{"id":"e-c735f431717d17a0","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-95a09e5daac73697","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":"abbb76f3e6f212f0414fb767ec1e24fd78f72988a48b3b9d8806678cfa73fc10","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":15845,"certificate_compressed_size":3077,"conclusion_name":"shift_x_plus_one of E2(n, k), the second-order Eulerian numbers with E2(0, k) = \u03b4_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a36b2e24b7a3dbb7","role":"sequence","name":"Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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-95a09e5daac73697","claim":"one_signed_coefficients"},{"node_id":"n-95a09e5daac73697","claim":"hadamard_kernel"},{"node_id":"n-95a09e5daac73697","claim":"real_rooted"}]}