{"id":"e-37ac46811d9937f6","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-b6aca576a689e9f0","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity; with lag L = 2, the induction runs in each residue class of n modulo L from its start row","lean_route":null,"lean_declarations":[],"certificate_hash":"37abffe6d51a5af3c35ba77ae8ec4cf2896fa7c72979192b4c7830e96829bcf4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3800,"certificate_compressed_size":1224,"conclusion_name":"Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-b6aca576a689e9f0","claim":"real_rooted"},{"node_id":"n-b6aca576a689e9f0","claim":"one_signed_coefficients"},{"node_id":"n-b6aca576a689e9f0","claim":"hadamard_kernel"}]}