1 files changed, 10 insertions, 17 deletions
diff --git a/theories/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v
index 6fcb4cf91..2d9eb395a 100644
--- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v
+++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v
@@ -574,27 +574,19 @@ Section ZModulo.
  generalize (Z_mod_lt [|x|] 2); omega.
  Qed.
 
- Definition sqrt x := Zsqrt_plain [|x|].
+ Definition sqrt x := Zsqrt [|x|].
  Lemma spec_sqrt : forall x,
        [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2.
  Proof.
  intros.
  unfold sqrt.
  repeat rewrite Zpower_2.
- replace [|Zsqrt_plain [|x|]|] with (Zsqrt_plain [|x|]).
- apply Zsqrt_interval; auto with zarith.
+ replace [|Zsqrt [|x|]|] with (Zsqrt [|x|]).
+ apply Zsqrt_spec; auto with zarith.
  symmetry; apply Zmod_small.
- split.
- apply Zsqrt_plain_is_pos; auto with zarith.
-
- cut (Zsqrt_plain [|x|] <= (wB-1)); try omega.
- rewrite <- (Zsqrt_square_id (wB-1)).
- apply Zsqrt_le.
- split; auto.
- apply Zle_trans with (wB-1); auto with zarith.
- generalize (spec_to_Z x); auto with zarith.
- apply Zsquare_le.
- generalize wB_pos; auto with zarith.
+ split. apply Z.sqrt_nonneg; auto.
+ apply Zle_lt_trans with [|x|]; auto.
+ apply Z.sqrt_le_lin; auto.
  Qed.
 
  Definition sqrt2 x y :=
@@ -602,7 +594,7 @@ Section ZModulo.
   match z with
    | Z0 => (0, C0 0)
    | Zpos p =>
-      let (s,r,_,_) := sqrtrempos p in
+      let (s,r) := Zsqrtrem (Zpos p) in
       (s, if Z_lt_le_dec r wB then C0 r else C1 (r-wB))
    | Zneg _ => (0, C0 0)
   end.
@@ -618,11 +610,12 @@ Section ZModulo.
  remember ([|x|]*wB+[|y|]) as z.
  destruct z.
  auto with zarith.
- destruct sqrtrempos; intros.
+ generalize (Zsqrtrem_spec (Zpos p)).
+ destruct Zsqrtrem as (s,r); intros [U V]; auto with zarith.
  assert (s < wB).
   destruct (Z_lt_le_dec s wB); auto.
   assert (wB * wB <= Zpos p).
-   rewrite e.
+   rewrite U.
    apply Zle_trans with (s*s); try omega.
    apply Zmult_le_compat; generalize wB_pos; auto with zarith.
   assert (Zpos p < wB*wB).