Definition congruentZ (x y n : Z) := divide Z IdZ Zmult 0%Z n (x + - y)%Z.

(* Specifications: *)
Lemma divide_selfZ : forall x : Z, divide Z IdZ Zmult 0%Z x x.
Theorem chinese_remaindering_theorem : forall a b x y : Z, gcdZ a b = 1%Z -> {z : Z | congruentZ z x a /\ congruentZ z y b}.