Library Coq.Numbers.Natural.Abstract.NParity

Require Import Bool NSub NZParity.

Some additional properties of even, odd.

Module Type NParityProp (Import N : NAxiomsSig')(Import NP : NSubProp N).

Include NZParityProp N N NP.

Lemma odd_pred n : n ~=0 -> odd (P n) = even n.

Lemma even_pred n : n ~=0 -> even (P n) = odd n.

Lemma even_sub n m : m <=n -> even (n -m) = Bool.eqb (even n) (even m).

Lemma odd_sub n m : m <=n -> odd (n -m) = xorb (odd n) (odd m).

End NParityProp.