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

1 files changed, 7 insertions, 8 deletions

diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v
index f9490cc9c7..099554cd0b 100644
--- a/theories/Numbers/Integer/BigZ/ZMake.v
+++ b/theories/Numbers/Integer/BigZ/ZMake.v
@@ -338,15 +338,14 @@ Module Make (N:NType) <: ZType.
   | Neg nx => Neg N.zero
   end.
 
- Theorem spec_sqrt: forall x, 0 <= to_Z x ->
-   to_Z (sqrt x) ^ 2 <= to_Z x < (to_Z (sqrt x) + 1) ^ 2.
+ Theorem spec_sqrt: forall x, to_Z (sqrt x) = Zsqrt (to_Z x).
  Proof.
- unfold to_Z, sqrt; intros [x | x] H.
-   exact (N.spec_sqrt x).
- replace (N.to_Z x) with 0.
-   rewrite N.spec_0; simpl Zpower; unfold Zpower_pos, iter_pos;
-    auto with zarith.
- generalize (N.spec_pos x); auto with zarith.
+  destruct x as [p|p]; simpl.
+  apply N.spec_sqrt.
+  rewrite N.spec_0.
+  destruct (Z_le_lt_eq_dec _ _ (N.spec_pos p)) as [LT|EQ].
+  rewrite Zsqrt_neg; auto with zarith.
+  now rewrite <- EQ.
  Qed.
 
  Definition div_eucl x y :=