{"id":"e-38de1ebb8f2e73b6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-86ae49c0879d58f0","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":"9a365f1b5a7a19cc5c225841583156ff2735268aa0831f295af22099361256ff","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":19357,"certificate_compressed_size":3767,"conclusion_name":"basis-map coefficients of A Jacobsthal triangle.","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-7"},"upper":{"closed":true,"value":"1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ba196dcd01800741","role":"sequence","name":"A Jacobsthal triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]"}],"cited_by":[{"node_id":"n-86ae49c0879d58f0","claim":"real_rooted"},{"node_id":"n-86ae49c0879d58f0","claim":"interlacing"}]}