induction x; intuition; simpl in *.
rewrite modNat_plus.
rewrite modNat_arg_eq.
rewrite plus_0_l.
eauto.
Qed.

Lemma modNat_add_same_l : forall x y z p,
modNat (x + y) p = modNat (x + z) p ->
modNat y p = modNat z p.