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
349
350
351
352
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Zbinary.v 9245 2006-10-17 12:53:34Z notin $ i*)
(** Bit vectors interpreted as integers.
Contribution by Jean Duprat (ENS Lyon). *)
Require Import Bvector.
Require Import ZArith.
Require Export Zpower.
Require Import Omega.
(** L'évaluation des vecteurs de booléens se font à la fois en binaire et
en complément à deux. Le nombre appartient à Z.
On utilise donc Omega pour faire les calculs dans Z.
De plus, on utilise les fonctions 2^n où n est un naturel, ici la longueur.
two_power_nat = [n:nat](POS (shift_nat n xH))
: nat->Z
two_power_nat_S
: (n:nat)`(two_power_nat (S n)) = 2*(two_power_nat n)`
Z_lt_ge_dec
: (x,y:Z){`x < y`}+{`x >= y`}
*)
Section VALUE_OF_BOOLEAN_VECTORS.
(** Les calculs sont effectués dans la convention positive usuelle.
Les valeurs correspondent soit à l'écriture binaire (nat),
soit au complément à deux (int).
On effectue le calcul suivant le schéma de Horner.
Le complément à deux n'a de sens que sur les vecteurs de taille
supérieure ou égale à un, le bit de signe étant évalué négativement.
*)
Definition bit_value (b:bool) : Z :=
match b with
| true => 1%Z
| false => 0%Z
end.
Lemma binary_value : forall n:nat, Bvector n -> Z.
Proof.
simple induction n; intros.
exact 0%Z.
inversion H0.
exact (bit_value a + 2 * H H2)%Z.
Defined.
Lemma two_compl_value : forall n:nat, Bvector (S n) -> Z.
Proof.
simple induction n; intros.
inversion H.
exact (- bit_value a)%Z.
inversion H0.
exact (bit_value a + 2 * H H2)%Z.
Defined.
End VALUE_OF_BOOLEAN_VECTORS.
Section ENCODING_VALUE.
(** On calcule la valeur binaire selon un schema de Horner.
Le calcul s'arrete à la longueur du vecteur sans vérification.
On definit une fonction Zmod2 calquee sur Zdiv2 mais donnant le quotient
de la division z=2q+r avec 0<=r<=1.
La valeur en complément à deux est calculée selon un schema de Horner
avec Zmod2, le paramètre est la taille moins un.
*)
Definition Zmod2 (z:Z) :=
match z with
| Z0 => 0%Z
| Zpos p => match p with
| xI q => Zpos q
| xO q => Zpos q
| xH => 0%Z
end
| Zneg p =>
match p with
| xI q => (Zneg q - 1)%Z
| xO q => Zneg q
| xH => (-1)%Z
end
end.
Lemma Zmod2_twice :
forall z:Z, z = (2 * Zmod2 z + bit_value (Zeven.Zodd_bool z))%Z.
Proof.
destruct z; simpl in |- *.
trivial.
destruct p; simpl in |- *; trivial.
destruct p; simpl in |- *.
destruct p as [p| p| ]; simpl in |- *.
rewrite <- (Pdouble_minus_one_o_succ_eq_xI p); trivial.
trivial.
trivial.
trivial.
trivial.
Qed.
Lemma Z_to_binary : forall n:nat, Z -> Bvector n.
Proof.
simple induction n; intros.
exact Bnil.
exact (Bcons (Zeven.Zodd_bool H0) n0 (H (Zeven.Zdiv2 H0))).
Defined.
Lemma Z_to_two_compl : forall n:nat, Z -> Bvector (S n).
Proof.
simple induction n; intros.
exact (Bcons (Zeven.Zodd_bool H) 0 Bnil).
exact (Bcons (Zeven.Zodd_bool H0) (S n0) (H (Zmod2 H0))).
Defined.
End ENCODING_VALUE.
Section Z_BRIC_A_BRAC.
(** Bibliotheque de lemmes utiles dans la section suivante.
Utilise largement ZArith.
Mériterait d'être récrite.
*)
Lemma binary_value_Sn :
forall (n:nat) (b:bool) (bv:Bvector n),
binary_value (S n) (Vcons bool b n bv) =
(bit_value b + 2 * binary_value n bv)%Z.
Proof.
intros; auto.
Qed.
Lemma Z_to_binary_Sn :
forall (n:nat) (b:bool) (z:Z),
(z >= 0)%Z ->
Z_to_binary (S n) (bit_value b + 2 * z) = Bcons b n (Z_to_binary n z).
Proof.
destruct b; destruct z; simpl in |- *; auto.
intro H; elim H; trivial.
Qed.
Lemma binary_value_pos :
forall (n:nat) (bv:Bvector n), (binary_value n bv >= 0)%Z.
Proof.
induction bv as [| a n v IHbv]; simpl in |- *.
omega.
destruct a; destruct (binary_value n v); simpl in |- *; auto.
auto with zarith.
Qed.
Lemma two_compl_value_Sn :
forall (n:nat) (bv:Bvector (S n)) (b:bool),
two_compl_value (S n) (Bcons b (S n) bv) =
(bit_value b + 2 * two_compl_value n bv)%Z.
Proof.
intros; auto.
Qed.
Lemma Z_to_two_compl_Sn :
forall (n:nat) (b:bool) (z:Z),
Z_to_two_compl (S n) (bit_value b + 2 * z) =
Bcons b (S n) (Z_to_two_compl n z).
Proof.
destruct b; destruct z as [| p| p]; auto.
destruct p as [p| p| ]; auto.
destruct p as [p| p| ]; simpl in |- *; auto.
intros; rewrite (Psucc_o_double_minus_one_eq_xO p); trivial.
Qed.
Lemma Z_to_binary_Sn_z :
forall (n:nat) (z:Z),
Z_to_binary (S n) z =
Bcons (Zeven.Zodd_bool z) n (Z_to_binary n (Zeven.Zdiv2 z)).
Proof.
intros; auto.
Qed.
Lemma Z_div2_value :
forall z:Z,
(z >= 0)%Z -> (bit_value (Zeven.Zodd_bool z) + 2 * Zeven.Zdiv2 z)%Z = z.
Proof.
destruct z as [| p| p]; auto.
destruct p; auto.
intro H; elim H; trivial.
Qed.
Lemma Pdiv2 : forall z:Z, (z >= 0)%Z -> (Zeven.Zdiv2 z >= 0)%Z.
Proof.
destruct z as [| p| p].
auto.
destruct p; auto.
simpl in |- *; intros; omega.
intro H; elim H; trivial.
Qed.
Lemma Zdiv2_two_power_nat :
forall (z:Z) (n:nat),
(z >= 0)%Z ->
(z < two_power_nat (S n))%Z -> (Zeven.Zdiv2 z < two_power_nat n)%Z.
Proof.
intros.
cut (2 * Zeven.Zdiv2 z < 2 * two_power_nat n)%Z; intros.
omega.
rewrite <- two_power_nat_S.
destruct (Zeven.Zeven_odd_dec z); intros.
rewrite <- Zeven.Zeven_div2; auto.
generalize (Zeven.Zodd_div2 z H z0); omega.
Qed.
Lemma Z_to_two_compl_Sn_z :
forall (n:nat) (z:Z),
Z_to_two_compl (S n) z =
Bcons (Zeven.Zodd_bool z) (S n) (Z_to_two_compl n (Zmod2 z)).
Proof.
intros; auto.
Qed.
Lemma Zeven_bit_value :
forall z:Z, Zeven.Zeven z -> bit_value (Zeven.Zodd_bool z) = 0%Z.
Proof.
destruct z; unfold bit_value in |- *; auto.
destruct p; tauto || (intro H; elim H).
destruct p; tauto || (intro H; elim H).
Qed.
Lemma Zodd_bit_value :
forall z:Z, Zeven.Zodd z -> bit_value (Zeven.Zodd_bool z) = 1%Z.
Proof.
destruct z; unfold bit_value in |- *; auto.
intros; elim H.
destruct p; tauto || (intros; elim H).
destruct p; tauto || (intros; elim H).
Qed.
Lemma Zge_minus_two_power_nat_S :
forall (n:nat) (z:Z),
(z >= - two_power_nat (S n))%Z -> (Zmod2 z >= - two_power_nat n)%Z.
Proof.
intros n z; rewrite (two_power_nat_S n).
generalize (Zmod2_twice z).
destruct (Zeven.Zeven_odd_dec z) as [H| H].
rewrite (Zeven_bit_value z H); intros; omega.
rewrite (Zodd_bit_value z H); intros; omega.
Qed.
Lemma Zlt_two_power_nat_S :
forall (n:nat) (z:Z),
(z < two_power_nat (S n))%Z -> (Zmod2 z < two_power_nat n)%Z.
Proof.
intros n z; rewrite (two_power_nat_S n).
generalize (Zmod2_twice z).
destruct (Zeven.Zeven_odd_dec z) as [H| H].
rewrite (Zeven_bit_value z H); intros; omega.
rewrite (Zodd_bit_value z H); intros; omega.
Qed.
End Z_BRIC_A_BRAC.
Section COHERENT_VALUE.
(** On vérifie que dans l'intervalle de définition les fonctions sont
réciproques l'une de l'autre. Elles utilisent les lemmes du bric-a-brac.
*)
Lemma binary_to_Z_to_binary :
forall (n:nat) (bv:Bvector n), Z_to_binary n (binary_value n bv) = bv.
Proof.
induction bv as [| a n bv IHbv].
auto.
rewrite binary_value_Sn.
rewrite Z_to_binary_Sn.
rewrite IHbv; trivial.
apply binary_value_pos.
Qed.
Lemma two_compl_to_Z_to_two_compl :
forall (n:nat) (bv:Bvector n) (b:bool),
Z_to_two_compl n (two_compl_value n (Bcons b n bv)) = Bcons b n bv.
Proof.
induction bv as [| a n bv IHbv]; intro b.
destruct b; auto.
rewrite two_compl_value_Sn.
rewrite Z_to_two_compl_Sn.
rewrite IHbv; trivial.
Qed.
Lemma Z_to_binary_to_Z :
forall (n:nat) (z:Z),
(z >= 0)%Z ->
(z < two_power_nat n)%Z -> binary_value n (Z_to_binary n z) = z.
Proof.
induction n as [| n IHn].
unfold two_power_nat, shift_nat in |- *; simpl in |- *; intros; omega.
intros; rewrite Z_to_binary_Sn_z.
rewrite binary_value_Sn.
rewrite IHn.
apply Z_div2_value; auto.
apply Pdiv2; trivial.
apply Zdiv2_two_power_nat; trivial.
Qed.
Lemma Z_to_two_compl_to_Z :
forall (n:nat) (z:Z),
(z >= - two_power_nat n)%Z ->
(z < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z) = z.
Proof.
induction n as [| n IHn].
unfold two_power_nat, shift_nat in |- *; simpl in |- *; intros.
assert (z = (-1)%Z \/ z = 0%Z). omega.
intuition; subst z; trivial.
intros; rewrite Z_to_two_compl_Sn_z.
rewrite two_compl_value_Sn.
rewrite IHn.
generalize (Zmod2_twice z); omega.
apply Zge_minus_two_power_nat_S; auto.
apply Zlt_two_power_nat_S; auto.
Qed.
End COHERENT_VALUE.
|
