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, 9 insertions, 6 deletions

diff --git a/theories/Arith/Mult.v b/theories/Arith/Mult.v
index fed1bbca5..479138a98 100644
--- a/theories/Arith/Mult.v
+++ b/theories/Arith/Mult.v
@@ -175,19 +175,22 @@ Qed.
 Lemma mult_S_lt_compat_l : forall n m p, m < p -> S n * m < S n * p.
 Proof.
   induction n; intros; simpl in *.
-    rewrite <- 2! plus_n_O; assumption.
+    rewrite <- 2 plus_n_O; assumption.
     auto using plus_lt_compat.
 Qed.
 
 Hint Resolve mult_S_lt_compat_l: arith.
 
+Lemma mult_lt_compat_l : forall n m p, n < m -> 0 < p -> p * n < p * m.
+Proof.
+ intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp).
+ now apply mult_S_lt_compat_l.
+Qed.
+
 Lemma mult_lt_compat_r : forall n m p, n < m -> 0 < p -> n * p < m * p.
 Proof.
-  intros m n p H H0.
-  induction p.
-  elim (lt_irrefl _ H0).
-  rewrite mult_comm.
-  replace (n * S p) with (S p * n); auto with arith.
+  intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp).
+  rewrite (mult_comm m), (mult_comm n). now apply mult_S_lt_compat_l.
 Qed.
 
 Lemma mult_S_le_reg_l : forall n m p, S n * m <= S n * p -> m <= p.