{"id":"e-2abd4179002ed34f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-e0e035772684cc42","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":"d06bd6f1e12d56347919818fdf2ab0a2fbf76170fa4be74ccdd1a108c0f1ba1e","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":12701,"certificate_compressed_size":2497,"conclusion_name":"shift_x_plus_one of Triangle of coefficients of cos(x)^n in polynomial for cos(nx).","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-dbf182f3ecfeda2a","role":"sequence","name":"Triangle of coefficients of cos(x)^n in polynomial for cos(nx).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}],"cited_by":[{"node_id":"n-e0e035772684cc42","claim":"real_rooted"}]}