{"id":"e-7ddc15d309ad2a75","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2de752bc9bc14cd4","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":"3daff7e5b2fdadecd801855b63777a5bb1642e0ca5e64619982562ccb1902f3d","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":18146,"certificate_compressed_size":3200,"conclusion_name":"shift_x_plus_one of Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"3"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f7e7f08472ec987a","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"4"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 4)"}],"cited_by":[{"node_id":"n-2de752bc9bc14cd4","claim":"real_rooted"},{"node_id":"n-2de752bc9bc14cd4","claim":"interlacing"}]}