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
|
From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint rat.
From mathcomp Require Import lra.
Local Open Scope ring_scope.
Lemma test (F : realFieldType) (x y : F) :
x + 2%:R * y <= 3%:R -> 2%:R * x + y <= 3%:R -> x + y <= 2%:R.
Proof.
lra.
Qed.
(* Print test. *)
(* Print Assumptions test. (* Closed under the global context *) *)
Lemma test_rat (x y : rat) :
x + 2%:R * y <= 3%:R -> 2%:R * x + y <= 3%:R -> x + y <= 2%:R.
Proof.
lra.
Qed.
Lemma test_realDomain (R : realDomainType) (x y : R) :
x + 2%:R * y <= 3%:R -> 2%:R * x + y <= 3%:R -> x + y <= 2%:R.
Proof.
lra.
Qed.
Lemma test_realDomain' (R : realDomainType) (x : int) (y : R) :
x%:~R + 2 * y <= 3 -> (2 * x)%:~R + y <= 3 -> x%:~R + y <= 2.
Proof.
lra.
Qed.
Section Tests.
Variable F : realFieldType.
Implicit Types x y : F.
Lemma test_cast : 0 <= 2 :> F.
Proof.
lra.
Qed.
Example test_div x y : x / 2 + y <= 3 -> x + y / 2 <= 3 -> x + y <= 4.
Proof.
lra.
Qed.
Example test_div_mul x : 1 / (2 * x) <= 1 / 2 / x + 1.
Proof.
lra.
Qed.
Example test_div_inv x : 1 / x^-1 <= x + 1.
Proof.
lra.
Qed.
Example test_div_opp x : (- x)^-1 <= - x^-1 + 1.
Proof.
lra.
Qed.
Example test_div_exp x : (x ^+ 2) ^-1 <= x ^-1 ^+ 2 + 1.
Proof.
lra.
Qed.
Lemma test_lt x y :
x + 2%:R * y < 3%:R -> 2%:R * x + y <= 3%:R -> x + y < 2%:R.
Proof.
lra.
Qed.
Lemma test_eq x y :
x + 2%:R * y = 3%:R -> 2%:R * x + y <= 3%:R -> x + y <= 2%:R.
Proof.
lra.
Qed.
Lemma test_eqop x y :
x + 2%:R * y == 3%:R -> 2%:R * x + y <= 3%:R -> x + y <= 2%:R.
Proof.
lra.
Qed.
Lemma test_prop_in_middle (C : Prop) x :
x <= 2%:R -> C -> x <= 3%:R.
Proof.
lra.
Qed.
Lemma test_opp x : x <= 2%:R -> -x >= -2%:R.
Proof.
lra.
Qed.
Lemma test_iff x : x <= 2%:R <-> -x >= -2%:R.
Proof.
lra.
Qed.
Lemma test_eq_bool x : x <= 2%:R = (-x >= -2%:R).
Proof.
lra.
Qed.
Lemma test_not x : x <= 2%:R -> ~ (x > 2%:R).
Proof.
lra.
Qed.
Lemma test_negb x : x <= 2%:R -> ~~ (x > 2%:R).
Proof.
lra.
Qed.
Lemma test_and x : x <= 2%:R -> (x <= 3%:R /\ x <= 4%:R).
Proof.
lra.
Qed.
Lemma test_andb x : x <= 2%:R -> (x <= 3%:R) && (x <= 4%:R).
Proof.
lra.
Qed.
Lemma test_or x : x <= 2%:R -> (x <= 3%:R \/ x <= 1%:R).
Proof.
lra.
Qed.
Lemma test_orb x : x <= 2%:R -> (x <= 3%:R) || (x <= 1%:R).
Proof.
lra.
Qed.
Lemma test_exfalso x (xle2 : x <= 2%:R) (xge3 : x >= 3%:R) : bool.
Proof.
lra.
Qed.
Lemma test_rat_constant x : 0 <= x -> 1 / 3%:R * x <= 2%:R^-1 * x.
Proof.
lra.
Qed.
Lemma test_rfstr (x : rat) : (x <= 2%:R) || true = true.
Proof.
lra.
Qed.
End Tests.
(* Examples from the test suite of Coq *)
Section TestsCoq.
Variable F : realFieldType.
Implicit Types x y : F.
Lemma plus_minus x y : 0 = x + y -> 0 = x - y -> 0 = x /\ 0 = y.
Proof.
lra.
Qed.
Lemma plus_minus' x y : 0 = x + y -> 0 = x - y -> 0 = x /\ 0 = y.
Proof.
move=> *.
lra.
Qed.
Lemma cst_test : 5%:R^+5 = 5%:R * 5%:R * 5%:R * 5%:R * 5%:R :> F.
Proof.
lra.
Qed.
Goal forall x y, x <> x -> x > y.
Proof.
move=> *.
lra.
Qed.
Lemma binomial x y : (x + y)^+2 = x^+2 + 2%:R * x * y + y^+2.
Proof.
move=> *.
lra.
Qed.
Lemma hol_light19 x y : 2%:R * y + x = (x + y) + y.
Proof.
lra.
Qed.
Lemma vcgen_25 (n m jt j it i : F) :
1 * it + -(2%:R) * i + -(1%:R) = 0 ->
1 * jt + -(2%:R) * j + -(1%:R) = 0 ->
1 * n + -(10%:R) = 0 ->
0 <= -(4028%:R) * i + 6222%:R * j + 705%:R * m + -(16674%:R) ->
0 <= -(418%:R) * i + 651%:R * j + 94 %:R * m + -(1866%:R) ->
0 <= -(209%:R) * i + 302%:R * j + 47%:R * m + -(839%:R) ->
0 <= -(1%:R) * i + 1 * j + -(1%:R) ->
0 <= -(1%:R) * j + 1 * m + 0 ->
0 <= 1 * j + 5%:R * m + -(27%:R) ->
0 <= 2%:R * j + -(1%:R) * m + 2%:R ->
0 <= 7%:R * j + 10%:R * m + -(74%:R) ->
0 <= 18%:R * j + -(139%:R) * m + 1188%:R ->
0 <= 1 * i + 0 ->
0 <= 121%:R * i + 810%:R * j + -(7465%:R) * m + 64350%:R ->
1 = -(2%:R) * i + it.
Proof.
move=> *.
lra.
Qed.
Lemma l1 x y z : `|x - z| <= `|x - y| + `|y - z|.
Proof.
Fail intros; split_Rabs; lra. (* TODO should work *)
Abort.
Lemma l2 x y :
x < `|y| -> y < 1 -> x >= 0 -> - y <= 1 -> `|x| <= 1.
Proof.
Fail intros; split_Rabs; lra. (* TODO should work *)
Abort.
(* Bug 5073 *)
Lemma opp_eq_0_iff x : -x = 0 <-> x = 0.
Proof.
lra.
Qed.
(* From L. Théry *)
Goal forall x y, x = 0 -> x * y = 0.
Proof.
move=> *.
nra.
Qed.
Goal forall x y, 2%:R * x = 0 -> x * y = 0.
Proof.
move=> *.
nra.
Qed.
Goal forall x y, - x * x >= 0 -> x * y = 0.
Proof.
move=> *.
nra.
Qed.
Goal forall x, x * x >= 0.
Proof.
move=> *.
nra.
Qed.
Goal forall x, -x^+2 >= 0 -> x - 1 >= 0 -> False.
Proof.
move=> *.
(* Requires CSDP *)
(* psatz 3. *)
(* Qed. *)
Abort.
Goal forall x, -x^+2 >= 0 -> x - 1 >= 0 -> False.
Proof.
move=> *.
nra.
Qed.
Lemma motzkin' x y :
(x^+2 + y^+2 + 1) * (x^+2 * y^+4 + x^+4*y^+2 + 1 - 3%:R * x^+2 * y^+2) >= 0.
Proof.
move=> *.
(* Requires CSDP *)
(* psatz 3. *)
(* Qed. *)
Abort.
Goal forall x, -x^+2 >= 0 -> x - 1 >= 0 -> False.
Proof.
move=> *.
nra.
Qed.
Goal 1 / (1 - 1) = 0 :> F.
Proof.
Fail lra. (* division by zero *)
Abort.
Goal 0 / (1 - 1) = 0 :> F.
Proof.
lra. (* 0 * x = 0 *)
Qed.
Goal 10%:R ^+ 2 = 100%:R :> F.
Proof.
(* pow is reified as a constant *)
lra.
Qed.
Goal ratr (1 / 2%:R) = 1 / 2%:R :> F.
Proof.
lra.
Qed.
Goal 1 ^+ (2 + 2) = 1 :> F.
Proof.
lra.
Qed.
(* Instance Dplus : DeclaredConstant addn := {}. *) (* TODO should work *)
Goal 1 ^+ (2 + 2) = 1 :> F.
Proof.
lra.
Qed.
End TestsCoq.
Example test_abstract_rmorphism (R : realDomainType) (f : {rmorphism R -> R})
(x y : R) : f y >= 0 -> f x + 2 * f (y + 1) <= f (3 * y + x) + 2.
Proof. lra. Qed.
Example test_concrete_rmorphism (R : realFieldType) (x y : rat) :
ratr y >= 0 :> R -> ratr x + 2 * ratr (y + 1) <= ratr (3 * y + x) + 2 :> R.
Proof. lra. Qed.
|
