Numbers and bitwise functions.

 See NatInt/NZBits.v for the common axiomatization of bitwise functions
 over naturals / integers. Some specs aren't pretty, but easier to
 prove, see alternate statements in property functors {N,Z}Bits.
 Negative numbers are considered via the two's complement convention.

 We provide implementations for N (in Ndigits.v), for nat (quite dummy,
 just for completeness), for Z (new file Zdigits_def), for BigN
 (for the moment partly by converting to N, to be improved soon)
 and for BigZ.

 NOTA: For BigN.shiftl and BigN.shiftr, the two arguments are now in
 the reversed order (for consistency with the rest of the world):
 for instance BigN.shiftl 1 10 is 2^10.

 NOTA2: Zeven.Zdiv2 is _not_ doing (Zdiv _ 2), but rather (Zquot _ 2)
 on negative numbers. For the moment I've kept it intact, and have
 just added a Zdiv2' which is truly equivalent to (Zdiv _ 2).
 To reorganize someday ?

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13689 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 28 insertions, 1 deletions

diff --git a/theories/ZArith/Zdiv.v b/theories/ZArith/Zdiv.v
index a14f29308..c9397db8b 100644
--- a/theories/ZArith/Zdiv.v
+++ b/theories/ZArith/Zdiv.v
@@ -356,7 +356,7 @@ Proof. intros a b. zero_or_not b. intros; rewrite Z.div_opp_r_nz; auto. Qed.
 
 (** Cancellations. *)
 
-Lemma Zdiv_mult_cancel_r : forall a b c:Z,
+Lemma  Zdiv_mult_cancel_r : forall a b c:Z,
  c <> 0 -> (a*c)/(b*c) = a/b.
 Proof. intros. zero_or_not b. apply Z.div_mul_cancel_r; auto. Qed.
 
@@ -521,6 +521,33 @@ Proof.
  split; intros (c,Hc); exists c; auto.
 Qed.
 
+(** Particular case : dividing by 2 is related with parity *)
+
+Lemma Zdiv2'_div : forall a, Zdiv2' a = a/2.
+Proof.
+ apply Z.div2_div.
+Qed.
+
+Lemma Zmod_odd : forall a, a mod 2 = if Zodd_bool a then 1 else 0.
+Proof.
+ intros a. now rewrite <- Z.bit0_odd, <- Z.bit0_mod.
+Qed.
+
+Lemma Zmod_even : forall a, a mod 2 = if Zeven_bool a then 0 else 1.
+Proof.
+ intros a. rewrite Zmod_odd, Zodd_even_bool. now destruct Zeven_bool.
+Qed.
+
+Lemma Zodd_mod : forall a, Zodd_bool a = Zeq_bool (a mod 2) 1.
+Proof.
+ intros a. rewrite Zmod_odd. now destruct Zodd_bool.
+Qed.
+
+Lemma Zeven_mod : forall a, Zeven_bool a = Zeq_bool (a mod 2) 0.
+Proof.
+ intros a. rewrite Zmod_even. now destruct Zeven_bool.
+Qed.
+
 (** * Compatibility *)
 
 (** Weaker results kept only for compatibility *)