{"id":"e-62101e32b433f3e2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-614d2749fe4b2963","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"2945aa52da16279590a98d78f7598f2c8b6d71b1e861e620081a6fa5882bad31","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":8270,"certificate_compressed_size":2391,"conclusion_name":"mobius_x_over_x_plus_one of Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2a657d152d421a92","role":"sequence","name":"Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.","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-614d2749fe4b2963","claim":"real_rooted"},{"node_id":"n-614d2749fe4b2963","claim":"one_signed_coefficients"},{"node_id":"n-614d2749fe4b2963","claim":"hadamard_kernel"}]}