{"chains":[{"edge":{"id":"e-3bcf5053bbc59f78","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-7c641d81e21514ce","premise_count":0,"theorem":"constant current multiplier: P_0 = 1, P_1 = s, P_(k+2) = a P_(k+1) + (b t + c) P_k with a, s >= 0, b > 0.  P_k(t) = Q_k(b t + c) for the rows Q_k with b = 1, c = 0, whose map sends PF inputs to PF outputs (an interlacing-preserving 2 x 2 matrix, Branden, arXiv:1410.6601, Theorem 8.5), so every PF input has a real-rooted image with roots <= -c/b; PF inputs (also those with roots in any [-R, 0]) have PF images if and only if every P_k(0) >= 0, if and only if a^2 + 4c >= 0 and (2s >= a or c >= s^2 - a s) (RealRooted orderTwoBasis_pf_equivalences, basisTransform_orderTwoBasis_preserves_bounded_pf_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"92c7b0bd6a91258079ee952f8ba3794f2caa6db78a2ba3737f13e3cc3d076da3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, -1/2]","statement":"every PF input has a real-rooted image with every root <= -c/b = -1/2"},{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, -1/2]","statement":"every PF input has a PF image (zero allowed)"}],"premises":[]},"premises":[]},{"edge":{"id":"e-363ec3019ccc29a9","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-7c641d81e21514ce","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"bc953de90d6126a0fb424adc5d8633e01177fe6b5f347a33523da03b2357bb7d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},"premises":[]}],"truncated":false,"expanded":2}