(* ArithCompl.v *)
(**********************************)
(* Some complements of arithmetic *)
(**********************************)
Require Export Wf_nat.
Require Export ZArith.
Require Export Znumtheory.
Require Export Reals.
Open Scope Z_scope.
Unset Standard Proposition Elimination Names.
(***************)
(* Regarding Z *)
(***************)
Definition is_sqr (n : Z) : Prop :=
0 <= n /\ exists i : Z, i * i = n /\ 0 <= i.
Definition both_odd (x y : Z) := Zodd x /\ Zodd y.
Definition distinct_parity (a b : Z) :=
(Zeven a) /\ (Zodd b) \/ (Zodd a) /\ (Zeven b).

Lemma Zodd_mult : forall u v : Z,
Zodd u -> Zodd v ->
(exists s : Z,
(exists w : Z, (u - v = 4 * s /\ u + v = 2 * w) \/
(u - v = 2 * w /\ u + v = 4 * s))).