{"id":"e-78e2985691766434","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":2}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-d66b602a2fc4bda7","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"a4e61d07ddca107159f0fe6e046d66beb627885f68b3c2364cc76164d254b46a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":5243,"certificate_compressed_size":1572,"conclusion_name":"mobius_x_over_x_plus_one of Even-numbered rows of Pascal's triangle.","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-08f49539099949d5","role":"sequence","name":"Even-numbered rows of Pascal's triangle.","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-d66b602a2fc4bda7","claim":"real_rooted"},{"node_id":"n-d66b602a2fc4bda7","claim":"one_signed_coefficients"},{"node_id":"n-d66b602a2fc4bda7","claim":"hadamard_kernel"}]}