{"id":"e-9ac4433d08f91ae9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c7f2f3338041a8d8","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":"a8f6f97fd63391368077d2b31938a4192f2d2f2e5bc7f46b1805f82fde00c07d","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":19435,"certificate_compressed_size":3784,"conclusion_name":"basis-map coefficients of Riordan array (1, 3+3x).","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1/3"},"upper":{"closed":true,"value":"1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+3x).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4/3"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]"}],"cited_by":[{"node_id":"n-c7f2f3338041a8d8","claim":"real_rooted"},{"node_id":"n-c7f2f3338041a8d8","claim":"interlacing"}]}