Add sqrt in Numbers

 As for power recently, we add a specification in NZ,N,Z,
 derived properties, implementations for nat, N, Z, BigN, BigZ.

  - For nat, this sqrt is brand new :-), cf NPeano.v

  - For Z, we rework what was in Zsqrt: same algorithm,
    no more refine but a pure function, based now on a sqrt
    for positive, from which we derive a Nsqrt and a Zsqrt.
    For the moment, the old Zsqrt.v file is kept as Zsqrt_compat.v.
    It is not loaded by default by Require ZArith.
    New definitions are now in Psqrt.v, Zsqrt_def.v and Nsqrt_def.v

  - For BigN, BigZ, we changed the specifications to refer to Zsqrt
    instead of using characteristic inequations.

 On the way, many extensions, in particular BinPos (lemmas about order),
 NZMulOrder (results about squares)

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13564 85f007b7-540e-0410-9357-904b9bb8a0f7

3 files changed, 11 insertions, 14 deletions

diff --git a/theories/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v
index 38855a85da..4f88008be4 100644
--- a/theories/Numbers/Integer/Abstract/ZAxioms.v
+++ b/theories/Numbers/Integer/Abstract/ZAxioms.v
@@ -9,7 +9,7 @@
 (************************************************************************)
 
 Require Export NZAxioms.
-Require Import NZPow.
+Require Import NZPow NZSqrt.
 
 (** We obtain integers by postulating that successor of predecessor
     is identity. *)
@@ -76,15 +76,20 @@ Module Type ZPowSpecNeg (Import Z : ZAxiomsMiniSig')(Import P : Pow' Z).
  Axiom pow_neg : forall a b, b<0 -> a^b == 0.
 End ZPowSpecNeg.
 
+(** Same for the sqrt function. *)
+
+Module Type ZSqrtSpecNeg (Import Z : ZAxiomsMiniSig')(Import P : Sqrt' Z).
+ Axiom sqrt_neg : forall a, a<0 -> √a == 0.
+End ZSqrtSpecNeg.
 
 (** Let's group everything *)
 
 Module Type ZAxiomsSig :=
   ZAxiomsMiniSig <+ HasCompare <+ HasAbs <+ HasSgn <+ Parity
-   <+ NZPow.NZPow <+ ZPowSpecNeg.
+   <+ NZPow.NZPow <+ ZPowSpecNeg <+ NZSqrt.NZSqrt <+ ZSqrtSpecNeg.
 Module Type ZAxiomsSig' :=
   ZAxiomsMiniSig' <+ HasCompare <+ HasAbs <+ HasSgn <+ Parity
-   <+ NZPow.NZPow' <+ ZPowSpecNeg.
+   <+ NZPow.NZPow' <+ ZPowSpecNeg <+ NZSqrt.NZSqrt' <+ ZSqrtSpecNeg.
 
 
 (** Division is left apart, since many different flavours are available *)
diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v
index 25989b2d40..1010a0f2fb 100644
--- a/theories/Numbers/Integer/Abstract/ZMulOrder.v
+++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v
@@ -92,14 +92,7 @@ Qed.
 
 Notation mul_nonpos := le_mul_0 (only parsing).
 
-Theorem le_0_square : forall n, 0 <= n * n.
-Proof.
-intro n; destruct (neg_nonneg_cases n).
-apply lt_le_incl; now apply mul_neg_neg.
-now apply mul_nonneg_nonneg.
-Qed.
-
-Notation square_nonneg := le_0_square (only parsing).
+Notation le_0_square := square_nonneg (only parsing).
 
 Theorem nlt_square_0 : forall n, ~ n * n < 0.
 Proof.
diff --git a/theories/Numbers/Integer/Abstract/ZProperties.v b/theories/Numbers/Integer/Abstract/ZProperties.v
index 7b9c6f4528..8b34e5b2db 100644
--- a/theories/Numbers/Integer/Abstract/ZProperties.v
+++ b/theories/Numbers/Integer/Abstract/ZProperties.v
@@ -11,6 +11,5 @@ Require Export ZAxioms ZMaxMin ZSgnAbs ZParity ZPow.
 (** This functor summarizes all known facts about Z. *)
 
 Module Type ZProp (Z:ZAxiomsSig) :=
- ZMaxMinProp Z <+ ZSgnAbsProp Z <+ ZParityProp Z <+ ZPowProp Z.
-
-
+ ZMaxMinProp Z <+ ZSgnAbsProp Z <+ ZParityProp Z <+ ZPowProp Z
+ <+ NZSqrt.NZSqrtProp Z Z.