proposition: Lemma div_mod_pos {n m: nat} : S m / (1 + n) + S m mod (1 + n) <> 0.
proof: Proof. move=> ?. have /Nat.div_small_iff : S m / (1 + n) = 0 by lia. move /(_ ltac:(lia)). have /div_exact : S m mod (1 + n) = 0 by lia. by lia. Qed.
imports: Arith Lia ZArith.Znat ZArith.Zdiv ssreflect ssrbool ssrfun