proposition: Theorem list_root_mult_succ_if : ∀ la d c md n, list_root_multiplicity acf c la d = S n → lap_mod_div_deg_1 R la c = md → d ≠ O ∧ ac_is_zero (lap_mod_deg_1 la c) ∧ list_root_multiplicity acf c (lap_div_deg_1 la c) (pred d) = n.
proof: Proof. intros la d c md n Hn Hmd. destruct d; [ discriminate Hn | simpl in Hn ]. split; [ intros H; discriminate H | idtac ]. unfold ac_is_zero. destruct (fld_zerop (lap_mod_deg_1 la c)) as [Hz| Hnz]. split; [ constructor | idtac ]. apply eq_add_S; assumption. discriminate Hn. Qed.
imports: Utf8 QArith NPeano ArithRing Misc Field2 ConvexHull Newton Fsummation Fpolynomial Puiseux_base Puiseux_series CharactPolyn