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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/CSums".
21 (* $Id: CSums.v,v 1.8 2004/04/23 10:00:54 lcf Exp $ *)
23 (*#* printing Sum0 %\ensuremath{\sum_0}% #∑<sub>0</sub># *)
25 (*#* printing Sum1 %\ensuremath{\sum_1}% #∑<sub>1</sub># *)
27 (*#* printing Sum2 %\ensuremath{\sum_2}% #∑<sub>2</sub># *)
29 (*#* printing Sum %\ensuremath{\sum}% #∑# *)
31 (*#* printing Sumx %\ensuremath{\sum'}% #∑'&*)
33 include "algebra/CAbGroups.ma".
37 %\begin{convention}% Let [G] be an abelian group.
45 alias id "G" = "cic:/CoRN/algebra/CSums/Sums/G.var".
47 (* Sum1 and Sum use subtraction *)
49 inline "cic:/CoRN/algebra/CSums/Sumlist.con".
51 inline "cic:/CoRN/algebra/CSums/Sumx.con".
54 It is sometimes useful to view a function defined on $\{0,\ldots,i-1\}$
55 #{0, ... i-1}# as a function on the natural numbers which evaluates to
56 [Zero] when the input is greater than or equal to [i].
59 inline "cic:/CoRN/algebra/CSums/part_tot_nat_fun.con".
61 inline "cic:/CoRN/algebra/CSums/part_tot_nat_fun_ch1.con".
63 inline "cic:/CoRN/algebra/CSums/part_tot_nat_fun_ch2.con".
65 (*#* [Sum0] defines the sum for [i=0..(n-1)] *)
67 inline "cic:/CoRN/algebra/CSums/Sum0.con".
69 (*#* [Sum1] defines the sum for [i=m..(n-1)] *)
71 inline "cic:/CoRN/algebra/CSums/Sum1.con".
73 inline "cic:/CoRN/algebra/CSums/Sum.con".
77 (*#* [Sum2] is similar to [Sum1], but does not require the summand to be
78 defined outside where it is being added. *)
80 inline "cic:/CoRN/algebra/CSums/Sum2.con".
82 inline "cic:/CoRN/algebra/CSums/Sum_one.con".
85 Hint Resolve Sum_one: algebra.
88 inline "cic:/CoRN/algebra/CSums/Sum_empty.con".
91 Hint Resolve Sum_empty: algebra.
94 inline "cic:/CoRN/algebra/CSums/Sum_Sum.con".
97 Hint Resolve Sum_Sum: algebra.
100 inline "cic:/CoRN/algebra/CSums/Sum_first.con".
102 inline "cic:/CoRN/algebra/CSums/Sum_last.con".
105 Hint Resolve Sum_last: algebra.
108 inline "cic:/CoRN/algebra/CSums/Sum_last'.con".
111 We add some extensionality results which will be quite useful
112 when working with integration.
115 inline "cic:/CoRN/algebra/CSums/Sum0_strext.con".
117 inline "cic:/CoRN/algebra/CSums/Sum_strext.con".
119 inline "cic:/CoRN/algebra/CSums/Sumx_strext.con".
121 inline "cic:/CoRN/algebra/CSums/Sum0_strext'.con".
123 inline "cic:/CoRN/algebra/CSums/Sum_strext'.con".
125 inline "cic:/CoRN/algebra/CSums/Sum0_wd.con".
127 inline "cic:/CoRN/algebra/CSums/Sum_wd.con".
129 inline "cic:/CoRN/algebra/CSums/Sumx_wd.con".
131 inline "cic:/CoRN/algebra/CSums/Sum_wd'.con".
133 inline "cic:/CoRN/algebra/CSums/Sum2_wd.con".
135 inline "cic:/CoRN/algebra/CSums/Sum0_plus_Sum0.con".
138 Hint Resolve Sum0_plus_Sum0: algebra.
141 inline "cic:/CoRN/algebra/CSums/Sum_plus_Sum.con".
143 inline "cic:/CoRN/algebra/CSums/Sumx_plus_Sumx.con".
145 inline "cic:/CoRN/algebra/CSums/Sum2_plus_Sum2.con".
147 inline "cic:/CoRN/algebra/CSums/inv_Sum0.con".
150 Hint Resolve inv_Sum0: algebra.
153 inline "cic:/CoRN/algebra/CSums/inv_Sum.con".
156 Hint Resolve inv_Sum: algebra.
159 inline "cic:/CoRN/algebra/CSums/inv_Sumx.con".
161 inline "cic:/CoRN/algebra/CSums/inv_Sum2.con".
163 inline "cic:/CoRN/algebra/CSums/Sum_minus_Sum.con".
166 Hint Resolve Sum_minus_Sum: algebra.
169 inline "cic:/CoRN/algebra/CSums/Sumx_minus_Sumx.con".
171 inline "cic:/CoRN/algebra/CSums/Sum2_minus_Sum2.con".
173 inline "cic:/CoRN/algebra/CSums/Sum_apzero.con".
175 inline "cic:/CoRN/algebra/CSums/Sum_zero.con".
177 inline "cic:/CoRN/algebra/CSums/Sum_term.con".
179 inline "cic:/CoRN/algebra/CSums/Sum0_shift.con".
182 Hint Resolve Sum0_shift: algebra.
185 inline "cic:/CoRN/algebra/CSums/Sum_shift.con".
187 inline "cic:/CoRN/algebra/CSums/Sum_big_shift.con".
189 inline "cic:/CoRN/algebra/CSums/Sumx_Sum0.con".
196 Implicit Arguments Sum [G].
200 Implicit Arguments Sum0 [G].
204 Implicit Arguments Sumx [G n].
208 Implicit Arguments Sum2 [G m n].
212 The next results are useful for calculating some special sums,
213 often referred to as ``Mengolli Sums''.
214 %\begin{convention}% Let [G] be an abelian group.
222 alias id "G" = "cic:/CoRN/algebra/CSums/More_Sums/G.var".
224 inline "cic:/CoRN/algebra/CSums/Mengolli_Sum.con".
226 inline "cic:/CoRN/algebra/CSums/Mengolli_Sum_gen.con".
228 inline "cic:/CoRN/algebra/CSums/str_Mengolli_Sum_gen.con".
230 inline "cic:/CoRN/algebra/CSums/Sumx_to_Sum.con".
237 Hint Resolve Sum_one Sum_Sum Sum_first Sum_last Sum_last' Sum_wd
238 Sum_plus_Sum: algebra.
242 Hint Resolve Sum_minus_Sum inv_Sum inv_Sum0: algebra.