{"id":"e-159e2ac1fa72d770","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-5cf6acfdac607d6d","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":"5e906b2f2247479995b03d44ee0306b660a712a7ea20e4ef08cf8bf394d3b20b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1907,"certificate_compressed_size":756,"conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","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, 12]","output_interval":"(-inf, 0]","statement":"an input root r <= 12 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-5cf6acfdac607d6d","claim":"basis_map_preserver"}]}