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, 19 insertions, 7 deletions

diff --git a/theories/Numbers/Natural/Binary/NBinary.v b/theories/Numbers/Natural/Binary/NBinary.v
index 0d6c379e05..65b166df60 100644
--- a/theories/Numbers/Natural/Binary/NBinary.v
+++ b/theories/Numbers/Natural/Binary/NBinary.v
@@ -10,14 +10,14 @@
 
 Require Import BinPos Ndiv_def.
 Require Export BinNat.
-Require Import NAxioms NProperties NDiv.
+Require Import NAxioms NProperties.
 
 Local Open Scope N_scope.
 
 (** * Implementation of [NAxiomsSig] module type via [BinNat.N] *)
 
 Module Type N
- <: NAxiomsSig <: UsualOrderedTypeFull <: TotalOrder <: DecidableTypeFull.
+ <: NAxiomsMiniSig <: UsualOrderedTypeFull <: TotalOrder <: DecidableTypeFull.
 
 (** Bi-directional induction. *)
 
@@ -144,6 +144,19 @@ Program Instance mod_wd : Proper (eq==>eq==>eq) Nmod.
 Definition div_mod := fun x y (_:y<>0) => Ndiv_mod_eq x y.
 Definition mod_upper_bound := Nmod_lt.
 
+(** Odd and Even *)
+
+Definition Even n := exists m, n = 2*m.
+Definition Odd n := exists m, n = 2*m+1.
+Definition even_spec := Neven_spec.
+Definition odd_spec := Nodd_spec.
+
+(** Power *)
+
+Definition pow_0_r := Npow_0_r.
+Definition pow_succ_r n p (H:0 <= p) := Npow_succ_r n p.
+Program Instance pow_wd : Proper (eq==>eq==>eq) Npow.
+
 (** The instantiation of operations.
     Placing them at the very end avoids having indirections in above lemmas. *)
 
@@ -164,14 +177,13 @@ Definition min := Nmin.
 Definition max := Nmax.
 Definition div := Ndiv.
 Definition modulo := Nmod.
+Definition pow := Npow.
+Definition even := Neven.
+Definition odd := Nodd.
 
-Include NPropFunct
+Include NProp
  <+ UsualMinMaxLogicalProperties <+ UsualMinMaxDecProperties.
 
-(** Generic properties of [div] and [mod] *)
-
-Include NDivPropFunct.
-
 End N.
 
 (*