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
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Zpow_facts Qfield Qreduction.
Lemma Qpower_positive_1 : forall n, Qpower_positive 1 n == 1.
Proof.
induction n;
simpl; try rewrite IHn; reflexivity.
Qed.
Lemma Qpower_1 : forall n, 1^n == 1.
Proof.
intros [|n|n]; simpl; try rewrite Qpower_positive_1; reflexivity.
Qed.
Lemma Qpower_positive_0 : forall n, Qpower_positive 0 n == 0.
Proof.
induction n;
simpl; try rewrite IHn; reflexivity.
Qed.
Lemma Qpower_0 : forall n, (n<>0)%Z -> 0^n == 0.
Proof.
intros [|n|n] Hn; try (elim Hn; reflexivity); simpl;
rewrite Qpower_positive_0; reflexivity.
Qed.
Lemma Qpower_not_0_positive : forall a n, ~a==0 -> ~Qpower_positive a n == 0.
Proof.
intros a n X H.
apply X; clear X.
induction n; simpl in *; try assumption;
destruct (Qmult_integral _ _ H);
try destruct (Qmult_integral _ _ H0); auto.
Qed.
Lemma Qpower_pos_positive : forall p n, 0 <= p -> 0 <= Qpower_positive p n.
Proof.
intros p n Hp.
induction n; simpl; repeat apply Qmult_le_0_compat;assumption.
Qed.
Lemma Qpower_pos : forall p n, 0 <= p -> 0 <= p^n.
Proof.
intros p [|n|n] Hp; simpl; try discriminate;
try apply Qinv_le_0_compat; apply Qpower_pos_positive; assumption.
Qed.
Lemma Qmult_power_positive : forall a b n, Qpower_positive (a*b) n == (Qpower_positive a n)*(Qpower_positive b n).
Proof.
induction n;
simpl; repeat rewrite IHn; ring.
Qed.
Lemma Qmult_power : forall a b n, (a*b)^n == a^n*b^n.
Proof.
intros a b [|n|n]; simpl;
try rewrite Qmult_power_positive;
try rewrite Qinv_mult_distr;
reflexivity.
Qed.
Lemma Qinv_power_positive : forall a n, Qpower_positive (/a) n == /(Qpower_positive a n).
Proof.
induction n;
simpl; repeat (rewrite IHn || rewrite Qinv_mult_distr); reflexivity.
Qed.
Lemma Qinv_power : forall a n, (/a)^n == /a^n.
Proof.
intros a [|n|n]; simpl;
try rewrite Qinv_power_positive;
reflexivity.
Qed.
Lemma Qdiv_power : forall a b n, (a/b)^n == (a^n/b^n).
Proof.
unfold Qdiv.
intros a b n.
rewrite Qmult_power.
rewrite Qinv_power.
reflexivity.
Qed.
Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z ('p))^n.
Proof.
intros n p.
rewrite Qmake_Qdiv.
rewrite Qdiv_power.
rewrite Qpower_1.
unfold Qdiv.
ring.
Qed.
Lemma Qpower_plus_positive : forall a n m, Qpower_positive a (n+m) == (Qpower_positive a n)*(Qpower_positive a m).
Proof.
intros a n m.
unfold Qpower_positive.
apply pow_pos_add.
apply Q_Setoid.
apply Qmult_comp.
apply Qmult_assoc.
Qed.
Lemma Qpower_opp : forall a n, a^(-n) == /a^n.
Proof.
intros a [|n|n]; simpl; try reflexivity.
symmetry; apply Qinv_involutive.
Qed.
Lemma Qpower_minus_positive : forall a (n m:positive),
(m < n)%positive ->
Qpower_positive a (n-m)%positive == (Qpower_positive a n)/(Qpower_positive a m).
Proof.
intros a n m H.
destruct (Qeq_dec a 0) as [EQ|NEQ].
- now rewrite EQ, !Qpower_positive_0.
- rewrite <- (Qdiv_mult_l (Qpower_positive a (n - m)) (Qpower_positive a m)) by
(now apply Qpower_not_0_positive).
f_equiv.
rewrite <- Qpower_plus_positive.
now rewrite Pos.sub_add.
Qed.
Lemma Qpower_plus : forall a n m, ~a==0 -> a^(n+m) == a^n*a^m.
Proof.
intros a [|n|n] [|m|m] H; simpl; try ring;
try rewrite Qpower_plus_positive;
try apply Qinv_mult_distr; try reflexivity;
rewrite ?Z.pos_sub_spec;
case Pos.compare_spec; intros H0; simpl; subst;
try rewrite Qpower_minus_positive;
try (field; try split; apply Qpower_not_0_positive);
assumption.
Qed.
Lemma Qpower_plus' : forall a n m, (n+m <> 0)%Z -> a^(n+m) == a^n*a^m.
Proof.
intros a n m H.
destruct (Qeq_dec a 0)as [X|X].
rewrite X.
rewrite Qpower_0 by assumption.
destruct n; destruct m; try (elim H; reflexivity);
simpl; repeat rewrite Qpower_positive_0; ring_simplify;
reflexivity.
apply Qpower_plus.
assumption.
Qed.
Lemma Qpower_mult_positive : forall a n m,
Qpower_positive a (n*m) == Qpower_positive (Qpower_positive a n) m.
Proof.
intros a n m.
induction n using Pos.peano_ind.
reflexivity.
rewrite Pos.mul_succ_l.
rewrite <- Pos.add_1_l.
do 2 rewrite Qpower_plus_positive.
rewrite IHn.
rewrite Qmult_power_positive.
reflexivity.
Qed.
Lemma Qpower_mult : forall a n m, a^(n*m) == (a^n)^m.
Proof.
intros a [|n|n] [|m|m]; simpl;
try rewrite Qpower_positive_1;
try rewrite Qpower_mult_positive;
try rewrite Qinv_power_positive;
try rewrite Qinv_involutive;
try reflexivity.
Qed.
Lemma Zpower_Qpower : forall (a n:Z), (0<=n)%Z -> inject_Z (a^n) == (inject_Z a)^n.
Proof.
intros a [|n|n] H;[reflexivity| |elim H; reflexivity].
induction n using Pos.peano_ind.
replace (a^1)%Z with a by ring.
ring.
rewrite Pos2Z.inj_succ.
unfold Z.succ.
rewrite Zpower_exp; auto with *; try discriminate.
rewrite Qpower_plus' by discriminate.
rewrite <- IHn by discriminate.
replace (a^'n*a^1)%Z with (a^'n*a)%Z by ring.
ring_simplify.
reflexivity.
Qed.
Lemma Qsqr_nonneg : forall a, 0 <= a^2.
Proof.
intros a.
destruct (Qlt_le_dec 0 a) as [A|A].
apply (Qmult_le_0_compat a a);
(apply Qlt_le_weak; assumption).
setoid_replace (a^2) with ((-a)*(-a)) by ring.
rewrite Qle_minus_iff in A.
setoid_replace (0+ - a) with (-a) in A by ring.
apply Qmult_le_0_compat; assumption.
Qed.
Theorem Qpower_decomp p x y :
Qpower_positive (x#y) p = x ^ Zpos p # (y ^ p).
Proof.
induction p; intros; simpl Qpower_positive; rewrite ?IHp.
- (* xI *)
unfold Qmult, Qnum, Qden. f_equal.
+ now rewrite <- Z.pow_twice_r, <- Z.pow_succ_r.
+ apply Pos2Z.inj; rewrite !Pos2Z.inj_mul, !Pos2Z.inj_pow.
now rewrite <- Z.pow_twice_r, <- Z.pow_succ_r.
- (* xO *)
unfold Qmult, Qnum, Qden. f_equal.
+ now rewrite <- Z.pow_twice_r.
+ apply Pos2Z.inj; rewrite !Pos2Z.inj_mul, !Pos2Z.inj_pow.
now rewrite <- Z.pow_twice_r.
- (* xO *)
now rewrite Z.pow_1_r, Pos.pow_1_r.
Qed.
|
