Deletion of useless Zsqrt_def

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14245 85f007b7-540e-0410-9357-904b9bb8a0f7

5 files changed, 12 insertions, 12 deletions

diff --git a/theories/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v
index 3bdbca44a8..d039fdcbf6 100644
--- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v
+++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v
@@ -574,15 +574,15 @@ Section ZModulo.
  generalize (Z_mod_lt [|x|] 2); omega.
  Qed.
 
- Definition sqrt x := Zsqrt [|x|].
+ Definition sqrt x := Z.sqrt [|x|].
  Lemma spec_sqrt : forall x,
        [|sqrt x|] ^ 2 <= [|x|] < ([|sqrt x|] + 1) ^ 2.
  Proof.
  intros.
  unfold sqrt.
  repeat rewrite Zpower_2.
- replace [|Zsqrt [|x|]|] with (Zsqrt [|x|]).
- apply Zsqrt_spec; auto with zarith.
+ replace [|Z.sqrt [|x|]|] with (Z.sqrt [|x|]).
+ apply Z.sqrt_spec; auto with zarith.
  symmetry; apply Zmod_small.
  split. apply Z.sqrt_nonneg; auto.
  apply Zle_lt_trans with [|x|]; auto.
@@ -594,7 +594,7 @@ Section ZModulo.
   match z with
    | Z0 => (0, C0 0)
    | Zpos p =>
-      let (s,r) := Zsqrtrem (Zpos p) in
+      let (s,r) := Z.sqrtrem (Zpos p) in
       (s, if Z_lt_le_dec r wB then C0 r else C1 (r-wB))
    | Zneg _ => (0, C0 0)
   end.
@@ -610,8 +610,8 @@ Section ZModulo.
  remember ([|x|]*wB+[|y|]) as z.
  destruct z.
  auto with zarith.
- generalize (Zsqrtrem_spec (Zpos p)).
- destruct Zsqrtrem as (s,r); intros [U V]; auto with zarith.
+ generalize (Z.sqrtrem_spec (Zpos p)).
+ destruct Z.sqrtrem 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).
diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v
index 173a8f1777..0bcf22e328 100644
--- a/theories/Numbers/Integer/BigZ/ZMake.v
+++ b/theories/Numbers/Integer/BigZ/ZMake.v
@@ -383,13 +383,13 @@ Module Make (N:NType) <: ZType.
   | Neg nx => Neg N.zero
   end.
 
- Theorem spec_sqrt: forall x, to_Z (sqrt x) = Zsqrt (to_Z x).
+ Theorem spec_sqrt: forall x, to_Z (sqrt x) = Z.sqrt (to_Z x).
  Proof.
   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.
+  rewrite Z.sqrt_neg; auto with zarith.
   now rewrite <- EQ.
  Qed.
 
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
index 390b52ebee..dd83b65da1 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
@@ -311,12 +311,12 @@ Qed.
 Lemma sqrt_spec : forall n, 0<=n ->
  (sqrt n)*(sqrt n) <= n /\ n < (succ (sqrt n))*(succ (sqrt n)).
 Proof.
- intros n. zify. apply Zsqrt_spec.
+ intros n. zify. apply Z.sqrt_spec.
 Qed.
 
 Lemma sqrt_neg : forall n, n<0 -> sqrt n == 0.
 Proof.
- intros n. zify. apply Zsqrt_neg.
+ intros n. zify. apply Z.sqrt_neg.
 Qed.
 
 (** Log2 *)
diff --git a/theories/Numbers/Natural/BigN/NMake.v b/theories/Numbers/Natural/BigN/NMake.v
index aabbf87f28..66b39aca9b 100644
--- a/theories/Numbers/Natural/BigN/NMake.v
+++ b/theories/Numbers/Natural/BigN/NMake.v
@@ -771,7 +771,7 @@ Module Make (W0:CyclicType) <: NType.
   intros x. rewrite sqrt_fold. destr_t x as (n,x). exact (ZnZ.spec_sqrt x).
  Qed.
 
- Theorem spec_sqrt: forall x, [sqrt x] = Zsqrt [x].
+ Theorem spec_sqrt: forall x, [sqrt x] = Z.sqrt [x].
  Proof.
   intros x.
   symmetry. apply Z.sqrt_unique.
diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
index 57277b4893..9bed794f29 100644
--- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
+++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
@@ -263,7 +263,7 @@ Qed.
 Lemma sqrt_spec : forall n, 0<=n ->
  (sqrt n)*(sqrt n) <= n /\ n < (succ (sqrt n))*(succ (sqrt n)).
 Proof.
- intros n. zify. apply Zsqrt_spec.
+ intros n. zify. apply Z.sqrt_spec.
 Qed.
 
 Lemma sqrt_neg : forall n, n<0 -> sqrt n == 0.