{"id":"e-c52c3082cda5f9b6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-10\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-1b374fdc6d596a8d","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":"91e0d050f91fee296e96f473e12128806a09674464e4e7cb6118b6afdc6b09df","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":19529,"certificate_compressed_size":3821,"conclusion_name":"basis-map coefficients of Riordan array (1, 3+x).","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"86/9"},"upper":{"closed":true,"value":"10"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5d39164c4108e043","role":"sequence","name":"Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4/9"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]"}],"cited_by":[{"node_id":"n-1b374fdc6d596a8d","claim":"one_signed_coefficients"},{"node_id":"n-1b374fdc6d596a8d","claim":"interlacing"},{"node_id":"n-1b374fdc6d596a8d","claim":"real_rooted"}]}