{"chains":[{"edge":{"id":"e-33c9a461744a57dc","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-ffeb6f77baa8bcee","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":"13112f92adb1b0f2fed3b6d13a1adb1d6a4299f3f17c7bbe115ce9504f6f1530","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":2,"region":{}}],"premises":[{"position":0,"node_id":"n-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","uses":{"claim":"real_rooted","from_index":2,"region":{}},"uses_text":"Every row P_n, n >= 2, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","uses":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3"}]},"premises":[{"premise":{"position":0,"node_id":"n-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","uses":{"claim":"real_rooted","from_index":2,"region":{}},"uses_text":"Every row P_n, n >= 2, is zero or real-rooted with all roots in (-inf, +inf)"},"chains":[{"edge":{"id":"e-9baf8deff42ffdae","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-3ffff24fcb4f40a0","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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"halfGrowth","lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"certificate_hash":"447d3e61ec8dd9e298d858fbb4b916ce19bdd0e612e5095f4b4d11a30fadbca9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}]},{"premise":{"position":1,"node_id":"n-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","uses":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3"},"chains":[]}]}],"truncated":false,"expanded":2}