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
348
|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2019 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import QArith Qabs.
Require Import ConstructiveReals.
Require Import ConstructiveAbs.
Local Open Scope ConstructiveReals.
(**
Definition and properties of finite sums and powers.
*)
Fixpoint CRsum {R : ConstructiveReals}
(f:nat -> CRcarrier R) (N:nat) : CRcarrier R :=
match N with
| O => f 0%nat
| S i => CRsum f i + f (S i)
end.
Fixpoint CRpow {R : ConstructiveReals} (r:CRcarrier R) (n:nat) : CRcarrier R :=
match n with
| O => 1
| S n => r * (CRpow r n)
end.
Lemma CRsum_eq :
forall {R : ConstructiveReals} (An Bn:nat -> CRcarrier R) (N:nat),
(forall i:nat, (i <= N)%nat -> An i == Bn i) ->
CRsum An N == CRsum Bn N.
Proof.
induction N.
- intros. exact (H O (le_refl _)).
- intros. simpl. apply CRplus_morph. apply IHN.
intros. apply H. apply (le_trans _ N _ H0), le_S, le_refl.
apply H, le_refl.
Qed.
Lemma sum_eq_R0 : forall {R : ConstructiveReals} (un : nat -> CRcarrier R) (n : nat),
(forall k:nat, un k == 0)
-> CRsum un n == 0.
Proof.
induction n.
- intros. apply H.
- intros. simpl. rewrite IHn. rewrite H. apply CRplus_0_l. exact H.
Qed.
Definition INR {R : ConstructiveReals} (n : nat) : CRcarrier R
:= CR_of_Q R (Z.of_nat n # 1).
Lemma sum_const : forall {R : ConstructiveReals} (a : CRcarrier R) (n : nat),
CRsum (fun _ => a) n == a * INR (S n).
Proof.
induction n.
- unfold INR. simpl. rewrite CRmult_1_r. reflexivity.
- simpl. rewrite IHn. unfold INR.
replace (Z.of_nat (S (S n))) with (Z.of_nat (S n) + 1)%Z.
rewrite <- Qinv_plus_distr, CR_of_Q_plus, CRmult_plus_distr_l.
apply CRplus_morph. reflexivity. rewrite CRmult_1_r. reflexivity.
replace 1%Z with (Z.of_nat 1). rewrite <- Nat2Z.inj_add.
apply f_equal. rewrite Nat.add_comm. reflexivity. reflexivity.
Qed.
Lemma multiTriangleIneg : forall {R : ConstructiveReals} (u : nat -> CRcarrier R) (n : nat),
CRabs R (CRsum u n) <= CRsum (fun k => CRabs R (u k)) n.
Proof.
induction n.
- apply CRle_refl.
- simpl. apply (CRle_trans _ (CRabs R (CRsum u n) + CRabs R (u (S n)))).
apply CRabs_triang. apply CRplus_le_compat. apply IHn.
apply CRle_refl.
Qed.
Lemma sum_assoc : forall {R : ConstructiveReals} (u : nat -> CRcarrier R) (n p : nat),
CRsum u (S n + p)
== CRsum u n + CRsum (fun k => u (S n + k)%nat) p.
Proof.
induction p.
- simpl. rewrite Nat.add_0_r. reflexivity.
- simpl. rewrite (Radd_assoc (CRisRing R)). apply CRplus_morph.
rewrite Nat.add_succ_r.
rewrite (CRsum_eq (fun k : nat => u (S (n + k))) (fun k : nat => u (S n + k)%nat)).
rewrite <- IHp. reflexivity. intros. reflexivity. reflexivity.
Qed.
Lemma sum_Rle : forall {R : ConstructiveReals} (un vn : nat -> CRcarrier R) (n : nat),
(forall k, le k n -> un k <= vn k)
-> CRsum un n <= CRsum vn n.
Proof.
induction n.
- intros. apply H. apply le_refl.
- intros. simpl. apply CRplus_le_compat. apply IHn.
intros. apply H. apply (le_trans _ n _ H0). apply le_S, le_refl.
apply H. apply le_refl.
Qed.
Lemma Abs_sum_maj : forall {R : ConstructiveReals} (un vn : nat -> CRcarrier R),
(forall n:nat, CRabs R (un n) <= (vn n))
-> forall n p:nat, (CRabs R (CRsum un n - CRsum un p) <=
CRsum vn (Init.Nat.max n p) - CRsum vn (Init.Nat.min n p)).
Proof.
intros. destruct (le_lt_dec n p).
- destruct (Nat.le_exists_sub n p) as [k [maj _]]. assumption.
subst p. rewrite max_r. rewrite min_l.
setoid_replace (CRsum un n - CRsum un (k + n))
with (-(CRsum un (k + n) - CRsum un n)).
rewrite CRabs_opp.
destruct k. simpl. unfold CRminus. rewrite CRplus_opp_r.
rewrite CRplus_opp_r. rewrite CRabs_right.
apply CRle_refl. apply CRle_refl.
replace (S k + n)%nat with (S n + k)%nat.
unfold CRminus. rewrite sum_assoc. rewrite sum_assoc.
rewrite CRplus_comm.
rewrite <- CRplus_assoc. rewrite CRplus_opp_l.
rewrite CRplus_0_l. rewrite CRplus_comm.
rewrite <- CRplus_assoc. rewrite CRplus_opp_l.
rewrite CRplus_0_l.
apply (CRle_trans _ (CRsum (fun k0 : nat => CRabs R (un (S n + k0)%nat)) k)).
apply multiTriangleIneg. apply sum_Rle. intros.
apply H. rewrite Nat.add_comm, Nat.add_succ_r. reflexivity.
unfold CRminus. rewrite CRopp_plus_distr, CRopp_involutive, CRplus_comm.
reflexivity. assumption. assumption.
- destruct (Nat.le_exists_sub p n) as [k [maj _]]. unfold lt in l.
apply (le_trans p (S p)). apply le_S. apply le_refl. assumption.
subst n. rewrite max_l. rewrite min_r.
destruct k. simpl. unfold CRminus. rewrite CRplus_opp_r.
rewrite CRplus_opp_r. rewrite CRabs_right. apply CRle_refl.
apply CRle_refl.
replace (S k + p)%nat with (S p + k)%nat. unfold CRminus.
rewrite sum_assoc. rewrite sum_assoc.
rewrite CRplus_comm.
rewrite <- CRplus_assoc. rewrite CRplus_opp_l.
rewrite CRplus_0_l. rewrite CRplus_comm.
rewrite <- CRplus_assoc. rewrite CRplus_opp_l.
rewrite CRplus_0_l.
apply (CRle_trans _ (CRsum (fun k0 : nat => CRabs R (un (S p + k0)%nat)) k)).
apply multiTriangleIneg. apply sum_Rle. intros.
apply H. rewrite Nat.add_comm, Nat.add_succ_r. reflexivity.
apply (le_trans p (S p)). apply le_S. apply le_refl. assumption.
apply (le_trans p (S p)). apply le_S. apply le_refl. assumption.
Qed.
Lemma cond_pos_sum : forall {R : ConstructiveReals} (un : nat -> CRcarrier R) (n : nat),
(forall k, 0 <= un k)
-> 0 <= CRsum un n.
Proof.
induction n.
- intros. apply H.
- intros. simpl. rewrite <- CRplus_0_r.
apply CRplus_le_compat. apply IHn, H. apply H.
Qed.
Lemma pos_sum_more : forall {R : ConstructiveReals} (u : nat -> CRcarrier R)
(n p : nat),
(forall k:nat, 0 <= u k)
-> le n p -> CRsum u n <= CRsum u p.
Proof.
intros. destruct (Nat.le_exists_sub n p H0). destruct H1. subst p.
rewrite plus_comm.
destruct x. rewrite plus_0_r. apply CRle_refl. rewrite Nat.add_succ_r.
replace (S (n + x)) with (S n + x)%nat. rewrite sum_assoc.
rewrite <- CRplus_0_r, CRplus_assoc.
apply CRplus_le_compat_l. rewrite CRplus_0_l.
apply cond_pos_sum.
intros. apply H. auto.
Qed.
Lemma sum_opp : forall {R : ConstructiveReals} (un : nat -> CRcarrier R) (n : nat),
CRsum (fun k => - un k) n == - CRsum un n.
Proof.
induction n.
- reflexivity.
- simpl. rewrite IHn. rewrite CRopp_plus_distr. reflexivity.
Qed.
Lemma sum_scale : forall {R : ConstructiveReals} (u : nat -> CRcarrier R) (a : CRcarrier R) (n : nat),
CRsum (fun k : nat => u k * a) n == CRsum u n * a.
Proof.
induction n.
- simpl. rewrite (Rmul_comm (CRisRing R)). reflexivity.
- simpl. rewrite IHn. rewrite CRmult_plus_distr_r.
apply CRplus_morph. reflexivity.
rewrite (Rmul_comm (CRisRing R)). reflexivity.
Qed.
Lemma sum_plus : forall {R : ConstructiveReals} (u v : nat -> CRcarrier R) (n : nat),
CRsum (fun n0 : nat => u n0 + v n0) n == CRsum u n + CRsum v n.
Proof.
induction n.
- reflexivity.
- simpl. rewrite IHn. do 2 rewrite CRplus_assoc.
apply CRplus_morph. reflexivity. rewrite CRplus_comm, CRplus_assoc.
apply CRplus_morph. reflexivity. apply CRplus_comm.
Qed.
Lemma decomp_sum :
forall {R : ConstructiveReals} (An:nat -> CRcarrier R) (N:nat),
(0 < N)%nat ->
CRsum An N == An 0%nat + CRsum (fun i:nat => An (S i)) (pred N).
Proof.
induction N.
- intros. exfalso. inversion H.
- intros _. destruct N. simpl. reflexivity. simpl.
rewrite IHN. rewrite CRplus_assoc.
apply CRplus_morph. reflexivity. reflexivity.
apply le_n_S, le_0_n.
Qed.
Lemma reverse_sum : forall {R : ConstructiveReals} (u : nat -> CRcarrier R) (n : nat),
CRsum u n == CRsum (fun k => u (n-k)%nat) n.
Proof.
induction n.
- intros. reflexivity.
- rewrite (decomp_sum (fun k : nat => u (S n - k)%nat)). simpl.
rewrite CRplus_comm. apply CRplus_morph. reflexivity. assumption.
unfold lt. apply le_n_S. apply le_0_n.
Qed.
Lemma Rplus_le_pos : forall {R : ConstructiveReals} (a b : CRcarrier R),
0 <= b -> a <= a + b.
Proof.
intros. rewrite <- (CRplus_0_r a). rewrite CRplus_assoc.
apply CRplus_le_compat_l. rewrite CRplus_0_l. assumption.
Qed.
Lemma selectOneInSum : forall {R : ConstructiveReals} (u : nat -> CRcarrier R) (n i : nat),
le i n
-> (forall k:nat, 0 <= u k)
-> u i <= CRsum u n.
Proof.
induction n.
- intros. inversion H. subst i. apply CRle_refl.
- intros. apply Nat.le_succ_r in H. destruct H.
apply (CRle_trans _ (CRsum u n)). apply IHn. assumption. assumption.
simpl. apply Rplus_le_pos. apply H0.
subst i. simpl. rewrite CRplus_comm. apply Rplus_le_pos.
apply cond_pos_sum. intros. apply H0.
Qed.
Lemma splitSum : forall {R : ConstructiveReals} (un : nat -> CRcarrier R)
(filter : nat -> bool) (n : nat),
CRsum un n
== CRsum (fun i => if filter i then un i else 0) n
+ CRsum (fun i => if filter i then 0 else un i) n.
Proof.
induction n.
- simpl. destruct (filter O). symmetry; apply CRplus_0_r.
symmetry. apply CRplus_0_l.
- simpl. rewrite IHn. clear IHn. destruct (filter (S n)).
do 2 rewrite CRplus_assoc. apply CRplus_morph. reflexivity.
rewrite CRplus_comm. apply CRplus_morph. reflexivity. rewrite CRplus_0_r.
reflexivity. rewrite CRplus_0_r. rewrite CRplus_assoc. reflexivity.
Qed.
(* Power *)
Lemma pow_R1_Rle : forall {R : ConstructiveReals} (x : CRcarrier R) (n : nat),
1 <= x
-> 1 <= CRpow x n.
Proof.
induction n.
- intros. apply CRle_refl.
- intros. simpl. apply (CRle_trans _ (x * 1)).
rewrite CRmult_1_r. exact H.
apply CRmult_le_compat_l_half. apply (CRlt_le_trans _ 1).
apply CRzero_lt_one. exact H.
apply IHn. exact H.
Qed.
Lemma pow_le : forall {R : ConstructiveReals} (x : CRcarrier R) (n : nat),
0 <= x
-> 0 <= CRpow x n.
Proof.
induction n.
- intros. apply CRlt_asym, CRzero_lt_one.
- intros. simpl. apply CRmult_le_0_compat.
exact H. apply IHn. exact H.
Qed.
Lemma pow_lt : forall {R : ConstructiveReals} (x : CRcarrier R) (n : nat),
0 < x
-> 0 < CRpow x n.
Proof.
induction n.
- intros. apply CRzero_lt_one.
- intros. simpl. apply CRmult_lt_0_compat. exact H.
apply IHn. exact H.
Qed.
Lemma pow_mult : forall {R : ConstructiveReals} (x y : CRcarrier R) (n:nat),
CRpow x n * CRpow y n == CRpow (x*y) n.
Proof.
induction n.
- simpl. rewrite CRmult_1_r. reflexivity.
- simpl. rewrite <- IHn. do 2 rewrite <- (Rmul_assoc (CRisRing R)).
apply CRmult_morph. reflexivity.
rewrite <- (Rmul_comm (CRisRing R)). rewrite <- (Rmul_assoc (CRisRing R)).
apply CRmult_morph. reflexivity.
rewrite <- (Rmul_comm (CRisRing R)). reflexivity.
Qed.
Lemma pow_one : forall {R : ConstructiveReals} (n:nat),
@CRpow R 1 n == 1.
Proof.
induction n. reflexivity.
transitivity (CRmult R 1 (CRpow 1 n)). reflexivity.
rewrite IHn. rewrite CRmult_1_r. reflexivity.
Qed.
Lemma pow_proper : forall {R : ConstructiveReals} (x y : CRcarrier R) (n : nat),
x == y -> CRpow x n == CRpow y n.
Proof.
induction n.
- intros. reflexivity.
- intros. simpl. rewrite IHn, H. reflexivity. exact H.
Qed.
Lemma pow_inv : forall {R : ConstructiveReals} (x : CRcarrier R) (xPos : 0 < x) (n : nat),
CRpow (CRinv R x (inr xPos)) n
== CRinv R (CRpow x n) (inr (pow_lt x n xPos)).
Proof.
induction n.
- rewrite CRinv_1. reflexivity.
- transitivity (CRinv R x (inr xPos) * CRpow (CRinv R x (inr xPos)) n).
reflexivity. rewrite IHn.
assert (0 < x * CRpow x n).
{ apply CRmult_lt_0_compat. exact xPos. apply pow_lt, xPos. }
rewrite <- (CRinv_mult_distr _ _ _ _ (inr H)).
apply CRinv_morph. reflexivity.
Qed.
Lemma pow_plus_distr : forall {R : ConstructiveReals} (x : CRcarrier R) (n p:nat),
CRpow x n * CRpow x p == CRpow x (n+p).
Proof.
induction n.
- intros. simpl. rewrite CRmult_1_l. reflexivity.
- intros. simpl. rewrite CRmult_assoc. apply CRmult_morph.
reflexivity. apply IHn.
Qed.
|
