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
|
(************************************************************************)
(* 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: Zcomplements.v 10617 2008-03-04 18:07:16Z letouzey $ i*)
Require Import ZArithRing.
Require Import ZArith_base.
Require Export Omega.
Require Import Wf_nat.
Open Local Scope Z_scope.
(**********************************************************************)
(** About parity *)
Lemma two_or_two_plus_one :
forall n:Z, {y : Z | n = 2 * y} + {y : Z | n = 2 * y + 1}.
Proof.
intro x; destruct x.
left; split with 0; reflexivity.
destruct p.
right; split with (Zpos p); reflexivity.
left; split with (Zpos p); reflexivity.
right; split with 0; reflexivity.
destruct p.
right; split with (Zneg (1 + p)).
rewrite BinInt.Zneg_xI.
rewrite BinInt.Zneg_plus_distr.
omega.
left; split with (Zneg p); reflexivity.
right; split with (-1); reflexivity.
Qed.
(**********************************************************************)
(** The biggest power of 2 that is stricly less than [a]
Easy to compute: replace all "1" of the binary representation by
"0", except the first "1" (or the first one :-) *)
Fixpoint floor_pos (a:positive) : positive :=
match a with
| xH => 1%positive
| xO a' => xO (floor_pos a')
| xI b' => xO (floor_pos b')
end.
Definition floor (a:positive) := Zpos (floor_pos a).
Lemma floor_gt0 : forall p:positive, floor p > 0.
Proof.
intro.
compute in |- *.
trivial.
Qed.
Lemma floor_ok : forall p:positive, floor p <= Zpos p < 2 * floor p.
Proof.
unfold floor in |- *.
intro a; induction a as [p| p| ].
simpl in |- *.
repeat rewrite BinInt.Zpos_xI.
rewrite (BinInt.Zpos_xO (xO (floor_pos p))).
rewrite (BinInt.Zpos_xO (floor_pos p)).
omega.
simpl in |- *.
repeat rewrite BinInt.Zpos_xI.
rewrite (BinInt.Zpos_xO (xO (floor_pos p))).
rewrite (BinInt.Zpos_xO (floor_pos p)).
rewrite (BinInt.Zpos_xO p).
omega.
simpl in |- *; omega.
Qed.
(**********************************************************************)
(** Two more induction principles over [Z]. *)
Theorem Z_lt_abs_rec :
forall P:Z -> Set,
(forall n:Z, (forall m:Z, Zabs m < Zabs n -> P m) -> P n) ->
forall n:Z, P n.
Proof.
intros P HP p.
set (Q := fun z => 0 <= z -> P z * P (- z)) in *.
cut (Q (Zabs p)); [ intros | apply (Z_lt_rec Q); auto with zarith ].
elim (Zabs_dec p); intro eq; rewrite eq; elim H; auto with zarith.
unfold Q in |- *; clear Q; intros.
apply pair; apply HP.
rewrite Zabs_eq; auto; intros.
elim (H (Zabs m)); intros; auto with zarith.
elim (Zabs_dec m); intro eq; rewrite eq; trivial.
rewrite Zabs_non_eq; auto with zarith.
rewrite Zopp_involutive; intros.
elim (H (Zabs m)); intros; auto with zarith.
elim (Zabs_dec m); intro eq; rewrite eq; trivial.
Qed.
Theorem Z_lt_abs_induction :
forall P:Z -> Prop,
(forall n:Z, (forall m:Z, Zabs m < Zabs n -> P m) -> P n) ->
forall n:Z, P n.
Proof.
intros P HP p.
set (Q := fun z => 0 <= z -> P z /\ P (- z)) in *.
cut (Q (Zabs p)); [ intros | apply (Z_lt_induction Q); auto with zarith ].
elim (Zabs_dec p); intro eq; rewrite eq; elim H; auto with zarith.
unfold Q in |- *; clear Q; intros.
split; apply HP.
rewrite Zabs_eq; auto; intros.
elim (H (Zabs m)); intros; auto with zarith.
elim (Zabs_dec m); intro eq; rewrite eq; trivial.
rewrite Zabs_non_eq; auto with zarith.
rewrite Zopp_involutive; intros.
elim (H (Zabs m)); intros; auto with zarith.
elim (Zabs_dec m); intro eq; rewrite eq; trivial.
Qed.
(** To do case analysis over the sign of [z] *)
Lemma Zcase_sign :
forall (n:Z) (P:Prop), (n = 0 -> P) -> (n > 0 -> P) -> (n < 0 -> P) -> P.
Proof.
intros x P Hzero Hpos Hneg.
induction x as [| p| p].
apply Hzero; trivial.
apply Hpos; apply Zorder.Zgt_pos_0.
apply Hneg; apply Zorder.Zlt_neg_0.
Qed.
Lemma sqr_pos : forall n:Z, n * n >= 0.
Proof.
intro x.
apply (Zcase_sign x (x * x >= 0)).
intros H; rewrite H; omega.
intros H; replace 0 with (0 * 0).
apply Zmult_ge_compat; omega.
omega.
intros H; replace 0 with (0 * 0).
replace (x * x) with (- x * - x).
apply Zmult_ge_compat; omega.
ring.
omega.
Qed.
(**********************************************************************)
(** A list length in Z, tail recursive. *)
Require Import List.
Fixpoint Zlength_aux (acc:Z) (A:Type) (l:list A) {struct l} : Z :=
match l with
| nil => acc
| _ :: l => Zlength_aux (Zsucc acc) A l
end.
Definition Zlength := Zlength_aux 0.
Implicit Arguments Zlength [A].
Section Zlength_properties.
Variable A : Type.
Implicit Type l : list A.
Lemma Zlength_correct : forall l, Zlength l = Z_of_nat (length l).
Proof.
assert (forall l (acc:Z), Zlength_aux acc A l = acc + Z_of_nat (length l)).
simple induction l.
simpl in |- *; auto with zarith.
intros; simpl (length (a :: l0)) in |- *; rewrite Znat.inj_S.
simpl in |- *; rewrite H; auto with zarith.
unfold Zlength in |- *; intros; rewrite H; auto.
Qed.
Lemma Zlength_nil : Zlength (A:=A) nil = 0.
Proof.
auto.
Qed.
Lemma Zlength_cons : forall (x:A) l, Zlength (x :: l) = Zsucc (Zlength l).
Proof.
intros; do 2 rewrite Zlength_correct.
simpl (length (x :: l)) in |- *; rewrite Znat.inj_S; auto.
Qed.
Lemma Zlength_nil_inv : forall l, Zlength l = 0 -> l = nil.
Proof.
intro l; rewrite Zlength_correct.
case l; auto.
intros x l'; simpl (length (x :: l')) in |- *.
rewrite Znat.inj_S.
intros; elimtype False; generalize (Zle_0_nat (length l')); omega.
Qed.
End Zlength_properties.
Implicit Arguments Zlength_correct [A].
Implicit Arguments Zlength_cons [A].
Implicit Arguments Zlength_nil_inv [A].
|
