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, 0 deletions

diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v
index f4649be04..437ce5726 100644
--- a/theories/Bool/Bool.v
+++ b/theories/Bool/Bool.v
@@ -555,6 +555,21 @@ Proof.
   destr_bool.
 Qed.
 
+Lemma negb_xorb_l : forall b b', negb (xorb b b') = xorb (negb b) b'.
+Proof.
+ destruct b,b'; trivial.
+Qed.
+
+Lemma negb_xorb_r : forall b b', negb (xorb b b') = xorb b (negb b').
+Proof.
+ destruct b,b'; trivial.
+Qed.
+
+Lemma xorb_negb_negb : forall b b', xorb (negb b) (negb b') = xorb b b'.
+Proof.
+ destruct b,b'; trivial.
+Qed.
+
 (** Lemmas about the [b = true] embedding of [bool] to [Prop] *)
 
 Lemma eq_iff_eq_true : forall b1 b2, b1 = b2 <-> (b1 = true <-> b2 = true).