simpl in |- *.
unfold before, after in |- *.
intros.
case i; case i'.
simpl in |- *.
intros x l x0 l0.
clear l0 l i' i N N'.
induction  x as [| x Hrecx]; induction  x0 as [| x0 Hrecx0];
unfold gt in |- *; auto with arith.
case Hrecx.
auto with arith.
intro.
unfold gt in |- *; case H.
intro; left.
rewrite H0.
auto.
unfold gt in |- *.
intro.
right.
case H0.
auto.
intros.
cut (S x < S m \/ S x = S m).
intro.
case H2.
auto.
auto.
left.
auto with arith.
Qed.

(* Specifications: *)
Lemma decide_before :
forall (N N' : nat) (i : fin N) (j : fin N'), before i j \/ notbefore i j.