# Library Coq.Numbers.Integer.Abstract.ZParity

Require Import Bool ZMulOrder NZParity.
Some more properties of even and odd.
Module Type ZParityProp (Import Z : ZAxiomsSig')
(Import ZP : ZMulOrderProp Z).
Include NZParityProp Z Z ZP.
Lemma odd_pred : forall n, odd (P n) = even n.
Lemma even_pred : forall n, even (P n) = odd n.
Lemma even_opp : forall n, even (- n) = even n.
Lemma odd_opp : forall n, odd (- n) = odd n.
Lemma even_sub : forall n m, even (n - m) = Bool.eqb (even n) (even m).
Lemma odd_sub : forall n m, odd (n - m) = xorb (odd n) (odd m).
End ZParityProp.