{"id":"e-337faa044bd16e1a","operation":{"index_shift":1,"operation":"preserver_image","operator":"{\"shape\":\"derivative_of_power_multiple\",\"power\":1,\"derivative_order\":2}"},"operation_detail":"derivative_of_power_multiple, shift 1","conclusion_id":"n-2c8bb380cc75972b","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":null,"lean_declarations":["RealRooted.DegreeDropReversal.splits_X_pow_mul_iff","Polynomial.Splits.iterate_derivative"],"certificate_hash":"2e328668c98fc5fa271037fd38c6fffaad254d8a0fdd48e4d0d75d8d6f083ad4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":12253,"certificate_compressed_size":3145,"conclusion_name":"Triangle related to the asymptotic expansion of E(x,m=3,n).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-efe23a76953cf69d","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-2c8bb380cc75972b","claim":"real_rooted"},{"node_id":"n-2c8bb380cc75972b","claim":"one_signed_coefficients"},{"node_id":"n-2c8bb380cc75972b","claim":"hadamard_kernel"},{"node_id":"n-2c8bb380cc75972b","claim":"interlacing"}]}