{"id":"e-610623b6dc347e01","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-24e260a0c0fb324c","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"464f97b812a665d7176e62c2c539e79a202d6014d58745817cad24b89f741a83","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":10415,"certificate_compressed_size":2644,"conclusion_name":"reversal of basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0b43db26e266f660","role":"sequence","name":"basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","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-24e260a0c0fb324c","claim":"one_signed_coefficients"},{"node_id":"n-24e260a0c0fb324c","claim":"hadamard_kernel"},{"node_id":"n-24e260a0c0fb324c","claim":"real_rooted"}]}