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
|
(************************************************************************)
(* * 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) *)
(************************************************************************)
Require Import BinInt.
Local Open Scope Z_scope.
(** An alternative power function for Z *)
(** This [Zpower_alt] is extensionally equal to [Z.pow],
but not convertible with it. The number of
multiplications is logarithmic instead of linear, but
these multiplications are bigger. Experimentally, it seems
that [Zpower_alt] is slightly quicker than [Z.pow] on average,
but can be quite slower on powers of 2.
*)
Definition Zpower_alt n m :=
match m with
| Z0 => 1
| Zpos p => Pos.iter_op Z.mul p n
| Zneg p => 0
end.
Infix "^^" := Zpower_alt (at level 30, right associativity) : Z_scope.
Lemma Piter_mul_acc : forall f,
(forall x y:Z, (f x)*y = f (x*y)) ->
forall p k, Pos.iter f k p = (Pos.iter f 1 p)*k.
Proof.
intros f Hf.
induction p; simpl; intros.
- set (g := Pos.iter f 1 p) in *. now rewrite !IHp, Hf, Z.mul_assoc.
- set (g := Pos.iter f 1 p) in *. now rewrite !IHp, Z.mul_assoc.
- now rewrite Hf, Z.mul_1_l.
Qed.
Lemma Piter_op_square : forall p a,
Pos.iter_op Z.mul p (a*a) = (Pos.iter_op Z.mul p a)*(Pos.iter_op Z.mul p a).
Proof.
induction p; simpl; intros; trivial. now rewrite IHp, Z.mul_shuffle1.
Qed.
Lemma Zpower_equiv a b : a^^b = a^b.
Proof.
destruct b as [|p|p]; trivial.
unfold Zpower_alt, Z.pow, Z.pow_pos.
revert a.
induction p; simpl; intros.
- f_equal.
rewrite Piter_mul_acc.
now rewrite Piter_op_square, IHp.
intros. symmetry; apply Z.mul_assoc.
- rewrite Piter_mul_acc.
now rewrite Piter_op_square, IHp.
intros. symmetry; apply Z.mul_assoc.
- now Z.nzsimpl.
Qed.
Lemma Zpower_alt_0_r n : n^^0 = 1.
Proof. reflexivity. Qed.
Lemma Zpower_alt_succ_r a b : 0<=b -> a^^(Z.succ b) = a * a^^b.
Proof.
destruct b as [|b|b]; intros Hb; simpl.
- now Z.nzsimpl.
- now rewrite Pos.add_1_r, Pos.iter_op_succ by apply Z.mul_assoc.
- now elim Hb.
Qed.
Lemma Zpower_alt_neg_r a b : b<0 -> a^^b = 0.
Proof.
now destruct b.
Qed.
Lemma Zpower_alt_Ppow p q : (Zpos p)^^(Zpos q) = Zpos (p^q).
Proof.
now rewrite Zpower_equiv, Pos2Z.inj_pow.
Qed.
|
