# Library Coq.Numbers.NatInt.NZParity

Require Import Bool NZAxioms NZMulOrder.
Parity functions
Module Type NZParity (Import A : NZAxiomsSig').
Parameter Inline even odd : t -> bool.
Definition Even n := exists m, n == 2*m.
Definition Odd n := exists m, n == 2*m+1.
Axiom even_spec : forall n, even n = true <-> Even n.
Axiom odd_spec : forall n, odd n = true <-> Odd n.
End NZParity.
Module Type NZParityProp
(Import A : NZOrdAxiomsSig')
(Import B : NZParity A)
(Import C : NZMulOrderProp A).
Morphisms
Instance Even_wd : Proper (eq==>iff) Even.
Instance Odd_wd : Proper (eq==>iff) Odd.
Instance even_wd : Proper (eq==>Logic.eq) even.
Instance odd_wd : Proper (eq==>Logic.eq) odd.
Evenness and oddity are dual notions
Lemma Even_or_Odd : forall x, Even x \/ Odd x.
Lemma double_below : forall n m, n<=m -> 2*n < 2*m+1.
Lemma double_above : forall n m, n<m -> 2*n+1 < 2*m.
Lemma Even_Odd_False : forall x, Even x -> Odd x -> False.
Lemma orb_even_odd : forall n, orb (even n) (odd n) = true.
Lemma negb_odd : forall n, negb (odd n) = even n.
Lemma negb_even : forall n, negb (even n) = odd n.
Constants
Lemma even_0 : even 0 = true.
Lemma odd_0 : odd 0 = false.
Lemma odd_1 : odd 1 = true.
Lemma even_1 : even 1 = false.
Lemma even_2 : even 2 = true.
Lemma odd_2 : odd 2 = false.
Parity and successor
Lemma Odd_succ : forall n, Odd (S n) <-> Even n.
Lemma odd_succ : forall n, odd (S n) = even n.
Lemma even_succ : forall n, even (S n) = odd n.
Lemma Even_succ : forall n, Even (S n) <-> Odd n.
Parity and successor of successor
Lemma Even_succ_succ : forall n, Even (S (S n)) <-> Even n.
Lemma Odd_succ_succ : forall n, Odd (S (S n)) <-> Odd n.
Lemma even_succ_succ : forall n, even (S (S n)) = even n.
Lemma odd_succ_succ : forall n, odd (S (S n)) = odd n.
Parity and addition
Lemma even_add : forall n m, even (n+m) = Bool.eqb (even n) (even m).
Lemma odd_add : forall n m, odd (n+m) = xorb (odd n) (odd m).
Parity and multiplication
Lemma even_mul : forall n m, even (mul n m) = even n || even m.
Lemma odd_mul : forall n m, odd (mul n m) = odd n && odd m.
A particular case : adding by an even number
Lemma even_add_even : forall n m, Even m -> even (n+m) = even n.
Lemma odd_add_even : forall n m, Even m -> odd (n+m) = odd n.
Lemma even_add_mul_even : forall n m p, Even m -> even (n+m*p) = even n.
Lemma odd_add_mul_even : forall n m p, Even m -> odd (n+m*p) = odd n.
Lemma even_add_mul_2 : forall n m, even (n+2*m) = even n.
Lemma odd_add_mul_2 : forall n m, odd (n+2*m) = odd n.
End NZParityProp.