(* ArithCompl.v *)
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(* 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.
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(* Regarding Z *)
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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.
Definition R_fact (x y : Z) :=
1 < x /\ 1 < y /\ exists q : Z, y = q * x /\ 1 < q.
Definition R_p4 (x y : Z) :=
0 <= x /\ 1 < y /\ exists d : Z, y = d * d * x /\ 1 < d.

Lemma prop4b : forall p q : Z, 0 <= p -> 0 <= q -> p <= q -> rel_prime p q ->
is_sqr (p * (q * (q * q - p * p))) ->
is_sqr p /\ is_sqr q /\ is_sqr (q * q - p * p).