proposition: Theorem nth_r_add : ∀ f L i j, nth_r (i + j) f L = nth_r i (nth_pol j f L) (nth_ns j f L).
proof: Proof. intros f L i j. revert f L j. induction i; intros. simpl. apply nth_r_n; [ reflexivity | reflexivity | symmetry ]. apply nth_c_n; reflexivity. rewrite Nat.add_succ_l, <- Nat.add_succ_r; simpl. rewrite IHi; simpl. f_equal; [ eapply nth_pol_n; reflexivity | idtac ]. eapply nth_ns_n; try eassumption; reflexivity. Qed.
imports: Utf8 QArith NPeano Sorting Misc SlopeMisc Slope_base QbarM NbarM Field2 Fpolynomial Fsummation Newton ConvexHullMisc ConvexHull PolyConvexHull NotInSegment Power_series Puiseux_series Ps_add Ps_mul Ps_div PSpolynomial Puiseux_base AlgCloCharPol CharactPolyn F1Eq PosOrder F1Prop InK1m Q_field RootHeadTail