{"chains":[{"edge":{"id":"e-836bdd982143692b","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-9c5c208df8605708","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":"11922840a912415135fc7b24d3b2d84490683ed44d79b8629b9e7f36f0f5dd81","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 = 4, and U_g"}],"premises":[]},"premises":[]},{"edge":{"id":"e-381fad74fe44e78a","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-9c5c208df8605708","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":"c16a58eeba924fd57150f645200f43efd77078331b98fc59add8d0d17eb010b8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},"premises":[]},{"edge":{"id":"e-dad3570ee638379f","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-9c5c208df8605708","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"aa312638377415374b33604f1021b2041d69fbb4868d5b31e4dac1464304ca5f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}],"truncated":false,"expanded":3}