Numbers: new functions pow, even, odd + many reorganisations

 - Simplification of functor names, e.g. ZFooProp instead of ZFooPropFunct
 - The axiomatisations of the different fonctions are now in {N,Z}Axioms.v
   apart for Z division (three separate flavours in there own files).
   Content of {N,Z}AxiomsSig is extended, old version is {N,Z}AxiomsMiniSig.
 - In NAxioms, the recursion field isn't that useful, since we axiomatize
   other functions and not define them (apart in the toy NDefOps.v).
   We leave recursion there, but in a separate NAxiomsFullSig.
 - On Z, the pow function is specified to behave as Zpower : a^(-1)=0
 - In BigN/BigZ, (power:t->N->t) is now pow_N, while pow is t->t->t
   These pow could be more clever (we convert 2nd arg to N and use pow_N).
   Default "^" is now (pow:t->t->t). BigN/BigZ ring is adapted accordingly
 - In BigN, is_even is now even, its spec is changed to use Zeven_bool.
   We add an odd. In BigZ, we add even and odd.
 - In ZBinary (implem of ZAxioms by ZArith), we create an efficient Zpow
   to implement pow. This Zpow should replace the current linear Zpower
   someday.
 - In NPeano (implem of NAxioms by Arith), we create pow, even, odd functions,
   and we modify the div and mod functions for them to be linear, structural,
   tail-recursive.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13546 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 9 insertions, 9 deletions

diff --git a/theories/Numbers/Rational/BigQ/QMake.v b/theories/Numbers/Rational/BigQ/QMake.v
index d850f927a..6bb9c2ac4 100644
--- a/theories/Numbers/Rational/BigQ/QMake.v
+++ b/theories/Numbers/Rational/BigQ/QMake.v
@@ -937,8 +937,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType.
 
  Definition power_pos (x : t) p : t :=
   match x with
-  | Qz zx => Qz (Z.power_pos zx p)
-  | Qq nx dx => Qq (Z.power_pos nx p) (N.power_pos dx p)
+  | Qz zx => Qz (Z.pow_pos zx p)
+  | Qq nx dx => Qq (Z.pow_pos nx p) (N.pow_pos dx p)
   end.
 
  Theorem spec_power_pos : forall x p, [power_pos x p] == [x] ^ Zpos p.
@@ -946,25 +946,25 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType.
  intros [ z | n d ] p; unfold power_pos.
  (* Qz *)
  simpl.
- rewrite Z.spec_power_pos.
+ rewrite Z.spec_pow_pos.
  rewrite Qpower_decomp.
  red; simpl; f_equal.
  rewrite Zpower_pos_1_l; auto.
  (* Qq *)
  simpl.
- rewrite Z.spec_power_pos.
+ rewrite Z.spec_pow_pos.
  destr_eqb; nzsimpl; intros.
  apply Qeq_sym; apply Qpower_positive_0.
- rewrite N.spec_power_pos in *.
+ rewrite N.spec_pow_pos in *.
  assert (0 < N.to_Z d ^ ' p)%Z by
   (apply Zpower_gt_0; auto with zarith).
  romega.
- rewrite N.spec_power_pos, H in *.
+ rewrite N.spec_pow_pos, H in *.
  rewrite Zpower_0_l in H0; [romega|discriminate].
  rewrite Qpower_decomp.
  red; simpl; do 3 f_equal.
  rewrite Z2P_correct by (generalize (N.spec_pos d); romega).
- rewrite N.spec_power_pos. auto.
+ rewrite N.spec_pow_pos. auto.
  Qed.
 
  Instance strong_spec_power_pos x p `(Reduced x) : Reduced (power_pos x p).
@@ -979,10 +979,10 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType.
  revert H.
  unfold Reduced; rewrite strong_spec_red, Qred_iff; simpl.
  destr_eqb; nzsimpl; simpl; intros.
- rewrite N.spec_power_pos in H0.
+ rewrite N.spec_pow_pos in H0.
  rewrite H, Zpower_0_l in *; [romega|discriminate].
  rewrite Z2P_correct in *; auto.
- rewrite N.spec_power_pos, Z.spec_power_pos; auto.
+ rewrite N.spec_pow_pos, Z.spec_pow_pos; auto.
  rewrite Zgcd_1_rel_prime in *.
  apply rel_prime_Zpower; auto with zarith.
  Qed.