Numbers: axiomatization, properties and implementations of gcd

 - For nat, we create a brand-new gcd function, structural in
   the sense of Coq, even if it's Euclid algorithm. Cool...
 - We re-organize the Zgcd that was in Znumtheory, create out of it
   files Pgcd, Ngcd_def, Zgcd_def. Proofs of correctness are revised
   in order to be much simpler (no omega, no advanced lemmas of
   Znumtheory, etc).
 - Abstract Properties NZGcd / ZGcd / NGcd could still be completed,
   for the moment they contain up to Gauss thm. We could add stuff
   about (relative) primality, relationship between gcd and div,mod,
   or stuff about parity, etc etc.
 - Znumtheory remains as it was, apart for Zgcd and correctness proofs
   gone elsewhere. We could later take advantage of ZGcd in it.
   Someday, we'll have to switch from the current Zdivide inductive,
   to Zdivide' via exists. To be continued...

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13623 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 33 insertions, 1 deletions

diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
index 6689875cb0..999e7cd03e 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
@@ -20,7 +20,7 @@ Hint Rewrite
  spec_0 spec_1 spec_2 spec_add spec_sub spec_pred spec_succ
  spec_mul spec_opp spec_of_Z spec_div spec_modulo spec_sqrt
  spec_compare spec_eq_bool spec_max spec_min spec_abs spec_sgn
- spec_pow spec_log2 spec_even spec_odd
+ spec_pow spec_log2 spec_even spec_odd spec_gcd
  : zsimpl.
 
 Ltac zsimpl := autorewrite with zsimpl.
@@ -349,6 +349,38 @@ Proof.
 intros a b. zify. intros. apply Z_mod_neg; auto with zarith.
 Qed.
 
+(** Gcd *)
+
+Definition divide n m := exists p, n*p == m.
+Local Notation "( x | y )" := (divide x y) (at level 0).
+
+Lemma spec_divide : forall n m, (n|m) <-> Zdivide' [n] [m].
+Proof.
+ intros n m. split.
+ intros (p,H). exists [p]. revert H; now zify.
+ intros (z,H). exists (of_Z z). now zify.
+Qed.
+
+Lemma gcd_divide_l : forall n m, (gcd n m | n).
+Proof.
+ intros n m. apply spec_divide. zify. apply Zgcd_divide_l.
+Qed.
+
+Lemma gcd_divide_r : forall n m, (gcd n m | m).
+Proof.
+ intros n m. apply spec_divide. zify. apply Zgcd_divide_r.
+Qed.
+
+Lemma gcd_greatest : forall n m p, (p|n) -> (p|m) -> (p|gcd n m).
+Proof.
+ intros n m p. rewrite !spec_divide. zify. apply Zgcd_greatest.
+Qed.
+
+Lemma gcd_nonneg : forall n m, 0 <= gcd n m.
+Proof.
+ intros. zify. apply Zgcd_nonneg.
+Qed.
+
 End ZTypeIsZAxioms.
 
 Module ZType_ZAxioms (Z : ZType)