Fact iA_A n : iA n ⊠ A n = MZ_one.
Fact A_minus u v : (v <= u)%nat -> A (u-v)%nat = A u ⊠ iA v.
Section alpha_nat_coprime.
Let A_eq_3_12 n : exists u v, u*α (S n) + v*α n = 1.

(* Specifications: *)
Lemma alpha_nat_coprime n : is_gcd (alpha_nat (S n)) (alpha_nat n) 1.
End alpha_nat_coprime.
Theorem find_odd_alpha u : exists n, (u <= alpha_nat (S n) /\ rem (alpha_nat (S n)) 2 = 1)%nat.
Theorem find_odd_alpha' u : exists n, (u <= alpha_nat n /\ rem (alpha_nat n) 2 = 1)%nat.
Notation expoZ := (mscal Zmult 1).