{"chains":[{"edge":{"id":"e-3733e878156287d8","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-8cae8db3d01b43b1","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":"7cd8528e9c687612035cb44a8a36cbaa243f627c01f4092120f5d278718ee1d0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]},"premises":[{"premise":{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},"chains":[{"edge":{"id":"e-59b849691f316c6f","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-4f6e747b794bcc2e","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"e3ca27eb149afbc47bcf26fd3c0386aee7356c92be2a7256e0e354d00bb50c26","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-ef36674c01230a86","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-4f6e747b794bcc2e","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"bd81984973c141465bfe548abc4318a92c337a288677a982958e5d7ceab17a48","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},"premises":[]}]},{"premise":{"position":1,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"},"chains":[]}]}],"truncated":false,"expanded":3}