helm/software/matita/contribs/CoRN-Decl/algebra/CSums.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   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                             *)
   9 (*      \   /                                                             *)
  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/algebra/CSums".
  18
  19 (* $Id: CSums.v,v 1.8 2004/04/23 10:00:54 lcf Exp $ *)
  20
  21 (*#* printing Sum0 %\ensuremath{\sum_0}% #&sum;<sub>0</sub># *)
  22
  23 (*#* printing Sum1 %\ensuremath{\sum_1}% #&sum;<sub>1</sub># *)
  24
  25 (*#* printing Sum2 %\ensuremath{\sum_2}% #&sum;<sub>2</sub># *)
  26
  27 (*#* printing Sum %\ensuremath{\sum}% #&sum;# *)
  28
  29 (*#* printing Sumx %\ensuremath{\sum'}% #&sum;'&*)
  30
  31 (* INCLUDE
  32 CAbGroups
  33 *)
  34
  35 (* INCLUDE
  36 Peano_dec
  37 *)
  38
  39 (*#* * Sums
  40
  41 %\begin{convention}% Let [G] be an abelian group.
  42 %\end{convention}%
  43 *)
  44
  45 (* UNEXPORTED
  46 Section Sums.
  47 *)
  48
  49 inline cic:/CoRN/algebra/CSums/G.var.
  50
  51 (* Sum1 and Sum use subtraction *)
  52
  53 inline cic:/CoRN/algebra/CSums/Sumlist.con.
  54
  55 inline cic:/CoRN/algebra/CSums/Sumx.con.
  56
  57 (*#*
  58 It is sometimes useful to view a function defined on $\{0,\ldots,i-1\}$
  59 #{0, ... i-1}# as a function on the natural numbers which evaluates to
  60 [Zero] when the input is greater than or equal to [i].
  61 *)
  62
  63 inline cic:/CoRN/algebra/CSums/part_tot_nat_fun.con.
  64
  65 inline cic:/CoRN/algebra/CSums/part_tot_nat_fun_ch1.con.
  66
  67 inline cic:/CoRN/algebra/CSums/part_tot_nat_fun_ch2.con.
  68
  69 (*#* [Sum0] defines the sum for [i=0..(n-1)] *)
  70
  71 inline cic:/CoRN/algebra/CSums/Sum0.con.
  72
  73 (*#* [Sum1] defines the sum for [i=m..(n-1)] *)
  74
  75 inline cic:/CoRN/algebra/CSums/Sum1.con.
  76
  77 inline cic:/CoRN/algebra/CSums/Sum.con.
  78
  79 (* Sum i=m..n *)
  80
  81 (*#* [Sum2] is similar to [Sum1], but does not require the summand to be
  82 defined outside where it is being added. *)
  83
  84 inline cic:/CoRN/algebra/CSums/Sum2.con.
  85
  86 inline cic:/CoRN/algebra/CSums/Sum_one.con.
  87
  88 (* UNEXPORTED
  89 Hint Resolve Sum_one: algebra.
  90 *)
  91
  92 inline cic:/CoRN/algebra/CSums/Sum_empty.con.
  93
  94 (* UNEXPORTED
  95 Hint Resolve Sum_empty: algebra.
  96 *)
  97
  98 inline cic:/CoRN/algebra/CSums/Sum_Sum.con.
  99
 100 (* UNEXPORTED
 101 Hint Resolve Sum_Sum: algebra.
 102 *)
 103
 104 inline cic:/CoRN/algebra/CSums/Sum_first.con.
 105
 106 inline cic:/CoRN/algebra/CSums/Sum_last.con.
 107
 108 (* UNEXPORTED
 109 Hint Resolve Sum_last: algebra.
 110 *)
 111
 112 inline cic:/CoRN/algebra/CSums/Sum_last'.con.
 113
 114 (*#*
 115 We add some extensionality results which will be quite useful
 116 when working with integration.
 117 *)
 118
 119 inline cic:/CoRN/algebra/CSums/Sum0_strext.con.
 120
 121 inline cic:/CoRN/algebra/CSums/Sum_strext.con.
 122
 123 inline cic:/CoRN/algebra/CSums/Sumx_strext.con.
 124
 125 inline cic:/CoRN/algebra/CSums/Sum0_strext'.con.
 126
 127 inline cic:/CoRN/algebra/CSums/Sum_strext'.con.
 128
 129 inline cic:/CoRN/algebra/CSums/Sum0_wd.con.
 130
 131 inline cic:/CoRN/algebra/CSums/Sum_wd.con.
 132
 133 inline cic:/CoRN/algebra/CSums/Sumx_wd.con.
 134
 135 inline cic:/CoRN/algebra/CSums/Sum_wd'.con.
 136
 137 inline cic:/CoRN/algebra/CSums/Sum2_wd.con.
 138
 139 inline cic:/CoRN/algebra/CSums/Sum0_plus_Sum0.con.
 140
 141 (* UNEXPORTED
 142 Hint Resolve Sum0_plus_Sum0: algebra.
 143 *)
 144
 145 inline cic:/CoRN/algebra/CSums/Sum_plus_Sum.con.
 146
 147 inline cic:/CoRN/algebra/CSums/Sumx_plus_Sumx.con.
 148
 149 inline cic:/CoRN/algebra/CSums/Sum2_plus_Sum2.con.
 150
 151 inline cic:/CoRN/algebra/CSums/inv_Sum0.con.
 152
 153 (* UNEXPORTED
 154 Hint Resolve inv_Sum0: algebra.
 155 *)
 156
 157 inline cic:/CoRN/algebra/CSums/inv_Sum.con.
 158
 159 (* UNEXPORTED
 160 Hint Resolve inv_Sum: algebra.
 161 *)
 162
 163 inline cic:/CoRN/algebra/CSums/inv_Sumx.con.
 164
 165 inline cic:/CoRN/algebra/CSums/inv_Sum2.con.
 166
 167 inline cic:/CoRN/algebra/CSums/Sum_minus_Sum.con.
 168
 169 (* UNEXPORTED
 170 Hint Resolve Sum_minus_Sum: algebra.
 171 *)
 172
 173 inline cic:/CoRN/algebra/CSums/Sumx_minus_Sumx.con.
 174
 175 inline cic:/CoRN/algebra/CSums/Sum2_minus_Sum2.con.
 176
 177 inline cic:/CoRN/algebra/CSums/Sum_apzero.con.
 178
 179 inline cic:/CoRN/algebra/CSums/Sum_zero.con.
 180
 181 inline cic:/CoRN/algebra/CSums/Sum_term.con.
 182
 183 inline cic:/CoRN/algebra/CSums/Sum0_shift.con.
 184
 185 (* UNEXPORTED
 186 Hint Resolve Sum0_shift: algebra.
 187 *)
 188
 189 inline cic:/CoRN/algebra/CSums/Sum_shift.con.
 190
 191 inline cic:/CoRN/algebra/CSums/Sum_big_shift.con.
 192
 193 inline cic:/CoRN/algebra/CSums/Sumx_Sum0.con.
 194
 195 (* UNEXPORTED
 196 End Sums.
 197 *)
 198
 199 (* UNEXPORTED
 200 Implicit Arguments Sum [G].
 201 *)
 202
 203 (* UNEXPORTED
 204 Implicit Arguments Sum0 [G].
 205 *)
 206
 207 (* UNEXPORTED
 208 Implicit Arguments Sumx [G n].
 209 *)
 210
 211 (* UNEXPORTED
 212 Implicit Arguments Sum2 [G m n].
 213 *)
 214
 215 (*#*
 216 The next results are useful for calculating some special sums,
 217 often referred to as ``Mengolli Sums''.
 218 %\begin{convention}% Let [G] be an abelian group.
 219 %\end{convention}%
 220 *)
 221
 222 (* UNEXPORTED
 223 Section More_Sums.
 224 *)
 225
 226 inline cic:/CoRN/algebra/CSums/G.var.
 227
 228 inline cic:/CoRN/algebra/CSums/Mengolli_Sum.con.
 229
 230 inline cic:/CoRN/algebra/CSums/Mengolli_Sum_gen.con.
 231
 232 inline cic:/CoRN/algebra/CSums/str_Mengolli_Sum_gen.con.
 233
 234 inline cic:/CoRN/algebra/CSums/Sumx_to_Sum.con.
 235
 236 (* UNEXPORTED
 237 End More_Sums.
 238 *)
 239
 240 (* UNEXPORTED
 241 Hint Resolve Sum_one Sum_Sum Sum_first Sum_last Sum_last' Sum_wd
 242   Sum_plus_Sum: algebra.
 243 *)
 244
 245 (* UNEXPORTED
 246 Hint Resolve Sum_minus_Sum inv_Sum inv_Sum0: algebra.
 247 *)
 248