{"id":"e-d4a7bd772f22ef34","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-6fea5aff4f44fa54","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":"18d4ed472ce4f1e761fc4708332bdcfd56a87ea225a01b47adc2cdad870fc2df","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1865,"certificate_compressed_size":717,"conclusion_name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","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, 0]","output_interval":"(-inf, 0]","statement":"an input root r <= 0 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-6fea5aff4f44fa54","claim":"basis_map_preserver"}]}