Cleanup of files related with power over Z.

 - Zpow_def, Zpower, Zpow_facts shortened thanks to stuff in BinInt.Z

 - The alternative Zpower_alt is now in a separate file Zpow_alt.v,
   not loaded by default.

 - Some more injection lemmas in Znat (pow, div, mod, quot, rem)

 - Btw, added a "square" function in Z, N, Pos, ... (instead of
   Zpow_facts.Zsquare).

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14253 85f007b7-540e-0410-9357-904b9bb8a0f7

4 files changed, 20 insertions, 5 deletions

diff --git a/theories/Numbers/Natural/Abstract/NAxioms.v b/theories/Numbers/Natural/Abstract/NAxioms.v
index 45a2cf3e1..ca6ccc1bf 100644
--- a/theories/Numbers/Natural/Abstract/NAxioms.v
+++ b/theories/Numbers/Natural/Abstract/NAxioms.v
@@ -34,11 +34,11 @@ End NDivSpecific.
 
 Module Type NAxiomsSig := NAxiomsMiniSig <+ OrderFunctions
   <+ NZParity.NZParity <+ NZPow.NZPow <+ NZSqrt.NZSqrt <+ NZLog.NZLog2
-  <+ NZGcd.NZGcd <+ NZDiv.NZDiv <+ NZBits.NZBits.
+  <+ NZGcd.NZGcd <+ NZDiv.NZDiv <+ NZBits.NZBits <+ NZSquare.
 
 Module Type NAxiomsSig' := NAxiomsMiniSig' <+ OrderFunctions'
   <+ NZParity.NZParity <+ NZPow.NZPow' <+ NZSqrt.NZSqrt' <+ NZLog.NZLog2
-  <+ NZGcd.NZGcd' <+ NZDiv.NZDiv' <+ NZBits.NZBits'.
+  <+ NZGcd.NZGcd' <+ NZDiv.NZDiv' <+ NZBits.NZBits' <+ NZSquare.
 
 
 (** It could also be interesting to have a constructive recursor function. *)
diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v
index b6b26363e..d5df6329a 100644
--- a/theories/Numbers/Natural/Peano/NPeano.v
+++ b/theories/Numbers/Natural/Peano/NPeano.v
@@ -53,6 +53,11 @@ Proof. reflexivity. Qed.
 Lemma pow_succ_r : forall a b, 0<=b -> a^(S b) = a * a^b.
 Proof. reflexivity. Qed.
 
+Definition square n := n * n.
+
+Lemma square_spec n : square n = n * n.
+Proof. reflexivity. Qed.
+
 Definition Even n := exists m, n = 2*m.
 Definition Odd n := exists m, n = 2*m+1.
 
@@ -737,6 +742,9 @@ Definition pow_succ_r := pow_succ_r.
 Lemma pow_neg_r : forall a b, b<0 -> a^b = 0. inversion 1. Qed.
 Definition pow := pow.
 
+Definition square := square.
+Definition square_spec := square_spec.
+
 Definition log2_spec := log2_spec.
 Definition log2_nonpos := log2_nonpos.
 Definition log2 := log2.
diff --git a/theories/Numbers/Natural/SpecViaZ/NSig.v b/theories/Numbers/Natural/SpecViaZ/NSig.v
index cfe842b9d..aaf44ca64 100644
--- a/theories/Numbers/Natural/SpecViaZ/NSig.v
+++ b/theories/Numbers/Natural/SpecViaZ/NSig.v
@@ -83,7 +83,7 @@ Module Type NType.
  Parameter spec_pred: forall x, [pred x] = Z.max 0 ([x] - 1).
  Parameter spec_sub: forall x y, [sub x y] = Z.max 0 ([x] - [y]).
  Parameter spec_mul: forall x y, [mul x y] = [x] * [y].
- Parameter spec_square: forall x, [square x] = [x] *  [x].
+ Parameter spec_square: forall x, [square x] = [x] * [x].
  Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n.
  Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n.
  Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n].
diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
index adbd8223a..a0ad7643d 100644
--- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
+++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v
@@ -15,8 +15,8 @@ Module NTypeIsNAxioms (Import NN : NType').
 Hint Rewrite
  spec_0 spec_1 spec_2 spec_succ spec_add spec_mul spec_pred spec_sub
  spec_div spec_modulo spec_gcd spec_compare spec_eqb spec_ltb spec_leb
- spec_sqrt spec_log2 spec_max spec_min spec_pow_pos spec_pow_N spec_pow
- spec_even spec_odd spec_testbit spec_shiftl spec_shiftr
+ spec_square spec_sqrt spec_log2 spec_max spec_min spec_pow_pos spec_pow_N
+ spec_pow spec_even spec_odd spec_testbit spec_shiftl spec_shiftr
  spec_land spec_lor spec_ldiff spec_lxor spec_div2 spec_of_N
  : nsimpl.
 Ltac nsimpl := autorewrite with nsimpl.
@@ -256,6 +256,13 @@ Proof.
  intros. now zify.
 Qed.
 
+(** Square *)
+
+Lemma square_spec n : square n == n * n.
+Proof.
+ now zify.
+Qed.
+
 (** Sqrt *)
 
 Lemma sqrt_spec : forall n, 0<=n ->