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
|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(**********************************************************************)
(** The integer logarithms with base 2. *)
(** THIS FILE IS DEPRECATED.
Please rather use [Z.log2] (or [Z.log2_up]), which
are defined in [BinIntDef], and whose properties can
be found in [BinInt.Z]. *)
(* There are three logarithms defined here,
depending on the rounding of the real 2-based logarithm:
- [Log_inf]: [y = (Log_inf x) iff 2^y <= x < 2^(y+1)]
i.e. [Log_inf x] is the biggest integer that is smaller than [Log x]
- [Log_sup]: [y = (Log_sup x) iff 2^(y-1) < x <= 2^y]
i.e. [Log_inf x] is the smallest integer that is bigger than [Log x]
- [Log_nearest]: [y= (Log_nearest x) iff 2^(y-1/2) < x <= 2^(y+1/2)]
i.e. [Log_nearest x] is the integer nearest from [Log x] *)
Require Import ZArith_base Omega Zcomplements Zpower.
Local Open Scope Z_scope.
Section Log_pos. (* Log of positive integers *)
(** First we build [log_inf] and [log_sup] *)
Fixpoint log_inf (p:positive) : Z :=
match p with
| xH => 0 (* 1 *)
| xO q => Z.succ (log_inf q) (* 2n *)
| xI q => Z.succ (log_inf q) (* 2n+1 *)
end.
Fixpoint log_sup (p:positive) : Z :=
match p with
| xH => 0 (* 1 *)
| xO n => Z.succ (log_sup n) (* 2n *)
| xI n => Z.succ (Z.succ (log_inf n)) (* 2n+1 *)
end.
Hint Unfold log_inf log_sup.
Lemma Psize_log_inf : forall p, Zpos (Pos.size p) = Z.succ (log_inf p).
Proof.
induction p; simpl; now rewrite ?Pos2Z.inj_succ, ?IHp.
Qed.
Lemma Zlog2_log_inf : forall p, Z.log2 (Zpos p) = log_inf p.
Proof.
unfold Z.log2. destruct p; simpl; trivial; apply Psize_log_inf.
Qed.
Lemma Zlog2_up_log_sup : forall p, Z.log2_up (Zpos p) = log_sup p.
Proof.
induction p; simpl log_sup.
- change (Zpos p~1) with (2*(Zpos p)+1).
rewrite Z.log2_up_succ_double, Zlog2_log_inf; try easy.
unfold Z.succ. now rewrite !(Z.add_comm _ 1), Z.add_assoc.
- change (Zpos p~0) with (2*Zpos p).
now rewrite Z.log2_up_double, IHp.
- reflexivity.
Qed.
(** Then we give the specifications of [log_inf] and [log_sup]
and prove their validity *)
Hint Resolve Z.le_trans: zarith.
Theorem log_inf_correct :
forall x:positive,
0 <= log_inf x /\ two_p (log_inf x) <= Zpos x < two_p (Z.succ (log_inf x)).
Proof.
simple induction x; intros; simpl;
[ elim H; intros Hp HR; clear H; split;
[ auto with zarith
| rewrite two_p_S with (x := Z.succ (log_inf p)) by (apply Z.le_le_succ_r; trivial);
rewrite two_p_S by trivial;
rewrite two_p_S in HR by trivial; rewrite (BinInt.Pos2Z.inj_xI p);
omega ]
| elim H; intros Hp HR; clear H; split;
[ auto with zarith
| rewrite two_p_S with (x := Z.succ (log_inf p)) by (apply Z.le_le_succ_r; trivial);
rewrite two_p_S by trivial;
rewrite two_p_S in HR by trivial; rewrite (BinInt.Pos2Z.inj_xO p);
omega ]
| unfold two_power_pos; unfold shift_pos; simpl;
omega ].
Qed.
Definition log_inf_correct1 (p:positive) := proj1 (log_inf_correct p).
Definition log_inf_correct2 (p:positive) := proj2 (log_inf_correct p).
Opaque log_inf_correct1 log_inf_correct2.
Hint Resolve log_inf_correct1 log_inf_correct2: zarith.
Lemma log_sup_correct1 : forall p:positive, 0 <= log_sup p.
Proof.
simple induction p; intros; simpl; auto with zarith.
Qed.
(** For every [p], either [p] is a power of two and [(log_inf p)=(log_sup p)]
either [(log_sup p)=(log_inf p)+1] *)
Theorem log_sup_log_inf :
forall p:positive,
IF Zpos p = two_p (log_inf p) then Zpos p = two_p (log_sup p)
else log_sup p = Z.succ (log_inf p).
Proof.
simple induction p; intros;
[ elim H; right; simpl;
rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
rewrite BinInt.Pos2Z.inj_xI; unfold Z.succ; omega
| elim H; clear H; intro Hif;
[ left; simpl;
rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
rewrite (two_p_S (log_sup p0) (log_sup_correct1 p0));
rewrite <- (proj1 Hif); rewrite <- (proj2 Hif);
auto
| right; simpl;
rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
rewrite BinInt.Pos2Z.inj_xO; unfold Z.succ;
omega ]
| left; auto ].
Qed.
Theorem log_sup_correct2 :
forall x:positive, two_p (Z.pred (log_sup x)) < Zpos x <= two_p (log_sup x).
Proof.
intro.
elim (log_sup_log_inf x).
(* x is a power of two and [log_sup = log_inf] *)
intros [E1 E2]; rewrite E2.
split; [ apply two_p_pred; apply log_sup_correct1 | apply Z.le_refl ].
intros [E1 E2]; rewrite E2.
rewrite (Z.pred_succ (log_inf x)).
generalize (log_inf_correct2 x); omega.
Qed.
Lemma log_inf_le_log_sup : forall p:positive, log_inf p <= log_sup p.
Proof.
simple induction p; simpl; intros; omega.
Qed.
Lemma log_sup_le_Slog_inf : forall p:positive, log_sup p <= Z.succ (log_inf p).
Proof.
simple induction p; simpl; intros; omega.
Qed.
(** Now it's possible to specify and build the [Log] rounded to the nearest *)
Fixpoint log_near (x:positive) : Z :=
match x with
| xH => 0
| xO xH => 1
| xI xH => 2
| xO y => Z.succ (log_near y)
| xI y => Z.succ (log_near y)
end.
Theorem log_near_correct1 : forall p:positive, 0 <= log_near p.
Proof.
simple induction p; simpl; intros;
[ elim p0; auto with zarith
| elim p0; auto with zarith
| trivial with zarith ].
intros; apply Z.le_le_succ_r.
generalize H0; now elim p1.
intros; apply Z.le_le_succ_r.
generalize H0; now elim p1.
Qed.
Theorem log_near_correct2 :
forall p:positive, log_near p = log_inf p \/ log_near p = log_sup p.
Proof.
simple induction p.
intros p0 [Einf| Esup].
simpl. rewrite Einf.
case p0; [ left | left | right ]; reflexivity.
simpl; rewrite Esup.
elim (log_sup_log_inf p0).
generalize (log_inf_le_log_sup p0).
generalize (log_sup_le_Slog_inf p0).
case p0; auto with zarith.
intros; omega.
case p0; intros; auto with zarith.
intros p0 [Einf| Esup].
simpl.
repeat rewrite Einf.
case p0; intros; auto with zarith.
simpl.
repeat rewrite Esup.
case p0; intros; auto with zarith.
auto.
Qed.
End Log_pos.
Section divers.
(** Number of significative digits. *)
Definition N_digits (x:Z) :=
match x with
| Zpos p => log_inf p
| Zneg p => log_inf p
| Z0 => 0
end.
Lemma ZERO_le_N_digits : forall x:Z, 0 <= N_digits x.
Proof.
simple induction x; simpl;
[ apply Z.le_refl | exact log_inf_correct1 | exact log_inf_correct1 ].
Qed.
Lemma log_inf_shift_nat : forall n:nat, log_inf (shift_nat n 1) = Z.of_nat n.
Proof.
simple induction n; intros;
[ try trivial | rewrite Nat2Z.inj_succ; rewrite <- H; reflexivity ].
Qed.
Lemma log_sup_shift_nat : forall n:nat, log_sup (shift_nat n 1) = Z.of_nat n.
Proof.
simple induction n; intros;
[ try trivial | rewrite Nat2Z.inj_succ; rewrite <- H; reflexivity ].
Qed.
(** [Is_power p] means that p is a power of two *)
Fixpoint Is_power (p:positive) : Prop :=
match p with
| xH => True
| xO q => Is_power q
| xI q => False
end.
Lemma Is_power_correct :
forall p:positive, Is_power p <-> (exists y : nat, p = shift_nat y 1).
Proof.
split;
[ elim p;
[ simpl; tauto
| simpl; intros; generalize (H H0); intro H1; elim H1;
intros y0 Hy0; exists (S y0); rewrite Hy0; reflexivity
| intro; exists 0%nat; reflexivity ]
| intros; elim H; intros; rewrite H0; elim x; intros; simpl; trivial ].
Qed.
Lemma Is_power_or : forall p:positive, Is_power p \/ ~ Is_power p.
Proof.
simple induction p;
[ intros; right; simpl; tauto
| intros; elim H;
[ intros; left; simpl; exact H0
| intros; right; simpl; exact H0 ]
| left; simpl; trivial ].
Qed.
End divers.
|
