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, 3 insertions, 5 deletions

diff --git a/theories/Numbers/NatInt/NZAdd.v b/theories/Numbers/NatInt/NZAdd.v
index bceea42b7..7d7cc4ac5 100644
--- a/theories/Numbers/NatInt/NZAdd.v
+++ b/theories/Numbers/NatInt/NZAdd.v
@@ -10,8 +10,7 @@
 
 Require Import NZAxioms NZBase.
 
-Module Type NZAddPropSig
- (Import NZ : NZAxiomsSig')(Import NZBase : NZBasePropSig NZ).
+Module Type NZAddProp (Import NZ : NZAxiomsSig')(Import NZBase : NZBaseProp NZ).
 
 Hint Rewrite
  pred_succ add_0_l add_succ_l mul_0_l mul_succ_l sub_0_r sub_succ_r : nz.
@@ -76,8 +75,7 @@ Qed.
 
 Theorem add_shuffle2 : forall n m p q, (n + m) + (p + q) == (n + q) + (m + p).
 Proof.
-intros n m p q.
-rewrite 2 add_assoc, add_shuffle0, add_cancel_r. apply add_shuffle0.
+intros n m p q. rewrite (add_comm p). apply add_shuffle1.
 Qed.
 
 Theorem sub_1_r : forall n, n - 1 == P n.
@@ -85,4 +83,4 @@ Proof.
 intro n; now nzsimpl.
 Qed.
 
-End NZAddPropSig.
+End NZAddProp.