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, 15 insertions, 4 deletions

diff --git a/theories/Numbers/Natural/Abstract/NPow.v b/theories/Numbers/Natural/Abstract/NPow.v
index 275a5c4f5..68976624e 100644
--- a/theories/Numbers/Natural/Abstract/NPow.v
+++ b/theories/Numbers/Natural/Abstract/NPow.v
@@ -50,10 +50,21 @@ Proof. wrap pow_mul_l. Qed.
 Lemma pow_mul_r : forall a b c, a^(b*c) == (a^b)^c.
 Proof. wrap pow_mul_r. Qed.
 
-(** Positivity *)
+(** Power and nullity *)
 
-Lemma pow_nonzero : forall a b, a~=0 -> a^b~=0.
-Proof. intros. rewrite neq_0_lt_0. wrap pow_pos_nonneg. Qed.
+Lemma pow_eq_0 : forall a b, b~=0 -> a^b == 0 -> a == 0.
+Proof. intros. apply (pow_eq_0 a b); trivial. auto'. Qed.
+
+Lemma pow_nonzero : forall a b, a~=0 -> a^b ~= 0.
+Proof. wrap pow_nonzero. Qed.
+
+Lemma pow_eq_0_iff : forall a b, a^b == 0 <-> b~=0 /\ a==0.
+Proof.
+ intros a b. split.
+ rewrite pow_eq_0_iff. intros [H |[H H']].
+  generalize (le_0_l b); order. split; order.
+ intros (Hb,Ha). rewrite Ha. now apply pow_0_l'.
+Qed.
 
 (** Monotonicity *)
 
@@ -143,7 +154,7 @@ Qed.
 
 Lemma odd_pow : forall a b, b~=0 -> odd (a^b) = odd a.
 Proof.
- intros. now rewrite <- !negb_even_odd, even_pow.
+ intros. now rewrite <- !negb_even, even_pow.
 Qed.
 
 End NPowProp.