proposition: Lemma lagrange_poly_part_0 {R: fieldType} (x: R) l m r: uniq (unzip1 (l ++ (m :: r))) -> x \in unzip1 l -> (\sum_(j <- subseqs_rec l m r) lagrange_poly_part j).[x] = 0.
proof: Proof. elim: r l m => [|m' r IHr] l [x0 y0] Huniq Hin. 1: by rewrite big_seq1 hornerZ lagrange_basis_0 ?GRing.mulr0. rewrite /= big_cons hornerD hornerZ. rewrite lagrange_basis_0 ?IHr ?GRing.mulr0 ?GRing.add0r //. 1: by rewrite -catA cat_cons. all: by rewrite /unzip1 map_cat -/unzip1 mem_cat Hin. Qed.
imports: Relational(OrderEnrichedCategory GenericRulesSimple) mathcomp(all_ssreflect all_algebra reals distr realsum ssrnat ssreflect ssrfun ssrbool ssrnum eqtype choice seq) Mon(SPropBase) Crypt(Axioms ChoiceAsOrd SubDistr Couplings UniformDistrLemmas FreeProbProg Theta_dens RulesStateProb pkg_core_definition choice_type pkg_composition pkg_rhl Package Prelude) extructures(ord fset fmap) Equations(Equations)