{"id":"e-415345b4c6061d69","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-2e0409c0014ec240","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":"fbe709d2c5014be775aafd765f7cc14e8fe56c0f9ad3b06a79921b31df49762b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1864,"certificate_compressed_size":723,"conclusion_name":"Triangle of coefficients in expansion of (1+4x)^n.","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-2e0409c0014ec240","claim":"basis_map_preserver"}]}