induction p.
exists 0...
destruct IHp.
exists (S x0).
subst...
Qed.

(* Specifications: *)
Lemma lt_exists_plus (x y: nat) (p: x < y): exists d, y = S (x + d).
Lemma n_lt_n_plus_Sm n m: n < n + S m.
Lemma minus_lt_compat_l x y z: (y <= x -> z < y -> x - y < x - z)%nat.
Proof with auto.