{"id":"e-408f6d5a7bdb6c51","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-f48bed27dbc29c5f","premise_count":0,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"76295ed2272bb99746804c632a5c2b7c94e30fd6c5817df0cc8cbbe1fca1efaf","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1929,"certificate_compressed_size":766,"conclusion_name":"The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 1]","output_interval":"(-inf, 0]","statement":"an input root r <= 1 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-f48bed27dbc29c5f","claim":"basis_map_preserver"}]}