{"id":"e-00d47151fcd2720e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-4689fb1386d440ad","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":"86a8a8d2fd5e94cfe906298f0344dbff1a624452ad431c8f61052d1e863969fa","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":14066,"certificate_compressed_size":3217,"conclusion_name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-78ba05b4a3dd925d","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-4689fb1386d440ad","claim":"one_signed_coefficients"},{"node_id":"n-4689fb1386d440ad","claim":"hadamard_kernel"},{"node_id":"n-4689fb1386d440ad","claim":"real_rooted"}]}