{"id":"e-941e1ded9756e612","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_of_power_multiple\",\"power\":1,\"derivative_order\":1}"},"operation_detail":"derivative_of_power_multiple, shift 0","conclusion_id":"n-6299862621ff8810","premise_count":1,"theorem":"Rolle's theorem: x^m p is real-rooted when p is, and every derivative of a real-rooted polynomial is real-rooted or zero, with its roots in the convex hull of the roots; coefficients of one sign stay of one sign","lean_route":"derivativeXMul","lean_declarations":["RealRooted.DegreeDropReversal.splits_X_pow_mul_iff","Polynomial.Splits.iterate_derivative"],"certificate_hash":"37fa516d810ad8e49ffd5e1e0eefa711a9aca0836a30e3a04b8746082cb51a73","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":6437,"certificate_compressed_size":1731,"conclusion_name":"Expansion of the rotational partition function for a heteronuclear diatomic molecule.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-39194b37db596ae8","role":"sequence","name":"a(n) = 1 if n is a triangular number, otherwise 0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-6299862621ff8810","claim":"real_rooted"},{"node_id":"n-6299862621ff8810","claim":"one_signed_coefficients"},{"node_id":"n-6299862621ff8810","claim":"hadamard_kernel"}]}