{"chains":[{"edge":{"id":"e-d8ead6c583c2763e","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-c322146fd9758ee0","premise_count":1,"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":"a85d8be5fba42abd9c76c3ae4bdcc2964f539709be87955b99472d3e344bf0d6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ea2139b8dbf6bf8d","role":"sequence","name":"Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-ea2139b8dbf6bf8d","role":"sequence","name":"Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-4d02ac33f5b2fbc9","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-ea2139b8dbf6bf8d","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"5c3bd295408c4be71cfe79410d65d0f4e5e15078c08c1fffe66e8717b4a8a220","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},"premises":[]}]}]}],"truncated":false,"expanded":2}