Proof using.
intros a l; elim l; simpl in |- *; auto.
intros H; case H.
intros a0 l0 H [H0| H0]; auto.
exists (snd a0); left; rewrite <- H0; case a0; simpl in |- *; auto.
case H; auto; intros l1 Hl1; exists l1; auto.
Qed.

Theorem length_map : forall l, length (map f l) = length l.
Proof using.