From d2204e78c2d8bfc4f46ba541749b07dedd5e7969 Mon Sep 17 00:00:00 2001
From: letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7>
Date: Fri, 1 Jul 2011 17:38:17 +0000
Subject: Cleanup of files related with power over Z.

 - Zpow_def, Zpower, Zpow_facts shortened thanks to stuff in BinInt.Z

 - The alternative Zpower_alt is now in a separate file Zpow_alt.v,
   not loaded by default.

 - Some more injection lemmas in Znat (pow, div, mod, quot, rem)

 - Btw, added a "square" function in Z, N, Pos, ... (instead of
   Zpow_facts.Zsquare).

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14253 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/PArith/BinPos.v    | 10 ++++++++++
 theories/PArith/BinPosDef.v |  9 +++++++++
 2 files changed, 19 insertions(+)

(limited to 'theories/PArith')

diff --git a/theories/PArith/BinPos.v b/theories/PArith/BinPos.v
index 956421785..2e4d52a29 100644
--- a/theories/PArith/BinPos.v
+++ b/theories/PArith/BinPos.v
@@ -629,6 +629,16 @@ Proof.
  unfold pow. now rewrite iter_succ.
 Qed.
 
+(** ** Properties of square *)
+
+Lemma square_spec p : square p = p * p.
+Proof.
+ induction p.
+ - rewrite square_xI. simpl. now rewrite IHp.
+ - rewrite square_xO. simpl. now rewrite IHp.
+ - trivial.
+Qed.
+
 (** ** Properties of [sub_mask] *)
 
 Lemma sub_mask_succ_r p q :
diff --git a/theories/PArith/BinPosDef.v b/theories/PArith/BinPosDef.v
index b6b7ab7a5..7916511a4 100644
--- a/theories/PArith/BinPosDef.v
+++ b/theories/PArith/BinPosDef.v
@@ -210,6 +210,15 @@ Definition pow (x y:positive) := iter y (mul x) 1.
 
 Infix "^" := pow : positive_scope.
 
+(** ** Square *)
+
+Fixpoint square p :=
+  match p with
+    | p~1 => (square p + p)~0~1
+    | p~0 => (square p)~0~0
+    | 1 => 1
+  end.
+
 (** ** Division by 2 rounded below but for 1 *)
 
 Definition div2 p :=
-- 
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