summaryrefslogtreecommitdiff
path: root/plugins/fourier/Fourier_util.v
blob: b10c304c875f5b790c7a4ba9630797e673d9eaf3 (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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

Require Export Rbase.
Comments "Lemmas used by the tactic Fourier".

Open Scope R_scope.

Lemma Rfourier_lt : forall x1 y1 a:R, x1 < y1 -> 0 < a -> a * x1 < a * y1.
intros; apply Rmult_lt_compat_l; assumption.
Qed.

Lemma Rfourier_le : forall x1 y1 a:R, x1 <= y1 -> 0 < a -> a * x1 <= a * y1.
red.
intros.
case H; auto with real.
Qed.

Lemma Rfourier_lt_lt :
 forall x1 y1 x2 y2 a:R,
   x1 < y1 -> x2 < y2 -> 0 < a -> x1 + a * x2 < y1 + a * y2.
intros x1 y1 x2 y2 a H H0 H1; try assumption.
apply Rplus_lt_compat.
try exact H.
apply Rfourier_lt.
try exact H0.
try exact H1.
Qed.

Lemma Rfourier_lt_le :
 forall x1 y1 x2 y2 a:R,
   x1 < y1 -> x2 <= y2 -> 0 < a -> x1 + a * x2 < y1 + a * y2.
intros x1 y1 x2 y2 a H H0 H1; try assumption.
case H0; intros.
apply Rplus_lt_compat.
try exact H.
apply Rfourier_lt; auto with real.
rewrite H2.
rewrite (Rplus_comm y1 (a * y2)).
rewrite (Rplus_comm x1 (a * y2)).
apply Rplus_lt_compat_l.
try exact H.
Qed.

Lemma Rfourier_le_lt :
 forall x1 y1 x2 y2 a:R,
   x1 <= y1 -> x2 < y2 -> 0 < a -> x1 + a * x2 < y1 + a * y2.
intros x1 y1 x2 y2 a H H0 H1; try assumption.
case H; intros.
apply Rfourier_lt_le; auto with real.
rewrite H2.
apply Rplus_lt_compat_l.
apply Rfourier_lt; auto with real.
Qed.

Lemma Rfourier_le_le :
 forall x1 y1 x2 y2 a:R,
   x1 <= y1 -> x2 <= y2 -> 0 < a -> x1 + a * x2 <= y1 + a * y2.
intros x1 y1 x2 y2 a H H0 H1; try assumption.
case H0; intros.
red.
left; try assumption.
apply Rfourier_le_lt; auto with real.
rewrite H2.
case H; intros.
red.
left; try assumption.
rewrite (Rplus_comm x1 (a * y2)).
rewrite (Rplus_comm y1 (a * y2)).
apply Rplus_lt_compat_l.
try exact H3.
rewrite H3.
red.
right; try assumption.
auto with real.
Qed.

Lemma Rlt_zero_pos_plus1 : forall x:R, 0 < x -> 0 < 1 + x.
intros x H; try assumption.
rewrite Rplus_comm.
apply Rle_lt_0_plus_1.
red; auto with real.
Qed.

Lemma Rlt_mult_inv_pos : forall x y:R, 0 < x -> 0 < y -> 0 < x * / y.
intros x y H H0; try assumption.
replace 0 with (x * 0).
apply Rmult_lt_compat_l; auto with real.
ring.
Qed.

Lemma Rlt_zero_1 : 0 < 1.
exact Rlt_0_1.
Qed.

Lemma Rle_zero_pos_plus1 : forall x:R, 0 <= x -> 0 <= 1 + x.
intros x H; try assumption.
case H; intros.
red.
left; try assumption.
apply Rlt_zero_pos_plus1; auto with real.
rewrite <- H0.
replace (1 + 0) with 1.
red; left.
exact Rlt_zero_1.
ring.
Qed.

Lemma Rle_mult_inv_pos : forall x y:R, 0 <= x -> 0 < y -> 0 <= x * / y.
intros x y H H0; try assumption.
case H; intros.
red; left.
apply Rlt_mult_inv_pos; auto with real.
rewrite <- H1.
red; right; ring.
Qed.

Lemma Rle_zero_1 : 0 <= 1.
red; left.
exact Rlt_zero_1.
Qed.

Lemma Rle_not_lt : forall n d:R, 0 <= n * / d -> ~ 0 < - n * / d.
intros n d H; red; intros H0; try exact H0.
generalize (Rgt_not_le 0 (n * / d)).
intros H1; elim H1; try assumption.
replace (n * / d) with (- - (n * / d)).
replace 0 with (- -0).
replace (- (n * / d)) with (- n * / d).
replace (-0) with 0.
red.
apply Ropp_gt_lt_contravar.
red.
exact H0.
ring.
ring.
ring.
ring.
Qed.

Lemma Rnot_lt0 : forall x:R, ~ 0 < 0 * x.
intros x; try assumption.
replace (0 * x) with 0.
apply Rlt_irrefl.
ring.
Qed.

Lemma Rlt_not_le_frac_opp : forall n d:R, 0 < n * / d -> ~ 0 <= - n * / d.
intros n d H; try assumption.
apply Rgt_not_le.
replace 0 with (-0).
replace (- n * / d) with (- (n * / d)).
apply Ropp_lt_gt_contravar.
try exact H.
ring.
ring.
Qed.

Lemma Rnot_lt_lt : forall x y:R, ~ 0 < y - x -> ~ x < y.
unfold not; intros.
apply H.
apply Rplus_lt_reg_r with x.
replace (x + 0) with x.
replace (x + (y - x)) with y.
try exact H0.
ring.
ring.
Qed.

Lemma Rnot_le_le : forall x y:R, ~ 0 <= y - x -> ~ x <= y.
unfold not; intros.
apply H.
case H0; intros.
left.
apply Rplus_lt_reg_r with x.
replace (x + 0) with x.
replace (x + (y - x)) with y.
try exact H1.
ring.
ring.
right.
rewrite H1; ring.
Qed.

Lemma Rfourier_gt_to_lt : forall x y:R, y > x -> x < y.
unfold Rgt; intros; assumption.
Qed.

Lemma Rfourier_ge_to_le : forall x y:R, y >= x -> x <= y.
intros x y; exact (Rge_le y x).
Qed.

Lemma Rfourier_eqLR_to_le : forall x y:R, x = y -> x <= y.
exact Req_le.
Qed.

Lemma Rfourier_eqRL_to_le : forall x y:R, y = x -> x <= y.
exact Req_le_sym.
Qed.

Lemma Rfourier_not_ge_lt : forall x y:R, (x >= y -> False) -> x < y.
exact Rnot_ge_lt.
Qed.

Lemma Rfourier_not_gt_le : forall x y:R, (x > y -> False) -> x <= y.
exact Rnot_gt_le.
Qed.

Lemma Rfourier_not_le_gt : forall x y:R, (x <= y -> False) -> x > y.
exact Rnot_le_lt.
Qed.

Lemma Rfourier_not_lt_ge : forall x y:R, (x < y -> False) -> x >= y.
exact Rnot_lt_ge.
Qed.