intuition.
unfold modNatAddInverse.
rewrite <- H.
rewrite modNat_plus.
rewrite minus_add_assoc.
rewrite (plus_comm).
rewrite <- minus_add_assoc.
rewrite minus_diag.
rewrite plus_0_r.
apply modNat_arg_eq.
trivial.
assert (modNat x p < p)%nat.
apply modNat_lt.
omega.
Qed.

(* Specifications: *)
Lemma modNatAddInverse_correct : forall n p,
modNat (n + modNatAddInverse n p) p = O.