{"chains":[{"edge":{"id":"e-9b2a752e203a2a8f","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-f425461cbf5769b3","premise_count":0,"theorem":"interval basis maps: for L f = (a + b t) f + kappa (t - l)(t - h) f', l < h, kappa > 0, b + kappa deg g > 0, and g with positive leading coefficient, real-rooted in [l, h] with endpoint multiplicities m_l, m_h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in [l, h] if and only if S lies in [a + b l - kappa (h - l) m_l, a + b h + kappa (h - l) m_h] (RealRooted intervalBasisMap_maps_realRootedIn_iff; Ma and Wang, EJC 15 (2008) R17, Theorem 2 and Corollary 3); a negative leading coefficient of g only changes the sign of T","lean_route":null,"lean_declarations":[],"certificate_hash":"b9198b30bbbb0b0351af1ac3670d3d84b6f391758c01df6fdb6e717da56f6138","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-1, 3]","output_interval":"[-1, 0]","statement":"source [-1, 3], target [-1, 0]"}],"premises":[]},"premises":[]},{"edge":{"id":"e-a7b6cc9ddce470c2","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-f425461cbf5769b3","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"391ff0ed14e025c496dd6d95b6a8f24639f8c554170bc0dccda5455a46f0fc9a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}],"truncated":false,"expanded":2}