{"id":"e-8e6ae20eb3f6019c","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-a679863398df5985","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":"ca33fe945d4089c7a1af8c08729d8e3b6b1da33fcc70fa4269bcd65f7a3dcffe","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1873,"certificate_compressed_size":727,"conclusion_name":"Triangle of coefficients in expansion of (14 + x)^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, 14]","output_interval":"(-inf, 0]","statement":"an input root r <= 14 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-a679863398df5985","claim":"basis_map_preserver"}]}