```
```   121 subsection \<open>Rules about the Operators @{term greatestP}, @{term isLb} and @{term isGlb}\<close>
```
```   122
```
```   123 lemma greatestPD1: "greatestP P x \<Longrightarrow> P x"
```
```   124   by (simp add: greatestP_def)
```
```   125
```
```   126 lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x"
```
```   127   by (simp add: greatestP_def)
```
```   128
```
```   129 lemma greatestPD3: "greatestP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<ge> y"
```
```   130   by (blast dest!: greatestPD2 setleD)
```
```   131
```
```   132 lemma isGlbD1: "isGlb R S x \<Longrightarrow> x <=* S"
```
```   133   by (simp add: isGlb_def isLb_def greatestP_def)
```
```   134
```
```   135 lemma isGlbD1a: "isGlb R S x \<Longrightarrow> x: R"
```
```   136   by (simp add: isGlb_def isLb_def greatestP_def)
```
```   137
```
```   138 lemma isGlb_isLb: "isGlb R S x \<Longrightarrow> isLb R S x"
```
```   139   unfolding isLb_def by (blast dest: isGlbD1 isGlbD1a)
```
```   140
```
```   141 lemma isGlbD2: "isGlb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
```
```   142   by (blast dest!: isGlbD1 setgeD)
```
```   143
```
```   144 lemma isGlbD3: "isGlb R S x \<Longrightarrow> greatestP (isLb R S) x"
```
```   145   by (simp add: isGlb_def)
```
```   146
```
```   147 lemma isGlbI1: "greatestP (isLb R S) x \<Longrightarrow> isGlb R S x"
```
```   148   by (simp add: isGlb_def)
```
```   149
```
```   150 lemma isGlbI2: "isLb R S x \<Longrightarrow> Collect (isLb R S) *<= x \<Longrightarrow> isGlb R S x"
```
```   151   by (simp add: isGlb_def greatestP_def)
```
```   152
```
```   153 lemma isLbD: "isLb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
```
```   154   by (simp add: isLb_def setge_def)
```
```   155
```
```   156 lemma isLbD2: "isLb R S x \<Longrightarrow> x <=* S "
```
```   157   by (simp add: isLb_def)
```
```   158
```
```   159 lemma isLbD2a: "isLb R S x \<Longrightarrow> x: R"
```
```   160   by (simp add: isLb_def)
```
```   161
```
```   162 lemma isLbI: "x <=* S \<Longrightarrow> x: R \<Longrightarrow> isLb R S x"
```
```   163   by (simp add: isLb_def)
```
```   164
```
```   165 lemma isGlb_le_isLb: "isGlb R S x \<Longrightarrow> isLb R S y \<Longrightarrow> x \<ge> y"
```
```   166   unfolding isGlb_def by (blast intro!: greatestPD3)
```
```   167
```
```   168 lemma isGlb_ubs: "isGlb R S x \<Longrightarrow> lbs R S *<= x"
```
```   169   unfolding lbs_def isGlb_def by (rule greatestPD2)
```
```   170
```
```   171 lemma isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::'a::linorder)"
```
```   172   apply (frule isGlb_isLb)
```
```   173   apply (frule_tac x = y in isGlb_isLb)
```
```   174   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
```
```   175   done
```
```   176
```
```   177 lemma bdd_above_setle: "bdd_above A \<longleftrightarrow> (\<exists>a. A *<= a)"
```
```   178   by (auto simp: bdd_above_def setle_def)
```
```   179
```
```   180 lemma bdd_below_setge: "bdd_below A \<longleftrightarrow> (\<exists>a. a <=* A)"
```
```   181   by (auto simp: bdd_below_def setge_def)
```
```   182
```
```   183 lemma isLub_cSup:
```
```   184   "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> (\<exists>b. S *<= b) \<Longrightarrow> isLub UNIV S (Sup S)"
```
```   185   by  (auto simp add: isLub_def setle_def leastP_def isUb_def
```
```   186             intro!: setgeI cSup_upper cSup_least)
```
```   187
```
```   188 lemma isGlb_cInf:
```
```   189   "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> (\<exists>b. b <=* S) \<Longrightarrow> isGlb UNIV S (Inf S)"
```
```   190   by  (auto simp add: isGlb_def setge_def greatestP_def isLb_def
```
```   191             intro!: setleI cInf_lower cInf_greatest)
```
```   192
```
```   193 lemma cSup_le: "(S::'a::conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> S *<= b \<Longrightarrow> Sup S \<le> b"
```
```   194   by (metis cSup_least setle_def)
```
```   195
```
```   196 lemma cInf_ge: "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> b <=* S \<Longrightarrow> Inf S \<ge> b"
```
```   197   by (metis cInf_greatest setge_def)
```
```   198
```
```   199 lemma cSup_bounds:
```
```   200   fixes S :: "'a :: conditionally_complete_lattice set"
```
```   201   shows "S \<noteq> {} \<Longrightarrow> a <=* S \<Longrightarrow> S *<= b \<Longrightarrow> a \<le> Sup S \<and> Sup S \<le> b"
```
```   202   using cSup_least[of S b] cSup_upper2[of _ S a]
```
```   203   by (auto simp: bdd_above_setle setge_def setle_def)
```
```   204
```
```   205 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"
```
```   206   by (rule cSup_eq) (auto simp: not_le[symmetric] setle_def)
```
```   207
```
```   208 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"
```
```   209   by (rule cInf_eq) (auto simp: not_le[symmetric] setge_def)
```
```   210
```
```   211 text\<open>Use completeness of reals (supremum property) to show that any bounded sequence has a least upper bound\<close>
```
```   212
```
```   213 lemma reals_complete: "\<exists>X. X \<in> S \<Longrightarrow> \<exists>Y. isUb (UNIV::real set) S Y \<Longrightarrow> \<exists>t. isLub (UNIV :: real set) S t"
```
```   214   by (intro exI[of _ "Sup S"] isLub_cSup) (auto simp: setle_def isUb_def intro!: cSup_upper)
```
```   215
```
```   216 lemma Bseq_isUb: "\<And>X :: nat \<Rightarrow> real. Bseq X \<Longrightarrow> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   217   by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   218
```
```   219 lemma Bseq_isLub: "\<And>X :: nat \<Rightarrow> real. Bseq X \<Longrightarrow> \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   220   by (blast intro: reals_complete Bseq_isUb)
```
```   221
```
```   222 lemma isLub_mono_imp_LIMSEQ:
```
```   223   fixes X :: "nat \<Rightarrow> real"
```
```   224   assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
```
```   225   assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
```
```   226   shows "X ----> u"
```
```   227 proof -
```
```   228   have "X ----> (SUP i. X i)"
```
```   229     using u[THEN isLubD1] X
```
```   230     by (intro LIMSEQ_incseq_SUP) (auto simp: incseq_def image_def eq_commute bdd_above_setle)
```
```   231   also have "(SUP i. X i) = u"
```
```   232     using isLub_cSup[of "range X"] u[THEN isLubD1]
```
```   233     by (intro isLub_unique[OF _ u]) (auto simp add: SUP_def image_def eq_commute)
```
```   234   finally show ?thesis .
```
```   235 qed
```
```   236
```
```   237 lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
```
```   238
```
```   239 lemma real_le_inf_subset: "t \<noteq> {} \<Longrightarrow> t \<subseteq> s \<Longrightarrow> \<exists>b. b <=* s \<Longrightarrow> Inf s \<le> Inf (t::real set)"
```
```   240   by (rule cInf_superset_mono) (auto simp: bdd_below_setge)
```
```   241
```
```   242 lemma real_ge_sup_subset: "t \<noteq> {} \<Longrightarrow> t \<subseteq> s \<Longrightarrow> \<exists>b. s *<= b \<Longrightarrow> Sup s \<ge> Sup (t::real set)"
```
```   243   by (rule cSup_subset_mono) (auto simp: bdd_above_setle)
```
```   244
```
```   245 end
```