{"id":"e-45f39b72ccd15e60","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-288ee4e6a60f9c49","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"8f5a26bc17f09b82bb19037d3b9ba7689096abce219958a3896d8621cd0611cf","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":11387,"certificate_compressed_size":2563,"conclusion_name":"obreschkoff_combination of Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}],"cited_by":[{"node_id":"n-288ee4e6a60f9c49","claim":"real_rooted"}]}