Proof with auto using Z_lt_neq.
intros H H1. apply Z.mod_divide...
rewrite Zminus_mod.
rewrite H1. rewrite <- (Z.mod_small c b) at 1.
- rewrite Z.sub_diag, Z.mod_0_l...
- subst. now apply Z.mod_pos_bound.
Qed.

Lemma Zdivide_mod_minus a b c:
0 <= c < b -> (b | a - c) -> a mod b = c.