From fc2613e871dffffa788d90044a81598f671d0a3b Mon Sep 17 00:00:00 2001
From: letouzey
Date: Thu, 5 Jul 2012 16:56:16 +0000
Subject: ZArith + other : favor the use of modern names instead of compat
 notations

 - For instance, refl_equal --> eq_refl
 - Npos, Zpos, Zneg now admit more uniform qualified aliases
   N.pos, Z.pos, Z.neg.
 - A new module BinInt.Pos2Z with results about injections from
   positive to Z
 - A result about Z.pow pushed in the generic layer
 - Zmult_le_compat_{r,l} --> Z.mul_le_mono_nonneg_{r,l}
 - Using tactic Z.le_elim instead of Zle_lt_or_eq
 - Some cleanup in ring, field, micromega
   (use of "Equivalence", "Proper" ...)
 - Some adaptions in QArith (for instance changed Qpower.Qpower_decomp)
 - In ZMake and ZMake, functor parameters are now named NN and ZZ
   instead of N and Z for avoiding confusions

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15515 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/Numbers/Integer/Abstract/ZBits.v     |  2 +-
 theories/Numbers/Integer/Abstract/ZDivFloor.v |  2 +-
 theories/Numbers/Integer/Abstract/ZPow.v      | 11 +++++++++++
 3 files changed, 13 insertions(+), 2 deletions(-)

(limited to 'theories/Numbers/Integer/Abstract')

diff --git a/theories/Numbers/Integer/Abstract/ZBits.v b/theories/Numbers/Integer/Abstract/ZBits.v
index 92afbcb537..7c3a99215b 100644
--- a/theories/Numbers/Integer/Abstract/ZBits.v
+++ b/theories/Numbers/Integer/Abstract/ZBits.v
@@ -292,7 +292,7 @@ Proof.
 Qed.
 
 (** Hence the number of bits of [a] is [1+log2 a]
-    (see [Psize] and [Psize_pos]).
+    (see [Pos.size_nat] and [Pos.size]).
 *)
 
 (** For negative numbers, things are the other ways around:
diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v
index 14003d8926..24c8dc8b07 100644
--- a/theories/Numbers/Integer/Abstract/ZDivFloor.v
+++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v
@@ -16,7 +16,7 @@ Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv.
 
     [a = bq+r /\ 0 <= |r| < |b| /\ Sign(r) = Sign(b)]
 
-    This is the convention followed historically by [Zdiv] in Coq, and
+    This is the convention followed historically by [Z.div] in Coq, and
     corresponds to convention "F" in the following paper:
 
     R. Boute, "The Euclidean definition of the functions div and mod",
diff --git a/theories/Numbers/Integer/Abstract/ZPow.v b/theories/Numbers/Integer/Abstract/ZPow.v
index 53d84dce1c..c9c98b1e5c 100644
--- a/theories/Numbers/Integer/Abstract/ZPow.v
+++ b/theories/Numbers/Integer/Abstract/ZPow.v
@@ -18,6 +18,17 @@ Module Type ZPowProp
 
  Include NZPowProp A A B.
 
+(** A particular case of [pow_add_r], with no precondition *)
+
+Lemma pow_twice_r a b : a^(2*b) == a^b * a^b.
+Proof.
+ rewrite two_succ. nzsimpl.
+ destruct (le_gt_cases 0 b).
+ - now rewrite pow_add_r.
+ - rewrite !pow_neg_r. now nzsimpl. trivial.
+   now apply add_neg_neg.
+Qed.
+
 (** Parity of power *)
 
 Lemma even_pow : forall a b, 0<b -> even (a^b) = even a.
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