{"id":"e-5a8f629e61d609b9","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-c27b11147fce8bfa","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":"0d4c38a70084d94a19cc54e1a0f36299b04467cfbaccf70316626f03dfd26f61","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4476,"certificate_compressed_size":1537,"conclusion_name":"Riordan array (1+2x,x(1+x)).","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-c27b11147fce8bfa","claim":"one_signed_coefficients"},{"node_id":"n-c27b11147fce8bfa","claim":"hadamard_kernel"},{"node_id":"n-c27b11147fce8bfa","claim":"real_rooted"},{"node_id":"n-c27b11147fce8bfa","claim":"interlacing"}]}