{"chains":[{"edge":{"id":"e-3a8d1c447a607f39","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-df97cd413ec843e8","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":"24415c785dea991d07306967675be1012c5bc574b317da450d072e9e8bc4b650","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 = 3, and U_g"}],"premises":[]},"premises":[]},{"edge":{"id":"e-e78bf90f7e218157","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-df97cd413ec843e8","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":"5c97090458284fa72d411a9d030fbb1db5ad204153626c51d79335ee00cb5f96","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}],"truncated":false,"expanded":2}