(* ArithCompl.v *)
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(* Some complements of arithmetic *)
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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.
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(* Regarding R *)
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Lemma sqr_inv : forall a b : Z, b <> 0 ->
(1 + IZR a * / IZR b * (IZR a * / IZR b) <> 0)%R.