{"id":"e-c8c4101b2f8f8c16","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-1\",\"intercept\":\"0\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-50902364601606aa","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"3c74c5ad69fc1899f4d918b494ecb6cf61ff531fc217a2ee43f6954ee266777c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":4786,"certificate_compressed_size":1493,"conclusion_name":"basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-74c1e94669ff21ff","role":"sequence","name":"basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 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-50902364601606aa","claim":"real_rooted"},{"node_id":"n-50902364601606aa","claim":"one_signed_coefficients"}]}