blob: 9442eff0743bfd0e18b9d2bd71acdaec1f6ccf4a (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Cauchy_prod.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*)
Require Rbase.
Require Rfunctions.
Require Rseries.
Require PartSum.
V7only [ Import nat_scope. Import Z_scope. Import R_scope. ].
Open Local Scope R_scope.
(**********)
Lemma sum_N_predN : (An:nat->R;N:nat) (lt O N) -> (sum_f_R0 An N)==``(sum_f_R0 An (pred N)) + (An N)``.
Intros.
Replace N with (S (pred N)).
Rewrite tech5.
Reflexivity.
Symmetry; Apply S_pred with O; Assumption.
Qed.
(**********)
Lemma sum_plus : (An,Bn:nat->R;N:nat) (sum_f_R0 [l:nat]``(An l)+(Bn l)`` N)==``(sum_f_R0 An N)+(sum_f_R0 Bn N)``.
Intros.
Induction N.
Reflexivity.
Do 3 Rewrite tech5.
Rewrite HrecN; Ring.
Qed.
(* The main result *)
Theorem cauchy_finite : (An,Bn:nat->R;N:nat) (lt O N) -> (Rmult (sum_f_R0 An N) (sum_f_R0 Bn N)) == (Rplus (sum_f_R0 [k:nat](sum_f_R0 [p:nat]``(An p)*(Bn (minus k p))`` k) N) (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N))).
Intros; Induction N.
Elim (lt_n_n ? H).
Cut N=O\/(lt O N).
Intro; Elim H0; Intro.
Rewrite H1; Simpl; Ring.
Replace (pred (S N)) with (S (pred N)).
Do 5 Rewrite tech5.
Rewrite Rmult_Rplus_distrl; Rewrite Rmult_Rplus_distr; Rewrite (HrecN H1).
Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r.
Replace (pred (minus (S N) (S (pred N)))) with (O).
Rewrite Rmult_Rplus_distr; Replace (sum_f_R0 [l:nat]``(An (S (plus l (S (pred N)))))*(Bn (minus (S N) l))`` O) with ``(An (S N))*(Bn (S N))``.
Repeat Rewrite <- Rplus_assoc; Do 2 Rewrite <- (Rplus_sym ``(An (S N))*(Bn (S N))``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r.
Rewrite <- minus_n_n; Cut N=(1)\/(le (2) N).
Intro; Elim H2; Intro.
Rewrite H3; Simpl; Ring.
Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))) (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N))).
Replace (sum_f_R0 [p:nat]``(An p)*(Bn (minus (S N) p))`` N) with (Rplus (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N)) ``(An O)*(Bn (S N))``).
Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N))); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r.
Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus (S N) l))`` (pred (minus (S N) k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) (Rmult (Bn (S N)) (sum_f_R0 [l:nat](An (S l)) (pred N)))).
Rewrite (decomp_sum An N H1); Rewrite Rmult_Rplus_distrl; Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym ``(An O)*(Bn (S N))``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r.
Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym (Rmult (sum_f_R0 [i:nat](An (S i)) (pred N)) (Bn (S N)))); Rewrite <- (Rplus_sym (Rmult (Bn (S N)) (sum_f_R0 [i:nat](An (S i)) (pred N)))); Rewrite (Rmult_sym (Bn (S N))); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r.
Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))) (Rmult (An (S N)) (sum_f_R0 [l:nat](Bn (S l)) (pred N)))).
Rewrite (decomp_sum Bn N H1); Rewrite Rmult_Rplus_distr.
Pose Z := (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))); Pose Z2 := (sum_f_R0 [i:nat](Bn (S i)) (pred N)); Ring.
Rewrite (sum_N_predN [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)).
Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred (pred N))) with (sum_f_R0 [k:nat](Rplus (sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) ``(An (S N))*(Bn (S k))``) (pred (pred N))).
Rewrite (sum_plus [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) [k:nat]``(An (S N))*(Bn (S k))`` (pred (pred N))).
Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r.
Replace (pred (minus N (pred N))) with O.
Simpl; Rewrite <- minus_n_O.
Replace (S (pred N)) with N.
Replace (sum_f_R0 [k:nat]``(An (S N))*(Bn (S k))`` (pred (pred N))) with (sum_f_R0 [k:nat]``(Bn (S k))*(An (S N))`` (pred (pred N))).
Rewrite <- (scal_sum [l:nat](Bn (S l)) (pred (pred N)) (An (S N))); Rewrite (sum_N_predN [l:nat](Bn (S l)) (pred N)).
Replace (S (pred N)) with N.
Ring.
Apply S_pred with O; Assumption.
Apply lt_pred; Apply lt_le_trans with (2); [Apply lt_n_Sn | Assumption].
Apply sum_eq; Intros; Apply Rmult_sym.
Apply S_pred with O; Assumption.
Replace (minus N (pred N)) with (1).
Reflexivity.
Pattern 1 N; Replace N with (S (pred N)).
Rewrite <- minus_Sn_m.
Rewrite <- minus_n_n; Reflexivity.
Apply le_n.
Symmetry; Apply S_pred with O; Assumption.
Apply sum_eq; Intros; Rewrite (sum_N_predN [l:nat]``(An (S (S (plus l i))))*(Bn (minus N l))`` (pred (minus N i))).
Replace (S (S (plus (pred (minus N i)) i))) with (S N).
Replace (minus N (pred (minus N i))) with (S i).
Ring.
Rewrite pred_of_minus; Apply INR_eq; Repeat Rewrite minus_INR.
Rewrite S_INR; Ring.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply INR_le; Rewrite minus_INR.
Apply Rle_anti_compatibility with ``(INR i)-1``.
Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring].
Replace ``(INR i)-1+((INR N)-(INR i))`` with ``(INR N)-(INR (S O))``; [Idtac | Ring].
Rewrite <- minus_INR.
Apply le_INR; Apply le_trans with (pred (pred N)).
Assumption.
Rewrite <- pred_of_minus; Apply le_pred_n.
Apply le_trans with (2).
Apply le_n_Sn.
Assumption.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Rewrite <- pred_of_minus.
Apply le_trans with (pred N).
Apply le_S_n.
Replace (S (pred N)) with N.
Replace (S (pred (minus N i))) with (minus N i).
Apply simpl_le_plus_l with i; Rewrite le_plus_minus_r.
Apply le_plus_r.
Apply le_trans with (pred (pred N)); [Assumption | Apply le_trans with (pred N); Apply le_pred_n].
Apply S_pred with O.
Apply simpl_lt_plus_l with i; Rewrite le_plus_minus_r.
Replace (plus i O) with i; [Idtac | Ring].
Apply le_lt_trans with (pred (pred N)); [Assumption | Apply lt_trans with (pred N); Apply lt_pred_n_n].
Apply lt_S_n.
Replace (S (pred N)) with N.
Apply lt_le_trans with (2).
Apply lt_n_Sn.
Assumption.
Apply S_pred with O; Assumption.
Assumption.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply S_pred with O; Assumption.
Apply le_pred_n.
Apply INR_eq; Rewrite pred_of_minus; Do 3 Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite minus_INR.
Ring.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply INR_le.
Rewrite minus_INR.
Apply Rle_anti_compatibility with ``(INR i)-1``.
Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring].
Replace ``(INR i)-1+((INR N)-(INR i))`` with ``(INR N)-(INR (S O))``; [Idtac | Ring].
Rewrite <- minus_INR.
Apply le_INR.
Apply le_trans with (pred (pred N)).
Assumption.
Rewrite <- pred_of_minus.
Apply le_pred_n.
Apply le_trans with (2).
Apply le_n_Sn.
Assumption.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply lt_le_trans with (1).
Apply lt_O_Sn.
Apply INR_le.
Rewrite pred_of_minus.
Repeat Rewrite minus_INR.
Apply Rle_anti_compatibility with ``(INR i)-1``.
Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring].
Replace ``(INR i)-1+((INR N)-(INR i)-(INR (S O)))`` with ``(INR N)-(INR (S O)) -(INR (S O))``.
Repeat Rewrite <- minus_INR.
Apply le_INR.
Apply le_trans with (pred (pred N)).
Assumption.
Do 2 Rewrite <- pred_of_minus.
Apply le_n.
Apply simpl_le_plus_l with (1).
Rewrite le_plus_minus_r.
Simpl; Assumption.
Apply le_trans with (2); [Apply le_n_Sn | Assumption].
Apply le_trans with (2); [Apply le_n_Sn | Assumption].
Ring.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply simpl_le_plus_l with i.
Rewrite le_plus_minus_r.
Replace (plus i (1)) with (S i).
Replace N with (S (pred N)).
Apply le_n_S.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_pred_n.
Symmetry; Apply S_pred with O; Assumption.
Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Reflexivity.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply lt_le_trans with (1).
Apply lt_O_Sn.
Apply le_S_n.
Replace (S (pred N)) with N.
Assumption.
Apply S_pred with O; Assumption.
Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus (S N) l))`` (pred (minus (S N) k))) (pred N)) with (sum_f_R0 [k:nat](Rplus (sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) ``(An (S k))*(Bn (S N))``) (pred N)).
Rewrite (sum_plus [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) [k:nat]``(An (S k))*(Bn (S N))``).
Apply Rplus_plus_r.
Rewrite scal_sum; Reflexivity.
Apply sum_eq; Intros; Rewrite Rplus_sym; Rewrite (decomp_sum [l:nat]``(An (S (plus l i)))*(Bn (minus (S N) l))`` (pred (minus (S N) i))).
Replace (plus O i) with i; [Idtac | Ring].
Rewrite <- minus_n_O; Apply Rplus_plus_r.
Replace (pred (pred (minus (S N) i))) with (pred (minus N i)).
Apply sum_eq; Intros.
Replace (minus (S N) (S i0)) with (minus N i0); [Idtac | Reflexivity].
Replace (plus (S i0) i) with (S (plus i0 i)).
Reflexivity.
Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring.
Cut (minus N i)=(pred (minus (S N) i)).
Intro; Rewrite H5; Reflexivity.
Rewrite pred_of_minus.
Apply INR_eq; Repeat Rewrite minus_INR.
Rewrite S_INR; Ring.
Apply le_trans with N.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Apply le_n_Sn.
Apply simpl_le_plus_l with i.
Rewrite le_plus_minus_r.
Replace (plus i (1)) with (S i).
Apply le_n_S.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring.
Apply le_trans with N.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Apply le_n_Sn.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Replace (pred (minus (S N) i)) with (minus (S N) (S i)).
Replace (minus (S N) (S i)) with (minus N i); [Idtac | Reflexivity].
Apply simpl_lt_plus_l with i.
Rewrite le_plus_minus_r.
Replace (plus i O) with i; [Idtac | Ring].
Apply le_lt_trans with (pred N).
Assumption.
Apply lt_pred_n_n.
Assumption.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Rewrite pred_of_minus.
Apply INR_eq; Repeat Rewrite minus_INR.
Repeat Rewrite S_INR; Ring.
Apply le_trans with N.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Apply le_n_Sn.
Apply simpl_le_plus_l with i.
Rewrite le_plus_minus_r.
Replace (plus i (1)) with (S i).
Apply le_n_S.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring.
Apply le_trans with N.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Apply le_n_Sn.
Apply le_n_S.
Apply le_trans with (pred N).
Assumption.
Apply le_pred_n.
Rewrite Rplus_sym.
Rewrite (decomp_sum [p:nat]``(An p)*(Bn (minus (S N) p))`` N).
Rewrite <- minus_n_O.
Apply Rplus_plus_r.
Apply sum_eq; Intros.
Reflexivity.
Assumption.
Rewrite Rplus_sym.
Rewrite (decomp_sum [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N)).
Rewrite <- minus_n_O.
Replace (sum_f_R0 [l:nat]``(An (S (plus l O)))*(Bn (minus N l))`` (pred N)) with (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N)).
Apply Rplus_plus_r.
Apply sum_eq; Intros.
Replace (pred (minus N (S i))) with (pred (pred (minus N i))).
Apply sum_eq; Intros.
Replace (plus i0 (S i)) with (S (plus i0 i)).
Reflexivity.
Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring.
Cut (pred (minus N i))=(minus N (S i)).
Intro; Rewrite H5; Reflexivity.
Rewrite pred_of_minus.
Apply INR_eq.
Repeat Rewrite minus_INR.
Repeat Rewrite S_INR; Ring.
Apply le_trans with (S (pred (pred N))).
Apply le_n_S; Assumption.
Replace (S (pred (pred N))) with (pred N).
Apply le_pred_n.
Apply S_pred with O.
Apply lt_S_n.
Replace (S (pred N)) with N.
Apply lt_le_trans with (2).
Apply lt_n_Sn.
Assumption.
Apply S_pred with O; Assumption.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply simpl_le_plus_l with i.
Rewrite le_plus_minus_r.
Replace (plus i (1)) with (S i).
Replace N with (S (pred N)).
Apply le_n_S.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_pred_n.
Symmetry; Apply S_pred with O; Assumption.
Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring.
Apply le_trans with (pred (pred N)).
Assumption.
Apply le_trans with (pred N); Apply le_pred_n.
Apply sum_eq; Intros.
Replace (plus i O) with i; [Reflexivity | Trivial].
Apply lt_S_n.
Replace (S (pred N)) with N.
Apply lt_le_trans with (2); [Apply lt_n_Sn | Assumption].
Apply S_pred with O; Assumption.
Inversion H1.
Left; Reflexivity.
Right; Apply le_n_S; Assumption.
Simpl.
Replace (S (pred N)) with N.
Reflexivity.
Apply S_pred with O; Assumption.
Simpl.
Cut (minus N (pred N))=(1).
Intro; Rewrite H2; Reflexivity.
Rewrite pred_of_minus.
Apply INR_eq; Repeat Rewrite minus_INR.
Ring.
Apply lt_le_S; Assumption.
Rewrite <- pred_of_minus; Apply le_pred_n.
Simpl; Symmetry; Apply S_pred with O; Assumption.
Inversion H.
Left; Reflexivity.
Right; Apply lt_le_trans with (1); [Apply lt_n_Sn | Exact H1].
Qed.
|
