proposition: Theorem points_in_newton_segment_have_nat_abscissa : ∀ f L, newton_segments f = Some L → ∀ pt, pt ∈ [ini_pt L … oth_pts L ++ [fin_pt L]] → ∃ h αh, pt = (Qnat h, αh).
proof: Proof. intros f L HL pt Hpt. destruct Hpt as [H| H]. rewrite <- H. eapply exists_ini_pt_nat; eassumption. apply List.in_app_or in H. destruct H as [H| H]. eapply exists_oth_pt_nat; eassumption. destruct H as [H| ]; [ idtac | contradiction ]. rewrite <- H. eapply exists_fin_pt_nat; eassumption. Qed.
imports: Utf8 QArith Sorted NPeano ConvexHull ConvexHullMisc Misc NbarM QbarM Newton PolyConvexHull Field2 Fpolynomial PSpolynomial Puiseux_base Power_series Puiseux_series Ps_div Slope_base