helm/software/matita/contribs/CoRN-Decl/reals/CauchySeq.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 set "baseuri" "cic:/matita/CoRN-Decl/reals/CauchySeq".
  18
  19 (* $Id: CauchySeq.v,v 1.8 2004/04/23 10:01:04 lcf Exp $ *)
  20
  21 (*#* printing IR %\ensuremath{\mathbb R}% *)
  22
  23 (*#* printing PartIR %\ensuremath{\mathbb R\!\not\rightarrow\!\mathbb R}% *)
  24
  25 (*#* printing ZeroR %\ensuremath{\mathbf0}% #0# *)
  26
  27 (*#* printing OneR %\ensuremath{\mathbf1}% #1# *)
  28
  29 (*#* printing AbsIR %\ensuremath{|\cdot|_{\mathbb R}}% *)
  30
  31 (* INCLUDE
  32 CReals
  33 *)
  34
  35 (*#* *Real Number Structures
  36 %\begin{convention}% Let [IR] be a structure for real numbers.
  37 %\end{convention}%
  38 *)
  39
  40 (*
  41 Require Export R_CReals.
  42 Definition IR : CReals := Concrete_R.
  43 Opaque IR.
  44 *)
  45
  46 inline cic:/CoRN/reals/CauchySeq/IR.var.
  47
  48 (* INCLUDE
  49 Transparent_algebra
  50 *)
  51
  52 (* begin hide *)
  53
  54 (* end hide *)
  55
  56 (* UNEXPORTED
  57 Section CReals_axioms.
  58 *)
  59
  60 (*#* ** [CReals] axioms *)
  61
  62 inline cic:/CoRN/reals/CauchySeq/CReals_is_CReals.con.
  63
  64 (*#* First properties which follow trivially from the axioms.  *)
  65
  66 inline cic:/CoRN/reals/CauchySeq/Lim_Cauchy.con.
  67
  68 inline cic:/CoRN/reals/CauchySeq/Archimedes.con.
  69
  70 inline cic:/CoRN/reals/CauchySeq/Archimedes'.con.
  71
  72 (*#* A stronger version, which often comes in useful.  *)
  73
  74 inline cic:/CoRN/reals/CauchySeq/str_Archimedes.con.
  75
  76 inline cic:/CoRN/reals/CauchySeq/CauchySeqR.con.
  77
  78 (* UNEXPORTED
  79 End CReals_axioms.
  80 *)
  81
  82 (* UNEXPORTED
  83 Section Cauchy_Defs.
  84 *)
  85
  86 (*#* ** Cauchy sequences
  87 *** Alternative definitions
  88 This section gives several definitions of Cauchy sequences. There
  89 are no lemmas in this section.
  90
  91 The current definition of Cauchy ([Cauchy_prop]) is:
  92
  93 %\[\forall \epsilon>0. \exists N. \forall m\geq N. |seq_m - seq_N|\leq\varepsilon\]%
  94 #for all e&gt;0 there exists N such that for all m&ge; N |seqm-seqN|&le; e#
  95
  96 Variant 1 of Cauchy ([Cauchy_prop1]) is:
  97
  98 %\[\forall k. \exists N. \forall m \geq N. |seq_m - seq_N|\leq1/(k+1)\]%
  99 #for all k there exists N such that for all m &ge; N |seqm-seqN| &le; 1/(k+1)#
 100
 101 In all of the following variants the limit occurs in the definition.
 102 Therefore it is useful to define an auxiliary predicate
 103 [Cauchy_Lim_prop?], in terms of which [Cauchy_prop?] is defined.
 104
 105 Variant 2 of Cauchy ([Cauchy_prop2]) is [exists y, (Cauchy_Lim_prop2 seq y)]
 106 where
 107 [[
 108 Cauchy_Lim_prop2 seq y := forall eps [>] Zero, exists N, forall m >= N, (AbsIR seq m - y) [<=] eps
 109 ]]
 110
 111 Note that [Cauchy_Lim_prop2] equals [seqLimit].
 112
 113 Variant 3 of Cauchy ([Cauchy_prop3]) reads [exists y, (Cauchy_Lim_prop3 seq y)]
 114 where
 115 [[
 116 Cauchy_Lim_prop3 seq y := forall k, exists N, forall m >= N, (AbsIR seq m - y) [<=] One[/] (k[+]1)
 117 ]]
 118
 119 The following variant is more restricted.
 120
 121 Variant 4 of Cauchy ([Cauchy_prop4]): [exists y, (Cauchy_Lim_prop4 seq y)]
 122 where
 123 [[
 124 Cauchy_Lim_prop4 seq y := forall k, (AbsIR seq m - y) [<=] One[/] (k[+]1)
 125 ]]
 126
 127 So
 128 [[
 129 Cauchy_prop4 -> Cauchy_prop3 Iff Cauchy_prop2 Iff Cauchy_prop1 Iff Cauchy_prop
 130 ]]
 131 Note: we don't know which formulations are useful.
 132
 133 The inequalities are stated with [[<=]] rather than with [<] for the
 134 following reason: both formulations are equivalent, as is well known;
 135 and [[<=]] being a negative statement, this makes proofs
 136 sometimes easier and program extraction much more efficient.
 137 *)
 138
 139 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop1.con.
 140
 141 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_prop2.con.
 142
 143 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop2.con.
 144
 145 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_prop3.con.
 146
 147 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop3.con.
 148
 149 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_prop4.con.
 150
 151 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop4.con.
 152
 153 (* UNEXPORTED
 154 End Cauchy_Defs.
 155 *)
 156
 157 (* UNEXPORTED
 158 Section Inequalities.
 159 *)
 160
 161 (*#* *** Inequalities of Limits
 162
 163 The next lemma is equal to lemma [Lim_Cauchy].  *)
 164
 165 inline cic:/CoRN/reals/CauchySeq/Cauchy_complete.con.
 166
 167 inline cic:/CoRN/reals/CauchySeq/less_Lim_so_less_seq.con.
 168
 169 inline cic:/CoRN/reals/CauchySeq/Lim_less_so_seq_less.con.
 170
 171 inline cic:/CoRN/reals/CauchySeq/Lim_less_Lim_so_seq_less_seq.con.
 172
 173 (*#* The next lemma follows from [less_Lim_so_less_seq] with [y := (y[+] (Lim seq)) [/]TwoNZ].  *)
 174
 175 inline cic:/CoRN/reals/CauchySeq/less_Lim_so.con.
 176
 177 inline cic:/CoRN/reals/CauchySeq/Lim_less_so.con.
 178
 179 inline cic:/CoRN/reals/CauchySeq/leEq_seq_so_leEq_Lim.con.
 180
 181 inline cic:/CoRN/reals/CauchySeq/str_leEq_seq_so_leEq_Lim.con.
 182
 183 inline cic:/CoRN/reals/CauchySeq/Lim_leEq_Lim.con.
 184
 185 inline cic:/CoRN/reals/CauchySeq/seq_leEq_so_Lim_leEq.con.
 186
 187 inline cic:/CoRN/reals/CauchySeq/str_seq_leEq_so_Lim_leEq.con.
 188
 189 inline cic:/CoRN/reals/CauchySeq/Limits_unique.con.
 190
 191 inline cic:/CoRN/reals/CauchySeq/Lim_wd.con.
 192
 193 inline cic:/CoRN/reals/CauchySeq/Lim_strext.con.
 194
 195 inline cic:/CoRN/reals/CauchySeq/Lim_ap_imp_seq_ap.con.
 196
 197 inline cic:/CoRN/reals/CauchySeq/Lim_ap_imp_seq_ap'.con.
 198
 199 (* UNEXPORTED
 200 End Inequalities.
 201 *)
 202
 203 (* UNEXPORTED
 204 Section Equiv_Cauchy.
 205 *)
 206
 207 (*#* *** Equivalence of formulations of Cauchy *)
 208
 209 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop1_prop.con.
 210
 211 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop2_prop.con.
 212
 213 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_prop3_prop2.con.
 214
 215 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop3_prop2.con.
 216
 217 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop3_prop.con.
 218
 219 inline cic:/CoRN/reals/CauchySeq/Build_CauchySeq1.con.
 220
 221 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_prop4_prop3.con.
 222
 223 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_prop4_prop2.con.
 224
 225 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop4_prop3.con.
 226
 227 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop4_prop.con.
 228
 229 inline cic:/CoRN/reals/CauchySeq/Build_CauchySeq4.con.
 230
 231 inline cic:/CoRN/reals/CauchySeq/Build_CauchySeq4_y.con.
 232
 233 inline cic:/CoRN/reals/CauchySeq/Lim_CauchySeq4.con.
 234
 235 inline cic:/CoRN/reals/CauchySeq/Build_CauchySeq2.con.
 236
 237 inline cic:/CoRN/reals/CauchySeq/Build_CauchySeq2_y.con.
 238
 239 inline cic:/CoRN/reals/CauchySeq/Lim_CauchySeq2.con.
 240
 241 (*#* Well definedness. *)
 242
 243 inline cic:/CoRN/reals/CauchySeq/Cauchy_prop_wd.con.
 244
 245 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_prop2_wd.con.
 246
 247 inline cic:/CoRN/reals/CauchySeq/Lim_wd'.con.
 248
 249 inline cic:/CoRN/reals/CauchySeq/Lim_unique.con.
 250
 251 (* UNEXPORTED
 252 End Equiv_Cauchy.
 253 *)
 254
 255 (* UNEXPORTED
 256 Section Cauchy_props.
 257 *)
 258
 259 (*#* *** Properties of Cauchy sequences
 260
 261 Some of these lemmas are now obsolete, because they were reproved for arbitrary ordered fields$\ldots$#...#
 262
 263 We begin by defining the constant sequence and proving that it is Cauchy with the expected limit.
 264 *)
 265
 266 inline cic:/CoRN/reals/CauchySeq/Cauchy_const.con.
 267
 268 inline cic:/CoRN/reals/CauchySeq/Lim_const.con.
 269
 270 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_plus.con.
 271
 272 inline cic:/CoRN/reals/CauchySeq/Cauchy_plus.con.
 273
 274 inline cic:/CoRN/reals/CauchySeq/Lim_plus.con.
 275
 276 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_inv.con.
 277
 278 inline cic:/CoRN/reals/CauchySeq/Cauchy_inv.con.
 279
 280 inline cic:/CoRN/reals/CauchySeq/Lim_inv.con.
 281
 282 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_minus.con.
 283
 284 inline cic:/CoRN/reals/CauchySeq/Cauchy_minus.con.
 285
 286 inline cic:/CoRN/reals/CauchySeq/Lim_minus.con.
 287
 288 inline cic:/CoRN/reals/CauchySeq/Cauchy_Lim_mult.con.
 289
 290 inline cic:/CoRN/reals/CauchySeq/Cauchy_mult.con.
 291
 292 inline cic:/CoRN/reals/CauchySeq/Lim_mult.con.
 293
 294 (* UNEXPORTED
 295 End Cauchy_props.
 296 *)
 297