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
|
(************************************************************************)
(* *)
(* Micromega: A reflexive tactic using the Positivstellensatz *)
(* *)
(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
(* *)
(************************************************************************)
Require Import ZArith.
Require Import Psatz.
Require Import Ring_normalize.
Open Scope Z_scope.
Require Import ZMicromega.
Require Import VarMap.
(* false in Q : x=1/2 and n=1 *)
Lemma not_so_easy : forall x n : Z,
2*x + 1 <= 2 *n -> x <= n-1.
Proof.
intros.
lia.
Qed.
(* From Laurent Théry *)
Lemma some_pol : forall x, 4 * x ^ 2 + 3 * x + 2 >= 0.
Proof.
intros.
psatz Z 2.
Qed.
Lemma Zdiscr: forall a b c x,
a * x ^ 2 + b * x + c = 0 -> b ^ 2 - 4 * a * c >= 0.
Proof.
intros ; psatz Z 4.
Qed.
Lemma plus_minus : forall x y,
0 = x + y -> 0 = x -y -> 0 = x /\ 0 = y.
Proof.
intros.
lia.
Qed.
Lemma mplus_minus : forall x y,
x + y >= 0 -> x -y >= 0 -> x^2 - y^2 >= 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma pol3: forall x y, 0 <= x + y ->
x^3 + 3*x^2*y + 3*x* y^2 + y^3 >= 0.
Proof.
intros; psatz Z 4.
Qed.
(* Motivating example from: Expressiveness + Automation + Soundness:
Towards COmbining SMT Solvers and Interactive Proof Assistants *)
Parameter rho : Z.
Parameter rho_ge : rho >= 0.
Parameter correct : Z -> Z -> Prop.
Definition rbound1 (C:Z -> Z -> Z) : Prop :=
forall p s t, correct p t /\ s <= t -> C p t - C p s <= (1-rho)*(t-s).
Definition rbound2 (C:Z -> Z -> Z) : Prop :=
forall p s t, correct p t /\ s <= t -> (1-rho)*(t-s) <= C p t - C p s.
Lemma bounded_drift : forall s t p q C D, s <= t /\ correct p t /\ correct q t /\
rbound1 C /\ rbound2 C /\ rbound1 D /\ rbound2 D ->
Z.abs (C p t - D q t) <= Z.abs (C p s - D q s) + 2 * rho * (t- s).
Proof.
intros.
generalize (Z.abs_eq (C p t - D q t)).
generalize (Z.abs_neq (C p t - D q t)).
generalize (Z.abs_eq (C p s -D q s)).
generalize (Z.abs_neq (C p s - D q s)).
unfold rbound2 in H.
unfold rbound1 in H.
intuition.
generalize (H6 _ _ _ (conj H H4)).
generalize (H7 _ _ _ (conj H H4)).
generalize (H8 _ _ _ (conj H H4)).
generalize (H10 _ _ _ (conj H H4)).
generalize (H6 _ _ _ (conj H5 H4)).
generalize (H7 _ _ _ (conj H5 H4)).
generalize (H8 _ _ _ (conj H5 H4)).
generalize (H10 _ _ _ (conj H5 H4)).
generalize rho_ge.
psatz Z 2.
Qed.
(* Rule of signs *)
Lemma sign_pos_pos: forall x y,
x > 0 -> y > 0 -> x*y > 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_pos_zero: forall x y,
x > 0 -> y = 0 -> x*y = 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_pos_neg: forall x y,
x > 0 -> y < 0 -> x*y < 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_zer_pos: forall x y,
x = 0 -> y > 0 -> x*y = 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_zero_zero: forall x y,
x = 0 -> y = 0 -> x*y = 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_zero_neg: forall x y,
x = 0 -> y < 0 -> x*y = 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_neg_pos: forall x y,
x < 0 -> y > 0 -> x*y < 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_neg_zero: forall x y,
x < 0 -> y = 0 -> x*y = 0.
Proof.
intros; psatz Z 2.
Qed.
Lemma sign_neg_neg: forall x y,
x < 0 -> y < 0 -> x*y > 0.
Proof.
intros; psatz Z 2.
Qed.
(* Other (simple) examples *)
Lemma binomial : forall x y, (x+y)^2 = x^2 + 2*x*y + y^2.
Proof.
intros.
lia.
Qed.
Lemma product : forall x y, x >= 0 -> y >= 0 -> x * y >= 0.
Proof.
intros.
psatz Z 2.
Qed.
Lemma product_strict : forall x y, x > 0 -> y > 0 -> x * y > 0.
Proof.
intros.
psatz Z 2.
Qed.
Lemma pow_2_pos : forall x, x ^ 2 + 1 = 0 -> False.
Proof.
intros ; psatz Z 2.
Qed.
(* Found in Parrilo's talk *)
(* BUG?: certificate with **very** big coefficients *)
Lemma parrilo_ex : forall x y, x - y^2 + 3 >= 0 -> y + x^2 + 2 = 0 -> False.
Proof.
intros.
psatz Z 2.
Qed.
(* from hol_light/Examples/sos.ml *)
Lemma hol_light1 : forall a1 a2 b1 b2,
a1 >= 0 -> a2 >= 0 ->
(a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) ->
(a1 * b1 + a2 * b2 = 0) -> a1 * a2 - b1 * b2 >= 0.
Proof.
intros ; psatz Z 4.
Qed.
Lemma hol_light2 : forall x a,
3 * x + 7 * a < 4 -> 3 < 2 * x -> a < 0.
Proof.
intros ; psatz Z 2.
Qed.
Lemma hol_light3 : forall b a c x,
b ^ 2 < 4 * a * c -> (a * x ^2 + b * x + c = 0) -> False.
Proof.
intros ; psatz Z 4.
Qed.
Lemma hol_light4 : forall a c b x,
a * x ^ 2 + b * x + c = 0 -> b ^ 2 >= 4 * a * c.
Proof.
intros ; psatz Z 4.
Qed.
Lemma hol_light5 : forall x y,
0 <= x /\ x <= 1 /\ 0 <= y /\ y <= 1
-> x ^ 2 + y ^ 2 < 1 \/
(x - 1) ^ 2 + y ^ 2 < 1 \/
x ^ 2 + (y - 1) ^ 2 < 1 \/
(x - 1) ^ 2 + (y - 1) ^ 2 < 1.
Proof.
intros; psatz Z 3.
Qed.
Lemma hol_light7 : forall x y z,
0<= x /\ 0 <= y /\ 0 <= z /\ x + y + z <= 3
-> x * y + x * z + y * z >= 3 * x * y * z.
Proof.
intros ; psatz Z 3.
Qed.
Lemma hol_light8 : forall x y z,
x ^ 2 + y ^ 2 + z ^ 2 = 1 -> (x + y + z) ^ 2 <= 3.
Proof.
intros ; psatz Z 2.
Qed.
Lemma hol_light9 : forall w x y z,
w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1
-> (w + x + y + z) ^ 2 <= 4.
Proof.
intros; psatz Z 2.
Qed.
Lemma hol_light10 : forall x y,
x >= 1 /\ y >= 1 -> x * y >= x + y - 1.
Proof.
intros ; psatz Z 2.
Qed.
Lemma hol_light11 : forall x y,
x > 1 /\ y > 1 -> x * y > x + y - 1.
Proof.
intros ; psatz Z 2.
Qed.
Lemma hol_light12: forall x y z,
2 <= x /\ x <= 125841 / 50000 /\
2 <= y /\ y <= 125841 / 50000 /\
2 <= z /\ z <= 125841 / 50000
-> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= 0.
Proof.
intros x y z ; set (e:= (125841 / 50000)).
compute in e.
unfold e ; intros ; psatz Z 2.
Qed.
Lemma hol_light14 : forall x y z,
2 <= x /\ x <= 4 /\ 2 <= y /\ y <= 4 /\ 2 <= z /\ z <= 4
-> 12 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z).
Proof.
intros ;psatz Z 2.
Qed.
(* ------------------------------------------------------------------------- *)
(* Inequality from sci.math (see "Leon-Sotelo, por favor"). *)
(* ------------------------------------------------------------------------- *)
Lemma hol_light16 : forall x y,
0 <= x /\ 0 <= y /\ (x * y = 1)
-> x + y <= x ^ 2 + y ^ 2.
Proof.
intros ; psatz Z 2.
Qed.
Lemma hol_light17 : forall x y,
0 <= x /\ 0 <= y /\ (x * y = 1)
-> x * y * (x + y) <= x ^ 2 + y ^ 2.
Proof.
intros ; psatz Z 3.
Qed.
Lemma hol_light18 : forall x y,
0 <= x /\ 0 <= y -> x * y * (x + y) ^ 2 <= (x ^ 2 + y ^ 2) ^ 2.
Proof.
intros ; psatz Z 4.
Qed.
(* ------------------------------------------------------------------------- *)
(* Some examples over integers and natural numbers. *)
(* ------------------------------------------------------------------------- *)
Lemma hol_light19 : forall m n, 2 * m + n = (n + m) + m.
Proof.
intros ; lia.
Qed.
Lemma hol_light22 : forall n, n >= 0 -> n <= n * n.
Proof.
intros.
psatz Z 2.
Qed.
Lemma hol_light24 : forall x1 y1 x2 y2, x1 >= 0 -> x2 >= 0 -> y1 >= 0 -> y2 >= 0 ->
((x1 + y1) ^2 + x1 + 1 = (x2 + y2) ^ 2 + x2 + 1)
-> (x1 + y1 = x2 + y2).
Proof.
intros.
psatz Z 2.
Qed.
Lemma motzkin' : forall x y, (x^2+y^2+1)*(x^2*y^4 + x^4*y^2 + 1 - 3*x^2*y^2) >= 0.
Proof.
intros.
psatz Z 1.
Qed.
Lemma motzkin : forall x y, (x^2*y^4 + x^4*y^2 + 1 - 3*x^2*y^2) >= 0.
Proof.
intros.
generalize (motzkin' x y).
psatz Z 8.
Qed.
|
