{"chains":[{"edge":{"id":"e-e739032c2ab562e9","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-60c7e36cbe887b7e","premise_count":0,"theorem":"index-dependent scalar term: P_(k+1) = (L + k s) P_k with L f = (a + b t) f + kappa (t - h) f' gives T = U_g o R_s (RealRooted basisMap_factorization_of_scalarIndex_recurrence), where R_s(u^k) = u (u + s) ... (u + (k-1) s) keeps every root <= 0 for s >= 0 (Su, Yang and Zhang, Australas. J. Combin. 56 (2013), Theorem 4; Brenti, Mem. AMS 413 (1989), Theorem 2.4.3; RealRooted NewtonParametersAffine.preservesNonposRooted) and U_g(u^k) = L^k g sends (-inf, a + b h + kappa m] into (-inf, h] (RealRooted rayBasisMap_maps_realRootedIn_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"cc304d0568fbfb78f21e30d4602a76b8abbaffdafc405fa1aec5663edd910f40","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"(-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g"}],"premises":[]},"premises":[]},{"edge":{"id":"e-6aa1e3cc7e408461","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-60c7e36cbe887b7e","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"339dfe43528e11d671a5f12e4dfc02f39846fa5b372fe1a1b7cfd7c7bfc65751","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},"premises":[]}],"truncated":false,"expanded":2}