(** * Euclidean division  - arities: (2,0) *)
Fixpoint div' (q y:nat)(x:nat) : nat :=
match q with
| 0 => 0
| S q' => if leb (q' * y) x then q' else div' q' y x
end.

Lemma div'_coq_correct : forall (q x y : nat) (Hy:y>0),
(q > proj1_sig (quotient y Hy x) ->
div' q y x = proj1_sig (quotient y Hy x)) /\
(q <= proj1_sig (quotient y Hy x) ->
div' q y x = Peano.pred q).