{"id":"e-f5bf3eab46ece5d6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-8\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-35239c46f20a8933","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":"e51fb6951761f6fea3cef305f7a8f4ff720374f0f65278f7f115943ba575cfec","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":19637,"certificate_compressed_size":3907,"conclusion_name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"3"},"upper":{"closed":true,"value":"7"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]"}],"cited_by":[{"node_id":"n-35239c46f20a8933","claim":"real_rooted"},{"node_id":"n-35239c46f20a8933","claim":"one_signed_coefficients"},{"node_id":"n-35239c46f20a8933","claim":"interlacing"}]}