(* 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).
Definition R_prime (x y : Z) := 1 < x /\ 1 < y /\ x < y.
Definition f_Z (x : Z) := Z.abs_nat x.

Lemma not_prime_gen : forall a b : Z, 1 < a -> 1 < b -> b < a -> ~ prime a ->
(forall c : Z, b < c < a -> rel_prime c a) ->
exists q : Z, exists b : Z, a = q * b /\ 1 < q /\ 1 < b.