{"id":"e-d8c18795222de68a","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-51b4f5e7901c2b50","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":"0f800e1d5cfd95db6becd3ce5a061d16cb62ec6cfb921ce6078723f9e98e27a4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1979,"certificate_compressed_size":802,"conclusion_name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","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, 2]","output_interval":"(-inf, 0]","statement":"an input root r <= 2 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-51b4f5e7901c2b50","claim":"basis_map_preserver"}]}